ELECTRICAL AND OPTICAL 
 WAVE-MOTION 
 
CAMBRIDGE UNIVERSITY PRESS 
 
 C. F. CLAY, MANAGER 
 ILotrtJon:- FETTER LANE, B.C. 
 flFtJinburgf): 100 PRINCES STREET 
 
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 THE MARUZEN-KABUSHIKI-KAISHA 
 
 All riyhtt reserved 
 
THE MATHEMATICAL ANALYSIS 
 
 OF 
 
 ELECTRICAL AND OPTICAL 
 WAVE-MOTION 
 
 ON THE BASIS OF MAXWELL'S EQUATIONS 
 
 BY 
 H. BATEMAN, M.A., Ph.D. 
 
 Late Fellow of Trinity College, Cambridge ; 
 Johnston Research Scholar, Johns Hopkins University, Baltimore 
 
 Cambridge : 
 
 at the University Press 
 
 1915 
 
Cambridge: 
 
 PRINTED BY JOHN CLAY, M.A. 
 AT THE UNIVERSITY PRESS 
 
PREFACE 
 
 THIS book is intended as an introduction to some recent 
 developments of Maxwell's electromagnetic theory which 
 are directly connected with the solution of the partial differential 
 equation of wave-motion. The higher developments of the 
 theory which are based on the dynamical equations of motion are 
 not considered at all. Even with this limitation the subject is a 
 vast one, and to bring the work of perusing the literature within 
 my power I have omitted an account of the modern theory of 
 relativity which has been expounded very clearly in several recent 
 publications. 
 
 For a thorough understanding of the present subject a very 
 extensive knowledge of mathematics is necessary, but there are 
 parts of the subject in which a reader with only a limited 
 mathematical equipment may soon feel at home and perhaps do 
 useful original work. With the idea of enabling suoh a reader to 
 obtain a quick grasp of the nature of the subject and the results 
 obtained, I have thought it advisable to state without proof a 
 number of relations of which adequate demonstrations can only 
 be obtained by means of complicated and difficult analysis. 
 I have also endeavoured to keep the analysis as elementary as 
 possible, but in some places where the work is perfectly straight- 
 forward a few details are omitted. 
 
 The book is far from being a complete treatise on the subject, 
 for I have not given any existence theorems to show that the 
 solutions of certain problems exist and are unique, and no 
 attempt has been made to enter into the details of numerical 
 computations. There are many parts of the subject indeed to 
 which a pure mathematician might make useful additions; in 
 particular, I might direct attention to p. 21, line 2, and p. 101, 
 where there are one or two matters which require further 
 discussion. 
 
VI PREFACE 
 
 Chapter VIII and paragraph 5 contain some of my own con- 
 tributions to the subject. At present there seem to be several 
 different directions in which future developments may be made, 
 and so it seems unwise to give a hasty judgment concerning the 
 physical significance of the results. Ideas which naturally 
 present themselves are that the aether can be regarded as built 
 up from singular curves of the type considered in 43, and that 
 41 and 44 may throw some light on the question of the difference 
 between positive and negative elementary electric charges. I 
 hope to discuss an hypothesis relating to the first idea in a future 
 note, but am unable to give any support at present to the 
 second idea. 
 
 I gratefully acknowledge my indebtedness to Sir Joseph 
 Larmor who read the manuscript before it was revised and made 
 some helpful suggestions, to Prof. Ames who read the greater 
 portion of the manuscript, to Prof. Morley and Mr Hass6 who 
 helped me with their advice and vigilance in reading the proof- 
 sheets, and to the officers and staff of the University Press for 
 their careful work and constant consideration shown in matters 
 connected with the printing. For the correctness of the new 
 formulae and examples I alone am responsible ; if any errors are 
 discovered I shall be grateful if my readers will inform me. 
 
 HARRY BATEMAN. 
 
 October, 1914. 
 
CONTENTS 
 
 CHAP. PAGE 
 
 I. FUNDAMENTAL IDEAS 1 
 
 ADDITIONS AND CORRECTIONS 
 
 p. 28. Formula (30) is due to Lame". Cf. A. E. H. Love, The Mathematical 
 
 Theory of Elasticity, 2nd edition, p. 55. 
 p. 101. An asymptotic expression for T m n (s) when n is a large positive integer 
 
 can be derived from a formula given by L. Feje"r in 1909. This 
 
 formula is accessible in a paper by 0. Perron, Arkiv der Mat. u. 
 
 Phys. (1914). 
 
 p. 118. The factor c in front of the double integrals should be omitted, 
 p. 120. Delete the minus sign in the second of equations (277). 
 p. 127. Line 8. This statement is incorrect, the equations are poristic, the 
 
 special case is the only one which can occur, 
 p. 132. Line 20. On account of the porism just mentioned, the hope may be 
 
 abandoned. 
 
 p. 150. Ex. 13. For equations (10) of 5 read equations (2) of 2. 
 p. 154. Ex. 24. The equation should read 
 
 9 <r . dV . dV~ d r. . d 
 
VI PREFACE 
 
 Chapter vni and paragraph 5 contain some of my own con- 
 tributions to the subject. At present there seem to be several 
 different directions in which future developments may be made, 
 and so it seems unwise to give a hasty judgment concerning the 
 physical significance of the results. Ideas which naturally 
 present themselves are that the aether can be regarded as built 
 up from singular curves of the type considered in 43, and that 
 41 and 44 may throw some light on the question of the difference 
 between positive and negative elementary electric charges. I 
 hope to discuss an hypothesis relating to the first idea in a future 
 T^fo Knt. am nnahlft t.n jrivft anv suDDort at Dresent to the 
 
CONTENTS 
 
 CHAP. PAGE 
 
 I. FUNDAMENTAL IDEAS 1 
 
 II. GENERAL SURVEY OF THE DIFFERENT METHODS OF SOLVING 
 
 THE WAVE-EQUATION 25 
 
 III. POLAR COORDINATES 35 
 
 IV. CYLINDRICAL COORDINATES 69 
 
 V. THE PROBLEM OF DIFFRACTION 82 
 
 VI. TRANSFORMATIONS OF COORDINATES APPROPRIATE FOR THE 
 
 TREATMENT OF PROBLEMS CONNECTED WITH A SURFACE OF 
 
 REVOLUTION . 95 
 
 VII. HOMOGENEOUS SOLUTIONS OF THE WAVE-EQUATION . . .110 
 VIII. ELECTROMAGNETIC FIELDS WITH MOVING SINGULARITIES . 115 
 
 IX. MISCELLANEOUS THEORIES . . . . . .141 
 
 LIST OF AUTHORS QUOTED . . . . . . . 155 
 
 INDEX . 158 
 
CHAPTER I 
 
 FUNDAMENTAL IDEAS 
 
 1. The fundamental equations for free aether. 
 
 In Maxwell's electromagnetic theory the state of the aether 
 in the vicinity of a point (x, y, z) at time t is specified by means 
 of two vectors E and H which satisfy the circuital relations* 
 
 
 and the solenoidal or sourceless conditions 
 div#=0. 
 
 If right-handed rectangular axes are used the symbol f rot H 
 denotes the vector whose components are of type 
 
 the three components of H being H x , H y) H z respectively. 
 The symbol div# denotes the divergence of H, i.e. the 
 quantity 
 
 dffz dHy dH z 
 
 W~Hy"^W 
 
 The vector E is called the electric displacement or electric 
 force and H the magnetic force. The quantity c represents the 
 
 * The equations are written in the symmetrical form in which they were 
 presented by 0. Heaviside, Electrical Papers, Vol. 1, 30, and H. Hertz, 
 Electric Waves, p. 138. Sir Joseph Larmor points out that a set of equations 
 equivalent to these was first used by MacCullagh in 1838 as a scheme 
 consistently covering the whole ground of Physical Optics, Collected Works of 
 James MacCullagh (1880), p. 145. 
 
 t We use here the units and notation employed in Lorentz's The Theory of 
 Electrons, Ch. i, except that large letters are used to denote vectors and E is 
 written in place of D. Maay writers "*} IA avmho^ ff rZ instead of rot. 
 
 B. 1 
 
... . , ^FUNDAMENTAL IDEAS [CH. 
 
 velocity of propagation of homogeneous plane waves and is 
 commonly called the velocity of light ; we shall assume it to be 
 a constant, although in the most recent speculations it is treated 
 as variable*. 
 
 Some of the modern writers on the theory of relativity 
 maintain that the introduction of the idea of an aether is 
 unnecessary and misleading. Their criticisms are directed 
 chiefly against the popular conception of the aether as a kind 
 of fluid or elastic solid which can be regarded as practically 
 stationary while material and electrified particles move through 
 it. This idea has been very helpful as it presents us with 
 a vivid picture of the processes which may be supposed to take 
 place, it also has the advantage that with its aid we can attach 
 a meaning to the term absolute motion, but herein lies its 
 weakness. Larmor, Lorentz and Einstein have shown, in e fact, 
 that the differential equations of the electron theory admit of 
 a group of transformations which can be interpreted to mean 
 that there is no such thing as absolute motion. 
 
 If this be admitted, the popular idea of the aether must be 
 regarded as incorrect, and so if we wish to retain the idea of a 
 continuous medium to explain action at a distance we must 
 frankly acknowledge that the simplest description we can give 
 of the properties of our medium is that embodied in the 
 differential equations (1). 
 
 If we abandon the idea of a continuous medium in the 
 usual sense only two ways of explaining action at a distance 
 readily suggest themselves. We may either think of the 
 aether as a collection of tubes or filaments attached to the 
 particles of matter as in the form of Faraday's theory which has 
 been developed by Sir Joseph Thomson and N. R. Campbell ; 
 or we may suppose that some particle or entity which belonged 
 to an active body at time t belongs to the body acted upon at a 
 later time t + T. From one point of view these two theories are 
 the same, for if particles are continually emitted from an active 
 
 * A. Einstein, Ann. d. Phys. Vol. 35 (1911), p. 898 ; Vol. 38 (1912), pp. 355 
 and 443. M. Abraham, Phys. Zeitschr. (1912), pp. 15, 310314, 793797 ; 
 Ann. d. Phys. (1912), pp. 444 and 1056 ; Fifth International Congress of Mathe- 
 maticians, Proceedings, Vol. 2, p. 256. 
 
I] THE FUNDAMENTAL EQUATIONS FOB FREE AETHER 3 
 
 body they will form a kind of thread attached to it. The first 
 form of the theory is, however, more general than the second. 
 
 At present we are unable to form a satisfactory picture of 
 the processes that give rise to, or are represented by, the vectors 
 E and H. We believe, however, that some points may be made 
 clear by studying the properties of solutions of our differential 
 equations. 
 
 It will be seen from the investigations of Chapter VIII 
 that the mathematical analysis connected with these equations 
 is suitable for the discussion of three distinct theories of the 
 universe, which may be described briefly as follows: 
 
 Aether Matter 
 
 Continuous medium. Aggregates of discrete par- 
 
 ticles. 
 
 Discontinuous medium con- An aggregate of discrete 
 
 sisting of a collection of tubes particles attached to the tubes, 
 or filaments. 
 
 Continuous medium. An aggregate of discrete 
 
 particles to which tubes are 
 attached. 
 
 The last theory may be supposed to include that form of 
 the emission theory of light in which small entities are projected 
 from the particles of matter under certain circumstances and 
 produce waves in the surrounding medium. This theory might 
 be justly ascribed to Newton*. 
 
 For other theories of the aether the reader is referred to 
 Prof. E. T. Whittaker's recent workf A History of the Theories 
 of the Aether. 
 
 In the first part of this book the analysis is adapted almost 
 entirely to the first theory, the high development of which we owe 
 to the pioneer work of Maxwell, FitzGerald, Hertz, Rayleigh, 
 Heaviside, J. J. Thomson, Lorentz and Larmor. The other 
 theories have not yet received much attention but it is hoped 
 
 * A form of the theory in which the entities are electric doublets has been 
 developed by W. H. Bragg and applied to the X and y rays. British Association 
 Reports (1911), p. 340. 
 
 t Dublin Univ. Press ; Longmans, Green and Co. (1910). 
 
 12 
 
4 FUNDAMENTAL IDEAS [CH. 
 
 that the analysis of Chapter vm will lead to further develop- 
 ments so that a comparison can be made between the different 
 theories. It is quite likely that one theory will be enriched by 
 the developments of another. 
 
 2. Electromagnetic fields. 
 
 For many purposes it is convenient to work with a complex 
 vector* M=HiE, where i = V"^l and the ambiguous sign 
 is independent of the ambiguity which occurs in the determi- 
 nation of V 1. The differential equations (1) may then be 
 replaced by the simpler equations 
 
 rotJf= + -'^, divJf=0 (2). 
 
 % c ot 
 
 When a solution of these equations has been found a pair 
 of vectors E and H satisfying equations (1) may be obtained by 
 equating coefficients of the ambiguous sign. In working with 
 an ambiguous sign it must be remembered that when two 
 ambiguous signs are multiplied together the ambiguity is 
 removed. The chief advantage in using the two independent 
 ambiguities + and V 1 is that we can assume that the vectors 
 E and H are the real parts of expressions of the fonn Ae iut and 
 we are at liberty to equate the coefficients of either i or in 
 any of our equations. 
 
 Definition. A solution of the differential equations (2) or (1), 
 which provides us with single-valued vector functions E and H 
 for each space-time point (x, y, z, t) belonging to a certain 
 domain D, is said to define an electromagnetic field in the 
 domain D. 
 
 Since the differential equations are linear the sum of any 
 number of solutions is also a solution. The physical meaning 
 of this is that when two electromagnetic fields are superposed, 
 they are together equivalent to an electromagnetic field. 
 
 Two superposed electromagnetic fields can of course be 
 related to one another in some way. When electromagnetic 
 
 * The use of a complex vector H-iE is recommended by L. Silberstein, Ann. 
 d. Phys. Vols. 22 and 24 (1907) ; Phil. Mag. (6), Vol. 23 (1912), p. 790. He does 
 not, however, use the ambiguous sign. 
 
l] THE FLOW OF ENERGY 5 
 
 waves fall upon an obstacle, a secondary disturbance is produced 
 which depends in character upon the nature of both the primary 
 waves and the obstacle. 
 
 We shall find that in some cases it is possible to find two 
 fields in which the vectors (E, H), (E', H') are connected by the 
 two relations embodied in the equation 
 
 (MM') = M X M X ' + MyM y ' + M z Mz =0 ......... (3) 
 
 for all values of (x, y, z, t) belonging to some domain. 
 
 When this is the case the fields are said to be conjugate 
 within this domain. 
 
 If we use the notation 
 
 we may write 
 
 (if 2 ) = (H*) - (E 2 ) 2i (EH) = /! 2i / 2 , 
 
 where I I and / 2 are two quantities which we shall call the 
 invariants*. It is easy to see that when two conjugate fields 
 are superposed the invariant I I for the total field is the sum of 
 the invariants 7j for the two component fields. Similarly for 
 the invariant / 2 . 
 
 When the invariants are zero over a given domain the field 
 may be called self-conjugate for this region f. 
 
 3. The flow of energy. 
 
 An entity whose volume density! p is a function of (x, y, z, t) 
 will vary in a manner which can be described as a simple flow 
 with component velocities (u, v, w) if the equation of continuity 
 
 is satisfied. This equation implies in fact that there is no 
 
 * They are invariants for the group of linear transformations which leave 
 the electromagnetic equations unaltered in form. Cf. H. Minkowski, Gott. 
 Nachr. (1908); E. Cunningham, Proc. London Math. Soc. (2), Vol. 8(1910), 
 p. 89 ; H. Poincare, Rend. Palermo (1906) ; M. Planck, Ann. d. Phys. Vol. 26 
 (1908). Other invariants are given by these authors. 
 
 f Silberstein calls it a pure electromagnetic wave. 
 
 The limitations to which the idea of density is subject and the question of 
 the continuity of the function p are discussed by J. G. Leathern, "Volume 
 integrals and their use in physics," Cambridge Mathematical Tracts (1905). 
 
6 FUNDAMENTAL IDEAS [CH. 
 
 creation or annihilation of the entity in the neighbourhood of 
 (x, y, z, t). 
 
 Now it is easy to see that the equation of continuity is 
 satisfied in virtue of equations (1) if we put 
 
 and two similar equations. We shall regard p in this case as 
 the volume density of the energy contained in the electro- 
 magnetic field. The vector S whose components are of the 
 type c (EyH z E z H y ) can then be supposed to indicate the rate 
 at which energy flows through the field. Since 
 
 p 2 (c 2 - w 2 - v 2 - w 2 ) = J c 2 (E 2 - H*Y + c 2 (EH) 2 , 
 it appears that energy travels through the field with a velocity 
 which is less than the velocity of light. The velocity c is 
 attained only in the case of a self-conjugate field. 
 
 The vector S was introduced by Prof. Poynting* and is 
 usually called Poynting's vector. The idea of describing the 
 transfer of energy in this way also occurred to Prof. Lamb 
 before the publication of Poynting's work. 
 
 Example. Prove that the equation of continuity may be satisfied by 
 putting 
 
 18(9 
 
 j, rr.TT * F 
 
 pw= c Wt E *~ te ffv+ fy ff *> pc= ~ a# *~ fy *r K* 
 
 where 6 is an arbitrary function. Obtain a similar solution by replacing 
 -K 
 
 4. First solution of the fundamental equations. 
 
 Let us use the symbol flu to denote the Dalembertian*f* 
 of M, viz. 
 
 _ia 2 w__a 2 w a^ &u_id*u 
 
 c*W~W + df + fo* c*dt*' 
 
 * Phil. Trans. A, Vol. 175 (1884), p. 343. See also H. A. Lorentz, The 
 Theory of Electrons, p. 22. 
 
 t This is the name suggested by Lorentz, loc. cit. p. 17. Many writers use 
 Cauchy's symbol D to denote the Dalembertian, but I think is preferable 
 because its form suggests a wave. Murphy's symbol A is also used here in place 
 of the usual symbol V 2 . E. B. Wilson and G. N. Lewis use the symbol O 2 ** 
 to denote the Dalembertian of u. Cf. Proc. Amer. Acad. of Arts and Sciences, 
 Vol. 48 (1912), p. 389. 
 
l] FIRST SOLUTION OF THE FUNDAMENTAL EQUATIONS 7 
 
 and the symbol grad U to denote the vector whose components 
 
 zu du du 
 
 dx ' dy ' dz 
 
 respectively. Let us also use HA, where A is a vector with 
 components A x , A y , A z , to denote the vector whose components 
 are HA r , tlA y , I1A 2 . The equation lu = Q will be called the 
 wave-equation and a solution of this equation a wave-function. 
 A vector function A will be said to satisfy the wave-equation 
 when each of its components is a wave-function, i.e. if HA = 0. 
 We may now satisfy equations (1) and (2) by writing 
 
 ............... (5), 
 
 c ct 
 
 where the scalar potential A = ^ + i<I> and the vector potential 
 L = B + iA satisfy the equations 
 
 nA = 0, n = 0, div + i|^ = ......... (6). 
 
 C ut 
 
 The last three equations may be solved in a general way by 
 writing 
 
 . ,. ~ . 
 
 A = - div G - - -5- 
 c c) 
 
 where the vector (7 = F + ill and the scalar /if satisfy the wave- 
 equation 
 
 lu=0 ........................... (8). 
 
 The solution of equations (1) which is embodied in (5), 
 (6) and (7) is a simple extension of Hertz's solution* and is 
 suggested by Whittaker's solution ( in terms of two scalar 
 potentials. It is clear that the function K drops out when we 
 differentiate to find M and so the electric and magnetic forces 
 depend ony on the vector G. The form of this vector indicates 
 that the electromagnetic field can be regarded as the sum of 
 two partkl fields ; one of these is derived from the vector II and 
 
 * Ann. i. Phys. Vol. 36 (1888), p. 1. The general solution is given by 
 Eighi, Bokgna Mem. (5), t. 9 (1901), p. 1 ; IlNuovo Cimento (5), t. 2 (1901), p. 2. 
 He finds siitable expressions for the vectors II and T in a number of cases. 
 
 t Proc. London Math. Soc. Ser. 2, Vol. 1 (1903). 
 
8 FUNDAMENTAL IDEAS [CH. 
 
 will be called a field of electric type, the other is derived from 
 the function T and will be called a field of magnetic type. 
 
 This resolution of an electromagnetic field into two partial 
 fields is analogous to the one used by H. M. Macdonald* in the 
 study of the effect of an obstacle on a train of electric waves. 
 The component fields are then of such a type that in one case 
 the magnetic force normal to the obstacle vanishes over the 
 surface of the latter, in the other case it is the eleciric force 
 normal to the obstacle that vanishes. The same idea has been 
 used recently by Mief and DebyeJ in the treatment of the case 
 of a spherical obstacle. 
 
 In Hertz's solution we have F = 0, K = and II has 
 components (0, 0, S). 
 
 The components of E and H are consequently given by the 
 formulae 
 
 rr -^ 
 
 ~ 
 
 98 IPS 
 
 Ez = W-c*~W' Hz = 
 
 Hertz uses Euler's wave-function 
 
 S = - sin K (r - ct\ r* = a? + f 
 
 J 
 
 and obtains in this way a theory of his oscillator||. The electric 
 and magnetic forces become infinite at the origin which is there- 
 fore a singularity of the electromagnetic field. A singularity of 
 this type is called a vibrating electric doublet and is regarded as 
 the simplest model of a source of light or electromagnetic waves. 
 
 * Electric Waves, Ch. vi. 
 
 t Ann. d. Phys. Vol. 25 (1908), p. 382. + Ibid. Vol. 30 (1*09), p. 57. 
 
 Periodic solutions representing a disturbance sent out from n-fold poles 
 had been used previously by H. A. Eowland and applied to the elucidation of 
 optical phenomena. Amer. Journal of Mathematics, Vol. 6, p. 359 ; Phil. Mag. 
 Vol. 17 (1884), p. 423. Cf. also Stokes, Cambr. Phil. Trans. (1849). 
 
 || To deal with the case in which the vibrations are damped we assume 
 
 S=- +"(-<*) sin K (r - ct). Cf. K. Pearson and A. Lee, Phil. Trans. 1, Vol. 193 
 (1900), p. 159. 
 
l] FIRST SOLUTION OF THE FUNDAMENTAL EQUATIONS 9 
 
 The solutions of equations (1) which are obtained by superposing 
 elementary solutions of this type are of great importance in 
 physical optics. 
 
 When r is very great the most important terms in the 
 expressions (9) are 
 
 where s = sin K (r ct). All the other terms are of order l/r s or 
 1/r 3 . These expressions give 
 
 Hence at a very great distance from the origin the field is 
 practically a self-conjugate field and so the energy travels with 
 a velocity very nearly equal to the velocity of light. The 
 expressions indicate that Poynting's vector is ultimately along 
 the radius from the origin; now the electric and magnetic 
 forces are at right angles to Poynting's vector and so the 
 vibrations of the light-vector, whether we take it to be the 
 electric or magnetic force, are at right angles to the radius. 
 The waves sent out from the source have, then, the character 
 of monochromatic light at a great distance from the origin*. 
 The amplitudes of the vibrations at points on the same radius 
 are proportional to the quantities 1/r when r is large, and so if 
 the intensity of the light be measured by the square of the 
 amplitude the inverse square law is fulfilled. 
 
 Since the electric force is ultimately at right angles to the 
 radius there is no total charge associated with the singularity, 
 for the charge is equal to the surface -integral of the normal 
 electric force over a large sphere concentric with the origin 
 and this integral is evidently zero. We are consequently 
 justified in regarding the singularity as a doublet and in fact 
 
 * For a fuller discussion see Larmor, Phil. Mag. (5), Vol. 44 (1897), p. 503 ; 
 Aether and Matter, Chap, xiv, where it is shown that energy is radiated from a 
 moving charge only when the velocity of the charge alters in either magnitude 
 or direction. 
 
10 
 
 FUNDAMENTAL IDEAS 
 
 [CH. 
 
 as a simple electric doublet of varying moment as is indicated 
 by the way in which the electric and magnetic forces become 
 infinite*. The axis of the doublet is along the axis of z. 
 
 The electric lines of force due to a vibrating electric doublet 
 have been drawn by Hertz f for various stages of the motion. 
 The general character of the lines of force is indicated in Fig. 1. 
 
 It will be noticed that the. lines are all at right angles to a 
 plane perpendicular to the axis of the doublet. M. Abraham 
 has used a Hertzian doublet to obtain a model of the electro- 
 magnetic field produced by the oscillations in a vertical antenna, 
 the plane just mentioned being supposed to represent the earth 
 which is regarded as a perfect conductor. Zenneck|| has, 
 however, pointed out that when the imperfect conductivity of 
 the earth is taken into account the circumstances of the 
 
 * See 42. 
 
 t Ann. d. Phys. Vol. 36 (1888), p. 1. The case of damped vibrations is 
 considered by K. Pearson and A. Lee, loc. cit. 
 
 I am indebted to the Macmillan Company and A. Gray, Esq., for permis- 
 sion to reproduce this diagram. 
 
 Phys. Zeitschr. Vol. 2 (1901), p. 329 ; Theorie der Elektrizitat, Vol. 2, 
 34; Encyklop. d. Math. Wiss. Band 5, 18. 
 
 || Ann. d. Phys. Vol. 23 (1907), p. 846 ; Phys. Zeitschr. Vol. 9 (1908), p. 50 ; 
 Ibid. p. 553. 
 
I] FIRST SOLUTION OF THE FUNDAMENTAL EQUATIONS 11 
 
 propagation are somewhat different. The spreading of electro- 
 magnetic waves over the earth's surface has been investigated 
 thoroughly by A. Sommerfeld* and his pupil H. v. Hoerschle- 
 mannf, and their results seem to indicate that the imperfect 
 conductivity of the earth is an important factor in directing 
 electric waves and in enabling their effects to be detected at 
 great distances. The ionisation of the air by sunlight is also 
 an important factor, as has been pointed out by J. J. Thomson, 
 W. H. EcclesJ and J. A. Fleming. Marconi's experiments have 
 indicated that the circumstances of propagation are not yet 
 thoroughly understood. No good reason has been given to 
 explain why communications by means of electric waves can be 
 made more easily when the receiving station is in a north or 
 south direction than when the direction is east or west. The 
 curious contrasts in the results obtained with waves of different 
 frequencies in day and night communications are also un- 
 explained ||. 
 
 The use of the vector II instead of the scalar S was 
 recommended by Abraham IF. Von Hoerschlemann has obtained 
 in this way a model of Marconi's bent antenna which gives 
 a directed effect to the radiation. A number of arrangements 
 of Hertzian doublets that can be used to imitate the action 
 of antennae have been described by Fleming**, Larmorf^*, 
 Sommerfeld and Macdonald. 
 
 In the theory of FitzGerald's magnetic oscillator we have 
 
 n = o, r = (o, o, N), 
 
 N being Euler's wave-function. Whittaker's solution is 
 obtained by adding the solutions of Hertz and FitzGerald. 
 
 * Ann. d. Phys. Vol. 28 (1909), p. 665. 
 
 t Jahrb. d. draht. Teleg. Vol. 5 (1912). 
 
 J Proc. Roy. Soc. A, Vol. 87, p. 79. 
 
 British Association Reports, Dundee (1912). See also 0. J. Lodge, 
 Phil. Mag. Vol. 25 (1913), p. 775. 
 
 || See Marconi's address to the Royal Institution, June, 1913. 
 
 H Theorie der Elektrizitdt, Vol. 2,'Ch. i. See also Righi, loc. cit. 
 ** Proc. Roy. Soc. A, Vol. 78, p. 1. 
 tt Ibid, in a footnote to Fleming's paper. 
 t Proc. Roy. Soc. A, Vol. 81, p. 394. 
 Trans. Roy. Dublin Soc. Vol. 3 (1883) ; Scientific Writings, p. 122. 
 
12 
 
 FUNDAMENTAL IDEAS 
 
 [CH. 
 
 5. Second solution of the fundamental equations. 
 It is easy to see that equations (2) will be satisfied if we 
 can find two functions (a, /3) such that 
 
 a(y,*)-*0 3(, 
 
 _ . 
 ~ 
 
 .(10). 
 
 3 (,)- 1> .0.(y,) 
 
 -"*2 == o~7 \ = i 1 "o7 IT 
 
 An electromagnetic field that is specified in this way is 
 necessarily a self-conjugate field, for if we multiply together 
 the two expressions for M x and do the same for M v , M z , we find 
 that M* = 0. A particular pair of functions a, /3 is obtained by 
 putting 
 
 a = x cos + y sin 6 + iz, /5 = x sin 6 y cos ct ...... (11), 
 
 where 6 is an arbitrary constant. To generalise this field we 
 multiply the expressions for M x , M y , M z by an arbitrary 
 function* of a, /3, 6 and integrate with regard to 6\ we thus 
 obtain a very general electromagnetic field in which 
 
 Jf fl 
 
 = + i [ 
 Jo 
 
 , ft, 6) cos 6 dd 
 
 - 
 
 Jo 
 
 .(12). 
 
 The components of the electric and magnetic forces are 
 obtained by equating the ambiguous and unambiguous parts in 
 these equations; it is easy to verify that they are all wave- 
 functions. 
 
 It should be remarked that these definite integrals may 
 give a representation of the electromagnetic field, required for 
 the solution of a problem, only in a certain limited domain of 
 
 * When we speak of an arbitrary function it must be understood that the 
 function may be subject to certain limitations which render the integration and 
 differentiation under the integral sign intelligible operations. 
 
I] SECOND SOLUTION OF THE FUNDAMENTAL EQUATIONS 13 
 
 the variables as, y, z, t ; the integrals may in fact represent 
 discontinuous functions. 
 
 The limits of integration could have been taken to be any 
 other constants instead of and 2?r ; they can also be taken to 
 be functions of #, y, z> t of the type o>, where o> is defined by an 
 equation of the form 
 
 x sin to y cos &> ct = F(co), 
 F being an arbitrary function. 
 
 A suitable pair of functions a, ft is also obtained by putting 
 
 and in this case Poynting's vector is along the radius from the 
 origin. A more general type of electromagnetic field in which 
 this is true is obtained by multiplying the above expressions 
 for the components of M by an arbitrary function of a and @. 
 
 Other pairs of functions a, j3 of a very general nature are 
 obtained in Chap. vm. It should be remarked that in all cases 
 the functions (a, j3) are of such a nature that if F (a, $) is an 
 arbitrary function of a and fi, F satisfies the partial differential 
 equation 
 
 which is of fundamental importance in geometrical optics* and 
 may be called Hamilton's equation. It is found that this 
 equation is also satisfied in many cases by the functions of 
 x, y, z, t which are the limits of a definite integral representing 
 a wave-function, when the function under the integral sign is 
 a wave-function for all values of the parameter with regard to 
 which we are integrating. Thus the function w just defined and 
 
 the function t - (r + r ) which will be used later are solutions 
 c 
 
 of this equation. 
 
 * For another connection between this equation and the electromagnetic 
 equations see A. Sommerfeld and J. Bunge, "Grundlagen der geometrischen 
 Optik," Ann. d. Phys. Vol. 35 (1911), p. 277. 
 
14 FUNDAMENTAL IDEAS [CH. 
 
 6. The fundamental equations for a material medium. 
 
 For a material medium which is stationary relative to the 
 axes of coordinates, the equations (1) must be replaced by the 
 more general equations* 
 
 rot # = --, divtf = 
 
 c ot 
 
 where D is the electric displacement, E the electric force or 
 field strength, H the magnetic force and B the magnetic induc- 
 
 tion. The quantity J 4- -^- represents the total current which 
 
 ot 
 
 is made up of a conduction-current (7, a displacement-current 
 
 -3- and a convection-current pv, p being the volume density of 
 ot 
 
 electricity. 
 
 Various notations have been used for the different vectors 
 of an electromagnetic field. Most English writers use (a, b, c) 
 for the components of the magnetic induction, (a, /?, 7) for those 
 of the magnetic force, (/, g, h) for the components of the electric 
 displacement and (P, Q, R) or (X, Y, Z) for those of the 
 electromotive intensity or electric force f. This is not to be 
 confused with the mechanical force F of electromagnetic origin, 
 whose components are sometimes denoted by (X, Y, Z). 
 
 * Lorentz (1892 1895) and Larmor (1895) have derived these equations and 
 a corresponding set of equations for moving bodies by a process of averaging, 
 starting from the fundamental equations of the theory of electrons in which we 
 have = H, D=E, J=pv. Cf. H. A. Lorentz, A kad. van Wetenschappen te 
 Amsterdam (1902), p. 305 ; Encykl. d. Math. Wiss. Bd. 5, 14, pp. 200210. This 
 method of averaging has been developed so as to give results in accordance with 
 the Theory of Eelativity by M. Born, Math. Ann. Bd. 68 (1910) andE. Cunning- 
 ham, Proc. London Math. Soc. Ser. 2, Vol. 10 (1911), p. 116. The Born- 
 Minkowski equations differ slightly from those of Lorentz and indicate the 
 existence of an electrostatic field due to the motion of a magnetised body. 
 
 It has been realised by the foregoing writers and others that the principle of 
 relativity alone is not sufficient to determine a complete set of equations for 
 moving bodies, a theory of the constitution of matter is needed. Cf. H. K. Hasse", 
 Phil. Mag. Jan. (1914). 
 
 t Clerk Maxwell, Electricity and Magnetism, 3rd edition (1892), Vol. 2, 
 p. 257. 
 
I] FUNDAMENTAL EQUATIONS FOR A MATERIAL MEDIUM 15 
 
 In any material medium there are certain constitutive 
 relations connecting the vectors D, E, B, H, J. In moving 
 media, crystalline media and ferromagnetic bodies the relations 
 are rather complicated, but for an isotropic medium in which 
 p = the relations can be represented to a good degree of 
 approximation by the simple equations 
 
 (16), 
 
 where e, a, JJL are scalar quantities which are generally regarded 
 as constants; they will be regarded in fact as the optical 
 constants of the medium. The quantity or is called the con- 
 ductivity, IJL the permeability, and e the dielectric-inductive 
 capacity. 
 
 The units that are used here are the so-called modified 
 units*, in which Heaviside's suggestion of eliminating a factor 
 4-7T has been adopted. We can pass to electrostatic units or 
 electromagnetic units by replacing our quantities E, H, etc. by 
 aE, (3H, etc., where a, ft are certain factors which are given in 
 the following table : 
 
 
 *D,J 
 
 E 
 
 B 
 
 H 
 
 m 
 
 Electrostatic system 
 
 x/4^ 
 
 1 
 
 c 
 
 1 
 
 C V^r 
 
 /s/4^ 
 
 V^r 
 
 C\/47r 
 
 Electromagnetic system 
 
 cV4^ 
 
 1 
 
 1 
 
 ^4^ 
 
 1 
 
 /s/4^ 
 
 C X/47T 
 
 x/4^ 
 
 We use e here to denote a quantity of electricity, and m a 
 quantity of magnetism. 
 
 7. The energy equation for a material medium. 
 If we use 2 as before to denote the vector whose components 
 are of type c (E y H z - E z H y ), we find that 
 
 92, 8e 
 
 * Of. H. A. Lorentz, Encyklopddie der Math. Wiss. Bd. 5, 13, pp. 8387. 
 
16 FUNDAMENTAL IDEAS [CH. 
 
 If /A is a constant and ^eE' 2 -\- ^/juH 2 be regarded as the 
 energy per unit volume, the change in the distribution of the 
 energy can be described by means of a flow a and a loss per 
 unit volume of magnitude o-E* due to the transformation of 
 electric energy into heat (Joule's heat)*. If B does not depend 
 on the instantaneous value of H so that /JL is not a constant 
 there is a loss of energy due to hysteresis. B may depend 
 upon H alone but not be a single-valued function of H, con- 
 sequently in a cycle of changes j(H.dB) is not zero and may be 
 taken as the heat per unit volume developed during the 
 description of the cycle. Notice that 
 
 $(H.dB) = -f(B. dH) round a cycle, 
 
 and is always positive since the value of B for a given value of 
 H is greater when H is increasing. The experimental analysis 
 and the accompanying theory are due to E. Warburg f and 
 independently in much greater development to J. A. EwingJ by 
 whom the name hysteresis was applied to such phenomena. 
 
 8. Solution of the fundamental equations for a 
 material medium. 
 
 Let us assume that <r, /JL, e are constants and that E, H are 
 the real parts of expressions of the form Ae~ iut , where A is 
 a complex quantity independent of t. Then if we write 
 
 e/^ + J^r v = _k^ M = H + ivE ..... (17) 
 
 C 2 /-wo 
 
 and regard E, H now as the complex vectors of which they were 
 formerly the real parts, the differential equations to be satisfied 
 
 by M are 
 
 rotM=kM, divJf=0 ............... (18). 
 
 These may be solved by putting 
 
 Jf=rotn igraddiv II MI ............ (19), 
 
 K 
 
 where II is a solution of the equation Au + k" 2 u 0, and may be 
 of the form UiV. 
 
 * For a fuller discussion see E. Gans, Einfiihrung in die Theorie der 
 Magnetisms; H. A. Lorentz, Encykl. d. Math. Wiss. Bd. 5, 14, p. 240, Heft 1 
 (1903) ; Heaviside, Electrical Papers, Vol. 1, pp. 437450. 
 
 t Ann. Phys. Chem. (3), Vol. 13 (1881), p. 141. 
 
 t Phil. Trans. A, Vol. 176 (1885), pp. 523640; Proc. Roy. Soc. Vol. 34 
 (1883), p. 39 ; Phil. Mag. Vol. 16 (1883), p. 381 ; Magnetic Induction in Iron 
 and other -Metals, London (1892). 
 
I] SOLUTION OF THE FUNDAMENTAL EQUATIONS 17 
 
 Problems which deal with the effect of small obstacles upon 
 light or electric waves require the taking into account of the 
 properties of the material of which the obstacle is composed : it 
 is only for long waves and conducting media that the obstacle 
 can be treated as a perfect reflector. This is illustrated by the 
 work of Maxwell Garnett* and Mief on the optical properties 
 of colloidal suspensions of metals and in the work of Sommerfeld 
 to which we have already referred. 
 
 Unfortunately, however, the analytical difficulties are very 
 great when imperfect conductivity is taken into account J. The 
 simple-looking problem of the reflection of the disturbance 
 produced by a moving charge, when the obstacle is an infinite 
 plane sheet of metal or other conducting substance, has not yet 
 been solved accurately and there are many similar problems 
 that have completely baffled mathematicians. 
 
 Much more progress has been made with problems dealing 
 with perfect reflectors. These problems are to some extent 
 ideal but some of the characteristics of actual physical problems 
 are often preserved ||. Apart from this, such problems are of 
 considerable mathematical interest, and have been studied by 
 some writers simply on this account. 
 
 9. Boundary- Conditions. 
 
 The conditions to be satisfied at a surface separating two 
 different media are obtained by integrating equations (15) across 
 a thin layer of transition^. Taking the axis of z along the 
 
 * Phil. Trans. A, Vol. 203 (1904), p. 385 ; Vol. 205 (1905), p. 237. 
 
 t Ann. d. Phys. Vol. 25 (1908), p. 377. 
 
 t An important solution of the equations has been given by M. Brillouin, 
 "Propagation dans les milieux conducteurs," Comptes Rendus, t. 136 (1903), 
 pp. 667, 746. (See Ex. 20, Ch. ix.) For other references see Ex. 5, p. 23. 
 
 An approximate solution was suggested by Maxwell and has been developed 
 by Larmor, Proc. London Math. Soc., Ser. 2, Vol. 8, p. 1. An accurate solution 
 for the case in which the sheet is treated as infinitely thin and the charge moves 
 with uniform velocity parallel to the plane has been given by G. Picciati, Rom. 
 Ace. Line. Rend. (5), 11 2 (1902), p. 221. 
 
 || A metal behaves practically as a perfect conductor to electric waves when 
 the oscillations are rapid but slow compared with waves of light. Cf . J. Larmor, 
 "Electric vibrations in condensing systems," Proc. London Math. Soc. Ser. 1, 
 Vol. 26, p. 119. 
 
 H Bayleigh, Scientific Papers, Vol. 1; Phil. Mag. Vol. 12 (1881), p. 81; 
 H. Hertz, Electric Waves, pp. 207, 238; Larmor, Phil. Trans. (1895), 
 
18 FUNDAMENTAL IDEAS [CH. 
 
 normal to the surface we shall assume that E, H, D, B and 
 their derivatives with regard to x, y, t are finite within the 
 layer and that the conductivity <r is also finite. The equations 
 dE y dE z _ldB x dH z dH y _ 
 
 ~~~~~ ~~" '" 
 
 then show that -^ , -^ and J x + -^ are finite and so their 
 oz oz dt 
 
 integrals with regard to z across a thin layer of thickness are 
 less than aO where a is a finite positive quantity independent of 6. 
 This means that the tangential components of the electric force, 
 magnetic force and electric current are continuous in crossing 
 the surface. 
 
 Again, the equation 
 
 dB x dB y dB z _ 
 
 ~7C - "T "^ i ~f\ ~~ V 
 
 dx dy oz 
 
 OD 
 
 shows that -~ is finite and so the normal component of the 
 
 magnetic induction is continuous. This result may, however, 
 be regarded as a consequence of the previous one. 
 
 The normal component of the electric displacement may be 
 discontinuous, for the equation div D p gives 
 
 f 9< dD [ 
 
 d z = I -^ dz = pdz + terms of order 0, 
 Jo oz Jo 
 
 where d z is the discontinuity in the electric displacement. Hence 
 if a is the surface-charge of electricity per unit area we have d z =a. 
 When the media are both conducting we have <r = and the 
 normal component of the electric displacement is continuous. 
 
 It is known that in the case of a good conducting body 
 rapidly alternating currents are confined within a very thin 
 layer close to the surface*. In the ideal case of a perfect 
 conductor or perfect reflector the field-vectors are zero within 
 
 p. 733; H. M. Macdonald, Electric Waves, p. 14; H. A. Lorentz, Zeitschrift 
 /. Math. u. Phys. Bd. 22 (1877). 
 
 * Of. H. Lamb, Phil. Trans. A (1883) ; 0. Heaviside, Electrical Papers, Vol. 2, 
 p. 168; J. J. Thomson, Recent Researches, p. 281. For some recent work on 
 the subject see a paper by E. F. Northrup and J. E. Carson, Journ. of the 
 Franklin Institute, Feb. (1914), p. 125. The results of many other researches 
 on the skin-effect and alternating current resistance are given in J. A. Fleming's 
 Th4 principles of Electric Wave Telegraphy. 
 
I] BOUNDARY-CONDITIONS 19 
 
 the body of the conductor. This case is characterised by <r = oo 
 and the tangential components of the magnetic force are no 
 longer continuous as a point moves across the surface, we have 
 in fact if h is the discontinuity in the magnetic force 
 
 C e dH 1 f* 
 
 h = I __^ dz a-E x dz -f- terms of order 0. 
 Jo 0* cJ 
 
 Hence if K is the surface-current and h the discontinuity 
 of the magnetic force, we have 
 
 h y =-~K X) h x =^Ky. 
 
 c c 
 
 At the surface of a perfect conductor the tangential com- 
 ponents of the electric force must vanish and as a consequence 
 of this we can say that the normal component of the magnetic 
 induction must also vanish. If the medium outside the con- 
 ductor is free aether the surface-conditions are simplified on 
 account of the relations D = E, B H. 
 
 In the case of a very thin conducting sheet it is convenient 
 to treat the thickness of the sheet as negligible and regard the 
 tangential components of the magnetic force as discontinuous 
 when a point moves along the normal from one side of the 
 sheet to the other. 
 
 The boundary-condition is then* 
 
 /e 
 o~dz is the conductivity of the sheet. 
 ) 
 
 If we wish to extend the idea of Green's equivalent layer to 
 electrodynamics we must consider electromagnetic fields in free 
 aether with surfaces at which the tangential components of the 
 electric and magnetic forces are discontinuous ; this requires an 
 electric current sheet and a magnetic current sheet on the 
 
 * Cf. T. Levi-Civita, Rend. Lincei (5), 11 2 (1902), p. 75. These conditions 
 are used by Picciati in his solution of the problem of an electric charge moving 
 parallel to a conducting sheet. In some previous papers, Rend. Lincei (5), llj 
 (1902), pp. 163, 191, 228, Levi-Civita had used the electromagnetic potentials to 
 determine the effect of a conducting sheet on an alternating current flowing 
 along a straight wire parallel to the sheet ; the boundary-conditions are then 
 determined by the discontinuities of the potentials due to the induced current. 
 
 22 
 
20 FUNDAMENTAL IDEAS [CH. 
 
 surface*. For a complete generalisation we ought to consider 
 the cases when the normal components are also discontinuous 
 and when the surface is in motion. The last circumstance 
 alters matters to some extent and must now be discussed. 
 
 In the case of electric waves in free aether, the vectors 
 E and H may be discontinuous at a wave-boundary. If this 
 can be regarded as the limit of a thin layer of transition within 
 which equations (1) are satisfied and the vectors E, H are 
 finitef, the values of these vectors on the two sides of the 
 boundary must satisfy certain conditions which may be found 
 as follows. 
 
 Let the equation of the moving boundary be expressed in 
 the form 
 
 t=f(x,y,z) ........................ (20). 
 
 If now we apply Green's theorem to the integral 
 
 4- ( My + icM z r- icM x g-J dzdx 
 
 icM x to M y <b<fy ...... (21), 
 
 which is supposed to be taken over a closed surface, we find 
 that it vanishes on account of the equations (2) provided t is 
 supposed to be expressed in terms of x, y, z according to some 
 definite law which we shall take to be that expressed by (20). 
 
 We now apply this theorem to a disc-shaped surface whose 
 two faces very nearly coincide. We shall suppose that on one 
 side of the disc the vector M represents the field of the advancing 
 waves and that on the other side it represents the field obtain- 
 ing just before the arrival of the waves. We shall also suppose 
 
 * Cf. J. Larmor, Proc. London Math. Soc. (2), Vol. 1 (1903), p. 11; 
 H. M. Macdonald, Electric Waves, p. 16 ; Proc. London Math. Soc. (2), Vol. 10 
 (1911), p. 91. 
 
 t The idea is practically due to Stokes, Math, and Phys. Papers, Vol. 2, 
 p. 275, but was not worked out in detail. The different possible types of dis- 
 continuity are discussed with some care by Love. The case in which E and H 
 are continuous but some of their derivatives are discontinuous at the moving 
 boundary may be discussed more simply by analysis analogous to that given in 
 Hadamard's Lemons sur la Propagation des Ondes, Paris (1903), Ch. 2. See also 
 Ex. 2, p. 23, and the references to Duhem and Silberstein on the next page. 
 
l] BOUNDARY-CONDITIONS 21 
 
 that the derivatives of M are finite or behave in such a way 
 that an application of Green's theorem is justifiable. Now 
 let M be the discontinuity of M, i.e. the difference in the values 
 of M at two neighbouring points on opposite sides of the wave- 
 boundary. Then when the two faces of the disc coincide we 
 find that a certain surface-integral over one face of the disc is 
 zero. The surface-integral is of the same type as (21) except 
 that M is written in place of M. Since the face of the disc can 
 be chosen arbitrarily the integrand must vanish and so we 
 obtain three equations of the type* 
 
 jr,TfcJT,||t0jr,|?-0 ............ (22). 
 
 These equations give 
 
 i7. A 
 
 M* = Q and - 
 
 dy 
 
 Hence the wave-front advances with the velocity of light 
 and the difference between the two electromagnetic fields at 
 the wave-boundary behaves as a self-conjugate field in which 
 Poynting's vector is along the normal to the moving 
 boundary. 
 
 If the equation of the wave-boundary be expressed in the 
 form 
 
 F(*,y,*,t)-Q, 
 
 we find on calculating the values of 
 
 dt dt dt 
 fa' <jy' dz' 
 
 * 
 
 that 
 
 This is the differential equation of the characteristics, it 
 expresses that the moving boundary moves normally to itself 
 with the velocity of light. According to the theory of Stokes )* 
 
 * Equations equivalent to these are obtained in a different manner by 
 0. Heaviside, Electrical Papers, Vol. 2, p. 405 ; A. E. H. Love, Proc. London 
 Math. Soc. Ser. 2, Vol. 1 (1903), p. 37 ; L. Silberstein, Ann. d. Physik, Vol. 26 
 (1908), p. 751 ; P. Duhem, Comptes Rendm, t. 131 (1900), p. 1171. 
 
 t Proc. Camb. Phil. Soc. (1896) ; Manchester Memoirs (1897) ; Math, and 
 Phys. Papers, Vol. 4. 
 
22 FUNDAMENTAL IDEAS [CH. 
 
 and Wiechert*, Rontgen rays consist of pulses travelling through 
 the aether, the energy in a pulse being confined within a thin 
 shell. The above theory indicates that the front and rear 
 surfaces of the shell move forward with the velocity of light. 
 A slight modification of the preceding method can be used 
 to find the conditions to be satisfied at a moving surface which 
 is the "boundary between two different media. If we write 
 L = B iD, N = E + iH and assume that the surface-charge and 
 surface-current can be neglected, the six boundary-conditions 
 can be expressed by saying that the three quantities of type 
 
 L - AT ^AT 
 
 must be continuous as the boundary is crossed. The equation 
 of the moving boundary is expressed as before in the form (20). 
 When the moving boundary is the surface of a perfect conductor 
 or perfect reflector, the boundary-conditions are simply that the 
 three quantities of type 
 
 B _ -E - E 
 
 should vanish f. 
 
 In the last two cases the boundary-conditions do not imply 
 that the boundary moves normally to itself with the velocity of 
 light ; in fact, the motion of the boundary can be quite arbitrary. 
 
 EXAMPLES. 
 
 1. The surface of discontinuity is the sphere rct and the electro- 
 magnetic field within this surface is expressed by the equations 
 
 / 8 2 n 8 2 n 8 2 n 
 
 where UAr~ 1 e~ q ( et ~ r )sinp (ct 
 
 * Abh. d. Phyg.-bkon. Ges. zu Kimigsberg, 1 Pr. (1896), p. 1; Ann. Phyg. 
 Chem. Bd. 59 (1896), p. 283. See also J. J. Thomson, Phil. Mag. (5), Vol. 45 
 (1898), p. 172. 
 
 t These conditions are equivalent to the condition that the mechanical force, 
 which would act on a charge moving with the normal velocity of the surface, 
 must be along the normal to the surface. Cf. Heaviside, Electrical Papers, 
 Vol. 2, p 514; Electromagnetic Theory, Vol.1, p. 273; Hertz, Electric Wavet, 
 p. 257. 
 
l] EXAMPLES 23 
 
 The constants p and q being known determine the constants A and 
 in order that the field outside the surface r = ct may be the electrostatic 
 
 field for which the potential $ is ~- ( - J . 
 
 (Cambr. Math. Tripos, Part II, 1904.) 
 
 2. If the vectors E and H are known for all points (x, y, z, t) of a 
 moving surface 
 
 F(*,y,*,<)=0 
 
 the values of all the derivatives of E and 3, and consequently values of E 
 and H at points not on the moving surface can generally be found pro- 
 vided F does not satisfy the differential equation 
 
 (Havelock, Proc. London Math. Soc. Ser. 2, Vol. 2, p. 297.) 
 
 3. An electromagnetic field is conjugate to an electrostatic field. 
 Prove that the flow of energy in the electromagnetic field takes place along 
 the lines of electric force in the electrostatic field. 
 
 4. Let the line OC of length c be drawn in the direction of Poynting's 
 vector at each space-time point of a self-conjugate electromagnetic 
 field and let 0V represent a velocity v associated with the point 0. 
 Prove that if this electromagnetic field is conjugate to another field 
 (E t H} in which cH is the vector product of v and E, the direction of E is 
 parallel to VC. 
 
 5. Prove that when <r, e, /n are constants, the vectors E, H of 6 
 satisfy the differential equations 
 
 ,, =f-a (Maxwell.) 
 
 Solutions of these equations for the case in which p = have been given by 
 
 0. Heaviside, Phil. Mag., Jan. (1889), p. 30 ; Electrical Papers, Vol. 2, 
 p. 478. 
 
 H. Poincare, Comptes Rendus (1893), p. 1030 ; TMorie analytique de 
 la propagation de la chaleur, Ch. 8. 
 
 J. Boussinesq, Comptes Rendus (1894), pp. 162 223 ; Theorie 
 analytique de la chaleur, t. 2 (1903), p. 538. 
 
 Kr. Birkeland, Archives des sciences physiques, Geneva (1895), p. 5. 
 
 0. Tedone, Rend. Lincei, Mar. 31st (1913), Jan. 18th (1914). 
 
 M. I. Pupin, Trans. Amer. Math. -Soc., Vol. 1 (1900), p. 259. 
 
 6. Prove that the function 
 
 =7v e 
 
24 FUNDAMENTAL IDEAS [CH. I 
 
 satisfies the equation 
 
 d z u d^u du 
 
 7. If u satisfies the partial differential equation 
 
 , 2 /8w\ 2 
 
 and we make the transformation 
 
 \ _/ ^L 
 
 where / is an arbitrary function, then u also satisfies the partial differential 
 equation 
 
 8. Plane electromagnetic waves fall on the convex surface of an infinite 
 paraboloid of revolution x=a r, whose surface is a perfect reflector. If 
 the incident waves are given by expressions of the type 
 
 JT _L^ // ox 8 fa*/ 3 ) i // 
 H x + iE x =f(a, ft) d ^ = -/(a, 
 
 where a=y+&, ^=^+c^ the boundary-conditions at the surface of the 
 paraboloid may be satisfied by subtracting from the primary field a 
 secondary field represented by expressions of a similar type but with 
 
 a = a , 
 
 If waves represented by the above expressions with a=y iz, $=x-ct, 
 fall on the concave surface of the paraboloid the boundary-conditions at 
 the surface of the paraboloid may be satisfied by supposing that the 
 secondary disturbance is of the form -M'-M" + M'", where the fields 
 M'j M", M'" are represented by expressions of the above type, where a, 3 
 have the values 
 
 P"=a-r-ct, 
 
 respectively. With these suppositions the forces are finite at the focus, 
 when f is independent of a. 
 
CHAPTER II 
 
 GENERAL SURVEY OF THE DIFFERENT METHODS OF 
 SOLVING THE WAVE-EQUATION 
 
 10. The object to be attained. 
 
 It has been shown in Chapter I that the solution of 
 Maxwell's equations can be made to depend upon the solution 
 of a single partial differential equation which is either the 
 wave-equation flu = or the equation Aw + l&u = which 
 is satisfied by wave-functions of the form u = e ikct /(x, y y z). 
 The properties of functions satisfying these equations must 
 accordingly be studied at some length. It is desirable, 
 also, that all types of such functions should be studied 
 and not merely those which admit readily of application to 
 physical problems. If certain solutions of the fundamental 
 equations must be rejected in the treatment of the boundary 
 problems of mathematical physics, a knowledge of their 
 behaviour is at any rate useful as it gives a clear indication of 
 the reason why such solutions must be rejected. There is, 
 however, another reason why the scope of the inquiry should 
 not be restricted. The theory of wave-functions forms a 
 natural extension of the theory of functions of a complex 
 variable* and may consequently lead to results of great value 
 for the general theory of functions. 
 
 * This point of view is adopted, for instance, by Volterra, Rend. Lined (4), 
 ra 2 , pp. 225-330, 274-287 (1887); r^, pp. 107-115, 196-202 (1889); v^pp. 158-165, 
 291-299, 599-611, 630-640 (1889); Eend. Palermo, Vol. 3, pp. 260-272. See 
 also Appell, Acta Math. t. 4, p. 313 (1884) ; Painleve, Toulouse Ann., t. 2B 
 (1888) ; Bdcher, Bull. Amer. Math. Soc. Vol. 9, p. 455 (1903). The theory of 
 functions of two complex variables is closely connected with the theory of wave- 
 functions. Cf. H. Poincare", Acta Math. Vol. 2 (1883) ; Vol. 22 (1898) ; H. F. 
 Baker, Camb. Phil. Trans. Vol. 18 (1899), p. 431; Proc. London Math. Soc. 
 Ser. 2, Vol. 1 (1903), p. 14. 
 
26 METHODS OF SOLVING THE WAVE-EQUATION [CH. 
 
 We shall now describe briefly some of the principal methods 
 of solving the wave-equation. 
 
 11. Reduction to ordinary differential equations. 
 The aim of this method is to determine elementary solutions 
 of the form 
 
 * =/. ()/. <)/. (y)/4 (S) ............... (23) 
 
 where /i,/ 2 ,/ 3 ,/4 are particular functions of their arguments and 
 a, /3,7, S are particular functions of #,y, z, t. This method was 
 used by D. Bernoulli in 1732 in the treatment of the vibrations 
 of a hanging chain, the partial differential equation being how- 
 ever in this case different. 
 
 The general theory of elementary solutions is due to Lame * 
 who transformed Laplace's equation into curvilinear coordinates. 
 For a historical account of the development of the theory we 
 may refer to Prof. Bocher's book Die Reihenentwickelungen 
 der Potentialtheorie, Leipzig (1894) and to Byerly's Fourier 
 Series and Spherical Harmonics. 
 
 A simple elementary solution of the wave-equation is 
 obtained by putting a = x, $ = y, 7 = z, S t; we can then 
 take 
 
 where the constants I, ra, n, p satisfy the relation 
 
 ^ = c 2 (J 2 + m 2 + n 2 ) .................. (25) 
 
 and can be either real or complex quantities. 
 
 When p is a purely imaginary quantity and I, m, n unre- 
 stricted, the solution is periodic and more general periodic 
 solutions may be derived from this one by summation, I, m, n 
 being regarded as variable parameters subject to the relation 
 (25). When I, m, n are purely imaginary the solution (24) is 
 appropriate for the representation of plane waves of mono- 
 chromatic light, the intensity and phase of which are the same 
 at all points of any plane perpendicular to the direction of 
 propagation. 
 
 * Liouville's Journal, t. 2, pp. 147-183 ; Lemons sur les coordonnees 
 curviligne$, Paris (1859). See also E. Mathieu, Court de physique mathematique 
 (1873). 
 
II] ELEMENTARY SOLUTIONS 27 
 
 Taking m = n = Q, I = - = v so that the axis of x is in the 
 direction of propagation, we may write 
 
 H z = acosv(x-ct) .................. (26) 
 
 where a is a constant. These waves will be said to be linearly 
 polarised in a direction parallel to the axis of y and will be 
 called homogeneous because E and H do not depend on y and z. 
 It will be noticed that the constant c represents the velocity of 
 propagation of a phase of the disturbance. 
 A wave-function of the type 
 
 n = sin vx cos vet 
 
 is appropriate for the representation of standing waves. Ex- 
 pressions for E and H may be written down by analogy with 
 the above. To obtain a representation of plane waves in a 
 conducting medium, we must use a solution of 
 
 where J<? has the complex value given in 8. Putting V = 0, 
 U= (0,0, eifcc-iwt) we find that 
 
 c 
 i<t ..................... (27) 
 
 where the real parts of the quantities are retained. If 
 
 -{+* 
 
 where 77 is positive, the oscillations of the vector E are damped 
 owing to the exponential factor e - v x . 
 
 The elementary electromagnetic fields that have just been 
 found are fundamental in the theory of the reflection and 
 refraction of light at a plane surface. This theory is given in 
 the text-books on Physical Optics* and need not be reproduced 
 here. Various attempts have been madef to prove that any 
 
 * See for instance, Wood's Physical Optics (1911), Ch. 13 ; Jeans, Elec- 
 tricity and Magnetism (1911), Ch. 18. 
 
 f See for instance a series of papers by G. Johnstone Stoney, Phil. Mag. 
 (5), Vol. 43 (1897), pp. 139, 273, 368 ; Vol. 44, pp. 98, 206 ; Brit. Assn. Reports 
 (1902), p. 539 ; Phil. Mag. Feb. 1903. The idea is probably due to Stokes. 
 
28 METHODS OF SOLVING THE WAVE-EQUATION [CH. 
 
 electromagnetic disturbance in the aether can be represented as 
 the sum of a finite or infinite number of elementary disturbances 
 of the character of plane-waves travelling in various directions. 
 Such a representation is generally only suitable within a 
 restricted domain of the variables x, y,z,t\ nevertheless, it may 
 sometimes be employed with advantage. 
 
 When waves of all directions and frequencies are considered, 
 the method of summation leads to Whittaker's formula * 
 
 (* f 
 
 = / I 
 
 f[x sin a. cos /3 + y sin a sin ft -f z cos a ct, a, /3] 
 
 xsina. dad/2 ........................ (28) 
 
 for a wave-function. The case when 
 
 = f >B 
 
 has been used by Debyef in a discussion of the behaviour of 
 waves of light in the vicinity of a focus. In order that an 
 integral of the type (28) may represent a wave-function it is 
 not necessary for the limits of integration to be those chosen. 
 The limits for a. may, for instance, be and 6 where 6 is a 
 root of an equation of type 
 
 x sin cos @ + y sin 6 sin + z cos 6 - ct = F (0). 
 
 In order to obtain other types of elementary solutions it is 
 necessary to transform our differential equations to a system of 
 orthogonal coordinates (u, v, w) for which the linear element is 
 given by 
 
 , dv? dv* dw* 
 
 If H u , H v , H w are the three components of a vector H in 
 directions normal to the surfaces u = const., v = const., w = const. 
 through a point (x, y, z\ the corresponding components of rot H 
 are of the type J 
 
 VW 
 
 v ~ 
 
 * Math. Ann. (1903). See also G. N. Watson, Mess, of Math. Vol. 36 
 (1906), p. 98. 
 
 t Ann. d. Phys. Vol. 30 (1909), p. 735. 
 
 J See for instance, H. M. Macdonald, Electric Waves, Ch. 6 ; M. Abraham, 
 Math. Ann. Bd. 52, p. 81. Some very general transformation-formulae are 
 
Il] TRANSFORMATIONS OF COORDINATES 29 
 
 the new expression for div H is 
 
 \ d f H u \ d ( H v \ d (H v \-\ 
 V [85 (Vw) + dv(wu) + ^v (UV)\~ 
 
 and the wave-equation becomes* 
 
 p f u 3^ *_iv_**\ + L(^L ^ 
 
 y + + 
 
 __ + 
 
 VTW Su) + Sv\WUdv) + dw\ UVSw 
 
 It is also sometimes advantageous to transform the wave 
 equation to a system of coordinates for which 
 
 dx* + df + dsfi - c*dt* = A*d? + B*drf + C*d? - D*dr*, 
 the wave-equation then becomes 
 d (BCD cty\ ^ (CD A <ty\ d_ (DAB ty\ 
 
 a|V~^~a|J" i "^V~5~8^/ + 8rv c ds) 
 
 a /ABCd4>\ 
 
 (33) - 
 
 12. The generalisation of wave-functions, 
 
 When a solution of the wave-equation has been" obtained 
 other solutions may be derived from it in various ways. For 
 instance, the function obtained by differentiating the given 
 wave-function any number of times with regard to the coordinates 
 x, 2/, z, t, is also a wave-function. By adding together arbitrary 
 constant multiples of all the wave-functions obtained in this 
 way we may obtain a very general type of wave-function. 
 
 Another method of generalisation is to make an arbitrary 
 change of rectangular axes. The wave-equation is a covariant 
 for such a transformation and so is a wave-function of the new 
 coordinates. A number of arbitrary constants can be introduced 
 into the solution in this way. We can also make a linear 
 transformation of coordinates for which the expression 
 
 contained in the papers of V. Volterra, Rend. Lined, Ser. 4, Vol. 5, pp. 599, 
 630 (1889), and J. Larmor, Cambr. Phil. Trans. Vol. 14 (1885), p. 121. 
 
 * Lame, Journ. de VEcole Poly technique, Cah. 23 (1833), p. 215; Legons 
 sur les coordonnees curvilignes, t. 2. A simplified proof was published by Lord 
 Kelvin, Cambr. Math. J&urn. Vol. 4 (1843). 
 
30 METHODS OF SOLVING THE WAVE-EQUATION [CH. 
 
 remains unaltered in form and the preceding remarks still hold 
 good. 
 
 To illustrate this let us first of all add together two particular 
 
 cases of Euler's wave-function - f(r ct), viz. 
 
 1 1 
 
 -7 -r and , 
 
 r (r ct) r(r + ct) 
 
 we then see that (r 2 c^ 2 )" 1 is a wave-function. Generalising 
 this by writing x x , y y Q , z z , t t Q in place of x, y, z, t, 
 
 we obtain the wave-function 
 
 1 
 
 (34). 
 
 When we have obtained a wave-function involving one or 
 more arbitrary parameters we may obtain others from it by 
 differentiating with regard to the parameters, or by integrating 
 with regard to them after having multiplied the expression by 
 an arbitrary function of the parameters. For instance, from 
 the above wave-function we may derive the more general wave- 
 function 
 
 f /fr)<fr / 35 ^ 
 
 J \x - # ) 2 + (y - 2/o) 2 + (? - -So) 2 - c 2 (t - r) 2 * * 
 
 where the integration is between constant limits. It is not, 
 however, really necessary for the limits to be constant, we may 
 
 for instance take them to be oo and t (r + r ), where 
 
 c 
 
 r 2 = # 2 + y 2 + z Q \ The resulting integral is then a wave-function 
 providedy(r) behaves in a suitable manner. If we takey(r) = 1, 
 we obtain a wave-function 
 
 where R 2 = (x - # ) 2 + (y - 2/o) 2 + (e - ^ 
 
 This function is independent of t and so must be a solution of 
 Laplace's equation Aw = 0. It is closely connected with the 
 function used on p. 3 of Basset's Hydrodynamics, Vol. 2. 
 
II] THE METHOD OF TRANSFORMATION 31 
 
 13. Transformations. 
 
 In addition to the linear transformations that have already 
 been mentioned there are certain other transformations which 
 enable us to pass from one wave-function to another*. 
 
 The first transformation, which is analogous to inversion, is 
 defined by the equations 
 
 where s 2 = r 2 cH 2 . 
 
 It is easy to verify that if /(#, y, z, t) is a wave-function, 
 then 
 
 'x v z 
 
 is also a wave-function f. 
 
 The second transformation is} 
 
 ,_-- 
 
 ' ' 
 
 _ 
 z-ct' ' z-ct' 2(z-ct) ' ' 2c(*-c<) 
 
 ......... (39). 
 
 It is easy to verify that if f(x, y, z, t) is a wave-function, 
 then 
 
 * y ga - 1 
 
 z-ct \_z-ct' z-ct' 2(z-ct)' 
 
 is also a wave-function. Since e~ (z+ct) is a wave-function we 
 may deduce in this way that 
 
 1 __?L 
 e z-ctf 
 
 z ct 
 is a wave-function. 
 
 It should be noticed that if we put 4r = z ct, o- z-\-ct, 
 a function of the type 
 
 (41) 
 
 * For a general account of these transformations see a paper by the author, 
 Proc. London Math. Soc. Ser. 2, Vol. 7 (1908). 
 
 t This of course is a simple generalisation of Kelvin's theorem for Laplace's 
 equation. The generalisation to the corresponding equation in n variables is 
 mentioned by Bocher, Bull, of the Amer. Math. Soc. Vol. 9 (1903), p. 459. 
 
 t Proc. London Math. Soc. Ser. 2, Vol. 7 (1909). This transformation is 
 equivalent to a conformal transformation of a space of four dimensions which 
 was discovered by Cremona. Of. Darboux, Lemons sur les systemes orthogonaux 
 et les coordonnees curvilignes, Paris (1910). 
 
32 METHODS OF SOLVING THE WAVE-EQUATION [CH. 
 
 is a wave-function if F satisfies the differential equation 
 
 The wave-function we have just obtained indicates that 
 
 I x*+y* 
 
 e *r = F ..................... (43) 
 
 is a solution of the above equation. This solution is fundamental 
 in the theory of the conduction of heat. It is evident that any 
 solution of the equation of the conduction of heat in two 
 dimensions can be used to construct a wave-function. 
 
 Our second transformation theorem for the wave-equation 
 also tells us that if F(x, y, T) is a solution of the equation (42) 
 the function 
 
 is also a solution. This result is due to J. Brill* and Appellf. 
 The corresponding theorems for the two-dimensional wave- 
 equation 
 
 i 
 
 4. _ - 
 8^ + ty* c 2 
 
 are first that if f(x, y, t) is a wave-function and 
 
 5 2 = ^ + 7/2 _ C 2 2j 
 
 the function 
 
 /Aa \ 
 
 (46) 
 
 is also a wave-function. This is equivalent to Lord Kelvin's 
 theorem for Laplace's equation if we simply replace ict by z. 
 
 The second theorem is that if f(x, y, t) is a wave-function, 
 then 
 
 iii /T * g2 ~ 1 g2 + 1 1 (47) 
 
 J -' ' -' 
 
 is also a wave-function. Writing 4r = y c, cr = y + c^ as 
 before we find that u e~"F (x, r) is a wave-function if 
 
 &F dF 
 
 W = TT ........................ (48) - 
 
 * Messenger of Mathematics (1891). 
 
 f Liouville's Journal, Ser. 4, t. 8 (1894). 
 
II] TRANSFORMATIONS 33 
 
 The second theorem can now be used to show that 
 
 00 
 
 (49) 
 
 is a solution of this equation and that if F (a, r) is one solution 
 the function 
 
 *"*"**(? -j) .................. < 5 ) 
 
 is also a solution. This theorem is likewise due to Brill and 
 Appell, it can evidently be generalised to the equation in 
 n variables. 
 
 EXAMPLES. 
 
 1. Prove that it is possible for a train of plane electric waves 
 to travel along an infinite isolated slab of dielectric material without 
 being dissipated by spreading out into the adjacent empty space. Show 
 that if 2a is the thickness and K the inductivity of the slab, the velocity 
 of propagation of such waves of length X along the slab, when polarised so 
 that their magnetic vector is parallel to it, is 
 
 where 6 is the lowest real or the pure imaginary root of the equation 
 
 -K U 
 
 -- ' 
 
 (Larmor, Cambr. Math. Tripos, Part II, 1906.) 
 2. Prove that with the notation of 13, the function 
 
 is a wave-function, o? , yo> z<)> *o being arbitrary constants. 
 
 3. If F(x, y, z, t} is a solution of the wave-equation, the function 
 
 <TCt 
 
 o \r' r' T J T V 
 
 is, under suitable conditions, a solution of the equation 
 
 c dt' 
 
 The quantity s 2 has the meaning assigned to it in 13 and is supposed in 
 the present case to be negative. 
 
 B. 3 
 
34 METHODS OF SOLVING THE WAVE-EQUATION [CH. II 
 
 4. The function 
 
 _^ 
 
 V= j= e d\ 
 
 V 
 
 satisfies the equation 
 
 ?!T _ = _ 
 
 of the conduction of heat in two dimensions : it is zero over a semi-infinite 
 line which covers part of the axis of x and moves in the positive direction 
 with uniform velocity v. The isothermal lines at any given instant are 
 confocal parabolas. 
 
 5. Prove that, under suitable limitations, the function 
 
 is a solution of 
 
 Obtain in this way the particular solution 
 
 P 
 
 6. Prove that if x 2 +y 2 >t 2 *, the integral 
 
 e~ a<r da 
 
 I 
 
 o 
 
 8 2 F 8 2 F 9 2 F 3F 
 
 satisfies W + W = ~W* <r 2t' 
 
 and if # 2 >* 2 , the integral 
 
 r ;" a<r f -F 
 
 32F 9 2 F 8F 
 satisfies W = W +<r ^t' 
 
 7. Prove that if p 2 = x* +y 2 , p 2 = ^o 2 +yo 2 , the integral 
 
 satisfies the equation 
 
CHAPTER III 
 
 POLAR COORDINATES 
 
 14. The elementary solutions. 
 If we make Laplace's transformation 
 
 x = r sin cos <, y = r sin 6 sin <j>, z = rcos0 ...(51), 
 the equation AM + k*u = becomes 
 
 This is satisfied by a function of the form 
 u-R(r)B(ff)<b(<f>) 
 
 gad) 
 if +m'*.0 ..................... (53), 
 
 The first equation is satisfied by <3> = cos (m<f> + a), the 
 second by P n m (cos 0) and Q n m (cos 0), where these are the 
 associated Legendre functions. The third equation may be 
 written in Bessel's form 
 
 1 dw rk 2 , . _! . 
 
 (5o), 
 
 where w = ?*^R,and is satisfied by J n+ (kr) and J_^ n+ ^ (kr). 
 
 In these solutions m and n can have any constant values. 
 It should be noticed that when n + ^ is an integer the Bessel 
 functions that have just been written down are not independent 
 and the second solution Y n+ ^ (kr) of Bessel's equation must be 
 used. 
 
 32 
 
36 POLAR COORDINATES [CH. 
 
 % In dealing with a problem such as the effect of an obstacle 
 on a train of electric waves, the secondary waves sent out from 
 the obstacle must have the character of diverging waves at a 
 great distance from the obstacle. In the case n = the differen- 
 tial equation for R is satisfied by 
 
 (57), 
 
 and if the real part of k has the same sign as o> when the 
 electric and magnetic forces are the real parts of expressions 
 of the form Ae ~ l(at , we can obtain a solution appropriate for the 
 representation of a diverging wave by taking the positive sign, 
 for then we have a function of the form 
 
 I^f(fcr~f4 
 r 
 
 Neither of the given solutions of Bessel's equation has the 
 required form in fact 
 
 / 2 /~2 
 
 *" V i3r sin kr > J ~ * ( kr ) " V ~^kr C S kr ' 
 We may, however, obtain solutions of the form (57) by taking a 
 suitable combination of the preceding solutions. 
 
 In the case of electromagnetic fields in the free aether the 
 physical interpretation of the elementary wave-functions when 
 n is zero is as follows* : 
 
 - cos k(r ct) Progressive divergent waves. 
 
 - cos k (r + ct) Progressive convergent waves. 
 
 - cos kr . cos kct Standing forced waves, source at origin. 
 r 
 
 i sin Ar.cos kct Standing free waves. 
 r 
 
 To obtain the solution of (56) appropriate for divergent 
 waves when n has any value we write f 
 
 - A fuller discussion is given by A. Sommerfeld, Jahresbericht der deutsch. 
 math. Verein, Bd. 21 (1913). 
 
 t The theory is due to Stokes, Phil. Tram. Vol. 158 (1868), p. 447; 
 Collected Papers, Vol. 4, p. 321. See also Rayleigh's Sound, Vol. 2, p. 304. 
 It should be mentioned that different notations are used by different writers. 
 This is the notation used by Debye, Ann. d. Phys. Vol. 30 (1909), p. 57. 
 
Ill] 
 
 ELEMENTARY SOLUTIONS 
 
 37 
 
 ...(58). 
 
 These new functions ?; n , f n are connected with Hankel's 
 cylindrical functions* by the relations 
 
 *) = \/ 
 
 
 \/ 
 
 
 When the real part of x is large and positive we have the 
 asymptotic expansion 
 
 (\ / '\W + 1 ix I i . * it' \ it ~f~ A ) 
 )"(-*) * l+ ^x n~ 
 
 1 (n - 1) n (n + 1) ( 
 
 2 
 
 '" ( 
 
 The series terminates when n is an integer and then gives 
 a true representation of the function. To get (#) we change 
 the sign of i. Various other notations have been used for the 
 solution of Bessel's equation that is suitable for the representa- 
 tion of diverging waves. Lamb uses D v (x), (z>= n + ) to denote 
 the solution of equation (56) which has the asymptotic value 
 
 while many other English writers use K v (ix) to denote the 
 solution with the asymptotic form 
 
 ()'--> If, .,": 
 
 The following formulae will be found useful : 
 
 * See Nielsen's Handbuch der Cylinderfunktionen, p. 16. 
 
38 POLAR COORDINATES [CH. 
 
 _ (n-l)n(n + l)(n + 2) (2n)l 
 
 ~ 2! ' ( l) ' 
 
 We have written down the last term in the series on the 
 supposition that n is an integer. In this case 
 
 = i |>' n+1 e~ ikr + (- i) n ^ e ikr l | r \ large . . .(62), 
 
 = (n + 1) ^ (*) - n ^ +1 (*) . . .(63), 
 
 (2n + 1) ^r (a?) = a? [>-! (a;) + ^ n+1 (a?)] ......... (64), 
 
 f 
 #(#) = e-so^-coshva.da ............ (65), 
 
 Jo 
 
 f 
 
 + ) J 
 
 cos cos sn " a * a> - i/ > - 
 
 ............ (66), 
 
 (67). 
 
 In the last formula n and p are supposed to be integers. 
 For further properties of Bessel functions the reader should 
 consult Gray and Mathews' Treatise on Bessel Functions, 
 Nielsen's Handbuch der Cylinderfunktionen, and Whittaker's 
 Analysis. Tables are given in the first work and in Jahnke 
 and Emde's Funktionentafeln, Leipzig (Teubner). A few ad- 
 ditions to the tables have been made recently by J. W. Nicholson, 
 Proc. London Math. Soc. Ser. 2, Vol. 11, p. 104 ; Dinnik, Archiv 
 der Mathematik und Physik (3), Bd. 20, Heft 3, 1912 ; J. R. Airey, 
 Phil. Mag. Vol. 22 (1911), p. 85, Brit- Ass. Reports (1911); 
 A. Lodge, Brit Ass. Reports (1909) ; J. G. Isherwood, Manchester 
 Memoirs (1904). 
 
 The best definitions of the generalised Legendre functions 
 for unrestricted values of m and n are those given by Hobson*. 
 
 * Phil. Trans. A, Vol. 187 (1896), pp. 443531. E. W.Barnes has recently 
 given new definitions of the functions as integrals involving Gamma Functions 
 which make it possible for the principal formulae to be proved very quickly. 
 His definition of Q n m (x) differs from that of Hobson by a numerical factor which 
 becomes rather troublesome when n is an integer and m is not. 
 
Ill] 
 
 USEFUL FORMULAE 
 
 39 
 
 We are interested here in the case when the variable is cos 
 and 6 is a real angle, such that < 6 < TT. We may then put, 
 with the usual notation of the Gamma and hypergeometric 
 functions, 
 
 P n (cos 0) = r ( 1 1 _ cot ^F (- n, n + 1 ; 1-ro; sin|) 
 
 ...... (68), 
 
 &W (C S *> 
 
 /i + m)7rcot TO 2 ^f w, Ti + 1 ; 1 m; sinO 
 
 ......... (69), 
 
 n - (cos 0) = 
 ; 
 
 C S 
 
 ^o (2 cos $ - 2 cos 
 
 When m is a positive integer, we have 
 
 ...... (70). 
 
 (VI), 
 
 and when n is a positive integer 
 
 ^ 
 
 * ~ 
 
 (cos (?) = 
 
 2 2 .4 2 
 
 .(73), 
 
40 POLAR COORDINATES [CH. 
 
 The last two formulae illustrate the method of deriving 
 more complicated wave-functions from simple ones by differen- 
 tiation. The functions 
 
 1 1 , r+z 
 
 - and jr- log - 
 r 2r & r z 
 
 are in fact solutions of (52) when k 0. 
 The formula 
 
 +l p n m (x) P (x) dx = (n + v) 
 -i 
 
 in which m, n, v are positive integers or zero, enables the 
 coefficients in an expansion of a function in series of functions 
 Pn m (#) to be determined by a simple integration. 
 
 For further properties of the Legendre functions we must 
 refer to Heine's Kugelfunktionen, Byerly's Fourier Series and 
 Spherical Harmonics, Whittaker's Analysis, Nielsen's Handbuch 
 der Cylinderfunktionen and Theorie des fonctions metaspheriques, 
 and to memoirs by E. W. Hobson* and E. W. Earnest. 
 
 Tables of the Legendre functions have been published by 
 J. W. L. GlaisherJ, J. Perry and A. Lodge ||; some tables of 
 the functions P n m (//,) have been given by H. TallquistlF. 
 
 For the history of the functions of Legendre and Bessel 
 the reader should consult the article by A. Wangerin in the 
 Encyklopddie der Mathematischen Wissenschaften, Bd. II. 1, 
 Heft 5 (1904), p. 695. 
 
 15. Relations between various solutions. 
 
 We have already remarked that when a wave-function or 
 a solution of equation (52) involving arbitrary constants has 
 been found, other solutions may be derived from it by the 
 method of summation or integration. By choosing our sum or 
 integral so that it represents certain simple solutions of the 
 
 * Proc. London Math. Soc. Ser. 1, Vol. 22 (1891), p. 431, and op. cit. 
 
 t Quarterly Journal, Vol. 39 (1908), p. 97. 
 
 J Brit. Ass. Report (1879). 
 
 Phil. Mag., Dec. (1891). See also Byerly, loc. cit. 
 
 || Phil. Trans. A, Vol. 203 (1904). 
 
 IT Acta Societatis Fennicae, Vols. 32, 33. 
 
Ill] RELATIONS BETWEEN VARIOUS SOLUTIONS 41 
 
 fundamental equation, a number of important identities may 
 be obtained. A few formulae will be written down to illustrate 
 this*. 
 
 If R 2 = r* + ri* - 2rr x cos 0, 
 
 (Heine and Hobson) ...... (77), 
 
 ) ^ (M) P n (cos 6) (r, > r) 
 (Heine and Macdonald) ...... (78), 
 
 n (Ar) ^ (M) P n (cos 6) (r, > r) 
 H 
 
 ......... (79). 
 
 We may illustrate the peculiar behaviour of certain definite 
 integral solutions of our fundamental equation by the following 
 
 example, in which k is supposed to be real and positive. 
 
 rm 
 Let f m ( x )= \ cosxtx(t)dt, 
 
 J o 
 
 rm 
 
 where ^ (t) is a function such that I | ^ (t) \ dt is convergent, 
 
 ./o 
 
 then it may be proved by means of Fourier's double integral 
 theorem thatf 
 
 1 f 00 sin k (r - r,) ,. , x , / / \ /j ^ 
 
 r,)dr^f m (r) (k>m) 
 
 =/*(*") 
 
 sin kR f , , , 
 Hence U= ~~ f ^ * 
 
 is a solution of (52) which reduces to either f m (r) or /* (r) 
 when 6 0. Solutions of (52) which are derived from the 
 elementary solutions by integrating with regard to n have 
 been employed by H. W. March J. He makes use of an inversion- 
 formula 
 
 * Some very general formulae are given by L. Gegenbauer, M&natsh. /. Math. 
 Bd. 10 (1899), p. 189. 
 
 t This equation is obtained in a different manner by G. H. Hardy, Proc. 
 London Math. Soc. (2), Vol. 7 (1909), p. 445. 
 
 t Ann. d. Physik, Bd. 37 (1912). See also H. Poincare", Comptes Rendui, 
 t. 154 (1912), p. 795 ; W. v. Rybcynski, Ann. d. Phys. Bd. 41 (1913). 
 
 This is in some respects analogous to the inversion-formula given by 
 F. G. Mehler, Math. Ann. Bd. 18 (1881), p. 161. 
 
42 POLAR COORDINATES [CH. 
 
 (80), 
 
 where 
 
 (81). 
 
 7T COS 
 
 It is sometimes instructive to find how a wave-function, 
 depending on an arbitrary function, can be expressed in terms 
 of elementary wave-functions. Now in the second example 
 of 5 the electric and magnetic forces are all of the form 
 
 or are the sums of terms of this form. Consequently, a function 
 of this type may be expected to be a wave-function and it is 
 easy to verify that this is the case*. We may now deduce that 
 
 
 
 is a solution of (52) and consequently it follows that tan m ^ 
 /a 
 
 and cot m ^ are solutions of equation (54) when n = 0. We 
 2i 
 
 have in fact 
 
 (/j H\ 
 
 cos WITT . cot m ^ tan m ^ J 
 
 In the last formula m must not be zero or a negative integer. 
 
 16. The convergence of series of elementary solutions. 
 
 When k our fundamental equation (52) reduces to 
 Laplace's equation Aw = and we have the familiar elementary 
 solutions 
 
 * The theorem also follows immediately from a result given by A. E. 
 Forsyth, Messenger of Mathematics (1898), p. 114, and E. W. Hobson, Joe. eit. 
 
7 -, 
 
 III] CONVERGENCE OF SERIES 43 
 
 r n P n m (cos 0) cos m (<f> - fa), r n Q n m (cos 0) cos m (<f> - fa) \ 
 ^ P n m (cos 0) cos m (<t> - fa), -^ Q n m (cos (9) cos m (< - </> )J 
 
 (84). 
 
 A pair of series of the type 
 
 2 - 
 
 which converge when r a are suitable for representing harmonic 
 functions inside and outside the sphere r = a because the first 
 converges absolutely when r < a and the second when r > a. 
 
 The case in which k is very similar. When n is large 
 the function ty n (kr) may be replaced by 
 
 (kr) n+l 
 
 and so a series of the form 
 
 ^ft(!r)/.(#,>) 
 
 converges like a power series. Again, when kr is real, we have 
 1 1 
 
 2 ' (/b-) 2 
 
 1.3...(2n-l) 1 ,.,, 
 
 ......... ( 
 
 2.4...(2n) 
 
 n being a positive integer. It is clear from this equation that 
 | n (kr) \ decreases as r increases, hence if a series of the form 
 
 =0 
 
 converges absolutely for any value of r it converges absolutely 
 for all greater values of r. 
 
 For a discussion of the convergence of series of spherical 
 harmonics we may refer to C. Neumann's book Ueber die nach 
 Kreis-, Kugel- und Cylinder-Fun ktionenfortschreitenden Entwicke- 
 lungen, Leipzig (1881); to Heine's Kugelfunktionen, Bd. 1, p. 435, 
 Bd. 2, p. 361, and to papers by IT. Dini, Ann. di Mat (2), t. 6 
 (1874); H. Poincare, Comptes Rendus, t. 118 (1894), p. 497; 
 S. Chapman, Quarterly Journal, Vol. 43 (1912), p. 1 ; T. H. 
 
44 POLAR COORDINATES [CH. 
 
 Gronwall, Math. Ann. Vol. 74 (1913), Vol. 75 (1914), Comptes 
 Rendus (1914), Amer. Trans. Jan. (1914), Vol. 15; C. Jordan, 
 Gours d' Analyse, 2nd ed., Vol. 2, p. 252 ; and B. H. Camp, Bull, 
 of the Amer. Math. Soc. Vol. 18 (1912), p. 236. The con- 
 vergence of series of Legendre polynomials has been discussed 
 very thoroughly by G. Darboux, Liouvilles Journal (2), t. 19 
 (1874), p. 1; (3), t. 4, p. 393; O. Blumenthal, Dissertation, 
 Gottingen (1898); E. W. Hobson, Proc. London Math. Soc. (2), 
 Vol. 7 (1909); L. Fejer, Math. Ann. Bd. 67, p. 76; D. Jackson, 
 Amer. Trans. Vol. 13 (1912). 
 
 17. The scattering of plane homogeneous electro- 
 magnetic waves by a spherical obstacle. 
 
 The effect of small particles in scattering incident radiation 
 has been discussed very thoroughly by Lord Rayleigh * who has 
 used it as the basis of a mathematical theory of the blue colour 
 of the sky. The action of a single spherical particle is of 
 fundamental importance and so the electromagnetic theory 
 of the scattering of light by a dielectric sphere has been worked 
 out by Lord Rayleigh f, Prof. Love* and other writers. This 
 theory can also be developed so as to cover the mathematical 
 theory of the rainbow. 
 
 The more general theory of the scattering of incident radia- 
 tion by a spherical obstacle with arbitrary optical properties || 
 admits of some very interesting applications in the study of the 
 colours exhibited by metal glasses, metallic films and colloidal 
 solutions or suspensions of metals. The electromagnetic theory 
 of these colours has been developed by J. Maxwell GarnettH", 
 G. Mie**, R. Gansff and Happelff, who have considered 
 
 * Phil. Mag. Vol. 41 (1871), pp. 107, 274, 447; Vol. 12 (1881), p. 81; 
 Collected Papers, Vol. 1, pp. 87, 104, 518. 
 
 t Phil. Mag. Vol. 44 (1897), pp. 2852; Collected Papers, Vol. 4, p. 321; 
 Proc. Boy. Soc. Vol. 84 (1910), p. 25 ; Vol. 90 (1914), p. 219. 
 
 J Proc. London Math. Soc. Vol. 30 (1899), p. 308. 
 
 The work of Stokes, Camb. Trans. Vol. 9 (1849), p. 1, with later appli- 
 cations, Collected Papers, Vol. 4, and of L. Lorenz, Wied. Ann. Vol. 2 (1880), 
 p. 70, opened up the subject. 
 
 || The case of small conductivity was discussed by G. W. Walker, Quart. 
 Journ. Vol. 30 (1899), p. 204; Vol. 31 (1900), p. 36. 
 
 IF Phil. Trans. A, Vol. 203 (1904), p. 385; Vol. 205 (1905), p. 237. 
 
 ** Ann. d. Phys. Vol. 25 (1908), p. 377. 
 
 ft Ibid. Vol. 29 (1909), p. 280 ; Vol. 37 (1912), p. 881. 
 
Ill] SPHERICAL OBSTACLE 45 
 
 the cases of spheres and ellipsoids endowed with the optical 
 constants 6, /JL, a. 
 
 The particular case of a perfectly conducting sphere was 
 worked out by J. J. Thomson* and has been discussed in 
 greater detail by J. W. Nicholson^. 
 
 The problem is also of importance in connection with the 
 theory of comets' tails which has been developed by Euler, 
 FitzGeraldij: and Arrhenius. The pressure of light on a 
 perfectly conducting spherical obstacle has accordingly been 
 calculated by K. Schwarzschild|| and J. W. Nicholson U. The 
 more general case of a sphere with the optical constants e, ^ <r 
 has been treated very fully by P. Debye**. 
 
 Let us assume that the electric and magnetic forces E', H' 
 at any point of space are the real parts of vectors E, H of the 
 form Ae"* 1 , where A is a complex quantity independent of t. 
 We then write 
 
 H ivE = Me***, where ^ = ^^ la ' ..... (86) 
 
 /L6CD 
 
 and we find that the differential equations satisfied by M in a 
 medium whose optical constants are e, p, a- can be written in 
 the form 
 
 ...... (87) ' 
 
 * Recent Researches, p. 437. 
 
 t Proc. London Math. Soc. (2), Vol. 9 (1910), p. 67 ; Vol. 11 (1912), p. 277. 
 
 Scientific Writings, pp. 108, 531. 
 
 Phys. Zeitschr. Vol. 2 (1901), pp. 8197 ; Das Werden der Welten, Leipzig 
 (1907), p. 85. 
 
 || Sitzungsber. d. Kgl. Bayer. Akad. d. Wiss. Vol. 31 (1901), p. 293. 
 
 IT British Association Reports (1910), p. 544; Monthly Notices of the Royal 
 Astronomical Society, Vol. 70, p. 544. See also J. Proudman, Ibid. Vol. 73 (1913), 
 p. 535. 
 
 ** Ann. d. Physik, Vol. 30 (1909), p. 57. 
 
46 POLAR COORDINATES [CH. 
 
 These equations may be satisfied by putting 
 
 I 8 2 (rfl) k 8(rH) 
 ~r 8r80 rsinl9 86 
 
 .(88), 
 
 If - 1 d 2 (rQ) - fc 8 (rQ) 
 ~~ + r 8(9 
 
 where the function l=U iV isa, solution of (52). 
 
 The electric and magnetic forces may be derived at once 
 from these expressions by equating the ambiguous and un- 
 ambiguous parts as explained in 2. 
 
 We shall now assume that the incident wave of plane 
 homogeneous monochromatic polarised light is represented by 
 
 (M r , M 9 , M+) = <p"**** (sin 0, cos 6, i) ...... (89), 
 
 the electric vector being parallel to the axis of y. 
 
 The corresponding function O is obtained by solving the 
 equation 
 
 o = sin 6 . 
 
 We easily find that 
 
 rft = 7<r 2 e** [cosec 6 . e ikrco * e +/" (0, <f>) e ikr +/ 2 (0, <j>) e~ ikr ] 
 
 ...... (90). 
 
 Choosing the unknown functions so that rfl is finite for 6 = 
 and 6 IT we obtain finally 
 
 f d 
 
 e** 2 cosec . e^ 008 ' - cot e ikr 
 
 G ~\ 
 - tan ^ IT*" 
 
 We may now assume an expansion for rH of the form 
 * I a n ^ n (kr)P n l (GOsO). 
 
 n=l 
 
 To determine the coefficients a n we multiply by sin 6 and 
 differentiate both sides of the equation. Then since 
 
 [sin . P n l (cos 0)] = - w (n + 1) sin . P n (cos 6) (92), 
 
Ill] 
 
 SPHERICAL OBSTACLE 
 
 47 
 
 the coefficients may be determined at once with the aid of 
 Lord Rayleigh's expansion* 
 
 ikr sin (9 . e ikrco *' = 2 i n+l (2n + 1) ^ n (kr) P n (cos 6) sin 6 
 =o 
 
 (93). 
 
 We thus obtain 
 
 (n+l) 
 
 (94). 
 
 Now let Q!, X1 2 be the functions from which the electric 
 and magnetic forces in the scattered light and transmitted 
 light may be derived respectively. The appropriate forms are 
 given by equations of the type 
 
 rV^ ZBnZn 
 
 n 1 (COS 6) COS 
 
 n 1 (cos 6) sin <f> 
 
 (95), 
 
 n=l 
 
 (cos (9) cos 
 
 r F 2 = D n \lr n (hr) P n l (cos 6) sin 
 
 (96), 
 
 where h, ij are the values of k, v respectively within the sphere. 
 It is easy to deduce from (88) that the tangential components 
 of the electric and magnetic forces are continuous in crossing 
 the sphere r a, if when r = a 
 
 .(97). 
 
 * Theory of Sound, Vol. 2, p. 272. The expansion was also obtained 
 independently by Heine, Kugelfunktionen (1878), Bd. 1, p. 82. 
 
48 POLAR COORDINATES 
 
 These conditions give 
 
 kA n &' (lea) - hO n ^n' (ha) = t+' 
 
 Solving these we get 
 
 r.j.! i _o 2tt + 1 H n 
 
 JL ,'n+i 
 
 D _ 
 
 __ 
 
 * tt 
 
 where 
 
 [CH. 
 
 ...(98). 
 
 E n 
 
 (ha) - i^r n ' (te) t (ha) 
 (ka) 1r n ' (ha) - rj^ n f (ka) + n (ha) 
 
 F n = Sn (ka) tn (ka) - $n (ka) ^ n f (ka) 
 S n = V+n (ha) f n (ka) - v& (ka) + n (ha) 
 L n = v^ (!M) n (ka) - *i& (ka) + n (ha) 
 
 We can prove that our series all converge absolutely and 
 uniformly at about the same rate as a power series of the form 
 
 . . .(99). 
 
 ka? 
 where x is either kr or . The proof depends on the fact 
 
 that when n is large we have approximately 
 
Ill] FREE VIBRATIONS 49 
 
 18. Free damped vibrations for the space outside the 
 sphere. 
 
 It should be noticed that some of the terms of our series 
 become infinite when either G n = or L n = : fortunately, 
 however, the roots of these equations turn out to be complex 
 and so when k is real no values of k need be excluded from the 
 discussion. The damped vibrations determined by the roots of 
 the equations L n = may be distinguished as the electric 
 vibrations, those determined by equations of type G n = as the 
 magnetic vibrations. Some of the roots of the equations have 
 been calculated for the case of a totally reflecting sphere by 
 Sir J. J. Thomson*, who finds that the roots are all complex. 
 The vibrations for the space inside a totally reflecting sphere 
 have been discussed by Prof. J. W. Nicholson f, those for the 
 space between two concentric spheres by Sir J. J. Thomson}, 
 Sir Joseph Larmor, Prof. H. M. Macdonald|| and A. Lampal". 
 
 P. Debye, who has calculated some of the roots for a case of 
 a dielectric sphere, finds that the roots are complex and of two 
 types. When the index of refraction is large, the imaginary 
 part of a root of the first type varies very little with the index 
 of refraction N and approaches a limit different from zero when 
 N-+ oo . If on the other hand p is a root of the second type, 
 Np tends to a finite real limit, viz. a root of i/r n (Np) 0, when 
 N * oo and so the imaginary part of a root of this type must 
 be very small when N is large. 
 
 The vibrations belonging to the space outside a sphere 
 must be in all cases damped on account of the loss of energy by 
 radiation; when the refractive indices of the outside medium 
 and sphere are very nearly equal, they are clearly very strongly 
 damped ; thus it is only when the refractive index is large that 
 some of them are durable. It is doubtful whether a substance 
 exists which has a large refractive index and does not absorb 
 light to a marked extent. 
 
 * Proc. London Math. Soc. Ser. 1, Vol. 15 (1884), p. 197; Recent Researclies, 
 p. 361. 
 
 t Phil. Mag. 1906, p. 703. 
 $ Recent Researches, p. 373. 
 
 Proc. London Math. Soc. Ser. 1, Vol. 26 (1894), p. 119. 
 i| Electric Waves, Chapters 6-7. IT Wien. Ber. 112 (1903), p. 37. 
 
 B. 4 
 
50 POLAR COORDINATES [CH. 
 
 It will be seen later that the characteristic vibratibns play 
 an important part in determining the size of the sphere on 
 which the pressure of a given type of incident radiation has 
 a maximum value. 
 
 Prof. Love* has used the solutions corresponding to the 
 characteristic vibrations to discuss the mode of decay of an 
 arbitrary initial disturbance. He makes use, in fact, of the 
 functions % n (kr\ where k is one of the roots of one of the 
 equations G n = 0, L n = ; only the sphere is treated as a perfect 
 conductor. 
 
 This method can easily be extended so as to provide us 
 with a method of discussing the problem of the scattering of an 
 arbitrary primary disturbance by a spherical obstacle. In this 
 method we assume that the total disturbance outside the sphere 
 can be represented by 
 
 rU=2 2 S^ln.m.p'fn (k p r) P n m (cos 0) cos m (</>-< 
 
 w =0 w=0p 
 
 r F = 2 2 2 #,,,,> ^ (V* 1 ) p n m ( cos 0) sin m (<t> - 
 
 .'..(<), 
 
 where the k p s are roots of one of the equations of the type 
 ,' (to) ^ n (ka) - ^ (ka) ir n (ha) = 0) 
 ,' (to) ^ M (ka) - 1)^ ' ""-^ '- ""-^ - "' 
 
 The coefficients must then be chosen so that this total disturb- 
 ance has the same character as the primary disturbance at its 
 singularities outside the sphere and at an infinite distance, 
 taking into account of course the presence of diverging waves 
 from the spherical obstacle the effect of which is, however, 
 negligible at infinity. 
 
 The field inside the sphere is represented by equations 
 similar to (o>), one with h p written in place of k p . The boundary 
 conditions are satisfied in virtue of (100). 
 
 If a series of type (co) should fail to represent the disturbance 
 outside the sphere, it may be necessary to add terms cor- 
 responding to the free characteristic vibrations. These are of 
 the form (&>) with the function ? n written in place of i/r n and 
 numbers k n determined by equations of type L n = 0, G n = 0. 
 * Proc. London Math. Soc. Ser. 2, Vol. 2 (1904), p. 88. 
 
Ill] SMALL OBSTACLE 51 
 
 19. The case of a very small obstacle. 
 
 When a, the radius of the sphere, is very small compared 
 with the wave-length X of the incident radiation, we may treat 
 ka and ha as small quantities. We may then obtain some idea 
 of the relative magnitudes of the different coefficients in our 
 series by using the expansions (60) and (61). 
 
 It is easy to see that the values of A ny B n decrease very 
 rapidly in absolute magnitude as n increases. The disturbance 
 radiated from the sphere can consequently be represented 
 approximately by superposing a small number of partial waves, 
 the effect of the others being negligible. 
 
 Remembering that vh rjk, we find that when H is finite 
 vk n h n+l (k* - h?) a 
 
 r . (h\ n _ ika 
 ^ \jk) ' 
 
 (n + 1) k n h n ~ l r) (k* - h 2 ) a +l 
 
 and it is easy to see that all our series converge. 
 
 It appears from these expressions that the nth magnetic 
 wave, i.e. the disturbance due to the nth term in the expansion 
 for U 1} is of the same order of magnitude as the (n + l)th 
 electric wave, i.e. the disturbance due to- the (n + l)th term in 
 the expansion of Fj. This is in sharp contrast with the result 
 obtained by Sir J. J. Thomson for the case of the totally 
 reflecting sphere wherein the nth electric wave and the nth 
 magnetic wave are of the same order of magnitude. 
 
 The first electric wave is clearly of chief importance and 
 Mie has proposed to call this Rayleigh's radiation. We easily 
 find that 
 
 A~~(& 2 -^ 2 )a 5 
 
 The following diagrams, which are taken from Mie's paper, 
 indicate the character of the electric lines of force for the first 
 four partial vibrations of each type. For the magnetic waves 
 
 42 
 
52 
 
 POLAR COORDINATES 
 
 [CH. 
 
 Electric vibrations. Fig 2. 
 
 Magnetic vibrations. 
 
Ill] 
 
 SMALL OBSTACLE 
 
 53 
 
 E r = and so the electric lines of force are spherical curves. 
 In the case of the electric waves the lines of force lie on 
 certain cones and the diagrams represent the intersections of 
 a sphere with these cones, the vertices of the cones being at 
 the centre of the sphere*. 
 
 20. Polarisation of the scattered light. 
 Let us now look for cases when the light scattered by the 
 sphere is linearly polarised. It is easy to see that E 9 and M+ 
 
 These conditions are 
 
 both vanish when ^ = and -TT-T = 0. 
 do o<f> 
 
 both satisfied by <f> = 0, i.e. when the observer looks in a direction 
 at right angles to the electric vibration in the incident wave 
 (Fig. 3). 
 
 It appears from the figure that the component of the electric 
 vector of the scattered light, which 
 is at right angles to the direction 
 in which the observer is looking, 
 is parallel to the electric vector 
 in the incident wave. In a similar 
 way it is found that E+ and M e 
 
 both vanish when <f> = -= , i.e. 
 
 when the observer looks in a 
 
 direction at right angles to the H~ 
 
 magnetic vibration in the incident Fi 8- 3 - 
 
 wave. The magnetic vibrations in the incident and scattered 
 
 waves are now found to be parallel. 
 
 The experiments of Steubingf with different kinds of col- 
 loidal gold solutions have shown that when the solution is 
 illuminated with polarised light and viewed in the manner 
 described, there is always a small quantity of unpolarised light 
 sent out from the particles, but the greater portion of the 
 scattered light is polarised in the way the theory requires. 
 The slight disagreement between the theory and observations 
 is attributed to the fact that the metallic particles are probably 
 
 * The author is indebted to the publisher of the Annalender Physi fc.Herr Johann 
 Ambrosius Barth, for permission to reproduce the figures on pp. 52, 59 and 64, 
 t Diitertation, Greifswald (1908) ; Ann. d. Phys. Vol. 26 (1908), p. 329. 
 
54 POLAR COORDINATES [CH. 
 
 not all spheres*, that some may have developed into crystals 
 perhaps of octahedral form. The mathematical theory of the 
 scattering of waves has not yet been fully developed. The 
 problem is, however, one of great importance in meteorological 
 optics. The case of a regular distribution of atoms or molecules 
 has recently been brought into prominence t by experimental 
 work on the scattering of Rontgen rays by a crystal J. Approxi- 
 mate mathematical theories have been given by several writers . 
 
 21. Intensity of the scattered light. 
 
 When Rayleigh's radiation alone is considered, we have 
 
 H r = 0, E r = 
 
 3(rF) 
 
 B 
 
 5<f> 
 
 _ikd(rV) 
 
 * 
 
 r 30 vrsmff 
 
 where 
 
 -fc Sn 
 
 At a great distance from the origin the radial electric force 
 is of order while the transverse electric and magnetic forces 
 
 are of order -. Hence the intensity of the scattered light 
 
 diminishes ultimately according to the inverse square law when 
 points on the same radius are considered. It also varies as the 
 square of the volume of the particle. 
 
 * This remark is made by both Maxwell Garnett and Mie. 
 
 f It had previously been considered by Lord Rayleigh, " On the influence of 
 obstacles arranged in rectangular order on the properties of a medium," Phil. 
 Mag. (5), Vol. 34 (1892), p. 481; Scientific Papers, Vol. 3, p. 19; and by 
 T. H. Havelock, Proc. Roy. Soc. A, Vol. 77 (1906), p. 170. 
 
 J Laue, Friedrich, und Knipping, Sitzungsber. der Konigl. Bayerischen 
 Akad. d. Wiss. June 1912. 
 
 See, for instance, W. L. Bragg, Proc. Camb. Phil. Soc. Vol. 17 (1913), 
 p. 43 ; Proc. Roy. Soc. A, Vol. 88 (1913), p. 428 ; M. Laue, MUnchener Ber 
 (1912), p. 363; Ann. d. Phys. Bd. 41 (1913), p. 989, Bd. 42 (1913), p. 397; 
 P. P. Ewald, Phyt. Zeitschr. (1913), p. 465; L. S. Ornstein, Amsterdam Proc. 
 (1913) ; M. Born u. T. v. Karman, Phys. Zeitschr. (1912), p. 297. 
 
Hi] INTENSITY OF THE SCATTERED LIGHT 55 
 
 An approximate formula for the intensity is 
 
 7 = * 
 
 the intensity of the incident radiation being . 
 
 If o- = for the medium outside the sphere, the quantity 
 k is inversely proportional to the wave-length X of the incident 
 radiation. Hence when h is large compared with k, we have 
 Lord Rayleigh's result that the intensity of the scattered light 
 varies inversely as the fourth power of the wave-length. The 
 short waves are on this account scattered far more profusely 
 than the long ones and so we have an explanation of the blue 
 colour of the sky. 
 
 The above formula for the intensity of Rayleigh's radiation 
 indicates that there is no light of this type in a direction for 
 
 which 6 = <f) = -= , i.e. when the observer is looking in a direction 
 
 parallel to the direction of the electric vibration in the incident 
 wave. To obtain an expression for the intensity of the light 
 sent out in this direction we must take into account the second 
 electric wave and the first magnetic wave. Referring back to 
 the expressions for B. 2 and A lt we find that when cr is neglected 
 both inside and outside the sphere, the intensity of the scattered 
 radiation varies inversely as the eighth power of the wave- 
 length *. This corresponds to Tyndall's " residual blue " which 
 is purer than the blue seen under other conditions. 
 
 22. The absorption of light by a spherical obstacle. 
 
 The total energy absorbed from the incident radiation by 
 a single particle consists of two parts ; first of all the energy 
 scattered and secondly the energy which flows into the particle 
 and is transformed into heat or chemical energy. Both these 
 quantities may be calculated by the following method which is 
 a simplification of the one given originally by Mie. 
 
 The flow of energy across unit area of a very large sphere 
 concentric with the spherical particle takes place at a rate 
 measured by the radial component of Poynting's vector, i.e. 
 E 'Ht E^H e f . 
 
 * Lord Kayleigh, Phil. Mag. Vol. 12 (1881), p. 81. 
 
56 
 
 POLAR COORDINATES 
 
 [CH. 
 
 Now if E, E and H, H are conjugate complex quantities, 
 this component is equal to 
 
 Integrating with regard to t so as to obtain the mean value 
 of this quantity over a period, we obtain an expression which 
 may be written in the formf 
 
 8 = ~ \M 9 St - M 9 Mt - M 9 *Sf + MfMf], 
 
 where M, M* are the values of M corresponding to the signs 
 +, - respectively in (86) and M, M* are derived from them by 
 changing the sign of i. 
 
 The function 1 for the outer space is the sum of H and fl lt 
 hence we may write 
 
 rfl = 2 
 
 =i 
 
 n=l 
 
 + v n er* + 
 
 rft* = S 
 
 (cos 
 (cos 
 [ (cosO)[u n e^ + v n e-^ 
 
 ...(106), 
 
 where 
 
 B n ) f 
 
 When the expression S is integrated over the spherical 
 surface, the result can be expressed in the form 
 
 ^o ~t~ * i -*oi> 
 
 where / depends only on the incident radiation, / x depends 
 only on the scattered radiation and represents the amount of 
 energy absorbed by scattering, / O i depends on both types of 
 radiation and represents the total absorption of energy from 
 
 f We assume now that <r = outside the sphere. 
 
Ill] 
 
 ABSORPTION OF LIGHT 
 
 57 
 
 the incident wave. The sum of the three terms with its sign 
 changed represents the amount of energy that flows into the 
 sphere and is truly absorbed by it. 
 
 Performing the integration with regard to <f> and collecting 
 the different terms together, we find that the surface integral 
 can be written in the form 
 
 ^- 1 1 rrfpn 
 
 2t?-lmlU* > 
 
 dd ysintf 
 
 f/dM n dv w 
 \(dr + dr 
 
 dw n dw n 
 dr dr 
 
 + kk (U n + V n ) (tt m + V m ) - kkw n W m l . 
 
 This expression can be simplified with the aid of the relations 
 
 = 
 
 - ) - -A = mn 
 /sm# 
 
 2n + l 
 
 m = n 
 
 The first of these is evident since the integrand is the 
 differential coefficient of a function which vanishes at both 
 limits. To prove the second we make use of the formulae 
 
 7 
 
 sin P w i (cos 6) = - cos . P,, 1 (cos 0) 
 
 
 SU1 
 
 + n (n + 1) cos . PJf 
 
 ...(109). 
 
58 POLAR COORDINATES 
 
 The integral is thus equivalent to 
 
 n (n + 1) m (m + 1) I * P n (cos 0) P m (cos 0) sin 
 J o 
 
 and so has the value we have assigned to it*. 
 
 [CH. 
 
 IT 
 
 2, 
 
 Our surface integral now reduces to 
 
 dv n \ 7 dw n 
 
 -j-2 l+kWn-j^ 1 
 
 drj " dr 
 
 
 Since <r = outside the sphere, we have k k. Also when 
 r is very large we may use the approximations 
 
 - TT) , 2 . nr _ 1 2^ + 1 
 
 ...(110). 
 
 u n ~ cos 
 
 \kr 
 \ 
 
 u n ~ cos r - n + 
 
 1) 
 
 We thus find that 7 = and 
 
 
 
 
 n (n 
 
 )] . . .(Ill), 
 
 When v = 1 these expressions are just half those given by 
 Mie on p. 436 of his memoir. The reason why they must be 
 doubled is that our expressions for the incident light give an 
 intensity . 
 
 In the case of a solution containing N spherical particles 
 per unit volume the total absorption coefficient is thus 
 
 A = ^ 2 n (n + 1) [* (I n + B n ) - (- i) n (A n + n )]...(113). 
 w n= i 
 
 It is interesting to study the variation of this quantity with 
 the wave-length for particles of different sizes. Mie has drawn 
 * Cf. L. Lorenz, Oeuvres tcientifique*, p. 526. 
 
Ill] 
 
 ABSORPTION OF LIGHT 
 
 59 
 
 the absorption curves for particles of gold varying in size from 
 20 pp to ISO/a/*. 
 
 20 
 
 JO 
 
 WO 
 
 5OO SSO GOO 
 
 Fig. 4. 
 
 G5O 
 
 L .' 
 Tfsoj 
 
 It will be seen that for ruby red gold solutions containing 
 very fine particles there is an absorption maximum in the 
 green corresponding to a wave-length of about 525 /JL/J,. 
 
 Mie has also drawn curves showing the pure absorption in 
 colloidal gold solutions. 
 
 The colours of silver particles in colloidal silver solutions 
 have been discussed with the aid of the mathematical theory 
 by E. Mliller*. The particles of a silver solution show beautiful 
 colour phenomena, all colours of the spectrum from the extreme 
 
 * Ann. d. Phys. Bd. 35 (1911). 
 
60 POLAR COORDINATES [CH. 
 
 blue to the extreme red being present*. For other applications 
 of the theory we must refer to the papers of Lord Rayleigh, 
 Garnett and Mie and to Prof. Wood's Physical Optics. 
 
 The mathematical theory for a large number of particles 
 has been developed further on certain simplifying assumptions 
 by F. Hasenohrlf, A. SchusterJ, W. H. Jackson, L. V. King||, 
 A. Einstein If and M. v. Smoluchowski**. It has been used 
 recently by W. J. Humphreys ff in a study of the effect on 
 climate of large quantities of volcanic dust in the upper 
 atmosphere. 
 
 23. The pressure of radiation on a spherical obstacle. 
 
 We shall now calculate the pressure of radiation on a spherical 
 obstacle, following the work of Debye except in some of the 
 details. The pressure is calculated on the assumption that the 
 force exerted by an electromagnetic field on a unit charge 
 moving with velocity v has components of typeJJ 
 
 F x = E x + v -*H z - V -lH y (114), 
 
 c c 
 
 and that the equations of the field can be derived by a process 
 of averaging from the electron equations 
 
 1 dE \ 1 
 
 Now if 
 
 X = E * E + - - - ._.. ( n 6)) 
 
 = c(E y H,-E,H y ) 
 
 * A coloured reproduction of an ultramicroscopic picture of a silver solution 
 is given by H. Siedentopf , Ber. d. Deutsch. Phys. Ges. (1910), p. 6. 
 
 f Wien. Berichte (1902). $ Astrophysical Journal, Vol. 21 (1908). 
 
 Bull, of the Amer. Math. Soc. Vol. 16 (1910), p. 473. 
 II Phil. Trans. A, Vol. 212 (1913), p. 375. 
 1T Ann. d. Phys. Bd. 38 (1910), p. 1275. 
 
 ** Boltzmann Festschrift (1904), p. 626 ; Ann. d. Phys. Bd. 25 (1908), p. 205 ; 
 Phil. Mag. Vol. 23 (1912). 
 
 ft Bull, of the Mount Weather Observatory (1913) ; Journal of the Franklin 
 Institute, Aug. (1913). 
 J: We assume now that o-=0, =/*=! for the external medium, so that v = l. 
 
Ill] PRESSURE OF RADIATION 61 
 
 etc., we have identically on account of (115) 
 
 dX x dX y dX z 1 dS x f ^ 
 
 pf x -^ -- r -K -- h-s --- ; -57- ......... V.U-O- 
 
 dx 3y 3s c 2 3$ 
 
 Hence by Green's theorem, if (I, m, n) are the direction 
 cosines of the outward drawn normal to a surface cr enclosing 
 all the charges in the field, 
 
 fj(lZ. + mX y + nX z ) &a = jff (pF. + 1 ^ dxdydz. . .(118). 
 
 The ^-component of the force exerted by the electromagnetic 
 field on the obstacle is thus the same as if there were tractions 
 X, F, Z at each point of the surface and a volume force 
 
 -^-: similarly for the other components. 
 
 Now when we integrate with regard to t so as to obtain the 
 
 ocr 
 
 mean value of the pressure over a period, the term -^ 
 
 contributes nothing on account of the periodicity of 8. The 
 pressure may consequently be calculated from Maxwell's tractions 
 X, Y, Z. In the present case the mean value of the pressure 
 in the direction of the axis of z is derived from the quantity* 
 (IZ X + mZ y + nZ g ) = - 4 cos [E ' 2 + Ef + #/ 2 -f H+* 
 
 - E r '* - H;*] - sin 6 [E r 'E 9 ' + H r 'H 9 ']. 
 
 The surface integral of the mean value of this quantity over 
 a period is accordingly 
 
 - T I 
 
 4JoJO 
 
 + sin 6 (E r E e + E r E e + H r H + S r H 9 )] sin 6 . dd d<f>. 
 
 When the incident radiation only is taken into account, this 
 integral becomes simply 
 
 r 2 fw r2ir 
 
 - 
 
 *J J 
 
 and vanishes completely. Again, it is easy to see that when r 
 is large the components E r , H r for the scattered field are small 
 
 * We assume now that the surface <r is a sphere whose centre is at the origin. 
 Our expression is easily obtained by writing down the expressions for E x ', E y ', E,' 
 in terms of E r ', Ee', E$'. 
 
62 POLAR COORDINATES [CH. 
 
 compared with the transverse components, a product such as 
 H r H 9 being of order r~ 3 may be neglected when r is large, and 
 so we have only to consider the integral 
 
 o o 
 
 where E=E Q + E l} H^H^ + H^ The terms which depend 
 only on the incident field may, moreover, be disregarded. 
 To evaluate P z we need the values of the integrals 
 
 JP l dP i\ 
 
 'P TO > + sin' 6 t- 5L cot . W, 
 
 By using the relations (109) we may transform the first of 
 these into 
 
 n (n + 1) m (m + 1) / * P n (cos 6) P m (cos 0) sin (9 cos d0 
 Jo 
 
 - * [cos ^ . P n i . P m i] cos . d0. 
 
 Now 
 (2 + 1) cos ^ . P n (cos ^) = (n + 1) P n+1 (cos 0) 4- nP^ (cos 
 
 and 
 
 (2n + 1) cos ^ . P,, 1 (cos 6) = r^P 1 n+1 (cos 0) + (w + 1) PVi (cos ^) 
 
 ......... (120); 
 
 hence when the second integral is integrated by parts we 
 obtain two integrals of the type (76) and so we have finally 
 
 /i = m = n 
 
 ' (121). 
 
 When the second integral 7 2 is integrated by parts it becomes 
 
 (2n - 
 
Ill] PRESSURE OF RADIATION 68 
 
 Writing P 2 in the form 
 
 - 5 r r 
 
 o JO Jo 
 
 O o 
 
 + M^M^ + MfMf) sin cos dddd<t> 
 and making use of equations (121), (122) and (106), we obtain 
 
 du n+l dv n+l \ /du dv n 
 * dr d 
 
 (du n dvn\ ,du n+l dv n+l \ dw n dw n+1 
 Ur + dr A dr dr )+ dr ~W~ 
 
 ^ (u n + v n ) (u n+1 + v n+1 ) + k*w n w n+l 
 
 dv 
 
 We now use the asymptotic expressions for u n , v n , w n when 
 r is large and omit the terms that depend only on the ua. 
 We also write 
 
 and obtain after some simplifications 
 
 ......... (123). 
 
 This expression turns out to be negative, i.e. the pressure 
 acts in the direction in which the incident light is moving. If 
 the constants in the incident light are chosen so that its 
 intensity is unity, the above expression must be doubled. The 
 numerical value of the pressure has been calculated from the 
 above formula by Debye in a number of cases. When the 
 radius of the spherical obstacle is small compared with the 
 
64 
 
 POLAR COORDINATES 
 
 [CH. 
 
 wave-length of the incident light, the functions ty n (ka), f n (ka), 
 -^n(Aa), fn(Aa) occurring in the expressions for a n , /3 n can be 
 expanded in ascending powers of a. This method, however, 
 fails when ka approaches unity and numerical values of the 
 functions must be used. 
 
 In the case of a totally reflecting sphere 
 
 - 
 
 , 124) 
 
 ( >' 
 
 and Debye finds that if 
 
 2 
 
 where X is the wave-length of the incident light, L denotes the 
 light-pressure and W = ^TTO? is the energy of the incident train 
 of waves per unit length of a cylinder circumscribing the 
 spherical obstacle and having its axis parallel to the direction 
 of motion of the waves ; then 
 
 The first term of the series was given by Schwarzschild. 
 The convergence of the series is slow as p approaches unity and 
 several terms of the series (123) must be taken into account. 
 By using numerical values of the functions ty n (p), n(p) and 
 their derivatives for n=l, 2... 5, Debye has succeeded in 
 
 drawing a curve for . 
 
 W 
 
 I J 
 
 Fig. 5. 
 
 > 
 
Ill] PRESSURE OF RADIATION 65 
 
 It will be seen that the pressure has a maximum value for 
 a certain value of p, approximately equal to 1. When p is 
 
 large the ratio -^. approaches asymptotically the value 1. 
 
 Debye compares the light pressure so obtained with the 
 gravitational attraction for a spherical particle of specific gravity 
 s under the influence of the sun's radiation. He finds that if G 
 is the gravitational attraction, 
 
 Z4800 L 
 G 
 
 and to get a numerical estimate he takes X = 600 JJL/JL, s = l. 
 
 It appears that the ratio vanishes both for small and large 
 values of p : it has a maximum value of about 20 for p = 1. 
 
 In the case of a dielectric sphere with refractive index n, 
 the expansion corresponding to (125) is 
 
 L^ 8 /yi 2 -iy 4 r _ p2 n * _ 29^4 + 34^2 + 12Q -i 
 TF~3U 2 + iy p |_ "15* (n 2 + 2)(2n' + 3) 
 
 ...... (126), 
 
 and is suitable for calculations only when np is small. 
 
 Debye has drawn curves for -^r T in the cases n = oo , n 2, 
 
 W 
 
 n 1*5 and n = 1*33. When n 2 the curve appears to have 
 three maxima and two minima between the values p = 1 and 
 /> = 3. 
 
 The greatest value of -^ is now about 2*6 ; the following 
 
 table indicates when the light pressure exceeds the gravitational 
 attraction, the numbers p Q and p give the extreme values of p 
 belonging to the range in which this is the case. 
 
 (I), 
 
 Po Pi 
 
 x 20 3 8 
 
 2 13 -6 5 
 1-5 3 -8 
 
 The maximum light pressure is just balanced by the 
 
 gravitational action when n is about equal to 1*33, the value 
 for water : for smaller values of n gravitation prevails. 
 
 B. 5 
 
66 POLAR COORDINATES [CH. 
 
 In the case of an absorbing spherical particle, the equation 
 which takes the place of (126) is 
 
 (127). 
 
 When a is small the light pressure and gravitational action 
 are both of order a 3 and their ratio tends to a finite limit, hence 
 for certain types of absorbing material there is no lower limit 
 in the size of a particle below which gravitation exceeds the 
 
 light pressure. Debye has drawn a curve for ^ for the case of 
 
 a gold particle and finds that there is a maximum value for 
 p = 1*5 nearly. 
 
 The existence of a maximum value for ^ in the cases that 
 
 have been discussed appears to be due to the fact that the 
 value of p for which the maximum occurs is very nearly equal 
 to the real part of the complex value of p corresponding to one 
 of the free damped vibrations*. The first electric vibration 
 seems to be of chief importance in determining the position of 
 the maximum. 
 
 The determination of the limiting value of -^ for very small 
 
 wave-lengths, i.e. for large values of p, is a matter of some 
 difficulty, it depends on some expressions giving the behaviour 
 of the Bessel functions for large values of n and p. These have 
 been found by J. W. Nicholson f and P. Debye J. 
 If p is real and n + $ < p, we have when p -* oo 
 
 .-<(*-;) 
 r (P) = -T- 
 
 (sm T )* 
 
 (128), 
 
 * Cf. Debye, loc. cit., and the similar remarks for the case of optical 
 resonance by F. Pockels, Physik. Zeitschr. Bd. 5 (1904), p. 152. 
 
 f British Association Reports, Dublin, 1908, p. 595 ; Phil. Mag. Vol. 13 (1906), 
 p. 195 ; Vol. 14 (1907), p. 697 ; Vol. 16 (1908), p. 271 ; Vol. 18 (1909), p. 6. 
 
 t Math. Ann. Bd. 67 (1909), p. 535. 
 
Ill] PRESSURE OF RADIATION 67 
 
 where T O is an angle lying between and for which 
 
 cos T = 
 
 P 
 and / = sin T O T O cos T O . 
 
 When n + J > p and p -* oo , we have 
 
 -ipfo \ 
 
 rnvr/- . i 
 
 (i sm T )2 
 
 where T O is now the root of the equation 
 
 cos TO = - 
 
 whose imaginary part has a negative sign. 
 
 When n and z are very nearly equal the values of f n (p) and 
 i/r n (p) can be made to depend on Airy's integral and are much 
 more complicated ; for these we must refer the reader to the 
 original memoirs. 
 
 24. Other problems which may be treated with the 
 aid of polar coordinates. 
 
 The diffraction of electric waves travelling round the earth 
 is a problem of some importance which has been discussed by 
 H. M. Macdonald*, Lord Rayleighf, H. PoincareJ, J. W. 
 Nicholson and other writers. 
 
 The calculations are very long and depend on the use of the 
 formulae to which we have just referred. Rybcynski|| has 
 recently treated the problem by a method due to March and 
 has taken into account the finite conductivity of the earth. As 
 we have already mentioned this was done by Zenneck and 
 Sommerfeld for the case in which the earth's surface is treated 
 
 * Proc. Roy. Soc. Vol. 71 (1903), p. 251 ; Vol. 72 (1904), p. 59 ; Vol. 90 (1914), 
 p. 50; Phil. Trans. A, Vol. 210 (1909), p. 113. 
 
 t Proc. Roy. Soc. Vol. 72 (1904), p. 40. 
 
 J Rend. Palermo (1910) ; Proc. Roy. Soc. Vol. 72 (1904), p. 42. 
 
 Phil. Mag. Vol. 19 (1910), pp. 276, 435, 516, 757; Vol. 20 (1911), p. 157; 
 Vol. 21 (1911), pp. 62, 281 ; Jahrb. d. draht. Telegr. Bd. 4 (1910), p. 20. 
 
 || Ann. d. Phys. Bd. 41 (1913). 
 
 52 
 
68 POLAR COORDINATES [CH. Ill 
 
 as a plane. The results of Nicholson and Poincare indicate 
 that diffraction round a perfectly conducting surface is not 
 sufficient to explain the apparent bending of the electric waves 
 round the earth's surface. A generally accepted opinion is that 
 the ionisation of the atmosphere by the sun's rays is a very 
 important factor in producing the observed effects*. 
 
 The diffraction of a solitary wave or pulse by a spherical 
 obstacle might be discussed with advantage. The evaluation 
 of certain definite integrals involving Bessel functions, however, 
 presents some formidable difficulties which probably account 
 for the fact that the problem does not appear to have been 
 solved. 
 
 The scattering of electric waves by a perfectly conducting 
 conical obstacle has been treated very briefly by H. S. Carslawf . 
 
 EXAMPLES. 
 1. Prove that 
 
 (C. Neumann.) 
 2. Prove that 
 
 (Jf sin a)== / J ( 2 fi+ 1) i* Jn+ ^ p n (cos 6} P n (cos a). 
 v ^ i xjkr 
 
 (E. W. Hobson.) 
 
 * Cf. the discussion at the British Association meeting, Dundee (1912), and 
 an article by W. H. Eccles in the Year Book of Wireless Telegraphy (1913). 
 Some quantitative experiments on long distance telegraphy have been made 
 recently by L. W. Austin, who obtains an empirical relation between the 
 magnitude of the current received and the distance between the two stations r 
 Bulletin of the Bureau of Standards, Vol. 7 (1911). 
 
 f Phil. Mag. Vol. 20 (1910), p. 690. 
 
CHAPTER IV 
 
 CYLINDRICAL COORDINATES 
 
 25. The wave-equation in Cylindrical Coordinates. 
 If we put x = p cos <f>, y = p sin <, the wave-equation becomes 
 
 at* idu la* aHi_i3H_ 
 
 dp* + pdp + P *c><t>* + dz* c 2 9* 2 ~ 
 
 Two particular solutions of this equation are suggested at 
 once by the general solution of 5 : they are* 
 
 u=l F Iz + ipcosa, t -sin a Ida. ...... (131), 
 
 and u \ F\z-\-ip cos a, t - sin a \ da. ...... (132), 
 
 respectively. The first of these represents a wave-function 
 which is symmetrical round the axis of z and which reduces to 
 %irF(z, t) when p = 0. It gives us at once the formulae 
 
 da. 
 
 where M is an integer f, and many other interesting formulae 
 may be written down by simply choosing different wave- 
 functions that are symmetrical round the axis. 
 
 * The first of these is an obvious generalisation of a formula given by 
 D. Edwardes, Educational Times, Oct. (1904). 
 
 f The formula is also true under certain limitations when n is not an integer. 
 See Hobson, Phil. Trans. A, Vol. 187 (1896). 
 
70 CYLINDRICAL COORDINATES [CH. 
 
 For instance, 
 
 1 f-togfr + frcOBg) 1 *_ 
 
 TT Jo z-\- ip cos 6 r & r + z 
 
 log (z + ip cos 0) - log ZQ JQ _ JL , r + ^o + ^ 
 
 ~~ 
 
 1 p*- sin k (z + t/o cos a) ikpa i na sin AT , qtn 
 
 =; I - - - : - 6 (101 - . . . ( 1 o i ) 
 
 2-7T J o z + ip cos a r 
 
 There is another general formula for a wave-function sym- 
 metrical about an axis, viz. 
 
 u=] jHtf-^cosha, ^r-psinha Ida ...(138), 
 
 where the function F is of such a nature that 
 
 1 . , dF , , dF 
 - smh a ^7 + cosh a ^5- 
 
 G Ot OZ 
 
 vanishes when a = oo . As a particular instance of this we 
 have the function* 
 
 .(139), 
 
 which may be regarded as the cylindrical wave-potential for a 
 line source of strength F (t) along the axis of z. 
 
 A peculiarity f of the two-dimensional propagation of waves 
 is the existence of a " tail " to the disturbance when F (t) is zero 
 
 for t < and t > T, for if t > r + - we have 
 
 c 
 
 -L I*" F(t-cosha]da ......... (140), 
 
 27T J s \ C / 
 
 where p cosh 5 = c (t r). It is clear that this expression for 
 II does not generally vanish. The wave-function (139) is thus 
 essentially different from Euler's wave-potential for a point 
 source, viz. 
 
 * Cf. H. Lamb, Hydrodynamics, pp. 281, 500 ; V. Volterra, Acta Math. 
 t. 18 ; Levi-Civita, Nuovo Cimento (4), t. 6 (1897). The formula is a particular 
 case of a more general one given by Dr Hobson in 1891. 
 
 t This was discussed by O. Heaviside, Phil. Mag. (5), t. 26 (1888); Elec- 
 trical Papers, Vol. 2. See also Lamb, Hydrodynamics, p. 282. 
 
IV] DEFINITE INTEGRALS 71 
 
 for in this case H is zero for t > r + , provided the source is 
 
 only active when < t < r. 
 
 V. Volterra* has obtained a number of elementary wave- 
 functions of the form 
 
 u = t n F(~}=t n F(s) . ...(141). 
 
 \pJ 
 
 He finds that F must satisfy the differential equation 
 s *(l-s*)^ +s (2n- S *)^ + n(n-l)F=0 ...(142). 
 
 If we try to solve (130) by means of a function of the form 
 u = Wp m cos ra (< < ), 
 
 where W is independent of <f>, we find that W must satisfy the 
 equation 
 
 __ 
 
 i O'ooo **\j.*> ~~~ ^ ...... I ATJtM. 
 
 / p 9/5 d2* c 2 8tf 2 
 
 . Solutions of this which are independent of z may be derived 
 from the following formulae f, in which ra > J, 
 
 -?- cos oA sin 2 " 1 a. da ......... (144), 
 
 W = ^ F (t ^ cosh 17^ sinh 211 * y . dij ...... (145). 
 
 There are, of course, certain limitations concerning the 
 behaviour of F (t) at infinity. The first formula enables us to 
 determine the value of W when its value is known for p = 0. 
 
 26. Elementary solutions of AM + k z u = 0. 
 The differential equation 
 
 f^i|_%I 2 g + 3X^ = ......... (146) 
 
 dp 2 p dp p 2 d<f>* oz* 
 
 possesses elementary solutions of the types 
 
 J m (p VF+T 2 ) e * hz cos m (cf> - < ) ...... (147), 
 
 /> -<) ...... (148), 
 
 * Acta Math. Vol. 18. 
 
 t E. W. Hobson, Proc. London Math. Soc. Vol. 22 (1891), p. 431. The 
 first solution is due to Poisson. 
 
72 CYLINDRICAL COORDINATES [CH. 
 
 where X, h and m are real or complex arbitrary constants. The 
 first solution may of course be generalised into 
 
 u= J m (p*J+Ti*)e hs! f(h)hdh ......... (149), 
 
 J a 
 
 and a similar remark applies to the second. If we wish to 
 express a given wave-function in the first of these forms, the 
 following inversion formula due to Hankel is particularly 
 useful*. 
 
 If F(x)= J m (xt}f(t)tdt 
 
 Jo 
 
 then f(t) = ^ J m (xt)F(x) xdx 
 
 .(150). 
 
 Let us use this formula to express - e ikr in the form (149) 
 
 when m 0. Since the representation should be valid for z 0, 
 we find on putting k 2 + h? = X 2 , a k, that 
 
 = T J (\p)f(h)\d\; 
 P Jo 
 
 o 
 
 k>0. 
 
 Hence we obtain Sommerfeld's formula j" 
 
 the upper or lower sign being taken according as z ^ 0. A 
 more complete proof is obtained by applying Hankel's inversion 
 formula to the equation 
 
 I J (XQ) e ikr - = 6 l ^^ X 2 >& 2 
 J ^^ J....(152), 
 
 X 2 <A; 2 
 
 which is established in Prof. Lamb's paper. 
 
 * See Gray and Mathews, Treatise on Bessel Functions (1895), p. 80; 
 N. Nielsen, Handbuch der Theorie der Cylinderfunktionen (1904), p. 366. 
 
 t Ann. d. Physik, Bd. 28 (1909), p. 683. Some analogous formulae are 
 given by H. Lamb, Proc. London Math. Soc. Ser. 2, Vol. 7 (1909), p. 140 ; 
 0. Heaviside, Electrical Papers, Vol. 2, p. 478 ; N. Sonin, Math. Ann. Bd. 16. 
 
IV] ELEMENTARY SOLUTIONS 73 
 
 A few more formulae will now be written down to illustrate 
 the method of generalisation by integration with regard to 
 a variable parameter 
 
 , _ f pikR 
 
 Ki(p*/\*-fr)coa\(z-b) = t] --cos\(a-6)da 
 
 (X 2 > k 2 , R 2 = p*+(z- a) 2 ). 
 If ??i is zero or a positive integer and n is a positive integer, 
 
 - f (cos 0) - 
 
 (Hobson.) 
 
 27. The propagation of electric waves on a semi- 
 infinite solid bounded by a plane surface*. 
 
 In this problem the surface of the earth is regarded as an 
 infinite plane and the waves are supposed to be generated by an 
 antenna, of which one portion is vertical and the other horizontal. 
 
 Let us assume that the electric and magnetic forces are the 
 real parts of vectors of the form Ee~ iut , He~ iut respectively, 
 
 then if M H ivE, where v 2 = - , we may satisfy Max- 
 
 fut 
 
 well's equations by putting 
 
 Jtf=rotn + igraddivIIMI ......... (153), 
 
 where AII + & 2 n = 0, &fc = e/iw 2 4- t>o><r ...... (154). 
 
 To imitate the action of the antenna we shall place two 
 vibrating doublets at a point at distance a from the plane. If 
 one of these vibrates vertically and the other horizontally, we 
 may put for the primary radiation n o = (Pa., 0, P z ), where 
 
 X , , J 
 
 and the axis of z is vertical. We write B with a negative sign 
 to signify that the horizontal branch of the antenna is drawn 
 in the negative ^-direction. 
 
 * A. Sommerfeld, Ann. d. Phys. Bd. 28 (1909), p. 665 ; H. v. Hoerschlemann, 
 Jahrb. d. draht. Teleg. Bd. 5 (1912), pp. 14, 188; Dissertation, Munich (1911). 
 
74 CYLINDRICAL COORDINATES [CH. 
 
 A convenient expression for TT is obtained by using Sommer- 
 feld's equation 
 
 ft Jo I 
 
 where 1= Vx 2 k 2 . Appropriate functions H lt II 2 for the re- 
 flected and transmitted disturbances are obtained by putting 
 
 11!= 2 cosn<f> I J n (Xp)e-*(*+> F n (\)d\ 
 
 n=0 Jo 
 
 /.oo 
 
 I Jn (Xp) e m(z ~ a) G n (\)d\ 
 Jo 
 
 00 
 
 n=0 JO 
 
 where m = A/X 2 h* and h is the value of k in the second 
 medium. The functions F n (X), G n (X) are vectors with com- 
 ponents [/ n (X), 0, ^n(X)], [g n (\)> 0, x(X)] respectively. 
 
 We can satisfy the condition that the tangential components 
 of the electric and magnetic forces should be continuous at the 
 surface of separation of the two media by putting for z = 
 
 rot i = 2 .(158), 
 
 where JJL has been taken to be unity for both media. 
 
 Substituting the integral expressions forII , H lt n 2 in these 
 equations and equating to zero the coefficients of functions 
 of type Jn(Xp) in the resulting integral equation, we obtain 
 the system of equations 
 _ la B _ la _ 
 
 A _ la _ 
 ~L e ' ~ : 
 - If a e~ la - Be^ = mg, e-, - lf n e~ la = mg n 
 
 I . _ la __ m 
 
IV] THE SPREADING OF WAVES OVER AN INFINITE PLANE 75 
 
 where the last equation has been simplified with the aid of the 
 relations / n = gr n = 0(n>0) which are a consequence of the 
 previous equations. 
 
 Solving these equations we eventually find that if 
 
 n, = (&. o, &), n,=(R x , o, B,), 
 
 The " directed effect " depends on the presence of the terms 
 involving cos (f> in the expressions for Q z and R z . Now when 
 o- = oc for the second medium, h = oo , and these terms vanish 
 altogether ; hence the possibility of directing the energy of the 
 radiation sent out from the bent antenna is due to the im- 
 perfect conductivity of the earth. Von Hoerschlemann has 
 given a numerical discussion of the above formulae but the 
 investigation is too long to be inserted here. 
 
 28. Propagation of electromagnetic waves along a 
 straight wire of circular cross-section*. 
 
 Let us consider the symmetrical case when the electric 
 force at any point is in a plane through the axis of the wire 
 
 * H. Hertz, Electric Waves ; J. J. Thomson, Proc. London Math. Soc. 
 Vol. 17 (1886), p. 310; Recent Researches, 259; A. Sommerfeld, Ann. d. 
 Physik, Bd. 67 (1899), p. 233; Gray and Mathews, Bessel Functions, Ch. 13; 
 M. Abraham, Encykl. d. Math. Wiss. Band V. 2, Heft 3 (1910), p. 526; 
 J. Larmor, Proc. of the 5th Int. Congress of Mathematicians, Vol. 1 (1912), p. 206. 
 The problem considered in this last paper is chiefly that of alternating currents, 
 viz. the forced alternations of flow produced by a uniform periodic electric force. 
 
76 CYLINDRICAL COORDINATES [CH. 
 
 and the magnetic force is in circles at right angles to this 
 plane. The field equations are then of the types 
 
 
 
 a^_a^ 
 
 c dt dz dp ) 
 
 These may be satisfied by putting 
 
 1 8 / an 
 
 >. (160), 
 
 """ c dtdp c dp } 
 
 where IT satisfies the equation 
 
 8 / 8IT 
 
 Putting IT = e^* w, we find that Aw + & 2 w = 0, where 
 
 7.2 _ e/Afl)' - tfMPO" 
 
 ~^~~ 
 We now assume that for points outside the wire* 
 
 and that for points inside the wire 
 
 J" ( 
 
 where /i is the value of A; inside the wire. These assumptions 
 are made for the purpose of determining the periods and rates 
 of decay of electric waves that can travel along the wire and 
 maintain their own field. The first solution is chosen so as 
 to make the flow of energy negligible at an infinite distance 
 from the wire so that the system is self-contained; and to 
 ensure this it is necessary to suppose that the real part of 
 Vx 2 A? is positive. The second solution is chosen so as to 
 make the electric and magnetic forces finite on the axis of 
 the wire (p = 0). 
 
 Let p = a be the equation of the surface of the wire, then 
 
 * We follow here the work of Sommerfeld in which it is supposed that there 
 are no conductors outside the wire. Sir J. J. Thomson allows for the presence 
 of external conductors by supposing the dielectric surrounding the wire to be 
 bounded by a cylindrical conductor having the same axis as the wire. 
 
IV] PROPAGATION OF WAVES ALONG A WIRE 77 
 
 the continuity of the tangential components of the electric and 
 magnetic forces requires that when p = a 
 
 _ (162), 
 
 JtVv-A' 
 
 op 
 
 v and v 1 being the values of the quantity (r + icoe at points 
 outside and inside the wire. The elimination of A and B gives 
 rise to a transcendental equation for the determination of X. 
 The total current flowing along the wire is 
 
 ra 
 
 J= 27rl <?E z pdp 
 J o 
 
 B~-J (ip v X 2 h?) for p = a. 
 
 On the other hand the electric force E z at the surface 
 of the wire is 
 
 = _ B(\*- 
 Hence, if we put 
 
 where R and L are the resistance and self-induction of the wire, 
 we get 
 
 The roots of the transcendental equation have been discussed by 
 Sommerfeld who used a method of successive approximations. 
 The values of X are complex as the waves which travel along 
 the wire are damped owing to the imperfect conductance of the 
 wire. It appears that when the disturbance does not penetrate 
 far into the wire, the damping is small and so the velocity of 
 propagation is very nearly equal to the velocity of light. 
 When, however, the field does soak into the wire to some 
 extent, the damping is of course considerable and so the wave 
 travels with a velocity a little less than that of light. In the 
 first case the real part of Vx 2 h 2 is large, in the second case it 
 is small. 
 
78 CYLINDRICAL COORDINATES [CH. 
 
 In Lecher's arrangement* there are two conjugate parallel 
 wires between which the waves travel, consequently the field is 
 not symmetrical round the axis of one of the wires. This case 
 has been discussed by G. Mief with the aid of bi -polar 
 coordinates, and the lines of force have been studied by W. B. 
 Morton | . The latter also considers the case of n parallel wires 
 passing through the corners of a regular polygon. 
 
 The mathematical analysis for the case of a curved or twisted 
 wire has not yet been fully developed. The important case of 
 a spiral wire has, however, been discussed by H. C. Pock- 
 lington|| and J. W. Nicholson 1. The latter gives numerous 
 references to the literature of the subject. D. Hondros** has 
 recently discussed the propagation of some types of unsym- 
 metrical waves along a single wire. The electromagnetic 
 theory of an electric cable has been given by Sir Joseph 
 Thomson ff and F. Harms JJ. The latter considers the case 
 when the outer conductor of a cable is replaced by air. 
 
 29. Other problems which may be treated with 
 cylindrical coordinates. 
 
 The diffraction of electromagnetic waves by a cylindrical 
 obstacle has been discussed by Lord Rayleigh, W. Seitz||||, 
 W. IgnatowskylTU, P. Debye*f, C. Schaefer*J, and J. W. Nichol- 
 son*^ Schaefer has made an extensive study of the case of a 
 
 * Ann. d. Phys. Bd. 41 (1890), p. 850. See also D. Mazzotto, II Nuovo 
 Cimento (4), t. 6 (1897), p. 172. 
 
 t Ann. d. Phys. Bd. 2 (1900), p. 201. The case in which the capacity 
 is small is discussed at length by J. W. Nicholson, Phil. Mag. Feb. Sept. (1909). 
 The current is supposed to flow along one wire and return along the other. 
 
 J Phil. Mag. Vol. 50 (1900), p. 605 ; Vol. 4 (1902), p. 302. 
 
 Phil. Mag. Vol. 1 (1901), p. 563. 
 
 || Proc. Camb. Phil. Soc. Vol. 9 (1897), p. 324. 
 
 f Phil. Mag. Vol. 19 (1910), p. 77. 
 
 ** Ann. d. Phys. Bd. 30 (1909), p. 905; Dissertation, Munich (1909). 
 
 ft Proc. Roy. Soc. Vol. 46 (1889), p. 1 ; Recent Researches, p. 262. 
 
 JJ Ann. d. Phys. Bd. 23 (1907), p. 44. 
 
 Phil. Mag. Vol. 12 (1881), p. 81. 
 
 Ill) Ann. d. Phys. Bd. 16 (1905), p. 746. 
 
 fir Ann. d. Phys. Bd. 18 (1906), p. 495. 
 
 *t Phys. Zeitschr. (1908), p. 775. 
 
 * Ann. d. Phys. Bd. 31 (1910), p. 462. 
 
 * Proc. London Math. Soc. Ser. 2, Vol. 11, p. 104. 
 
IV] PROBLEMS CONNECTED WITH CYLINDERS 79 
 
 dielectric cylinder and his results have been tested experimentally 
 by Grossmann. This is one of the few cases in which the 
 influence of the material properties of the obstacle has been 
 taken into account in the mathematical treatment of a diffrac- 
 tion problem; the necessity of doing this has been clearly 
 indicated by some of the experimental results. 
 
 P. Debye has studied the diffraction problem with reference 
 to the theory of the rainbow. This had been done for the case 
 of the sphere by L. Lorenz long ago*. 
 
 Electrical vibrations in regions bounded by cylinders of 
 various shapes have been studied by Sir J. J. Thomsonf, Lord 
 RayleighJ, Sir J. Larmorg, R. H. Weber ||, A. KalahneU, and 
 J. W. Nicholson**. The latter has also calculated the pressure 
 exerted by a train of plane electromagnetic waves on a perfectly 
 conducting cy Under j"f. 
 
 EXAMPLES. 
 
 1. An infinitely long metal cylinder of specific conductivity <r 
 and permeability /x, bounded by the surface r=R, is surrounded by a 
 dielectric of specific inductive capacity e. A train of waves, in which the 
 electric force is perpendicular to the cylinder and to magnetic force and 
 if undisturbed would be represented by the real part of M e i(*>t + kr<x>*<t>\ 
 is passing in the dielectric. Prove that the magnetic force H z inside 
 the cylinder and the part HI of the magnetic force outside representing 
 the scattered wave, are given by the real parts of 
 
 #! = e iut 2 a m K m (kr) cos m<, ff 2 = e iv>t 2 b m J m (hr] cos mf, 
 m m 
 
 where A 2 = -47rxo-to) and 
 
 - a m K m (kR) = 
 
 
 * Oeuvres scientifiques, pp. 405 502. See also Gans and Happel, loc. cit. 
 (p. 44). The most recent paper on the rainbow is by W. Mobius, Ann. d. Phys. 
 Bd. 33 (1910). 
 
 t Recent Researches, p. 344. 
 
 Phil. Mag. (5), Vol. 43, p. 125. 
 
 Proc. London Math. Soc. (1), Vol. 26, p. 119. 
 
 || Habilitationsschrift, Heidelberg (1902) ; Ann. d. Phys. (4), Bd. 8 (1902), 
 p. 721. 
 
 IT Ann. d. Phys. (4), Bd. 19 (1906), pp. 80, 879 ; Bd. 18 (1905), p. 92. 
 
 ** Phil. Mag. Aug. (1905), May (1906). 
 
 ft Proc. London Math. Soc. Ser. 2, Vol. 11, p. 104. 
 
80 CYLINDRICAL COORDINATES [CH. 
 
 for positive integral values of m. The constants here have reference to the 
 electromagnetic system of units. 
 
 (Cambr. Math. Tripos, Part II, 1905.) 
 
 2. Plane electromagnetic waves represented by 
 
 E x =E v =H x =ff t =0, H v =e ik(z+ct \ E e =e ik (* +ct \ 
 
 fall upon the perfectly conducting cylinder p=a. Prove that in the 
 scattered field 
 
 (P. Debye.) 
 3. The wave-potential for a circular ring of point sources is given by 
 
 o-f 
 
 Jo 
 
 (A. G. Webster.) 
 
 4. The wave-function 
 
 Q = e ikct ("e^ J Q (p VF+X*) cos aX d\ 
 Jo 
 
 is zero at points on the plane z=0 which lie inside the circle p 2 <a 2 ; for 
 3=0, p 2 >a 2 its value is e tkct (p 2 a 2 )~*cos(/s/p 2 -a 2 ). (Sonin.) 
 
 5. If J ft 2 =p 2 + (2-f-asinhw) 2 , = e u , the integral 
 
 "dt 
 
 f I" 'JJ 
 
 e ;*/z 
 represents the function -Q- . e~ a tt outside a paraboloid of revolution 
 
 whose focus is at the singularity z -a sinhw, p = 0, and which passes 
 through the circle p = a, z = 0. It is zero inside the paraboloid. 
 
 6. The circuital relations in cylindrical coordinates are 
 
 pdE p _dH z ^ dffift p dE<f> d.ffp %H t 
 
 and similar equations in which E is replaced by H and H by E. 
 
 7. If ^^tfcosa^-ysmotf, F=# sina>+ycosG>, Z=z-vt, where v 
 and <o are constants, the function Q = F(X, F, Z) is a wave-function if .P 
 satisfies the partial differential equation 
 
 2 /~ap ^aj 
 
IV] EXAMPLES 81 
 
 Obtain particular solutions of the form 
 
 ( V 2\ nfia> 2 
 1 2 H -- 2" an d ^j ni,A,B are arbitrary constants. 
 
 8. An oscillatory current is induced on a circular wire of radius a 
 excited by a uniform electric force R^t acting on its surface from the 
 surrounding medium. Obtain expressions for the inductance and resistance 
 of the wire per unit length when the wire is regarded as straight and no 
 disturbing conductor is near. 
 
CHAPTER Y 
 
 THE PROBLEM OF DIFFRACTION 
 
 30. Multiform solutions of the wave- equation*. 
 
 The wave-functions required to solve many of the boundary 
 problems of Mathematical Physics are not single-valued func- 
 tions of x, y, z, t in an ordinary space. We may, however, 
 regard them as single-valued functions in a Riemann's space. 
 This is a simple generalisation of the Riemann's surface of the 
 theory of functions of a complex variable ) ; every plane 
 section of the Riemann's space is in fact a Riemann's surface. 
 Instead of branch lines and branch points we have branch 
 membranes and branch curves. Thus in the physical problem 
 of the diffraction of light through a circular hole in a screen, 
 the boundary of the shadow of the screen is the branch mem- 
 brane and the edge of the hole the branch curve. 
 
 We shall commence by finding a multiform solution of 
 the equation 
 
 The fundamental solution u = e ik < x C09 a + * sina > = e ik * * ( * ~ a) is 
 of period 2?r and can be expanded in the form 
 
 e ik P cos(*-) = j Q (kp) + 2 2 i n J n (kp) cos n (0 - a). . .(164). 
 
 * This theory is due to A. Sommerfeld, Math. Ann. Bd. 45 (1894), Bd. 47 
 (1896) ; Zeitschr. fur Math. u. Phys. Bd. 46 (1901) ; Proc. London Math. Soc. 
 (I), Vol. 28 (1897), p. 417. It has been developed by H. S. Carslaw, Proc. 
 London Math. Soc. (1), Vol. 30, p. 121 : (2), Vol. 8, p. 365 ; Phil. Mag. Vol. 5 
 (1903), p. 374: Vol. 20 (1910), p. 690 ; Fourier's Series and Integrals, Ch. 18; 
 W. Voigt, Gdtt. Nachr. (1899) ; E. W. Hobson, Camb. Phil. Trans. Vol. 18 
 (1900), p. 277. Different methods have been used by H. M. Macdonald, Electric 
 Waves, Appendix D; K. Schwarzschild, Math. Ann. Bd. 55 (1902), p. 177; 
 Proc. London Math. Soc. Vol. 26 (1895), p. 156; C. W. Oseen, Arkiv for mat. 
 Bd. 1 (1904), Bd. 2 (1905). 
 
 f See Harkness and Morley's Theory of Functions (1893), Ch. 6- 
 
CH. V] MULTIFORM SOLUTIONS OF THE WAVE-EQUATION 83 
 
 A solution of period 2m?r may evidently be constructed by 
 writing down a series of the form 
 
 _2 a n J"n(fy>) cos (<-), 
 
 n = co OT 
 
 where the a n 's are suitable constants. The solution that seems 
 the most natural extension of (164) is 
 
 oo n 
 
 F (p> </>> <M JQ (kp) + 2 2 i m J n (kp) cos (</> </> ) . . .(165). 
 
 1 m 
 
 To sum this series when m 2, we transform the terms for 
 which n is odd by means of the equation 
 
 n 
 
 J n (kp) = ^ P e^ C08 a sin n a . da. 
 
 Summing the two series separately we find that 
 
 F t (p, 4>, #>) 
 where 
 
 [ 
 
 with 2W2V~cos-, 
 
 Now 
 
 and 
 
 hence we may write 
 
 F z (p, <#>, </>) = (--)* e^ cos <* - *> J T ^-^ dX . . .(166), 
 
 where T = V2 /?/> cos ^ (< - <#> )- 
 
 With the aid of the function F 2 we can solve some problems 
 on the diffraction of plane electromagnetic waves by a semi- 
 infinite plane bounded by a straight edge. Let us consider 
 
 62 
 
84 THE PROBLEM OF DIFFRACTION [CH. 
 
 the case of a totally reflecting screen*. If the electric force 
 in the incident wave is parallel to the edge of the screen, the 
 electric force u E z for the total disturbance must vanish over 
 both faces of the screen and must satisfy the differential equa- 
 tion (163). These conditions are fulfilled by taking 
 
 E^F^^^-F^p,^-^} ......... (167). 
 
 This value of E z also satisfies the right conditions at infinity. 
 To prove this we must find an asymptotic expression for F 2 
 when p is large. 
 
 Now when r > 0, we have the asymptotic expansion f 
 
 while when T< 0, I has a similar asymptotic expansion with the 
 
 J -00 
 
 sign changed. This means that when cos (</> </> ) > e > we 
 have 
 
 .4 
 
 cos 
 
 while when cos (<f> </> ) < e < there is a similar asymptotic 
 expansion in which the first term is missing. It thus appears 
 that the electric force in the geometric shadow vanishes at 
 infinity to the order p~$ . 
 
 If the magnetic force in the incident wave is parallel to the 
 edge of the screen, the magnetic force u H z must satisfy the 
 
 differential equation (163) and be such that ^r =0 over both 
 faces of the screen. The conditions are fulfilled by putting 
 
 #* = ^(?,<Mo) + ^ 2 (p><k-<W .......... (168). 
 
 Prof. H. M. Macdonald has shown that the solution of 
 a problem concerning a perfectly absorbing body can be made 
 to depend on the solution of two allied problems J. " A per- 
 
 * The incident waves are supposed to come in a direction for which 
 = 7r + . An approximate solution of this problem was given by H. Poincare", 
 Acta Mathematica, Bd. 16, p. 297 ; Bd. 20, p. 313. 
 
 f Bromwich's Infinite Series, p. 328. 
 
 I Phil. Trans. A, Vol. 212 (1912), p. 337 ; Proc. London Math. Soc. (2), 
 Vol. 12 (1913). 
 
V] SCREEN WITH A STRAIGHT EDGE 85 
 
 fectly absorbing body may be regarded as a body which is 
 incapable of supporting either electric or magnetic force ; 
 hence if C is the electric current distribution on the surface 
 of the body when it is supposed to be perfectly conducting, 
 and C' is the magnetic current distribution on the surface of 
 the body when it is supposed to be incapable of supporting 
 magnetic force, the superposition of these two distributions 
 gives the electric and magnetic current distributions on the 
 surface of the body when it is perfectly absorbing and the 
 amplitude of the incident waves is doubled." 
 
 Now if we suppose our screen to be incapable of supporting 
 magnetic force, the boundary condition is that the tangential 
 component of the magnetic force should vanish. When the 
 
 ?)E 
 
 electric force is parallel to the axis of z, -^- must vanish over 
 
 o<p 
 
 the screen. Hence 
 
 E z = F z (p,<t>,<l>*) + F,(p,<l>,-4>) (169). 
 
 The solution for a totally absorbing screen is thus simply * 
 
 E z = F 2 ( P ,<t>,<l> Q ) (170). 
 
 Similarly, it can be shown that when the magnetic force 
 is parallel to the axis of z, the solution for a perfectly 
 absorbing screen is 
 
 H,=F 2 (p,$,<t>,) (iri). 
 
 Prof. Lamb t has discussed the case of perpendicular incidence 
 with the aid of the parabolic substitution 
 
 f = p* cos <, if = p* sin %<}> 
 
 2 + ^|...(172). 
 
 The curves f = const., 77 = const, are confocal parabolas, 77 = 
 is the screen. A solution of Maxwell's equations is obtained 
 by writing 
 
 H<r = 0, H,, = 0. H* = u \ 
 
 x > y I 
 
 3 ft* 3/if 3 C 7 2<>j 3 J5 7 /"I'TQN 
 
 OJ^f OU 0-&V OU OfLiz r. r...lllO), 
 
 _- Q y_ * Q I 
 
 dt "by ' 9i 3 ' dt 
 
 * C/"lV W J. W / -* hv 4 \ 
 
 where ^ + = (174). 
 
 * W. Voigt, Gott. Nachr. (1899), p. 1, discusses the case of an absorbing 
 screen. 
 
 f Proc. London Math. Soc. (2), Vol. 4 (1907), p. 190 ; Vol. 8 (1910), p. 422. 
 
86 THE PROBLEM OF DIFFRACTION [CH. 
 
 Starting with Poisson's wave-function* ^ = p~% cos J < .f(ct p\ 
 let us put 
 
 du _ 
 
 fa~ X ' 
 Transforming to the coordinates f, 77, we obtain 
 
 Solving this partial differential equation by Lagrange's method 
 and adjusting the complementary function so that the boundary 
 
 condition ^ = for rj is satisfied, we obtain 
 
 where F is an arbitrary function. If the boundary condition is 
 u = for i? = the sign of the second term must be changed. 
 It is easy to verify that each of the integrals represents a wave- 
 function. 
 
 2 f 
 
 Let us now put f(x) = - F' (x - tf) dv, 
 
 trj o 
 
 and make the substitution f=o-cosa, = 0- sin a, then after 
 a little reduction we find that-f 
 
 u=F(ct + y) -- 2 F[ct + y (p + y) sec 2 a] da. 
 
 TTJo 
 1 f 2~ 
 
 + - F[ct-y-(p-y)sec*a]da x<0, y^O 
 
 Trj o 
 
 1 f? 
 
 u = F(ct+y) + F(ct-y) -- F [ct + y - (p 4- y) sec 2 a] da. 
 
 1 f 
 
 = l 
 
 TTJo 
 
 
 (177). 
 
 * Journal de VEcole Poly technique, Cah. 19, t. 12 (1823). See also V. 
 Volterra, A eta Math. t. 18 ; Hantzschel, Reduction der Potentialgleichung , Ch. 1. 
 t These formulae are not given in Prof. Lamb's paper. 
 
V] SCREEN WITH A STRAIGHT EDGE 87 
 
 Each of the integrals represents a wave-function provided 
 jP(oo) = o and 
 
 *7T 
 
 tan a. F' \ct y (p y) sec 2 aj > as a -^ . 
 
 In these circumstances we have the solution of the diffraction 
 problem for the case when the initial disturbance is represented 
 by u = F (ct + y\ the magnetic force being parallel to the 
 axis of z. By suitably choosing F we can deal with the case 
 of a solitary wave. 
 
 A new method of solving the problem of diffraction by 
 a straight edge has been given recently by Oseen*. 
 
 Problems connected with a wedge have been treated success- 
 fully by Sommerfeld and other writers by using a certain type 
 of contour integral. The fundamental solution of (163) is now 
 
 __ e ikp cos (* - ) III ZZ 
 ~' rf' V 
 
 where v = e n e n and the path of integration is a simple con- 
 tour which starts from oo i + 7 and goes to oo i 4- 7' without 
 crossing the real axis. The quantities 7, 7' are subject to the 
 inequalities 
 
 2-7T > 7 > 7T, > 7' > 7T. 
 
 This function u is multiform and of period 2n7r, but on an 
 n-sheeted Riemann's surface with the origin as branch-point 
 and the line <f> = - (TT < ) as branch-section, it is uniform. 
 
 With the aid of this function a number of diffraction 
 problems may be solved. 
 
 Thus in the case of a perfectly conducting prism of angle 
 2-7T a if the electric force in the incident waves is parallel to 
 the edge of the screen and is represented by the real part 
 of the expression 
 
 it can be shown by an extension of the method of images, that 
 for the total disturbance the electric force is the real part of 
 the expression t 
 
 * Arkiv for matematik, astronomi ochfysik (1912). 
 t Macdonald, Electric Waves, p. 192 (1902). 
 
88 THE PROBLEM OF DIFFRACTION [CH. 
 
 where 
 
 7TL 7T / , , v 7TC 7T / , 
 
 W = COS COS - ($ </>o), MI = COS -- COS (< -f 9 ), 
 
 and the path of integration is the same as before. 
 
 In the associated problem when the magnetic force in 
 the incident wave is parallel to the axis of z, the magnetic 
 force for the total disturbance is the real part of the expression 
 
 H g = -J-. e ikct j e ik o cos f ~ log (wwj d ...... (180). 
 
 These solutions and the solutions of analogous problems have 
 been discussed by W. H. Jackson *, H. M. Macdonald f , F. 
 Reiche J, A. Wiegrefe , and other writers. 
 
 31. Elliptic coordinates ||. 
 If we put 
 
 x cosh a) cos %, y = sinh to sin ^ ...... (181), 
 
 the differential equation (163) becomes 
 
 ,-cos 2 % )^ = ...... (182). 
 
 The elementary solutions are now of the form 
 
 where E and F satisfy the equations of the elliptic cylinder 1F 
 
 4- (fc 2 cosh 2 a) + p)E = o] 
 
 I ............ (183). 
 
 Appropriate solutions of these differential equations have been 
 obtained recently by Prof. Whittaker **. 
 
 * Proc. Lvndvn Math. Soc. Ser. 2, Vol. 1 (1904), p. 393. 
 
 f Ibid. Vol. 12 (1913), p. 430. 
 
 J Ann. d. Phys. Bd. 37 (1912), p. 131. 
 
 Ibid. Vol. 39 (1912), p. 449. 
 
 II H. Weber, Math. Ann. Vol. 1 (1869) ; Mathieu, Louville's Journal, Ser. 2, 
 Vol. 13 (1868) ; Hartenstein, Hoppe'g Archiv (2), t. 14, p. 170; K. C. Maclaurin, 
 Cambr. Phil. Trans. Vol. 17 (1898), p. 41. 
 
 IT For this equation see Heine, Handbuch der Kugelfunktionen ; Lindemann, 
 Math. Ann. Bd. 22 ; Hantzschel, Zeitschr. /. Math. u. Phys. Vol. 31, p. 25 
 (1883); Mathieu, Liouville's Journal, Ser. 2, Vol. 13 (1868). 
 
 ** Math. Congress, Cambridge (1912). 
 
V] ELLIPTIC COORDINATES 89 
 
 Elliptic coordinates are appropriate for the solution of 
 problems connected with the scattering of electromagnetic 
 waves by an elliptic cylinder*. 
 
 W. Wien has suggested f that the problem of the diffraction 
 of light through a straight slit in a screen J: may be treated 
 with the aid of elliptic coordinates by regarding the screen as a 
 limiting case of a hyperbolic cylinder. 
 
 H. Weber has shown that when k the elliptic and 
 parabolic substitutions are the only transformations which lead 
 to elementary solutions of the equation (163). For further 
 properties of this differential equation we may refer to Pockels, 
 Uber die partielle Differentialgleichung Aw -f k*u = 0, Teubner, 
 Leipzig (1891), and to Lord Rayleigh's Theory of Sound. 
 
 32. Other diffraction problems. 
 
 The diffraction of light and electric waves by a grating of 
 wires is a problem of importance, but the mathematical treat- 
 ment is very difficult and the theories that have been given so 
 far are of an approximate character. Sir J. J. Thomson || has 
 discussed the theory of Hertz's grating IF which consists of 
 a number of parallel equidistant metal wires. When electric 
 waves whose wave-length is large compared with the distance 
 between the wires fall normally on the grating, they pass 
 through if the electric force is at right angles to the wires but 
 are reflected if the electric force is parallel to the wires. Prof. 
 Lamb ** has considered the case of a grating which consists of 
 parallel strips of metal ; his theory has been supported by the 
 
 * See, for instance, K. Aichi, Proc. Tokyo Math. Phy$. Soc. (2), 4, p. 266 
 (1908) ; B. Sieger, Ann. d. Physik (4), Bd. 27 (1908), p. 626. 
 
 t Jahresbericht d. deutsch. Math. Verein, Bd. 15 (1906), p. 42. 
 
 J For this problem see K. Schwarzschild, loc. cit. ; Lord Kayleigh, Phil. 
 Mag. Vol. 43 (1897), p. 259 ; Scientific Papers, Vol. 4, p. 283 ; Proc. Roy. Soc. 
 A, Vol. 89 (1913), p. 194. An interesting experimental result has been obtained 
 recently by P. Zeeman, Amsterdam Proc., Nov. 28 (1912), p. 599. 
 
 Math. Ann. Bd. 1. See also Hantzschel, Reduction der Potentialgleichung , 
 p. 137. 
 
 || Recent Researches (1893), p. 425. 
 
 IF Collected Works, Vol. 2, p. 190. For some recent experimental work see 
 H. du Bois and H. Eubens, Ann. d. Phys. Bd. 35 (1911), p. 243 ; A. D. Cole, 
 Phys. Review, Jan. (1913). 
 
 ** Proc. London Math. Soc. Ser. 1, Vol. 29 (1898), p. 523. 
 
90 THE PROBLEM OF DIFFRACTION [CH. 
 
 experimental work of 01. Schaefer*, J. Langwitz*, and G. H. 
 Thomson f. 
 
 Lord Rayleigh J has given an approximate electromagnetic 
 theory of the action of a grating on waves of light and this 
 theory has been extended by W. Voigt so as to take into 
 account the properties of the material of which the grating 
 is made. Voigt's theory has been tested experimentally by 
 B. Pogany|| who gives an account of previous experimental 
 work on the subject. 
 
 The diffraction of light through a circular hole in a screen 
 is a problem of interest to mathematicians which has yet to be 
 solvedlf. A promising method of attack is to regard the screen 
 as the limiting case of a hyperboloid of revolution of one sheet. 
 
 Examples. 1. Prove that 
 
 2. If x+iy = a cosh (o> 
 
 J (kp) e iv *=2 J 
 
 n 
 
 where the summation extends over all even integral values of n if v is even 
 and over all odd integral values of n if v is odd. 
 
 (J. H. Hartenstein, Grun&rts Archiv (2), 1. 14, p. 170.) 
 
 32a. The introduction and elimination of discontinuities. 
 
 Wave-functions with singular lines or with singularities 
 travelling along straight lines with the velocity of light may 
 sometimes be employed with advantage in the solution of 
 diffraction problems. To illustrate the method to be adopted 
 we shall consider the diffraction of waves of sound by an 
 
 * Ann. d. Phys. Bd. 21 (1906), p. 587. The theory is developed from a new 
 point of view by C. Schaefer and F. Reiche, Ibid. Bd. 32 (1910), p. 577 ; Bd. 35 
 (1911), p. 817. 
 
 f Ibid. Vol. 22 (1907), p. 365. 
 
 t Proc. Eoy. Soc. A, Vol. 79 (1907), p. 532. 
 
 Gott. Nachr. (1911). || Ann. d. Phys. B4. 37 (1912), p. 257. 
 
 H Approximate solutions have been given by G. G. Stokes, Camb. Phil. 
 Trans. (1849); H. Lorenz, Vidensk. Selsk. Skr., Copenhagen (1890); H. A. 
 Rowland, Amer. Journ. Vol. 6 ; A. Grimpen, Diss. Kiel (1890) ; A. E. H. Love, 
 Phil. Trans. A, Vol. 197 (1901). 
 
V] DIFFRACTION OF SOUND WAVES 91 
 
 infinitely thin semi-infinite plane bounded by a straight edge*. 
 We shall suppose that the waves are sent out from a stationary 
 source and that the screen acts as a perfect reflector f. 
 
 Let the axis of z be taken along the edge of the screen and 
 the axis of x in the plane of the screen at right angles to the 
 edge. Let P be the source of sound and Q an arbitrary point 
 on the edge of the screen. If the disturbance issuing from P 
 were reflected according to the laws of geometrical optics, the 
 total disturbance would be discontinuous in crossing two semi- 
 infinite planes, each of which is bounded by the edge of the 
 screen. The first of these planes is a boundary of the geometrical 
 shadow, when continued across the edge of the screen it passes 
 through P. The second plane is the boundary of the geometrical 
 shadow for the optical image of P, viz. Pj . 
 
 To obtain the correct solution of the diffraction problem we 
 must add to the disturbance just described a second one having 
 discontinuities which will annul the above-mentioned dis- 
 continuities, the new disturbance must also be chosen so that 
 the boundary condition is satisfied at the two faces of the 
 screen. We shall now show that the required disturbance can 
 be built up by superposition from elementary disturbances 
 with singularities along lines such as PQ and P l Q produced. 
 
 Let R be the distance of an arbitrary point (x, y, z, t) 
 from Q, then if (0, 0, ) are the coordinates of Q and c is the 
 velocity of sound, we know that a function of type 
 
 satisfies the wave-equation. Let us choose the arbitrary function 
 yin such a way that the expression becomes infinite along the 
 line PQ produced and returns to its initial value when the 
 point x, y, z is rotated twice round the edge of the screen. 
 This last condition is added so as to enable us to satisfy the 
 boundary condition. 
 
 * This problem has been solved by H. S. Carslaw, Proc. London Math. Soc. 
 Vol. 30 (1898), p. 121. A transformation of his solution suggested the method 
 described here. 
 
 f This assumption is usually justifiable. In Prof. A. G. Webster's experi- 
 ments on the reflection of sound from the ground (Phys. Review, Vol. 28 (1909) 
 p. 65) it was found that the reflection is more than 90 / . 
 
92 THE PROBLEM OF DIFFRACTION [CH. 
 
 Now if we write 
 
 where u and <j> are real quantities, and use R Q , u , </> to denote 
 the values of R, u, cj> for the point P, it is easy to see that the 
 function 
 
 satisfies the requirements, for it is periodic in </> with period 
 4?r and is infinite along PQ produced, where 
 
 U=U<>, < = </> -f7T. 
 
 We now imagine sources corresponding to wave-functions of 
 this type to be associated with each element d of the edge, 
 and suppose the strength and phase of the source at Q to 
 depend on its position relative to P in such a way that 
 
 where f(t) is the strength of the source at P at time t. 
 In this way we obtain an integral 
 
 which will be shown to be discontinuous in the way required 
 as the point (x, y, z) crosses the boundary of the shadow for P. 
 In a similar way we can construct an integral 
 
 which can be shown to be discontinuous in the way required as 
 the point (x, y, z) crosses the boundary of the geometrical 
 shadow for P l . 
 
 Now let r, r be the distances of the point x t y, z respectively 
 from P and P lt then the velocity potential V of the total 
 disturbance is given by the following expressions in the different 
 regions of space 
 
 ^ in,,, 
 
 in&, 
 
 V 
 
 F=-7 l -F a 
 
V] DIFFRACTION OF SOUND WAVES 93 
 
 The space S 1 is bounded by the screen and the limiting 
 plane of the geometrical shadow for P l} 8 2 is bounded by the 
 limiting planes of the shadows for P and P lf S 3 is bounded 
 by the screen and the limiting plane of the shadow for P. 
 
 dV 
 
 The boundary condition is that ^-r should be zero over the 
 
 d<p 
 
 two faces of the screen and it is easy to verify that this 
 condition is satisfied. To show that V is continuous for the 
 whole of the space outside the screen and vanishes at infinity 
 when the function / is finite, we shall transform the integrals 
 V lt V 2 to the forms given by Prof. Carslaw. To do this we put 
 u + u = b, then if p is the distance of a point from the axis 
 of z, we have 
 
 pe u = z - Z+R, p e> = z,-S+ R , 
 
 pp Q cosh (u + MO) = (z - ?) (^ ?) + 
 + p* +(z- ztf + Zpp, cosh (i* + tio) = (R + ) 2 , 
 
 Hence it follows that 
 
 On substituting the expression for R + R in terms of 6 we 
 obtain an integral which is equivalent to the one given by 
 Prof. Carslaw. To see that it is discontinuous* we use V + and F_ 
 to denote the values of the integral for <j> = TT + < + e and 
 (f> = TT + <^> e respectively, where e is a small quantity. The 
 difference between these quantities may be regarded as a 
 contour integral and can be evaluated by Cauchy's theorem 
 We may write 
 
 Taking the residue for f = 0, i.e. (f> TT + </> , b = 0, we get 
 * A more careful proof is given in Prof. Carslaw's paper. 
 
94 THE PEOBLEM OF DIFFRACTION [CH. V 
 
 for the conditions < = 7r4-< , 6 = imply that the radii 
 'R and R are in one straight line and so give r when added 
 together. 
 
 It is now clear that the integral V l possesses the right type 
 of discontinuity and a similar remark holds for the integral F 2 . 
 The method can no doubt be modified so as to give solutions of 
 other types of diffraction problems, the chief difficulty arises 
 in the choice of a function which will satisfy the boundary 
 conditions. At any rate the method suggests an interesting 
 type of boundary problem in which the desired wave-functions 
 have specified discontinuities instead of being continuous 
 everywhere. This type of problem ought to be studied more 
 completely. 
 
 In the general problem of the diffraction round a moving 
 object of the waves issuing from a moving source, the wave- 
 functions that are derived by the methods of geometrical optics 
 have discontinuities at a certain boundary which is the locus of 
 points travelling along straight lines with the velocity of light. 
 The points in question start from certain points of the 
 moving object and move along tangents to the surface of the 
 object, their paths being in fact continuations of the paths of 
 particles that may be considered to have been emitted from the 
 source. Indeed, if we imagine the source to emit particles in 
 all directions as it moves about, the particles which just graze 
 the moving object will, when they continue their rectilinear 
 motion with the velocity of light, form the boundary at which 
 the discontinuities arise. 
 
 In Chapter vm we shall obtain a class of wave-functions 
 with singularities moving along straight lines with the velocity 
 of light. These functions seem to be just the ones that are 
 required for the building up of wave-functions with dis- 
 continuities of the type just described. The problem of forming 
 in this way the functions which will enable us to complete the 
 solution of the diffraction problem is one which awaits solution. 
 
CHAPTER VI 
 
 TRANSFORMATIONS OF COORDINATES APPROPRIATE FOR 
 THE TREATMENT OF PROBLEMS CONNECTED WITH A 
 SURFACE OF REVOLUTION 
 
 33. Spheroidal coordinates. 
 
 Problems in which there is symmetry round the axis of z 
 can often be treated with the aid of a substitution of the form* 
 
 p + iz=f(a + il3) (184). 
 
 Taking a, /3, <f> as orthogonal coordinates, we have 
 
 da? + dy* + dz* = (da? + 
 and equations (18) of 8 become 
 
 (186), 
 
 Sa 
 
 9(a, 
 
 where ^ < 
 
 * This substitution has been used in other branches of mathematical 
 physics by C. Neumann, Theorie der Elektricitats- und Wdrme-Vertheilung in 
 einem Ringe (1864) ; E. Mathieu, Cours de physique mathematique (1873) ; 
 A. Wangerin, Berliner Monatsberichte (1878) ; Hantzschel, Reduction der 
 Potentialgleichung ; Michell, Mess, of Math. (1890) ; Basset, Hydrodynamics, 
 Vol. 2, p. 8 ; F. H. Safford, Amer. Journ. Vol. 21 ; Archiv der Math. Bd. 13 
 (1908), p. 22. The important developments on which the following analysis is 
 founded are contained in papers to which we shall refer presently. 
 
96 TRANSFORMATIONS OF COORDINATES [CH. 
 
 These equations may be satisfied by putting 
 
 -* k ~ 
 
 where l=UiV is a solution of the partial differential 
 equation 
 
 i /dp an dp an^ 
 
 The problem of finding the periods of free electrical oscilla- 
 tions on a conducting spheroid is of considerable interest 
 because a straight rod of circular cross-section can be regarded 
 as approximately equivalent to a prolate spheroid whose major 
 axis is relatively much longer than the minor axis. This 
 problem has been treated very fully by M. Abraham*, 
 R. C. Maclaurinf, M. BrilloumJ, F. Ehrenhaft and J. W. 
 Nicholson ||. The effect of a spheroidal obstacle on a train of 
 waves has been studied by K. F. HerzfeldH. 
 
 For prolate spheroids the appropriate substitution is** 
 
 or p = a sinh a sin (3, z = a cosh a. cos 0, 
 
 o /_ \ 
 
 = -a 2 (cosh 2 a- cos 2 ft) (190). 
 
 d (a, p) 
 The partial differential equation is now 
 
 ...... (191), 
 
 and there are elementary solutions of the form H = A (a) B ( 
 where A and B satisfy the differential equations 
 
 - coth - (^ 2 cosh 2 a + X) A = 
 
 - cot/3 ~ 
 
 * Dissertation, Berlin (1897) ; Ann. d. Phys. Bd. 66 (1898), p. 435 ; Math. 
 Ann. Bd. 52 (1899), p. 81. 
 
 t Cambr. Phil. Trans. Vol. 17 (1898-9), pp. 41108. 
 
 J Propagation de Vtlectricite (1904), Ch. vi. 
 
 Wiener Berichte (1904), p. 273. 
 
 || Phil. Mag. (1906). IT Wiener Berichte (1911), p. 1587. 
 
 ** Cf. Heine, Crelle, Bd. 26 (1843), p. 185 ; Kugelfunktionen, Bd. 2, 38 ; 
 Lamb's Hydrodynamics, p. 132. 
 
VI] SPHEROIDAL COORDINATES 97 
 
 These equations are discussed in some detail in the papers 
 to which we have just referred, they may be reduced to a special 
 form of an equation obtained by Prof. C. Niven* in a study of 
 the conduction of heat in ellipsoids of revolution. 
 
 For oblate spheroids the appropriate substitution is 
 
 p +iz= a cosh (a -f- ift) ............... (193), 
 
 giving p = a cosh a cos ft, z = a sinh a sin ft, 
 
 d (a, ft) 
 
 The surfaces ft const, are now hyperboloids of one sheet, 
 the surface ft = can be regarded as the surface of a screen 
 which is pierced by a circular hole of radius a. 
 
 The partial differential equation for H is now 
 
 - tanh a + tan + ^ a2 (sinh 2 + sin 2 ) fl = 0, 
 
 and there are elementary solutions of the form fl = A (a) B (ft) 
 where 
 
 ~ tanh a . + (X + a 2 #> sinh 2 a) 4 = 
 
 When H is independent of , & = and the elementary 
 solutions are of the form 
 
 = /P w (f ) df JPn (17) d*, f = cosh a, rj = cos /9 . . .(195) 
 for prolate spheroids, and of the form 
 
 fl = /P()df/PnO?)di7, f = isinha, r} = smft... (196) 
 for oblate spheroids. In either of these solutions a function P n 
 can be replaced by Q n . The corresponding solutions of Laplace's 
 equation are of the type 
 
 F= [AP n (?) + BQ n ($)] [CP n (,) + DQ (,)]. . .(197), 
 where A, B, C y D are arbitrary constants. 
 
 * Phil. Trans. (1880), p. 138. 
 B. 7 
 
98 SPHEROIDAL COORDINATES [CH. 
 
 Examples. 1. Prove that 
 [(cosh a cos j8 - cos y) 2 -I- sinh 2 a sin 2 /3] ~ i 
 
 (C. Neumann.) 
 
 2. Prove that a function B (/3) which satisfies the differential equation 
 (192) and is zero for /3=0, /3 = ?r is a solution of the homogeneous integral 
 equation 
 
 o 
 
 where]/* is determined by the condition that the integral equation should 
 possess a continuous solution which is not identically zero. 
 
 (M. Abraham.) 
 3. If A (a) be denned by the equation 
 
 o 
 
 it* satisfies the differential equation (192). A second solution of this 
 equation is given by 
 
 F 
 Jo 
 
 and is suitable for the representation of divergent waves. 
 
 (M. Abraham.) 
 
 34. Paraboloidal coordinates. 
 If we write 
 
 so that the transformation is 
 
 ^ = _o-^, p = 2V^o ............ (198), 
 
 the differential equation (143) becomes* 
 
 and is satisfied byf 
 
 F = 4 (a 
 
 * Cf H. J. Sharpe, Quarterly Journal, Vol. 15 (1878) ; Proc. Camb. Phil. 
 Soc. Vol. 10 (1899), p. 101; Vol. 13 (1905), p. 133; Vol. 15 (1909), p. 190; 
 H. Lamb, Proc. London Math. Soc. Ser. 2, Vol. 4 (1907), p. 190. 
 
 t The existence of elementary solutions for the paraboloid and certain 
 other surfaces is established in Bocher's Die Reihenentwickelungen der Potential- 
 theorie. See the table on pp. 256-7. 
 
VI] PARABOLOIDAL COORDINATES 
 
 if 
 
 (199), 
 
 where h is arbitrary. 
 
 Putting 2ikn ik(m + \) h we find that the differential 
 equations are satisfied by putting 
 
 A = e~ ik *F m n (2ik*\ B = e~ ik ^F m n (2^/3), 
 where F m n (s) satisfies the differential equation* 
 
 - + ^O ......... (200). 
 
 When n is a positive integer, one solution of this equation 
 is furnished by Sonin's polynomial f T m n (s\ which may be 
 defined with the aid of the expansion 
 
 (1 + t)-- 1 &+* = 2 T (m + n + 1) t n T m n (s). . .(201). 
 =o 
 
 A few properties of this function are given here for the sake 
 of reference. 
 
 [ e~*s m T m n (s) T m v (s) ds = 
 
 v = n 
 
 vwr-Jw .................. (204) - 
 
 t (s) ............ (205), 
 
 * This is a slight modification of Weiler's canonical form for an equation of 
 Laplace's type, Crelle's Journal, Bd. 51 (1856), p. 105. The equation is discussed 
 for real values of m and n by O. Schlomilch, Hotieren Analysis, Bd. 2 (1874), p. 517. 
 
 + Math. Ann. Bd. 16. Further properties of the function are given by 
 L. Gegenbauer, Wien. Ber. (1887), p. 274, who proves that the roots of the equation 
 T m Ti (*) = 0, considered as an equation of the nth degree in *, are all real, 
 positive and unequal. This is a generalisation of the result obtained by 
 Laguerre for the case m = 0. A geometrical proof has been given by Bocher, 
 Proc. of the Amer. Acad. of Arts and Sciences, Vol. 40 (1904). 
 
 72 
 
100 SONIN'S POLYNOMIALS [CH. 
 
 / _ 1 \n p s fin 
 
 - 5 cos2 Sn2m * < 
 
 Equations (201), (202) and (203) were given by Abel* and 
 Murphy f for the case m = : the polynomial is then equivalent 
 to the polynomial of TchebychefFJ and Laguerre which occurs 
 in the theory of interpolation and also in the theory of continued 
 fractions. When m + ^, the polynomial can be expressed in 
 terms of the polynomial U n (x) discussed by Tchebycheff|| and 
 HermitelF, or in terms of the function of the parabolic cylinder, 
 discussed by Weber**, Whittakerff and others {f. 
 
 The above analysis indicates the existence of a wave-function 
 of the form 
 
 ft = e ik ( ZC V {w * T m n (ZiJca) T m n (2ifc) p m . . .(208). 
 
 This function can be expressed as an integral of the form 
 used in 5, we have in fact the equation 
 
 cos i + 
 
 ...... (209), 
 
 from which the required representation can be immediately 
 
 derived. In this formula m is either zero or a positive integer. 
 
 The convergence of a series of terms of type (208) in which 
 
 * Memoires de mathematique par N. H. Abel, Paris (1826) ; Oeuvres, Sylow 
 and Lie, t. 2. 
 
 f Cambr.Phil. Trans. (1833). 
 
 J Mem. de VAcad. de St Petersburg (1860). 
 
 Bull, de la Soc. math, de France, t. 7 (1879) ; Oeuvres de Laguerre, t. 1, 
 p. 428. 
 
 || Loc. cit. See also Sturm, Liouville's Journal, Vol. 1. 
 
 If Comptes Rendus, t. 58 (1864), p. 93. The Hermite functions have been 
 generalised by Curzon, Proc. London Math. Soc. Vol. 13 (1914), p. 417. The 
 generalised functions are intimately connected with the functions considered here. 
 
 ** Math. Ann. Bd. 1 (1869), p. 1. 
 
 ft Proc. London Math. Soc. Ser. 1, Vol. 35 (1903), p. 417. 
 
 JJ Baer, Diss. Cilstrin (1883) ; Hantzschel, Zeitschr. fiir Math. Bd. 33 (1888) ; 
 Adamoff, Annales de St Petersbourg, t. 5 (1906) ; G. N. Watson, Proc. London 
 Math. Soc. Ser. 2, Vol. 8 (1910), p. 393. 
 
VI] SECOND SOLUTION OF THE DIFFERENCIAL f 
 
 n takes different integral values can be partially discussed with 
 the aid of the equation 
 
 n(n-I)(m+n + I)(m + n + 2) 1 
 
 )(m + 4) '"J " 
 
 which shows that the modulus of T m n (ix) increases with #. 
 Hence if a series of terms of type (208) converges absolutely 
 for any given value of a, it converges absolutely for all smaller 
 values of a. 
 
 For a fuller discussion of the convergence it would be useful 
 to have, an asymptotic expression for T m n (s) when n is large. 
 Suitable asymptotic expressions have already been found for 
 the case m = -J by Adamoff and Watson. 
 
 The differential equation (200) has been studied for general 
 values of m and n by Pochhammer*, Jacobstahlf, Whittakert 
 and Barnes. It usually possesses two distinct solutions which 
 can be expanded in power series converging for all finite values 
 of s. If, however, m and n are positive integers, there is only 
 one solution which can be represented by a convergent power 
 series in s, the other may be defined by the equation 
 
 ** (s ~ 
 
 it contains a logarithmic term. For negative integral values of 
 n we may adopt the definition 
 
 It should be noticed that when | a | is large, U m n (Zika.) has 
 an asymptotic expansion of which the first term is 
 
 (2ika. 
 The solutions of type 
 
 e ik (z+ct) im4> U 
 
 * Math. Ann. Vol. 36, p. 84 ; Vol. 46, p. 584. f Ibid. Vol. 56, p. 129. 
 
 J Bull. Amer. Math. Soc. (1904). 
 
 Cambr. Phil. Trans. Vol. 20 (1906), p. 253. 
 
oV;{.,} /SQNJN'S POLYNOMIALS [CH. 
 
 are consequently suitable when a is large for the representation 
 of waves diverging to infinity in the positive direction of the 
 axis of z. 
 
 It may be worth while to mention here that the functions 
 Tm n (s), U m n (s) both satisfy Gegenbauer's difference equations 
 
 F:~ l (s) = (m + n) <;1 (s) + Fl (s) 
 
 n (m + n) F% (s) - {s - (m + 2n - 1)} F^ 1 (s) + F^ 2 (s) = 
 (n - 1) F^ 1 (s) = {s-(m + n-l)} F^ (s) - F^l (s) 
 (n - 1) F^ 1 (s) = {s-(m + 1)} F^ (s) - s F^\ (s) 
 
 ...... (213). 
 
 The function U m n (s) also satisfies an equation analogous to 
 (204). 
 
 35. Relations between different solutions. 
 
 Many useful formulae may be obtained by expanding known 
 wave-functions in series of elementary wave-functions of type 
 (208) and by identifying our elementary wave-functions with 
 certain definite integrals which are known to represent wave- 
 functions. For instance, we have the expansion | tan \ co | <i 1 
 
 ^ (-l) w nl (m + n) I 
 
 (214) 
 
 which enables us to represent a plane wave with the aid of 
 a double series of solutions of the form (208). 
 
 Further identities may be obtained by deriving wave- 
 functions from Cunningham's solutions* of the equations 
 
 du _ d^_ du _ &u d*u . 
 
 dr~~w a^~8^ + a^ 
 
 The first equation possesses the polynomial solutions 
 
 * Proc. Hoy. Soc. Ser. A, Vol. 81 (1908), p. 310. See also Wera Lebedeff, 
 Dm. Gottingen (1906) ; Math. Ann. (1907). The first result is given by Appell, 
 Liauville't Journal, Ser. 4, t. 8 (1892), p. 187. . 
 
VI] RELATIONS BETWEEN DIFFERENT SOLUTIONS 103 
 
 and also the solutions 
 
 n+2 
 
 The second equation possesses the polynomial solutions of 
 type 
 
 7*prT m *(-fymnm(<l>-b) ......... (218) 
 
 and also the solutions of type 
 
 . v in ] 
 
 -^) ...(219). 
 
 Wave-functions may be derived from these solutions by the 
 method of 13. 
 
 We add here a few relations which are obtained by ex- 
 pressing the solutions thus formed in terms of old solutions. 
 
 cos co] 
 = 2 (-imro + n+l^V^OZU-fo) **""... (220), 
 
 m--n 
 
 n 
 
 _ 1 ^n Orn+i 
 
 ' ...... (222) ' 
 
 = F (n + 1) T (n + m + 1) 
 
 ...... (223). 
 
 The proofs of these are left to the reader. 
 
 Prof. G. D. Birkhoff has remarked to me that the differential 
 equation (200) can be regarded as a limiting case of the hyper- 
 geometric equation when two of the singularities coincide at 
 infinity*, consequently many properties of the solutions can 
 be derived from known properties of hypergeome trie functions f. 
 It should be noticed that when W is independent of t there 
 
 * Cf. Bocher, Die Reihenentwickelungen der Potentialtheorie, p. 137. 
 f This method was used in a particular case by Kummer, Crelle's Journal, 
 Bd. 15 (1836), p. 138. 
 
104 PARABOLOIDAL COORDINATES [CH. 
 
 are elementary solutions of equation (143) of the form W = AB 
 where 
 
 J m 
 
 ~ 
 
 We thus obtain elementary solutions of Laplace's equation 
 of the form 
 
 (0-</> ) ......... (224), 
 
 where m, X, < are arbitrary parameters. 
 
 36. Toroidal coordinates. 
 If we put 
 fc = pcos(f>, y = psm(f>, z cosh &>, ct % sinh o>...(225), 
 
 p + i a coth =p2T t 
 
 a sinh cr asin->/r / 226) 
 
 cosh cr cosi/r ' * cosh cr cos ^ """" ^ 
 the wave-equation becomes 
 
 _3_ f sinh cr sin i|r 3^) 9 | sinh cr sin ijr du] 
 
 dor ((cosh a COS >/r) 2 9cr) 9i|r ((cosh a COS ^r) 2 9i|r) 
 
 sin ir 9 2 w sinh cr d*u _ 
 
 8 
 
 sinh cr (cosh cr cos >/r) 2 9<^> 2 sin ^ (cosh cr - cos ^) 2 9o) 8 
 
 ...... (227). 
 
 This is satisfied by 
 
 u = F(o-)G (i|r) (cosh cr - cos T|T) e A;w cos m (< - < ). . .(228) 
 
 if cosech cr -j- (sinh <r-j-} \n(n+ 1) + . ^ 1^=0 
 c^o- V do-J { smh 2 crj 
 
 coseci/r ^-- (sini/r ) 4. J|t( n +l)- . I ^=0 
 
 r ttA/r \ C?i/r/ ( Sin 2 -^] 
 
 ...... (229). 
 
 Hence we obtain wave-functions of the form 
 
 (cosh cr - cos T/T) P n m (cosh cr) P n * (cos >/r) e A:w cos m (<j> <) 
 
 ...... (230). 
 
 Other solutions of the wave-equation may be obtained by 
 replacing the functions P n m , P n k by Q n m , Q n k . 
 
 Many useful formulae may be obtained by expanding 
 particular wave-functions in series of wave-functions of type 
 
Vl] RELATIONS BETWEEN DIFFERENT SOLUTIONS 105 
 
 (230). The expansion of unity, for instance, gives rise to 
 Neumann's expansion 
 
 cosh.- cos + =, (2?l + T) " < C 8h ff > P " < C S - (231) ' 
 It should be noticed that when we make the substitution 
 (225) the wave-equation becomes 
 
 d*u Idu l&u Vu Idu 1^_ XOOON. 
 8p 2 + p dp + ? ^ + d? + ? 8f " ? 2 9o>* ~ 
 it thus possesses elementary solutions of the forms 
 
 m (V) epw cos m (0 ~ *o) ...... (233), 
 
 (V) * p " sm(<t>-<t> ) ......... (234). 
 
 The expression of solutions of type (230) in terms of the 
 solutions just found leads to some interesting identities. Thus 
 we have the equation 
 
 (Xf ) J m (\p) J 
 
 x(cosh a- - cos *}P~ p P +m _ n (cos ^r) P; W _ W (cosh IT) 
 
 p> 1, m> 1, p + m> I ......... (235). 
 
 Many important formulae connected with Bessel functions 
 are simply particular cases of this one*. It should be remem- 
 bered that 
 
 -*--JT-!(*) (236). 
 
 The corresponding integral in which K p (Xf) is replaced by 
 J-p (Xf ) can also be evaluated in terms of Legendre functions, 
 but the formulae are more complicated. The case p m is 
 discussed by Macdonald f. 
 
 It should be noticed that if we write 
 cosh (a a)) = i cot -^r, cos (0 <f>) = coth G \ 
 sinh (a co) = i cosec iff, sin (& </>)= i cosech a- J ' 
 three relations of type 
 
 (238) 
 
 * See, for instance, the formulae given by H. M. Macdonald, Proc. London 
 Hath. Soc. Ser. 2, Vol. 7, p. 147, and by the author, ibid. Vol. 12, Abstracts. 
 t Loc. cit. p. 142. 
 
106 TOROIDAL COORDINATES [CH. 
 
 are satisfied and so the functions , /3 can be used to obtain 
 an electromagnetic field by the method of 5. It is easy 
 to verify that the function 
 
 u = (cosh <7- cos ^)/(a,) (239) 
 
 satisfies the wave-equation, / being an arbitrary function. 
 
 We add here a few formulae for P n m (cosh o-), Q n m (cosh er) ; 
 these and other formulae will be found in the memoirs of 
 Dr Hobson and Dr Barnes to which we have already referred. 
 
 x F \ n, n+\\ 1 m ; sinh 2 -^ I 
 
 T(l-m) v 
 
 xF{$-m,l+n-m; I - 2m ; l-e~ 2<r } 
 
 <7>0, 
 
 n m (C0sh 
 
 xJP{ra + J,w + m + l; w + f , e-^} <r > 0. 
 
 Various asymptotic expansions for these functions are given 
 by the authors just named and by Dr Nicholson*. 
 
 It should be mentioned that the solutions of the wave- 
 equation that have just been obtained are not directly useful 
 for the treatment of the boundary problems of mathematical 
 physics. They may, however, be used to construct useful 
 solutions of the equation AM + l&u = by means of various 
 artifices. If, for instance, we multiply one of our wave-func- 
 tions by e ikct and integrate with regard to t between z and oo , 
 the resulting function will often be a solution of Av + k*v = 0. 
 This may be illustrated by taking the wave-function 
 
 u = J (tXf) J m (Xp) cos m (< - </><>) 
 and using the formula 
 
 T. 
 
 At = ~ > W 
 
 /c 
 
 ...... (240). 
 
 * British Association Reports, Winnipeg (1909), p. 391. 
 
VI] TOROIDAL COORDINATES 107 
 
 The integration can be taken between other limits in 
 certain cases; for instance, the integral 
 
 e ikte dt 
 
 ......(241) 
 
 represents the solution of Av -f &v = corresponding to a 
 circular ring of sources. In this case our wave-function u 
 is a constant multiple of cosh a- cos i|r. 
 
 The theory of electrical oscillations on a conducting anchor 
 ring has been treated by H. C. Pocklington*, W. McF. Orrf 
 and Lord RayleighJ, without the use of toroidal coordinates ; 
 the results are of course only approximate. 
 
 37. Solutions of Laplace's equation. 
 If we put 
 
 a sinhcr asin ^ /942^ 
 
 P i r > * 1. r \^*/i 
 
 cosh cr cos ty cosh a cos \jr 
 
 the angle ^r may be interpreted as the angle which two fixed 
 points A, B whose coordinates are z 0, p = a, subtend at 
 
 PA. 
 
 a point P (p,z); the quantity cr may be interpreted as log . 
 
 The surfaces ty = const, are spheres having a real circle (p = a, 
 z = 0) in common, the surfaces cr = const, are anchor rings. 
 
 If we use the toroidal coordinates cr, -^r, c, Laplace's equation 
 becomes 
 
 _ 9 f sinho- du} 9 f sinhcr du\ 
 da) dty (( 
 
 do- (cosh <7 cos i|r da-} dty (cosh a cos -^ 9^r J 
 
 sinh cr (cosh cr cos 
 
 * Proc. Camb. Phil. Soc. Vol. 9 (1897), p. 324. 
 
 t Phil. Mag. Vol. 6 (1903), p. 667. 
 
 J Proc. jRoy. Soc. Ser. A, Vol. 87 (1912), p. 93. See also C. W. Oseen, 
 Phyg. Zeitschr., Dec. 1st (1913) ; Arkiv for Mat. Ast. och Fysik, Bd. 9 (1913). 
 
 B. Riemann, PartielleDifferentialgleichungen,Hattendort'8 edition (1861); 
 C. Neumann, Theorie der Elektricitats- und Wdrme-Vertheilung in einem Ringe, 
 Halle (1864); W. M. Hicks, Phil. Trans. (1881), p. 609; A. B. Basset, Amer. 
 Journ. Vol. 15, Hydrodynamics, Vol. 2. For an alternative method see F. H. 
 Safford, Annals of Mathematics, Vol. 12 (1898), p. 27. 
 
108 BIPOLAR COORDINATES [CH. 
 
 and possesses solutions of the form 
 
 P 
 
 u (cosh a cos T/r) a cos n (i|r - -Jr ) cos m (</> (/> ) 
 
 ( 
 
 (243), 
 
 which are suitable for the treatment of problems connected 
 with the anchor ring, circular disc and spherical bowl *. 
 
 For problems connected with two spheres bipolar coordinates 
 may be used ; the appropriate substitution isf 
 
 asim/r _ asinhcr /944A 
 
 ^ ~~ cosh cr cos ^ ' cosh cr cos ty 
 
 The surfaces cr = const, are now coaxal spheres with imaginary 
 common circle. The radius of the sphere <r = cr is a \ cosech <T O \ 
 and the distance of its centre from the origin is a \ coth cr 1 . 
 
 The ratio of the distances of a point from the limiting points 
 of the system of coaxal spheres is e" and the angle between the 
 radii from these points is -fy. 
 
 The appropriate solutions of Laplace's equation are now of 
 the typej 
 
 u = (cosh cr cos ^)- [A cosh (n + -J) cr -f B sinh (n + 1) cr] 
 
 x cos m (</> - < ) [fP n (cos ^) + g Q n (cos ^)]. . .(245). 
 It should be noticed that when we are using toroidal 
 coordinates the function 
 
 u = (cosh cr cos ty)^f \<f> i log tanh cos ^ (^ ^ ) 
 
 (246) 
 
 satisfies Laplace's equation and that when we use bipolar 
 coordinates the corresponding solution is 
 
 (cosh a- - cos ^f \ <j> i log tan |j cosh (cr - o- ). . .(247). 
 
 * See for instance E. W. Hobson, Cambr. Phil. Trans. Vol. 18 (1899) ; C. 
 W. Oseen, Arkivfor Matematik, Bd. 2, No. 5; H. C. Pocklington, Phil. Trans. 
 A, Vol. 186 (1895), p. 603. 
 
 f W. Thomson (Lord Kelvin), Liouville's Journal (1847). 
 
 J G. B. Jeffery, Proc. Roy. Soc. Ser. A, Vol. 87 (1912) ; G. R. Dean, Phys. 
 Review (1912) ; G. Darboux, Bull, des Sciences math. t. 31 (1907), p. 17. 
 Another method of dealing with problems connected with two spheres is 
 described by A. Guillet and M. Aubert, Journal de Physique, t. 3 (1913). 
 
VI] SOLUTIONS OF LAPLACE'S EQUATION 109 
 
 It should be mentioned here that other simple ["solutions 
 may be obtained by using the formulae 
 
 _ (cosh <r) = - - cosh ner 
 
 EXAMPLES. 
 
 1. If with the notation of 36 we write 
 
 (+ip=a cosh (a + 1 
 the wave-equation becomes 
 
 Hence show that there are wave-functions of the form 
 
 u = A(a}B 03) e k " cos m (0 - < ), 
 where a, , m and </> are arbitrary constants. 
 
 2. Prove that if p + iz=f(a + ip) the wave-equation becomes 
 
 . ft 
 
 3 2 c 2 2 ' 
 
 and obtain elementary solutions of type .4 (a) .6 (/3) e* 7 " 4 when 
 
 z + ipa cosh (a+z'^3). Notice that the solutions of equation (188) are not 
 wave-functions, they are analogous to the stream-line functions of hydro- 
 dynamics. 
 
CHAPTER VII 
 
 HOMOGENEOUS SOLUTIONS OF THE WAVE-EQUATION 
 
 38. The method of Stieltjes*. 
 
 Wave- functions which are homogeneous functions of x t y, z, t 
 may be studied with the aid of the substitution 
 
 x = s cos d cos </>, 2 = ssin 0cos%| 
 
 y = scos6 sin <, ict = s sin sin ^ j " 
 
 The wave-equation in these coordinates has the form 
 
 8Ht 3du I&u I _ ^w_ 
 ds 2 sds s*d0* s*co&0d<f) z 
 
 * cot#-tan0du 
 ~ ~~ 
 
 Putting cos 20 fi y we find that there are elementary solutions 
 of degree 2n of the form 
 
 ............... (249), 
 
 ......... (250). 
 
 This equation is satisfied by 
 
 ......... (251), 
 
 with the usual notation of the hypergeometric function t. 
 
 * Comptes Rendm, t. 95 (1882), p. 901 ; Liouville's Journal, Ser. 4, t. 5 
 (1889), p. 55. See also Tisserand, Traite de mecanique celeste, Paris (1889); 
 H. Bateman, Proc. London Math. Soc. Ser. 2, Vol. 3 (1905), p. 111. 
 
 t This gives a polynomial if either n + 1 H -- -^ or - n is zero or a 
 
 2 2 
 
 negative integer. 
 
CH. VII] THE METHOD OF STIELTJES 111 
 
 It should be noticed that if we write 
 cos 6 sec a, sin 6 = i tan a, sin % = cosh w, cos % = i sinh w, 
 
 yu, = sec 2 a + tan 2 a, 
 we obtain real wave-functions of the form 
 
 -<o) ......... (252), 
 
 the variable in the hypergeometric function is now tan 2 a. 
 When m p the equation for is the equation satisfied by the 
 associated Legendre functions. We thus obtain wave-functions 
 of the form 
 
 x> ......... (253). 
 
 Comparing this with the elementary solution of Laplace's equa- 
 tion in polar coordinates (r, 0, (/>), we see that if f(r, 6, <f>) is a 
 solution of Laplace's equation / (s 2 , 20, <f> + ^) is a wave-function. 
 We may thus derive wave-functions from harmonic functions ; 
 
 in particular, the fundamental harmonic function gives us the 
 
 fundamental wave-function = - - - - - . We have 
 
 already remarked in 13 that Lord Kelvin's method of inver- 
 sion may be extended to wave-functions, it is easy to see that 
 the result is an immediate consequence of the fact that the 
 differential equation (250) ia, unaltered when (n + 1) is 
 written in place of n. 
 
 It is easy to see that there are (n + I) 2 linearly independent 
 polynomial solutions of degree n, for a general polynomial of 
 
 degree n contains ^ (n + 1) (n + 2) (n + 3) coefficients and when 
 this is operated on with Q, the vanishing of the resulting 
 polynomial of degree n 2 gives -*(n I)n(n + l) conditions, 
 
 The difference between these two numbers is (n 4- 1) 2 . 
 
 A. polynomial solution of degree n is given by the integral 
 
 u = I (x cos a + y sin a -f iz)* (x sin a - y cos a - ct) n ~* e ima da. 
 
 Jo 
 
 A set of (n + 1) 2 linearly independent polynomials is obtained 
 
112 HOMOGENEOUS SOLUTIONS OF THE WAVE-EQUATION [CH. 
 
 by allowing m and p to take the values (0, 1 ... n). A better 
 set of solutions is obtained by using the integrals of type 
 
 u = I [(# - iy) e*" + i(z ct)] p 
 Jo 
 
 x [(x -f iy} e~ ia + i (z + ct)] n ~ p e im * da. 
 
 Polynomial solutions may also be obtained by differentiating 
 
 the fundamental wave-function and using generalised inver- 
 
 s 
 
 sion*. The polynomial solutions were first discussed by 
 Cayleyf. WaelschJ has recently studied them from a new 
 point of view. 
 
 Example. Prove that when n is a positive integer 
 
 1 ' 3 "' 2 ^ 1 P n (cos 20) = i r I (cos 6 cos (j> + i sin 6 cos x ) 2n d$ d x . 
 
 39. The method of Green. 
 
 Homogeneous solutions may also be investigated with the 
 aid of Green's substitution 
 
 x = s sin a sin ft cos c, y s sin a sin /3 sin <f>) 
 
 r\ > 
 
 z s sm a cos p, ict s cos a j 
 
 The wave-equation now becomes 
 
 &u 3 du 2 du 1 
 
 + + C< a + ~ 
 
 _ A , . 
 
 2 2 2 ~ 2 ~ 
 
 s 2 sin 2 a sin d/3 dp s 2 sin 2 a sin 2 
 
 and possesses elementary solutions of the form 
 u = s n A (a) 5 OS) cos m (< - </> 
 
 * Cf. F. Didon, Annales de VEcole Normale (1), t. 5 (1868), p. 229; t. 6 
 (18GD), p. 7; t. 7 (1870), pp. 89, 247 ; P. Appell, Rend. Palermo, t. 36 (1913) ; 
 K. de F&iet, Comptes Rendus, Nov. 17th (1913). 
 
 t Liouville's Journal, t. 13 (1848) ; Phil. Trans. Vol. (165) n. (1875), p. 675. 
 See also Hermite, Oeuvres, t. 2. 
 
 + Deutsche Math. Verein, Bd. 19, p. 90. 
 
 Cambr. Phil. Trans. Vol. 5 (1835), p. 395; Collected Papers, p. 187; 
 Cayley, loc. cit. See also Heine, Handbuch der Kugelfunktionen, Bd. 1, p. 449 ; 
 Crelle, Bd. 60, 61, 62 (18621863) ; E. W. Hobson, Proc. London Math. Soc. 
 Ser. 1, Vol. 24, p. 67, Vol. 25 ; F. G. Mehler, Progr. Danzig (1864) ; Crelle, 
 Bd. 66 (1866), p. 161; C. Neumann, Zeitschr. Math. Phys. Bd. 12 (1867), 
 p. 116 ; V. Giulotto, Gior. d. Mat. 39 (1901), p. 162. 
 
vn] GREEN'S METHOD 113 
 
 where 
 
 We may thus take 
 
 J. = Vcos^ P; (cosa)+6 2 C 
 where Cj, C 2 , 61, 6 2 are arbitrary constants. 
 
 40. Wave-functions of degree zero. 
 
 If n is a wave-function of degree 2, the formulae 
 
 an an _ #an an 
 
 H 9 = y, -- o~ > E x = - -r + ct -~ 
 y dz dy c dt fa 
 
 an an y aa ^ an 
 
 Z ^ -- ^~ Ey=--^- +Ct ^~ 
 
 dx dz c dt dy 
 
 an an ., z an an 
 
 X ^ -- 2/^~ &z - ~^r + ct ^r- 
 
 dy y dx c dt dz 
 
 ,...(257) 
 
 give a solution of Maxwell's equations. 
 
 A homogeneous wave-function of degree 2 can, of course, 
 be derived from a homogeneous wave-function of degree zero. 
 If F is an arbitrary function of two variables subject to suitable 
 restrictions, the integral 
 
 c* 1 ^ ...... 
 
 iz J 
 
 Q _ 
 
 \_x cos a + y 
 
 represents a wave-function of degree zero, and when this is 
 multiplied by a wave-function of degree 2 is obtained. We 
 
 S" 
 
 add here a few particular wave-functions of degree zero : 
 
 Electromagnetic fields which are derived from this type of wave- 
 function of degree 2 may be generalised by writing x f (T), 
 y V ( T )> z ~ ? ( T )> * ~ T instead of x, y, z, t respectively and 
 integrating round a closed contour in the complex r plane. 
 The integrals thus obtained can generally be evaluated by 
 B. 8 
 
114 HOMOGENEOUS SOLUTIONS OF THE WAVE-EQUATION [CH. VII 
 
 means of Cauehy's theorem. Many of the results given in the 
 next chapter are suggested at once by this method and may 
 be thoroughly established by a method of direct verification. 
 
 It is worthy of note that if we write r in place of ct in a 
 wave-function of degree zero the resulting function is a solution 
 of Laplace's equation AM =0. A general solution of Laplace's 
 equation of degree zero can be derived at once in this way from 
 the first of the solutions (259). We thus obtain Donkin's 
 formula* 
 
 /^ /-M ............ 
 
 \z + r ] y \z + rj 
 
 A similar result is that if 
 
 l=F(x,y ) z ) w, t) 
 is a homogeneous function of degree J satisfying the equation 
 
 and s be written in place of w, the resulting function is a 
 wave-function. Now if f(x, y, z) is a solution of Laplace's 
 equation, the function 
 
 *Jw ct \w-ct' w ct' w 
 
 z \ 
 
 -ct) 
 
 satisfies the requirements, consequently we may conclude that 
 the function 
 
 ......... (261 > 
 
 is a wave-function f. Other wave-functions may be derived 
 from this by generalised inversion or by interchanging the 
 variables x t y, z, ict. 
 
 * Phil. Trans. (1857). This solution may be obtained at once from Jacobi's 
 theorem that if p, q, r are three functions of u which satisfy the equation 
 1> 2 + q 2 + r 2 = and u is denned by the equation au = xp(u) + yq (u) + zr (u), 
 then an arbitrary function of w is a solution of Laplace's equation, Werke, 
 Bd. 2, p. 208. See also Forsyth, Mess, of Math. (1898). 
 
 t This result is obtained in another way by Pockels, Uber die partielle 
 Differentialgldchung Aw + k*u - 0. Teubner, Leipzig (1891). 
 
CHAPTER VIII 
 
 ELECTROMAGNETIC FIELDS WITH MOVING 
 SINGULARITIES 
 
 41. An electromagnetic field with a simple singularity 
 or electron, first model of a corpuscle*. 
 
 We shall now derive a family of wave-functions from the 
 fundamental wave-function 1/s 2 , where 
 
 and T is a variable parameter, which is at first independent of 
 x, y, z, t. Using a method invented by Prof. A. W. Conway t 
 we consider the integral 
 
 0~iM 
 
 2?ri J 
 
 s 2 
 
 taken round a closed contour in the plane of the complex vari- 
 able T. If this contour contains only one root T of the equation 
 s z = 0, the value of the integral is 
 
 where 
 
 V=?(r)(*-t)+V'(T)(y-l)+C(T)(*- & 
 
 (263) 
 
 and T is the root in question. 
 
 * I have ventured to use Johnstone Stoney's term " electron " to denote the 
 simple point singularity and Sir Joseph Thomson's term " eorpuscle " to denote 
 the elementary charged particle which has been discovered by experimental 
 work. 
 
 f Proc. London Math. Soc. Ser. 2, Vol. 1 (1903). Integrals over complex 
 paths had been employed previously in electromagnetic theory by Sommerfeld 
 and other writers. 
 
116 FIELD OF A MOVING ELECTRON [CH. 
 
 This function v vanishes when a? = f , y = v, = ?> t = r and 
 so the wave-function H has a singularity which moves along the 
 curve F represented by 
 
 *=?to, y = *7(T), *=?(T) (264). 
 
 If, moreover, the velocity of this singularity E is always less 
 than the velocity of light, it is easy to see that v does not vanish 
 for any real values of (a?, y, z, t) other than those just mentioned. 
 
 When the velocity of the singularity E is always less than c, 
 there is only one value of r less than t for which the equation 
 s 2 = is satisfied : a?, y, z, t being supposed to be given. 
 
 To prove this we surround each point E on F by a sphere of 
 radius c (t r) having E as centre ; then it is clear that each 
 sphere lies entirely within the neighbouring one corresponding 
 to a smaller value of r, provided r<t and d 2 + dvf + d 2 <c*dr 2 . 
 
 This shows that one and only one of these spheres passes 
 through a given point of space and so there is only one value of 
 T < t for which the equation 
 
 [* - t to? + [y~1 W? + [* - f (T)? = C" (t - T)*. . .(265) 
 is satisfied*. 
 
 Now let a point Q (a?, y, z, t) move with a velocity less than c 
 along a curve G and let us consider the variation of r with t. 
 As t increases from t to t + dt the radius of the sphere associated 
 with each point E will increase by cdt and since Q moves a 
 distance less than cdt in the interval dt, its new position will 
 lie within the new sphere associated with the time r. Conse- 
 quently the new position of Q lies on a sphere associated with a 
 greater time r. 
 
 Hence if Q moves in any manner with a velocity less than the 
 velocity of light, r increases with t. 
 
 Things are quite different when the velocity of E is greater 
 than c. The spheres then have a real envelope and there may 
 be more than one sphere through a given point in space, also T 
 may sometimes decrease when t increases. 
 
 In this case, however, v vanishes for values of x, y, z, t other 
 than x = f , y = r),z=, t = r, and so fl has oo J singular lines 
 
 * Cf. A. W. Conway, loc. dt. ; H. Bateman, Manchester Memoirs (1910) ; 
 G. A. Schott, Electromagnetic Radiation (1912). The theorem is due to Lienard. 
 
vin] LI^NARD'S POTENTIALS 117 
 
 through each point E. These singular lines form a right 
 circular cone whose axis is along E*$ direction of motion : each 
 singular line is described by a singular point that travels with 
 the velocity of light. 
 
 When the velocity of the singularity is less than c-we can 
 obtain a solution of Maxwell's equations having the moving 
 singularity by using the potentials* 
 
 _F(r) _erj'(r) _<(r) _ e^_ 
 
 ~4^T' A * ~*^T' Az ~ 47TI/ "' ~4 
 
 (266). 
 
 It is easy to verify that they satisfy the relation 
 
 divA+-^-=0 (267). 
 
 c oi 
 
 When the electric and magnetic forces are calculated from 
 these potentials with the aid of the formulae 
 
 > (268) 
 
 I/ Vl/ 
 
 it is found thatf 
 
 H _?_ ^ilif ) ^ _ e <* ( T > ff ) 
 
 4>7T d (y, z) ' 4f7rc d (x, t) 
 
 where 
 
 (270). 
 
 It is clear from these equations that the magnetic force is 
 
 * A. Lie"nard, L'eclairage tlectriqtie, Vol. 16 (1898), pp. 5, 53, 106. See also 
 E. Wiechert, Arch, neerlandaises (2), Vol. 5 (1900), p. 54 ; K. Schwarzschild, 
 Gott. Nachr. (1903). The potentials are usually written in the form 
 
 A = , , where the square bracket indicates 
 
 that the quantity enclosed is to be calculated at time r = t-. Cf. H. A. 
 
 Lorentz, The Theory of Electrons, p. 50. To obtain a model of a corpuscle we 
 must write de instead of e and integrate over a small region. 
 
 t These expressions for the components of E and H were communicated to 
 me by Mr E. Hargreaves in 1909 ; they should be of some historical interest 
 in connection with the general theory of 5. This was, however, the outcome 
 of some independent work. Cf. Proc. London Math. Soc. (2), Vol. 10 (1911), 
 p. 96. 
 
118 CONSTANCY OF THE ELECTRIC CHARGE [CH. 
 
 perpendicular to the electric force and also to the radius 
 from the effective position of E\ for we have the relations 
 
 dr c. 9r _ 
 
 , (271). 
 
 dr dr 
 
 It also follows from these relations that r satisfies the char- 
 acteristic equation 
 
 This is to be expected because, as Jacobi has remarked* for the 
 case of Laplace's equation, the argument r of an arbitrary func- 
 tion occurring in the solution of a partial differential equation 
 must satisfy the partial differential equation of the character- 
 istics!. 
 
 To prove that there is a constant charge e associated with 
 the singularity of our electromagnetic field we shall calculate 
 the integral of the radial component of E over a sphere having 
 the singularity as centre. We have to evaluate the integral 
 
 ce fr/3o-9r do- dr d<rdr 1 dor dr\ , 
 4m J J \dx dx + ty dy + Yz te ~ <? dt 'dt) ' 
 
 which is easily transformed into 
 
 //- 
 
 * 
 
 Transforming the axes so that the axis of z is in the direction 
 of motion of the singularity, we may put 
 
 v = r (v cos 6 c), dS r* sin 
 and our integral becomes 
 
 ce 
 
 e. 
 
 * Werke, Bd. 2, p. 208 ; Crelle's Journal, Bd. 36 (1848). 
 
 f For the general theory of characteristics see Hadamard, Propagation de 
 Ondes (1903), Chapters vn. and vin. ; J. Coulon, Comptes Rendus, t. 128 
 (1899), p. 1386 ; A. V. Backhand, Math. Ann. Bd. 13 (1878), p. 411 ; J. Beudon, 
 Comptes Rendus, t. 124 (1897), p. 124. 
 
vui] WHITTAKER'S POTENTIALS 119 
 
 Prof. E. T. Whittaker* has calculated potentials T, II from 
 which A and 4> can be derived by using (7). He finds that 
 
 n = (o,o,S), r=(o,o,JV), 
 
 where 
 
 = e sinh- 1 - birN =-e tan 
 
 V(- # + (y - 
 
 _ 
 - e log V(a? - f ) 2 + (y - 1?) 2 
 
 ...... (273). 
 
 It may be verified without difficulty that the functions are 
 wave-functions. This result is a particular case of the 
 following general theorem. 
 
 Iff(x, y, z) is a homogeneous function of degree zero satisfy- 
 ing Laplace s equation Aw = 0, the function 
 
 0-/[*-(T),y-i,*-f(T)] ...... (274) 
 
 is a wave-function. 
 
 42. The electromagnetic field due to a moving doublet. 
 
 Let us now derive an electromagnetic field by superposing 
 two electromagnetic fields of the type just described wherein 
 the singularities move along the two neighbouring curves 
 
 ^ = ?(T), y = *!(*), * = ?(T), 
 
 = f ( T + ( T I)> y = */ ( T I) + e/9 (TJ), = ? (rO + 67 ( TI ), 
 e being a quantity whose square may be neglected. 
 If T! is defined in terms of x, y, z, t by the equation 
 
 [* - f (T,) - * (r,)] 2 + [y-i (T,) - e/3 (rO] 2 
 
 and T! = T + e#, we easily find that 
 
 Also if Vi is the quantity corresponding to v, we have 
 v l v 4- e [0v<r 4- a' (sc f ) 4- ft' (y rj) 4- 7' (z f) 
 
 ik."r / A / v I cT \ ^i / "T" G ( T*i 1 1 * / 6/C 
 
 Now if J./ = ls v \ 7 i-iJ 4) r -, 
 
 4wv, 
 
 * Proc. London Math. Soc. Ser. 2, Vol. 1. 
 
120 FIELD OF A MOVING DOUBLET [CH. 
 
 we find that 
 o. = i [A X '-A X ] 
 
 <M-J*'-*] = 
 
 But 
 
 j, a ' _ p? + " _ ^00-f ' = n '(y.^- m ' (*-)- cV (t - r) 
 
 + c z ct-r) f n- "m -(r{a(t-r)- 
 where J=f -y,/, mssy '- a 
 
 Hence we may write 
 
 a /\ a '^ * .,-., -(27s). 
 
 The electromagnetic field derived from these potentials is due 
 to a moving electric doublet. It should be noticed that we 
 have the relations 
 
 m 
 
 We can write down by analogy the potentials for an electro- 
 magnetic field due to a moving magnetic doublet. They are 
 
 
 where m 
 
 and cr , yg , 7 , ^ , m , w are functions of r. When ot, ft, 7, , m, 
 n are functions of r which are not connected by the relations 
 (276) the potentials (275) can be used to construct an electro- 
 magnetic field which must be regarded as that due to an 
 electric doublet and magnetic doublet which move together. 
 
 43. Electromagnetic fields in which singularities are 
 projected from a moving point or curve and travel with the 
 velocity of light. 
 
 We shall now develop some mathematical analysis of con- 
 siderable interest whose physical significance is not yet fully 
 
VIII] STRUCTURE OF THE AETHER 121 
 
 understood. At first sight it seems appropriate for a discussion 
 of an emission theory of light in which waves in the aether are 
 either produced or guided by small particles which move in 
 straight lines with the velocity c. After further study I have 
 thought it may be useful for a discussion of the question " Has 
 the aether a structure ? " 
 
 This question has already been raised by Sir Joseph 
 Larmor* and Sir Joseph Thomson f. The latter has, indeed, 
 developed a theory in which the aether has a kind of atomic 
 structure of which the elements are Faraday tubesj. In the 
 most recent form of the theory it is assumed that the electric 
 and magnetic forces are zero outside the tubes and that a 
 certain amount of work is performed when one corpuscle 
 crosses a tube of force attached to another. In an application 
 of the present analysis to Sir Joseph Thomson's theory the 
 aim would be to build up his discontinuous electromagnetic 
 fields from electromagnetic fields with certain types of sin- 
 gularities, making use of discontinuous definite integrals. To 
 illustrate the possibility of doing this it will be sufficient to 
 mention the definite integral 
 
 T7 f 2jr r da da] 4?r 
 
 V= --r- -. : + - = - Z > 0, 
 
 Jo \_z + w cos a + ly sin a r J r 
 
 = z < 0. 
 
 The electrostatic field derived from the function V is zero on 
 one side of the plane z and has the character of the field 
 due to a point charge on the other side. It should be noticed 
 that the integrand is a potential function which becomes infinite 
 along the line z 0, x cos a -f y sin a. = 0, and as a varies this 
 line sweeps out the plane of discontinuity of our electrostatic 
 field. 
 
 To generalise this result we must endeavour to solve the 
 
 * Aether and-Matter (1900), p. 188. The question as to whether the aether 
 is continuous or discontinuous is discussed by H. Witte, Ann. d. Phys. (4), 
 Bd. 26 (1908). 
 
 t Presidential Address, British Association Reports, Winnipeg (1909). 
 
 % Recent Researches on Electricity and Magnetism-, Electricity and Matter \ 
 Phil. Mag. Vol. 19 (1910), p. 301, Oct. Dec. (1912). See alsoN. R. Campbell, 
 The New Quarterly (1909); Phil. Mag. Vol. 19 (1910), p. 181. 
 
122 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 general problem of finding electromagnetic fields whose 
 singularities lie on moving curves*. 
 
 A partial solution of this problem may be obtained by con- 
 sidering first of all the field represented by equations (10) 
 of 5. We may obtain a suitable pair of functions a, ft by 
 solving the equations 
 
 dy. 
 
 (279); 
 
 dy dz ~ c 2 dt 
 
 oadft dadff dudff = l_dad^ 
 dx d& dy dy dz dz c 2 dt dt 
 
 for clearly 
 
 - [7 
 L 
 
 dy dz dz dy 
 
 ??Y 4. i^Yl [7MY 4. fMYl _ fa M 4. ?? MY 
 va^/ J LW/ va^y J W a y + a^ a^ ; 
 
 i /aay _ /a^y] ri. /a^y __ /a^yl _ /i a a^ __ a a/sy 
 c 3 Va * J laj J [c 2 l a< ) \dx) J U 2 ae dt a# a^J 
 
 1 ^ M _ ?? MY 
 
 c 2 (dx dt dt die) ' 
 
 Two other equations can be obtained in a similar way and so it 
 follows that if we make a suitable choice of an ambiguous sign 
 which is involved in the definitions of the functions a and 
 y&, the equations (10) will be a consequence of equations (279). 
 A more general electromagnetic field is obtained by 
 multiplying the components of M in equations (10) by an 
 arbitrary function f(a.,j3). Since the components of M are 
 necessarily solutions of the wave-equation it follows that, if 
 g \lf t an expression of the type 
 
 is a solution of the wave-equation (8). 
 
 * The aim must be not only to obtain a complete generalisation of Green's 
 equivalent layer which will be applicable to the case of a moving surface, but to 
 obtain, if possible, an analysis of the electric and magnetic current sheets which 
 are required. Cf. p. 29. 
 
VIII] VELOCITY OF MOTION LESS THAN THAT OF LIGHT 123 
 
 The electromagnetic field which has just been obtained 
 generally has singularities at space-time points for which 
 /(a, ) is zero. Let us write f=u(x, y, z, t) 4- iv (x, y, z, t) where 
 u and v are real when x, y, z, t are real ; then the points for 
 which f = lie on the moving curve, denned by the equations 
 u = 0, v = 0. Now it follows from (279) that / is a solution of 
 the equation 
 
 and consequently 
 
 /8iA 2 fdu\ 2 1 /9A 2 fdv\* fdv\ 2 /dv\* 1 /8v\ 2 
 
 ^(ty) + (fa) &(to) vw + (dy) + (dz) ~c 2 l^y ' 
 
 du dv du dv du dv _ 1 du dv 
 dx dx dy dy dz dz c 2 dt dt ' 
 
 Now let F(u,v) = be the equation of a moving surface which 
 always contains the moving curve, then if 
 
 __dFdv 
 P ~~dudx 
 
 dFdv 
 
 ~~ 0-. 3~ ^ ~ 
 
 du dz 
 
 af8w 
 at- dx ' 
 
 a^aw 
 a 8^' 
 
 aF8v_aF8w 
 
 8w 8y 9 1' 9y ' 
 
 ^dFto^dFdu 
 S ~dudi dvdt' 
 
 we have 
 
 dF* 
 
 dF dF 
 
 SF SF' 
 
 consequently 
 
 ^y /a^- /a^y i/apy 
 
 to) + (d) + \dz) *9\*/' 
 
 This means that the component velocity of the surface F = in 
 the direction of the normal at (x, y, z, t) is less than the velocity 
 of light, it is equal to the velocity of light only in an excep- 
 tional case. Since this is true for any surface that always 
 contains the moving curve it follows that the curve can be 
 
124 SOLUTION OF THE FUNDAMENTAL EQUATIONS [CH. 
 
 regarded as moving with a velocity less than that of light. It 
 should be understood that the curve generally changes in shape 
 as it moves but this is not necessarily the case. 
 
 It often happens that the moving curve /=0 can be 
 regarded as made up of the instantaneous positions of a series 
 of points which move in straight lines with the velocity of 
 light. To see this let us suppose that a function </> (a, /3) can 
 be found such that <f> is a real function of x, y, z, t. It is evident 
 from (279) that 
 
 = 
 
 da das cy dy dz dz c 2 fit dt ' 
 
 and this means that if a point starts from (x, y, z, t) and moves 
 with the velocity of light along a straight line whose direction 
 
 cosines I, m, n are proportional to ?? , ?? , J- , the function 
 
 ox oy oz 
 
 f will remain constant along its path, and consequently if the 
 point once lies on the moving curve /= it will always lie on 
 this curve. It should be noticed that the function </> and its 
 first derivatives with regard to a?, y, z, t all remain constant along 
 the path of the moving point. 
 
 The case in which no such function < exists may be of 
 importance in future developments of the subject; this case 
 has not yet been discussed in any detail. 
 
 Two methods of solving equations (279) are known, but 
 they are not really distinct. In the first method the functions 
 a, ft are defined by equations 
 
 [* - f (> )? + [y - V (, ft)? + - f (, )] 2 = ^ \t - r (a, /3)] 2 | 
 I (a, P)[x-% (a, 0)] + m (, ) [y - rj (a, )] 
 
 + n (a, /3) [z - ? (, 0)] = ?P (, ) [* - T (a, )]) 
 
 ...... (280), 
 
 where f , 77, f, r, Z, m, n, p are arbitrary functions satisfying the 
 relation 
 
 The functions a, / may evidently be replaced by two other 
 functions a', ft' defined by equations such as 
 a'=F(a,p\ ' = (,); 
 consequently we may without loss of generality introduce a 
 
VIII] SPECIFICATION OF THE ELECTROMAGNETIC FIELDS 125 
 
 further relation connecting the functions f, 77, f,r, I, m, n,p. The 
 relation we shall choose is 
 
 When a and j3 are defined in this way we can obtain the simple 
 specifications of two types of electromagnetic fields ; one type 
 is obtained by writing 
 
 /fcflJt *M. <fi ......... (282), 
 
 d(y,z) ~ c d(ac,t) 
 
 and two similar equations. A second type of electromagnetic 
 field may be derived from the potentials* 
 
 _ ISg (a, ) _ mSg(*,j3) } 
 
 X ~PS-QR' y ~ PS-QR 
 
 __ t 
 ~' 
 
 PS-QR' PS-QR J 
 
 where 
 
 (283), 
 
 , , . 
 
 = / ^ + m ^ + n ^ - C 2 5- - 5- (a? - 
 
 9a 9a da ^ da da ^ 
 
 9r 
 5- 
 da 
 
 dn 
 
 It is easy to see that these potentials satisfy the relation (267). 
 To prove that they are wave-functions we remark that 
 
 9 (a, ) 9w 9(,/3) dp 9 (a, 
 '9(a ; , 2 /) + 9^'a^^) + 9 y 8' 9(*, 
 Now it has already been proved that an arbitrary function 
 of a and # multiplied by one of the Jacobians represents a 
 
 * We can also call these potentials L x , L v , L g , A and derive an electro- 
 magnetic field from them by the method of 4. 
 
126 CONJUGATE FIELDS [CH. 
 
 wave-function, hence A x is a wave-function. In a similar way 
 it can be shown that the other potentials are wave-functions. 
 
 It should be noticed that the electromagnetic field specified 
 by the potentials (283) is conjugate to the field given by (282). 
 To prove this we observe that 
 
 a. *, 
 
 ' -- PS-QR mu te + v te ^ 
 and there are similar expressions for A y , A z , <I>. Hence for the 
 electromagnetic field specified by the potentials (283), we have 
 
 , _ 8 (t*, a) d (v, ft) i d (u, a) i 3(^/8) 
 
 ~ + 
 
 The relation (3) is now seen to be satisfied in virtue of two 
 equations of type 
 
 d(x,t) d(z,x) 9(y,t) a(*,y) d(z,t) 
 
 ,> 8(a,/3)8(M,q) 8 (a, ft) 8 (^ a) 
 8(*,0 3(y,jr) + 8(y,0 9(^,4?) "^ 8(^,0 3(,y) 
 
 ...... (284). 
 
 It should be remarked that the vectors E, H in both fields 
 generally become infinite when PS QR = 0. This equation 
 is certainly satisfied when (#, y, z, t) lies on the moving curve 
 defined by the equations 
 
 In some cases this curve may reduce to a moving point, 
 as for instance when f, rj, f, T are independent of /9. 
 
 It is evident that the quantity PS QR is usually zero for 
 space-time points which do not lie on the moving curve (285). 
 If, however, we regard I, ra, n, p as complex functions of the 
 type < (a, /3) + fy (a, P), the equation PS = QR will generally 
 give rise to two distinct equations connecting x, y, z, t, when 
 we equate real and imaginary terms on both sides. Hence 
 all the real singularities that are defined by PS = QR will 
 generally lie on one or a number of moving curves. 
 
 So far we have said nothing about the choice of a suitable 
 pair of roots of equations (280). In general we cannot expect 
 
VIII] THE DETERMINATION OF O. AND ft 127 
 
 them both to be real and it is difficult to lay down rules which 
 will enable us to pick out just one a. and just one ft in all cases. 
 To proceed further we must consider some particular examples ; 
 before we do this, however, it will be worth while to point out 
 that if we assign given complex values to a and & the equations 
 (280) will generally determine two real points x, y, z, t, but 
 in special cases they may give co x space-time points which can 
 _.be. regarded as the consecutive positions of a moving point. 
 Thus we have an interesting specification of the real points 
 in space by means of two complex quantities. If we assign 
 a complex value to ft the corresponding space-time points 
 x, y, z, t generally lie on a moving curve which travels with 
 a velocity not greater than that of light. Hence in the para- 
 metric representation of x, y, z, t in terms of the complex 
 quantities a, ft the loci a = const., ft = const, are generally 
 moving curves which may alter in shape as they move but 
 never travel with a velocity greater than that of light. 
 
 Matters are somewhat different if a or some function of 
 a and ft is always real when x, y, z, t are real. This case will 
 now be illustrated by a particular example. 
 
 Examples. 1. Prove that the ratios of the Jacobians 
 9 (a, ft) 9(0,0) 8(a,0) 
 
 are functions of a and 0. 
 
 2. Prove that the ratio Q/S depends only on a and 0. 
 
 3. Obtain the general solution of equations (279) by taking x, y, a, 
 as new independent variables. 
 
 44. Projection of singularities from a moving point, 
 second model of a corpuscle. 
 
 Let us now suppose that f , 77, f, r are independent of ft and 
 that T = a. We may then define a uniquely by restricting 
 it to be real and introducing the inequalities 
 
 
 
 To obtain a single value of ft we may assume I, m, n, p 
 to be linear functions of ft. Consequently we may put 
 
 - f) + mi(y-i/) + n, (z- )- 
 
128 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 where 1 , l lt etc. are functions of a which satisfy the equations 
 
 l<? + 2 4- n 2 = c'po* ) 
 
 > ............ (288). 
 
 J 
 
 It is easy to verify that the wave-equation is satisfied by 
 a function of type 
 
 thus we have a generalisation of the theorem of 41. 
 
 We shall assume that 1 , ?w , n Q , p Q are real and that some 
 or all of the quantities ^, m lf %, ^ are complex. It is easy 
 to see that if we assign a real value to a and a complex value 
 to ft the corresponding space-time points (a, y, z, t) can be 
 regarded as the successive positions of a point which starts 
 from the point 
 
 * = () V = *(*), *=?()> * = ...... (289) 
 
 and moves with the velocity of light along a straight line 
 through this point. There is clearly just one line through 
 this point for each complex value of (3 and vice versa. If we 
 consider all the points in space at a particular time t we can 
 specify each point uniquely by a real parameter a and a real 
 or complex parameter y3. 
 
 Let us now consider the electromagnetic field which is 
 specified by the potentials 
 
 * = ...... (290), 
 
 where I = y% - 1 , m - ^m l -m , n = ^ ?i , p = ^ p Q and 
 f is an arbitrary function of a and /3. These potentials are 
 derived from (283) by putting Q = 0. 
 
 After a long calculation we find that the component of the 
 electric force along the radius from (f, ij, f, a) to (x, y, z, t) is 
 
 To obtain an electromagnetic field in which there is a 
 constant electric charge associated with the singularity 
 (ft *?> a )> we assume that p^ = p\f^- and that 
 
VIIl] SECOND MODEL OF A CORPUSCLE 129 
 
 ...(291). 
 
 <+A+*| = ffl)'+ (?)'+(' 
 
 tfa 3a 9a \9a/ \da/ V9a/ 
 The expression for the radial electric force then becomes 
 
 _ fc'2_ '2 _ V'2 
 
 r, f J . 
 
 Comparing this with the expression for the radial electric 
 force in the case of an electromagnetic field with a simple 
 singularity (, 77, f, r), we see that there is a constant electric 
 charge 4?r/c associated with the singularity (, 97, f, a). 
 
 It should be mentioned that the second of equations (291) 
 is a consequence of the other equations satisfied by Z , m , n 9 . 
 To prove this we take the axis of x in a direction parallel to 
 the velocity of the singularity (f, 77, f) at time a. We then 
 
 O j\ c* 
 
 have for this instant = ^ = 0. If, moreover, we choose the 
 
 Ctt COL 
 
 axis of y in such a way that n^ 0, we may satisfy the first 
 of equations (291) by writing 
 
 ~ = c cos 0, /! = c sec 0, m^ ic tan 6, I c cos 0, 
 
 da 
 
 m = 0, n = + c sin 0, 
 
 and then it is clear that the second of equations (291) is 
 satisfied. 
 
 Let us now write 1 = c cos 6, m^ = 0, WQ = + c sin 0, 
 ^ = c sec 6, m^ = - ic tan 0, Wj = 0, x-j; = X, y - 77 = F, 
 z %=Z, tr=T; then it is easy to see that if 
 
 we have cos 2 0SS=Ul?, U+U=- 2P. 
 
 Now since /3S = V it follows that the potentials (290) become 
 
 infinite when U ' 0, i.e. when 
 
 Z = F = ^ = T 
 1 MO no 1 
 
 When o is given these equations are satisfied by a point which 
 
 starts at (f, ??, , a) and moves with the velocity of light along 
 
 B. 9 
 
130 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 a straight line whose direction cosines are proportional to 
 1 , w , n . This line makes an angle 6 with the direction of 
 motion of the point (f, ?;, f, a). 
 
 The electric and magnetic forces in the electromagnetic 
 field derived from the potentials (290) are generally infinite 
 for U = and so our field possesses a number of singular points 
 which are projected from the moving point (, ??, f, a) and travel 
 along straight lines with the velocity of light. It should be 
 remarked, however, that if we retain only the real parts of the 
 potentials (290), the projected singularities disappear as soon 
 as the singularity , 17, f, a moves in a straight line with uniform 
 velocity and I, m, n are independent of a. The field then 
 becomes identical with that derived from Lienard's potentials. 
 To prove this we shall show that, on the above assumptions, 
 , the field derived from the potentials 
 
 ,1/=R, c&^R-p- ......... (292), 
 
 is everywhere null, R being used to denote the real part of a 
 quantity following it. In the first place we remark that 
 we now have 
 
 Hence 
 
 (* + 8 g - ?' )} 
 
 U d , SS rt 9 
 
 pto log ^- 2 
 
 us. u a 
 
 Now since U + U + 2P = 0, it follows that U/P is a function of 
 J7/Z7 and so A x , A y } A z , - - 3> are the derivatives of a single 
 
VIII] SECOND MODEL OF A CORPUSCLE 131 
 
 function, consequently the electromagnetic field derived from 
 these potentials is everywhere null. 
 
 Summing up our results we can say that when the 
 conditions (291) are satisfied, the electromagnetic field derived 
 from the potentials (290) contains a point charge which moves 
 with a velocity less than that of light ; attached to this point 
 charge there is a certain curve which becomes the locus of 
 a series of moving point singularities whenever its form differs 
 in any portion from a straight line or its direction changes. 
 The form of the curve at any instant is subject to the condition 
 that the points of the curve can be regarded as having been 
 projected from the moving charge at different instants, the 
 direction of projection being partially determined by the law 
 cos 6 = v/c where v is the velocity of the point charge, and 9 is 
 the angle between the direction of projection and the direc- 
 tion of motion of the point charge. 
 
 We may now obtain a new model of a corpuscle by con- 
 sidering an aggregate of elementary fields of the type just 
 described, the point charges and exceptional curves being 
 nearly coincident. If we write de for the charge associated 
 with one of the elementary fields we may obtain a field in which 
 the electric and magnetic forces are finite by a suitable process 
 of integration. According to this idea a corpuscle has a kind 
 of tube or thread attached to it. When the motion of the 
 corpuscle changes a wave or kink runs along the thread; the 
 energy radiated from the corpuscle spreads out in all directions 
 but is concentrated round the thread so that the thread acts as 
 a guiding wire. This theory of radiation is in some respects 
 similar to that given by Sir Joseph Thomson in his theory 
 of the Rontgen rays*. It is in accordance with his idea that 
 the energy may be concentrated round certain points of the 
 wave-front. 
 
 The following figure indicates roughly the changes in 
 the form of a tube which always lies in one plane and is 
 attached to a corpuscle performing a simple harmonic motion ; 
 it is seen that a type of progressive wave travels along the tube. 
 
 * Electricity and Matter, London (1904) ; Phil. Mag. Vol. 19 (1910). 
 
 92 
 
132 
 
 FIELDS WITH MOVING SINGULAR CURVES 
 
 [CH. 
 
 In Sir Joseph Thomson's theory of the Rontgen rays the 
 kink in the tube of force becomes longer and longer as it 
 recedes from the charge. A similar remark applies to the 
 
 oscillations of the thread attached to our point charge. This 
 phenomenon may be due entirely to the fact that the tube of 
 force and thread extend to infinity. If we suppose that the 
 tube or thread does not extend to infinity but ends at some 
 other point charge, the circumstances of the motion will be 
 different. If in this case we treat the thread as a singular 
 line of an electromagnetic field and suppose that it is given by 
 an equation of the form 
 
 /(,* o 
 
 where a and ft are functions which satisfy (279), we must 
 conclude that there is no function of type F (a, ft) which is 
 a real function of x, y,z,t\ for if this were the case the moving 
 thread would be the locus of points travelling in straight lines 
 with the velocity of light and would consequently extend to 
 infinity. 
 
 Electromagnetic fields with moving point charges joined by 
 singular curves which do not extend to infinity have not yet 
 been obtained, but- 1 -think there is some hope of deriving them 
 by the general methods of 43, when the quantities or, ft are 
 bo 
 
 fV. /M 
 
 Examples. 1. Discuss the properties of the electromagnetic fields 
 that can be derived from the potentials 
 
 
 s 
 
 is 
 
 respectively and determine the lines of electric and magnetic force. 
 
VIIl] VELOCITY GREATER THAN THAT OF LIGHT 133 
 
 2. Prove that if a and |3 are defined as in 44, the electromagnetic 
 field specified by the equations (282) is conjugate to the field specified by 
 the potentials (290), or to the field specified by Lienard's potentials. 
 
 45. Electromagnetic fields with singularities moving 
 with velocities greater than that of light. 
 
 Some of the preceding analysis holds and provides us with 
 solutions of Maxwell's equations when the velocity of the 
 primary singularity (f , 77, f, r) is greater than that of light, but 
 in the case of a field specified by potentials of type (266) 
 a transition from a velocity less than that of light to a velocity 
 greater than that of light does not seem to be physically pos- 
 sible on account of the occurrence of infinite values of the 
 electric and magnetic forces in the critical case. Moreover, it 
 is difficult in the general case to give a rule which will enable 
 us to pick out just one root of the equation (265). An interest- 
 ing type of field may, however, be obtained by a process of 
 summation over some of the roots of th equation*. 
 
 The case of infinite velocity is of some interest, for then 
 we obtain electromagnetic fields with singularities along a 
 fixed curve at a given instant of time. The following example 
 indicates that the case in which the primary singularities are 
 imaginary may be associated with another case in which they 
 are real. 
 
 Consider the two equations 
 
 (x a cos a) 2 + (y a sin a) 2 + z* = c 2 2 , 
 
 z ~ 
 
 and write in analogy with (263) 
 
 v = a cos a (y a sin a) a sin a (x a cos a)=a(y cos a x sin a), 
 
 v 1 = ia sinh ft (z ia cosh ft) t'acosh ft (ct ia sinh ft) 
 
 = ia (z sinh ft - ct cosh ft). 
 We evidently have 
 
 4j/ 2 = 4a 2 (x* + f) - (x* + f + z*- c*t 2 + a 2 ) 2 , 
 41/j 2 = 4a 2 (* - c 2 * 2 ) + (x 2 + f + z* - c*t* - a 2 ) 2 , 
 hence i/ 2 + v? = 0. 
 
 * A more complete discussion is given in G. A. Schott's Electromagnetic 
 Radiation. 
 
134 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 It is easy to verify that the function 
 
 is a wave-function and that the functions a, /3 are suitable for 
 obtaining electromagnetic fields by the method of 5 ; they are, 
 in fact, the functions considered in 36. 
 
 46. Second solution of the fundamental equations. 
 The fundamental equations (279) are also satisfied when 
 the functions (a, ft) are defined by the relations 
 
 l(a,ft)x + M (, 0) y+N(a, ft)z - c*P (a, ft) t + G(a,ft) = 
 
 ......... (293) 
 
 where I, L, etc. are arbitrary functions satisfying the relations 
 p + m + w 1 = c*p\ L 2 + M 2 + N 2 = c 2 P 2 , 
 
 lL + mM+nN = c*pP ............... (294). 
 
 If now 
 
 dl dm dn d da 
 \ = # + y =- -4-^5--c a <^ + /, 
 da. * da da da da 
 
 dl dm dn d da 
 
 dL dM dN dP , dG 
 = x-^- + y -^- + z-^ -- 0-^-5-4-3--, 
 da y da da da da 
 
 dL dM dN dP dG 
 
 we can derive an electromagnetic field from the potentials 
 
 Xtar - 
 .:.... (295) 
 
 (296). 
 
 4.i*^-, -te^cf 3 ^! 4,-;-^-, *^ 
 
 X-BT /AV X-CT yU-l^ 
 
 where / is an arbitrary function of a and /, and 
 
 az m dN dp 
 
VIII] SECOND SOLUTION OF THE FUNDAMENTAL EQUATIONS 135 
 
 To verify that these potentials satisfy the wave-equation and 
 the relation (267), it is sufficient to remark that 
 
 d (a, ft) dN 9 (a, 0) dP 9 (a, ft) 
 9/3 d (x, y) 9/3 d (x, z) 9/3 3 (x, t) 
 
 , . 
 - 
 
 c a/3 d(*,t) c dp 9 ( y , t) - 9 9 (y, *) 
 
 When similar expressions are obtained for A y , A z , <I> it is clear 
 that the relation (267) is satisfied. Other types of electro- 
 magnetic fields are obtained by writing a instead of /3 in 
 equation (296) or by writing I, m, n, p in place of Z, M, N, P 
 in (295) and (296). If A x ', A y ' t A g ' t <' are the potentials 
 obtained in either of these ways we have clearly 
 
 A x A x ' + Ay Ay +A Z A Z ' -<&<$>' = ......... (298). 
 
 I have noticed that this relation is often satisfied by the 
 potentials of two conjugate fields. 
 
 Another type of electromagnetic field may be obtained 
 by the method of 5. 
 
 Some particular cases of the preceding theorems may be 
 deduced by contour integration. To illustrate the method let 
 us suppose that I, m, ... L y M,... are functions of a parameter a 
 which satisfy the relations (294). If we regard these quantities 
 as independent of x, y, z, t, the contour integral 
 
 JL [ 
 
 27Ti J 
 
 Ix + my + nz c?pt + g 
 will represent a wave-function. Now let us suppose that the 
 contour encloses only one root of the equation 
 
 xl (a) + ym (a) + en (a) - cHp (a) + g (a) = . . .(299), 
 and that the numerator is finite and single- valued within the 
 contour and on its boundary, then by Cauchy's theorem the 
 value of the integral is generally 
 
 ) ........................ (300) 
 
 where = xL (a) + yM (a) + zN (a) - cHP (a) + G (a). . .(301) 
 and X is defined in the same way as before. 
 
 We have then the result that the function (300) satisfies 
 the wave-equation. This is a particular case of the general 
 
136 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 theorem of 43 and is a generalisation of a theorem due to 
 Forsyth*. In Forsyth's work the functions L, M, N t P are 
 assumed to be the derivatives of /, ra, n y p with regard to a. 
 The functions a, /3 are evidently particular cases of the func- 
 tions that have already been denned and so may be used to 
 construct an electromagnetic field by the method of 5. An 
 interesting electromagnetic field may also be derived from the 
 potentials 
 
 A.-*, A,-*. A.-*&. * = ^>...(302); 
 
 it is easy to verify that the relation (267) is satisfied. The 
 case in which I, m, n, p, g are all real functions of a is un- 
 interesting because then our potentials become infinite for 
 oo 3 space-time points which lie in oo * planes. When, however, 
 I, m, n, p, g are complex functions of the type <f> (a) 4- ty (a), 
 the singularities of the electromagnetic field generally lie on 
 a moving curve. 
 
 It should be remarked that when a and /9 are defined by 
 the equations (299) and (301) a function of the type 
 
 satisfies the wave-equation, and at the same time satisfies the 
 differential equation 
 
 This is a generalisation of a theorem due to Forsyth f and 
 JacobiJ. A solution of Maxwell's equations may be derived 
 from the potentials 
 
 ...(303) 
 
 if + 
 
 80 82 da r da 
 
 , dl dm dn dp 
 
 * Messenger of Mathematics, Vol. 27 (1898), p. 138. The theorem is 
 obtained in another manner by Forsyth. 
 + Loc. cit. 
 J Werke, Bd. 2, p. 208. 
 
VIII] STATIONARY SINGULAR CURVES 137 
 
 One way of satisfying these equations is to put 
 
 1 =1, m = m, n-o = n, p Q = p. 
 
 An interesting type of wave-function may be obtained by 
 a generalisation of a method due to Schottky *. 
 Let 
 a o ( a ) = / J ( a ) da, b (a) = / ra (a) da, c (a) = / w (a) da, 
 
 do () = j> (a) da, e (a) = /# (a) da, 
 then H = #a (a) + yb (a) + 2C (a) - cHd Q (a) + e (a) 
 is a wave-function whose derivatives with regard to x, y, z, t 
 are all functions of the single variable a. The function a is 
 supposed to be defined by equation (299). 
 
 Example. If =1 a, w=s + a, n=i( a), cp = l+sa, <7 = a 2 , where s 
 is a constant, the function X vanishes when (#, y, 2, f ) lies on the moving 
 curve 
 
 1_ S 2 352_ V 2 25 s 2 ~V 2 -2* 
 
 A point for which v is constant moves in a straight line with the 
 velocity of light. 
 
 47. A wave-function with a fixed curve of singu- 
 larities. 
 
 Let T be defined in terms of x, y, z by the equation 
 
 ar = xp (T) + yq (T) + zr (T) 
 where p a + c + r 2 = and a is a constant. 
 
 Let v = a - xp (T) - yq' (T) - zr (T), 
 
 then H = - f(0, T) is a wave-function (. 
 
 A solution of Maxwell's equations may be derived from the 
 potentials 
 
 * BerZin. Sitzungsberichte (1909), p. 1152. 
 
 t This result was derived from a theorem given by Prof. Forsyth, loc. cit. 
 If we put 0=/3, r=a the functions a, /3 can be used to obtain an electromagnetic 
 field by the method of 5. These functions are, of course, particular cases of 
 the functions defined at the beginning of 46. 
 
138 FIELDS WITH MOVING SINGULAR CURVES [CH. 
 
 To obtain a real electromagnetic field we must retain only 
 the real parts of these expressions. 
 It can be shown by putting 
 
 p = P l (u, v) -f i'P 2 0, v), q = Qi (u, v) + iQ 2 O, v), 
 r R^ (u, v) + iR 2 (u, v), a = AI (u, v) + iA z (u, v), 
 
 that the electromagnetic field has generally a fixed curve of 
 singularities. In the special case when 
 
 p = l-T 2 , 2=2r, r = i(l + T 2 ), A 1 = h ) A 2 = k, 
 the fixed curve is the circle a? + 2 2 = \ k 2 , y = \ h. 
 
 48. Cylindrical wave-functions with moving singu- 
 larities. 
 
 If we define r in terms of x, y by the equation 
 [* - f (r)J + [y - r, (r)f = <? (t - rf, 
 and define I (T), ra (T), p (T), so that 
 
 the function v = I (x - % ) + m (y tj) - c?p (t - r) 
 
 is such that ~>rf( T ) 
 
 is a wave-function. 
 
 In particular, if = 77 = 0, 1 = 1, m = i, p = Q we obtain 
 Poisson's wave-function 
 
 Va? + iy 
 
 Another interesting result is that if F (x, y, t) is a homo- 
 geneous function of degree ^ which satisfies the wave-equation, 
 the function 
 
 also satisfies the wave-equation. 
 If <T = ?(T)(x-t) + r,'( 
 
 the function 
 
VIIl] EXAMPLES 139 
 
 satisfies the wave-equation only when f ' 2 + rj' 2 = c 2 . When 
 
 iz is written in place of ct these wave-functions can all be 
 regarded as solutions of Laplace's equation. 
 
 EXAMPLES. 
 1. Let 
 
 '= 
 
 where T is defined in terms of #, y, z, t by means of equation (265) and the 
 inequality r^t. Prove that if an electromagnetic field (E, H) is such 
 that 
 
 I8r/l dr _ .9r ,, .dr ,. \ 8r /3r 8r ,- 3r 
 
 with similar equations, where M=HiE, then an electromagnetic field in 
 the variables #, y, 2, if can be found such that we have identically 
 
 (*, y) + tc JT,rf (ar, *) 
 
 where c?(y, 2) denotes dybz-dzty and c&c, S^7, etc. are two independent 
 sets of increments of the variables. 
 
 2. Prove that iff'(r) is always positive the variable t increases with t. 
 Show also that if (|, ij, , f) correspond to (|, ?;, ^, r), the point (|, ^, ^) 
 moves along a curve T with a velocity less than that of light and that the 
 velocities at two corresponding points of T and f are the same in magnitude 
 and direction. 
 
 3. Prove that the conditions imposed upon the electromagnetic field 
 in Example 1 are all satisfied in the case of the field specified by the 
 potentials (266). Hence show that the transformation transforms the 
 field of an electron moving along the curve T into the field of an electron 
 moving along the curve T. 
 
 4. Prove that the conditions of Example 1 are also satisfied for any 
 electromagnetic field of type (282) where a and /3 are defined as in 44. 
 Hence show that a field of this type is transformed into another field 
 of the same type associated with the curve T. 
 
140 FIELDS WITH MCmNG SINGULAR CURVES [CH. VIII 
 
 5. Prove that the conditions of Example 1 are equivalent to only two 
 conditions which imply that the field (E 9 H] is conjugate to any electro- 
 magnetic field of type (282). 
 
 6. Prove that electromagnetic fields of types (302) and (303) can be 
 transformed into fields of the same general types with the aid of the 
 transformation 
 
 2/=y+im(s)f(s}ds, t = t + i p (*)/() <fo. 
 
 7. If r be defined in terms of #, y, z, t by the equations 
 
 ft r)P+0-7(a, ft r)] 2 + 1> - (a, ft r)] 2 = c 2 [* - (a, ft r)] 2 , 
 
 it satisfies the partial differential equation 
 
 8. If # 
 
 and r ~ a 
 
 the function [j7 2 +y 2 + (z a) 2 ] 
 
 satisfies Laplace's equation. 
 
 9. Prove that if in the last example we write 
 
 and integrate with regard to u between and 2?r we can obtain a potential 
 function which is zero outside the tube | < + ty | k. 
 
 10. Particles are projected in certain directions from the different 
 positions of a moving electron and travel along straight lines with the 
 velocity of light. Prove that if the law, according to which the direction 
 of projection varies with the velocity of the electron, be suitably chosen 
 the particles will at each instant form a line of electric force in the 
 electromagnetic field due to the moving electron. 
 
CHAPTER IX 
 
 MISCELLANEOUS THEORIES 
 
 49. KirchhofFs formula and its extensions. 
 
 An important solution of the wave-equation is embodied in 
 KirchhofFs formula* which is usually interpreted as the mathe- 
 matical expression of the principle of Huygens. This formula 
 has been extended by Lovef and MacdonaldJ so as to give 
 a representation of an electromagnetic field outside a surface 
 in terms of the electric and magnetic forces tangential to 
 the surface. In Macdonald's formula it is the time derivatives 
 of E and H that are so expressed. Tonolo has given a 
 formula in which E and H are expressed in terms of their 
 surface values. The formulae are given in examples 3 5 at 
 the end of this chapter. 
 
 When the surface is a sphere Kirchhoffs formula reduces to 
 the formula of Poisson|| (Ex. 5) which enables us to find a wave- 
 function which satisfies the conditions 
 
 */ \ ^u . . 
 
 * /(* v> *)> ft =9 ( x > y> z )- 
 
 Poisson's formula may be used to derive the theorem If that the 
 mean value of a wave-function u over a sphere of radius CT at 
 time t is equal to the mean value of u at the centre of the 
 
 * Berlin. Ber. (1882), p. 641; Wied. Ann. Bd. 18 (1883) ; Get. Abh. t. 2, 
 p. 22. Simple proofs of the formula have been given by Beltrami, Rend. Ace. 
 Line. Rom. (5), t. 4 (1895) ; Larmor, Proc. London Math. Soc. Ser. 2, Vol. 1, p. 1 ; 
 Love, Ibid. p. 37 (1903); Lamb, Hydrodynamics, 2nd edition (1906), p. 477; 
 H. A. Lorentz, The Theory of Electrons, p. 233 ; E. Laura, II Nuovo Cimento 
 (1913). 
 
 f Phil. Trans. A, Vol. 197 (1901). 
 
 J Electric Waves, p. 16; Proc. London Math. Soc. Ser. 2, Vol. 10 (1911), 
 p. 91; Phil. Trans. A, Vol. 212 (1912), p. 295. This theorem gives an 
 analytical specification of a generalised Green's equivalent layer. See p. 29. 
 
 Annali di Matematica, Ser. 3, t. 17 (1910). 
 
 || The details of the calculation are given by Love, loc. cit. A simple proof 
 of Poisson's formula is given by Lamb, loc. cit. p. 471. 
 
 H Cf. Eayleigh's Sound, appendix, and H. Bateman, Amer. Journ. (1912), 
 where some other theorems of a similar kind are given. 
 
142 MISCELLANEOUS THEORIES [CH. 
 
 sphere during the interval t r to t + r. The function u is 
 subject to the conditions in Kirchhoffs theorem. 
 
 When the function u is independent of z, Poisson's 
 formula reduces to Parseval's formula for a cylindrical wave- 
 function. Volterra* has extended Parseval's formula so as 
 to obtain a two-dimensional analogue of Kirchhoffs formula. 
 His formula indicates that the propagation of cylindrical 
 waves is essentially different in character from that of 
 spherical waves. In the three-dimensional case the value of 
 a wave-function u (x, y, z, t) at a point (x, y, z) at time t 
 
 is completely determined by the values of u and over a 
 
 ot 
 
 concentric sphere of radius cr at time t r. In the two- 
 dimensional case, on the other hand, the value of u (x, y, t) at 
 a point (#, y) at time t is not determined by the values of u and 
 
 j5T over a concentric circle at time t r. To find u (x, y, t) we 
 
 must know the values of u and ^- over a series of such circles 
 
 ot 
 
 in which the radius cr varies from to some other value CTJ. 
 The essential difference between the two cases may be attri- 
 buted to the fact that in the three-dimensional case the wave- 
 
 1 / T\ 
 
 function for a source is of type -f[t J, while in the two- 
 dimensional case it is of type I fit cosh^J and a wave 
 
 does not leave the region undisturbed after it has passed, but 
 has a tail or residue (. 
 
 When u is a periodic function of t, Kirchhoff's formula may 
 be replaced by the simple formula of HelmholtzJ. In this case 
 there is an analogous formula for cylindrical wave-functions, the 
 function K Q (ipk) taking the place of e ikr /r. 
 
 50. Green's Functions. 
 
 The solution of a problem in which a periodic wave-function 
 is to be determined from a knowledge of its behaviour at 
 
 * Acta Math. t. 18 ; Lectures at Clark University (1912), p. 38. 
 t See Lamb's Hydrodynamics, p. 474. 
 
 J See also J. Hadamard, Butt, de la Societe math, de France, t. 28 (1900), 
 p. 69 ; J. Larmor, Proc. London Math. Soc. (2), Vol. 1 (1903), p. 13. 
 
ix] GREEN'S FUNCTIONS 143 
 
 certain boundaries can be made to depend on that of an 
 auxiliary problem, viz. the determination of the Green's 
 function*. 
 
 Let G (x,y,z\ x l} y lt z^) be a solution of Aw + &u = with 
 the following properties: It is to be finite and continuous, 
 as also its first and second derivatives, in a region bounded 
 by a surface S, except in the neighbourhood of the point 
 ( x i> y\> &i)t where it is to be infinite like cos &r/47rr, when 
 r -+- 0. At the surface S, G satisfies some boundary condition 
 
 such as (l)w=0 or (2)^=0. 
 
 Adopting the notation of Plemelj f and KneserJ we shall 
 denote the values of a function <f> (%, rj, f) at the points (x, y, 2), 
 (i, 2/i, z\) respectively by </> (0) and <#>(!). The Green's function 
 is then denoted by the symbol 0(0, 1). The importance of the 
 Green's function depends chiefly on the following theorem. 
 
 Let (f) be a solution of 
 
 A</> + & 2 </>+/O,2/,s) = 0, 
 
 which is finite and continuous, together with its first and second 
 derivatives, through the interior of the region and satisfies the 
 same boundary condition as G(Q, 1), then 
 
 This theorem is proved by applying Green's theorem to the 
 region between a small sphere 2, whose centre is at (sc lt y lt z-^ t 
 and the surface 8. For since 
 
 -//(*-* 
 
 we obtain the required relation by making 2 - and using the 
 boundary conditions. 
 
 * This function was first used by Green in the solution of a problem of 
 electrostatics, Essay on the application of mathematical analysis to the theories 
 of electricity and magnetism, Nottingkam (1828) ; Math. Papers, p. 31. 
 
 t Monatsheftefiir Math. u. Phys. (1904) and (1907). 
 
 $ Die Integralgleichungen und ihre Anwendung in der mathematischen Physik, 
 31, Brunswick (1911). 
 
144 MISCELLANEOUS THEORIES [CH. 
 
 If g (2, 0) is the Green's function for the same boundary 
 condition as (2, 0) but for k = <r, we must also surround the 
 point (2) by a small sphere when we apply Green's theorem 
 with $ (0) = g (2, 0). We then obtain the equation 
 
 g (2, 1) = G (2, 1) - (k* - * 2 )ffjg (2, 0) G (0, 1) dxdydz. 
 
 This important relation indicates that #(2,1) is the solving 
 function of an integral equation of which G (0, 1) is kernel and 
 vice versa. The theory of integral equations tells us that when 
 G (0, 1) is given there may be certain singular values of er* 
 for which #(2, 1) is not finite. These are the values of o- 2 for 
 which the homogeneous integral equation 
 
 </> (1) = (<?- - &)<t> (0) G (0, 1) dxdydz 
 
 possesses a continuous solution <j> (0) which is different from 
 zero. Formula (2) indicates that for such values of <r 2 the 
 differential equation A$ + & 2 c/> = possesses a solution satisfy- 
 ing the boundary condition and the other conditions imposed 
 on <. The solutions of this type are of great importance in the 
 theory of sound and have been discussed by many writers*. 
 
 If we put /(O) = (<r 2 & 2 )#(0, 2) and proceed as before, 
 Green's theorem gives 
 
 g (1, 2) = 6 (2, 1) -(If- ) fjfg (0, 2) G (0, 1) dxdydz. 
 
 Putting a k and comparing this with the previous equation 
 we get 
 
 g (1,2) = fir (2,1). 
 
 Hence the Green's function is a symmetric function of the 
 coordinates of the points 1, 2. When the boundary condition is 
 
 ^\ 
 
 ^- = this result is equivalent to Helmholtz's theorem f. 
 
 Since G is a real symmetric function when k = it follows 
 from the general theory of integral equations that there is 
 
 * See especially Lord Kayleigh , Theory of Sound, Vol. 2 ; Pockels, Die partielle 
 Differentialgleichung Aw+fc% = 0; A. Sommerfeld, Encyklopadie der Math. 
 Wiss. Band n. 1, Heft 4, p. 540. 
 
 t Cf. Kayleigh's Sound, Vol. 2, p. 131. 
 
ix] GREEN'S FUNCTIONS 145 
 
 at least one real singular value of cr 2 ; that all the singular 
 values are positive may be deduced at once from the 
 equation 
 
 cr" 1 1 1 </> 2 dxdydz = - 1 1 1 $A$ dxdydz 
 
 777 
 
 ^ + hr- + ^ dxdydz. 
 bx) \by) \dzj J 
 
 The Green's function is usually obtained in practice by finding 
 a suitable expansion in terms of elementary solutions of the 
 equation A?/ + k z u = 0. This method is explained in Heine's 
 Kugelfunktionen and many examples of Green's functions are 
 given for the case k = 0. The general case has been discussed 
 at length by A. Sommerfeld* who also obtains a number of 
 definite integrals which represent Green's functions. These 
 expressions lead to interesting generalisations of Fourier's 
 theorem. 
 
 The problem of electrical oscillations in a cavity has been 
 discussed by Weylf. With the aid of a generalisation of the 
 Green's function, viz. a Green's tensor, he obtains a number 
 of inequalities satisfied by the periods of vibration. 
 
 The Green's function for the equation Aw + k*u = can 
 theoretically be found when the corresponding Green's function 
 for the equation Aw = is known. Considerable progress has 
 been made in the theory since the appearance of Heine's work 
 and so a few references to recent literature will be useful J. A 
 
 * Phys. Zeitschr. Bd. 11 (1910), p. 1087 ; Jahresbericht der deutsch. math. 
 Verein, Bd. 21 (1913). 
 
 t Math. Ann. Bd. 71, p. 441 ; Crelle, Bd. 141 (1912). 
 
 For the determination of special Green's functions see E. W. Hobson, 
 Cambr. Phil. Trans. Vol. 18 (1899), p. 277; H. M. Macdonald, Ibid. p. 292, 
 Proc. London Math. Soc. Vol. 26 (1895), p. 161 ; A. G. Greenhill, Proc. Cambr. 
 Phil. Soc. Vol. 3 (1880); J. Dougall, Proc. Edinburgh Math. Soc. (1900); 
 H. S. Carslaw, Ibid. (1912), Proc. London Math. Soc. (2), Vol. 8, p. 365; C. W. 
 Oseen, Arkiv for mat. Bd. 2; C. Neumann, Ldpziger Berichte, Bd. 58 (1906), 
 Bd. 62 (1910) ; W. Burnside, Proc. London Math. Soc. Vol. 25 (1894), p. 94. 
 For the general theory H. Poincare, Eend. Palermo, t. 8 (1894), p. 57; 
 S. Zaremba, Ibid. t. 19 (1905) ; E. R. Neumann, Studien iiber die Methode 
 von C. Neumann und G. Eobin zur Losung der beiden dwertaufgaben der 
 Potentialtheorie, Leipzig (1905) ; D. Hilbert, Gdtt. Nachr. f 1904) ; M. Mason, 
 Newhaven Math. Colloquium (1910) ; E. Picard, Ann. de VEcole Normale (1906), 
 p. 509. 
 
 B. 10 
 
146 MISCELLANEOUS THEORIES [CH. 
 
 good account of the developments up to 1900 is given in 
 Sommerfeld's article in the Encyklopddie der Mathematischen 
 Wissenschaften. 
 
 51. The transformation of the electromagnetic 
 equations. 
 
 The transformations which can be used to transform any 
 solution of the wave-equation into another solution or any 
 electromagnetic field into another belong to a group which 
 is characterised by a relation of the form* 
 
 dx'* + dy' 2 + dz">- c*dt'* = A, 2 (dx 2 + dy z + dz* - c 2 dt z ). 
 The linear transformations belonging to this group are of great 
 importance in the modern theory of relativity f ; two of the 
 non-linear transformations have been mentioned in 13. 
 
 In addition to these transformations there are other trans- 
 formations, involving arbitrary functions in their specification, 
 which can be applied to certain types of wave-functions, and to 
 certain types of electromagnetic fields. There are often two 
 families of wave-functions to which a given transformation can 
 be applied, when the transformation is of a suitable character ; 
 each of these families may be defined by a linear relation which 
 exists between the wave-function and its derivatives, sometimes 
 between the derivatives alone. Some idea of the theory may be 
 derived from the examples. It also happens that there is often 
 a family of electromagnetic fields to which a given transforma- 
 tion can be applied and this family is defined by means of two 
 linear relations between E and H, which can be interpreted to 
 mean that the field is conjugate to some definite electromagnetic 
 field or family of electromagnetic fields determined by the 
 transformation. In some cases these last fields are self-con- 
 jugate and the transformation is applicable to them also. 
 
 * H. Bateman, Proc. London Math. Soc. Ser. 2, Vol. 7 (1909), Vol. 8 (1910) ; 
 E. Cunningham, Ibid. Vol. 8 (1910). 
 
 t For this see A. Einstein, Ann. d. Phys. Bd. 17 (1905) ; Laue, Das 
 Relati'Oitdtsprinzip, Brunswick (1911); E. Cunningham, British Association 
 Reports (1911) ; H. Minkowski, Gott. Nachr. (1908) ; E. B. Wilson and G. N. 
 Lewis, Proc. Amer. Acad. of Arts and, Sciences, Vol. 48 (1912), p. 389 ; J. Ishiwara, 
 "Bericht iiber die Belativitatstheorie," Jahrbuch der Radioaktivitdt und 
 Elektronik,Edi. 9 (1912), pp. 560648; L. Silberstein, The Theory of Relativity , 
 Macmillan and Co. (1914). 
 
IX] TRANSFORMATIONS 147 
 
 The fact that the condition of conjugacy between two 
 electromagnetic fields often implies the existence of one or 
 more transformations depending on arbitrary functions, may 
 be regarded as of some philosophical interest. 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. Show that the most general periodic solution ^e^t (valid for all 
 space outside a given closed surface) of the wave-equation is 
 
 where P and Q are arbitrary functions, r is the distance from the element 
 of surface dS to the point where is estimated, and \//> is the angle between 
 r and the outward drawn normal. Show further that the necessary and 
 sufficient condition that the value of |, given by the same analytical 
 
 expression, should vanish for points inside the surface, is that P= ~- . 
 
 (Cambr. Math. Tripos, Part II, 1904.) 
 
 2. Let Q be a function which satisfies the wave-equation and is such 
 that its differential coefficients of the first order are continuous functions 
 of x, y, 2, t within a region bounded by a closed surface S. If either O or 
 
 5 be given for points on the surface S there is only one function Q which 
 on 
 
 reduces to a given function/ (x, y, z) for t = t Q . 
 
 (A. E. H. Love, Proc. London Math. Soc. Ser. 2, VoL 1, p. 42 ; 
 J. Hadamard, Bull, de la Soc. Math, de France, t. 28 (1900).) 
 
 3. If throughout a specified region of space and a specified interval of 
 time u and its differential coefficients of the first order are continuous 
 functions of .r, y, z and of t, if also the differential coefficients of the 
 
 9 2 w d z u 
 second order such as ~- 2 , ^ are finite and integrable, then a solution of 
 
 the equation 
 
 y, z, t= 
 which is valid for this region is given by the formula 
 
 where r 2 =(#-.r ) 2 + (#-,yo) 2 -K 2:< - >2; o) 2 > n denotes the normal to dS drawn 
 into the specified region and the integration is taken throughout this 
 
 102 
 
148 MISCELLANEOUS THEORIES [CH. 
 
 region and over its boundary. The function a- is supposed to be finite and 
 integrable and a quantity within square brackets is calculated at time 
 
 t=t --. (G. Kirchhoff.) 
 
 c 
 
 4. If an electromagnetic field is such that the specified region does 
 not contain any charges or convection currents and M=H+iE, the value 
 M of the vector M at (# , #o> b, *o) is given by the formula 
 
 47r M x = f f{[M r ] cos nx - [M n ] cos rx - [ M x ] cos rn} ^ 
 + - x / / {[ M r ] cos nx [M n ] cos rx [M x ] cos rn 
 +t [M e ] cos Tiy - i [M v ] cos 712} . (A. Tonolo.) 
 
 5. Prove that in the same circumstances 
 
 where a=ft[J/,] v [M v ], etc. and (X, /*, v) are the direction cosines of the 
 normal drawn into the region bounded by S. 
 
 (H. M. Macdonald.) 
 
 6. If u is a wave-function independent of z and periodic in t like e ikt 
 
 7. A wave-function which satisfies the conditions 
 
 u=f(x,y,z) ~=g(x,y )Z ) 
 is given by the formula 
 
 where /, g denote the mean values of /, g respectively over the surface of 
 a sphere of radius ct having the point #, y, z as centre. 
 
 (S. D. Poissori.) 
 
 8. If u satisfies ^-5 + 5-= = or, and has finite second derivatives within 
 ox* vy* ot 
 
 a suitable domain 
 
 where 
 
IX ] EXAMPLES 149 
 
 and a- denotes the area within the circle cut out on the plane T= t by the 
 cylinder 
 
 (Parseval and Volterra.) 
 
 9. Prove that if a transformation of variables from (#, y, z, t} to 
 (#, y, s, t) is such that 
 
 - c 2 di 2 = \ 2 (dx 2 + dy 2 + cfe 2 - c 2 dt 2 } + (Idx + mdy+ndz - cpdt) 
 
 where l 2 + m 2 +n 2 =p 2 , J 2 + m 2 +V == ? > o 2 ; ^ can ^ used to transform an 
 electromagnetic field (#, H] into another electromagnetic field (E, H) 
 with an identical relation of the same type as that used in Ex. 1, Ch. vin, 
 if the two conditions embodied in the relation 
 
 M x (mn - m n) + M y (nl - 
 
 + iMy (mpo - m p) + iM a (np Q - n^p) = 
 are satisfied. 
 
 Prove that the conditions can also be thrown into the form 
 
 y i (m 
 =1 Q (Uft+m 
 
 and similar equations. 
 
 10. In the last example if I =lj mQ=m, no*=n, p = p, the conditions 
 are satisfied if Poynting's vector is in the direction (I, m, n). Show, in 
 particular, that the transformation 
 
 can be applied to an electromagnetic field in which Poynting's vector is 
 along the radius from the origin and that in the resulting electromagnetic 
 field Poynting's vector is parallel to the axis of z. Apply the transformation 
 to the electromagnetic field derived from the functions a, /3 given by 
 equations (13), 5. 
 
 11. If a transformation of coordinates is such that 
 
 Cti? + dy 2 +d2*-c 2 di 2 = \ 2 (dx 2 + dy 2 +dz 2 -<?dt 2 ), 
 where X is a function of #, y, 2, t ; there is also a relation of type 
 
 (J. Liouville and S. Lie.) 
 12. Prove that the differential equation 
 
 is covariant for a transformation of the type considered in Ex. 11. 
 
 (S. Lie.) 
 
150 MISCELLANEOUS THEORIES [CH. 
 
 13. Prove that if , ;, , r are functions of #, y, z, * such that 
 - rjM, icrM x ) dx + (M e - M X icrM v ) dy 
 
 is an exact differential and e is a quantity whose square may be neglected, 
 the value of Jfat the point d=x+ , y'=y+rj, z'=z + f(, t' = t+fr may 
 be calculated by assuming that the integral form 
 
 M x (dybz- dz dy) + M y (dzbx- dx 8z) + M e (dx by - dy 8x) 
 
 + icM x (dx 8t - dt dx) + icM y (dy &t-dtdy) + icM z (dzdt- dt dz) 
 is an invariant for the infinitesimal transformation, it being supposed 
 that the function M satisfies equations (10) of 5. 
 
 14. Let a transformation from the coordinates (a/, /, 2', t) to (#,y, 2, t) 
 be such that 
 
 Q' 
 
 + =(ldx+mdy+ndz-c 2 pdt) 2 
 VA 
 
 2$ 
 
 - (I dx + mdy + n dz - c 2 p dt) (I'dx + m'dy + n'dz - tfp'dt) 
 VA 
 
 + -?= (I'dx + m'dy + n'dz - 
 VA 
 
 where e 
 
 and o-, I, m, n, ^>, I', m\ ri, p' are functions of #, y, 0, t ; then if satisfies 
 the equations 
 
 32^ 92^ 32^ x 32^ 
 it also satisfies ^* + g* + 87 2 " ? 8^ 
 
 In the preceding equation we have 
 
 as a# az ap 8.R 
 
 8P 
 
 L = lp' - I'p, M= mp' m'p, N= np' - n'p. 
 
 Show that in certain cases a function of type A0 is a solution of the 
 wave-equation in consequence of the two equations imposed on 6. Discuss 
 
/ 
 
 IX] EXAMPLES 151 
 
 the case of the transformation of Ex. 1, p. 139 : also the case of a trans- 
 formation which leaves the functions X, Y, Z of Ex. 7, p. 80 unaltered in 
 form. 
 
 15. Prove that if a function V satisfies the equation 
 
 vanishes at infinity and has continuous derivatives except at points of the. 
 curve <r where its normal derivative is discontinuous in such a way that 
 
 the symbol Q being used to denote the coordinates of a point Q, then 
 
 where z = 
 
 (Levi-Civita, Nuovo Cimento, 1897.) 
 
 16. Prove that 
 1 
 
 and deduce that the integral F= / / / -p (^, ;, C) dgdrfdg satisfies Poisson's 
 
 equation AF+47rp(ar, y, )0. 
 
 (J. Weingarten.) 
 
 17. Prove that the equation A^^-^ + S ~ is satisfied by 
 
 r 
 
 7 <- 
 
 where ^ =(< _^_ T) 2_ r2j r 2 =(a; 
 
 and /o(^) is the Bessel's function J Q (i6). .r , y , Z Q and ^ are arbitrary 
 constants. 
 
 (M. Brillouin, Comptes Rendus, 1903. 
 
 18. Prove that a solution of the differential equation 
 
 1HT: W'-, 
 
 ~ 
 
 is given by 
 
 tr= f " ^ [/ (X) 
 J -> L 
 
152 MISCELLANEOUS THEORIES [CH. 
 
 3 2 V d V 3 2 V 
 hence obtain a solution of the equation ^2 + ^^ sss ^~z bv P uttin g 
 
 V= Ufa t) e~*. If U=f (x\ -ft=g(x) when t = 0, the functions F, G may be 
 determined by the equations 
 
 g (*) = I e** G (A) dX, 
 
 J -oo 
 
 with the aid of Fourier's theorem. 
 
 (H. Poincare', Comptes Rendus, 1893-4.) 
 
 19. Prove that with the conditions of the last example 
 
 / 
 
 where G fa ; t, r) = J Q V(^OMf )'- 
 
 (Laplace (1779) ; E. Picard, Bull. soc. math. t. 22 (1894), p. 2.) 
 
 3 2 $ 3$ 
 
 20. Prove that a solution &=e~ t V of the equation A$=^r+2 ^~- 
 
 ot* ot 
 
 is given by the following extension of Kirchhoff's formula : 
 
 ~~^~ 
 
 dt dr 
 
 where r2 = ^-^ )2 + (y-y ) 2 + (^-^o) 2 , ^-(*-*i)-*, ^'-jgAW 
 and / (0) is the Bessel's function J (i6}. 
 
 The first integral extends over the volume enclosed by both the sphere 
 r=ti and the surface 2; when this sphere cuts the surface the second 
 integral extends over the part of the spherical surface inside 2, the last 
 two integrals extend over the part of 2 which lies inside the sphere. The 
 normal n is supposed to be drawn into the region of integration. If 2 is a 
 closed surface and (# 0) #0? ^o) lies outside, the region of integration is the 
 space outside 2 and inside the sphere. 
 
 (M. Brillouin, Comptes Rendus, 1903.) 
 
 21. Let a, /3, a> be denned in terms of x, y, z by the equations 
 
 y 2 #=cosacos(/34-&o>), y 2 y = cosasin(/3+<B), y 2 z=sinacosa>, 
 where -y 2 =l sin a sin a> and Tc is a constant. A solution of Laplace's 
 
IX J EXAMPLES 153 
 
 equation Aw = is then given by u=yF(a,ft) provided F satisfies the 
 partial differential equation 
 
 3 
 
 _ . 
 
 sm 20-5-? +4 --- a -332 a 5 -- -; 
 da 2 sin2a 8/3 2 da 4 
 
 (U. Amaldi, ./fend. Palermo (1902), p. 1.) 
 
 22. Prove that the following transformations of coordinates lead to 
 binary potentials, i.e. to solutions of Laplace's equation of typett=/ T (|, 17) : 
 
 (1) *=& y=i/, *=; 
 
 (2) # 
 
 (3) # 
 
 (4) # = sin cos 77, y = f sin sin ?;, 
 
 (5) tf^sinije^cosf, y=smrje m *sm 
 
 where wi is an arbitrary constant. The differential equations satisfied by 
 F in cases (3) and (5) are 
 
 = 
 
 and 
 
 -^2+|2 ^ + ^ (2 + m 2 cosec 2 ^) + - C ot7;^-=0 
 
 respectively. In the other cases the differential equations are already 
 familiar. 
 
 (T. Levi-Civita, Turin Memoirs, (2) t. 49 (1900).) 
 
 32 y fpy 92 y 
 23. If the differential equation ^ + -^ = -^ is satisfied by an ex- 
 
 pression of type V=yf(0) where /is an arbitrary function, 6 must satisfy 
 the differential equation 
 
 Prove that if we write /icW = cos adx+sin ady dt, x\ 
 y = ^sina+v, there is a relation between a, u t v. Discuss the cases in 
 which a is a function of 6 and a constant respectively, and obtain the 
 general value of y in each case. 
 
 24. If in the last example u=f(a, 0), v=g( a , 6} and we write 
 
154 MISCELLANEOUS THEORIES [CH. IX 
 
 2 y 32 tr 32 y 
 the new form of the equation -^ + ^-^ = ^ when a, 0, t are taken as 
 
 independent variables is 
 
 Prove that this equation can only possess a solution of type V=yF(6), 
 
 with F arbitrary, if ^- = 0, and in this case 6 is defined by an equation of 
 type 
 
 [x 
 
 while the most general value of y is 
 
 where I, m, n satisfy the relation 
 
 25. Show that wave-functions of type yf(B} may be derived from 
 solutions of Laplace's equation of this type by means of the results given 
 on pp. Ill, 114. Hence show that there are wave-functions of type 
 which are not particular cases of a more general wave-function of 
 ?/(> ) where a, /3 are defined by equations (280). 
 
LIST OF AUTHOES QUOTED 
 
 Abel 100 
 
 Abraham 2, 10, 11, 28, 75, 96, 
 
 Adamoff 100, 101 
 
 Aichi 89 
 
 Airey 38 
 
 Airy 67 
 
 D'Alembert 6 
 
 Appell 25, 32, 33, 102, 112 
 
 Aubert 108 
 
 Austin 68 
 
 Backlund 118 
 
 Baer 100 
 
 Baker 25 
 
 Barnes 38, 40, 101, 106 
 
 Basset 30, 95, 107 
 
 Bateman 110, 116, 141, 146 
 
 Bernoulli 26 
 
 Bessel 35 
 
 Beudon 118 
 
 Birkeland 23 
 
 Birkhoff 103 
 
 Blumenthal 44 
 
 Bocher 25, 26, 31, 98, 99, 103 
 
 Bois 89 
 
 Born 14, 54 
 
 Boussinesq 23 
 
 Bragg, W. H. 3 
 
 Bragg, W. L. 54 
 
 Brill 32, 33 
 
 Brillouin 17, 96, 151, 152 
 
 Bromwich 84 
 
 Burnside 145 
 
 Byerly 26, 40 
 
 Gamp 44 
 
 Campbell 2, 121 
 
 Carslaw 82, 91, 93, 145 
 
 Carson 18 
 
 Cauchy 6, 114, 135 
 
 Cayley 112 
 
 Chapman 44 
 
 Cole 89 
 
 Con way 116 
 
 Coulon 118 
 
 Cunningham 5, 14, 102, 146 
 
 Curzon 100 
 
 Darboux 31, 44, 108 
 Dean 108 
 
 Debye 8, 28, 36, 45, 49, 60, 63, 64, 
 
 65, 66, 78, 79, 80 
 Didon 112 
 Dini 43 
 Dinnik 38 
 Donkin 114 
 Dougall 145 
 Duhem 21 
 
 Eccles 11, 68 
 Edwardes 69 
 Ehrenhaft 96 
 Einstein 2, 60, 146 
 Emde 38 
 
 Euler 8, 11, 30, 45 
 Ewald 54 
 Ewing 16 
 
 Faraday 2, 121 
 
 Fe>r 44 
 
 de Feriet 112 
 
 FitzGerald 3, 11, 45 
 
 Fleming 11, 18 
 
 Forsyth 42, 114, 136, 137 
 
 Fourier 41, 145 
 
 Friedrich 54 
 
 Gans 16, 44 
 
 Garnett 17, 44 
 
 Gegenbauer 41, 99, 102 
 
 Giulotto 112 
 
 Glaisher 40 
 
 Gray 10, 38, 72, 75 
 
 Green 19, 21, 61, 112, 122, 141, 142, 
 
 143, 144, 145 
 Greenhill 145 
 Grimpen 90 
 Gronwall 44 
 Grossmann 79 
 Guillet 108 
 
 Hadamard 20, 118, 142, 147 
 Hamilton 13 
 Hankel 37, 72 
 Hantzschel 88, 89, 95, 100 
 Happel 44 
 Hardy 41 
 Hargreaves 117 
 Harkness 82 
 Harms 78 
 
156 
 
 LIST OF AUTHORS QUOTED 
 
 Hartenstein 88, 90 
 
 Hasenohrl 60 
 
 Basse" 14 
 
 Havelock 23, 54 
 
 Heaviside 1, 3, 15, 18, 21, 22, 23, 
 
 70, 72 
 
 Heine 40, 41, 43, 47, 88, 96, 112, 145 
 Helmholtz 142, 143 
 Hermite 100, 112 
 
 Hertz 1, 3, 7, 8, 10, 11, 17, 22, 75, 89 
 Herzfeld 96 
 Hicks 107 
 Hilbert 145 
 Hobson 38, 40, 41, 42, 44, 68, 69, 
 
 70, 71, 82, 106, 108, 112, 145 
 Hoerschlemann 11, 73 
 Hondros 78 
 Humphreys 60 
 Huygens 141 
 
 Ignatowsky 78 
 Isherwood 38 
 Ishiwara 146 
 
 Jackson, D. 44 
 Jackson, W. H. 88 
 Jacobi 114, 118, 136 
 Jacobstahl 101 
 Jahnke 38 
 Jeans 27 
 Jeffery 108 
 Jordan 44 
 Joule 16 
 
 Kalahne 79 
 Karman 54 
 Kelvin 29, 31, 32 
 King 60 
 
 Kirchhoff 141, 142, 148 
 Kneser 143 
 Knipping 54 
 Kummer 103 
 
 Lagrange 86 
 Laguerre 100 
 Lamb 6, 18, 37, 70, 72, 85, 86, 89, 
 
 96, 98, 141, 142 
 Lame" 26, 29 
 Larnpa 49 
 Langwitz 90 
 Laplace 26, 32, 35, 152 
 Larrnor 1, 2, 3, 9, 14, 17, 20, 29, 
 
 33, 49, 75, 79, 81, 121, 141, 142 
 Laue 54 
 Laura 141 
 Leathern 5 
 Lebedeff 102 
 Lecher 78 
 Lee 8, 10 
 Legendre 35, 105 
 
 Levi-Civita 19, 70, 151 
 
 Lewis 6, 146 
 
 Lie 149 
 
 Lienard 116, 117, 133 
 
 Lindemann 88 
 
 Liouville 149 
 
 Lodge, A, 40 
 
 Lodge, 0. J. 11 
 
 Lorentz 1, 2, 3, 6, 14, 15, 18, 117, 
 
 141 
 
 Lorenz 44, 79, 90 
 Love 20, 21, 44, 50, 90, 141, 147 
 
 MacCullagh 1 
 
 Macdonald 8, 11, 18, 20, 28, 41, 49, 
 
 67, 82, 84, 87, 88, 105, 145, 148 
 Maclaurin 88, 96 
 March 41, 67 
 Mason 145 
 Mathews 38, 72, 75 
 Mathieu 26, 88, 95 
 Maxwell 1, 3, 14, 17, 23, 61 
 Mazzotto 78 
 Mehler 41, 112 
 Michell 95 
 Mie 8, 17, 44, 51, 53, 55, 58, 59, 60, 
 
 78 
 
 Minkowski 5, 14, 146 
 Mobius 79 
 Morley 82 
 Morton 78 
 Miiller 59 
 Murphy 6, 100 
 
 Neumann, C. 43, 68, 95, 98, 105, 107, 
 
 112, 145 
 
 Neumann, E. B. 145 
 Newton 3 
 Nicholson 38, 45, 49, 66, 67, 68, 78, 
 
 79, 96, 106 
 
 Nielsen 37, 38, 40, 72 
 Niven 97 
 Northrup 18 
 
 Ornstein 54 
 
 Orr 107 
 
 Oseen 82, 87, 107, 108, 145 
 
 Painleve" 25 
 Parseval 142 
 Pearson 8, 10 
 Perry 40 
 Picard 145, 152 
 Picciati 17, 19 
 Planck 5 
 Plemelj 143 
 Pochhammer 101 
 Pockels 66, 89 
 Pocklington 78, 107, 108 
 Pogany 90 
 
LIST OF AUTHORS QUOTED 
 
 157 
 
 Poincare 5, 23, 25, 41, 43, 67, 68, 
 
 84, 145, 152 
 
 Poisson 71, 86, 138, 141, 142, 149 
 Poynting 6, 9, 13, 21, 55 
 Proudman 45 
 
 Bayleigh 3, 17, 36, 44, 47, 54, 55, 
 
 60, 67, 78, 79, 89, 90, 107, 141 
 Eeiche 88, 90 
 Eiemann 82, 87, 107 
 Eighi 7, 11 
 Eontgen 22, 131, 132 
 Eowland 8, 90 
 Eubens 89 
 Eunge 13 
 Eybcynski 41 
 
 Safford 95, 107 
 
 Schaefer 78, 90 
 
 Schlomilch 99 
 
 Schott 116, 133 
 
 Schottky 137 
 
 Schuster 60 
 
 Schwarzschild 45, 64, 82, 89, 117 
 
 Seitz 78 
 
 Sharpe 98 
 
 Sieger 89 
 
 Silberstein 4, 5, 20, 21, 146 
 
 Smoluchowsky 60 
 
 Sommerfeld 11, 13, 17, 36, 67, 72, 
 
 73, 75, 76, 77, 82, 115, 145 
 Sonin 72, 80, 99 
 Steubing 53 
 Stieltjes 110 
 
 Stokes 8, 20, 21, 27, 36, 44, 90 
 Stoney 27, 115 
 Sturm 100 
 
 Tallquist 40 
 
 Tchebycheff 100 
 
 Tedone 23 
 
 Thomson, G. H. 90 
 
 Thomson, J. J. 2, 3, 11, 18, 22, 45, 
 
 49, 51, 75, 76, 78, 79, 89, 115, 121, 
 
 131, 132 
 Tisserand 110 
 Tonolo 141, 148 
 Tyndall 55 
 
 Voigt 82, 85, 90 
 
 Volterra 25, 29, 70, 71, 142, 149 
 
 Waelsch 112 
 
 Walker 44 
 
 Wangerin 40, 95 
 
 Warburg 16 
 
 Watson 28, 100, 101 
 
 Weber, H. 88, 89, 100 
 
 Weber, E. H. 79 
 
 Webster 80, 91 
 
 Weiler 99 
 
 Weingarten 151 
 
 Weyl 145 
 
 Whittaker 3, 7, 11, 28, 38, 88, 100, 
 
 101, 119 
 
 Wiechert 22, 117 
 Wiegrefe 88 
 Wien 89 
 Wilson 6 146 
 Witte 121 
 Wood 27, 60 
 
 Zaremba 145 
 Zeeman 89 
 
INDEX 
 
 (The numbers refer to the pages) 
 
 Absorption of light 55 
 Aether, equations for free 1 
 structure of 3, 121 
 
 BessePs functions 35 
 Boundary conditions 17 
 
 Characteristics, differential equation 
 
 of 21, 118 
 Conjugate electromagnetic fields 5, 
 
 23, 126, 133 
 Convergence of series of elementary 
 
 solutions 42, 100 
 Coordinates, cylindrical 69 
 paraboloidal 98 
 polar 35 
 spheroidal 95 
 toroidal 104, 107 
 transformations of 28, 29 
 Corpuscle, first model of 114 
 
 second model of 127 
 
 Degree of wave-function 110 
 Dielectric sphere, scattering of waves 
 
 by a 44 
 Diffraction by a screen with a straight 
 
 edge 83 
 
 of electric waves 67 
 of sound 90 
 
 Discontinuity, surfaces of 20, 22, 90, 
 121, 140 
 
 Electric doublet 8, 119 
 
 Electric waves, propagation of 10, 11, 
 
 67, 73, 75 
 diffraction of 67, 83 
 Electromagnetic fields 4 
 
 conjugate 5 
 self -conjugate 5 
 Electron, field due to a moving 113, 
 
 140 
 Elementary solutions 35, 71, 100, 104, 
 
 105 
 
 Energy equation fora material medium 
 15 
 
 Faraday tubes 121 
 Flow of energy 5 
 
 Green's equivalent layer 19, 122 
 functions 142 
 
 Hamilton's equation 13, 21, 24, 123, 
 
 140 
 Hysteresis 16 
 
 Intensity of scattered light 54 
 
 Invariants 5 
 
 Inversion, generalised 31, 111, 112 
 
 Kirchhoff s formula 140, 147 
 
 extensions of 140, 147 
 
 Laplace's equation 97, 107, 114, 119, 
 
 139 
 
 Legendre functions 38, 106 
 Light, absorption of 55 
 
 diffraction of 83 
 
 model of a source of 9 
 
 polarisation of 53 
 
 scattered 54 
 
 Magnetic oscillator 11, 119 
 
 vibrations 49 
 Material medium 14, 16 
 Multiform solutions of wave-equation 
 82 
 
 Obstacle, cylindrical 79, 80 
 small 51 
 spherical 44 
 spheroidal 96 
 
 Paraboloidal coordinates 98 
 
 mirror, reflection by 24 
 Poisson's formula 140, 147 
 
 wave-function 86, 138 
 Polarisation of light 53 
 Polynomial solutions of wave-equation 
 
 102, 111 
 Poynting's vector 6, 13 
 
INDEX 
 
 159 
 
 Pressure of radiation 60 
 Projection of singularities 
 moving point 127 
 
 from a 
 
 Kayleigh's Radiation 51 
 
 Relations between various solutions 
 
 40, 68, 70, 72, 100, 103, 105 
 Residual blue 55 
 Rontgen rays 22, 54, 131 
 
 Screen, diffraction by a 83 
 Singular curves 123 
 Singularities, moving 115 
 Sonin's polynomials 99 
 Sound, diffraction of 90 
 Surface, Kiemann 82 
 
 Transformations of coordinates 28 
 
 Transformations of electromagnetic 
 equations 2, 139, 
 
 146, 149 
 wave-equation 31, 150 
 
 Velocity of light 2 
 
 of moving singular curves 123 
 Vibrations, free damped 49 
 
 electric and magnetic 49 
 
 Wave-equation 7 
 function 7 
 
 function of degree zero 113 
 Waves, propagation of, along a wire 75 
 over earth's surface 
 
 11,67, 73 
 scattering of 44 
 Wedge, diffraction by a 87 
 
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