MR, A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
13 
means of graphical units and lines of different sorts, can, by the introduction of 
additional or auxiliary graphical units, be represented by means of graphical units 
and links only. It will be convenient for the purpose of description to suppose 
that the lines of different sorts are of different colours. In the place of any red 
line substitute a red unit (i.e., one coloured red, instead of having a letter inside), 
this unit being connected by two links to the two units which were joined by 
the red line (fig. 20). Effect this change in the case of all the red lines ; and 
similarly in the place of blue, yellow, &c., lines, substitute blue, yellow, &c., units, 
joined by links to the units which were connected by the several coloured lines. The 
red units and links will have precisely the same effect in rendering pairs distinguishable 
as the red lines had. We may, if we please, substitute letters for the various colours 
in the added units, and so obtain a diagram consisting of graphical units and links, 
representing a collection of which the original one is only a component. This 
component is, however, of the same form as before. 
69. In the case of an unsymmetrical pair, pq, of undistinguished graphical units 
which would ordinarily be joined by a barbed line, we must have two additional 
undistinguished graphical units in some cases, as in fig. 21, where u, v are the 
auxiliary units. 
70. In the case of unsymmetrical pairs of distinguished units we have seen that 
no barbs are necessary (sec. 62), and we need therefore only one additional graphical 
unit, as in sec. 68. 
71. All systems of which the forms are determined by the distinguishableness and 
undistinguishableness of units and pairs only, may be represented by the methods 
we have been considering. But there are also systems, the forms of which are not 
so determined, but depend upon the distinguishableness and undistinguishableness 
of triads, tetrads, &c. Such systems may at first sight require other expedients for 
their graphical representation, but we shal] see, as already stated (secs. 33, 60), that 
these are not necessary. 
72. Diagrams such as those we have been discussing may come under our 
consideration in other ways than as representing graphically systems of different 
forms. In such cases it must not be assumed that the units under consideration 
