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



Hence the coefficient of P, l+m _ 2r+1 in (m+ 1) Q m+l + mQ m . 1 (2m+ 1) p.Q m will 

 be 



A(m r+l)A (r)A(n-r) , . 



A i i nnn ,. i i \ > ' 



2m-2r + 



_ A(m-r+l)A(r-l)A(n-r+l) . . / n + m-2r + 2 \ 



A / /yj 1 fyyi 'I* 1 1 i * ' \ ^<y) I Oivy) '? / y* I Q / 



./i i /t ~r **fr ^^ * i^ *- / \ Aft/ \^ ^j 1 1 >j ^^ A i ^^ o / 



_ ^(m-r)^l(r)^ (w-r) / 2m + ^ / n + m-2r+l \ 

 A(n + m r) ' \2n + 2m 2r+l/ 



A(m-r)A(r-l)A(n-r+l) 



J. _ i _ L - 5 - L ifft I - 



A(n + m r) \2n + 2m 2r+ I 



The sum of the first two lines of this expression is 



A (m-r+l)A (r-l ) A(n-r) 

 A(n + m-r+ l)'(2n + 2m - 2r + 3 ) 



x -I- -(m+ l)(2n + 2m 4?- + 3) (2m+l)(-/ 



(^ r w ? + 1 



Suppose for a moment that n r + 1 = q, then the quantity within the 

 brackets becomes 



^' J- / . -, \ / ^ . -. . ~ ^ \ ^^ J- /~ - \ / \ 



1 + g-r). 



Now this quantity evidently vanishes when q = r, and therefore it is 

 divisible by q r. It also vanishes when m + 1 = r, and therefore it is 

 likewise divisible by m r + 1. 



Hence it is readily found that this quantity 



_ q r, 

 qr ^ 



n-2r+l 



or 



of P +7n _ 2r+1 



So that the sum of the first two lines of the expression for the coefficient 



A(m-r+l)A(r-l)A(n-r) f(m-r+ 1) (n-2r+l)} 

 A(n + m r+l) \ r(n r+l) )' 



