UNIVERSITY OF CALIFORNIA AT LOS ANGELES SPHERICAL HARMONICS. AN ELEMENTARY TREATISE ox SPHERICAL HARMONICS AND SUBJECTS CONNECTED WITH THEM. REV. N. M. FERRERS, M.A., F.R.S., FELLOW AND TUTOB OP GONVILLE AND CAIUS COLLEGE, CAMBRIDGE. Hon&on: MACMILLAN AND CO. 1877 [All Rights reserved.] CTamliri&ge: PRINTED BY C. 3. CLAY, 31. A., AT THE UNIVERSITY PRESS. GIFT OF MRS, FRANK MORL.EY - / Sciences \ (c Library PEEFACE. THE object of the following treatise is to exhibit, in a concise form, the elementary properties of the expressions known by -- the name of Laplace's Functions, or Spherical Harmonics, and of some other expressions of a similar nature. I do not, of course, profess to have produced a complete treatise on these functions, but merely to have given such an introduc- tory sketch as may facilitate the study of the numerous works and memoirs in which they are employed. As Spherical Harmonics derive their chief interest and utility . from their physical applications, I have endeavoured from the outset to keep these applications in view. I must express my acknowledgments to the Rev. C. H. Prior, Fellow of Pembroke College, for his kind revision of the proof-sheets as they passed through the press. N. M. FERRERS, GONVILLE AND C-VIUS COLLEGE, August, 1877. F. H. 444685 CONTENTS. CHAPTER I. INTRODUCTORY. DEFINITION OF SPHERICAL HARMONICS. CHAPTER II. ZONAL HARMONICS. AET. PAGE 1. Differential Equation of Zonal Harmonics 4 2. General solution of this equation 3. Proof that P i is the coefficient of h l in a certain series . . 6 5. Other expressions for Pi 11 6. Investigation of expression for P 4 in terms of fi, by Lagrange's Theorem 12 7- The roots of the equation P, = are all real .... 13 8. Rodrigues' theorem ib. 10. Proof that /"* P i P a dp. = 0, and f * P? dp = ^ . . . 16 2i + 1 /-i J-i 12. Expression of P 4 in ascending powers of /t 151 15. Values of the first ten zonal harmonics 22 16. Values of | l i m P i dfj. 25 Jo 17. Expression of fj in a series of zonal harmonics .... 26 viu CONTEXTS. ART. TAfiK 18. Expression of P t in a series of cosines of multiples of . . 29 19. Value of P i cos mO sin 6 dO ....... fft. J 7o 20. Expression of cos mO in a series of zonal harmonics ... 33 21. Development of sin 6 in an infinite series of zonal harmonics . 35 22. Value of - in a series of zonal harmonics .... 37 mjt 24. Value of I PiPdn ......... 38 /'" JM 25, 26. Expression of Zonal Harmonics by Definite Integrals . . 39 '27. Geometrical investigation of the equality of these definite integrals 41 28. Expression of P t in terms of cos 6 and sin 6 . . . . 42 CHAPTER III. APPLICATION OF ZONAL HARMONICS TO THE THEORY OF ATTRAC- TION. REPRESENTATION OF DISCONTINUOUS FUNCTIONS BY SERIES OF ZONAL HARMONICS. 1. Potential of an uniform circular wire 44 2. Potential of & surface of revolution 46 3. Solid angle subtended by a circle at any point . . . . 47 4. Potential of an uniform circular lamina 49 5. Potential of a sphere whose density varies as R~ 5 ... 51 6. 9. Relation between density and potential for a spherical surface 54 10. Potential of a spherical shell of finite thickness .... .vs 12. Expression of certain discontinuous functions by an infinite series of zonal harmonics 01 14. Expression of a function of n, infinite for a particular value of fj., and zero for all other values 65 lo. Expression of any discontinuous function by an infinite series of zonal harmonics 66 CONTENTS. IX CHAPTER IV. SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SECTORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXES IN ANY POSITION. POTENTIAL OF A SOLID NEARLY SPHERICAL IN FORM. ART. PAGE 1. Spherical Harmonics in general 69 2. Relation between the potentials of a spherical shell at an inter- nal and an external point ib. 3. Eelation between the density and the potential of a spherical shell 70 4. The spherical harmonic of the degree i will involve 2i + l arbi- trary constants 72 ' 5. Derivation of successive harmonics from the zonal harmonic by differentiation ......... ib. 6. Tesseral and sectorial harmonics 74 7. Expression of tesseral and sectorial harmonics in a completely developed form 75 8. Circles represented by tesseral and sectorial harmonics . . 77 9. New view of tesseral harmonics 78 10. Proof that f f 2 " Y t Y m dfjid=Q . 80 J-iJo 11. If a function of ft and can be developed in a series of surface harmonics, such development is possible in only one way . 82 12. Proof that [ 21r w= i r ( 2w m 1-Jo PtYi 13. Investigation of the value of / I Y^^pd^ .... J-iJo 14. Zonal harmonic with its axis in any position. Laplace's co- efficients 87 15. Expression of a rational function by a finite series of spherical harmonics 90 X CONTENTS. ART. PAGE 16. niustrations of this transformation 91 17. Expression of any function of JJL and 93 18. Examples of this process 95 19. Potential of homogeneous solid nearly spherical in form . . 97 20. Potential of a solid composed of homogeneous spherical strata . 99 CHAPTER V. SPHERICAL HARMONICS OF THE SECOND KIND. 1. Definition of these harmonics 101 2 and 3." Expressions in a converging series 102 4. Expression for the differential coefficient of Qi , 105 5. Tesseral Harmonics of the second kind IOC CHAPTER VI. ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 1. Introduction of Ellipsoidal Harmonics 108 2. Definition of Elliptic Co-ordinates ib. 3. Transformation of the fundamental equation . . . .109 4. Further transformation 110 5. Introduction of the quantities E, H 113 6. 7. Number of values of E of the degree n ib. 8. Number of values of the degree n + i 117 9,10,11. Expression of EHH' in terms of x, y, z .... ib. 12. Potential for an external point 121 13. Law of density 123 14. Fundamental Property of Ellipsoidal Harmonics . . . l'2t; 15. Transformation of 1 1 el^VadS to elliptic co-ordinates . . . 128 COOTEXTS. xi AST. PAGB 16. Modification of equations when the ellipsoid is one of revolution about the greatest axis ISO 17. Interpretation of auxiliary quantities introduced . . . 133 18. Unsymmetrical distribution 134 19. Analogy with Spherical Harmonics 135 20. Modification of equations when the ellipsoid is one of revolution about the least axis 136 21. Unsymmetrical distribution 139 22. Special examples. Density varying as Pi(p) .... ib. 23. External potential varying inversely as distance from focus . 142 24. 25. Consequences of this distribution of potential . . . 143 26. Ellipsoid with three unequal axes 145 27. Potential varying as the distance from a principal plane . . 146 28. Potential varying as the product of the distances from two prin- cipal planes . . ib. 29. Potential varying as the square on the distance from a principal plane 147 30. Application to the case of the Earth considered as an ellipsoid . 150 31. 32. Expression of any rational integral function of x, y, z, in a series of Ellipsoidal Harmonics 152 33. On the expression of functions in general by Ellipsoidal Har- monics 153 EXAMPLES . . 155 ERRATA. 1 p. 113 line 8, for V read E. p. 136 line 11, for tp read vr. p. 142 line 6, for point read axis. CHAPTER I. INTRODUCTORY. DEFINITION OF SPHERICAL HARMONICS. 1. IF V be the potential of an attracting mass, at any point x, y, z, not forming a part of the mass itself, it is known that V must satisfy the differential equation v da? T d? or, as we shall write it for shortness, V 2 F"= 0. The general solution of this equation cannot be obtained in finite terms. We can, however, determine an expression which we shall call V t , an homogeneous function of x, y, z of the degree i, i being any positive integer, which will satisfy the equation ; and we may prove that -to every such solution V { there corresponds another, of the degree (i + 1), Y expressed by -^ , where r 2 = x s + y* + z 2 . For the equation (1) when transformed to polar co-ordi- nates by writing x = r sin 8 cos } z = r cos 6, becomes 1 d /, a dV\ , 1 tfV -. sin H^^ -4 - *-L _ . . = () ...( L I. * _7 2 I ' l JQ \ "*" " 1Q I > ' ' dr sm a rfc' \ du I sin And since V satisfies this equation, and is an homo- geneous function of the degree i, V i must satisfy the equa- tion t - (l -+i) r i+ i ^f sln ^') + i ^ = o, siu u dv \ do 1 sm u d

<*' *' INTRODUCTORY. since this is the form which equation (2) assumes when V is an homogeneous function of the degree ". Now, put V i = r s>+1 U it and this becomes or ({ + !) E/: + -T .Afsinfl^pU-r-Va ^ = (2). sm 6 do\ do J sin a

only, w r e shall call a Surface Spherical Harmonic of the same degree. A very important form of spherical harmonics is that which is independent DEFINITION OF SPHERICAL HARMONICS. 3 of (f). The solid harmonics of this form will involve two of the variables, x and y, only in the form x* + y*, or will be functions of ie a + 7/ 2 and z. Harmonics independent of < are called Zonal Harmonics, and are distinguished, like spherical harmonics in general, into Solid and Surface Harmonics. The investigation of their properties will be the subject of the following chapter. The name of Spherical Harmonics was first applied to these functions by Sir W. Thomson and Professor Tait, in their Treatise on Natural Philosophy. The name "Laplace's Coefficients" was employed by "Whewell, on account of Laplace having discussed their properties, and employed them largely in the Me'caniqiie Celeste. Pratt, in his Treatise on the Figure of the Earth, limits the name of Laplace's Coefficients to Zonal Harmonics, and designates all other spherical harmonics by the name of Laplace's Functions. The Zonal Harmonic in the case which we shall consider in the following chapter, i.e., in which the system is symmetrical about the line from which 9 is measured, was really, however, first introduced by Legendre, although the properties of spherical harmonics in general were first discussed by Laplace; and Mr Todhunter, in his Treatise, on this account calls them by the name of "Legendre's Coefficients," applying the name of "Laplace's Coefficients" to the form which the Zonal Harmonic assumes when in placo of cos 6, we write cos OcosO' + sin #sin#'cos( <'). The name " Kugelfunctionen " is employed by Heine, in his standard treatise on these functions, to designate Spherical Harmonics in general. 12 CHAPTER II. ZONAL HARMONICS. 1. WE shall in this chapter regard a Zonal Solid Har- monic, of the degree i, as a homogeneous function of (x* + y)^> and z, of the degree i, which satisfies the equation _ * dx* djf dz Now, if this be transformed to polar co-ordinates, by writing r sin 6 cos for x, r sin 6 sin for y, r cos 6 for z, we observe, in the first place, that x 2 + y* r 2 sin 2 6. Hence V will be independent of , or will be a function of r and only. The differential equation between r and 6 which it must therefore satisfy will be J(rF) . I d (. a dV\ r -TJ-* + -v 7. -jz sin 6 -^ = 0. dr sin 6 d6 \ dd J Now V, being a function of r of the degree i, may be expressed in the form r*P,, where P t is a function of 6 only. Hence this equation becomes or, putting cos 6 = In accordance with our definition of spherical surface harmonics, P< will be the zonal surface harmonic of the ZONAL HARMONICS. 5 degree t. When it is necessary to particularise the variable involved in it, we shall write it P i (/*). The line from which 9 is measured, or in other words for which /^ = 1, is called the Axis of the system of Zonal Harmonics; and the point in which the positive direction of the axis meets a sphere whose centre is the origin of co-ordinates, and radius unity, is called the Pole of the system. Any constant multiple of a zonal harmonic (solid or surface] is itself a zonal harmonic of the same order. 2. The zonal harmonic of the degree i, of which the line /J, = 1 is the axis, is a perfectly determinate function of /j,, having nothing arbitrary but this constant. For the expression r i P i may be expressed as a rational integral homogeneous function of r and #, and therefore P i will be a rational integral function of cos 0, that is of /z, of the degree i, and will involve none but positive integral powers of //,. But PJ is a particular integral of the equation '('+ i)/00 = o (3), and the most general form of /(/i) must involve two ar- bitrary constants. {Suppose then that the most general form of/(/i) is represented by P i Ivdfj,. We then have a\ > -/ d Hence, adding these two equations together, and ob- serving that, since 2\ satisfies the equation (3), the coefficient ZONAL HARMONICS. of Ivclfj, will be identically equal to 0, we obtain, for the de- termination of v, the equation whence P t (1 - /*) |? + 2 [(1 - ^ ^-'^- or - i /-< the integral of which is log v + log-P/ (1 - yu, 2 ) = log <7 t = a constant ; Hence \vdii = C+ C. I -775 J J f t and we obtain, for the most general form of /*(//.), Now, P, being a rational integral function of yu- of i r 7 dimensions, it may be seen that /^ -- ^\ pa w i^ assume the J (I ft ) 1 { form of the sum of z + 2 logarithms and i fractions, and therefore cannot be expressed as a rational integral function of fi. Expressions of the form P, I j- -- . 2 pa[ are called Kuyel- J (LJA ) /| functionen der zweiter Art by Heine, who has investigated their properties at great length. They -have, as will hereafter be seen, interesting applications to the attraction of a sphe- roid on an external point. We shall discuss their properties more fully hereafter. 3. We have thus shewn that the most general solution of equation (2) of the form of a rational integral function of a ZONAL HARMONICS. 7 involves but one arbitrary constant, and that as a factor. We shall henceforth denote by P., or P. (/A), that particular form of the integral which assumes the value unity when //. is put equal to unity. We shall next prove the following important proposition. J/h be less than unity, and if (1 - 2/zh + h 2 )"^ be expanded in a series proceeding by ascending powers of h, the coefficient oftiwillbe^. Or, (1 - 2/ih + h 2 )-' = P + P t h -f . . . + P,h' + . . . We shall prove this by shewing that, if H be written for (1 2/Jt + /i 2 )"-, H will satisfy the differential equation d ( ,. dH\ . j 2 1 ./ T o 7^ "- /i ~" *jjLt/i ~f" X - ~7~7*t j 1 T-/^ or -^ = , A /. the above equation becomes h d* n rr^ d f,, . d f H ^3r,{*fi) + X-^C 1 -X)j 2 a/i x "/^ i a/A \3 ,(f 2 (A^) rf f n or A I -^ + J(l- dA 2 a/i ( v 4. Having established this proposition, we may proceed as follows: If Pi be the coefficient of 1C in the expansion of //, . h ~, (hH) = 1 .2p,A + 2.3p^ 8 + ... +i (i Ct/t Also, the coefficient of 7i* in the expansion of ifa-rtfrUz-Jfl-tt^ a/A [ x J a/ij rf/i i v ' rf/J Hence equating to zero the coefficient of A*, 10 ZONAL HARMONICS. Also jt?i is a rational integral function of /*. And, when p, = I, 11= (I - 2h + k 2 }^ = l+7* + / t 2 +...+A'' + ... Or when /* = 1, p t = 1. Therefore p i is what we have already denoted by P,. We have thus shewn that, if h be less than 1, If h be greater than 1, this series becomes divergent. But we may write P, since ,- is less than 1, It p p p - J> + f J 4. J. A ~ A + tf + + A i+1 + Hence P i is also the coefficient of 7t-(' +1 ) in the expan- sion of (1 2/i/i + 7r) "^ in ascending powers of r when /i is greater than 1. We may express this in a notation which is strictly continuous, by saying that This might have been anticipated, from the fact that the fundamental differential equation for P t is unaltered if (i + 1) be written in place of i ; for the only way in which i appears in that equation is in the coefficient of P,, which is i (i + 1). Writing (i + 1) in place of i, this be- comes (i + 1) { ({+!) + 1} or (i + 1) i, and is therefore unaltered. ZONAL HARMONICS. 11 5. "We shall next prove that where ? >2 = a? + t/ 2 + z 2 . Let * = (* 2 + 2/ 2 + z) 4 =/(*), and let k be any quantity less than r. Then (a; 2 + f + (z - k?}-* =/ (z - k), and, developing by Taylor's Theorem, the coefficient of lc is Also {ic 2 + 7/ 2 + (^ - &)T } = ( r2 - 27 ^ + ^T 4 since z = fir, in the expansion of which, the coefficient of k' is Equating these results, we get The value of P 4 might be calculated, either by expanding (1 2/i/i + A 2 )"^ by the Binomial Theorem, or by effecting the r m d* fl\ differentiations in the expression ( 1)* .,- .-r ' 1 .2. 3 ... idz\rj & and in the result putting - = p. Both these methods how- ever would be somewhat laborious ; we proceed therefore to investigate more convenient expressions. 12 ZONAL HARMONICS. 6. The first process shews, by the aid of Lagrange's Theorem, that Let y denote a quantity, such that h being less than 1. Then 2/ 1 Also -_ = l_^ + _ ; Hence, by Lagrange's Theorem, therefore, differentiating with respect to /i and observing that ZONAL HARMONICS. 13 7. From this form of P { it may be readily shewn that the values of /j,, which satisfy the equation P 4 = 0, are all real, and all lie between 1 and 1. For the equation (^ l) f = has i roots = 1, and i roots = 1, .'. -r- (/i 2 - 1)' = has i - 1 roots = 1, (i - 1) roots = - 1, and one root = 0, ^^ -T-2 (/** l) i= =0 has (i 2) roots = 1, one root between 1 and 0, one between and = 1, and (i 2) roots = 1, and so on. Hence it follows that x7* n n j . (/A 2 1) = has = roots between 1 and 0, and r roots be- ClfJb ' ' tween and 1, if i be even, 7 " -" I 7( r T and - - roots between 1 and 0, - - roots between and 1, and one root = 0, if i be odd. It is hardly necessary to observe that the positive roots of each of these equations are severally equal in absolute mag- nitude to the negative roots. 8. We may take this opportunity of introducing an im- portant theorem, due to Rodrigues, properly belonging to the Differential Calculus, but which is of great use in this subject. The theorem in question is as follows: Ifmle any integer less than i, (J l ~ m 1 9 (I vn\ fj i+m " 14 ZONAL HARMONICS. It may be proved in the following manner. If (x~ 1)' be differentiated i -m times, then, since the equation has ?' roots each equal to 1, and i roots each equal = 1, it follows that the equation djc*' has i (i 7?i) roots (i. e. m) roots each = 1, and m roots each = 1, in other words that (a; 2 I/" is a factor of AVe proceed to calculate the other factor. For this purpose consider the expression Conceive this differentiated (I) i m times, (II) times. The two expressions thus obtained will consist of an equal number of terms, and to any term in (I) will corre- spond one term in (II), such that their product will be (x + aj (x + aj ... (+,) (* + &) (# + &) ... (*+ ft), i.e. the term in (II) is the product of all the factors omitted from the corresponding term in (I) and of those factors only. Two such terms may be said to be complementary to each other. Now, conceive a term in (II) the product of p factors of the form x + a, say x + a!, x + a" ... x + a * 1 , and of q factors of the form x+fi, say x+/3fX + /3 /t ...x + /3 (i) . We must have p + q = i m. The complementary term in (I) will involve p factors x + ff, x + /3" ... x + "*, q factors x + y /y x + a. tl . . . x -f a ta; . ZONAL HARMONICS. 15 Now, every terra in (I) is of i + m dimensions. We have accounted for p + q (or i TO) factors in the particular term we are considering. There remain therefore 2?n factors to be accounted for. None of the letters a', a"...a w , /3, ,,... ^ can appear there. Hence the remaining factor must involve m a's and m /3's, say, There will be another term in (II) containing (ar + P) (x + /3") . . . (x + /3 W ) (x + a) (x + aj . . . (x + J. The corresponding term in (I) will be, as shewn above, (x + a ) (x + a") ...(*+ cJ pl ) (aj + ,) (x + /3J . . . (a; Hence, the sum of these two terms of (I) divided by the sum of the complementary two terms of (II) is Now, let each of the a's be equal to 1, and each of the /3's equal to 1, then this becomes (# 2 1)'". The same factor enters into every such pair of the terms of (I). Hence JcL (V TV (II) - (x ft 1 - ( x * _ I Y ft 1 *" (x 2 IV Or - -^ = (x 9 - l) m - - ^ ^- , to a numerical dx dx factor pres. The factor may easily be calculated, by considering that fl i ~ m ('T^ 1 V the coefficient of x i+m in - ^ ^-is2{(2i-l)... ^ /+m (. x 2 1)* . and that the coefEcient of x i ~ m in : ^- - is aa; 2 (2i - 1) . . . (i + m + 1) (i + m} . . . (i - m + 1). 16 ZONAL HARMONICS. Hence the factor is 1 1.2...(i-ro) - or (i+m) (i + m-1) ... (i-m + iy 1.2... (t+m) ' 9. This theorem affords a direct proof that (7-^ (jj? 1)', (7 being any constant, is a value of / (/*) which satisfies the equation For from above, ^ F/ 2 IN ^ f d 1 . 2 - Nl )~| .,. ^ w ^ } I \ = * ( * dp, or Hence, the given differential equation is satisfied by put- Introducing the condition that P i is that value of f (ji) which is equal to 1, when /* = 1, we get P (u? IV ^'~^ 10. We shall now establish two very important proper- ties of the function P ( ; and apply them to obtain the develop- ment of P in a series. ZONAL HARMONICS. 17 The properties in question are as follows : If i and m be unequal positive integers, And '" 7 The following is a proof of the first property. We have Multiplying the first of these equations by P m , the second by P it subtracting and integrating, we get d . ~dP , f d + {i (i + 1) - m (m + 1)} JFf^ft = 0. Hence, transforming the first two integrals "by integration by parts, and remarking that i (i+ 1) m (m + 1) = (i m) (i+ m + 1), we get /* = 0, or i -V.. / ~ \ J since the second term vanishes identically. F. H. 18 ZONAL HARMONICS. Hence, taking the integral between the limits 1 and 4- 1, we remark that the factor 1 y? vanishes at both limits, and therefore, except when i m, or i + m + 1 = 0, -i We may remark also that we have in general a result which will be useful hereafter. 11. We will now consider the cases in which i m, or i + m + 1 = 0. We see that i -j- m + 1 cannot be equal to 0, if i and m are botl^positi^ integers. Hence we need only discuss the casein which ni i. We may remark, however, that since P 4 = P.U+D, the determination of the value of / P 2 dp will also . n . '-* give the value of I P, P^$ dp. J "~ 1 ri The value of Pfdp may be calculated as follows : J -i Integrate both sides with respect to p, ; then since f -i 1 j z.h> we get, taking this integral between the limits 1 and + 1, all the other terms vanishing, by the theorem just proved. ZONAL HARMONICS. 19 Hence = T J -1 J -I J -I Hence, equating coefficients of h", -7~1 O T ri 12. From the equation I P t P m dfji, = 0, combined with J -i the fact that, when fi = 1, P 4 = 1, and that P t is a rational integral function of ft, of the degree i, P 4 may be expressed in a series by the following method. We may observe in the first place that, if ra be any ri integer less than i, /j, m P i d/j, = 0. J -i For as P m , P m _ l . . . may all be expressed as rational in- tegral functions of fi, of the degrees m, m 1 . . . respectively, it follows that fj, m will be a linear function of P m and zonal harmonics of lower orders, p. m ~ l of P m _ l and zonal harmonics of lower orders, and so on. Hence Ip^P^fj, will be the sum of a series of multiples of quantities of the form I P m P t dfi, f 1 m being less than z', and therefore I ^ m P i dp l = 0, if m be any J -i integer less than i. Again, since it follows, writing h for h, that / i 2 J-^ = P -P 1 / i +... + (-l) < P^+... 22 20 ZONAL HARMONICS. And writing /A for n in the first equation, P ', P/...P/... denoting the values which P , P,...-^, respectively assume, when p, is written for p. Hence P/ = P 4 or P t , according as i is even or odd. That is, P. involves only odd, or only even, powers of i, according / as i is odd or even*. ' Assume then Our object is to determine A t , A t _ 2 .... Then, multiplying successively by /xT 2 , /i'" 4 , ... and inte- grating from 1 to + 1, we obtain the following system of equations : 2-4-... J O ^ ** ~ O-' V 1 9' -^ 9 9 - 1 I V * ' ' ( *-O -1- -4, i ""*- i i _ _^-2! i _ A T^OJ e ~ " - o~ o * v ' 2-3 2i-5 2*'-2s-3 <-g 1 J d * C^ f I I t~i * ' 4s 1 And lastly, since P 4 = 1, when p = 1, the last terms of the first members of these several equa- tions being 13. The mode of solving the class of systems of equa- tions to which this system belongs will be best seen by considering a particular example. * This is also evident, from the fact that P i is a constant multiple of ZONAL HARMONICS. 21 Suppose then that we have a + a 6+ a c -f a ~~ ' a + (o b-\- G) c + ft) G)* From this system of equations we deduce the following, being any quantity whatever, a? y z _ _!_ (0d)(8(S} (a + to) (5 + to) (c + ft)) /i ~^~ 7 . /i '" . /i ~~ / \ / /">\ / i /i\ TT I /i\ / " yTT For this expression is of 1 dimension in a, b, c, a, /3, 7, 0, w; it vanishes when Q a, or # = & and for no other finite value of d, and it becomes = , when 6 = . ta We hence obtain lf.-j.in/JL g ~* ; o> o>-aw-/ and therefore, putting = a, (a + o>) (ft + o>) (c + _ (a &) (a c) (a) a) (to /3) with similar values for y and g. And, if &) be infinitely great, in which case the last equation assumes the form x+y + z=l, we have _ (a + a) (a ~ (a-b)(a-c) ' with similar values for y and z. 14. Now consider the general system 22 ZONAL HARMONICS. .** + *'-* +...+ -^ +...=o, (0-or,) (^ and, multiplying by a^, 4- ^, and then putting = a ,-- - < *- ~ a *-s - a i or a o 15. To apply this to the case of zonal harmonics, we see, by comparing the equations for x with the equations for A, that we must suppose o> = GO ; and < = i, a^ 2 = i- S,...^, = i - 2s ... a, = i-l, a,_ 2 = i - 3,...a,_ w = i - 2s- 1... Hence _ (2* -2^-1) (2 - 2s - 3). ..{2 (t - 2g) - 1|... **- _2 S -2*-2...i-2*-l or i- 2s (2s - 2).. .2 x 2 . 4... (i - 2s - 1) or (i - 2s) ' ZONAL HARMONICS. 23 Or, generally, if i be odd, _ (2i 2.*r;;(i-3)x2 ' . 2.4...(i-5)x2.4 ' _ 1-1 ; And, if i be even, '~ 2.4...(*-2)x2 . _(2i-5)(2i-7).(i-3) ^ 2. 4.. ..(*'- 4) x 2. 4 ' We give the values of the several zonal harmonics, from P e to P 10 inclusive, calculated by this formula, - 2 2 ' 2 ' 5/a 3 - 3/a 2 ' 24 ZONAL HARMONICS. 7> 5J 3.1 p _9.7 5 7.5 " ~ _ ~ 8 11.9.7 9.7.5 7.5.3 5.3.1 2.4.6^~2.4x^ + 2x2.4 / * 2.4.6 - 315/I 4 + 105/A 2 - 5 16 p = 13 - n -9 7 H.9.7 5 9.7.5 ,7.5.3 2.4.6 ^ 2.4x2^ + 2x 2. 4^ 2.4.6^ _ 429/^7 - 693^ 5 + 315/a, 3 - 35^ 16 _15.13.11.9 8 13.11.9.7 6 11.9.7.5 4 8= 2.4.6.8 ^ 2.4.6x2^ + 2. 4x2. 4^ 9.7.5.3 , , 7.5.3.1 2x2.4.6^ ' 2.4.6.8 _ 6435^ 8 - 12012/^ 6 + 6930/* 4 - 1260/t 2 + 35 128 17.15.13.11 9 15.13.11.9 7 ,13.11.9^7 6 fl= 2.4.6.8 / *" 2. 4. 6x2 p 2. 4x2. 4^ 11.9.7.5 3 9.7.5.3 ""2x2.4.6^ 2. 4. 6. 8^ _ 12155/t 9 - 25740/I 7 + 18018/ - 4620^ + 315/A 128 P -L 9 ^ 1 ^!?- 1 ! 10 17.15.13.11.9 8 15.13.11.9.7 6 w ~ 2.4.6.8710"^ 2.4.6.8x2 ^ + 2. 4. 6x2. 4^ ZONAL HARMONICS. 25 13.11.9.7.5 4 11.9.7.5.3 , 9.7.5.3.1 ~ 2. 4x2. 4. 6^ 2x2.4.6.8^ 2.4.6.8.10 _ 46189/* 10 - 109395^ 8 + 90090^'- 30030//+ 3465/t 2 - 63 256 It will be observed that, when these fractions are reduced to their lowest terms, the denominators are in all cases powers of 2, the other factors being cancelled by correspond- ing factors in the numerator. The power of 2, in the denominator of P t , is that which enters as a factor into the continued product 1 . 2...Z. // * i* l *T**-' e *^o <> t> ~ 2 16. We have seen that / p m P t .dfji, = 0, if m be any integer less than i. It will easily be seen that if m + i be an odd number, the values of I// 1 P i . dp are the same, whether p be put = 1 or 1 ; but if m + i be an even number, the values of I//,"* P i . dp corresponding to these limits are equal and opposite. Hence, (m + i being even) J -i I Jo P i .dfj, = Q, if m = *- 2, *" 4 We may proceed to investigate the value of / //" P. . d/j, Jo if m have any other value. For this purpose, resuming the notation of the equations of Art. 13, we see that, putting 6 = m + 1, and o> = oo , we have a, + m + 1 a,_ 2 + m + 1 a^ + m + 1 _ ( m + 1 g<) (ra + 1 a,..,) . . . (m + 1 Cf { _ v } ... ~ K + m + 1) (a^ + m + 1) ... (a^ + m + 1) ...' 2G ZONAL HARMONICS. and therefore, putting x i =A i ... J a,= *..., a,= i 1..., we get I A A A uTP..du. = - 7 * ' A <~* ' ' -** - 2+m+l i-2s+m+l . . ,, (m t + 2) (m 1*+4) ... m . ... and = ; - . TW -- -IN -- 5* sr-p -- ITT- if 4 be even. (m + * + 1) (m + 1 - 1) ... (m + 3) (m + 1) In the particular case in which m = i, we get f Jo 17. We may apply these form alas to develope any positive integral power of fj> in a series of zonal harmonics, as we proceed to shew. Suppose that m is a positive integer, and that fi m is de- veloped in such a series, the coefficient of P i being C i} so that then, multiplying both sides of this equation by P 4 and inte- grating between the limits 1 and 1, all the terms on the right-hand side will disappear except C { P a i dp, which will J -i 2 become equal to ^-. - (7 4 . Hence which is equal to 0, if m + i be odd. Hence no terms appear unless m 4- i be even. In this case we have ZONAL HARMONICS. 27 Hence the formula just investigated gives if i be odd, and - /? -i- ... m ' (m+i+ 1) (m+i- 1) ... (m + 3) (m+ 1) if i be even. Therefore if m be odd, m _ 2.4.6...(m-l) _ p ' 1} (2m+ l)(2m - 1) ... (w + 4) ' + (m + 4) (m + 2) 3 m+2 If m be even, _ i\__ _ 2. 4. 6... TO _ p ' ) (2w+l) (2m- 1) ... (m + 3) (m+l) m + , g m Pi 1 (m + 3) (m + 1) 2 m+ 1 Hence, putting for T?Z successively 0, 1, 2 ... 10, we get IL P ^ J o r 'o^j^-s^ts ^fc7=^P s + ?P. I O 94. i 4 - 7.5 ZONAL HARMONICS. _2^6_ _4.6_ 6 1 * d 13.11.9.7 6+ 11.9.7 4 + s + 231 6 77 * 21 2 7 * T , 2.4.6 p 4.6 p 7 6 p 3 p 15 . 13 . 11 . 9 7 + 13 . 11 . 9 5 + 11 . 9 s + 9 ' "429 7 39 6 33 3 3 ' 2.4.6.8 4.6.8 * ^^^.lo.ia.ll^^ 84 " ^15.13.11.9^ 19 6 ' 8 P 15 8 y !3.11.9 4+ H.9 128 64 48 40 64 p 48 40 p 1 " t "495 "^US *" t "9 ( J 2 9 0> "6435 8 ""495 "US *""9 ( J 2 9 2.4.6.8 4.6.8 + 19.17.15.13.11 9 17. 15. 13. 11 8 128 192 16 56 3 12155 9 2431 7 ^65 5 ^ 143 ^11 ' 2.4.6.8.10 4.6.8.10 21.19.17.15.13.11 ""^ 19.17.15.13.il +9 -Ai + 17.15.13.11 15.13.11 * T 13.11 46189 PW + 2717 Pfl + 187 PS + 143 P < ^ ZONAL HARMONICS. 29 18. Any zonal harmonic P i may be expressed in a finite series of cosines of multiples of 0, these multiples being *0, (i-2)0.... Thus therefore, writing cos 6 for p, and observing that 1 - 2 cos 6h + h* = (1 - Ae vC ~i e ) (1 - he- vz i*), we obtain (1 -he^ 6 )-* (1 - fte-^T* = P + P i h+...+ PJJ +..,.. or whence, equating coefficients of h\ the last term being ) - ~ : > if i be even, and ^ 2 . 4t ... J 1.3...(i+l) 1.3...(t-2)_ .,., ,. , 1N -TS T-; 2 cos 0, if i be odd. 2.4... (i + 1) 2 . 4 ... (i 1) 19. Let us next proceed to investigate the value of T P,cos7n0sin0c70. Jo ^^XA cr ^JLV SO ZONAL HARMONICS. This might be done, by direct integration, from the above expression. Or we may proceed as follows. The above value of P i when multiplied by cos md sin (that is by ~ (sin (ra+ 1) sin (m 1) 0}) will consist of a series of sines of angles of the form {i 2n (m I)} 0, that is of even or odd multiples of 0, as i + m is odd or even. Therefore, when integrated between the limits and IT it will vanish, if i + m be odd. We may therefore limit our- selves to the case in which i + m is even. Again, since cos mO can be expressed in a series of powers of cos 0, and the highest power involved in such an expression is cos 6, it follows that the highest zonal harmonic in the development of cos mO will be P m . Hence / P 4 cos md sin dd Jo will be = 0, if m be less than i. Now, writing P i = C t cos id + C^ cos (i- 2) + ... we see that P t cos md sin dd will consist of a series of sines of angles of the forms (m + i + 1) 0, (m + i 1) . . . down to (m i 1)0, there being no term involving md, since the coefficient of such a term must be zero. Hence /' Jo P t cos md sin dd, will consist of a series of fractions whose denominators in- volve the factors m + i + 1, m + i 1 ... m i 1 respectively. Therefore when reduced to a common denominator, the result will involve in its denominator the factor (ro + t+1) (m+t-1)... O + l)(w-l) ... (ro-i-1) if 7H, be even, and (ro + * + l)(m + *-l)... (ra + 2) (TO -2) ... (m-t-1) if m be odd. For the numerator we may observe that since r P i cos m sin Odd ZONAL "HARMONICS. 31 vanishes if m be less than *, it must involve the factors m (i 2), m (i 4) . . . m + (i 2), and that it does not change sign with m. Hence it will involve the factor {m - (i- 2)} [m - (i - 4)} ... (m - 2) m* (m + 2) ... (m + f - 2) if m be even, and if m be odd. To determine the factor independent of m, we may pro- ceed as follows : P t = C t cos id + C t _ z cos (* - 2) + . . . ; .'. P t cos mO = x C. [cos (m + i) + cos (m i) 6} 2 + g ^2 ( cos ( m + *- 2 ) + cos (m - t + 2) 0} + ... ; 1 .'. P 4 cos ra# sin = -7 6' 4 {sin (m + i + 1) sin (m + i 1) 4 + sin (m i + 1) sin (m i 1) 0} + T CLj {sin (m + i 1) sin (w -M 3) + sin (m t + 3) sin (m i + 1) 0} + . . . ; .'. I P, cos m6 sin J0 ^0 {/ . \ _ t J | 2 |m + i+l m + i 1 t + 1 m i Ij - - - + - -- - + . 2 (m. + 1 1 m + i 3 m i +3 m i+1 , rf _ , - '--- -+- 32 ZONAL HARMONICS. Now, when m is very large as compared with i, this be- comes m m since C t + C^ z + ... = 1, as may be seen by putting Q = 0. / 2 Hence I P, cos ra# sin 6 dd tends to the limit , , as m Jo ^ m" is indefinitely increased. The value of the factor involving m has been shewn above to be {m - (i - 2)} (m - (i- 4)} ... (m - 2) m 2 (m + 2) ... (m + i- 2) {m -( + !)} {m- (t-1)} ...(m-l)(ra + l) ... (m + i+ 1) if w be even, and {m - (i- 2)} {m - (a - 4)] ... (m - 1) (m + 1) ... (m + i- 2) jw - 3 + IT}! - (* - 1)} ...(- 2) (* + 2) ... (m + '+ 1) if m be odd. Each of these factors contains in its numerator two factors less than in its denominator. It approaches, therefore, when m is indefinitely increased, to the value 5 . Hence r I J P t cos md sin d9 o _ {m-(i+l}} [m-(i'-l)} ... (m - 1) (m + 1) ... {m + (i if m and * be even, and |m~( {7W-(i if m and t be odd. In each of these expressions i may be any integer such that m i is even, i being not greater than m. Hence they will always be negative, except when i is equal to m. ZONAL HARMONICS. S3 20. We may apply these expressions to develop cos 1116 in a series of zonal harmonics. Assume Multiply by Pi sin 0, and integrate between the limits and TT, and we get ...2)}_ 2 {m - (i + l)J [m - (i- 1)} ... (m + (/+!)} 2 + 1 '' Hence Hence, putting m successively = 0, 1, 2, ... 10, cos00=P ; 3*. 3-., 2.4 ._ p _ _ p " K Q If < cos 4^ - - Q ' ' P - -^ -1.1.3.5.7.9 + 1.3.5.7 2 JL 3.5 -1- P * ) - * o __ 35 4 21 2 15 F. H. 34 ZONAL HARMONICS. 2.4.6.8 ,_ 4.6 _128 8 1 " 63 6 9 s 7 *' 2. 4. 6 8 . 8. 10 p COS UC7 = 10 i ^ ^ n T^j T7i -t 6 -1.1.3.5.7.9.11.13 1.3.5.7.9.11 P4 ~ 5 3T5 . 7.9 * 2 ~" 577 o 4.6 2 .8 p_ 5 __6L P _J_p 4 2 e 231 385 21 2 35 ' 2.4.6.8.10.12 -l. 1.3. 5. 9. 11. 13.15 P ' _ 4.6.8.10 6.8 p __3_ P 1.3.5.9.11.13 5 3.5.9.11 3 5.9 ' _1024 128 112 1 " 429 7 117 6 495 3 15 " ^ 2.4.6.8M0.12.14 17 -1.1.3.5.7.9.11.13.15.17 P8 4. 6. 8 2 . 10. 12 p q 6.8MO 1.3.5.7.9.11.13.15 6 3.5.7.9.11.13 4 Q2 -1 p. - _ 5.7.9.11 2 7.9 _ 16384 4096 p 256 p _ _64 p _ _1_ p 6435 8 3465 6 1001 4 693 * 63 ; U 2.4.6.8.10.12.14.16 -1.1.3.5.7.11.13.15.17.19^" 4.6.8.10.12.14 6.8.10.12 1.3.5.7.11.13.15.17 7 3.5.7.11.13.15 7 8-10 p_ 3 _ 3 ___ 5.7.11.13 3 7.11 ZONAL HARMONICS. 35 _ 32768 3072 128 16 p 3 "12155 ' 2431 7 455 5 143 8 77 *' .1 2.4.6.8.10M2.14.16.18 1 -1.1.3.5.7.9.11.13.15.17.19.21 P 4.6.8.10M2.14.16 7 1.3. 5. 7. 9. 11. 13. 15. 17. 19 8 6.8.10M2.14 p 8.10M2 3.5.7.9.11.13.15.17 6 5.7.9.11.13.15 4 - 10* p 1 p 7.9.11.13 9.11 _ 131072 _ 32768 _ j>12 JL28 __ 500 " 46189 * 24453 8 1683 s 1001 4 9009 2 99 ' 21. The present will be a convenient opportunity for investigating the development of sin# in a series of zonal harmonics. Since sin 6 = (1 /x 2 )^, it will be seen that the series must be infinite, and that no zonal harmonic of an odd order can enter. Assume then _ a /^PL/^P^I i/rpj_ sm = G r + O 2 r; + ... + (^ i J J i + ... i being any even integer. Multiplying by P i? and integrating with respect to \L between the limits 1 and + 1, we get rl 2 p . * , X j -1 supposing P< expressed in terms of the cosines of 6 and its multiples 32 :>(J ZONAL HARMONICS. Hence, putting i = 0, 1 3 Putting i = 2, and observing that P 2 = 7 + 7 cos TE TJ ~ 5 r (1 + 3 cos 20) (1 - cos 20) Ja (j = -T \ - - at) 4Jo 4 J^ 32 ^ For values of i exceeding 2, we observe, that if we write for Pj the expression investigated in Art. 18, the only part of the expression I P< (1 cos 20) d0 which does not vanish Jo will arise either from the terms in Pj which involve cos 20, or from those which are independent of 0. We have therefore .... ....- g ~ 2 4 '2.4... 2.4... (t- 2) [' firJ: + *^ 2 cos 2^ (1 - cos 20) d0 Jo V * * + 2 / .3... (t-3) i- 4 2.4... t 2.4...(*-2) V * 2/+11.3... (t-1) 1.3... (i'-3) 2 2.4...t( + 2)2.4...(-2)t" Hence sin0 = ^P -gp 2 - ... 2t + lw 1.3... /-J 1.3... /- 2 2. 4. .. (i + 2) 2. 4... (t-2) : i being any even integer. * ZONAL HARMONICS. 37 dP. 22. It will be seen that -~ , being a rational and integral function of /A*" 1 , //.'~ 3 ..., must be expressible in terms of Pj-u Pi_3"' To determine this expression, assume then multiplying by P m , and integrating with respect to f from 1 to + 1, L / ^ 9 -^t ^ r rZP r ^P And P m ^dp = P n P t - P^d/t. j dfj, j ap XT . . Now, since i>m, /. P. ^ = [P^.] 1 - [P.PJ-* = 2, since either m or i must be odd, and therefore either P m or P 4 = 1, when ^ = 1 ; .-. = (2i - 1) P^ + (2i - 5) P 1H1 + (2i - 9) Hence -' - - 2 = (2 - 1) P,.,. rfjtt dp 23. From this equation we deduce the limits /A and 1 being taken, in order that P 4 - P_ 8 may be equal to at the superior limit. 444685 ZONAL HAEMONICS. Now, recurring to the fundamental equation for a zonal harmonic, we see that p p * ^-*- 24. We have already seen that I P t P m dp, = 0, i and m J -i being different positive integers. Suppose now that it is required to find the value of I P t P m dp. J u. We have already seen (Art. 10) that P j P w rf/* = And, from above, dp, P ^ ZONAL HARMONICS. SO 25. "We will next proceed to give two modes of ex- pressing Zonal Harmonics, by means of Definite Integrals. The two expressions are as follows : P<= 1 {/* (/*-!) * cos*} P,=- 7T JO These we proceed to establish. Consider the equation -f TT Jo a 6 cos* (tf-V}^' The only limitation upon the quantities denoted by a and b in this equation is that 6 2 should not be greater than a 2 . For, if 6 2 be not greater than a 3 , cos * cannot become equal to -j- while * increases from to IT, and therefore the expression under the integral sign cannot become infinite. Supposing then that we write z for a, and V 1 p for 6, we get 1 We may remark, in passing, that r d Jo s V and is therefore wholly real. Supposing that p 2 = a; 2 + 2/ 2 , and that jc 2 +^ 2 + 3 2 = r a , we thus obtain ^ i 40 ZONAL HARMONICS. Differentiate i times with respect to z, and there results 1 ^L t * ^ 7T flfe'Jo Z V 1 _ p r* 1 (-1)' tf p Hence /*, _- V - . -5 4 TT 1 . 2 .3.. .1^2* Jo z- v- 1 = r^ f- ^ 7T Jo ( S _V~1 d& In this, write fir for z, and (1 /A 2 )- r for p, and we get which, writing TT ^ for ^-, gives 26. Again, we have i i r (a 8 J 2 * TT J & ( In this write 1 - jdi for a, and + (/x 2 - 1)^ li for &, which is admissible for all values of h from up to fi (/i 2 1)*, and we obtain, since a 2 6 2 becomes 1 2/A + A", + {/x Gu, 2 -!)^ cosf } * + ... + * + 2 -l lr cos ZONAL HARMONICS. Hence, equating coefficients of h\ 41 The equality of the two expressions thus obtained for P i is in harmony with the fact to which attention has already been directed, that the value of P 4 is unaltered if (i + 1) be written for *. 27. The equality of the two definite integrals which thus present themselves may be illustrated by the following geometrical considerations. Let be the centre of a circle, radius a, C any point within the circle, PCQ any chord drawn through C, and let OC=b,COP = *r,COQ = ^. Then OP 2 = a 2 + 6 2 - CQ* = a 2 + 6 2 - 2a& cos ^r. Hence (a 2 + 6 2 - 2a6 cos &) (a 2 + 6 2 - 2a5 cos f ) = (a 2 - 6 2 ) 2 ; sin ^ cfer sn - a cos _ _ ~ ' 42 ZONAL HARMONICS. Again, since the angles OP G> OQO are equal to one another, sm_0 sin OPC_ sin OQG _ sin^ CP = 00 00 '' OQ ' sin ^ sin-r (a 2 + 6 2 2a6 cos^) (a* + 6 2 - 2a& cos . = 0. (a 2 + 6 2 - 2a& cos ^)* (a* + 6 2 - 2a6 cos ' (<* In this, write a 2 + 6 2 = /i, 2a&=+(^, 2 1) $ , which gives a 2 6 2 = 1, and we get (,.0>' -!)*.}" >* ^ ^ C We also see, by reference to the figure, that as ^ in- creases from to TT, ^ diminishes from TT to 0. Hence 28. From the last definite integral, we may obtain an ex- pansion of Pj in terms of cos 6 and sin 9. Putting p = cos 6, we get 1 f P t = g- J o [{cos + V- 1 cos i/r sin 0}}* + (cos 6 V 1 cos ty sin 6}*] d-fy 7T Jt X * ZONAL HAKMOXICS. 43 (cos yry* (cog }i - 2 (gin ^ m Now , i(i-I)...(i-2m + l) (2w- 1) (2m-3)...l 1.2.. .2m 2w(2m-2)...2 a (2.4...2m) a , = (cos 6}* - *-^^- (cos 0)** (sin 0) a + . . . CHAPTER III. APPLICATION OF ZONAL HARMONICS TO THE THEORY OF ATTRACTION. REPRESENTATION OF DISCONTINUOUS FUNCTIONS BY SERIES OF ZONAL HARMONICS. 1. WE shall, in this chapter, give some applications of Zonal Harmonics to the determination of the potential of a solid of revolution, symmetrical about an axis. When the value of this potential, at every point of the axis, is known, we can obtain, by means of these functions, an expression for the potential at any point which can be reached from the axis without passing through the attracting mass. The simplest case of this kind is that in which the attracting mass is an uniform circular wire, of indefinitely small transverse section. Let c be the radius of such a wire, p its density, k its transverse section. Then its mass, M, will be equal to lirpck, and if its centre be taken as the origin, its potential at any point of its axis, distant z from its centre, will be - - r . (c 2 + z 2 )* Now, this expression may be developed into either of the following series : 1 !!^. 1 - 3 ! 4 - . V a + 4 '" + ( ' 2. 4.. . We must employ the series (1) if z be less than c, or if the attracted point lie within the sphere of which the ring is a great circle, and the series (2) if z be greater than c, or if the attracted point lie without this sphere. APPLICATION OF ZONAL HARMONICS. 45 Now, take any point whose distance from the centre is r, and let the inclination of this distance to the axis of the ring be 0. In accordance with the notation already em- ployed, let cos 6 =*= p, Then, the potential at this point will be given by one of the following- series : 2 V 2.4 2. 4.0... 2i ' 2.4.6...2 p For each of these expressions, when substituted for V, satisfies the equation V 2 V = 0, and they become respectively equal to (1) and (2) when 6 is put = 0, and consequently 7* = z. The expression (2') also vanishes when r is infinitely great, and must therefore be employed for values of r greater than c, while (!') becomes equal to (2') when r = c, and will therefore denote the required potential for all values of r less than c. These expressions may be reduced to other forms by means of the expressions investigated in Chap. 2, Art. 25, viz. P t = + ^l cos t = i [ vr Jo or P = * + 1 cos Substitute the first of these in (!') and (observing, that r = z] we see that it assumes the form . 4 274 46 APPLICATION OF ZONAL HARMONICS which is equivalent to b/ : The substitution of the last form of P i in the series (2') brings it into the form 1O which is equivalent to M fir IT Jo 2. Suppose next that the attracting mass is a hollow shell of uniform density, whose exterior and interior bounding surfaces are both surfaces of revolution, their common axis being the axis of z. Let the origin be taken within the interior bounding surface ; and suppose the potential, at any point of the axis within this surface, to be Then the potential at any point lying within the inner bounding surface will be A P + AJPf + Aff + . . . + A.P/ + ... For this expression, when substituted for V, satisfies the equation V 2 F=0; it also agrees with the given value of the potential for every point of the axis, lying within the inner bounding surface, and does not become infinite at any point within that surface. Again, suppose the potential at any point of the axis without the outer bounding surface to be TO THE THEORY OF ATTRACTION. 47 Then the potential at any point lying without the outer bounding surface will be *>+? + *+ + s < p <+ r r* r 3 " r m "" For this expression, when substituted for V, satisfies the equation V 2 V= ; it also agrees with the given value of the potential for every point of the axis, lying without the outer bounding surface, and it does not become infinite at any point within that surface. By the introduction of the expressions for zonal har- monics in the form of definite integrals, it will be found that if the value of either of these potentials for any point in the axis be denoted by < (z), the corresponding value for any other point, which can be reached without passing through any portion of the attracting mass, will be , f , of A ^ -> cos , 3. We may next shew how to obtain, in terms of a series of zonal harmonics, an expression for the solid angle subtended by a circle at any point. We must first prove the following theorem. The solid angle, subtended by a closed plane curve at any point, is proportional to the component attraction perpendicular to the plane of the curve, exercised upon the point by a lamina, of uniform density and thickness, bounded by the closed plane curve. For, if dS be any element of such a lamina, r its distance from the attracted point, 6 the inclination of r to the line perpendicular to the plane of the lamina, the elementary solid angle subtended by dS at the point will be dS cos And the component attraction of the element of the lamina corresponding to dS in the direction perpendicular to its plane will be pk - cos 6, 48 APPLICATION OF ZONAL HARMONICS p being the density of the lamina, Ic its thickness. Hence, for this element, the component attraction is to the solid angle as pk to 1, and the same relation holding for every element of the lamina, we see that the component attraction of the whole lamina is to the solid angle subtended by the whole curve as pk to 1. Now, if the plane of acy be taken parallel to the plane of the lamina, and V be the potential of the lamina, its component attraction perpendicular to its plane will be -T-. Now since Fis a potential we have V 2 F=0, whence , or V=0. Hence ~ is itself a potential, dz \dz ) dz and satisfies all the analytical conditions to which a potential is subject. It follows that, if the solid angle subtended by a closed plane curve at any point (x, y, z) be denoted by w, o) will be a function of x, y, z, satisfying the equation V 2 = 0. Hence, if the closed plane curve be a circle it follows that the magnitude of the solid angle which it sub- tends at any point may be obtained by first determining the solid angle which it subtends at any point of a lint- drawn through its centre perpendicular to its plane, and then deducing the general expression by the employment of zonal harmonics. Now let be the centre of the circle, Q any point on the line drawn through perpendicular to the plane of the circle, E any point in the circumference of the circle. With centre Q, and radius QO, describe a circle, cutting QE in L. From L draw LN, perpendicular to Q 0. Then EL = (c* + *>)* - ,, ON- ~ {(c' + rf - z} ~ TO THE THEORY OF ATTRACTION. 49 And the solid angle subtended by the circle at (^ ON = 4<7T --- -Irl, To obtain the general expression for the solid angle sub- tended at any point, distant y from the centre, we first dcvelope this expression in a converging series, proceeding by powers of z. This will be c 2c 3 2.4c 5 if z be less than c, and I A V A * V V x - x^J-.tF^BBl^v JL I \s )9^ 2 ~~9~~rr 4+ '"~~^ ' 2.4...2i ? s j" if 2 be greater than c. Hence, by similar reasoning to that already employed, we get, for the solid angle subtended at a point distant r from the centre, Yf . 2 c 3 2.4 c 5 _ 1^.. (2i-l) P^r^ v. / 94, 9," ^'^ T _ . T? . . . t c/ if r be less than c, and oflilV: * ' ** ^ c i / -ty 1 . 3...(2t l)P 8M c'' *[2 r a 274 7- 4 l ' 2.4...2 r" if ?* be greater than c. 4. We may deduce from this, expressions for the potential of a circular lamina, of uniform thickness and density, at an external point. For we see that, if V be the potential of such a lamina, k its thickness, and p its density, we have for a point on the axis, F. H. 50 APPLICATION OF ZONAL HARMONICS whence V if M be the mass of the lamina. Expanding this in a converging series, we get T7 _J/f Is? l.lz 4 1.1. 3-8" -" 4 ~~ 8 + s ~ _ (_ 1 V 'f>"-\^- -*) * , ^ ' 2.4.6...2i c^ 1 if z be less than c, and l/[lc' Lie 4 1.1. 3c 8 c z \Zz 2. 4 z 3 2.4.63 s .....*- 1 ' 2.4.6...2i if z be greater than c. Hence we obtain the following expressions for the po- tential of an uniform circular lamina at a point distant r from the centre of the lamina : v .....- 1 ; " 2.4.6...2 " c"- 1 " if r be less than c, and F= W^- 1 -! 1 P ^ t 1 ' 1 - 3 5!- C 2 12 r 2.4 r 3 + 2.4.6 r* " 1.1.3...(2/-3)P 2 ^c 2i "^ ; 2.4.6...2i r 2 '" 1 if r be greater than c. TO THE THEORY OF ATTRACTION. 51 It may be shewn that the solid angle may be expressed in the form d0, and the potential of the lamina in the form 2 JJ/" 2 f* -r- J 5. As another example, let it be required to determine the potential of a solid sphere, whose density varies inversely as the fifth power of the distance from a given external point at any point of its mass. It is proved by the method of inversion (see Thomson and Tait's Natural Philosophy, Vol. 1, Art. 518) that the potential at any external point P' will be equal to , p> , 0' being the image of in the surface of the sphere, and M the mass of the sphere. We shall avail ourselves of this result to determine the potential at a given internal point. Let C be the centre of the sphere, the given external point. Join CO, and let it cut the surface of the sphere in A, and in CA take a point 0', such that 00.00= CA*. Then 0' is the image of 0. Let P be any point in the body of the sphere, then we wish to find the potential of the sphere at P. Take as pole, and OC as prime radius, let OP = r, POC = 0. Also let CA = a, CO = c. Let the density of the sphere at its centre be p, then its density at P will be p 6 . Hence 42 .")2 APPLICATION OF ZONAL HARMONICS the limits of r being the two values of r which satisfy the equation of the surface of the sphere, viz. and those of 6 being and sin" 1 - . c Hence, if r,, ?* 8 be the two limiting values of r, w r e have 2 J f \jr* r') K 1 1 2ceos0 /I 1\ Now -, a = -, r ) . 'a i\ c a \r a rj 1 1 2c cos 6 Also + = - c' - a 8 ' 1 1 f * t I'S r,r Q c a _ * c'-a" = 2 ^-" 2cin2 " 1f 2-rrpC 5 2c 2 fsin-V rf. I/ . t*\l .'. M = - . -y cos ^ sin 6 (a" c sin 0) 5 v ^>* ^* /* /-f* f ^ ^ -^ C d C ft j o 47TDC 6 /"sin- 1 ^ = 7-3- 2 \i> I cos 6* sin a (a c sm (c-- 2 ) 2 7 > / ~i *N? o (c a ) Now, if F be the potential at P, we have (see Chap. I. Art. 1) cT(rV] 1 d f . a dV\ 4-Trpc 5 TO THE THEORY OF ATTRACTION. 2 7TOC 5 This is satisfied by V= -= j-. Assume then, as the complete solution of the equation, It remains to determine the coefficients A , A^^A.^.B^, jPj...^, so that this expression may not become infinite for any value of r corresponding to a point within the sphere, and that at any point P on the surface of the sphere it may be equal to 77-^, where O'P : OP :: a : c, and therefore, at the surface, a OP 3( 2 85, this expression be- comes Or, the density may be obtained by dividing the alge- braic sum of the normal component attractions on two points, one external and the other internal, indefinitely near the sphere, and situated on the same normal, by -ivr x thickness of the shell. 7. It follows from this that if the density of a spherical shell be expressed by the series C , (7j, C^ ... C { ... being any constants, its potential (FJ at an internal point will be" . f " '" b 3~P~ 5 ~" and its potential ( F 2 ) at an external point will be (C ICjy IC a PJ>* I G t P}t \ J \ r + 3 ^ + 5~^~" + 2l+I r i+1 ~ '")' In the last two Articles, by the word " density" is meant "volume density," i.e. the mass of an indefinitely small element of the attracting sphere, divided by the volume of TO THE THEORY OF ATTRACTION. 57 same element. The product of the volume density of element of the shell, into the thickness of the shell in the i/eighbourhood of that element, is called "surface den- sity^' We see from the above that, if the surface density expressed by the series *\ p "o ( ~ + K Hr") a* an external point ; S/y z 1 t \^ T* ~rt 9 OUi ~- 4- o OS ~~ / i , or, since P/ 2 = - - ^ r = - , we obtain JL L 7T /I 1 *J^2 _ 2\ ' ) for the potential at an internal point, r + ^ A- r ^ T 7, <- 3 [r 4- =-{ 3 H r ) r for that at an external point. 5 V r r J) 9. As an example of the case in which the density is re- presented by an infinite series of zonal harmonics, suppose we Avish to investigate the potential of a spherical shell, whose density varies as the distance from a diameter. Taking this diameter as the axis of z, the density will be represented by p sin 9, or p (1 /A 2 ) 2 . We have investigated in Chap. II. Art. 21, the expansion of sin 6 in an infinite series of zonal harmonics. Employing this expansion, we shall obtain for the potential __ 2 b 2 16 2 b* 2.4.. . (i+2) ' 2 A . . (i-2>" or 7T r 7 flP _J L p& 2 _ 1.3...(t-l) 1.3...f*-3) p ft* 2 p (2 r 16 V ' 2.4...t(i+2)'2A..(t-2)i V m '"] according as the attracted point is internal or external to the spherical shell, i being any even integer. All these expres- sions may be obtained in terms of surface density, by writing, instead of p U, 47rcV. 10. We may next proceed to shew how the potential of a spherical shell of finite thickness, whose density is any solid zonal harmonic, may be determined. Suppose, for instance, that we have a shell of external radius a, and internal radius a, whose density, at the distance c from the centre, is -j-i P 4 c', h being any line of constant length. Dividing the sphere into concentric thin spherical shells, of thickness dc } the potential of any one of these shells, of TO THE THEORY OF ATTRACTION. 59 radius c, at an internal point distant r from the centre will DC* be obtained by writing c for b, jr for C, ^jrc'dc for U, in the first result of Art. 6. This gives P 4>7TC 2 dc PjCV 4-7T P -p, < 7 or T li 1 A* To obtain the potential of the whole shell, we must inte- grate this expression, with respect to c, between the limits a and a. This gives 27T pP 2i +1 A,* Again, the potential of the shell of radius c, at an external point, will be _ * . QJ. _ p ^Q Integrating as before, we obtain for the potential of the whole shell, 4?r p p (a z ' +3 a /arl ~ 3 ) \IS)' i Q\ T"i m \+l ' Suppose now that we wish to obtain the potential of the whole shell at a point forming a part of its mass, distant r from the centre. We shall obtain this by considering sepa- rately the two shells into which it may be divided, the extern al radius of the one, and the internal radius of the other, being each r. Writing r for a, in the first of the fore- going results, we obtain 2-7T pP i . 2 2 . , x- . T ^TY (a r ) r. 2i + l h* ^ And writing r for a in the other result, we obtain _ 3) A* " r' +1 * Adding these, we get for the potential of the whole sphere 69 APPLICATION OF ZONAL HARMONICS It is hardly necessary to observe that the corresponding results for a solid sphere may be obtained from the foregoing, by putting a = 0. If the density, instead of being ~ P i c l , be ^ P t c m , similar reasoning will give us, for the potential of the thin shell of radius c and thickness dc at an internal and external point respectively, 4T P -n i ri -m j j 4?r p D c <+m+2 , p > r c dc > and *'-"-*- And, integrating as before, we obtain for the potential of the whole shell, p ' a """ ~ a """ r> at an intcraal 4nr p a m * i+3 a'" >+i+3 - f(t . , 1W , , ^i^Pi - - at an external point. (2t + 1) (m + 1 + 3) h r" And, at a point forming a part of the mass, ti r m+l+3 _ 2i + 1 "P" V m-i + 2 m + i + 3 11. Suppose, for example, that we wish to determine, in each of the three cases, the potential of a spherical shell whose external and internal radii are a, a, respectively, and whose density varies as the square of the distance from a diametral plane. Taking this plane as that of xy, the density may be ex- pressed by pz 2 , or -c> 2 - Now /i, 2 = ^ . Hence the density of this sphere may be expressed as The several potentials due to the former term will be, 2 writing 2 for i and multiplying by ^ , TO THE THEOEY OF ATTRACTION. 01 ~ - - -- 1.5 "1? '105/i 2 2 r s ' 15 A* , 2 3 ' And for the latter term, writing for i, and 2 for m, and multiplying by K, f jx A' I* M \AJ *-l -*- 14 N* / W- """" / / r f 4 ^4\ r r I I 12 /t 2 ^ ^ ^' 15 A 2 r 3 A* V 4 5r And, since P/ 2 = , we get for the potential at an 3! internal point co P I -^ f 2 11 at an external point -^TT / o (\ /o o o\ , / 4 / 16 (a " ~ a ^ ( ~ ^ + 3 ^ at a point forming a part of the mass p J47T /a 2 - r 2 r 7 -a'^ f n 2N ^TT /a 4 -r 4 r s -a^ J ~~ ^~ ( T } + ~ 15 12. We may now prove that by means of an infinite series of zonal harmonics we may express any function of p what- ever, even a discontinuous function. Suppose, for instance, that we wish to express a function which shall be equal to A from /z, = 1 to /i = X, and to B from p,=\ to /A= 1. Consider what will be the potential of a spherical shell, radius c, of uniform thickness, whose density is equal to A for the part corresponding to values of /n between 1 and X, and to B for the part corresponding to values of p between A, and 1. Divide the shell, as before, into indefinitely narrow strips by parallel planes, the distance between any two successive planes being G2 APPLICATION OF ZONAL HAEMONICS We have then, for the potential of such a sphere at any point of the axis, distant z from the centre, for the first part of the sphere and for the latter part d/j, f -i(c 2 +* 2 -: These are respectively equal to c at an internal point ; and to at an external point. Now it follows from Chap. II. (Art. 23) that if i be any positive integer, whence, since I P 4 cZyti = 0, it follows that '-i TO THE THEORY OF ATTRACTION. 63 Also f P d/* = l-X, pp ^= J A. J _i Hence the above expressions severally become : For the potential at an internal point on the axis 27rc 2 5c r -A- 13 z c . 3 2 c A B s 2 and for the potential at an external point on the axis { P,(X) -P O( X)) , sa Hence the potentials at a point situated anywhere are respectively at an internal point; 04 APPLICATION OF ZONAL HARMONICS and [{A (1 - X) + B(l + X)} at an external point. Now, if we inquire what will be the potential for the following distribution of density, - X) + 5(1 + X) - (A-3)[P,(\) - P (X)}P we see by Art. 6 that it will be exactly the same, both at an internal and for an external point, as that above in- vestigated for the shell made up of two parts, whose densities are A and B respectively. But it is known that there is one, and only one, dis- tribution of attracting matter over a given surface, which will produce a specified potential at every point, both ex- ternal and internal. Hence the above expression must represent exactly the same distribution of density. That is, writing the above series in a slightly different form, the expression -^ X + TO THE THEORY OF ATTRACTION. 65 is equal to A, for all values of /tt from 1 to X, and to B for all values of p from X to 1. 13. By a similar process, any other discontinuous function, whose values are given for all values of p from 1 to 1, may be expressed. Suppose, for instance, we wish to express a function which is equal to A from fju = 1 to p = \, to B from p = \ to fj, = X 2 , and to C from //. = X 2 to /z. = 1. This will be obtained by adding the two series 2^-^ir t\+ t p A) -PO c\)) p + - +], For the former is equal to A B from //, = 1 to /u- = X t , and to from ^ = \ to /A = 1 ; and the latter is equal to B from p = 1 to p = X 2 , and to C from p = X 2 to p, = 1. By supposing A and (7 each = 0, and B = 1, we deduce a series which is equal to 1 for all values of /x from ft = \ to fj, = X 2 , and zero for all other values. This will be ix - \ This may be verified by direct investigation of the potential of the portion of a homogeneous spherical shell, of density unity, comprised between two parallel planes, distant respectively c\ and cX 2 from the centre of the spherical shell. 14. In the case in which \ and X 2 are indefinitely nearly equal to each other, let X a = X, and \ = X + d\. We then have, ultimately, F. H. 66 APPLICATION OF ZONAL HARMONICS Hence P^^XJ - P,. +1 (X 2 ) - P^XJ - P^XJ (^(X) dP<_,(\n , = 1 -JT. ^T r "X (^ aA. a\ Hence the series 5 ^ {1 + 3P 1 (X)P + 5P S (X)P + pU yj is equal to 1 when /* = X (or, more strictly, when- fj, has any value from X to X + cX) and is equal to for all other values of fj,. We hence infer that i + UVNP.OO + ... + (2 + i)P,(x)P,(M) -f ... v is infinite when /A = X, and zero for all other values of p. 15. Representing the series for the mo-ment, we see that p(\)d\ is equal to p when /i = X, and to zero for all other values. Hence the expression is equal to p t when /4 = X^ to p 2 when //, = X,;'.. Supposing now that \ t X a ... are a series of values varying continuously from 1 to 1, we see that this expression becomes f / * -i p being any function of X, continuous or discontinuous. Hence, writing (X) at length, we see that I If 1 is equal, for all vAlues of /A from 1 to -f 1, to the same function of /n that p is of X. TO THE THEOEY OF ATTRACTION. 67 16. The same conclusion may be arrived at as follows : The potential of a spherical shell, whose density is p, and volume U, at any point on the axis of z, is -f 2J-,, pd\ TJ z which is equal to ^- -I I pdX + - + ^l l pP i (\)d\+...\, c J-i ) for an internal point, and to ft /* for an external point. It hence follows that the potential, at a point situated anywhere, is for an internal point, and to r i for an external point. And these expressions are respectively equal to those for the potentials, at an internal and external point re- spectively, for matter distributed according to the following law of density : 52 68 APPLICATION OF ZONAL HAKMONICS, &C. It will be observed, in applying this formula, that if p be a discontinuous function of \, each of the expressions of the form I pPj(\)d\ will be the sum of the results of a series of J -i integrations, each integration being taken through a series of values of X, for which p varies continuously. CHAPTER IV. SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SEC- TORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXIS IN ANY POSITION. POTENTIAL OF A SOLID NEARLY SPHERICAL IN FORM. 1. WE have hitherto discussed those solutions of the equation V 2 F=0 which are symmetrical about the axis of z, or in other words, those solutions of the equivalent equation in polar co-ordinates- which are independent of <. We propose, in the present Chapter, to consider the forms of spherical harmonics in general, understanding by a Solid Spherical Harmonic of the I th degree a rational integral homogeneous function of x, y, z, of the I th degree which satisfies the equa- tion V V= 0, and by a Surface Spherical Harmonic of the i th degree the quotient obtained by dividing a Solid Sphe- rical Harmonic by (of + ?/ 2 -f z*)' 2 . Such an expression, as we see by writing sc = r sin 6 cos , y r sin 9 sin <, z = r cos 6, will be of the *** degree in sin cos $, sin 6 sin <, cos 0; and will satisfy the differential equation in Y if 1 d /. ~dY\ L 1 tf Y. , .,. _.._, -! - Q -j- Q sm 0-^ ) + -7-^ -j +t(t+i) r**o. sin ^ ad \ ^ / Ean 1 6 dp or, writing /i for cos 0, d It will be convenient, before proceeding to investigate the algebraical forms of these expressions, to discuss some of their simpler physical properties. 2. We will then proceed to shew how spherical har- monics may be employed to determine the potential, and 70 SPHERICAL HARMONICS IN GENERAL. consequently the attraction, of a spherical shell of indefinitely small thickness. We will first establish an important theorem, connecting the potential of such a shell on an external point with that on a corresponding internal point. The theorem is as follows : If be the centre of such a shell, c its radius, P any in- ternal point, P' an external point, so situated that P' lies on OP produced, and that OP . OP' = c 2 , and if OP = r, OP' = r', then the potential of the shell at P is to its potential at P 7 as c to r, or (which is the same thing) as r' to c. For, let A be the point where OP' meets the surface of the sphere, Q any other point of its surface. Then, by a known geometrical theorem, QP: QP ::AP:AF::c-r: r'-c. c r _ cr r* _cr r 2 r c ~ ; ^ T, == ^ ~~j , r c rr cr c cr c r V> - * Again, considering the element of the shell in the im- mediate neighbourhood of Q, its potential at P is to its potential at P as QP is to QP, that is, as c to r, or (which is the same thing) as r' to c, which ratio, being independent of the position of Q, must be true for every element of the spherical shell, and therefore for the whole shell. Hence the proposition is proved. 3. Now, suppose the law of density of the shell to be 7*' such that its potential at any internal point is F(p, <) -^. r 1 Then F(ii, ) must be a solid harmonic of the degree i. c Hence F (JJL, <) must be a surface harmonic of the degree i. Let us represent it by Y { . By the proposition just proved, the potential at any external point, distant ?' from the centre, must be <+1 Y * * TESSERAL AXD SECTORIAL HARMONICS. 71 Hence, the component of the attraction of the sphere on the internal point measured in the direction from the point inwards, i. e. towards the centre of the sphere, is -**;-,-. And the component in the same direction of the attraction on the external point, measured inwards, is (N-l) r,;. Now suppose the two points to lie on the same line passing through the centre of the sphere, and to be both indefinitely close to the surface of the sphere, so that r and r are each indefinitely nearly equal to c. And the attraction on the external point exceeds the attraction on the internal point by Y ff)S i 1\ ^ * Now, supposing the shell to be divided into two parts, by a plane passing through the internal point perpendicular to the line joining it with the centre, we see that the at- traction of the larger part of the shell on the two points will be ultimately the same, while the component attractions of the smaller portions, in the direction above considered, will be equal in magnitude and opposite in direction. Hence the Y. difference between these components, viz. (2i + 1) , will be c equal to twice the component attraction of the smaller portion in the direction of the line joining the two points. But if p t be the density of the shell, So its thickness, this component attraction is 2 < 7rp i c. - 7 Hence (2t + l) '- 1 c or 72 SPHERICAL HARMONICS IN GENERAL. And, if cr { be the corresponding surface density, * ~^d^> It hence follows that if the getenbtafrof a spherical shell, of indefinitely small thickness, be a siirface harmonic, its potential at any internal point will be proportional to the corresponding solid harmonic of positive degree, and its po- tential at any external point will be proportional to the corresponding solid harmonic of negative degree. That is, the proposition proved for zonal harmonics in Chap. in. Art. 6, is now extended to spherical harmonics in general. 4. The spherical harmonic of the degree i will involve 2i + 1 arbitrary constants. LJ-, For the solid spherical harmonic, ^Y^ being a rational integral A function of x, y, z of the i th degree, will consist of ~ terms. Now the expression V s F, being a rational integral function of x, y, z of the degree i 2, will (i 1) i consist of - ~- terms ; and the condition that it must be 2 = for all values of x, y, z, will give rise to \ - relations 25 (i+ i) f{ + 2) among the - ~ coefficients of these terms, leaving ~ , or 2i + 1, independent coefficients. 5. We proceed to shew how the spherical harmonic of the degree i may be arranged in a series of terms, each of which may be deduced by differentiation from the Zonal Harmonic symmetrical about the axis of z. The solid zonal harmonic, which, in accordance with the notation already employed, is represented by r i P i (//.), is a function of z and r of the degree i, satisfying the equation V 2 V= 0, or -7-^ + ^- 2 - + -j-^ = 0. dx cly dz Now, if we denote this expression by P t (z}, we see that TESSERAL AND SECTORIAL HARMONICS. 73 since it is a function of z and r, it is a function of the dis- tance (z) from a certain plane passing through the origin, and of the distance (r) from the origin. Further, if we write for z the distance from any other plane passing through the origin, leaving r unaltered, the equation -r-$ + -7-2- + -1-3 = will continue to be satisfied. Now z + a(x + *Jly), a being any quantity whatever, represents the distance from a certain plane passing through the origin, since in this expression, the sum of the squares of the coefficients of z, x, y is equal to unity. Hence P i {z + a (# + */ !#)} is a solid zonal harmonic of the CC 7/ degree i, its axis being the imaginary line - = ^= z - Therefore the equation cfo 2 dy* dz* is satisfied by V=P { {z + a (x + V !#)}, that is, expanding by Taylor's Theorem, it is satisfied by /\, / T ^iz / r- P, (z) +a (x + V-ly) ^- ; + j-^ (* + V- a-(a; + /or aZ? vaZwes o/ a. Hence, since the equktion in V is linear, it follows that it is satisfied by each term separately, or that, besides P i (z) itself, each of the i expressions, satisfies the equation V=Q. By similar reasoning we may shew that each of the i ex- pressions, satisfies the same equation. 74 SPHERICAL HARMONICS IX GENERAL. Now each of the 2& solutions, thus obtained, is imaginary. But the sum of any two or more of them, or the result obtained by multiplying any two or more by any arbitrary quantities, and adding the results together, will also be a solution of the equation. Hence, adding each term of the first series to the corresponding term of the second, we ob- tain a series of i real solutions of the equation. Another such series may be obtained \>j subtracting each term of the second series from the corresponding term of the first, and dividing by V 1. We have thus obtained (including the original term P t (z)) a series of 2/+1 independent solutions of the given equation, which will be the 2i + 1 independent solid harmonics of the degree t. 6. We may deduce the surface harmonics from these by writing r sin 6 cos for x, r sin 6 sin for y, r cos 9 for z, and dividing by r*. Then, putting cos 9 = p, and observing that P t (a) = r'P, (/), = r < . . . . we obtain the fol- lowing series of 2 + 1 solutions : , , ,.. , dp dp' Off Q dPM . . , a d?PM . ., . ta d l PM sin 6 sin 6 - -~^- , sin 26 sm 2 6 , -^ , . . . sm z< sin* 6 ' \ f - . dp dp dp* Expressions of .the form or S sin (/*) or ^(cos(9). It will be convenient, for the purpose of comparison with the forms of Tesseral Harmonics given in the Mecanique Celeste, and elsewhere, to obtain T^ in a completely de- veloped form. 1 d'tu? IV Now, since P,0) =157; s~^ - ^~* > we see that !j . 1 . Z . O . . . uLt dp? 2'. 1.2. 3...* /^ i+, by the symbol P (/&), and calls these expres- sions by the name Zugeordnete Functioned Erster Art (Hand- buck der Kugelfunctionen, pp. 117, 118) which Todhunter translates by the term "Associated Functions of the First Kind," which we shall adopt. Heine also represents the series (i-o-Xt'-g-lHi-o 2)C-a 3) ._^ 4 2.4(2t-l)(2i-3) ^ by the symbol f^(^), (p. 117). The several expressions, 2^'J, 0j), ^>, P^, -JJJ, are con- nected together as follows : - g-fa) _ / -i\ a pi _ a _ a 1 *" 8. It has been already remarked that the roots of the equation P i = are all real. It follows also that those of the dP equations -j-^O, -y-^ =0... are real also. Hence we may arrive at the following conclusions, concerning the curves, traced on a sphere, which result from our putting any one of these series of spherical harmonics = 0. By putting a zonal harmonic =0, we ob taint small circles, whose planes are parallel to one another, perpendicular to 78 the axis of the zonal harmonic, and symmetrically situated with respect to the diametral plane, perpendicular to this axis. If i be an odd number this diametral plane itself becomes one of the series. By putting the tesseral harmonic of the order cr=0, we obtain i a small circles, situated as before, and cr great circles, determined by the equation cos crc/> = 0, or sin cr = 0, as the case may be, their planes all intersecting in the axis of the system of harmonics, the angle between the planes of any two consecutive great circles being - . By putting the sectorial harmonic = 0, we obtain i great circles, whose planes all intersect in the axis of the system, the angle between any two consecutive planes being IT 9. The tesseral harmonic may be regarded from another point of view. Suppose it is required to determine a solid harmonic of the degree i, and of the form Yp', such that Y t shall be the product of a function of /*, and of a function of c/>, which functions we will denote by the symbols M it <& i} respec- tively. The differential equation, to which this will lead, is d Now this will be satisfied, if we make 3I t and , satisfy the following two equations : The latter equation gives 4> 4 = C cos cr< + C' sin + C' T& sin a. 10. In Chap. II. Art. 10 we have established the fundamental property of Zonal Harmonics, that if * and m be two unequal positive integers, ! P t P m diJ, = 0. This is a particular case J "~1 of the general theorem that if Yj, Yj,, be two surface har- monics of the degrees i and m respectively, TESSERAL AXD SECTORIAL HARMONICS. 79 Y (J \i+* former is satisfied by M t = T<"\ i. e. (1 - p?}*( f- (1 - /i')' , \<*/v as we proceed to prove. We know that Differentiate will be investigated here- after. 11. We may hence prove that if a function of fi and < can be developed in a series of surface harmonics, such de- velopment is possible in only one way. For suppose, if possible, that there are two such develop- ments, so that and also Then subtracting, we have 0= Y 9 - F '+ ?;- F/+...+ F f - Y; + ... identicaUy. Now, each of the expressions Y 9 - Y ', Y t -Y{... Y t - Y t ' being the difference of two surface harmonics of the degree 0, 1, ... i... is itself a surface harmonic of the degree 0, 1, ........ Denote these expressions for shortness by Z ...Z... so that ... identically. Then, multiplying by Z t and integrating all over the surface of the sphere, we have f 1 f 2i 0= J -iJ o That is, the sum of an infinite number of essentially positive quantities is = 0. This can only take place when each of the quantities is separately = 0. Hence Z. is identi- cally = 0, or Y t ' = Y { , and therefore the two developments are identical. We have not assumed here that such a development is always possible. That it is so, will be shewn hereafter. TESSERAL AND SECTORIAL HARMONICS. 83 12. By referring to the expression for a surface har- monic given in Art. 4, we see that each of the Tesseral and Sectorial Harmonics involves (1 //)^, or some power of (1 fj?"fi, as a factor, and therefore is equal to when fj, = 1. From this it follows that when //, = + !, the value of the Surface Harmonic is independent of , or that if Y (/*, ) repre- sent a general surface harmonic, Y( 1, <) is independent of tf>, and may therefore be written as F(+ 1). Or F(l) is the value of Y(JJS>, <) at the pole of the zonal harmonic P^}, Y( 1) at the other extremity of the axis of P t (/*). We may now prove that r2* 'o For, recurring to the fundamental equation, a f Jo Now, if we integrate this equation with respect to , between the limits and 2?r, we see that, since and the value of Y i only involves $ under the form of cosines or sines of and its multiples, and therefore the values of --r* are the same at both limits, it follows that /: Hence trl *. fZn Hence Y t d^> is a function of p which satisfies the Jo fundamental equation for a zonal harmonic, and we therefore have 62 84? SPHERICAL HARMONICS IN GENERAL. f Jo '0 C being a constant, as yet unknown. To determine C, put /*=!, then by the remark just made, Y t becomes F,(l), and is independent of 0. Hence, when /* = !, P r F 4 d0 = 27rr j (l). AlsoP 4 (/a) = l. We have there- fore o It follows from this that 13. We may now enquire what will be the value of ri rz* J -iJo Y i} Z. being two general surface harmonics of the degree i. Suppose each to be arranged in a series consisting of the zonal harmonic P, whose axis is the axis of z, and the system of tesseral and sectorial harmonics deduced from it. Let us represent them as follows : iTp cos + C t TV> cos 20 + ... -f C T^ cos o-< + ... + CiTWcosty ^ sin 20 + ... + 8 9 T

sino-0 + ... on cos + c 2 r) cos 20 + . . . t CeTjd cos o-0 + ... sn + . . . + s v sn cr Hence the product Y]^ will consist of a series of terms, in which will enter under the form cos d=\ si Jo Jo = TT. Hence the question is reduced .to the determination of the value of Now T.W = (1- /-I _ 2\li ^ ; But, by the theorem of Eodrigues, proved in Chap. II. Art. 8, we know that Hence T^ may also be expressed under the form _?. fji- = j 2 . sii TESSERAL AND SECTORIAL HARMONICS. 87 lfB,. [ a* (Ch 14. We have hitherto considered the Zonal Harmonic under its simplest form, that of a " Legendre's Coefficient " in which the axis of z, i. e. the line from which 6 is measured, is the axis of the system. We shall now proceed to consider it under the more general form of a "Laplace's Coefficient," in which the axis of the system of zonal harmonics is in any position whatever, and shall shew how this general form may be expressed in terms of P i (/u.) and of the system of Tesseral and Sectorial Harmonics deduced from it. Suppose that ff, are the angular co-ordinates of the axis of the Zonal Harmonic, i.e. that the angle between this axis and the axis of z is ff, and that the plane containing these two axes is inclined to a fixed plane through the axis of z which we may consider as that of zx, at the angle '. In accordance with the notation already employed, we shall represent cos & by //. The rectangular equations of the axis of this system will be * = y i = z sin & cos <>' sin & sin <' cos & ' Hence the Solid Zonal Harmonic of which this is the axis is deduced from the ordinary form of the solid zonal har- monic expressed as a function of z and r by writing, in place of z, x sin & cos <' + y sin & sin + z cos 6 ' . To deduce the Surface Zonal Harmonic, transform the solid zonal harmonic to polar co-ordinates, by writing r sin cos (j) for x, r sin 6 sin for y, r cos 6 for z, and divide by r*. The transformation from the special to the general form of surface zonal harmonic may be at once effected, by substituting for p, or cos 6, cos cos 0'+ sin sin ^cos(0 <'). Now, in order to develope P t (cos cos ff + sin 9 sin ff cos ( $')} 88 SPHERICAL HARMONICS IN GENERAL. in the manner already pointed out, assume Pf (cos 6 cos ff + sin 9 sin & cos (0 0')} = AP t (fi) + (GW cos + S sin 0) ZJ + ( C cos o-0 + <> sin o-0) T J -i/o and therefore f 1 f 2 ' P, (cos e cos ^ + sin 6 sin 0' cos (A - <')} cos TV>dpd J -L/O Hence or (7^ = 2 + (f Similarly S& = 2 L^^ sin o-f ^*> (//). And to determine -4, we have ; f IP, (cos cos 0' + sin 6 sin ^ cos ( - <') } P, 0*) dpd J -iJ o or A = PM. Hence, P i (cos cos 0' + sin 9 sin ^ cos (<^> <' = p, M p, (/,) + 2 cos (0 - ') 1 T~ X 90 SPHERICAL HARMONICS IN GENERAL. 15. We have already seen (Chap. n. Art. 20) how any rational integral function of /z, can be expressed by a finite series of zonal harmonics. We shall now shew how any rational integral function of cos 6, sin cos , sin 6 sin , can be expressed by a finite series of zonal, tesseral, and sectorial harmonics. For any power of cos or sin , or any product of such powers, may be expressed as the sum of a series of terms of the form cos cr<, or sin cr<, the greatest value of a- being the sum of the indices of cos $ and sin , and the other values diminishing by 2 in each successive term. Hence any rational integral function of cos 9, sin 6 cos , sin 6 sin , will consist of a series of terms of the form cos B sin" 6 cos to the sum of a series of terms of the form cos 1 " sin " cos o-^>, or, writing cos = p, of the . Similarly cos m sin" sin c is reduced to a series of IT terms of the form p? (1 p*) 2 sin o-<. and fj, p+0 can be developed in a series of terms of the form of multiples of P p+v> P p + ff -z .... (Chap. n. Art. 17.) Hence \i? can be expressed in a series of the form d" (A Q TESSERAL AND SECTORIAL HARMONICS. 91 A , A t representing known numerical constants, and therefore a fjf (1 /i 2 )* assumes the form (A Tp+ and ^ (1 /J?} * sin a$ in series of tesseral harmonics. 16. We will give two illustrations of this transformation. First, suppose it is required to express cos 2 6 sin 2 sin < cos in a series of Spherical Harmonics. Here we have sin cos < = ^ sin 2<. 9 Hence cos 2 sin 2 sin $ cos < = - cos 3 sin 2 # sin 2<. 2 Comparing this with cos"* sin" sin cr<, we see that n is not greater than cos < 92 SPHEEICAL HARMONICS IN GENERAL. Next, let it be required to transform cos 3 # sin 3 6 sin cos 2 < into a series of Spherical Harmonics. Here sin cos 2 < = ^ sin 2$ cos < = -r (sin 3< + sin <). 4 Now cos 8 sin 3 sin 3< = //. 8 (1 - /i 2 ) ' sin 3< Also cos 3 6 sin 3 sin = /*' (1 - ^ 2 ) (1 - ^ 2 )^ sin Also (Chap. Ill Ait. 17) ^ = |P 4 ^P 2 + |P , ^.ilp.^p.io i "231 6 + 77 4+ 21 2 + 7 * Hence cos 3 6 sin 3 sin 3 I6^ i 24^PA } + 1 ^ J ' 385 ^ 63 9 4, _ _ Td)- - V693 6 385 4 63 - sn ' / c 9 4, _1 _ ci) __ _ Td)- - Td)] sin6- 4 2 _ y (D - T W - T (1) 1 sin [693 2 * 770 * 63 2 j TESSERAL AND SECTORIAL HARMONICS. 93 17. The process above investigated is probably the most convenient one when the object is to transform any finite algebraical function of cos 6, sin 6 cos <, and sin 6 sin <, into a series of spherical harmonics. For general forms of a function of ft and , however, this method is inapplicable, and we proceed to investigate a process which will apply universally, even if the function to be transformed be discon- tinuous. We must first 'discuss the following problem. To determine the potential of a spherical shell whose surface density is F (JJL, <), .F denoting any function whatever of finite magnitude, at an external or internal point. Let c be the radius of the sphere, r' the distance of the point from its centre, & ', <' its angular co-ordinates, V the potential. Then fj, being equal to cos r r / J -j ] o *_ 2cr' (cos cos ff + sin 6 sin & cos (< - <')} + c 2 ] * ' The denominator, when expanded in a series of general zonal harmonics, or Laplace's coefficients, becomes for an internal and an external point respectively, P 4 (//-, ) being written for Pj (cos 6 cos & + sin 6 sin 6' cos (< <')}. Hence, V l denoting the potential at an internal, F 2 at an external, point, (f 1 f 2jr r f 1 f 27r r t = c 4\ F (ft, <^>) dnd -\ Pj (ft, (j -i J o c J _i J o c* J.iJo ' ) 94 SPHERICAL HARMONICS IN GENERAL. = If ' (J -l J f I r J -IJ It will be observed that the expression P t (/it, <) involves /x and pf symmetrically, and also < and $'. Hence it satisfies the equation And, since p and ^> are independent of p and ^>', this differential equation will continue to be satisfied after P i has been multiplied by any function of //. and 0, and integrated with respect to /u. and <. That is, every expression of the form f J -i JO is a Spherical Surface Harmonic, or "Laplace's Function" with respect to // and <' of the degree i. And the several terms of the developments of F t are solid harmonics of the degree 0, 1, 2...1... while those of F 2 are the corresponding functions of the degrees 1, 2, 3... (i+ 1), ... And these are the expressions for the potential at a point (r, /u/, $') of the distribution of density F(/jf } <') at a point (c, /*', $'). Now, the expressions for the potentials, both external and internal, given in the last Article, are precisely the same as those for the distribution of matter whose surface density is f f ^ J-1./0 or, as it may now be better expressed, TESSERAL AND SECTORIAL HARMONICS. 95 ri rZir + 3 P l (cos0 cos0'+ sin0 sintf' J-iJO /i r2jr P j J -IJ cos <- And, since there is only one distribution of density which will produce a given potential at every point both external and internal, it follows that this series must be identical with F(IM, <'). We have thus, therefore, investigated the development of F(p, ) in a series of spherical surface harmonics*, The only limitation on the generality of the function F(/J>', (j>) is that it should not become infinite for any pair of values, comprised between the limits 1 and 1 of /x, and and 2-7T of <>. 18. Ex. To express cos 2^>' in a series of spherical har- monics. For this purpose, it is necessary to determine the value of 27r Now P i {cos 6 cos & + sin 6 sin & cos (< - (0080)^(0080') 2 . .dP^cosfl) . /y . 2 . B1 oS^') ,, . 7i - cos 2 (A -9) 2 /27T Now I cos a-((j> 6') cos 2 cZ^> = 0, Jo for all values of a- except 2. * In connection with the subject of this Article, see a paper by Mr G. H. Darwin in the Messenger of Mathematics for March, 1877. 96 SPHERICAL HARMONICS IN GENERAL. /"2ir And I cos 2 ( rf>') cos 2^> dtfj = TT cos 2^>' JO O Also And 7*2 / 2 _ I \i JH-1 / 2 _ 1 y ^LZ ^ ~ ^ ' - -~ i "\ - -2M 2 3 ^P -2.1.2.3..^^ Now when /i = 1, _ i V /7*" 1 fi* 2 _ 1 V - 1 ^Q ttP-1 ^ - 1 -0 +1 ' /ijr< ~ 1> - 1 And when /* = !, i+1 a ) cos dp = 4?r cos 2<' or 0, as i is even or odd ; cos 2<' . 2 ,,^P 2 (cos^) 4 sm ^ - -T - " cos If,, 2 T- 1 T oo 4?r ( 1 . 2 . 3 . x 4 2 . 4 ,y 4 . / sm* ^ - *j-75 - v cos 2

S ^ U-2.3.4 c 2 + 3.4.5.6 c 4 + BJf.f37+"~F and at an internal and external point respectively. 19. We will now explain the application of Spherical Harmonics to the determination of the potential of a homo- geneous solid, nearly spherical in form. The following investigation is taken from the Me'canique Celeste, Liv. in. Chap. II. Let r be the radius vector of such a solid, and let r = a + a (a, Y 1 + a 2 F 2 + ... + a, Y t + ...), a being a small quantity, whose square and higher powers may be neglected, a lt a 2 ,...a,... lines of arbitrary length, and F t , Y t ,..+ Y t ... surface harmonics of the order 1, 2,...i... re- spectively. 4 The volume of the solid will be vra 3 . o For it is equal to UJ. J ' -1 > -i-/ 4 = ^ ira , since for all values of t. F. H. 98 SPHERICAL HARMONICS IN GENERAL. Again, if the centre of gravity of the solid be taken as origin, a x = 0. For if 2 be the distance of the centre of gravity from the plane of xy, 4 p fi f2w 77a 3 2= rV<2r< 3 J J -i J o r 1 f 2ir a fj,Y J -i J o Similarly 4 3 4 ^ Tra v= 4a a. a, | | (i ju,*) 2 sm Now Fj is an expression of the form ' (1 /A*;* cos + (7(1 /z, 2 )- sin , r 1 r 2ir = 4a s a.a. (1 -^f- cosY.dfjid, J -iJ o r 1 r 2 * . a t I (1 - J-l.'O and therefore all the expressions x, y, ~z cannot be equal to 0, unless a t = 0. We may therefore, taking the centre of gravity as origin, write as the equation of the bounding surface of the solid. Now this solid may be considered as made up of a homo- geneous sphere, radius a, and of a shell, whose thickness is The potential of this shell, at least at points whose least distance from it is considerable compared with its thickness, will be the same as that of a shell whose thickness is oa, and density T' fl TESSERAL AND SECTORIAL HARMONICS. 99 p being the density of the solid. Therefore the potential, for any external point, distant R from the centre, will be The potential at any internal point, distant R from the centre, will be made up of the two portions A I jr Po R* + 27r Po (a 2 - 2 ) or for the homogeneous sphere, for the shell, and will therefore be equal to 20. If the solid, instead of being homogeneous, be made up of strata of different densities, the strata being concentric, and similar to the bounding surface of the solid, we may s* deduce an expression for its potential as follows. Let - r be the radius vector of any stratum, p its density, r having the same value as in the last Article, and p being a function of c only. Then, Sc being the mean thickness of the stratum, that is the difference between the values of c for its inner and outer surfaces, the potential of the stratum at an ex- ternal point will be 8c c 2 Sc/rt. 2 F 2 c 2 a,F 8 c 8 + ^-rrpa. - - ^- 2 yg + - 8 ^ 75 + ... R a \ 5 xr 7 R To obtain the potential of the whole solid at an external point we must integrate this expression with respect to c, between the limits and a, remembering that p is a func- tion of c. 7-2 100 SPHERICAL HARMONICS IN GENERAL. Again, the potential of the stratum, above considered, at an internal point will be To obtain the potential of the whole solid at an internal point we must integrate the expression (1) with respect to c between the limits and R, and the expression (2) with respect to c between the limits R and a, remembering in both cases that p is a function of c, and add the results together. CHAPTER V. SPHERICAL HARMONICS OF THE SECOND KIND. 1. WE have already seen (Chap. u. Art. 2) that the differential equation of which P i is one solution, being of the second order, admits of another solution, viz. Now if /j, between the limits of integration be equal to + 1, or to any roots of the equation P t = (all of which roots lie between 1 and 1), the expression under the integral sign becomes infinite between the limits of inte- gration. We can therefore only assign an intelligible meaning to this integral, by supposing //, to be always be- tween 1 and oo , or between 1 and oo . We will adopt the former supposition, and if we then put (7= 1, the Of- 3L \ -n i. expression -nrr\ -- 2\ I 1 - 6 - m / a T\l W1 ^ " e a ^ wa y s posi- "t (*- ~ P ) \ " i (p ~ *)/ tive. We may therefore define the expression P.V-1)' as the zonal harmonic of the second kind, which we shall denote by Q if or Q. (//,), when it is necessary to specify the variables of which it is a function. It will be observed that, if jj, be greater than 1, P f is always positive. Hence, on the same supposition, Q i is always positive. We see that Q n 102 SPHERICAL HARMONICS OF THE SECOND KIND. And, in a similar manner, the values of Q 2 , * ~ I j dv (v fJL} c/fj, _ ~ P I ~j~ Now, let : - be expanded in a series of zonal harmonics , PM.../M, so that Then - (1-^) by the definition of P (//,). SPHERICAL HARMONICS OF THE SECOND KIND. 103 d And. also , -\\*. * > . r . . . -r 7 ~ dv (/ ' dv } dv And these two expressions are equal. Hence, equating the coefficients of P t (/A), d_ dv Hence fa(v} satisfies the same differential equation as P t and Q t . But since U=- when v = , it follows that fa (v) =0 when v= co . Hence fa(v) is some multiple of Q i (v)=AQ l (v*) suppose. It remains to determine A. Now, fa(v) may be developed in a series proceeding by ascending powers of , as follows. Wehave ^. !+ + .. .+-+.. V fJ, V V V and also = (v) P (p>) +fa(v} -f^C/*) +-" + fa(v) P 4 (/i) + ... Now, by Chap. II. Art. 17, we see that, if m be any integer greater than i, the coefficient of P 4 in p, m is (m i-\- 2) (m i and (2t + ] m being always even. Hence, writing for m successively i, i + 2, i + 4, ... we get 6.8...(t -c-i, ,. - ^ + - If ' be odd ' 104 SPHERICAL HARMONICS OF THE SECOND KIND. /n * . i \ I **.-*... and = !i + l)(2-l). ..( + !) ^ 4. 6.. .(1 + 2) J_ "*" (2i + 3) (2t + !)...(* + 3) i/ +3 R ft f*'.i.A\ 1 *\ if i be even. Now, recurring to the equation Q (jO-Pi we see that, if Q^v] be developed in a series of ascending powers of -, the first term will be ~ . . m , where G is the coeflficient of yu.' in the development of P 4 (ytt) ; . ...- .. ., that is <7= g .. v _. - if i be odd, 2. 4.6...i 1 an d = ^- iJl^v^Jviiixr: v if i be even 2.4. 6...t Hence the first term in the development of Q. (v) is 9 A. K (! 1\ r if * be odd, 2.4.6...* ..., and = 7^ Z-T-T-: irr ^r-. ^. ... r-r it i be even, (*+ 1) (* + 3)...(2t- 1) (2* + 1) which is the same as the first term of the development of P t (v), divided by . . Hence A = 2i+ 1, and we have V-fJ, 3. The expression for Q, may be thrown into a more convenient form, by introducing into the numerator and de- SPHERICAL HARMONICS OF THE SECOND KIND. 105 nominator of the coefficient of each term, the factor neces- sary to make the numerator the product of i consecutive integers. We shall thus make the denominator the product of i consecutive odd integers, and may write 1.2. 3.. a J_ 3.4.5...(t + 2) 1 tyi( V ) 1 X I 0/_i_ 1 \ ,.1-t-l 7 , Oo i Q\ ,.* n 5.7.9...(2i+5) i/ m f (2* + 1) (2& + 3) . . . (2 + 2& + 1) T*^ 1 + "" whether i be odd or even. 4. We shall not enter into a full discussion of the pro- perties of Zonal Harmonics of the Second Kind. They will be found very completely treated by Heine, in his Handbuch der Kugelfunctionen. We will however, as an example, investi- gate the expression for -^- in terms of Q i+1 , Q l+3 ... Recurring to the equation we see that Now we have seen (Chap. II. Art. 22) that Hence = (2* + 1) P t (fi + (2- 3) P<_M +. 106 SPHERICAL HARMONICS OF THE SECOND KIND. ^^=(2* + 5) P I4 + (2< ^ = (2. + 9) P i+ + (2i+ 5) And therefore the coefficient of P^} in the expansion r d 1 Of -j- - 18 tt/M. V fJ, Again, i_i_ = ^w <^,w __ aw p dv dv + (2i + 1) And -_L + _L = . Hence, comparing coefficients of P t J^ = - (2i + 3) Q , +1 (v) - (2i + 7) < Hence it follows that and therefore that 5. By similar reasoning to that by which the existence of Tesseral Harmonics was established, we may prove that there is a system of functions, which may be called Tesseral Har- monics of the Second Kind, derived from T, ( . I ~ 7 v . I ~ _ > . I ~ 7 v . I ~ 2 , / -"- a + v b + v c + v Thus a 2 + e, Z> 2 -f e, c 2 -f e are the squares on the semiaxes of the confocal ellipsoid passing through the point x, y, z. a? + v, 6 2 + v, c 2 + v, the squares on the semiaxes of the confocal hyperboloid of one sheet. a 2 + v, Z> 2 + v, c* + v, the squares on the semiaxes of the confocal hyperboloid of two sheets. Thus, e is positive if the point x, y, z be external to the given ellipsoid, negative if it be internal. And, if a 2 be the greatest,' c 2 the least, of the quantities a', b\ c\ e will lie between SPHEROIDAL HARMONICS. -t _~ dydy. dydy. dydz_ dx dx dy dy dz dz dx dx dy dy dz dz Then o (9) dx dx dy dy dz dz = dVdy dx dy. dx dfi dx dy dx' _d?V/da\* tfV(d$\* tfVfdyV ~ ~dl (dx) + dff (dx) + dy 9 (da.) -- 2 _ + 9 dftdy dx dx dyd2 dx dx " dzdft dx dx da.dx* dfidx* ^7 dx 1 ' d*V d*V .- and -y^ being similarly formed, we see that, when the three expressions are added together, the terms involving -777, -v will disappear by the conditions (1), and those ou dp ay mvolving - f7rT - , -'j- , -7 ^ by the conditions (2). Hence D dydi daap 4. Now, let v) ~ a 2 + o) ~~ 6 2 + a> ~ c 2 + a> ~~ + 4 +' P + c 2 ) ' c 2 ) w being any quantity whatever. For this expression is of dimensions in to, e, v, v, it vanishes when to = e, u, or u', and for those values of &> only, it becomes infinite when o) = a 2 , & 2 , or c 2 , and for those values of o> only, and it is = 1 when o) = x . From this, multiplying by a* + a>, and then putting 0) = a 2 , we deduce 2 ^(e + a 2 ) (u + a 2 )(i/ + a 2 ) (a* -?)(, and then putting a) = __ 6-v- (o* + e)" (6 a + e) 8 + (c z +e) a (e + a?) (e H- 6") (e + c') de\ /de y / A y _ (e + a 8 ) (e + 6') (e + c s ) *) " (e-v)(e-v) .-. V 2 F= . -- ,^-7-, ^-. -- N - v T - ^02 (u v ) (v e) (e v ) { d'j. dp + ( The equation V 2 V= is thus transformed into . 2 ) . . , (v J (v-a> 2 ) ... (t/-&> n ). Hence EHH' = (e - J (w - oO (V -,)... (e - o>J (w - .) (v' - ) Now we have shewn (see Art. 4 of the present Chapter) that (e twj (u wj (i/ wj = K + a 2 ) (a>. + 6 s ) fa + c 2 ) f-i^ + r^ 2 - + 3^- - 1 y '-- Each of the factors of EHH' being similarly transformed, we see that EHH' is equal to the continued product of all expressions of the form ( + ) ( the several values of w being the roots of the equation o) n + n/yo"-* + TC ^ ~ 1 ^X H> + +^ = 0. As this equation has been already shewn to have (n + 1) distinct forms, we obtain (n + 1) distinct solutions of the equation V 2 F=0, each solution being the product of n expressions of the form That is, there will be n + 1 independent solutions of the degree 2n in x, y, z, each involving only even powers of the variables. 10. To complete the investigation of the number of solu- tions of the degree 2n, let us next consider the case in which E ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 119 The object here will be to transform the product (e + b 2 )* (v + b^ (v' + b^ (e + c s ^ (v + c 2 )* (v' + c 2 )*, since the other factors will, as already shewn, give rise to the product of ?i 1 expressions of the form m? if z* I " | z i ' C 2 + 0) Now, by comparison of the value of x* given in Art. 4, Ave see that (e + b 2 ) (v + b 2 } (v + V) (e +c 2 ) (v + c 2 ) (i/ + C 2 ) = (b 2 - c 2 ) (b 2 - a 2 ) (c 2 - a 2 ) (c 2 - b 2 ) y 2 z\ Hence, we obtain a system of solutions of the form of the product of (n 1) expressions of the form T 2 r* ' - multiplied by y0. Of these there will be n, and an equal number of solutions in which zx, xy, respectively, take the place of yz. Thus, there will be 4w + 1 solutions of the degree 2w in the variables of which n + 1 are each the product of n expressions of the form 2 z* a n are each the product of (ft 1) such expressions, multiplied zx, xy. 11. We may next proceed to consider the solutions of the degree 2?i + 1 in the variables x, y, z. Consider first the case in which E = (e + a 1 120 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. Here the product (e + a 2 )^ (v + a 2 )* (v + a 2 )'^ will, as just shewn, give rise to a factor x in the product EHH'. Hence we obtain a system of solutions each of which is the product of n expressions of the form --- a 8 + w b* + o> multiplied by x. Of these there will be n + l, and an equal number of solutions in which y, z, respectively take the place of the factor x. Lastly, in the case in which E=(e + a 2 )* (e -I- J 2 )* (e + c 2 )* e- 1 + (n - 1) Pl e*-* -- "* we see that in EHH' the product )* (u+a 2 ) 4 (v '+a 2 )* (e+ V)* (v+b*)* (v + J 2 )^ (e+c 2 )^ will give rise to a factor xyz. Hence we obtain a system of solutions each of which is the product of (n 1) expressions of the form multiplied by a;y^. Of these there will be n. Thus there will be 4n + 3 solutions of the degree 2/i + 1 in the variables, of which (n, + 1) are each the product of n expressions of the form a; 2 ?/ 2 z* -i --- H TT^ -- H -s -- 1 multiplied by or, a* + o> b' + w c 2 -f 2 ) (e + c 2 ))* ^ E . jvde 122 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. Now, since by supposition, the equation for the determi- nation of U is satisfied by putting U=E, it follows that when E Ivde is substituted for U, the terms involving Ivde will cancel each other, and the equation for the determina- tion of v will be reduced to Idv 2dE 1 f 1 1 r ~ J T "S"5~ T H I : ~t + . 72 "1" e+a" e + If 1 v de ^ E de """ 2 \e + a* whence log v + 2 log E + log { (e + a 2 ) (e + 6 2 ) (e = log v + 2 log E + log a&c, r and E being the values of v and J, corresponding to e = 0. E* abc Hence v = v. -^ , : We may therefore take, as a value of the potential at any external point, , F= v E* abc EHH' f de ^ . For this obviously vanishes when e = oo . It remains so to determine v that this value shall, at the surface of the ellipsoid, be equal to the value C. EHH', already assumed for an internal point. This gives de f Jo Hence, putting v . E* . abc F , we see that to the value of the potential de ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 123 for any internal point, corresponds the value a* V n EHH' for any external point. 13. We proceed to investigate the law of distribution of density of attracting matter over the surface of the ellipsoid, corresponding to such a distribution of potential. Now, generally, if &n be the thickness of a shell, p its volume density, the difference between the normal compo- nents of the attraction of the shell on two particles, situated close to the shell, on the same normal, one within and the other without will be 4 && being the thick- ness of the shell at the extremity of the greatest axis ; V, a I HH' *'* p ~2TrSaabc E Q ' and this is proportional to the value of V corresponding to any specified value of e, since HH 1 is the only variable factor in either. Hence functions of the kind which we are now considering possess a property analogous to that of Spherical Harmonics quoted at the beginning of this Chapter. On account of this property, we propose to call them Ellipsoidal Harmonics, and shall distinguish them, when necessary, into surface and solid harmonics, in the same manner as spherical harmonics are distinguished. They are commonly known as Lame's Functions, having been fully discussed by him in his Lecons. The equivalent expressions in terms of a;, y, z have been con- sidered by Green in his Memoir mentioned at the beginning of this chapter, and for this reason Professor Cayley in his ' Memoir on Prepotentials," read before the Royal Society on June 10, 1875, calls them " Greenians." We may observe that the factor 4vr Sa abc is equal to -. JK- , and therefore also to -. ^ or 126 ELLIPSOIDAL AXD SPHEROIDAL HARMONICS. Hence, it is equal to 4-TT - (bcBa + caSb + abSc) o or to volume of shell ' and the potential at any internal point = i volume of shell x JEE n . p I P Jo jE"a 2 and the potential at any external point r ^7 = \ volume of shell x EE Q . p where for p must be substituted its value in terms of v and v. 14. "We will next prove that if Fj, F 2 be two different ellipsoidal harmonics, dS an element of the surface of the ellipsoid, I le V^dS^O, the integration being extended all over the surface. We have generally . And throughout the space comprised within the limits of integration, V 2 V l = 0, V 2 F 2 = 0. Hence Now it has been shewn already that V lt V z are each of the form EHH' ', where E is a function of e only, H the same function of v, H' of v. We may therefore write and similarly F 8 =/ 8 (e)/ 2 (v)/ 8 (v). ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 127 Hen V?*-VV&Q KI de ~ * . dV t _ v ,_ FF . V* ' K * rfe - K r UW Now, all over the surface, e = 0. Hence f (0) f (0) Hence, unless J * ' J l :. ' = 0, which cannot happen /2 W /I \^/ unless the functions denoted by ^ and f t are identical*, or only differ by a numerical factor, we must have Now e is proportional to the thickness of the shell at any point. Calling this thickness Be, we have therefore Hence, adding together the results obtained by integrating successively over a continuous series of such surfaces, we get Fj , F" 2 now representing solid ellipsoidal harmonics, and the integration extending throughout the whole space comprised within the ellipsoid. * This may be shewn more rigorously by integrating through the space bounded by two confocal ellipsoids, denned by the values X and /* of e. We then get, as in the te xt, Now the factor within { } cannot vanish for all values of X and p., unless the functions devoted by f t and / 2 be identical, or only differ by a numerical factor. 128 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 15. It will be well to transform the expression to its equivalent, in terms of v, v. For this purpose we observe that if ds, ds be elements of the two lines of curvature through any point of the ellipsoid, dS=dsds. Now, ds* is the value of dx* + dy* + dz* when e and v are constant, rfs' 2 ... ... ... e and v (6* -*)(<;* -a 2 ; therefore if e and v do not vary, 2dx _ dv ~x v + a 2 ' , 1 x , *"= 2 ^r/"- Similarly ^-lJL.fcj _-_ Again, differentiating with respect to o> the expression x 1, we get (V 6)) (v G>) a z obtained for -j --- h -^- -- h - --- 1, we get a+w b" + to c +to (y+"^) a (c* + o>) 2 ~ (a 2 (v G>) f e w) (e w) (v ft)) (v r o) ~" ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 129 therefore, putting a* = v, ds* - - ~ A similar expression holding for ds* we get i/-v'e-i;) g-i/ _ 16 (a a +v) (F + i;) (c 2 +v) (a f +i/) (6 2 +t/) (e'+i/) i_ _^_ _^_ -+ ^vriting e for &> in the expression above ; . _ - , 16 (a'+v) (V+v) (c'+v) (a'+v')(b'+v') (c'+v') a It has been shewn that, integrating all over the surface, the limits of v are - c 2 and V, those of v, b* and a 2 . Hence, V lt F 2 , denoting two different ellipsoidal har- monics 0::^ The value of the expression 1 1 1 V*dxdydz, or its equiva- lent alc r e ' r 6 ' _ v(v'- J -b*J -a* {(a 2 + v) (6 2 +u) (c 2 +v) (a in any particular case, is most conveniently obtained by expressing V as a function of x, y, z. r. H. 9 130 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 16. Before proceeding further with the discussion of ellip- soidal harmonics in general, we will consider the special case in which the ellipsoid is one of revolution. We must enquire what modification this will introduce in the quantities which we have denoted by a, ft, 7, viz. -f- J, (a - 7 and in the differential equation __ _ __ . We will first suppose the axis of revolution to be the greatest axis of the ellipsoid, which is equivalent to supposing b* = c 2 . To transform a and 7, put a 2 + i/r = 0", a 2 + = if, a* + v = to 2 ; we then obtain = To transform ft, we must proceed as follows. Put -f = - c * cos 2 tj - 6 2 sin 2 w, i = - c 2 cos 2 - 6 2 siu 2 <^, we then get generally (^ _ c ) C o S V, c 2 + ^ = (c 2 - & 2 ) sin 2 ^ ; d-='2 c 2 6 2 cos OT sin CT C?CT ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 131 Hence, ^..\ (,'- a * + y) | , w "^ / 9 O 7O\^ -j- = - (a> _ a* + &') -T-, ay '2^ ' day d (a'-V)* d Also, e = 77 2 -a 2 , v '=w*-a?, v = -b 2 , and our differential equation becomes - a 2 + Z> 2 ) j( w 2 - a 2 + 5 2 ) -f- I' F ( tttuj or (a, 2 - a 2 + V] L 2 - a 2 + 5 2 ) ^}' F I ttl fj This equation may be satisfied in the following ways. First, in a manner altogether independent of <, by sup- posing F to be the product of a function of rj and the same function of &>, this function, which we will for the present denote by/ (77) or /(to), being determined by the equation ' 132 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. Secondly, by supposing -v- a a constant multiple of V, = a* V, suppose. Our equation may then be written d1 l) - (rf - a 2 + J 2 ) j(a> 2 - a 2 + Z> 2 ) -f- [' V ( aca) - a 2 (a 2 - J 2 ) {(a> 2 - a 2 + & 2 ) - (7? 2 - a 2 + ft 2 )} F= 0, which may be satisfied by supposing the factor of F inde pendent of < to be of the form F (rj] ^(w), where - o- 2 (a 2 - 6 2 ) F(n) = m (^ 2 -a 2 2 - a 2 + 6 2 ) (ft)) -cr 2 (a 2 -6 2 ) jF(o)) =TW ( will be of the form A cos o-< + ^ sin a(j). Now, returning to the equation we see that, supposing the index of the highest power of 17 involved in/(^) to be i, we must have m = i (i + 1). Now, it will be observed that 77 may have any value however great, but that o> 2 , which is equal to a 2 + v, must lie between a 2 6 2 and 0. Hence, putting o> 2 = (a 2 & 2 ) /w, 2 , where /*,* must lie between and 1, we get - + i (i + l)/{(a' - V} V] = 0. ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 133 Hence this equation is satisfied by /f(a 2 & 2 )*//,} = CP { , C being a constant ; and supposing C = 1 we obtain the following series of values for / (o>), t = 0, /()=!, 2 (a* - 6") ,. .- 5w s -3ft>(a 2 -6 2 ) = *> J W = 7T7~9 77* Exactly similar expressions may be obtained for/ (17), and these, when the attraction of ellipsoids is considered, will apply to all points within the ellipsoid. But they will be inadmissible for external points, since i) is susceptible of in- definite increase. The form of integral to be adopted in this case will bo obtained by taking the other solution of the differential equation for the determination of f(rf), i.e. the zonal har- monic of the second kind, which is of the form Q where Or, putting tf = (a 2 - 5 a ) v z , ff> = (a 2 - Z> 2 ) X 2 , we may write 17. We may now consider what is the meaning of the quantities denoted by 77 and 03. They are the values of t ^- which satisfy the equation 134 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. and are therefore the semi-axes of revolution of the surfaces confocal with the given ellipsoid, which pass through the point x, y, z. One of these surfaces is an ellipsoid, and its semi-axis is 77. The other is an hyperboloid of two sheets whose semi-axis is o>. Now, if be the eccentric angle of the point x, y, z, measured from the axis of revolution, we shall have a* = ff cos 2 6. But also, since 77*, &> 2 , are the two values of ^ 2 which satisfy the equation of the surface, Hence a> 2 = (a 2 - 6 2 ) cos 2 0, and we have already put whence the quantity which we have already denoted by ft is found to be the cosine of the eccentric angle of the point x, y t z considered with reference to the ellipsoid confocal with the given one, passing through the point x, y, z. We have thus a method of completely representing the potential of an ellipsoid of revolution for any distribution of density symmetrical about its axis, by means of the axis of revo- lution of the confocal ellipsoid passing through the point at which the potential is required, and the eccentric angle of the point with reference to the confocal ellipsoid. For any such distribution can be expressed, precisely as in the case of a sphere, by a series of zonal harmonic functions of the eccentric angle. 18. When the distribution is not symmetrical, we must have recourse to the form of solution which involves the factor A cos (r + B sin 2 = a 2 . We see then that 97 will become equal to the radius of the concentric sphere passing through the point, and rf a? 4- V will become equal to 17*. Hence the equation for the determination of/ (77) will become which is satisfied by putting f(rf] = ?/, or rj~ (i * l \ The former solution is adapted to the case of an internal, the latter to that of an external point. ' With regard to /*(), it will be seen that the confocal hyperboloid becomes a cone, and therefore at becomes inde- finitely small. But u,, which is equal to - , , remains (a*-V}V cc finite, being in fact equal to - or cos 6. Hence /(/A) becomes the zonal spherical harmonic. Again, the tesseral equations, for the determination of F (77), F (&>), become which are satisfied by F(ij) =rf or ri~ (i+l \ And, writing for tw 2 , (a 2 & 2 ) /x- 2 , we have, putting F(w) which gives % (yu,) = T^ (/JL). 136 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 20. We will next consider the case in which the axis of revolution is the least axis of the ellipsoid, which is equi- valent to supposing a 2 = 6 2 . To transform a and /3, put c 2 .+ ^r = 0* y c 2 + e = if, c 2 + v = a> s , we thus obtain f = 2 J.r a _ c *+0* (a 2 _ c 2 , 2 tan" To transform 7, we must proceed as follows : Put >/r = a 2 sin 2 ts V cos 2 OT, v = a 2 sin 2 < Z> 2 cos 2 0, we then get, generally, , a s + ^ = (a 2 - Z> 2 ) cos 2 r, 6 2 + -f = - (a 2 - 6 2 ) sin a *r, c 2 -f i/r=c 2 a 2 sin 2 ^> J 2 cos 2 0, cty=% (a 2 6 2 ) sinw cos or C?CT. Hence = 2 r ^ = 2# . f 2 = J*(a 2 sinV + 6 2 cos*OT- J. / a ... o\ tt also, = rf c 2 , v=o> 2 -c 2 , i/ = -a 2 , and our differential equation becomes ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 137 We will first consider how this equation may be satisfied by values of V independent of (j>. We may then suppose V to be the product of a function of 77, and the same function of or 9.i( v }> which will (a 2 -c 2 )*) be equal to It is clear that /"() may be expressed in exactly the same way. But it will be remembered that if and o> 2 are the two values of ^ 2 which satisfy the equation *, 2 Hence 77, as before, is the semi-axis of revolution of the confocal ellipsoid passing through the point (x, y, z}. But if a? = (a? c 2 ) 3 2 , an essentially negative quantity, since a 2 is greater than c 2 . Hence to 2 is essentially negative. Now, if 6 be the eccentric angle of the point (x, y, z} measured from the axis of revolution, we have = vf cos 2 #. Hence and therefore to 8 = (a? c 2 ) cos 2 6 = (a 2 c 2 ) /A 2 , suppose. Hence the equation for the determination of/(&>) assumes the form the ordinary equation for a zonal spherical harmonic. Hence we may write IJL being the cosine of the eccentric angle of the point x, y, z, considered with reference to the confocal ellipsoid passing through it. ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 139 21. We have thus discussed the form of the potential, corresponding to a distribution of attracting matter, sym- metrical about the axis. When the distribution is not symmetrical, but involves in the form A cos -f B sin acf), \ve replace, as before, P t (/u,) by T^ (/A), and p t (fj.) by a function t^(v} determined by the equation ,0 and q. (v) by t^ (v) J V )*. d\ 22. As an application of these formulae, consider the fol- lowing question. Attracting matter is distributed over the shell whose z? v 2 _|_ / surface is represented by the equation + , 2 = 1, so Ct (J that its volume density at any point is P i (M), i*> being the cosine of the eccentric angle, measured from the axis of revolution ; required to determine the potential at any point, external or internal. The potential at any internal point will be of the form CP t (rtPM (i), and at an external point, of the form VP t (tiQ t (v) (2), where (a 2 b 2 )^ v = the semi-axis of the figure of the con- focal ellipsoid of revolution passing through the point (p, v). Now the expressions (1) and (2) must be equal at the surface of the ellipsoid, where v = L . Hence 140 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. But generally Hence \ * Upf a } r & < l(a 2 - b*$ ' l(a 2 - 6'^j J -_ P< (\)i 2 (\ 2 - 1) ' (a 2 -6")* _ P, (\), 2 (X 2 - 1) s-fe 2 ) 4 We may therefore, putting C' = AP i \ - j[ , write ((a - b )*J and we thus express the potentials as follows : AP i (/A) P i (v) QA ij- at an internal point, I (a o j2j AP i (fi} Q. (v) PA- rf at an external point. I (a 2 V)*) Or, substituting for Q. its value in terms of P i} at an internal point, V^APMP^l at an external point. Now, to determine A, we have, Sa being the thickness of the shell at the extremity of the axis of revolution, ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 141 4?r 8a . (3-6 2 ) a P> a 2 - 6 s -1 Hence, if p = P< (/A), we obtain And we thus obtain F> (a 2 - \ -2 , _2 . I / ^ l\ 1 (12 Z2 l cZ\ P. M ' V ; If the shell be represented by the equation a' c a it may be shewn in a similar manner that we shall have 142 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. C V, = torada P t q, V Pi 23. We may apply this result to the discussion of the following problem. // the potential of a shell in the form of an ellipsoid of revolution about the greatest jxrirtfT be inversely proportional to the distance from one focus, find the potential at any internal point, and the density. If the potential at P'be inversely proportional to the distance from one focus 8, and H be the other focus, we have, = 2rj, HP-SP=2a>, Hence if M be the mass of the shell, F 2 the potential at any external point, M V ' o 77 ft) M (jf-fttytir-/! M Now, by what has just been seen, the internal potential, corresponding to P t (yu,). Q i (v), is Hence, if F 2 be the potential at any internal point, ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 143 And the volume density corresponding to P i (/z) Q t (v) is Hence the density corresponding to the present distri- bution is M * ra a - If F 2 had varied inversely as HP, we should have had M and our results would have been obtained from the foregoing by changing the sign of o>, and therefore of p. 24. Now, by adding these results together, we obtain the distributions of density, and internal potential, corre- sponding to Y- M M -M *n ' > - iu a a > ij a) 77 + w 1) a>* or, in geometrical language, V - M - ~ + ~ SPHP~ SP.HP ' = ^^ multiplied by the axis of revolution of the confocal ellipsoid, and divided by the square on the conjugate semi- diameter. We may express this by saying that the potential at any point on the ellipsoid is inversely proportional to the 144 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. square on the conjugate semi-diameter, or directly as the square on the perpendicular on the tangent plane. Corresponding to this, we shall have, writing 2k for i, since only even values of i will be retained, 4 being 0, or any positive integer. Again, subtracting these results we get Jf 2m F _ 9 1 CO f] + (0 f] CO = M multiplied by the distance from the equatoreal plane, and divided by the square on the conjugate semi- diameter. This gives, writing 2k + 1 for i, Q* bSf 25. In attempting to discuss the problem analogous to this for an ellipsoid of revolution about its least axis, we see that since its foci are imaginary, the first problem would re- present no real distribution. But if we suppose the external potential to be the sum or difference of two expressions, each inversely proportional to the distance from one focus, we ELLIPSOIDAL AND SPHEROIDAL HARMONICS, 145 obtain a real distribution of potential in the first case inversely proportional to the square on the conjugate semi-diameter, in the latter varying as the quotient of the distance from the equatoreal plane by the square on the conjugate semi-diameter. It will be found, by a process exactly similar to that just adopted, that the distributions of internal potential, and density, respectively corresponding to these will be : In the first case 4-rr (a 2 - c 9 ) 4 aSa Jc being 0, or any positive integer. In the second case ** 7c being 0, or any positive integer. 26. We may now resume the consideration of the ellip- soid with three unequal axes, and may shew how, when the potential at every point of the surface of an ellipsoidal shell is known, the functions which we are considering may be employed to determine its value at any internal or external point. We will 'begin by considering some special cases, F. H. 10 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. by which the general principles of the method may be made more intelligible. 27. First, suppose that the potential at every point of T/" /-> the surface of the ellipsoid is proportional to x = - suppose. In this case, since x when substituted for V, satisfies the equation y 2 V= 0, we see that V - will also be the potential CL at any internal point. But this value will not be admissible at external points, since x becomes infinite at an infinite distance. Now, transforming to elliptic co-ordinates And the expression F ((6+a 2 )(u+a s )(u'+a 2 )p a | (a 2 -6 2 )(a 2 -c 2 ) J . ' ' _ ) (t + &*) (f +C 1 )}* satisfies, as has already been seen, the equation v 2 ^^^ ' ls fp equal to V at the surface of the ellipsoid, and vanishes Cb at an infinite distance. This is therefore the value of the potential at any external point. It may of course be written a + a 2 ) (f + 6 2 ) 28. Next, suppose that the potential at every point of 7/2J the surface is proportional to yz=V -.~, suppose. In this ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 147 case, as in the last, we see that, since yz when substituted for V, satisfies the equation y 2 y=0, the potential at any internal point will be V v- ; while, substituting for y, z their values in terms of elliptic co-ordinates we obtain for the potential at any external point be J e fy + i*) ty + c 2 ) [fy + a ) ^ + j") (^ + C 2j ji 29. We will next consider the case in which the po- Cu tential, at every point of the surface, varies as cc 2 = F -, ot suppose. This case materially differs from the two just con- sidered, for since a? does not, when substituted for V, satisfy the equation y 2 F"= 0, the potential at internal points cannot in general be proportional to a; 2 . We have therefore first to investigate a function of x, y, z, or of e, v, v which shall satisfy the equation y 2 F=0, shall not become infinite within the surface of the ellipsoid, and shall be equal to a? on its surface. Now we know that, generally (6 2 + w) (c 2 + w) x 2 + (c 2 + to) (a 2 + co) f + (a 2 + to) (& 2 + w) - (a 2 + a>) (b* + a>) (c 2 + ) = 0...(l), we see that V 2 (e-^)(u-^)(u'-^)=0, and V 2 (^ ~ ) ( - ^ ( - 2 ) = 0. And, by properly determining the coefficients A , A lt A 2 , it is possible to make when 6'cV + cV/ + -a'JV - a"6V = 0. 102 148 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. Hence, the expression (2) when A , A lt A 2 are properly determined will satisfy all the necessary conditions for an internal potential, and will therefore be the potential for every internal point. Now, we have in general (V+ ^(c* + 630?+^ + 0^+03 tf+ (a* + e i Y(V+0 1 }z> - (a 2 + 0J (6 2 + 0,) (c 2 + 0J = (e - OJ (v - ej (v - 0,} and, over the surface 6 s cV + cV/ + a s 6 2 2 2 - a 2 6 2 c 2 = 0. Hence, ^ being any quantity whatever, we have, all over the surface, and therefore, putting ^ = a 2 , Hence, the right-hand member of this equation possesses all the necessary properties of an internal potential. It satisfies the general differential equation of the second order, does not become infinite within the shell, and is proportional to ic 2 all over the surface. We observe, by equation (1), that 2 +a)) (a 2 +o>)-f (a -ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 14>9 identically, and therefore, writing a 2 for a>, Hence, over the surface of the shell, x z = a 2 ' and we therefore have, for the internal potential, F ((e-^-^Mu'-flJ ( 6 -0,)(y-0 2 )(,/- 2 1 "3 " f " ' This is not admissible for external points, as it becomes infinite at an infinite distance. We must therefore substi- tute for the factor e t - &iY [ty + a ' 2 ) (^ + ^ 2 ) (^ + c2 )}* ' J (+-6. with a similar substitution for e # g , thus giving, for the external potential, (f -^.{(t+ 2 )(^+^) (f + C S ) 150 .ELLIPSOIDAL AND SPHEROIDAL HARMONICS. The distribution of density over the surface, correspond- ing to this distribution of potential, may be investigated by means of the formula or its equivalent in Art. 13 of this Chapter. We thus find that i a v TT da 3abc [_ 0* (0 l - 2 ) (a* + t ) ~L (^lr-0. (v-ow-ej . r Wt-*,) ("+^J ' V.-fl +lJ .r _ *+ i ' {(t + ^Mt + ^Ht + c 2 )]*-!* 30. The investigation just given, of the potential at an external point of a distribution of matter giving rise to a potential proportional to ce 2 all over the surface, has an in- teresting practical application. For the Earth may be re- garded as an ellipsoid of equilibrium (not necessarily with two of its axes equal) under the action of the mutual gravi- tation of its parts and of the centrifugal force. If, then, V denote the potential of the Earth at any point on or with- out its surface, and fi the angular velocity of the Earth's rotation, we have, as the equation of its surface, regarded as a surface of equal pressure, .'. V + ^ O 2 (a? + 2/ s ) = a constant, II suppose. Hence, if a, I, c denote the semi-axes of the Earth, -we have, for the determination of F, the following conditions ; ELLIPSOIDAL AND SPHEEOIDAL HAKMONICS. 151 d*V d*V V= at an infinite distance ............... (2), when a? if The term II will, as we know, give rise to an external potential represented by n f + ; +t- + * : .C 1 1 The two terms - HV, s^-V w ^l gi ye r i se to terms which may be deduced from the value of F 2 just given by successively writing for F , -^HV, and ~ft 2 6 2 , and (in ^2 the latter case) putting 6 s for a 2 throughout. We thus get a 5* v Pi Vi- *) (f - ^)" {(^ + a 2 ) (^ + 6 2 ) (^ + J. - c s )] 152 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 31. Any rational integral function V of x, y, z, which satisfies the equation y 2 F=0, can be expressed in a series of Ellipsoidal Harmonics of the degrees 0, 1, 2...1 in x, y, z. For if V be of the degree * the number of terms in V be . Now the conditi(m = r=0 is o equivalent to the condition that a certain function of x, y, z of the degree i 2, vanishes identically, and this gives rise to - - 7^ - - conditions. Hence the number of inde- D pendent constants in V is (t + 1) (t + 2) (i+ 3) (> - 1) i ('+ 1) 6 6 or (i+ I) 2 . And the number of ellipsoidal harmonics of the degrees 0, 1, 2...i in x, y, z or of the degrees 0, ^, 1, -^-^ in e, v, v, is, as shewn in Arts. 6 to 10 of this Chapter, 1 + 3 + 5+.. . or (t + 1) 2 . Hence all the necessary conditions can be satis- fied. 32. Again, suppose that attracting matter is distributed over the surface of an ellipsoidal shell according to a law of density expressed by any rational integral function of the co-ordinates. Let the dimensions of the highest term in this expression be i, then by multiplying every term, except those of the dimensions i and i I by a suitable power of a? 7/ 2 z* __ L __ i __ a 2 + 6 a c" we shall express the density by the sum of two rational inte- gral functions of x, y, z of the degrees i t i 1, respectively. The number of terms in these will be (ilL(S) +(+!) or (,-+!). ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 153 And any ellipsoidal surface harmonic of the degree i t & 2... in x, y, z, may, by suitably introducing the factor be expressed as a homogeneous function of x, y, & of the degree i ; also any such harmonics of the degree i 1, i 3... in x, y, z may be similarly expressed as a homogeneous function of x, y, z of the degree i \. And the total number of these expressions will, as just shewn, be (i + I) 2 , hence by assigning to them suitable coefficients, any distribution of density according to a rational integral function of as, y, z may be expressed by a series of surface ellipsoidal harmonics, and the potential at any internal or external point by the corresponding series of solid ellipsoidal harmonics. 33. Since any function of the co-ordinates of a point on the surface of a sphere may be expressed by means of a series of surface spherical harmonics, we may anticipate that any function of the elliptic co-ordinates v, v may be expressed by a series of surface ellipsoidal harmonics. No general proof, however, appears yet to have been given of this proposition. But, assuming such a development to be possible at all, it may be shewn, by the aid of the proposition proved in Art. 15 of this Chapter, that it is possible in only one way, in exactly the same way as the corresponding proposition for a spherical surface is proved in Chap. IV. Art. 11. The development may then be effected as follows. De- noting the several surface harmonics of the degree i in x, y, z, or I in u, t/, by the symbols Vp\ F/ 2 >, ...F/ 2i+1 >, and by 2t F(v, v) the expression to be developed, assume k Then multiplying by eV^ and integrating all over the surface, we have j eF (v, t/) F/'> dS = Cpe (F/*>)' dS. 154; ELLIPSOIDAL AND SPHEROIDAL HARMONICS. The values of feF(v, v^V^dS, and of je "be ascertained by introducing the rectangular co-ordinates x, y, z, or in any other way which may be suitable for the particular case. The coefficients denoted by C are thus determined, and the development effected. EXAMPLES. 1. Prove that (sin 0) = A P - 1? P 8 + A P 4 . , Why cannot (sin 0) 3 be expanded in a finite series of spherical harmonics ] , 1 1 1 2. Prove that 1 + P,+ P 8 + P 3 + ... =log 3. Establish the equations 4. If JM - cos 6, prove that and also that 2\(p.) = (- 1)' + (- !)*> i(i + 1) cos 2 + ... + T / (-!)'+-' -( (\m)*-m \ 5. Prove that, if a be greater than c, and i any odd integer greater than m t 6. Prove that T (^ V = * ( + !) ' ^ 156 EXAMPLES. 7. Prove that, when p. = 1, -r^ - 8. Prove that IP P i, .TJ ... jr i P P P f y f i * <+i P p P * * t+1 ' * Ut-l \i m 2 I m is a numerical multiple of -/ 9. Prove the following equation, giving any Laplace's co- efficient in terms of the preceding one : ' n dp + C, where Cp - pp.' + Jl -p? J\ - ///* cos (w - a/) and C is zero if n be even, and (-i)" ~*n if n be odd. 10. If i, J, ^ be three positive integers whose sum is even, prove that 1.3... (j+yfc-t'-l) 2.4... (j + k-i) 2.4 ... (i+j-k) 1.3 ... (i + j + k 1) i+j+k+1' Hence deduce the expansion of P i P J in a series of zonal harmonics. 11. Express x*y + y 3 + yz + y + z as a sum of spherical harmonics. 12. Find all the independent symmetrical complete harmonics of the third degree and of the fifth negative degree. 13.' Matter is distributed in an indefinitely thin stratum over the surface of a sphere whose radius is unity, in such a manner V. that the quantity of matter laid on an element (&S) of the surface is 8>S (1 + ax + by + cz +fxr + gif + hz*) t EXAMPLES. ] 57 where x, y, z are rectangular co-ordinates of the element $S re- ferred to the centre as origin, and a, b, c, f, g, h are constants, Find the value of the potential at any point, whether internal or external. 14. If the radius of a sphere be r, and its law of density be p = ax + by + cz, where the origin is at the centre, prove that its 47TT 5 potential at an external point (, 17, ) is (a + br)+c) where It is the distance of (, t], ) from the origiit 15. Let a spherical portion of an infinite quiescent liquid be separated from the liquid round it by an infinitely thin flexible membrane, and let this membrane be suddenly set in motion, every part of it in the direction of the radius and with velocity eq"ual to S a a harmonic function of position on the surface. Find the velocity produced at any external or internal point of the liquid. State the corresponding proposition in the theory of Attraction. 16. Two circular rings of fine wire, whose masses are M and J.T, and radii a and a', are placed with their centres at distances b, b', from the origin. The lines joining the origin with the centres are perpendicular to the planes of the rings, and are in- clined to one another at an angle 0. Shew that the potential of the one rin.^r> /Z- 158 EXAMPLES. 18. Of two spherical conductors, one entirely surrounds the other. The inner has a given potential, the outer is at the potential zero. The distance between their centres being so small that its square may be neglected, shew how to find the potential at any point between the spheres. 19. If the equation of the bounding surface of a homo- geneous spheroid of ellipticity c be of the form \ f prove that the potential at any external point will be tr n i *L C - A P n * . r 3' where C and A are the equatoreal and polar moments of inertia of the body. Hence prove that F will have the same value if the spheroid be heterogeneous, the surfaces of equal density differing from spheres by a harmonic of the second order. 20. The equation R = a (1 + ay) is that of the bounding surface of a homogeneous body, density unity, differing slightly in form and magnitude from a sphere of radius a; a is a small quantity the powers of which above the second may be neglected; and y is a function of two co-ordinate angles, such that where Y , F, ... Z , Z^ ... are Laplace's 'functions. Prove that the potential of the body's attraction on an external particle, the distance of which from the origin of co-ordinates is r, is given by the equation Q 3r a ", ir ", + +-, - * ~T ^H. + ' ' ( ' 2r 4/i + 2 r n " J EXAMPLES. 159 21. If M be the mass of a uniform hemispherical shell of radius c, pi'ove that its potential, at any point distant r from the centre, will be 2c 2 c s V2 ' 2 . 3 p r 5 3.5 p r 7 \ + 2.4.6 V~2.4.6.8 7 c 9 + "7' g+.i'fi'.i-ri'.a 2?* 2 \2 f* 2.4 7* or zr ^ \^ * r" 2.4 - r" 3 , c s 3.5 according as r is less or greater than c; the vertex of the hemi- sphere being at the point at which p. = 1. 22. A solid is bounded by the plane of xy, and extends to infinity in all directions on the positive side of that plane. Every point within the circle x* + y 2 = a*, z = is maintained at the uniform temperature unity, and every point of the plane xy without this circle at the uniform temperature 0. Prove that, when the temperature of the solid has become permanent, its value at a point distant r from the origin, and the line joining which to the origin is inclined at an angle to the axis of z will be r I r 3 1 3 r 5 P P 4.-P- P JL. * ^'a 2 V 274 V ....- "^ ' 2.4...2i **"* if r < a, and 1 L3 p a , 1.3...(2i-l) 'a 1 ^ J7I ?* "^ ; 2.-4...2 **+'? + if r > . 23. Prove that the potential of a circular ring of radius c, whose density at any point is cos mif/, cij/ being the distance of the point measured along the ring from some fixed point, is 160 EXAMPLES. 2, cos m* (sin ^ 2. 4. 6. ..(2m + 2) */* ..... where r is greater than c. If r be less than c, r and c must be interchanged. 24. A solid is bounded by two confocal ellipsoidal surfaces, and its density at any point P varies as the square on the perpendicular from the centre on the tangent plane to the confocal ellipsoid passing through P. Prove that the resultant attraction of such a solid on any point external to it or forming a part of its mass is in the direction of the normal to the confocal ellipsoid passing through that point, and that the solid exercises no attraction on a point within its inner surface. '/* CAMBRIDGE : PKINTED BY c. J. CLAY, M,A. AT THE UNIVERSITY PRESS. a L. MAY 20 me "'. JAN16 W7 MAY 2 REM JAN 2 19SL JAN 2 4 1951 Afn< ** 1AR 111953 JUL 8 1964 V MAY 2 5 19*1 SEP 17 1964 I) EEB 14 195? APR 6 1955 MAR i 2 1962 Form L-fl 23m-2, '43(5203) MAR 8 1965]