OF THE FORM d m P n jdp. m x dPP q /dpP. 395 



And since q=p + 2, this is 



= , _ v r (g+p)l(2n-2r)l(n + q-r)l 



' r\(q-r)l(n-r)\(2n + 2q-2r+l)l 



(n + m) I (n m + q +p 2r + 2)1 (n + m + 1) ! (n m + q + p 2r + 1 ) ! ~" 

 (n m) I (n + m + q2^ 2r 2)1 (n m 1)! (n + m + qp 2r 1)1 



X 



(n + m + 2) I (n m + q +p 2r) '. 

 (n m 2)1 (n + m + q p 2r) I 



This last square bracket is equivalent to 



' (n + m) ! (n - m + q+p-2r + 2)\ _ (n + m +1)1 (n - m + q +p-2r+l) ! " 

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



(n + m + 2) I (n-m + q+p- 2r) ! 

 (n m 2)l (n + m 2r + 2) I 



NOTE. This square bracket differs from the corresponding square bracket 

 in the value of 2D m P n x D"P q only in the sign of m, and the law of 

 formation of the terms is clearly seen. 



13. Adopting the form for D m P n xD v P q as given in Arts. 6 and 7 

 above, we arrive at the conclusion that 



(q -p) ! D m P n x D"P q = 2 (2 + 2q - 4r + 1) D m+ "P, l+g _, r 



. y (q +p) I (2n - 2r) ! (n + q - r) I (2g - 2r) I (n + in) ! 

 ' (q+pl 2n~2rl rl -r\ n-rl 2n + 2q-2r+ 1)1 



s\ (q-p-s)\ (q+p-2r + s)\ (n + m-s}\\\ 



\ (q+p + s)\ 1 

 \ (n + m-s}\\ 



\ 1} 



= % (2n + 2q-4r + l) D^P n+ ^ r x ( - l)" - 



v ' rl (q r)l (n r)l 



f (q-p)\ (q+p + s)l (2n-2r 



*! (<1-P- S ) 1 (q+p-2r + s)\ 



-2r + q-p-s)\ 



From note to Art. 12 we see that another form of this coefficient 

 to (2n + 2q-4r+l)D m+t> P n+q _ w is 



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

 ' r\ (q-r}\ (n-r)\ (2n + 2q-2r + l)\ (n-m)\ 



- IV (g-p)! (n-m + s)\ (n + m + 2q-2r-s)l 

 ' s ! (q p s)\(n m 2r + s)\ (n + m s) I ' 



502 



