THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 221 



All M. Mathieu's groups of (Ni+ i)Nt . (N'— i), when N 

 is any prime number, can be thus discussed and constructed 

 without the aid of congruences. And the triplets of all the 

 didymous factors are reducible to duads. This may perhaps 

 ripen into a complete tactical theory of groups. 



In order that the construction of the group of 8 . 7 . 6 

 should evidently follow from the notation of the above mui- 

 tiplets, it would be necessary (and it would not be difficult) 

 to treat the matter from the beginning in a manner some- 

 thing like the following mode of discussing two groups of 

 considerable interest, of 7 . 6 . 4 and 11 . 10.6. 



From the seven triads which exhaust the duads of seven 

 elements, namely 



157. 261, 372, 413, 524, 635, 746, 



we can form twenty-one triads thus, each containing a 

 capital and two small figures, 



157^ 571. 315. 261, 612, I26, &c. 

 We can collect the triplets of these triads which contain 

 the same small figures thus, the order of the small figures 

 being indificrent : — 



I57 I57 517 517 571 571 75n 



327 237 327 237 431 341 431 .&c. ... (A) 

 467 647 647 467 26i 621 621 J 



where every first vertical row is one of the fundamental 

 triads. We can thus form twenty-eight triplets of triads, 

 and exhaust 3.7.4 of the 21.10 couplets possible with 

 the twenty-one triads I57, 571^ &c. 



We can next form a quadruplet upon each of the twenty- 

 one triads thus : — 



on (157) 517 on (524) 254 on (663) 653 



I26 517 617 



751. 452 365 



I34 563 524 



In the first of these, I57, 126,134 are three triads with 



