254 CHAPTER XL 



We proceed to prove that the total number of substitutions of 6r 

 Of canonical form C with a, /3, y distinct is, for d = 1 or 3, 



, JT ff2n 



- 2 " 



By 220, the only ternary homogeneous substitutions commutative 

 with C with a, /5, y distinct are the (p n I) 2 substitutions 



T: x'=ax, y'=by, z'=cz 



For <$ = !, each set of unequal multipliers therefore leads to 



conjugate substitutions, so that we obtain the number 244). For 



^ = 3, the substitutions T give only ~^(p n I) 2 distinct substitutions 



in 6r. Furthermore, by 102, C can be transformed into 0(7 if, 

 and only if, the multipliers a, /3, y form a permutation of 1, 6 7 6 2 . 

 The special substitution (7, 



is transformed into C, 0(7 or 2 (7 by exactly the 3(y I) 2 products 

 Tj (xyz) r L, (xzy)T. The corresponding substitution is therefore one 

 of N! (p n I) 2 distinct conjugate substitutions under 6r. Each of 

 the remaining substitutions (7 with unequal multipliers is one of a 



set of N -T- ~o-(p n I) 2 conjugate substitutions under 6r. 



Corresponding to the p n d 1 sets of multipliers a, /3, y of 

 which two are equal, there are ~^(p n d 1) substitutions C" of 6r, 

 no two of which are conjugate. Such a substitution 



(7': x'=ax, y' = ay, 2'=ys (a 2 y = 1, y 4= a) 



cannot be transformed into 0(7'. By 218, the most general ternary 

 linear homogeneous substitution which transforms C' into itself is 



x^ax + by, y' a'x + &'?/, 2 r =c f 2. 

 The number of such substitutions in the 6rJP[jO n ] of determinant 



Unityis 



Hence the total number of substitutions in 6r of the canonical form C' is 



1 / -4 \ -iV 



245) 



_ 



233. As a check upon the accuracy of our enumeration of the 

 substitutions of 6r, we may verify that the numbers given by the 

 formulae 236), 237), 239), 241), 243), 244) and 245), together with 

 unity, to count the identical substitution, give as total sum the 

 order N of the group 6r. 



