MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
49 
282. We have collections of triads connecting S A S M S„* in which each pah’ connecting 
and appears once in a triad, and once only, e.g., if S contains two units a, b, only, 
we have such collections as 
«A % «v 
«a K 
^A % «, 
K K a „ 
The number of such collections in the general case is n n \ each containing n 2 triads. 
When the triads are regarded as units we get n n2 collections each containing n 2 units 
of E. 
283. When these collections are unified we get a system A of n n ‘ units, which will 
be a multiple system. 
284. The form of A is in part independent of that of S and in part dependent, 
certain components being distinguished whatever be the form of S, while it depends 
on what the form of S is whether others are distinguished or not. 
285. When considering the multiplicity of the system A, or those peculiarities of 
form it possesses which arise from the mode of its construction apart from any special 
peculiarities in the form of S, we regard S as a single heap system. 
286. If S be a single heap system, certain units of A are undistinguished from 
each other, while others are distinguished, If S be not a single heap system, so 
that its self-correspondences are more restricted than they would be in the former 
case, some units of A originally undistinguished become distinguished, so that a 
unit which was one of several undistinguished units, may become unique. Thus 
certain units of A have definite relations to S, and are of use as auxiliaries to the 
latter. 
287. A convenient mode of conceiving of the system E is to regard its units as 
small cubes composing a big one, the axes of the cube being represented by X, g, 
and v. Any layer of the cube normal to X, say a X layer, contains small cubes repre¬ 
senting triads (unified) all containing the same unit of S A , and similarly in the cases 
of g and v. If a K be the unit of S A in the triads which are represented when unified by 
a X layer, the layer may be termed the a K layer, and when unified represents a k . We 
may also conveniently term a row of cubes parallel to the axis Xap row ; it contains 
cubes representing triads all of which contain the same connecting pair of and S„, 
say the pair aj) v , and when unified is represented by (a^b,). The X layers thus repre¬ 
sent the system S A , and must be supposed to admit of substitutions among themselves 
characteristic of the form of S A (sec. 276), and similarly in the case of g and v. Here 
the collection of units, which when unified gives a unit of A, is represented by cubes 
* In strictness this should be “ triads connecting the systems which when unified are Sa, S m , S,. respec¬ 
tively ; ’ but the abbreviation is convenient and will not lead to misconception (see sec. 20). 
MDCCCLXXXVI. h 
«A 
% 
b v 
«A 
% 0, 
«A 
K 
a v 
or 
«A 
7 
Kb, 
^A 
% 
K 
h 
% b, 
K 
K 
a. 
b„ a, 
