42 The Rev. Thos. P. Kirkman on 



If we name the 23 substitutions following unity, 

 010*03 ... 023, we get, for && = Q£x&\ G Ba = a G D 2 - 1 , &c. 



tO G k23 = O^Gd^ 1 , 



the 23 groups here under written after G D = 234561 : 



2 = aftyyyy. 



G D (2345 6l )> 235164, 236514, 234615, 236145, 235641 ,» 



G E i (246315), 254631, 265143, 245361, 264513, 256134 BM 



G E2 (261345), 265314, 246531, 251364, 256341, 245613 R3 J 



G E3 (254163), 241635, 251643, 264135, 241563, 261534 K o_, 



These 23 are marked out by G D as all giving (3£ " '. 

 That G D is 60- valued is plain if we put y = o in G D ; for 



aft aft aft a/3 a/3 a/3 a/3 a/3 aft aft aft aft 



12 + 21 + 23 + 32 + 34 + 43 + 45 + 54 + 56 + 65 + 6l + 16 = <3„, 



is a 60-valued function, and the 23 equivalents must be all 

 60- valued also. The function on G^ " is retained. 



There are 60 — 1 2 — 24 = 24 groups yet unmarked. We 

 take 342615 for G 2D . It has no 9 in common with L 4 

 On this we form, by the same AiX 2 . • X 2 3, 



XlG 2D \i =GaElj X 2 G2rA2~ 1 = G 2E2 , &c, 

 which are thus written after G 2 „ which marks the 23 out. 



2 = aftyyyy. 



G 2D = (3 4 26l5), 352641, 362145, 342561, 362514, 352164, E20 



G 2E i, (435 2 6i), 53 6l2 4> 634512, 436215, 635142, 534621, E2t 



G 2K2 , (654132), 641532, 45 I2 63, 564123, 541623, 461235, E22 



G 2E3 , (561324), 465213, 546321, 651342, 456231, 645312, E js 



The function of this group under S, 

 G 2D = 123456 165432 



is that under 2 of its derivates by 125346, 124536, 

 124356, 125436, 123546, which are in \ U) but not in 



