48 The Rev. Thos. P. Kirkman on 



we get upon them by did,- -d ly the three sevens here below 

 written under the standards which mark them out. 

 2 = na/3/3yy. 

 G r> = 1 23456, G 2D = 1 23456, 

 234561, 245613, 



345612, 461325, 



Gei, 415362, G 2El , 316524, 



Gb2, 245361, G. 2E 2, 416532, 



Gb 3) 3 r 45 6 2, G 2E3 , 246531, 



Ge*, 246315, G 2E4 , 235641, 



G EB , 314625, G 2E5 , 236514, 



G E6 , 234615, G 2E6 , 415623, 



G E7 , 416325, G 2E7 , 315642, 



We have thus given account of our 60 groups under 

 S = act/3/3yy. The reader will easily satisfy himself that no 

 substitution C of I 8 can make 



CG = 6 (a, art. 3). 



true of any one of our 1 1 retained standards except the 

 first, O cl . art. ii x . 



We have proved that 1 1 distinct functions are given 

 under aa/3/3yy; viz., putting = for given upon, 



(3^ = 436521 has 6 terms and 15 values ; 



0^=254631, ($^ = 316245, and 6^ = 462315 have each 6 



terms and 60 values ; 

 (3^ = 356142, 025 = 456231, and 633 = 462531 have each 6 



terms and 60 values ; 

 (3d =234561, ^20 = 245613, and <Bsn = 342615, and & id 



= 354612 have each 12 terms and 60 values. 

 S = aaa/3/3/3 



12. Our index group is I 33 . The group (456312), viz. : 

 G d = 123456+ 13254663 



456312 546213 



6,312645 21365465 



645231 654321 



©1231564 32146564 



564123 465132 



