83 



Let tv equivalent groups be eonstructible of / 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^), whose form is, by hypothesis, 



/Pi = A,a, + Km, + K,a, + • • (= a) 



+ B,3i + B,&, + ^A + • • (= + ^) 



+ C,c, + C,e, + C3C3 + • • (= + c) 



+ jji + ^-2h + ^zj\ + • • (= +y) 



this being the partition of « + 6 + . . . 4- 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 ILL A A 111 A -^ — 



Ai ' A2 A3 Bi B2 • • • Ci a 



whose denominators are the orders of the circular factors of the 

 complete group {g^)^ being Ac of the a elements. Be of the b 

 elements, &c. 



Let the substitution Q following be formed :— 

 ABC HE L 



Q =. 





when (a /3 7 • • ^ 1/ • • • • c • • • . is) every one in turn of the 

 / — 1 substitutions of the group (^^), a /3 7 ' * ' 6, being the 

 a elements, jj • • • • being the 6 elements, £ • • • being thee elements, 

 &c. ; and where the terms of the numerator are those of the 

 denominator in different order, so that B = ^^ if /3 = 3, &c. 



