14 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
are the small circles, or only those; in general this will not be so ; the units may 
be the links, or the ways in which a number of small circles or spots can he connected 
by links, &c. If we ascertain exactly what the units really dealt with are, we may 
construct a graphical representation of the system they constitute, and this may 
be of a very different character from that of the diagram from which it is derived. 
Aspects. 
73. If two collections of units are undistinguished, they may be regarded as 
corresponding to each other in one or more ways, in each of which correspondences to 
each unit, and therefore to each pair, triad, &c., of one collection there corresponds in 
the other a counterpart unit, pair, triad, &c., undistinguished from the former in dress 
or other circumstance. In any one of these correspondences, two corresponding units 
are regarded as occupying corresponding places, or, as we may express it, places of the 
same sort. Furthermore, each of the corresponding units is regarded as belonging to 
one or the other of the two collections, each of which is regarded also in the aggregate 
as a single unit. 
74. Now the unit A which is dealt with when we thus regard a unit a as occupying 
a place of a particular sort g in a particular collection, is a different unit from a; it may 
be called an aspect of a. 
75. Thus when we consider a correspondence of two undistinguished collections 
a, b, c, d, ... and p, q, r, s, . . . where 
a corresponds with p 
^ >> >i Q 
C >; jj V 
and so on, we deal with a collection of units A, B, C, D,.P, Q, R, S,. 
where any unit R is the unit arrived at by considering the unit r as occupying a place 
of a particular sort, the unit C being that arrived at by considering the unit c as 
occupying a place of the same sort; and similarly in the case of the other corresponding 
units. Each of the collections A, B, C, D, . . . and P, Q, R, S, . . . . is an aspect, 
the former of the collection a, b, c, cl, . . . and the latter of the collection^, q, r, s, . . . 
We also regard A, B, C, D, . . . in the aggregate as a single unit Y, which may be 
termed a unified aspe-ct of a, b, c, d, . . . and so also in the case of P, Q, R, S, . . . 
76. The collections A, B, C, D, . . . and P, Q, R, S, . . . maybe different aspects of 
the same collection l, m, n, o, as a collection may be self-correspondent, and the 
number of units a, b, c, cl, . . . be accordingly the same as that of the number of sorts 
of places considered. 
77. We may have a number of undistinguished collections, each of n units, all 
corresponding to each other. If there be m such collections, in any correspondence in 
