GROUPS AND MANY-VALUED FUNCTIONS. 359 



which sets will be found in any other equivalent. We 

 obtain, therefore, the number of equivalent groups by 

 dividing by p that given in theorem B (12) of the equiva- 

 lent groups of powers of a substitution of the N"' order. 

 73. The correction required in theorem D, (14), is 



simply to write — ^^ instead of — ^s ^ in the first 



line of itj adding the definition, thai p is the number of 

 integers, unity included, which are <N and prime to it, 

 and ivhich are primitive roots of the congruences 



x^^i and ^ o mod N, 



X— 1 



in which m is any divisor > 1 0/ N, 



If the paragraph beginning at line nine and ending at 

 line fifteen of page 288 be erased, the fourteenth and 

 fifteenth articles are corrected. 



The groups of theorem E are all equivalent to the 



simple group 



G"=S(^i + c), 



in which p is every power of any primitive root of the 

 congruence chosen 



^'■-1^0 (mod. N). 

 The correction required at page 289 is to write in the 

 nineteenth, twenty-fifth, thirtieth and thirty -first lines, 



n(N-i) ^^^ n(N-i) ^ 



and to add the definition, that p is the number of the 

 integers 1 pp"^ • •p^~^, which are powers of the primitive 

 root employed of the congruence x'' -x^ o, when r is a 

 divisor of N, and which are roots of no congruence 



^37 — ° (n^od-N), 



in which m<N; and that p= i, when r is not a divisor of 

 N. 



"We have |0= 1, if N is a prime number, in both theo- 

 rems. 



