136 
The first is made with four elements, the other two with 
eight. The term 6, shews that there are six substitutions 
having each a circular factor of the fourth order. The term 
6 21 2 shews six substitutions, having each one circular factor of 
the second order, and two elements undisturbed ; the term 
832 j2 exhibits eight substitutions, each having two circular- 
factors of the third order, and two elements undisturbed. 
Exponents in the signature ( = sub-index) are merely co- 
efficients; 3 2 1 2 meaning the sum of 3311, and 2‘ meaning 
that of 2222. 
The clear terms of a title (clear of units) as 6., , 8 6 , , are 
those in which no units are read; these have no element 
undisturbed: the unblear items, 'hs^a 2 , 3m, shew units, that 
is, undisturbed elements, in the signature. 
III. From the title of any group K, transitive or intran- 
sitive, made with n — y — 1 elements, and of the order It, 
the unclear terms of the title of every transitive group T, 
made with n letters, and of the order nk, which contains Iv, 
are all written at once by the general theorem following. 
Theorem. The number of substitutions of a given form 
and of any order p in the group K, is always the term of 
the title, II,, being the number of integers, unity included, 
which are less than p and prime to it, and i being the 
signature, containing l" 1 , or shewing m (>0) undisturbed 
letters : the whole unclear term .1, of the title of T is deter- 
mined by (aR ; ,)„ and is always — where the 
signature,/ differs from i only by having y + 1 more units, or 
l m+y+1 for l’\ 
7LCC 
If the number — T is not an integer thus obtained 
m + y + 1 
from every term of the title of lv, no transitive group T made 
with n elements exists containing tire base Iv. 
IV. The unclear terms of the title of T being thus alj 
found, the clear terms are next required. First, we know, 
