94. REV. T. P. KIRKMAN ON THE 
mainly the difficulty of stating the conditions of the pro- 
blem in algebraical propositions, 7.e. in equations. In 
this paper the reader will find that this statement is 
effected, and that nothing remains to be done for the 
complete solution except a certain amount of laborious 
elimination. 
The first step is to invent a representation of a polyedron 
on paper in symbols that algebra can handle. The con- 
sideration of the solids in space will help us little. I have 
tried to refer the figure to its amplest, 7.e. most angled, 
face as a base, and the figure has obstinately refused to 
acknowledge one face rather than another for a foundation. 
Its amplest face may be a g-gon, but it may have a hun- 
dred of these g-gons, about every one of which may stand 
a different configuration of summits and faces. I have 
endeavoured to generate the figures by defined processes 
from pyramids; but not only is the solid in general dedu- 
cible from different pyramids, but often in a great number 
of ways from the same pyramid, no one of which will allow 
another to be a better way than itself. And when I have 
attempted by authority to put the work into the hands of 
some oue of these methods, all equally clamorous and con- 
fident, I have invariably found that, in spite of all I could 
do to prevent it, the operator would persist in carving the 
same polyedron over and over again in different postures, 
so that it was impossible to know how many really distinct 
ones had been generated. 
The true method of looking at the question turns out, 
as is usually the case, to the mortification of wasted inge- 
nuity, to be a very simple one. 
All that we require is to have a clear statement of the 
summits that are about every face, and of the faces about 
every summit, and of the order about each in which they 
stand. This order is fixed by a correct account of the 
