MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
15 
which they all correspond we have mn unit aspects such as A, n sorts of places, and 
each sort of place occupied by m units. 
78. A, the aspect of a in a correspondence, bears definite relations (sec. 7.9) to three 
other units, viz. :— 
(1.) a of which it is an aspect. 
(2.) The unified aspect V arrived at by regarding as a unit the aspect of which 
it is a component. 
(3.) The sort g of the place a occupies in the correspondence. 
79. The “ definite relations ” of the preceding section consist merely in this, viz., 
that the pairs which A makes with a, V, and g, are all distinguished from all the other 
pairs which A makes with other units. (See secs. 143^, on Associates.) 
80. The tetrad A, a, V, g, may be graphically represented as in fig. 22 ; the 
graphical unit A being joined to no other graphical units besides a, V, and g. Since 
A 
Fig. 22. 
the units A, a, V, and g, are obviously distinguished from each other, the graphical 
units are made so also. 
81. It appears, then, that in studying the form of a system S by means of 
correspondences of undistinguished components, we really regard S as a component of 
a more extensive system, containing 
(1.) the system S ; 
(2.) the system X composed of units which are conceived of as sorts of places; 
(3.) the system Y composed of units which are conceived of as unified aspects of 
all the various component collections of S ; 
(4.) the system Z composed of units which are conceived of as aspects of single 
units of S. 
The pairs connecting Z and the joint system S, X, Y, break up into two systems, 
graphically represented by linked and unlinked pairs of graphical units respectively. 
82. A graphical representation in which the units of S, X, Y, Z, are all represented 
by graphical units, and links connecting the graphical units are drawn in the manner 
indicated in the last two sections, fully defines the form of S, as it completely 
indicates what components are undistinguished and what are not. Such a graphical 
representation is sometimes termed a “linkage,” so that we may say that every 
system may be graphically represented by a linkage (secs. 194, 195). 
