tip of a b-node in a non-triangular by-wall A is marked 3', (10), the b-node is not reducible at the by-tip a.

15. No b-node of a p-edron K can be more than twice made. If A, as well as in K, is a b-node, it is possible that K may be four times built in our processes, on four different (p-4)-edra. No such K can be more than four times made.

16.

Construction of a b-node. This is the converse of the process in (14).

At either side of either node of any autopolar (p-2)-edron K', make between ' and A' a new A=3, augmenting ' and adding y=3 to the by-wall A' of K'. Then between ' and ' make a new λ = 3, augmenting ', and adding = 3 to the I by-tip a' of '. K' has become an autopolar p-edron K with a b-node.

1

If K' has four different nodal sides, four different b-nodes can be thus built of as many different p-edra K, whether ' in K' be a triangle or not. K' from symmetry may have fewer than four different sides.

17. Definition and reduction of a w-node, or a wall-tipbuilt node. A w-node has one tip y a triace, and the irreducible (11), by reason of a

other no triace, but
triangular section across the side x.

angle whose tips are y and λ, so in the wall r

aw may be termed the wall-angle, of which and w are

the tips, and in the wall A = y v.. Ө we may call and v the wall-tips. But whenever we speak of a wall-tip we mean always either w or v, or the corresponding summits read from (8) in the walls of the node A.

As the tip-built node was reduced (11) by lowering a tip, and the by-tip-built node by lowering a by-tip (14); so we are about to see that the wall-tip-built node is reduced by lowering a wall-tip.

18. The nodal signature of a w-node in the p-edron

L, is

℗ = 0 31 μ . . . y', (y > 3); (℗ > 3);

1

гWB..F3,

0

where the i over y denotes that the side y is indelible by reason of a triangular section (10) of the solid across it. We may speak of such a tip y', (y > 3) as of an indelible tip.

19. It can be demonstrated:--1st, that the by-wall W is no triangle; 2nd, that along the edge 3, = W O there is no triangular section of L, and hence, 3rd, that 3,μ can convanesce (11), whereby W and each lose an edge, and consequently that we 3 M can be deleted, whereby w and each lose an edge; also the summits 3, and μ coalesce into a (+1)-ace tip, while 3, and M unite into a (M+1)gonal wall.

=

The p-edron L, by the loss of the tip 3, and the wall 3,, has become an autopolar (p-1)-edron L', whose nodal signature at its reduced ', which may or may not be a triangle, is

which is a t-node having no triace tip, but an indelible tip > 3, over which is written the wall-tip (17) in the wall

For, in L, 3, is

(M + 1) = o'y (w-1)

O.

г W, that is, 3, is 0 y w, collateral with is an edge of . The deletion of the edge Ow makes the enlarged face

M, because 3, 3 M

=

(M + 1) = (0−1) y (w−1) .

The reciprocal (W-1) of this wall-tip of A' = (M+1) in is the tip-wall of x' = (+1), in the (p-1)-edron L'. When above is a triangle, B and F are the same bywall (9).

The triangular section across by in L has not been disturbed by the vanishing of the edge 3 M Ow; it has

=

';

merely become a triangular section across '(M+1)=0'y, of which the trace lies in (M+1) in which 3 of is merged. That is, there is still the triangular section across 'y in L', and the tip is still y' indelible in L'.

20. In (17) we said that the w-node of L was reduced by lowering a wall-tip. We have lowered w, a by-tip of L. In every wall we can find both a wall-tip and a by-tip, which, when the wall is a triangle, and in that case alone, are the same summit of the solid.

"

This wall-tip (w-1) over y in L', on which the w-node of L is constructed, is always given in the t-nodal signature in L", the subject on which ' in L' is built in our process. We never build a t-node 'but by augmenting a summit m which m are written two faces

of a subject node, under containing m, of which the one that is not made a by-wall in the new t-node is the tip-wall, whose reciprocal is the wall-tip (w-1) above named.

We call the above t-node ', carrying the wall-tip (w−1), a t-node under a wall-tip. On such a t-node every w-node is constructed. We shall presently discuss the construction of w-nodes, and learn when the wall-tip is or is not to be superscribed.

21. There can be no other node-forms than three, viz., the t-node, which has at least one reducible tip, and may be a triangle ; the b-node, which has two triace tips and is no triangle ; and the w-node, no triangle, which has one triace tip and one indelible tip (18).

Since every autopolar can be reduced without changing the number of its nodes, none can have more than two nodes; because the pyramid to which reduction must bring us has only two nodes.

It is possible so to name the faces of certain symmetrical autopolars that there shall seem to be more than two nodes. A 4-edron can have its faces so named. But this is a false

signature, made by misnaming faces A, A, A, . . . which have the same configuration.

It is important to remark that, after the construction of a node of any form, every feature of the subject signature which is augmented by the operation, and is also a feature of A, must be augmented in the subject ▲; and if another feature augmented by an operation on the corrected ▲ is read in the just constructed, the like augment must be given after the operation to this feature in that . This is perfectly simple when the nodal signatures are before us.

22. The nine species of autopolars. The names of the species are tt, t(t), bb, b(b) ww, w(w), tb, tw, bw. The first six have and ▲ of the same node-form; the other three have and ▲ of different node-forms.

23. Definition, reduction, and construction of the species (tt). The definition is, that either t-node of a p-edron 'tt can be (11) reduced without destroying the t-form of the other node. Such a p-edron H can be reduced (11) by operations at both nodes, no matter in which order, to a (p-2)-edron H', which may be of any species; and on this H', H can be built (12) in one way only by operations at both nodes of H'..

We make the rule, that no „tt shall be constructed but by operation (12) (21) at both nodes of a (p-2)-edron. If H has more than one pair y, ◊, of reducible tips, it can be reduced to more than one H'. If γΦ, γ π, λφ, λπ are all reducible pairs, H can be reduced to H', H', H', and H1⁄4, and on each of these four (p-2)-edra the p-edron H can be built in one way only.

49

24. The only possible reasons why ptt the p-edron H should not be reducible at the pair of tips y, π, to a (p-2)-edron, are

1st. That one or both of y, π, is a triace;

2nd. That one or both is indelible (18);

3rd. That, neither y nor

being triace nor indelible, there is a quadrilateral section across by and dτ.

The deletion in such case of both will introduce a linear section, Q. E. A.

The sides by and dπ are in this case called a pair of semiindelible sides, and we call y and π semi-indelible tips, and write y and in the nodal signatures. There may be two pairs of semi-indelible tips, yπ and yo, or у and λ, in which case we write y", ", and "'; or y", "", and x", 4". The first condition allows only λ and λ4, the second allows only yo and λ, to be reducible pairs.

φ.

25. There is a number of little theorems about indelible and semi-indelible sides, with which I shall not here trouble the reader. It may suffice to lay down the leading principles.

1st. The conditions necessary and sufficient that yi shall stand in a nodal signature is, that there be an edge e or a face F of the solid, which has a summit of and a summit of A. For a triangular section across by Amust traverse For contain e.

=

2nd. A quadrilateral section across y=OA and dπ=AÞ is a section through the faces A ▲ & in that order, or else in the order Θ Λ Φ Δ.

The condition necessary and sufficient that there shall be the former section is that A and A shall have a common summit, and that ✪ and Þ shall have a common summit.

The condition necessary and sufficient that there shall be the latter section is, that A and shall have, and that ▲A and shall have, a common summit.

26. There is never any doubt about the right number or names of indelible or semi-indelible tips, in a constructed p-edron ptt, H. The nodal signatures of the (p-2)-edron on which H is built always supply the required information. If there be a suspension of judgment, to be indicated by a (?)