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XI. On Autopolar Polyedra, 
By the Rev. Thomas P. Kiekman, A.M., F.R.S., Rector of Croft with Southworth. 
Communicated by Aethue Cayley, Esy.., F.R.S. 
Eeceived and Eead June 19, 1856. 
I.* As any face of a^-edron N is named from the number m of its edges an m-gon., so 
from the number n of its edges I call a summit of N an n-ace ; and a polyedron having 
k faces and I summits may be denominated a \.-edron or an \-acron., as may be most con- 
venient. 
By an autopolar polyedron, I mean one in which the number, rank, and collocation 
with respect to an a-gon A of its summits and remaining faces, are exactly the number, 
* The following note, “ On the Analytic Problem of the Polyedra,” is kindly placed at my disposal by 
Abthtie Catiet, Esq., who has wisely judged that to this investigation of a subject so new and intricate, some 
such statement should appear by way of introduction. I doubt not that the reader will approve of my 
appending it as he has written it. The note comprises a clearer statement of some things which may be found 
in my memoir “ On the Eepresentation and Enumeration of Polyedra,” in the twelfth volume of the Memoirs 
of the Literary and Philosophical Society of Manchester, wdth some matter which has arisen in our corre- 
spondence ; and particularly it supplies a defect in my statement of analytic conditions in art. 22 of that 
memoir, which IMr. Catley with his rare penetration was the first to poiut out and amend. I have there 
laid down, that “ multiplets are to be made with a symbol, under these two conditions : first, that every 
contiguous pair of symbols in any multiplet shall be a contiguous pair in some one other ; and secondly, that 
no three symbols in any multiplet shall occur in any other.” It should have been laid down, as Mr. Cayley 
here states it, that no contiguous duad shall occur non-contiguously, and that no non-contiguous duad shall 
be twice employed. By the words, in some one other, any reader of my memoir will see that I meant, in 
some one other only. 
“ The Problem of Polyedra. 
“ Let a, h, c, d, e,f g, h, &c. represent the vertices of a polyedron, then a face will be represented e.g. by 
abode, where the contiguous duads, viz. ab, be, cd, de, ea are the edges of the face ; and calling the face K, 
we may write 
1^.=. abode (1.) 
“ It is to be noticed that the letters of a face-symbol may be taken forwards or backwards from any letter 
without altering the meaning of the symbol. Thus, abode, bodea, &c., edoba, &c. might any of them be 
taken to denote the face K. The diagonal of a face cannot be either an edge or a diagonal of any other face, 
i. e. a non-contiguous duad such as ao in face-symbol K cannot be a duad contiguous or non-contiguous of 
any other face-symbol. But each edge of a face must be an edge of one and only one other face, i. e. each 
contiguous duad such as ab in the face-symbol K must be a contiguous duad of one and only one other face- 
symbol L. And moreover two faces cannot have more than a single edge in common, i. e. two face-symbols 
cannot contain more than a single contiguous duad the same in each symbol. 
“ The face K contains the edges ab, ac, i. e. the edge ab is contained in the face K ; it will also be contained 
in one and only one other face, suppose L ; this face will contain another edge through the vertex a, suppose 
