324 The Rev. Thos. P. Kirkman on 



Now, the derivate under S of any group G^ has instantly 

 become under S a value of (Bg, 2>., of G^ under S. 



Equation (ii) represents t-m groups, 



Gel = 0r^Gd0i ; Ge2 = ^^^Gd02 , Ge3 = 63" ^Gd^s 5 + &c. ; (12) 



to which correspond /-w equations 



among which is the above, (lo), 



Whether these t-m groups (12) G^i, G^2 . •• ^ ^t{t-m) are 

 all different, we shall discover presently. We shall see that 

 there are only t' <^t- in of them different. Hence we are 

 bound to mark out this G^i from our table (A,^), along with 

 t-m — i others, as in art. 8 under (5). 



1 1. This we may conceive as follows. Our standard 

 group Go, comes forward with equations 1 2 and 1 3 in his 

 hand, to make t—m charges against certain of our Q equiva- 

 lents in (A„). The charge in each case is, that the accused 

 under S has a value algebraically identical with (3^. 



We ask — which of the Q— i groups do you first call up? 

 Ga answers — " my first equivalent by 0i, 0iG(^0i~^" We 

 look into (A J and remark — that is our group G/ ; which of 

 the values of (^f do you accuse ? " That given by the derivate 

 under S, 0iG/." We examine that, and see that, under 2, 0iG/ is 

 nothing else algebraically than the L products of G^^. We, 

 therefore, write d opposite in the line G/, to show that G/ has 

 been expelled by the standard G^^. 



In the same way we proceed to listen to G^^'s other 

 charges, and are compelled by the truth of them to mark 

 out with d a number of other groups in (AJ. Presently, 

 we hear a like accusation against QmGdQm\ and on finding it 

 we exclaim, — that is our friend G/ again. "Very likely,'* 

 says Gd, "look at the derivate d^Gf" We look, and see 

 that it is none other than 0iG/ above found, a derivate con- 



