270] Spherical Harmonics 231 



and this remains true however many of the roots a sy a 4 ... coincide among 

 themselves, so long as they do not coincide with the root a l8 Thus, in 

 expression (185), the value of c s is 



9 ( (a-arf 



Putting 



we find that 



= a [ i 1 a f i ] 



8/i ((1 - /* 2 ) { 0*)} 2 J M=ag 9 s 1(1 - etf) { (a.)}'} 

 Since (//, - a g )R(fj,) is a solution of equation (182), we find that 



On putting p = a S) this reduces to 



|- f(l - tf) Ji (.)} + (1 - .) ^- } = 0, 

 giving, on multiplication by R (a,), 



Hence c s = 0. 



Equation (185) now becomes 



i . s 



so that, on integration, 



f dp . , IJL + 1 ^ d s 



J^(^- 1) {P n (/z)} 2 = * g ^1 + y^cT/ 



On multiplying by P n (fi\ we obtain from equation (184), 



where T^-! is a rational integral function of JJL of degree n 1. 



It is now clear that Q n (/&) is finite and continuous from /A = 1 to /* = + 1, 

 but becomes infinite at the actual values /*= + !. 



To find the value of "R^ we substitute expression (186) in Legendre's 

 equation, of which it is known to be a solution, and obtain 



W log 



