44 



CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



This is extended by T\% C\ to a G s , which 71 3 Ci extends 

 to the total group : 



G: 



24 



i , G C 2 (three), 7"i 2 71 3 (two), T 12 T 13 C { Cj (six), 

 7i2 Cj (nine), 7] a G C 2 C 3 (three), 



of which the first line constitutes our group H. 



2. For fl n =$, m = 3, the groups Q and O are the 

 above <9 4 and G 2i respectively. The latter is extended to 

 the total group £120 by the substitution 



R = 



112 

 121 

 211 



R 2 =i. 



Indeed the left-hand multipliers in a rectangular table for 

 £120 with £24 as first row are 



1, R, C,C 2 R, CxC z R, C%CzR. 



This follows from the relations, 



C x C 2 C 3 R = RC l C 2 C 3 , CxR=C % C 3 RCx C 2 C 3 , 



T n C\R = C 2 RT 23 , T 12 C S R=C S RT 2S , 



TuT n R = RT 13 T 12 , etc. 



We might express the £120 by the formula 



(R C 3 y { Gn} Cx (• = *,- -4). 



Indeed (R C 3 ) 5 = Cx C\ C S} 



while no lower power of R C s is contained in * G 2i \ Cx . 

 Thus 



(i?C 3 ) 2 = 



2 

 1 

 1 



1 1 



1 —2 



2 1 



3. For f l =n, m = 3, the Q y are C A Cj or 6^;f. 



The group Qi>2 is of order six, since 0££is of period 

 three. The left hand multipliers extending Q to the total 

 group are of the types (where y = 5 if) : 



