416 
REV. T. P. KIRKMAN ON THE REPRESENTATION OF POLYEDRA. 
The triangles made by three adjoining letters in all the three upper examples, as 
A^d, T)ab, correspond ; but this is the case only in the second of the lower ones. 
I add the following examples : — 
a 1357 b 2617 c 3527 d 4137 e 5847 
A 7531, B 7162, C 7253, D 7314, E 7485, 
which forms 
A d 
a 
D 
G 
b 
f 
c 
s 
B 
F 
b' 
B 
C 
D 
E 
A 
G 
E 
F 
d 
f 
an aiitopolar octaedron, having a pentagonal, two quadrilateral, and five triangular 
faces. 
and 
give 
T1 
lis can 
be folded 
to 
lay a upon 
A, &c. 
a 
1228 
b 1382 
c 
1864 
d 
1473 
e 
4657 
/ 5668 
s 
3758 
A 
8122, 
B 2138, 
C 
4186, 
D 
3147, 
E 
7465, 
F 8566, 
G 
8375, 
a 
1 138 
b 1473 
c 
1824 
d 
2574 
e 
2765 
/ 5 
667 
s 
3728 
A 
8113, 
B 3147, 
C 
4182, 
D 
4257, 
E 
5276, 
F7 
566, 
G 
8372, 
B D 
C . 
. A 
a 
. B 
C 
. A 
a 
A . 
. 
, 
. b 
• 
D 
E 
G C 
b 
. 
. 
d G 
A 
, * 
b G 
d 
. 
. 
c 
E . 
b 
c 
d . 
. 
. 
E 
G F 
• 
d . 
, 
e 
F . 
. 
. 
e / 
F 
. G 
. 
• 
/ 
F 
E . 
• 
O' 
■ 
D e 
. 
• 
e g 
B 
D 
/ 
. 
c 
a B 
• s 
/ 
. 
c 
g « 
• 
. 
These are both autopolar octaedra having, like the preceding, a pentagonal, two 
quadrilateral, and five triangular faces. They are all three different figures. In the 
first, the pentagon bcdea has every other face either collateral or synacral with it, 
for yBC and DEG, the only ones not collateral with it, have the summits fCae, 
and oDGb common with it, as is evident from the duads fC and DG. And its two 
quadrilaterals Ac?gB and AEFc have the common side A. 
The second octaedron has also its pentagon either collateral or synacral with every 
other face, but its two quadrilaterals have not a common side. 
The thiid has a crown triangle AbG, neither collateral nor synacral with the 
pentagon y'egBD, for none of the angles Ab, bG, GA are in the summits of that 
pentagon. 
