732 MR DUNCAN M. Y. SOMMERVILLE ON 
m1 ” 
Let p’, p”, p’” be the number of n’-, n”-, n’’-gons meeting at each point; N, 
the number of nodes, N,’, N.”, N.” the number of n’-, ”’-, n’”-gons in the whole 
network. 
Then each n’-gon has n’ angles ; but if we count up the whole number of angles con- 
tributed by all the n’-gons, each is counted p’ times. 
Hence fe en Ne 
Similarly g Nope 
ne No = p Ns 
Therefore N,’: N,”:N,” = ae Ae eee 
Hore dso = be — eel Hor S05, 19 30 alm 
Wiel, Ibs lets ¢ Il DS SE Easel eo 
Td bes pea Tol (hs 11 Sel, Gelels I DW=Be Ds ii 
Da a Die 
§ 12. Let us investigate the analogous formulee for composite types. Let ,N, be the 
number of nodes at which there are p,’ n’-gons, p," n’-gons, p;" n”’-gons, and p;'" n*-gons, 
where one at least of the quantities p, is zero, and let 
NG sig Ne tastes oe NG ete trates foe 
Then Ww No =p, Not Pe aNot <3] Pra NG 
=(p; +hypo + .... +h,_-)p-),Ny 
, 1 
Therefore Ny =—( py + haps + ey Eine 
Let N,, N,, No have their usual meanings, then 
No =,N + eens N= tht of. +h) NG 
NENG +N," +N," +N,". 
Also, by the analogue of EuuEr’s polyhedral formula, 
N,-N,+N,=1. 
N, can also be expressed in terms of ;N,,....,,No thus: at each of the ,N, points there are 
p, lines, and each line joins two points, hence 
2N,=71;,N p+ .... +9;-No, 
whence we get 
t=4 A=r A=r 
1 i 1 
(Daan) Dh, -1P) Sor 1 TANG ‘ 
vi A A=1 
Now, since the number of species of nodes in the whole network is finite, one at least of 
the quantities ,N,,....,,N, must be infinite. Hence we may put ,N)=o. The equa- 
tion then becomes 
A=? , F ” pause iv 
> (ee Pr. ee 1 )ina=0. 
ee n n n 2 
But this is an identity on account of the fundamental relation A=0. Without further 
