44 
ME, A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
257. Every unit a is accompanied by one unit a , which makes with it a diagonal 
pair. We may term a the companion of a. We have (a'Y=a. 
258. Let each pair of the diagonal network be represented by a it, then if a be an 
unsymmetrical pair of any simple network of unsymmetrical pairs, we have 
oc — tt, aA—ct.', aA— 1. 
Also if (3 be a symmetrical pair we have /3 2 — 1. 
259. The network ( n ) is commutative with all the others of the group. For if a 
be any unsymmetrical pair we have a 2 —tt, and therefore a7r=aaa = 7ra; and if (3 be 
any symmetrical pair, either there is no unsymmetrical pair in the group, every circuit 
being of period 2, in which case every pair is commutative with every other, or else 
there are unsymmetrical pairs XX ... pn ... , &c., in which case let 
(3y=X, y/3=fx, 
then 
7r/3 = X 2 /3 =(3y.(3y.f3 — (3.yf3.y(3 =/3p, 2 = (3n. 
260. If a, (3 be any pairs, a/3=/3a or /3a n. For let a/3=\, and a(3=/3aa , then 
X' = a/3.a./3=a/3/3a.cr=ala.cr or a7ra.cr = a 2 cr or arncr 
= cr or 7tct. 
But X” — 7i or 1, therefore <r=7r or 1. 
261. We have 
according as 
a/3yS = ay/3S or ay/j7rS 
= ay/3S or ay/3§77 
(3y = y/3 or y(3rr 
i.e., if we start from any given unit and proceed along a chain composed of pairs 
belonging to a given collection of networks, and then starting from the same unit 
proceed along a chain composed of pairs of the same kind but in a different order, we 
shall either arrive at the same unit as in the previous case or at its companion. 
2(52. If we substitute for n the symbol —1, so that 
a' = Tra = —a 
we have, if /3 be a symmetrical pair, (3-= 1 ; if unsymmetrical, /3 2 = — 1 . Also, if a 
and /3 be any two pairs, we have 
a/3 = (3a or — /3a. 
