MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 0 
Pairs.—Graphical Representation. 
50. There are three different forms of pairs; viz., if a, b be two units, we can 
have:— 
(1) a distinguished from b , and therefore also ab distinguished from ba; 
(2) a undistinguished from b, but ab distinguished from ba; 
(3) a undistinguished from b, and ab undistinguished from ba. 
51. The two units in case (1) belong to different single systems, and thus the pair 
is a pair connecting two single systems. 
52. In cases (2) and (3) both units belong to the same single system, and the 
pairs are components of that system. The units in case (2) may be said to form 
an unsymmetrical pair, those in case (3) being said to form a symmetrical pair. It 
will be convenient in speaking of undistinguished unsymmetrical pairs ab, ccl, to 
say that ab and cd are of the same polarity, and that ab and dc are of opposite 
polarities. Of course in this case ab and ba are of opposite polarities. In the 
case of a symmetrica] pair e, f ef and fe will be of the same polarity. 
53. The expression “unsymmetrical” may also be applied to pairs falling under 
case (1). We may also in the case of such pairs say that ab and cd if undistinguished 
are of the same polarity, ab and dc of opposite polarities. 
54. If in a diagram consisting of a number of graphical units, some pairs of those 
units are joined by plain lines or links, as in fig. 6, while others are not so joined, the 
Fig. 6. 
pairs will be divided into two systems. Pairs which are thus joined by links will not 
be necessarily undistinguishable, nor will pairs which are not joined by links be 
necessarily undistinguishable, but we merely have pairs which are joined by links 
distinguished from pairs which are not so joined. Thus the two systems into which 
the pairs are divided may each be either a single or multiple system. 
55. A large number of systems may be completely defined by diagrams consisting 
of graphical units and links only. We have already seen that an extensive class 
of systems may be represented by diagrams consisting of different sorts of graphical 
units only ; there are also systems which may be fully defined when we indicate 
by links a dichotomy of their component pairs, without defining further what units 
are distinguished and what are not. Thus the system of fig. 7 must be a discrete 
MDCCCLXXXVI. C 
