84 
and where the exponents of the factors p r are anything we please 
Z A + 1, the exponents of the factors q r are anything we 
please, z B + 1, &c., and where the same system of exponents 
is employed in the denominator through all the l — 1 substitu- 
tions Q to be formed. 
The l — 1 derived groups C^G, Q 2 G, QjG, form with 
G a grouped group of KZ substitutions, and the number of 
such grouped groups constructible on the given partition of 
N is 
r a -m — x b i ^ri~ • • x . . x j j ~ J rr * *) 
Iv (\ K + (l—\) R A ) 
where X is the number of the principal substitutions of the 
group g, when their form is 
A (?! (=a) 
+B i, (= + 4) 
+J Ji (4l?)5 
and where ^ =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) R a - 
The elementary groups ( 0 ) of these grouped groups are 
groups of K substitutions ; and each group (p) is composed 
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, •whose elements (p) 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 
g of the order l 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. 
