50 The Rev. Tiios. P. Kirkman on 



There remain 60-6- 18 = 36 groups unmarked which 

 have no substitution of I 3 e. One of these is 

 Ga= 123456+ 146253 



264135 43 2r6 5 



6 S I2 43 35 J 426 



532614 564312 _ 



346521 623541 ' 



415362 215634 



If we add to H 18 , above written, its derivate by 321654, 

 we have I 36 before us, which is I 36 = 1 + Xi + X 2 + ' ' " + A 3 5- 



On Ga = 264135 we get by these \ x . X 2 ,' ' X 36 the 35 

 equivalents marked out by Ga. They are all that remain 

 of the 60 groups, and it is not necessary to write them 

 down. They all give the function (3 A - ". 



We have proved that under 2aaa/3/3/3, 

 there are three functions obtained : 



(3d has 2 terms and 10 values ; 

 (3d has 6 terms and 60 values ; 

 (3 a has 12 terms and 60 values. 



S = an/7/3/3/3. 



13. The Group Ga = 234561 has 216543 in common with 



I 4 8=Ia4x 12345 6 =H 2 J 2 



216543 = 8 



where H 24 is exactly H 4 = I 24 of art. g 2 . 



Ga here marks out the 23 groups, which under 

 2 = a/3yyyy it marked in that art. g 2 , and which are there 

 headed by it as G D , which had no 9 in common with I. 2t 



under a/3yyyy. 



(3a is now a 60-valued function of 6 terms. 



The group 342615 = G 2D of art. g. 2 is here our standard 

 Goa, and marks out by the same H 24 in that article the 23 

 groups that therein stand under it as G 2d , which had no 9 



of I 2t under a/3yyyy. 



(3 2 a is a function here of 6 terms only. 



G.>A is under this 2 in its 6 terms, as its twelve terms 



