THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 217 



Let US suppose b = S in d ; then k=2: what is m in the 

 circle $Smy . . ? It follows 8 ; wherefore m=jj andj is the 

 letter preceding 7 in the circle 5 Sjy . . . Now^^ in d= 38 ighij2 

 and in the circle 58^7. . . , can be none of 123578. lij=6 

 precedes 7, 2=7^ and 4 precedes 7; ify=5 precedes '],h—'jj 

 and 2 precedes 7 ; \ij=^ precedes 7, ^=7, and 6 precedes 

 7 : all absurd. 



Therefore h is not 8 in dj and in the circle ^brnj . . . 

 Try 6= 6 : then i= 2, and m precedes 7 in the circle ^6my . . ._, 

 wherefore m follows 6, or m =ff. We have now d= 36 imh2jk. 

 If m=\=g in B^ we have the absurd vertical circle 6424 . . ; 

 and if m = 8 =gj A: = 4, and A = 5 and j = j of necessity ; for 

 df hj and c must have each two elements undisturbed. 

 This gives 



35162487= A 

 36i85274 = </ 

 38176542 = 6 

 37148625=6 

 34127856=0 

 32154768 = 0. 



We have to examine df and ce : these are 



d = 36185274 c = 32154768 



f - 65432178 e = 42615387 



k = 52861473 k = 52861473 



z = 21354876 V = 12786534 



y = 14628375 2^ = 62378145 

 »i= 48513672 §'=82437651 

 A = 83246571, j = 72543816, 



systems of seven as they ought to be. We have now all the 

 49 elements in our power; We thus complete the list of 

 capitals and small letters, of which we yet require 



gilnprstuwxa^, DEFGHIJKLMPTUV. 



