60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 493 



stated, the coefficient of the corresponding term will vanish. Hence prac- 

 tically we may consider r to be unrestricted in value. 



Similarly, let Q m _^ denote the quantity of which the general term is 



A (m - r)A (r - 1 ) A (n - r + 1 ) /2n + 2m - 4r + 3\ 



A(n + m-r) ' \2n + 2m-2r+l) ^+- > + 



writing m I for m and r 1 for r in the general term given above. 

 Also let Q m+1 denote the quantity of which the general term is 



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



' 



writing m + I for m in the general term first given. In consequence of 

 the evanescence of A (m) when m is negative, we may in all these general 

 terms suppose r to vary from to m + 1 . 



Let us assume that Q m ^ 1 = P m - l P n , an d a l so that Q m P m P n , then we 

 have to prove that Q m+1 = P m+1 P n - 



As before, (m + l)P m+1 + mP^ - (2m + 1) fiP m = 0, 



.-. (m+l)P m+1 P n + mP m _ l P n -(2m+l)f J .P m 



Hence our theorem will be established if we prove that 



Now Q m = ...... 



A (m-r+\)A(r-l)A(n-r+l) /2n + 2m - 4r + 5\ p 



~~A~(n+~rri-r+l) ~ \2n + 2m-2r + 3/ n+m ' 



A(m r)A (r) A(n r) /2n + 2m 4r_ 

 n.. . ~i_ r \ ' \2n + 2m- 



Multiplying by ^ and substituting for /x.P n+m _.>r +2 and p.P n + m -*-> & c -> m 

 terms of P m+m _ 2r+1 , &c., we find the coefficient of P n+m ^ +1 in p-Q m to be 



A(m-r+l)A (r-l)A(n-r+l) in + 2m-2r + 2\ 



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

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



