MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
43 
(S) B =( M 8 . g 5 = 
cc 
T 5 =« bcdefghijJcl 
heal j k d e f h i g 
c a b g h i l j k e f d 
defabcjklg hi 
e f d i g h a b c k l j 
f d e j kliglibca 
g h i c a b e f d l j k 
h i g k l j cab f d e 
i g In, c fdkljabc 
j k l fdebcaigh 
kljhigf decab 
l j kb cahigdef 
Some General Forms of Groups. 
253. The graphical representations of the preceding forms suggest others which 
exist in the case of larger numbers of units. Thus the form in which a simple circuit 
passes through all the units of the group appears for all the given values of n; it 
obviously also exists as one form of group for every other value of n. 
254. So if n be even, a form such as that given by the first figure of G 3 when 
7i= 4, 6, 8, and 10, and by the first figure of G 3 when n = 12, in which all the units 
are connected by a continuous chain of non-polar lines of two kinds, clearly exists 
whatever value n has. 
255. If n=2 m we have the form such as G when n= 2 
G 3 >> 4 
G x „ 8 
in which all pairs are symmetrical. This form is closely related to the important one 
considered in Logic (sec. 381), which may be derived from it by ignoring the 
differences between pairs constituting a pure complete network. 
A Family of GroupsA 
256. Let us consider groups in which every circuit is of period 2 or 4. Some of 
the symmetrical pairs of these groups are component pairs of sets of four units 
composing simple circuits of period 4, say are diagonal pairs (ae in fig. 43 is a diagonal 
pah'); while some (e.g., cf in fig. 43) are not diagonal pairs of any simple circuit. We 
might study these groups generally, but I propose here to restrict myself to groups in 
which all diagonal pairs are undistinguished from each other, so that they all belong 
to the same one-way simple network, say the diagonal network. 
* The investigation of sections 256-269 was suggested by the late Professor Clifford’s paper on 
“ Grassman’s Extensive Algebra,” in the ‘American Journal of Mathematics,’ vol. i., pp. 350-358. 
G 2 
