MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
37 
of each of the original groups. We may fully define the group S by a pure complete 
network composed of a number of pure networks such that each detached portion of 
one is composed of units constituting a group of the form of one of the original groups. 
Groups containing from one to twelve Units . 
240. In the following twelve sections I shall denote the number of units in a group 
by n; f will denote the number of forms of groups for any value of n; G the 
graphical representation; T the tabular; and (S) will be the symbol described in 
section 204. Where f is greater than unity, the symbols G, T, and (S) will have 
numerical suffixes corresponding to the different forms. In the graphical representa¬ 
tion, only such lines will be drawn as are necessary to completely define the group, 
and in some cases alternative representations will be given, obtained by taking 
different circuits. The values of n taken are from 1 to 12 inclusive. 
241. If n— 1 we have f— 1, 
(S) = p, G= T =a. 
1 Fig. 30. 
242. If n —2 we have f— 1, 
243. If n — 3 we have f— 1, 
(S) — -, G— ®d— a 
Fig. 31. b 
(S)=\, G= | S 
, T=a b 
b c 
c a 
Fig. 32. 
244. If n— 4 we have f— 2, 
12 a. b 
b c 
bed 
c d a 
d a b 
W 
c/ c 
Fig. 33. 
r cf 
Fig. 34. 
T 3 =a b c 
bad 
c d a 
deb 
b 
a. 
c 
a 
b. 
d 
a 
b 
c. 
d 
c 
b 
a. 
