MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
31 
181. If m = y, and m ^ m, m' will be a factor of m, and 8' may be said to be a 
factor system of S. We have here m = m'k = y. 
182. If S' be a factor system of S we have k r = 1 ; i.e., there is no self-corre¬ 
spondence of S, S', other than the identical-correspondence, in which S is identically - 
correspondent. If S' is not a factor system of S, we have k > 1, and there are 
self-correspondences of S, S', other than the identical-correspondence, in which S is 
identically-correspondent. 
183. Hence if S' be a factor system of S, there are no units of S' which are 
duplicates with respect to S. And if there are units of S' which are duplicates with 
respect to S, then S' is not a factor system of S. Further, if S' be not a factor system 
of S, there are units of S' which are duplicates with respect to S, unless S and S' 
are replicas of each other. And if there are no units of S' which are duplicates with 
respect to S, then S' is a factor system of S, unless S and S' are replicas of each other. 
184. If S' be a factor system of S, each single system of S' must be so also. For if 
m/ be the number of self-correspondences of S 1 ' a single system of S', we have 
w/ m, i.e., m/m jy mm < y, i.e., m/m < y. 
185. If S' be a factor system of S, it is also a factor system of A, the single system 
composed of the unified aspects of S. This follows immediately from the fact that the 
combined system S, A has the same number of self-correspondences as S has. 
186. Hence also every single system of S', a factor system of S, must be a factor 
system of A. 
187. A system which has no factor systems, except systems of one unit, may be 
termed a prime system. 
188. If S" is a factor system of S', and S' is a factor system of S, then S" is a factor 
system of S. For let a", b" units of S" be duplicates with respect to S ; then we 
have a"S >-<6"S. Now if c be any unit of S', there must be a unit d' such that 
a"c'S >-< b"d' S ; and since S' is a factor of S, we cannot have c'S >-< d' S, unless 
c is d'. Thus we have a"c'S>-<6"c'S, i.e., we have ct"c'>-< b"c, whatever 
unit c of S we take, i.e., a ' is b", and there are no units of S" duplicates with respect 
to S, i.e., S" is a factor system of S. 
Three Modes of Compounding Systems . 
189. There are three modes of deriving a system from two or more other systems to 
which a passing reference may here he made. In the first the compound system 
is arrived at by regaining the w-ads connecting n independent or related systems as 
units. 
190. Again, we may have a system which may be regarded as composed of n 
independent (and therefore detached) undistinguished sets, each of a given form F, 
and such that their unified aspects compose a system of the form F'. 
191. An important special case of such a compound system is that in which the form 
