488 THE PRODUCT OF ANY TWO LEGENDRE'S COEFFICIENTS [60 



Now -Pi = p, 



pp "--r -L p n p 



" 2TO+1 2n + l 



p_3 p 1 



Again, we have 2 ~ 2 ^ "~ 2 ' 



.-. P.P^-l/iP.P.-ip,, 



3 n+l .3 _ p _1 p 



1^ "- 1 2 *' 



Substitute for pP n+1 and /nP,,,, their equivalents obtained by writing 

 n+l and n 1 successively for TO in the above formula, 



PP _3 ( 

 B ~ 



2(2+l)(2w 



_ 3 



_ _ 



2 (2 + 1) (2w +3) 22 (2 - 1) (2w + 1)J 



(n-l)n 



"- : 



By a slight reduction the coefficient of P n becomes 



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



_3 

 H ~2(2n+l)(2n n 



I 3 (n-l)n 

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



Again, putting n = 2 in our original formula, we have 



PP --uPP --^PP 



-^a-Ln o r 1 2-* n n j -i- r n 



_5 (TO+l)(TO + 2) 



^ n 



5 (TO-I)TO p 2 TO+! p 2 TO 



1 "2(2TO-1)(2TO+1)' 1 - 2 32TO+1 " +1 32TO+1 " 



