OF THE FORM d m P n ldp. m x dfP^dpf. 385 



Comparing these expressions we see that 



/ 1Y ~ -s)! _, y 2rl (2n-2r)\(n + q-r)\ 



I s \ (q-s)l (n-s)l (q-2r + s)l ^ n\ r! (n-r)l (q-r)l 



which is a remarkable expression. 



It will be well to test this formula numerically. 

 For instance let n = 3, q = 3, r = 0. 

 Then the series is 



1.2.3.1.2.3.4.5.6.7.8.9 _ 1.2. 3.4. 1.2.3.4 ._5_._ 6 . 7 . 8 

 1.2.3.1.2.3.1.2.3 1.1.2.1. 2TT. 2T3 .~4 ~ 



1.2.3.4.5.1.2.3.4.5.6.71.2.3.4.5.6.1.2.3.4.5.6 

 ~T7 2 . T7T. 1.2.3.4.5 1.2.3.1.2.3. 4^75". 6~ 



= 20 . 7 . 8 . 9 - 30 . 6 . 7 . 8 + 60 . 6 . 7 - 120 = 2400. 

 Also the other expression 



1.2.3.4.5.6.1.2.3.4.5.6 

 1.2.3.1.2.3. 1.2.3 



= 20 . 120 = 2400, which agrees. 

 Next let r=l. 

 Then the series is 



1.2.3.1.2.3.4.5.6.7 1.2.3.4.1.2.3.4.5.6 





1.2.3.1.2.3.1 1.1.2.1.2.1.2 



1.2.3.4.5.1.2.3.4.5 1.2.3.4.5.6.1.2.3.4 



1.2.1.1.1.2.3 1.2.3.1.2.3.4 



= 20 . 6 . 7 - 108 . 20 + 60 . 20 - 120 

 = 20 [42 -108 + 60 -6]= -240, 



1.2.1.2.3.4.1.2.3.4.5 

 and the other expression = --- - - - - - - - - -- = 12.4.5= 240, 



l.^.O.l. l.^j.l.Z 



which agrees. 



Next let r 2. 



Then s = gives zero since /( I) is infinity; and the series is 



1.2.3.4.1.2.3. 4 1.2.3.4.5.1.2.3 _ 1.2.3.4.5.6.1.2 

 1.1.2.1.2 " 1.2.1.1.1 1.2.3.1.2 



= -144 + 360-120 



= 96; 

 A. II. 49 



