146, 147] DEVELOPMENT IN SPHERICAL HARMONICS. 401 



and applying it to the case that F is a spherical harmonic 



fiv 

 V m = r">Y m , |I = _ mr - ir,,, 



we obtain, since 



118) F M (P) - 



00 



If the coordinates of P he r f , -9-', cp', we have, 



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

 definite integrals as coefficients. Since the equality must hold for 

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



119) o o 



*m(&> ( P') = 



o o" 

 that we have for the values of the integral 



120) / IY m (&, <p)P m ([i)sm&d&d(p = ^r^Y n 







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

 obtained by spherical trigonometry, 



121) IL = cos (rr') = cos # cos &' -f sin # sin &' cos (<jp qp'). 



By means of the above integral expressions, 119) and 120) we 

 may find the development of a function of #, qp, assuming that the 

 development is possible. Suppose we are to find the development 



122) /(#, 9 >) = r + r 1 + y 2 +.... 



Multiply both sides by P n (p) sinftdftdcp, and integrate over the 

 surface of the sphere and since every term vanishes except the n ih 

 we obtain 



Tt iTt 



123) fff(, V) P.GO sin 9 d dy = ^ Y n (', p' 







124) Y n (', J) = ~f(^, 9) P W sin * d d<p. 





 , Dynamics. 26 



