EEV. T. P. KIEKMAN ON AUTOPOLAE POLTEDEA. 
211 
polars (S), having no r-gonal O, is to be added to (Q)" in XXXIII., completing the list 
of purely enodal antopolars. And thus the question of (XXXI.) is determined. 
We have exhausted all the methods of adding two faces and two summits to the 
(?"+l)-edral ppamid. By adding two faces in the base Q, and their two poles about 
the vertex a, we formed the (r+3)-edra (Q). By adding a face and a summit in Q, and 
their polars about cu, we formed the (r+3)-edra (S). 
Collecting, now, our results from XIII., XVIII., XXIX., XXX., XXXIII., we find 
the numbers IIj of (r+2)-edra and Ha of (r+3)-edra, nodal, enodal, arid utral, which are 
generable firom the (r+I)-edral pjoramid, to be the following: 
n.=i{r^-3r+2.4,_a+(^-3).2,_J; 
U,=-i^{r%r-4:)(r—2)}2, 
-{-^g{2r . {r— 3)(r— 4)(r + 1) — 3 . (r — 3)(r— 1)+ 3 . (r — 3)(r — 3) }2,_, 
+i(/^-3r+2)4-^4-(/’-3)(7'-5)(r-I).2,_.+i(r-5).2,_,-0(’-'‘)^-'’-7)^-2.0(>-=^ 
or 
na=^{(/’"-6r®+20r^-36r+24)2,+(r-3)(r®-2r"+5r+8)2,_,-0(’-‘‘«’-7)’-2.0(’-®>^}. 
It has been proved that these Hi and Ha, and these only, are the autopolar (r+2)- 
edra and (r+3)-edra generable from the (r+I)-edral pyramid, that none of them is 
reducible by vanescible pahs to a higher pyramid, nor any one a repetition of another. 
The problem of enumeration of the ^-edra may, by a shght extension of the meaning 
of the word partition, be stated thus : to find the 'k-partitions of a pyramid. This 
depends on another : to find the k-partitions of a polygon ; which is also thus : to find 
the k-partitions of a pencil. By the ^-partitions of a j9-gon, I mean the number of ways, 
none a repetition or reflexion of another, in which k lines can be drawn in a j?-gon, none 
crossing another, so as to make the system of 1 face and p summits into a system of 
h-\-\ faces and^+^ summits (Ji-\-i—k)^ the k hues being terminated either by summits 
of the^-gon, or by i points chosen either on its edges or within its area ; with the under- 
standing that at least three hnes shall meet in each of the i points, two of which will 
always be a side and its segment, when ^ is chosen on a side, the segment counting 
among the k hnes. 
VTienever a ^-gon and a ^-ace are similarly ^-partitioned, a certain number of auto- 
polar (^+^+l)'edra are obtained by different ways of applying the^-ace to the^-gon; 
when they are dissimilarly ^-partitioned, or when the ^-gon is ^-partitioned and the 
j)-ace (^+Z)-partitioned, a certain number of heteropolars will arise from different ways 
of applying the ^-ace to the ^-gon. The direct or reverse manner of applying the ^-ace 
to the ^-gon will give nodal or enodal autopolars. 
The way seems now clearly indicated, and partly laid open to the solution of a geome- 
trical problem, which, while it seems at first sight almost elementary, has lain for centu- 
ries before mathematicians unanswered. The enumeration of the ^-edra is a question of 
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