EEV. T. P. KTRKMAN ON AUTOPOLAE POLTEDEA. 
197 
become triangles, and oo\ oo'\ which are their edges out of the base, are neither of them 
convanescible. Therefore ode cannot be united. 
There may be a convanescible edge ee', which is e.(e+l), or FO'. If F be a penta- 
gon, for a reason just given, it adjoins two triangles upon (e—l)e and (e-|-l)(^+2), so 
that oe nor oe', one of the sides of it, cannot be a convanescible edge. If F be a quadri- 
lateral oo'ee', its sides oe', o'e will be in the triangle ode when ed convanesces. Therefore 
oee' cannot be united into one summit. 
Our three sought summits will therefore be (e—1), e, (e-\-l). That (e—l)e may con- 
vanesce, its faces F and O must not be triangles, neither can(F — 1) and O' about e(e-\-l) 
be triangles. Therefore O and O' together have at least six summits of Q,, which, sup- 
plying at least one summit more to O", cannot be simpler than a heptagon. Neither 
can it be ampler', for e is at most a pentace, adjoining two triaces (e—1), (e+l), for a 
reason above given. And no diagonals have been drawn to make a third tessarace among 
(e — 1), e, (e+1), supposing two of them tessaraces. That is, these three can unite at 
most to form a heptace a". We are then to examine whether two diagonals can be drawn 
in the heptagonal Cl to make two adjoining convanescible edges. The only way of 
doing this is by drawing either 62 and 63, or 72 and 53: in either case 65 and 67 are 
both convanescible, as well as do" and do. The faces of the decaedron in the first 
r0Slllt 
3456, 362, 2671, lo"2, 2o"o'o3, 3o4, 4o5, 5oo'6, 6o'o"7, 7o"l ; 
and in the second, 
345, 35672, 721, 7o"l, lo"o'2, 2o'3, 3o'o4, 4o5, 5oo'6, 6o'o"7. 
These are identical, if we exchange the signatures in the edges 5o, 6o', and 7o". 
XVIII. This decaedron is the only one of the autopolar (r+l)-edra Q above con- 
structed which has two leading systems, i. e. two pau’s of leading convanescible edges, 
and two pairs their gamics evanescible; and in this case alone have the operations (62 
and 63) that gave us Q' brought out the same polyedron vrith the operations (72 and 
53) that gave us Q". It is therefore necessary to deduct one from the (Q) enumerated, 
namely, one for the value r=7. 
This number is consequently (r>3), (XV. XVI.), 
n,+n>^.ri.(r-4)(r-2).2, 
+^{2.r.(r-3)(r-4)(r-l-l)+3(r-3)(r-l)+3(f- 3)(r-5)}.2,_,-0(’-^)l 
XIX. Our next step is to take up one of the (r-f-2)-edra P (XIII.), obtained by draw- 
ing a diagonal ef in the base O of the pyramid. P has two quadrilaterals E and F, in 
either of which F we can draw a diagonal, making a triangle F', a side of which is od, 
the gamic of 00' in the base. This new diagonal in F is either F^o) or (F^o'); and the 
triangle is either F^oo' or bfid. Thus a new triangle is introduced between F and F-j-1, 
or else between F and F — 1, about the line od. Consequently there must be a new 
triace between /and (/-}-l), or between /and (/—I) on the edge 00'. Thus the gene- 
