GROUPS AND MANY-VALUED FUNCTIONS. 361 



example of Art. (19)^ that when N = 8, 5 is the only 

 primitive root > 1 of the congruence 



^o (mod. 8), 



x-i ^ ' 



and we have 



5^^i (mod. 8) j 



Tin 



whence /3 = 2, and there are (theorem D) -^^ equivalent 



o 



groups of 32, each comprising two sets of powers of sub- 

 stitutions 6^ of the eighth order, which have however 

 common powers 



when a is a factor of 8. 



Tin 



It follows by theorem A, (9), that there are ~ equi- 



JT8 -8 

 valent groups each of -j^ — =64 made with ^ght elements. 



When N=9 we have only j9 = 4 and i?i = 7 primitive 

 roots > 1 of the congruences 



^^^1 and ^o (mod. 9). 



7JO 



Hence p = 3, and there are ^ — (theorem D) equivalent 



n8 



groups of 54 ; whence by theorem A there are 7 — equi- 

 valent modular groups^ each of — ffo~^ — 9 * ^ • 3j ™ade 

 with nine elements. 



When N=i6 we have only ^ = 5, Pi = 9) P2=^3> 

 primitive roots > 1 of the congruence 



/v>16 J 



a?*^i and ^o (mod. 16); 



X- 1 



whence p = 4., and we have (theorem D) -^ — ^ equivalent 

 groups each of 16 -8 substitutions, and consequently by 

 theorem A, ^ — - equivalent modular groups each of 



1115 -^ 



SER. III. VOL. I. 3 A 



