24 The Rev. Thos. P. Kirkman on 



The title of our 60 equivalent groups of order 62 is 

 6-2 = l + 2 6 + 2 33 + 4 222 + 3 2211 Q = G0. 



That they are maximum groups appears from QL = «/, 

 i.e., 6o'i2 = 6!, L being the order, and Q the number of the 

 equivalents : (F.G. 4). 



I shall refer by (F.G. a, b) to article a and equation b, 

 in my paper "Functions given by groups," in the Proceedings 

 and Memoirs of this Society, 1 890- 1 . 



The index system S may be any of the nine (F.G. 5, 2) 



afiydet, afiyecc, afiyylc, afiyyyy, afifiyyy, aa/3/3yy, aaa/3/3/3, 

 aa/30/3/3, a/3/3/3/3/3. 



We choose the first, 



2 = afiySee. 



2. The index group determined by 2 is (F.G. 6), 



I<+i = 12=123456 =i+0, 

 123465 = 



of which the title is 2 = 1 + i 2Uu . 



Since our title (art. 1) has no subscript 21111, shewing 

 four elements undisturbed, none of our 60 equivalent groups 

 under S has any 9 in common with I (+1) (F.G. 12, 15). 



Denoting by (234561), a principal substitution of it, the 

 cyclical group G + RG, we take this (234561) for G d , and we 

 form (F.G. 10, 12) by in I, +1 the equivalent of G d , 



0G rf 0- 1 = G e = ( 2 3 4 6i5), 

 which determines G e by one of its two principal substitutions. 

 This G e is (F.G. 8) to be marked out of our table (A„) of 

 60 equivalents. 



We take any third group (235164) = G dd , or, better, G 2d , 

 for our second standard, which marks out 

 0G 2(Z 1 ~ = (236145) = G 2c 

 from the table (A„). 



We thus employ 30 standards, 



^d> "2d> ^3rf- • • • ^30 d» 



