liii iir I mv<^ CORNELL UNIVERSITY LIBRARIES Mathematict Library White Hall CORNELL UNIVERSITY LIBRARY 924 059 551 386 DATE DUE ^H^^X •^ /(tlo^ ^-^r/ncu 'AJ'in ^ ^ '996 ■Ilk I ilUN ' !lTO ocro6 'MS lilUJ^ CAYLORO PRINTED INU.S.A. Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059551386 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the commission on Preservation and Access and the Xerox Corporation. 1991. BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Sienrg IB. Sage 1891 .A.,:^.U9. "'"'^Al/^^^ UTILITY OF QUATEENIONS IN PHYSICS. UTILITY OF QUATEENIONS IN PHYSICS BY A. M^AULAY, M.A., LECTURER IN MATHEMATICS AND PHYSICS IN THE nNIVEEBITT OF TASMANIA. Hontion : MACMILLAN AND CO. AND NEW YOBK. 1893 [All Rights reserved.] A. 5'ZL 2_Q .^j*^' Ci-" CBmbtiligf : PRINTED BY C. J. CLAT, M.A., AND SONS, AT THE UNIVKSSm PBE88. PEEFACE. The present publication is an essay that was sent in (De- cember, 1887) to compete for the Smith's Prizes at Cambridge. To the onlooker it is always a mournful thing to see what he considers splendid abilities or opportunities wasted for lack of knowledge of some paltry common-place truth. Such is in the main my feeling when considering the neglect of the study of Quaternions by that wonderful corporation the University of Cam- bridge. To the alumnus she is apt to appear as the leader in all branches of Mathematics. To the outsider she appears rather as the leader in Applied Mathematics and as a ready welcomer of other branches. If Quaternions were simply a branch of Pure Mathematics we could understand why the study of them was almost confined to the University which gave birth to them, but as the truth is quite otherwise it is hard to shew good reason why they have not struck root also in Cambridge. The prophet on whom Hamilton's mantle has fallen is more than a mathematician and more than a natural philosopher — he is both, and it is to be noted also that he is a Cambridge man. He has preached in season and out of season (if that were possible) that Quaternions are especially useful in Physical applications. Why then has his Alma Mater turned a deaf ear ? I cannot believe that she is in her dotage and has lost her hearing. The problem is beyond me and I give it up. But I wish to add my little efforts to Prof Tait's powerful advocacy to bring about another state of affairs. Cambridge is the prepared ground on which if anywhere the study of the Physical applications of Quaternions ought to flourish. VI PREFACE. When I sent in the essay I had a faint misgiving that per- chance there was not a single man in Cambridge who could understand it without much labour — and yet it is a straightfor- ward application of Hamilton's principles. I cannot say what transformation scene has taken place in the five years that have elapsed, but an encouraging fact is that one professor at any rate has been imported from Dublin. There is no lack in Cambridge of the cultivation of Quater- luons as an algebra, but this cultivation is not Hamiltonian, though an evidence of the great fecundity of Hamilton's work. Hamilton looked upon Quaternions as a geometrical method, and it is in this respect that he has as yet failed to find worthy followers resident in Cambridge. [The chapter contributed by Prof. Cayley to Prof Tait's 3rd ed. of 'Quaternions' deals with quite a different subject from the rest of the treatise, a subject that deserves a distinctive name, say, Cayleyan Quaternions.] I have delayed for a considerable time the present publication in order at the last if possible to make it more effective. I have waited till I could by a more striking example than any in the essay shew the immense utility of Quaternions in the regions in which I believe them to be especially powerful. This I believe has been done in the 'Phil. Trans.' 1892, p. 685. Certainly on two occasions copious extracts have been published, viz. in the P. R. S. K, 1890-1, p. 98, and in the ' Phil. Mag.' June 1892, p. 477, but the reasons are simple. The first was published after the subject of the ' Phil. Trans.' paper had been considered sufficiently to afford clear daylight ahead in that direction, and the second after that paper had actually been despatched for publication. At the time of writing the essay I possessed little more than faith in the potentiality of Quaternions, and I felt that something more than faith was needed to convince scientists. It was thought that rather than publish in driblets it were better to wait for a more copious shower on the principle that a well-directed heavy blow is more effective than a long-continued series of little pushes. Perhaps more harm has been done than by any other cause to the study of Quaternions in their Physical applications by a silly superstition with which the nurses of Cambridge are wont to PREFACE. vii frighten their too timorous charges. This is the behef that the subject of Quaternions is difficult. It is difficult in one sense and in no other, in that sense in which the subject of analytical conies is difficult to the schoolboy, in the sense in which every subject is difficult whose fundamental ideas and methods are different from any the student has hitherto been introduced to. The only way to convince the nurses that Quaternions form a healthy diet for the young mathematician is to prove to them that they will " pay" in the first part of the Tripos. Of course this is an impos- sible task while the only questions set in the Tripos on the subject are in the second part and average one in two years. [This solitary biennial question is rarely if ever anything but an exercise in algebra. The very form in which candidates are invited, or at any rate were in my day, to study Quaternions is an insult to the memory of Hamilton. The monstrosity " Quaternions and other non-commutative algebras" can only be paralleled by " Cartesian Geometry and other commutative algebras." When I was in Cam- bridge it was currently reported that if an answer to a Mathe- matical Tripos question were couched in Hebrew the candidate would or would not get credit for the answer according as one or more of the examiners did or did not understand Hebrew, and that in this respect Hebrew or Quaternions were strictly analogous.] Is it hopeless to appeal to the charges ? I will try. Let me suppose that some budding Cambridge Mathematician has fol- lowed me so far. I now address myself to him. Have you ever felt a joy in Mathematics ? Probably you have, but it was before your schoolmasters had found you out and resolved to fashion you into an examinee. Even now you occasionally have feelings like the dimly remembered ones. Now and then you forget that you are nerving yourself for that Juggernaut the Tripos. Let me implore you as though your soul's salvation depended on it to let these trances run their utmost course in spite of solemn warnings from your nurse. You will in time be rewarded by a soul-thrilling dream whose subject is the Universe and whose organ to look upon the Universe withal is the sense called Quaternions. Steep yourself in the delirious pleasures. When you wake you will have forgotten the Tripos and in the fulness of time will develop into a Vlll PKEFACE. financial wreck, but in possession of the memory of that heaven- sent dream you will be a far happier and richer man than the millionest millionaire. To pass to earth— from the few papers I have published it will be evident that the subject treated of here is one I have very much at heart, and I think that the publication of the essay is likely to conduce to an acceptance of the view that it is now the duty of mathematical physicists to study Qituternions seriously. I have been told by more than one of the few who have read some of my papers that they prove rather stiff reading. The reasons for this are not in the papers I believe but in matters which have already been indicated. Now the present essay reproduces the order in which the subject was developed in my own mind. The less complete treatment of a subject, especially if more diffuse, is often easier to follow than the finished product. It is therefore probable that the present essay is likely to prove more easy reading than my other papers. Moreover I wish it to be studied by a class of readers who are not in the habit of consulting the proceedings, &c., of learned societies. I want the slaves of examination to be arrested and to read, for it is apparently to the rising generation that we must look to wipe off the blot from the escutcheon of Cambridge. And now as to the essay itself. But one real alteration has been made. A passage has been suppressed in which were made uncomplimentary remarks concerning a certain author for what the writer regards as his abuse of Quaternion methods. The author in question would no doubt have been perfectly well able to take care of himself, so that perhaps there was no very good reason for suppressing the passage as it still represents my con- victions, but I did not want a side issue to be raised that would serve to distract attention from the main one. To bring the notation into harmony with my later papers dv and V which occur in the manuscript have been changed throughout to dS and A respectively. To facilitate printing the solidus has been freely introduced and the vinculum abjured. Mere slips of the pen have been corrected. A formal prefatory note required by the conditions of competition has been omitted. The Table of Con- tents was not prefixed to the original essay. It consists of little PREFACE. IX more than a collection of the headings scattered through the essay. Several notes have been added, all indicated by square brackets and the date (1892 or 1893). Otherwise the essay remains absolutely unaltered. The name originally given to the essay is at the head of p. 1 below. The name on the title-page is adopted to prevent confusion of the essay with the 'Phil. Mag.' paper referred to above. What in the peculiar caligraphy of the manuscript was meant for the familiar ///()rfs has been consistently rendered by the printer as JJJ()d%. As the mental operation of substituting the former for the latter is not laborious I have not thought it necessary to make the requisite extensive alterations in the proofe. I wish here to express my great indebtedness to Prof Tait, not only for having through his published works given me such knowledge of Quaternions as I possess but for giving me private encouragement at a time I sorely needed it. There was a time when I felt tempted to throw my convictions to the winds and follow the line of least resistance. To break down the solid and well-nigh universal scepticism as to the utility of Quaternions in Physics seemed too much like casting one's pearls — at least like crying in the wilderness. But though I recognise that I am fighting under Prof. Tait's banner, yet, as every subaltern could have conducted a campaign better than his general, so in some details I feel compelled to differ from Professor Tait. Some two or three years ago he was good enough to read the present essay. He somewhat severely criticised certain points but did not convince me on all. Among other things he pointed out that I sprung on the unsuspicious reader without due warning and explanation what may be considered as a peculiarity in symbolisation. I take this opportunity therefore of remedying the omission. In Quaternions on account of the non-commutative nature of multiplication we have not the same unlimited choice of order of the terms in a product as we have in ordinary algebra, and the same is true of certain quaternion operators. It is thus inconvenient in many cases to use the familiar method of indicating the connection between an operator and its operand by placing the former im- X PREFACE. mediately before the latter. Another method is adopted. With this other method the operator may be separated from the operand, but it seems that there has been a tacit convention among users of this method that the separated operator is still to be restricted to precedence of the operand. There is of course nothing in the nature of things why this should be so, though its violation may seem a trifle strange at first, just as the tyro in Latin is puzzled by the unexpected comers of a sentence in which adjectives (operators) and their nouns (operands) turn up. Indeed a Roman may be said to have anticipated in every detail the method of indicating the connection now under discussion, for he did so by the similarity of the sufiixes of his operators and operands. In this essay his example is followed and therefore no restrictions except such as result from the genius of the language (the laws of Quaternions) are placed on the relative positions in a product of operators and operands. With this warning the reader ought to find no difficulty. One of Prof. Tait's criticisms already alluded to appears in the third edition of his ' Quaternions.' The process held up in § 500 of this edition as an example of " how not to do it " is contained in § 6 below and was first given in the ' Mess, of Math.,' 1884. He implies that the process is a " most intensely artificial applica- tion of" Quaternions. If this were true I should consider it a perfectly legitimate criticism, but I hold that it is the exact reverse of the truth. In the course of Physical investigations certain volume integrals are found to be capable of, or by general considerations are obviously capable of transformation into surface integrals. We are led to seek for the correct expression in the latter form. Starting from this we can by a long, and in my opinion, tedious process arrive at the most general type of volume integral which is capable of transformation into a surface integral. [I may remark in passing that Prof. Tait did not however arrive at quite the most general type.J Does it follow that this is the most natural course of procedure ? Certainly not, as I think. It would be the most natural course for the empiricist, but not for the scientist. When he has been introduced to one or two volume integrals capable of the transformation the natural course of the mathematician is to ask himself what is the most general volume PREFACE. XI integral of the kind. By quite elementary considerations he sees that while only such volume integrals as satisfy certain conditions are transformable into surface integrals, yet any surface integral which is continuous and applies to the complete boundary of any finite volume can be expressed as a volume integral throughout that volume. He is thus led to start from the surface integral and deduces by the briefest of processes the most general volume integral of the type required. Needless to say, when giving his demonstration he does not bare his soul in this way. He thinks rightly that any mathematician can at once divine the exact road he has followed. Where is the artificiality ? Let me in conclusion say that even now I scarcely dare state what I believe to be the proper place of Quaternions in a Physical education, for fear my statements be regarded as the uninspired babblings of a misdirected enthusiast, but I cannot refrain from saying that I look forward to the time when Quaternions will appear in every Physical text-book that assumes the knowledge of (say) elementary plane trigonometry. I am much indebted to Mr G. H. A. Wilson of Clare College, Cambridge, for helping me in the revision of the proofs, and take this opportunity of thanking him for the time and trouble he has devoted to the work. ALEX. McAULAY. University of Tasmania, HOBART. March 26, 1893. CONTENTS. Section I. Introduction. PAOE General remarks on the place of Quaternions in Physics . . 1 Cartesian form of some of the results to follow . 6 Section II. Quaternion Theorems. 1. Definitions . . 12 2. Properties of f . . 15 5. Fundamental property of a . ... 18 6. Theorems in Integration . . . 19 9. Potentials 22 Section III. Elastic Solids. 11. Brief recapitulation of previous work in this branch .... 25 12. Strain, stress-force, stress-couple .26 14. Stress in terms of strain . . 27 16. The equations of equilibrium 32 16a. Variation of temperature . 35 17. Small strains 37 20. Isotropic bodies .40 22. Particular integral of the equation of equilibrium ... 42 24. Orthogonal coordinates . . ... 44 27. Saiut-Veuant's torsion problem .... 46 29. Wires .49 Section IV. Electricity and Magnetism. 34. Electrostatics — general problem .... . . 55 41. The force in particular cases 63 43. Nature of the stress . . 65 46. Magnetism — magnetic potential, force, induction ... .67 49. Magnetic solenoids and shells 69 54. Electbo-maonetism— general theory ... ... 71 60. Electro-magnetic stress 74 XIV CONTENTS. Skhtion V. Hydrodynamics. 61. Preliminary 62. Notation ... 63. Euler's equations . . .... 68. The Lagrangian equations 69. Canchy's integrals of these equations 71. Flow, circulation , vortex-motion 74. Irrotational motion . ... 76. Motion of a solid through a liquid 79. The velocity in terms of the convergences and spins 83. Viscosity PAGE 76 76 77 80 81 82 84 8.5 88 91 Section VI. The Vortex-atom Theory. 85. PreUminary . .... . . .94 86. Statement of Sir Wm. Thomson's and Prof. Hicks's theories . . 94 87. General considerations concerning these theories . . 95 88. Description of the method here adopted .... . . 96 89. Acceleration in terms of the convergences, their time-flnxes and the spins 97 91. Sir Wm. Thomson's theory 98 93. Prof. Hicks's theory . . 100 94. Consideration of all the terms except - v (ff")/2 . 101 96. Consideration of the term - v . (SVn.(iwA = 0. The Cartesian exact equivalent consists of Thomson and Tait's equations (7), viz. d {^dw /da. , ■,\ .^w da^ dw da\ dx \ dA \dx ) ~dh dz dc dy) d { dw da dw da dw fda dy \ dB dy da dz dc \dx d f „ dw da dw da dAi^ fda ,-.][_ r. '^d^\dCd^'^d~^d^db[di'^'^~' and two similar equations. 1—2 4 INTRODUCTION. Many of the equations indeed in the part of the essay where this occurs, although quite simple enough to be thoroughly useful in their present forai, lead to much more complicated equations than those just given when translated into Cartesian notation. It will thus be seen that there are two statements to make good : — (1) that Quaternions are in such a stage of development as already to justify the practically complete banishment of Carte- sian Geometry from Physical questions of a general nature, and (2) that Quaternions will in Physics produce many new results that cannot be produced by the rival and older theory. To establish completely the first of these propositions it would be necessary to go over all the ground covered by Mathematical Physical Theories, by means of our present subject, and compare the proofs with the ordinary ones. This of course is impossible in an essay. It would require a treatise of no small dimensions. But the principle can be followed to a small extent. I have there- fore taken three typical theories and applied Quaternions to most of the general propositions in them. The three subjects are those of Elastic Solids, with the thermodynamic considerations necessary, Electricity and Magnetism, and Hydrodynamics. It is impossible without greatly exceeding due limits of space to consider in addi- tion. Conduction of Heat, Acoustics, Physical Optics, and the Kinetic Theory of Gases. With the exception of the first of these subjects I do not profess even to have attempted hitherto the desired applications, but one would seem almost justified in argu- ing that, since Quaternions have been found so applicable to the subjects considered, they are very likely to prove useful to about the same extent in similar theories. Again, only in one of the subjects chosen, viz.. Hydrodynamics, have I given the whole of the general theory which usually appears in text-books. For instance, in Electricity and Magnetism I have not considered Elec- tric Conduction in three dimensions which, as Maxwell remarks, lends itself very readily to Quaternion treatment, nor Maonetic Induction, nor the Electro-Magnetic Theory of Light. Again, I have left out the consideration of Poynting's theories of Elec- tricity which are very beautifully treated by Quaternions, and I felt much tempted to introduce some considerations in connection with the Molecular Current theory of Magnetism. With similar reluctance I have been compelled to omit many applications in the Theory of Elastic Solids, but the already too large size of the essay admitted of no other course. Notwithstanding these omis- INTEODUCTION. 5 sions, I think that what I liave done in this part will go far to bear out the truth of the first proposition I have stated above. But it is the second that I would especially lay stress upon. In the first it is merely stated that Cartesian Geometry is an anti- quated machine that ought to be thrown aside to make room for modern improvements. But the second asserts that the improved machinery will not only do the work of the old better, but will also do much work that the old is quite incapable of doing at all. Should this be satisfactorily established and should Physicists in that case still refuse to have anything to do with Quaternions, they would place themselves in the position of the traditional workmen who so strongly objected to the introduction of ma- chinery to supplant manual labour. But in a few months and synchronously with the work T have already described, to arrive at a large number of new results is too much to expect even from such a subject as that now under dis- cussion. There are however some few such results to shew. I have endeavoured to advance each of the theories chosen in at least one direction. In the subject of Elastic Solids I have ex- pressed the stress in terms of the strain in the most general case, i.e. where the strain is not small, where the ordinary assumption of no stress-couple is not made and where no assumption is made as to homogeneity, isotropy, &c. I have also obtained the equations of motion when there is given an external force and couple per unit volume of the unstrained solid. These two problems, as will be seen, are by no means identical. In Electrostatics I have con- sidered the most general mechanical results flowing from Maxwell's theory, and their explanation by stress in the dielectric. These results are not known, as might be inferred from this mode of statement, for to solve the problem we require to know forty-two independent constants to express the properties of the dielectric at a given state of strain at each point. These are the six coefficients of specific inductive capacity and their thirty-six differential coeffi- cients with regard to the six coordinates of pure strain. But, as far as I am aware, only such particular cases of this have already been considered as make the forty-two constants reduce at most to three. In Hydrodynamics I have endeavoured to deduce certain general phenomena which would be exhibited by vortex-atoms acting upon one another. This has been done by examination of an equation which has not, I believe, been hitherto given. The result of this part of the essay is to lead to a presumption against 6 INTRODUCTION. Sir William Thomson's Vortex-Atom Theory and in favour of Hicks's. As one of the objects of this introduction is to give a bird's-eye view of the merits of Quaternions as opposed to Cartesian Geome- try, it will not be out of place to give side by side the Quaternion and the Cartesian forms of most of the new results I have been speaking about. It must be premised, as already hinted, that the usefulness of these results must be judged not by the Cartesian but by the Quaternion form. Elasticity. Let the point {x, y, z) of an elastic solid be displaced to {x, y, z). The strain at any point that is caused may be supposed due to a pure strain followed by a rotation. In Section III. below, this pure strain is called t/t. Let its coordinates be e, f, g, a/2, 6/2, c/2; i.e. if the vector (^, r], ^) becomes (^', rj', f) by means of the pure strain, then &c., &c. Thus when the strain is small e,f,g reduce to Thomson and Tait's 1 -I- e, 1 +f, 1+g and a, b, c are the same both in their case and the present one. Now let the coordinates of '^j § 16 below, be E, F, G, AI2, B/2, 0/2. Equation (15), § 16 below, viz. pves in our present notation E = e' + c'li + b'j4> = {dx'jdxf + {dy'ldxf + {d^ldxj, &c., &c. A=a{f+g) + be 12 = 2 {(dx'/dy) (dx'/dz) + (dy'/dy) (dy'/dz) + (dz'/dy)(dz'ldz)], &c., &c. which shew that the present E, F, G, A/2, B/2, C/2 are the A, B, C, a, b, c of Thomson and Tait's Nat. Phil., App. C. * This result is one of Tait's {Quaternions § 365 where he has '=w'). It is given here for completeness. INTRODUCTION. Let us put \x y zi y'z JP^ =/,., &c., &c., \yz J J (y,) = J.. J (f^) = J.V &C., &C., &C., &C. I have shewn in § 14 below that the stress-couple is quite independent of the strain. Thus we may considei* the stress to consist of two parts — an ordinary stress PQRSTU as in Thomson and Tait's Nat. Phil, and a stress which causes a couple per unit volume L'M'N'. The former only of these will depend on strain. The result of the two will be to cause a force (as indeed can be seen from the expressions in § 13 below) per unit area on the x- interface P, U+N"/2, T-M'/2, and so for the other interfaces. If L, M, N be the external couple per unit volume of the unstrained solid we shall have L' = - LI J, M' = - Ml J, N' = - NjJ, for the external couple and the stress- couple are always equal and opposite. Thus the force on the a;-interface becomes P, U-NI2J, T + MI-2J and similarly for the other interfaces. To express the part of the stress (P &c.) which depends on the strain in terms of that strain, consider w the potential energy per unit volume of the unstrained solid as a function of E &c. In the general thermodynamic case w may be defined by saying that w X (the element of volume) = (the intrinsic energy of the element) - (the entropy of the element x its absolute temperature X Joule's coefficient). Of course w may be, and indeed is in §§ 14, 15 below, regarded as a function of e &c. The equation for stress is (15 6) § 16 below, viz., J^a) = ^x^Qwx"' = 2p^Sp^coSV^t(jwV^. The second of the expressions is in terms of the strain and the third in terms of the displacement and its derivatives. In our present notation this last is INTRODUCTION. JP /dx'\''dw . [dx'\^dw fdx'\^dw fdxV dw 2 ~\dxl dE^ \dy) dF^ \dz ) dG dx' dx' dw dx' dx' dw dx' dx dw dy dz dA dz dx dB dx dy dC &c., &e. JS _ dy' dz' dw dy' dz dw dy' dz' dw ~¥~dxdx dE'^ dy dy dF'^ dz dz dG dy' dz dy dz"\ dw f dy' dz' dy' dz'\ dw dy dz dz dy) dA \ dz dx dx dz j dB idy' dz dy' dz'^^ dw \dx dy dy dx) dC &c., &c. In § 14 I also obtain this part of the stress explicitly in terms of e, f, g, a, b, c, of w as a function of these quantities and of the axis and amount of rotation. But these results are so very com- plicated in their Cartesian shape that it is quite useless to give them. To put down the equations of motion let X^, Y^, Z^ be the force due to stress on what before strain was unit area per- pendicular to the axis of x. Similarly for Xy, &c. Next suppose that X, Y, Z is the external force per unit volume of the un- strained solid and let Z) be the original density of the solid. Then the equation of motion (15 n) § 16 a below, viz. Dp' = F + TA, gives in our present notation X + dXJdx + dXJdy + dXJdz = x'B, &c., &c. It remains to express X^ &c. in terms of the displacement and LMN. This is done in equation (15 § 16 below, viz. TO, = - 2p/sv^^,(ii(;» -H 3 FM Vp;p'S(oV,vj2Ssi;v.y^Sp^'p^'p^:* In our present notation this consists of the following nine equations : * The second term on the right contains in full the nine terms corresponding to (Jy^ - J-i^M)j2J. Quaternion notation is therefore here, as in nearly aU cases which occur in Physics, considerably more compact even than the notations of determinants or Jacobians. INTRODUCTION. ^ KdEdx^ dCdy^dBdz)'^ 2/ „ (dw dz dw dz dw dz'\ J^^M — J^^L {dE dx ^ dCdy^ dBdzj^ 1J and six similar equations. We thus see that in the case where LMN are zero, our present X^,X^, X, are the PQR of Thomson and Tait's Nat. Phil. App. C {d), and therefore equations (7) of that article agree with our equations of motion when we put both the external force and the acceleration zero. These are some of the new results in Elasticity, but, as I have hinted, there are others in §§ 14, 1.5 which it would be waste of time to give in their Cartesian form. Electricity. In Section IV. below I have considered, as already stated, the most general mechanical results flowing from Maxwell's theory of Electrostatics. I have shewn that here, as in the particular cases considered by others, the forces, whether per unit volume or per unit surface, can be explained by a stress in the dielectric. It is easiest to describe these forces by means of the stress. Let the coordinates of the stress be PQRSTU. Then F^F^F^ the mechanical force, due to the field per unit volume, exerted upon the dielectric where there is no discontinuity in the stress, is given by F^ = dPjdx + dU/dy + dTjdz, &c., &c. and {I, m, n) being the direction cosines of the normal to any surface, pointing away from the region considered F; = -[lP + mU+ nT], -[]„ &c., &c., where a, b indicate the two sides of the surface and F^, F^', F^' is the force due to the field per unit surface. It remains to find P &c. Let X, Y, Z be the electro- motive force, a, ^8, 7 the displacement, w the potential energy per unit volume and K„, K„, K„, K^„ K„, K^ the coefficients of specific 10 INTRODUCTION. inductive capacity. Let 1 + e, 1 +/, 1+g, a/2, b/2, c/2 denote the pure part of the strain of the medium. The K's will then be functions of e &c. and we must suppose these functions known, or at any rate we must assume the knowledge of both the values of the ^s and their differential coefficients at the particular state of strain in which the medium is when under consideration. The relations between the above quantities are 47ra = K^j: + K^Y + KJZ, &c., &c. = {K^^X' + K^Y' + K^' + 2K,JZ + 2K^X + 2K^^Y)/8-rr. It is the second of these expressions for w which is assumed below, and the differentiations of course refer only to the K's. The equation expressing P &c. in terms of the field is (21) § 40 below, viz. ^c^ aj)^c,) (i.e. i = aj, + bj + cjc, &c.). ,j,Q* is defined as a symbolic linear vector function whose coordi- nates are (d/da^, d/db^, d/dc^, djda^, d/db^, djdc^, djda^, d/db^, djdc^, and to ^(j is to be applied exactly the same system of suffixes as in the case of V. Thus q being any quaternion function of he a, self-conjugate linear vector function it has only six coordinates. If these are PQRSTU (i.e. i = Pi + Uj + Tk, &c.) ^a is defined as a self-conjugate linear vector function whose coordinates are (d/dP, d/dQ, djdR, \djdS, \djdT, id/dU). We shall frequently have to compare volume integrals with integrals taken over the bounding surface of the volume, and again surface integrals with integrals taken round the boundary of the surface. For this purpose we shall use the following nota- tions for linear, surface and volume integrals respectively JQdp, * I have used an inverted D to indicate tlie analogy to Hamilton's inverted A. § 3.] QUATERNION THEOREMS. 16 JjQdl,, fJJQds where Q is any function of the position of a point. Here dp is a vector element of the curve, d2 a vector element of the surface, and ds an element of volume. When comparisons between line and surface integrals are made we take dX in such a direction that dp is in the direction of positive rotation round the element dX close to it. When comparisons between surface and volume integrals are made d2 is always taken in the direction away from the volume which it bounds. Properties of f. 2. The property of f on which nearly all its usefulness depends is that if o- be any vector o- = - ^S^a-, which is given at once by equation (1) of last section. This gives a useful expression for the conjugate of a linear vector function of a vector. Let be the function and a, t any two vectors. Then (f>' denoting as usual the conjugate of ^ we have ScOCpT = >SiT<^ (O, whence patting on the left t = — ^S^t we have St (- ^S(o'(o, or since t is quite arbitrary (f>'(o = - ?/Sfw^So>^ - ^Sco,},^ = 25a) ) And all the other well-known relations between ' are at once given e.g. S^cj)^ = S^(f)' ^, i.e. the ''convergence" of =the " convergence " of '. 3. Let Q (\, fji) be any function of two vectors which is linear in each. Then if <^&) be any linear vector function of a vector tu given by ^w = - 2^/Sf(Ba ) ,^. we have Q (?, ?) = SQ (- ^Sfa, ^) = %Q{a,^)\ ^ '' 16 QUATERNION THEOREMS. [§ 3. or more generally Q(l'f>xD = ^Q(x''^,^) (4«)- To prove, it is only necessary to observe that and that - f-Sfx'" = x'«- As a particular case of eq. (4) let have the self-conjugate value (^o) = - ^ 2 i^Scoa + aSco^) Then Q(r,<^f) = i2{Q(«,/3) + Q(/3,a)} j ^^^- or if Q (\, /i) is symmetrical in \ and fi Q(r,<^r) = 2Q(a,/3) (6). The application we shall frequently make of this is to the case when for a we put Vj and for 0, o-,, where a is any vector function of the position of a point. In this case the first expression for (f> is the strain function and the second expression the pure strain function resulting from a small displacement a- at every point. As a simple particular case put Q (X, fi) = SXfj. so that Q is sym- metrical in X and fi. Thus 0-Q{'^,0 (6a). This is quite independent of the form of (f>. To prove, observe that by equation (2) and that — ?^^. Thus we get rid of ^ and may now drop the suffix of fj and so get eq. (6a). [Notice that by means of (6a), (4a) may be deduced from (4) ; for by (6a) Q (?, xO = Q ix% 'f>0 = ^Q (X'a, /3) by (4).] 3a. A more important result is the expression for ^~^co in terms of (j>. We assume that Sfi^^^^cj>^^ = iim (66). which gives m in terms of . That S^.^^S^i^^ = 6 is seen by getting rid of each pair of ^s in succession thus : — «?, («?. y^.) u, = - svu, . ^^ = s (cr, - m.o r, = - sr,?, = 6. Next observe that Multiplying by V^^i ^^^ again on the right getting rid of the f s we have V^.^S4>co^^4,^^ = -2mm (6c), whence from equation (66) or changing a into ^"'to By equation (6c) of last section we can also put this in the form so that <^"'a) is obtained explicitly in terms of or ^'. Equation (6c) or (6(Z) can be put in another useful form which is more analogous to the ordinary cubic and can be easily deduced therefrom, or *less easily from (6d), viz. = (6/). • [Note added, 1892. The cubic may be obtained in a more useful form from the equation (<>5fif2i'3S0f,^f20f3= - 3rfifo.S0u0f,0i-3 thus Again ^fiS . 0w0f2Ff,f2=0ii5 . ^uF. ^f2T'fif2= 0fiS0" ( - fiSfi0f2 + faSfi-^-fo) = - * (fiSii^") 'Si-2^f2 + {fiSfi0f2) Sfo^" Hence ^'(i)-m"^''(i) + m'0ai-7)!fc(=O, where 6m=Sfif2f3S^i'i0f20i'3 2m'=-SFf,i-2F^fi^f2 M. 2 IS QUATERNION THEOREMS. [§ 3a. As a useful particular case of equation (&d) we may notice that by equation (4) § 3 if (f)(0 = — tTjlStoV, cfr'a, = - 3 VVy^Scoa^aJSV;^y^a^,T,'-', yfr be two linear vector functions of a vector. Then if where ;)^ is a quite arbitrary linear vector function for we may put X?~ tS^co where t and to are arbitrary vectors, so that St^ a linear vector function SQ = -Q,s^e>a,r (7). The property is proved in the same way as for V, viz. by ex- panding (SS^f^Qif in terms of the coordinates of ^(j. First let ^ be not self-conjugate, and let its nine coordinates be {afi^c,aj)^c^aj}^c^). * [Note added, 1892. The following sUghtly more general statement is a practi- cally mneh more convenient form of enunciation: if Sxf^f=Sxi'^f, where Y is a perfectly arbitrary Belf-conjngate and

-a)=o T(p — a) QU'(p — a) is zero or ^nite, the added line integral will be zero or finite respectively (of course including in the term finite a possibility of zero value). If this expression be infinite, the added line integral will generally also be infinite. This leads to the consideration of potentials which is given in§!). 8. Some particular cases of equations (8) and (9) which (except the last) have been proved by Tait, are very useful. First put Q = a simple scalar P. Thus fPdp=JJVdtVP (10), JJPdt=JJfVFd% (11). If P be the pressure iu a fluid —JJPdX is the force resulting from the pressure on any portion and equation (11) shews that — VP is the force per unit volume due to the same cause. Next put Qa = S(oa and Vaa. Thus JSdpa- =JJSd^Va- (12), JVdpa =//F(FdSV. ~ ScoV . 7] (1), where y is the displacement that gives rise to the strain. Let X consist of a pure strain ■sjr followed by a rotation q{ )q'^ as explained in Tait's Quaternions, § 365 where he obtains both q and i|r in terms of x- Thus Xw = # = a> — JSmV . t] — ^V^So)Tj^ (3), by equation (3) §2 above. Similarly g( )q~^ becomes Vd{ ) where •2,6 = V^T]. The truth of these statements is seen by putting in equation (2) for q, 1 + dj2 and therefore for q'^, 1 — 6j2 for i/r, x and then neglecting all small quantities of an order higher than the first. 13. Next let us find the force and couple per unit volume due to a stress which varies from point to point. Let the stress function be \ for the volume considered in the equation may be taken as the element d%. Again the moment round any arbitrary origin is jjVpdX =ljjvp,i>s/^d%+jjjvp.i>y^d%, § 14.] ELASTIC SOLIDS. 27 by equation (9) § 6. The second term on the right is that due to the force ^A just considered, and the first shews that in addition to this there is a couple per unit volume = V^cf)^ = twice the " rotation " vector of ^. Let then ^ be the pure part and Ve ( ) the rotatory part of (j>. Thus Force per unit volume = ^A = ^A- We (4), Couple per unit volume = Vm=2e (5). These results are oi course equivalent to those obtained by Tait, Quaternions, §§ 487 — 8. The meanings just given to rj, x, X' '^> 1' •^> ^ ^^^ ^ ^^^^^ ^^ retained throughout this Section. In all cases of small strain as we have seen we may use i/r or t^ indifferently and whenever we wish to indicate that we are considering the physical phenomenon of pure strain we shall use i/r, y^ being regarded merely as a function of x- We shall soon introduce a function in- which will stand towards somewhat as i/r towards ■y^ and such that when the strain is small w = ^. It is to be observed that ^w is the force exerted on a vector area, which when strained is to, not the stress on an area which before strain is a. Similarly in equations (4) and (5) the in- dependent variable of differentiation is p + 'q so that strictly speaking in (4) we should put 0p+,A — Fp+,V6. In the case of small strain these distinctions need not be made. Stress in terms of strain. 14. To express stress in terms of strain we assume any displacement and. consequent strain at every point of the body and then give to every point a small additional displacement Si; and find in terms of t^ and ^ the increment JJJSwd%^ in the potential energy of the body, wds^ being the potential energy of any element of the body whose volume before strain was d%^. Thus JJJSwdi^ = (work done by stresses on surface of portion considered), — (work done by stresses throughout volume of same portion). 28 ELASTIC SOLIDS. [§ 14. Thus, observing that by § 12 the rotation due to the small dis- placement Sr] is Fp+,V8»;/2, we have JJJBwdi, = - fJS8r,dt + fJJSBvcf>,,+,V^d% + JfJSe,^,VSvd%. The first of the terms on the right is the work done on the surface of the portion of the body considered ; the second is — (work done by stress-forces ^p+,A) ; and the third is — (work done by stress- couples 2e). Thus converting the surface integral into a volume integral b}' equation (9) § 6 above fJJSwd%, = - fJJSBv, being self-conjugate Bw = — mSBy}rx'^^q~^4>^q- This can be put into a more convenient shape for our present purpose. First put for yf^ its value '' by means of the rotation qvr'to . o)'. * Observe that we do not make this assumption. We really shew that it is true. 30 ELASTIC SOLIDS. [§ 1*- But ft)' = qaq^, so that ■or'to = q~^^ (qeoq'^) q = taw. Thus we see that ta is as it were <^ with the rotation undone. Before proceeding further with the calculation let us see what we have assumed and what equation (9) teaches us. The one thing we have assumed is that the potential energy of the body can be taken as the sum of the potential energies of its elements, in other words that no part of the potential energy depends conjointly on the strains at P and Q where F and Q are points separated by a finite distance. This we must take as an axiom. By it we are led to the expression for Sw in equation (9). This only involves the variation of the pure strain i|r hut not the space differential coefficients of -v/r. This is not an obvious result as far as I can see but it is I believe always assumed without proof. We may now regard w as a function of yfr only. Therefore by equation (7) § 5 above therefore by equation (9) of this section In equation (6a) § 3 above putting ^ = i/r"' the right-hand member of this equation becomes »n(SST/rfEri/r"'f. Now putting in equation (6a) §30 = ■BTi/r"' this member becomes mS^S-^i}r''zj^ or mSS-\jr^\jr'''ar^. Thus we have 2SBf^^(J^^=mSByjr^(7!ry]r-' + -^"V) f. Now iD-\|r"' 4- yfr'^is is self-conjugate. Hence by § 4 above m(i!ryjr~^+ylr~^-iiT) = 2^(JW (11). This equation can be looked upon as giving zr in terms of the strain. We can obtain to- however explicitly for i|r~'tT is the conju- gate of ■Bj'\|r~\ Hence from equation (11) (because (i|r~'t3- -|- wi^"')/2 is the pure part of ■Ej\/r"') rms\jr~^m = ^QW(o + Vda> where ^ is a vector to be found. Changing ;Sa,V.V^/SV,V,V,S/,,»3'...(150. It is unnecessary to write down what equation (15^) becomes wrhen we substitute for t, changing e into — M/2 by equation (15A). In the important case however when M = 0, the equation is quite simple, viz. F = 2/3;SV,<,QwA (15m). Addition to % 16, Bee, 1887 (sent in with the Essay). [The following considerations occurred just before I was obliged to send the essay in, so that though I thought them worth giving I had not time to incorporate them in the text. It is interesting to consider the case of an isotropic body. Here w is a function of the three principal elongations only and therefore we may in accordance with §§ 14 and 15 suppose it a function of a, b, c or in accordance with § 16 of A, B, C where a ^?tf, 6 = -SrVr=r/2, c = -;Srt'?/3 {A). A = - -S?^?, B = - S?^'r/2, C = - Sr*»f/3 {B). Let us use x, y, z, X, Y, Z for the differential coefficients of w with respect to a, b, c, A, B, C respectively. Thus dw = xda H- ydb +zdc=- Sd-^^ (x + y^ + z^f/') f as can be easily proved by means of equation (6a) § 3 above. But dw = —Sd^^^Qwl^. ^Qw = X + y^ + z^'' (0), by § 4 above. Similarly we have *aw = X+ F^ + Z^' {D). § 16a.] ELASTIC SOLIDS. 35 (Notice in passing that to pass to small strains is quite easy for ^Qw is linear and homogeneous in -i/r — 1 so that ^qw = a; + y^ where y is a c(mstant and x + y a, multiple of a — 3.)* From equation ((7) we see that in equation (12) | 14 is zero and therefore that from equation (13) mw = ^(JW\jr = X'xfr + ytfr^ + Z\}r^ {E). Again from equation (14) § 15 Ta> = xqmq'^ + y)(a> + zjc\lrco + V€')^~^os (F). Similarly from equations (156) and (15/) § 16 we may prove by equation (D) that im4> = Xxx'+Y(xxy + Zixxy (»)- and T = 2Xp<; + 2 Yxx'x + '^HxXXX + y^x" ( ) (^- If we wish to neglect all small quantities above a certain order the present equations pave the way for suitably treating the subject. I do not however propose to consider the problem here as I have not considered it sufficiently to do it justice.] Variatiov of Temperature. 16a. The w which appears in the above sections is the same as the w which occurs in Tait's Thermo-dynamics, § 209, and therefore all the above work is true whether the solid experience change of temperature or not. w will be a function then of the temperature as well as i/r. To express the complete mathematical problem of the physical behaviour of a solid we ought of course instead of the above equations of equilibrium, to have corresponding equations of motion, viz. equations (15A), {\ol) and (instead of (15gf)) Z)/5'=F + tA (15n), where D is the original density of the solid at the point con- sidered. Further we ought to put down the equations of con- duction of heat and lastly equations (16/) and (16d) below. We do not propose to consider the conduction of heat, but it will be well to shew how the thermo-dynamics of the present question are treated by Quaternions. * [Note added, 1892. In the original essay there was a slip here which I have corrected. It was caused by assuming that ^ instead of ^ - 1 was small for small strains. In the original I said " where ?/ is a constant and i is a multiple of n."] 3—2 36 ELASTIC SOLIDS. [§ 16a. Let t be the temperature of the element which was originally dS„ and Ed%„ its intrinsic energy. Let where M, N are a linear self-conjugate function and a scalar respectively, both functions of ^fr and t. Here BHd%^ is the beat required to be put into the element, to raise its temperature by St and its pure strain by S-^. Now when t is constant BH must be a perfect differential so that we may put where y is some function of i/r and t. Thus BH=-tSBf^^af^+mt (16). Now we have seen in § 14 that the work done on the element during the increment Bijr, divided by d%^ Thus by the first law of Thermo-dynamics BE = JBH - m/SfS^|r■f-•?•■=rf = Jmt - JtSBf^^aJX- mSBff-' fwf, where / is Joule's mechanical equivalent. Thus JN=dE/dt (16a), and ^ (wi/r"' + -«|r"* ra-) = ,^(jw (166), where w = E - Jtf (16c). To apply the second law we go through exactly the same cycle as does Tait in his Thermo-dynamics, § 209, viz. (t, if + Bf, t) if + Bf,t + Bt) if, t + Bt) if. t). We thus get* or Jf=-dw/dt (16d). the arbitrary function of t being neglected as not affecting any physical phenomenon. Substituting for w from equation (16c), Jtdfldt = dEldt = JN (16e) by equation (16a). Thus from equation (16) BH=tBf (16/), * [By asBrnning from the second law that the work done by the elemeut in the cycle, i.e. the sum of the works done by it during the first and third steps is Jitjt multiplied by the heat absorbed by the element in the third step. Note added, 1893.] § 17.] ELASTIC SOLIDS. 37 SO that in elastic solids as in gases we have a convenient function which is called the " entropy ''. Thus the intrinsic energy Edio, the entropy /d%^ and the stress in- have all been determined in terms of one function w of yfr and t which function is therefore in this general mathematical theory supposed to be known. If instead of regarding w (which with the generalised meaning it now bears may still conveniently be called the " potential energy" per unit volume) as the fundamental functiim of the substance we regard the intrinsic energy or the entropy as such it will be seen that one other function of -v^ must also be known. For suppose / the entropy be regarded as known. Then since dw/dt = — Jf w=W-JJfdt (IQg). where the integral is any particular one and Ti'^ is a function of ^(r only, supposed known. Again E=w + Jtf or E=W+Jitf-Jfdt} (16A). Thus all the functions are given in terms of y and W. Similarly if j^ be taken as the fundamental function w = tiW'-}Edtlf) (16i), Jf=Elt-f-fEdt/t'-W' (16;), where as before the integral is some particular one and H^ is a function of i^r only. Small strains. 17. We now make the usual assumption that the strains are so small that their coordinates can be neglected in comparison with ordinary quantities such as the coefficients of the linear vector function ^(jw. We can deduce this case from the above more general results. To the order considered q = l so that by equation (10) § 14 above d> = w. We shall use the symbol ot rather than ^ for the same reasons explained in § 13 above as induce us to u.se yfr rather than ^. Remembering that a>„ and (o are now the same and that i/r may be put = 1 so that V^^a'^ylr^= V^^(J'W^ = and therefore of equation (12) = 0, we have from equation (13), •era) = ^(jwo) (17). 38 ELASTIC SOLIDS. [§ 17. Of course we do not require to go through the somewhat com- plicated process of §§ 14, 3 5 to arrive at this result. In fact in equation (6) § 1-1, we may put m=l and p+,V =V so that Ew — — SBrjj^V^ = — /SSt/jTO-Vj, and therefore by | 3 above Sw = — (SSj^fcrf = — SSy}r^zT^. But by § 5 above and therefore by § 4 we get equation (17). 18. It is convenient here to slightly change the notation. For i/r, X we shall now substitute ■yfr + 1, x+^ respectively. This leads to no confusion as will be seen. With this notation the strain being small the stress is linear in i|r i.e. ^Qw is linear and therefore w quadratic. Now for any such quadratic function w = -Sf^^.(lw^/2 (18), for we have by § 5, dw = - Sd'^fr^ ^QW^. Put now yp- = 7iy}r', Then because ^Qw is linear in \|r where ^q'w is put for the value of ^(jw when the coordinates of yjr' are substituted for those of yfr. Thus keeping -i^' constant and varying n, dw = — ndnS-\lr'^ ^d'w^, whence integrating from n = to n = 1 and changing yfr' into i^ we get equation (18). From equation (18) we see that for small strains we have W = - Syjr^'ST^/i (19). Now w is quadratic in yp and therefore also quadratic in -sr, so that regarding w as a function of zr we have as in equation (18) lu = - S^^ ^aw^l2, so that by equation (19) and § 4 above V^ = oraw (20). All these results for small strains are well-known in their Car- tesian form, but it cannot be bias that makes these quaternion § 19.] ELASTIC SOLIDS. 39 proofs appear so much more natural and tlierefore more simple and beautiful than the ordinary ones. 19. Let us now consider (as in § 16 is really done) w as a function of the displacement. Now w is quadratic in i|r, and >/r is linear aud symmetrical in V^ and t?,. In fact from equation (3) § 12 above, remembering that the ^^ of that equation is our present i/r + 1, we have Therefore we may put w = w{'n„V^,r,,^,rj,) (21), where w (a, /3, 7, S) is linear in each of its constituents, is sym- metrical in a and /S, and again in 7 and S, and is also such that the pair a, /9 and the pair 7, 8 can be interchanged. [This last statement can be made true if not so at first, by substituting for w (a, /3, 7, S), w (a, /3, 7, S)/2 + w (7, S, a, /3)/2, as this does not affect equation (21).] Such a function can be proved to involve 21 inde- pendent scalars, which is the number also required to determine an arbitrary quadratic function of ijr, since a/t involves six scalars. Thus we have the two foUowiag expressions for Bw, which we equate - ;,, V,, g,,^, V,) or* by § 3 above. Now let us put Br] = (o'Smp where w' and (o are arbitrary constant vectors. We thus get 8(o'iffa> = -2w (a, to, 77,, V,) = 2Sa>'^w (f, u>, i;,, V,). Whence since a' is quite arbitrary, ^a) = 2rw(r,a>,77.,V.) (22). The statical problem can now be easily expressed. As we saw in § 14, equations (7) and (8), it is simply F+FVM/2-f«7A = (22a), * [Note added, 1892. Better thus :— by § 3 above, -S5xl/iwi=2w (S (f, u, 7;,, Vi) for fu; (f, d), T/i, Vi) regarded as a function of u is clearly self conjugate. 40 ELASTIC SOLIDS. [§ 19- throughout the mass; and at the surface F,-rUdlM/2-'STUdX = (226), where P, M are the given external force and couple per unit volume and F, is the given external surface traction per unit surface. Substituting for w from equation (22) F+FVM/2 + 2rw(?,A,7;„V,)=0 ] .gS) F,- FC^dSM/2-2?w(?, ra2,7,„V,) = 0] ^ Isotropic Bodies. 20. The simplest way to treat these bodies is to consider the (linear) relations between w and i/r. In the first place notice that ijr can always be decomposed into three real elongations (contractions being of course considered as negative elongations). Thus i being the imit vector in the direc- tion of such an elongation, yp-w = — "ZeiSia. The elongation - eiSiio will cause a stress symmetrical about the vector i, i.e. a tension Ae in the direction of i and a pressure Be in all directions at right angles ; A and B being constants (on account of the linear relation between ^ and yjr) independent of the direc- tion of i (on account of the isotropy of the solid). This stress may otherwise be described as a tension {A + B)e in the direction of i and a hydrostatic pressure Be. Thus i^Tio = — {A + B) 'ZeiSio) - Bate ^{A+B)^w+ BwS^yjr^. To obtain the values of A and B in terms of Thomson and Tait's coefficients k and n of cubical expansion and rigidity respec- tively ; first put yfra) = eto and oro) = Skeca, and then put yjro) = V\o)fi and ■sro) = 2n VXutfi, \ and /x being any two vectors perpendicular to each other. We thus get A+B = 2n, B=^-{k-2n/S), whence uro) = 2nyfrw - {k - 2n/3) ayS^yjr^ (24). § 21.] ELASTIC SOLIDS. 41 From this we have ■\jrto = vTajin + ta^fi/r^ {k - 2n/S)j2n, but from the same equation Equation (24) gives stress in terms of strain and (25) the converse. 21. We can now give the various useful forms of w for isotropic bodies for from equation (19) § 18, w = - Sf^Ts^l2. Therefore from equations (24) and (25) respectively W = -n (yjr^y + i(k- 2n/3) S^,yjr^^S^,ylr^, (26), -' = -1 (-«" - i (^ - 9^) ^r.SfV7; (30), - P. + VUd%fa./2 = nSU'dl,'^ . v + wV.Sjy.fTdS + (m - n) fArfS-SfVT? (31). We now proceed to apply these results for small strains in isotropic bodies to particular cases. These particular cases have all been worked out by the aid of Cartesian Geometry and they are given to illustrate the truth of the assertion made in the Introduction that the consideration of general problems is made simpler by the use of Quaternions instead of the ordinary methods. 42 ELASTIC SOLIDS. [§ 22 Particular integral of equation (30)* 22. Since from equation (30) (F being put for simplicity in- stead of F + FVM/2) we have nV»77 = F - mVSVr) we obtain as a particular case by equations (18) and (19) § 10, 'i^iTnrj = JJJu (F - mVS <^r}) ds, where u has the meaning explained in § 9, and the volume inte- gral extends over any portion (say the whole) of the body we may choose to consider. To express JJfaVS\/r]ds as a function of F put in this term w = — ^fTp^V, where p is taken for the p — p' of § 9, and apply equation (9) § 6. Thus 4.Trnrj = JJJuFdS + ^m JJJUpyyS^^ridS = JJfuTdS - ^mJJJUpV'SV7,dS + a surf. int. = fJJuFdi - ^ -JJJUpS\IFd% + the surf int. for by equation (30) (SvT = {m + n) V^S^rj. Now (in order to get rid of any infinite terms due to any discontinuity in F) apply equation (9) § 6 to the second volume integral. Thus 4:7rnv = fffuFdi + o / "^ n IfJSTV . Upd% + a surf. int. The surface integral may be neglected as we may thus verify. Call the volume integral ^irnr]'. Thus ^irnVS = 47rF + „ , "^^ jjjSFV . V'Upd% 2 {m + n) '^ iTrmVS^V = ^ IIISFV .Vuds+ „ ."^^ . JJJSFV . VS^ Upds. n 2n{m + n) '^ so that putting VUp = — 2uwe get whence we have as a particular solution of equation (30) 77 = 17' or V = ,^J//«F^S 4- 8— (^,)///^FV . Upd, (32). * [Note added, 1892. For a neater quateruion treatment of this problem see Phil. Hag. June, 1892, p. 493.] § 23.] ELASTIC SOLIDS. 43 This is generally regarded as a solutioD of the statical problem for an infinite isotropic body. In this case some law of con- vergence must apply to F to make these integrals convergent. Thomson and Tait {Nat. Phil. § 730) say that this law is that Fr converges to zero at infinity. This I think can be disproved by an example. Put *Fr" = r~°X, where X is a constant vector and a a positive constant less than unity. Equation (32) then gives for the displacement at the origin due to the part of the integral extend- ing throughout a sphere whose centre is the origin and radius R m + 3?i R^-" .^ 6n (m + 71) I —a Putting R= 00 , T) also becomes oo . The real law of convergence does not seem to me to be worth seeking as the practical utility of equation (32) is owing to the fact that it is a particular integral. The present solution of the problem has only to be compared with the one in Thomson and Tait's Nat. Phil. §§ 730-1 to see the immense advantage to be derived from Quaternions. It is easy to put our result in the form given by them. We have merely to express /SFV . Up in terms of P and r'SFV . Vu where r is put for the reciprocal of u. Noting that Vu = -u''p, Up = up, SFV.p = -F, we have at once (SFV .Up=-uF- u'pSFp SFV .Vu = u'F + 3u'pSFp therefore eliminating pSFp, STV .Up = - MP2/3 - r'/SFV . Vm/3, .: V = {247rn (m +n)]-'JJJd% {2(2m-t-3m) MF-mr''>SFV.VM)...(33), which is the required form. 23. Calling the particular solution t}' as before and putting r]=r] +T] the statical problem is reduced to finding tj" to satisfy * [Note added, 1892. This is not legitimate since it makes F= oo for )=0. The reasoning is rectified in the Phil. May. paper just referred to by putting F = from r=0 to r = b and Tr=r-''\ from r=b to r = x .] 44 ELASTIC SOLIDS. [§ 23. and the surface equation either t) + rj' = given value, i. e. t)" = given vahie, or vsJIdi^ = given surface traction, i.e. by equation (29) § 21 above, nSUdXV . 7," + nV^Sv" Udl, + {m-n) UdXS^v" = known value. This general problem for the spherical shell, the only case hitherto solved, I do not propose to woik out by Quaternions, as the methods adopted are the same as those used by Thomson and Tait in the san:ie problem. But though each stop of the Cartesian proof would be represented in the Quaternion, the saving in mental labour which is effected by using the peculiarly happy notation of Quaternions can only be appreciated by him who has worked the whole problem in both notations. The only remark necessary to make is that we may just as easily use vector, surface or solid, harmonics or indeed quaternion harmonics as ordinary scalar harmonics. Orthogonal coordinates. 24. It is usual to find what equations (22a) of § 19 and (3) of § 12 become when expressed in terms of any orthogonal coordi- nates. This can be done much more easily by Quaternions than Cartesian Geometry. Compare the following investigation with the corresponding one in Ibbetson's Math. Theory of Elasticity, Chap. V. Let X, y, z be any orthogonal coordinates, i.e. let a; = const., 2/ = const., ^= const., represent three families of surfaces cutting everywhere at right angles. Particular cases are of course the ordinary Cartesian coordinates, the spherical coordinates r, Q, (f> and the cylindrical coordinates r, (p, z. Let D^, Dy, D, stand for differentiations per unit length perpendicular to the three coordi- nate surfaces and let \, fi, v be the unit vectors in the correspond- ing directions. Thus Thus, using the same system of suffixes for the D'& as was ex- plained id connection with V in § 1, ^^ = i)„<^.\ + Z>„.<^.^ + D^^i^^v (34), or <^A = Z), (iM) + A (") - (D^ + A/^ + A"). • .(35). § 26.] ELASTIC SOLIDS. 45 25. Now to put equation (22a) § 19 into the present coordi- nates all that is required is to express ctA in terms of those co- ordinates. Let the coordinates of ra- be PQRSTU. Thus from equation (35) we have - vr (D^\ + Dyfi + D^v). The first thing then is to find D^, Bxfi, DyX &c. Let^j, ^j be the principal curvatures normal to a; = const., i.e. (by a well-known property of orthogonal surfaces) the curvatures along the lines of intersection of « = const., with z = const., and y = const, p^, p^ will be considered positive when *the positive value of dx is on the convex side of the corresponding curvatures. Similarly for q^q^r^r^. Thus for the coordinates r, d, , z as our coordinates, the axis of z beiug parallel to the generating lines of the cylinder. Let \, /i, v be the unit vectors in the directions of dr, dcf>, dz respectively and let 7] = u\ + Vfi + wv as before. We shall follow Thomson and Tait's lines of proof — i.e. we shall first find the effect of a simple torsion and then add another displacement and so try to get rid of stress on the curved surface. Holding the section z = fixed let us give the cylinder a small torsion of magnitude r, i.e. let us put 7] = Tzr/j. (40), for all points for which tz is small. The practical manipulation of such expressions as this is almost always facilitated by considering the general value of ^(V,, rjj where Q is any function linear in each of its constituents. Thus in the present case Q (^r '?>) = r {zQ (\, fi) - zQ (/., X) + rQ (v, ii)]. [If Q is symmetrical in its constituents, e.g. in the case of stress below this reduces to the simple form Q(Vj, ??,) = Tr'Q(i/, ^).] From this we at once see that rj satisfies the equation of internal equilibrium for putting Q (a, /3) = a^ Vt) = T ('2zv - rX) = tV(z'- r72) so that both SVrj and V't; = 0. Again the value for Q at once gives us the stress for BTO) = — M/SmV . 9j — nV^Smrji — (m — n) (oSVt), or HTQ) = — nrr {fiScov + vScofi) (41), which is a shearing stress nrr on the interfaces perpendicular to H and V. Putting (o = the unit normal of the curved surface we have for the surface traction ■B7a» = — mrvSwii. In the figure let the plane of the paper be s = 0, the origin, P a point on the curved surface and OM the perpendicular from 48 ELASTIC SOLIDS. [§ 27. on the tangent at P. Thus - rStofi = OP cos 0PM = PM, PM being reckoned positive or negative according as? it is in the positive or negative direction of rotation round Oz. Thus we see that the surface traction is parallel to Oz and = mPM. Hence in the case of a' circular cylinder a torsion round the axis satisfies all the conditions of our original problem, but this is true in no other case. The surface traction at any point on the plane ends necessary to produce this strain is uv^mrfi by equation (41) so that its moment round the origin is nr^^r^dA, where dA is an element of area and the integral extends over the whole cross-section. 28. Let us now assume a further displacement rj = wv (42), where w is a function of r, only, and let us try to determine w so that there is still internal equilibrium and so that the stress on the curved surface due to w shall neutralise the surface traction already considered. In the present case Q(V.,^,) = Q(V«;, ,;). Thus SVr) = (since v is perpendicular to Vw) and therefore the equation of internal equilibrium gives V'w = 0. Again ■sra = — nSaV . t) — nV^Scori^ , or vra> = — n{vSc0Vw + ^wSo}v) (43), a shea.T = nTVw on the interfaces perpendicular to Vw and v. § 29.] ELASTIC SOLIDS. 49 Thus putting « for the unit normal to the curved surface, the present surface traction will neutralize the former if SwVw = — rrScofj- = — rStov (\r) = J rSvmV (r'), i-e. dw/dn = d(Try2)jds, where d/dn represents differentiation along the normal outwards and d/ds differentiation along the positive direction of the bound- ing curve. We leave the problem here to the theory of complex variables and Fourier's Theorem. Observe however that the surface traction at any point on the plane end =cti/ = mVw by equation (43), and therefore that the total couple = - nfJrSfj.VwdA = nfj(dw/d) dA. This leads to the usual expression for torsional rigidity. Wires. 29. In the following general treatment of Wires some of the processes are merely Thomson and Tail's translated into their shorter Quaternion forms ; others are quite different. The two will be easily distinguished by such as are acquainted with Thom- son and Tait's Nat. Phil. The one thing to be specially careful about is the notation and its exact meaning. This meaning we give at the outset. The wires we consider are not necessarily naturally straight but we assume some definite straight condition of the wire as the " geometrically normal " condition. The variable in terms of which we wish to express everjrthing is s the distance along the wire from some definite point on it. Any element of the wire, since it is only slightly strained, may be assumed to have turned as a rigid body from its geometrically normal position. This rotation is expressed as usual (Tait's Quaternions, § 354) by the quaternion q ; the axis of q being the axis of rotation, and the angle of q, half the angle turned through. a is taken so that the rate of this turning per unit length of the wire is qmq'^ so that w is the rate of turning per unit of length when the whole wire is moved as a rigid body so as to bring the M. 4 50 ELASTIC SOIJDS. [§ 29. element under consideration back to its geometrically normal position. Of course to is a fanction of q and its derivative with reference to s. This function we shall investigate later. The resolved part of qcoq''^ parallel to the wire is the vector twist and the resolved part perpendicular to the wire is in the direction of the binormal and equal to the curvature. In fact m is the vector whose coordinates are the k, \, t of Thomson and Tait's Nat. Phil. § 593. When we are given q or a for every point we know the strain of the wire completely. (»„ is defined as the naturally normal value of eu, i.e. the value of to when the wire is unstressed. As usual we take p as the coordinate vector of any point of the wire, p like the rest of the functions being considered as a func- tion of s. We shall denote (after Tait, Quaternions, Chap, ix.) differentiations with regard to s by dashes. We now come to the dynamical symbols. F and M are the force and couple respectively exerted across any normal section of the wire on the part of the wire which is on the negative side of the section by the part on the positive side. Finally let X, L be the external force and couple per unit length exerted upon the wire. 30. When the wire is strained in any way let us impose a small additional strain represented by an increment Seo in to and an increment he in the elongation at any point. Then the work done on the element d8 by the stress-force = — SeSFp'ds and that done by the stress-couple = — SqS(oq''^JI/lds. If (as we assume, though the assumption is not justified in some useful applications of the general theory of wires) F and M to be of the same order of magnitude the former of these expressions can be neglected in comparison vnth the latter for Be is a quantity small compared with Sw. Now the work done on the element by the stress = the increment of the element's potential energy = Swds where w is some function of the strain. Hence Sw = — SSmq'^TO.q. Thus w is a function of to only and Sw = — SSa)J^w, whence we see that q-^Ta.q = J^w (46). Notice that M is Thomson and Tait's f, ?/, ^and q'^tlLq their KLM (Nat Phil. §§ 594, 614). § 32.] ELASTIC SOLIDS. 51 •SI. Now since the strain is small, q'^ftLq is linear in terms of the strain and therefore in terms of w. Hence we see that w is quadratic in terms of (w-co,)/2 + w^ (47), [as can be seen by a comparison of the last three equations w„ being the function of the temperature] and q-'fllq = {o>-w,) (48). When the natural shape of the wire is straight these become w = - Sa>a)/2 + w„ (49), q-'Mq = 4>co (50), and when further the wire is truly uniform <^ and w„ are constant along the wire. 32. Assuming the truth of these restrictions let us conceive a rigid body moving about a fixed point which, when placed in a certain position which we shall call the normal position, has, if then rotating with any vector angular velocity a> where (o,-{2Vq-'q'-a>,)q-' (59), ♦ This could be deduced from Tait's QuatemUms, § 356, equation (2). His e is our gwj"' and his dots our dashes. 54 ELASTIC SOLIDS. [§33. which gives M also explicitly. Substituting for M and p in equations (55) and (56) we have 4I,X,-T+F,xr(l.+| Sq^jf"- {. I.+ ds = (60), ) = (61), q,j>i2Vq-'q-o,,)q-' which are sufficient equations to determine T and q, whereupon M is given by equation (59), w by equation (58) and p' and therefore also p by equation (57). Spiral springs can be treated very simply by means of the above equations, but we have already devoted sufficient space to this subject. SECTION IV. Electricity and Magnetism. Electrostatics — General Problem. 34. Merely observing that all the theorems in integration given in the Preliminary and ivth chapters of Maxwell's treatise on Electricity and Magnetism, Part i., are easy particular cases of equations (8) and (9) § 6 above, we pass on to the one application of Quaternions that we propose to make in Electrostatics. This is to find the most general mechanical results arising from Maxwell's theory of Electrostatics, and to see if they can be explained by stress in the dielectric. This problem as far as I am aware has not hitherto in all its generality been attacked though the most, important practical cases have been, as we shall see, considered by Maxwell, Helmholtz, Korteweg, Lorberg and Kirchhoff. It is necessary first of all to indicate as clearly as possible what I take to be Maxwell's theory of Electricity. He assumes* all space to be uniformly filled with a certain substance called Electricity. Whatever electrical actions take place * Prof. J. J. Thomson in hia paper on Electrical Theories, B.A. Reports, 1885, p. 125, does not credit Maxwell with such a definite and circumscribed theory as that described in the text, and he is thereby led to find fault with Maxwell's term "Displacement" and points out that there is an assumption made with reference to the connection between the true current and this polarisation (displacement). He says moreover, " It is rather difficult to see what is meant in Maxwell's Theory by the phrase ' Quantity of Electricity.' " None of these remarks are called for if the view I take of Maxwell's theory be correct, and these grounds alone I consider sufficient for taking that view. The paper of Thomson's here mentioned I shall frequently have to refer to. [Note added, 1892. In the text I have given much too rigid a form to Maxwell's theory. What I have called his theory I ought rather to have called his analogy. StUl I think the present foot-note is in the main just. In my opinion it is no more and no less difficult to see what is meant in Maxwell's Theory by " Quantity of Electricity " than by " displacement" since the two are 56 ELECTRICITY AND MAGNETISM. [§ 3-t. depend on the continued or past motion of this substance as an incompressible fluid. If electricity is brought from a distance by any means and placed in a given space there must be a displace- ment of the original electricity outwards from that space and the quantity of foreign electricity is conveniently measured by the surface integral of that displacement. Dielectrics are substances in which this displacement tends to undo itself, so to speak, i.e. the original electricity tends to go back to its primitive position. In conductors, on the other hand, there is no property distinguishing any imported electricity from the original electricity. The rate of variation of displacement, whether in dielectrics or conductors, of course constitutes an electric current as it is conve- niently called. We have next to consider a vector at each point of space called the electro-motive force, which depends in some way at present undefined on the distribution of the displacement in the dielectrics, the distribution of currents whether in dielectrics or conductors, and on extra-electrical or semi-electrical action, e.g. chemical or mechanical. If at any point the electro-motive force be multiplied by a scalar the medium at the point remaining (except electrically) unchanged, the current in the case of conductors and the displace- ment in the case of dielectrics is altered in the same ratio. In other words the current or the displacement, as the case may be, is a linear vector function of the electro-motive force, and the coordinates of the linear vector function* at any point depend connected by perfectly definite equations. Of course it is wrong to define " dis- placement" as "displacement of quantity of electricity" and then to define "quan- tity of electricity" in terms of "displacement," but Maxwell does not seem to me eTen tacitly to do this. Bather he says — the dielectric is polarised ; this polarisa- tion can be represented by a vector D ; electrical quantity can be expressed in terms of S; the mathematical connections between electrical quantity and I> are the same as those between quantity of matter in a space and the displacement out of that space made by other matter to make room for the given matter ; we will impress this useful analogy firmly on our minds by calling D the displacement. But I have expressed my present views on the meaning of Maxwell's theory much more fully in Phil. Trans. 1892, p. 685.] * This frequently recurring cumbrous mode of description must be tolerated unless a single word can be invented for "a linear vector function of a vector." Might I suggest the term " Hamiltonian ? " Thus we should say that the dis- placement is a Hamiltonian of the electro-motive force, the HamUtonian at any point being a function of the state of the medium. § 35.] ELECTRICITY AND MAGNETISM. 57 solely on the state of the medium (whether fluid, solid, &c., or again strained or not) at that point. To complete the theory we have to explain how the part of the electro-motive force which is a function of the distribution of dis- placement and current depends on this distribution. This expla- nation is obtained by making the assumption that the electro- motive force bears to electricity defined as above exactly the same energy relation as ordinary force does to matter, i.e. — (wk. dn. on electricity moved during any displacement) = (total displacement of elect.) x (resolved part of E.M.F. in the direction of displacement). In the ivth part of Maxwell's treatise he gives complete inves- tigations of the mechanical results flowing from this theory so far as it refers to currents, but he has not given the general results in the case of Electrostatics. Nor has he shewn satisfactorily, it seems to me, that the ordinary laws of Electrostatics flow from his theory. It is these investigations we now propose to make. 35. Our notation will be as far as possible the same as Maxwell's. Thus for the displacement at any point we use D, and for the E.M.F. E. From the connection explained in last section between D and E we have D = ZE/47r (1), where K at any point is some linear vector function depending on the state of the medium at the point. If the medium change in any manner not electrical, e.g. by means of ordinary strain K will in general also suffer change. Let w be the potential energy per unit volume due to the electrical configuration. Thus if a small increment SD be given to D at all points, the increment fJJSwds in JJfwds, the poteutial energy of the electrical configuration in any space, = work done on the electricity in producing the change, . • . JfJ8wd% = - JJJSESDdS, by the relation stated in § 34 existing between E and D. Thus limiting the space to the element dS Bw = -SEBjy (2). 58 ELECTRICITY AND MAGNETISM. [§ 35. Now suppose D = nD' so that by equation (1) E = nE' where E', D' are corresponding e.m.f. and displacement respectively. Thus Bw = - nBnSE'D'. Integrating from n = to n=l, and finally changing D', E' into D, E we get w = -SDE/2 (3). From this we get Sw = - »SE8D/2 - SDSE/2, so that by equation (2) SE8D = SUBE or by equation (1) SEKBE = SBEKE. Hence (because E and SE are quite arbitrary) K is self- conjugate and therefore involves only six instead of nine coordinates*. In electrostatics the line integral of E round any closed curve must be zero, for otherwise making a small conductor coincide with the curve we shall be able to maintain a current by § 34, and so (by the same section) constantly do work on it (i.e. as a matter of fact create heat) without altering the statical configuration. Hence E must have a potential, say v. Thus E = -Vv (4). Since in an electrostatic field there is no current in a conductor, E = throughout any such conductor and therefore v = const. 36. The charge in any portion of space is defined as the amount of foreign electricity within that space. Thus the charge in any space is the surface integral of the displacement outwards. Thus if there be a charge on the element dS of a surface in the dielectric this charge = Sd^aPa + /SieZSjDj where a, b denote the two faces of the element (so that dS^ = — dSj) and in accordance with § 1 above dXa points away from the region in which the dis- placement is D(,. Thus o- being the surface density a=[SOUdXU, (.5), where the notation []a+b is used for []»+ [ ]&. Similarly if there be finite volume density of foreign electricity, i.e. finite volume density of charge in any space, the charge = — JJSlidl = - JjJjSVDdg, so that if D be the volume density B = -SVI3 (6). * We see from thia that D = jjVio or E=jjVM' according as w is looked upon as a function of E or D. § 37.] ELECTRICITY AND MAGNETISM. 59 [The reason for having + ScflaT^a before and - /SdSD here is that in the former case we were considering a charge outside the region where D^ is considered — between the regions a and b in fact — whereas in the latter case we are considering the charge inside the region where D is considered. The same explanation applies to the sign of — SD Ud% for the surface of a conductor given below.] In conductors, as we saw in § 34, the displacement has virtually no meaning (except when it is changing and so the phenomenon of a current takes place) for the foreign electricity and the original electricity are not to be distinguished. Not so however with the surface of the dielectric in contact with the conductor. We may therefore regard the electricity within the body of the conductor as the original electricity so that the charge is entirely at the surface. Thus the surface density will be — S'DUd'2 where d2 points away from the conductor and D is the displacement in the dielectric. This may be regarded as a particular case of equation (5) D being in accordance with what we have just said considered as zero in the conductor. 37. All the volume integrals with which we now have to deal may be considered either to refer to the whole of space or only to the dielectrics, as the conductors (except at their surfaces) in all cases contribute nothing. The boundary of space will be con- sidered as a surface at infinity and all surfaces where either D or E is discontinuous. Putting W = JJJwd% we have already found one expression for W, viz. 2W=-JJJSDEd%. We now give another. By equatiou (4) 2W = JJJSD^vdS = JJvSdlJy - JJJvS^I)d%, by equation (9) § G above. Thus by equations (.5) and (6) § 36, 2W = JjvV^d% (9), we shall have the following expression for F the force per unit volume due to the system T=A (10), and the following expression for Fg the force per unit surface at any surface of discontinuity in cf) F, = -[UdXUi (11), the notation being the same as in equation (.5) § 36. Moreover if be self-conjugate the forces both throughout the volume and at surfaces of discontinuity are producible by the stress dX -f //jS8i7(^.V,ds. where of course the element dS is taken twice, i.e. once for each face. But 8 W = - (work done by the system F, F, of forces) = JJSSnT^s + JIJSBvFds, where the element ds is taken only once. Equating the coefiicients of the arbitrary vector St) for each point of space we get the required equations (10) and (11). 39. We must then put B W where W is given by equation (8) in the form given in equation (9). We must first define B when applied to a function of the posi- tion of a point. Suppose by means of the small displacement St; § 40.] ELECTRICITY AND MAGNETISM. 61 any point P moves to F. Then Q being the value at P, before the displacement Sij, of a function of the position of a point, Q + 8Q is defined as the value of the function at P' after the displace- ment. Thus even in the neighbourhood of a surface of discon- tinuity BQ is a small quantity of the same order as Brj. Now the charge within any space, that is the quantity of foreign electricity within that space will not be altered by the strain. .-. S(Dd%)=0, S( in equations (10) and (11) § 38, (fiO)==- FDa)E/2 - ^Qww (21). This is a self-conjugate function so that as we saw in § 38 it is a stress which serves to explain forces both throughout the volume of the dielectric and over any surfaces of discontinuity in D or E*. The force in particular cases. 41. Let us first consider that part — FBAE/2 of the force (equations (10) § 38 and (21") § 40) which does not depend on the variation of K with the shape of the body. Suppose our dielectric is homogeneous and electrically isotropic so that K is a simple constant scalar. In this case 4^-7rD = -KVv (22), by equations (1) and (4) § 35. Therefore by equations (5) and (6) § 36, ^■n-D = KV\ (23), 47r<7 =-K[SUdl'^v]a+i, (24). From these we at once get by the theory of potential that Kv = JJJuDds + ffuads (25). From this we know by the theory of potential that at the surface where the charge a resides Vv is discontinuous only with regard to its normal component and at all other points is continuous. Thus VVa = '^Vi, + xU'd2a and by equation (24) x = iira/K so that (WrK {Vva - VVi) = crUdta, whence (4SfE {dKjd log m) E = fcEVSTr. Thus the force - ^Q_wi^ [equations (21) § 40 and (10) § 38] result- ing from the change of K with pure strain is in the case we are now considering -iVEV8^ (30), and is* therefore, according as A; is positive or negative, in the direction of or that opposite to that of the most rapid increase of the square of the electromotive force. Thus even in the case of a fluid dielectric which has no internal charge but which forms part of a non-uniform field of (electromotive) force the surfaces of equal pressure and therefore the free surface will if originally plane no longer remain so. * This is the same result as Helmholtz's on the same assumption Wiss. Ahli. I. p. 798. § 44.] ELECTRICITY AND MAGNETISM. 65 Nature of the Stress. 43. We have seen that the stress which serves to explain the electrostatic forces is that given by equation (21) § 40, viz. ^0) = - FDwE/2 - ^Qwo) (21). Let us first consider the part — FD(uE/2 which does not depend on the variation of K. Putting w first = Z7D and then = He we get UB = TDTE. C/'E/2, ^UE = TDTE.UI)I2. Therefore putting w first = any multiple of UH + UE and then = any multiple of CTD — UE we get o) = TDrEw/2, ^6) = - TTtTEw/2. Lastly, since - FD«E = coSDE - D/SfoE - EStoTi, we see that if we put w = any multiple of FDE 4>o) = to8DEj2 = - caw. Thus we see that the stress now considered is a tension along one of the bisectors of D and E (the bisector of the positive direc- tions or the negative directions of both) = 2'DTE/2, an equal pressure along the other bisector and a pressure = w perpendicular to both these directions. When D is parallel to E this at once reduces to Maxwell's case, viz. a tension in the direction of E and a pressure in all directions at right angles each = w. 44. We have now to consider the other part of the stress, viz. (o = ^a^a)SEK,E/8'ir (31). If we assume that if is a function of the density (m) of the medium only we shall have dK/de = dK/df= dK/dg = - dKjd log m = - k, say, and dK/da = dK/db = dK/dc = 0, as in § 42. Here however k is not in general a mere scalar but a self- conjugate linear vector function. We have then in this case a) = - toSEkElSv, M. 66 ELECTRICITY AND MAGNETISM. [§ 44. which is a hydrostatic pressure or an equal tension in all direc- tions according as SEkE is positive or negative. In this case the 36 coordinates of ^Q[ia> . K^ reduce to the 6 oi k for each point of space. A more general assumption is that SKg (§ 40) depends only on the elongations in the directions of the principal axes of K. Taking i, j, k as unit vectors in these directions we again have dKlda = dK/db = dK/dc = 0, and thus i = iSE (dK/de) E/Stt, and similarly for j and k, so that the principal axes of the stress now considered are the principal axes of K. 45.* The most natural simple assumption for solid dielectrics seems to me to be that the medium is electrically isotropic before strain, and also isotropic with regard to the strain in the sense that if the strain be, so to speak, merely rotated, SKg will sufifer exactly the same rotation. We may treat this problem exactly as we did (§ 20) that of stress in terms of strain for an isotropic solid. Thus splitting up B^fr into its principal elongations, i.e. putting B^|ra> = — XBeiSlO), we shall get, as in § 20, BK,o} = - (a - /3) tSeiSiay + /Sto28e = (O - /3) Byjrco - ^(oS^Syfr^. But SKgO) = — SB^fr^^(JJ^^.K^a by equation (16) § 40, so that from equation (31) - 87r + j8E' X (solid angle subtended by shell at point) (46). Thus if P be a point on the positive side of the shell and P' a point infinitely near P but on the negative side Potential at P — Potential at P' = 47r^, or what comes to the same thing -j SUdp = i-ir^. This integral may be taken along any path, e.g. along a path which nowhere cuts the shell. The same integral is true if H be the magnetic force due to a whole field of which the shell is only one of several causes, for the part contributed by the rest of the 70 ELECTRICITY AND MAGNETISM. [§ 50. field is zero on account of the infinite proximity of P and P'. For future use in electro-magnetism observe that this statement cannot be made if for H in the integral be substituted B. 51. The condition that any magnet can be divided up into such shells is at once seen to be that I can be put in the form I = V<^ (47), where is some scalar. In this case the potential is by equation (33) n = - JJJSlVuds = -JJJSV<}>Vuds, or by equation (9) § 6 n = - fJ,f,SdXVu + JfJ'!7\d%, i.e. n = -JJ,f>SdlVu+4!'n-Vud%, or by equation (9) § 6 A=jJVdlVu (49). 52. The potential energy of a magnetic shell of strength tj) placed in a field of potential £1 is of importance. We see by equation (35) that it is w=- jfSdi.vn. If then the magnets which cause il do not cut the shell anywhere, so that — Vfl = VA, we shall have Tf = JjSdXB (50), or W=JSdpA (51), by equation (8) § 6. Suppose now that A is caused by another shell of strength '. Then by equation (49) A = f JjVdt'Vu = f Jvdp', by equation (8) § 6. Thus finally the potential energy M of these two shells is given by M = ,f>4>' JJuSdpdp (52). 53. The general theory of induced magnetism when once the proposition (given in equation (42) § 48) that [SdSB]a+j is zero is § 55.] ELECTRICITY AND MAGNETISM. 71 established, is much the same whether treated by Quaternion or Cartesian notation. We shall therefore not enter into this part of the subject. Electko-magnetism. General Theory. 54. We now propose to prove the geometrical theorems con- nected with Maxwell's general theory of Electro-magnetism by means of Quaternions. We assume the dynamical results of Chaps. V., VI. and VII., and the first six paragraphs of Chap. VIII. of the fourth part of his treatise. These assumptions amount to the following. Connected with any closed curve in an electro-magnetic field there is a function p = -JSAdp (53), where A is some vector function at every point of the field. The function p has the following properties. If any circuit be made to coincide with the curve the generalised force acting upon the electricity in the circuit is E = -p (54). Again, if there be a current of electricity 7 flowing round this circuit, the generalised force X, corresponding to any coordinate x of the position of the circuit due to the field acting upon the con- ductor, is X = ydplda; (55). 55. The first thing to be noticed is that p can be transformed into a surface integral by equation (8) § 6 above. Thus p = -JJSBdl (56), where B = FVA (57), so that (SVB = 0. Next we see by the fundamental connection (§ 34 above) between the e.m. F. E and electricity, that £ must equal the line integral of E round the circuit, or E=-JSEdp (58). 72 ELECTRICITY AND MAGNETISM. [§ 55. We are now in a position to find E in terms of A and B, i.e. of A. 56. The rate of variation of p is due to two causes, viz. the variation of the field (A) and the motion of the circuit {p). In the time ht then there will be an increment 8A in A and an incre- ment Sd2 in d2 to be considered. Thus pht = -JSBAdp - fJSBM^. [This amounts to assuming that JSSAdp = JJSSBdl,, which of course is true by equation (8) § 6.] Now when the circuit changes slightly we may suppose the surface over which the new integral extends to coincide with the original surface and a small strip at the boundary traced out by the motion (pSt) of the boundary. Thus Sd2 ig zero everj'where except at the boundary and there it = V(pBt) dp, so that pSt = - JSBAdp + Bt JSpBdp, whence dividing by Bt p=JSi-A+VpB)dp (59), but by equations (54) and (58) p =jSEdp. Thus E = - A + F/.B - V-f (60), where i^ is a scalar and — Vi/r is put as the most general vector whose line integral round any closed curve is zero. 57. We now come to the mechanical forces exerted on an element through which a current C per unit volume flows. We see by equation (55) that the work done by the mechanical forces on any circuit through which a current of magnitude y flows in any small displacement of the circuit equals y x the increment in p caused by the displacement. Give then to each element dp of the circuit an arbitrary small displacement Bp and let F' be the mechanical force exerted by the field upon the element. Thus as in last section -JSF'Bp = yBp = - yJJSBBd'Z = - yJSBpdpB. Thus the force F' on the element dp is yVdpB. But we may suppose this element to be an element ds of volume through which the current C flows. Thus if for ydp we write Cds, and for F', Td%, where F is the force per unit volume exerted by the field, we get F= FCB (61). § 58.] ELECTRICITY AND MAGNETISM. 73 58. So far we have been able to go by considering the electric field as a mechanical system, but to go further (as Maxwell points out) and find how B or A depends on the distribution of current and displacement in the field we must appeal to experiment. It has been shewn by experiment that a small circuit produces exactly the same mechanical effects on magnets as would a small magnet, at the same point as the circuit, placed with its positive pole pointing in the direction of the positive normal to the plane of the circuit when the positive direction round the circuit is taken as that of the current*. Moreover the magnetic moment of the magnet which must be placed there is proportional to the strength of the current x the area of the circuit. Further, the effect of this circuit upon other such small circuits is the same as the mutual effects of corresponding magnets. We have now only to consider a finite circuit as split up in the usual way into a number of elementary circuits to see that a finite circuit will act upon magnets or upon other circuits exactly like a magnetic shell of strength proportional to the strength of the current and boundary coinciding with that of the circuit. The unit current in the electro-magnetic system is so taken as to make this propor- tionality an equality. The one difference between the circuit and the magnetic shell is that there is no discontinuity in the magnetic potential in going round the circuit, so that by § 50 above the line integral of H round the circuit will be 47r x the strength of the current. In symbols fSHdp = 4!-7rJJSCdl for any curve, so that by equation (8) § 6 47rC = FVH (62), whence 5fVC = (63), which of course is a direct result of our original assumption that electricity moves like an incompressible fluid. Maxwell tacitly assumes this by making the assumption that only one coordinate is required to express the motion of electricity in a circuit. * See § 1 above for the convention with respect to the relation between the positive side of a surface and the positive direction round its boundary. Hitherto we have had no reason for choosing either the right-handed or the left-handed screw as the type of positive and negative rotation. But to make the statement in the text correct we must take the former. 74 ELECTRICITY AND MAGNETISM. [§ 59. 5D. We are now in a position to identify the B we are now using with the magnetic induction for which we have already used the same symbol. We see by equation (50) § 52 that the mechanical force on the shell corresponding to any coordinate x is -(ftdj^SdfB'jdx, where B' is the magnetic induction ; and by equations (53) and (55) that the force on the corresponding electric circuit is - <}>dfJ8dtB/dx, therefore B = B' wherever there is no magnetism. And where there is magnetism B is not = H for jSVB = 0, as we have seen. Thus B = B' at all points. In other words the two vectors are identical and we are justified in using the same symbol for the two. This practically ends the general theory of electro-magnetism. We content ourselves with one more application of Quaternions in this subject. We give it because it exhibits in a striking manner the advantages of Quaternion methods. Electro-magnetic phenomena explained by Stress. 60. In § 46 equation (35) we have seen that the potential energy of a magnetic element = Ids in a magnetic field is SlFLds when H has a potential. Maxwell assumes that the same ex- pression is true whether H have a potential or not. Assuming this point* with him we can find the force and couple acting on the medium and a stress which will produce that force and couple. The force and couple due to the magnetism of an element * I do not defend the legitimacy of this assumption. It seems to me bold to assume that a magnet possesses any such thing as potential energy in a field which has no potential. If we assume S and its derivatives to be continuous throughout our typical element ds of volume containing a great number of molecules (both material and magnetic) the force on a magnetic molecule fi consisting of two poles is - Sjnv- H and the force per unit volume - Sly. H, which is only identical with the expression - ViSlHj obtained below when H has a potential. With this expression a stress cannot be found that produces the force. If, however, B and its derivatives be not assumed continuous in this manner the force on the magnet fi. is quite inde- terminate whether the magnetic pole or the molecular current view of magnetism betaken, unless it be specified in what way the poles and currents are distributed in the element of volume. § 60.] ELECTRICITY AND MAGNETISM. 7 is obtained by giving the element an arbitrary translation and rotation and assuming that the work done by this force and couple = the decrement in the potential energy of the element. Thus the force per unit volume is — V^/SIHi for the decrement in the potential energy due to a small translation Sp is SBpV^SlHi. Similarly the couple M is given by M=7IH (64) for the decrement — (SMSco in the potential energy due to a small rotation Sto is - SVSmI . H = - SSw FIH. The total force F per unit volume is the sum of that just given and that given by equation (61), so that F= FCB-Vi/SIH,. Therefore by equation (62) 4nrT = FF^H . B - 47rV,SlH, = - H,SV,B + V.SBH. - 47rV,,SlH, = -Hi/SV,B + Vj,SfHH,. Now /SVB = so that H^SV^B = H/SfAB, and again V^,SHHj = V(ff)/2, so that P = ^A (65), where Stt^w = - 2H5fa)B + bH" (66). From this we get 87rV^^= 2FBH = SttFIH, so that M = F?^^ (67). From these two results (equations (65) and (67)) we see by § 13 above that the stress /dt + R) = 0, so that djdt + R = H (12), where His a, function of t only. In next section we shall obtain in the case of an infinite fluid a generalisation of this which I believe has not hitherto been obtained. Here we have made the assumption that if Wa = at one epoch it will be so always. This we shall prove later. 66. Greenhill next considers the case of steady motion. In this case da/dt = 0, so that equation (11) becomes 2Ve(T + VB = (13), and therefore the surface R = const, contains both vortices and stream-lines and the relation dRjdn = Iqm sin 6 given on p. 446 of Greenhill's article is the natural interpretation of our equation. So far we have been going over much the same ground as § 67.] HYDRODYNAMICS. 79 Hicks, but now we enter upon applications of Quaternions that I think have not been made before. G7. Greenhill next considers rotating axes and finds the form of the equations of motion when referred to these. Let ==-SS7ff/D, and we get d{p + cVVa .jD)dt = a -cSVaV .a .jB (26). This shews that p' = p+c WajD is the variable vector of a particle, for the equation asserts that the time-flux of p' = the velocity at p'. Thus again we get the laws of vortices given in §70. Irrotational Motion. 74. We use this heading merely to connect what follows with what Greenhill has under the same on p. 449. It is not very appropriate here. For irrotational motion we have seen that we may put a = V (27). If the fluid be incompressible we have further = SV (28). Let T be the kinetic energy of this liquid. Thus -2T= DfJI{V4>yd% = DJJ8d^V - DfJf(f>V'4>d% by equation (9) § 6 above. Thus by equation (28) T=-^DJJ<},Sdl.V(l> (29). Hence we see that if we have a vector o- which satisfies the equations Wa = and SVa = throughout any singly-connected space, or more generally satisfies the second of these equations owing wholly or in part to my being indebted to other old papers of Prof. Tait, or whether in writing his 3rd ed. he has arrived independently at the same treatment, but am inclined to the latter belief.] § 76.] HYDRODTKAMICS. 85 throughout any space, and also has zero circulation round any closed curve in that space*; and further satisfies the equation Sd^a = at the boundary ; then the function or o-= at all points of the space, for each element of the integral being essentially negative must be zero. 75. The following important theorem now follows : If there are given ; at every point of any region the convergence S^a and the spin Wa; at every point of the boundary the normal velocity (and therefore SdXa), and the circulation round every cavity which increases the cyclomatic number of the region ; then the motion (as given by be the linear and angular velocities respectively of the moving solid. Let us put ^ = -Sa-y}r-Smx (30), where <^ is the required velocity potential of the liquid, and i^ and j^ are two vector functions of the position of a point independent of a and lo. Let us see whether i/r and x can be found so as to satisfy these conditions, a and a are of course assumed as quite arbitrary. The conditions are first the equation (28) of continuity V'(j, = (28), * To insure that is single valued and so, that the part of the surface integral ol equation (29) due to " barriers " is zero. 86 HYDRODYNAMICS. [§ 76. which gives and second, the equality of the normal velocity —SUd^V^ of the liquid at any point of the boundary with that of the boundary at the same point. Thus for a fixed boundary So- (Sd-ZV .y!r)+Sm {SdXV . x) = 0, whence on account of the arbitrariness of a- and &> SdSV.-f = £fdSV.x=0 (32). For a moving boundary again we have Sdl,Vp), or So- (dS + Sd%V .f) + S(o( VpdZ + 5dSV . x) = 0, which gives dX+SdtV .ylr = (33), rpd%+SdtV.x = (34). Now it is well known that yjr and ^ can be determined to satisfy all these conditions. In fact x being a coordinate of either ifr or ;^ these conditions amoupt to : — V^x = throughout the space, and SdSNx = given value at the boundary. 77. We do not propose to find yfr and ^ in ^'^y particular case. We leave p. 454 of Greenhill's article and go on to p. 455, i.e. we proceed to find the equations of motion of the solid. For this purpose we require the kinetic energy of the system. Calling this T we shall have T=- Sffla-12 - Sco^a- - 80)0,0,12 (35), where S, il and . When yfr and x are known 2, * and fl are all known. We will obtain the expressions for them, and for simplicity will assume that the origin is at the centre of gravity of the moving solid. Let M be the mass of this last and fiw (where, as is well-known, fj, is a, self-conjugate linear vector function) the moment of momentum. Thus by equation (29) 2T=- DJJSdtV(p - Ma' - 8w/j,w. * For let A{ff,fia> where the surface integral must be taken only over the moving boundary. Thus noting that 2 and II are self-conjugate SX. = if\ - ^DjJ (fSXdl. + dlS\yfr) (36), n.\ = fi\- ^DfJ ixSXpdl, + VpdlSXx) (37), - SX^X = ^DJJ {SkxSK'd^ + SXpdl^SX'yp-) (38). These surface integrals can be simplified, for in each of these equations the first surface integral equals the second. In equation (36) in order to prove this we have merely to put for d%, — Sd^V.-\jr (equation (33)) when we shall find by equation (9) § 6 above (since we may now suppose the integrations to extend over the whole boundary), that JJyIrSXdl, = - JJJyfr,SVy.^Xy{r^d% =JJdtSXf. Similarly for equation (37). Again in equation (38), jfSXxS^'d^ = -JjfSV,V,SXx,SX'y{r^d$ =}JSXpdlSX'ylr. Thus finally for 2, 11 and $ ^X = MX- DJJfSXdX = MX - DjjdXSXylr (39), nx = fiX- DjJxSXpdl = ixX- DjJVpdXSXx ■ . .(40), - ^\*\' = DJJSXxSX'dt = BJJSXpdXSX'yIr (41). This last may be put in the following four forms $'\ = - DfJdtSXx = - DJH SXpdX (42), 4>\ = - Dffx^XdX = - DJSVpdXSXf (43). Thus assuming that ■yjr and x are determined we have found T as a quadratic function of a and to. 78. If P, Cr be the linear and angular impulses of the system respectively our equations of motion are P+FbP = F, G+FwG-M, by the footnote to § 67 above. Here F and M are the external force and couple applied to the body. Now at the instant under consideration T = J^T,G = J7T. But if the origin had been at the point - p, P would still be „VT whereas G would be J?T+ VpF. 88 HYDRODYNAMICS. [§ 78. Thus dififerentiation with regard to t does not affect the form of P but does that of G. In fact using the last stated values of P and G along with the last two equations and eventually putting /5 = ff, p = we get dJ^TIdt+Vm,VT=T (44), dJ7T/dt+ro,J^T+ Va,VT=m (45). Now from equation (35) we see that ^VT=%a + ^'a> (46), J^T=<^a + nco (47), whence from equations (44) and (45) Sd- + *'«+Fa)2o-+F(a4>'(u = P (48), * (56), where <^ satisfies the equations V=^ = (57), and Sd2V<^ = (Sd2( can be determined so as to satisfy 90 HYDRODYNAMICS. [§ 80. these two equations. Therefore the problem under discussion always admits of solution. The above is equivalent to Lamb's §§ 128 — 131. His § 130 is the natural interpretation of the equation i-rra = - Iff WuVad%. His § 132 is seen at once from equation (50) above. For in the case he considers we, in accordance with § 7 above, take each side of the surface of discontinuity as a part of the boundary. Now [Sdl,a]a+b = so that for this part of the boundary we can leave out the part Sd%a and we get 47ro- = fJVVuVdXa- - JfJV^uVffdB (59), so that if we regard [FdScr]^^ as — 2 x an element of a vortex, we get the same law for these vortex slieets as for the vortices in the rest of the fluid. 81. The velocity potential due to a single vortex filament of strength dd is at once obtained by putting in equation (50) for FVffds, 2d6dp. Thus calling o-' the part of the velocity due to the filament 27r Putting fj. and v for any two vectors perpendicular to a we see that the normal Ua to any interface becomes Ux"'o hy the strain. Hence the interface « experiences a shear (strain) which equals the resolved part of x^<^ perpendicular to the vector x'"'« equals resolved part of (x - x") U<» perpendicular to x'"'^. Now when X is the strain function due to a small displacement adt, by § 12 X(o = (o-Sq}V .adt (66), • KeUand and Tait's Quaternions, chap. x. equation («). We have already used a particular case of this in § 15, above. 92 HYDRODYNAMICS. [§ 83. whence x'~^a) = m + V^Seoaj^dt (67), and we see that the shear is the resolved part of perpendicular to co, i.e. parallel to the interface o). From this we see by our assumption concerning viscosity that any element of the fluid is subject to a stress <^ given by o} = - Rq} - fj, (S(oV . a- + Vj/Sojo-,) R being a scalar. Now we define the pressure p by putting 3^ = S?t= SB + 2fj.SVa- .-. a) = -po) + ^fKo8^y^, so that f=- JffDSffFd% - ///,Sdl + JJJSa^4,V^d% (70). If now if) be expressed in terms of p and fi we have f depend- ing on P, p and fi. The part of t depending on F and p repre- sents energy stored up as potential energy of position and potential § 84.] HYDRODYNAMICS. 93 energy of strain respectively, but the part depending on /a repre- sents a loss of energy to the system we are considering the energy being converted into heat. Thus putting <^ =p 4- sr we have by equation (68) where i/r is given by equation (75) below. Thus we see that the rate of loss of energy is JJScrwdt - fJJSa^vrV^dS (72). The surface integral is the work done by viscosity against the moving fluid at the boundary and the volume integral is con- sidered due to the work done against the straining of the fluid. Thus we put i^=-iS'