SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 737 
different kinds of polygons, but not more. The species of nodes thus fall under three 
classes, Class A. consisting of the homogeneous nodes. We shall take the other two 
classes separately, and find those nodes which give simple types. 
§ 19. Class B. Let there be p, 7,-gons and p, %-gons at a point; then, if a, a, are 
the angles of the 7,- and n,-gon, 
Mya, > (n, — 2). 
Also Pia, + Pod. = 27. 
9 
Therefore a,( = =) 4 pi a =) Ze 
ny Ny 
or, NyN(2 — p) + 2(pyny + pom) =A>0. 
Giving A positive integral values we get a series of equations to solve under the 
following conditions: 1+, > 3, 3+p45, therefore p,, p.>4. Also, if p,=2, p.=38, 
the smallest values of n, and n, which are possible are n;=4, %.=3, but these make 
A=0, hence the only possible sets of values of », and p, are 
Further, if p,=1, p,=2, in order that the node may give a simple type, m) must 
be even, for the n,-gon must be surrounded alternately with n,-gons and 1,-gons. 
We have then 
2n,p, — A 
Nn, => SOIC GIREG 
1 n,(p - 2) — 2p» 
We shall take each set of values of p,, p, in turn. 
net _2m-A_, 8-A 
P,=2 woe: Ny — 4 Ta ies 4° 
Since 7, is even, A must be even. 
A=2. n,=6, 10 A=6. n,=6 
n,=5, 3 n,=3 
A=4, n,=6, 8 ANS I9) 5 Ny = 4 
m=4, 3 m, =any integer. 
ope 2, Ay 3—4A 
Do=8 pO eo 3 
A must be even. 
A=2, n=4 A=6. ,=3 
nN, =3 nm, = any integer. 
A=4 impossible 
Pe _ 2n,—A 
po=4S 1” 3n, — 8 
16-3A 
or 3n,=2+ 3, —8 
p=) i [A=3. n,=3=n,, excluded. | 
n,=5 
(=D Dp A> 3 
