34 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
showing that there are three different simple networks w r hich divide S into p sets of 
a units, and so on. We may in such a case write the equation thus :— 
205. Two distinguished simple networks, components of a single system S, either 
divide S into the same detached sets, or the pairs of one of the simple networks are 
pairs connecting the detached continuous portions of the other. For pairs belonging 
to one simple network must either be all component pairs of the sets into which S is 
divided by the other simple network or all pairs connecting them. 
206. A network composed of two or more simple networks may be called a com¬ 
pound network, and may be called double, treble, &c., according to the number of the 
component simple networks. A compound network will consist of one or more undis¬ 
tinguished detached continuous compound portions. The units of each of these 
portions constitute a set. 
207. Where a network simple or compound consists of only one detached portion, it 
may be said to be complete. 
208. If the pairs of each simple network of a compound network N connect 
detached portions of the compound network composed of the remaining simple net¬ 
works of N, then the compound network N may be said to be pure. 
209. Any two simple networks of a pure compound network may be said to be out¬ 
side each other. 
210. If each pair of a simple network connects units which are both components of 
the same detached portion of a simple or compound network, the latter network may 
be said to enclose the former. 
Chains. 
211. A succession of undistinguished pairs, ab, be, cd . . . may be termed a simple 
chain. Where the chain has no terminal units it may be said to be closed. Every 
simple chain is a portion of a closed chain, unless it contains one pair only, when it 
may be a pair connecting two systems. 
212. Compound chains are such as contain distinguished pairs, or undistinguished 
pairs of opposite polarities. 
213. In the case of the pairs connecting two systems we may have networks 
consisting entirely of undistinguished pairs ; but every chain in each consists of succes¬ 
sions of pairs alternately of opposite polarities. 
214. A complete network of a system contains simple or compound chains connect¬ 
ing every two units of the system. 
215. If a chain of pairs of a pure compound network contains one pair only from 
one of the component simple networks, it cannot be closed, for if it were the single 
