26 The Rev. Thos. P. Kirkman on 



These questions ought to have been put and answered 

 in (F.G. 12). We here consider them in turn. 



3. First, if Gy under 2 has fewer than 60 values there 

 must be in G/" under 2, i.e., in (B/* - , a repetition of the 

 value Q> f , and we must have 



K0 / = G / , (a) 



where KG is a derivate of G}, made by a substitution K ; 

 for the values of (3/ are obtainable only by substitutions 

 among the variables under barred or immovable indices : 

 that is, there is somewhere in the derivate KG 7 a substitution 

 C, which under 2 is Q, algebraically identical with unity 

 under 2 in (B/> thus making (a) true. This C must therefore 

 be somewhere in the index group, and is no 9 common to 

 l t+ i and G 7 ; for no derivant C of G f is in G f ; and not only 

 has every substitution of I m the value under 2 which 

 unity under 2 has, but no C having that property is 

 missing from l t+1 . Wherefore when (a) is true there is a 

 substitution 0j=C in I m , which makes it true, making 

 C0/=K0 / = Of, although 



CG / =G / 



is always false, if C be not in G f ; and 0/ i.e. G f under 2, 

 must be symmetrical in ^the variables transposed by G t = C. 

 Examples will occur. It is clear that none of our 60 maxi- 

 mum groups is, under 2 = a/3yoee, symmetrical in 5 and 6 

 disturbed by d in l t+1 in art. 2 ; for 5 and 6 carry different 

 indices all through every (3, except in two products. 



Thus we know that no one of our 30 standards under 2 

 has fewer than 60 values. 



Secondly, if 

 (a) (3 2< i = (3 d , or 0, i+l)d = e d (a") 



there must, if a' is true, be a derivate KG 2(1 , which under 2 

 has in it a product C algebraically identical with unity in 

 G d under 2 ; so that KG 2(! is CG 2d , which, by what precedes 



