86 



Let N=:A-rt + B i + C c + . . + Jy, 



A7B7C7..7J; K being the least common multiple 

 of A B C . . J, and ahc..j being any numbers. 



Let E < K be any one of the numbers A B C..J; 

 and let E— 1 be a primitive root of the congruence 



a'" ^ 1 (mod. M). 

 M being any one of the uumbers A B C . . J ; and such a 

 root that it is also a root, primitive, or not, of the congruence 



x'' ^ 1 (mod. X), 

 X being one each in turn of the numbers A B C . . . J. 



We can form by the aid of this root E — 1, on the given 

 partition of N, Rjc being as in theorem J, 



E'^TT N _ y 



R, Tra • 7rZ> • • • TTJ A« B* • • • J^ ^' 



equivalent groups of Kr substitutions, e being the multiplier 

 of E in the partition (Aa + BZ> + . .) of N. These groups 

 are all of the form 



G + 0iG+e2G+.. + e,_i G, 

 when G is any one of the W groups of theorem C* 

 For example, let 



N=:26=:10-l-f 8-2+4-2 (E=40). 



We have the primitive root 4—1 of the congruence 



x',^1 (mod. 10) (Ee=4-2) 

 which is also a root of the congruences 



x^^i (mod. 8) and 

 x^=^l (mod. 4). 



One of theW:- ''"^^^^ 



R,o(«-.2)-nO-8--4- 

 groups G of theo. C is 



1234567890 abed efg h ij hi m n ]) q =G 

 2345678901 bed efg h a j kli np q m 

 3456789012 c d efg hab hlij p q m n 

 4567890123 d efg hab c lijk qmnp 



of 40 substitutions (1 A A" . . . A'^l) 



