where hifh...h> are the axes of Yn. Two harmonics of the same
degree are said to be conjugate, when the surface integral of their
product vanishes; if Yâ€ž, ZÂ» are two such harmonics, the addition
of conjugacy is

Mi?**) 2 -****- -

Lord Kelvin has shown how to express the conditions that 2Â« + l harmonics of degree n form a conjugate system (see B. A. Report, 1871).

16. Expansion of a Function in a Series of Spherical Harmonics. â€” It can be shown that under certain restrictions as to the nature of a function F(m, <t>) 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(/u, <p) be represented by

_ VoU^+ViU *) + ... +V.&*, *) + ...; change n, <t> into n', <Â£' and multiply by

we have then

Pâ€ž(cos 9 cos 0'+sin 9 sin 9' cos <t>â€”<t>'),

J " J F(ji', tfOPnCcos 9 cos 0'+sin 9 sin 9' cos <t> â€” <t>')dii'd<t>'

= j Â«* ( *VÂ»0i', â– *>')?,, (cos e cos 0'+sin 9 sin 9' cos <t> â€” <j>')dti'd<l>'

= -4=-Vâ€ž(Â», *), 2n + i '

hence the series which represents F(/n, tj>) is

CO

i*^ (2n + i) f"Jlf(l*', <t>')Pn(cos COS B'

+sin 9 sin 0' cos <j> â€” 4>')dii'd<j>'. (24) A rational integral function of sin 9 cos <f, sin 9 sin <j>, cos 9 of degree n may be expressed as the sum of a series of spherical har- monics, by assuming

/â€ž(*, y, z) = Yâ€ž+rÂ»Y_+r<Y_ < + ... and determining the solid harmonics Yâ€ž, Yâ€ž_ 2 , . . . and then letting r= 1, in the result.

Since V , (> a *YÂ»_j,) =2s{2n â€” 2s + i)r 1 '~ i Y n -u, we have

V 2 /Â» = 2(2n-i)Y^ 2 +4(2Â»- 3 )r 1 Y^+6(2n-5)^Y n _,+ . . . vyÂ» = 2.4(2n-3)(2Â»-5)Y_,-|-4.6(2Â»-5)(2n-7)r 2 Y^,-l-. . . the last equation being

Vfn = n(n + i)(n â€” 2)(Â» â€” 1). . .Y , if n is even, or

V" _1 /Â» = (m â€” i)(Â»+2)(nâ€” 3)n. . .Yi, if Â» is odd from the last equation Yo or Yi is determined, then from the pre- ceding one Y 2 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{l-\-tu), the body being made of homogeneous material of density unity, and u being a given function of 9, <t>, the quantity e being so small that its square may be neglected. The potential is given by

jrA/o^V+r' 2 -^' cos Tl-'rfr'W,

where y is the angle between r and r' \ now let u' be expanded in a series

Y (m', *')+Y,(m', *') + ... +Yâ€ž(/, Â«') + â€¢â– â€¢ of surface harmonics; we may write the expression for the potential

J0J-J0 |-+pPi(cos T ) + ...

+ 0i?n{cos y) + ... J r'Hr'du'd*'

which is,

/;-/â– .

u

1 o'

\f r (i+Mu')+~y 2 d+W)P l +...

+^3 ^(i+Â«+3Â«Â«')PÂ»(cos t) I <W

on substituting for u' the series of harmonics, and using (22), (23), this becomes

4JTIJ'

L3 f ( 3'

yiYiU+J+^Oi, *) + ...

+ (2n ; n ;; rn - 1 Y n (M,^)+...i]

) / h, dV\ d 1 h 2 gy\ a / h 3 ay\ _ n , .

which is the required potential at the external point (r, 9, <t>).

17. The Normal Solutions of Laplace's Equation in Polars.- â€” If hi, hi, 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 â€” rr^dh 2 + rndh\ + rndh\, where Hi, H 2 , H s are

functions of hi, h\, h s , which depend on the form of these parameters; it is known that Laplace's equation when expressed with hi, hi, h as independent variables, takes the form

In case the orthogonal surfaces are concentric spheres, co-axial, circular cones, and planes through the axes of the cones, the para- meters are the usual polar co-ordinates r, 9, 4>, and in this case

Hi = i, H 2 =-, H 3 = â€” , â€” - thus Laplace's equation becomes

r r sin 9 * ^

&_ 1 2 sv\ i 9 ( . dv\ , i_ m

dr V dr) + sin d9\ Sm 69 J + sin 2 d^ Assume that V = Re* is a solution, R being a function of r only, 9 of 9 only, * of <j> only ; we then have

1 d ( 2 dK\ , 1 d I â– A<S\ , 1 <P4>_ n Rdr\ r dr) ^esm9d9 \ smv de) + sin 2 .* d<t?~^

This can only be satisfied if -tt-j- ( f-r- ) is a constant, say Â«(n + l), X-JT5 is a constant, sayâ€” m 2 , and satisfies the equation

if we write u for 9, and ft for sin 0, this equation becomes

From the equations which determine R, 9, u, it appears that Laplace's equation is satisfied by

r n cos . _ _*_, â€¢ nob . u m r^ 1 l sin T Â»

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. â€” If in the above normal solution we

consider the case m = 0, we see that

r n

Â»-n~lWn

(27)

is the normal form, where Â«n satisfies the equation

i! (i -* 2) S+ M(M+i)M=0 ' â€¢

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 Pn(/u), and to find the complete primitive we must find another particular integral ; in considering the forms of solution, we shall consider /a to be not necessarily real and between Â±1. If we assume

U =--n m + aiti m - 2 + a i n m ~ i + â– . . as a solution, and substitute in the equation (27), we find that m =Â«, orâ€” n â€” I, and thus we have as solutions, on determining the ratios of the coefficients in the two cases, 1

1 LÂ»_J1(!LZI) M Â»Â«4....[ C 2 . 2nâ€”i* ' )

and

( 1 . (w + i)(n+2) 1 , (n+i)(Â»+2)(Â»+3)(Â»+4) 1 , J

P I ^n+lT 2 2 n+3 ^" +3 " r 2 . 4 . 2Â»+3 . 2W+5 |i"+ 5 ^ ' " ' J

the first of these series is (Â« integral) finite, and represents PÂ«(m), the second is an infinite series which is convergent when mod n > 1.

If we choose the constant /3 to be

1 â– 2 â– 3 â–

the second

3.5... 2Â» -resolution may be denoted by QÂ»(m), and is called the Legendre's function of the second kind, thus

Q.G0

1-2. 3

3-5- . .n { 1

- n + i I /i n+

(n+l)(n+2) 1

^+5+--- \

â€žn+l I 2.2W-(-3 y.

1.2.3 . . .n _ 1 r- f n+i n+2 2n+3

P ( n+i n+2 2n+3 i\ , â€ž

3.5...2W+I m" +1 \ 2 ' 2 2 '/*Â»/â€¢ k Â°>

This function QÂ»(m), thus defined for mod n > I, is of considerable importance in the potential theory. When mod ju < I, we may in a similar manner obtain two series in ascending powers of n, one of which represents PnOu), and a certain linear function of the two series represents the analytical continuation of Q..(m), as defined above. The complete primitive of Legendre's equation is

M^AP.OO+BQnGO.

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

P.W1 2 '

the lower limit being arbitrary.