390 REV. T. p. KIllKMAN ON THE THEORY OP 



{234567890«i}u{369i470258a}5{38i9507246«}4 



X {65i893472oa}3=:E 

 {23456789oai}n{369i470258«}5{2407i85936a}4 

 x{5629834i7oa}3=Ei. 

 The three last factors of F or of Fi are a modular group 

 (9) of twenty-four made with six elements (234567). 



The three last factors of E or of E^ are a non-modular 

 group of sixty made with (1234567890). The last two 

 factors of any of the four groups are modular groups also, 

 of eight or of twelve, given by the application of the pre- 

 ceding theorems. 



All these groups of 8 • 7 • 3 and 1 2 • 1 1 • 5 are maximum 

 and non-modular, whose equivalents can be easily enume- 

 rated, as can those of the inferior groups exhibited. 



The group F is formed by adding to G=S{2''i + c), of 

 the twenty-first order, 7 • 7 • 3 substitutions of the form 



The group F' is formed by adding to the same G, 7 • 7 • 3 

 substitutions of the form 



Xi = 5 {2'T^a)-^ + 3 [2''T^af + b, 

 where a ox h may have any of seven values 0123456. 



The group E is formed by adding to G'=S(3''«i-|-c) of 

 the fifty-fifth order, 1 1 • 1 1 • 5 substitutions of the form 



and Ej is made by adding to the same G' as many of the 

 form 



where a and h have each any of eleven values. 



The groups F and Ej are those given by M. Hermite at 

 page 63 of his Theorie cles Equations modulaires, Paris, 

 1859. It may not have been observed before, that there 

 is one, and one only, of the equivalents of F or of E which 

 has for a factor the same cyclical group of 7 • 3 or of 1 1 • 5 

 substitutions. The functions constructed on F or E will 

 have no value in common with those given by F^ or Ei . 



