SECTION III. 



ON THE DEFINITE INTEGRAL OF THE PRODUCT OF 

 TWO LEGENDRE'S COEFFICIENTS. 



1. LET V = - = (l-2nr+i*)~*. 



Then - = 1 + Pp + P.r + &c. + P tt i " + &c. , 



P 



and VP m = ^? = P M + P^r + P.P m /~ + &c. + PJV" + &c. 



P 



Integrating from ^ = to ju,= l, we get 



f VP m dp. = I' P> " dp = PP,,,^ + r fpfjlp +... + r n \ l P m PJb + &c - 



Jo Jo p Jo Jo Jo 



f 1 pp 



Hence P n P m dit is the coefficient of r' 1 in the expansion of -dp. or 



Jo Jo p 



r dx 1 1 



P m dx. where u. = x + -(lx 2 ) and ^- = - j = - ; (see p. 245) 



4* (l-2^r + r^ /> 



x is the value of x when /A = ; 



fl 



P m P n du, is the coefficient of r m r" in the expansion of 

 Jo 



f 1 



J o ( 1 - 2 



__ 



( 1 - 2r/u, + r 1 )* ( 1 - 2r^ + -r,") 4 ' 

 in powers of r and ?v 



2. Now if P,/ be what P m becomes when x is substituted for /x, 



we get 



r clP x 



P -P X A-- l\ -r^ ~^- 4- (- 



f m -r m +l x h 



