REV. T. P. KIRKMAN ON THE REPRESENTATION OF POLYEDRA. 
415 
a 
1126 
b 2536 
c 3446 d 4356 
e 
5226 
A 6112, 
B 6253, 
C 6344, D 6435, 
E 6522, 
giving the paradigms 
• 
• 
A d 
• 
B 
a E 
0 
A 
• 
• 
. B 
e 
C 
A . 
e 
B 
a 
• 
• 
c 
D 
, * 
d B 
C 
D 
b 
, * 
• 
E 
c 
C . 
D 
• 
E 
c 
A 
. b 
D 
* , 
E 
b 
c 
d e 
a 
. 
e a 
b 
c d 
The first arrangement may be so folded that A shall fall upon a, B upon b, &c. 
The second cannot. Every e-gonal face is polar to an e-edral summit, the face and 
summit showing letters on their edges of like names and succession. And every con- 
tiguous duad, as aD, in a horizontal line, is contiguous also in a vertical line, if we 
observe that the extremes of any multiplet are a contiguous duad. This shows that 
any angle «D in a face is also an angle in a summit, a property which the paradigm 
of course always has, whether of autopolar or heteropolar figure. It is observable, 
that in the first arrangement no edge a meets its garnic A in a point ; whilst in the 
second we see the angles ah. and cC, which may be denominated nodal angles, in 
the nodal face aEA, at the nodal summit ahe. 
The 2m-gonal pyramid can only be represented by pairs of quadruplets of the 
second form, abed, dabc. Thus for 7w=3, the system 
a 1127 b 2637 c 3547 d 4457 e 5367 / 6217 
A 7112, B 7263, C 7354, D 7445, E 7536, F 7621 
gives this paradigm, showing two nodal gamic pairs ah and dY), 
a F . . ... A 
A . . . . / B 
. . . . e B C 
. . , d C . J) 
c D . . E 
. E . . . F 
f a b c d e 
The reason why the (2m-l- l)-g’onal pyramid has the first arrangement as well as the 
second, is, that every base-summit may be taken for the pole of the wall-face opposite 
it. In the 2m-gonal no summit is opposite to a face, nor can the interval between 
a base summit and its polar-wall face be constant. This is best seen by inspection 
of the schemes 
ABCDEAB, ABCDEAB, ABCDEAB. 
d e a b c d c b a e d c b a e d c b 
ABCDEFAB, ABCDEFAB. 
e f a b c d e c b a f e d c 
3 I 2 
