XI. On autopolar polyedra

Abstract

I. As any face of a p -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 l summits may be denominated a k- edron or an l- acron as may be most convenient. By an autopolar polyedron , I mean one in which the number, rank, and collocation with respect to an α -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 Arthur Cayley, 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 Representation and Enumeration of Polyedra,” in the twelfth volume of the Memoirs of the Literary and Philosophical Society of Manchester, with some matter which has arisen in our correspondence; and particularly it supplies a defect in my statement of analytic conditions in art. 22 of that memoir, which Mr. Cayley with his rare penetration was the first to point 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 .