iPllli , 1 ■sitis *** HlM£%^: vfeS **m ^C^K*< \vl/-' .!#» &&&» 15. ^V- r r- * §Sfi| • :^ ', a£ 0§m 1 : il^^»«€^ W^&^m^MMK %h SSwM #& • M ■•^Wr^ 1 mmmm ^r. k f^ f S&$$5£$ft •.r'V/ zgm^rm pitill ■>^ ■\* ■». &&> mM$. fM^t m&s W* t*is£& v> : to« '-» ■v-aii ■-; ^-> &.L.3? - TRANSACTIONS CAMBEIDGE PHILOSOPHICAL SOCIETY. ESTABLISHED NOVEMBER 15, 1819. VOLUME THE NINTH. CAMBRIDGE: 3Printt& at tj)e 58nfomttg prtss; AND SOLD BY DEIGHTON, BELL AND CO. AND MACMILLAN AND CO. CAMBEIDGE; BELL AND DALDY, LONDON. M.DCCC.LVI. CONTENTS. PART I. Page N°. I. On the Dynamical Theory of Diffraction. By G. G. Stokes, M.A. Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge ] II. Criticism of Aristotle's Account of Induction. By W. Whewell, D.D. Master of Trinity College, Cambridge 63 III. On Impact of Elastic Beams. By Homehsham Cox, Esq. B.A. of Jesus College 73 IV. On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula, and the application of the Theory of Probabilities to some questions of Evidence. By Augustus De Morgan, Sec. R.A.S. of Trinity College, Cambridge, Professor of Mathematics in University College, London 79 V. Mathematical Exposition of some Doctrines of Political Economy. Second Memoir. By W. Whbweu, D.D. Master of Trinity College, Cambridge 128 VI. Second Memoir on the Intrinsic Equation of a Curve, and its Application. By W. Whewell, D.D. Master of Trinity College, Cambridge 150 VII. On the Knowledge of Body and Space. By H. Wedgwood, Esq. Christ's College, Cambridge 157 VIII. On the numerical Calculation of a Class of Definite Integrals and Infinite Series. By G. G. Stokes, M.A. Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge. . 166 CONTENTS. PAET II. Pagb N°. IX. Mathematical Exposition of Certain Doctrines of Political Economy. Third Memoir. By W. Whewell, D.D. Master of Trinity College. \_\~\ X. On the Effect of the Internal Friction of Fluids on the Motion of Pen- dulums. By G. G. Stokes, M.A. Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge . . . [8] XI. On some points of the Integral Calculus. By Augustus Db Morgan, of Trinity College, Secretary of the Royal Astronomical Society, and Professor of Mathematics in University College, London ^107] XII. Of the Transformation of Hypotheses in the History of Science. By W. Whewell, D.D. Master of Trinity College l l3 9~] XIII. On the Colours of Thick Plates. By G. G. Stokes, M.A. Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge [1^7] XIV. The Deflexion of Imperfectly Elastic Beams and the Hyperbolic Law of Elasticity. By Homersham Cox, B.A. Jesus College, Cambridge . . [177] CONTENTS. PAET III. N°. XV. On the Oscillations of a Suspension Chain. By J. H. Ron us, M.A. late Fellow of Jesus College, Cambridge 379 XVI. On the Composition and Resolution of Streams of Polarized Light from different Sources. By G. G. Stokes, M.A. Fellow of Pembroke College, and Lucusian Professor of Mathematics in the University of Cambridge 3QQ XVII. On some recent Improvements in Clock-Escapements. By Edmund • Beckett Denison, Esq. M.A. of Trinity College, Cambridge ... 417 Appendix. The Theory of the Long Inequality of Uranus and Neptune, depending on the Near Commensurability of their Mean Motions : An Essay, which obtained the Adams Prise for the Year 1850, in the University of Cambridge. By R. Peirson, M.A. Fellow of St John's College. CONTENTS. N°. XVIII. XIX. XX. XXI. XXII. XXIII. XXIV. XXV. XXVI. XXVII. XXVIII. XXIX. PAET IV. Paoi On the Geology of some parts of Suffolk, particularly of the Valley of the Gipping. By J. B. Phear, M.A. F.G.S. Fellow and Assistant Tutor of Clare College 431 On the Transformation of Surfaces by Bending. By James Clerk Maxwell, B.A. Trinity College , 445 On Self-repeating Series. By Henry Warbukton, M.A 471 On the Determination of the Longitude of the Observatory of Cambridge by Galvanic Signals. By the Rev. J. Challis, M.A. F.R.S. F.R.A.S. Plumian Professor of Astronomy and Experimental Philosophy 487 On some points in the Theory of Differential Equations. By Augustus De Morgan, of Trinity College, Secretary of the Royal Astronomical Society, and Professor of Mathematics in University College, London . . . 515 On the Purbeck Strata of Dorsetshire. By the Rev. Osmond Fisher, F.G.S. Fellow and Tutor of Jesus College 555 On Plato's Survey of the Sciences. By W. Whewell, D.D. Master of Trinity College 582 On Plato's Notion of Dialectic. By W. Whewell, D.D. Master of Trinity College 590 Of the Intellectual Powers according to Plato. By W. Whewell, D.D. Master of Trinity College 598 Remarks on the Fundamental Principle of the Theory of Probabilities. By R. L. Ellis, M.A. late Fellow of Trinity College 605 On the Singular Points of Curves, and on Newton's Method of Co-ordinated Exponents. By Augustus De Morgan, of Trinity College, Vice- President of the Royal Astronomical Society, and Professor of Mathe- matics in University College, London 6'08 On the External Temperature of the Earth, and the other Planets of the Solar System. By W. Hopkins, M.A. F.R.S. of St Peter's College, and Vice-President of the Society 6*28 Index to Vol. IX 673 ADVERTISEMENT. The Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. The Society takes this opportunity of expressing its grateful acknowledgments to the Syndics of the University Press, for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. ERRATA. I— 1 Page 471, line 1 of the second paragraph, for — read ^=± 1.2. ..a? 1.2...* Page 472, line 18, for (-l)C read (-l) r C -X -X Page 638, line 14,/or 273°(C) read -273°(C) Page 653, line 5, for u read u n Page 653, line 11, for «° read u Page 573, line 3 from bottom, for 11 feet read 3 feet Pages 573, 574, dele beds cxv. cxvi. TRANSACTIONS CAMBRIDGE PHILOSOPHICAL SOCIETY. Volume IX. Part I. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESSt AND SOLD BY JOHN WILLIAM PARKER, WEST STRAND, LONDON; J. DEIGHTON ; AND MACMILLAN & CO., CAMBRIDGE. M.UCCC.LI. I. On the Dynamical Theory of Diffraction. By G. G. Stokes, M.A., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge. [Read November 26, 1849.] When light is incident on a small aperture in a screen, the illumination at any point in front of the screen is determined, on the undulatory theory, in the following manner. The incident waves are conceived to be broken up on arriving at the aperture ; each element of the aperture is considered as the centre of an elementary disturbance, which diverges spherically in all directions, with an intensity which does not vary rapidly from one direction to another in the neighbourhood of the normal to the primary wave ; and the disturbance at any point is found by taking the aggregate of the disturbances due to all the secondary waves, the phase of vibration of each being retarded by a quantity corresponding to the distance from its centre to the point where the disturbance is sought. The square of the coefficient of vibration is then taken as a measure of the intensity of illumination. Let us consider for a moment the hypotheses on which this process rests. In the first place, it is no hypothesis that we may conceive the waves broken up on arriving at the aperture : it is a necessary consequence of the dynamical principle of the superposition of small motions ; and if this principle be inapplicable to light, the undula- tory theory is upset from its very foundations. The mathematical resolution of a wave, or any portion of a wave, into elementary disturbances must not be confounded with a physical breaking up of the wave, with which it has no more to do than the division of a rod of variable density into differential elements, for the purpose of finding its centre of gravity, has to do with breaking the rod in pieces. It is an hypothesis that we may find the disturbance in front of the aperture by merely taking the aggregate of the disturbances due to all the secondary waves, each secondary wave proceeding as if the screen were away ; in other words, that the effect of the screen is merely to stop a certain portion of the incident light. This hypothesis, exceedingly probable a priori, when we are only concerned with points at no great distance from the normal to the primary wave, is confirmed by experiment, which shews that the same appearances are presented, with a given aperture, whatever be the nature of the screen in which the aperture is pierced, whether, for example, it consist of paper or of foil, whether a small aperture be divided by a hair or by a wire of equal thickness. It is an hypothesis, again, that the intensity in a secondary wave is nearly constant, at a given distance from the centre, in different directions very near the normal to the primary wave ; but it seems to me almost impossible to conceive a mechanical theory which would not lead to this result. It is evident that the difference of phase of the various secondary waves which agitate a given point must be determined by the difference of their radii ; and if it should afterwards be found necessary to add a constant to all the phases the results will not be at all affected. Vol. IX. Part I. 1 2 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. i Lastly, good reasons may be assigned why the intensity should be measured by the square of the coefficient of vibration ; but it is not necessary here to enter into them. In this way we are able to calculate the relative intensities at different points of a diffrac- tion pattern. It may be regarded as established, that the coefficient of vibration in a secondary wave varies, in a given direction, inversely as the radius, and consequently, we are able to calculate the relative intensities at different distances from the aperture. To complete this part of the subject, it is requisite to know the absolute intensity. Now it has been shewn that the absolute intensity will be obtained by taking the reciprocal of the wave length for the quantity by which to multiply the product of a differential element of the area of the aperture, the reciprocal of the radius, and the circular function expressing the phase. It appears at the same time that the phase of vibration of each secondary wave must be accele- rated by a quarter of an undulation. In the investigations alluded to, it is supposed that the law of disturbance in a secondary wave is the same in all directions ; but this will not affect the result, provided the solution be restricted to the neighbourhood of the normal to the primary wave, to which indeed alone the reasoning is applicable; and the solution so restricted is sufficient to meet all ordinary cases of diffraction. Now the object of the first part of the following paper is, to determine, on purely dynamical principles, the law of disturbance in a secondary wave, and that, not merely in the neighbourhood of the normal to the primary wave, but in all directions. The occurrence of the reciprocal of the radius in the coefficient, the acceleration of a quarter of an undulation, and the absolute value of the coefficient in the neighbourhood of the normal to the primary wave, will thus appear as particular results of the general formula. Before attacking the problem dynamically, it is of course necessary to make some suppo- sition respecting the nature of that medium, or ether, the vibrations of which constitute light, according to the theory of undulations. Now, if we adopt the theory of transverse vibra- tions — and certainly, if the simplicity of a theory which conducts us through a multitude of curious and complicated phenomena, like a thread through a labyrinth, be considered to carry the stamp of truth, the claims of the theory of transverse vibrations seem but little short of those of the theory of universal gravitation — if, I say, we adopt this theory, we are obliged to suppose the existence of a tangential force in the ether, called into play by the continuous sliding of one layer, or film, of the medium over another. In consequence of the existence of this force, the ether must behave, so far as regards the luminous vibrations, like an elastic solid. We have no occasion to speculate as to the cause of this tangential force, nor to assume either that the ether does, or that it does not, consist of distinct particles ; nor are we directly called on to consider in what manner the ether behaves with respect to the motion oi solid bodies, such as the earth and planets. Accordingly, I have assumed, as applicable to the luminiferous ether in vacuum, the known equations of motion of an elastic medium, such as an elastic solid. These equations contain two arbitrary constants, depending upon the nature of the medium. The argument which Green has employed to shew that the luminiferous ether must be regarded as sensibly incompressible, in treating of the motions which constitute light *, appears to me of great force. • Camb. Phil. Trans. Vol. vn. p. 2. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 3 The supposition of in compressibility reduces the two arbitrary constants to one; but as the equations are not thus rendered more manageable, I have retained them in their more general shape. The first problem relating to an elastic medium of which the object that I had in view required the solution was, to determine the disturbance at any time, and at any point of an elastic medium, produced by a given initial disturbance which was confined to a finite portion of the medium. This problem was solved long ago by Poisson, in a memoir contained in the tenth volume of the memoirs of the Academy of Sciences. Poisson indeed employed equations of motion with but one arbitrary constant, which are what the general equations of motion become when a certain numerial relation is assumed to exist between the two constants which they involve. This relation was the consequence of a particular physical supposition which he adopted, but which has since been shewn to be untenable, inasmuch as it leads to results which are contradicted by experiment. Nevertheless nothing in Poisson's method depends for its success on the particular numerical relation assumed ; and in fact, to save the constant writing of a radical, Poisson introduced a second constant, which made his equations identical with the general equations, so long as the particular relation supposed to exist between the two constants was not employed. I might accordingly have at once assumed Poisson's results. I have how- ever begun at the beginning, and given a totally different solution of the problem, which will I hope be found somewhat simpler and more direct than Poisson's. The solution of this problem and the discussion of the result occupy the first two sections of the paper. Having had occasion to solve the problem in all its generality, I have in one or two in- stances entered into details which have no immediate relation to light. I have also occasionally considered some points relating to the theory of light which have no immediate bearing on diffraction. It would occupy too much room to enumerate these points here, which will be found in their proper place. I will merely mention one very general theorem at which I have arrived by considering the physical interpretation of a certain step of analysis, though, properly speaking, this theorem is a digression from the main object of the paper. The theorem may be enunciated as follows. If any material system in which the forces acting depend only on the positions of the particles be slightly disturbed from a position of equilibrium, and then left to itself, the part of the subsequent motion which depends on the initial displacements may be obtained from the part which depends on the initial velocities by replacing the arbitrary functions, or arbitrary constants, which express the initial velocities by those which express the correspond- ing initial displacements, and differentiating with respect to the time. Particular cases of this general theorem occur so frequently in researches of this kind, that I think it not improbable that the theorem may be somewhere given in all its generality. I have not however met with a statement of it except in particular cases, and even then the subject was mentioned merely as a casual result of analysis. In the third section of tbis paper, the problem solved in the second section is applied to the determination of the law of disturbance in a secondary wave of light. This determination forms the whole of the dynamical part of the theory of diffraction, at least when we confine ourselves to diffraction in vacuum, or, more generally, within a homogeneous singly refracting 1—2 4 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. medium : the rest is a mere matter of integration ; and whatever difficulties the solution of the problem may present for particular forms of aperture, they are purely mathematical. In the investigation, the incident light is supposed to be plane-polarized, and the follow- ing results are arrived at. Each diffracted ray is plane-polarized, and the plane of polari- zation is determined by this law ; The plane of vibration of the diffracted ray is parallel to the direction of vibration of the incident ray. The expression plane of vibration is here used to denote the plane passing through the ray and the direction of vibration. The direction of vibration in any diffracted ray being determined by the law above mentioned, the phase and coefficient of vibration at that part of a secondary wave are given by the formulae of Art. 33. The law just enunciated seems to lead to a crucial experiment for deciding between the two rival theories respecting the directions of vibration in plane-polarized light. Suppose the plane of polarization, and consequently the plane of vibration, of the incident light to be turned round through equal angles of say 5° at a time. Then, according to theory, the planes of vibration of the diffracted ray will not be distributed uniformly, but will be crowded towards the plane perpendicular to the plane of diffraction, or that which contains the incident and diffracted rays. The law and amount of the crowding will in fact be just the same as if the planes of vibration of the incident ray were represented in section on a plane perpen- dicular to that ray, and then projected on a plane perpendicular to the diffracted ray. Now experiment will enable us to decide whether the planes of polarization of the diffracted ray are crowded towards the plane of diffraction or towards the plane perpendicular to the plane of diffraction, and we shall accordingly be led to conclude, either that the direction of vibration is perpendicular, or that it is parallel to the plane of polarization. In ordinary cases of diffraction, the light is insensible at such a small distance from the direction of the incident ray produced that the crowding indicated by theory is too small to be detected by experiment. It is only by means of a fine grating that we can obtain light of considerable intensity which has been diffracted at a large angle. On mentioning to my friend, Professor Miller, the result at which I had arrived, and making some inquiries about the fineness, &c. of gratings, he urged me to perform the experiment myself, and kindly lent me for the purpose a fine glass grating, which he has in his possession. For the use of two graduated instruments employed in determining the positions of the planes of polarization of the incident and diffracted rays I am indebted to the kindness of my friend Professor O'Brien. The description of the experiments, and the discussion of the results, occupies Part II. of this Paper. Since in a glass grating the diffraction takes place at the common surface of two different media, namely, air and glass, the theory of Part I. does not quite meet the case. Nevertheless it does not fail to point out whereabouts the plane of polarization of the diffracted ray ought to lie, according as we adopt one or other of the hypotheses respecting the direction of vibration. For theory assigns exact results on the two extreme suppositions, first, that the diffraction takes place before the light reaches the grooves; secondly, that it takes place after the light has passed between them ; and these results are very different, according as we suppose the vibrations to be perpendicular or parallel to the plane of polarization. Most of PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 5 the experiments were made on light which was diffracted in passing through the grating. The results appeared to be decisive in favour of Fresnel's hypothesis. In fact, theory shews that diffraction at a large angle is a powerful cause of crowding of the planes of vibration of the diffracted ray towards the perpendicular to the plane of diffraction, and experiment pointed out the existence of a powerful cause of crowding of the planes of polarization towards the plane of diffraction ; for not only was the crowding in the contrary direction due to refraction overcome, but a considerable crowding was actually produced towards the plane of diffraction, especially when the grooved face of the glass plate was turned towards the incident light. The experiments were no doubt rough, and are capable of being repeated with a good deal more accuracy by making some small changes in the apparatus and method of observing. Nevertheless the quantity with respect to which the two theories are at issue is so large that the experiments, such as they were, seem amply sufficient to shew which hypothesis is discarded by the phenomena. The conclusive character of the experimental result with regard to the question at issue depends, I think, in a great measure on the simplicity of the law which forms the only result of theory that it is necessary to assume. This law in fact merely asserts that, whereas the direction of vibration in the diffracted ray cannot be parallel to the direction of vibration in the incident ray, being obliged to be perpendicular to the diffracted ray, it makes with it as small an angle as is consistent with the above restriction. This law seems only just to lie beyond the limits of the geometrical part of the theory of undulations. At the same time I may be permitted to add that, for my own part, I feel very great confidence in the equations of motion of the luminiferous ether in vacuum, and in that view of the nature of the ether which would lead to these equations, namely, that in the propagation of light, the ether, from whatever reason, behaves like an elastic solid. But when we consider the mutual action of the lumi- niferous ether and ponderable matter, a wide field, as it seems to me, is thrown open to conjecture. Thus, to take the most elementary of all the phenomena which relate to the action of transparent media on light, namely, the diminution of the velocity of propagation, this diminution seems capable of being accounted for on several different hypotheses. And if this elementary phenomenon leaves so much room for conjecture, much more may we form various hypotheses as to the state of things at the confines of two media, such as air and glass. Accordingly, conclusions in favour of either hypothesis which are derived from the comparison of theoretical and experimental results relating to the effects of reflection and refraction on the polarization of light, appear to me much more subject to doubt than those to which we are led by the experiments here described. In commencing the theoretical investigation of diffraction, I naturally began with the simpler case of sound. As, however, the results which I have obtained for sound are of far less interest than those which relate to light, I have here omitted them, more especially as the paper has already swelled to a considerable size. I may, perhaps, on some future occasion bring them before the notice of this Society. PART I. THEORETICAL INVESTIGATION. Section I. PRELIMINAEY ANALYSIS. 1. In what follows there will frequently be occasion to express a triple integration which has to be performed with respect to all space, or at least to all points of space for which the quantity to be integrated has a value different from zero. The conception of such an integration, regarded as a limiting summation, presents itself clearly and readily to the mind, without the consideration of co-ordinates of any kind. A system of co-ordinates forms merely the machinery by which the integration is to be effected in particular cases ; and when the function to be integrated is arbitrary, and the nature of the problem does not point to one system rather than another, the employment of some particular system, and the analytical expression thereby of the function to be integrated, serves only to distract the attention by the introduction of a foreign element, and to burden the pages with a crowd of unnecessary symbols. Accordingly, in the case mentioned above, I shall merely take dV to represent an element of volume, and write over it the sign fff, to indicate that the integration to be performed is in fact triple. Integral signs will be used in this manner without limits expressed when the integration is to extend to all points of space for which the function to be integrated differs from zero. There will frequently be occasion too to represent a double integration which has to be performed with reference to the surface of a sphere, of radius r, described round the point which is regarded as origin, or else a double integration which has to be performed with reference to all angular space. In this case the sign ff will be used, and dS will be taken to represent an element of the surface of the sphere, and da- to represent an elementary solid angle, measured by the corresponding element of the surface of a sphere described about its vertex with radius unity. Hence, if dV, dS, da denote corresponding elements, dS = r*da, dV = drdS = r"drda. When the signs fff and ff, referring to differentials which are denoted by a single symbol, come together, or along with other integral signs, they will be separated by a dot, as for example fff. ffUd Vda. d 2 d? d? 2. As the operation denoted by - — - + — ■ + — - will be perpetually recurring in this paper, I shall denote it for shortness by V- This operation admits of having assigned to it a geometrical meaning which is independent of co-ordinates. For if P be the point {x, y, z), T a small space containing P, which will finally be supposed to vanish, dn an element of a normal drawn outwards at the surface of T, U the function which is the subject of the d? d* d 2 operation, and if V be defined as the equivalent of — + — ; + , it is easy to prove that da? dy i dz" PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 7 rr d £ ds . "J dn V tf= limit of - , (1) the integration extending throughout the surface of T, of which dS is an element. In fact, if I, m, n be the direction-cosines of the normal, we shall have "dU ,„ rrf,dU dU dU ff—dS- ff(l~ + m~~ n—-)dS J J dn J J \ dx dy dz) -IJfjy d%+ IS d ^ dxdx+ lS d TJ a,dy (2)> We have also, supposing the origin of co-ordinates to be at the point P, as we may without loss of generality, — = (— ] + ( — - ] x + [- — - ) y + [- — — ] * + terms of the 2nd order, &c. (3), dx \dxj \dx 1 \dxdyj \dxd«/ where the parentheses denote that the differential coefficients which are enclosed in them have rrdU the values which belong to the point P. In the integral -r-dy dz, each element must J J dx be taken positively or negatively, according as the normal which relates to it makes an acute or an obtuse angle with the positive direction of the axis of w. If we combine in pairs the elements of the integral which relate to opposite elements of the surface of T, we must write II — — - - — — dydz, where the single and double accents subscribed refer respectively to the first and second points in which the surface of T is cut by an indefinite straight line drawn parallel to the axis of x, and in the positive direction, through the point (o, y, z). We thus get by means of (3), omitting the terms of a higher order than the first, which vanish in the limit, rrfdU dll\ , (d*U\ rr Uw: - -ix") ** *• = u$ ifa - •-) * **> But ff(x /t - x t ) dy dz is simply the volume T. Treating in the same manner the two other integrals which appear on the right-hand side of equation (2), we get Dividing by T and passing to the limit, and omitting the parentheses, which are now no longer necessary, we obtain the theorem enunciated. If in equation (l) we take for T the elementary volume r 2 sin 6 drd9d(p, or rdrdOdz, according as we wish to employ polar co-ordinates, or one of three rectangular co-ordinates combined with polar co-ordinates in the plane of the two others, we may at once form the expression for yd, and thus pass from rectangular co-ordinates to either of these systems without the trouble of the transformation of co-ordinates in the ordinary way. 8 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 3. Let /be a quantity which may be regarded as a function of the rectangular co- ordinates of a point of space, or simply, without the aid of co-ordinates, as having a given value for each point of space. It will be supposed that / vanishes outside a certain portion T of infinite space, and that within T it does not become infinite. It is required to determine a function U by the conditions that it shall satisfy the partial differential equation VUmf. (4). at all points of infinite space, that it shall nowhere become infinite, and that it shall vanish at an infinite distance. These conditions are precisely those which have to be satisfied by the potential of a finite / mass whose density is ; and we shall have accordingly, if be the point for which the 47T value of U is required, and r be the radius vector of any element drawn from 0, tt. a —km"- (5) - In fact, it may be proved, just as in the theory of potentials, that the expression for U given by (5) does really satisfy (4) and the given conditions; and consequently, if U + U' be the most general solution, U' must satisfy the equation y V = at all points, must nowhere become infinite, and must vanish at an infinite distance. But this being the case it is easy to prove that V' cannot be different from zero. The solution will still hold good in certain cases when / is infinite at some points, or when it is not confined to a finite space T, but only vanishes at an infinite distance. But such instances may be regarded as limiting cases of the problem restricted as above, and therefore need not be supposed to be excluded by those restrictions. 4. Let U be a quantity depending upon the time t, as well as upon the position of the point of space to which it relates, and satisfying the partial differential equation d*U d -=a=y£7. ........ (6). It is required to determine U by the above equation and the conditions that when t = 0, U and — — shall have finite values given arbitrarily within a finite space T, and shall vanish outside T. Let O be the point for which the value of U is sought, r the radius vector of any element drawn from O; f(r), F(r) the initial values of V, — . By this notation it is not meant that these values are functions of r alone, for they will depend likewise upon the two angles which determine the direction of r; but there will be no occasion to express analytically their dependance on those angles. The solution of the problem is U=~ffF(at)d ff + ^-~tfff(at)da (7). 47T 47T at See a memoir by Poisson Mem. de fAcadimie, Tom. HI. p. 130, or Gregory's Examples, p. 499. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 9 5. Let He a function which has given finite values within a finite portion of space, and vanishes elsewhere ; and let it be required to determine three functions £, »/, £ by the conditions i= 0; (8) - dy dt] d% di( dt] dz dz das da; dx dy dz (9). The functions £, t], £ are further supposed not to become infinite, and to vanish at an infinite distance. To save repetition, it will here be remarked, once for all, that the same supposition will be made in similar cases. By virtue of equations (8), £ dx + r/dy + t£dz is an exact differential d\}/, and (9) gives y \|/ = $. Hence we have by the formula (5) *=-hlII~r dV - CM), and \// being known, £, t], £ will be obtained by mere differentiation. To differentiate \j/ with respect to x, it will be sufficient to differentiate § under the integral sign. For draw 00' parallel to the axis of x, and equal to A*, let P, P' be two points similarly situated with respect to 0, O', respectively, and consider the part of \^ and that of yjs + A \i/ due to equal elements of volume d V situated at P, P' respectively. For these two elements r has the same value, since OP = O'P", and in passing from the first to the second S is changed into 3 + AS, and therefore the increment of ^ is simply d V. To get the complete increment of \^ we have only to perform the triple integration, an integration which is always real, even though r vanishes in the denominator, as may be readily seen on passing momentarily to polar co-ordinates. Dividing now by Ax and passing to the limit, we get dS dV e-tt— J- m*l*i dt). * dx 4tt JJJ dx r By employing temporarily rectangular co-ordinates in the triple integration, integrating by parts with respect to x, and observing that the quantity free from the integral sign vanishes at the limits, we get ^-hJIS^ C0 ^ rx)dv - (12) ' as might have been readily proved from (10), by referring Oto a fixed origin, and then differen- tiating with respect to x. The expressions for rj and £ may be written down from symmetry. 6. Let •ar', w ", nr"' be three functions which have given finite values throughout a finite space and vanish elsewhere ; it is required to determine three other functions £, v, t by the conditions dt dt] , dP dt „ dt] dP „, ,_ -T -^~ = 2 *r, -A" 7^ = 27«r", - r L--±=2 W "\ (13). dy dz dz dx dx dy dP dn dT , . dx dy dz Vol. IX. Part I. 2 10 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. It is to be observed that •&', tst", nr" are not independent. For differentiating equations (13) with respect to x, y, z, and adding, we get d-ar d-ar" dw"' , v dm dy dz ■ Hence •&', ■&"> w" must be supposed given arbitrarily only in so far as is consistent with i the above equation. Eliminating £from 0*)> ano ^ ^ e second of equations (13), we get d fdf dr,\ d?£ dsr" 1 + — t = 2- d td% dr)\ Ix \dx dy) dx \dx dy) dz 2 dz „ d Idn df\ dip" or v I +-T- h -- -r\ = 2 ^r~' dy \dx dy) dz which becomes by the last of equations (13) Consequently, by equation (5), Idsr" dw"\ d V i-hm dy dz j v Transforming this equation in the same manner as (11), supposing x, y, z measured from O, and writing down the two equations found by symmetry, we have finally, £=^///(^"'-*^")^ (16). . 7. Let d, tst', w", "&"' be as before ; and. let it be required to determine three functions £, rj, £ from the equations (9) and (13). From the linearity of the equations it is evident that we have merely to add together the expressions obtained in the last two articles. 8. Let £ , r) , £ be three functions given arbitrarily within a finite space outside of which they are equal to zero: it is required to decompose these functions into two parts £„ jj„ £, and £21 »?a. £2 suc h tnat %\dx + r) x dy + £i4m may be an exact differential d\|/„ and f 2 , t) 2 , £ 2 may satisfy (14). Observing that £ 2 = £, - £, ~ = % - ^ £ 2 = £ - £„ expressing (•„ m , £, in terms of f u and substituting in (14), we get where S is what 5 becomes when £ , t; , £ are written for £, >;, £. The above equation gives whence £„ >;„ £„ and consequently £ 2 , % , £ 2 , are known. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 11 Section II. PROPAGATION OF AN ARBITRARY DISTURBANCE IN AN ELASTIC MEDIUM. 9. The equations of motion of a homogeneous uncrystallized elastic medium, such as an elastic solid, in which the disturbance is supposed to be very small, are well known. They contain two distinct arbitrary constants, which cannot be united in one without adopting some particular physical hypothesis. These equations may be obtained by supposing the medium to consist of ultimate molecules, but they by no means require the adoption of such a hypothesis, for the same equations are arrived at by regarding the medium as continuous. Let x, y, z be the co-ordinates of any particle of the medium in its natural state ; £, r),t the displacements of the same particle at the end of the time t, measured in the directions of the three axes respectively. Then the first of the equations may be put under the form dt 2 \dx< dy* dz 2 ) K ' dx \dx dy dz)' where a 2 , b 2 , denote the two arbitrary constants. Put for shortness dp dn dt k dx dy dz and as before represent by v£ tne quantity multiplied by b 2 . According to this notation, the three equations of motion are or ay (18). It is to be observed that 8 denotes the dilatation of volume of the element situated at the point (x, y, z). In the limiting case in which the medium is regarded as absolutely incom- pressible 8 vanishes; but in order that equations (18) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 8 by a new function of the co- ordinates. If we take — p to denote this function, we must replace the last terms in these dp dp dp equations by — — , — — , — -— , respectively, and we shall thus have a fourth unknown dx dy dz J function, as well as a fourth equation, namely that obtained by replacing the second member of (17) by zero. But the retention of equations (18) in their present more general form does not exclude the supposition of incompressibility, since we may suppose a to become infinite in the end just as well as at first. 10. Suppose the medium to extend infinitely in all directions, and conceive a portion of it occupying the finite space T to receive any arbitrary small disturbance, and then to be left to itself, the whole of the medium outside the space T being initially at rest ; and let it be required to determine the subsequent motion. 2 — 2 12 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. Differentiating equations (18) with respect to w, y, «, respectively, and adding, we get by virtue of (17) %-^- <•»■ Again, differentiating the third of equations (18) with respect to y, and the second with respect to z, and subtracting the latter of the two resulting equations from the former, and treating in a similar manner the first and third, and then the second and first of equations (18), we get ~SF'^^ -JT^-"' ^~ = bvw ' ■ • (20) - where ■&', w", ■&"' are the quantities defined by equations (13). These quantities express the rotations of the element of the medium situated at the point {x, y, z) about axes parallel to the three co-ordinate axes respectively. Now the formula (7) enables us to express $, w', ■&", and w'" in terms of their initial values and those of their differential coefficients with respect to t, which are supposed known ; and these functions being known, we shall determine f, y, and £ as in Art. 7- Our equations being thus completely integrated, nothing will remain but to simplify and discuss the formulae obtained. 11. Let be the point of space at which it is required to determine the disturbance, r the radius vector of any element drawn from O ; and let the initial values of $, — be represented at by f(r), F(r), respectively, with the same understanding as in Art 4. By the formula (7), we have *-$- RF (?*)*«+ ±-%-tfff{at)d* . . . .(21). 47T 47T at The double integrals in this expression vanish except when a spherical surface described round O as centre, with a radius equal to at, cuts a portion of the space T. Hence, if O be situated outside the space T, and if r ir r 2 be respectively the least and greatest values of the radius vector of any element of that space, there will be no dilatation at until at = »y The dilatation will then commence, will last during an interval of time equal to a -1 (r 2 - r,), and will then cease for ever. The dilatation here spoken of is understood to be either positive or negative, a negative dilatation being the same thing as a condensation. Hence a wave of dilatation will be propagated in all directions from the originally dis- turbed space T, with a velocity a. To find the portion of space occupied by the wave, we have evidently only got to conceive a spherical surface, of radius at, described about each point of the space T as centre. The space occupied by the assemblage of these surfaces is that in which the wave of dilatation is comprised. To find the limits of the wave, we need evidently only attend to those spheres which have their centres situated in the surface of the space T. When t is small, this system of spheres will have an exterior envelope of two sheets, the outer of these sheets being exterior, and the inner interior to the shell formed by the assemblage of the spheres. The outer sheet forms the outer limit to the portion of the medium in which the dilatation is different from zero. As t increases, the inner sheet con- tracts, and at last its opposite sides cross, and it changes its character from being exterior, PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 13 with reference to the spheres, to interior. It then expands, and forms the inner boundary of the shell in which the wave of condensation is comprised. It is easy to shew geometrically that each envelope is propagated with a velocity a in a normal direction. 12. It appears in a similar manner from equations (20) that there is a similar wave, propagated with a velocity b, to which are confined the rotations •&', w", -nr"'. This wave may be called for the sake of distinction, the wave of distortion, because in it the medium is not dilated nor condensed, but only distorted in a manner consistent with the preservation of a constant density. The condition of the stability of the medium requires that the ratio of b to a be not greater than that of ^/3 to 2 *. 13. If the initial disturbance be such that there is neither dilatation nor velocity of dilatation initially, there will be no wave of dilatation, but only a wave of distortion. If it be such that the expressions %d% + r,dy + X^dss and -j-d,v + ——dy + — - dz are initially exact differ- UV (JbZ (XZ entials, there will be no wave of distortion, but only a wave of dilatation. By making 6 = we pass to the case of an elastic fluid, such as air. By supposing a = eo we pass to the case of an incompressible elastic solid. In this case we must have initially $ = and — = ; but in order €LZ that the results obtained by at once putting a = oo may have the same degree of generality as those which would be obtained by retaining a as a finite quantity, which in the end is supposed to increase indefinitely, we must not suppose the initial disturbance confined to the space T, but only the initial rotations and the initial angular velocities. Consequently, outside . T the expression %dx + rjdy + "^dx must be initially an exact diffei-ential d\|/, where \j/ satisfies the equation y\^ = o derived from (14), and the expression — dx + -—dy+-^ dz must be CLZ u Z CLZ initially an exact differential d^r x , where \^/ l satisfies the equation y \^, = 0. So long as a is finite, it comes to the same thing whether we regard the medium as animated initially by certain velocities given arbitrarily throughout the space T, or as acted on by impulsive accelerating forces capable of producing those velocities ; and the latter mode of conception is equally applicable to the case of an incompressible medium, for which a is infinite, although we cannot in that case conceive the initial velocities as given arbitrarily, but only arbitrarily in so far as is compatible with their satisfying the condition of incompressibility. It is not so easy to see what interpretation is to be given, in the case of an incompressible medium, to the initial dis- placements which are considered in the general case, in so far as these displacements involve dilatation or condensation. As no simplicity worth mentioning is gained by making a at once infinite, this constant will be retained in its present shape, more especially as the results arrived at will thus have greater generality. 14. The expressions for the disturbance of the medium at the end of the time t are linear functions of the initial displacements and initial velocities; and it appears from (21), and the corresponding equations which determine •&', -^' ', and •ar'", that the part of the disturbance which is due to the initial displacements may be obtained from the part which is due to the • See a memoir by Green, On the reflexion and refraction of Light. Camb. Phil. Trans. Vol. vn. p. 2. See also Camb. Phil. Trans. Vol. vm. p. 319. 14 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. initial velocities by differentiating with respect to t, and replacing the arbitrary functions which represent the initial velocities by those which represent the initial displacements. The same result constantly presents itself in investigations of this nature : on considering its physical interpretation it will be found to be of extreme generality. Let any material system whatsoever, in which the forces acting depend only on the posi- tions of the particles, be slightly disturbed from a position of equilibrium, and then left to itself. In order to represent the most general initial disturbance, we must suppose small initial displacements and small initial velocities, the most general possible consistent with the connexion of the parts of the system, communicated to it. By the principle of the superposi- tion of small motions, the subsequent disturbance will be compounded of the disturbance due to the initial velocities and that due to the initial displacements. It is immaterial for the truth of this statement whether the equilibrium be stable or unstable ; only, in the latter case, it is to be observed that the time t which has elapsed since the disturbance must be sufficiently small to allow of our neglecting the square of the disturbance which exists at the end of that time. Still, as regards the purely mathematical question, for any previously assigned interval t, how- ever great, it will be possible to find initial displacements and velocities so small that the disturbance at the end of the time t shall be as small as we please ; and in this sense the prin- ciple of superposition, and the results which flow from it, will be equally true whether the equilibrium be stable or unstable. Suppose now that no initial displacements were communicated to the system we are con- sidering, but only initial velocities, and that the disturbance has been going on during the time t. Let/(£) be the type of the disturbance at the end of the time t, where f(t) may represent indifferently a displacement or a velocity, linear or angular, or in fact any quantity whereby the disturbance may be defined. In the case of a rigid body, or a finite number of rigid bodies, there will be a finite number of functions f(t) by which the motion of the system will be defined : in the cases of a flexible string, a fluid, an elastic solid, &c, there will be an infinite number of such functions, or, in other words, the motion will have to be defined by functions which involve one or more independent variables besides the time. Let « be in a similar manner the type of the initial velocities, and let t be an increment of t, which in the end will be supposed to vanish. The disturbance at the end of the time t + t will be represented by fit + t) ; but since by hypothesis the forces acting on the system do not depend explicitly on the time, this disturbance is the same as would exist at the end of the time t in consequence of the system of velocities v a communicated to the material system at the commencement of the time — t, the system being at that instant in its position of equilibrium. Suppose then the system of velocities « communicated in this manner, and in addition suppose the system of velocities - v communicated at the time 0. On account of the smallness of the motion, the disturbance produced by the system of velocities v will be expressed by linear functions of these velocities; and consequently, if/(£) represent the disturbance due to the system of velo- cities v , - f(t) will represent the disturbance due to the system - v . Hence the disturbance at the end of the time t will be represented by f[t + t) -f(t). Now we may evidently regard the state of the material system immediately after the communication of the system of velocities - v as its initial state, and then seek the disturbance which would be produced by the initial disturbance. The velocities v going on during the time t will have produced PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 15 by the end of that time a system of displacements represented by rv . By hypothesis, the system was in a position of equilibrium at the commencement of the time - t ; and since the forces are supposed not to depend on the velocities, but only on the positions of the particles, the effective forces during the time r vary from zero to small quantities of the order t, and therefore the velocities generated by the end of the time — t are small quantities of the order t 2 . Hence the velocities — v g communicated at the time destroy the previously existing velocities, except so far as regards small quantities of the order t 2 , which vanish in the limit, and therefore we have nothing to consider but the system of displacements tv . Hence the disturbance produced by a system of initial displacements rv Q is represented by f(t + r) -fit), ultimately ; and therefore the disturbance produced by a system of initial displacements v is represented by the limit of ■r~ 1 {f(t + t) - /(/) } , or by f (t). Hence, to get the disturbance due to the initial displacements from that due to the initial velocities, we have only to differen- tiate with respect to t, and to replace the arbitrary constants or arbitrary functions which express the initial velocities by those which express the corresponding initial displacements. Conversely, to get the disturbance due to the initial velocities from that due to the initial displacements, we have only to change the arbitrary constants or functions, and to integrate with respect to t, making the integral vanish with t if the disturbance is expressed by displace- ments, or correcting it so as to give the initial velocities when t = if the disturbance is ex- pressed by velocities. The reader may easily, if he pleases, verify this theorem on some dynamical problem relating to small oscillations. 15. Let us proceed now to determine the general values of £, n, Y in terms of their initial values, and those of their differential coefficients with respect to t. By the formulae of Section I., £, t], £ are linear functions of $, •&', tu", and ■ar'", and we may therefore first form the part which depends upon $, and afterwards the part which depends upon -Br', ■&", -Br"', and then add the results together. Moreover, it will be unnecessary to retain the part of the expressions which depends upon initial displacements, since this can be supplied in the end by the theorem of the preceding article. Omitting then for the present -ar', tst", sr'", as well as the second term in equations (2l), we get from equations (10) and (21), *~^su-ifT*™° » To understand the nature of the integration indicated in this equation, let be the point of space for which the value of \f/ is sought ; from O draw in an arbitrary direction OP equal to r, and from P draw, also in an arbitrary direction, PQ equal to at. Then F (at) denotes the value of the function F, or the initial rate of dilatation, at the point Q of space, and we have first to perform a double integration referring to all such points as Q, P being fixed, and then a triple integration referring to all such points as P. To facilitate the transformation of the integral (22), conceive PQ produced to Q', let PQ' = s, let d V be an element of volume, and replace the double integral ffF.da by the triple integral h~ l fffF.s' 2 dV, taken between the limits defined by the imparities at at. Substituting in (22), and omitting the accents, which are now no longer necessary, we get + = -^~fff F - dV ( rat) ■ • • (23) - where the limits of integration are defined by the imparities written after the integrals, as will be done in similar cases. 16. Let u , v , w , be the initial velocities; then du dv dw dx dy dss Substituting in the first term of the right-hand member of equation (23), and integrating by parts, exactly as in Art. 5, we get -^affI F - dr ^ at ^^ jfifiw-- w.l d y d * dV - — fff( u o a! + %y + "><>*) -5- ( r > °0- The double integrals arising from the transformation of the second member of equation (23) destroy one another, and we get finally PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 17 ' rrr, . d V \// = III (u x + v y + w z) ~{r>af) (24). 17. To obtain the part of the displacement £ due to the initial velocity of dilatation, we have only to differentiate \|/ with respect to x, and this will be effected by differentiating u , v , w under the integral signs, as was shewn in Art. 5. Treating the resulting expres- sion by integration by parts, as before, and putting I, m, n for the direction-cosines of the radius vector drawn to the point to which the accents refer, and £, for the part of ? due to F, we get £i = v. //{('«• + i""o+ nw o)„ ~ ( lu o + mv o + nw ) } dy dz 4 wO t J J t rrrf d x d y d ss\ + 7~ /// \ u °7--3 + "»r^ + M, ori \dV(r>at). 47r JJJ \ das r 3 dwr door 1 } Let q 9 be the initial velocity revolved along the radius vector, so that q =*lu + mv + nw , and let (q )at be the value of q at a distance a t from O ; then ff{(lu + mv + nw a ) it - (lu + mv + «w ),} dy dss = ffl (q^ dS = effffl (q )at d d x d y d % u — 3lq and u — - + v — - + w — - = . °dxr % dxr 3 dxr* r Substituting in the expression for f u we get finally ^ = l r If l ^)a t da + ~fff(u Q -3lq ) d ^(r>at) (25), 18. Let us now form the part of f which depends on the initial rotations and angular velocities, and which may be denoted by £ 2 . The theorem of Art. 14 allows us to omit for the present the part due to the initial rotations, which may be supplied in the end. Let <*>o'> w o " ■> w o" be the initial angular velocities. Then £ 2 is given in terms of w" and •ar'" by the first of equations (16), and -ar", ■ar"' are given in terms of w ", w '" by the formula (7), in which however b must be put for a. We thus get The integrations in this expression are to be understood as in Art. 15, and w ", w '" are supposed to have the values which belong to the point Q, but PQ is now equal to bt instead of at. The quintuple integral may be transformed into a triple integral just as before. We get in the first place i-Mffhy r ff"-£-fif« ir ff"-£)- ■ ■ ■ ™- The double integration in this expression refers to all angular space, considered as extend- ing round Q; x, y, % are the co-ordinates, measured from 0, of a point P situated at a distance bt from Q, and r = OP. If dS - {btfda, the expressions for the integrals ffxr~ 3 dS, ffyr~ 3 dS, ffzr~ 3 dS Vol. IX. Part I. 3 18 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. may be at once written down by observing that these integrals express the components of the attraction of a spherical shell, of radius bt and density 1, having Q for centre, on a particle situated at 0. Hence if w, y', z' be the co-ordinates of Q, measured from O, and r = OQ, the integrals vanish when r < bt, and are equal to ^{btfx r'~\ i,r(btyy'r'- 3 , ^{btf z'r'- 3 , respectively, when r>bt. Hence we get from (26), omitting the accents, which are now no longer necessary, since we have done with the point P, & = £fff(<»°"y-«>o'*)^(r>bt) (27). Now , dw dv „ du dw „, dv du 2w = -- — ; 2w =— — ; 2a> = - — , ay dz dz dx dx dy du Substituting in (27), and adding and subtracting x — — under the integral signs, we get dx t rrr\ f d d d\ j du dv dw \\ dV ^rj}I\-{ X Tx + y Ty^d7.) U ^{ V Jx- + y Jx +Z ^))^ r>ht) - d d d d But x— - + y-— + z— is the same thine as r -— , and we get accordingly dx * dy dz 8 dr 8 ° J The second part of £ a is precisely the expression transformed in the preceding article, except that the sign is changed, and b put for a. Hence we have ^ = —ff(u -lq ) bt dbt). . . (28). 19. Adding together the expressions for £, and £>, we get for the disturbance due to the initial velocities l m ^}J l (lo)atdv + ^ffao ~ l 1oXt d • ( 2 9>- The part of the disturbance due to the initial displacements may be obtained immediately by the theorem of Art. 14. Let £ , ij , £ be the initial displacements, p the initial dis- placement resolved along a radius vector drawn from 0. The last term in equation (2.9), it will be observed, involves t in two ways, for t enters as a coefficient, and likewise the limits depend upon t. To find the part of the differential coefficient which relates to the variation of the limits, we have only to replace dV hy r'drda, and treat the integral in the usual way. We get for the part of the disturbance due to the initial displacements ?=s//H ( M4:)-E-b»s//H^'f-'(v.*»f)l'« dV + ^fjJ(* l f*-fa-jr(i>t and an arbitrary radius vector drawn from 0, whose direction varies from one element da of angular space to another, and that the at or bt subscribed denotes that r is to be supposed equal to at or bt after differentiation. To obtain the whole displacement parallel to x which exists at the end of the time t at the point O, we have only to add together the second members of equations (29) and (30). The expressions for r\ and £ may be written down from symmetry, or rather the axis of x may be supposed to be measured in the direction in which we wish to estimate the displacement. 20. The first of the double integrals in equations (29), (30) vanishes outside the limits of the wave of dilatation, the second vanishes outside the limits of the wave of distortion. The triple integrals vanish outside the outer limit of the wave of dilatation, and inside the inner limit of the wave of distortion, but have finite values within the two waves and between them. Hence a particle of the medium situated outside the space T does not begin to move till the wave of dilatation reaches it. Its motion then commences, and does not wholly cease till the wave of distortion has passed, after which the particle remains absolutely at rest. 21. If the initial disturbance be such that there is no wave of distortion, the quantities •ar', "sr", "&'", w, to", w" must be separately equal to zero, and the expression for P will be reduced to £ 15 given by (25), and the expression thence derived which relates to the initial displacements. The triple integral in the expression for (-, vanishes when the wave of dilatation has passed, and the same is the case with the corresponding integral which depends upon the initial displacements. Hence the medium returns to rest as soon as the wave of dilatation has passed ; and since even in the general case each particle remains at rest until the wave of dilatation reaches it, it follows that when the initial disturbance is such that no wave of distortion is formed the disturbance at any time is confined to the wave of dilatation. The same conclusion might have been arrived at by transforming the triple integral. 22. When the initial motion is such that there is no wave of dilatation, as will be the case when there is initially neither dilatation nor velocity of dilatation, £ will be reduced to £ 2 , given by (28), and the corresponding expression involving the initial displacements. By referring to the expression in Art. 17, from which the triple integral in equation (28) was derived, we get d V rrr I d w d y d Now fff^-"^-fr-ffj('-ii?*"Si?->-'"r.?) ir - ■ ■ (31) - W+ S ? dv - Iff"- S ? <"><~ff{l?} «"-fff£ ? "<>«• the parentheses denoting that the quantity enclosed in them is to be taken between limits. By the condition of the absence of initial velocity of dilatation we have 3—2 20 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. Substituting in the second member of equation (31), and writing down for the present only the terms involving « , we obtain !SS{t?+*k ?)<"><•< ^dy , . , . & y da. which, since = — — , becomes dx r 3 dy r 3 filly- ^*»**** or Jf &)*"**'' Treating the terms involving w in the same manner, and substituting in (31), we get fff(u,-S lq „ ) d -I-ff^)^ + ff(^)^>ff{^)^ ! ,. Now the integration is to extend from r = bt to r = SO. The quantities once integrated vanish at the second limit, and the first limit relates to the surface of a sphere described round as centre with a radius equal to bt. Putting dS or b 2 t 2 dtr for an element of the surface of this sphere, we obtain for the value of the second member of the last equation - (bt)-*ff(lu + mv + nw )JdS, or - ffl \q a ) bt da ; and therefore the triple integral in equation (28) destroys the second part of the double integral in the same equation. Hence, writing down also the terms depending upon the initial displacements, we obtain for P the very simple expression d f = he ffM*** + ^ ' tf^» d °- This expression might have been obtained at once by applying the formula (7) to the first of equations (18), which in this case take the form (6), since h = 0. 23. Let us return now to the general case, and consider especially the terms which alone are important at a great distance from the space to which the disturbance was originally con- fined ; and, first, let us take the part of £ which is due to the initial velocities, which is given by equation (29). Let the three parts of the second member of this equation be denoted by £ a , £ 6 , £ c , respec- tively,and replace da by (at)~ Q dS or (bt)~ 2 dS, as the case may be; then t-'^iff'toW- (32) " Let Oi be a fixed point, taken within the space T, and regarded as the point of reference for all such points as 0. Then when is at such a distance from O y that the radius vector, drawn from O, of any element of T makes but a very small angle with 00 15 we may regard I as constant in the integration, and equal to the cosine of the angle between OO t and the direction in which we wish to estimate the displacement at O. Moreover the portion of the surface of a sphere having O for centre which lies within T will be ultimately a plane per- PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 21 pendicular to OO x , and q will be ultimately the initial velocity resolved in the direction 00^. Hence we have ultimately where, for a given direction of O x O, the integral receives the same series of values, as at increases through the value 00 x , whatever be the distance of from O v Since the direction of the axis of w is arbitrary, and the component of the displacement in that direction is found by multi- plying by I a quantity independent of the direction of the axes, it follows that the displacement itself is in the direction OO x , or in the direction of a normal to the wave. For a given direc- tion of OiO, the law of disturbance is the same at one distance as at another, and the magni- tude of the displacements varies inversely as at, the distance which the wave has travelled in the time t. We get in a similar manner lb^^jf t jji. u o-ho) b idS, (33). where I, and the direction of the resolved part, g , of the initial velocity are ultimately constant, and the surface of which dS is an element is ultimately plane. To find the resolved part of the displacement in the direction 00,, we must suppose a; measured in that direction, and therefore put I = 1, q = u , which gives £ 6 = 0. Hence the displacement now considered takes place in a direction perpendicular to 00 l} or is transversal. For a given direction of X 0, the law of disturbance is constant, but the magnitude of the displacements varies inversely as bt, the distance to which the wave has been propagated. To find the displacement in any direction, OE, perpendicular to 00„ we have only to take OE for the direction of the axis of w, and therefore put I = 0, and suppose u to refer to this direction. Consider, lastly, the displacement, £ c , expressed by the last term in equation (29). The form of the expression shews that £ c will be a small quantity of the order — or — , since t is of the same order as r ; for otherwise the space T would lie outside the limits of integration, and the triple integral would vanish. But £ a and % b are of the order - , and therefore £ c may be neglected, except in the immediate neighbourhood of T. To see more clearly the relative magnitudes of these quantities, let v be a velocity which may be used as a standard of comparison of the initial velocities, R the radius of a sphere whose volume is equal to that of the space T, and compare the displacements f a , % b , £ c which exist, though at different times, at the same point 0, where t ■ r. These displacements are comparable with which are proportional to vR? vR 2 vRH ar or r* ' 1 1 /.' a b r t r 22 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. But, in order that the triple integral in (29) may not wholly vanish, - must lie between the T limits - and - , or at most lie a very little outside these limits, which it may do in conse- nt 6 quence of the finite thickness of the two waves. Hence the quantity neglected in neglecting £ R is of the order — compared with the quantities retained. r The important terms in the disturbance due to the initial displacements might be got from equation (30), but they may be deduced immediately from the corresponding terms in the dis- turbance due to the initial velocities by the theorem of Art. 14. 24. If we confine our attention to the terms which vary ultimately inversely as the distance, and which alone are sensible at a great distance from T, we shall be able, by means of the formulae of the preceding article, to obtain a clear conception of the motion which takes place, and of its connexion with the initial disturbance. From the fixed point 0„ draw in any direction the right line 0,0 equal to r, r being so large that the angle subtended at by any two elements of T is very small ; and let it be required to consider the disturbance at 0. Draw a plane P perpendicular to 00 a and cutting 00i produced at a distance p from 0,. Let - p u + p 2 be the two extreme values of p for which the plane P cuts the space T. Conceive the displacements and velocities resolved in three rectangular directions, the first of these, to which £ and u relate, being the direction 00,. Let f u (p>, f v (|>), f a (p) be three functions of p defined by the equations /. (P) = jfadS, /, (p) = ffv dS, /. (p) - ffw dS, . . (34) and /j (p), f v (p), / f (p) three other functions depending on the initial displacements as the first three do on the initial velocities, so that ft GO = fftodS, /„ (J>) = //„, dS, ft (p) = //£, dS. . . . (35). These functions, it will be observed, vanish when the variable lies outside of the limits - p, and + p 2 . They depend upon the direction 0,0, so that in passing to another direction their values change, as well as the limits of the variable between which they differ from zero. It may be remarked however that in passing from any one direction to its opposite the functions receive the same values, as the variable decreases from + pi to — p 2 , that they before received as the variable increased from — p, to + p 2 , provided the directions in which the displacements are resolved, as well as the sides towards which the resolved parts are reckoned positive, are the same in the two cases. The medium about remains at rest until the end of the time a~ l (r-p l ), when the wave of dilatation reaches 0. During the passage of this wave, the displacements and velocities are given by the equations 4 irar %irr « - T— A' (at -r) + ~f $ "(at -r), v = 0, w = 0. 47TP %7TT (36). PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. 23 The first term in the right-hand member of the first of these equations is got from (32) by putting 1=1, introducing the function /„, and replacing at in the denominator by r, which may be done, since at differs from r only by a small quantity depending upon the finite dimensions of the space T. The second term is derived from the first by the theorem of Art. 14, and u is of course got from (• by differentiating with respect to t. Had t been re- tained in the denominator, the differentiation would have introduced terms of the order t~ 2 , and therefore of the order r -2 , but such terms are supposed to be neglected. The wave of dilatation will have just passed over O at the end of the time a~ l (r + p.^). The medium about O will then remain sensibly at rest in its position of equilibrium till the wave of distortion reaches it, that is, till the end of the time 6 _1 (r - p x ). During the passage of this wave, the displacements and velocities will be given by the equations ,fAbt-r)+-±-fi(bt-r), (37). 4nrbr " 47rr" U = °' * ' j£? fJ (6 ' " r) + ^r f "" ibt ~ r) ' 471-r 47tf After the passage of the wave of distortion, which occupies an interval of time equal to b' 1 (p\ + pi), the medium will return absolutely to rest in its position of equilibrium. 25. A caution is here necessary with reference to the employment of equation (30). If we confine our attention to the important terms, we get e- r^ ff l ( d ^r) ^+-^7 /y#-'£l *& ■ ( 38 )- 4>irat JJ \dr) at 4,TrbtJJ\dr drf bt v ' Now the initial displacements and velocities are supposed to have finite, but otherwise arbitrary, values within the space T, and to vanish outside. Consequently we cannot, without unwarrant- ably limiting of the generality of the problem, exclude from consideration the cases in which the initial displacements and velocities alter abruptly in passing across the surface of T. In particular, if we wish to determine the disturbance at the end of the time t due to the initial disturbance in a part only of the space throughout which the medium was originally disturbed, we are obliged to consider such abrupt variations ; and this is precisely what occurs in treating the problem of diffraction. In applying equation (38) to such a case, we must consider the abrupt variation as a limiting case of a continuous, but rapid, variation, and we shall have to add to the double integrals found by taking for -£- and —2 the finite values which refer to the space T, certain single integrals referring to the perimeter of that portion of the plane P which lies within T. The easiest way of treating the integrals is, to reserve the differen- tiation with respect to t from which the differential coefficients just written have arisen until after the double integration, and we shall thus be led to the formulae of the preceding article, 24 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. where the correct values of the terms in question were obtained at once by the theorem of Art. 14. 26. It appears from Arts. 11 and 12, that in the wave of distortion the density of the medium is strictly the same as in equilibrium ; but the result obtained in Art. 23, that the dis- placements in this wave are transversal, that is, perpendicular to the radius of the wave, is only approximate, the approximation depending upon the largeness of the radius, r, of the wave compared with the dimensions of the space T, or, which comes to the same, compared with the thickness of the wave. In fact, if it were strictly true that the displacement at due to the original disturbance in each element of the space T was transversal, it is evident that the crossing at of the various waves corresponding to the various elements of T under finite, though small angles, would prevent the whole displacement from being strictly perpendicular to the radius vector drawn to O from an arbitrarily chosen point, O lt within T. But it is not mathematically true that the disturbance proceeding from even a single point 0,, when a dis- turbing force is supposed to act, or rather that part of the disturbance which is propagated with the velocity b, is perpendicular to 00j, as will be seen more clearly in the next article. It is only so nearly perpendicular that it may be regarded as strictly so without sensible error. As the wave grows larger, the inclination of the direction of displacement to the wave's front decreases with great rapidity. Thus the motion of a layer of the medium in the front of a wave may be compared with the tidal motion of the sea, or rather with what it would be if the earth were wholly covered by water. In both cases the density of the medium is unchanged, and there is a slight increase or decrease of thickness in the layer, which allows the motion along the surface to take place without change of density : in both cases the motion in a direction perpendicular to the surface is very small compared with the motion along the surface. 27. From the integral already obtained of the equations of motion, it will be easy to deduce the disturbance due to a given variable force acting in a given direction at a given point of the medium. Let 0, be the given point, T a space comprising 0,. Let the time t be divided into equal intervals t; and at the beginning of the n th interval let the velocity tF(ht) be com- municated, in the given direction, to that portion of the medium which occupies the space T. Conceive velocities communicated in this manner at the beginning of each interval, so that the disturbances produced by these several velocities are superposed. Let D be the density of the medium in equilibrium; and let F(nr) = (2)7')" 1 /(»t), so that t/(«t) is the momentum communicated at the beginning of the n th interval. Now suppose the number of intervals t indefinitely increased, and the volume T indefinitely diminished, and we shall pass in the limit to the case of a moving force which acts continuously. The disturbance produced by given initial velocities is expressed, without approximation, by equation (29), that is, without any approximation depending on the largeness of the distance O0 a ; for the square of the disturbance has been neglected all along. Let 00i = r ; refer the displacement at to the rectangular axes of x, y, % ; let I, m, n be the direction- PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 25 cosines of 00! ; l\ m, n those of the given force, and put for shortness k for the cosine of the angle between the direction of the force and the line 00, produced, so that k = It + mm' + nri . Consider at present the first term of the right-hand side of (29). Since the radius vector drawn from O to any element of T ultimately coincides with 00,, we may put I outside the integral signs, and replace da by r~ 2 dS. Moreover, since this term vanishes except when at lies between the greatest and least values of the radius vector drawn from to any element of T, we may replace t outside the integral signs by a~ i r. Conceive a series of spheres, with radii ar, 2ar...nar,... described round 0, and let the n th of these be the first which cuts T. Let S u Sn... be the areas of the surfaces of the spheres, beginning with the » th , which lie within T; then ff(9o) at dS - krF(t - nr) S, + krF \t - (n + 1) T } S, 4- ... But F(t - nr), F {t - (n + 1) r] ... are ultimately equal to each other, and to *(»-3- ° r < Br >-''K> and arSi + arS 2 + ... is ultimately equal to T. Hence we get, for the part of f which Ik t t\ arises from the first of the double integrals, / \t — 1 . The second of the double ^■n-Da'r \ a) integrals is to be treated in exactly the same way. To find what the triple integral becomes, let us consider first only the impulse which was communicated at the beginning of the time t — nr, where m lies between the limits a^ l r and 6~'r, and is not so nearly equal to one of these limits that any portion of the space T lies beyond the limits of integration. Then we must write nr for t in the coefficient, and 3lq — u becomes ultimately {3lk — I') rF(t - nr), and, as well as r, is ultimately constant in the triple integration. Hence the triple integral ultimately becomes (3 Ik - I') T iTrr 3 nr . rF(t - nr), and we have now to perform a summation with reference to different values of n, which in the limit becomes an integration. Putting nr = if, we have ultimately t = c^, l.nr.rF(t-nr) = f r ] t' F {t - t') dt'. a It is easily seen that the terms arising from the triple integral when it has to be extended over a part only of the space T vanish in the limit. Hence we have, collecting all the terms, and expressing F (t) in terms o{f(t), > lk j.L r \ l'-lk ,/ r\ Slk-l' r l ,, , , To get tj and £, we have only to pass from /, t to m, m' and then to n, n. If we take 00 x for the axis of x, and the plane passing through 00i and the direction of the force for Vol. IX. Part I. 4 26 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. the plane x%, and put a for the inclination of the direction of the force to 00, produced, we shall have /=1,m = 0, n =0, I' = k= cos a, m! = 0, ri = sin a ; whence cos a ' / r\ cos a rl , „ , K , , v = o, ?™/H)-^/ ? <'/<<-6 2 r 3 X X V 2/ \ (41) and we see that the most important term in {• is of the order — compared with the 71-r leading term in £, which represents the transversal vibrations properly so called. Hence £, and the second and third terms in £, will be insensible, except at a distance from O x comparable with X, and may be neglected ; but the existence of terms of this nature, in the case of a spherical wave whose radius is not regarded as infinite, must be borne in mind, in order to understand in what manner transversal vibrations are compatible with the absence of dilatation or condensation. 30. The integration of equations (18) might have been effected somewhat differently by first decomposing the given functions £ , tj , £ > an d ^V^o* w a i nto two parts, as in Art. 8, and then treating each part separately. We should thus be led to consider separately that part of the initial disturbance which relates to a wave of dilatation and that part which relates to a wave of distortion. Either of these parts, taken separately, represents a disturbance which is not confined to the space T, but extends indefinitely around it. Outside T, the two disturb- ances are equal in magnitude and opposite in sign. Section III. DETERMINATION OP THE LAW OP THE DISTURBANCE IN A SECONDARY WAVE OP LIGHT. 31. Conceive a series of plane waves of plane-polarized light propagated in vacuum in a direction perpendicular to a fixed mathematical plane P. According to the undulatory theory of light, as commonly received, that is, including the doctrine of transverse vibrations, the light in the case above supposed consists in the vibrations of an elastic medium or ether, the vibra- tions being such that the ether moves in sheets, in a direction perpendicular to that of propa- gation, and the vibration of each particle being symmetrical with respect to the plane of polarization, and therefore rectilinear, and either parallel or perpendicular to that plane. In order to account for the propagation of such vibrations, it is necessary to suppose the existence of a tangential force, or tangential pressure, called into play by the continuous sliding of the 4 — 2 28 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. sheets one over another, and proportional to the amount of the displacement of sliding. There is no occasion to enter into any speculation as to the cause of this tangential force, nor to entertain the question whether the luminiferous ether consists of distinct molecules or is mathe- matically continuous, just as there is no occasion to speculate as to the cause of gravity in calculating the motions of the planets. But we are absolutely obliged to suppose the existence of such a force, unless we are prepared to throw overboard the theory of transversal vibrations, as usually received, notwithstanding the multitude of curious, and otherwise apparently inex- plicable phenomena which that theory explains with the utmost simplicity. Consequently we are led to treat the ether as an elastic solid so far as the motions which constitute light are concerned. It does not at all follow that the ether is to be regarded as an elastic solid when large displacements are considered, such as we may conceive produced by the earth and planets, and solid bodies in general, moving through it. The mathematical theories of fluids and of elastic solids are founded, or at least may be founded, on the consideration of internal pressures. In the case of a fluid, these pressures are supposed normal to the common surface of the two portions whose mutual action is considered : this supposition forms in fact the mathematical definition of a fluid. In the case of an elastic solid, the pressures are in general oblique, and may even in certain directions be wholly tangential. The treatment of the question by means of pressures presupposes the absence of any sensible direct mutual action of two portions of the medium which are separated by a small but sensible interval. The state of constraint or of motion of any element affects the pressures in the surrounding medium, and in this way one element exerts an indirect action on another from which it is separated by a sensible interval. Now the absence of prismatic colours in the stars, depending upon aberration, the absence of colour in the disappearance and reappearance of Jupiter's Satellites in the case of eclipses, and, still more, the absence of change of colour in the case of certain periodic stars, especially the star Algol, shew that the velocity of light of different colours is, if not mathematically, at least sensibly the same. According to the theory of undulations, this is equivalent to saying that in vacuum the velocity of propagation is independent of the length of the waves. Consequently the direct action of two elements of ether separated by a sensible interval must be sensibly if not mathematically equal to zero, or at least must be independent of the disturbance; for, were this not the case, the expression for the velocity of propagation would involve the length of a wave. An interval is here considered sensible which is comparable with the length of a wave. We are thus led to apply to the luminiferous ether in vacuum the ordinary equations of motion of an elastic solid, provided we are only considering those disturbances which con- stitute light. Let us return now to the case supposed at the beginning of this section. According to the preceding explanation, we must regard the ether as an elastic solid, in which a series of rectilinear transversal vibrations is propagated in a direction perpendicular to the. plane P. The disturbance at any distance in front of this plane is really produced by the disturbance continually transmitted across it ; and, according to the general principle of the superposition of small motions, we have a perfect right to regard the disturbance in front as the aggregate of the elementary disturbances due to the disturbance continually transmitted across the PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 29 several elements into which we may conceive the plane P divided. Let it then be required to determine the disturbance corresponding to an elementary portion only of this plane. In practical cases of diffraction at an aperture, the breadth of the aperture is frequently sensible, though small, compared with the radius of the incident waves. But in determining the law of disturbance in a secondary wave we have nothing to do with an aperture ; and in order that we should be at liberty to regard the incident waves as plane all that is necessary is, that the radius of the incident wave should be very large compared with the wave's length, a condition always fulfilled in experiment. 32. Let 0, be any point in the plane P; and refer the medium to rectangular axes passing through 0„ x being measured in the direction of propagation of the incident light, and * in the direction of vibration. Let/ (bt - x) denote the displacement of the medium at any point behind the plane P, x of course being negative. Let the time t be divided into small intervals, each equal to t, and consider separately the effect of the disturbance which is transmitted across the plane P during each separate interval. The disturbance transmitted during the interval t which begins at the end of the time t' occupies a film of the medium, of thickness br, and consists of a displacement/^/) and a velocity bf (bt'). By the formulae of Section II. we may find the effect, over the whole medium, of the disturbance which exists in so much only of the film as corresponds to an element dS of P adjacent to X . By doing the same for each interval t, and then making the number of such intervals increase and the magnitude of each decrease indefinitely, we shall ultimately obtain the effect of the disturbance which is continually propagated across the element dS. Let O be the point of the medium at which the disturbance is required ; I, m, n the direction-cosines of X measured from 0„ and therefore — /, — m, — n those of 00 x measured from O; and let 00^ = r. Consider first the disturbance due to the velocity of the film. The displacements which express this disturbance are given without approximation by (2.Q) and the two other equations which may be written down from symmetry. The first terms in these equations relate to normal vibrations, and on that account alone might be omitted in considering the diffraction of light. But, besides this, it is to be observed that t in the coefficient of these terms is to be replaced by a~ x r. Now there seems little doubt, as has been already remarked in the introduction, that in the case of the luminiferous ether a is incomparably greater than b, if not absolutely infinite*; so that the terms in question are insensible, if not absolutely evanescent. The third terms are insensible, except at a distance from O] comparable with X, as has been already observed, and they may therefore be omitted if we suppose r very large compared with the length of a wave. Hence it will be sufficient to consider the second terms only. In the coefficient of these terms we must replace t by b~ x r ; we must put w = 0, v Q = 0, w = bf (bt - r), write - I, — m, - n for I, m, n, and put q = —nw = — nbf (bt — r). The integral signs are to be omitted, since we want to get the disturbance which corresponds to an elementary portion only of the plane P. " I have explained at full my views on this subject in a paper On the constitution of the luminiferous ether printed in the 32nd volume of the Philosophical Magazine, p. 349. 30 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. It is to be observed that da represents the elementary solid angle subtended at by an element of the riband formed by that portion of the surface of a sphere described round 0, with radius r, which lies between the plane yz and the parallel plane whose abscissa is br. To find the aggregate disturbance at corresponding to a small portion, S, of the plane P lying about O x , we must describe spheres with radii ... r - 26r, r - br, r,r + 6t, r + 26t ..., describing as many as cut S. These spheres cut S into ribands, which are ultimately equal to the corresponding ribands which lie on the spheres. For, conceive a plane drawn through OO x perpendicular to the plane yss. The intersections of this plane by two consecutive spheres and the two parallel planes form a quadrilateral, which is ultimately a rhombus ; so that the breadths of corresponding ribands on a sphere and on the plane are equal, and their lengths are also equal, and therefore their areas are equal. Hence we must replace da by r~ 2 dS, and we get accordingly IndS^,^ . mndS „ ,,_ . „ (l-w 2 )dS. , t [ £.- T _/'(6*-r), *--— — f(bt-r), £= ' fQ>t-r). . .(42) btrT 4u-r *7rr Since 1% + mr) + n £= 0, the displacement takes place in a plane through perpendicular to L 0. Again, since £ : i\ :: I : m, it takes place in a plane through Y and the axis of ss. Hence it takes place along a line drawn in the plane last mentioned perpendicular to 00,. The direction of displacement being known, it remains only to determine the magnitude. Let 1' TIT f^ bt ~ r ^ ?■ . r -f'(bt-r) iirt 4nrr 47rr PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 31 The displacement represented by these equations takes place along the same line as before ; and if we put ^ 2 for the displacement, and write cos 9 for I, we get £ 2 = cos 6 sin (pf'(b( - r) (44). 33. By combining the partial results obtained in the preceding article, we arrive at the following theorem. Let £ = 0, ti = 0, £ = f(bt - at) be the displacements corresponding to the incident light ; let O l be any point in the plane P, dS an element of that plane adjacent to Oj ; and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let be any point in the medium situated at a distance from the point O x which is large in comparison with the length of a wave ; let 0^0 = r, and let this line make angles 6 with the direction of propagation of the incident light, on the axis of x, and (p with the direction of vibration, on the axis of %. Then the displacement at O will take place in a direction perpendicular to O t O, and lying in the plane zO^O; and if £' be the displacement at 0, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive, In particular, if we shall have . Then dS = r'dr'da> - rdrdw, since r 8 = p 2 + r' 2 , and p is constant. Let £' = s sin (p. The displacement <^ takes place in the plane ssOx 0, and perpendicular to X ; and resolving it along and perpendicular to x z, we get for resolved parts s sin 2 (p, s sin (p cos (p, of which the latter is estimated in the direction OM, where M is the projeetion of O t on O'y. Let MOO' = Y, % ^ e ' n g reckoned positive when M falls on that side of O 1 on which y is reckoned positive ; then, resolving the displace- ment along OM parallel to O'x', O'y', we get for resolved parts — s sin

, x y, O x %, O x 2 , O x O to a sphere described round O x with radius unity. Then x0 2 forms a spherical triangle, right angled at 2 , and x0 2 = w, 2 0= — + 0, 0% = — + ■%, whence we get from spherical trigo- nometry, cos d> = — sin 6 sin w, sin ^> cos ^ = cos 6, sin

, >? = — s sin 2 9 sin a> cos a>, £ = s (l — sin 2 sin 2 a>). To find the aggregate disturbance at O, we must put for s its value, and perform the double integrations, the limits of m being and 2ir, and those of r being \Zp* and eo . The positive and negative parts of the integrals which give £ and t) will evidently destroy each other, and we need therefore only consider £. Putting for s its value, and expressing in terms of r, we get £= — //(r +p) (r 2 cos 2 u> + p 2 sin 2 w) cos— - (bt - r) — - — . . (47) Let us first conceive the integration performed over a large area A surrounding 0', which we may afterwards suppose to increase indefinitely. Perform the integration with respect to r first, put for shortness F (r) for the coefficient of the cosine under the integral signs, and let R, a function of w, be the superior limit of r. We get by integration by parts fF (r) cos — (bt - r) dr - - — F (r) sin — (bt - r) + (—) * F' (r) cos— (bt - r) + ... X 27r X \2 ir I X Now the terms after the first must be neglected for consistency's sake, because the formula (46) is not exact, but only approximate, the approximation depending on the neglect of terms which are of the order X compared with those retained. The first term, taken between limits* gives ~F(±p)si a ^(btTp)-~F(R)sin^(bt-R), 2w X 27T X where the upper or lower sign has to be taken according as lies in front of the plane P or behind it. We thus get from (47) t--0 ±l)sin— (bt*?p) - — f ' F(R) sin— (bt-R) da,. 2 X 47T^0 X When R becomes infinite, F (R) reduces itself to cos 2 w, and the last term in £ becomes 4 t J 2tt .ecs 2 u> sin — (bt — R) dw. Suppose that no finite portion of the perimeter of A is a circular arc with O' for centre, and let this perimeter be conceived to expand indefinitely, remaining similar to itself. Then, for any finite interval, however small, in the integration with respect to m, the function sin — (bt - R) will change sign an infinite number of times, having a mean value which is X PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 33 ultimately zero, and the limit of the above expression will be rigorously zero. Hence we get in the limit £ = c sin — (bt - p), or = 0, • A according as p is positive or negative. Hence the disturbance continually transmitted across the plane P produces the same disturbance in front of that plane as if the wave had not been broken up, and does not produce any back wave, which is what it was required to verify. It may be objected that the supposition that the perimeter of A is free from circular arcs having 0' for centre is an arbitrary restriction. The reply to this objection is, that we have no right to assume that the disturbance at O which corresponds to an area A approaches in all cases to a limit as A expands, remaining similar to itself. All we have a right to assert a priori is, that if it approaches a limit that limit must be. the disturbance which would exist if the wave had not been broken up. It is hardly necessary to observe that the more general formula (45) might have been treated in precisely the same way as (46). 35. In the third Volume of the Cambridge Mathematical Journal, p. 46, will be found a short paper by Mr. Archibald Smith, of which the object is to determine the intensity in a secondary wave of light. In this paper the author supposes the intensity at a given distance the same in all directions, and assumes the coefficient of vibration to vary, in a given direction, inversely as the radius of the secondary wave. The intensity is determined on the principle that when an infinite plane wave is conceived to be broken up, the aggregate effect of the secondary waves must be the same as that of the primary wave. In the investigation, the difference of direction of the vibrations corresponding to the various secondary waves which agitate a given point is not taken into account, and moreover a term which appears under the form cos co is assumed to vanish. The correctness of the result arrived at by the latter assumption maybe shewn by considerations similar to those which have just been developed. If we suppose the distance from the primary wave of the point which is agitated by the secondary waves to be large in comparison with \, it is only those secondary waves which reach the point in question in a direction nearly coinciding with the normal to the primary wave that produce a sensible effect, since the others neutralize each other at that point by inter- ference. Hence the result will be true for a direction nearly coinciding with the normal to the primary wave, independently of the truth of the assumption that the disturbance in a secondary wave is equal in all directions, and notwithstanding the neglect of the mutual inclination of the directions of the disturbances corresponding to the various secondary waves. Accordingly, when the direction considered is nearly that of the normal to the primary wave, cos# and sin

a be the azimuths of the planes of polarization of the incident and diffracted rays, both measured from planes perpendicular to the plane of diffraction, we should expect to find these angles connected by the equation tan a = sec0 tan-ar in the former event, and tana = cosfl tan-zsr in the latter. If the law and 5—2 36 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. amount of the crowding agree with theory as well as could reasonably be expected, some allowance being made for the influence of modifying causes, (such as the direct action of the edge of the diffracting body,) whose exact effect cannot be calculated, then we shall be led to conclude that the vibrations in plane-polarized light are perpendicular or parallel to the plane of polarization, according as the crowding takes place towards or from the plane of diffraction. In all ordinary cases of diffraction, the light becomes insensible at such a small angle from the direction of the incident ray produced that the crowding indicated by theory is too small to be sensible in experiment, except perhaps in the mean of a very great number of observa- tions. It is only by means of a fine grating that we can obtain strong light which has been diffracted at a large angle. I doubt whether a grating properly so called, that is, one consist- ing of actual wires, or threads of silk, has ever been made which would be fine enough for the purpose. The experiments about to be described have accordingly been performed with the glass grating already mentioned, which consisted of a glass plate on which parallel and equi distant lines had been ruled with a diamond at the rate of about 1300 to an inch. Although the law enunciated at the beginning of this section has been obtained for diffrac- tion in vacuum, there is little doubt that the same law would apply to diffraction within a homogeneous uncrystallized medium, at least to the degree of accuracy that we employ when we speak of the refractive index of a substance, neglecting the dispersion. This is rendered probable by the simplicity of the law itself, which merely asserts that the vibrations in the diffracted light are rectilinear, and agree in direction with the vibrations in the incident light as nearly as is consistent with the necessary condition of being perpendicular to the diffracted ray. Moreover, when dispersion is neglected, the same equations of motion of the luminiferous ether are obtained, on mechanical theories, for singly refracting media as for vacuum ; and if these equations be assumed to be correct, the law under consideration, which is deduced from the equations of motion, will continue to hold good. In the case of a glass grating however the diffraction takes place neither in air nor in glass, but at the confines of the two media, and thus theory fails to assign exact values to a. Nevertheless it does not fail to assign limits within which, or at least not far beyond which, a must reasonably be supposed to lie; and as the values comprised within these limits are very different according as one or other of the two rival theories respecting the direction of vibration is adopted, experiments with a glass grating may be nearly as satisfactory, so far as regards pointing to one or other of the two theories, as experiments would be which were made with a true grating. The glass grating was mounted for me by Prof. Miller in a small frame fixed on a board which rested on three screws, by means of which the plane of the plate and the direction of the grooves could be rendered perpendicular to the plane of a table on which the whole rested. The graduated instruments lent to me by Prof. O'Brien consisted of small graduated brass circles, mounted on brass stands, so that when they stood on a horizontal table the planes of the circles were vertical, and the zeros of graduation vertically over the centres. The circles were pierced at the centre to admit doubly refracting prisms, which were fixed in brass collars which could be turned round within the circles, the axes of motion being perpendicular to the planes of the circles, and passing through their centres. In one of the instruments, which I used for the polarizer, the circle was graduated to degrees from 0° to 360°, and the PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 37 collar carried simply a pointer. To stop the second pencil, I attached a wooden collar to the brass collar, and inserted in it a Nicol's prism, which was turned till the more refracted pencil was extinguished. In a few of the latest experiments the Nicol's prism was dispensed with, and the more refracted pencil stopped by a screen with a hole which allowed the less refracted pencil to pass. In the other instrument, which I used for the analyzer, the brass collar carried a vernier reading to 5'. In this instrument the doubly refracting prism admitted of being removed, and I accordingly removed it, and substituted a Nicol's prism, which was attached by a wooden collar. The Nicol's prism was usually inserted into the collar at random, and the index error was afterwards determined from the observations themselves. The light employed in all the experiments was the sun light reflected from a mirror placed at the distance of a few feet from the polarizer. On account of the rotation of the earth, the mirror required re-adjustment every three or four minutes. The continual change in the direction of the incident light was one of the chief sources of difficulty in the experiments and inaccuracy in the results ; but lamplight would, I fear, be too weak to be of much avail in these experiments. The polarizer, the grating, and the analyzer stood on the same table, the grating a few inches from the polarizer, and the analyzer about a foot from the grating. The plane of diffraction was assumed to be parallel to the table, which was nearly the case ; but the change in the direction of the incident light produced continual small changes in the position of this plane. In most experiments the grating was placed perpendicular to the incident light, by making the light reflected from the surface go back into the hole of the polarizer. The angle of diffraction was measured at the conclusion of each experiment by means of a protractor, lent to me for the purpose by Prof. Miller. The grating was removed, and the protractor placed with its centre as nearly as might be under the former position of the bright spot formed on the grating by the incident light. The protractor had a pair of opposite verniers moveable by a rack ; and the directions of the incident and diffracted light were measured by means of sights attached to the verniers. The angle of diffraction in the different experiments ranged from about 20° to 60°. The deviation of the less refracted pencil in the doubly refracting prism of the polarizer, though small, was very sensible, and was a great source both of difficulty and of error. To understand this, let AB be a ray incident at B on a slip of the surface of the plate contained between two consecutive grooves, BC a diffracted ray. On account of the interference of the light coming from the different parts of the slip, if a small pencil whose axis is AB be incident on the slip, the diffracted light will not be sensible except in a direction BC, determined by the condition that AB + BC shall be a minimum, A and C being supposed fixed. Hence AB, BC must make equal angles with the slip, regarded as a line, the acute angles lying towards opposite ends of the slip, and therefore C must lie in the surface of a cone formed by the revolution of the produced part of AB about the slip. If AB represent the pencil coming through the polarizer, it will describe a cone of small angle as the pointer moves round, and therefore both the position of the vertex and the magnitude of the vertical angle of the cone which is the locus of C will change. Hence the sheet of the cone may sometimes fall above or below the eye-hole of the analyzer. In such a case it is necessary either to be content to 38 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. miss one or more observations, corresponding to certain readings of the polarizer, or else to alter a little the direction of the incident light, or, by means of the screws, to turn the grating through a small angle round a horizontal axis. The deviation of the light which passed through the polarizer, and the small changes in the direction of the incident light, I regard as the chief causes of error in my experiments. In repeating the experiments so as to get accurate results, these causes of error would have to be avoided. At first I took for granted that the instrument-maker had inserted the doubly refracting prism in the polarizer in such a manner that the plane of polarization of the less refracted pencil was either vertical or horizontal, (the instrument being supposed to stand on a horizontal table,) when the pointer stood at 0°, having reason to know that it was not inserted at random ; and having determined which, by an exceedingly rough trial, I concluded it was vertical. Meeting afterwards with some results which were irreconcilable with this supposition, I was led to make an actual measurement, and found that the plane of polarization was vertical when the pointer stood at 25°. Consequently 25° is to be regarded as the index error of the polarizer, to be subtracted from the reading of the pointer. The circumstance just mentioned accounts for the apparently odd selection of values of tjt in the earlier experiments, the results of which are given in the tables at the end of this section. On viewing a luminous point or line through the grating, the central colourless image was seen accompanied by side spectra, namely, the spectra which Fraunhofer called Spectra of the second class. After a little, these spectra overlapped in such a manner that the individual spectra could no longer be distinguished, and nothing was to be seen but two tails of light, which extended, one on each side, nearly 90° from the central image. On viewing the flame of a spirit lamp through the grating, the individual spectra of the second class could be seen, where, with sun-light, nothing could be perceived but a tail of light. The tails themselves were not white, but exhibited very broad impure spectra ; about two such could be made out on each side. These spectra are what were called spectra of the first class by Fraunhofer, who shewed that their breadth depended on the smaller of the two quantities, the breadth of a groove, and the breadth of the polished interval between two consecutive grooves. In the grating, the breadth of the grooves was much smaller than the breadth of the intervals between*. In the experiments, the diffracted light observed belonged to a bright, though not always the brightest, part of a spectrum of the first class. The compound nature of the light was easily put in evidence by placing a screen with a vertical slit between the grating and the eye, and then viewing the slit through a prism with its edge vertical j. A spectrum was then seen which consisted of bright bands separated by dark intervals, strongly resembling the appear- • On viewing the grating under a microscope, the grooves were easily seen to be much narrower than the intervals be- tween ; their breadth was too small to be measured. On look- ing at the flame of a spirit lamp through the grating, I counted sixteen images on one side, then several images were too faint to be seen, and further still the images again appeared, though they were fainter than before. I estimated the direction of zero illumination to be situated about the eighteenth image. If we take this estimation as correct, it follows from the theory of these gratings that the breadth of a groove was the eighteenth part of the interval between any point of one groove and the corresponding point of its consecutive, an interval which in the case of the present grating was equal to the l-1300th part of an inch. Hence the breadth of a groove was equal to the l-23400th part of an inch. f To separate the different spectra, Fraunhofer used a small prism with an angle of about 20°, fixed with its edge horizontal in front of the eye-piece of the telescope through which, in his experiments, the spectra were viewed. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 39 ance presented when a pure spectrum is viewed through a pinhole, or narrow slit, which is half covered by a plate of mica, placed on the side at which the blue is seen. At a consider- able angle of diffraction as many as J 5 or 20 bands might be counted. In the first experiment the grating was placed with its plane perpendicular to the light which passed through the polarizer, the grooved face being turned from the polarizer. The light observed was that which was diffracted at emergence from the glass. It is only when the eye is placed close to the grating, or when, if the eye be placed a few inches off, the whole of the grating is illuminated, that a large portion of a tail of light can be seen at once. When only a small portion of the grating is illuminated, and the eye is placed at the distance of several inches, as was the case in the experiments, it is only a small portion of a tail which can enter the pupil. The appearance presented is that of a bright spot on the grooved face of the glass. The angle of diffraction in the first experiment was large, 57° 5' by measure- ment. Besides the principal image, or bright spot, a row of images were seen to the left : the regularly transmitted light lay to the right, right and left being estimated with reference to the position of the observer. These images were due to internal diffraction and reflec- tion, as will be better understood further on. They were separated by small angles, depending on the thickness of the glass, but sufficient to allow of one image being observed by itself. The observations were confined to the principal or right-hand image. In the portion of a spectrum of the first class which was observed there was a predomi- nance of red light. In most positions of the pointer of the polarizer the diffracted light did not wholly vanish on turning round the analyser, but only passed through a minimum. In passing through the minimum the light rapidly changed colour, being blue at the minimum. This shews that the different colours were polarized in different planes, or perhaps not strictly plane-polarized. Nevertheless, as the intensity of the light at the minimum was evidently very small compared with its intensity at the maximum, and the change of colour was rapid, it is allowable to speak in an approximate way of the plane of polarization of the diffracted light, just as it is allowable to speak of the refractive index of a substance, although there is really a different refractive index for each different kind of light. It was accordingly the angular position of the plane which was the best representative of a plane of polarization that I sought to determine in this and the subsequent experiments. In the first experiment the plane of polarization of the diffracted light was determined by six observations for each angle at which the pointer of the polarizer was set. This took a good deal of time, and increased the errors depending on changes in the direction of the light. Accordingly, in a second experiment, I determined the plane of polarization by single obser- vations only, setting the pointer of the polarizer at smaller intervals than before. Both these experiments gave for result that the planes of polarization of the diffracted light were dis- tributed very nearly uniformly. This result already points very decidedly to one of the two hypotheses respecting the direction of vibration. For according to theory the effect of dif- fraction alone would be, greatly to crowd the planes either in one direction or in the other. It seems very likely that the effect of oblique emergence alone should be to crowd the planes in the manner of refraction, that is, towards the perpendicular to the plane of diffraction. If then we adopt Fresnel's hypothesis, the two effects will be opposed, and may very well be 40 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. supposed wholly or nearly to neutralize each other. But if we adopt the other hypothesis we shall be obliged to suppose that in the oblique emergence from the glass, or in something else, there exists a powerful cause of crowding towards the plane of diffraction, that is, in the manner of reflection, sufficient to neutralize the great crowding in the contrary direction produced by diffraction, which certainly seems almost incredible. The nearly uniform distribution of the planes of polarization of the diffracted light shews that the two streams of light, polarized in and perpendicular to the plane of diffraction respectively, into which the incident light may be conceived to be decomposed, were diffracted at emergence from the glass in very nearly the same proportion. This result appeared to offer some degree of vague analogy with the depolarization of light produced by such sub- stances as white paper. This analogy, if borne out in other cases, might seem to throw some doubt on the conclusiveness of the experiments with reference to the decision of the question as to the direction of the vibrations of plane-polarized light. For the deviation of the light from its regular course might seem due rather to a sort of scattering than to regular diffraction, though certainly the fact that the observed light was very nearly plane-polarized does not at all harmonize with such a view. Accordingly, I was anxious to obtain a case of diffraction in which the planes of polarization of the diffracted light should be decidedly crowded one way or other. Now, according to the explanation above given, the approximate uniformity of distribution of the planes of polarization in the first two experiments was due to the antago- nistic effects of diffraction, (according to FresnePs hypothesis respecting the direction of vibration,) and of oblique emergence from the glass, or irregular refraction, that is, refraction produced wholly by diffraction. If this explanation be correct, a very marked crowding towards the plane of diffraction ought to be produced by diffraction at reflection, since dif- fraction alone and reflection alone would crowd the planes in the same manner. To put this anticipation to the test of experiment, I placed the grating with its plane perpendicular to the incident light, and the grooved face towards the polarizer, and observed the light which was diffracted at reflection. Since in this case there would be no crowding of the planes of polarization in the regularly reflected light, any crowding which might be observed would be due either to diffraction directly, or to the irregular reflection due to diffraction, or, far more probably, to a combination of the two. The experiments indicated indeed a marked crowding towards the plane of diffraction, but the light was so strong at the minimum, for most positions of the pointer of the polarizer, that the observations were very uncertain, and it was evidently only a rough approximation to regard the diffracted light as plane-polarized. The reason of this was evident on consideration. Of the light incident on the grating, a portion is regularly reflected, forming the central image of the system of spectra produced by diffraction at reflection, a portion is diffracted externally at such an angle as to enter the eye, a small portion is scattered, and the greater part enters the glass. Of the light which enters the glass, a portion is diffracted internally at such an angle that after regular reflection and refraction it enters the eye, a portion diffracted at other angles, but the greater part falls perpendicularly on the second surface. A portion of this is reflected to the first surface, and of the light so reflected a portion is diffracted at emergence at such an angle as to enter the eye. Thus there are three principal images, each formed by PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 41 light which has been once diffracted and once reflected, the externally diffracted light being considered as both diffracted and reflected, namely, one has been diffracted internally, and then regularly reflected and refracted, a second in which the light has been regularly refracted and reflected, and then diffracted at emergence, and a third in which the light has been diffracted externally. Any other light which enters the eye must have been at least twice diffracted, or once diffracted and at least three times reflected, and therefore will be comparatively weak, except perhaps when the angle of incidence, or else the angle of diffraction, is very large. Now when the grating is perpendicular to the incident light the second and third of the principal images are necessarily superposed ; and as they might be expected to be very dif- ferently polarized, it was likely enough that the light arising from the mixture of the two should prove to be very imperfectly polarized. To separate these images, I placed a narrow vertical slit in front of the grating, between it and the polarizer, and inclined the grating by turning it round a vertical axis so that the normal fell between the polarizer and the analyzer. As soon as the grating was inclined, the image which had been previously observed separated into two, and at a certain inclination the three principal images were seen equidistant. The middle image, which was the second of those above described, was evidently the brightest of the three. The three images were found to be nearly if not perfectly plane-polarized, but polarized in different planes. The third image, and perhaps also the first, did not wholly vanish at the minimum. This might have been due to some subordinate image which then appeared, but it was more probably due to a real defect of polarization. The planes of polarization of the side images, especially the first, were greatly crowded towards the plane of diffraction, or, which is the same, the plane of incidence. Those of the middle image were decidedly crowded in the same direction, though much less so than those of the side images. The light of the first and second images underwent one regular refraction and one regular reflection besides the diffraction and the accompanying irregular refraction. The crowding of the planes of polarization in one direction or the other produced by the regular refraction and the regular reflection can readily be calculated from the known formula;*, and thus the crowding due to diffraction and the accompanying irregular refraction can be deduced from the observed result. The crowding of the planes of polarization of the third image is due solely to diffraction and the accompanying irregular reflection. The crowding in one direction or the contrary, according as one or other hypothesis respecting the direction of vibrations is adopted, is readily calculated from the dynamical theory, and thus is obtained the crowding which is left to be attributed to the irregular reflection. In the absence of an exact theory little or no use can be made of the result in the way of confirming either hypothesis ; but it is sufficient to destroy the vague analogy which might have been formed between the effects of diffraction and of irregular scattering. The crowding of the planes of polarization of the middle image, after the observations had * It is here supposed that the regularly reflected or re- I ing to a system of spectra is affected as to its polarization in fracted light which forms the central colourless image belong- ' the same way as if the surface were free from grooves. Vol. IX. Part I. 6 42 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. been reduced in the manner which will be explained in the next section, appeared somewhat greater than was to have been expected from the first two experiments. This led me to suspect that the crowding in the manner of reflection produced by diffraction accompanying the passage of light from air, across the grooved surface, into the glass plate, might be greater than the crowding had proved to be which was produced by diffraction accompanying the passage from glass, across the grooved surface, into air. I accordingly placed the grating with its plane perpendicular to the incident light, and the grooved face towards the polarizer, and placed the analyzer so as to receive the light which was diffracted in passing across the first surface, and then regularly refracted at the second. I soon found that the planes of polarization were very decidedly crowded towards the plane of diffraction, and that, notwithstanding the crowding in the contrary direction which must have been produced by the regular refraction at the second surface of the plate, and the crowding, likewise in the contrary direction, which might naturally be expected to result from the irregular refraction at the first surface, con- sidered apart from diffraction. This result seemed to remove all doubt respecting the hypothesis as to the direction of vibration to which the experiments pointed as the true one. On account of the decisive character of the result just mentioned, I took several sets of observations on light diffracted in this manner at different angles. I also made two more careful experiments of the same nature as the first two. The result now obtained was, that there was a very sensible crowding towards the plane of diffraction when the grooved face was turned from the polarizer, although there was evidently a marked difference between the two cases, the crowding being much less than when the grooved face was turned towards the polarizer. Even the first two experiments, now that I was aware of the index error of the polarizer, appeared to indicate a small crowding in the same direction. Before giving the numerical results of the experiments, it may be as well to mention what was observed respecting the defect of polarization. I would here remark that an investigation of the precise nature of the diffracted light was beside the main object of my experiments, and only a few observations were taken which belong to such an investigation. In what follows, ■sr denotes the inclination of the plane of polarization of the light incident on the grating to a vertical plane passing through the ray, that is, to a plane perpendicular to the plane of dif- fraction. It is given by the reading of the pointer of the polarizer corrected for the index error 25°, and is measured positive in the direction of revolution of the hands of a watch placed with its back towards the incident light. Whether the diffraction accompanied reflection or refraction, external or internal, the diffracted light was perfectly plane-polarized when tst had any one of the values 0°, 90°, 180°, or 270°. The defect of polarization was greatest about 45° from any of the above positions. When the diffracted light observed was red or reddish, on analyzation a blue light was seen at or near the minimum ; when the diffracted light was blue or blueish, a red light was seen at or near the minimum. When the angle of diffraction was moderately small, such as 15° or 20°, the defect of polarization was small or insensible ; when the angle of diffraction was large, such as 50° or 60°, the defect of polarization was considerable. For equal angles of diffraction, the defect of polarization was much greater when the grooved face was turned towards the polarizer than when it was turned in the contrary direction. By the term angle of diffraction, PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 43 as applied to the case in which the grooved face was turned towards the polarizer, is to be understood the angle measured in air, from which the angle of diffraction within the glass may be calculated from a knowledge of the refractive index. The grating being placed perpendicularly to the incident light, with the grooved face towards the polarizer, the light diffracted at a considerable angle, (59° 52' by measurement,) to the left of the regularly transmitted light was nearly white. When the pointer of the polarizer stood at 70°, so that -zsr = + 45°, on turning the Nicol's prism of the analyzer in the positive direction through the position of minimum illumination, the light became in succession greenish yellow, blue, plum colour, nearly red. When sr was equal to — 45°, the same appearance was presented on reversing the direction of rotation. Since the colours appeared in the order blue, red, when tst = + 45°, and in the order red, blue, when w = - 45°, the analyzer being in both cases supposed to turn in the direction of the hands of a watch, the deficiency of colour took place in the order red, blue, when w m + 45°, and in the order blue, red, when w = - 45°. Hence the planes of polarization, or approximate polarization, of the blue were more crowded towards the plane of diffraction than those of the red. On placing a narrow slit so as to allow a small portion only of the diffracted light to pass, and decomposing the light by a prism, in the manner already described, so as to get a spectrum consisting of bright bands with dark intervals, and then analyzing this spectrum with a Nicol's prism, it was found that at a moderate angle of diffraction all the colours were sensibly plane- polarized, though the planes of polarization did not quite coincide. At a large angle of diffraction the bright part of the spectrum did not quite disappear on turning round the Nicol's prism, while the red and blue ends, probably on account of their less intensity, appeared to be still perfectly plane-polarized, though not quite in the same plane. On treating in the same manner the diffracted light produced when the grooved face of the glass plate was turned from the polarizer, all the colours appeared to be sensibly plane-polarized. In the former case the light of the brightest part of the spectrum was made to disappear, or nearly so, by using a thin plate of mica in combination with the Nicol's prism, which shews that the defect of plane polarization was due to a slight elliptic polarization. The numerical results of the experiments on the rotation of the plane of polarization are contained in the following table. In this table ST is the reading of the polarizer corrected for the index error 25°. A reading such as 340° is entered indifferently in the column headed " tst" as +315° or - 45°, that is, 340° - 25° or - (360° - 340°) - 25°. a is the reading of the analyzer, determined by one or more observations. The analyzer was graduated only from - 90° to + 90°, and any reading such as - 20° is entered indifferently as - 20°, + l60°, or + 340°, being entered in such a manner as to avoid breaking the sequence of the numbers. On account of the light left at the minimum, the determination of a was very uncertain when the angle of diffraction was large, except when tst had very nearly one of the values 0°, 90°, 380°, or 270°. In the most favourable circumstances the mean error in the determination of a was about a quarter of a degree. In some of the experiments a red glass was used to assist in rendering the observations more definite. This had the advantage of stopping all rays except the red, but the disadvantage of considerably diminishing the intensity of the light. The minutes in the given value of 9, the angle of diffraction, cannot be trusted ; in fact, during any experi- 6—2 44 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. ment 6 was liable to changes to at least that extent in consequence of the changes in the direction of the light. The same remark applies to i, the angle of incidence, in experiments 11 and 12. In these experiments the three principal images already described were observed separately. The angle of diffraction is measured from the direction of the regularly reflected ray, so that i is the angle of incidence, and i + 9 the angle of reflection, or, in the case of the images which suffered one internal reflection, the angle of emergence. The eleven experiments which are not found in the following tables consist of five on diffraction by reflection, which did not appear worth giving on account of the superposition of different images; one on diffraction by refraction, to which the same remark applies, the grating having been placed at a considerable distance from the polarizer, so that the spot illuminated was too large to allow of the separate observation of different images ; one on diffraction by reflection, in which the grating was placed perpendicularly to the incident light, with the grooved face turned from the polarizer, but the errors of observation, though much smaller than the whole quantity to be observed, were so large on account of the large angle of diffraction, (about 75°,) with which the observations were attempted, that the details are not worth giving; one on diffraction by refraction, in which the different observations were so inconsistent that the experiment seemed not worth reducing; one which was only just begun ; and two qualitative experiments, the results of which have been already given. I mention this that I may not appear to have been biassed by any particular theory in selecting the experi- ments of which the numerical results are given. The following remarks relate to the particular experiments : No. 1. In this experiment each value of a was determined by six observations, of which the mean error* ranged from about 15' to 55'. So far the experiment was very satisfactory, but it was vitiated by changes in the direction of the light, sufficient care not having been taken in the adjustment of the mirror. No. 2. a determined by single observations. No. 13. a determined by two observations at least, of which the mean error ranged from about 10' to nearly 1°, but was usually decidedly less than 1°. At and about the octants, that is to say, when ■& was nearly equal to 45°, or an odd multiple of 45°, the light was but very imperfectly polarized in one plane. No. 14. a determined by two observations. Marked in note book as " a very satisfactory experiment." The mean of the mean errors was only ll'. No. 15. a determined by three observations at least. The light was very imperfectly polarized, except near the standard points, that is to say when ■&■ was equal to 0° or 90°, or a * The difference between each individual observation and the mean of the six is regarded as the error of that observa- tion, and the mean of these differences taken positively is what is here called the mean error. When two observations only are taken, the mean error is the same thing as the semi- difference between the observations. Since, for a given position of the pointer of the polarizer, the readings of the analyzer were usually taken one immediately after another, the mean error furnishes no criterion by which to judge of the errors produced by the small changes in the direction of the light incident on the grating, but only of those which arise from the vagueness of the object observed. The reader will be much better able to judge of the amount of probable error from all causes after examining the reduction of the experiments given in the next section. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 45 multiple of 90°. This rendered the observations very uncertain. About the octants the mean error in a set of observations taken one immediately after another amounted to near 2°. No. 17. a determined by two observations. The light was very imperfectly polarized, except near the standard points. Yet the observations agreed very fairly with one another. The mean of the mean errors was 25', and the greatest of them not quite 1°. No. 18. a determined by two observations, which, generally speaking, agreed well with one another. For •& = - 90° and W = + 225° the light observed was rather scattered than regularly diffracted, the sheet of the cone of illumination having fallen above or below the hole of the analyzer. No. 21. a determined by two observations at least. In this experiment the polarizer was covered with red glass. No. 22. a determined by two observations. Marked in note book as " a very satisfactory experiment, though the light was not perfectly polarized." No. 23. a determined by two observations at least. The hole in a screen placed between the polarizer and the grating was covered with red glass. This appears to have been a good experiment. No. 11. a determined by two observations, which agreed well with one another. In the table, a (l), a (2), a (3) refer respectively to the first, second, and third of the three principal images already mentioned. In this experiment the polarizer was reversed, that face being turned towards the mirror which in the other experiments was turned towards the grating, which is the reason why a and •zzr increase together, although the light observed suffered one reflection. The same index error as before, namely 25°, is supposed to belong to the polarizer in its reversed position. No. 12. ct determined by three observations. The largeness of the angle of diffraction rendered the determination of a very uncertain. TABLE I. •sr a •2T a ■sr a ■sr a "ST a(l) a (2) a (3) Experiment, No. 1. Grooved face from No. 13, - 30° continued. + 25° No. 15, - 20° continued. - 92°23' No. 21, - 45° continued. + 12°35' Experiment, No. 11. Polarizer. - 20° + 33°25' - 30° -115°55' - 30° + 27°52' i = 14°50' ; 6 = 22°30'. e = 57V. - 10° + 46° 5' - 40° -124°25' - 15° + 44°47' 0° + 56°35' - 50° -133°4l' 0° + 6l°40' -105° -113°35' -117°50' -115° - 76°4l' + 10° + 67°50' - 60° -140°29' + 15° + 78°25' - 85° -103° 5' -101° -102°2 - 921° - 52°56' + 20° + 76°58' - 70° -148°18' + 30° + 92°1S' - 65° - 90° - 83° 5' - 89° - 70° + 30° + 87°55' - 80° -152°50' + 45° + 107°25' - 45° - 78°40' - 63°55' - 74 n 5 -471° - 6°52' + 40° + 99°27' - 90° -158°30' + 60° + 122°30' - 25° - 58°50' - 44° - 53°1 — 25° + 14°5l' + 37°51 + 50° + 60° + 10S°30' + 120 n 35' n A - 5° + 15° - 25°5' + 13°15' - 21°10' + 1°25' - 23°1 + 7°5 - 2l° Z 2 Experiment, No. 17. + 75" + 90° + 137" + 151°32' + 20° + 42l° + 65° + 61° 5 + 82°54 + 106°46 + 70° + 80° + 90° + 129° 2' + 137°42' + 146°57' + 35° + 55° + 38°35' + 53°50' + 24° 5' + 43°10' + 32° + 51°3 Grooved face towards Polarizer. Experiment, No. 22. = S0°4/>'. Grooved face/rom Polarizer. Experiment, No. 12. Experiment, No. 2. Experim ent, No. 14. - 90° + 77*15' Grooved face from Polarizer. Grooved Poll face from irizer. - 80° + 85°30' -180° -187° 2' i = 9°i'; 9 = 53°39 . 9 - 50°23'. 9 = - 50° 29°57' 4- 22°25' - 70° - 6o° - 50° - 40° - .30° + 93° 12' + 101°15' -165° -150° -170°37' -154°30' - 25° - 45° + 5°35' + 15° - 32° - 9°40' - 13°4 4- 2° -105° - 95° - 85° - 75° - 80° - 70°25' - 6l°15' - 51°30' - 40° - 30° - 20° - 10° + 31°15' + 41°40' + 51°55' + 62°37' + 109°47' + 117°12' + 129°57' -135° -120° -105° - 9& -140°25' - 124°45' -110°40' - 96°55' - 9(f -135° + 26°15' + 34°30' + 26°15' + 65° + 26°1 + 51°1 - 65° - 41°10' 0° + 71°10' Experiment, No. 18. - 75° - 83°32' - 55° - 29°15' + 10° + 81°47' Grooved face towards - 60° - 69° 7' - 45° - 20° 5' + 20° + 93°47' Polarizer. - 45° - 54°50' - 35° - 9°55' + 30° + 103°10' 9 = 21°39'. - 30° - 38°55' - 25° + 0°20' + 40° + n3°15' - 90° -103"23' - 15° - 22°50' - 15° + 10 n 15' + 50° + 122°42' - 45° - 59°53' - 5° + 20°20' + 60° + 132°42' 0° - 12°58' Experiment, No. 23. + 5° + 15° + 30°55' + 70° + 143° + 45° + 33°37' Grooved face towards + 40°55' + 80° + 152°47' + 90° + 77°27' Polarizer. + 2.6° + 50°45' + 90° + l6l°57' + 135° + 120° 2' Red glass used. + 35° + 45° + 6l°45' + 70°55' + 100° + 110° + 171°52' + 182°52' + 180° + 225° + 167°57' +214°10' 9 = 54°53'. + 55° + 82°15' + 120° + 130° + 19l°47' +202"12' 0° -h 15° - 6°30' + 11° 5' Experiment, No. 13. + 140° +211°42' Experiment, No. 21. Grooved face towards Polarizer. + 30° + 27°55' Grooved face towards + 45° + 42°30' Polarizer. Experime nt, No. 15. + 60° + 58°22' , 9 = 39°50'. Grooved f Pols ace towards rizer. Red glass used. 9 = 28°26'. + 75° + 90° + 71° 5' + 83°22' - 60° - 6°5' = 59°52'. - 90° - 29° + 105° + 96°12' - 50° + 4°53' 0° - 68°10' - 75° - 16° 2' + 120° + 108°30' - 40° + 15°52' - 10° - 81° - 60° - 2°12' + 135° + 122°45' PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 47 Section II. DISCUSSION OF THE NUMERICAL RESULTS OF THE EXPERIMENTS, WITH REFERENCE TO THEORY. According to the known formulae which express the laws of the rotation of the plane of polarization of plane-polarized light which has undergone reflection or refraction at the surface of a transparent uncrystallized medium, if sr, a be the azimuths of the planes of polarization of the incident and reflected or refracted light, both measured from planes perpendicular to the plane of incidence, they are connected by the equation tan a = m tan •&, (48) where m is constant, if the position of the surface and the directions of the rays be given, but is a different constant in the two cases of reflection and refraction. According to the theory developed in this paper, the same law obtains in the case of diffraction in air, or even within an uncrystallized medium, but m has a value distinct from the two former. It seems then extremely likely that the same law should hold good in the case of that combination of diffraction with reflection or refraction which exists when the diffraction takes place at the common surface of two transparent uncrystallized media, such as air and glass. If this be true, it is evident that by combining all the observations belonging to one experiment in such a manner as to get the value of m which best suits that experiment, we shall obtain the crowding of the planes of polarization better than we could from the direct observations, and we shall moreover be able in this way easily to compare the results of different experiments. It seems reasonable then to try in the first instance whether the formula (48) will represent the observations with sufficient accuracy. In applying this formula to any experiment, there are two unknown quantities to be determined, namely, m, and the index error of the analyzer. Let e be this index error, so that o = a + e. The regular way to determine e and m would no doubt be to assume an approximate value e x of e, put e = e, + A e 19 where A e L is the small error of ei, form a series of equations of which the type is tan (a - ei) - sec s (a — e,) Aei = m tan sr, and then combine the equations so as to get the most probable values of Ae { and m. But such a refinement would be wholly unnecessary in the case of the present experiments, which are confessedly but rough. Moreover e can be determined with accuracy, except so far as relates to errors produced by changes in the direction of the light, by means of the observations taken at the standard points, the light being in such cases perfectly polarized. By accuracy is here meant such accuracy as experiments of this sort admit of, where a set of observations giving a mean error of a quarter of a degree would be considered accurate. Besides, when- ever the values of w selected for observation are symmetrically taken with respect to one of the standard points, a small error in e would introduce no sensible error into the value of m which would result from the experiment, although it might make the formula appear in fault when the only fault lay in the index error. 48 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. Accordingly I have determined the index error of the analyzer in a way which will be most easily explained by an example. Suppose the values of a to have been determined by experiment corresponding to the following values of bt, - 15°, 0°, +.15°, ... + 75°, + 90°, + 105°. The value of a for -ur = 0°, and the mean of the values for ■ar = — 15° and -ar = + 15°, furnish two values of e ; and the value of a for sr = + 90°, and the mean of the values for •& = + 75° and sr m + 105°, furnish two values of e + 90°. The mean of the four values of e thus deter- mined is likely to be more nearly correct than any of them. In some few experiments no two values of w were symmetrically taken with respect to the standard points. In such cases I have considered it sufficient to take proportional parts for a small interval. Thus if a r , a 2 be the readings of the analyzer for sr = - 10°, ■& = + 5°, assuming eti = e - 10° - 2 a?, a 2 = e + 5° + x, we get 3x = a 2 — a, — 15°, whence e, which is equal to a 2 — 5° — x, is known. The index error of the analyzer having been thus determined, it remains to get the most probable value of m from a series of equations of the form (48). For facility of numerical calculation it is better to put this equation under the form log m = log tan a - log tan w, (49) where it is to be understood that the signs of a and •& are to be changed if these angles should lie between and — 90°, or their supplements taken if they should lie between + 90° and + 180°. Now the mean of the values of log m determined by the several observations belonging to one experiment is not at all the most probable value. For the error in log tan a produced by a small given error in a increases indefinitely as a approaches indefinitely to 0° or Q0°, so that in this way of combining the observations an infinite weight would be attributed to those which were taken infinitely close to the standard points, although such observations are of no use for the direct determination of log m, their use being to determine e. Let a + A a be the true angle of which a is the approximate value, a being deduced from the observed angle a corrected for the assumed index error e. Then, neglecting (A a') 2 , we get for the true equation which ought to replace (49), , 2i!/Aa' , log m = log tan a H — ; ; log tan tst, sin 2 a M being the modulus of the common system of logarithms. Since the effect of the error A a' is increased by the division by sin 2 a', a quantity which may become very small, in com- bining the equations such as (49) I have first multiplied the several equations by sin 2 a, or the sine of 2 (a — e) taken positively, and then added together the equations so formed, and determined log m from the resulting equation. Perhaps it would have been better to have used for multiplier sin 8 2 a, which is what would have been given by the rule of least squares, if the several observations be supposed equally liable to error ; but on the other hand the use of sin 2a' for multiplier instead of sin 2 2a' has the effect of diminishing the comparative weight of the observations taken about the octants, where, in consequence of the defect of polarization, the observations were more uncertain. The following table contains the result of the reduction of the experiments in the way just explained. The value of e used in the reduction, and the resulting value of log m, are written down in each case. The first column belonging to each experiment gives the value of a — "sr PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 49 calculated from (49) with the assumed value of log m, and is put down for the sake of com- parison with the value of a — iff deduced from the difference, a — w, of the observed angles a, •&, corrected for the assumed index error e. The upper and lower angles belonging to any pair found in the second and third columns refer respectively to the upper and lower signs in the first column, and in the column headed "ar". In the table, the experiments are arranged in classes, according to their nature, and those belonging to the same class are arranged according to the values of 9. The first three experiments in the table relate to diffraction at refraction, in which the grooved face of the grating was turned from the polarizer, the next six to diffrac- tion at refraction, in which the grooved face was turned towards the polarizer, and the last two to the experiments in which the grating was a little inclined, and the three principal images were observed. The result of Experiment No 1, is here given separately, on account of the different values of iff there employed. Experiment No 1. 6 = 57°5'; assumed index error e = 40°5'. W a — •& - 115° - 1°46' - 92^° - 0°3l' - 70° - 47£° + 0°33' - 25° - 0°14' - 2£> + 0°16' + 20° + 1° + 42l° + 0°19' + 65° + l°4l' The values of a for w = - 115° and 73- = + 65° ought to differ by 180°, whereas they differ by 3°27' more. This angle is so large compared with the angles a - •ar given just above, that it seems best to reject the experiment. The experiment is sufficient however to shew that the crowding of the planes of polarization, be it in what direction it may, is very small. On com- bining all the observations belonging to this experiment in the manner already described, a small positive value of log rn, namely + .002, appeared to result. This value, if exact, would indicate an extremely small crowding in the manner of reflection. Vol. IX. Pakt I. 50 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. TABLE II. Experiment, No. 14. Experiment, No. 2. Experiment, No. 22. Experiment, No. 18. Experiment, No. 21. 6 = 29°57'; 9 = 50°23'; 6 = 55°38'; 6 = 2i°39'; 6 = 28°26'; 6= + 72°23'; e = + 24°12'; e = - 7°27'; 6 = - 12°44'; e = + 60°49'; log m = + .009. log m = + .010. log m = + .035. log m m + .029. log m = + .039. a ' —Tp (in tenths of 1°). a- tzt a — w a — nr a — "sr ■w calc. obs. diff. calc. obs. dift'. calc. obs. diff'. calc. obs. diff'. calc. obs. diff. 0° -12 -12 f ** 0-2-2 0+8+8 + 0+5 ± 5° f s* ± 1 i -0-8 1 + 1 ± 10° ± 2 < 1 + 2+4 + 6+3 ( + 26 +13 ± 15° ± 3 ■ ± 12 \ ± 13 { + 14 -10 0+3 [-8+4 1-10 +3 ± 20° ± 4 J -5-1 ' + 4 - 1 l ± 25° + 8+3 ± 5 . 0+5 f + 15 _8 ± 30° ± 5 ± 20 J ± 23 { ° ± 35° -7-2 ± 6 . ? ? 1-19 +1 1-29 - 6 ± 40° + 9+3 -2+4 ± 6 J -11-5 '+4-3 f + 13 - 6 [ + 16 -10 ± 45° ± 7 < =fc 23 I =•= 19 < ± 26 < T + 3-3 -4+3 [-28 - 5 ^1-21 - 2 1-32 - 6 ± 50° ± 6 < 0+6 ; + 19 +13 ± 55° '+3-2 ± 6 J + 4 +10 , f+ 17 - 5 =t 60° ± 5 . ± 20 | (-21 - 1 ± 22 | T ( - 30 - 8 ± 65° + 6+2 ± 5 . - 15 -10 =fc 70° ± 4 . ( + 12 ± 75° ± 3 . . ± 11 < ± 12 { T " + 4+2 - 18 -15 [-15 -4 1- 16 - 4 ± 80° ± 2 - V ± 85° ± 1 . =fc 90° • -4-4 - 15 -14 °\ o{ + l + l f+ 7 + 7 A 95° -5-3 =F 1 • - 7 - 8 1+1+1 1-6-6 1-1-2+2 ±100° T 2 • : ±105° + 5+9 T 3 -3-6 T n 1 1+13 +2 ±110° =F 4 • ±115° '-6-1 T 5 ±120° =f 5 ■ =F2o| ±125° [-2+4 t6 1+22 +2 ±130° t6 f_ 22 - 3 ±135° ±140° =f6 - -7-1 t7 [+16-7 =Fl9 {+19 ±145° «p6 ±150° =f 5 T 20 J ±155° =F5 1+25 +5 ±160° T 4 ±165° =F 3 =f 12 | "' ±170° T 2 1+14+2 ±175° T 1 180° 0+7+7 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 51 TABLE II, continued. Experiment, No. 13. Experiment, No. 17. Experiment, No. 23. Experiment, No. 15. 6 = 39°50'; 6 = 50°45'; 9 = 54°53'; d = 59°52'; 6 = + 56°50'; e = + 167°15'; e = - 7°27'; e = - 68°15'; log »J = + .034. log »» = + .122. log m = + .082. log m = + .225. isr a — nr « - W a — ar a — w calc. obs. diff. calc. obs. diff. calc. obs. diff. calc. obs. diff. 0° 0-2-2 0+2+2 0+1+1 ± 5° ( + 10 +2 ± 10° ± 8 ±31 f „ ± 65 ' ± 15° 1-7+1 (+ 1 -14 ±2 9 ( +27 " 2 - 27 +38 ± 20° ±i5 r„. ± 57 I ±114- ± 25° [-34 -19 - 41 +73 |+ 11-9 f + 46 +3 ± 30° ± 20 { \- 18+2 ±74^ - 73+1 ±43 j ±141 . - 177 -36 ± 35° f + 26 +4 V ± 40° ±22< ±80^ f — ±146. I- 10 -12 - 100 -20 (+42 -12 - 162 -16 ± 45° f+ 17 - 5 ±54< ± 50° =•= 22 { ±76. I. ±134 . ± 55° 1-19 +3 - 75+1 - 154 -20 f+37 +18 (+50 +6 ± 60° ±l9 \- 9 -10 ±64- -60+4 ±44< ±110. - 122 -12 ± 65° f + 22 +8 \ ± 70° ± 14 j ±46. -40+6 f+ 35 +10 ± 78 - 100 -22 zfc 75O [+9+2 ±25 | ± 80° ± 7 i ±24. I ± 40 < ± 85° 1+1+1 -17+7 f - 46 - 6 ± 90° ° 1 ■ - 2-2 ± 95° I I ±100° f- 31 - 6 ±105° =f 25 { ±110° I ±115° f_ 48 - 4 ±120° =F 44 < ±125° I ±130° f- 56 - 2 ±135° T54{ ±140° I ±145° ±150° =F 43 ±155° ±160° ±165° =p29 ±170° ±175° 180° 7—2 52 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. TABLE II, continued. Experiment, No. 11. Experiment, No. 12. i = 14°50'; = 22°30'; e = - 15°30'. i = 9°i' ; 9 = 53°39'; e = - 63°45'. Fiist Image. Second Image. Third Im8ge. First Image. Second Image. Third Image. log m = + .289. log TO = + .061 . log m = + .209. log m = + .756. logm = + .122. log m = + .366. a — ar a — IB" a — W a — "sr a — tst a — tsr •2T calc. obs. dirt'. calc. obs. diff. calc. obs. ditf. calc. obs. diff. calc. obs. diff. calc. obs. diff. 0° f [ ± 5° ± 10° ± 15° ±46 {- * o * M- 7 ± 30 < - 27+3 ±12 5 ( +137+12 ±2! +19 " 2 ± 84 < ' + 84 ± 20° ± 25° ± 30° ± 35° I ±172 {-183-11 ±18 7 { +191+4 ±178 | \-182 - 4 1 ±32 {-35 -3 ±39 H +7 ±120- ±136 • - 128 - 8 '+125 -11 f+443 - 1 ±444< ± 67 { + 67 ° ±223{ + 25 ° +27 ± 40° ± 45° ± 50° ± 55° I ± 40 | } - 34 +6 ±133 • -143 -10 ±351 { +337 ~ 14 ± 79 { +91 +12 ±21 7 { +217+ ° ±152{ +U3 - 9 ±3 7 ( +37 ° ±116' + 120 + 4 ± 60° I r I r ± 65° i 70° ±115 > \~ 95 +10 ± 29 [_ 26 + 3 ± 89' - 85+4 ± 75° ± 80° r ( r ± 85° ± go* ± 24 { \- 26-2 ± 7| (-5+2 ±19 {- 18+1 of ° ° f oi ° ° ± 95° I I 1 ±100° r r ±105° ±110° T 71 { + 6 9 - 2 =Fl9 |+27 +8 ±115° ±120° ±125° ±130° ±135° ±140° T 35l{" 367 - 16 T 79 j" 62 +1? f-200 +17 =p217j ±145° ±150° ±155° ±160° ±165° ±170° ±175° 180° PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 53 The reader will please to remark that in order to follow the observed values of a — 73- beginning with sr = 0°, and going in the positive direction, it will be necessary to begin at the top of the table and go downwards, taking the upper number in each bracket. In order to go in the negative direction from sr = 0°, it would be necessary to begin at the top and go downwards, taking the under number in each bracket. A nearly constant error appearing in the table of differences would indicate merely that the value of e used in the reduction was slightly erroneous. A slight error in e, it is to be remembered, produces no sensible error in log m, whenever the observations are balanced with respect to one of the standard points. In the first two experiments entered in the table, the crowding of the planes of polarization is so small that it is masked by errors of observation, and it is only by combining all the ob- servations that a slight crowding towards the plane of diffraction can be made out. In all the other experiments, however, a glance at the numbers in the second column is sufficient to shew in what direction the crowding takes place. From an inspection of the numbers found in the columns headed "diff." it seems pretty evident that if the formula (49) be not exact the error cannot be made out without more accurate observations. In the case of experiment No. 15, the errors are unusually large, and moreover appear to follow something of a regular law. In this experiment the observations were extremely uncertain on account of the large angle of dif- fraction and the great defect of polarization of the light observed, but besides this there appears to have been some confusion in the entry of the values of ■&. This confusion affecting one or two angles, or else some unrecorded change of adjustment, was probably the cause of the apparent break in the second column between the third and fourth numbers. Since the value of log m is deduced from all the observations combined, there seems no occasion to reject the experiment, since even a large error affecting one angle would not produce a large error in the value of log m resulting from the whole series. In the entry of experiment No. 12 the signs of "or have been changed, to allow for the reversion produced by reflection. This change of sign was unnecessary in No. 11, because in that experiment the polarizer was actually reversed. The results of experiment No. 12 would be best satisfied by using slightly different values of the index error of the analyzer for the three images, adding to the assumed index error about — 1^°, + 1 J n , + 2°, for the first, second, and third images respectively. The largest error in the third columns, 2.7°, is for ■ar = + 25°, third image. The three readings by which a was determined in this case were - 1.5°, - 13°30', - 12° ? Hence the error + 2.7°, even if no part of it were due to an index error, would hardly be too large to be attributed to errors of observation. Since the formula (49), even if it be not strictly true, represents the experiments with sufficient accuracy, we may consider the value of log m which results from the combination of all the observations belonging to one experiment as itself the result of direct observation, and proceed to discuss its magnitude. Let us consider first the experiments on diffraction at refraction, in which the light was incident perpendicularly on the grating. Although the theory of this paper does not meet the case in which diffraction takes place at the confines of air and glass, it leads to a definite result on each of the three following supposi- tions : 54 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. First, that the diffraction takes place in air, before the light reaches the glass : Second, that the diffraction takes place in glass, after the light has entered the first surface perpendicularly : Third, that the diffraction takes place in air, after the light has passed perpendicularly through the plate. On the first supposition let a„ a.,, a be the azimuths of the plane of polarization of the light after diffraction, after the first refraction, and after the second refraction respectively, and 9' the angle of refraction corresponding to the angle of incidence 9, so that sin 9 = p. sin 9',/u being the refractive index of the plate : and first, let us suppose the vibrations of plane-polarized light to be perpendicular to the plane of polarization. Then by the theory of this paper we have tan cti = sec 9 tan w, and by the known formula applying to refraction we have tan ct 2 = cos {9 - 9') tan oti, tan a = cos (9 — ff) tan a 2 » whence tan a = m tan zzr, where m = sec 9 cos 51 (9 - 9'). On the second supposition, if a t be the azimuth after diffraction at an angle 9' within the glass, we have tan a, = sec 9' tan -ar, tan a = cos {9 — 9') tan a„ whence tan a = m tan sr, where m = sec 9' cos (9 - 9 1 ). On the third supposition we have tan a = m tan tst, where m = sec 9. If we suppose the vibrations parallel to the plane of polarization, we shall obtain the same formulas except that cos 9, cos 9' will come in place of sec 9, sec 9', the factor cos (9 — 9') being unaltered. Theory would lead us to expect to find the value of log m deduced from observations in which the grooved face was turned from the polarizer lying between the values obtained on the second and third of the suppositions respecting the place of diffraction, or at most not much differing from one of these limits. Similarly, we should expect from theory to find the value of logm deduced from observations in which the grooved face was turned towards the polarizer lying between the values obtained on the first and second suppositions, or at most not lying far beyond one of these values. The following table contains the values of log m calculated from theory on each of the hypotheses respecting the direction of vibration, and on each of the three suppositions respect- ing the place of diffraction. The numerals refer to these suppositions. The table extends from 9 = to 9 = 90°, at intervals of 5°. When 9 = 0, m = l, and logm = 0, in all cases. In calculating the table, I have supposed /x — 1. 52, or rather equal to the number, (1.5206,) whose common logarithm is .182. This table is followed by another containing the values of logrra deduced from experiment. PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 55 TABLE III. Values of log»» from theory, n being supposed equal to 1.5206. Vibrations supposed Vibrations supposed perpendicular to the p ane parallel to the p ane of e of polarization. polarization. I II III I II III 5° +.001 +.001 + .002 - .002 -.001 - .002 10° + .005 + .002 + .007 - .008 -.004 - .007 15° + .011 + .004 + .015 - .019 -.008 - .015 20° +.020 + .008 + .027 - .034 -.015 - .027 25° +.032 + .012 + .043 - .053 -.023 - .043 30° + .047 +.017 + .062 - .078 -.033 - .062 35° + .065 + .022 + .087 - .109 -.044 - .087 40° + .086 + .028 + .116 - .146 -.058 - .116 45° + .111 +.033 + .150 - .190 -.073 - .150 50° + .139 + .037 + .192 - .244 -.090 - .192 55° + .173 +.040 + .241 - .310 -.109 - .241 60° + .214 + .040 + .301 - .388 -.129 - .301 65° + .262 + .039 + .374 - .486 -.151 - .374 70° + .324 + .034 + .466 - .608 -.175 - .466 75° + .408 + .022 + .587 - .766 -.202 - .587 80° + .533 + .005 + .760 - .987 -.231 - .760 85° + .773 -.022 + 1 .060 -1.347 -.265 -1.060 90° + o° -.059 + 00 — 00 -.305 — 00 TABLE IV. Values of log m from observation. Nature of experiment. No. e log m Diffraction at refraction. Incidence perpendicular. Grooved face of glass plate turned from the incident light. 14 2 22 29°57' 50°23' 54°38' +.009 +.010 +.035 Diffraction at refraction. Incidence perpendicular. Grooved face of glass plate turned towards the incident light. 18 21 13 17 23 15 21°39' 28°26' 39°50' 50°45' 54°53' 59°52' + .029 + .039 + .034 + .122 +.082 + .225 56 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. A comparison of the two tables will leave no reasonable doubt that the experiments are decisive in favour of Fresners hypothesis, if the theory be considered well founded. In con- sidering the conclusiveness of the experiments, it is to be remembered that on either the first or second supposition respecting the place of diffraction, (and the third certainly cannot apply to the case in which the grooved face is turned towards the incident light,) the planes of polar- ization of the diffracted light are crowded by refraction towards the perpendicular to the plane of diffraction, and therefore the observed crowding towards the plane of diffraction does not represent the whole effect of the cause, be it what it may, of crowding in that direction. If /3 be the value of a - nr for ■zsr = 45°, /3 = 1° when log to = .015, nearly; and when log m is not large, /3 is nearly proportional to log to. In this case /3 is nearly the maximum value of a - w. Hence the greatest value of a — w, expressed in degrees, may be obtained approximately from Table IV, and, within the range of observation, from Table III, by regarding the decimals as integers and dividing by 15. Thus, for logm= - .388 the real maximum is 24°.8, and the approximate rule gives 25°.9, so that this rule is abundantly sufficient to allow us to judge of the magnitude of the quantity by which the two theories differ. For 6 = 60°, the two columns in Table III headed " I", as well as those headed "III", differ by .602, and those headed "II", differ by .169, so that the values assigned to /3by the two theories differ by about 40° or 11°, according as we suppose the diffraction to take place in air or in glass. For = 40°, the corresponding differences are 15° and 6°, nearly. These differences, even those which belong to diffraction within the glass plate, are large compared with the errors of observation ; for the probable cause of the large errors in experiment No 15, has been already mentioned. In the following figure the abscissas of the curves represent the angle of diffraction, and the ordinates the values of log to calculated from theory. The numerals refer to the three supposi- tions respecting the place of diffraction, and the letters E, A, (the first vowels in the words perpendicular and parallel,) to the two hypotheses respecting the direction of vibration. The dots represent the results of the experiments in which the grooved face of the glass plate was turned towards the polarizer, and the crosses those of the experiments in which it was turned in the contrary direction. III. E. • I. E. II. E. II. A. III. A. I. A. The smallness of log to in experiment No. 23, to which the 5th dot belongs, is probably due PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 57 in part to the use of the red glass, since, as has been already remarked, the planes of polarization of the blue were more crowded towards the plane of diffraction than those of the red. On this account the dot ought to be slightly raised to make this experiment comparable with its neigh- bours. On the other hand it will be seen by referring to Table II, that No. 23 was a much better experiment than No. 15, which is represented by the 6th dot, and apparently also better than No. 17, which is represented by the 4th dot. No. 21, represented by the 2nd dot, seems to have been decidedly better than No. 13, which is represented by the 3rd. Nos. 14 and 22, repre- sented by the 1st and 3rd crosses respectively, were probably much better, especially the latter of them, than No. 2, which is represented by the 2nd cross. Now, bearing in mind the character of the experiments, conceive two curves drawn with a free hand, both starting from the origin, where they touch the axis, and passing, the one among the dots, and the other among the crosses. The former of these would apparently lie a little below the curve marked I. E, and the latter a very little below the curve II. E. Hence the observations are very nearly represented by adopting Fresnel's hypothesis respecting the direction of vibration, and, whether the grooved face be turned towards or from the incident light, suposing the wave broken up before it reaches the grooves. I think a physical reason may be assigned why the supposition of the wave's being broken up before it reaches the grooves should be a better representation of the actual state of things than the supposition of its being broken up after it has passed between them. Till it reaches the grooves, the wave is regularly propagated, and, according to what has been already remarked in the introduction, we have a perfect right to conceive it broken up at any distance we please in front of the grooves. Let the I figure represent a section of the grooves, &c, by the plane of diffrac- ff.JP v.h.... tion. Let a A, Bb be sections of two consecutive grooves, AB being ( y NL-^ . |/_ the polished interval. Let eh be the plane at which a wave incident in the direction represented by the arrow is conceived to be broken up. Let O be any point in eh, and from O draw ORS in the direction of a ray proceeding regularly from O and entering the eye ; so that OR, RS are inclined to the normal at angles 9, 9\ or 9', 9, according as the light is passing from air into glass or from glass into air. The latter case is represented in the figure. Of a secondary wave diverging spherically from O, which is only partly represented in the figure, those rays which are situated between the limits OA, OB, and are not inclined at a small angle to either of these limiting directions, may be regarded as regularly refracted across AB. In a direction inclined at a small angle only to OA or OB, it would be necessary to take account of the diffraction at the edge A or B. Let ' w $e F, to KaOeKacrToi' naicfjofiiov, otov avOpunrov, K.CU iiriro's, icai >}/Uioi/os). And it is afterwards said that all these are acholous : (d\\a Kal to B, to fjni] e^ov yo\riv, wavri inrdpyei t«S T.) But man has a gall-bladder: and the fact was well known in Aristotle's time, for instance, to Hippocrates; so that it is not likely that Aristotle would have made the mistake which the text contains. But at any rate, it is a mistake ; if not of the transcriber, of Aristotle ; and it is impossible to reason about the passage, without correcting the mistake. The substitution of e\e(pa at] V o\ eirl tov tjiraTos, evioi d ovk eyovTes. Ato Kal ylveTai d/n.(pio-(i^TriaK vepl oXov to? yevovs' oi yap evTvyovTei oiroTepwaovv eyovai Trent 7rai/Tcoj/ viroXa/xfiavovffiv ws airavTwv eyovTwv (D) lb. fill. Aio Kal yapieoTara Xeyovai twv apyalwv o'i (paaKOVTes diTiov elvai tov TrXelw Xr)v yjiovov to /xt] e^etv yo\ijv, (3\e\}/avTes eirl to ixoovvya Kal to? eXdcpovs' TavTa yap dyoXu T€ Kal £»} iroXvv y^povov. Etj de Kal Ta nr/ ewparxeva vir eKe'ivwv oti ovk eyei yoXrjv, olov SeX(ph Kal KanrjXos, Kal toiIto Tvyyavei naKpofiia ovTa. EvXoyov yap, k.t.X. (E) The elephant and man are mentioned together as long-lived animals (De Long, et Brev. Vitce, iv. 2, and De Generat. Animal, iv. 10. 2.) The following is the import of these passages : (A) "Of viviparous quadrupeds, the deer, roe, horse, mule, ass, seal, and some of the swine have not the gall-bladder. ... The elephant also has the liver without gall-bladder, &c." (B) "The external parts of man are well known: the internal parts are far from being so. The parts of man are in a great measure unknown; so that we must judge concerning them by reference to the analogy of other animals. ... ,1 (C) "Some animals are altogether destitute of gall-bladder, as the horse, the mule, the ass, the deer, the roe. ..But in some kinds it appears that some have it, and some have it not, as the mice kind. And among these is man ; for some men appear to have a gall-bladder on the liver, and some not to have one. And thus there is a doubt as to the species in general ; for those who have happened to examine examples of either kind, hold that all the cases are of that kind." (D) "Those of the ancients speak most plausibly, who say that the absence of the gall- bladder is the cause of long life ; looking at animals with uncloven hoof, and deer : for these are destitute of gall-bladder, and live a long time. And further, those animals in which the ancients had not the opportunity of ascertaining that they have not the gall-bladder, as the dolphin, and the camel, are also long-lived animals." It appears, from these passages, that Aristotle was aware that some persons had asserted 72 Dr. WHEWELL'S CRITICISM OF ARISTOTLE'S ACCOUNT OF INDUCTION. man to have a gall-bladder, but that he also conceived this not to be universally true. He may have inclined to the opinion, that the opposite case was the more usual, and may have written avOpcviros, in the passage which I have been discussing. Another mistake of his is the reckon- ing deer among long-lived animals. It appears probable, from the context of the passages (C) and (D), that the conjecture of a connection between absence of the gall-bladder and length of life was suggested by some such notion as this : — that the gall, from its bitterness, is the cause of irritation, mental and bodily, and that irritation is adverse to longevity. The opinion is ascribed to "the ancients," not claimed by Aristotle as his own. W. W. Trinity Lodge, April 13, 1850. III. On Impact on Elastic Beams. By Homersham Cox, Esq., B.A. of Jesus College. [Read Dec. 10, 1849.] The modern use of iron girders for Railway Bridges has given great interest and im- portance to the investigation of the strength of Elastic Beams when subjected to various statical and dynamical tests. Unfortunately the theoretical examination of the dynamical question is very difficult, and to supply additional knowledge respecting them an extensive series of experiments has been instituted under the authority of the Royal Commission ap- pointed to inquire into the Application of Iron to Railway Structures. Part of these experiments, related to the effects of impact in producing the deflection and fracture of beams. In most cases the impact was given in a horizontal direction by means of heavy balls moving in circular arcs as pendulums. The ball was suspended by a fine inextensible cord from a point of support vertically over the centre of the beam which was fixed in a horizontal position by bearings near its two ends. The ball descended by its own weight, and when it reached the lowest point of its arc, struck the centre of the beam hori- zontally in a direction perpendicular to its length. Care was taken to prevent the ends of the beam from yielding to the blow, and to carefully mark the degree of deflection produced at the centre. The velocity acquired by the ball before impact, was computed by a well known formula from the chord of the arc of descent. The course of experiments comprised great variations of the strength and size of the beam, the velocities of impact, and the relative masses of the beam and ball. There were two ways in which these experiments might be made practically useful : either (firstly) as confirmations of an independent theoretical investigation of the laws of im- pact on beams — or (secondly) supposing such an independent investigation impossible, as empirical data susceptible of theoretical generalisation. It is to the first of these purposes that the experiments will here be applied, and it will be attempted to be shewn that the observed results might have been predicted with a very great degree of confidence and accuracy from known dynamical principles. A most elaborate paper by Poisson, entitled Sur VEquilibre et le Mouvement des Corps FAastiques, is given in the Memoirs of the French Academy of Sciences, Tome Till., and the eighth chapter of the Traite de Mecanique of the same author, relates to vibrations of elastic rods. It is assumed however in these investigations, that the initial velocity and position of every particle are known. But the essential difficulty of the problem here to be discussed, is to determine that which, for the purposes of the illustrious writer referred to, might be considered as known — namely, the initial effects upon the several points of the system produced by assigned external causes of motion. Vol. IX. Part I. 10 74 H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. The subject of the present investigation may be stated as follows : An elastic beam of uniform density and section throughout its length, abuts at each extremity against a fixed vertical prop, and is impinged upon at its centre by a ball moving horizontally, with an assigned velocity in a direction perpendicular to the length of the beam before collision, and subsequently moving in contact with the beam throughout its deflection. It is required to determine the deflection of the beam produced by the impact. The dynamical circumstances of the problem may be divided into two stages. The first consists in the sudden alteration of the velocity of the ball, at the instant of collision ; the second, the effect of the elastic forces developed in the beam by deflection, in destroying the vis viva which the system has immediately after collision. (1) In order to determine the first part of the problem, it will be assumed for the present that the ends of the beam remain in contact with the fixed abutments, and that the side of the beam which is struck, begins to take the form of a curve concave in every part. The case in which the ends of the beam recoil from their bearings after impact, requires a different method of investigation. Now in the first case referred to, the curve will not, while the deflection is small, differ considerably from the elastic curve of a beam deflected by statical pressure at its centre. As the beam begins to take a curvilinear form, it begins to move in different parts with different velocities. But while the deflection is indefinitely small, the velocities are parallel to the direction of impact and proportional to the spaces described. D'Alemberfs Principle holds for simultaneous percussions as well as for finite forces and, as it reduces every dynamical problem to a statical form, may here be combined with the principle of virtual velocities. The legitimacy of combining the two principles is specially shewn by Poisson in the ninth chapter of his Traite de Mecanique, numero 535. Let it be supposed that the arbitrary displacement is that which actually occurs during motion. To construct the equation of virtual velocities, this displacement of each particle must be multiplied by its quantity of motion ; i. e. by its initial velocity multiplied by its mass. The sum of the products so formed must be put equal to the external impulsive force of the ball multiplied by its virtual velocity. If d,v be an element of the length of the beam and /udx its mass, initially at rest, and y the small distance through which it moves in the indefinitely small time t, parallel to the direction of impact, - is the velocity, fxd.v.- is the "quantity of motion," and is the * ' t product of the quantity of motion by the virtual velocity. Also let P be the force of impact, and / the value of y at the centre of the beam. Pf is the product of the blow and its virtual velocity. Hence combining the principles above referred to, we have the equation ^fi-jy (i). where the integral includes the product of the quantity of motion of every particle of the beam by its virtual velocity. In forming the equation of the equilibrium of the quantities of motion and the force H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. 75 of impact, the quantities of motion produced by the finite elastic forces do not appear as, that period being indefinitely small, those quantities are indefinitely small also and disappear in the limit. In order to effect the summation of the products of the quantities of motion of the several particles by their respective virtual velocities, it is necessary to ascertain the form which the beam assumes at the instant after impact. We set out with the hypothesis, that the side of the beam, which is struck, begins to take the form of a curve concave in every part. The ordinates of this curve are the distances which the several parts of the beam described in an indefinitely short time. Consequently the ordinates are proportional to the initial velocities of the several particles, and by substituting their values from the equation to the curve, the summation required is reduced to the summation of a function of a single variable. The curve in question is here assumed to be the elastic curve, as determined by Poisson and Prof. Moseley to be the form assumed by a uniform beam deflected by a pressure applied perpen- dicularly at its centre. The accuracy of the computation does not, however, depend essen- tially on the selection of this particular curve, for the quantity to be computed from it is involved in such a manner, that if the curve had been assumed to be a portion of a circle or parabola, the final results would not be very widely different. Poisson has shewn (Traite de Mecanique, Tom. i. p. 641) the equation to the elastic curve to be f y = — 3 (3a?x -4>a?), where a is the whole length of the beam, and x the ordinate measured from one end along the beam when undeflected. This equation is the same, mutatis mutandis, with that arrived at by different methods by Professor Moseley, and given in his Principles of Engineering. Squaring both sides of the equation, and integrating between limits x = ^a, and x = 0, I 17 , 7(T m For the whole length of the beam the integral will have double the above value, if *tfdm = ^ffxa (2), o So and therefore equation (l) becomes 17 f- /ma 35 •* -*/ Or, dividing by /, putting v for the velocity at the centre of the beam, and /xa the mass of the beam = M, P=- Mv (3). 35 Now by the ordinary principles of impact, if u be the velocity of the ball before impact, and m its mass, the blow is equal to the quantity of motion lost by the ball, or since v is the same for both beam and ball after impact, P = m (u — v). 10—2 76 H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. Substituting this value of P in equation (3), And consequently m(u — v) = — Mv. ' 35 m v = — — — . u (4). m + — M 35 From (2) it appears that the total vis viva after impact or •■I f#|Mf_17/M B H w f 35 f 35 and the total vis viva of the beam and ball together is mv" + — Mv 2 = (from 4) — — — mu? (5). 35 v ' 17 K ' m + —M 35 Adopting, then, the elastic curve to represent the initial velocities of the several parts of the system, effecting the integration and supplying the numerical calculations, we find ulti- mately that rather less than one half the inertia of the beam may be supposed to act initially to resist the ball ; or, to speak more precisely, that at the instant after impact the impinging ball loses as much of its motion as it would have done if it had impinged on another free ball having 17-35ths of the mass of the beam. From this conclusion it is easy to infer, as in the accompanying formula, the total vis viva of the system after impact. (2) The second part of the problem consists in determining the effect of the elastic forces developed in the beam by deflection. At the end of the deflection the whole system is sup- posed to be brought to rest simultaneously, and by the principle of the conservation of vis viva, the whole work done by the elastic forces is equal to half the vis viva destroyed. It seems safe to assume that the elastic forces are functions of the distances between the particles of the beam and not of their velocities. This assumption is made in investigations of vibrating cords and rods, of which the results are confirmed by experiment. If, then, the elastic forces of the beam vary as the extension and compression directly, the work done in bending the beam into a particular form will be the same whether the particles move intermediately with a greater or less velocity. Now when the beam is deflected statically through a certain distance at its centre, the deflecting pressure is to the distance of deflection in a nearly constant ratio which is usually determined by ascertaining experimentally the number of pounds weight which will statically maintain a deflection of one inch. Let a be that weight, f the deflection in inches, af is the pressure necessary to maintain, and ^a/ 2 the work necessary to produce the deflection. Consequently 1 a/ 2 will be the work done in deflecting the beam after impulse, if / be the central deflection, and the final form of the beam be that which it would statically assume. Therefore from (5) by the principle of vis viva as explained ^ .m« 2 = a/ 2 (6). ft M+m H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. 77 Now u is the velocity of the ball before impact : if this velocity be produced by a ver- tical descent h, u* - 2gh, where g is the force of gravity. Whence (6) becomes m 2 ^-ip>^ =a/2 (7) - Also by the geometrical properties of the circle, if r be the radius and c the chord, c 1 h ■ — . So that 8r »»£• - . rim- = «/ ........ (8). The principal mathematical formula arrived at by the above methods may be enunciated as follows. Divide the weight of the ball by itself + ^yths of the weight of the beam. Multiply the resulting fraction by twice the product of the weight of the ball in pounds by the vertical distance of descent ; the result is equal to the square of the deflection multiplied by the number of pounds which statically maintain one inch deflection. Hence we conclude that for a beam of assigned mass and elasticity struck by a ball of given weight the deflection varies as 1st. The velocity of impact directly ; from (6), 2nd. The square root of the vertical distance of the ball's descent ; from (7), 3rd. The chord of impact directly ; from (8). All these results are confirmed by the experiments above referred to. The comparison of theory and observation in the accompanying table is extremely satisfactory, and has been made for beams and balls of very different dimensions, and the agreement of the results under widely- varying circumstances is so close as to leave nothing to be desired. When the beam is very flexible and subjected to great velocity of impact, parts of it will recede with the blow and parts move in contrary directions. In this case the above investiga- tions do not apply, and the problem becomes excessively difficult : but the difficulty is the less to be regretted because in practice beams of great rigidity are always employed. 78 H. COX, Esq. ON IMPACT ON ELASTIC BEAMS. COMPARISON OF THEORY AND EXPERIMENT. The length of the wire suspending the ball in all cases = 210 inches. (Table I. Report on Railway Structures, p. 39.) Statical pressure to maintain 1 inch deflection = 500 lb. Weight of beam 403 lb. Defti. inches Weight of Ball 151 lb. Chord of Impact. Difference Formula. Experiment. nearly. Weight of Ball 75J Defn. Chord of Impact, inches. Formula. Experiment lb. Difference nearly. Defn. inches. Weight of Ball 6001b. Chord of Impact. Difference Formula. Experiment. nearly. * 20.207 20.812 - Jj L 35.82 35.437 + irk 2 7.64 7.875 - Jj i 40.41 40.625 - ^ 1 71.65 70.375 + 7iV 1 15.28 15.750 - ^L- i* 60.62 58.875 + £g \ 107.47 105.625 + A »i 22.91 23.125 - jfr 2 80.83 79.000 + J ¥ 2 30.55 30.750 - fa H 101.04 99.125 + -^ 2l 38.19 38.500 - -fa 3 121.24 119.750 + ^ 3 H 4 45.83 53.46 61.10 46.125 - jfc 53.812 - ^ 61.187 - ^ (Table II. Report, p. 42.) Weight of beam 262 lb. Statical pressure to maintain 1 inch deflection 157 lb. Defn. inches. Weight of Ball 75J lb." Chord of Impact. Formula. Experiment difference nearly. Defn inches Weight of Ball 151 lb. Chord of Impact. Formula. Experiment. Difference nearly. * 17.03 15.5 + tV 1 20.12 20.1875 - 287 i 34.06 30.3125 + i 2 40.23 39-875 + 10(5" »* 51.09 44.125 + i 8 3 60.35 58.875 + 40 2 68.12 59-75 + 1 4 80.46 79-625 + ; 1 100 H 85.15 73.875 + 1 7 5 100.58 99-5 + 1 100 3 102.18 87.625 + i 7 6 120.70 118.5 + -1^ 5 7 4 119.21 101.75 + 1 7 4 136.24 1 16.75 + 7 (Ibid.) Bars half the length of the preceding. Weight of bar 130 lb. Statical pres- sure to maintain 1 inch deflection 1275 lb. Weight of Ball 6031b. Defn. Chord of Impact. Difference inches. Formula. Experiment. nearly. Defn. inches M r eight of Ball 75Jlb. Chord of Impact Difference Formula. Experiment. nearly. Weight of Ball 151£ lb. Defn. Chord of Impact. Difference inches. Formula. Experiment nearly. 1 11.23 11.3437 - yla J. 2 41.22 41.875 1_ 68 2 25.54 26 - tV | 16.84 17.125 - -gL. 3. 4 61.83 63.125 1_ 52 3 4 38.31 39 _ _1_ 55 1 22.46 22.719 - Jg- 1 82.45 84.3125 ~ 3T0" 1 51.08 51.75 - tV lj 28.07 28.156 - ^ li 103.06 105.531 - J* 42 ii 63.86 65.125 — 5T 1^ 33.61 33.625 - -j-l-g *i 123.67 126.0625 1_ 52 *i 76.63 77.625 " tV '1 144.28 147.875 " IT 2 89.40 102.17 89.875 100.75 17S + tV * In this case the velocities were so great, and the bar was so flexible that the experimental results do not agree accurately with those of the formula, which is applicable only when the beam is very rigid. IV. On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula, and the application of the Theory of Probabilities to some questions of evidence. By Augustus De Morgan, Sec. R.A.S., of Trinity College, Cambridge, Professor of Mathematics in University College, London. [Read February 25, 1850.] Three years ago I communicated to the Society some developements of the theory of the Syllogism, which I have since embodied, with additions, in a work * on Formal Logic. I now proceed to consider the subject still further, with particular reference to the application of symbols, and the tendency which such application has to develope what I must call the algebra of the laws of thought. It will be necessary for me to refer, in various ways, to the literary topics of a controversy in which the paper above-mentioned involved me : and this I can do without dwelling on the part of it which is personal to myself or to my opponent. That controversy turned upon the connexion between two systems of syllogism. The first, Sir William Hamilton's alteration of the Aristotelian system by the invention of forms of predication, so as to assign either of the two modes of quantity, universal or particular, to either of the terms of a proposition, subject or predicate, in either of the different species of propositions, affirmative or negative. The second, my own numerically definite system, in which the number of objects of thought that are spoken of, whether under subject or predicate, and also all that exist, are numerically signified, either by the specific or general symbols of arithmetic. But in the present paper I have nothing to do with the numerically definite system, except as it may be alluded to in illus- tration of the others. Still, I shall have to compare two systems. The first, that of Sir William Hamilton above alluded to. The second, the other and prior of my own two systems, in which the extension of the Aristotelian system is made by the application of contrary terms to all the usual forms of predication, without any direct invention of modes of applying quantity. And I may further state, that the methods of this paper have nothing in common with that of Professor Boole, whose mode of treating the forms of logic is most worthy the attention of all who can study that science mathematically, and is sure to occupy a prominent place in its ultimate system. * Referred to, throughout this paper, by the initials F . L. 80 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Section I. ON THE APPROXIMATION OP LOGICAL AND ALGEBRAICAL MODES OF THOUGHT. Throughout this paper I use the word logic in the purely technical sense. The progress of algebra as distinguished from arithmetic, is marked by the gradual approach to the following theorem, that every pair of opposite relations is undistinguishable from every other pair, in the instruments of operation which are required. So early did this principle gain some practical acceptance, that no attempt was ever persisted in (even if made, which is more than I know) to signify different oppositions by different pairs of symbols : + and - were found instrumentally adequate to all the wants of the mechanism of the science. I do not say that this was a benefit : I only state it as a fact. An algebraist, who may be required, should the proper problem occur, to interpret - (- ( + (-«))) as the removal from an expression of all traces of a loss in- curred in an ascent made at a time prior to a certain epoch — may have gained the power of making such interpretation very slowly, in consequence of never having sufficiently distinguished differences, as a preliminary to, or a concurrent with, abstraction by observation of resem- blances. There may be many for whom it would have been better that the above symbol had been - (*(+ (J«)))> or the like, until identity of rules had suggested identity of symbols : and I am sure that I was of the number. The forms of thought which have not immediate relation to magnitude have been otherwise treated : consideration of differences has predominated, that of resemblances has been almost ignored. In the single case in which algebra was forestalled, the old maxim that two negatives make an affirmative, so loose was the treatment that the penalty of algebra was incurred. The two negatives, which are only instrumentally the equivalent of an affirmative, are two signs as different in origin and character as the two negative signs in — (—a). In " man is not (not-animal)," the first negative disconnects, the second describes the predicate disconnected. In many cases, the difference of symbols, so much wanted by the beginner in algebra, far from encouraging abstraction of resemblances, stimulated differences of interpretation, as in • not unwilling,' which means less than willing : the double negative has, in common language, deteriorated into an affirmative of a lower degree. The suggestions of symbolic notation have led me to more recognition than is usually made of harmonies which exist among various pairs of opponent notions common in logical thought. I select the following ; — affirmative and negative — universal and particular — the subjective distinction of possible and impossible — the objective distinction of existent and non-existent necessary and not necessary — sufficient and insufficient — conjunctive and disjunctive — con- vertible and inconvertible — conclusive and inconclusive — singular and plural — definite and indefinite : omitting true and false, the most general of all, as most obviously capable of forming one element of the distinctive definition of any pair. I believe that any of these oppositions might be interchanged and used for each other : but not always without what would be called forcing. This, however, is not a conclusive objection : a forced analogy may only deserve that name because we have not been accustomed to the comparisons which it suggests, or THE THEORY OF SYLLOGISM, ETC. 81 to the language, or to the order of ideas, &c. The following phrase of Sir William Hamilton's system, ' All A is not some B ' is very forced, both in order and phraseology ; one who sees it for the first time finds it hard to make either English or sense of it. The meaning is, ' Each A is not any one among certain of the Bs :' and in its place in the system alluded to, the uncouth expression helps to produce system, and the perception of uniform law of inference. I now take an instance from the preceding list, in which it will appear that an obscurity of expression, if not absolute error, which has often occured, would have been avoided if the mind had been forced to an analogy or an analogy had been forced until the mind readily saw it. I say that the distinction of universal and particular may be replaced by that of conclusive and inconclusive. In comparing • every X is Y ' and ' some -Ys are Fs,' the first is conclusive with respect to every case which can be brought to the proposition for settlement by it. Cer- tain Xs, determinable or indeterminable, are proposed ; are they Fs, or not ? The universal is conclusive on this point, the particular inconclusive. Had this comparison been always made, we should not have had so many* writers who have expressed themselves as if the 'some 1 of a particular proposition excluded every other some. I will now go through all the cases I have named, making universal and particular the standard case which is to be compared with all the rest. 1. The distinction of universal and particular may be replaced by that of affirmative and negative. The universal affirms the right to assign one or other or both of two names to every object of thought in the universe of the proposition : the particular denies it. Thus, x being the contrary name of X, the proposition ' Every X is V affirms that F and so fill the universe ; and ' some Xs are not Fs ' denies it. 2. The distinction of universal and particular is that of possible and impossible, or of existent or non-existent, according as the proposition speaks subjectively or objectively. Let PQ signify the name compounded of P and Q, and belonging to every object which has a right to both names. Then ' every X is F' sets down Xy as impossible or non-existent, and • some ^Ys are not Fs 1 sets down Xy as possible or existent. 3. On the connexion of universal and particular with necessary and not-necessary (contin- gent) see the doctrine of modals, passim : on the correlative connexion with sufficient and not- sufficient, see F. L. p. 73. 4. The distinction of universal and particular is that of conjunctive and disjunctive; the universal speaks conjunctively, the particular disjunctively, of the same set. The Xs being * No small number of the elementary writers do this. But | prium of man is to have as much as is wanted. The same that their leaders had no such error in their heads, is clear enough. Ludovicus Vives, speaking of the maxim that Differentia et proprium non accipiunt magis et minus, re- marks that to be fit for a physician or a sailor are propria of man, and yet one man may have more fitness for either than another. But this is wrong, and the error consists in introduc. ing into the proprium that which is not part of it, as proprium: for one beast is more fit for a sailor than another ; and the pro- Vol. IX. Paet I. maxim, non aecipit magis et minus, was practically and con- sistently applied to particular quantity, though not stated of it, that I am aware of: but the writers I speak of use language which, but for their subsequent proceedings, would lead any one to suppose that they accepted magis et minus as indefinitely applicable to particulars, and definitely to the relation of uni- versal and particular. 82 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, distinguished as X% X 2 X 3 &c, the universal 'Every X is Y' affirms that X l is F, and that X 2 is F, and that X 3 is P", e£ ccetera. But the particular ' some Xs are not Fs' only declares that either X x is not F, or that X 2 is not F, or that X 3 is not F, aut ccetera. Nor do I here narrow the meaning of the particular : as used in logic, this species of proposition does not necessarily affirm nor deny of more than one. 5. The distinction of universal and particular may be made that of convertible and incon- vertible. This is the only case in which I have had to search for a meaning to make the system good : in all the other cases, perception of the instance preceded that of the general analogy ; I believe my work on logic will shew this of nearly all. Now convertibility and inconvertibility are only expressions of identity and non-identity : and it may be easily shewn that the universal is the identity of (followed by the right to convert) two names; and the particular the non-identity. Let U be the name of everything in the universe of the proposition : and let X, Y be a name which includes everything that is either X, or F, or both. Then the universal t. Every X is F' affirms the convertibility of w,Y and £7; while ' some Xs are not Fs ' denies it. 6. On the connexion of universal and particular with conclusive and inconclusive, I have already spoken. 7. The distinction of universal and particular is that of singular and plural. All books of logic affirm that the singular proposition is universal. But the manner in which logi- cians have treated the universal proposition as singular, in effect, if not in name, will be the material of a curious chapter in the history of logic, when written. Some of them have seen their own tendency, and have made ' Man is animal ' to be a proposition of a distinct species from ' Every man is an animal.' The universal proposition treats the subject col- lectively, and makes one singular notion of the whole : the particular makes, or may make, two groups, of indefinite proportions to the whole, and affirms or denies of one, neither affirming nor denying of the other. 8. Sir William Hamilton has very effectively forced the attention of logicians to the manner in which their universal and particular are definite and indefinite. I shall presently insist on this same distinction as that of indefinite and definite, and that with particular relation to Sir William Hamilton's system. If, instead of taking universal and particular as the standard relation, I had chosen affirmative and negative, the principle for which I contend would have appeared more clearly, perhaps: but at the same time it would have appeared to state nothing but what everybody knows. Surely, one would remark, all oppositions stand to one another in an affirmatory and negatory relation, so that affirmation and negation are the root of them all, and as things which are connected with the same are connected with one another, it follows that all opposite relations are connected with one another. This is perfectly true, and fully admitted : never- THE THEORY OF SYLLOGISM, ETC. 83 theless, true and easy as it is, writers on logic are not yet masters of it, nor were writers on algebra till recently. In this last science, all oppositions are instrumentally reducible to addition and subtraction: let gain, ascent, prior time, ....give or require addition, then loss, descent, posterior time, ....give or require subtraction. This easy key to the generalization of the meanings of + and - , is modern as to clear perception and full acceptance : D'Alembert de- nied its universality. I think it reasonably probable that the advance of symbolic logic will lead to a calculus of opposite relations, for mere inference, as general as that of + and — in algebra. On the advan- tages or disadvantages of its introduction it would be vain to speculate beforehand. I now proceed to another point of the approximation of logic and algebra. When the dry and lifeless instrumental forms of syllogism are placed before a student who has already familiarized himself with their use without thinking about them, it may easily happen that they are received with disgust, and it often has happened. That the noble act of the mind called by us inference, should be defined as consisting in mere transformation and substitution, appears* ridiculous. And the definition is truly so, unless it be confined to the instrumental part of inference, the part of the process which might be done by a machine. Algebra might be just as unworthily treated, by confining it to those few general rules in which its operative part really consists, and elevating this part to the dignity of a whole. At the highest, we can but compare the forms of logic in reasoning with the laws of linear perspective in painting : and the presence of these forms with the incidental lines which perspective requires, and which are rubbed out, not merely before the design is finished, but before the higher art of the process begins. And the parallel holds still further. Many great painters have disfigured their work by too much neglect of the instrumental laws of perspective; many have wilfully and skilfully violated them to produce the effect they wanted ; — and so has it been with reasoners. Speaking instrumentally, what is called elimination in algebra is what is called infer- ence in logic. If there be four " in ,° . involving any number of objects of ° assertions logic ° J J , it is possible from the four to produce one " . , excluding three obiects of assertion r r assertion ° J ^ ■. from among those in the originals. The . „ is free from the three assertion ° ° inference eliminated quantities middle terms The logicians confine themselves in the first instance to the simple syllogism, which is the elimination of one middle term between two assertions. In like manner the algebraist asserts that all elimination may be reduced to successions of eliminations of one quantity between two equations. And just as all direct power of elimination, exclusive of what are called artifices, depends upon our being able to find one quantity in terms of others with which it is involved in an equation — so all our power of expressing inference depends upon our being able to describe one object of thought in terms of others, by means of an assertion in which they are all in- The two propositions Omnis dives est sapiens, and Solus sapiens est dives are logical equivalents ! 11- 84 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, volved. When I say "'John met Thomas in the street' — 'John shook hands with Thomas' — that is, 'John shook hands with what John met in the street'" — there is an elimination of ' Thomas' perfectly answering in process to 'a + = b, y = c + x, therefore y = c + b - a'. Several remarkable matters connected with inference may be made to arise out of this view, of which some will be noticed. But I now proceed to observe that this perfect sameness of logical and algebraical process does not continue. Whenever close resemblances exist which are rarely or never noted, we may be pretty sure that the attention has been diverted by differences as remarkable or more so. In algebra, in which the result is quantity, equations are perfect identities. If x be 10, every x that can be produced under precisely the same circumstances, is entitled to the copular sign (=) in connexion with any abstract 10 that we can imagine. The consequence is, that a complete conversion of all the processes of elimination can always be made. If y = y or xy, as compared with x = y. Nevertheless, elimination between inequalities is sometimes required: and then we know that in x>y our right of sub- stitution is, that we may for x write an equal or a greater, for y an equal or a less. In x>y, THE THEORY OF SYLLOGISM, ETC. 85 y is used after the manner of a universal term in logic, a> after the manner of a particular. We must know the whole of y ; but we may be enabled to make the assertion by knowing a part of ,v. A little consideration suggests as a necessary rule of inference, the right to sub- stitute a larger term used particularly for a smaller one, however used, and a smaller, used in either way, for a larger used universally. What we may affirm or deny of some or all men, we may affirm or deny of some animals: what we may affirm or deny of all animals, we may affirm or deny of all or some men. The second part of the rule is the dictum de omni et nulla ; the first part has not, within my reading, been added to it : both might well be incor- porated in one under the name of the dictum* de majore et minore. Observing that every inference was frequently declared to be reducible to syllogism, with no exception unless in the case of mere transformation, as in the deduction of ' No X is Y' from • No Y is XJ I gave a challenge in my work on formal logic to deduce syllogistically from ' Every man is an animal' that ' every head of a man is the head of an animal.' From the total absence of attempt to answer this challenge, I conclude f that no one has succeeded in whose way it has fallen. I shall presently have occasion to pursue this subject a little further : I now proceed to a new section. Section II. ON THE FORMATION OF SYMBOLIC NOTATION FOR PROPOSITIONS AND SYLLOGISMS. The commonly received method of denoting the affirmative and negative universals by A, E, and the particulars by /, O, is rather mnemonical than instrumental : it suggests to nothing but the memory. Of all the systematic deductions of the valid forms of syllogism, not one came into general use : inductive selection and exclusion were employed, when any- thing more than a mere declaration of results was given. The figure being given, the above symbols are sufficient : thus AEI in the first figure can be nothing but the invalid mode ' Every Y is Z, no X is Y, therefore some Xs are Zs.' In my former paper, and in my work since published, I borrowed from the above notation the use and meaning of A, E, I, O, and added symbolic distinctions. So far I have nothing to change: any system must use some mode of expressing its relation to the language of so many centuries. I also adopted a more detailed mode of expressing propositions : and here it would have been better if the detail had been greater at first, that it might have been ulti- mately made still less. A fundamental symbol should not be of compound meaning : that is, should not expressly signify more than one thing. Composite expressions should be reserved for symbols which are * Eveiy syllogism in the Aristotelian system is a direct use of this dictum. + This would be a very unsafe conclusion from the absence of printed answer. But any one who writes on a controverted subject gains a number of private correspondents, with and without names. When I advanced, in my discussion with Sir William Hamilton, that a person kept close to Aristotle's forms could not prove that some must have both coats and waistcoats, if a majority have coats, and a majority waistcoats, I had various ingenious attempts to disprove my assertion. 86 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, avowedly in abbreviation of combinations of fundamental symbols. Again, the complex symbol should not be prior in invention to the simple ones ; nor should it be invented until the simple ones have had their chance of good suggestion. For instance, in the old notation, the letters A, &c. are of compound meaning ; A is universal and affirmative. If A and E had stood simply for affirmative and negative, and two consonants, as B and N, for universal and particular, the distinction of figure might have been symbolized. Barbara does not suggest the first figure, nor Camestres the second, except by memory. But if the premise-consonants had been made to imitate the middle term in location, the figure would have been seen in the word which would have resulted, whether unmeaning letters had been added for euphony or not. Thus AE occurring as premises in the second figure, in which the middle term is predicate of both premises, would have given ABEBEB ; and so on. If some choice of liquids had been allowed for designation of par- ticulars, and of other consonants for designation of universals, euphonic words might have been invented, which would have been, by this time, as venerable as Bokardo or Felapton. Nor would it have been difficult to have fitted on letters symbolic of the method of reduction into the first figure. This suggestion, however, comes a few centuries too late : the following one is more to our purpose. Symbols in which relative position is the whole or part of the symbol, whatever their advantages may be in other respects, lie under one great disadvantage : abstraction is not sug- gested, and can only be done awkwardly. The exponential symbol in algebra has this defect : we cannot describe it independently of others, except by ( ) ( ', or some such contrivance. In the more detailed notation of my former paper this fault was committed : thus XY, by mere position of the letters, was made to indicate ' Some Xs. are Fs.' The distinctive characters of the proposition are made to be, usually, the terms, the copula (affirmative or negative), and the quantity of the subject. But if the quantity of the predicate be also symbolized, notice of the terms is not distinctively necessary : for every proposition used has neither more nor less than two terms, and a term need not enter except to have its quantity noted. The symbols of the terms, in fact, are only pegs on which to hang distinc- tions: so that it is desirable that they should not be essential, though capable of introduction. The quantities of the terms give name to the proposition : which is usually called universal or particular after its subject. This is arbitrary ; and it is open to us, by the same license, to make the proposition take the name of its predicate. The enlargement of the proposition, whether Sir William Hamilton's or my own, will probably require a new word to express the distinction of propositions : for both have their double universals, and their double particulars, which are not in the Aristotelian set. At present, however, I am not prepared to suggest a term which would apply to both systems. Let the subject and predicate, when specified, be written before and after the symbols of quantity. Let the inclosing parenthesis, as in X) or (X, denote that the name-symbol X, which would be inclosed if the oval were completed, enters universally. Let an excluding parenthesis, as in )X or X(, signify that the name-symbol enters particularly. Let an even number of dots, or none at all, inserted between the parentheses, denote affirmation or agree- THE THEORY OF SYLLOGISM, ETC. 87 ment ; let an odd number, usually one, denote negation or non-agreement. Thus X )) Y means that all Xs are Fs ; X(.( Y means that some Xs are not Fs : but )) and (.( specify the charac- ters of the propositions ; as do also (( and ).). We must conceive ourselves at liberty to read either way : thus X)) Y and Y((X both denote that every X is F. A syllogism may be denoted by juxtaposition of the symbols of the premises, taking the 'order XY, YZ, XZ. Thus 'Every AT is F, some Zs are not Fs, therefore some Zs are not Xs ' may be stated thus, X))Y).)Z gives X).)Z or ))).) gives).). In the Aristotelian system, and in my extension, the canon of formation of the inference, when there is one, is ; — Erase the symbols of the middle term, the remaining symbols shew the inference. Thus, if there be a valid inference from ).( (.) it is ). .) or )). Thus also X()Y))Z expresses the premises 'Some Xs are Fs, and every F is Z: 1 erase F and its accompaniments, and we have XQZ for the conclusion, or ' Some Xs are Zs.' In Sir William Hamilton's system, this law is not quite universal in the symbolic deduction of the inference ; but a certain variation, which I shall presently suggest, will make it so. If we wish to read by distinction of figure, that is, by Aristotelian figure, we may contrive it thus. Let the subject of each proposition have its quantity denoted by a thicker or larger parenthesis. Then the first figure, in which we read through the concluding terms, would present the appearance | ] ; remembering that when the distinction of major and minor term of conclusion is preserved, we read the second premise first. Thus X))Y))Z = X))Z, which is now symbolized in the first figure, is Y))Z + X)) Y = X))Z. . The second figure, in which we read to the middle term, is as seen in | |; the third, in which we read from the middle term, is as seen in |||| ; and the fourth, in which we read through the middle term, is as seen in ||||. All notation, no doubt, is both pictorial and arbitrary : nevertheless there are cases in which one or the other character decidedly predominates. The arbitrary character decidedly predominates in the preceding notation : but the syllogism admits of a graphical representation which is as suggestive as a diagram of geometry. This was partially adopted by Lambert and Euler (F. L. p. 323), and may be more completely shewn, and without new types or wood- cuts, in the following way. Let all the instances in the universe of the syllogism be represented by the points of a definite straight line : but to avoid confusion, let this straight line be repeated as often as it is necessary to introduce a name. Let the division of this straight line into a continuous and a dotted portion signify the distribution of the universe into a name and its contrary. When a proposition is asserted, let a second line run over so much of the extent of each name as is declared by the proposition to be in agreement or disagreement with the whole or part of the other : extents which fall under one another being taken as in agreement. Thus in the follow- ing diagrams we see the propositions ' Every X is F,' and ' Some things are neither Xs nor Fs, 1 X L= X Y U== Y 88 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Let the middle name of a syllogism be placed in the middle, and when the two premises are formed, let the extents which enter into the conclusion be signified by thicker lines, or thicker dots. Thus it will appear what the conclusion is, and also whether the quantities in the conclusion be those which entered into the premises, or whether, by the character of the inference, either be curtailed. Thus in the diagram before us X Y Z (•))')-() we see the pictorial and arbitrary notation for the following syllogism : — Everything is either Zor F; some Zs are not Ys; therefore some Zs (as many as entered the premise) are Xs (not necessarily as many as in the premise). We also see that the real middle term of agree- ment is a portion (or what may be only a portion) of the extent of y : and that the affirmative form of the syllogism is X((y()Z = X()Z. References to figure might easily be added. I should here close this section, if it had not been that Sir William Hamilton's scheme of notation has been published by an acute writer, with such commendations*, that I must not appear to shun the comparison. This system is certainly so simple, that a person who knows the premises and inference well, would write down any case of it immediately. I exhibit one case of it in the three figures : premising that Sir William Hamilton rejects the distinction of major and minor, and draws two conclusions in the second figure and in the third : but does not permit the fourth figure to append itself to the first, nor to appear in any way. X ,Y: Z X : >,Y: First Figure. Some Y is all X. Some Z is all Y. Some Z is all X, or Second Figure. All X is some Y. Some Z is all F. Some Z is all X. All X is some Z, or Third Figure. Some Y is all X. All Y is some Z. Some Z is all X. All X is some Z. ,Z Negation is expressed by drawing a vertical line through the sign of predication. When the thin end of this sign is made the subject, the syllogism is read by intension. It would appear at first that this notation is almost identical with what I have proposed above, as to principle. Leave out the lines of predication, and the above syllogism would be " " A mode of notation proposed by Sir William Hamil- ton, is, beyond doubt, one of the most important contributions to pure Logic which has ever been made since the science was put forth ; and I am fortunate in being permitted to annex it. Its excellences are — that it is very simple, that it shews the equivalent syllogisms in the different figures at a glance, that it shews as readily the convertible syllogisms in the same figure, that it enables us to read each syllogism with equal facility according to extension and intension, " Outline of the Necessary Laws of Thought. By William Thomson, M.A. (2nd. Edition, 1849, p. 265.) A work to be strongly recom- mended. THE THEORY OF SYLLOGISM, ETC. 89 X: , Y:, Z=X: , Z, instead of X))Y))Z = X))Z. And I am bound to attribute* to Sir William Hamilton the expression of the two distinct quantities, universal and particular, as two distinct matters of phraseology and thence of notation, when the quantities are not numeri- cally conceived and expressed. (F. L. p. 301.) If then the preceding notations were as much alike as they appear to be, I should call mine '(which was first used in trying Sir W. Hamilton's system, F. L. p. 302), so far as affirmative syllogisms are concerned, Sir William Hamilton's with what I judge to be a more convenient mode of expression. But there is, in fact (independently of the omission of term-symbols), a very material difference, and one which makes the notation I have given more suggestive T, and its symbolic rules more easy to some extent in my own system, to a greater in Sir William Hamilton's. In his mode of notation, the symbols of universal and particular are absolute (: and ,) : in mine, the universality of the subject has the same symbol as the particularity of the predicate, and vice versa: thus in X))Y, the same symbol [)] is applied to the universal subject, and the particular predicate. The notion on which this mode of symbolizing quantity was tried and found to succeed, was as follows : The most natural mode of predication, because the easiest premise for inference, is the affirmative, and the syllogism of affirmative premises is the one to which all other cases are naturally reduced. And here the predicates are always particular, while the subjects are either universal, or, which is the same reality in inference, take the whole extent named in the premise into the conclusion. Whenever this is the case, the invention of a name will shew that the inference is of the same kind as one in which the term of the premise and of the conclusion are both universal. In representing " All X is in Y, all Y is in Z, therefore all X is in Z? by X))Y))Z there is something pictorial, and the particular character of the predicate has its symbol of particularity invented out of the relation to the subject from which we deduce that character. * And I may also note an inaccuracy of expression used by Mr Thomson (pp. 265, 266): "Many of the different ele- ments of the notation are not new, but the novelty lies in the completeness and simplicity of the whole scheme." Not so; for though the notation had failed entirely both in complete- ness and simplicity, there would have remained the most re- markable novelty which there now is, namely, notation of quantity in both subject and predicate. t It would seem that the forms given by Sir William Ha- milton have not suggested any rules. " In the negative modes the distribution of terms will remain exactly the same as it was in the affirmatives from whence they were respectively formed, with some few exceptions in which the conclusion has a term distributed which was not when it was affirmative." Thomson, Op. Cit. p. 267. Under a suggestive notation, the rule which regulates the exceptions will be visible; and it will be seen that these so called exceptions are in the rule, and that cases of the affirmative syllogism are the exceptions. Vol. IX. Part I. 12 90 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Section III. ON THE SYMBOLIC FORMS OF THE EXTENSION OF THE ARISTOTELIAN SYSTEM IN WHICH CONTRARIES ARE ADMITTED. This system is, by consequence, not by assumption, one in which any term, be it subject or predicate, may have either kind of quantity, universal or particular, in any proposition, affirmative or negative. Sir William Hamilton's system has the same peculiarity, as the basis of invention for the forms of predication : that is, the accidental form of my system is the substantial form of his, so far as these terms are applicable. I did not make this point of agreement between the two systems prominent in my work on formal logic, for the following reasons. In the memoir printed by this Society, in which all my interest in novelty of quantification was directed to the algebraical form of numerically definite propositions, this complete distribution of all the quantifications, existing in the system of contraries, was overlooked. So much so, that no one could conclude from my words* more than that, with eight forms of predication, and knowledge of the doctrine of combinations, I must have seen the necessity of this alternative — either two forms of predication with the same terms and the same quantities, or a distribution of all possible pairs of quantifications. But I have no remembrance of even this alternative suggesting itself. When the discussion with Sir William Hamilton turned all my attention to the question whether he had or had not the numerical system, and, subsequently, to the comparison of his system (F. L. pp. 300 — 302) with the numerical one, it became evident of course, that the complete distribution of quantifications is incidental to the system of contraries. But I did not mention this explicitly in my work (pp. 63, 293) : because, as the controversy was then unfinished, I neither wished to dwell upon an irrelevant quantification (that which is assumed and constructed, not that which can be derived, being the subject-matter of the dispute), such as might mislead the reader of the controversy, nor to appear as insinuating that I had published to the Society, before I had had any correspondence with my opponent, a system containing by derivation the whole extent of quantification, the invention of which was in the subject-matter of the discussion. Such insinuation would have been untrue : for though the system I now write upon does contain that extent of quantification, and though it was published (to the Society) before I had any knowledge even of the fact of Sir William Hamilton having a system of his own, yet I can most distinctly affirm that all my perception of complete quanti- fication of both terms was derived from the algebraical form of numerical quantification. The universal and particular affirmative may be made the bases of all the modes of predication : the others arising out of the various substitutions of contraries in them. The following are then the eight forms, with reference to the order XY. * As follows, in the fourth page of my paper "... every proposition speaks in different ways of each term and its con- trary; making one particular or universal, according as the other is universal or particular.. ..And of the two terms and their contraries, each proposition speaks universally of two, and particularly of two." THE THEORY OF SYLLOGISM, ETC. 91 Universals. Former Memoir. Notatic Work on Logic. n of my Both. Notation now proposed. Proposition expressed in common language. A 4 X)Y X))Y Every X is Y a J 1 ))y or X((Y Every Y is X E Ei X)yov X.Y X))y or X).(Y No X is Y e E x x) Y or at . y a?)) For X{.)Y Particulars. Everything is X or Y or both I h XY X{)Y Some Xs are Ys i r wy w()yovX)(Y Some things are neither Xs nc Or Xy or X : Y x()y or xur Some Xs are not Is O 1 .vYor Y:X w()Yor X).)Y Some Ys are not Xs. I think, that the words universal and particular can be better described for this system than in the usual way, both as to terms and propositions. A term enters universally, when, in order to verify the proposition by induction, every instance of the name must be examined : particularly, when the verification may be attained without it. A proposition is universal, when every instance in the universe must be examined before it can be inductively verified : particular, when the verification may be attained without. So that the phrases • universal proposition, 1 and ' particular proposition, 1 refer to all things in the universe of the proposition ; while • universal term 1 and ' particular term 1 refer to all things contained in a term, or portion of that universe. The introduction of contraries does, in fact, introduce a third term into the proposition ; the universe, or summum genus, be it the whole universe of thought, or a conceivably separable portion of it. And it is to be particularly remembered, that every term is supposed to be part only of the universe : that is, to have an existing contrary in that universe. In the inductive examination above alluded to, we suppose that we do not know before- hand which instances of the universe are Xs and which are Ys ; nor, in selecting* an instance from the universe, can we say that that instance is not an X, till we have examined it with all the Xs. In verifying X))Y, we have then to examine every instance of the universe to see whether it be X, but only when it is X need we examine further to see whether it be F: Ys may never be ascertained to be such, since, by reason of their not being Xs, the examination may never take in those Fs. Hence X))Y is a universal proposition in which X is universal and F particular. Again, in X)(Y or 'some things are neither Xs nor Fs, 1 the^rs^ instance of the universe which is carried through all the Xs and all the Fs, and found to agree with no one instance of either, verifies all that the proposition asserts : hence the proposition is particular, and its terms both universal. I can now correct a symbolic want of my former writings, though the thing to be signified * That is, we do not suppose, as it were, that every instance is ticketed in its place in the universe as X or x, Y or y, but that each instance taken from the universe has its own uni- verse-mark, and that we must then, in order to find if that instance be an X, examine a separate index of Xs, to see if that mark occur. The ticketing of the instances, first men- tioned, is the representative of the supposition on which some have discovered that the syllogism concludes in what was already known before the premises could be asserted. 12—2 92 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, was often matter of explanation. Expression for the contrary of a contrary was wanting in the subject of the proposition : whence it arose that only three equivalents appeared in the detailed symbols, though four existed in thought and in the more compressed symbols. Thus I had A { of XY, E l of Xy, A 1 of xy, E l of xY, as equivalents: while only X)Y = X.y =y)x appeared in the detailed symbols. I now have jr))'r-jr).te-*«y-»(.)r X(.(Y=X0y = x).)y=*)(Y. The rule of transformation is ; — To use the contrary of a term, without altering the import of the proposition, alter the curvature of its parenthesis, and annex or withdraw a negative point. We may now say that the quantitative contrary of 'every X" 1 is 'some *•,' and of ' some Xs," 1 ' every x.' Thus, when I say, ' some Xs are Fs ' I deny something of every x : namely, that any one of them is one of those Fs. Again, ' Every X is F 1 denies of some xs that they are Fs : for Fs must not fill the universe. And so on. The distinction of affirmative and negative, in the usual sense, is abandoned : for any affirmative proposition, as XQY, is also negative, as seen in its equivalent X(.(y. My two new forms of predication were properly called, (.) negative, and )( affirmative, and were derived in the forms x).(y and xQy as 'no not-^ is not-F 1 and 'some not-^Ts are not- Fa.' Never- theless, stated in reference to X and F, the first appears affirmative ' Everything is either X or F', and the second negative, ' Some things are neither Xs nor Fs.' But the first obeys the rules for the indisputable negatives, and the second those for the affirmatives. There enters an extension of an old maxim ; — it is that three negatives make a negative. There are three positive ideas, X, F, affirmation ; opposed to the three correlative negatives, x, y, negation. Negative propositions present an odd number of the negatives ; positive ones an even number (or none). Thus in all the equivalent forms of X(.)Y, nothing but an odd number of the negatives will occur ; as seen in X(.)Y=x))Y=X((y = x).(y. There is yet one more opposition ; the quantitative parentheses may turn the same or different ways : which are to be considered as positive and negative cases. By this, and the opposi- tion of affirmation or negation, the extent of the proposition is determined. When these oppositions are none or two, the proposition is universal : when one, particular. Thus X)) Y having none, and X(.) Y having two, are universals: but XQY having one, is particular. In a universal proposition, any one quantity may be altered, either from universal to par- ticular, or from particular to universal; and the result is always a true deduction, though not an equivalent. Thus X))Y gives both X() Y and X){Y. Contrary propositions (usually called contradictory) of which one must be true and one false, differ both in quantities and copula. Thus X))Y and X(.(Y are contraries. The concomitants of a universal, to which it is perfectly indifferent, differ from it in quantities, or in copula, not in both. Thus X))Y coexists either with X((Y or X).)Y. The superior universals of a particular, or the inferior particulars of a universal, are made by altering one quantity only: thus X){Y has X((Y and X )) F for its superior universals; and X(.)Y has X).) Y and X(.(Y for its inferior particulars. The alteration of one quantity, and the THE THEORY OF SYLLOGISM, ETC. 93 copula, turns a universal into another and inconsistent universal, and a particular into another and a consistent particular : as A")) F into A(.)F, or A(.)F into X).)Y. An alteration of either of the terms into its contrary, must count as alteration of the quantity of that term, and the copula. To carry this a little further, observe that though all the particulars consist with one another, or may exist together, yet they take distinctions, when looked at as to probability, which have some resemblance to those existing among the universals either as probabilities or as certainties. Take the universal X))Y, and change both quantities : it becomes A((F. If the latter be true, it is in favour of X))Y rather than otherwise. Do the same with AQF, it becomes A)(F: if the second be true, it is also rather a presumption for X()Y than otherwise. The more things there are which are neither X nor Y, the smaller the number of instances of the universe within which all the As and all the Ys are contained ; and the greater the proba- bility of X()Y. Now take the universal X))Y and alter one quantity and the copula: we have AQF and A).(F, both absolutely inconsistent with X))Y. Do the same with A()F: we have X).)Y and A(.(F, neither inconsistent with A()F, but both diminishing its pro- bability. This observation is of some wortli in classification : it may justify us in extending the general name of concomitants to particulars in which both quantities differ, and opponents to those in which one quantity and the copula differ ; and will help us to system. Now let a proposition be considered as having two sides, on each of which a change of quantity counts as one, and a change of copula as one for each side. Changes of quantity then may be represented by l|o and o|l, change of copula is l|l, and changes of term are 2|l and l|2. The utmost amount of change is 4J4, which restores the original : thus o?(..(y is X))Y affected by 4J4, and it is X))Y itself. And it will be found that l|o and o|l always pro- duce a superior universal or an inferior particular, that l|l produces a concomitant, 2|l or l|2 an opponent, 2J2 a contrary ; and so on. When contraries are allowed, all inference may be made to consist in declaration of agree- ment. The disagreement of two things, because one agrees and the other disagrees with a third, may be made to fall under the other and more simple case ; not indeed, as the easiest rule of thought, but as the best basis of a classification. This X agrees with this F; this Z does not agree with this Y ; therefore this X does not agree with this Z — may be stated as, This X agrees with this Y, something not this Z agrees with this Y, therefore this X agrees with something not this Z. And the universe, explicitly divided into this Z and other things, is a provision for the definite understanding of the manner in which the second proposition is transformed. Accordingly, agreement between Xs and Ys, Ys and Zs leads to agreement between As and Zs, if the number of agreements be altogether more than there are Ys. Restricted as we are now to the case of the universal and the indefinite particular, one of the propositions must mention all the Fs. So that the fundamental syllogisms, from which all the rest are derived by every introduction of contraries, seem to be J l A 1 A l X))Y))Z = X))Z or )))) = )) A' A' A' A((F((Z = A((Z or (((( = (( A()F))Z = A()Z or ())) = () 7,^,7, X((Y()Z = X()Z or ((() = () ^'7,7, A l AJ t X((Y))Z = X()Z or (()) = (). 94 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, But in truth this basis is excessive, in three distinct ways. 1. (((( and )))) really belong to one of the sets which is to be formed; and the pro- ceeds of the two will be identical. One only of them should be taken. 2. The sets from ())) and ((() only differ in respect to the species of figure* which I employ. I keep to XY, YZ, XZ; and one of these sets is the other referred to ZY, YX, ZX : the refusal of all distinction of figure obliges me to retain both sets. 3. In (()) = () we see only a strengthened syllogism : that is, stronger in the premises than is required to produce the conclusion. It is either ( ))) or ((( ) with the particular middle term universalized. The number of variations produced by combining X or x, Y or y, Z or z, is eight, taking one out of each pair. And as these variations do not affect the character of the proposition, we must have as follows. Eight universal syllogisms, with two universal premises and a universal conclusion, derived from )))) = )). Sixteen particular syllogisms, each with a particular conclusion, derived from one universal and one particular premise ; eight commencing with a particular, from ())) = (); eight commencing with a universal, from ((( ). Eight strength- ened syllogisms, in which two universals conclude with a particular, derived from (()) = ( ). The canon of validity is as follows. The change of Y into y will alter the middle paren- theses by changing them both, and therefore will not affect their relative character. Conse- quently, if the middle parentheses turn the same way, ((or )), any two propositions which give this arrangement allow of an inference, if one at least be universal ; that is, if the number of oppositions in the parentheses, and the number of negative dots, put together, be an odd num- ber in all, or an even number (0 included) in each proposition. But when the middle parentheses turn contrary ways, there need but be two universals, or an even number (0 included) on each side. The canon of inference is merely this ; — Strike out the middle parentheses, and two negative dots, if there be two ; the remaining symbol shews the inference. * I thought I had sufficiently expressed this in my work by stating that 1 abandoned the distinction of figure. But I sup- pose it was hardly clear: for a learned reviewer observes, "That is not new — but is Sir William Hamilton's avowed rule." This is not correct : Sir William Hamilton's method is equally indifferent to all figures ; mine holds by one, and recognizes no other, in its classifications. I have one establish- ment, and tolerate no distinction of sect. Sir William Hamil- ton has none, but tolerates three sects : Aristotle had three state-churches, Galen (they say) founded a sect of dissenters, which, after some centuries of toleration, was made a fourth establishment. THE THEORY OF SYLLOGISM, ETC. 95 In the following table, the syllogisms are arranged in the manner described in the headings. Terms employed. Particular syllogism with first premise particular. Universal syllogism strengthened by the first quantity of the particular. Particular syllogism strengthened by the second quantity of the particular. X'Y Z x y z x Y Z X y z X y Z x Y z X Y z x y Z I.A.I, ())) = () PAT )(((=)( 0'A,0' ).)))=).) 0,A'0, (.(((=(.( O.E'I, (.((.)=() O'E.I' ).)).(=)( I.E.O, ())•(=(•( I'E'O' )((.)=).) A,A,A, ))))=)) A'A'A' ((((=(( E'A,E" (.))) = (.) E,A'E, ).(((=).( E.E'A, ).((.)-)) E'E.A' (.)).(=(( A,E,E, ))).(-).( A'E'E' (((.) = (.) A'A.I, (())=() A.AT ))((=)( E,A,0' ).())=).) E'A'O, (.)((=(.( E'E'I, (.)(.)=() E,E,I' ).().(=)( A'E.O, (().(=(.( A.E'O' ))(.)=).) Terms employed. Particular syllogism with second premise particular. Universal syllogism strengthened by the second quantity of the particular. Particular syllogism strengthened by the first quantity of the particular. X Y Z x y z x Y Z X y z X y Z x Y z X Y z x y Z A'1,1, ((() = () A,rr »)(=)( E,I,0' ).(()=).) E'I'O, (.))(=(.( E'O'I, (.)).)=() E,0,I' ).((.(=)( A'0,0, (((.(=(.( A.O'O' ))).)=).) A'A'A' ((((=(( AAA, ))))=)) E,A'E, ).(((=).( E'A,E' (.))) = (.) E'E,A' (.)).(=(( E.E'A, ).((.)=)) A'E'E' (((.)=(.) A,E,E, ))).(=).( A'A.1, (()) = () A.A'I 1 ))((=)( E,A ( 0' ).())=).) E'A'O, (.)((=(.( E'E'I, (.)(.)=() E,E,I' ).().(=)( A'E.O, (().(=(•( A.E'O' ))(.) = (.( The universal and strengthened syllogisms appear twice, once from each set of particular syllogisms. The contranominals, or those in which all the terms used are contraries, are written together : they are formed from each other by alteration of all the parentheses. Altering one quantity in a particular proposition, even though a universal term be changed into a particular one, strengthens the proposition, or converts it into a universal. In fact, in this system strength and weakness are terms which, like affirmation and negation, have nothing but opposition of character left : the less the strength of a proposition with respect to a term, the greater with respect to its contrary : so that what I have called strengthening a proposition, in only altering the distribution of strength. The opponents of a syllogism, meaning those in which one premise and the contrary of the conclusion produce the contrary of the other premise, are thus formed. Substitute the con- trary of the conclusion for one premise and change the order of reading in the other, or let the symbol of the other revolve about an axis. Thus the two opponents of ).)).( are (.((.) and (.)).(; and those of ())) are ()).( and ).((( 96 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, The opponents of the strengthened syllogisms are universals weakened in their conclusions. The change of the figure from XY YZ XZ to ZY YX ZX is merely the writing of the symbol from the other end : it turns )))) into (((( and ()).( into ).((). Those syllogisms which agree in their parentheses and differ only in their affirmations and negations are the con- comitants of my work (pp. 88, 89, 94), or the coexistents in a complex syllogism. Thus A X A X A X , O A x O x and A x ] 1 are )))) ).))) and ))).). These syllogisms coexist in my complex syllo- gism X),i)iX)| ; and perhaps, then, the best notation for this last might be derived from ).)).), indicative of the only one of the four combinations which is not valid as a simple syllogism. I need not enter further into this subject, as what is here given on notation may be easily applied throughout my work. Section IV. ON THE SYMBOLIC FORMS OF THE SYSTEM IN WHICH ALL THE COMBINATIONS OF QUANTITY ARE INTRODUCED BY ARBITRARY INVENTION OF FORMS OF PREDICATION. This system, which belongs to Sir William Hamilton, has not yet been published in detail by its learned author, except in lectures; in which, I believe, it was first published in 1840 or 1841. There is some account of its forms, communicated by him, and printed with his sanc- tion, in Mr. Thomson's Outlines, already cited. See also F. L. pp. 300 — 302. The modes of predication in this system, are, by hypothesis, as follows, at least when the language of extent, preferred by Sir William Hamilton, is changed into that of numeration of instances. The symbols attached are dictated by the quantities of the terms, with reference to the order XY. All Xs are all Ys Some Xs are some Ys All Xs are some Ys Some Xs are all Fs Some authors had gone so far (F. L. Appendix n.) as to adapt expressed quantity to the predicate, for the purpose of procuring convertible forms : and Mr. Thomas Solly (Syllabus of Logic, 1839, p. 47) gave the above eight forms, with his reasons for reducing them to four. But Sir William Hamilton is the first who published the idea of taking all phases of usual quantification, and making them the basis of a system of syllogism. It will be observed that I have not called this system an extension of that of Aristotle. That it is more extensive, in one sense, I admit ; namely, in so far as it includes all which Aristotle included, and more. But a mathematician cannot therefore call it an extension, accustomed as he is to a very precise use of that term. With him enlargement is not exten- sion, unless the wider extent be governed by the laws of the narrower one. The name multi- plication, conceived by aid of integer numbers, is properly allowed to be extended to fractional )( A 1 + A>. No Xs are Fs ).( Ex () I» Some Xs are not some Fs (.) )) A v No Xs are some Fs ).) O l (( A\ Some Xs are no Fs (.( 0,. THE THEORY OF SYLLOGISM, ETC. 97 ones, because, among other reasons, in every problem in which integers demand integer multi- plication, fractional data demand what is thence called fractional multiplication. Botany was once agriculture, but in its present state it cannot properly be called an extension of agricul- ture : the union of England and Scotland was not an extension of England. This refusal to use the word extension, in the present case, is not the assertion of any defect in the system, but rather the contrary : it is quite open to inquiry whether the best form of syllogism be what I call an extension of Aristotle, or contain the incorporation of new fundamental principles. Perhaps some may ask why I have called my own system an extension of Aristotle. I answer that no new laws are propounded, though the application of the old ones to an enlarged subject-matter of predication introduces some new forms of expression, and some striking points of view from which to look at the old ones. Every one of my syllogisms can be reduced to an Aristotelian form, without any addition except that of contraries to the matters of predication. For example, one of my new syllogisms, ))(( = )(> or ' -All As and all Zs are Ys, therefore some things (namely, all that are not Ys) are neither As nor Zs — is reducible to ordinary form. With X Y Z it is new : but with X y % it is Fesapo, being X).(y))x = X).)z. The syllogism ))(( = )( can thus be made Aristotelian : but in my system, a plain man who sees clearly that some things are proved to be neither men nor mice, were it only because they do not eat cheese, may rest content that his knowledge, even in the form of the light of nature, can be made science, without the necessity of having recourse to the following very venerable, but very unsatisfactory, form : No man is a non-eater of cheese All non-eaters of cheese are other things than mice Therefore some other things than mice are also not men. Now take one of Sir William Hamilton's peculiar syllogisms ; — Some men are soldiers, Some animals are not men, Therefore some soldiers are not some animals. This syllogism cannot in any way be made Aristotelian, either with the terms as they stand, or with any others derived from them by a method independent of the syllogism itself; for instance, the derivation of the contrary from the direct term. Sir William Hamilton's system is there- fore an independent addition to that of Aristotle ; and the addition must be discussed on its own merits. The forms )) (( ).) (.( ( ) ).( exist in the old system, in that of contraries, and in that of invention of predicates. The peculiar propositions of the second and third may therefore be compared as follows, and under the same* symbols : * Looking at the fact that the system of contraries admits premises both negative, and that of invention of predicates admits premises both particular, with other analogies which I do not describe here, — I strongly suspect that the two systems have some correlative formation in which the distinction of affirmative and negative appears in the first with the same laws of form under which that of universal and particular appears in the second; and vice versa. If this be the case, then (•) though it have a particular form in common language, will be a universal, and )( a particular. This is a hint for the conside- ration of the reader : I have not been able to make anything out of it, as yet. Vol. IX. Part I. 13 98 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Introduction of Contraries. (.) Universal negative, with particular terms, and affirmative form in common lan- guage. All things are either Xs or Ys. ) ( Particular affirmative, with universal terms, and negative form in common language. Some things are neither Xs nor Ys. Invention of predicates. (.) Particular negative with particular terms, not used in common language. Some Xs are not some Ys. ) ( Universal affirmative with universal terms, being declaration of identity in common language. All Xs are all Ys. My objections to this system as promulgated by Sir William Hamilton (F. L. p. 302) may be developed as follows : First, the fundamental propositions of a logical system should be independent of each other, so that no one of them should be a compound of two others. Now X)(Y, or ' X and Fare identical names,' is really compounded of X)) Y and X(( Y. If we once grant a complex propo- sition, why this one only, when there are others, out of which, as I have shewn, a separate system of complex syllogism may be constructed ? To say that the mode of inventing propositions yields no other, is not an answer : for it is the mode itself which is attacked in its results. Every syllogism in which )( occurs, is either a strengthened form, or the resultant of two other syllogisms. Secondly, one object of formal logic being to provide form of enunciation for all truth, and form of denial for all falsehood, it is clear that every falsehood which can be enunciated as a truth should be deniable within the forms of the science. Now the simple denial of )( is the disjunctive assertion ').) or (,(*. Though it happen that I can prove one of these to be true, without knowing which, yet the power of denying in an elementary form the elementary pro- position )( is refused me. A philologist asserts the Greek words A and B to be identical in meaning : he says " All A is all B." One passage of Homer, and one of Hesiod, both contain the doubtful word G, having two possible explanations, the first of which makes Homer assert that some As are not Bs, while the second makes Hesiod assert that some Bs are not As. The premises being admitted, the resulting denial of the simple proposition of Sir William Hamilton's system is only obtainable by a dilemma, or, as it were, metasyllogism. Thirdly, the proposition (.), or 'Some Xs are not some Fs,' has no fundamental proposition which denies it, and not even a compound of other propositions. It is then open to the above objection : and to others peculiar to itself. It is what I have called (F. L. p. 153) a spurious proposition, as long as either of its names applies to more than one instance. And the denial is as follows : ' There is but one X, and but one Y, and X is F.' Unless we know beforehand that there is but one soldier, and one animal, and that soldier the animal, we cannot deny that ' some soldiers are not some animals.' Whenever we know enough of X and F to bring forward ' Some Xs are not some Fs' as what could be conceived to have been false, we know more, namely, • No X is F,' which, when X and F are singular, is true or false with 'Some Xs are not some Fs.' THE THEORY OF SYLLOGISM, ETC. 99 These difficulties* lie on the surface; and the first objector is sure to seize them. I expect a powerful consideration of them in Sir William Hamilton's forthcoming work, from the known learning and acuteness of the author, with some weight given to his assertion that his system has been " adequately tested and matured." And I should not be surprised at a successful explanation : for, though I cannot give one myself, as long as the system stands on its in- ventor's ground, yet I can prevent the appearance of the objections by shifting that ground. The occurrence of eight forms, corresponding in their modes of quantification with those which I had obtained, and by coincidence which did not arise from any sameness in the path of investigation, struck me as exceedingly remarkable. I could not entirely declare against the possibility of sufficient reason for a system which, independently of the habitual ac- quaintance of its promulgator with the logician's mode of thinking in every age, had, as we shall see, strong symbolic claims to being something. Symbolic language gives the expression of the laws of thought in their purest forms : and it has never deceived those who were willing to be its servants before they claimed to be its masters. In the present case, there seemed something resembling a system of algebra with a singular form in it. Formal Logic must teach how to enunciate all definitely conceivable truth and falsehood, just as symbolic algebra must teach how to enunciate all definitely expressible quantity : and • some ATs are not some Fs' appeared to partake very much of the indeterminateness of ^. An algebraist has not profited by the history of his science, if he dogmatically reject what appears incapable of interpretation in connexion with the rest of its system. Thinking on this, I tried whether there might not be some view of predication which would make Sir William Hamilton's eight forms self-consistent : that is, make them contradict each other four and four. The thing required is that )), ((, ).), (.(, ).(, and ( ) should remain related to each other as at present : and that )( and (.) should be a double universal and a double particular, destructive each of the other. We might put this question to any person, When you say ' every man is an animal,' do you speak of all men, or of one man, of as many animals as there are men, or of one animal ? Is your proposition cumular, or what I will call exemplar ? I apprehend the general first answer would be in favour of the cumular view, but not the universal one. Some would say, I speak of one man, being any one I can select, and of one animal, but not any one I please, for upon what man I select, depends what animal I select. Some would say that the article an, which denotes one animal, confines the subject to one man : how else can every man be an animal ? And in truth they are etymologically right, for every is each, not all, in meaning. * Sir William Hamilton is, I have no doubt, the first advocate of the form (•) : but it, and the peculiar syllogism derived from it (with two particular premises) have been seen and rejected, by myself at least, probably by others. I men- tion the following as a curious coincidence. Sir William Hamilton states that he had in his own mind arrived at the form, ' most Ys are Zs, most Fs are Xs, therefore some Xs are Zs' before me, and thrown it away, unpromulgated, as a cumbrous and useless subtlety. He had thu9 made the ap- proach of a single instance towards the numerically definite syllogism. Now I, on my part, had made and published, as a true inference, but not within the forms of predication, one syl- logism in the new part of Sir William Hamilton's system ; in the following words : " The weakest syllogism from which such an inference [particular negative] can be drawn would then seem to be as follows. Some Xs axe Ys, some Zs are not Fs, therefore some Zs are not Xs....But here it will appear on a little consideration, that the conclusion is only thus far true, that those Xs which are Ys cannot be those Zs which are not Fs. ...(First Notions of Logic, 1839, reprinted (with slight alteration) as the introductory chapter of F. L.) 13—2 100 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Some would be inclined to say they were sometimes cumular, and sometimes exemplar : and a great many would hesitate to affirm perfect identity of mode of thought as existing between ' Every man is an animal' and ' all men are animals ; ' and would admit, when it was put to them, that there is more or less of this difference — the former tells off, the latter sums up : the former is a completed induction, the latter a transformation preparatory to deduction. The next question would be, Are the general propositions of any science, as actually proved, cumular or exemplar : and here the answer must be, that what is proved is the exemplar proposition. Euclid, meaning to prove that all isosceles triangles have equal angles at the bases, fulfils his meaning by proving that any one isosceles triangle is so, and the perception of the goodness of his proof is entirely dependent on the perception of the force of the word any before one. A student who looks at the definition of proportion sees nothing but a hopeless heap of conditions between him and the knowledge of any possible proportion, until he sees a case established in which any one multiple of the first is shewn to lie among the multiples of the second as that same multiple of .the third lies among those of the fourth. If the cumular proposition can, generally speaking, only be proved by the help of the exemplar, it follows that the exemplar proposition must precede in order of thought : and it is justifiable to propose it as the basis of a logical system. The distinction of the two modes exists in every language in which I can form the sentence : if there be one in which both forms do not exist, the study of the minds of those who speak that language would be curious. Of our language and many, I suppose most, others, it must be obvious that the exemplar and cumular forms of expression are much more apt to be confounded as to subject than as to predicate, as to affirmation than as to negation. The logician, who must have forms, has to make a choice ; and he has invented cumular expressions which do not suit the genius of com- mon thought or common language. " All man is not fish " is the form in which a logician denies that any man is a fish : Sir William Hamilton says, " All man is not all fish." Common language would deny the first by saying, " No, nor any part of him." Even " All men are not fishes " only means, in common language, " some men are not fishes," with emphasis upon the great number that are implied to be so ; and would therefore be held false. The predi- cate of a negative must be exemplar : it is, " Every man is not any one fish." The examina- tion of the following table will shew that there is much less forcing of common expression in a list of nothing but exemplars, than in a list of nothing but cumulars. Quantity now becomes mode of selection of the example : universal is replaced by wholly indefinite, particular by not wholly indefinite, having some, no matter how much nor how little, limitation on the right of selection, under conditions known or unknown. The opposi- tions of logic must always be mere contradictions, of which one must exist (F. L. p. 59). Accordingly, definite is to mean not wholly indefinite. The terms of selection may be any one and some one. Now apply the two selective forms in every way, and the following exemplar forms of pre- dication will occur, arranged in contrary pairs and presenting a system of predication free from the objections which I have urged against the cumular forms, so. far as contradiction is con- cerned. THE THEORY OF SYLLOGISM, ETC. 101 A' 4 o; A i A' a E' ) ( Any one X is any one Y giving (.) Some one X is not some one F ) ) Any one X is some one Y (.( Some one X is not any one Y ( ( Some one X is any one Y ).) Any one X is not some one Y ).( Any one X is not any one Y ( ) Some one X is some one Y giving giving giving giving giving giving giving There is but one X and one Y, and X is Y. There can be found some one X and some one Y which are not the same. All Xs are Fs. Some Xs are not Fs. Every Fis X. Some Fs are not Xs. No X is F. Some Xs are Fs. The detailed notation needs no explanation. The form given to the old notation may be explained thus. In the system of contraries, the accents refer to the relation of standard terms and contraries. Thus O t being X(.(Y, O' is a?(.(y or Y(.(X. In the exemplar system, the accent refers to the example, sw&accent to subject, superaccent* to predicate. Every accent whicli occurs shews that its term is named after the proposition. Thus in /' both subject and predicate are like the proposition, particular or not wholly indefinite : but in A only the subject is, like the proposition, universal or wholly indefinite. The E' and /' of the system of contraries cannot exist here, but E' and /' have double accents, and we have the new proposi- tions A' and O', which have similar selective forms in both terms. The non-entrance of contraries keeps the following rules of syllogism intact. The middle term must be indefinite in one of the premises — negative premises give no conclusion. But on the other hand, the system of contraries acknowledges the rule that particular premises give no conclusion — which is not a rule of the exemplar system — and does not acknowledge the two first named. Taking the 64 cases of combination of premises, in any one figure, we must then reject 16 cases of negative premises, and 12 others in which, the premises not being both negative, both the middle terms are not indefinite. There remain 36 cases for examination, and all admit of inference : so that the canon of validity is ; — one affirmative premise and one indefinite middle term. This is also the case when exemplar forms are abandoned in favour of cumular ones : but the conclusions in the second (or Sir William Hamilton's) form are not always the same as those in the first. The validity of every case in which there is one indefinite middle and one premise is affirma- tive, immediately appears : the any one Y of one premise may be taken to be the some one Y of the other : and the same is then compared with two others, both giving agreement, or one agreement and one disagreement. Further it will appear that in every case, the selective forms of the terms of the conclusion are the same as they had in the premises. Thus ' any one X is some one F, any one F is some one Z' gives ' any one X is some one Z,' for whatever some one F the first premise can allow, the second premise makes it some one Z : and similarly of others. * Subject, as used in reference to a proposition, is liable to some objections. He would be a bold man who would dare to substitute subdict and superdict for subject and predicate : or, retaining subject, to call the predicate the superject : but a word on the rashness of this imaginary individual may help the reader in remembering the use of the accents. 102 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Hence the symbolic canon of inference is the same as in the system of contraries, namely, the erasure of the middle parentheses gives the symbolic form of the conclusion. But in the cumular system, this canon of inference is modified. In the exemplar system, the proposition X)(Y or 'Any one X is any one Y,' though it appear to be of the widest character, is restricted by its double indefiniteness : doubly indefinite produces doubly singular. This will explain the mode in which conclusions are admissible in the exemplar form which are not admissible in the cumular. For example, in the former, )) )( gives )( or ' Any one X is some one Y, and any one Y is any one Z,' gives ' Any one X is any one Z.' Here, if all we knew were that Y and Z are identical, all we could infer would be, ' Any one X is some one Z;" but because we know from Y) {Z that there is but one Z, we may say that any one X is any one Z. In the cumular system, then, )) )( gives )) : and the modification of the canon of inference which the cumular system requires is ; — Erase the middle parentheses, but when they both turn one way, any parenthesis of indefiniteness which turns the other way must itself be turned, unless it be protected by a negative point. Thus ( ))( gives ( ), but ))).( gives ).( . To collect the syllogisms, we may observe that there are two doubly indefinite forms A[ and E', each of which has three less indefinite varieties; the first A /t A', and 7 ' ; the second, 0, 0', and O'. A syllogism then in which both premises are double, admits, in original and weakened forms, of 16 varieties; and there are three such double syllogisms which are valid. Classifying the syllogisms by the degrees of relaxation in their forms, we have the following list : a;a;a; )()(=)( a;e;e; )().(=>.( e;a;e; ).()(=>.( a,a;a; )))(=)( a'a;a' (()(=(( a/a a, )())=)) a;a'a; )(((=)( a e/e; »).(=).( a'e;o, (()•(=(.( A/O'O' )().)=).) a: ox )((.(=).( oa;e; ).))(=)•( oa;o, (•()(=(.( e;ao' ).())=).) e;a'e; ).(((=).( i;a;a' ())(=(( i/a,i; ())) = () AAA ))))=)) A'I'I' t t ((() = () A'AI' (())=() A'A'A' ((((=(( A'l'J, )(()=)) I/E/O, ())•(=(•( 1/00/ ())•) = (•) A, 00' )))•)=)•) A'O/O/ (((•) = (•) A'O'O' (()•)=(•) A'O i i (((•(=(•( A'O'O' 4 t )((•)=)•) o;A;o t (.))(=(•( O/AO/ (•)))=(•) O'AO' )•)))=)•) 0,1/0/ (•(()=(•) o,ao/ (•())=(•) o,a'o, (■(((=(•( e/i;o' )•(()=)•) The cases in which the cumular system requires the above-mentioned modification of the law of inference have their capital letters in Italics. The three fundamental syllogisms give, each of them, four cases with one relaxation, five with two, and two with three. Many different rules of formation might be produced, and many analogies, but it is not necessai^y for my present purpose to examine them. THE THEORY OF SYLLOGISM, ETC. KC5 One however may be mentioned, as giving the mode of arrangement preferred by Sir William Hamilton. The preceding is a table of double entry, and the passage from compart- ment to compartment in the same column shews that all the negative syllogisms are formed from the affirmative ones by simple insertion of the negative distinction into one or the other premise and the conclusion. Thus, in the second column, at the head of the three compart- ments, stand )))( = )( and ))).( = ).( and ).)) ( = ).(. In the cumular system, the modified modes of inference occur with certain affirmative propositions, which are therefore exceptional. Looking at the preceding as one way of meeting the formal difficulties of Sir William Hamilton's system, it will easily be conceived possible that there may* be others, for one or more of which we are to look to its inventor, and one more of which I shall presently give myself. But, in the manner in which it has been given out, up to the present time, the defects which I have pointed out exist unanswered. The exemplar system and that of contraries have 21 syllogisms in common. Of the re- maining 15 of the first, 7 correspond to 7 of the syllogisms with both premises negative in the second. The remaining 8 of the first belong, a pair to each of the four syllogisms of the second which have both middle terms particular, one of them being the remaining syllogism with premises both negative. When a symbol taken from the exemplar system is invalid if read in the other, first try it by annexing the negative sign to each premise which is affirmative : if this do not make it valid, alter the curvature of both the middle parentheses, and then either the addition or removal of a negative symbol will make it valid. Section V. ON THE THEORY OF THE COPULA AND ITS CONNEXION WITH THE DOCTRINE OF FIGURE. The analysis of the so called simple proposition, or judgment, shews that it is, in every one of its particulars, of a complex character. This is readily admitted in regard to the terms, and to the quantity in its connexion with them. But the copula has always been considered as the extreme both of simplicity and generality ; and any attempt at the resolution of the copular relation into its elements, is likely to be misunderstood. In algebra, as it now stands, the forms born and educated in arithmetic have left their parent and set up for themselves. Any meanings which obey certain specified laws may be adopted as the means of giving expression to the forms : and the results must be accepted as * Of course I might instance my own numerically definite system, which, as Sir William Hamilton acknowledges, con- tains his system, and which is not open to any charge of incon. sistency. The acknowledgment was made in the assertion that my numerical system, so far as it gives expressed quantity of all kinds to the predicate, was derived from information furnished by him. I am still waiting for two citations : one from Sir William Hamilton's communication, containing in- formation ; one from my own subsequent writings, containing use made of it. I want, in fact, an exemplar proposition of the form )( : having hitherto obtained nothing but ) ), or ( ) : I have had enough of single indefiniteness, and want an in- stance of the double character which gives singularity to the things to be compared. 104 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, true in every instance in which the combinations used are consistent and intelligible under the meanings given. Thus we may have, and have, a calculus in which the symbols are in frequent cases not magnitudes at all, but directions how to operate : and though this calculus, when pushed beyond a certain point, has, for the present, its unintelligible symbols, which play the part of the impossible quantities of the older algebra, we have good reason to think that some day the victory over these will be cited as encouragement under the difficulties of a yet more advanced stage of progress. In my work on Formal Logic (pp. 46 — 54) I followed the hint given by algebra, and separated the essential from the accidental characteristics of the copula, thereby shewing the con- ditions of invention for a copula different from the ordinary one, or for a copula which, being substituted for the ordinary one, shall leave all the forms and conditions of inference unaltered. On this a learned critic remarks that I claim the abstract copula as an improvement : adding, that some of my modes of making the copula are less abstract, none more so, than is and is not. As this remark either embodies or suggests the whole, or nearly the whole, of what would be said by a logician, technically so called, it will be worth while to dwell upon it. By an "abstract copula" of course is meant a formal mode of joining two terms which carries no meaning, and obeys no law except such as is barely necessary to make the forms of inference follow. Any concrete or actual copula, fulfilling other conditions, is, to the extent of those conditions, less abstract. The best proof of the perception of an abstraction, is the invention of an abstract designation, which must be a technical symbol, if convenience be to dic- tate. The mode of denoting terms by letters fully shews us that the abstract term is arrived at : but as there is no symbol for the copula except the verb is, we are left to ascertain from the use of that word how the matter stands with respect to the copula. If an abstract copular symbol had been used, the copular conditions would have been ex- pressly laid down. They are two in number ; together sufficient for all forms of inference, but not both necessary in all. The first is what I shall call transitiveness, symbolized in X V Z = X—Z ; meaning that if X stand in the relation denoted by — to Y, and Y to Z, X therefore stands in that relation to Z. Very many copulas exist in which this transitive relation is seen ; as is, — rules, — lifts, — draws, — leads to, — is superior to, — is ancestor of, — is brother of, — joins, — depends upon, — is greater than, — is equal to, — is less than, — agrees with (in a given par- ticular), &c. The second condition is convertibility, symbolized in X Y=Y X, in which the relation is its own correlation. Of those mentioned above, convertible relation is seen in — is, — is brother of, — joins (if a middle verb), — is equal to, — agrees with. As instances of the convertible without the transitive character we may take — converses with, — is in the habit of meeting, — is cousin of, — is in controversy with, &c. &c. The connexion between the affirmative and negative copula is merely that of contrariety : in X — Y and X Y it is supposed that one or the other must be. It is not necessary that one should be the denial of the other, as I may say, over the whole universe of thought : it is THE THEORY OF SYLLOGISM, ETC. 105 enough if within the universe of the syllogism, either X — Y or X Y must be true. Thus ' agrees in colour,' ' agrees in size, 1 may be alternative copula? in a universe the alternative of which is that any X is either of the colour or of the size of some Y. When contraries are introduced, the copular condition further required is that either X — Y or X y should hold for any X. The least abstract of all copula; is the is and is not of the logicians, when employed, whether subjectively or objectively, in the sense of identity. But logicians rarely confine it to that sense. It takes many meanings in their modes* of speaking and in their examples: and perhaps may be stated as generally signifying agreement in some understood and, pro vice, unvarying particular. I have never seen it carried further, formally at least : and I feel safe in affirming that is, — used as merely satisfying the transitive and convertible conditions, with a consequent right, for example, to treat it as standing for ' is brother to,' — is a more abstract copula than the is of the logicians. I should almost presume this from the silence of the writers who distinguish between reading the extension and reading the intension of a proposition, as to any other change of the copula being possible. ' Every man is an animal' is read either in extension, ' All man is animal,' or in intension, as by saying that the notion of animal forms part of the notion of man ; every attribute of an animal is an attribute of man. Without at all denying the existence of the distinction, or its metaphysical value, I feel convinced that it is not another reading of the proposition, but another proposition, derived infer entially, though not syllogistically, by aid of the dictum de majore et minore. For a term used universally we may substitute a lesser term ; and the attribute of all animals is the attribute of all men. The proposition only says that man is the lesser term ; and intensive reading is thus derived from the dictum. Invent a sub- stantive, A, such that eco vi termini, the 'i of an animal,' is one in which the word animal must be construed universally ; and then we may affirm that 'A of an animal' is 'A of a man.' And though, undoubtedly, the word attribute, used metaphysically, will be the one most often employed in place of A, yet the formal process of reading by intension will be only a case of that principle on which we say that ' Englishman who does not use tobacco' is included in ' Euro- pean who does not take snuff,' obtained by enlargement of the particular and diminution of the universal. Formally speaking, the change from extension to intension is as follows ; and it will be seen that a new distinction is introduced, and further, that the two modes of reading are not con- vertible ; the extensive mode gives the intensive, but not vice versa in all cases. * But not in their formal definitions and enunciations. The substantive verb is the most common definition. The Port Royal Logic, which I choose from those lying before me, as a work which dares to differ, or to revive, at pleasure, makes its definition very distinct. " Non sufBcit hos duos terminos percepisse; sed debent mente vel conjungi vel separari. Et hsec mentis Operatio in Propositione notatur verbo, Est..." To take two extremes of the part of time in which scholastic logic and printing flourished together, we have Paulus Venelus (1474), "Copula semper est verbum substantivum scilicet sum, es, est," — and Crackanthorp (4th ed. 1677), "Et haec [sc. copula] est in omni Propositione Verbum hoc Substanti- vum, Sum, es, fui, aut aliqua persona illius." Vol. IX. Part I. 14 106 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, Extension given. Intension deduced. Every X is F. Every existing part of all Ys is an existing part of all Xs. No X is Y. No sufficient* part of any Y is an existing part of any X. Some Xs are Ys. Every existing part of all Ys is an existing part of some Xs. Some Xs are not Fs. No sufficient part of any F is an existing part of some Xs. Intension given. Extension deduced. A sufficient part of some one Y is an existing part of every X. Every X is Y. An existing part of any Y is not an existing part of any X. No X is Y. A sufficient part of some Fs is an existing part of some Xs. Some Xs are Fs. An existing part of any F is not an existing part of some Xs. Some Xs are not Fs. By treating the predicate itself as one attributive notion, the modes of reading both kinds of propositions intensively may be assimilated : but not without losing sight of an important distinction. In the affirmative, any portion of the intension of the predicate may be affirmed of the subject ; in the negative, it is not true that any portion of the intension of the predicate may be denied of the subject. Thus ' no planet moves in a circle' gives us a right to deny any constitutive attribute of circular motion to that of a planet, but not any attribute ; not, for instance, the progression through every longitude. Leaving the distinction of extensive and intensive reading entirely out of view, as not connected with the theory of the copula, I proceed to inquire into the connexion of the doctrine of figure with that of the copula ; a connexion which I take to be the most im- portant part of the former doctrine. And first, I take the system from which contraries are excluded. Looking at the two copular conditions, the transitive and the convertible, I ask what inferences hold good when only the transitive condition holds : for this one must hold, as long as there is but one species of copula ; we shall presently see that it is the condition of permanence of the copula. As a representative of a transitive but inconvertible copula, let us take the verb give, assuming that he gives who gives that which gives. Then ' X gives F, F gives Z, therefore X gives Z' is legitimate, being the mere expression of the transitive hypothesis. And the opponent forms of this must be legitimate. Thus i X gives F, X does not give Z' leads to F does not give Z: for if F gave Z, X giving F, X would give Z : this is in the third figure. Again, 'AT does not give Z, F gives Z' leads to ' X does not give F:' for if X gave F, F giving Z, X would give Z : this is in the second figure. When these propositions are made cumulative, with quantities proper for inference, the usual inferences are obtained. Accordingly, + meaning affirmative, and — negative, all + + syllogisms in the first figure, h — syllogisms in the second figure, and — i- syllogisms in the third figure (reading premises in the Aristotelian order) do not need a convertible copula. They are I. II. III. Barbara )))) Camestres ).((( Felapton (().( Darii ())) Baroko (.((( Ferison ()).( Bokardo (((.( * Meaning part sufficient to determine it to be K. I do not say attribute, because the inference does not depend upon the metaphysical meaning of that word. THE THEORY OF SYLLOGISM, ETC. 107 Turning to Sir William Hamilton's system, I find that his peculiar syllogisms will not follow this rule : but in the exemplar system it is preserved throughout. The fourth figure is incapable of inference, as long as one inconvertible copula, and that only, is used. The copula being inconvertible, we may complete the modes of inference by allowing the correlative copula; as ' X gives V ' Y is given by AT'. That is, whenever conversion is necessary to turn the syllogism into + + of the first figure, H — of the second, or — + of the third, the correlative copula must somewhere be introduced. The following addition to the poetry of logic may be euphonized by any one to his own taste : the letter g, following a vowel, denotes that the premise (or conclusion) denoted by that vowel takes the correlative copula. Barbara, Celagrent, Darii, Ferigoque prioris Cesareg, Camestres, Festinog, Baroko secundas Tertia Darapgi, Disagmis, Datigsi, Felapton Bokardo Ferison habet. Quarta insuper addit Bramantigp, Camegnes, Dimarigs, Fegsapo, Fregsison. This means, for example, that Fesapo may be read in the fourth figure under the additional condition seen in Fegsapo, namely, that the copula of the first premise is to be correlative of that used in the second premise and conclusion : and Darapgi in the third, if the second copula used be correlative of the other. two. Thus 'all the piles supported arches; all the piles were supported on gravel, therefore gravel did then support arches' is a good syllogism, and not capable of reduction to an Aristotelian syllogism. The proposition ' support of support is support' is necessary to the inference, which inference can only be obtained from a sorites, and not then except by help of the dictum already quoted. It may be said that this is more than an Aristotelian syllogism ; I maintain it to be less, if either. The outstanding copular relation (always implied) of an Aristotelian syllogism is 'is that which is, gives is:' of the preceding case, ' support of support is support.' The former demands transitive and convertible meaning for is, or that is shall be its own correlative : the latter demands transitive meaning for support, and the allowance of its correlative. The following table shews the way in which any combination of premises is to be read in any figure, it being presumed, as in Sir William Hamilton's system, that every syllogism may be read in any figure. I. + + + ( + ) _ _ _ (+) - II. (+)++ + _ _ - +(-) III. + (+) + + -(-) - + - IV. + + (+) + (-)- (-) + - These cases may be remembered as follows. Let + +, + -, - +, and , be called the primitive forms of the four figures : the fourth figure taking its root in an inconclusive form. In every primitive form the correlative copula need not appear. When one premise of a primitive form is altered, the necessity of a correlative copula is thrown upon the other : when both, upon the conclusion. The principles laid down in Section I. enable us to make a still further enlargement. The 14—2 108 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, logician, followed by the algebraist, has restricted himself to one copula : the former uses is, the latter = ; and both are used in some variety of sense. The algebraist, indeed, sometimes goes a little further, and introduces the correlatives > and <, which might be generalized for the purposes of logic into symbols of correlative copulse in general. In every kind of logic, formal and applied, the transitive copula is insisted on. This however is not necessary : inference may be seen without it. The perception of relations by means of relations does not require us to use only one relation. If I can see that Every X has a relation to some Y and Every Y has a relation to some Z, it follows that every X has a compound relation to some Z. Be the premised relations what they may, there is a concluding relation, which may or may not be expressible by one word. Thus if John can persuade Thomas, and Thomas can command William, we cannot infer that John can either persuade or command William : but if we express by one word the process of gaining an end by persuading one who can command — say we choose to use the word control* for this purpose — then John can control William. We have then a bicopular syllogism, in which the intransitiveness of the individual copulae is supplied by the invention of a compound copula for the conclusion. This is the step by which we ascend to the general theory of the copula : but those who proceed from a general copula to a particular one, will merely see how to read the general conditions in the particular case. Can a bicopular syllogism be reduced to a compound process of unicopular syllogism ? In the case before us the conditions are Postulate. Control includes the influence exerted over the governed by one who can persuade the governor. Premises. John can persuade Thomas. Thomas can command William. Conclusion. John can control William. The only way in which this can be reduced to unicopular syllogism is by the following sorites. John is {one who can persuade Thomas}. {One who can persuade Thomas} is {one who can control all whom Thomas commands.} )ne who can control all whoml . ( One who can 1 Thomas commands. J (control William J ' John is {one who can control William}. The algebraical equation y = vris \6yov yivofievov — and the reason given is that it is measured by long and short syllables. On this I say first, that Aristotle would probably have seen what his determined disciple Crackanthorp after- wards saw, that the time of speaking is not speech; secondly, that it is incredible that he should have used a wrong word and have afterwards corrected himself by a synonyme instead of an erasure ; thirdly, that dpiQads Kal Xoyos, in the language of the time, are integer number, and fractional ratio or ratio of THE THEORY OF SYLLOGISM, ETC. Ill animal, and in another of animal as part of the notion man. The " integrate or mathematical whole" which, according to Sir William Hamilton, " the philosophers contemned," and which is seen in ' All men are animals', must take its proper place : and " philosophy", which " tends always to the universal and the necessary" must be taught that the universal belongs to the mathematical whole, as surely as the necessary to the metaphysical whole. The admission of relation in general, and of the composition of relation, tends to throw light upon the difference between the invented syllogism of the logicians and the natural syllogism of the external world. The logician, tied to a verb of identity, from which if he wander it is never quite out of sight, is bound to subject and predicate of the same class ; objective both, or subjective both. He cannot say the rose is red, for his is would require the inference that some red is the rose. He has nothing but a method of reducing his predicate to an object : the rose is a red thing; some red thing is a rose. The common man uses a copula which ties the object up in relation to a more subjective predicate; not reading inversely by intension, not dwelling on redness as an attribute of the rose, but directly by extension, thinking of the family rose as his external object, and the sensation red as one condition under which it appears to his senses. Again, an ordinary person says that the rose is red, and red is pretty, so that the rose is pretty. The logician's pupil, when not far advanced, will interpret him as saying that ' The rose is a red thing ; a red thing is a pretty thing, &c.' thus committing him to an opinion upon red cabbage which perhaps was not within his meaning. The logician himself will substitute another process : ' The colour of the rose is red colour ; red colour is pretty colour ; therefore the colour of the rose is pretty colour : ' he thus reduces it to the equation of two terms by comparison with a third ; when most probably the real form of thought (and logic professes to be the study of existing forms) is nothing but the composition of relations. The rose is (in its relation to one sense) red; red is (in the relation of the perceptions to the judgment) pretty; therefore the rose is (to one sense that which is to the judgment) pretty. This distinction of copulae is more nearly present to the mind than any one of the transformations by which the technical form is arrived at : it is visible, more or less distinctly, that the first premise is matter of fact, the second matter of taste ; here is a recognition of difference between is of the first premise and is of the second ; which leads, with proportionate strength, to the recognition of the composite character of is in the conclusion. Nor does the logician's form avoid it, except by a pure assumption that the copulas are the same ; an assumption not true in fact. Nor can an answer be given by saying that logic takes no cognizance of the matter, but only of the form; for it is bound, by the laws of form, not to represent differences by agreements, except when it is formally shown that the differences cannot affect the object to be gained. In one very material consideration, the logicians have evidenced the manner in which their commensurables. An editor who was not a mathematician, seeing \6ym, which to him was only communication in its most general sense, may have excogitated the absurdity above alluded to, by help of <$>wvr\. Proclus has been used in a similar way. Barocius makes him inform us that the prede- cessors of Euclid had written on inexplicables (qua non ei/ili- cari possuni) : Proclus says aXoycr, and means incommensu- rables. But how can Xoyos be called discrete ? Before Euclid it must have been so. Euclid's definition, which brings incom- mensurable;; under the same law with commensurables, gives the term ratio a title to continuity, which it had not before: just as in modern times, the interminable decimal, when ad- mitted as a distinct conception of arithmetic, gives continuity to number. 112 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, copula 'is 1 partakes of the verb of identity on which, as above remarked, they test all their inferences. When the copula of identity is employed, it is of necessity that one subject example can only agree with one predicate example : any one man IS, merely because he IS one man, only one animal. Extend the copula to a meaning short of perfect identity, and the copular relation may belong, in an affirmative, to more than one example of the predicate. For instance, when the copula is agreement in colour, ' Every X is F 1 may mean that each X is of the colour of one or more of the Fs. The particular character of the predicate, that is, the description of its extent as being ' one or more, possibly all ' may belong to it, independently of the number of instances of the subject, in right of, and by derivation from, one instance of the subject. And the proposition • one X Every V may be true without our being obliged to say that there is therefore only one F in existence, by reason of the relation indi- cated by existing between that one X and every F in existence. Hence follows an extended mode of interpreting Sir William Hamilton's system into con- sistency. Let the quantity of the predicate be determined by the least number of instances of the predicate which are declared to stand in the copular relation to each instance of the subject : it being understood that the declaration of the copular relation existing between a certain X and a certain Y does not deny that the relation may exist between that X and other Fs, or between other Xs and that Y. Thus, "All Xs all Fs," or X)(Y, now means that every X stands in the copular relation to each and every F. If there were ten Xs and ten Fs, Sir William Hamilton's proposition would assert ten agreements : the form here proposed declares a hundred. And it is now contradicted by X(.)Y or " some Xs some Fs :" for either each X and each F agree, or some Xs do not agree with some of the Fs. This view of the subject meets another objection of mine to the system as proposed. It can no longer be asserted that X) (F is a proposition compounded of two others of the system, X))Y and Y))X: for each X may agree with some of the Fs, and each F with some of the Xs, without every X agreeing with every Y. There is however a distinction to be drawn. When the quantity of the affirmative predi- cate is determined solely by the number of instances which each instance of the subject is related to, let us say that the predicate has exemplar quantity. But in the common system, in which the quantity of the predicate is cumulated from all the instances of the subject, let us say that it has cumular quantity : this second system permitting, like the first, each instance of the subject to be related to more than one instance of the predicate. Thus, quantity being exemplar, we must say of ' Every man is an animal ' that the predicate is singular : quantity being cumular, that the predicate is arithmetically coextensive with the subject. But when it is " Every man — agrees with in being carnivorous — a brute," the predicate is not singular, even when its quantity is exemplar. [An extension of the numerically definite system here arises, which I shall not stop to investigate the consequences of. If j- be the whole number of A"s in the universe, r\ the num- ber of Fs, and n the number of Fs (at least) to which each of mXs is related ; them mn is the least number of relations asserted. If mn be greater than (£ - l)rj, then mn - (£ - l)»y at least of the r/ Fs must be related to £ Xs, that is, to all the Xs. Now as n cannot exceed r\, m must be £ : and a necessarily convertible proposition may then exist, if n exceed t] - q -j- £.] THE THEORY OF SYLLOGISM, ETC. 113 By merely allowing a complete exemplar form of quantification we procure the extension of Sir W. Hamilton's system which I have described in the last section : we now annex to this admission, a predicate quantity of more than one example. And the forms with their meanings are as follows, each universal with its particular con- tradiction. iX)(Y Each X is related to all the Fs. (^Y(.)F Some Xs are not related to some of the Fs. {X) ) Y Each X is related to one or more Fs. X(.(Y Some Xs are not related to any Fs. {X((Y Some Xs are (among them) related to all the Fs. X).)Y No ^Yis related to some one or more Fs. {X).{Y No Xh related to any one F. X()Y Some Xs are related to some one or more Fs. This system as it stands, is free from all objections which I can raise. Every proposition has its contradictory, the form (.) is not generally spurious, no numerical dictum is established* by the mere assertion of a proposition. The symbolic law of inference does not present any modified cases ; and, transitive copular relation existing, though inconvertibly, the use of the correlative relation, in the places previously marked out, will allow of inference. The manner in which the introduction of more general copular relation allows of more general inference, and the alterations in the import of the forms )( and (.), with the value of those alterations in the expression of the forms of thought, would admit of considerable discussion, if the length to which this paper has attained did not forbid. Taking the predicate as both exemplar and cumulative in quantity, the copular relation as both transitive and convertible, we see a system which differs from that of Sir William Hamil- ton only in the admission of a wider alternative of copular agreement : instead of ' one F or no F at all ' it is ' one or more Fs or no F at all.' Circumstances require me, I think, to point out my reasons for concluding that this extension of alternatives has not been made by Sir Wil- liam Hamilton. They are first, his description of his own use of quantity, made in termsf so technical, that it is impossible to suppose he varies from the ancient modes of quantity, in extending the application of them : secondly, that the author of the " Outlines," an accredited expounder of the details of this system, not only does not give the smallest hint to the contrary, but adopts the conclusions! suggested by the old modes, as "of course." I shall now proceed to the application of correlative copulae to the system of predication * If indeed we were to declare that the number of correlated instances should never exceed 10, then )( would imply that there are not more than 10 X& nor more than 10 Vs. It is the refusal to admit of more than one instance of each in relation which reduces )( to double singularity in the exemplar form first discussed . t " The first scheme is that which logically confines all expressed quantity to the Subject, presuming the Predicate to be taken — in negative propositions, always determioately in its greatest and least extension (universally and singularly), in affirmative propositions, always indeterminately in some part of Vol. IX. Part I. its extension (particularly). The second scheme is that which logically — extends the expression of quantity to both the pro- positional terms, and allows the Predicate to be of any quan- tity, in propositions of either quality.... The first doctrine is the common or Aristotelic; the second is mine;..." "Letter, &c" subsequent to my " Statement," pp. 31, 32. J Mr Thomson agrees with me as to the spuriousness of "Some Xs are not some Fs" in Sir Wm. Hamilton's sys- tem, except "of course" as a denial of singularity and iden- tity: "Except of course they represent individuals." "Out- lines, &c." pp. 188, 189. 15 114 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, by contraries. I insert a slight and preliminary notice of this extension more as an example of the study of extension than because of its representation of any very common modes of thought. At the same time it should be remembered that we cannot appeal to existing modes, and to their agreement with the Aristotelian forms, as conclusive evidence of the natural character of those forms, or of any restriction which results from their use. For many centuries, and in particular during those centuries which saw the relations of the modern and ancient languages finally settled, every man who learnt to read was alphabetically drilled in the Aristotelian syllo- gism. If technical forms could produce such an effect as either to supplant more natural ones, or to prevent the natural growth of the most natural ones, the Aristotelian forms have had every advantage given them, and may have done it. Consequently, no modes of thinking which are not easily reconcileable with the Aristotelian forms ought to be suppressed because they do not seem to have any usage in their favour. The copula to be extended is the word is, in its widest sense : the extended copula may be said to be has relation to, the relation being expressed or understood, but distinctly conceived as transitive. The correlative copula may be expressed by has correlation to : and it is demonstrable, or is rather seen to accompany than to follow, that all transitive relations have transitive correlatives. As a short mode of speaking, we may say that our relations are gives to and receives from. The terms active and passive might be generalized into words of description for the correlatives ; and middle into a word of description for relations which are their own correlatives. In the common system, if a certain X be a certain Y, that Y is that same X : but the correlative (then called converted) relation need not be much dwelt upon. Of the two ' X is Y" 1 and ' Y is X," 1 either may be called direct, and the other inverted. In the extension, each instance may stand both in direct and inverted relation to something : so that every X gives something, every Y gives something ; every X receives from something, every Y receives from something. And it must be that no X gives to nor receives from, any contrary, x, either directly or by transition. If, for instance, a certain X give to one or more Ys, no one of those Ys can give to any tc : but as each of those Ys must give, and either to Xs or #s, it must be that they give to Xs, but not necessarily to those from which they received. Nor must the same X give to both Y and y, &c. We have then for any one X either XY' and YX' or X t y' and y t X' : meaning that either an X selected gives to Ys, and each of those Ys to Xs ; or to ys, and each of those ys to Xs. And the subscript and superscript accents may be applied to the parentheses : thus X t ))'Y may stand for 'every X gives to one or more Ys." 1 [Accordingly*, if we adopt this hypothesis, namely, that a direct and inverted relation always exists, it follows that all syllogisms are valid, put what marks of relation we may. For X t ))' Y never exists without X'))^: and the same of all the other forms. And X,))' Y has X t ).{'y and y ))'« and w (.)' Y among its equivalents. From this most complete of all possible assignments of relation, we may descend by two steps. First, we may dismiss the condition that every existing relation has its correlative : and then we may ask under what relations each syllogism is valid, there being one existing * The paragraphs between [] are substituted for those originally given : their date is July 1, 1850. THE THEORY OF SYLLOGISM, ETC. 115 between each of the pairs XY, YZ, and one (inferred) between XZ. Here each relation is supposed to extend over the terms and their contraries : thus from X ) )' Y we are to infer X t ).(y, and may use it. Secondly, we may dismiss the condition that the relation extends over the contrary, and then we arrive at the modification of the common or Aris- totelian syllogism, already discussed : in this case, the inference can be obtained without any use of the contrary, or extension of the relation. I* now give the heads of the results which arise from the hypothesis that one of the correlations is assigned or inferred, and the other not admitted. The equivalent propositions- are those of the ordinary system, with the relations unaltered : thus X i {)'Y=X i (. or ■^,)('^- ^° P rove tne second, observe that every y supplies a x, and if any such w supply a Z, the y which supplies that x also supplies that Z; but every Z is supplied by a F. Consequently «,()'*, or X)('Z, as before. In each syllogism, one or other of its two cases must involve the direct entrance of contraries: so that the Aristotelian form never allows two cases. For instance, F))'Z + ^ / ))'F = X / ))'Z 15—2 116 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, is obviously true without consideration of contraries ; every X supplies a Y, and every Y a Z. But Y t ))'Z + X')),Y = X t ))'Z must be proved thus. If any X supplied a z (here the contrary enters), a Y would supply that z, and would therefore supply both a Z and a z. The remarkably simple modification which occurs in the Aristotelian forms depends upon those forms being confined to particular predicates in affirmatives, and universal ones in negatives. When all relations have their correlatives, the reading by figure is a matter of indifference ; when the two correlatives are not allowed in the same proposition, the figures are grouped in pairs, the first and third, the second and fourth ; when no term is allowed its alternative, or contrary, the four figures become distinct in their properties. The primitive forms of the four figures being + +,+—, — 1-, , the mode of reading by thickened parentheses may be connected with them as follows : i- Mil n. | | || in. | || | iv. |" | \\ in which, over the thick lines in I, occur + + ; in II, + - ; in III, - + ; in IV, . All these relations may, no doubt, be connected ; but a general demonstration of the law of relations, when two correlatives are not admitted in one proposition, would be requisite : this demonstration is very easy. In my written communication to the Society, having seen that the + + , + - , and - + syllogisms of the first three figures required no correlative, I concluded that, on in- vestigation, the syllogisms of the fourth figure would require none. And as this turned out to be true, I looked no further, at the time. But I afterwards found that, though this be all correct, the entrance of the contraries under which alone a syllogism gains existence, gives rise to the validity of every syllogism without any correlative.] When an intermediate relation exists, which is equally related to both the correlatives, as in the case of greater, equal, and less, that intermediate relation may be employed for either. The effect, if any, upon the conclusion, is easily connected with the law by which the strength- ening or weakening one of the premises produces its effect on the conclusion. But the length of this paper compels me to omit the detail of this and other points. The combination of the system of invention of predicates with that of contraries remains for consideration. Section VI.* APPLICATION OF THE THEORY OF PROBABILITIES TO SOME POINTS CONNECTED WITH TESTIMONY. Every application of the numerical theory of probabilities requires and presupposes an hypothesis on the cases enumerated in the problem : they are, or they are not, equally pro- * This section was forwarded as a separate communication, and was dated Nov. 19, 1849. THE THEORY OF SYLLOGISM, ETC. 117 bable. If not, the assignment of their several probabilities is, as it were, equivalent to a further subdivision of the cases, each of which is made to consist of several : all the individuals of the ultimate resolution being supposed equally probable. This last arrangement I call the primary distribution. Not only do a great many acknowledged errors arise from mistaken modes of making this primary distribution : but it is a fair matter of inquiry whether diversity of method, without error, may not be forced upon the mind by its own legitimate act. A little consideration will shew, with regard to the mathematical part of the theory of probabilities, that the primary distribution is out of the subject, as much as the matter of a premise is out of the subject in logic, or the material substance of a solid out of the subject in geometry. Numerical application may be made to a false distribution, as well as to a true one. The well-known mistake once* made by D'Alembert, was one of primary distribution : it could not be in the power of a mathematician, as such, to convince him of his error. The replies of Lacroix and Laplace both amount to nothing more than a perfectly correct denial of D'Alembert's primary distribution, and the proposal of another. The primary distribution is a mental act. It matters nothing that the circumstances of the problem appear to dictate it. When it is stated that an urn contains 100 white and 100 black balls, and that therefore there is an even chance of drawing a white ball, it is the want of sufficient reason for any other allotment which produces a provisional assent. Experience may shew sufficient reason, and may dictate a different distribution. Thus, should it turn out that 2000 drawings produce 1800 black balls, that circumstance alone would demand the change. Both distributions may be true : that is, true exponents of the rational result of the existing knowledge of the party whose mind is addressed, at two different times. I was led to consider the following question; — What is the primary distribution of the mind in regard to a proposition and its contradiction, antecedently to the production of any evidence in favour of either. In the writings of logicians, although no formal exposition of their ideas upon probability is made, I thought I had detected a leaning to the notion that ' Every X is Y," 1 and ' some Xs are not YsJ are a priori of equal probability. And by a priori, I mean antecedently to the production of the specific subject and predicate. Say that opposite to X and Y are to be written at hazard, by two persons selected at hazard, and not in communication, the verbal descriptions of two objects of thought. Which is most likely to turn out true, that every X is Y, or that some Xs are not Ys ? We should pronounce with- out hesitation in favour of the latter, and should even say perhaps, that its extreme case, no X is Y, far exceeds in probability all the others put together. Nevertheless, writers on logic, in their tacit references to authority and its effects, seem to " 1 say once made, because, though never mentioned, it is pretty certain he saw his error before he died. The second edition of his Opuscules (and the first also, I believe) contains the reflexions on the Theory of Probabilities prefixed to his dissertation on the effects of inoculation for the small-pox. Herein is contained, as cited by Lacroix and others, the cele- brated argument that the probabilities for head at or before the second toss are two to one, the three possible, (and according to D'Alembert, equally probable) cases being H, TH, and TT. But in the collection of D'Alembert's works, published by Bastien (1805, Paris, 18 vols. 8vo.) in the fourth volume of which the memoir on inoculation and its preliminaries are somewhat recast and augmented, though the paradox of the Petersburg problem and some other objections are reproduced, all allusion to the problem above described, and all objection to its ordinary solution, are struck out. 118 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, admit that a given authority, producing a universal proposition, has his weight in no material degree lowered by the very great antecedent improbability of his statement. Their admission is quite correct, but, so far as I know, not yet explained by the theory. On looking at this theory, we find it appear that almost every sound process of mathe- matical application has results counterpart to its own among the results of the operations of unassisted thought. I should wonder at no one who suspected that a manifestation of the secrets of the brain would exhibit something more like a calculating machine for the apprecia- tion of probabilities than any one, as matters stand, could venture to mention without ridicule, or to maintain without deserving it. Kant has gone so far as to call our mental organization, in this respect, a weighing machine with unstamped weights. The verifications of theory above mentioned will lead all who feel their force to have confi- dence in the converse, namely, in the theory ultimately confirming every widely observed result. Now we have before us the following phenomena, gathered from observation : — 1. That an authority, or a witness, of whose value we have a previously acquired notion, produces an effect upon us by his testimony which partly depends upon the preconceived pro- bability of his statement : so that, whatever his general credit may be (short of imputed infalli- bility), his particular credit with regard to any one statement is a function of the proba- bility of that statement, as well as of his previous character. 2. That the universal proposition is, a priori, very much less probable than the contrary particular : that, X and Y being terms on the connexion of which evidence has never been offered, it is very much (not to say infinitely) more probable that some Xs are not Ys than that every X is Y. 3. That an authority, stating a proposition on the connexion of the terms of which we have no previous opinion, does not produce any very marked difference in our disposition to trust him, by stating the particular rather than the universal. We receive his statement, one of us with another, with much the same reliance, whether it be ' every X is Y,' or • some Xs are not Ps. 1 The difficulty derived from the apparent incompatibility of the third phasnomenon with the first two, and the necessity of admitting that the universal and its contrary particular are each made to be of an even chance a priori, was with me long antecedent to the explanation. This admission I imagined to have been made by the logicians : nor was I singular here. I have heard more than one person versed both in mathematics and in technical logic, express himself for the superior antecedent probability of the particular proposition, in terms which implied that he thought himself in opposition to general opinion. . An observation which contains the spirit of the true answer would be triumphantly met by many from the common maxims of the theory. Suppose a person in whose accuracy (accuracy of statement, a compound function of veracity and judgment) we have ordinary confidence, to draw a card from the pack and to announce that it is the seven of spades. We cannot conceal from ourselves that we believe he has made a correct statement as much as we believed that he was THE THEORY OF SYLLOGISM, ETC. 119 going to make one : and yet he has ventured an assertion against the coming truth of which it was 51 to 1. But a common person says at once, Why not the seven of spades as well as any other ? against which the student of the theory is tempted perhaps to retort, Yes, but why the seven of spades rather than some one or another out of the 51 others ? The observation, Why not the seven of spades as well as another, is a sound one : it reminds us, that in our absence of all knowledge of motive or bias, it is as hard to believe in error having fallen exactly on the seven of spades, as it is to believe in the seven of spades having been actually drawn : if I may speak so chemically, these difficulties combine and neutralize each other, and disengage our original belief in the witness. The reply is fallacious: it rubs out the distinctive marks from the other 51 cards, and writes on each of them ' not the seven of spades ' as its only exponent. The difference of these two cases is admirably elucidated by Laplace, in two successive problems (Th. des Prob. 3rd edit. pp. 446 — 451,) but the effect of the contrast is destroyed by a strange remark. First, there are ri counters, each of which is marked with a number ; and a witness of veracity p and judgment r announces that n° i was drawn. The probability that it was so drawn is made to be (l-p)(l-r) r n-\ in which it will be observed that how great soever the number of counters, that is, how improbable soever the event announced, a priori, the probability which the testimony gives cannot be less than pr. In the second problem, n — 1 balls are black and one white, and the same witness announces that the white ball has been drawn. The result is, q representing pr + (l — p)(l — r), that the probability of the event is 9 + (l - 9) (» - ' so that the best witness living might be incredible, if n were great enough. On this last problem Laplace remarks as follows ; — ' Ainsi Ton voit comment les faits extra- ordinaires affaiblissent la croyance due aux temoins ; le mensonge ou l'erreur devenant d'autant plus vraisemblable, que le fait atteste est plus extraordinaire en lui-meme.' Without denying all the conclusion, we may see that a comparison of these two problems shows it by no means sufficiently arrived at. If the counters were 10 10 in number, all differently marked, the production of, say n° 5000, is just as improbable beforehand, as the production of the white ball when one only is white, and 10 10 - 1 black. And the first case more nearly represents our mode of primary distribution than the second. If an astronomer were to tell us that he had seen in the telescope a dragon fly off the moon, we should certainly never think, pro hac vice, of dividing all possible events into that of a dragon flying from the moon — and others. From among other events, we should select and give prominence to the possibilities of a dream, a defect in the object glass, an atmospheric phenomenon, a fly in the telescope, &c. &c. &c. The case in which error of judgment is distinguished from wilful falsehood, need hardly 120 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, be entered upon, except in a manner subsequently mentioned. Let the general credibility of a witness, with reference to a particular event, be the measure of our previous belief that, should that event happen, he will state it to have happened, if he make any state- ment at all. Should his general credibility be the same with reference to each event which can happen, we have the case to which calculation is usually applied. But it is necessary to recognize the distinctions which exist in fact, between our opinion of a witness as to one event, and as to another. We may have reason to think we know beforehand that, according as A t or A 2 shall happen, the narrator will be a willing or unwilling witness, a sagacious or a foolish expositor. Let the particular credibility of the witness be the measure of our belief in his statement, after it has been made, and we know what it is. Let J.,, A 2 , ...A a be the names of the events any one of which may have happened, and one of which, A k , the witness asserts to have hap- pened. Let vi, v 2 , •■■"„ be the probabilities of these several events, before the assertion, in the mind of the receiver of the testimony. Let p q signify, in the same mind, the previous proba- bility that if A q should happen, the witness will state A p . Hence the general credibility of the witness, before any statement, is fi = v,l, •+• i> 2 2 2 +...+v n n n ; after statement of A k , it isk k - But the previous probability of his asserting A k is v l k l + v 2 k 2 +... + v n k n . The particular credibility, after assertion of A k , is thus found. Either A k has happened, and he has announced it, of which the previous probability was v k k k ; or A l (if k be not = l) has happened and he announces A k , of which the previous probability was v^ ; or A 2 {k not being = 2) has happened, &c. &c. Hence the particular credibility of the witness is P , v j& . * vik 1 + v 2 k 2 + + i',A If there be only two cases, then, if 1, - 2 2 , we have n= 1,; and 1, + Z t = 1, 1 2 + 2 2 = 1, give 1 2 = 2 X = 1 — fi. Hence the particular credibility is, for the statement A„ r t /u divided by vin + (l - v t ) (1 — n), which agrees with the result of Laplace's second problem, in which Vl = l -h n, ,u = q. The general credibility as to A k remains unaltered by the statement, if v k = ~S,v s k s . That is to say, the witness is unaltered by the statement, with reference to the event stated, if the asserted event, and the assertion of that event, be a priori equally probable ; if what he says be just as likely as that he should have said it. And according as the first is more or less likely than the second, the witness is raised or lowered by his assertion. Again, let Hfv,k, stand for ~2v s k s with v k k k omitted. Then v k = "2>'v s k s -H (l - k k ). When k k = ix, particularly when the immediate source of this equation is 1, = 2 2 = ... =w„, this last result is of a well known and admitted character. For 2'i»,&, -r- (l — fi) then represents the probability that of all erroneous announcements any given one shall be A k : and we thus see that the credit of the witness is unaltered by his statement if the previous probability of the event stated be equal to the previous probability of its erroneous statement; that is, its previous pro- bability in the minds of those who know that some error is coming. When a witness asserts an extraordinary thing, his credit may be maintained, in spite of his assertion, by our previous feeling of the great unlikelihood of his making such an THE THEORY OF SYLLOGISM, ETC. 121 assertion. The strangeness of the assertion is, as it were, balanced by the strangeness of his making it. In the formula v k = ~2v s k s , if v k be very small, and Ar 15 k 2 ...k n also very small, these circumstances may produce the balance mentioned, so that P k = k k . If l, = 2 2 &c. = fx, the particular' credibility is v k ix + Zv s k s If there be no particular bias towards inaccuracy, then, 2's m being 1 - m m , s m is (1 - m m ) -T- (n - 1), or (1 - ju) ^ (n - 1), except only when s = m. Hence, when the chance of accuracy is the same for every event, and no bias whatever towards one inaccuracy rather than another, the particular credibility, after the assertion of A k , is (2'^ being 1 — v k ) p _ »*M "AM + (» - 1)"' (1 - v h ) (1 - M) Here P k = n when v k = 1 -5- n, the mean probability. So that, when we know nothing of any particular bias, the particular credibility exceeds, equals, or falls short of the general credibility, according as the previous probability of the assertion exceeds, equals, or falls short of, the mean probability. When n is very great, the preceding is very near to unity, unless v k fi be of the cor- responding order of smallness. For a given value of /x, the supposition of absence of particular bias will make a very bad witness almost an infallible authority. Practically, we allot more than his general credibility to the particular statements of any witness, when we see no reason to suppose a particular bias. But experiments upon extreme cases cannot be made : for in fact, the want of particular bias is almost the sufficient reason for a growth of general habit of accuracy : so that the preceding case is almost always one in which /u. is not small. In human affairs it generally happens that a great majority of the cases have probabilities of the same order of magnitude as 1 -~ n ; and these are the ordinary events. Cases of a probability much differing from 1 H- n are comparatively few. Hence, when we disbelieve the dragon flying from the moon, above supposed as an assertion, it is not because the probability is small, for so, generally speaking, is that of an ordinary event. But the probability of the asserted event is small compared with I -~ n; or n v k is small. Any one event which we were not expecting, and for reasons, will be such as, a priori, we should call improbable. And in the common run of occurrences, things improbable (but still ordinary) are happening one after another. D'Alembert pronounced the occurrence of 100 suc- cessive tosses of head to be metaphysically possible, and physically impossible. In our day, we should translate his phrases into subjectively possible, and objectively impossible; conceivable, but unattainable. In the remarks made upon this assertion, whether with or without reference to D'Alembert, there are several different points to notice; and some matters irrelevant to my main subject must be touched upon, to clear the way for the consideration of the manner in which the question of primary distribution enters. Vol. IX. Part I. 16 122 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, The objective impossibility has disappeared before thought and experience. All who either think or try are now satisfied that repetition of trials will produce any event, however rare. Buffon, in 2048 trials of the Petersburg problem, found only one case in which the first head was deferred as late as the 9th throw. In the other instance of as many trials, mentioned in my Formal Logic, p. 185, head was once deferred till the 16th throw. A third trial of 2048 sets was made by a gentleman in the country, who communicated the results to me : he also had one case in which head was deferred to the 16th throw. Both these cases are extraordinary : for, a priori, more than 22,000 trials must be undertaken, to have an even chance of seeing head deferred beyound the fifteenth throw. It is sometimes replied that any given order of head and tail in 100 throws is just as unlikely as a hundred consecutive heads : and is therefore objectively as impossible. But this reply does not attend to the circumstance that the fulfilment of precedent conditions is the extra- ordinary event counted upon as impossible. And it is to be noted that, in looking out for the way of judging the probability of what may take place in 100 throws, the primary distribution of our minds does not take in 2 100 cases, on the one hand ; neither do we divide into the case of 100 heads — and others. We group the cases in thought into something between these two extremes : in fact, we imagine collections of the same degree of remarkability. Different minds will do this in different ways : if we could stamp the weights in some one mind, we might find perhaps only 2 5 groups, many of them involving each more than 2 95 cases. We should thus have ordinary events, each of a probability of 2 ~ 5 nearly ; and cases more or less extraordinary, varying in probability down to 2" 100 . Those who have not attended to arithmetic would make the primary distribution in a manner essentially wrong. Nothing would be called ordinary which very much differs from alternate head and tail : so that they would be surprised at almost any thing that might happen. This method of distribution into groups, the members of each group being of equal notability, is that which prevails in our methods of judging : and it serves to explain the conclusions which we actually, though unconsciously, draw on the subject of evidence. The preceding remarks may now be applied to the case of an authority asserting a universal proposition. If we suppose n - 1 to be the number of Xs in existence, and if we admit into our thoughts all the varieties of the numerically definite proposition, from • Every X is Y' through ' Every X but one is V &c. down to 'no AT is Y : ' then, if all these n cases be of equal probability a priori, that is, before any consideration whatever of the connexion of the terms, it follows that the credit of the witness is neither raised nor lowered by the assertion of any one of them. And the same, if we make groups of equal notability, m in number, provided that the probability of the universal a priori, is looked upon as being exactly or nearly l -r m. The two universals would each form a group, the idea of necessity, as distinguished from that of contingency, not only securing them this character, but perhaps giving them rather a higher share of probability than deliberate consideration would approve of. I do not profess THE THEORY OF SYLLOGISM, ETC. 123 here to stamp the weights : but only to show that the application of the theory explains the apparent inconsistency of the three phaenomena above noted. Let us now suppose that the statement of the witness is a denial that A k happened, and nothing more. Then either A k happened and he denies it, of which the previous probability is v k (1 - k k ) : or A r (k not being = 1) happened, and he denies A k , of which the previous probability is v\ (l — &i), &c. So that, from his assertion, " k C 1 - K) and 1 ~ "* - s^A are the particular probabilities that A k did and did not happen. The first is the probability of his inaccuracy after the statement ; and the previous probability that he shall inaccurately deny Ak is 1 — k k . So that, looking at the previous force of ~Zv s k s when equal to v k , we see that his previous reputation for inaccuracy as to A k , is increased, unaltered, or diminished, according as the probability of the event denied is greater than, equal to, or less than, the probability against the event, the affirmation of which would have left his reputation for accuracy unaltered. The particular proposition being considered as more probable than the universal, the denial of the universal A k , which is the affirmation of the contrary particular, generally makes the universal still more improbable. But we can hardly suppose a formal affirmation of the particular, except in opposition to some amount of belief gained for the universal, of a larger amount than its natural probability. About such an hypothesis there is nothing parti- cular to examine. The following is the most plausible supposition as to the bias of inaccuracy. The receiver of the testimony supposes that the probabilities of the n events are Xj, X 2 , ... X„ in the mind of the witness ; and, k k being still the previous probability of accuracy when k happens, he divides 1 — k k , the probability of inaccuracy, among the events which may be inaccurately stated, in proportion to the presumed probabilities of these events in the mind of the witness. That is, since 1„, + 2 m + ... + n m m 1, he makes p m = \ p (l - m m )-i-(l - X m ). On these sup- positions, the particular credibility of the assertion A k is v k K + X t 2' &) Here, for a given probability in the receiver's mind, both as to the event and the witness stating it if it happen, the more incredible he is supposed to think it, the more must the receiver be inclined to believe it. If X s = s s , for all values of s, that is, if the previous probability of the witness asserting an event when it happens, be exactly the previous probability in his own mind that it shall happen, then P k = v k , and his testimony is nothing either way. According to the point of view from which this result is looked at, it appears exceedingly rational or exceedingly absurd. That the more extraordinary he thinks an event, the less likely is he to state it, even if it happen, would by itself make us trust him : but then, on the other hand, the more likely he thinks an event, the more likely is he to state it when it does not happen ; which by itself 16—2 124 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, lowers his credit. The balance of the two is perfect, on the definite supposition above : and thus an indisposition to state what he thinks extraordinary, is not, by itself, always a prima facie indication of a good witness to an extraordinary event. If instead of \, = s, we take s s = 6\ s , we find that P k = v k when This is the case in which the disposition to accurate statement varies as the witness's pre- vious expectation of the event. But if 1 — s, = 6 (1 — X s ), that is, if the bias against inaccuracy vary as the previous expectation that the event shall not happen, we find * (i-e)v k + e\ This is v k when 9=1, as before. Let there be two distinct sources of mistatement, by the double action of which an incorrect inaccuracy may be a correct statement : as in intentional and unintentional inaccuracy. Let all the preceding symbols apply to error of judgment, that is, let p q signify the chance of his believing A p when A q happens ; and let p q ' represent the previous probability in the mind of the receiver, that the witness will wilfully state A p when he believes A q happens. The state- ment A k having been made, the possible originators of this statement are the compound events of which the probabilities are thus stated. Let A m happen, let the witness judge that A v happened, and state that A k happened. The previous probability of this combination is v m v m kj: and the denominator of the particular probability P k is 22 (y m v n k v ') for all combined values of m and v. Select only those cases in which m = k, and we have 2 iy k v k k^) for the numerator. Accordingly If s s and »/ be always p and r, as in Laplace's case, and v s = 1-r-n, and if the biases both to falsehood and error of judgment be all equal, we have m, = (1 - p) -T- (n - 1), and m s ' = (1 - r) H- (n - l) ; 2v k v k kJ = n~ l pr + (n - 1) {n' 1 (1 - p) (l - r) (n - J) -2 } the rest of 22 ^M = (n - l) {li^ (r + ^(l - r))} + ^^ (1 - r) = |(l-p)r + (l -r)p+^-J(l-p)(l-r)Ji, whence the whole denominator is 1 -f- n, and (1 - p) (1 - r) . . _ . P k = pr + £i-J , as given by Laplace. n — 1 There is, however, but one case of the above which need be especially considered. When we suspect intentional falsehood in an assertion, it is usually because there is some particular THE THEORY OP SYLLOGISM, ETC. 125 reason why want of veracity should lead to that one particular assertion. Taking the extreme case, let us suppose A k to be the only assertion which has any probability of being falsely made, so that m/ = 0, except only when m = k. So far the formula is unaffected : but as it generally happens that the previous bias towards falsehood is towards the assertion itself, and neai'ly or altogether independent of what may really have happened ; let k k = 1, and &/ = k\ = ... = k- Then we have p = »> {K + * - K)} = (* ~ K ) "A + KV k (1 -(t)2. !/,*, + «' lying, as of course might have been predicted, between v k k k -r-'2v s k s and v k . If there be no probability of error of judgment, k k = 1, and k, = when s is not = k. We have then P k = V * . (1 - K) V k + K The formula for any number of witnesses may be constructed as follows : Let p q m represent the previous probability that the m th witness, A having happened, will judge that A p has happened ; and p q ' m , the previous probability that the m th witness, judging A q to have happened will state A p . If the statements made by the several witnesses be A k , A„ A m , &c, the denomi- nator of the probability thence arising in favour of A t , is the sum of all terms of the form v «w« k J l x x uC x y» m * x for all combinations of values of u, w, m, y, &c. : the numerator is the collection of all the terms of the denominator in which u = t. If all the witnesses be of the same character in every respect, and all agree in asserting A k , the coefficient of v u is for all combinations of values of w, x, y, &c. In all that has preceded, it is to be remembered that application can immediately be made to the case in which the n events are not the only possible ones : that is, in which 2i/, is not equal to unity. For this is but the supposition of one case more, namely, no event at all, with the probability 1 - 2i/,. A. DE MORGAN. University College, London, January 7, 1850. 126 PROFESSOR DE MORGAN, ON THE SYMBOLS" OF LOGIC, ADDITION. It may be useful to advert to the manner in which logicians, who all contend more or less explicitly for the sufficiency of ordinary syllogism, meet the cases in which premises give inference which cannot be reduced to one ordinary syllogism with those premises. I pass over the following, as extra-logical. First, giving the inference a bad name, as call- ing it a subtlety, or the like ; logic is the science of the necessary laws of thought, subtle and not subtle. Secondly, affirming that the inference is very easy : which is just as much the case with the favoured modes of syllogism. I proceed to one which is both logical and sufficient, but which again applies just as much to the ordinary syllogism as to the cases which will not fall under it. In every inference, there is an act of the mind, which we may perform with or with- out consciousness of reference to the general, self-evident, and indemonstrable postulate under which the validity of that act might be maintained. Useful as reference to the postulate may be, it is not, or need not be, formally necessary, since the act of the mind by which we refer the instance of inference to the postulate is, logically considered, of the same kind as that by which we draw the inference at once from the premises. Never- theless, the formal syllogism, in the first case, is never anything but that easiest of all cases, the syllogism of principle and example, (F. L. p. 257). And thus we have for the ordinary syllogism, two forms of argument : the common one, and the reference to the postulate. Now this postulate always has had, perhaps always must have, a composition of rela- tions, such as I have generalized in Section V. Consequently, I assert that the logicians have, when compelled to declare the ordinary syllogism incapable, had recourse to instances of the composition of relations out of which I have constructed the bicopular syllogism ; though in truth they have not seen the extension of the theory, merely because their reference of the example to the principle may be made under the form Barbara. First, I take a common syllogism, say X)) Y).(Z = X).(Z. Using the terms of my work (F. L. ch. xiv., in which every syllogism is thus reduced, though composition of re- lations is not, as it ought to have been, the leading idea,) we see that X is a species of Y which is an external of Z, so that X is a species of an external of Z. Now if we merely compound the relations, we see that species of external is external ; whence X is an external of Z. If we choose to make our last step in the greatest form, we have Every species of an external (of any notion) is an external (of that notion), X is a species of an external of Z; Therefore X is an external of Z. If I were to attack the syllogistic theory in a manner analogous to that in which lo- gicians have, in certain isolated cases, supplied its deficiencies, the attack would involve the assertion that the preceding transformation is the legitimate form. I now give instances of the supplementary use of this method. Reid justly remarks that "A is equal to B, and B to C, therefore A is equal to C, cannot be brought into any syllogism in figure and mode. ,, On which Sir W. Hamilton's note (p. 702) is as fol- THE THEORY OF SYLLOGISM, ETC. 127 lows ; — " Not as it stands ; for, as expressed, this reasoning is elliptical. Explicitly stated, it is as follows : — What are equal to the same, are equal to each other; A and C are equal to the same (B) ; Therefore, A and C are equal to each other." I am quite at a loss to see how the second is an expanded form of the first. But I see distinctly the composition of relation, * equal of equal is equal,' expressing the tran- sitiveness of the copula equals, which, with its convertibility, renders it of equal validity with is as the copula to be employed in a syllogism. Again, a recent and learned editor of Aldrich, the Rev. H. L. Mansel, observes of the preceding, that "the reasoning is elliptical, and therefore, as it stands, material; though owing to the suppressed premise being self-evident, its deficiency is apt to be overlooked ." This suppressed premise is the major given above. But does it not then follow that the ordinary syllogism is elliptical ? Is there not always the suppressed premise (as I feel bound, on these authorities, to call it), " What are the same are each other ?" There is some want of distinctness in the use of the word material, as distinguished from formal. When the last named writer makes it " material and therefore extra-logical " that Alexander was the son of Philip because Philip was his father, he uses the word historically. The formal connexion of relation and correlation exists, though Aristotle did not recognize it as copular except when the relation is its own correlation, and then only in a limited case. Certainly the matter, in the case of father and son, supplies the knowledge of the correlative relation existing, but not the mode of using it in inference, when known to exist. In this and a great many other instances, matter is opposed by writers, not to form, but to what is recognized as form in the school of Aristotle : the assumption of course being that that school exhausts the forms of thought. Historically speaking, the copula has been material to this day : this I must continue to believe until it be pointed out where the formal conditions have been separated from the matter, and made the instru- ments of inference, independently of the separated matter. A certain indifference to close description of the copular relation is manifest in many logical works, accompanied by negligence in its application ; and this may produce strange consequences in literal translations. The proposition ' X is F, 1 may bear expression in French as ' X doit etre F,' and in Latin as ' X debet esse F,' though both are rather negligent forms for a strict work : but the English ' X ought to be F' is of a different signification. Both* the English translations of the Port Royal Logic give a paralogism for the instance cited in illustration of the definition of the word reasoning: " having judged that true virtue ought to be referred to God, and that the virtue of the heathens was not referred to him, we thence conclude that the virtue of the heathens was not true virtue." Had the copula been as fully treated of as the terms, such inaccuracies could hardly have occurred. * A new one has recently been published by T. S. Baynes, Edinburgh, 1850. July 3, 1850. Mathematical Exposition of some Doctrines of Political Economy. Second Memoir. By W. Whewell, D.D., Master of Trinity College. [Read April 15, 1850.] 1. There have appeared of late years many works upon the subject of Political Economy in which the reasoning, illustrated by numerical examples or in other ways, is of such a kind that a person of mathematical habits of mind, in reading the works, is naturally led to reflect upon the possibility of putting the reasoning into a general algebraical form, and upon the con- sequences which would result from such a mode of treating the subject. It is evident that such a mathematical mode of investigation would, when the fundamental principles of the subject were once distinctly stated and expressed in algebraical symbols, give the results with a certainty and simplicity of method which would be more satisfactory than special numerical examples, — at least to a mathematician. And among many other advantages of such a process, we may expect that we should obtain these : — that we should see both how the doctrines may be generalized, and how they must be limited, much more easily and clearly than we could do by the light of special numerical examples only. Accordingly, attempts have not been wanting to make such an application of mathematical methods to Political Economy : among which I may refer to a paper of my own, published in the Transactions of this Society*. 2. It would, however, be to take a very erroneous view of the consequences of this applica- tion of mathematics to Political Economy, to suppose that it can add anything to the certainty of the fundamental principles. There is perhaps in some persons a propensity to believe that any subject, when clothed in a mathematical shape, acquires something of mathematical demon- strative character; and that by applying mathematics to assumed principles of knowledge, we in some measure create a science. I must beg leave very distinctly to repudiate all pretensions of this kind. By stating distinctly our fundamental principles, which such an undertaking as the present requires us to do, we may bring them more clearly under notice and examination than would otherwise be done ; but we add nothing whatever to the evidence of the principles. All that we pretend to say is, that if the conclusions be false, the fallacy must be in the prin- ciples, if the process of deduction be truly mathematical. 3. The questions which I now intend to consider are some which relate to the connection of demand, supply, and price, whether in the same country, or in different countries. And, first, in the same country. That the price of any commodity depends upon the relation of the demand and the supply is commonly and truly stated. But in order to express this dependance " Mathematical Exposition of some Doctrines of Political Economy. 1829. SOME DOCTRINES OF POLITICAL ECONOMY. 129 in a mathematical manner, we must define with precision the quantities which enter into the relation. 4. Let p be the price of a commodity 0, and q the quantity effectually demanded, that is, bought, at that price ; p being of course expressed in units of money, and q in terms of some unit which measures the commodity C. The demand for C depends (cwteris paribus) upon the price : q depends upon p. Let p increase in any ratio : the demand, that is, the quantity which buyers are willing to purchase, will commonly diminish : but it may diminish either in the same ratio in which the price increases, or less rapidly. If the price of corn be increased by one-fourth, the consumption may be diminished by one-fifth, one-eighth, or one-tenth. In the first case, the money demand, that is, the money expended on the commodity, would be unaltered : (for it would be 5 4 5 7 35 7 = price x quantity = - x - = l). In the second case, it would become =-x- = — = 1-| — , T) {} T* O O 4 \J/i or would be increased between one-fourth and one-fifth. In the third case, the money demand 5 9 9 1 would become =-x — = - = 1 + -, and would be increased by one-eighth. 4 10 8 8 5. In order to express such a relation generally, let p become p' = p (l + x), and let q become q , such that p q = pq (l + moo), where m is a certain coefficient. This expresses the cases iust mentioned according as m = 0, m = -, or m = -. J & 8 2 6. But the price may also be considered as depending on the supply, that is, on the quantity supplied for sale : and the quantity supplied to buyers at a given price is the same as the quantity bought at that price; therefore the equation just given, p q =pq (l + mx), expresses the relation between supply (q) and price (p'). If the quantity supplied of any given commodity be diminished by one-fourth (for example), the price may increase by one-third, or it may increase one-fifth, or it may increase one-half. 3 4 In these cases the money demand respectively becomes = - x - = 1, that is, remains unaltered ; 3 6 18 1 j. • • , , , , 3 8 9,1 becomes = - x - = — =1 , or diminishes by one-tenth ; becomes = — x- = -=l+-, or 45 20 10 4288 increases by one-eighth. And since a is in these cases = -,-,- respectively, and mx = 0, — -,- 1 1 respectively ; we have in the three cases, m = 0, m = , m = -, respectively. If the quantity supplied be increased by one-fourth (for example), the price may be diminished by one-fifth, or one-tenth, or one-half. In these cases, the money demand becomes 5 4 5 9 9 1,. respectively = - x - = 1, that is, remains unaltered ; becomes =-x — =-=1+-, that is, 4 5 4 10 8 8 increases by one-eighth ; — becomes = -x-=- = l , that is, diminishes by three-eighths. 4 2 8 8 Vol. IX. Paht I. 17 130 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 11,1 . , , 13, 4 Since x is in these cases = -, — , and - respectively, and mx = 0, -, — , we nave m = 0, m = -, 5 10 2 r ; 8 8 5 3 m = — in the three cases respectively. 4 7- Which of these classes of cases is likely to exist in practice ? It will be apparent, on consideration, that one or other will occur according to the nature of commodity, and thus, that m has different values for different commodities. We shall endeavour to indicate certain classes of commodities according to this relation. 8. (A) There may be some commodities on which, in a given society, the same sum is expended whatever be the price of the article (per unit) ; a smaller quantity being bought exactly in proportion as the price is higher. Such would be the case with ornamental attire, for instance, if each person, or if persons on an average, were to spend upon it a determinate sum every year ; — an allowance for dress, as it might be termed. In this case, when p becomes p (1 + x), p'q', the money demand, remains unaltered, whence pq (1 + mx) = pq. Here m = 0. 9. (-B) There may be other commodities of which the quantity bought is the same whatever be the price : such, for example, may be articles which are looked upon as necessary by rich persons ; as, for instance, official dresses, and conventional appendages of persons in office, and the like. Here, when p becomes p (1 + x), q remains unaltered. Therefore pq (l + mx) = pq (l + x). Here m = 1. 10. (C) There are other commodities of which the price increases more rapidly than the quantity supplied diminishes : for instance, the general necessaries of life. It has been supposed that a deficiency of one-fifth in the supply of corn will raise the price four-fifths. Supposing 4 9 36 11 .4 11 this true, the money demand becomes = -x- = — = 1 -\ — . And since x = - and mx = — , ' 5 5 25 25 5 25 11 m = — . 20 11. The more we suppose prices to rise for a given diminution of supply, the more will m approach to 1. If we suppose that a diminution of one-fifth in the supply will treble the 4 12 7 . 7 price, we have the money demand =-x3= — = 1+-. And since mx = - and x = 2, 5 5 5 5 7 14 m = — = — . 10 20 In all these cases m is between and 1. 12. (D) There may be other commodities of which the price increases in a less pro- portion than the supply diminishes : or, as the case is perhaps more evident, there may be com- modities, of which, when the price diminishes, the demand increases, and in so great a proportion that the whole sum expended on them is greater than before. This may be the case with SOME DOCTRINES OF POLITICAL ECONOMY. 131 some luxuries : for instance, when tea or coffee fall to half-price, they may come into use with new and numerous classes of persons, so that the whole sum expended on them may be double what it was. In this case, it is evident that the quantity bought must be four times what it was. In this case, m ■ , mx = 1 ; hence m = — 2. 2 This may be the case in a smaller degree : if we suppose that a fall to half-price increases the money demand by one-half, we have x = — , mx = - ; whence m = — \. If we suppose that a fall to half-price increases the money demand by one quarter, we have x = , mx = - ; hence m m . 2 4 2 In the last two cases, the quantity bought is, in the former case trebled, in the latter, increased in the ratio - . 2 In all these cases m is negative. 13. We must suppose that the above formulas apply also in the cases in which x is negative, or the price falls. Applying this remark to the four classes of conditions A, B, C, D, we see that For class A of commodities, (m = 0,) if the price fall, the money demand remains un- altered, and the quantity sold increases inversely as the price. For class B, (rn = 1,) if the price fall, the money demand falls in the same proportion ; and the residue is saved by the purchasers for other employments. For class C, (m> and < 1,) if the price fall, the money demand falls also, but in a less proportion, the quantity sold being increased. Thus, if for any commodity for which m = - , 4 the price fall one-half, the money demand will fall one-eighth. If when m = - , the price fall one-half, the money demand will fall one-fourth. To produce the former effect the quan- 7 "- tity supply must increase in the ratio - : to produce the latter the quantity supply must . , . 3 increase in the ratio - . 2 Q 14. If m be very nearly = l, for instance if m = — , and the price fall one-tenth, the 9 1 money demand will fall : to produce this effect, an increase in the quantity of — will 18 suffice. If the price fall two-tenths, the money demand will fall — : to produce this effect, an increase in the quantity of — will suffice. If the price fall one-half, the money demand will 9 . 1 . fall by — : to produce this effect, an increase in the quantity supplied of — will suffice. 17—2 132 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF Hence it appears that when m is very nearly equal to I, a small increase in the quantity supplied, ( — , — , — , J will produce a large diminution in the price ( — ,-,-). In such cases we may say that the price is very susceptible of change, (by alteration of the supply), and since, as m is larger, this susceptibility is greater, we may take m to measure the suscepti- bility of change of price of each commodity. 15. But if we consider the demand as varying with the price, it is evident that for a given increase of price, a greater increase of money demand indicates a stronger effectual demand : and as this is greater according as m is greater, m may measure the strength of demand for each commodity, as shewn when the price changes. 16. We may include these two ways of regarding m, by calling it the specific rate of change of each commodity ; meaning thereby, both the change of price when the supply varies, and the change of demand when the price varies. 17. For an example of the value of m, we may take data given by Mr. Tooke {High and Low Prices, p. 285). He says that to diminutions in the supply of corn we have correspond- ing augmentations of price, in the following proportion : 12 3 4 5 Diminution of supply — , — , — , — , — ; rr J 10 10 10 10 10 „ . 3 8 16 28 45 Increase of price — , — , — , — , — . r 10 10 10 10 10 Now if we take m = - , we have the following correspondencies : 12 3 4 5 Diminution of supply — , — , — , — , — ; rr/ 10 10 10 10 10 t * ■ H 6 3 15 40 • - • Increase of price -» , -« , — , — , infinite. r 10 10 10 10 For a deficiency of supply of one-half, the prices would be infinite. This shews that the formula with the value m = -, is not applicable for so great a deficiency of supply : but upon the whole, m = - , appears to be near the value of the specific rate of change for corn. 1 3 17 If we take the diminution of supply = — , and increase of price = — , we have m = — . rr-7 10 r 10 30 5 . 45 7 If we take the diminution of supply = — , and increase of price = — .we have m = — . r * J 10 ^10' 18 The former is greater than - , the latter is less than - *. 2 2 " It appears that when the diminution of supply is small, I is nearly one-half, the value m = % is too large. Therefore the the value m = J is too small, and when the diminution of supply | function pq(l+mx) is not really the true formula of the SOME DOCTRINES OF POLITICAL ECONOMY. 133 18. For the class of commodities D, (m negative,) if the price rises, the quantity effectually demanded diminishes in a higher ratio. In the case supposed before, in which a fall of price to one-half produced a fourfold amount of sale, we might at first, perhaps, suppose that the rule applies in the inverse order of change, and that if the price be again doubled, the quantity demanded will again fall to one-fourth, and the money demand to one-half. But this result will not be given by the formula. For in this case m was = - 2 ; and if when p'q = pq{\ - 2a?) we suppose q to be -q, we have 1 + x = 4(1 - Zoo), which would give a? = - , denoting an increase of price of 4 ^ one-third, and a money demand diminished to one-third. 19. This apparent inconsistency arises from the formula p'q =pq (l + rnx) being made to rest on a given medium standard price and quantity, p and q, and to express the changes by an increment or decrement x. Hence large changes are not proportionally the same above and below the standard point. 1 / x\ When m = , we have the money demand = pq I 1 I , and hence, since 1 - (X\ 2 1 I ■> P = P + ">)y we nave 9 ~ 1 • 5 When the quantity increases by one and one-half times its value, we have this = - , whence x = — , and the price is reduced to one-half, as in the case above stated. When the quantity m l - - 2 1. .1 decreases by one-half, we have — ^ = - , which would give x = - , and the price is in- 1 T X >£ ~ creased by one-half. x 1 2 2 2 When the quantity decreases by one-third, we have — — = - ; whence x = - , the price .9 .5 is increased in the ratio - : the money demand is diminished in the ratio - . 20. It appears from what has been said, that we have four classes of commodities, which differ according to different values of m, the susceptibility of change in the price by change money demand. It would be easy to devise a formula which should more nearly represent Mr. Tooke's progression; but even if his numbers were derived from facts, they would, of themselves, be insecure grounds for generalization. And the nature of the progression makes it probable that the progression is hypothetical merely : the third difference being constant : as appears thus: 3 8 16 28 45 3 5 8 12 17 2 3 4 5 1 1 1 The general term of the series 3, 8, 16, 28, 45 is n(w+l)(w+2) |2n _ 6 134 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF of supply, or the rate of change in the money demand for a change of price. The classes occur as »» is 1, is between 1 and 0, is 0, or is negative. So far as these classes of com- modities are exemplified by the instances above adduced, we may call them Conventional Necessaries, General Necessaries, Articles of Fixed Expenditure, and Popular Luxuries. For the first class, the quantity sold is the same whatever be the price. For the second class, when the price rises the quantity sold diminishes, but the money demand increases. For the third class, the money demand is always the same, and therefore the quantity sold is inversely as the price. For the fourth class, when the price falls the quantity sold is augmented, so that the money demand also is augmented. 21. I suppose that there are no commodities of which a greater quantity would be sold if the price were increased, and a less quantity sold if the price were diminished. It is conceivable that this might be, as a matter of caprice or fashion. For instance, we may conceive that diamonds might in some way (by the discovery of abundant mines or the like) become so common as to grow out of use, so that a less quantity might be sold than at present. If there should be such commodities, they would correspond to values of m greater than 1. 22. If it were possible to arrange commodities according to the value of to, the specific rate of change, (as is done hypothetically for the sake of example in the above instances) so that we should for every quantity know the value of to, we might solve a great variety of problems respecting the variations of price, of demand, and of supply, so far as these quan- tities depend on each other. And so far as the formulae are applicable, we have the equations 1 + mx p'=p(l+x), p'q = pq(l +mx), q = q 1 + x 23. It is well observed by Mr. J. S. Mill (Polit. Econ. i. 529) that instead of saying, as writers have often said, that the price depends upon the ratio of demand and supply, we ought rather to say that the price depends upon the equation of demand and supply. And we may apply the term, the equation of demand and supply to the equation p'q' = pq(l + mx). 24. As examples of the above formula?, let it be supposed that an increase of — in the supply (the whole being sold) produces a fall of - in the price. 5 TO 1 ™, J j 5 U 3 Then x = , and — — = — , whence to = - . 5 1 10 5 1 -- 5 This being the case, what effect on the price would be produced by an increase, and what by a diminution, each of - , in the supply ? SOME DOCTRINES OF POLITICAL ECONOMY. 135 3d? 1 + — 5 5 5 . . .8 For the increase = - ; no = : the price falls in the ratio — . 1 + ao 4 13 13 Sx ^,, + T 3 5 .... . .8 For the decrease — — = - ; « «■ - : the price rises in the ratio - . 1 + x 4 3 3 25. Hitherto we have not considered the manner in which the extent of the supply is determined by its causes. But it is evident that in general there is, between the quantity sup- plied and the price, a relation depending upon the conditions which govern the supply, as being the produce of agriculture or of manufacture, or of some other agency. 26. So far as this dependance goes, there are three main classes of commodities (J. S. Mill's Polit. Econ. i. 524). (a) Articles of absolutely limited and fixed supply, in which no increase of price can increase the supply, as old pictures, peculiar wines, building ground in a town. (/3) Articles of unlimited supply with proportional labour ; as, in general, manufactured articles, cottons, woollens, linens, which might be produced in unlimited abundance by a proportional application of capital and labour. In these, the price or cost, in the long run, and independently of temporary fluctuations, is constant ; except so far as it may be diminished by improvements in manufacture or cultivation, or increased by the increased cost of the raw material ; or raised for a time, and its ultimate value delayed, by want of capital or want of labourers. (7) Articles of increased supply at increasing cost, as agricultural produce in a given limited country. In such articles, every addition to the quantity produced increases the cost of some portions, and therefore the price of all. To add a million quarters of corn to the annual produce of a country, would raise the cost-price, it may be, 1 shilling a quarter. To add a second million quarters to the produce, would raise the price still further, it may be a shilling more, or two shillings above the original standard ; and so on. 27. But it may be that the increase of cost-price goes on in a higher ratio than the increased amount of produce. Thus the original standard quantity of produce being q millions, and the cost-price of a unit of that quantity being p shillings, let it be supposed that, in order to add to q respectively l million, 2 million, 3 million units, the price must rise at each step successively by 1 shilling, 3 shillings, 5 shillings, &c. at the successive steps of extended and more expensive cultivation. Then the whole rise of price at these steps above the original standard will be 1,4,9, &c which are as the squares of 1, 2, 3, &c. the increase of quantity. We may express this relation thus; n being a coefficient hereafter to be determined, we may suppose q' = q(l+y), p=p(l+ny e ), and we may call this the equation of increased cost of production. 136 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 28. It is not pretended that the law of the increased cost of increased quantity is exactly represented, for agricultural or any other produce, by the formula just given; but only that such a formula may serve to exemplify a dependance having such a character as has been described. 29- In commodities of the first of the three classes above-mentioned, (a), (commodities of Fixed Supply,) the price depends entirely upon the intensity of the demand. As the amount of supply is invariable, the price can vary only from some variation in the demand ; and this must arise from changes not expressed in our formulae. In commodities of the second class, (/3), (commodities of Fixed Cost) the quantity supplied will depend upon the amount of capital, labour and skill employed in the production. When this quantity varies for a unit of the commodity, the price will vary. And the relation will be given by the equation of demand and supply (23). If p, the selling price, be greater than the cost price with the usual rate of profits, capital and labour will flow into the employment, till p is reduced down to cost price. If by any cause, for instance, improvement in the process of manufacture, the cost price is diminished, the diminution may be expressed by x, in that equation ; and (m being known) the equation will give the quantity which will be sold at the new price. In commodities of the third class, (y), (commodities of Increasing Cost) the quantity sup- plied will be determined by the cost price which purchasers are willing to give. The selling price, in the long run and on the large scale, will be the cost price with the ordinary profit. If the purchasers are willing to give a higher price, (as for instance, in consequence of the increase of wealth and population) the quantity produced will be increased by extended or more expensive cultivation. And the amount of increase will be given by the equation of increased cost (27). (If by improved methods of cultivation the cost price p be diminished, this equation no longer applies to that change. The case then comes into the second class and the equation of demand and supply is to be used). 30. For example ; let the price which purchasers are willing to pay rise from 9 to 10 shillings, and let n = 1. Then ny 2 = -, and y = -. Therefore q will be increased in the . 4 ratio - . 3 An increase of - in the population, with a proportional increase in the means of living, 3 would on these suppositions, correspond to an increase of price from 9 to 10. On the same suppositions, what rise of price would take place on the next equal increase of population, under the like conditions ? 2 / 4A Herey = -, whence p' = p I 1 +-) . Hence the price would be nearly half as much again as at first. If the original population be doubled under the like conditions, y m l t p' = 2p, and the price is doubled, though for the increase of - in the population, the price was only increased-. 3 9 SOME DOCTRINES OF POLITICAL ECONOMY. 137 31. But it must be recollected that these suppositions, of a population which, during its increase in numbers, increases in the same proportion in money demand for corn, while the increased production of corn is not accompanied by any improvement of cultivation occasioning a diminution of cost, is an arbitrary hypothesis, made in order to apply the formulas, and not likely to be realized in fact. 32. And on the other hand, if the quantity of produce of this class has been increased while the price has not been increased, or has been diminished, that circumstance shews that in the country and during the period in question, the commodity has not been one of increasing cost of production. In this case improvements in cultivation must have balanced or more than balanced the increased difficulties which, without them, would have made the cost of every added portion of produce greater than the preceding. To this case our equation does not apply. 33. The preceding formulas apply to prices as affected by demand and production within the limits of our country. Prices within such a circle are governed ultimately by the cost of production. The prices of a unit of each of two commodities C and D, are as the cost of production of the two ; that is, as the labour (including skill estimated in labour) by which each is produced and brought to market. For if the ratio of the prices were different from this, labour would be transferred from the production of the one to that of the other, so as to tend to restore the equality. But between foreign countries there is no such tendency to equilibrium between price and labour, because labour is not transferred from one country to another when the prices are in a different ratio. A pound of tea, if produced in China by the same labour which produces a yard of cloth in England, may nevertheless exchange for two yards, or for half a yard : for there will not be a transfer of tea-producing labour to produce cloth, in the first case, or of cloth-producing labour to tea in the second. Hence then the relation of prices of commodities, native and imported, is not governed by the equations already given. By what then is it governed ? What is the principle which regulates inter- national values? (Mill, Polit. Econ. n. 121.) 34. The principle which regulates such values is (in addition to the principle of supply and demand already spoken of,) this: — that when the international trade has been established, the relative value of all commodities which are exported and imported is the same in the two countries (omitting for the present the cost of carriage). This we may call the principle of uniformity of international prices. It is evident that if tea and cloth are exchanged between China and England, the rate of exchange of the two must be the same in the two countries : for if it were not, the current of trade would be determined one way or other, and would, by increasing the import of the one commodity or the other, tend to restore the equilibrium. 35. In order to apply this principle, let there be two commodities (C) and (D) (cloth and linen for example) : and let C and D represent the value of a unit (yard) of each in terms of any other commodity. Vol. IX. Part I. 18 138 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF Suppose that in England p is the price of D in terms of C ; and let q be the quantity of (/)) consumed (that is, bought) in England at that price. D Then D=pC, C = — in England. P Suppose that in Germany P is the price of C in terms of D, and let Q be the quantity of (C) consumed in Germany at that price. C Then C = PD, D = - in Germany. Now D in England costs p C when there is no international trade ; but if obtained from C Germany by exporting C, would there cost — . Therefore there would be a gain for England in obtaining D by exporting C rather than Q producing it at home if pC > — ; that is, if Pp > 1 : for the cost would be less. In like manner, there would be a gain for Germany in obtaining C by exporting D rather D than by producing it at home, if PD > — ; that is, if Pp > 1 : for the cost would be less. Hence on the supposition that England exports C and imports D, and Germany exports D and imports C, both countries gain. 36. What will be the amount of the exports and imports, and the prices, when the inter- national trade 6x5818? In order to solve this problem, we must introduce another principle; namely this: that in the long run, and in the permanent condition of the trade, the value of the exports of each country must equal the value of its imports. For each country pays for its imports by its exports. Under the trade let p, the price of D in terms of C, in England, become p' ; and q, the quantity of (D) consumed in England, become q', the whole being imported from Germany. And let P, the price of C in terms of D in Germany become P, and let Q, the quantity of (C) consumed in Germany, become Q', the whole being imported from England. In England p'C = D, in Germany, P"D = C ; and since these equations express prices under the trade, by the principle of uniformity of international prices, the relation of C and D is the same in the two countries. Therefore multiplying together the two equations, Pp = 1, which is the equation of uniformity of international prices. 37. England exports Q' of (C) and imports q of (D) ; and of this last the value is p q in terms of C : therefore, by (36) p'q'=* Q'. In the same manner Germany exports q of (D) and imports Q' of (C) ; and of this last the value is P'Q' in terms of D: therefore PQ'= q. The equation p'q'=Q', or PQ' = q', is the equation of import and export. The two equations are identical in virtue of the equation P'p' = 1. SOME DOCTRINES OF POLITICAL ECONOMY. 139 38. Now to find the quantities of the imports and the prices. The consumption of (D) in England varies with the price (22). When the price falls from pC to p'C, let the quantity consumed be increased from q to q. Let p' = p (1 - x), and let the law of altered money demand be, as before, p'q 1 ' = pq(l - mx). Hence q = q . 1 - x In like manner the demand for (C) in Germany varies with the price. Let P = P(l - X), PQ' = PQ (l - MX) ; whence Q' = Q " 1 -X Since p (l - x) = p\ P (l - X) - P, we have Pp (1 - Jf) (1 - a?) = Pp' = i, by (36). Hence (l - X) (l - x) = -— - = l - A, suppose, A; being a fraction ; since by (36) Pp is greater than 1. The equation (1 - X) (l - x) = 1 - k, gives X = — ^— . The equation 9 '= P'Q', (37), gives q ~ ™ X = PQ (l - 3f^). ] — a? Put for .y its value , and solve the equation in x ; and we find PQ (1 - Mk) - q X ~ PQ(l-M)-mq' 39. The values of x and X depend upon the ratio existing between PQ and q originally, before the trade : that is, upon the relative value of (C) consumed in Germany and of (D) consumed in England : and also upon m and M, the specific rate of change of each com- modity. In general let PQ = nq; and we have n (1 - Mk) - 1 1 - n (l - Mk) x = , or x = . n (1 - M) -m m - n(\ - M) 40. I will apply these formula; to the numerical examples given by Mr. Mill, (Polit. Econ. ii. 123). g He supposes that originally, in England, C = -D, and in Germany, C = 22). Hence 2 4 p = - , P = 2, Pp = - which being > 1, the trade is advantageous ; 3 3 — - =i 1 — k : whence k = - . Pp 4 4 2 Let m and JW be each = - . Hence we have 3 6 - 5n x = . 4 -2« 18—2 140 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 8 1 If we suppose n to be - , we find x = — . Hence the price of D in terms of C falls - in England ; that is, the price of D becomes 2 5 10 - x - C = — C, which is one of Mr. Mill's suppositions. 3 6 18 rr 1 ■■ tYt ffl 1 On this supposition, since q ' = q = q — , the quantity of (D) consumed in England is greater by — than it was before the trade. 3 Q 1,1- MX 28 Also (1 - x) (1 - X) = -. Hence i-X= — , X = — ; Q'=« ~ = Q— • v ' v ' 4 10 10 J - X 27 The quantity of (C) consumed in Germany is — greater than it was before the trade. 41. If we suppose n to be — , we find x = - , -3f = 0. 9 4 Hence the price of D in terms of C in England falls - ; that is, the price of D becomes 3 2 10 - x - Cor ^C The quantity of (Z?) consumed in England becomes — q. In this case the relative price of C and D in Germany is not altered by the trade, and Q is not altered. If we suppose n to be - , we find x = 0, X = - , Q' = — Q. In this case the relative price of C and D in England is not altered by the trade, and q is not altered. 6 . .5 42. If we suppose n to be greater than - , for instance, if x = — , we should find from 5 4 1 5 the formulae, x = — - , A" = — . 6 14 But this solution, implying that the price of D in England is increased by the trade, is of course inapplicable. In like manner if n be less than — we shall find X negative, and have an inapplicable solution. 43. We can now trace the gain of each country by the trade on the above suppositions. 8 8 4 Let n = - and P = 2 as above (40). Then PQ = -q, and since P = 2, Q = -q. In this case, before the trade, England produces for her own consumption a quantity q 28 of (X)) and a certain quantity of (C). During the trade she exports — Q of (C) with SOME DOCTRINES OF POLITICAL ECONOMY. 141 which she obtains — q of (D). The — Q of (C) is equal to — q of (C) which, in England, ID £ I £ I Q is produced by the same labour as -q of (D). Therefore, of the labour before employed in producing q of (D), - is liberated, and may be employed in producing (C) or any other com- modity : and this, along with the — q of (Z>), is the gain resulting to England from the trade. Before the trade, Germany produces for home consumption a quantity Q of (C) and a certain quantity of (D). During the trade she exports — q of (Z>), with which she obtains 15 98 ifi 28 — Q of (C). The — q of (Z>) is equal to — Q of (D) which, in Germany, costs the same 14 1 labour as — Q of (C). Therefore Germany obtains an addition of — in the amount of (C), and of so much labour as would produce — of the (C) originally consumed, which may be 15 employed in producing (D) or any other commodity. « 10 5 44. Let n = — ; therefore Q = -a. 9 9 In this case during the trade, England exports Q of (C) with which she obtains in 10 5 Germany 2 Q or — q of (D). The Q of (C) is equal to -q of (C) which in England is pro- 5 1 duced by the same labour as -q of (D). Hence England gains - q of (D) and gains as much labour as produced -q of (Z>), which is ready for any new employment. Germany 10 5 exports— q of (D), with which she obtains in England, -q of (C) that is, Q of (C). The y j 10 5 —q of (O) is produced with the same labour as -q of (C). Therefore she gains nothing by y y the trade. But still the trade will go on, for England offering (C) in Germany at the same price as (C) produced in Germany, this supply of (C) will be sold along with the rest ; and if not, a tem- porary depression of its price in relation to (Z>) will find it a market ; and then, by the principle of transferable capital, there will be a transfer of German capital from (C) to (D) till the whole quantity requisite to produce the — q of (D) is transferred. 6 3 In the same manner if n m -, Q = -q. Germany exports q of (D) with which she purchases 5 5 2 10 1 in England ~q of (C), which is — Q of (C). Therefore she has - more of (C) than she had 142 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 1 5 before. Also the q of (D) cost in Germany the same labour as -q of (C), that is, as -Q of (C). * o Therefore - of the labour at first employed in producing (C) is set free for any new 6 employment. England gains nothing, consuming, as before, q of (D) and a certain quantity of (C) : but 2 the labour which produced q of (£>) is transferred to the production of -q of (C) for the German customer. 6 5 5 5 45. If n be > -, as, if n = -, PQ = -7, Q = -9. In this case, Germany wants Q of (C) 5 4 4 8 3 which she can procure in England by exporting -Q of (D) instead of producing it at home where it would cost 2Q of {D). Hence she gains -Q of (D), and this may be expended on (C), (D), or any new commodity. 2 5 England exports Q of (C), which requires as much cost as -Q, that is, — a, 01(D). Hence the labour or cost which produced — q of (D) is transferred to produce (C), and produces -q of (C), in addition to the quantity required for home consumption. And the production of (D) S 15 in England is diminished by -Q or — q which is brought from Germany. 46. The case in which x or X is negative by the formulae, is the case in which the inequality of mutual demand is such that no relative value of C and D intermediate between that in Germany and that in England will satisfy the equation of import and export (37). In that case, the gain is all on one side, as in the case last examined. To determine this gain in general in such cases, we must consider that Germany wants Q Q of (C) which she can procure in England by exporting — of (D) instead of producing so much (C) at home, where it would cost PQ of (X)). Hence her gain is Q Pp-1 kQ PQ -3 = _£ Q = — of (J>), or kQ of (C), P. P P which may be spent in adding to (C) or to (D), or on any other commodity. 2 47. It appears by what has been said that in the case supposed, where p = -, P = 2, 3 (Mr. Mill's example), there is no possibility of a trade in which both countries shall gain, except 10 , ( n be between — and - 9 I England alone gains. 10 6 6 10 n be between — and -. If n be greater than - , Germany alone gains : if n be less than — SOME DOCTRINES OF POLITICAL ECONOMY. 143 2 This is on the supposition that m =■ -. In general, since x must be less than k, and _ l - n(l - Mk) ~ m-n{\ - M)' we must have 1 > n(l — Mk) in order that the numerator may be positive. And since .r — , and n < 1 - k ' l-Mk It « = -, m = -, this gives n > — and n < - as above. 4 3 & 9 5 11 8 7 48. If m = -, 3f = -. Then n > - and n < -. 2 2 7 6 In this case a?= — . Since n must be between - and -, that is, between — and — 4 - 4>n 7 6 84 84' 97 7 6 let n = —. Then x = —, w = — . 84 52 45 , 1 - raw I 1 \ . \ - Mx ( 2 \ Hence q' = q = 1 + - } g ; Q' = Q - 1 + - Q. 1-a; V 21/^ l-« V 39/ Both q and Q are increased by a fraction, and both countries gain. 8 . 7 If n be less than -, England alone gains : if n be greater than -, Germany alone gains. 49- If we have to use the formula x = — — - , w(l -M)~ 1 1 - mk 1 we must have n < - — — , and n > l - k ' l-Mk 50. Let now England produce another exportable commodity (£), and let the price of E, a unit of (£), be = rC in England. Let Germany produce another exportable commodity (F), and let the price of F be i?X) in Germany ; and let S be the quantity of (E) required in Germany, and s the quantity of (F) required in England at the original prices. It is required to find the amount and prices of the imports and exports. 51. Let the trade be established: let (E) be exported and (F) imported by England. In this case, when p becomes p = p (l — x), and P becomes P' — P(l — X), let r become r, and R become i?', * become s', and S become S'. If England be entirely supplied with (F) by Germany, and Germany entirely supplied with (E) by England, the import of England is now q of (D) and s' of (F); and the export of England is now Q' of (C) and & of (E). 144 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 52. But we must introduce an additional consideration. The relative price of (C) and (E), both of which are produced in England, will depend on their relative cost in England, and will not be altered by the trade. And, in like manner, the relative price of (D) and (F), which are produced in Germany, will depend upon their relative cost in Germany, and will not be altered by the trade. This we may term the principle of the relation of cost in the pro- ducing country. Hence the ratio of r to p' is the same as the ratio of r to p. Let r = yp, then r' = yp'. In the same way if R = TP, R 1 = YP". And hence j? is the same for p' and r' ; X, for P' and R'. 53. The quantity s of (F) is equivalent, in Germany, to IV of (D) : hence the whole import of England is equivalent to q + TV of (D), which in terms of (C) is p'q' + Tp's'. The export of England is Q' of (C) and & of (F), equivalent to Q' + yS' of (C). Hence the equa- tion of import and export, (37), becomes p'q' + Fp's' = Q' + 7-S^. In like manner the equation of import and export for Germany is q + Ts' = P'Q' + yP'S'. Hence, if the equation of supply and demand for (E) and (F) (22) be respectively, r's' = rs (1 - nx), R'S' = RS (X - NX), we shall have q 1 ~ mx + rs 1 — n - = PQ(l - MX) + yPS(\ - NX). But we have, as before, under the trade, by the uniformity of international prices, (36) P'p' = 1. And if. as before, — — = 1 - k, Pp (l - X) (1 - w) m l - *, whence X = . 1 — 00 Hence q(\-mx) + T*(l -m)= PQ{l - Mk-(\ - M)x\ + y PS\l - NK - (l -N)»\, W hence . = j_ + F« - «l - JfAQ PQ + (1 - Mfry iff mq + nrs - { (l - M)PQ + 7 (l - N)PS\ If we suppose that n = m and N = M, this becomes q + Ts - (l - Mk)P(Q + ys) m(q+ Vs) - (1 - M) P(Q + ys) ' And if P(Q + ys) = n'(q+ Ts), 1 -» w'(l - il/A) W = m-»'(l - it^) ' as before (39). And the applicable values of oo will occur only for values of n' between certain limits, as before. That is, the trade will be profitable to both countries only when the inequality of mutual demand is within certain moderate limits. / SOME DOCTRINES OF POLITICAL ECONOMY. 145 54. It is evident that if we had any other commodities (£,) (E 2 ) exported by England, and any other commodities (Pi) (P 2 ) exported by Germany, we should have equations of the same form, to any number of terms. 55. If Pp be considerably greater than l (35), the range of values of n within which the trade is possible with mutual advantage will be wider. Thus, let a yard of cloth (C) exchange for l bushel of corn (Z>) in England; but for 113 1 4 bushels in Poland. Here p = 1, P = 4. Pp = 4 ; .\ — = - , k = - . Let m = M = - . Pp 4 4 2 , , 1 . 1 - mk , x , , 8,5 Hence «, (3Q), which must be between — - and — , (48), must be between - and - . v ' 1 - Mk 1 - k 5 2 ■I h. (7? 1 56. For instance, if n = 2, w = -, whence X = = - ; ' 2 1 - a? 2 and (38) q = , T — - = -,, Q= T — - = i Q, p - ^ P' - ^ Here, since PQ = nq, and P = 4, « <= 2, 2Q = §. In this state of the trade, a yard of cloth will exchange for 2 bushels of corn. Poland will produce and export -q of corn, besides her own consumption. England will produce and 3 3 export -Q of cloth, which is -q, besides her own consumption. And in this case England has half as much more corn, Poland half as much more cloth, as before. 57. But this mutual advantage arises from the mutual demand being nearly equal. Poland is supposed to demand, at first, half as many yards of cloth as England demands bushels of corn, for Q = -q. 2 If this be otherwise : if, for example, Poland demand at first only one-fourth as many yards of cloth as England demands bushels of corn, or Q = -q : since P = 4, PQ = q ; there- 4 fore n = 1. In this case I supposing m and M each = - ) by (38) q' = q = q , Eng- land's import, also P'Q' = England's export ; k _ x 3 — 4# 5 — 4* 5 — 4a? and since X =- = , P'Q' = PQ (l - MX) = PQ - q . !_# 4 - 4a? 8 - 8a? *8-8a? And the export will be less than the import for every positive value of a? ; that is, there cannot be any fall of the price of corn, with relation to cloth in England, so small as to make the imports and exports equal. In this case the relative price of cloth and corn in England w^U under the trade be ulti- mately what it was before the trade ; and therefore ultimately, the relation of the prices will Vol. IX. Paet I. 19 146 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF also be the same in Poland. When this condition has been attained Poland will obtain the -q yards of cloth which she requires, by exporting -q bushels of corn, which will purchase -q yards in England, though the cloth would have cost q bushels in Poland. Poland saves -q bushels of corn : England gains nothing. 4 58. Since England gains nothing by the trade, why does she carry on the trade ? Because otherwise the Polish corn would be sold in England at a lower price than p = 1, (a bushel for a yard of cloth). And thus, capital and labour in England would be driven from corn growing to cloth making. When this transfer has begun, it will go on till capital has been transferred enough to produce the -q yards of cloth which Poland wants, and then p will 4 a^ain become 1. There is a temporary depression of the price of corn, but the price of cloth is brought down to agree with it by the transfer of producing capital. The depression of the price of corn relative to cloth is only temporary, and has no permanent value which satis- fies the equations of international demand and supply. In this case England imports only part of her supply of corn, namely, -q, the remaining 4 — are produced at home. 59- Upon the principles here laid down, can a country lose by foreign trade ? Upon the principles here laid down, a country cannot lose by foreign trade. But among the principles here laid down is (44) the principle of transferable capital ; that is, the principle that capital and labour can be transferred, without loss, from the pro- duction of one commodity to another, when the state of trade produces, or threatens, a diminution of profits in any branch of production. This can never be exactly the case. In almost all transfer of capital there is loss. But the loss may be temporary: the gain, or saving of further loss is supposed to be enduring; therefore, on the assumption here made, there may still be a general saving in all foreign trade. 60. But if it be not true that capital and labour are transferable without enduring loss, the results here obtained are vitiated by the failure of that part of the foundation. If when a portion of a home-produced commodity is displaced in the home market by importation from a foreign country, the capital and labour thus set free cannot be employed in producing any other commodity for which there is a demand, the assumptions of our investigation fail. If, for instance, the labourers displaced become paupers, and have to be supported by the country in which the displacement occurs, there may be loss to the country arising from the trade. 61. We have hitherto left out of consideration the cost of carriage of the exported and imported commoditiea: what will be the result of introducing this element into the calculation ? SOME DOCTRINES OF POLITICAL ECONOMY. 147 In this case, the two commodities (C) and (Z>) (35) will no longer exchange at precisely the same rate in the two countries. The commodity (D) which has to be carried to England, will be dearer there by its cost of carriage ; and (C) will be dearer in Germany by the cost of carrying it from England. Suppose that the cost of carrying a unit C to Germany is a fraction u of the price of C, and the cost of bringing a unit D to England is a fraction U of the price of D- The price of C in Germany is P'D, if D be the price of a unit of (Z>). Therefore the PD price of a unit C in England is . Also the price of a unit D in England isD(l + U). 1 + u But the price of C in England is to the price of D as 1 to p. Therefore Pp'D i + u = D (1 + U) and P'p m (1 + U) (l + y). This equation takes the place of the equation of the uniformity of international prices, P'p' = 1, (36) in the former investigation. 62. If, as before, p' - P (l - *), P - P (l - X), and if we put (1 + U) (1 + u) Pp we shall have, as before, (39) the equations, 1 _ n (i _ M k') l-k\ m-n(l - M ) (1 - X) (l - a?) = 1 - k'. 63. There will be no export of C, and import of D, except (1 + U)(l+u) Pp <1. 13 1 Thus, if, as before, (40), — - = - , but u and U each = - , Pp 4 6 (1 + U) (1 + u) 49 Pp " 48 ' and there will be no importation. 64. This may be illustrated by a numerical example. Let the commodities be cloth and linen. Since u = - , U = - , for each 6 yards of C (cloth) manufactured in England, one yard must be manufactured, or the cost of one yard supplied, to provide for the expense of carriage to Germany : that is, the cost of 7 yards must be incurred to take 6 to Germany. In like manner the cost of 7 yards of D (linen) must be provided in order to take 6 to England. Hence 98 (= 2 x 7 x 7) yards of cloth in England which is (on our former supposition) equivalent to 147 (=3x7x7) yards of linen, being exported to Germany, is reduced to 84 (= 2 x 6 x 7) yards by the expense of carriage. And this in Germany exchanges for 19—2 148 Dr. WHEWELL, ON THE MATHEMATICAL EXPOSITION OF 168 (=4x6x7) yards of linen. But this quantity is reduced to 144 (=4x6x6) by the expense of carriage to England. Hence by exporting cloth and importing linen, England would only obtain 144 yards, instead of 147 which she could obtain by the same labour employed on linen at home. Hence, though the price of linen as compared with cloth is only - as great in Germany as 4 in England, the expense of carriage prevents the trade being profitable. 65. In Mr. Mill's example (Polit. Econ. 11. 125) the cost of carriage of each, Cand D, is supposed equal, nearly, to — of the price of D ; which is equal, nearly, to — of the price of C, or nearly to — . Hence u = — , U = — . '....../ 29 29 17 (1 + C7)(1+m) 30x18x3 1620 1 = 1 , nearly. Pp 29x17x4 1772 5 2 1 = - , and making k = - , 3 5 5 l - n (1 - Mk') 15 - 13ra 2 , 1 And taking, as in (40), m = M = - , and making k = - , m -n{\ - M) 10- 5n' In order that there may be an applicable value of x, we must have (47) n between 15 , 13 — and — . 13 12 If n be — , which is between those limits, 5 1 x = — = -. 40 8 Also for X, (1 - X) (1 - x) = 1 - k' = 1 - - ; 5 v * 8 32 j v 3 whence 1 — AT = -x- = — , and X = — . 5 7 35 35 7 7 2 The relative price of D and C in England is p' - p - - — , for p = - f (40). S 1 ~ o 32 64 13 The relative price of C and D in Germany isP^P— = — = — nearly, for P = 2, (40). In England 7 yards of cloth exchange for 12 of linen ; in Germany 7 yards of cloth for 13 of linen nearly ; the difference depends on the cost of carriage both ways. 66. In the preceding investigations, I have done little more than put into a general and algebraical form the reasonings which Mr. Mill has presented to his readers in numerical examples. But I think it will have appeared that by this mode of dealing with the subject, the limits of the truth of the theorems of the solutions of the problems are much more easily brought into SOME DOCTRINES OF POLITICAL ECONOMY. 149 view, at least for the mathematical reader, than in the numerical way. It has been seen, for instance, (57), that if there be a trade between any two countries, say Poland and England, Poland exporting, we will suppose, corn, and England, cloth, the case in which the trade is advan- tageous to both countries is (on the suppositions made) confined within narrow limits, namely, when the mutual demand is very nearly equal : and that if the demand of Poland for our commodity, cloth, be much smaller than our demand for theirs, corn, the advantage, according to the mathematical principles here applied, rests entirely with Poland. 67. But there is another consideration to be attended to. All the principles here applied are principles of equilibrium : that is, principles depending on the assumption of an enduring state of equality on the two sides : such are the principles of uniformity of international prices (36), the equality of import and export (37), transferable capital (44). These principles are each expressed by an equation, which enters into the investigation. But in fact, we never can have an absolute equilibrium. Even if the principles just mentioned be true, they exist as tendencies only, not as conditions attained : just as the principle that water finds its level governs the state of fluids, (rivers and oceans) not by producing equilibrium, but by producing motion, the level never being attained. The equilibrium theory of the tides is highly imperfect, and in many respects erroneous : the equilibrium theory of trade is probably no less so. Yet the equilibrium theory of the tides has been very useful in suggesting the form and laws of many of the principal features of tidal phenomena ; though it is absolutely necessary to obtain the amount and epoch of the inequalities by local observations. It is possible that, in like manner, the equilibrium theory of trade may suggest propositions respecting the natural progress of trade, which propositions may be afterwards confirmed, limited, and reduced to their due measure by experience. Trinity College, April 3, 1850. VI. Second Memoir on the Intrinsic Equation of a Curve and its Application. By W. Whewell, D.D., Master of Trinity College, Cambridge. [Read April 15, 1850.] 38. The method of deducing properties of curves which I employed in a former memoir on the Intrinsic Equation of Curves* is capable of many other applications. I the more readily proceed to point out some of these, in consequence of remarks which have been made to me upon the former memoir. Scruples have been entertained by some of my readers as to whether we rightly suppose the portion of curve, added after reaching a cusp, to be negative. It has been said that every cusp may be conceived to be the remnant left by a loop when the breadth of the loop vanishes ; and as in a looped curve the increment of the length of the arc could nowhere become negative, it ought not to do so in the ultimate form of the looped curve. 39- In order to see whether this remark is generally true, let the intrinsic equation to a curve be s = ad> + b sin 0. Therefore — = a + b cos b, the curve is looped, and the radius of curvature never vanishes. In this example the loop arises from the evanescence of two cusps ; and if we take the order of changes inversely, the evanescence of the loop gives rise to two cusps. In the curve s = ad) + 6 sin 0, the smaller sinus between two cusps is negative, and when a vanishes, the smaller and larger sinus become equal, which is the case of the cycloid ; and hence in that case the alternate sinuses are negative. 42. The cycloid may however be considered as the ultimate form of a looped curve, namely, of the protracted cycloid. If while the circle VB, fig. 5, rolls upon the straight line DB, a point P in the radius QC produced describe a curve AP, this curve is the protracted cycloid. To find the intrinsic equation to the protracted cycloid, let DA be the original position of QP; PM, and CE, perpendicular on AD; AM = a, MP = y, VCP = 9. And let CP = a, CB = ma. Then x = AM = EA - EM = CP - CN = a - a cos 9, y = MP = MN + NP = DB + NP=BQ+NP = ma9 + a sin 9. Also the curve at P is perpendicular to BP : and the curve at A is perpendicular to DA. Hence the angle between BP and DA, that is, angle PBN = (p. wu NP s ™ e W hence tan d> = = (1) r BN m + cos 9 w We have dm = a sin 9 . d9, dy = a (cos 9 + m) d9 : whence ds = y/(da? + dtf) m ad9^/(\ + wi 8 + 2m cos 9). ... (2) Also, from (l), sec 2 = ( ^ + mcos ^ dd T T (m + cos 9y i » , o , 1 + m 2 + 2mcos9 . , ^ but sec' m=l+ tan - * d> = — — : by (1) T T (m + cos 9y d9 l + m s -+2mcos# whence — - = -p. . a0 1 + mcost/ 152 Dr. whewells second memoir on the intrinsic equation ds And before — - = a W (l+m 2 + 2m cos 8), ad . ds a (1 + m 2 + 2 m cos 9)% whence — = ; . . . (3) d(p 1 + m cost) which is the intrinsic (differential) equation to the protracted cycloid. To find s in terms of 1. The curve becomes the common cycloid when m = 1, and the elliptical integration changes its nature for this case. OF A CURVE AND ITS APPLICATION. 153 45. I will take some other cases of the intrinsic equation. ds Let — = a sin + 6 sin 20 = a sin

. = /(a sin cos + 26 sin cos 2 0) d0. = C COS^ COS d G> 2 r 3 r a , 26 = - (1 — cos 2 0) + — (1 — cos 3 0). 46 When = tt, this = — = MC. Hence AM depends on a alone, and MC on 6 alone. ds 46. Again, let — — - = a sin + 6 sin 3 = a sin + 6 (3 sin - 4 sin 3 0) = {a + 36) sin 0—46 sin 3 0. There is a cusp when sin = 0, and when sin 2 = — , '46 dy = ds sin = (a + 36) sin 2 - 46 sin 4 0, dx = ds cos cp = (a + 36) sin cos0 - 46 sin 3 cos 0, (a + 3 6) . , X = sin 2 — 6 sin 4 0. When = - , # = : when = ir, * = 0. Form as in figure 8. 47. All such curves as have equations of these forms may be called generally cycloidal curves. They have cusps and sinuses which, after going through a certain cycle, recur. Vol. IX. Part I. 20 154 Dr. WHEWELL'S SECOND MEMOIR ON THE INTRINSIC EQUATION 48. If we have s = any series of integral powers of 0, we shall still have a curve of this cycloidal kind, as to its finite part. Thus let s = any integral function of [n + 1) dimensions of (p. ds Then — = (0 - a) (0 - /3) (0 - 7) &c. to n terms. And there is a cusp when = a, when = /3, when = y : And in the sinuses, s is alternately positive and negative. With regard to the infinite branches, ds If s = a

n dm And it is evident that if n be positive, - — goes on increasing as increases, that is, as s increases. Hence there is an infinite diverging spiral Therefore if s = any integral fraction of decreases, that is, as s increases; wherefore we have a diverging infinite spiral. • When n is negative and n + 1 positive, n is a negative proper fraction, and n + 1 a positive proper fraction : and if we get rid of fractional indices by involution, we shall have (p = some integral power of s ; which /equation, it appears, gives a converging spiral. Let (p = any series of integral powers of s. Then we shall have (p = (s - a) (s - b) (« - c) &c.

dd> , (b , /7^r^ = ^ +v/m, -^ 2( ^ +c ^ +jD ^ 5+&c - ); OF A CURVE AND ITS APPLICATION. 155 cos + Cd> 3 + J>0 5 + &c.) VV- s s/m'-cj) 2 \/»» ! -f + s/rrf-q? (B + 3 C

! + 5Z>m 2 4 + 7Em*(f) 6 + &c. - (50* + SC0 4 + 5Z>0 6 + &c.) 2 d>* 6 But cos = 1 - -2— + J- £■ . r 1.2 1.2.3.4 1.2.3.4.5.6 Hence equating coefficients; A + m*B=l, 2J5-3Cm 2 = , 4C-5Z>m 1! = , 6D-7Em ! = 1.2 1.2.3.4 1.2.3.4.5.6 1 lEm* Hence D = — - + — -— : 1.2.3.4.5.6.6 6 1 5Dm? 1 m» 5.7. Em* "~ 1.2.3.4.4 4 1.2.3.4.4 1. 2.3.4 s . 6 s 4.6™ 1 3Cm 2 _ 1 »re 2 m 4 3. 5. 7 Em? " 1 .2.2 2 " 1.8.. « ~ 1 . 2 8 . 4 2 + 1.2*. 4 s . 6* 2.4.6 = to 2 m 4 m 6 Taking the integral from = to (p = m, we have it = J. 51. In the same manner r sin (bdd> w, . J / g _ ^ 2 " a sin-1 ~ + V «* 2 - 2 (6 + c0 2 + d0 4 + e0 6 + &c.) sin a (h —/ —— - / - / (6 + c0 2 +d!0 4 +&c.) \/V-0 2 v/m 2 -0 2 ^/rri'-d? + \/»» 2 - 2 (2c0 + 4d0 3 + 6e0 5 + &c.) Hence sin ■» a - (60 + c0 3 + d0 5 + &c.) + 2cm* + 4dra 2 3 +6em 2 5 + &c. - (2c0 3 + 4d0 5 + &c.) But sin = - — + 2. &c. r ^ 1.2.3 1.2.3.4.5 Hence equating coefficients a = 0; 6 - 2cm 2 = 1 ; 3c-4d.wj 2 = ; 5d - 6em s «= ; &c. 1.2.3 1 .2.3.4.5 1 6em" Hence a = + 1.2. 3.4.5.5 5 20—2 156 Dr. WHEWELL'S SECOND MEMOIR ON THE INTRINSIC EQUATION, ETC. 1 4,dm 2 1 m 2 4.6.m 4 e 2 ° = 1 . 2 . 3 . 3 + 3 ~ 1 . 2 . 3* ~ 1 . 2 . 3* . 5* 3.5 ' m° m* Taking the integral from

3 — m w) d w, where m is pro- •'o 2 portional to the perpendicular distance of the point considered from the caustic, and is reckoned positive towards the illuminated side. Mr. Airy has also given a table of the numerical values of the above integral extending from m = — 4> to m = + 4, at intervals of 0.2, which was cal- culated by the method of quadratures. In a Supplement to the same paper -j- the table has been re-calculated by means of a series according to ascending powers of m, and extended to m = ± 5.6. The series is convergent for all values of m, however great, but when m is at all large the calculation becomes exceedingly laborious. Thus, for the latter part of the table Mr. Airy was obliged to employ 10-figure logarithms, and even these were not sufficient for carrying the table further. Yet this table gives only the first two roots of the equation W= 0, W denoting the definite integral, which answer to the theoretical places of the first two dark bands in a system of spurious rainbows, whereas Professor Miller was able to observe 30 of these bands. To attempt the computation of 30 roots of the equation W = by means of the ascending series would be quite out of the question, on account of the enormous length to which the numerical calculation would run. After many trials I at last succeeded in putting Mr. Airy's integral under a form from which its numerical value can be calculated with extreme facility when m is large, whether positive or negative, or even moderately large. Moreover the form of the expression points out, without any numerical calculation, the law of the progress of the function when m is large. It is very easy to deduce from this expression a formula which gives the i th root of the equation W = with hardly any numerical calculation, except what arises from merely (Tn\ i passing from I — J , the quantity given immediately, to m itself. The ascending series in which W may be developed belongs to a class of series which are of constant occurrence in physical questions. These series, like the expansions of e~*, sin at, cos a?, are convergent for all values of the variable at, however great, and are easily calculated numerically when