OF THE FORM (^ - If d n P n /d f i.'" x cFP q /dfiP. 



393 



The expression in large brackets may be arranged as follows, in 

 order to separate out the factors (2p + '2) and (2n + 2p 4r + 5), which are 

 factors of this expression 



(2p + 2) (2n + 2p - 4r + 5) (2p + 3) (p - r + 2) (n + m- 2r + 2) (n + m~2r+l) 



- (2p + 3) (2n + 2p-4r+5)2r(p- r + 2) (2n -2r+l)(n + m- 2r + 2) 



- (2p + 3) 2r (p -r + 2) (2 - 2r + l)2r (n + m - 2r + 2) 



-(2p + 3)2r(p-r + 2)(2n-2r+ l)(2n-2r + 3) 



X {(2n + 2p 4r + 5) (n + m 2r + 2)} 



+ (2p + 2) (2p + 3) r (2n - 2r + 1 ) (n - m) {(2n + 2p - 4r + 5) - (n + m - 2r + 2)} 

 + (2n + 2p - 4r + 5) 2r (p -r + 2) (2n - 2r + I) (2n + 2p - 2r + 5) 

 = (2p + 2) (2n + 2p - 4r + 5) r (2p + 3)(p-r+2)(n + m - 2r + 2)(n + m - 2r + 1)' 



+ (2p + 3) r (2n - 2r + I ) (n - ni) 



- 2r (p-r + 2)(2n - 2r + l)(2n -2r+2) 



- (2p + 3)r(n + m- 2r + 2) (n - m) 



= (2p + 2)(2n + 2p-4r+5)r(2p + 3)(p-r + 2)(n + m-2r + 2)(n + m-2r+l)~ 



also 



+ (2p + 3) r (n - m) (n -m-l) 



- 2r(p -r + 2) (2n - 2r + l)(2n - 2r + 2) 



' (2p + 3) (p + 2)(n + m-2r + 2)(n + m-2r+l) 

 - (2p + 3) ( + m - 2r + 2) 2r (2n-2r+l) 

 + r(2r-l)(2n-2r+])(2n-2r+2) 

 Hence the coefficient of 



(2n + 2p - 4r + 5) D m - p P n+p _. 2r+ , 

 in the expansion of 2 (^ - 1 ) p D m P n x D"P pJr ., 



. _ . r (2p + 2) ! (2n - 2r) I (n + m) ! (n - m + 2p -2r + 2)l (n +p - r + 2) ! 

 ' r! (p-r + 2)\ (n-r)l (n-m)\ (n + m-2r + 2)\ (2n + 2p-2r + 5)l 



2r (2r- 



- 2r + 2) 2r (2n - 2r 

 - 2r+l) (2n-2r + 2) 



A. 11. 



50 



