10 
MR. A. B. KEMPE OX THE THEORY OF MATHEMATICAL FORM. 
heap, each graphical unit admitting of separate definition. By the use of links 
in conjunction with graphical units of different sorts the number of representable 
forms is greatly increased. I shall give in the following four sections some examples 
of forms which may be represented by the use of graphical units and links. 
56. Figs. 8 and 9 illustrate the fact that a system may be defined in various ways; 
Z>@- 
s @——® r 
Fig. 8. 
they represent systems of the same form, 
fig. 8 are joined in fig. 9, and vice versa. 
57. I give the systems of figs. 10 and 
A© @Z 
r © © s 
Fig. 9. 
Pairs which are not joined by links in 
11 as examples of two varieties of single 
systems of the same number of units. In fig. 10 there are four sorts of pairs of 
which Im, In, lo, Ip, are types. In fig. 11 there are only three sorts, of which uv, 
uw, and ux, are types. It may be noticed that in each case the linked pairs compose 
a single system ; while the unlinked pairs in the former case compose a treble system, 
in the latter a double one. 
58. Fig. 12 is an example of a triple system containing unsymmetrical pairs of both 
sorts. We have p, q an unsymmetrical pair composed of two undistinguished units, 
and p>, r an unsymmetrical pair composed of two distinguished units. 
59. Ihe single system shown in fig. 13 is one of considerable interest; it is that 
dealt with in the case of the theorem that if two coplanar triangles are coaxial they 
