84 



and where the exponents of the factors /),- are anything- we please 

 Z A + 1, the exponents of the factors q,- are anything we 

 please, Z B + 1, &c., and where the same system of exponents 

 is employed in the denominator through all the / — I substitu- 

 tions Q to be formed. 



The I — 1 derived groups QiG, Q^G, QjG, form with 



G a grouped group of K/ substitutions, and the number of 

 such grouped groups constructible on the given partition of 

 ISIis 



Rj,M wt^(A«-rr • • X EM -' r • • X . . X J-^-rr • •) 



K (X K + (/— X) Kx) 

 where X is the number of the principal substitutions of the 

 group g^ when their form is 



A «i (=^) 



and where X =0 in every other case R/, is here the number 

 of integers less than K and prime to it, unity included. 



The number of principal substitutions in each of the S 

 grouped groups is 



X K + {l-X) \\^ 



The elementary groups (p) of these grouped groups are 

 groups of K substitutions ; and each group ((>) is composed 



K 



of a vertical column of— ^ square groups of powers of a sub- 

 stitution of A elements, or of a column of— ^ square groups 



of powers of substitutions of B elements, &c. 



These grouped groups, w'hose elements (^)) are made up of 

 groups of A powers, or of B powers, are grouped groups of the 

 first class. 



(70) In theorem H we have restricted the auxiliary groups 

 cj of the order / to such as have no circular factors, formed 

 with the a elements, whose order does not divide A, etc. By 

 this limitation we have exactly enumerated a definite and 

 large class of equivalent grouped groups. 



