36 
MR. A. B. KEMPE ON" THE THEORY OF MATHEMATICAL FORM. 
225. The table, or constituent of a table, representing a group of v units has n rows, 
a letter of any one sort appearing once and once only in each column. 
226. The simple networks of a group are all one-way (sec. 199) networks. The 
closed chains which constitute the detached portions of any network of a group may 
be called simple circuits. Any closed compound chain of a group may be called a 
compound circuit. 
22 7. If the component pairs of one simple circuit are undistinguished from those of 
another, the two circuits must have the same period. 
228. If the period of a simple circuit be rs where r and s are integers, there will 
be r simple circuits of period s, and s of period r, connecting the units of the circuit. 
229. Now the number of units in a simple circuit of a group of n units must be a 
factor of n (sec. 203); so that if n be a prime number, there is only one form of 
group; for one simple circuit must contain all the units, and this being given all the 
other circuits are given ; they are all of period n. 
230. In a group if the pairs of one circuit are distinguished from those of another, 
but both circuits are of the same period, they may be said to be similar. The pairs 
of such circuits may also be said to be similar. 
231. If a, b, c, d, e, . . . ., A, B, C, D, E,. . . . be units of a group, and if a, b, c, d,e,.. 
constitute a simple circuit in the order given, and if the pairs oA, 6B, cC, c/1), .... 
be all undistinguished from each other, then the units A, B, C, D, E, . . . . constitute 
a similar circuit in the order given, 
232. In the case of a group the terminals of any chain \p.vp constitute a pair of a 
definite simple network (a), i.e., we cannot have \p.vp=a, and \pvp = (3. 
233. We may use the symbol (a/3y) to represent the network (X) where X= a(3y. 
234. The network of which the two terminals of a chain of pahs are a component 
pah, may be termed the product of the networks of which, the pahs composing the 
chain are components ; the order of the networks in the product being of course the 
same as the order of their component pairs in the chain. 
235. A pure complete network furnishes chains connecting every pair of units of a 
group, and thus if a, (3, y, be unified pairs of the simple networks of such a 
network, we can with them make chains a/3, /3y'\ 8a, &c., whose terminals constitute 
pahs of every simple network of the group. A group is accordingly fully defined if 
one of its pure complete networks is given. 
236. We have, whatever pair a may be, aa=l (sec. 200), and if a be a symme¬ 
trical pair a—a, and thus a a =l. 
237. If a/3=/3a, the two networks (a) and (/3) may be said to be commutative. 
238. If a simple network is commutative with each of a number of others, it is 
commutative with every simple network which is enclosed in the compound network 
composed of the latter. 
239. The system S arrived at by compounding together a number of independent 
groups in the mode described in sec. 189 is a group having component sets of the form 
