139 
of substitutions having any signatures a b c g, whether clear 
or unclear ; and if o„ p h p c p g . . be the number of times that 
V„ V»V.V,.... are repeated in all the Q T equivalents of T, 
we readily prove, as in the 86th article of that Memoir, that 
/ . 7 r\ V aPa V* 6 1 
Vt — '^Vk — ~7 - 
B & “ C c 
Gr„ 
whence we obtain 
b- A a AB a I hi’ 7: blCj c 1' ,:{)<:> &C., 
F a F t F c being known numbers. Hence 
b L o = F„p„ + Fjp;, + F c p f + . . . 
where either h is known, or the possible factors of h arc 
known. In every case yet considered in which T is impos- 
sible, its non-existence, if not exposed before, is proved by a 
simple absurdity, as that h does and does not contain a 
certain factor g. If, after clearing the two members of the 
last equation of common factors, no such absurdity appears, 
we select values of p^ b pc consistent with the preceding con- 
siderations, thus determining admissible systems of values of 
ABC • • • to fulfil the condition 
L o = A„ + B,, + C c + . . . j 
and we complete with each system a title of a group T. 
VIII. It is sometimes a simpler consideration than (VII.) 
to take account of the products of the substitutions of the 
second order in the title. If it be a real one, we can account 
for every such product <*(3. These are either permutables, 
giving the product aft—y of the second order, or else didymous 
factors of higher substitutions of the title. The number of 
substitutions permutable with « of the second order, the 
number of different sets of didymous factors that <j> of any * 
order can have, the number of substitutions 0, \p, ■%, of like 
or different orders, under which a can be written as a didy- 
mous factor — all these are readily deducible from the theorems 
of my published Memoir. A title is thus often easily shewn 
to be impossible, by the fact, that no account can be given 
of the products of its square roots of unity. 
