241] 
ON POINSOTS FOUR NEW REGULAR SOLIDS. 
85 
which, however, does not apply to the two stellated figures where e' is different from 
unity; the general form is 
eS + e'H=A + W, 
which applies to all the nine figures. This applies to all polyhedra, regular or not, 
which are such that e has the same value for each vertex, and e the same value 
for each face. To prove it, we have only to further extend Legendre’s demonstration. 
If for any face, stellated or not, the sum of the angles is s, and the number of sides 
n, then, according to the foregoing mode of reckoning, the area of the face (measured 
in right angles) is 
s + 4e' — 2 n. 
Now the sum of all the faces is D times the spherical surface, = 8D. But the sum 
of the term s is equal to the sum of the angles about each vertex, = 4eS; the sum 
of the term 4e' is = 4e'H, the sum of the term 2n is four times the number of 
edges, = 4A. Hence 4eS + 4e'H — 4A = 8D, or eS + e'H = 2D. 
I remark that the small stellated dodecahedron and the great dodecahedron are 
descriptively the same figures, and that, if we represent the vertices by a, b, c, d, e, 
f, g> K j, P, g, and the faces by A, B, G, D, E, F, G, H, I, J, P, Q, then the 
relations of the vertices and faces is shown by either of the following Tables: 
a 
b 
c 
d 
e 
= p, 
A 
G 
E 
B 
D 
=P> 
P 
b 
i 
h 
e 
= m, 
P 
I 
E 
B 
H 
= a, 
P 
e 
j 
i 
a = 
= B, 
P 
J 
A 
G 
I 
= b, 
P 
d 
j 
b = 
-C, 
P 
F 
B 
D 
J 
= c, 
P 
e 
g 
f 
c = 
= D, 
P 
G 
G 
E 
F 
= d, 
P 
a 
h 
g 
d-- 
= E, 
P 
H 
D 
A 
G 
= e, 
j 
c 
d 
g 
g~- 
= F, 
J 
D 
Q 
G 
G 
=/ 
f 
d 
e 
h 
<r- 
-G, 
F 
E 
Q 
D 
H 
=g> 
g 
e 
a 
i 
g = 
-H, 
G 
A 
Q 
E 
I 
= K 
h 
a 
b 
j 
g = 
-I, 
H 
B 
Q 
A 
J 
= i 
i 
b 
c 
f 
g = 
-J, 
I 
C 
Q 
B 
F 
=j> 
f 
g 
h 
i 
j = 
-Q, 
F 
H 
J 
G 
I 
= ?> 
where it is to be noticed that in either Table each non-consecutive duad of any 
pentad occurs once, and only once, as a non-consecutive duad of another pentad. The 
restriction that a non-consecutive duad of any multiplet is not to occur as a duad, 
consecutive or non-consecutive, of any other multiplet (see my note appended to 
Mr Kirkman’s paper “On Autopolar Polyhedra,” Phil. Trans. 1857, p. 183 [259]), applies 
only to ordinary polyhedra, and not to the class here considered. 
2, Stone Buildings, W.G., January 13, 1859.