SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 741 
alternately with quadrilaterals and triangles. The boundary of this group is a regular octagon, 
and by turning it round through = we get the other variety (figs. 26, 27). 
Cra ce = 85 NG Sie 
i =n, No = 2 
N,=2n, N,=4n, N,=2(n+1) (fig. 25). 
La — 2p =: 
(Il) AV i =5 ING = 20) 
jh 1, IN ly 
N,=30, N,=60, N,=32. 
A certain great circle divides this network into two equal groups, By turning one of these 
through = a second variety is obtained (figs. 28, 29). 
(2) A=4, S65 WES 
A i 
Nj, =12, N,=24, N,=14. 
Like the preceding, this network has two varieties which may be obtained in a similar way, 
viz., by turning one of the groups through = (figs. 30, 31). 
i = 1, pT SA. 
(1) A=1, m =), N, =12 
irom NE 80 
N,=60, N,=150, N,=92 (fig. 32), 
(2) ee N 1G 
% = 9), Nj = 32 
N,=24, N,=60, N,=38 (fig. 33), 
These two networks are asymmetrical, Each exists in two forms which are enantiomorph. 
The one could be obtained from the other by turning the sphere inside out, supposing this to be 
possible, as it would be in space of four dimensions, 
eee Class CC 1 pt, p’=1, p= 1. 
(1) A=8. n’ =4, N,' =30 
n” =6, N,” =20 
n”=10, Nj” =12 
N,=120, N,=180, N,=62 (fig. 34). 
(2) A=16. n' =4, N,’ =12 
n’ =6, Nj” = 8 
nu!” =8, Ni” = 6 
N,=48, N,=72, N,=26 (fig. 35). 
A=4, n =3, N, =20 
Ch. NS 
dt APN 30) 
N,=60, N,=120, N,=62. 
Of this type there are five varieties, which may be obtained as follows:—In one of the 
varieties (fig. 36) there are no two quadrilaterals adjacent, Each pentagon has a quadrilateral on 
each of its sides and forms the centre of a group with a regular decagon as boundary. Let us call 
TRANS. ROY, SOC, EDIN., VOL. XLI. PART III. (NO, 29). 110 
