Let iv equivalent groups be constructive of l substitutions 
made with a + b + c + . . j elements, of which the a elements 
are consecutive in unity, the b elements consecutive, the c 
elements consecutive, &c. ; and such groups (g) that every cir- 
cular factor in them made of the a elements is a divisor of A, 
every circular factor made out of the b elements is a divisor of 
B, &c. ; and also such groups ( g ) that every vertical row is 
composed only of the a , or only of the b, &c. elements. 
Any one ( g *) of these groups ( g ) begins by the power of a 
principal substitution (P 1 ), whose form is, by hypothesis, 
/P 1 = Ajflj + A 2 ge 2 + A 3 a 3 + • * (= «) 
d* + Bo5 2 + 1*3^3 + ' ’ (— + 
d~ CjCj + C 2 c 2 -p C 3 c 3 -p • • (= -p c) 
+ 
d - J1J1 d* J2J2 d - J3J3 d~ ' ‘ (— d - ^) 
this being the partition of a -p b -p . . . d- j, in which the 
equivalent groups ( g ) are formed either by theo. C or by any 
of those which follow. 
Let M be the least common multiple of the integers — 
AAA 111 A A ill A A — 
A] 9 A 2 A 3 iq B 2 ■ C'i C 2 
whose denominators are the orders of the circular factors of the 
complete group (y 1 ), being A c of the a elements, B c of the b 
elements, &c. 
Let the substitution Q following be formed : — 
ABC HE L 
V*' r * . : * : 
^ p p m p n ■ ■ v i" y e ‘ * ’ 9 r * ' ' * 
12 3 a a-fl a- j-2 rr-f-& a-j-fr-fl 
when (a (3 y ’ * 9 r] • * • * £ • • • . is) every one in turn of the 
l — 1 substitutions of the group ( g l ), a j 3 y * * * 6 , being the 
a elements, rj • • • • being the b elements, t • * * being thee elements, 
&c. ; and where the terms of the numerator are those of the 
denominator in different order, so that B = n if /3 = 3 , &c. 
