THE SOLUTION OF THE PROBLEM OF THE AUTOPOLAR P-EDRA, WITH FULL CONSTRUCTIONS UP TO P = 10. BY THOMAS P. KIRKMAN, M.A., F.R.S. 1. In an autopolar p-edron, to every edge ABcd, whose faces are an A-gon and a B-gon, and whose summits are a c-ace and a d-ace, corresponds one edge CDab, whose faces and summits are reciprocals of the former, A and a, like B and b, etc., being the same number. These edges are a pair of reciprocals. 2. Every nodal autopolar has two, and only two, nodal faces, and A, each containing its reciprocal summit, viz., 0 and 8, which are the two nodal summits. An enodal autopolar has no nodal face or summit. 3. The nodal angle of is yθλ, and that of Ais πδ φ. The summits y and a are the nodal tips, or the tips of , and and are the tips of Δ. 4. The edges by and θλ are the two nodal sides, or the sides of, and 8 and do are the sides of A. 5. Of the faces collateral with, A which contains the side by, and I which contains θλ, are the two nodal walls, or the walls, of the node. Each wall, F, is the reciprocal of the opposite tip y, and each side is the reciprocal of the other side. In like manner the walls of the node A are П, containing 4, and I containing π. • λ, 6. If the summits of the face are in order θγμ the faces about the summit procals of those summits. 7. If the faces collateral with are in order AAB FT, the summits in the rays of the pencil 6 are in order yf... bax, reciprocals of those faces. K The faces A and F next to the walls A and I are the two by-walls of the node, and their reciprocals a and f in the pencil are the two by-tips of the node. The remaining faces about between A and F, are the off-walls, and the summits in the pencil 6, between a and f, are the off-tips, of the node. 8. Every autopolar p-edron is first found, and described in our tables, by its two nodal signatures, its tabular signature, and the specific statement of its symmetry or asymmetry. The nodal signature of > 3 is written : Θ = θγμ...λ ΛΑΒ.. FF where by is the common edge of and the wall Λ, γμ that and the by-wall A, etc, θλ being the common edge of and the wall Γ. of 9. We shall have little need to trouble the reader with nodal signatures, as the autopolars of ten and fewer faces are all figured (pp. 153 to 167). The dotted edges are in inferior faces. In every figure the two nodal angles are marked by a small curve in each nodal face. If the reader writes on any one of these marked faces, putting at the nodal summit, with y in the wall A, and in the wall I, at the tips, and if he then names the remaining summits abc in any order, he will find that the faces will denominate themselves so that each will answer exactly to its reciprocal summit. He can then write down from the lettered figure the nodal signatures. He will thus construct those of obbs, fig. 128, which happens to be drawn upon a nodal face as base, viz. : 10 Here has two triace tips, and consequently two triangular walls; it has for by-walls a 4-gon and a triangle; its 4-ace by-tip 4, but not its triace by-tip 3, is read in A. A is an off-wall of, and is a by-wall of A. The two nodal faces have a common edge 3,34, but not, as many have, a common tip. In the tabular signature 3' means 3333333. The nodal signatures of tts, in the same page are 31 The nodes have no feature in common. Each has a triace and a pentace tip. A being a triangle, has only one by-wall, which belongs alike to either tip. .. It is best to call the triaces 3,3,3,. the 4-aces 4,4... and the 5-aces 5. etc. The walls may be omitted, being given by the opposite tips. 10. Node-Forms. There are three kinds of node-forms, the t-node, or tip-built node; the b-node, or by-tip-built node; and the w-node, or wall-tip-built node. Definition of a t-node. A t-node has at least one reducible tip, that is, a tip whose side can be deleted without loss of the tip. Therefore this reducible tip is no triace, for that cannot lose an edge; nor can it be in a side across which is a triangular section of the solid; for, if such a side be deleted, there is a linear section, Q.E.A. A tipy > 3 in a t-node is always written y', i meaning indelible, when there is a triangular section of the solid across the side θy. A tip y=3 in a t-node is never marked indelible, whether there be or be not a triangular section across by. But in a b-node, to which we shall come presently, when such a section exists, the tip y=3 is always written 3. Every autopolar marked tt has two t-nodes; every one marked to or tw has one. 11. Reduction of a t-node. Every t-node of a p-edron H has at least one side ΘΛ = θγ (4) which can be deleted, while its reciprocal θλ = ΘΓ convanesces by the coalition of and λ. The two faces and A revolve into one, which is the new nodal face O', containing the new nodal summit θ', made by the union of and λ. H has thus become an autopolar (p-1)-edron H'. If the t-node in H has two reducible tips, θλ instead of by can be deleted, while by convanishes, and we have another (p-1)-edron H", different from H', if by and θλ in H are not identical by reason of a zonal trace between them. The new nodal face in H' is a (+1-3)-gon; that in H" is a (+Г-3)-gon. The p-edron H will be built by our processes both on H' and on H", and H is said to be twice made at its node 6. If H, being asymmetric, can be thus also twice reduced at its node 8, it is said to be four times made. No H can be more than four times made. 12. Construction of a t-node. This is the converse of the process (11). The (p-1)-edron H' or H" may be any autopolar whatever. We consider first the case of a triangular in the pedron H. At the '-node of the (p-1)-edron H', whatever θ' or H' may be, draw from either tipy a line meeting the opposite side (3) in the triace 0=3. This 4=3 is the new nodal summit in the new nodal triangle=3, in our new p-edron, H, whose tips are y=y'+1, and λ=0'=3. If ' in H' be a triangle, only two t-nodes can be built on it, of two p-edra H, one by the line just drawn, and another by such a line drawn from d' in H', constructing a different=3. 13. But if ' > 3 in H' we can construct, besides these two triangular, others not triangles. Let μ be any summit of O', not a tip. We can draw in 1 O'the diagonal θ'μ, and if we also draw its reciprocal, all will be autopolar. Instead of θ'μ, draw first Θ' Μ=0"q, partitioning θ', and adding an edge to '. We can take our choice of joining μ and θ", or μ and q, thus partitioning the augmented O' into ', and '2, and adding an edge to 6" or to q. The result is an autopolar p-edron, having for nodal face either >3, containing the nodal summit 0"+1=0', or containing the nodal summit q+1='2. In one case the new tips are μ+1 and q, in the other they are μ+1 and 0". The two sides in either case are the lines partitioning O' and θ' of H'. '>3, This will be clearly enough seen by the reader, if he effects in 10tb 33 and in 10tb34, figs. 207, 208, the reduction (11) of the 4-gonal t-node. He will see that a diagonal θμ has been drawn in two ways in the same subject 9-edron, to the same point μ. Two such t-nodes '1+1,>3, and '2+1,>3 can be obtained by a diagonal drawn from ' to every different summit μ, which is not a tip, in the subject '>3. 14. Definition and reduction of a b-node. A b-node has two triace tips y = λ = 3 (8). It can be demonstrated of 1st, that > 3 except in any b-node on a p-edron K the 4-edron; 2nd, that the two by-walls (8) A and F of the node are not both triangles; 3rd, that whether there be one triangular by-wall or none, there is one side whose tip is in a non-triangular by-wall F, across which side there is no triangular section of the solid; 4th, that this side a can be deleted, whence the by-wall F loses the triace λ, while θγ the reciprocal of a convanishes, whereby the by-tip f loses the triangle A; the result being an autopolar (p-2)-edron K', which may be of any species; 5th, that if K has no triangular by-wall, and if there be no triangular section of K across either θyorθλ, K is reducible in the same manner at the tip y = 3 to one, and to one only autopolar (p-2)-edron K", at the like cost to the by-wall A and the by-tip a. But if the |