394 Proceedings of Royal Society of Edinburgh. [sess. 
an Wj-gon, an ?i 2 -gon, and an ?i 3 -gon, n v n 2 , and n z must all be even, 
for the ftj-gon must be surrounded alternately with rc 2 -gons and 
%-gons. With the angles which remain there are thirteen simple 
types, two with two varieties each and one with five, and two 
infinite series of simple types, one corresponding to right prisms 
on a regular polygonal base, the other with triangles instead of 
quadrilaterals. 
Of composite types it is probable that none exist, if we make 
the condition that the angle of a regular polygon must be less 
than 180°. When a polygon occurs in a particular combination 
its angle is thereby determined, and if it occurs in another com- 
bination its angle must be the same, which is not in general the 
case. 
III. The Hyperbolic Plane. — The number of simple types 
here is infinite. For example, one w-gon and two 2m-gons at a 
point determine a simple hyperbolic network for all values of n 
and m for which the network is neither Euclidean nor Elliptic. 
As regards the composite types, the same considerations hold 
here as in the case of the spherical networks. 
( Issued separately February 1, 1905.) 
