MR. A. B. KEMPB ON THE THEORY OF MATHEMATICAL FORM. 
35 
pair would not connect two detached portions of the compound network composed 
of the simple networks from which the remaining pairs are taken, but two units 
of a continuous portion. 
216. The number of pairs in a closed chain may be termed its period. The period 
of a closed chain is of course the same as the number of units it connects. 
217. A symmetrical pair of units may be regarded as constituting a simple 
closed chain of period 2 ; thus 
Fig. 28. 
218. It may also be convenient in some cases to regard each unit as con¬ 
nected to itself by a single barbed line constituting a simple closed chain of period 
1, thus 
Fig. 29. 
219. We may represent a chain of pairs thus ctAbpcvdpe, where a, b, c, cl, e represent 
units, and the sorts of X, p, v, p the networks of which the pairs ab, be, cd, de, are 
respectively components ; X, p, v, &c., being the letters placed alongside of the plain 
lines if we employ the graphical method of sec. 64. In some cases it will not be 
necessary to employ specitic letters to represent the units, they may be represented 
by the spaces between the letters representing the unified pairs, e.g., we may represent 
the chain given above thus aApvpe, where only the terminal units are given. If we 
have aApvpe and air ere, we see that the two units a, e are connected by the two 
different chains Apvp and ncr. 
220. We may have equations such as 
Apvp — TT CT — K, 
denoting the fact that the three chains Apvp, it a, k can have the same terminals, and 
we may write a chain such as Apppv for shortness X/xV and similarly in other cases. 
221. We may employ as the symbol to be attached to the barbed line of sec. 218 
in accordance with the method of sec. 64 the symbol 1. An equation such as Apv= 1 
will then show that the chain Apv on the left hand side of the equation is closed. 
Gro ups — Give uits. 
222. Any single set which is such that each component unit is unique with respect 
to each of the others, may be termed a group. 
223. The single system A of unified aspects of any system (sec. 185) is a group, as 
each unified aspect is unique with respect to each of the others. 
224. Every component set of a group is a group. 
e 2 
