316 KEY. T. P. KIKKMAN ON THE THEORY OP 



denominator in a different order, so that B = w, if /3=3 

 &c. ; and where the exponents of the a factors p^ are any- 

 thing we please >o, < A+ i, those of the b factors q^. are 

 anything whatever > o, <B + i^&c. ; the same system of 

 exponents being used in all the /- 1 derivants Q, to be 

 formed. 



The I- X derived groups Q^G Q^G • • Q^-iG form with G 

 a grouped group of K/ substitutions ; and the number of 

 equivalent grouped groups constructible on the given par- 

 tition of N is 



K(XK+(/-A,))E^ 

 where the form of the principal substitution © of the 

 group ^ is 



/0 = Ai«i + A2a2 + A3«3+ . • ( = a) 



+ B161 + B262 + B353+.. ( = 6) 



+ : i 



+ Ji ii + ^Ji + 3Js + '• {=j); 

 where X is the number of these principal substitutions of 

 g, when their form is 



f& = Aai ( = a) 

 + Bb, { = b) 



+ 



and where \=o in every other case. 



E-;;. is here the number of integers < k and prime to it, 

 unity included. 



The number of principal substitutions in each of the 

 grouped groups is 



The elementary groups (p) of these grouped groups are 



groups of k substitutions, and each group {p) is composed 



k 

 of a vertical columrn of -j square groups of powers of a 



A. 



