Functions given by Groups. 327 



If there are still unmarked equivalents in (A„), they 

 must be h'{t-\-\) in number: for if not, our last standard 

 G^ will find that two or more of the groups that he 

 accuses, defined by Qj, and 0,„ . . . , are the same one in our 

 table (A„). Let us suppose that 



0.G^0ir^ = 0„.G^0;»^ (17) 



are identical. It follows that 



^m^^^G^ = G^0~^6fc, i.e. 



0pG^ = G^0p, (18) 



because in, I^+i, %~}Qu^Qp. 



This (18) is possible only on condition either that 



^pG^ = G^ = G^0p, 



showing that 0^, is in G^, or that Qfi^ is a derived de- 

 rangement of G^. The first is absurd, because G^ has no 

 substitution of I^+i but unity. The second is absurd, because 

 G^ is a maximum group. Therefore the unmarked groups 

 in ( A„), from which we have to choose Gd, Gdd . . . G^, are 

 in number h{t-\-\). 



Therefore all the t groups accused by G^ will be found 

 distinct among the unmarked in (A„). We mark out each 

 with a A ; and there are now no groups unmarked in our 

 table, besides our standards 



GdGddGddcz- • -GdGdi). . .G^. 



If the number of these is R, we have learned how to- 

 construct R distinct many-valued functions under the 

 same S, and by help of the same index-group I^+i. 



The number of groups marked out as useless under 

 this S is Q — R, which carry the marks d, d^^ d^ . . , not of 

 necessity in equal numbers, where Q is the r of M.M, 



P,342. 



We have it now in our power to dictate, with all their 

 values, R different functions, of which no one is a value of 



