OF THE FORM d m P n /dp m x tfPJdpP. 379 



Hence the general or (r+l)th term in (^- l)"D m+ "P n is 



, _ , ,. p\ (n-m+p-2r)l (n + m+p)\ 1.3.5... (2n-2r-l) 



' ?! (p-r}\ (n-m-p)\ (n + m+p-2r)\ 1.3.5 ... (2n + 2p-2r + 1) 



y _>! 2 J< '(2n 2r)\ (n m+p 2r)\ (n + m+p)\ (n+p r)l 



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



Hence putting mp for m, the general term in (^ 2 l) p D" l P n is 



,,. pi 2" (2n -2r)l (n-m + 2p- 2r) I (n + m) I (n +p - r) I 

 ' ri (p-^l (n - r) ! (n ^m) I (n + m- 2r) I (2n + 2p - 2r +1 ) ! 



5. From the expression for P n we get 



Now differentiating m + p times we get 



pi 1. 3. 5 ... (2n-2r-l) 



T)m p _ <? / _ -I \c _ 1 



rl (p-r)l 1. 3. 5 ... (2n + 2p-2r+l) 



We have also D"P P = 1 . 3 . 5 . . . (2p - 1), 



and DP p+l =l.3.5...(2p+l)n. 



Now by formulae obtained in an earlier part of the work (see p. 255) 

 we have 



and 



hence we have DP p+l -DP p _ 1 = (2p+ 1) P p . 



Putting p + I for p in this equation we get 



Differentiating this equation successively we get 



1 , 



482 



