Functions formed on Groups. 27 



in this article, must be 0,G 2rf , where 0; = C is in l [+l , and is 

 no 6 of G 2d ; and thus, if a' is true, 



algebraically. This makes G 2d one of the G e marked out by 

 G„ (KG. 8, 5). 



But this is impossible, because no new standard G nd was 

 ever selected (F.G. 12) from the groups marked out by 

 preceding standards. 



Wherefore neither a' nor a" is ever true of any two of 

 our retained standards. 



We have demonstrated that there are thirty and no more 

 different functions of 12 terms that have, under 2 = a/3yoee, 60 

 values. They are formed by writing this 2 over the 

 rectangles of the maximum groups above given, each 

 given by a principal substitution, viz., G d , G 2d , ■ • G 30d ; 

 which thus become 60-valued functions. 



4. We shall next show how all these 30 functions, with 

 all their values, can be easily dictated more rapidly than 

 they can be written down. 



Here follows the complete paradigm of the cyclical 

 maximum group (234561), or G^i = G d - above handled. 



The Paradigm of (234561)+ • ■ = G +- • 



