THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 209 



four substitutions of the fifth order, which complete the 

 non-modular group of 60. 



The triplets (F) and the quintuplets (G) are correctly 

 written thus, with changes of subindices only, 



ahjd^j ac^e^, aj)^e, a^c^d, ajbc, a^e^dj^f h^cjlj^j 

 ab^e^cdf ac^dj^e, a^hji^ce^y a J) cjl^e^y ajb^ej^d^c^y 



in the exact order of their succession as didymous radicals. 



It is remarkable that this group (J) of 60 diflPers not in 

 the number and form of its substitutions from the group of 

 60 made with 12345 written parallel in a column with the 

 same group made with 67890 ; but it is not, for all that, 

 equivalent to the latter. The latter is intransitive ; for i 

 could appear only in vertical circles of five made with 

 12345, but in this group last constructed i is found in a ver- 

 tical circle of five with every other element, and the group is 

 transitive as well as non-modular. 



This group of 60 is given analytically by M. Betti in M. 

 Hermite^s ' Theorie des Equations Modulaires,^ Paris. I 

 know not whether its first discovery is due to Galois, Betti, 

 or Kronecker. 



Take fifteen elements, a^a^a^^ hfijb^^ CiCzC^, d^d^^d^, ^i^z^j, 

 forming five triplets, and ten capitals, ABCDEFGHIJ. 

 It is easy to form the fifteen triplets following : — 



ABa, 



KM, BQd, FEe, IDe^ CJ)b^ GcU, 



Mc, BGe, FH^, IJc?, CEc^ GHc, EJ«^ HD« -J 



(R) 



where the fifteen small elements are once employed, where 

 the capitals are any fifteen duads showing each of the ten 

 letters AB ... J three times, and where the small letters 

 are combined with them in any way so as to answer the 

 condition that the five triplets in which X and Y occur shall 



SER. 111. VOL. II. p 



