40 The Rev. Thos. P. Kirkman on 



which has in common with l /+1 the subgroup J m+1 and no 

 more, the h groups ■ 



are different equivalents of G d , and the k groups 



ViGavT 1 + v&d07 l + •• +VkGdVk l 



are different equivalents of G,, ; but it is not hereby affirmed 

 that these h + k groups are h + k different equivalents of G d . 

 For let it be supposed that 



v a G d r)~ 1 = r) b G U )ii; 1 ; 

 it follows that 



m 1 iaG d =G d r]b l Va, i>e- v e Gd=G d T) c , 



which is impossible unless either 



VcG d = G d =G d r]c 



showing that v r is in G d , or 



VcG d = G d r) c , 

 shewing that i c G d is a derived derangement of G d . But the 

 first is untrue because G d has only 9's and has no v c m 

 common with I (+ i : and the second is impossible, because 

 G d is a maximum group. 



In like manner can be proved, exactly as in art. 6, that 

 no two of the aforesaid h equivalents of G d are alike. 



Thus theorem U is demonstrated. 



g. Returning to (356142) (art. 8 and 6) as our G d under 

 2 = 0/37777, we f° rr n with the X's of H c the equivalents 



XjGdXf , X 2 G,iXr , A 3 G d X ; 7 , X 4 G fZ X i , X B G,jX5 , i.e., 

 G < i = 435 6l2 > G c2 = 3 6 42i5,G c3 = 3465i2 ) G e 4 = 534i62,G,5 = 45 l6 3 2 



which five are marked out by G d , as all giving the same ($p . 

 Taking next (542631) art. 6 for G 2(/ , we get by the same 



H 6 



G 2 «i = 354261, G 2e2 = 452.3,61, G 2r3 = 436521, 



G 2 , 4 = 345 62I > G 2e5 = 536241- 



five groups marked out by G, d as all giving the same 6 2 V'- 



