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



the same capital, and I57, 671, 7 15 are three which have 

 the same figures ; and so on with the rest. 



The 2 1 quadruplets thus formed exhaust the 21.6 duads 

 not found in the 28 triplets, so that we have once and once 

 only employed the duads possible with the 21 triads in 

 these 21 quadruplets and 28 triplets. 



The whole of the triplets and quadruplets are thus 

 written : — 



lS7 I57 517 517 571 571 751 751 I75 I75 715 '"( 

 327 237 327 237 431 341 431 341 365 635 365 

 467 647 647 467 26 1 621 621 26i 425 245 245 



715 372 372 732 732 273 273 

 425 452 612 642 612 653 663 

 635 I62 542 I62 432 I43 4i3 



>. (A) 



7s3 723 674 674 764 764 746 746 476 476 



653 563 254 314 254 314 1^6 2i6 I26 2i6 



413 I43 I34 524 314 254 536 356 356 536^ 



These triplets may be denoted thus : (134)7, (126) 7^ &c. 



(157) 



(327) 



(467) 



(237) 



(647) 



(431) 



(26i) 



517 



237 



674 



372 



476 



314 



612 



126 



341 



413 



26i 



621 



467 



254 



751 



723 



746 



723 



764 



143 



126 



134 



365 



425 



254 



635 



452 



273 



(126) 



(134) 



(254) 



(341) 



(356) 



(5i7) 



(715) 



26i 



341 



542 



413 



663 



175 



157 



157 



126 



237 



327 



327 



524 



723 



612 



413 



425 



134 



635 



751 



571 



134 



157 



26i 



356 



341 



536 



746 



(723) 



(746) 



(612) 



(653) 



(452) 



(536) 



(542) 



237 



467 



126 



536 



624 



365 



425 



715 



723 



647 



612 



413 



524 



636 



372 



674 



26i 



365 



245 



653 



254 



746 



715 



653 



647 



476 



517 



517 



>■ ■ (B) 



' If we consider these 28 triplets and 24 quadruplets to be 

 systems of didymous factors of as many substitutions of the 

 third and fourth orders, % b^, &c., A4 B4, &c., we may de- 



