70 Prof. J. C. Adams on Legendres Coefficients. [Jan. 31, 



and therefore the sum of the last two lines of the expression for the 

 coefficient of ~P n+m _ 2r+1 is 



A(m — r) A(r— 1) A(n—r) 

 A{n+m—r) 



x J Q-2r + l) (»+m-r + l) (27K-2r + l) 1 

 1 r(«-r + l) ' 2?i + 2ra — 2r + l J 



Hence the whole coefficient of P M+m _ 2! . +1 is 



A(m— r) A(r— 1) A(>— r) w — 2r-f- 1) 

 AO + m— r + 1) r(> — r + 1 



x{2m-2r + l)-(2m-2r + l)} = 0. 



And the same holds good for the coefficient of every term. Hence 

 we finally obtain 



O+l) Q« + i + mQ m _ x - (2m + 1>Q W = 0, 



which establishes the theorem above enunciated. 



The principle of the process employed in the above proof may be 

 thus stated : 



Every term in the value of Q,„ gives rise to two terms in the value 

 of /iQ m or in that of (2??i + l)/iQ M ; one of these terms is to be sub- 

 tracted from the corresponding term in (m + l)Q m+1 , and the other 

 from the corresponding term in mQ m _ 11 and it will be found that the 

 two series of terms thus formed identically destroy each other. 



Hence we can find at once the value of the definite integral 



for if p = n + m — 2r we have 

 m n ' k(n + m+v\ n+m+p+1 1 



\-^) 



+ &c. 



n\ /n + m—p\ . fn+p — m\ 



-) A (~~2— ) A V~^~J gP±I f 1 (P y- 



n + m +p + 1 A j^ n + m+p ^ 



Hence 

 a _(m+p 



Jf-i 



