186 
EEV. T. P. KrRE3IA2s’ ON AETOPOLAE POLTEDEA. 
and whose faces are polar to the summits of H. Every edge of H is the gamic of 
an edge of H', and H and H' are either the sympolar of the other, and form a sym- 
polar pair. 
II. An edge of a polyedron is said to convanesce^ when its two summits run into one, 
and it is said to evanesce^ when its two faces revolve into one. 
An edge AB is said to be convanesmble, when neither A nor B is a triangle, and AB 
joins two summits, which have not besides A and B two faces, one in each summit, colla- 
teral nor having a common summit. 
An edge ab is said to be evanescihle^ when neither a nor 5 is a triace, and the two faces 
about ah are not, one in each, in two summits, besides a and 5, collateral nor in one face. 
If AB be convanescible, its gamic ab is evanescible, and mce versa ; and the pair AB. 
ab are vamscible gamics, when in an autopolar polyedron. 
III. Theorem. No polyedron, not a pyramid, has every edge both in a triangle and in 
a triace. 
For let it be supposed that P, which is not a pyramid, has every edge both in a tri- 
angle and in a triace, and let E be a face of P not less than the greatest ; the faces 
collateral with E are by hypothesis aU triangles. Let A and B he two of these having 
two contiguous edges of E. There will be a summit of E not a triace, otherwise P would 
be a pyramid ; let this be d, a summit of E, A and V (EAV), V bemg a face collateral 
with A, but not with E. As one extremity of the edge AV is not in a triace, the other 
is ; therefore AV will end at the triace AVW, the vertex of A. As P is no pp-amid, it 
has more summits than one not in its base E, which are all connected by edges ; there- 
fore the edge AW must pass from AVW to some summit not in E. 
The edge EA having an extremity d not in a triace, terminates at tho triace EAB, 
where AB is the only edge meeting E, and the triangles A and B have a common vertex 
AVB, the intersection of AV and AB. But this is the triace AVW ; therefore AW is AB, 
an edge passing through a summit of E, contrary to what has been proved. Q. E. A. 
Therefore P has an edge either not in a triace, or not in a triangle. Q. E. D. 
IV. Theorem. Every polyedron P, not a pyramid, has either a convanescible or an 
evanescible edge. 
For P has either the edge AB, whose faces A and B are neither of them triangles, or 
the edge cd, whose summits are neither of them triaces. Let it have AB, an edge not 
in a triangle, lying between the summits e and/*. By definition of a convanescible edge, 
if e and y have not besides A and B two collateral faces, C and D, AB is a convanescible 
edge of P. C and D may have a common edge gh, of any length, or a point. 
If e and /have two collateral faces, C through e, and D through/ intersecting in gh, 
then neither e nor /will be a triace, or else one of them, e, Avill be a triace. K, fii-stly, 
neither be a triace, AB=^is not in a triace, and is therefore evanescible, imless A and 
B about it are in two other summits Jc and I collateral or in one face. Now if a section 
be made through ef, and m, any point of gh, it -svill cut any edge kl drawn or di’awable 
upon P, from ^ in A to ^ in B, and will divide P into P', containing A, and P", contain- 
