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REV. T. P. KIRKMAN ON THE REPRESENTATION OF POLYEDRA. 
polygon of^ sides can be drawn on the faces of the figure, so as to have a side on 
every face, and to pass through no summit. 
There is no use in perplexing further the enunciation of B, as its truth follows from, 
and its extent is parallel with, that of A. Nor is it worth while to trouble the reader 
with demonstrations of these properties, since they are not properties of all polyedra, 
a negative which may however be worth the proving. It may suffice, and may per- 
haps be useful, to show the connexion between these closed polygons and a pleasing 
mode of representation. 
Let us suppose that in any ^-edral ^-acron we have traced the closed jo-gon through 
the faces, and the closed ^’-gon through the summits, and let the edges of the jo-gon 
be numbered in order 1 2...p, and the angles of the g'-gon 1 2...q. Thus are all 
the faces and summits of the jo-edral §'-acron numbered, in the circles 1 2 3...p\2.., 
and 1 2 3...q\2.. 
Any edge of the figure may be read abed, a and c being its left and right summits, 
and b and d its upper and lower faces, or edab, which is the same thing, turning the 
figure about. In the same face d we read, passing towards the right from c to the 
summit e, the edge cfed or edef, in the two faces d and f. Thus it appears that, in 
the reading of the edges, any consecutive and external duad, as cd in abed, will occur 
reversed and internal, as dc in edef, and vice versa. We can thus represent the 
edges of the jo-edral 5 ^-acron by as many quadruplets, so formed, that every 
contiguous internal or external duad shall occur again reversed as an external or 
internal duad ; the quadruplets being all read from left to right. 
Let xyx^y^, x^y.^x^y^ be two of these quadruplets. The former is an edge at the xth 
summit and in the yih face, and also at the .rAh summit and in the yfa face. The 
like is conceived of the latter. At the points xy and x^y^ referred to right axes, write a, 
at Xji/i and x^y^ write b, and so on with all the j»+g'-l -2 edges abed.... 
The result is a paradigm of the figure and its sympolar. The horizontal multiplets 
will be the faces, denoted by their edges, the vertical ones the summits so denoted, 
or viee versa. The edges in summit or face will stand in their true order. For the 
closed q-goi\ through the summits, if it leaves any face before it has completed the 
circuit thereof, must return to it to complete that in the same direction ; otherwise 
it would cross its own path and pass more than once through one or more summits, 
which is impossible, as it has only q sides. 
Every pyramid is autopolar. If the base be ( 2 w-l- l)-gonal, the system of edges is 
denoted by An-\-2 quadruplets in pairs of the form abed, deba; or as well by as many 
in pairs of the form abed, dabc. Either edge in any pair lies between the poles of the 
faces through the other. I call these two edges (aA) a gamic pair, and either is the 
gatnic of the other. Thus the pentagonal pyramid is represented by either of the 
systems, 
a 1356 b 2416 c 3526 d 4136 e 5246 
A 6531, B 6142, C 6253, D 6314, E 6425, 
