REV. T. P. KIRKMAN ON THE REPRESENTATION OF POLYEDRA. 
417 
The following gives a sympolar pair of octaedra, having each a pentagon, two 
quadrilaterals, and five triangles : — 
a 1228, A2338, c3344, dAbM, e5566, f7Q^7, ^8118, 
41142, /2243, j3456, 4 3678, Z8541, m6587, n77QS-, 
h . . 
a i 
b c 
• • J 
. : k 
gab 
I ... g 
h ... . 
i . . . 
c d . . . 
d c m . I 
• j e f . 
f n m 
k n. 
Here are two distinct heteropolars ; one has two quadrilaterals, deml and kjef, 
having a common side e ; the other has two quadrilaterals, cjkl and glnin, that have 
no common side. The reader will find no difficulty in drawing these octaedra, by 
joining the angles of a pentagon to three included wall-summits. 
It is to be observed, that in all these sets of quadruplets representing any jo-edral 
^--acron, if we collect those which contain any given numeral in the same place, we 
shall find that in the two adjoining places they exhibit circles of the same numbers 
differing by one cyclical step. Thus, collecting from the above the quadruplets con- 
taining 8 in the fourth place, 
8118, 1228, 2338, 3678, 7788, 
show the circle 81237 in the first and third places. And those containing 1 in the 
second place, 8118, 1142, 4185, show the circle 814 in the first and third. 
From this property of the closed jw-gon and gc-gon of props. A. and B, it is possible 
that some light may be thrown, when the matter is better handled, on the classifi- 
cation of polyedra, such as may lead to the solution of the problem of their enume- 
ration. 
It is easy to prove that there are polyedra on which the closed polygons cannot be 
drawn. 
For suppose the g'-gon of prop. A. drawn on a ^'-acron. In making the circuit of 
any face G which we enter across an edge FG, which is not an edge of the ^-gon, we 
add to the number of summits counted in F and other faces, all the summits of G, 
except two, these two having been enumerated in the circuit of F from which we 
enter G. That is, counting first all the summits of the base, we add to these for 
every m-gon whose circuit we proceed to make, m— 2 summits more. The num- 
ber of faces, connected with each other and with the base by edges, not part of 
the closed g'-gon, whose circuits the closed ^'-gon makes and includes, will be 
« 3 +« 4 +“ 5 + •••+«/(; vvhere is the number of m-gons among them, and a* that 
