135 
have been together printed in the Memoirs of this Society ; 
while there is a considerable portion of the positive, besides 
nearly all the negative results, the more difficult to establish, 
which are comprehended under no theorems elsewhere to be 
found, whether definitely expressed, or indefinitely by 
imperfect description of limiting cases. All that is necessary 
to be added to that treatise, including the supplement, for 
the complete mastery of the problem, is briefly given in the 
following nine articles. The tables which next follow give 
account of all transitive groups made with fewer than eleven 
elements. 
I. Every transitive group T, made with n elements, is of 
the order nk, and contains a group K of the order k, made 
with n — y — 1 elements, of which the nth, or final, is not 
one, which may be a transitive or intransitive group K, and 
which we call the base of the group T. Hence the problem 
is completely solved, if, when any such group K of the order 
k is given, we can construct every transitive group T made 
with n elements, which contains K, with an accurate descrip- 
tion of T, and with the enumeration of its equivalents, or 
prove that no such group T exists. We thus establish all 
the required results, both positive and negative. 
II. The group K made with n — y — 1 elements, is given 
at first sufficiently by its title, and by the number Q K of its 
equivalents, without its actual construction. 
Definition. The title of any group K exhibits the 
number of its substitutions of every form, i.e., having any 
circular factors, c.y. Here are the titles of three transitive 
groups K of the order k - 24, with the numbers Q of their 
equivalents : — 
24?= 1 + + G.-jp + 8^ + 3.^2 
24: — 1 + bgg + Gp + + 1 2 i 
24 ~lf ^39 ^3T" ^ 2‘ 
Q = 840. 
