60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 489 



Substitute for ^P n ^, pP n and ju,P n _ s their equivalents as before, 



_5 p 



n n+ 



f 5 (n+l)(n + 2) n + 2 5 n(n + l) n + l _ 2 n + l} 



\2(2n+l)(2n + 3)2n + 5 3(2w-l)(2w + 3)2n+l 32w+lJ n+l 



{5 n(n+l) n 5 (n l)n n l 2 n \ r> 



3(2n-l)(2n + 3)2n+l 2 (2w-l)(2n + l) 2w-3 ~ 3 2?i+ 1J *"' 



p 



By reduction the coefficient of P n+l in this expression becomes 



3 n (n+l) (n + 2) 



2 (2w-l)(2n+l)(2w + 5)' 



and similarly the coefficient of P n _ 1 becomes 



2 (2n-3)(2n+l)(2n+3)' 

 Hence we have 



pp _5 (n-ri^ v'tT^M'"" 1 ""/ p 

 ^^ n ~ 2 (2w+l)(2 + 3)(2w + 5) 



3 n(n+l)(n+2) p 

 3 (n-l)n(n+l) p 



\ ^ / c^ ii\/o^iO\ n - 1 



- p 



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



Again, since "^ >4 = 4 / A -^ > "~4^ >2 ' 



we have PtP n = ^P- (PaP n ) ~^\" *"*)' 



62 



