267-270] Spherical Harmonics 229 



or, multiplying by 7J and rearranging, 



or 



On integration this becomes 



(1 yit 2 ) PT? = constant. 

 We may therefore take 



in which the limits may be any we please. If we write 



the complete solution of equation (182) is 



269. The two solutions ^ and Q n can be obtained directly by solving 

 the original equation (182) in a series of powers of /x. 



Assume a solution 



S n = b p r + b^ 1 + 6 2 /Lt r+2 + . . . , 



substitute in equation (182), and equate to zero the coefficients of the 

 different powers of JJL. The first coefficient is found to be b r(r 1), so 

 that if this is to vanish we must have r = or r 1. The value r = leads 

 to the solution 



n(n + l) (n-2)n(n + l)( + 8) 

 1.2 ^ 1.2.3.4 -< 



while the value r = 1 leads to the solution 



(n-l)(n + 2) . (n-3)(n-l)(n + 2)(n + 4) 

 ^~ / "" 1.2.3 ^ 1.2.3.4.5 "^ 



The complete solution of the equation is therefore 



If n is integral one of the two series terminates, while the other does 

 not. If n is even the series u terminates, while if n is odd, the terminating 

 series is u lf But we have already found one terminating series which is 

 a solution of the original equation, namely R. Hence in either case the 

 terminating series must be proportional to P n , and therefore the infinite 

 series must be proportional to Q n . 



270. We can obtain a more useful form for Q n from expression (184). 

 The roots of ^(/*) = are, as we have seen, n in number, all real and 



