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IX. The groups T, whose titles are thus consistently 
described, are now considered possible : their reality is 
proved, if we can construct them. This construction is 
always indicated by the group K and by the completed title. 
We have in general to multiply K by a root or roots of one 
of its lowest substitutions. These roots are readily found by 
the tactical method of evolution, on which I believe that I 
have had the honour to write all that has been written, and 
what is equally important, in this theory, with uivolution. 
Sometimes we have to multiply K by a substitution 6, which 
has for one didymous factor one of the square roots of unity 
in K, and for the other a nermutable with another in K. 
This 0 is always most easily found by a tactical method. In 
every case in which a possible group is indicated by a com- 
pleted title, I have thus far found that the construction 
lollows by the most simple and rapid tactical methods. The 
construction of K is of course tactically known, as is that of 
every group transitive or intransitive in our register. And 
we can always give a clear account of the equivalent or non- 
equivalent groups T, which are built on the same base K. 
The determination of the titles of groups T is assisted 
throughout by the general theorems of my Memoir, on 
grouped, woven, and woven-grouped groups, of which we 
know the majority for all values of n, by Avell-defined 
constructions, as well as the enumeration of their equivalents. 
The greater number of intransitive groups are found to be 
inadmissible as bases K of transitive groups T, for all values 
of n, by simple rules deduced from (HI). Of those not thus 
excluded, the bases K, which form part of no 'I 1 made with 
n elements, are rapidly disposed of, chiefly by the theorem of 
(III). And the demonstrations, whether affirmative or 
negative, are always very briefly drawn from the preceding 
considerations and from what has before been written. 
The most remarkable thing in this method is, that we need 
no algebraical substitutions : avc are never conscious of their 
