OF THE FORM d m P n jdp. m x dfPJd^. 387 



Now 2Z) m P n xZ) J 'P i)+2 = (2^ + 3) / LtZ) J) P p+1 + terms which do not affect the 

 result wanted. 



Hence 2D m P, l xD"P p+ , 



Again 3lD m P, l xDP p+3 = (2p + 5)i J ,D m P n xI> 1 'P 1 , + ,_ + temns not required 



_5)_l_. 3 . 5_.. J _(2n - 1) 

 ~ 



. 3. 5 ... 



x ( w - HI+ 1 ) (n - m + 2) (n - TO + 3) D m+ "P n+p+3 + &c. 



_ / _ y 1-3. 5... (2p + 5)1.3. 5.. . (2n, - 2p - 3) 

 1.3.5 ... (2n+l) (2-2j)-3) 



x (n + ??i) (n + m - 1 ) (w + ?n, - 2) D m+p P n _ p _. r 

 This may be conveniently written in the form (q p] \ D m P n x D"P y 



p)S 



" +? 



1.3.5 ...(2n + 2gr-l)(n-m)l 



1.3.5...(2 7 -l)1.3.5...(2n-2g + l)(n + m)! 



.- 



1.3.5 ... 



Putting it in this form we see that this expression is general and 

 includes the previous results for q p = 2 and q p = 1 and q =p. 



Hence it is true for all values of q p. 

 Expressing this result in factorials we get 



(q -p) I . DP n x BV, = 2gI2nl(n + g )!(n-,n + g -. J >)l 



q!n!(2n + 2q)](n m)! 



- 

 ql(n-q)l(2n+l)l(n + m- 



10. The following is a simple method of determining the coefficient 

 of D m+ *P n+q in the product D m P n x D"P q . 



The coefficient of p"- in D m P n is --- 



(n m)l 



492 



