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



Let g be any model group of I substitutions formed 

 with 



« + 6 + c H Vj 



elements, such that the first a vertical rows of g written in 

 a column shall contain only the first a elements of unity, 

 the h nest vertical rows shall contain none but the next h 

 elements, &c. 



Let this group be 



where 



1 = 1234 • • a ' ' 



& = al3'yS • -67) • • e • • 

 Q' = KXfJiv • ' p • ' 



(00') = cr^x4' • • ^ ■ ' 

 where 7] is one of the b elements, e one of the c elements, 

 &c. 



We form with the circular factors of G the substitution 

 following, with the aid of © in ^ : 



Pi Pi Pa ' ' Pa 9a+l 9(1+2 ' • Qu+b ^a+b+l ' ' 



in which the factors of the numerator are those of the 

 denominator in a different order, so that B = ra, if yS=3 

 &c., and where the subindices of the numerator are the 

 elements of 0. 



29. The effect of the substitution Q, in the derived 

 group QG is merely to change the order of entire vertical 

 ranks of G; that is, QG is a derived derangement of G. 

 In fact we see that the A vertical ranks in which the 

 elements of the circular factor pa repeat themselves, in 

 QG, occupy exactly the places of those in which the cir- 

 cular factor pi repeats itself in G; the precise order of 

 those rows being determined by the substitution 



Pt^ 

 Pi' 



