OF TWO LEGENDRE'S COEFFICIENTS. 353 



This should evidently be the case since 



m P n + S n P m ) dp S m S n ; 



and since m n+l, one of the quantities m and n will be even and the 

 corresponding quantity S m or S n will be of odd dimensions in p, and will 

 therefore vanish when fj. = 0, and both S m and S n vanish when p, = 1 ; there- 

 fore S m S n taken between the limits vanishes, 



or 



I (S m Pn + S n P m )dfts=0, or S m P n dfj.= I S n P m diJ., as above found. 



Jo Jo Jo 



11. Now take the more general case in which m>n+l. We have 



mn 



Am + l 



here m 1 and n are not the same quantities, therefore by what we have 

 proved, both these definite integrals vanish unless one of the quantities 

 m + 1 and n is odd and the other even, i. e. unless m and n are both 

 even or both odd. 



First suppose m and n to be both even and m > (n + 1 ). Then by 

 what has been before proved we have 



/: 



2r) ! 



-' 



/ 



for all values of r from to - and 0! = 1. 



Similarly 



_, 



Now generally - *, f-^- r^ +!=?: 



J 4 m \ m \ 2m 



I- rll \-r+l] M'+l 



\2 / \2 / 2 



Hence 



m+n 



(-1)"" 2 (m-2?--2)! (rt + 2r)! ,_, 



? m Pdp = ^r- 2 1 ; ^ . \ , n r- -(7). 



A. II. 45 



