192 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



If the coordinates of P be r', #', <' we have, 



while on the right we have an infinite series in powers of /, with 

 definite integrals as coefficients. Since the equality must hold for 

 all values of r' less than r, we must have, collecting in terms in r' s 



Y m (6, 6) P s O) sin 0d0dd>=0, s^ m, 



o o 



(30) 



m 4- <? -1- 1 r /*2T 



' 



r /*2T 



Y m ( 



j o J o 



so that we have for the values of the integral 



fir [Sir 4_ 



(31) J o J o Y m (0, *) P m (A) sin d0d<f, = ^j- Y m (ff, f ). 



In performing the integration, we must put for //, the value 

 obtained by spherical trigonometry, 



p = cos (r/) = cos cos & + sin 6 sin ff cos (< <$>'). 



By means of the above integral expressions (30) and (31) we 

 may find the development of a function of 0, $, assuming that the 

 development is possible. Suppose we are to find the development 

 (32) 



Multiply both sides by P n (/i,)sin#c0d<, and integrate over 

 the surface of the sphere, and since every term vanishes except the 

 nth we obtain 



(33) /(* *> P " 0*) si 



__ 

 (34) F M (0', f ) = /(tf, 0) P n Ot) sin 



Accordingly to find the value of any term T n at any point 

 P, (0', <') we find the zonal surface harmonic whose axis passes 

 through the point P, multiply its value at every point of the 

 sphere by the value of / for that point, and integrate the product 

 over the surface. It remains to show that the development is 

 possible, that is that the sum of the series 



~ 2 (2 + l)j"J*Jf(0, <) P n Gt) sin 6d6 d$, 



actually represents the function f(0', $'). This theorem was 

 demonstrated by Laplace, but without sufficient rigor, afterwards 



