332 REV. T. p. KIRKMAN ON THE THEORY OF 



we write it in a column^ then affect the element x througli- 

 out with any exponent 7', not greater than the number of 

 vertical rows in the square group we intend to substitute, 

 and if we next so reduce the exponents in every vertical 

 row, that unity unaffected shall surmount all, we may 

 write everywhere for i™ the m"' cyclical permutation of 

 the first square group of R elements ; for 2* the q*^ per- 

 mutation of our second square group of R elements, &c. 

 The result is always a grouped group, and no two groups 

 thus obtained will be alike, though they will all be equi- 

 valents. 



The mode of stating these constructions, in the enuncia- 

 tion of theorem H, comprehends all forms of the results 

 for grouped groups of the first class. 



§ 8. 

 Grouped groups of a higher class, whose elementary groups 

 are of the order kr, comprising a group {g) of k powers 

 of a substitution and r- 1 derived derangements of [g) . 



40. Let 



A>2, A>B, B>C, 

 &c., abc- -j being any integers, and k being the least 

 common multiple of ABC- -J. 



The simplest of the W groups of theorem C (12) made 

 on this partition may be represented by 



G = («! + c) (a^ + c) o . (a„ + c) Oi + c) {^^ + c) • • 



{/3b + c){y, + c)-'{y, + c), 



where the variable «i is any one of the A first elements (i) 



of unity, a^ is any one of the next A elements &c., and 



where 8^ is any one (i) of the B elements 1 j, 2 j, 3 j • • Bj, 



where the subindex b affects not the numerical value, and 



/Sa any one (i) of the B following elements, &c. 



The constant c has any one of the values 



o 1 2. • k- 1 . 



