234 Methods for the Solution of Special Problems [OH. vm 



m 



or, multiplying by (1 yu, 2 ) 2 , 



(193). 



^ 3m+n 



(i-,,) 



U 



Then - 



3/4 8^ +n+2 a//, 



mu 

 (m + n + 1) (n m) -I- m ^ ^ , by equation (193), 



Thus v satisfies 



and this is the same as equation (188), which is satisfied by . 

 274. The solution of equation (188) has now been seen to be 



where - il = AP n + BQ n . 



Hence & = A (1 - 



The functions 



are known as the associated Legendrian functions of the first and second 

 kinds, and are generally denoted by P^(^)> Qn(^)' A S regards the former 

 we may replace P n , from equation (159), by 



1 d n 



