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by the elements of this theory, that no substitution can be 
in T, unless all its powers are T. The unclear powers of all 
substitutions in T are already written ; hence we know by 
inspection what clear substitutions having unclear powers are 
possible. This always reduces the admissible clear signatures 
to a small number. Let L, be the sum of the unclear terms 
of T. Then nh — L 4 = L 0 is the sum of the clear terms, and we 
have 
L 0 — A a + Bj + C c + . . 
a , b, c . . being the admissible clear signatures, and ABC . . 
being sought numbers. 
Y. Def. A positive substitution or signature has an even 
number (>0) of even circular factors. Thus, in the first of 
the three above written titles, the signatures of 1, 8 31 , and 
3,-* are positive. In the other two titles all are positive. 
A signature or substitution not positive, is negative , as 
in 6 21 s , 6 4 . 
A mixed group has as many positive as negative substitu- 
tions, and the former of themselves are always a group. A 
positive group has no negative substitutions. 
Hence if L t is all positive and greater than \nk, we know 
that L 0 has only positive signatures, and all its terms have to 
be found. If Li is mixed, we know that T is mixed, and we 
have its positive half in our tables of titles. It may be 
transitive or intransitive, and is easily recognised b)' com- 
parison of L 4 ; and w r e know that the negative half is the 
positive half multiplied by any negative substitution in L, ; 
i.e. the group T is completely given by construction, although 
we have not yet its title. We extract from the positive half 
in our tables tire positive clear terms in L 0 , and the negative 
terms of L„ alone have to be found. In any case -we have, 
(M5O), 
L'„=L 0 — M=A a +B t +C c + . . 
where ABC... are sought numbers whose sum is known, 
