MR. A. B. KEMPE OH THE THEORY OF MATHEMATICAL FORM. 
45 
263. The group (No. 5 of sec. 248 ; see fig. 4 6b) of 8 units given by the following- 
table :— 
u x y z u x y z 
x u' z y x u z y 
y z u x y z u x 
z y x u' z y x u 
u x y z u x' y z 
x u z y x u z y 
y z u x y' z! u x 
z y x u z y x u' 
is one of the species we are considering. The laws of multiplication of the component 
simple networks (sec. 234), viz. of 
( ux ), {uxj, {uy), (uyj, ( uz ), (uz ), ( uu ), (uu), 
(sec. 201) are the same as those of the quaternion expressions 
i, i, j, j, Jc, It, 1 , 1 . 
For example, we have uyy - <xz and uz> - <zu, and thus, just as we have 
ijk= — 1, so we have ( ux)(uy)(uz) = (ux)(xz)(zu') = (uu ). I shall accordingly term the 
group a quaternion group. 
264. A double netwmrk of which the detached portions (sec. 206) are each such as 
that shown in fig. 4 6b may be termed a 'pure quaternion network. The two component 
networks are non-commutative; each may be termed the conjugate of the other, 
and we may represent the conjugate of (a) by (a). 
265. I proceed to show that every group of the family now under consideration may 
be defined by a pure complete network (sec. 235) consisting of m pairs of conjugate 
networks (i.e. m distinct quaternion networks), and r other simple networks having no 
conjugates in the network, (2m-\-r simple networks in all) ; the 2ra+r simple net¬ 
works being such that each two are commutative unless they be a conjugate pair. 
266. Consider a pure compound network consisting of m distinct pure quaternion 
networks. If P be any chain of the pairs composing the 2m simple networks, each 
sort of pair entering only once, if at all, into the chain, we have Pa—aP7r or aP, 
i.e., P non-commutative or commutative with a, according as the chain contains or does 
not contain a pair a. Now if ( v ) be any simple network which is not one of the 2m. 
(r) will be commutative with some of the 2m, and non-commutative with others. Let 
N be a chain containing one pair from each of the conjugates of those of the 2m 
networks with which (v) is non-commutative, then the simple network (Nv) (sec. 233) 
3C Z' 
