Functions formed on Groups. 33 



As soon as the reader is familiar with these few considera- 

 tions, all easily verifiable on the paradigm, and has proceeded 

 a little way into this paper, he will find that he has broken 

 every bone in the body of the champion problem of many- 

 valued functions of n variables. For what has just been stated 

 about dictation, by a little girl from one paradigm, of all the 

 values of every function given by the first maximum groups 

 of n = 6 elements, is equally true and practicable from one 

 paradigm for 11 = 600 elements, if only the girl can pronounce 

 in order correctly the figures or letters of a line so long. 



And the one paradigm needs not be the cyclical. That 

 of any equivalent will serve equally well; but the cyclical 

 is most easily surveyed, because every substitution of it is, 

 backwards or forwards, in clear order 2n times cyclically 

 read. That is the only eminence or precedence which the 

 cyclical has over any other of the Q equivalents. It has 

 been deservedly preferred ; but has been very much over- 

 valued by writers who have rarely condescended to glance 

 at the equivalents. 



5. As our title (art. 1) has no subscript containing in, 

 showing three circles of order one under unrepeated 

 exponents, none of our 60 equivalents has a substitution of 

 the index group determined by 2. This group is, m = o, 



I (+ i = 10=123456 (F.G. 12, 15) 



123564 = 6! 

 123645 = 6)3 



i234 6 5 = e 8 

 123654 = 6*4 



123546 = 65 

 Taking (234561) for G,,, we form 



e 1 23456i6»r 1 =235i64 = G el ; 2 G d a~ 1 = 236514 = G e2 ; 



s GA -1 = 234615 = G c8 ; 6> 4 G rf 6V = 236145 = G e4 5 



05GA~ 1= = 235641 = G e5 ; 

 and we mark out the five G/s as giving all the same (3<t ' ' , 

 D 



