I. Relation between Spherical Harmonics of Positive and Negative Degrees

A function which is homogeneous in x, y, z, of degree n in those variables, and which satisfies Laplace's equation a 2 v 2 ay, az2 =0, or v 2 V =0, (I) is termed a solid spherical harmonic, or simply a spherical harmonic of degree n. The degree n may be fractional or imaginary, but we are at present mainly concerned with the case in which n is a positive or negative integer. If x, y, z be replaced by their values r sin 0 cos 43, r sin 0 sin 4), r cos 0 in polar co-ordinates, a solid spherical harmonic takes the form r n f,,,(B, 43); the factor f n (0, 0) is called a surface harmonic of degree n. If Vn denote a spherical harmonic of degree n, it may be shown by differentiation that p2(rmVn) =m(2n + m + I)r m - 2 V„, and thus as a particular case that V 2 (r2n -W„) =o; we have thus the fundamental theorem that from any spherical harmonic V n of degree n, another of degree - n - I may be derived by dividing V n by r 2n+i . All spherical harmonics of negative integral degree are obtainable in this way from those of positive integral degree. This theorem is a particular case of the more general inversion theorem that if F ( x, y, z) is any function which satisfies the equation (I), the function r Ia y ' r2' r2l also satisfies the equation.

The ordinary spherical harmonics of positive integral degree n are those which are rational integral functions of x, y, z. The most general rational integral function of degree n in three letters contains 2(n+I)(n+2) coefficients; if the expression be substituted in (I), we have on equating the coefficients separately to zero Zn(n - I) relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains 1(n+ I) (n +2) - Zn(n - I), or 2n+I independent constants. There exist, therefore, 2n+I independent ordinary harmonics of degree n; and corresponding to each of these there is a negative harmonic of degree - n-1 obtained by dividing by r2n+i. The three independent harmonics of degree I are x, y, z; the five of degree 2 are y 2 - z 2, z 2 - x 2, yz, zx, xy. Every harmonic of degree n is a linear function of 2n+i independent harmonics of the degree; we proceed, therefore, to find the latter.

2. Determination of Harmonics of given Degree

It is clear that a function f(ax+by+cz ) satisfies the equation (i), if a, b, c are constants which satisfy the condition a 2 +b 2 +c 2 =o; in particular the equation is satisfied by ( z-F-cx cos a+iy sin a)". Taking n to be a positive integer, we proceed to expand this expression in a series of cosines and sines of multiples of a; each term will then satisfy (I) separately. Denoting 1a by k, and y+cx by t, we have I (z + Y2 z2 cx cos a +cy sin a) n = (z+kt+ - _ ) which may be written as (2kt)-"{ (z+kt) 2 - r 2) } n. On expansion by Taylor's theorem this becomes 2 n l e t s as ( 2kt)- n s! dz s ' O the differentiation applying to z only as it occurs explicitly; the terms involving cos ma, sin ma in this expansion are COS ma j (' n + m )m a? +m r n + ( y n z 2 - (n)! n - m k ri+m)! m n +m m I - y+cx ) a / y +cx) sin_ = ma azn+m(z2r2)n (n - m)? azn'n (z2 - r2)m where m I, 2,. .. n; and the term independent of a is I an azn(2 On writing ( y+cx) m =c m r m (cos m4) - c sin m4))2 sin me, (y+ cx) - m= / i mr m (cos m4)+c sin mc6) sin-mO and observing that in the expansion of ( z+ex cos a+cy sin a)m the expressions cos ma, sin ma can only occur in the combination cos m(4) - a), we see that the relation m m an- imrm sin e a (z2 - r 2)' = Z m m sin -) a (z 2 - r 2)" ( n+. yr ?)I a n+m (n - m!

must hold identically, and thus that the terms in the expansion reduce to 2n_irm cos ma cos m4) sin meazn+rn(z2 - r2)n an+m 2 n_ i r m sin ma sin mcp sin mea2n+m(z2 - r2)n.

are ui the n+m rn cos mg s i n me d?n +m(! ? 2 - I)n where ,u denotes cos 0; by giving m the values 0, I, 2 ... n we thus have the 2n+I functions required. On carrying out the differentiations we see that the required functions are of the form cy ) - m (n - m)(n - m - I) (n - m)(n - m - I) (n - m2) (n - m-3) zn_m_4(x2 +y2 +z2)2 (2) 2.4.2n - I .2n-3 where m = o, i, 2, 3, ... n.

3. Zonal, Tesseral and Sectorial Harmonics

Of the system of 2n-1-I harmonics of degree n, only one is symmetrical about the z axis; this is n do Pn(12) = 2 7 n! dµn ( µ - I)n, we observe that P n (u) has n zeros all lying between i, consequently the locus of points on a sphere r=a, for which Pn(u) vanishes is n circles all parallel to the meridian plane: these circles divide the sphere into zones, thus P„(µ) is called the zonal surface harmonic of degree n, and r"P n (p,), r n -°Pn(i) are the solid zonal harmonics of degrees n and - n - I. The locus of points on a m n+m 2 n sphere for which sn mcp.sin O di n+m(, u - I) vanishes consists of n - m circles parallel to the meridian plane, and m great circles through the poles; these circles divide the spherical surface into quadrilaterals or TEVVepa, except when n = m, in which case the surface is divided into sectors, and the harmonics are therefore called tesseral, except those for which m =n, which are called 2 2mdmPn(µ) by P( ,), the tesseral sectorial. Denoting (I - µ) du n ' Y n surface harmonics are a°n m4) .1 3 7 (cos 0), where m = I, 2, ... n - I, and the sectorial harmonics are s n n4. Pn(cos 0). The functions P n (,u), P7 (p) denote the expressions (n+m) !

We thus see that the spherical harmonics of degree n form 2. 2n - I +y2 + writing (2n)!' n(n-I) ? n(n-I)(n-2)(17-3) - 4..

n 2. - I 2.4 .2n -I.2n -3 272 .

P72 (' µ ') = n -m)I (I -µ - )2m ( n -m)(n m - + 2.2n P 72 (µ) = 2"n!

"n !I(I -/a'2) Every ordinary harmonic of degree n is expressible as a linear function of the system of 272-1-I zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is n aoPn(12) +1(am cos mcp+bm Sin m4)P n t (/c). (5) In the present notation we have (z+cx COS a+cy Sin a) n = n j P. (A) +2?c'n (n n + ' m) ! P7(12) COS 112(4)(112(4) - a) ( I if we put a = o, we thus have n (cos 0+c sin 6 cos ( ?) n = n (cos 6 ) +2/ 7m (n +m)! P , m (cos 0 ) cos m¢, from this we obtain expressions for P, c (cos 0 ), P m n (cos 0 ) as definite integrals Pn(cos 0) = I, f o (c os 0+c sin 0 cos c6)nd4) c" (72+m) i P (cos 6) = J o (cos 0+c sin 0 cos 4) n cos m4d4. 4. Derivation of Spherical Harmonics by Differentiation. - The linear character of Laplace's equation shows that, from any solution, others may be derived by differentiation with respect to the variables x, y, z; or, more generally, if a a a f (x' ay' 3z) denote any rational integral operator, a a a f x, 8y' a) V is a solution of the equation, if V satisfies it. This principle has been applied by Thomson and Tait to the derivation of the system of any integral degree, by operating upon I /r, which satisfies Laplace's equation. The operations may be conveniently carried out by means of the following differentiation theorem. (See papers by Hobson, in the Messenger of Mathematics, xxiii. 115, and Proc. Lond. Math. Soc. vol. xxiv.) a a _ I fn ay az r - 27-117 ! r2n I 2.2n-I which is a particular case of the more general theorem { a a a dnF 2 „- 2 do-IF Jn (-57 ay , Oz) F (r) = 2n d(r 2 ) n + I I! d(r2)u-1v 2+ .

2 do-sF // + S ! d(r2)n-sv gs+. i n( x, y, z) where fn(x, y, z) is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree -n - I in the expression (7) is 2.2n 2 I + 2.4.20 y4 II .4202-3' ... f?L(x, y , z).

It can be verified that even when n is unrestricted, this expression satisfies Laplace's equation, the sole restriction being that of the convergence of the series.

5. Maxwell's Theory of Poles. - Before proceeding to obtain by means of (7), the expressions for the zonal, tesseral and sectorial harmonics, it is convenient to introduce the conception, due to Maxwell (see Electricity and Magnetism, vol. i. ch. ix.), of the poles of a spherical harmonic. Suppose a sphere of any radius drawn with its centre at the origin; any line whose direction-cosines are 1, m, 17 drawn from the origin, is called an axis, and the point where this axis cuts the sphere is called the pole of the axis. Different axes will be denoted by suffixes attached to the direction-cosines; the cosine ( lcx+mcy+ncz)/r of the angle between the radius vector r to a point (x, y, z) and the axis ( lc, Inc, nc ), will be denoted by Ac; the cosine of the angle between two axes is lcl

+m,my+ncny, which will be denoted by The operation l ax + mc a ay ay+ n a z performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis ( lc, mc, nc), and is denoted by a/ahc. The potential function Vo=eo/r is defined to be the potential due to a singular point of degree zero at the origin; eo is called the strength of the singular point. Let a singular point of degree zero, and strength eo, be on an axis h 1 , at a distance ao from the origin, and also suppose that the origin is a singular point of strength-co; let eo be indefinitely increased, and ao indefinitely diminished, but so that the product eoao is finite and equal to eo; the origin is then said to be a singular point of the first degree, of strength e l , the axis being h 1 . Such a singular point is frequently called a doublet. In a similar manner, by placing two singular points of degree, unity and strength, e l, -e 1 , at a distance a i along an axis h 21 and at the origin respectively, when e l is indefinitely increased, and a l diminished so that e 1 a 1 is finite and =e 2, we obtain a singular point of degree 2, strength e 2 at the origin, the axes being h 1, h 2 . Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength e n at the origin, the singular point having any n given axes h 1, h 2, ... hn. If en_i 4'n _ i (x, y, z) is the potential due to a singular point at the origin, of degree n - I, and strength en_i, with axes h 1, h2,...hn_1, the potential of a singular point of degree n, the new axis of which is hn, is the limit of en_1 472-1 ( x - lna, y - mna, z-nna)-en. -1 4)n_1 (x, y, z); La =0, Len_i = Laen_1 =e.; this limit is -- en l n ' a ax ' a a ' y 1 +nna az 1) or - en all ? n-1

n Since 0 0 =1,/r, we see that the potential V, due to a singular point at the origin of strength e n , and axes h1, h 2, ...h n is given by Vet=(-I) n en an I (8) ah l ah 2 ... ahnr 6. Expression for a Harmonic with given Poles. - The result of performing the operations in ( 8 ) is that V. is of the form n !enyi, where Y n is a surface harmonic of degree n, and will appear as a function of the angles which r makes with the n axes, and of the angles these axes make with one another. The poles of the n axes are defined to be the poles of the surface harmonics, and are also frequently spoken of as the poles of the solid harmonics Ynr n, Y n r n - 1 . Any spherical harmonic is completely specified by means of its poles.

In order to express Yn in terms of the positions of its poles, we apply the theorem (7) to the evaluation of V. in (8). On putting r=n fn(x, y, z) =II(l r x+m r y+n r z), we have (2 n! I ((1 Yn - 2 n n!n! n I 2.2 72 -I + 2.4.2n-I .2n-3 . .

/ X n II (lrx+mry+nrz). By /(µsXn-2s) we shall denote the sum of the products of s of the quantities µ, and 11-2S of the quantities X; in any term each suffix is to occur once, and once only, every possible order being taken. We find II(lx+ my +nz) =I(X')r n, A 2 H(lx+nmy+nz) and generally 6, 2m H ( Ix +my+nz) = 2mm ! (72mXn-2m)rn-2m ' .

thus we obtain the following expression for Y,, the surface har monic which has given poles h1, h 2, ...h n; n +i(-I)n . an 1 Y n - r n! ah i ah 2 ... ahn r (2n-2m)!

2n-mn! (n-m)!

m=0 where S denotes a summation with respect to m from m=o,to m = In, or 1(nI), according as n is even or odd. This is Maxwell's general expression (loc. cit. ) for a surface harmonic with given poles.

If the poles on a sphere of radius r are denoted by A, B, C..., we obtain from (9) the following expressions for the harmonics of the first four degrees: Y 1 =cos PA, 1(3 cos PA cos PB -cos AB), = cos PA cos PB cos PC -cos PA cos BC -cos PB cos CA -cos PC cos AB), Y4= 1 8 - (35 cos PA cos PB cos PC cos PD -51 cos PA cos PB cos CD + E cos AB cos CD).

7. Poles of Zonal, Tesseral and Sectorial Harmonics. - Let the n axes of the harmonic coincide with the axis of z, we have then by (8) the harmonic I) n r n +1 an I n! az n (7) ? f " (x ' y z) 2.4.2n-I .2n-3 (71), = ( 6) applying the theorem (7) to evaluate this expression, we have (- n n+1 a n I (an)! I r2v2 r4v4 n! az n r 2 n n!n! r n I 2.2n - I + . 4.2n -I .2n - 3 2n - (2n)! ? n(n n-2 + - 2 n n!n! 2.2nIII the expression on the right side is P n (u), the zonal surface harmonic; we have therefore (- I)' rn+1 an 1 Pn(7a) = n! a z n r

The zonal harmonic has therefore all its poles coincident with the z axis. Next, suppose n - m axes coincide with the z axis, and that the remaining m axes are distributed symmetrically in the plane of x, y at intervals 7r/m, the direction cosines of one of them being cos a, sin a, 0. We have m -I r7r a r7rl I ] +"` II ? (a +) x + sin (a + m !ay ? = 2R e ` (a ( 33) +m a a -Fe ' a ( + L) Let = x +ty, n = xty, the above product becomes

T eQ+ r,r) a e ?(a+ra) a  ?

11 m ?. m o as which is equal (to e (m - 1)7 n Z em ea (22.) m - emLa (- (1) l ! this becomes e(m - 1)'2 2 1 (A) m I m(a)m ] i - ( ) ayl e(mI)?2 C m+ I m ( a)9n - ) an (2n) ! I r2Q2 n - m 2nnl r 2ni 1 I 2.2 - I +

?y) -  ! = (- I)n 2nyl)? y +1+1(m?tt sin m4,) sin cOSn -m9 (nm) (n -m- I) 2.2n - I hence a m a a mI (n-m)!

a z n-m (ax e `ay ) = (I) n rn+1 (cos mit t sin (cos 0 ), as we see on referring to (4); we thus obtain the formulae COS + m Y = (- I) 2 -1 1 COS m? . PT (COS 0) n - (n)! ,rt n-m m m _ a2n - m (a an r (- I) 2 ,n it +1 sin mO. n (cos 0) It is thus seen that the tesseral harmonics of degree n and order m are those which have n - m axes coincident with the z axis, and the other m axis distributed in the equatorial plane, at angular intervals 7r/m. The sectorial harmonics have all their axes in the equatorial plane.

8. Determination of the Poles of a given Harmonic. - It has been shown that a spherical harmonic Y n (x, y, z) can be generated by means of an operator a a a I n ay' a z acting upon y, the function fn being so chosen that Yn I) n I - 2. + ... fn(x, y, z); i thit relation shows that if an expression of the form (x2 + y2 + 2)fn-2(x, y , z) is added to fn(x, y, z), the harmonic Y n (x, y, z) is unaltered; thus if Y. be regarded as given, f n (x, y, z) =0, is not uniquely determined, but has an indefinite number of values differing by multiples of x 2 +y 2 +z 2. In order to determine the poles of a given harmonic, f n must be so chosen that it is resolvable into linear factors; it will be shown that this can be done in one, and only one, way, so that the poles are all real.

If x, y, z are such as to satisfy the two equations Y n (x, y, z ) =0, x2+y2+z2= 0, the equation fn(x, y, z) is also satisfied; the problem of determining the poles is therefore equivalent to the algebraical one of reducing Y. to the product of linear factors by means of the relation x 2 +y 2 -{-z 2 = 0, between the variables. Suppose Yn(x, z) =AII (l s x + m s y + n s z) +(x2 +y 2)Vn -2(x, y, z), s - I we see that the plane l s x-Fmsy+n s z=0 passes through two of the 2n generating lines of the imaginary cone x 2 -Fy 2 -1-z 2 =0, in which that cone is intersected by the cone Y n (x, y, z ) =0. Thus a pole (1 s, ms, ns) is the pole with respect to the cone x 2 +y 2 +z 2= 0, of a plane passing through two of the generating lines; the number of systems of poles is therefore n(2n-I), the number of ways of taking the 2n generating lines in pairs. Of these systems of poles, however, only one is real, viz. that in which the lines in each pair correspond to conjugate complex roots of the equations Yn=0, x2+y2+ z 2= 0. Suppose x _ y z a1 t ?1 a2+432 a3 1433 gives one generating line, then the conjugate one is given by x y z a i - 13 i 2 -43 2 3 - 03' and the corresponding factor lx+my+nz is x R/ z al a2+432 a3 a 1 1 a 2 -t / 3 2 a3-103 which is real. It is obvious that if any non-conjugate pair of roots is taken, the corresponding factor, and therefore the pole, is imaginary. There is therefore only one system of real poles of a given harmonic, and its determination requires the solution of an equation of degree This theorem is due to Sylvester ( Phil. Mag. (1876), 5th series, vol. ii., " A Note on Spherical Harmonics "). 9. Expression, for the Zonal Harmonic with any Axis. - The zonal surface harmonic, whose axis is in the direction ' z' (xx'+yy'+zz') r, r', y,, I S n rr' or Pn(cos B cos B'+ sin 0 sin 0 cos cb -4'); this is expressible as a linear function of the system of zonal, tesseral, and sectorial harmonics already found. It will be observed that it is symmetrical with respect to ( x, y, z) and (x', y', z ), and must thus be capable of being expressed in the form aoP n (cos B)P n (cos 0) +EamPn (cos 0)P7 , (cos 0')cos m(4 -49'), and it only remains to determine the co-efficients ao, ...am...an. To find this expression, we transform (x'x+y'y-Fz'z)n, where x, y, z satisfy the condition x 2 +y 2 +z 2 =0; writing =x+ty, = x - ty, ' =x'+ty', n' = 'x - ty', we have +yy'+z2')n=(zn' 41E'77+2z')n which equals /a b, a b a (zz') +}.}. n' +77 (zz/)n - a - b a!b!(n - a - b)! 2a +6 ' the summation being taken for all values of a and b, such that a+bin, a>b; the values a=0, b=0 corresponding to the term (zz') n. Using the relation On = -z 2, this becomes b (xx' +yy +zz')n=(zz')n+MI 2,0_1,a!b!(n n a (-b)(,n/)bz/n-a-b 1 (77'() a-b + ('n) a-b }z n - ab, putting a - b=m, the coefficient of t m z n - m, on the right side is - I) b n! / /m /n-m 2m+2b b!(m+b)!(n - m - 2b)!(77) 7 z ' from b= 0 to b=2(n - m), or 2(n - m - I), according as n-m is even or odd. This coefficient is equal to i n. (x, n-m)(n-m - I) z x / 2 /2 2mm!(n - m)! - y) 2.2m (+ y) + (n - m)(n - m - I) ( n - m-2)(n - m -3) z , n - m- 4 2+ y /2)2 -; 2.4.2rn+2.2m+3 in order to evaluate this coefficient, put z = 1, x' = t cos a, y =I. sin a, then this coefficient is that of (t cos a+sin a) m, or of t m e - m wa in the exp

ansion n!' of (z'+tx' cos a+ty' sin a)) 1 in powers of ea and e ra , this has been already found, thus the coefficient is (n n! !e `m? Pn(cos B').r'n.

Similarly the coefficient of n m z n - m is (n n! ie +?m  ?'pn (cos 0')r'n; hence we have +yy'+22')n = z n Pn(COs 0') +n!/P n (COS O'){cos mt'(Im+71-a) .,n-m +L sin ml1:' (7 7 m - tm) } (n +m) 1.

In this result, change x, y, z into a a x' ay' az'; when a = o, and From (7), we find -m a amI r cOs n - m - 2 e +.. .

, (io) and let each side operate on I/r, then in virtue of (io), we have (rr')nPn (xx +zz') - Pn(cos 9 cos O +sin 0 sin B cos 0-0') = Pn(cos 0) Pn(cos 0')+2Z(n+m) ? Pn (cos 0) Pn (cos 0 ) cos m(0-0') (I I) which is known as the addition theorem for the function Pn. It has incidentally been proved that) m +t FT (cos 0) _ ,n ? sln'n° cosn_m9 2 (n-m). (((n-m)(n-m-I) cos n- ' n-2 Bsin 2 6+ (12) 2.2m+2 which is an expression for Pn (cos 0) alternative to (4)

10. Legendre's Coefficients. - The reciprocal of the distance of a point ( r, 9, 0 ) from a point on the z axis distant r from the origin is (r 2 µ+r'2)_1 which satisfies Laplace's equation, µ denoting cos 0. Writing this expression in the forms r r 2 ' ) - 3 I ` r' r ' 2 - y, I-2 ?,/.4, L - r i ( y y2 it is seen that when r < r', the expression can be expanded in a convergent series of powers of r/r', and when r' < r in a convergent series of powers of r'/r. We have, when h2 ( 2µ-h)2 (I -2hµ+h2) 2 =I +h(2µ - h) + 2:4h2(2µ-h)2+ .. .

+1 2 4...2n Ihn(2µ-h)n+...

and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of h n is then found to be I.3.5...2n-I(µn n(n- I)µ_+ n(n- I)(n-2)(n - 3)µn-4...

I.2.3...71 ` 2.2n - I 2.4.2n-I.2n-3 this is the expression we have already denoted by P n (µ); thus (I-2hµ+h 2) =Po(µ)+hPl(µ)+...+hnPn(1-0+..., (13) the function Pn(µ) may thus be defined as the coefficient of h n in this expansion, and from this point of view is called the Legendre's coefficient or Legendre's function of degree n, and is identical with the zonal harmonic. It may be shown that the expansion is valid for all real and complex values of h and such that mod. h is less than the smaller of the two numbers mod. (/M2-'). We now see that (r2 -2rr' +r'2) is expressible in the form ? yn Pn (l-?) when r < r', or y'n y n+I +I Pn () 0 when r' < r; it follows that the two expressions r n P n (µ), r n-1Pn(µ) are solutions of Laplace's equation.

The values of the first few Legendre's coefficients are Po() = 1, P1(µ) = µ, P2(µ) = Z(3µ2 - I), P3(1) = z (51.33µ) P 4(2) =g(35µ4-30µ2+3), P 5(2) =g(63µ5-70µ+l5µ) P 5(µ) = i(231µ6 _3 1 5µ 4 + 10 5µ 2 - 5), P 7(µ) = (429, - 69310 +315/23-35,u).

We find also Pn(I) = I, P n (- I =(- I)n P.(0)=0, or (-I);ni.3.5...nI 2.4...n according as n is odd or even; these values may be at once obtained from the expansion (13), by putting / 2=1, o,- i .

II. Additional Expressions for Legendre's Coefficients. - The expression (3) for P n (µ) may be written in the form (2n)! n n i-n i i t P n() = 2nn ? nµ F (2, 2 2 n+ 2/ with the usual notation for hypergeometric series. On writing this series in the reverse order P (µ) =(-),n n! F n n +I I n I 2' (' I n ) (-1-n)2 !

2 (--, 2, 2, or n_I I 2 n! F. (- nI n I+ 3 2) (-) 2n-1n- I n-I µ 2 2 + 2 µ ! j 2 2 according as n is even or odd.

From the identity (I - 2h cos 9 +h 2)-4 = (I - LO) - 1(1 - he o) I, it can be shown that I.3. _,)(2 cos (n -4) 0 +

(14) I. 2. (2n - I) (2n - 3) By (13), or by the formula do Pn(µ) -2nn!dµn(A2-I)n which is known as Rodrigue's formula, we may prove that P n (cos0) = I-(n i I)nsin 22 +(n+2) (I+22)n(nI)sin4-...

F (n+ I, -n, 1, singe). Also that) 2 P n (cos 0)

=COS 2r ?F (-n, -n, I, - tan 2 ? J

By means of the identity ((/ (12hµ+h2) -A =(I- hµ)-1 j 1 +? (II hu)) ( i, (17) transformed I(? d0 .J, 0(µ-l / µ 2 -I cosi,G)n+1 by means of the relation (,u+ ,u 2 - I COS 0)(µ-s1p.2-1 cos I.

Two definite integral expressions for Pn(µ) given by Dirichlet have been put by Mehler into the forms Pn(cos 0) _ 2 f B cos ( n+)4) = 2 (n sin (n+)4) do 7. .1/2 cos 4-2 cos 9 7r a-J2 cos 9 -2 cos 4) When n is large, and 0 is not nearly equal to o or to 7r, an approximate value of Pn(cos 0 ) is 12/n7r sin 9}; sin {(n+2)9+47 1.

12. Relations between successive Legendre's Coefficients and their Derivatives. - If (I -2hµ+h 2)-i be denoted by u, we find (I -2hµ+h2 ) ah+(h-µ)u =0; on substituting Eh n P n for u, and equating to zero the coefficient of h n , we obtain the relation nPn-(2n-I)MP2_1+(n-I)Pn_2=0.

From Laplace's definite integral, or otherwise, we find (µ 2 -I) Pn= n (µPn-P,,-1) = -(n+I)(µPn-pn+l). We may also show that dP n dP n _ 1 _ µ iµ - nPn ( n+I)Pn= - µdPn+dPn +1 dµ dµ (2n + I) Pn = dP n+1 - dPn_1 dµ dP„ ( n + I)d d u +nd dµ 1 (2n +I) (µ21) dPn =n(n+ I)(Pn4.1 - Pn-1) dPn= (2 n' - I)Pn-1+(2n-5) Pn-3+ (2n-9) Pn-5+..

d µ the last term being 3P 1 or Po according as n is even or odd.

13. Integral Properties of Legendre's Coefficients. - It may be shown that if P n (µ) be multiplied by any one of the numbers 1, µ, ... µn-1 and the product be integrated between the limits I,- I with respect to µ, the result is zero, thus 1 µkPn(µ)dµ=0, a =0, I, 2,. n - I. (18) To prove this theorem we have (?? 4 (/2 2 J AL n v./) dµ = 2 n n ! 1 l -1) -apt, 3 5 2n-I (I

n Pn(cOs }l 2.4.6...2n. cos I. Icos (n-2)0 it may be shown that Pn(cosO), =cos n 9 I - n( i)tan2 9 + n(n-1) 2 2 4 2 )(n- 3)tan 40-... =cos n eF(- 2n, 1-1n, I, -tan29).

Laplace's definite integral expression (6) may be into the expression (15) (16) on integrating the expression k times by parts, and remembering that CO- O n and its first n--- i derivatives all vanish when / 2= = I, the theorem is established. This theorem derives additional importance from the fact that it may be shown that AP n (u) is the only rational integral function of degree n which has this property; from this arises the importance of the functions P n in the theory of quadratures.

The theorem which lies at the root of the applicability of the P n to potential problems is that if n and n' are unequal 'Pni(M)Pn(/2)d/2= o, (19) which may be stated by saying that the integral of the product of two Legendre's coefficients of different degree taken over the whole of a spherical surface with its centre at the origin is zero; this is the fundamental harmonic property of the functions. It is immediately deducible from (18), for if n' P n ,(u) is a linear function of powers of u , whose indices are all less than n.

When n'=n, the integral in (19) becomes f I {P n (u)} 2 du; to evaluate this we write it in the form I n n I 2 2n n!n! _Idun (p 2 .. 1 ) n d u n (u 2 1)ndu; on integrating n times by parts, this becomes -)n I 2n ' (2n)! I 2 2n n!n? _I(/? 2 - I) n d 2 n (u - I ) n dA, or 2 2nn!n! f _ I (I which on putting 1 f I u = 2 (I - u, becomes 2 ( n! 2n) n! J _ I u n (I -u)ndu, hence ?II I {Pn(u)}2du- 2n+I. (20) 14. Expansion of Functions in Series of Legendre's Coefficients. - If it be assumed that a function f(A ) given arbitrarily in the interval 14= - I to +I, can be represented by a series of Legendre's coefficients ao+aiPi(u)+a2P2(2)+

.+anPf(u)+

..and it be assumed that the series converges in general uniformly within the interval, the coefficient a can be determined by using (19) and (20); we see that the theorem (19) plays the same part as the property f ir nO COS n'OdO = 0, (n * n) does in the theory of the expansion of functions in series of circular functions. On multiplying the series by Pn (u), we have f i an I { Pn (/-?) } 2 d/2 = f I I f (2) Pn (/2) dm hence 211+I fI 2 f(u) Pn (u) d/I, hence the series by which f(A) is in general represented in the interval is ?y 211+ I Pn(u) f I I f (A')Pn(A') du'

(21) 2 The proof of the possibility of this representation, including the investigation of sufficient conditions as to the nature of the function f(u), that the series may in general converge to the value of the function requires an investigation, for which we have not space, similar in character to the corresponding investigations for series of circular functions (see Fourier'S Series). A complete investigation of this matter is given by Hobson, Proc. Lond. Math. Soc., 2nd series, vol. 6, p. 388, and vol. 7, p. 24. See also Dini's Serie di Fourier. The expansion may be applied to the determination at an external and an internal point of the potential due to a distribution of matter of surface density A ) placed on a spherical surface r =a. If = E a + V we see that Vi, Vo have the characteristic properties of potential functions for the spaces internal to, and external to, the spherical surface respectively; moreover, the condition that Vi is continuous with Vo at the surface r=a, is satisfied. The density of a surface distribution which produces these potentials is in accordance with a known theorem in the potential theory, given by I a V i 3Vo = (ar - ar)r=a, hence = ra2 Z(2n+i)A n P n (u); on comparing this with the series (21), 4 we have A.= 27ra 2 f f (u') P n (u') du', hence [Vi = 27ra" - P n ( /2) J I If V) Pn (/2') d/2' 27r a an+l Pn ?") f (u ') Pn (u ') du' are the required expressions for the internal and external potentials due to the distribution of surface density AA). 15. Integral Properties of Spherical Harmonics. - The fundamental harmonic property of spherical harmonics, of which property (19) is a particular case, is that if Yn(x, y, z), Z n' (x, y, z) be two (ordinary) spherical harmonics, then, J[Yn(x, y, z) z n, (x, y, x) dS =0, (22) when n and n' are unequal, the integration being taken for every element dS of a spherical surface, of which the origin is the centre. Since v 2 Yn = 0, v 2 Z n, = 0, we have J (YnV 2 ZnZ n ,v 2 Y n ,) dxdydz = 0, the integration being taken through the volume of the sphere of radius r; this volume integral may be written JJf ax (YnaOx'zn, aa n) +15, (y n a a z; - z n, yn) a r aY) + az Yn Z "' azn dxdydz = 0; by a well-known theorem in the integral calculus, the volume integral may be replaced by a surface integral over the spherical surface; we thus obtain (y'_z,,)f r (Y1_ 'z) + r(Yn-5-17, a? +-Z.,ay) dS=0;: on using Euler's theorem for homogeneous functions, this becomes. n'71 = 0, YnZn'dS whence the theorem (22), which is due to Laplace, is proved.

The integral over a spherical surface of the product of a spherical harmonic of degree n, and a zonal surface harmonic P n of the same degree, the pole of which is at (x', y', z') is given by Yn(x, y, z) P n dS =2:+r 1 an+2YYn (x ', y', z'); (23) thus the value of the integral depends on the value of the spherical harmonic at the pole of the zonal harmonic.

This theorem may also be written 1 Vn(O, P n (cos a cos 0'+sin 0 sin 0' cos 0 - 4:1)44 7r = 2n,+, IV. ( 0', 4)') To prove the theorem, we observe that V n is of the form aoPn(u) +E(am cos m4)+bm sin m¢)Pn (/o); I to determine ao we observe that when 1.1.= I, P n(u) = I, Pn (u) =0; hence ao is equal to the value Vn(0) of V n ( 6, 4 )) at the pole 0=0' of Pn(u). Multiply by P n (u) and integrate over the surface of the sphere of radius unity, we then have 0 2n f 1 1 V. (8, 4 ) Pn(u)dud4= ao f o f P n ( /2) }2d/2dg5 2nd-1ao-2n+ 1 Vn(0), if instead of taking A= I as the pole of P n (u) we take any other point (u', (') we obtain the theorem (23).

If f(x, y, z) is a function which is finite and continuous throughout the interior of a sphere of radius R, it may be shown that Yn(x, 22! y, z)f( x , y, z)d5= 4?R ( 2n+ 1)! 1+ 2.2n+3 R4v4 + 2.4.2n+3. 2n +5+ ... Y n a x ' a y ' a)f(x, y z) where x, y, z are put equal to zero after the operations have been performed, the integral being taken over the surface of the sphere of radius R (see Hobson, " On the Evaluation of a certain Surface Integral," Proc. Lond. Math. Soc. vol. xxv.).

The following case of this theorem should be remarked: If f n (x, y, z) is homogeneous and of degree n Yn(x, y , z)fn(x, y, z)dS=47rR2n+' (2n1) ,Yn (ax' ay' az) fn(x,y,z) if f n (x, y, z) is a spherical harmonic, we obtain from this a theorem,. due to Maxwell ( Electricity, vol. i. ch. ix.), 1J 'Yn(x,' y , z)fn(x, y, z)dS = 2 RA +I 2 n! ah l ah 2. .ahfn(x, y , z) functions integers an = where h1h2...hn are the axes of Y. Two harmonics of the same degree are said to be conjugate, when the surface integral of their product vanishes; if Y n , Z. are two such harmonics, the addition of conjugacy is Y7,(-87 ay' Oz) y, z) =0.

Lord Kelvin has shown how to express the conditions that 212+I harmonics of degree n form a conjugate system (see B. A. Report, 1871).

16. Expansion of a Function in a Series of Spherical Harmonics.- It can be shown that under certain restrictions as to the nature of a function F(µ, 0 ) given arbitrarily over the surface of a sphere, the function can be represented by a series of spherical harmonics which converges in general uniformly. On this assumption we see that the terms of the series can be found by the use of the theorems (22), (23). Let F(µ, ¢) be represented by V0 Gu, 4) +V 1(µ, 4;) + ... +V n (µ, 0) + ...; change µ, 4> into µ', 4) and multiply by Pn(cos 0 cos 0 +sin 0 sin 0 cos 4.-0'), we have then f:f1F', 4,')Pn(COS 0 cos e +sin 0 sin 0 cos cp-(a')dµ'dct' V12(µ', 4)')Pn(cos o cos 0 +sin o sin 0 cos 49-0')dis'do' 4" V n (o, 4)), 212+1 hence the series which represents F(µ, ¢) is co ir (2n+I) f o 1 f I I F(u', q')P n (cos 0 cos o' 0 +sin 0 sin 0 cos 4,-4')dµ'd4'. (24) A rational integral function of sin o cos 4,, sin 0 sin 4,, cos o of degree n may be expressed as the sum of a series of spherical harmonics, by assuming fn(, z) = Yn+r 2 Yn-2 +r 4 Yn-4+ .. .

and determining the solid harmonics Y n, Yn_2, ... and then letting = 1, in the result.

Since V 2 (r 2s Yn_2 8) =2s(2n-2s+i)r 2s - 2 Y n _ 28j we have v 2 f7, = 2(2n-I)Yn-2+4(2n-3)r 2 Yn-4+6(2n-5)r 4 Yn- 6 +..

v 4 fn = 2.4(2n -3) (2n -5) Y n-4+4.6 (2n -5) (2n - 7)r 2 Yn_6+ .. the last equation being vnfn=n(n+ I) ( n-2)(n-I)...Yo, if n is even, v fn =(n-I)(n+2)(n-3)n...Y 1r if n is odd from the last equation Yo or Y 1 is determined, then from the preceding one Y2 or Y3, and so on. This method is due to Gauss (see Collected Works, v. 630).

As an example of the use of spherical harmonics in the potential theory, suppose it required to calculate at an external point, the potential of a nearly spherical body bounded by r =a(i +Eu), the body being made of homogeneous material of density unity, and u being a given function of 0, 4 ), the quantity E being so small that its square may be neglected. The potential is given by f 0 7r (I I 1 J {r 2 +r'2-err cos y} -=dr'dµ'dc¢', where -y is the angle between r and r'; now let u' be expanded in a series Yo(µ', 49') -1-Y'1(µ', 4)')+...+Y12(µ', 4)')+... of surface harmonics; we may write the expression for the potential I r' j I`0 +r2P1(cos)+... 0 n + T2 Pn(cos y) + ... r' dr'd,u'd?' 1.12 which is, oI ? 3r (1 +3Eu') +? Q 2(1 1 i ..

an-3 + 12 +. 3 r n+l+l (I +n - R €u') Pn (cos y) } dµ/d4' on substituting for u' the series of harmonics, and using (22), (23), this becomes + (2n+I)rn-ln (µ, 4;) T an+1. .

which is the required potential at the external point (r, 0, 4)). 17. The Normal Solutions of Laplace's Equation in Polars.-If h 1, h2, h 3 be the parameters of three orthogonal sets of surfaces, the length of an elementary arc ds may be expressed by an equation of the form ds = 2 dh; + 2 dh3 + 2 dh3, whe r e H1, H2, H3 are H H H3 functions of h1, h2, h3, which depend on the form of these parameters; it is known that Laplace's equation when expressed with h 1 , h2, h as independent variables, takes the form a H 1 aVl + a H2 aVI a H3 (5) ah 1 ' H2H 3 / aZ 2 CH 3 H 1 a/22) + ah 3 C H1H2 n, =U.= 0. 2 In case the orthogonal surfaces are concentric spheres, co-axial. circular cones, and planes through the axes of the cones, the parameters are the usual polar co-ordinates r, 0, 4, and in this case H1 = I, H2 = r ' H3 r s n e thus Laplace's equation becomes a aV I a

a _ V) I 02V (TTr2 ar) + sin o a8 ` si n B aB + sin 2 ai - 0' Assume that V = R04) is a solution, R being a function of r only, 0 of B only, 43 of 4) only; we then have d C r2 dR) I d / s in o i d24)-0.

R dr dr O sin 0 de d0 sin 2 0. d4)2 This can only be satisfied if R dr (r2 V/4) is a constant, say n(n+I), ) is a constant, say-m 2, and 0 satisfies the equation sins 5 si n g I) s n 22 0} O -0, if we write u for 0, and µ for sin 0, this equation becomes ?(I _t`2)d + i ? 2 u (26) From the equations which determine R, 0, u, it appears that Laplace's equation is satisfied by r n cos 'n 12_1 sin un where u is any solution of (26); this product we may speak of as the normal solution of Laplace's equation in polar co-ordinates; it will be observed that the constants n, m may have any real or complex values.

18. Legendre's Equation.-If in the above normal solution we consider the case m = 0, we see that rn r-n -1 u n is the normal form, where un satisfies the equation =0, known as Legendre's equation;- we shall here consider the special case in which n is a positive integer. One solution of (27) will be the Legendre's coefficient P.(µ), and to find the complete primitive we must find another particular integral; in considering the forms of solution, we shall consider µ to be not necessarily real and between =1. If we assume as a solution, and substitute in the equation (27), we find that m =n, or - n -I, and thus we have as solutions, on determining the ratios of the coefficients in the two cases, a µ 2.2n - I µ n+2 and I (n+I)(n+2) I (n--1)(n--2)(n--3)(n--?4) R +1+ 2.2n+3 µn +3 + 2.4. 2n+3.2n+5 µn +5 + the first of these series is. (n integral) finite, and represents P.(µ), the second is an infinite series which is convergent when mod µ > If we choose the constant # to be I. 2 " n, the second 3.5...212+1 solution may be denoted by Qn(µ), and is called the Legendre's function of the second kind, thus Q12(µ) = I.2.3...n ` I ( n+I)(n+2) 3.5... 2n I }t tin+t + 2.2n+3 1.0+3+ 1 F (n+I n+2 212+3 i (28) - 3.5...2n+I An +0222 This function Q.(µ), thus defined for mod ,u > I, is of considerable importance in the potential theory. When mod µ < I, we may in a similar manner obtain two series in ascending powers of µ, one of which represents Pn(A), and a certain linear function of the two series represents the analytical continuation of Q.(µ) as defined above. The complete primitive of Legendre's equation is /1= AP. (,u) + B Qn (µ)

By the usual rule for obtaining the complete primitive of an ordinary differential equation of the second order when a particular integral is known, it can be shown that (27) is satisfied by 12ca) _1µ (µ2- 1)}P.61)}2' the lower limit being arbitrary.

(27) or 47a2 L 3 Q?-E r2 Y 1(µ, ?) ?-5 r 3 Y r 2U1, 0) ?-'..

From this form it can be shown that 2 P. (µ) log AL + I -` ? %n-1(µ)r where W n _ 1 (µ) is a rational integral function of degree n-I in ,u; it can be shown that this form is in agreement with the definition of Q n (µ) by series, for the case mod µ > i. In case mod µ < i it is convenient to use the symbol Q n (µ) for - „(µ) log TS -, 1(µ), which is real when is real and between ti, the function Q,(µ) in this case is not the analytical continuation of the function Qn(µ) for mod µ> i, but differs from it by an imaginary multiple of P n (µ). It will be observed that WI), Q n (- I) are infinite, and Qn(cc) =o. The function W n _ 1 (µ) has been expressed by Christoffel in the form n - 3.27 - Pn-3 (µ) + 2 72 -92 Pn- ° (µ) + ..., 5 and it can also be expressed in the form n Po (µ) P n -1 (µ) -}- + I P1(µ) n-2 (µ) + -1(µ) Po ( µ)

n It can easily be shown that the formula (28) is equivalent to ? J µ (2 - I I ) n+1 which is analogous to Rodrigue's expression for Pn(µ). Another expression of a similar S character is Qn(µ) - ( i)n(2n)! dµn) µ I) f J F?(µ a I

+L .

It can be shown that under the condition mod {u--I (u 2 - i) } >mod {µl l (µ -I)}, the function I/(µ-u) can be expanded in the form E(2n+I)P n (u)Q n (u); this expansion is connected with the definite integral formula for Q n (µ) which was used by F. Neumann as a definition of the function Q,(f µ), this is Qn(F%) 2 J 1Pn( u dur which holds for all values of which are not real and between From Neumann's integral can be deduced the formula dk Qn(µ) = f 1 {µ +ll (µ2 - I).cosh tfi}"+lr which holds for all values of which are not real and between I, provided the sign of 1 l (m 2 - I) is properly chosen; when µ is real and greater than 1, A / (µ - i) has its positive value.

By means of the substitution.

+-1 COI).cosh }{µ1 l 2 I).cosh x}= I, the above integral becomes Q (µ) f o {µ ? ( µ 2 - I). cosh x} dx, where xo loge This formula gives a simple means of calculating Q n (µ) for small values of n; thus Q0 JXodx = 2lo ge µ - Q1(µ) = xo - -V (m - I) .sink xo A.

µ .2log + I I - .

Neumann's integral affords a means of establishing a relation between successive Q functions, thus nQ n - (2n-)/Qn_1 +(n - I) Qn-2 - If l nPn(u)+(n-I) Pn_2 (u)-(2n - I) 2 -1 µ- u µPh-1(u) du = -2f l (2n-I)P n _ 1 (u) =0. Again, it may similarly be proved that = (2n+I)Qn. dµ dµ 19. Legendre Associated Functions. - Returning to the equation (26) satisfied by un the factor in the normal forms r n-1 n mq5

un we shall consider the case in which n, m are positive integers, and n" m. Let u = (µ - I) v, then it will be found that v satisfies the equation (I -µ 2)aµ 2 m =o. If, in Legendre's equation, we differentiate m times, we find m+2u (I - µ2) d µm2- 2(m+ I)µdµm dm-Flu (n+m+I)d mu (I m imdmu it follows that v=-Ti.--,„,,,hence c u7 _ (µ -I) d m

The complete solution of (26) is therefore u= (µ - I) m A d? + d µn when µ is real and lies between i, the two functions (I - µ2) ?sm d" L Pn (µ), (1 - µ2)1md Q n (A) dµ dµ are called Legendre's associated functions of degree n, orthr m, of the first and second kinds respectively. When not real and between i, the same names are given to the functions (µ2 -I).imd ?p(µ) diurn in either case the functions may be denoted by P7(1.0, Q702). It can be shown that, when is real and between i (-O -± ]tm do 2 (n m)! I -µ) dµn{(µ- I)n +m(µ +I)n -m} _ I /1 -µ #m dn1 2 (n-m)! I+µ) dµn{(µ - I)n - 'n(µ+I)n+m}. In the same case, we find Pn (cos 0)-2(m+ I) cot 0 P7 (COS 0) +(n-m) (n+m+i)Pn (cos 0) =o, (n -m+ 2)P;+ 2 (cos 0)- (2n +3)P.Ki_i(cos 0) -(n+m+i)Pn (cos 0) =o. 20. Bessel's Functions. - If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are z, the cylindrical co-ordinates; in that case H 1 =1, H2 = i, H 3 = i /p, and the equation (25) becomes a a V I aV i a2v a2, 2 + ap2 +p ap + p = 0.

To find the normal functions which satisfy this equation, we put V=ZR4, when Z is a function of z only, R of p only, and 'F of 4), the equation then becomes I d Z i (d R i dR i i d24) dz2 + R apz + A dp -173, 24 -) 4 2 =0' d2Z That this may be satisfied we must have 2 dz2 constant, say = k2,. d2Z El constant, say = -m 2 , and R, for which we write u, must satisfy the differential equation d2u dp2 + p 4.-r- ( k2 P2) u=o, it follows that the normal forms are e  ? 1 m(u k  ?. p ), where satisfies the equation l dp2 p dp+(I - p2)u=o.

This is known as Bessel's equation of order m; the particular case d u i du apl P d p +u = O, corresponding to m =o, is known as Bessel's equation.

If we solve the equation (29) in series, we find by the usual process that it is satisfied by the series p 2

p I 2.2m+2 2.4.2m+2.2m+4 - the expression p p2 2"11(M) 2.2m+22.4. 2m+2

2m+4 or (_ I) npm+2n 2 H (m +n) II (n) is denoted by Jm()

n=o, When m=o, the solution 2 2+ 2 2.4 2-

of the equation (30) is denoted by Jo(p) or by J(p).

Qn(µ) = 2nn!

and .. µ is two t I.

(29) (30) The function J, n (p ) is called Bessel's function of order m, and Jo(p) simply Bessel's function; the series are convergent for all finite values of p. The equation (29) is unaltered by changing m into -m, it follows that J_ m (p ) is a second solution of (29), thus in general u = AJm(p) + BJ-m(P) is the complete primitive of (29). However, in the most important case, that in which m is an integer, the solutions J_m(p), J,n ( p ) are not distinct, for J -, n (p ) may be written in the form m- (p) -rn (- I) n p 2n II(n - m)II(n) ( 2) n=0 +(_ I)m 2 m II(m+p)II(P) 2) P=o now II(n -m ) is infinite when m is an integer, and n< m; thus the first part of the expression vanishes, and the second part is (- I) m J m (p), hence when m is an integer J_ m (p) _ (- I) m J, n (p), and the second solution remains to be found.

Bessel's Functions of the Second Kind. - When m is not a real integer, we have seen that any linear function of Jm(p), J-m(P) satisfies the equation of order m. The Bessel's function of the second kind of order m is defined as the particular linear function In a similar manner it can be proved that dP J. P(P) =2PpP-m d(P2)P {PmJm(P)}

From the definition of Y m (p), and applying the above analysis, we prove that Ym+P(P) _ ( -2) Ppm+P d (p2)P 3-MY Y =2PpP_"`d(PP)P{P"`Yr(P) As particular cases of the above formulae, we find J (- 2 P) P d(p2)P Jo(P), Yp(P) _ (- 2P) Pd(p2)PYo(P) J1 ( P) _ - dj dp p) ' Y I(P) = d dpp)

22. Bessel's Functions as Coe f ficients in an Expansion. - It is clear that ei p OOs =e, or 't =e« satisfy the differential equation (31), hence if these exponentials be expanded in series of cosines and sines of multiples of 4 ), the coefficients must be Bessel's functions, which it is easy to see are of the first kind. To expand e Lp sin 0, put e4' =t, we have then to expand elP(t - t-1 ) in powers of t. Multiplying together the two absolutely convergent series and -m ( P ) cos m7r. jm(p) sin 2mir m i t-1 P t m 2p _ m! 2, -(t m m 2p m, ipt e = and may be denoted by Y m ,(p). This definition has the advantage of giving a meaning to Yr(p ) in the case in which m is an integer, for it may be evaluated as a limiting form o/o, and the limit will satisfy the equation (29). The only failing case is when m is half an odd integer; in that case we take cos'm2r. Yr(p ) as a second finite solution of the differential equation.

When m is an integer, we ` have Y m(p) = (- m dJ _rn_ I)mdjd „ on carrying out the differentiations, and proceeding to the limit we find m(P) = Jm(P) log 3 -1-1 - 2 ( 122) n =o where t(n) denotes II'(n)/II(n). When m=o we have the second solution of (30) given by (- I)nt(n) ( p II(n)II(n) 2/ 0 21. Relations between Bessel's Functions of Different Orders. - Since ez sm mcp

um(p) satisfies Laplace's equation, it follows that

sin (p ) satisfies the differential equation 02, 32u axe ay. (31) The linear character of this equation shows that if u is any solution af is also one, f denoting a rational integral function of the operators. Let,, denote x+ty, x - cy, then since p m T m um (A n ) satisfies the differential equation, so also does m ' or d(P2 ) P i P 71 um(P) h thus we have u m+P C p m+P d (d p P 2) P where C is a constant. If um(p) = Jm(p), we have um+P=Jm+P(P), and by comparing the coefficients of p m+P , we find C = ( -2) P , hence Jm+P(P) _ (-2)Ppm+P d(P2)P !P mJr(P)), and changing m into -m, we find JP -m (p) _ - 2) PpP - md (P2 P) P iJ IP m J - m (P) i 1

we obtain for the coefficient of t m in the product ?j m 2 m m! (I 2.2M+2 + 2.4.2m+2.2M+ or J (P), hence e2p(1 - 1-1) = Jo(P) +tJi (P) + ... +t m J m (p) + ... l? - t-111(p)+...+(- I)mt - mjm(P) /+t m Jm (P) the Bessel's functions were defined by Schliimilch as the coefficients of the powers of t in the expansion of e2P('- t 1), and many of the properties of the functions can be deduced from this expansion. By differentiating both sides of (32) with respect to t, and equating the coefficients of t m - 1 on both sides, we find the relation Jm-1(P)+Jm+1(9) 2p2Jm(P), which connects three consecutive functions. Again, by differentiating both sides of (32) with respect to p, and equating the coefficients of corresponding terms, we find 2 d Jm(P) -Jm 1(P)- Jm+1(P)

dp In (32), let t=e" , and equate the real and imaginary parts, we have then cos ( p sin 4))=Jo(p)-1-2J2(p) cos 24)+2J 3 (p) cos 30+.. sin ( p sin 4,) =2J 1 (p) sin 4)+2J3(p) sin we obtain expansions of cos ( p cos 0), sin ( p cos 4)), by changing into z -4). On comparing these expansions with Fourier's series, we find expressions for J m (p) as definite integrals, thus Jo ( p) _ J 0 cos ( p sin 0)d4, Jm(p) =- 51 0 7 cos ( p sin 4>) cosm4dq (m even) Jm(p) = 7, f 0 sin ( p sin 4)) sin mod¢ (m odd).

It can easily be deduced that when m is any positive integer J m (p) _ 7, 1 - 0 7 cos (mq- p sin 4)d4). 23. Bessel's Functions as Limits of Legendre's Functions. - The system of orthogonal surfaces whose parameters are cylindrical coordinates may be obtained as a limiting case of those whose parameters are polar co-ordinates, when the centre of the spheres moves off to an indefinite distance from the portion of space which is contemplated. It would therefore be expected that the normal forms ±,(z Jm ( Xp)y:m4 ) would be derivable as limits of r _ n n 1 Pn (cos 0)s m4 ), and we shall show that this is actually the case. If 0 be the centre of the spheres, take as new origin a point C on the axis of z, such that OC =a; let P be a point whose polar co-ordinates are r, 0, (1> referred to 0 as origin, and cylindrical co-ordinates p, z, 4) referred to C as origin; we have p=r sin 0, z= r cos 0-a, hence 0) n Pn(cosO) =sec n 6 (i+ !) n Pn(cos 0). Now let 0 move off to an infinite distance from C, so that a 'becomes Yo(P) =Jo(P)log P + um (P)) [i(n) - 1 - t(m+n)1 IIn(m ? -jII(n) (2) 2' n =0 m ( 2)-m?II(In ) 1) 2n I( 2 (32) infinite, and at the same time let n become infinite in such a way that n/a has a finite value X. Then ?a L sec' B=L (sec /I) L (1 + 1) = e? Tz and it remains to find the limiting value of P n (cos 0). From the series (15), it may be at once proved that (n+i)n ( 9) 2 P n (cos 0 ) I - 12 sin 2 -?- .. .

n=  It may be shown that Yo ( p) is obtainable as the limit of Q n (cos the zonal harmonic of the second kind; and that  Ym(P) =Ln m Q (cos  n). 24. Definite Integral Solutions of Bessel's Equation. - Bessel's equation of order m, where  m is unrestricted, is satisfied by the expression  pm f etP t (t 2 - i) m-l dt, where the path of integration is either a curve which is closed on the Riemann's surface on which the integrand is represented, or is taken between limits, at each of which  e L P t (t 2 - Om-Hi is zero. The equation is also satisfied by the expression f  e 1t - t-1) t - m - 4 dt where the integral is taken along a closed path as before, or between limits at each of which  e1p(t-t-1)t-m-1  vanishes. 
 

26. The Asymptotic Series for Bessel's Functions. - It may be shown, by means of definite integral expressions for the Bessel's functions, that  Jr(P) = P cos ( +-) -{- Q sin (-/212-r-1:5--p)  Ym(P) = N/2pem7n sec mir j P sin (?+ -P) -Q cos (2  ?+ 4  - p) where P and Q denote the ser
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Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Spherical Harmonics'. 1911 Encyclopedia Britanica. https://www.studylight.org/​encyclopedias/​eng/​bri/​s/spherical-harmonics.html. 1910.
	








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