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



exhibit five difierent small letters, disregarding the sub- 

 indices. Thus the five triplets 



AF^, FEe^ FH^, CEc, EJ«,, 



in which F and E occur, exhibit five difierent small letters. 

 The triplets (R) give us the fifteen quadruplets 



A«,B«j B4C^3 ¥d,B.d^ CbJ)b^ Gcjic^ 

 Kb^b^ Bejje^ lej)e^ CcjEc, ^aja^ 

 Ac^lcj FCjEgj Id^Sdj^ Gcbjb^ Hafia^,, 



(Q) 



as also the ten sextuplets, formed from the three triplets 

 containing A or B, &c. 



BCiF«,I5, A.e^^cJ)d^ YcJeJ^a^ lajoid^b^- 

 Ae,Ca,Gd, BcJ)d,^b^ la^Ce.Kb^ ► . (S) 



Ad^^b.Ke^ Bc^Je,B.b^ ¥afidj)c^. 



In (Q) and (S) we have exhausted all the duads of the 

 ten capitals, and all duads made with a capital and any 

 small element. The duads of small letters not yet em- 

 ployed are found in the six quintuplets, 



afijC^d^e^ ajb^c^d^e^ a^^c^d^e^-,] 



and this is the only way in which they can be combined so 

 as to repeat no duad. 



We have exhausted all our duads of our 15+10 elements 

 in (Q,), (S), (T) . Let us define that the quadruplets (Q) are - 

 didymous radicals of fifteen groups of the fourth order, the 

 6-plets (S) those of ten groups of the sixth, and the 5-plets 

 (T) those of six groups of the fifth order. There will be (art. 

 79) a suhstitution of the second order, ^^4 or ^^5, permu- 

 table with any four in (Q), or any six in (S) ; and as we see 

 that «i«p «4«i5 «i«2, are all permutables in Q, we have in our 

 first triplets a^a^a^ = i = bj)^^ — kc, and each of these fifteen 

 small letters is 6^ permutable with a set of (Q) . And as 

 fl,AB= I in the first 4-plet, we have A and B for the sub- 



