382 REV. T. p. KIRKMAN ON THE THEORY OP 



89. The group o/N'(N-i)(N-2), when N-i is a prime 

 number, is always a maximum. It has no derived derange- 

 ment. 



The group 0/ N* (N- i)(N-3), when N- 1 is a power 

 greater than the first of a prime number, is never maxi- 

 mum. It has always derived derangements, which can 

 be enumerated by the application of the theorems of this 

 Memoir. 



Thus when 



N- 1=2^=8, 



there are, as has been shewn above, 240 equivalent groups 

 of 8 • 7 made with eight elements, and consequently 240 

 groups of fifty-six made with nine elements, in which the 

 ninth element is final and undisturbed. Each of these 

 240 determines a group of 9 -8 -7. We have then 240 

 equivalent groups of 9 • 8 • 7 ; wherefore by my theorem A 

 there are 240 equivalent groups each of 



^ = 9.8.7.3 

 2411 ^ ' -^ 



substitutions, of which each one contains one of M. 

 Mathieu's groups of 9 '8 '7. 



If in this group of 9 • 8 • 7 • 3 we collect the substitutions 

 terminating with 9, we shall, after erasing 9, have the 

 group of 8 • 7 • 3 which has been found before, (86), and 

 which is made by adding to the group of 8 • 7 two derived 

 derangements of it. 



The number of equivalent groups of nine containing 

 each eight substitutions of the third order is given by the 

 theorem H to be 840. There are, consequently, by the 

 theorem A 840 equivalent modular groups each of 



substitutions. 



M. Mathieu's group of 3^*(3'^-i) is part of this maxi- 

 mum group of 9 • 8 • 6, and is given with it. Each of these 



