1878.] Prof. J. C. Adams on Legendre's Coefficients. 69 



The sum of the first two lines of this expression is 



A(m — r + 1) A(r— 1) A(ji — r) 

 A^ + m^-f 1) ~(2n + Zm — 2r + 3 ) 



x i 2 ^^ + 1)(2» + 2m -4r + 3) - 2n ~ 2r+ i 1 (2m + 1 ) (n + m- 2r + 2) 

 [ r % — r+1 



Suppose for a moment that w— r + l = g r , then the quantity within 

 the brackets becomes 



2r ~ 1 (m + l) (2m -f l + 2tf-2r)- 2 il- L (2m + l) (m + l.+ g-r) 

 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 



= -2_Zlo_ r+ i) (2m+2g + l) 

 ^r 



or = - n ~ 2r + 1 ( m _ r +l) (2w+2m-2r + 3) 

 r(n — r + 1) 



So that the sum of the first two lines of the expression, for the co- 

 efficient of P n+m _ ir+l is 



_ A(m— r + 1) A(r— 1) A(n—r)f (m— r + 1) (n— 2r + l) "I 

 A(n + m— r + 1) (_ r(w — r + 1) J 



Again, the sum of the other two lines of the expression for the co- 

 efficient Of P n+Il ,_2r+1, ^ 



A(m— r) A(r— 1) A(n — r) 

 "A(% + m —7)^(2^+ 2 m — 2r + 1 ) 



x{_?!— L (2m + 1) (n + m-2r + l) + 2n ~ 2r "j" ] m(2w + 2 m - 4r + 3 ) 

 L r r + 1 



As before suppose n — r + l — q, and the quantity within the 

 brackets becomes 



- 2r ~ 1 (2m + 1) (m+g-r)+?illlm(2m + l + 2g-2r) 

 r 2 



Now this quantity evidently vanishes when q=r, so that it is 

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

 likewise divisible by m + q. 



Hence it is readily found that this quantity 



= 'LzL (q + m) (2w-2r+l) 



or = — — 2> (n-\-m — r + 1) (2m — 2r + l) 

 r{n— r+1) 



