OF TWO LEGENDRE'S COEFFICIENTS. 347 



If m = n, the last term becomes 



By integration by parts 



(2n + 



Continuing this process we find 



i\-t JL , / 1V . 2n(2n-2) ... 4.2 



' (2n + 1) (2i-l) ...5.3^' 



Hence between the limits and 1 



(u' - 1 Ydu. 1 V 2n(2n-2)...4.2 



^ ' V " ' 2-l)...5.3' 



{2"!} 2 1 



(2n + 1 ) ! = 2^+ 



[ 

 J 



6. We will now consider the case when one of the quantities m, n is 

 even and the other odd. 



First then let m the greater of the two quantities m and n be even 

 and n odd, and let m = 2p and n = 2q I, then q may be equal to p, but 

 cannot be greater than it. 



The general term on the left-hand side of the equation is now 



In order that this may not vanish when /x = 0, since all the powers 

 of fj. contained in (/* 2 l) 2p or (jit 2 I) 27 " 1 are even, we must have r odd and 

 therefore r+l and rl even, or 2p rl and 2q + r I even. 



.'. ( l) r = 1 in this case. 



442 



