138 
and a b c . . are either all positive or all negative clear 
signatures. 
VI. An important theorem is the following : — 
Theorem . — If d„ and <p„ he two substitutions of any order p 
having the same signature a, in any group T, and if Q a and 
(J) a are not one a power of the other, and have not a 
common power different from unity. T cannot contain 
(as A„) fewer than (2 -f- R ;) ) R y) substitutions having the 
signature a, where ly, has the value above given (III). 
If 0„ and (p a have a common power different from unity, 
that is, if they be roots of -the same substitution X, and 
not one a power of the other, the number A may he less than 
(2+Il ; ,)R^. If X is unclear, or clear and positive in a mixed 
group T, we have before us the number of substitutions in 
T of the form of X. The number of rth roots that any 
substitution X can have is easily deduced from the theorems 
of my published Memoir. Thus we have a conditioned 
limit of A. And, by the inferior groups already registered, 
which contain more than one rth root of X, we know in 
general, by the presence or absence of certain other substitu- 
tions, whether our group T can or cannot contain more than 
one rth root of X. 
VII. The above considerations in most cases enable us to 
determine the values of ABC .... If obscurity remains, we 
must have recourse to the consideration of the numbers 
Q k and Q t , of the equivalents of K and T. Q K is always 
given, and we easily prove that 
Qt=AQ k > 
and also that 
«.<?., that h is some divisor of a given number. When the 
group T is mixed, we know the value of h exactly, by virtue 
of a remark in Art. V. We know also, by the theorems of 
my printed Memoir, the entire numbers V„ V h V c V„ . . . . 
