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



Whence by substituting the values found above for P 3 P n and P 2 P n 

 and again for ftP n+3 , pP n+l , &c., we obtain 



5.7 (n+l)(n + 2)(n + 3) [n + 4 n + 3 p \ 



4 n ~2.4 (2n + l)(2n + 3)(2n + 5)\2n + 7 T 2n + 7 + 'J 



3. 7 ro(n+l)(n + 2) f w_+2 p n + !_ p \ 



~3 "J 



'2.4 (2n-l)(2n+l)(2n + 5)[2n + 3 2n + 



3. 7 (n-l)n(ro+l) f n p __ n-1 



' 2 . 4 (2n -^y (2n + 1 ) (2 + 3) \2w - 1 " 2n-l 



5^ (n-2)(n-l)n fra-2 p ~-3_ p 1 

 "2.4 (2n-8)(2n-l)(2+l) \2n-5 "" : 2?i-5 "~ 4 J 



3.3 



3 - 3 _ (n-l)n 



4 (2n- l)(2n + 



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



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



2 2n 



Similarly, the coefficient of P, 4 _ 3 becomes 

 5 ro-2n-l 



_ 

 2 (2w - 5) (2 - 1) (2n + 1 ) (2n + 3) ' 



and finally, the coefficient of P n becomes 



... p 



' n 



Hence, collecting the terms, we have 



1.3.5.7 



1.2.3.4 (2w+l)(2w + 3)(2n + 5)(2n+7) 



1.3.5 1 n(n+l)(n + 2)(n + 3) p 



+ 1.2.3 ' 1 (2 - 1) (2n + 1) (2w + 3) (2w + 7) " 



1. 3 1 . 3 _ (n-l)n(n+l)(n + 2) 



h ' 



+2 



-2 



.2 (2n-3)(2n-l)(2w 



1 1.3.5 (n-2)(n-l)n(M+l) 



h l ' 1 . 2. 3 (2-5)(2w-l)(2n+l)(2 + 3) * 



1.3.5.7 (w-3)(n-2)(n-l)n p 



h I . 2. 3. 4 (2n-5)(2n-3)(2n-l)(2n+l) n ~" 



