MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
11 
are also copolar (Theorem (1), sec. 357). The graphical units may be taken to 
represent either the ten straight lines of the theorem, or the ten points of intersection ; 
the form is the same in either case. Taking the former case, the pairs of graphical units 
which are joined by links correspond to pairs of lines whose points of intersection are 
points other than the ten considered in the theorem. It should be noticed that the 
two systems into which the pairs are divided (linked and unlinked), are each single. 
60. We shall see (secs. 81, 82), that by the addition of units to any system S, i.e., by 
regarding S as a component of a more extensive system, we can represent the form of 
S by the use of graphical units and links only. It will, however, be convenient in 
many cases to employ some further devices which will enable us, without employing 
additional graphical units, to represent forms which could not be exhibited by the use 
of plain links only. 
61. Thus we may have lines of various sorts joining two graphical units, viz., 
dotted, wavy (fig. 14), red, blue, &c., in addition to links, which will always be under- 
Fig. 14. 
stood to be plain lines. Pairs joined by unlike lines will be, as in the case of links, 
distinguished from each other. 
62. Where a pair is unsymmetrical an arrow-head or barb may be added to the link 
or other line joining the two graphical units which represent the pair (fig. 15). The 
arrow-head has the effect of making ab distinguished from ba. In the fig. the arrow- 
Fig. 15. 
heads make ab distinguished from fe, but not from ef. The relative directions of 
the arrow-heads in the case of pairs joined by unlike lines are immaterial. It is 
c 2 
