728 MR DUNCAN M. Y. SOMMERVILLE ON 
There are no unequal values of m, and n, satisfying this equation. 
ae =x, LO Bs 16 
Po=4 4 
whence ”,= 
No =3., 
p,=2 | — ee 4 
pPo=2! "9 7, <2 
whence ”,= 
Ny = = 
oe oe ORT oe 
p= 8 1 5 ee oR 
whence ,=4 
There are thus six species of nodes in this class. They may be denoted by TD,, 
SO,, Dec P,, TH, 1,03, 1.8... 
§ 5. Crass C. Here we have 
2 2 2 
p(d a +p x -) f p(1 a ae 2 
or A = 1 NN, (2 — p) + 2(p Mots + PoNgN, + PoNN.)=0. 
. 2PM No 
Solving for n,, Ne = Aen ene 
We have to solve this equation in integers under the following conditions : n,+n.+n;>8, 
Pi, P2, Pp > 0, P + 38, also p + 4 (for 3a,+0,+a, > 360°), therefore p,, p., Ps $ 2. 
Further, if p,=2, n, must be either 3 or 4, for 2a,+4,+a, > 360°. Again, we cannot 
have n,=5, m%=6, n;=7 together, for a, +o,+4, > 360°. 
The following are therefore the only possible sets of values of p,, 7%, 3, 71: 
p,=1, or 2; p,=p,=1; n,=3, or 4. 
We shall take these cases in succession. 
1 = 12) = 13> — bn. 36 
\ Me epee: Otten 
Whence 775— is a, sO ek), 
y= 42, 24, 18, 15. 
PaPe=Ps= 1 ji 4n, Hope 16 
hy — 3 ny — 4 N, — 4 
whence 7= 5, 6, 
n,=20, 12. 
py =2, Py=P3= Pa 3, = 9 
i — } aimee =e Ny — 3 
whence 7,=4, 
N,= 12. 
Als Sages Qe mt 
n= 4 ms feo Es 
whence 7,=3, 
n,=6. 
There are thus eight species of nodes in this class. 
