EEY. T. P. KIEKMAN ON AUTOPOLAE POLYEDEA. 
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so that r+3>-5-f-3, andr>>5. Also r<^5, because Q' cannot be generated, when r< 5. 
We have then to endeavour to draw ^ and eh in the base of the 6-edral pyramid, so that 
ef shall be evanescible, and e(e+l) then convanescible (for eo cannot vanish, since 
0 o' and o" are all triaces of necessity). It would be trifling with the reader to write a 
demonstration of what inspection of the pentagon proves in a moment, that of the three 
summits which e may be, 1, 2, and 5, 2 and 5 alone fulfil the conditions. Draw 25 and 
24 ; then 25 is evanescible, and by its evanescence 21 becomes convanescible. Again, 
draw 53 and 52 ; then by the evanescence of 53, 54 becomes convanescible. These two 
Q have therefore been enumerated both in the class (Q) and the class (S), from which 
latter we have then to subtract two when r=5. That is, we must add — 2.0'^’’“'^^ to 
IT before found in XXIX. This makes up 
the correct number of the (r+3)-edra (S) generable from (r4-l)-edral pyramid with 
nodal signatures. And thus the Problem (XIV.) is solved. 
XXXI. It is yet required that we consider (VI.) the pyramid on an odd-angled base 
with enodal signatures, and determine whether, by operating upon it, we can obtain 
(r+2)-edra or (r+3)-edra which we have not already constructed with nodal signatures. 
The nodal arrangement becomes enodal in Q the base of the pyramid, if without 
disturbing those of the faces we exchange the signatures at the extremity of every nodal 
parallel, including among them that edge of Q which is bisected by the nodal axis (VI.). 
After these exchanges every summit stands as the pole of the face opposite to it. And, 
conversely, an enodal signature of a pyi’amid becomes nodal, if all the diagonals parallel 
to any side of the enodal Q be drawn, and then the signatures be exchanged both in that 
side and in these diagonals. The line bisecting that side, and passing through the oppo- 
site summit, becomes the nodal axis. All this will be clear to the reader, if he mil have 
the goodness to draw a pentagon or a heptagon, and make these exchanges. 
Let r=2^-f-l. All the single generators that can be drawn in the enodal Q to 
generate a (r-l-2)-edral autopolar are among the parallels to the side [Jc, ^+1). Let 
{e, r—e) be one of these: it is perpendicular to the axis of symmetry (r, which 
passes through r and bisects Q. If E and (R — E) become 4-laterals, opposite to the 
4-aces, e, and (r — e), the edge ^E, (R — E)) is the gamic of {e, [r — efj, and the result is 
autopolar. 
Let now the signatures be exchanged in (e(;r — efj and on both sides the axis in lines 
parallel to (e{r — e)y The result is still autopolar with nodal signatures; from which 
the nodally autopolar pyramid may be obtained by the evanescence of (e, r—e) and the 
convanescence of (E, R — E) ; its nodal axis will be that axis of symmetry. 
Thus it appears that every enodally autopolar (r-|-2)-edron generable from the enodal 
(r-l-I)-edral pyramid is also nodally autopolar, and generable from the same nodal 
pyramid. That is, it is one of the (r-{-2)-edi’a P already enumerated. 
XXXII. Suppose that instead of drawing one diagonal {e, r—e) perpendicular to the 
