MR. A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
33 
197. Where a single system P of pairs connects two single systems S and t, the 
number of pairs of P which connect any unit a of S to units of 2 is the same as the 
number of pairs of P which connect any other unit b of S to units of 2. The number 
in the case of units of S is the same as the corresponding number in the case of units 
of 2 if S and. 2 contain the same number of units, but not otherwise. 
198. Where the pairs of a single system of pairs are components of a single system 
S of units, the number /x of pairs of the former system which connect any unit a of 
S to other units of S is the same as that in the case of any other unit b of S. Also 
the number of such pairs connecting units of S to a is the same as that of those 
connecting units of S to any other unit b of S, being in both cases also /x (sec. 136). 
199. A single system of component pairs of any single system of units constitutes 
a simple network of which the number /x of the last section may be called the ivay- 
nurnber. This network consists of one or more portions each continuous and detached 
from the other portions. Where there are two or more detached continuous portions 
of a simple network, each is undistinguished from the others, for component pairs of 
distinguished portions would be distinguished. 
200. Every simple network is accompanied by one in which the pairs connect the 
same units but in the reverse order, so that if ab is a pair of one network, ba is a pair 
of the other. The two networks may be called the reverses of each other. If we 
represent the unified pairs of the one by single letters a, a, a, a, we may represent 
those of the other by a! , a , oi , a'. Here the sorts of the letters may be regarded as 
representing the networks when regarded as units. 
201. We may use the symbol (a) to represent the network, of which a is a unified 
pair. Here (a) takes the place of the sort of a. Or we may use (ab) (sec. 263). 
202. The units of a continuous detached portion of a simple network form a set, for 
any two are connected by a chain of one or more pairs of the network, while no com¬ 
ponent unit is connected by such a chain with a detached unit, so that all component 
pairs are distinguished from all connecting pairs of the collection (sec. 135). 
203. Thus a simple network divides the units of a single system into one or more 
detached sets, each of the same number of units. If, then, p be the number of units 
in each set, and cr the number of sets, pcr=n the number of units in the system. Thus 
p and cr must each be integral factors of n. 
204. Equations such as 
p q r' s, 
abed 
wher e pa=qb=rc=sd=n, may be employed to denote the fact that there is a simple 
network which divides the units of S into p sets of a units, another which divides 
them into q sets of b units, and so on. We may have equations such as 
^ ' a a a b b’ 
MDCCCLXXXVI. 
F 
