SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 727 
a ae 
On 108 | 120 | 1284 | 135 | 150 
co | 99 
Taking the four simplest polygons, we find that the sum of the angles is 378", 7.2. >360°, 
Hence there cannot be more than three different kinds of polygons at a pot. The 
species of nodes therefore fall under three classes. 
Cuass A. contains the homogeneous nodes. Denoting the regular polygons by their 
initial letters, the nodes of Class A. can be denoted by 
1. Ty Tose Cem 
§ 4. Ciass B. Let there be at a point p; n,-gons and p, 1.-gons. Then we have 
PO + Po%_ = 277, 
ala—2)2(1-2)=2. 
Integralising, we obtain on the left the function 
hence from (1) 
MMo(2 — p) + 2(pyNq + Po) 
where p=p,+p,. We shall denote this by A. It is easily seen that the sign of A 
characterises the network as elliptic, hyperbolic, or Euclidean. For Huclidean networks 
A is always zero ; for elliptic and hyperbolic networks A 2 0 respectively. 
For the regular networks there is a corresponding function n(2—p)+2p, and for 
three kinds of polygons we shall find a similar function. Where there is no risk of 
confusion we shall call each of these A. The values of A for the regular networks are as 
follows: Tetrahedral 3, Hexahedral and Jee beta 2, Dodecahedral and Icosahedral 1. 
Solving now for n,, we have 
We have to find the integral solutions of this equation under the following conditions : 
Mn, >3,1,P2 >0,p +3 and p+} 5, therefore p,, p. + 4. 
The only possible sets of values of p,, p, are then 
Dy ally des 
Py = 2, 3, 4, 2, 3. 
- We shall take these cases in succession. 
oa, 5 ae 
po=?2 " ae we ey, 
whence 2”,— 3,4, 10, 
a= 125.8, 1O\. 
py = ! < BS 3 
Po= Me i, Se a 
