SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 745 
There are eight possible varieties. Let X denote the original group, R a group 
turned through x O the decagon replacing a group, then, if XY denotes two 
Borg g p g group 
opposite groups, a three groups mutually circumjacent, the varieties can be ex- 
PP groups, y7 group Vf. Ly) 
pressed as follows : 
(a) 45 (315,49) + 10 (4,5,10,) 
(a) O (8) OR 
O O 
*(y) RX (8) RR 
(b) 30 (3,5,4,) + 20 (4,5,10)) 
O O 
(2) OO 8 =(8) OX _~— (y) OR 
(¢) 15 (3,5)4_) + 30 (4,5,10)) 
O 
OO 
§ 29. Further, if we allow angles >180° we get the following : 
1. From 3,°, replacing four covertical triangles by a quadrilateral (fig. 45). 
(3,) +4 (8,4,). Angle of quadrilateral 180°, 
2. From 3,” (fig. 44). 
(3,) +5 (3,5,). Angle of pentagon 216°. 
. From 8,°4,"° (fig. 27 bounded by the heavy lines). 
4 (3,45) +8 (3,4,8,). iijAngle of octagon 196° 50’. 
4, From 3,°4,°, replacing the hexagonal group (fig. 31 bounded by the heavy lines). 
3 (8,4.) +6 (3,4,6,). Angle of hexagon 180°. 
5. From $,"5,”, replacing the decagonal group (fig.-29 bounded by the heavy lines). 
10 (3,5,) + 10 (3,5,10,). Angle of decagon 180°. 
6. From 3,%5,"4,” (fig. 37 deleting the part within the heavy lines). 
5 (3,5,4,) +10 (3,4,10,). Angle of decagon 204° 6’. 
co 
These are all the composite types obtainable from the simple types. It seems 
probable that there are no others. 
Ill. Tot Hyperporic PLANE. 
§ 30. This case does not admit of exhaustive treatment. The number of types of 
networks is evidently infinite, for there is no limit to the number of lines at a point. 
As a rule, the hyperbolic plane contains the types which cannot exist on the 
Huelidean or the elliptic plane. For example: one n-gon and two 2m-gons at a point 
* Asymmetrical. Two enantiomorphic forms, ux and er - 
