CORNELL UNIVERSITY LIBRARIES '<^ Mathematics Library \Nhhe Halt CORNELL UNIVERSITY LIBRARY 3 1924 059 551 287 DATE DUE ?.*.M "^"^ -opnrr-^ MAY 2 Mi GAVLORD ^RINTGDINU-CA. Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059551287 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volvune on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990. SPHEEICAL HARMONICS. AX ELEMENTARY TREATISE ON" SPHERICAL HARMONICS AND SUBJECTS CONNECTED WITH THEM. REV. N. M. FERRERS, M.A., F.R.S., FELLOW AND TOTOB OF GONVILLE ASD CAIDS COLLEGE, CAMUBIDOE. Uontron: MACMILLAN AND CO. 1877 [All Rights Tesen-ed.] QTambrtlige: PBINTED BV C. J. CLAT, M.A., AT THE UNIVEBSITT PREBB. PKEFACE. 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. GOKVILLE AND CaiUS CoIXEGE, August, 1877. F. H. CONTENTS. CHAPTER I. INTRODUCTOEY. DEFINITION OF SPHERICAL HARMONICS. CHAPTER II. ZONAL HARMONICS. ABT. TACE 1. Difiereutial Equation of Zonal Harmonics 4 2. General solution of this equation ... 5 3. Proof that Pi is the coefficient of hf in a certain series 6 5. Other expressions for Pi . . ... 11 6. Investigation of expression for Pi in terms of n, by Lagrange's Theorem 12 7. The roots of the equation Pj = are all real IS 8. Eodrigues' theorem .... iO. 10. Proof that /"' P^ P. d^ = 0, and ( J'*' <*M = gi^i • ^'^ 12. Expression of P^ in ascending powers of /j. . 19 15. Yalnes of the first ten zonal harmonics 22 16. Values of j /i-P, d/j. . .25 17. Expression of fif in a series of zonal harmonics .... 20 via CONTEXTS. At!T. riCE 13. Expression of Pj in a series of cosines of multiples of 9 . 29 19. \a\neol f P,coBmesinede .... . ib. lof r Jo 20. Expression of cos m9 in a series of zonal harmonics ... 33 21. Development of sin d in an infinite series of zonal harmonics . 35 dP 22. Value of -3-' in a series of zonal harmonies 37 dfi 24. Value of IPiPudu. .... . . 38 i of I PtPiid/j. 2.5, 26. Expression of Zonal Harmonics by Definite Integrals . . 39 27. Geometrical investigation of the equality of these definite integrals .41 23. Expression of Pi in terms of cos 8 and sin . . . . 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 a 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 B'' . 51 6. 9. Eelation between density and potential for a spherical surface 'A 10. Potential of a spherical shell of finite thickness .... TiS 12. Expression of certain discontinuous functions by an infinite series of zonal harmonics ... . . 61 14. Expression of a function of n, infinite for a particular value of /i, and zero for all other values ... . . 65 15. Expression of any discontinuous function by an infinite series of zonal harmonics . . . . 66 CONTENTS. IX CHAPTER IV. .SrHERICAL HARMONICS IN GENERAL. TESSERAL AND SECTORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXES IN ANY POSITION. POTENTIAL OF A SOUD NEARLY SPHERICAL IN FORM. ART. PAGE 1. SphehcaJ Harmonics in general . ... 69 2. Relation between the potentials of a spherical shell at an inter- nal and an external point .... . . ib. 3. Belation between the density and the potential of a spherical sheU . . 70 4. The spherical harmonic of the degree i will involve 2i+l arbi- trary constants 72 5. Derivation of gnccessive harmonica from the zonal harmonic by differentiation . ib. a. TesseraJ and sectorial harmonics . 74 7. Expression of tesseral and sectorial harmonics in a completely developed form 75 6. Circles represented by tesseral and sectorial faannonics . 77 9. New view of tesseral harmonics 78 10. Proof that [ [^' YJ^dixdtp^O 80 11. If a function of /i and ^ can be developed in a series of snrface harmonics, such development is possible in only one way . 82 12. Proof that f ^^ 7,^^ = 2t r.(l)PiO') Jo 2i+l and r 1^' P,Yidiid = ^ Y, (1) 83 13. Investigation of the value of | \ ^ Y,Zidnd(t> . . .84 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 CONTENTS. 16. Illastrations of this transformation 17. Expression of any function of n and

read sr. p. 142 line 6, for point read axis. CHAPTER I. INTEODUCTOET. 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 Fmust satisfy the differential equation d^V d^V d^ dof^ d7+ d^ " ^^^' or, as we shall write it for shortness, V'F= 0. The general solution of this equation cannot be obtained in finite terms. We can, however, determine an expression which we shall call F„ 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 F] there corresponds another, of the degree — (i + 1), V expressed by -jj^ > where r* = x' + ^* + a'. For the equation (1) when transformed to polar co-ordi- nates by writing 3; = r sin ^ cos ^, y = r sin 5 sin ^, z = r cos 6, becomes d^{rV)^ 1 d(. «rfF\ 1 d^V And since V satisfies this equation, and is an homo- geneous function of the degree i, F, must satisfy the equa- tion d'V, 'd d<^' F. H. ••(••+ l)''.+ii^^(™^f) + S^ -0, 2 INTHODUCTORT. since this is the form which equation (2)_ assumes when F is an homogeneous function of the degree i. Now, put F, = r"** Ui, and this becomes iU + 1 dr. „dZL\ 1 Now since Z7, is a homogeneous function of the degree or d{rU;) dr ' ar = -iUr, d'irUd dr' .dU, ' dr = i{i+i)^; d'irU:) i(i+l)L\: therefore equation (2) becomes d^irU) 1 d f . „dU,\ , 1 d'U, r - — ~ 4 , — I sin f) — .' 4- . ^ * : ur' shewing that CT, is an admissible value of F, as satisfying equation (2). It appears therefore that every form of Pj can be ob- tained from F,, by dividing by r*"*", and conversely, that every form of F, can be obtained from £/] by multiplying by r"**. Such an expression as F] we shall call a Solid Spherical Harmonic of the degree i. The result obtained by dividing F, by r', which will be a function of two inde- pendent variables 6 and <^ only, we shall call a Surface Spherical Harmonic of the same degree. A very important form of spherical harmonics is that which is independent DEFINITION OF SPHEBICAL HARMONICS. 3 of ^. The solid harmonics of this form will involve two of the variables, x and y, only in the form ae' + j^, or will be functions of x' + y' 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 Mecanique Cdeste. 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 6 is measured, was really, however, first introduced by Legendre, althpugh 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 place of COS&, we write cos ^ cos 5' + sin ^ sin ^' cos (^ — ^'). The name " Kugelfunctionen " is employed by Heine, in his standard treatise on these functions, to designate Spherical Harmonics in general. CHAPTER II. ZONAL HARMONICS. 1. We stall in this chapter regard a Zonal Solid Har- monic, of the degree i, as a homogeneous function of (»* +y^^> and ^> of tbe degree i, which satisfies the equation di* ■*" dy "*■ dz' ~ • Now, if this be transformed to polar co-ordinates, by writing r sin cos ^ for x, r sin d sin for y, r cos ^ for z, we observe, in the first place, that a;' -f- y* = r' sin' ^. Hence V will be independent of ,|;|(l_;.>} + 2(l-^-)^'..0, whe»<« i',Cl-,.-)|+2{(l-/'')f-''-f.l''-''' the integral of whicli is log V + log P,'.(l - /i') = log Ci = a constant ; _ ^, Hence Jz>cZ/i = C+C^j pa,^_ ». ; and we obtain, for the most general form offfjj,). Now, P,* being a rational integral function of fj, of i dimensions, it may be seen that I .. _ », pa ^i^l assume the form of the sum of i + 2 logarithms and i fractions, and therefore cannot be expressed as a rational integral function oifi. Expressions of the form P, I j- ^ p» are called Kugel- 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 (2j of the form of a rational integral function of u ZONAL HAEMONICS. 7 involves but one arbitrary constant, and that as a factor. We shall henceforth denote by P,, or P^ (fi), 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. Ifhbe less than unity, and if {1— 2/ih + h')"* he expanded in a series proceeding hy ascending -powers of h, the coefficient of h' will he P,. Or, (1 -2/ih + hT^ = P, + P.h + ... + P,h' + ... We shall prove this by shewing that, if H be written for (1 — 2/i/4 + A')"^, H will satisfy the differential equation For, since H= (1 - 2^ + hy^, .-. _^,= 1-2M + A'; 1 dH ••U'dp.'"' A dp. •■ hdp,X- '^'dp.) dpC i = -2pn' + 3{l-p')hH\ ^ dH , ^""^ H'dh^'"'^' ••• Th^^'^ = ^-^^'dh = ^ [w-^H'dh) = H'{l-2ph + h'' + h(fi-h)} =H'ii-phy, ZONAL HAKMONICa. UhE) = i{W{l-,^h)) = -3m-H''+ 3 {(1 - /t')/. + (1 - tih){ji-h)] E' = - SfiE' + 3 {m (1 + h') - 2/i'Al E' = -Sfi [E' - (1 - 2/i;i + h') E'] = 0, sincel-2/iA + A' = JI-^. Therefore -^ a/j, ,. dff ] , . (P (l-'^'^d^ir^jA'^^^^^- This may also be shewn as follows. If X, y, z he the co-ordinates of any pomt, z the distance of a fixed point, situated on the axis of i, from the origin, and B, be the distance between these points, we know that. and that '■©-<^ Now, transform these expressions to polar co-ordinates, by writing a: = r sin 5 cos ^, y = rein ^ sin <^, ^ = rcos^, and we get i2' = r^-2aVcos^ + a'*, and the differential equation becomes "^ dr" ^©+il^lh^|©} = ^' ZONAL HABMONICS. or, putting cos = fi, Now, putting r = z'h, we see that 1 H .'. the above equation becomes or ^^r-iio-"-)!}-- 4. Having established this proposition, we may proceed as follows: If p, be the coefficient of A' in the expansion of IT, .: hff= h + pji* +i>^t' + . . . + p,h'*^ + ... .-. h^,{hH) = 1.2p,h + 2.SpJi'+...+i(i + l)p^' + ... Also, the coefficient of h' in the expansion of Hence equating to zero the coefficient of h'. 10 ZONAL HArwMONICS. Also p, is a rational integral function of /*. And, when fi = l, U= {l-2h + hy^ = l + h + h' + ...+h' + ... Or when ^ = 1, p, = 1. Therefore p, is what we have already denoted by Pj. We have thus shewn that, if h be less than 1, (1 - 2M + /O'* = -P. + Pj^ + ... + Pfi' + - If h be greater than 1, this series becomes divergent. But wo may write 1 / P P. \ since , is less than 1, A Hence P, is also the coefficient of A-<*+W in the expan- sion of (1 — 2/iA + /*■) "^ in ascending powers of r when h is greater than 1. We may express this in a notation which is strictly continuous, by saying that Pj=P-(i+i)- This might have been anticipated, from the fact that the fundamental diiferential equation for Pj is unaltered if — (i + 1) b6 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) {_ (i + 1) + 1| or (i + 1) i, and is therefore unaltered. ZONAL HABMONICS. 11 0. We shall next prove that where r' = a;" + y* + «'. Let -].= (xHy* + 2')-* =/(.). and let k be any quantity less than r. Then [a?Arrf-{-{z-lcf]-^=f{z-k), and, developing by Taylor's Theorem, the coefficient of k' is Also {x^ + y' + {z-Tcy]-^ = {f-'2Jez + ky^ ' since z = fir, in the expansion of which, the coefficient of U is P, ■pi' Equating these results, we get P = (-iy ^'" 1(1] The value of P, might be calculated, either by expanding (1 — 2fih + hy^ by the Binomial Theorem, or by eifecting the differentiations in the expression (- 1)' . ^ ^ g '{d?\r) ' and in the result putting - = tt. Both these methods how- ever would be somewhat laborious ; we proceed therefore to investigate more convenient expressions. i^ ZONAL HARMONICS. 6. The first process shews, by the aid of Lagrange's Theorem, that -^' = 2'.1.2.3...i djL' ^' " '^^'• Let y denote a quantity, such that h being less than 1. Then 1 <7y _ h Also ^2,__j =!__ + _; Hence, by Lagrange's Theorem, + i.2...id/t'-'V 2 ;+■••' therefore, differentiating with respect to /* and observing that ZONAL HABMONICS. 13 7. From this form of P, it may be readily shewn that the values of fi, which satisfy the equation P, = 0, are all real, and all lie between — 1 and 1. For the equation (m* — ly = has t roots = 1, and i roots = - 1, ••• X (/** ~ ^)' =° ^ ^^ * ~ ^ ™o*s = 1, (i - 1) roots = - 1, and one root = 0, T-i (/** — !)' = has (t — 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 J—, (jj.* — 1)' = has 2 roots between 1 and 0, and - roots be- tween and — 1, if i be even, 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: Jfmle any integer less than i, dx*- _ ^ l•2■■.(^-m) _ d^ _ 14 ZONAL HAEMONICSi It may be proved in the following manner. If {x"— 1)* be differentiated i —m times, then, since the equation (x»_l)' = has I roots each equal to 1, and i roots each equal = — 1, it follows that the equation has i — {i — wi) roots (i. e. m) roots each = 1, and m roots each = — 1, in other words that (x' — 1/' is a factor of We proceed to calculate the other flictor. For this purpose consider the expression (a; + a,) {x+a^ ... {x + a) (x + fi^) (a; + /SJ ... (x + ^,). Conceive this differentiated (I) i—m times, (II) i + m 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 + a,) {x + a^) ... (x+a,) (a; + /3,) (a; + y8.) ... (x + fi,),le. 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 a; + a, say a; + a', x + a" ...x + a*^', and of q factors of the form x + ^, say x + ^^, x. + ^,^...x+ ^8,,,. We naust ha,\ep + q = i — m. The complementary term in (I) will involve p factors x + ^,x + ^" ... x +/3*', q factors x + a^, x + a^,...x+ ct,,,. ZONAL HAEMONICS. 15 Now, every terra in (I) is of i + m dimensions. We have accounted fov p+ q (or i — m) factors in the particular term we are considering. There remain therefore 2m factors to be accounted for. None of the letters a, a" ... &"', ^,, ^^, ...^,j^ ^', /3"...^'". a„ a„...a„„ can appear there. Hence the remaining factor must involve m a's and m j8's, — say, ,/S, ^...^. There will be another term in (II) containing (0. + ^) (^ + /3") ... (a; + /3^') {x + a)ix + a„) ... (x + aj. The corresponding term in (I) will be, as shewn above, (x + a') {x + a")...{x + a'") (x + /3) {x + /3J ...{x + ^J (x + ^a) (x + .t) ... (x + ^t) (a; + ,/S) {x + ^0) ... (« + jS). Hence, the sum of these two terms of (I) divided by the sum of the complementary two terms of (II) is (x + ^a) (x + ^a) ... (x + „a) {x + ^0) (x + ^fi) ... {x + JT). Now, let each of the a's be ieqiial to 1, and each of the /S's olqual to — 1, then this becomes (x^ — 1)'". The same factor enters into every such pair of the terms of (I). Hence ic'-" " ^ ~ ^ h^' ' ^ numerical factor pris. The factor may easily be calculated, by considering that j(-m /a _ 1 \< the coefiScient of a;'*™ in — ^.'^ is2i(2t-l)...(i+m+l), and that the cceflBcient of x*^ in -5-3^;; — - is 2i (2i- 1) ... (i + m + 1) (i + wi) ... (i-m + 1). 16 ZONAL HAKMONICS. Hence the factor is 1 1.2...(i-m) (t + m) [i+m-l)...(i-m + l)' 1.2... (i + m) ' 9. This theorem aflfords a direct proof that C-r-, (/*' — 1)', C being any constant, is a value of f (fi) which satisfies the equation d d/i {^-f^l^}+iii+^)fO^)=^- from above, or |[('-''')|i{|(-'-M]^-<*-^"{|'''-«'}-°- Hence, the given differential equation is satisfied by put- ting/(^)=(7|],0*'-l)'. Introducing the condition that P, is that value of / (jjl) which is equal to 1, when /* = 1, we get ■^* " 2M.2...t d/? ^' ~ ^^'• 10. We shall now establish two very important proper- ties of the function P, ; and apply them to obtain the develop- ment of Pj in a series. ZONAL HABMONICS. 17 The properties in question are as follows : If i and m he unequal positive integers. The following is a proof of the first property. We have Multiplying the first of these equations by P^ the second by P„ subtracting and integrating, we get + [i {i + 1) - tn (m + 1)} jPtPJ/i = 0. Hence, transforming the first two integrals by integration by parts, and remarking that i{i+l)—m{m + l) = {i-m) (i+m + 1), we get + (i - wi) (i + m + 1) jPtPJ/i = 0, or since the second term vanishes identically. F. H. /: /: 18 ZONAL EABUOKICS. Hence, taking the integral between the limits - 1 and + 1, we remark that the factor 1—fi' vanishes at both limits, and therefore, accept when i — m, or «'+ ni + 1 = 0, -1 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+m + 1 cannot be equal to 0, if i and m are both positive integers. Hence we need only discuss the case in which m = i. We may remark, however, that since -P( = -P_(i+i), the determination of the value of / F^d/j, wiU also give the value of / F,P_^^^^^d/i. The value of I P^^d/i may be calculated as follows : J —1, {l-2iih+h')-i = P^ + Pfi + ... + P/i*+...; .: {l-2^Lh + hY = {P, + PJi+... +F^h' + ...y = Po' + P'h''+... + P,Vi"+ ... + 2P„PJi + 2P,PJi* + . . . + 2P,P,A' + . .. Integrate both sides with respect to ji ; then since / (1 - 2M + hY d^ = -~ log (1 - 2M + hT). 2h we get, taking this integral between the limits — 1 and + I, all the other terms^ vanishing, by the theorem just pi07ed. ZONAL HABMONICS. 19 Hence, equating coefficients of h", I ' ■^*-25+T 12. From the equation I PiP„^d/i = 0, combined with the fact that, when fi=l, P, = l, and that P, is a rational integral function of fi, of the degree i, P^ may be expressed in a series by the following method. We may observe in the first place that, if m be any integer less than i, j fj.''P,d/j, = 0. For as P„, P„.j . . , may all be expressed as rational in- tegral functions of fi, of the degrees m, jn — 1 ... respectively, . it follows that ju." will be a linear function of P„ and zonal harmonics of lower orders, /*""* of P,^, and zonal harmonics of lower orders, and so on. Hence jfi^Pfd/i will be the sum of a series of multiples of quantities of the form I P„P,dfj,, m being less than i, and therefore I f^^Pid/i = 0, if m be any integer less than i. Again, since {l-2fih + hYi = P, + PJi+... + PJi' + ... it follows, writing — h for h, that (1 + 2fih+hT^=p,-p,h+... + (- iyp,4'+ ... 2—2 20 ZONAL HARMONICS. And writing — fiior fi in the first equation, (1 + 2^Ji+V)-^=P^ + P;h + ... + P/A' + ... P,', P,'...P/... denoting the values which P„, P, ... J|., respectively assume, when —ft, is written for /it. Hence P^=P^ or — Pj, 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 P. = ^ ) ^+^'^+'')[jl,+0 + c+0j ~ to ia>-a}{o>-^ {b+0){c+0) ' and therefore, putting = — a, 1 (tt+a)(a + /3)(o + M)(& + ft))(c + o)) " ft> (a — J) (a — c) (w — a) (w — ^8) ' with similar values for y and z. And, if «o be infinitely great, in which case the last equation assumes the form a; + y + ^ = 1, we have (ffi + a) (a + /3) • ^~ {a-h){a-c)' with similar values for y and z. 14, Now consider the general system a;, . a;,., .__?(-» a, + Hfc,, o^ + a^ a^^ + o,.. + ... = 0, 22 ZONAL HARMONICS. + ...+ •"'-" +-=0. a, + a^„ a,^ + a^„ a,_„ + a,_. ^ — + z — r-r + — + „ ■ „ + a>' ttt + a a^ + )-(g^,.+6>)-- and, multiplying by Oj.,, + 5, and then putting 6 = — a^„, ^ ^ 1 («,-« + g,) («,_,. + O • • • (««-» + "«-») • • • '-=• (i»-a,)(6>-«,.,)...(<»-a,.J... {a, + o) (a,.t + to)...(ai.„ + 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 « = oo ; and a, = i, a,_j = i — 2,,..a,_^ = i—2s... a, = » — 1, a^ = i — 3,...a^j, = i — 2s— 1... Hence . _ (2%-'2s-l)(2{-2s-3)...{2({-2s)-l]... *-"' (-2s){-(2s-2)}...{(i-2s-l) or(i-2s)} ^ , ,,. (2i-2s-l) {2i -2s -3)...{2 (1-28) -I]... ^ ' 2s(2s-2)...2x2.4...^i-2s-l)or (i-2s)" ZONAL HAKMONICS. 23 Or, generally, if i be odd, A ^(2i-lK2;-3)...(z + 2) 2.4...(i-i) ' •-* 2.4...(i-3)x2 ' ^ _ (2^-5)(2^-7)■..(^•-2) *-" 2.4...(i-5)x2.4 ' ^'-^ ^^ 2.4...(i-l)' And, if i be even, ^_ (2e-l)(2z-3)...Ct+l) ' 274::i • . _ (2^-3)(2^^-5)...(^•-l) '-* 2.4...(i-2)x2 . _ (2i-5)(2i-7)...(t-3) *-* 2.4...(i-4)x2.4 ' , _, ,J (t-l)(t-3)...l ^•-^"■^^ 271^:^ — • We give the values of the several zonal harmonics, from P, to P„ inclusive, calculated by this formula, ■tj 2'^ 2' 3/t'-l ~ 2 ' r, 5 , 3 ■P. = 2'*-2'* 5/t'-3^ ~ 2 ' 24 ZONAL HAEMONICS. 1\ 7.5 = 2.4" _35|u,*- , 5.3 3.1 2.2^*^2.4 -30/ + 3 8 P. 9.7 , = 2.4'^ _63/i'- 5 7.5 , . 5.3 ' 2.2'*+2.4'* -70/+ 15/4 8 Pe 11.9. ~ 2.4. 231/ 7 . 9.7.5 7.5.3 , 6^* 2.4x:i'*+2x2.4''- -315/ +105/ -5 5 2, ,3, .4. 1 ,6 16 p 13.11.9 , 11.9.7 , 9.7.5 , 7.5.3 ' ~ 2.4.6 '^ 2.4x2^* '*'2x2.4'* 2.4.6'* 429/ - 693/ + 315/ - 35/t - ^ 16 . 15.13.11.9 , 13.11.9.7 , 11.9.7.5 , »~ 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/ - 12012/ + 6930/ - 1260/ + 35 128 17.15.13.11 . 15.13.11.9 T , 13_ll_-9j^ 5 •~ 2.4.6.8 '* 2.4.6x2 '*■*' 2.4x2.4 '* 11.9.7.5 , 9.7.5.3 2x2.4.6'* ■^2.4.6.8'* _ 12155/- 25740/ + 18018/- 4620/ + 315/t 128 „ _ 19.17.15.13.11 ., 17.15.13.11.9 g . 1 5.13.11. 9.7 , " 2.4.6.8.10'* 2.4.6.8x2 '*'^2. 4. 6x2. 4'* ZONAL HAEMONICS. 25 13.11.9.7.5 .11.9.7.5.3 , 9.7.5.3.1 2.4x2.4.6'* "''2x2.4.6.8'* 2.4.6.8.10 _ 46189/*"'- 109395/+ 90090/t'- 30030/^*+ 3465/^*- 63 256 It will be observed tbat, wben 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„ is that which enters as a factor into the continued product 1 . 2...i. 16. We have seen that / /*" P, . rf/t = 0, if m be any integer less than i. It will easily be seen that if »n + 1 be an odd number, the values of jfi" P, . dfi are the same, whether fi be put = 1 or — 1 ; but if m + 1 be an even number, the values of jfi" P, . dfi corresponding to these limits are equal and opposite. Hence, (m + i being even) and then I /ii"P,.d/* = 0, if m = t — 2, i — 4 Jo We may proceed to investigate the value of I fi'^Pf d/i, Jo it m have any other value. For this purpose, resuming the notation of the equations of Art. 13, we see that, putting ^ = m + 1, and a> = oo , we have Ot+m + l a<.,+ m+l a^^ + m + l (m + 1 — a^(m + 1 — a^) ... (m + 1 — o,.J ... ~ (o, + m+'l) [at^ + m + 1) ...(a^ + m + 1) ... ' 20 ZONAL HARMONICS. /, /. and therefore, putting a;, = u4,..., a, = i..., tt, = i—l..., we get (m — i+2)(to — i + 4) ...(ot — 1) .. ., ,, = -; ^— : wr-, — -^^—. ^-^ —, ,. , — - — ;^ U t be Odd, (»i + « + l)(m + t-l) ... (ni + 4)(m + 2) "«"""> J (to — t + 2)(m — 1 + 4) ..,m ., ., *°<^ = / — , • , TV/ — , • TV , . Q^ / — nr II t be even. (m + « + 1) (to + z — 1) ... (m + 3) (to + 1) In the particular case in which to = i^ we get ^'^''^^ = (2^•f 1) (L-- 1) .^'.^•+4) (^•+2) (^''^'^'^)' "'^^ = (2^•4-l)(2.•-l^^■^.^^•+3)(^• + l) (*'^^^^)- 1 7. We may apply these formulae to develope any positive integral power of /it in a series of zonal harmonics, as we proceed to shew. Suppose that m is a positive integer, and that /t" is de- veloped in such a series, the coeflBcient of P^ being (7„ so that then, multiplying both sides of this equation by P, and inte- grating between the limits — 1 and 1, all the terms on the right-hand side will disappear except I C, P", d/t, which will 2 ' become equal to ^. — r G,. 2i + 1 Hence Ci = ^ J V'-P.f^A*. which is equal to 0, if m -|- 1 be odd. Hence no terms appear unless m + » be even. In this case we have = (2;+i)[V"P,cZ/t. •' ZONAL HARUONICS. 27 Hence the formula just investigated gives r rgiMI (wt-t + 2)(m-t + 4)...(TO-l) ' ^ "^ ''(m + t + l)(m + i-l)...(m + 4)(m + 2) if i be odd, and r ('^i\^\ (wI-^^+2)(CT-^ + 4)...OT ' ~ ^''*'^ -^^ (m + » + 1) (m+ 1- 1) ... (m + 3) (m+ 1) if I be even. Therefore if m be odd, „_. . 2.4.6...(m-l) p /* -C^'« + l;(2m+l)(2m-l)...(«i + 4;(m+2)^»+-" ■^' (m + 4)(m + 2; »'*'m + 2 "• If jn be even, „ _ ,„ ^ V 2. 4.6 ... TO p ^ (m + 3)(m+l) »^m+l •• Hence, putting for m successively 0, 1, 2 ... 10, we get 2 12 1 = ^P ±.i.v Ap 35 *^7 » 5 •• 28 ZONAL HARMONICS. 4^ + 1^ + 1^^' -i6p 24 10 1 "231 "^77 * 21^^7 "^ 7 ,. 2.4.6 p ,„ 4.6 p y 6 3 ^=^^ 15.13.11.9 ^^ + ^h3.11.9^' + ^llT9^' + 9^" . -iip+lp+l^p+lp "429 '39 * 33 » 3 " ,>. 2.4.6.8 p , ..^ 4.6.8 „ 17 . 15 . 13 . 11 . 9 ' 15 . 13 . 11 . 9 ' . _6^_8_p , 5_8_p , Ip ^ ^ 13 . 11 . 9 * ^ 11 . 9 -^o ^ 9 ■" ^m^ 48 40 1 6435 ^ ^ 495 ' ^ 143 * ^ 99 * ^ 9 <" „8 _ 2.4.6.8 „ . ,, 4.6.8 = 19 .^ ,.,•.•" „ P.+i5 ,„ ;• ,•" „ A 7 '^ ~ 19.17.15.13.11 »■*■ 17.15.13.11 ■*""l5.13.11-^'''''^i3TII-^»'^^ll-^" = _M-P +i92 1_6 _56 ^ 12155 • 2431 ' ^ 65 " ^ 143 ' ^ 11 >' 2.4. 6.8.10 4.6.8.10 '^ 21 . 19 . 17 . 15 . 13 . 11 "^""^ ^' 19.17.15.13.11 » ^^'*17.15.13.ir'+^15.13.11-^« + ^13.11^«+H^'> 256 128 32 48 50 1 46189 ">" 2717 • 187 • 143 *^ 143 »^ 11 "' ZONAL HARMONICS. 29 18. Any zonal harmonic P^ may be expressed in a finite series of cosines of multiples of 6, these multiples being id, {i^ 2)6.... Thus (l-2M + A')'* = P, + P,A + ... + P.;i,'+...; therefore, ■writing cos 6 for /*, and observing that 1 - 2 cos e/i + A» = (1 - Ae^i ») (1 - Ae-^^»), we obtain or -^^tS^^^'-— ■■) whence, equating coefficients of h\ the last term being | -' "' ■ [ if i be even, and l,3_,,4+2) 14^:1^^) 2 cos e, if i be odd. 19. Let us next proceed to investigate the value of /, P, cos mO sin 6 dO. 30 ZONAL HARMONICS. This might be done, by direct integration, from the above expression. Or we may proceed as follows. The above value of P^ when multiplied by cos md sin (that is by ^ {sin (m+ 1) ^ — sin (to — 1) 6]) will consist of a series of sines (if angles of the form {i — 2n + (m + 1)} 6, that is of even or odd multiples of 6, as i + m is odd or even. Therefore, when integrated between the limits and ir it will vanish, if i + m be odd. We may therefore limit our- selyes to the case in which t + m is even. Again, since cos md can be expressed in a series of powers of cos 6, and the highest power involved in such an expression is cos*"^, it follows that the highest zonal harmonic in the development of cos m6 will be P„. Hence / P. cos m^ sin dd Jo will be = 0, if m be less than {. Now, writing P,= <7,cos;^+ C^cos {t — 2)0 + ... we see that P, cos m6 sin 6 dd will consist of a series of sines of angles of the forms (m + i+1) 0, {m + t — 1) ... down to {m — {—V)0, there being no term involving m0, since the coefficient of such a term must be zero. Hence /■ Jo Pj cos m0 sin d0, '0 will consist of a series of fractions whose denominators in- volve the factors m + i+l,m + {—l...m—i—l respectively. Therefore when reduced to a common denominator, the result will involve in its denominator the factor (m + 1 + 1) (m + 1 - 1) . . . (ot + 1) (m - 1) . .. (m - 2 - 1) if m be even, and {m + i + l)(m + i-l)... (m + 2) (m-2) ... (»i_;_l) if m be odd. For the numeriator we may observe that since I Pf cos m0 sin 6d9 JQ ZONAL HARMONICS. 31 vanishes if m te less than i, it must involve the factors m — (i — 2), m — (i — 4) . . . «i + {{ — 2), and that it does not change sign with m. Hence it will involve the factor {m-(;-2)}{ni-(t-4)}... (TO-2)m''(»t + 2)...(m + i-2) if m be even, and [m - {{- 2)} {m - (i- 4)} ... (m - 1) (m + 1) ... {m + i- 2) if m be odd. To detennine the factor independent of m, we may pro- ceed as follows : P. = C,cos id + C^ cos (i- 2)6+ ... ; ,", Pf cos m^ = 5 C^ {cos (»ra + i)6-\- cos (»n — t) ^} + 2 ^'-» t°°^ (wi + i- 2) 5 + cos (m - i + 2) 6} + ... ; .•. P,cosm5sin^ = T (7, {sin (m. + i + 1) 5 — sin (ni+i— 1) ^ + sin (to — J +1)^ — sin (m — 4 — 1^^} + \g^ {sin (m + 1 - 1) 5 - sin (m + z- 3) 9 + sin(m-i + 3)^-sin(m-t+l)^] + ...; .-. {'p^cosmesmedd icj^ i__^_l L_l 2lTO + i + l m + i—l m — i + 1 m — i—l\ . c.J_J L_+_i 3^1 + ... 2\m + i-l m + i-3 m-i + 3 m— t + lj r i-1 t-1 ] = ^' i " 7«'-(i + l/ ^ m" - (» - 1)'J ■l)»'^»i''-(i-3rj 32 ZONAL HARMONICS. Now, when m is very large as compared with t, this be- comes = -2 C,+ C^, + ..._ _2 m m since C, + (7(_, + ... = 1, as may be seen by putting ^ = 0. Hence I F, cos m^ sin d dd tends to the limit — - , as to Jo »» is indefinitely increased. The value of the factor involving m has been shewn above to be {m-(i -2)]{m- {t-4)} ... (m-2) m'(m + 2) ... (m + t-2) {m - {i+l)\ (m - {i - 1)} ... (m - Ij (m + 1) ... (to + 1 + 1) if to be even, and {m - (i- 2)} {m - (t- 4)} . .. (ot - 1) (to + 1) ■ ■■ (m + i- 2 ) '{to - (i+l)] [m - {i- l)j ... (to - 2) (to + 2) ... (to + i+ 1) if TO be odd. Each of these factors contains in its numerator two factors less than in its denominator. It approaches, therefore, when TO is indefinitely increased, to the value — j . Hence /, F, cos md sin 6 dd ^ {to- (^•-2)HTO-(^-4)]...(OT-2)m'(TO+2)...{TO+(^-2)} {m-(i+l)}{m-(i-l)\...{m-l){m + l)...[m+(i+l)] if m aiid i be even, and ^ _ {TO-(t-2)}{TO-(t-4))...(TO-l)(TO + l)...{TO+(/-2)} {TO-(^•+l)}{TO-(^•-l)}...(TO-2)(TO+2)...{TO^-(^^-l)j if TO and t be odd. In each of these expressions i may be any integer such that TO — I is even, t being not greater than m. Hence they will always be negative, except when i is equal to m. ZONAL HAKMONICS. S3 20. We may apply these expressions to develop cos m0 in a series of zonal harmonics. . Assume cosme = £^P„ + B^,P^ + ... +B,P, + ... Multiply by P* sin 0, and integrate between the limits and IT, and we get „ {m-(^-2)}{m-(^•-4)}..■{m+(^-2)} 2_ ^{m-(^ + l)}lm-(^•-l)}...l»^ + (^ + l)} 2i + l'^'- Hence , {m-{i-2)}{m-ii-i)]...\m + (i-2)} A - k^^ + ^-l {^ _ (;+ i)j [m - (i- 1)} ... {™ + (i + 1)} Hence, putting m successively = 0, 1, 2, ... 10, cosO^ = Pj; cos6 = P,; J-p-lp- 3 » 3 "' cos8g=-7 _^yg y P.-3^P. 6 » 5 " *=°'*^ = -^-l. 1.3. 5.7.9^'" +1.3.6.7^' 3.5 -l^rr^. _64p 16p_lp. ~35 * 21 -^^ 15 •' r. H. 34 ZONAL HARMONICS. <=°s^^ = -"_l.l,3.7.9.11 ' '1.3.7.9 " -3-^P _128 8 1 p. -"63^'"9^» 7 " 2. 4. 6'. 8. 10 '-1.1.3.5.7.9.11.13' COS 6» = - 13 1 1 o E hr n n 1«J •'^e 4.6'. 8 6'__p_J^p ^1.3.5.7.9.11 * 3.5.7.9^ 5.7 ' _512 384 j^ J. p. "231 ' 385^ 21 ' 35 »' ^. ,. 2.4.6.8.10.12 p 4.6.8.10 6.8 3^ 1.3.5.9.11.13 » '3.5.9.11 » 5.9 ' 1024 p_128p_112p_J_p. COS 8^ = -17 "429 ' 117 » 495 « 15 2.4.6.8M0.12.14 -1.1.3.6.7.9.11.13.15.17 4.6.8M0.12 6.8M0 1.3.5.7.9.11.13.15 ' 3.5.7.9.11.13^ ■^5- 8' .7.9.11 P,- 1 7.9 P. 256 1001 -P.- 64 693 " 1 63 P.; » 16384 4096 p - 6435 » 3465 ' coaQg- 19 2-4.6.8. 10.12. 14.16 cosyp- -ly. 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 ' 3.5.7.11.13.15 = '5.7.11.13 " 7.11 • ZONAL HAKMONICS. 35 _ 32768 3072 128 16 „ 3 "12155 • 2431 ' 455 « 143 » 77^' infl- 2. 4. 6.8. 10M2. 14.16. 1.8 p *^°^^"''~~^^-1.1.3.5.7.9.11.13.15.17.19.21^» 4.6.8.10M2.14.16 1.3.5.7.9.11.13.15.17.19" 6.8.10M2.14 8.10M2 3.5.7.9.11.13.15.17 ' 5.7.9.11.13.15 * 5 10' p l_p 7.9.11.13 » 9.11 » _131072 32768 p 512 128 500 p ~ 46189 " 24453 ' 1683 ' 1001 * 9009 * 99 » 21. The present will be a convenient opportunity for investigating the development of Bind in a series of zonal harmonics. Since sin ^ = (1 — /*')* it will be seen that the series must be infinite, and that no zonal harmonic of an odd order can enter. Assume then sm0= G,P,+ (7,P, + ... + CiPi+ ... I being any even integer. Multiplying by Pi, and integrating with respect to /i between the limits — 1 and + 1, we get ^'P,sin5^/* = 2^a; ^ Jo supposing Pj expressed in terms of the cosines of 6 and its multiples = ?i+if'p.(i_cos25)dft 4 Jo 3—2 /: 3G ZONAL HAKMONICS. Hence, putting » = 0, c,=lf\i-cos2e)de=l. 1 3 Putting i= 2, and observing that P, = t + t cos 20, ^ 5 f ' (1 + 3 cos 20) (1 - cos 26) ^^ = ^ r jlH- 2 cos 25 - 1 (1 + cos 40)1 rf5 5 32''- For values of t exceeding 2, we observe, that if we write for Pi the expression investigated in Art. 18, the only part of the expression I Pj (1 — cos 20) d0 which does not vanish will arise either from the terms in P^ which involve cos 20, or from those which are independent of 0. We have therefore _ 2i-f 1 l■3...(^-l)l■3..■(^•-3) 4 '2.4... { 2.4... (t-2) /' (rr" "^ r~\ ^ ""^^ ^^) ^^ ~ ^^^ ^^ ^^ _2i + 11.3...{{-l)l.S...(i-S) fi-1 « + l\ 4 2.4... I 2;+ 11.3... 2.4... (;-l) 1 {i- .3. -2)" V i -3) i Hence 2 2.4... sin^ = ^P 4 ' (2» + l)7rl. «(t + 2)2 » 32 » 3... (;- .4. 1) 1.3. .2)i- -3) r* being any even integer. 2 2 .4 ... i(i + 2) 2.4 ... (1-2) i ^*~ ZONAL HARMONICS, S7 HP 22. It will be seen that -j-* , being a rational and integral function of /x", /«.***..., must be expressible in terms of -Pj-u -P(-8'" To determine this expression, assume ' = C^*-i -Pl-i + ^(-» -'^l-s + ••• + C'm-Pm + dfjL then multiplying by P„, and integrating with respect to ^ from — 1 to + 1, Now, since j>m^ 4j ^.^„.= (iy.* = [P„PJ^-[P.Pr = 2, since either m or i must be odd, and therefore either P„ or P, = — 1, when /i = — 1 ; •••2 = 2^^--^-=^"^-^^' .-. ^'= (2i-l) P^. + (2i- 5) P..3 + (2i- 9) P^+ ... Hence f -^' = (2-1)^-- 23. From this equation we deduce the limits (i and 1 being taken, in order that P^-Pi-i may be equal to at the superior limit. 3S ZONAL HARMONICS. Now, recurring to the fundamental equation for a zonal harmonic, we see that •••■P.-«-.-|^(^-''''%" -6^4t)(i-m-)°-- 24. We have already seen that | PjP„ dfi = 0, i and m being different positive integers. Suppose now that it is required to find the value of I P,P„ d/i. We have already seen (Art. 10) that i/'^- '^ (i-m)(i + m+l) • And, from above, ■ fpP t?ir- ^ f m(TO + l) p .p _^p . ■J^ ' " '^ (»-m)(i+TO+l)t2m + l ^'^-^"« -^-1^ i(i + l) _ I ZONAL HARMONICS. 39 25. We will next proceed to give two modes of ex- pressing Zonal Harmonics, by means of Definite Integrals. The two expressions are as foUows : p _ 1 r d^ ' tJo {/* + (/*» -l)i cos &P' p, = 2 r {/i + (^» - i)i cos fy df. These we proceed to establish. Consider the equation -f- TTJo a — d^ 6cos^ (o*-5')i' The only limitation upon the quantities denoted by a and b in this equation is that 6* should not be greater than a\ For, if b' be not greater than a', cos ^ cannot become equal to y- while S- increases from to ir, and therefore the expression under the integral sign cannot become infinite. Supposing then that we write z for a, and V— 1 p for b, we get 1 f» d^ _ 1 We may remark, in passing, that f' d^ ^ I' ^ Jo 8 — V^pCOS^ JQ Z + '^ — l Jo pcos^ zd^ /o z' + p'cos'a^' and is therefore wholly real. Supposing that p* = a? + y', and that a?+if' + s* = r\ we thus obtain 1 [' d^ 1 ttJo z-t/^pcos^ t' 40 • ZONAL HAEMONICS. Differentiate i times ■with respect to z, and there results Trcfo'Jo z-V-lpcos^ de'r r**'^ ' \ p cos ^ Hence P, = — ,- o~o — -Ti /— ^ TT Jo (2_V— 1 O (2_V-lpC0Sa^)'" In this, write fir for a, and (1 —/*')- r for />, and we get 1 T" d ■^* " ^ Jo {/*-0*'-l)*cos^l'« ' which, w^ting ?r— ^ for ^, gives d^ o{/t+0'-l)4co8^P' 26. Again, we have 1. 1 f' dyfr __L;__1 /■" (a'-i')i-,rio^ • 6 COS 1^ In this write 1-fih for a, and + (/*'- 1)* h for 6, which is admissible for all values of h from up to /* — (/*' — 1)*, and we obtain, since a' — 6' becomes 1 — 2jxh + A", 1 _ 1 f « C?1^ (l-2M + A')*~^Jo l-/iA+(/t»-l)4 _ 1 /■» d^Jr TT/o l-fyx + O**— ■ Acos-^ 'ol-{/t±0*'-l)*costlA' .•.l+P,A+... + P^' + ... + {/t ± (m'-1)* cos tl'A' +...]. ZONAL HARMONICS. 41 Hence, equating coefficients of h*, ■P. = ^/^"{/*±(/*'-l)*cos^}'di^. The equality of the two expressions thus obtained for P, is • in harmony with the fact to which attention has already been directed, that the value of P^ is unaltered if — (i + 1) be written for i. 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 OG=h, COP = %COQ = ir. Then CP* = a* + V-2abcoa^, CQ' = a' + b'-2abcos-^. Hence (a" + V- 2ab cos ^) (a" +V-2ab cos f) = (a» - 6»)» ; sin^d^ Bin^frdyfr _^ •'• a» + 6" - 2a6 cos ^ ■*" a' + i' - 2a6 cos ^ 42 ZONAL HABMONICS. Again, since the angles OPC, OQC sue equal to one another, sing Bin OPC sin OQO sin f ^ CP ~ 00 ~ 00 GQ ' ■ sin S^ sin'i^ ■■ (a=+6'-2a6cos^)4" (a" + 6'- 2a6 cos^)* ' whence -, j. ^—, ri — 0. (a' + 6" - 2a6 cos ^)4 ^ (a» + V- 2a6 cos f)* In this, write a* + b^ = /ji, 2a& = + (/*' — 1)*, which gives a^ — b' = 1, and we get {M±(^'-t*cos^r =-{^±^'-^)^-°-^^'^^- We also see, by reference to the figure, that as S^ in- creases from to TT, -^ diminishes from tt to 0. Hence •' {/* ± u* — ir cos »j J 28. From the last definite integral, we may obtain an ex- pansion of P, in terms of cos and sin 0, Putting ft = cos 0, we get Pi = 2^ r [{cos 6 + V^ cos ■f sin 0)}* + {cos — V^ cos ■^ sin g}*] d^ = i j"''{(cos 0)' - i^^ cos' t (cos g)'- (sin g)' +. . . ZONAL HARMONICS. 43 + ...]dyjr. XT f'/ .Nsmj, (2m-l)(2m-3)...l Now {cosfy"dy}r=ir ^ „ .y — »- ' . Jo 2m (aw — 2). ..2 , ^•(^-l)■■.(^•-2TO + l) (2m- 1) (2m-3)...l 1.2.. ,2m 2m(2m-2)...2 ^ ^•(^•-l)■■.(^•-2OT + l) . (2. 4... 2m)' ' /. P. = (cos ey - ^^^ (cos 0)" (sin &)»+... +(-ir ^'^^~g-i:.2,';'''^ (°osg)'-"(siagr+-. CHAPTER III. APPLICATION OF ZONAL HARMONICS TO THE THEOET OF ATTRACTION. REPRESENTATION OF DISCONTINUOUS FUNCTIONS BT 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 I'evolution, 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 2irpck, 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' + a')* Now, this expression may be developed into either of the following series : 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 « 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 ^ = /li. Then, the potential at this point will be given by one of the following series : .(!'). 1.3.5...(2i-l) ^' ^^ ' 2. 4. 6. ..24 -^•'c" ,,l. M( 1 c' 1.3pc* ,1.3.5...(24-1) c« 1 }...(2'). For each of these expressions, when substituted for F, satisfies the equation V'K = 0, and they become respectively equal to (1) and (2) when d is put = 0, and consequently r = z. ' The expression (2') also vanishes when r is infinitely g^eat, and must therefore be employed for values of r greater than c, while (l*) becomes equal to (2') when r = e, 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, = l W + J'^^^1 cos&)'d&, or P, = - f 0* + J7^ cos -f )-"*" d'^. IT Jo Substitute the first of these in (1') and (observing that fir = z) we see that it assumes the form 1.3f . + (.'-r^icos^r _ V ^2.4 c* J 46 APPLICATION OF ZONAL HARMONICS ■which is equivalent to Mr ^ TT j„ [c^ + {s + («"-»•')* cos &ni' The substitution of the last form of P, in the series (2') brings it into the form Mf'i 1 1 c' ^io tz + C^-r'^^cos^ 2{2 + (/_r»)4cos^}' 1.3 c^ \.^ ■which is equivalent to Mt' IT Jo [{3 + (^-r')icos^r + c"]i' 2. Suppose next that the attracting mass is a hollo^w 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 A^ + A^z + A^^ + ...+A/+... Then the potential at any point lying ■within the inner bounding surface will be AJP, + A,P,r + A^y+...+ A,Py + ... For this expression, when substituted for V, satisfies the equation V'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. 4? Then the potential at any point lying without the outer bounding surface will be For this expression, when substituted for V, satisfies the equation V 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 tbe value of either of these potentials for any point in the axis be denoted by t}> (z), the corresponding value for any other point, which can be reached without passing through any portion of the attracting mass, will be - f A{3 + (s'-r»)4cos^}d^. ""Jo 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, svhtended by a closed plane curve at any point, is proportimd to the component attraction perpendicidar 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, 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 9 And the component attraction of the element of the lamina corresponding to dS in the direction perpendicular to its plane will be 7 dS n pk —^ cos V, 48 APPLICATION OF ZONAL HARMONICS p being the density of the lamina, k 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 wy 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 —-J-. Now since Fis a potential we have V'F=0, whence -7-V'F=0, or V'(-j-]=0. Hence -5— is itself a potential, 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 (a?, y, z) be denoted by to, to will be a function of x, y, z, satisfying the equation V'A; {(c' + s')i - «} if 31 be the mass of the lamina. Expanding this in a converging series, we get •^""^f "•^■^2c 2. 4 c''*' 2. 4. 6c' •" ^ ^^ 2.4.6...2i c"*-'^ j if z be less than c, and ■¥[lc' l.lc^ 1.1.3c' c't2z 2. 4 z'"^ 2. 4. 6 3' ■" ^ '' 2.4.6...2i a«'-» "*■••••) if a 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 : Tr -^^fn B . 1^,»^ I.IP/ 1.1.3...(2t-3) Py I ^ ^ 2.4.6...2i c"'-' ■^••■•j if r be less than c, and T^_Jlf fl P/ 1. 1 P/ , 1 .1 .3 P/ c»t2 r 2.4 r' "*'2.4.6 / "■ ' ,; _ . 1.1.3...(2i-3) F^ . I ^ ^ 2.4. 6.. .2i • r«*7''^""J 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 2.-2 r ^ + (^'-r')'cos0 Jo[c'+{2 + (z»-r»)4cos^}']i and the potential of the lamina in the form filf[c'+{z+ (a" - o* cos ^n* de - ^. 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 M potential at any external point P' will be equal to 77-57, 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 CO . CO = CA*. Then O 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 OG 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 -5 . Hence Jfcf = 27r [fp p r' sin dr d0, 4—2 52 APPLICATION OF ZONAL HAKMOHICS the limits of r being the two values of r which satisfy the equation of the surface of the sphere, viz. r' + c' — 2or cos 6 = a\ and those of 6 being and sin"' - . c Hence, if rj, r, be the two limiting values of r, we have 2_2 _ 2c cos g /I 1\ Now ., 1 1 2c cos ^ Also - + _=_^ r^ r, c - o* ,^ii ^r, r,y c' — a* fa' -c* sin' ^) J = 2 c* - a' ,, 2Trpc' 2c 2 rsin-'e ^ . ^ . . . ■■■ ^^^ -9 ;;^^r« -.T ^ cos^sin^(a'-c'sin'^)*' = -r^rz-^2J cos^sin0(a'-c'sin'^)irf^ _ 4 '^PC* , 3(c^-a7 • Now, if r be the potential atP, we have (see Chap. i. Art. 1) ^^{rV) 1 d ( . .dV\ ^TTpc' TO THE THEOnr OF ATTRACTION. 53 This is satisfied by F= — -z- ^^. Assume then, as the complete solution of the equation, + {a,,'+P)p, + .... It remains to determine the coefficients A„, A^.-.A^.-.B^, B^...B^, 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 M be equal to yp-p, where (XP : OF :: a : c, and therefore, at the surface, „_ Mc 1 _ 4 irpcW a OF 3(c'— tt';-/- And, at the surface, we have r'- 2crcos0 + c' - a* = ; 1 _ 1 / 1 2ccosg \ c —a \ r r J + (A^r-\--A Pi + ... identically. and P„ 5„...5....^„ A^...A^ all = 0. APPLICATION OF ZONAL HARMONICS Hence since P. = 1, 2 frpc' f 2a^ ] and A = Q ;5- whence we obtain, as the expression for the potential at any internal point, „_2 7rpc° 5a^-c^ . 4 Trpc' cos0 _2vpc^ 3(c'-ay r ■'■3?-a=' / 3 /"* 6. We shall next proceed to establish the proposition that if the density of a spherical shell, of indefinitely small thick- ness, be a zonal surface harmonic, its potential at any internal point will he proportional to the corresponding solid har- monic of positive degree, and its potential at any external point will he proportional to the corresponding solid harmonic of negative degree. Take the centre of the sphere as origin, and the axis of the system of zonal harmonics as the axis of z. Let h be the radius of the sphere, hb its thickness, U its volume, so that U= 4!TrVBb. Let GP, be the density of the sphere, P, being the zonal surface harmonic of the degree i, and C any' con- stant. Draw two planes cutting the sphere perpendicular to the axis of z, at distances from the centre equal to f, ?+ df respectively. The volume of the strip of the sphere inter- cepted between these planes will be ^ C^ and its mass will be OP ^TT ' 26 Now ?= bfi, hence d^ = hdfi, and this mass becomes TO THE THEORY OF ATTKACTION. 55 Hence the potential of this strip at a point on the axis of s, distant z from the centre, will be CJJ P. , which may be expanded into ^P.(p. + P.? + ... + P,^+....)i/.if.5. To obtain the potential of the whole shell, we must inte- grate these expressions with respect to /t between the limits - 1 and +1. Hence by the- fundamental property of Zonal Harmonics, proved in Chap. II. Art. 10, we get for the po- tential of the whole shell GU a' . ^ , . 9"= — f liH ^t an internal point, „ . — 7 -Tfi at an external pomt. 2i + 1 z ^ From these expressions for the potential at a point on the axis we deduce, by the method of Art. 1 of the present Chapter, the following expressions for the potential at any point whatever : CU Pi* v. = ,-r^ — 5 TSi at an internal point, • "2.1 + \ h^^ ^ ^^ CU P.h' , , . , V, = TT- — T TT at an external pomt. * 2i + l r '■ From hence we deduce the following expressions for the normal component of the attraction on the point. Normal component of the attraction on an internal point, measured towards the centre of the sphere, dr 2i + l b in 56 APPLICATION OF ZONAL HARMONICS Normal component of the attraction on an external point, measured towards the sphere, dV„_i+l ^jjVl In the immediate neighbourhood of the sphere, where r is indefinitely nearly equal to h, these normal component at- tractions become respectively and their difference is therefore And writing for U its value, 47r&'85, this expression he- comes 4irSh. CP,. 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 iv x thickness of the shell. 7. It follows from this that if the density of a spherical shell be expressed by the series C^,+ C,P,+ C,P, + ... + O.P.-h ..., (/„, C,, Cj ... (7, ... being any constants, its potential (V^) at an internal point will be" and its potential ( FJ at an external point will be /^ 10^ 101^ 1 C,P,V \ 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 the same element. The product of the volume density of any element of the shell, into the thickness of the shell in the neighbourhood of that element, is called "surface den- sity." We see from the above that, if the surface density be expressed by the series o-„P„+ «r,P, + o-,P, + ... + 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-PiC', be ^P,c'", 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. And, integrating as before, we obtain for the potential of the whole shell, 47r p „ a"-''" - a'"'-"^» , . , • . (2t + l)(m + t + 3) F^' P^^ ^* *^ ^^*"™^^ P°^'^*- And, at a point forming a part of the mass, 2i + l A" V m-i + 2 '' ''' w + i + 3 r'W ' 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 xi/, the density may be ex- pressed by ^^^ or tjcV- Now /*' = — ^ — . 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 - , u ' TO THE THEORY OF ATTRACTION. 01 1.5 A« ^" " -''^ ' 105 A'^' ~^^' 15 -r It""^ + -w) • And for the latter term, writing for i, and 2 for m, and multiplying by s, o 4Tr p IF?.^"^ 47r p f a*-r* r'-a"\ 12'h'^'' a;. 15A'~r- ' ~3 /7 l,~r~"*" ~5r /• 3s' — r' And, since P^r* = — ^ — , we get for the potential at an internal point at an external point at a point forming a part of the mass p f47r /a» - r« , r^- a^X ,„ , ^ i-n- /a*-r* , r*-a"\\ /?iiil-F-+^(3.'-^^+T(-T-+^r 12. We may now prove that by means of an infinite series of zonal harmonics we may express any function of /* what- ever, even a discontinuous function. Suppose, for instance, that we wish to express a function which shall be equal to A from /A = l to /* = X, and to B froiii /t=X, to ft = — 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 fi between 1 and \/ and to B for the part corresponding to values of (i between \ and — 1. Divide the shell, as before, into indefinitely narrow strips, by parallel planes, the distance between any two successive planes being cdfi. 62 APPLICATION OF ZONAL HARMOmcS 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 ItrBeic These are respectively equal to at an internal point ; and to Z at an external point. Now it follows from Giap, IL (Art. 23) that if i be any positive integer, whence, since I ^,d/i = 0, it follows that TO THE THEORY OF ATTRACTION. 63 Also [ P^dn = 1 - \, rP,dfi= l+\\ Hence the above expressions severally become : For the potential at an internal point on the axis '^-^ 1 ^ (1 - X) + 5 (1 + X) -^{P^ix) -p„(x)] I [^ (1 - X) + 5 (1 + X) -^^{P^QO -P„(X)] ^~^{^.w-p.(x)i?!-, -^{pa>-)-Pi-.m and for the potential at an external point on the axis ^-3^{P.W-P.W}^. -^^{P.(X)-P,(X)}J-... ■^{P^.(X)-P,,(X)1^,-, Hence the potentials at a point situated anywhere are respectively ?^[{4(l-x,)+B(i+x)lP.M -^mw-p.w)^-.. at an internal point; C4, APPLICATION OF ZONAL HAEMONICS and 27rc=acr{^(l-X) + £(l+X)} PoW - 2t + l l^<+iW - ^i-iWl — m- - J at an external point. Now, if we inquire what will be the potential for the following distribution of density, ilA{l-\) + B(l + X) - (^ - B){P,{X) - P.(X))P,(m) -. (A - B) {P.(\) - P,(\)1P, (m) - • • • - {A-B){P^,{X) -P,_,(X)}P.(m) -...], 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 diiferent form, the expression + {P,(X)-P,(X)1P,0.)4-... ... + {P.,,(X)-P,.,(X)]P.(^)-h...] TO THE THEORY OF ATTRACTIOIT. 65 is equal to A, for all values 0(41 from 1 to \, and to B for all values of 7* from \ to — 1, 13. By a similar process, any other discontinuous function, whose values £^re given for all values of fi from 1 to — 1, may be expressed. Suppose, for instance, we wish to express a function which is equal to A from /x. = 1 to /t = \, to 5 from fi = \to /i = \,, and to G from fi=\toji = — l. This will be obtained by adding the two series .+ {P.„(\)-P^,(\)}P.(m) + ...J, ^-^,[\+{PA)-P.(\)l^»+- For the former is equal to A—B from /* = 1 ,to ^ = X,, and to from fj, = \ to /j. = —l; and the latter is equal to B from fi = It© /* = jtp and to C from /* = X, to /* s=— 1. By supposing A and G each = 0, and 5 = 1, we deduce a series wMch ^is equal to 1 for all values of /*'frem /i=\to fi = \ and zero for all other values. This will be This may be verified by direct investigation of the potential of the pprtion of a homogeneous spherical shell, of density unity, comprised between two .parallel planes, distant respectively c\ and cX, from the centre of the spherical shell. 14, In the case in which X, and X, are indefinitely nearly equal to each other, letX, = X, and\ = X + dX. We then have, ultimately, P.W-P.W=^dx- r. H. ^ 66 APPLICATION OF ZONAL HARMONICS Hence P^,(\J - PU\) - -fli(\) " A.,(\) Hence the series ^{1 + 3P.(X)P» + 5P,(X}P.(/x) + ... + {2i + l)P,(\)P,{^L) + ...} is equal to 1 when fi = X (or, more strictly, when fi has any value from X, to X + dX) and is equal to for all other values of /i. We hence infer that 1 + 3P,{X)P» + ... + {2i + 1)P,{\)PM + ... is infinite when fi = \, and zero'fer all other values of fi. 15. Bepresenting the series i{l + 3P.(X)P.W + ... + {2i+ 1)PA)PM+ ••-•} by ^(X) for the moment, we see that p^QCjdX is equal to p when fi = \ and to zero for all other values. Hence the expression is «qual to p^ when fi = \ to p, when p,=\... Supposing now that X,, X,... are a series of values varying continuously from 1 to — 1, we see that this expression becomes r p^{\)dX, p being any function of X, continaons or discontinuous. Hence, writing ^(X) at length, we see that I [fjd\ + ZP.Iji) f pP,{\)dk + ... + (2t + l)P,(/t) J* pP,(X)dK + ...I is equal, for all values of p, from — 1 to + 1, to the same function of p, that p is of X. TO "fSS tKSOET' Of IttRACTIdlt. 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 pdK whieh: is e-djual io^U p^+-j pP^QC}^ + ... +|j*pF,(\)) — must be a solid harmonic of the degree i. Hence FQi, ^) 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 TESSEHAL AITO SECTORIAL HAKMONICS. 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 (»'+i)yi;i. 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 (2i + l)-'. c 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- tractiou 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 difference between these components, viz. (2i+l) — , will be equal to twice the component attraction of the smaller portion in the direction of the line joining the two points. But if p, be the density of the shell, ho its thickness, this component attraction is 2irp,Sc. Y Hence (2t + 1) — = 47rp, Zc, c 2i + l ^ 72 SPHERICAL HABMONICS IN GENEEAL. And, if we ob- tain a series of i real solutions of the equation. Another such series may be obtained by 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 Pt{z)) a series of 2i + l independent solutions of the given equation, which will be the 2i + 1 independent solid harmonics of the degree i. _ 6. We may deduce the surface harmonics from lliese "bj writing r sin cos ^ for x, r sin d sin for y, r cos ^ for z, and dividing by r*. Then, putting ,cos 6 = fi,, and observing that ^(a) = r'P.(/*), ^|M = ^^^... we obtain the fol- lowing series of 2t + 1 solutions : cosshxe^^, cos2^sin«^'^4^. .... cosi^sin'^^^ , suK^smd—P^, sm2 2.4.(21-1) (2i- 3) '^ '")' And therefore -^^ 2M.2.3,..; ^^ /^^ r 2{2i-l) ^ (^•-^)(^•-^-l)(^•_^_2)(^•-a-3) ,_„_, } "^ 2.4(2i-l)(2j-3) '^ ■•■)■ The form given by Laplace for a Tesseral Surface Har- monic of the degree i and order a is (see Mecanique Celeste, Liv. 3, Chap. 2, pp. 40—47) ^ (1 - ^')^ 1^^-- (^^-^1^^:^-^) /.i-'- + ...,j cos ..^, A being a quantity independent of 6 and <^. The ft,ctor of this, involving /i, is denoted by Thomson and Tait {Natural Philosophy, Vol. 1, p. 14'9) by the symbol ©,<"). Thomson and Tait also employ a symbol &/'), adopted by Maxwell in his Treatise on Electricity and Magnetism, Vol. 1, p. 164, which is equal to (i+(T){% + <7-l)...{i-a+l)^ '^' d^k" ' 1 ? n- or 2' T-. r-n ' — rW (i+, by the symbol PJ(jj.), and calls these expres- sions by the name Zugeordnete Functionen Erster Art {Hand- huch der KugelfumcUonen, pp. 117, 118) which Todhunter translates by the term "Associated Functions of the First Kind," whiph we shall adopt. Heine also represents the series J.*-" — {{-a) {{-a - 1) ^^_j f^ 2(2i-l) ji-c) (i-^-l) (i-a-2) (i-a-S) , "*■ 2.4(2»-l)(2t-3) '* by the symbol ^^(/t), (p. 117). The several expressions, Ti'\ ®('\ &<»), Pt, ^'„ are con- nected together as follows : 2'.1.2.3...r 2i(2i-l)...(t-o- + l) r W = 0,W (i+ «r + l) (x + o' + Z;...^» 8. It has been already remarked that the roots of the equation P, = are all real. It follows also that those of the equations ^=0, Vs = 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 obtain t small circles, whose planes are parallel to one another, perpendicular to 78 SPHERICAL: HAKMOSriCiff W GENEKAH 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 tesaeral harmonic of the order , which functions we will denote by the symbols Jf^, ^j, respec- tively. The differential equation, to which this will lead, is Now this will be satisfied,, if we iMake M. and <£>, satisfy the following two equations : The latter equation gives ^^ = (7 cos a^ + (J sin ct^. And, taking SECXOBUL HABHONICS. 79 former is satisfied by M^= 2^W, i.e..(l-/t*)*(^j (l-/t')', as we proceed to prove. We know that Difierentiste a tames, and we get whence, by Leibnitz's Theorem, + »C*'+1)^'=«' or (1 -'*') i;?S-2(-+i)/*|i^'+(^'--K*+-+i)i^'-0' and, multiplying by (1 - ny, Now, putting (I-//fp=2'/'V we get 80 SPHERICAL HAKMONICS IN GENERAL. -{.(.+ 1) (1 - m')^- o^ (a - Mn^']^.' ■ And t(i+i)r/'')^i(t+i),(i-/*')^|7; . =^«(i-M^^"^'i)y(i) Hence the equation above given for M^, is satisfied by Mf = r/'), and the equation in Y, is satisfied by r; = Cr/') cos a + C T,"> an o-^. 10. In Chap. n. Art. 10 we have established the fundamental property of Zonal Harmonics, that if i and m be two unequal positive integers, I PtP^d/i = 0. This is a particular case of the general theorem that if Y„ Y„ be two surface har- monics of the degrees t and m respectively, fJ^Y,YJixd,f, = 0. TESSERAL AND SECTOBUL HARMONICS. 81 For, let Fp F„ be the corresponding solid harmonics, so that V,=r*Y„V„ = r''Y„. Then, by the fundamental pro- perty of potential functions, we have at every point at which no attracting matter is situated, cki'^df^ dz" "' da? ^. df ^ dz" ~^' and therefore '^*[dar' ^ df ^ dz'J '^"{da? ^ df^ dz')'^' or, in accordance with our notation, V^^^*V„— F„v*^ = 0. Now, integrate this expression throughout the whole space comprised within a sphere whose centre is the origin and radius a, a being so chosen that this sphere contains no attracting matter. We then have jjj{V,v'V„ - r^v'V,) dxdydz = 0. But also, when the integration extends over all space comprised within any closed surface, we have ///(F.V"F„- F^v'IQ dxdydz=\\{v,^^.vJl)d8=0, dS denoting an element of the bounding surface, and -p differentiation in the direction of the normal at any point. Now, in the present case, the bounding surface being a sphere of radius a, and FJ, F„ homogeneous functions of the degrees t, m, respectively, dS = a'dnd(f>. ^ = «i*-'F„ ^ = 7nar^Y„, and, the integration being extended all over the surface of the sphere, the limits of /it are — 1 and 1, those of ip, and 27r. Hence P.H. 6 82 SPHERICAL HARMONICS IN GENDBAL. whence, if va — i he not — 0, The value of / I Y'dfid 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, swh de- velopment is possible in only one way. For suppose, if possible, that there are two such develop- ments, so that F(ji,^)=Y,+ Y, + ... + T, + ... and also F{fi,4>)=Y:+Y,'+... + Y:+... Then subtracting, we have 0= i;- P.'+ F,- Y,'+ ... + Y,-Y; + ... identicaUy. Now, each of the expressions F, — F,', 1^ — F/... I^ — F/ being the difference of two surface harmonics of the degree 0, 1, ...i... is itself a surface harmonic of the degree 0, 1, ...I.... Denote these expressions for shortness by ^, ^...^... so that 0=Z^ + Z^ + ...+Zt+ ... identically. Then, multiplying by ^ and integrating all over the surface of the sphere, we have 0=1 ?" Z^dyd^. 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 F/ = F„ 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 — /»*)*, as a factor, and therefore is equal to when /*=+!. From this it follows that when /* = ±1, the value of the Surface Harmonic is independent of ^, or that if Y (jj., <})) repre- sent a general surface harmonic, Y (± 1, ^) is independent of ^, and may therefore be written as F (+ 1). Or Y (1) is the value of Y(ji, ^) at the pole of the zonal harmonic P, (/*), Y{—1) at the other extremity of the axis of P, (/*). We may now prove that r'r.(?0=2^r.(i)P.(/*). Jo For, recurring to the fundamental equation, Now, if we integrate this equation with respect to <}>, between the limits and 2ir, we see that, since / and the value of F, only involves ^ imder the form of cosines or sines of d> and its multiples, and therefore the values of dY -TT* are the same at both limits, it follows that /. Hence far ■ P 1 Hence I Fjt?^ is a function of n which satisfies the Jo fundamental equation for a zonal harmonic, and we therefore have 6—2 84 SPHKRICAL HAEMONICS IN GENERAX. r^ C being a constant, as yet unknown. To determine C, put n=l, then by the remark just made, Y, becomes Y,{1), and is independent of ^. Hence, when /i=l, I Ytd^ = 2TrY,{l). AlsoP,(/*) = l. We have there- fore * 27rr;(l) = C, L ^Y,d,l, = 27rY,il)P,{fi). It follows from this that 13. We may now enquire what will be the value of p_J''Y,Z,d^,d, Y„ 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 : r.= AP, + C,T(» cos <}> + C^T^^) cos 2^ + ... + C^T^^') cos o-^ + ... -i-Cirwcost^ + 8JPsaiif> + S^T® sin2(^ + ... + ^,2;w sinff<^ + ... + 8.Tpsai.i6; Z,= aP, + cJ'P cos + c,r/« COS 2^ + ... + c,r/') cos , or cos 0-^ sin o-'^. This expression when integrated between TfiSSERAL AND SECTORIAL HARMONICS. 85 the limits and 27r vanishes in all cases, except when a =(T and the expression consequently becomes equal to cos' a^, or svc^jrjt. In these cases we know that, a being any positive integer, rite rtit I cos* (r^d^=\ sin' y dfid =1^^ /""(r/'' sin a4>)' dfid^f, [i + f 2ff "[ ■i-g- 2i + l' It -will be observed that this result does not hold when o- =»0, in which case we have fi r^ 4i7r Hence I / YfZ,dnd«f» = 2iTl^A • In this case J eoa* c^id^i = J Bia^a^) dip =2r. TESSERAL AND SECTORIAL HARMONICS. 87 + rj= (C„c.r + )Sf,«,) + ... + [2i (C;c, + ^^jl . 14. We have hitherto considered the Zonal Harmonic under its silnplest form, that of a " Legendre's Coefl&cient " in which the axis of z, ie. 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,(^) and of the system of Tesseral and Sectorial Harmonics deduced from it. Suppose that 0", ' are the angular co-ordinates of the axis of the Zonal Harmonic, i.e. that the angle between this axis and the axis of « is ^, 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 0' by /*'. The rectangular equations of the axis of this system will be a; _ y _ z sin ff cos - 4>.)} 88 SPHERICAL HARMONICS IN GENERAL, in the manner already pointed out, assume P^ {cos 5 COS ^ + sin 6 sin ff cos (^ - ^')] = APt Oi) + (0« cos + fifW sin ) Tp + ( CM cos 2<^ + fi'® sin 2) T^^l + ... + ( CW cos o-^ + fi'W sin o-^) r.W + . . . + ( C» cos i^ + fi'W sin i^) Z;» the letters A, ... C^'\ 8'>'K.. denoting functions of fi and ^', to be determined. To determine C'''\ multiply both sides of this equation by cos aTf'^'1 and integi'ate all over the surface of the sphere, ie. between the limits — 1 and 1 of /*, and and Stt of (p. We then get I I P| {cos cos 6' + sin Oamff cos {<}> — ^')} cos o-^ 2y'> dfid^ = CMp f" (cos T('^y dfjid It remains to find the value of the left-hand member of this equation. Now cos T['^ is a surface harmonic of the degree i, and therefore a function of the kind denoted by 1^ in Art. 12. And we have shewn, in that Article, that f_£pMY,d^ = ^^Y,il). that is, that if any surface harmonic of the degree i be multi- plied by the zonal harmonic of the same degree, and the product integrated all over the surface of the sphere, the integral is equal to „■ .. into the value which the surface harmonic assumes at the pole of the zonal harmonic. Hence fl f2ir TESSEBAL AND SECTORIAL HABMONICS. 89 and therefore Li ^' {««« ^ cos ^ + sin 5 sin 6' cos (0 - ^')} cos a^J?-)rf^rf^ Hence Similarly ^^'^ = 2^sin .4>' T^'> (^'). And to determine ^, we have /_J^P,{cos^cos^' + siii^sin^cos(^-^')}-P.W^A*^^ or A = P, (,,'). Hence, P, {cos ^ cos ^ + sin ^ sin ^ cos (0 - f )} b*— 1 =^*O*')P*W+2^cos(0-^')r.tt)o*')^/'>W fi-2 +^^^°''2(0-^')2'.®Ot')^;®w+. 90 SFHEBICAL HABMONICS IN GENERAL. + 2[=COS«r(<^-f)r.W(M')2'/''>M + ... 1^ + g 15. We have already seen (Chap. n. Art. 20) how any rational integral function of /* can be expressed by a finite series of zonal harmonics. We shall now shew how any rational integral function of cos 6, sin 6 cos , sin 6 sin ^, can be expressed by a finite series of zonal, tesseral, and sectorial harmonics. For any power of cos j> or sin <^, or any product of such powers, may be expressed as the sum of a series of terms of the form cos , where n is not less than a: If n be greater than to the sum of a series of terms of the form cos*" sin' cos + or sin = a cos' sin' sin 2, Comparing this with cos"* sin" sin o-^, we see that n is not greater than 92 SPHEEICAL HARMONICS IN GENEEAL. Next, let it be required to transform cos'^ sin' 5 sin ). Now cos' sin» ^ sin 3^ = / (1 - fi")^ sin 3«^ Also cos" sin' 5 sin ^ = /*' (1 - /*") (1 - ^')i sin <}> = (At»-^')(l-^')4sin^ Also (Chap. II. Art. 17) «.= iip+24p. lOp Ip '^ 231 •^77 "^21 "^7 •■ Hence cos' 6 sin' 5 sin 3^ = 120(23-15;;? +77^j(^-'*^'^"^^'^ - ={3^5^«'^ + 34^*'"}^-3*- Andcos'^sin'5sin^=-f4^e+*^^ + 4^^ ^ \,693 dfi 77 d/i' 63 d/* 2 dP, 1 = ^—^ T») + ^ T w| sin3^ _ I A^ ya) _ _L y a) — -f (4 sin<6. (693 • 770 « 63 » J '^ TESSERAL AND SECTORIAL HARMOKICS. 93 17. The process above investigated is probably tbe most convenient one when the object is to transform any finite algebraical function of cos 0, sin 6 cos <^, and sin 6 sin <^, into a series of spherical harmonics. For general forms of a function of /t 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 Fiji, ^), JP" denoting any function whatever of finite magnitude, at an external or internal point. Let c be the radius of the sphere, / the distance of the point from its centre, ff, ^' its angular co-ordinates, F the potential. Then ^ being equal to cos 6 jr^r p F{,i,4>)i?d^d4, J_Jo [r"-2cr'{cos^co8^H-sin^sin(?'cos(^-^')} +c^4 ' The denominator, when expanded in a series of general zonal harmonics, or Laplace's coefficients, becomes ^|i+p,(/.,^)^+p>,^)5+...+p.(m,^)^+...}, for an internal and an external point respectively, P, (/*, ^)' being written for P, {cos 6 COB ff -If sin 6 sin ff cos (^ - j>')]. Hence, F, denoting the potential at an internal, F, at an external, point, '/-UO > (ji, ) dfid(}> + '^ f^ J"P, (/*, <}>) Fiji, ) dfid(j> -f- ... +^f fy.ijJ^. ) Fij^. ) diid + ...^, 94 SPHEBICAL HABHONICS IN GENEBAL. r' It will be observed that the expression P, (/*, (ft) involves fi and fi symmetrically, and also ^ and '. Hence it satisfies the equation And, since ft and are independent of fi and )F{ji,)dtid4> is a Spherical Surface Harmonic, or "Laplace's Function" with respect to ft and ' of the degree i. And the several terms of the developments of V^ are solid harmonics of the degree 0, 1, 2...1... whUe those of F, are the corresponding functions of the degrees —1, —2, — 3... — (i+ 1), ... And these are the expressions for the potential at a point (r', /*', ^') of the distribution of density Fyi, ^') at a point (c, fi, '). 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 ^{/ fo^^' *^ ^f^^'f^ + ^fJ^M ^) -^0*. 4>)dfid+... or, as it may now be better expressed. + + .. + TESSERAL Ain) SECTOBIAL HABMONICS. 95 Sj j P^{cos0(Ma0'-\-BmeBmff COB (if>-')F(/i,)diJid^ (2i+l) f j F,{cosecose'+smeamffcoa{-4>')]F(ji,(j))dtid!J>+... 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(ji', 0'). We have thus, therefore, investigated the development of F{ji, <])') in a series of spherical surface harmonics*. The only limitation on the generality of the function F(jj!, ') is that it should not become infinite for any pair of values comprised between the limits —1 and 1 of ft, and and 2ir 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 (2t+ 1) f j P, {costf coB^+sin^ sin^ cos (^-f )} cos 2dfid^. Now Pi {cos 6 cos ff + sin sin ff cos (^ — ^')} = P,(cos^)P,(cose') 2 . ^dP Jeosff) . ^dP, (cosff) .. ,,. + .,. ,. smg —^ ^sm^ — '-^ — ^cos(<^-^) i(i+l) dfi dfi 2 ^!^{£^,i,V^5(?^ cos2(0-^') + - Now I cos o- (^ - ^') cos 2 = 0, Jo for all values of o" except 2. • In connection with the subject of this Article, Bee a paper by Mr G. H. Darwin in the Metienger of Mathematics for March, 1877. 96 SPHERICAL HAHMONICS IN GENEEAIi. And I cos2(^ — ^')cos2<^ — 47r cos 2^' or 0, as i is even or odd ; .-, cos 2^' _L Q 2 , . .^«fP,(cos0') _,, TESSEKAL AND SECTORIAL HARMONICS. 97 ^ } -lcoBl (^1.2.3.4 + 3.4.5.6 + 5X778 + •••■;• Hence the potential of a spherical shell, of radius c and surface density cos 2(/)', will be «'^"''^2flof3:4?+374^?+5:6.V78 ?+■•••]' and ^'^°*'^2^Ai:fer-^+3:4t5:6/^+.5:6^r^'+ ••••)' at an internal and external point respectively. 19. We will now explain the application of Spherical Harmonics to the determination of tlie potential of a homo- geneous solid, nearly spherical in form. The following investigation is taken from the M€canique C^ste, Liv. ill. Chap. II. Let r be the radius vector of such a solid, and let r = a + a (a, F, + c, F, + ... + a, r; + ...), a being a small quantity, whose square and higher powers may be neglected, a,, Oj,...a,... lines of arbitrary length, and F,, Fj,... F,... surface harmonics of the order 1, 2,...i... re- spectively. The volume of the solid will be x "■«'• For it is equal to f f rr'drdftdif) = 1 r r{a' + 3a*(i{aJ^ + a,Y,+ ...+a,Y, + ...)]dfid = ^ Tra', since I I F,a/ia^ = 0, for all values of i. F. H. "^ 98 SPHEBICAL HAKMONICS IN GENERAL. Again, if the centre of gravity of the solid be taken as origin, a^ = 0. For if z be the distance of the centre of gravity from the plane of xy, = 1 r rv+4a'a (a,y;+«,F,+...+c,y;+ ...)tiM«?^ = 4ia^c^aA 1 fj.Y^diJLd(j>. Similarly K ira^x = 4a' a . a, (1 - mT cos ^ T^ cZ/i df^, ^ ■Tra'^ = 4a' a . a, j I (1 — ^*)2 sin Y^ d/*d(f>. Now r", is an expression of the form Ati. + B{l- fjJ'i^ cos 4> + C(l - /i,')* sin 0, and therefore all the expressions x, ^, a cannot be equal to 0, unless a, = 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 aa, and density ^.(t^.+-+!^.^--). TESSEBAL AND SECTORIAL HABMONICS. 99 p, being the density of the solid. Therefore the potential, for any external point, distant R from the centre, will be A a' . .fajr.a* . . a.Y, a' \ 47rp,3^^ + 4wa (^-^^ _, + .... +^_^^ + ....j • The potential at any internal point, distant R from the centre, will be made up of the two portions -37r|0,2J'+2,rp.(a'-iJ') or 27rp(a'-|') for the homogeneous sphere, for the shell, and will therefore be equal to „ / , R?\ , . ,fa,Y,R* ^ a,Y, R' ^ \ 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 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 tlie stratum at an ex- ternal point will be 47rflc% , , c'Sc /a,Y, c' a,r, c' + ^rl+ ] (1). 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 ^^i^--) (^)- 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 B and a, remembering in both cases that /> is a function of c, and add the results together. CHAPTER V. SPHERICAL HABUONICS OF THE SECOND KINS. 1. We have already seen (Chap. ii. Art. 2) that the differential equation of which P, is one solution, being of the second order, admits of another solution, viz. "^^j P.' (1 -/*")' Now if fi between the limits of integration be equal to + 1, or to any roots of the equation P, = fall of which roots Ke 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 ft to be always be- tween 1 and 00 , or between - 1 and — oo . We will adopt the former supposition, and if we then put C=-l, the expression „, .^ _ .,> fi.e. „, . 8_,J will be always posi- tive. We may therefore define the expression J u dfi P.'l^'-l)' as the zonal harmonic of the second kind, which we shall denote by ^„ or Q, {/i), when it is necessary to specify the variables of which it is a function. It will be observed that, if /* be greater than 1, P, is always positive. Hence, on the same supposition, Q, is always positive. Weseethat (?,= f -i^ = llog^ , J ^ fi — L i fl — X 102 SPHERICAL HARMONICS OF THE SECOND KIND. And, in a similax manner, the values of Q,, ^,, ... may be calculated. 2. But there is another manner of arriving at these functions, which will enable us to express them, when the variable is greater than unity, in a converging series, with- out the necessity of integration. This we shall do in the following manner. Let U= , V being not less, and fi not greater, than unity. Then ^=_— J_ ^=_1_, dv (v — fi)* ' d/A (i> — /i)' ' ,. ,.dU 1^-1 ,. ^dU 1-m' dv\^ ' dv]~[y-^')V-l V-fJLj~^{v-^L)" •••^{(-'4!}=|i{<'-'->f}- Now, let be expanded in a series of zonal harmonics ■P.W, ■P.W.-.P.M.sothat by the definition of P (/i). SPHERICAL HARMONICS OF THE SECOND KIND. 103 And these two expressions are equal Hence, equating the coefficients of F^ {fi), Hence ^,(i') satisfies the same differential equation as Pj and Q,. But since U= when i/ = oo , it follows that <^,(i/) =0 when j/= 00 . Hence ^^(1;) is some multiple of Q^{v)=AQ^(v) suppose. It remains to determine A. Now, ^j(i') may be developed in a series proceeding by ascending powers of - , as follows. We have 1 ^ fi* ' ^^ T ";« + •. «H ^XT + •••• r -• ■ V— fi V IT and also = ^,{v) P,(jjl) + ^.(i/) P^ +- + M P<(/^) + ... Now, by Chap. 11. Art. 17, we see that, if m be any integer greater than i, the coefficient of P, in /t" is /«. ,v (j» — 1+2) (7ra-t + 4)...(m-l) .»., ,, (2^ + 1) (^^:^^■^-l)(J■;^-l)...(^ + 4)(m^■2) ^ * ^' '^^' and {2i + l) , (rT: + ^V":^'T^^";r^i. if^'feeeven, ^ ''(m+t+l)(m + t-l)...(»?H-3)(»n+l) m — i being always even. Hence, writing for m successively i, i +2, t + 4, ... we get 2.4...fi'-l) j_ .(2i+l)(2i-l)...(t + 2) i;'^* 4.6...(i + l) , 1 + (2t + 3j(2i + l)...(i + 4) i/'^ 6.8...(t + 3). 1 (2t + 5)(2i + 3)...(t + 6) y'' 4... '-ff-^tr'l .■ l. + ....\iii^^oM. 104 SPHERICAL HAKMONICS OF THE SECOND KIND. 2.4...; 1 and = (2^'+l)t ..+1 ..<+3 :+l){2i-l)...{i+l)i^: 4.6.. .(1 + 2) 2 '^{2i+S){2i+l)...{i + 3) v' + 6-8-C^' + ^). 4+.. ..lift be even. ^ (2t+ 5) {2i+S)...(i+ 5) i;'*"^ j Now, recurring to the equation we see that, if Qi{v) be developed in a series of ascending powers of - . the first term will be ^,„. — tt— sr> where G is the coefficient of /a* in the development of P, (/i) ; XV. X- n (t + 2)(i + 4)...(22-l) .. ., ,, that IS C = - J . „ ,. ,, if * be odd, 2: 4.6...(t — 1) (i+1) (i + 3)(t + 5)...{2i-l) .. ., and = "-^-^— 2 4 6~t * ^^®°" Hence the first term in the development of Q, {v) is (i + 2j(i + 4)...(2i-lj (2i+l) 2 4 6 ..i (i+l) {i + 3)...(2i-l) (2t + l) which is the same as the first term of the development of P.H divided by gjqpy. Hence A = 2i+ 1, and we have 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 DomiQator 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...t 1 3.4.5...(t + 2) 1 ^*^' 1.3.5...(2j + l)j;'*'"^3.5.7...^2i + 3)i/'*' 5 .6. 7.. .(1 + 4) 1 {2k+l){2k+2)...{i+2k) 1 "^ (2^+1) (2ifc + 3)...(2j + 2k + 1) v'^-' "^ ••" 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 -j-^ in terms of ^^.,, Q^ Recurring to the equation 1 ii+f- = (?„(.) P» + 3(2, WP» + ... V- ft -K2i + 1)5Q.(,/)P.(^) + ... we see that + (2.- + l)5«M^+(2.-+3)«.„M^^ + .... Now we have seen (Chap. II. Art. 22) that ^^ = (2i - i) P... W + (2» - 5) P,.. (/*) + ... Hence ^^^ = (2» + 1) P.M + (2i- 3) P«(/*) +... 106 SPHERICAL HARMONICS OF THE SECOND KIND, ^^^=(2i+5)P^»+(2i + l)P,(;.) + ... ^^^ = (2/+ 9) P^(m) + (2i+ 5) P^(/x) + (2i + l)P» + ... And therefore the coefficient of P,Oii) in the expansion , d 1 . 01 -j IS dfi V — fi (2i+l) {(2i+3) Q^.W + (2i+7) (?,,3M + (2i+ll) (?<,,« + ...}. Again, And J 1- T = 0. av V — fi clfi V — fL Hence, comparing coefficients of P^ (/*), -(2t + ll)(2.„(.)-... ^i^) = -(2i + 3)Q,,.W-(2i + 7) or sino-^, will give an expression satisfying the differential equation |(1 - M-') l^y U+ {{ (i + 1) (1 - /*') -a'}U= 0. and which may be called the Tesseral Harmonic of the second kind, of the degree i and order a. CHAPTER VI, ELLIPSOIDAL AND SPHEEOIDAL HAKMONICS. 1. The characteristic property of Spherical Harmonics is thus stated by Thomson and Tait (p. 400, Art 637). "A spherical harmonic distribution of density on a spheri- cal surface produces a similar and similarly placed spherical harmonic distribution of potential over every concentric spherical surface through space, external and internal." The object of the present chapter is to establish the ex- istence of certain functions which possess an analogous pro- perty for an ellipsoid. They have been treated of by Lam^, in his Lemons stir les fonctions inverses des transcendantes et les fonctions isothermes, and were virtually introduced by Green, in his memoir On the Determination of the Exterior and Interior Attractions of Ellipsoids of Variable Densities, (Transactions of the Cambridge Philosophical Society, 1835). We shall consider them both as functions of the elliptic co- ordinates (as Lamd has done) and also as functions of the ordinary rectangular co-ordinates; and after investigating some of their more important general properties, shall pro- ceed to a more detailed discussion of the forms which they assume, when the ellipsoid is a surface of revolution. 2. For this purpose, it will be necessary to transform the equation d^v^d:'v^d^v . „. - into its equivalent, when the elliptic co-ordinates e, v, v are taken as independent variables. If a, h, c be the semiaxes of the ellipsoid, the two sets of independent variables are connected by the relations ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 109 ..3 -A- ^ -i. — 1 Thus a' + e, J' + e, c* + e are the squares on the semiaxes of the confocal ellipsoid passing through the point x, y, z. a +v, b* + v, c* + V, the squares on the semiaxes of the confocal hyperholoid of one sheet. a + v, J' + v, c' + v, the squares on the semiaxes of the confocal hyperboloid of two sheets. Thus, € is positive if the point as, y, z be external to the given ellipsoid, negative if it be internal And, if o* be the greatest, c" the least, of the quantities a*, 6', c', e will lie between — c* and oo , " „ „ -** „ -c', V „ „ -a' „ -b\ . ,, d'V cTV d'V . , 3. Now -j-^ + -T-^ + T-j = is the condition that taken throughout a certain region of space, should be a mini- mum. In the memoir by Green, above referred to, this expression is transformed into its equivalent in terms of a new system of independent variables, and the methods of the Calculus of Variations are then applied to make the resulting expression a minimum. We shall adopt a direct mode of transformation, as follows : Suppose 0, /8, 7 to be three functions of a;, y, z, such that V*a = 0, v'/9 = 0, V'7 = (1), such also that the three families of surfaces represented by the equations a = constant, /3 = constant, 7 = constant, inter- sect each other everj'where at right angles, i.e. such that 110 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. d^dy d^dy d/3dy _^ dy dt dydi dr/dz _ dx dx dy dy dz dz ' dx dx dy dy dz dz ' d^^_^_d2d^_^d2d0^^ dxdx dy dy dzdz ^ Then dV^dVda. dVd§ dVdy dx d% dx dli dx dydx' ? ~'dl[dx) "^ d0' \dx} "^ df \dx} d'Vd^dy d^Vdydx d'V dad^ dfidrf dx dx dyd'x dx dx dad^ dx dx dV^ dV^ dV^ dadai''^ d^dx''^ dydx'' d^V d'V . -y-f and -5-^ being similarly formed, we see that, when the three expressions are added together, the terms involving -J- . -jTT, -J- will disappear by the conditions (1), and those ^''^°^^'°S d^'d^x' d^^^ *^^ conditions (2). Hence v''-g|(DV(|)v©} 4. Now, let ^^r df ELLIPSOIDAL AND SPHEEOIUAL HARMONICS. Ill ^ /•-*' df All these expressions satisfy the conditions (1), for a is the potential of a homogeneous ellipsoidal shell, of proper density, at an external point, and /S and 7 possess the same analytical properties. Again, a is independent of v and v, and is therefore con- stant when e is constant. Similarly /3 is constant when v is constant, and 7 is constant when v is constant. Hence a, ;8, 7 satisfy the conditions (2). Now jj" ^» g« And -»-:- + i/r-+^-r- = l- a + e + e c + e 222 a; u r 1 ^ 1 de 2x dx o* + c ' with similar expressions for -,- and -j- . Hence, squaring and adding, But from the equations ^ ^j^+^-1 »' , y' I '' -1 i^T'e'^fi' + e'^c' + e" ' a'+i;^6'-(-w c' + w ' • /S is a purely imaginary quantity. We may, if we please, write J-lp" tor /S. 112 IXLIFSOIDAL AND SPHEBOIDAL HABMOXICS. ^— 4.-J^4- ^* -1 we deduce a!* _y* 3* _ (ft) — e) {a — v){a> — v) ~ W+m ~ b' + a ~ c» + ft) ~ (o) + a'^""(o> + J')(o> + c') ' ft) being any quantity whatever. For this expression is of dimensions in o, e, v, v, it vanishes when o) = e, u, or v, and for those values of to only, it becomes infinite when ft) = — a*, — ¥, or — c", and for those values of .6' = {me + r) |e" + np,r' + ^^^;^^ jj/" + • ■• +^ ■ Hence, equating coefficients of like powers of e, we get n r« + 2) = "*' n ^{n - 1) {(n - 2)p, + S/J + 1 ((« - 1) p, + 2/,}] = nmp, + r, .[(n-l){ ("-f^-^) j.. + 3(.-2)Xp, + 3/j n(n — l) n |(7i - 1)/a., + g/j),. j = rp^ or, as they may be more simply written, ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 115 n |(re - 1) ^71 - 2) p,' + 3nf\ = nmp^ + r, n(n — l) n |(« - l)/,p„_, + 2/»i>»-iJ = »•?». It thus appears that p^ is a rational function of r of the first degree, p^ of the second, p^ of the »"", ' and when the letters p,,/)jj...p„ have been eliminated, the resulting equa- tion for the determination of r will be of the (w + l)"" degree. Each of the letters p^, p^-.-p^ will have one determinate value corresponding to each of these values of r; and we have seen that m = n (n + -^j . There will therefore be (n + 1 ) values of E, each of which is a rational integral expression of the m"" degree, n being any positive integer. 7. But there will also be values of E, of the n* degree, of the form (e+6')i(e+c')i|e-+(n-l)?,6-+^^|=^2,e-»+...+2„.,}. We thus obtain de {(e + a')(e + i')(e + c')li^ = (e + a^i (e + b') (e + c') (n - 1) je""' + (n - 2) g.e""' + - — YT2 ^■■' +••• + ?--»}' 8—2 116 ELLIPSOIDAL ASD SPHEBOIDAL HABMONICa + (6 + o')i (6 + 6')} (n - 1) jc"^ +{n-2) q/^ (n-2)(7i-3) ] + f72 ?S^ +-"+?iMl| + (€+a')* (e+J') [€+(?) {n-1) (n-2) je"-+(n-3) y.e""' , ("-3)(n-4) n Hence i| (6 + 6*) (e + c") + (e + a') (e + c') + (e + a') (e + i*)! + (e + a') (6 + 6») (e + c=) (n-1) (n - 2) je"-* + (m - 3) q,e'^ (n-3)(w-4) 1.2 + "" 7^' "^ 2,e-» + . . . + j^.| = (me + r)|e"-»+(n-l),) (w - wj . . . (v - «*„), £-' = (v'-(bJ (V- 6),) ...(!/' -0)„). Hence EHH' = (e - o>,) (1/ - o).) (w -«,)... (6 - 6)„) {v - 6)„) (v - to,). Now we have shewn (see Art. 4 of the present Chapter) that (e — w,) (u — w,) (u - <»,) = K + a')K + 6=)(a,,+ c')(^ + ^^ + ^^-l). Each of the factors of EEE' being similarly transformed, we see that EHH' is equal to the continued product of all expressions of the form (a. + a')(o, + 5')(a, + o')(^-^ + ^+^-l), the several values of ca being the roots of the equation __, w (w — 1) __, 6)» + n^.«"-* + 12 -P' +• ••+!'» = 0. As this equation has been already shewn to have {n + 1) distinct forms, we obtain {n + 1) distinct solutions of the equation V'F=0, each solution being the product of n expressions of the form a* + u) V+co c' + to That is, there will be w + 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 2ra, let us next consider the case in which E •ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 119 The object here will be to transform the product (€ + 6»)4 (y + J')i (v' + 1")^ (e + c')* (v + c')i (v' + ^i, since the other factors will, as already shewn, give rise to the product of n — 1 expressions of the form Now, by comparison of the value of a? given in Art. 4, we see that (e + V) (y + b') (u' + b') (e +0*) (y + (T) (v' + (f) = (&* - c*) (6' - a') (c' - a") (c* - b") fz\ Hence, we obtain a system of solutions of the form of the product of in — 1) expressions of the form . ^^ 7.> 1 -. ~ «« i -. ' a'+o) J' +6) c' + <» multiplied by yz. 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 4» + 1 solutions of the degree In in the variables of which w + 1 are each the product of n expressions of the form n are each the product of (n— 1) such expressions, multiplied byj^a, n ... ... ... «a;, n ... ... ... ooy- 11. We may next proceed to consider the solutions of the degree 2n + 1 in the variables x, y, z. Consider first the case in which 120 ELLIPSOIDAL AND SPHEROIDAL HAEMONICS. Here the product (e + a')* {v + a')* («' + a")* 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 _^+^(L + _£!__i a'+o> i' + o) c" + w ' multiplied by a:. Of these there will be m + 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 + o*)* (e + Vf (e + c*)* |e""' + {n-\) p/-^ (n-l)(n-2) „_ 1 + 172 P»= +---+P<^ih we see that in UHH' the product (e+o=)i („+«')* (w'+a«)4 («+ J')^ (w+i')* (v' +6')* (e+c')* will give rise to a factor icyz. Hence we obtain a system of solutions each of which is the product of (n — 1) expressions of the form a' + w 6' + ft) c' + oj ' multiplied by xyz. Of these there will be n. Thus there will be 4w + 3 solutions of the degree 2ra + I in the variables, of which (n. + l) are each the product of n expressions of the form jb' «" «' -J- h Ta f- -^i 1 multiplied by x, (re + 1) are each the product of n such expressions, multiplied (m + 1) ... ... ... ... z, ELUPSOIDAL AND SPHEKOIDAL HARMONICS. 121 n are each the product of (n — 1) such expressions, multi- plied by xyz. 12. Now an expression of the form C . EHH', C being any arbitrary constant, is an admissible value of the potential a;" v' «' • • at any point within the shell -s + '^ + -5 = 1. But it is ^ a c not admissible for the space without the shell, since it becomes infinite at an infinite distance. The factor which becomes infinite is clearly JS, and we have therefore to enquire whether any form, free from this objection, can be found for this factor. We shall find that forms exist, bearing the same relation to S that zonal harmonics of the second kind bear to those of the first. Now considering the equation [{(e + a') (e + V) (e + c")}* ^] V = {me + r) U, which we suppose to be satisfied by putting U=E, we see that, since it is of the second order, it must admit of another particular integral. To find this, substitute for U, E \vde, we then have + {(€ + a')(€ + 6')(e + c')l4iJw; = [{(e + a") (e + 6') {.e + c^]^ ^^ E . ^vde + (e + a')(6 + J')(e + c')^.« + \ {(e+ i") (6 + + (6 + (6 + a») + (€ + a') (6 + J*)} Ev + (e + o^(. + J')(e + c')(^.t; + £j|). 122 ELLIPSOIDAL AND SPHEROIDAL HARMONICS. Now, since by supposition, the equation for the determi- nation of Z7 is satisfied by putting U= E, it follows that when Ejvde is substituted for U, the terms involving Ivde will cancel each other, and the equation for the determina- tion of i; will be reduced to de [ de 2\6+a e + b e + c J ) Idv 2dE 1/1 .1.1 \_^ *"■ vd€'^Ed^'^2[i+^'^e + b'^€ + cy~ ' whence log v + 2 log jF + log {(e -I- a*) (e + 6') {e + c*)}* = log «,+ 2 log E^+ log ahc, v„ and E^ being the values of v and E, corresponding to e = 0. Hence ,=.,^^ --^--^^^-^^; We may therefore take, as a value of the potential at any external point, F= v,E' ahc EHH' f r . 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 C=v,.E,*abcr- ^^ Jo . ^{(e + a')(6 + 6')(e + c')l*" ^0 J Hence, putting », . E* . aic = F,, we see that to the value of the potential de V EHH' '- E'{{n + a'){e + b'){e + c')}i' ELUPSOIDAL AND SPHEROIDAL HARMONICS. 123 for any internal point, corresponds the value 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 Sn be the thickness of a shell, p its volume density, the diflference 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 ^irphn. This is the attraction of the shell on the outer particle, minus the attraction on the inner particle. ' But the normal component of the attraction on the outer particle estimated inwards is -^. And, if V denote the potential of the shell on an in- ternal particle, the normal component of the attraction on it estimated inwards is --,— . an Hence 47rpS» = -^-^. dV_dVdx dVdy dVdz dn ~ dx dn dy dn dz dn ' /7i» And -i- is the cosine of the inclination of the normal at dn the point x, y, z to the axis of x, and is therefore generally equal to e—, — , e denoting the perpendicular from the centre on the tangent plane to the surface "^ V ^ n a'+e'^6' + 6 c'-f-e 124 ELLIPSOIDAL ASTD SPHEROIDAL HARMONICS. And we have shewn that , {a' + e){a' + v){a''+v') , 2dx 1 whence - t- = -. , a; de a + e X _ dx or =2j-; a'+e de' dx_ dx ' ' dn de' Similarly ^=2e?, ^ = 24', an de an de • ^=2e(~~ — ^ ^^"^ = 26 — ' ' dn \dx de dy de de de) de bimilarly -y— = 2e . dn de Now V ^ ""■"' ''" * = V,.EHH'\ -t; _Y -Q^, dE r ^e de » • de}^ E" {(e + o') (e + 6') (e + c"))* " de . And F= F„ . EEH' (' — therefore, generally, de • deJ,E' {{a' + e) {b' + e) (c' + e)}* - V^.EHH' ^ r- But, when the attracted particle is in the immediate neighbourhood of the surface, e = 0. Hence, the fii'st line ELLIPSOIDAL AlTD SPHEROIDAL HABXONICS. 125 of the value of — - becomes identical with the value of --i— . cte de and we have — — = V — J- de~ de~ • E, abc' E^ denoting the value which E assumes, when 6 = 0. Hence, 47rp8ra = 2e K ^ — . Mj^ abc But hn, being the thickness of the shell, is proportional to e, and we may therefore write ^ = r; i ^a being the thick- ness of the shell at the extremity of the greatest axis ; _ F, g 1 EH' " P 27r Sffl abc E, ' and this is proportional to the value of V corresponding to any speciiied value of e, since MM' 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 axe commonly known as Lamp's Functions, having been fully discussed by him in his Legons. The equivalent expressions in terms of x, 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 J_ « _L 4ir Za abc 126 ELLIPSOIDAL AND SPHEBOIDAL HAEMONICS. Hence, it is equal to -5- [bcBa + caZb + ahhc) or to volume of shell ' and the potential at any internal point = i volume of shell x EE^ . p 1 — -- — —rr ——. tte > and the potential at any external point r de = I volume of shell x EE^ . p — — — — — -r -tt ; J. ^{(a' + e)(6'-i-€)(c' + 6)}* where for p must be substituted its value in terms of v and v. 14. We ■will next prove that if T^, F, be two di£Ferent ellipsoidal harmonics, dS an element of the surface of the ellipsoid, lie V^V^dS=0, the integration being extended all over the surface. We have generally ///(F.V7.-F.VF,)& in the expression above ; • • ^'^^ 16 {a?+v) {b'+v) (c'+w) (o'+i;') (6"+i;') (c'+u) * It has been shewn that, integrating aU over the surface, the limits of u are - c' and —b', those of v, — b* and - a\ Hence, Fj, F,, denoting two different ellipsoidal har- monics l-i^J-ci' {{a'+v) (6*+v) (c'+w) (a'+i;') (6'+v ) (c'+i;')}* The value of the expression jlj V^dxdydz, or its equiva- lent /•-«» /•-»« F'(u'-v)dt; (a' + f)i (6' + f «)4 (c" + f )4 and in the differential equation („_,)_ + („'_,)_ + (,_„) _^0. We will first suppose the axis of revolution to be the greatest axis of the ellipsoid, which is equivalent to supposing J» = c". To transform a and 7, put a* + ■^ = ^, a* + e = rf, a' + w' = ft>* ; we then obtain ~ = 2r <^^ _ 1 1 (a'-6')^- a> To transform )8, we must proceed as follows. Put Vr = -c'cos'iir-ii>'sin'«r, v = - c* cos* ^ - 6' sin' A, we then get generally 6' + ^ = (6» _ c») cosV, c' + -f = (c* - 6») sin*w ; rf'^ = 2 (c'-i") cosw sincr dcr ; V-li*(a'-6»)4 (a'-&')i* ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 131 Also, e = ri'- a\ u = «» - re", y = - h\ and our differential equation becomes + (,» _ a« + £•) |(a,' - a« + V) ^X V -(o,'-,')(a«-5')^=0, or («• - a» + J') !(,,• - a' + J") ^ V -(a'-i')(,'-»')^=0. This equation may be satisfied in the following ways. First, in a manner altogether independent of ^, by sup- posing V to be the product of a function of rf and the same function of to, this function, which we will for the present denote hj/{ri) orf{a>), being determined by the equation dr} — da> {(a,«-a'+6')^}/(a,) = m/(a,). 9-2 132 ELLIPSOIDAL AND SFHEBOIDAL HARMONICS. (f F Secondly, by supposing -^-r. a constant multiple of V, = — ff'F, suppose. Our equation may then be written - tr' (a' - V) {(o)' - a» + V) - (rf' -a' + b')]V= 0, which may be satisfied by supposing the factor of V inde- pendent of to be of the form F (i?) F{(o), where |(,»_ a« + 1") ^F{7,) - o^ (a« - V) F{r,) = m (i,»-a'+ J') F{v), 1(6)'- a» + 6') ^}V(«) -o^ (a'-i") i^(Q>) =>« (a.'-a*+ J") i!'(<»). The factor involving ^ will be of the form A cos a^ + B sin o-^. Now, returning to the equation ||(,'-a' + J')|}/(,) = r«/(,), we see that, supposing the index of the highest power of »; involved in/'(9j) to be i, we must have m = i (i + 1). Now, it will be observed that ri may have any value however great, but that to*, which is equal to a' + v, must lie between a* — 6* and 0. Hence, putting w" = (o' — V) ^', where /t' must lie between and 1, we get ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 133 Hence this equation is satisfied by /{fa*— 6')V] = C'P,, C being a constant ; and supposing C = 1 we obtain the following series of values for / (<»), 36)* -(o'- 2(a'-& 56)'-36>(a'-y) . - ... 3a)*-(a'-y) ' = 2, /H 2(a'-6') ' ^■ = ^'^('")=-2(a*-J*)» ' Exactly similar expressions may be obtained for/(i7), 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 ij is susceptible of in- definite increase. The form of integral to be adopted in this case will be obtained by taking the other solution of the diflferential equation for the determination of /{ij), i.e. the zonal har- monic of the second kind, which is of the form Q, \ —- — — rf , [{a - ) V where dd 'l(a*-6*> (jg'-a'+h*) 0*i Or, putting V = (a* - i*) v\ ^ = (a* - i*) \*, we may write rd\. j>Wi'(^'-i) - 17, We may now consider what is the raeanin? of the quantities denoted by y and to. They are the values of ^ which satisfy the equation 134 ELLIPSOIDAL AND SPHEROIDAL HAHMONICS. 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 t}. The other is an hyperboloid of two sheets whose semi-axis is w. Now, if 6 be the eccentric angle of the point x, y, z, measured from the axis of revolution, we shall have a? = rf cos' d. But also, since -q*, w', are the two values of ^ which satisfy the equation of the surface, Hence to' = (a' -6*) cos' ^, and we have already put whence the quantity which we have already denoted by /* is found to be the cosine of the eccentric angle of the point X, y, 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 foim of solution which involves the factor ^ cos becomes inde- finitely small. But /*, which is equal to -,, remains finite, being in fact equal to - or cos B. Hence /(/i) becomes the zonal spherical harmonic. Again, the tesseral equations, for the determination of ^(17), F{u)), become which are satisfied by F{7J) =rf or 17""*". And, wilting for «', (a*-5*) /t', we have, putting -fW =x(m). |(A**-l)|^|\w + «r'xW = t(i+l)0*'-l)x(M). which gives X (/*) = ^j^'H/*)' 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' = 6'. To transform a and fi, put c' + •^ = ^', c^ + € = rf, c* + w = 10*, we thus obtain To transform 7, we must proceed as follows : Put -^ = — a* sin' ts — h^ cos' «r, w' = — a' sin' ^ — J' cos' 0, we then get, generally, o' + f = (a' - 6') cos' lir, 6' + f = - (o* - 6') sin' ct, c'+i/r=c'-a'sin'^-J'cos*0, ••-3. /(,). Vty-^, _ and generally We will denote the general value of /(17) by p, j ,_ J , or, writing 1; = (a' - ^v, by 'p^ (v). 138 ELLIPSOIDAL AND SPHEROIDAL HAKMONICS. For external points, we must adopt for / (17) a function ■which we will represent by q, \ » » 1 , ' °^ SM> ^^ich will be equal to d0 ,(a'-c')4JJ, f_J_) C^ + a'-c') It is clear that /(<») may be expressed in exactly the same way. But it will be remembered that rj* and «" are the two values of y which satisfy the equation Hence 1}, as before, is the semi-axis of revolution of the confocal ellipsoid passing through the point (a;, y, z). But v'fo* = — (a" — c') a*, an essentially negative quantity, since a' is greater than c*. Hence a)' is essentially negative. Now, if ^ be the eccentric angle of the point \x, y, z) measured from the axis of revolution, we have a' = rf cos'^. Hence i?V = -(a*-c^i7»cos'^, and therefore a* = — (a* — c') cos' Q = — (a* — c") /*', suppose. Hence the equation for the determination of /(<») assumes the form the ordinary equation for a zonal spherical harmonic. Hence we may write /I 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 a-<}> + B sin a-- Hence if M be the mass of the shell, V^ the potential at any external point, ' 7]— CO M 1 M (a:'-b')i X{2i + l)P,(p,)QM. Now, by what has just been seen, the internal potential, corresponding to P, (fi) $, {v), is -Pi 0) Pi W Pr _a_) Hence, if V^ be the potential at any internal point. ELLIPSOIDAL AND SPHEROIDAL HAEMONICS. 143 And the volume density corresponding to P, (/*) Q, {v) is 4wbBbpA ^—V Hence the density corresponding to the present distri- bution is P = M iiria'-b'jHSb ^ ' p '*W-6')iJ If F", had varied inversely as MP, we should have had M V,= »7 + o>' and our results would have been obtained from the foregoing by changing the sign of o), and therefore of fi. 24. Now, by adding these results together, we obtain the distributions of density, and internal potential, corre- sponding to * 17-ft) 1J + W Tf — tir or, in geometrical language, M M _ ^SP + EP ^*~SP^'SP~ SP.HP' = M 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, 2M ^„, . ,, W-^')4 p r ST, f^ h being 0, or any positive integer. Again, subtracting these results we get K= : — =M- » rj~(o ij + a 1)^—10^' = 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, 2Jlf ^.., . „^ ^•^'l(a'-b')4 „ ,,p ,. ^1 = T-r-iri 2 (4* -I- 3) / .„ ^ Pa+, W -Pa+i ("). ai+i 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 IILLIFSOIDAL AITD SFHEBOIDAL HABIfOKICS. 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 Jc being 0, or any positive integer. In the second case Te 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, 10 146 ELLTPSOIDAl AND SPHEROIDAL HAEMONICS. by wMch the general principles of the method may be made more intelligible. 27.- First, suppose that the potential at every point of the surface of the ellipsoid is proportional to aj = — ^ suppose. Of In this case, since x when substituted for V, satisfies the equation v*F= 0, we see that F„- will also be the potential 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 ( (e + a')(t; + a')(v' + «'m And the expression V, ae+a*)(v+ar){v'+a'') )h r' d^ a \ (a»-6')(a"-c') J J. (Vr+a'){(^+a')(V-+6')(t+c')]* <: satisfies, as has already been seen, the equation v'F=0, is equal to F,- at the surface of the ellipsoid, and vanishes at an infinite distance. This is therefore the value of the potential at any external point. It may of course be written a J, d±^ J (f + a') {(^ + a') (f + b") if + c')}^' 28. Next, suppose that the potential at every point of yz 'be' the surface is proportional to lfs=Vgj^, 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 ^'7"= 0, the potential at any internal point will be F, ?- ; while, substituting for y, z their values in terms of elliptic co-ordinates we obtain for the potential at any external point y,yzr j± di^ 29. We will next consider the case in which the po- 2 tentiaJ, at every point of the surface, varies as a;'=F„^ suppose. This case materially differs from the two just con- sidered, for since a? does not, when substituted for V, satisfy the equation v*F= 0, the potential at internal points cannot in general be proportional to a;'. We have therefore first to investigate a function of x, y, z, or of e, v, v which shall satisfy the equation v'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 Q? + oi) (c* + 0.) ^' + (c»+ 6)) (a« + «) / + (a' + «) (6' + «) a' -(a^ + wXi'-f-w) (c' + a)) = (e-a>)(u-«) (w'-w). And, if e^, 6^ be the two values of «* which satisfy the equation (6*+o.) (c'+o)) + (c'-l-a)) (a*+a)) + (a'+fl>) (6"+fi)) = 0...(l), we see that and Tj\e-e>,{v-0:){v'-e^ = (i. And, by properly determining the coefficients A^, A^, A^, it is possible to make = 14- when &'cV + c'aV + a'iV - a'6V = 0. 10—2 148 JXLIPSOIDAL AND SPHEROIDAL HAEMONICS, Hence, the expression (2) when A„, A^, -d., are properly determined will satisfy all the necessary conditions for an internal potential, and wiU therefore be the potential for every internal point. Now, we have in general (i' + e^) (c' + 0.) x' + (c' + 0,) (a*+ e^ f + (a» + 6^' (6'+ 6^ ^ - (a= + ^,) (6= + 6,) (c' + e,) = (6 - e,) {v - e,) (v- - 0,) {V + 0:) {) + {c'+(o) {a'+a>) + (a'+a) {b'+a)=3i0,-w) {0,-a) Ellipsoidal and spheroidal harmonics. 149 identically, and therefore, writing — o' for a, (a» - J^ (a" - c*) = 3 (a» + ^.) (a' + 6^. Hence, over the surface of the shell, and we therefore have, for the internal potential, This is not admissible for external points, as it becomes infinite at an infinite distance. We must therefore substi- tute for the factor e — 6. c-^jf d^ . r d± with a similar substitution for e - 6^, thus giving, for the external potential, Tr_FJ(e-g. )(»-^.)(»'-g.) r dVr ^r d± 'Jo{.f- 0,T {{f + a') if + 6') (t + O)^' ^, (^,-^i) («'+^,) J. if-e;)" [if+a') if+v) (t+c')}i ^f d± r dj>^ ^ r d± 1 150 JXLIPSOIDAL AND SPHEEOIDAL HARMONICS. The distribution of density over the surface, correspond- ing to this distribution of potential, may be investigated by means of the formula ^ 27r rfaVde fl?e/.=o' or its equivalent in Art. 13 of this Chapter. We thus find that 1 « F, r {v-e,)W-6:} •It dir ■ J. K^r + a") (t + &') (V^ + c'^^J ' 30. The investigation just given,, of the potential at an external point of a distribution of matter giving rise to a potential proportional to x' 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, F denote the potential of the Earth at any point on or with- out its surface, and £1 the angular velocity of the Earth's rotation, we have, as the equation of its surface, regarded as a surface of equal pressure, .*. F + 5 fi' (a? + ^) = a constant, 11 suppose. Hence, if a, J, c denote the semi-axes of the Earth, we have, for the determination of F, the following conditions : ELLIPSOIDAL AND SPHEROIDAL HAKMONICS. 151 d^V , d'V cPF „ 5^ + ^+^ = « (1)' F=0 at an infinite distance..., ,...,..,(2), F=n-ln'(a^ + /)when a' + F + c-^ = ^ (3). The term IT will, as we know, give rise to an external potential represented by n f ^^ . f ^^ J. {if+a'){^P'+b'){yJr+c')}i ' Jo {(^+a')(^+6")(t+c')}*' The two terms — - fIV, — ^ liy, will give rise to tei-ms which may be deduced from the value of F, just given by successively writing for F„ -^flV, and -lil'b', and (in the latter case) putting 6* for a* throughout. We thus get ^ r d^ fl v o» V \ {e-0,){v-e,){v'-e,) r d^ ^r (i>f^ nv o' 5' \ . r ^f "•'• (t-^,n(f +«')(t.+&')(t+oi*" 152^ ELLIPSOIDAL AKD SPHEROIDAL HABMOmCS. 31. Any rational integral function V of x, y, z, which satisfies the equation v'F= 0, can be expressed in a series of Ellipsoidal Harmonics of the degrees 0, 1, 2...t in x, y, z. For if F be of the degree {, the number of terms in V will be (^'+l)(* + 2)(* + 3) ^ ^^^ ^^^ condition ^W=0 is equivalent to the condition that a certain function of x, y, z of the degree i— 2, vanishes identically, and this gives rise to ^ —J^ conditions. Hence the number of inde- b pendent constants in F is (t+-l)(i+2)(i + 3) {{-\) {{{+!) 6 6 or (i + 1)*. And the number of ellipsoidal harmonics of the degrees 0, 1, 2...1 in x, y, z or of the degrees 0, ■=, 1, g.,.^ in e, V, v, is, as shewn in Arts, 6 to 10 of this Chapter, 1 + 3 + 5+.. . + 2t + l, or (i + 1)'. 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 — 1 by a suitable power of we shall express the density by the sum of two rational inte- gral functions of x, y, z of the degrees i, i — 1, respectively. The number of terms in these will be ELLIPSOIDAL AND SPHEROIDAL HABMONICS. 15^ And any ellipsoidal surface harmonic of the degree i,i — 2... in X, y, z, m&j, by suitably introducing the factor be expressed as a homogeneous function of x, y, e of the degree i ; also any such harmonics of the degree i— 1, t— 3... in X, y, z may be similarly expressed as a homogeneous function of x, y, z of the degree i — 1. And the total number of these expressions will, as just shewn, be (t + 1)', hence by assigning to them suitable coefficients, any distribution of density according to a rational integral function of x, 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 pogsible 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 V, v\ by the symbols F.«), F/«,...Fe'+i), and by F[v, v) the expression to be developed, assume Fiy. v')=C,V,+ C<-Wm + C^^ F,»)-J- C7WFW-I-... + c,'»rp +... + Ci^Fw + ... Then multiplying by eF/') and integrating all over the surface, we have jeF (v, v) F/'> dS = Q'> |e (F W)* dS. 154 ELLIPSOIDAL AND SPHEROIDAL HAKMONICS. The values o{jeF(y, v')F/'>d>Sf, and of fe ( F/'))' d>Sf must 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 fl)« = 1 f. - ^ i', + ^ P^. Why cannot (sin 6)' be expanded in a finite series of Bpherioal harmonics t 1 1 1 l+sms 2, Provethatl + ^P. + ^P, + ^P,+ ...=log. "^ 3. Establish the equations 4. If /I = cos ^, prove that /'.(^) = l-;(i + l)sin'- +... + (- l)-^^^^(sin»5)+... and also that i'.w = (- 1)' + (- ir »•(» + 1) «08' I + ... + (- i)'*"/tV^=- i'^'l) + ^ ' (I m)' | t - wi \ 2/ 5. Prove that, if a be greater than c, and i any odd integer greater than m, J_j^ '^ ' "^ g'^'c" ]2m'^ \t-ni a' (-j-Md'/t = i(i+l). 156 EXAMPLES. 7. Prove that, when /ix = * 1, -r-;'= ^ — ditT \i-m 2"|wi 8. Prove that 1, A .-. -P. P P •* O -^ 1+1 ••• * «l-l is a numerical multiple of 9. Prove the following equation, giving any Laplace's co- e£Scient in tenus of the preceding one : P.M = PP.+»f,PJP+0, where Cp^nn' + Jl -fi' Jl-fi" cos (u — w') and C is zero if n be even, and •+» |w+l 10. If i,j, k be three positive integers whose sum is even, prove that / P,P^,dl^ \.i...(J+h-i-\) \.Z...{k + i-j-\) \.Z...{i+j-h-\) 2.4... O'+A-*) 'ZA...{k + i-j) 2.4...(i+j-A) 2.4...(i+j + fc) 1 \.'i ...{i + j + k-\) i+j+k+\' Hence deduce the expansion of PJPj in a series of zonal harmonics. 11. Express «'y + y' + y»+y + « 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, iu such a manner that the quantity of matter laid on an element i^S) of the surface is 8/5(1 +a»+6y + «+/«* + ^y' + fe'). EXAMPLES. 357 where x, y, x are rectangular co-ordinates of the element iS re- ferred to the centre aa 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 he r, and its law of density be p = ax+hy + (iz, where the origin is at the centre, prove that its potential at an external point (f, 17, ^ '^T^va(M+ ^+ 2.4...2i if r < a, and 2 ' r" 2.4 "r* ^ ' 2,4...2i -^»*+i,.»* "^ ••• if r > a. 23. Prove that the potential of a circular ring of radius c, whose density at any point is cos m^, c^ being the distance of the point measured along the ring from some fixed point, is 160 :example8, ■^2.4. 6.. .(2m + 2) d/*"" j-"*''^"' , 1.3.5...(2^-1) drP^,^ jr** . 1 '*'2,4.6,..2(ot + * d/*- r"+"*' "'j ' 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 ita 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 BoUd exercises no attraction on a point within its inner surface. CAMBKIDOE : FBINTEO BY C. J. CLAT, U.A. AT THE UNITEBSITT FBESS.