270-273] Spherical Harmonics 233 



The solution of the former equation is single valued only when K is of the 

 form m 2 , where m is an integer. In this case 



<E> = C m cos m<f> + D m sin m^>, 

 and <*) is given by 



or, in terms of /JL, 



an equation which reduces to Legendre's equation when m = 0. 



273. To obtain the general solution of equation (188), consider the 

 differential equation 



........................ (189), 



of which the solution is readily seen to be 



f#) n .............................. (190). 



If we differentiate equation (189) s times we obtain 



If in this we put s = n, and again differentiate with respect to /x, we 

 obtain 



d n z 

 which is Legendre's equation with ^ as variable. Thus a solution of this 



equation is seen to be 



& - (!f <>-*> 



giving at once the form for P n already obtained in 249. The general 

 solution of equation (192) we know to be 



If we now differentiate (192) m times, the result is the same as that of 

 differentiating (189) m + n + l times, and is therefore obtained by putting 

 1 in (191). This gives 



