.8 
918] FOR TWO, THREE, FOUR, FIVE, SIX, SEVEN, AND EIGHT LETTERS. 119 
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upon the three letters; and so in other cases. In the case, however, of two letters, 
I write simply (ab) to denote the complete group (1, ab) of substitutions, and so 
also for any substitution such as ab. cd, where the complete group is (1, ab. cd), I denote 
the group by (ab. cd). Moreover, (abc) eye., denotes the group of cyclical substitutions 
(1, abc, acb) upon the three letters; and so in other cases. 
A group will in general contain positive and negative substitutions, and, when 
this is so, the positive substitutions will form a group which is denoted by the 
symbol, pos.; the negative substitutions (which of course do not form a group) are 
denoted in like manner by the symbol, neg. Thus, (abc) all, pos., or for shortness, 
(abc) pos., will denote the group formed by the positive substitutions of (abc) all; 
(abc) pos. is thus the same thing as (abc) eye., but obviously (abed) pos. and (abed) eye. 
have quite different meanings. It is to be noticed that, for any odd number of 
letters, the substitutions of a group (abc) eye. are all positive, and thus (abc) eye. pos. 
is the original group: for any even number of letters, the group (abed) eye. or 
(abedef) eye. contains positive and negative substitutions, but the positive substitutions 
thereof form a cyclical group, thus 
(abed) eye. pos. = (ab. cd) eye., (abedef) eye. pos. = (ace . bdf) eye., 
and the notation, ( ) eye. pos., is thus unnecessary, and it will not be used. 
Substitutions or groups which have no letter in common are said to be inde 
pendent. The product of two independent groups is of course a group, and the 
components may be called independent factors of the resultant group: we use for 
such a product the notation A . B, and call the group a composite group. If each 
of the groups A and B contain positive and negative substitutions, we thence derive 
a new group, (A. B) pos., viz. the substitutions hereof are the products of a positive 
substitution of A and a positive substitution of B, and the products of a negative 
substitution of A and a negative substitution of B, say 
(A . B) pos. = (A pos. B pos.) + (A neg.) (B neg.): 
obviously the number of substitutions or order of the group (A . B) pos. is one half 
of that of the group (A .B). 
A more general, but not perfectly definite, notation is that of (A . B) dim., the 
dimidiate of the group A . B. Suppose, for instance, that A is a group of substitu 
tions of the letters (a, b, c, d); and that B is the group (ef). Here if A is 
composed of two equal sets 1, P, Q, ...; B, S, T, ..., where the first set 1, P, Q, ..., 
is a group, then we have a group 1, P, Q, ..., ef. R, ef.S, ef.T, ..., which is a 
group of the form in question, (A. B) dim. But in some cases the group A can be 
in more than one way divided into two equal sets the first of which forms a group, 
and a further explanation of the notation is required. I do not give at present a 
more complete explanation, nor explain the analogous notions trisection (tris.), &c. 
Substitutions or groups having a letter or letters in- common are non-independent. 
If A, B are such groups, then the substitutions of A are not commutable with