on integrating the expression k times by parts, and remembering | hence
that (m 2 — i) n and its first » — I derivatives all vanish when n= ±i,
the theorem is established. This theorem derives additional
importance from the fact that it may be shown that AP„(m) is the
only rational integral function of degree n. which has this property ;
from this arises the importance of the functions P„ in the theory of
quadratures.
The theorem which lies at the root of the applicability of the functions P„ to potential problems is that if n and n' are unequal integers
f 1
P„,Gu)PnM<^-0,
(19)
to
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' <n, P„'(m) is a linear function of powers of m< whose indices are all less than n.
When n' =n, the integral in (19) becomes I {P„Gz)) ! <2m
evaluate this we write it in the form
on integrating n times by parts, this becomes
fel£f > - O-jgiO*- 0*M. or SD 1 -"»"". which on putting
u = -(i—p, becomes 2 \ '\ t u n (l— u)"du,
aence
/:
jP.00Nm =
2n+r
(20)
14. Expansion of Functions in Series of Legendre's Coefficients. — If it be assumed that a function /(/») given arbitrarily in the interval /i=— I to +1, can be represented by a series of Legendre's co- efficients oo+OiPi(m)+12P2(m)+- • -+a n P n (n)+. . .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
(n + n') does in the theory of the expansion of
On multiplying the series
Vo=2™2^P„(m)J^/U')P«UVm'
/:
COS COS '
unctions in series of circular functions, by P-. (ju), we have
hence
t fj PnOx) fdp = f^b) P.O.)^
a»=-
2n +
hence the series by which /(/») is in general represented in the interval is
-P.to/^/GOP.O'W-
(21)
The proof of the possibility of this representation, including the investigation of sufficient conditions as to the nature of the function f(ji), 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 investi- gation of this matter is given by Hobson, Proc. Land. 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 /(^i) placed on a spherical surface r = a. If
v.=.z\j£iP.OO, Vo=sA n ^jP„G,),
we see that Vt, V 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 /dVi dV a \ °^[-d7~dr-) r = a >
hence
<r = — — 22(2M-fi)A n Pn(A*); on comparing this with the series (21),
- have A„ = 2*0^/0*') P.OOV.
are the required expressions for the internal and external potentials due to the distribution of surface density /(m).
15. Integral Properties of Spherical Harmonics. — The fundamental harmonic property of spherical harmonics, of which property (19) is a particular case, is that if Y»(x, y, z), Z n >(x, y, z) be two (ordinary) spherical harmonics, then,
//
Y n (x, y, z)Z n ,{x, y, *)<2S=0,
(22)
when n and n' are unequal, the integration being taken for every element <2S of a spherical surface, of which the origin is the centre. Since v 2 Y» = 0, v 2 Z»/ =0, we have
JJJ(Y„v 2 Z n ,-Z^v 2 Y„,)dK^dz = 0,
the integration being taken through the volume of the sphere of radius r ; this volume integral may be written
jJJ I Tx V Y ""aF - ln '~dx~) +dy V'ly- ~ Zn '~dy~)
+ s ( Y 'H9F _z "4r) t 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 ) !«-«;
on using Euler's theorem for homogeneous functions, this becomes
n ^/)Y^dS = 0,
whence the theorem (22), which is due to Laplace, is proved.
The integral over a spherical surface of tfee product of a spherical harmonic of degree n, and a zonal surface harmonic P„ of the same degree, the pole of which is at (x', y', z') is given by
JjY„(*, y, z)P n dS=^~a^Y n (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
I (T I i Vn( - 9, *)P»( C0S e cos 0'+sin 9 sin 0' cos </> — <(> ')dij.d<j>
To prove the theorem, we observe that V„ is of the form n OoPnU) +2(o m cos m<t>+b m sin m<f)P n (/») ; i- to determine do we observe that when n = 1 ,
p.rjO-i.PrGO-0;
hence Oo is equal to the value V„(0) of V„ (8, </>) at the pole = of Pn(ju). Multiply by P„(m) and integrate over the surface of the sphere of radius unity, we then have
4?r = ^ZL
7^0
- V„(0),
2M+I ° 2n+i
if instead of taking y. = I as the pole of P»(ju) we take any other point (m', <t>') 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
ffY n{ x, y, z)f(x, y, z)dS = ^R^ T 0^, \ i +J ^
+;
R'V 4
+3
Y '(h'£? B fix ' y>s)
2-4- 2 "+3-2»+5 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. Land. Math. Soc. vol. xxv.).
The following case of this theorem should be remarked: If fn(x, y, z) is homogeneous and of degree n
//¥.(*, y, z) M x, y, s)d s^R*^0^Y.(§- x . |. |)/.(*,y,i)
if f n (x, y, z) is a spherical harmonic, we dbtain from this a theorem, due to Maxwell (Electricity, vol. i. ch. ix.),
Y.(*. y, «)/„(*, y, *)«*S— sj+r - x Midht ,, dh Mx, y, z)
