68 Prof. J. C. Adams on Legendre's Coefficients. [Jan. 31, 



writing m — 1 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 



W + 2m-2r + 3/ n+m ~ 2r 



A(m— r + 1) A(V) A(n — 

 A(n-\-m — r + 1) 



writing m + 1 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 _ a — P m __ 1 P n , and also that Q m = P m P n , then we 

 have to prove that Q m+1 = P TO+1 P n . 



As before, (m + l)P m+1 + mP m _ 1 -(2m + l) / iP m = 



(m + l)P M+1 P n + mP„ l _ 1 P n -(-2m + lVP m P n =0 



Hence our theorem will be established if we prove that 



(m + 1) Q m+ i + mQ^ - (2m + l>Q m = 



Now Q m = 



A(m— r + l)A(r-l)AQ-r + l) / 2u + 2m— 4r + 5 \ p 



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



A(m—r) A(r) A(n—r) / 2n + 2m — 4r + 1\ p 

 + AO + m-r) \2» + 2wj-2r + l y^" 3 '- 



+ 



Multiplying by /a and substituting for /*P w+m _ 2r+ 2 and /*P n+m . 2) . &c, 

 in terms of P n+m _ 2) . +1 &c, we find the coefficient of P n+m _ 2r+1 in ,uQ m 

 to be 



A(m-r + l) A(r-l) AQ-r + 1) / ??+2m-2r + 2 \ 

 A(> + m— r + 1) \2n+m— 2r+3/ 



A(m— r) A(r) A(w — r) / n-\-m — 2r + l \ 

 + A(n + m— r) \2w + 2m — 2r + lJ 



Hence the coefficient of P n+m _ 2r+1 in (m + l)Q m+1 +mQ M _ 1 — (2m + l) y aQ m 

 will be 



A(m-r + l)A(r)AQ — r) / 2^ + 2m— 4r + 3 \ 



A(w + m— r + 1) lm ; \2% + 2m-2r + 3/ 



A(m— r + l)A(r- 1)AQ — r + 1) ^ / n + m~2r + 2 \ 



~ A(rc+m— r + 1) 1 m ^ \2n + 2m-2r + 3/ 



^ A(m,-r)A(r)A(>-r) / 2m + 1 \ ( ^ + m-2r + l \ 

 A(w + m— r) \2w + 2m — 2r + l/ 



AQ— r)A(r — l)A(n — r + 1 ) / 2^ + 2m— 4r + 3 \ 

 + A(n + m— r) \2n + 2m — 2r + 1/ 



