GROUPS AND MANY-VALUED FUNCTIONS. 357 



tlie N"* order ; for, if not, we shall have 



{pi + c^)\=)pH +^^Ci(=)«, 

 for a value of h < N. But as 



' > o by hypothesis, and c^ is = i , or prime to N, 



^^_ V i = o (mod. N) 



is impossible. Therefore pi + c^ is of the N"' order. 



And we thus see that the entire number M of substitu- 

 tions of the W^ order in the group G'=^{pi -\-c) (15) is not 

 less (12) than Hjv times the number of terms in the series 



^PxV'iPz-- 



of integers < N and prime to it, including unity, which 



are primitive roots of the congruences 



»^- 1 

 p^'^iX and ^^- ^o (mod. N), 



in which m is a divisor > 1 of N. 



It is easy to shew that M is not greater than that 

 number. 



The derivatej^e'G is the derived derangement 



of 



G=i+^ + ^2_^..=S(i + c), 



and has exactly the same vertical rows ; that is, (Art. 6), 



^_P, 



^i~Pi 

 are the same substitution. If then Q and P^ be of the 

 same order, & and P^ will be of the same order. There 

 are then as many substitutions P^ of an order below the 

 N*'' as there are powers of Q below the N^'*, which number 

 is N-Rjv (12). 



72. Wherefore it is proved that the number of substitu- 

 tions of the N"' order in the group G' = S(j3/ + c) (Art. 15) 

 is 



^n9) 



