GROUPS AND MANY-VALUED FUNCTIONS. 323 



We understand by {a„^ + c) (A» + ^) (7« + c) the resi- 

 dues, 



«„,-{- c, (mod. A), 



^,n + c, (mod. B), 



7,„ + C; (mod. C), &c. 



k . 

 The substitution i + c of the group «,„ + c is -^ times 



repeated in the column 



i 



i+t 

 i + 2 



i + k- 1 



4- /• is 



B 



and every substitution i + c oi /3^ + c is ^ times repeated 



in a parallel vertical column of Gr. 



41. Let M be any one of the numbers 



AB-.J. 

 Let y < M be a primitive root of the congruence 



a^^'^i (mod. M) (r>2). 

 We can add to the group 



{f^m + C) 



r- 1 derived derangements Art. (16), which complete with 

 {[Jbjn + c) a regular group. 



Let us conceive that ?/ is a root either primitive or not 

 of the congruence 



af'^ 1 (mod. X), 

 X being every one in turn of the number AB • • J. 



Let us add to each of the a + b+ > > ■\-j elementary 

 groups of (a, in the same vertical column, the r- 1 derived 

 groups 



e being every one of the numbers 1, 2, 3 • • (r- 1) and 6^ 

 being every one of aj^^^^y,,^ • • • We add thus to 



G= (a„, + c) {/3^ + c) (7,„ + c) . . 

 &,Q = y'{a^ + c) y'{^„ + c) y'{ry^ + c) • •, 



