253-255] Spherical Harmonics 219 



The result is 



all products of the form P n P^. vanishing on integration, by equation (167). 



r+i 



Thus ]}*df* is the coefficient of h in 

 J -i 



J _! 1-2V + & 2 ' 

 i.e. in *wl^. 



and this coefficient is easily seen to be 

 We accordingly have 



255. We can obtain this theorem in another way, and in a more general form, by 

 using the expansion of 240, namely 



( JFP 8 (cos6)dS, 



where 6 is the angle between the point P and the element dS on the sphere. This 

 expansion is true for any function F subject to certain restrictions. Taking F to be a 

 surface harmonic S n of order n, we obtain 



' ] n P 8 (cos 0) dS 



all other integrals vanishing by the theorem of 237. Thus 



or s n PMda> = (S^ =l .............................. (169). 



This is the general theorem, of which equation (168) expresses a particular case. To 

 pass to this particular case, we replace S n by P n (fi) and obtain, instead of equation (169), 



(P n (,*)} sin 6d6dj> = P n (1), 



or, after integrating with respect to <, 



agreeing with equation (168). 



