60] EXPRESSED BY A SERIES OF LEGENDRE'S COEFFICIENTS. 495 



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

 efficient of P n+m _ 2r+1 is 



A(m-r) A(r-\}A (n-r) 

 A(n + m r) (2n+2m 2r + 1) 



As before suppose n-r + I=q, and the quantity within the brackets 

 becomes 



J r ( 2m +l)(m + q-r) + -^ - m (2m + 1 + 2q - 2r). 



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 



or = (n-Hft-r + 1) (2m-2r + I), 



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

 coefficient of -P H+m _, r+] is 



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



A(n + m r) [r(n r+l)' 2n + 2m 2r+l )' 



Hence the whole coefficient of P n+m _, r+l is 



A (m r)A (r 1) A (n r) (n 2r + l) 

 A (n + m r+ 1) r(n r+l) 



x {(2m - 2r + 1) - (2m - 2r + 1 )} = 0. 



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

 obtain 



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 m gives rise to two terms in the value 

 of fj.Q m or in that of (2m +1) p.Q m ; one of these terms is to be subtracted 

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

 responding term in mQ m _ l , and it will be found that the two series of 

 terms thus formed identically destroy each other. 



