12 
MR. A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
unnecessary to acid an arrow-head in the case of a pair of distinguished units; the 
difference between the graphical units indicates the absence of symmetry. 
63. We may in some cases of symmetrical pairs instead of employing a single line, 
use two barbed lines of opposite polarities, as in fig. 16. 
Fig. 16. 
64. In place of using a number of different sorts of lines, we may use plain lines, 
barbed or unbarbed according as the pairs are unsymmetrical or symmetrical, with 
letters alongside of them ; like letters being used where otherwise like lines would be 
used, and unlike where unlike lines would be used (fig. 17). Though this method is 
not so good as the previous one as regards its power of enabling us to visualize the 
systems represented, it has distinct advantages for purposes of description, as we can 
denote pairs of different sorts by the annexed letters, the sorts (sec. 84) of which 
denote the sorts of the pairs. 
65. If cr denotes an unsymmetrical pair ab, we may denote the pair ba by -o ; or, as 
this is in some instances awkward, by cr' where (cr')'=cr. 
66. Although, as we have seen, it is in many cases unnecessary to draw lines 
connecting all pairs of graphical units in order to completely define a system 
(secs. 55-59), it may in some cases be desirable to do so, especially where we wish to 
show how many different sorts of pairs there are under consideration. In such cases 
we may connect like pairs by like lines and unlike by unlike lines. Thus the diagram 
of fig. 18, which completely defines a system of six units, may be completed as in 
Fig. 18. 
Fig. 19. 
fig. 19 so as to show that there are two sorts of unsymmetrical pairs and one sort of 
symmetrical pairs. 
67. In general, in diagrams in which lines other than links are employed, either in 
conjunction with links or not, like pairs of graphical units will be joined by like 
lines. 
68. It can readily be shown that every form which admits of representation by 
