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XVIII. On the Representation of Polyedra. By the Rev. Thomas P. Kirkman, M.A. 
Communicated hy A. Cayley, F.R.S. 
Received August 6, — Read December 6, 1855. 
To every jo-edral ^'-acron, or solid having- p faces and q summits, corresponds a 
§'-edial ^-acron, whose q faces have the same order (of succession) and rank (as to 
number of edges) with the q summits, and whose ^ summits correspond in the same 
way to the p faces, of the (/-acron. 
Whenjo^g-, the corresponding pair will sometimes be identical figures, as to the 
number, rank and arrangement of their faces and summits ; whilst at other times, as 
will always be the case ifjo is not =q, the two figures will differ. When they differ 
they may be called a sympolar pair of heteropolars, or simply a sympolar pair-, when 
they are the same figure, it may be called an autopolar polyedron. 
An elegant method of representing an immense number of sympolar pairs and 
autopolars, may be deduced from the property enunciated in the theorems following, 
A. and B. 
Def. — Any most-angled face of a polyedron being taken as its base, the angles of 
the base may be called hase-summits, the remaining angles of the faces either colla- 
teral or synacral with the base may be termed wall-summits, and all summits lying 
only in faces neither collateral nor synacral with the base, we may name crown- 
summits. 
If a polyedron has base-summits a^a.yi^..., wall-summits and crown- 
summits besides, the latter may consist of a system c^c^c^... lying in faces that con- 
tain also some of b^bf^..., and an interior system d^df^... not lying in faces containing 
any of b^bj)^.... The summits dff^... may be looked at as wall-summits referred to 
C1C2C3... as base-summits, and as crown-summits referred to bfif^... as base-summits. 
And there may be any number of systems of crown-summits interior to dff^..., as 
616363... leading up iofff... &c. 
A. If in any g'-acron there are either no crown-summits, or if from each of the wall- 
summits bib^b^... tliere passes an edge to one of the crown-summits CjCgC,..., and from 
each of 616363... an edge to one of dff^..., and from each of dff^... an edge to one 
of 616363..., and so on, the q summits of the g'-acron are the angles of a closed g-gon, 
whose q sides are all edges of the g-acron. 
B. If in any p-edron there are either no faces yiy^yz..., of which none has any 
summit collateral or synacral with the most-angled summit S of the g?-edron, or if 
each of the faces about S is collateral with some one of 717273 ••••? a closed 
3 I 
MDCCCLVI. 
