$6 The Rev. Thos. P. Kirkman on 



H A+1 of orders in + i and h+ i, that have no common sub- 

 stitution ; and if G d be any maximum group of the n 

 elements, which has, in common with 



i, +1 = i + e + e + • • + e m (d) 



+ Xi + x 2 + ■ ■ + \ h 

 + 6, + 6 2 + • • + B t _ m _ ht 

 the subgroup 



Jm+i=i + 0i + ©2 + - - + e m , 



and no other substitution of I, +l ; then if 



H A+1 =i +\ 1 + X a + ■■ + X h , 

 the h groups 



XiG rf Xf , X 2 G d X r 1 ... X h G d Xh 1 , 



are h different maximum groups, equivalents of 

 For, be it supposed that two are equal, as 

 ^ a G d X~ l = X h G d \i x ; 

 it follows that 



XjT X a G d = G rf \4 A a , i.e , 

 X c G d = G d X c . 



This is possible only on condition either that 

 \G d = G d = G d X c 

 shewing that X c is in G rf , or that 



X e G d = G d X c , 

 shewing that X c G d is a derived derangement of G d 



The first is impossible, because G^ has only unity in 

 common with H h+1 ; and the second is impossib e, because 

 G rf is a maximum group (F.G. 4). 

 Wherefore all the 1 +/i groups 



GdiXjGtfAf 1 } XaGdXjT •-•AjG ( A« 



are different maximum groups 



The equation (d) above is (F.G. 8, 7) n more defined 

 form. 



In this theorem in is not limited, being ;;/>o. When 



m = o, h = t, I M 1 = H,_ 1 , J w+ i = Ji 1 



