SEMI-REGULAR NETWORKS OF THE PLANE IN ABSOLUTE GEOMETRY. 739 
§ 21. We have found, then, the following developable species of nodes : 
Class B. aaa: 2, = SOheSad 3503, 5, 6, 
Dy— 2) NNO Gono, G5 4. 4.4. . 
Ppa lh m= Sis be Gr, 
ea N=4; 3,3,3,. 
p= \ m,=5, 4 
Po=4) My=3, 3 
pPy=2 | m=3, 38 
Aa) m,=5, 4 
Class C. p,=1) m= 4,4. 
=} n= 6, 6. 
pj=1! n,=10, 8. 
Py) on, 3 
nai N= 
Pg=2) n=4. 
Each of these gives a simple type of network. 
§ 22. We shall find the number of polygons, lines, and points in the complete 
network. 
Let N,’, N.”, N.” be the number of n’-, n’’-, and n”’-gons, N, , N,, Ny the number 
of polygons, lines, and points respectively. 
Then (§§ 11, 12) 
Nea Ne IN NG” ae EMA et eed (1) 
HON ep Ni, 
nN,” =p'N, (2) 
nN ah =p) No 
oN, = 2N, (3) 
where p=p'+p" +p". 
Finally, Evxer’s polyhedral formula is 
NMI NENG SO, kee a tok ae C8) 
From (2) and (1) we get 
p p- ad 
Ny= (Stan tam No, 
and from (3) 
N,=5N, 
Substituting in (4), 
Pp p pp ky! 
(24528, ot 1)N=25 
whence 
’ An’ nln” 
= 
A 
Hence On! nn” p 
N,=——__—*.. 
AN 
N om 4p’ n” ni” N oe 4p” nl” n N ne 4p” n' “wr 
: ean Muli hue ee oe dk 
