198 
EEV. T. P. KIEKMAI^ ON AUTOPOLAE POLTEDEA. 
rator 00' is drawn from a summit of CL to the mid-point of a side of CL, dividing the 
faces 00 ' whose summits together make r+3. 
We are to consider how many (y+3)-edra S can be obtained by such an opera- 
tion in CL. 
The diagonal ef being drawn, becomes by the motion of the extremity c(/+D- 
e'f be the fellow-generator of ef, is that of e{f-\-V) when r is even, and 
e'{f — V) that of when r is odd. Two fellow-generators when drawn are sym- 
metrical with respect to the nodal line, which is a diameter when r is even, but not when 
r is odd. Consequently, e\f-fz\) will be the fellow-generator of e{f+\) when r is even, 
and of e{f^\) when r is odd. Even vfbenfe is the fellow of ef, or efis an unpaired 
generator, /(e+^) is a different line from either e{f+\) or e{f^\). Consequently, 
every line X that can be drawn in CL from a summit to the mid-point of an edge, has a 
fellow-generator, except only when e{f-\-l) is the fellow of ef, which happens when r is 
odd, and e{f-\-^) bisects CL, being the nodal axis perpendicular to the nodal line. And 
there is plainly but one such fellow of X, which being dra’WTi shall be symmetrical with 
it as to the nodal line. Therefore every (r-j-3)-edron S, except one, that is generated 
by a line X, will be generated also by a line X'. Of lines X there are r . (r — 2); hence the 
number of (r-l-3)-edra S is not greater than ^{r.{r — 2)-l-2,_i}. Nor are there fewer 
that can be reduced to a pyramid on the exact base CL. It is, however, to be inquired, 
whether some of these (S) may not have two faces not O and O', which can by the 
vanishing of two edges be united into an r-gon CL', giving a pyramid on the base CL'. 
But before we examine whether any of these (r-j-3)-edra S are repetitions of each 
other, it is desirable to ascertain how many of them are reducible to the (;’-}- 2 )-edral 
pyramid. If any one of them, s, is so reducible by the vanishing of a single gamic pair, 
it Avill be a repetition of a (r-l-3)-edron P, generated by a diagonal in the (r-|-I)-gonal 
CL ; and will have a convanescible edge g between summits whose united ranks a- and ^ 
give .r-l-^=y-{-3. That these two summits in s cannot be o and o' is certain; for the 
edge oo' is in a triangle. Then one only, or neither of these ^-ace and y-ace, will be 
0 or o'. 
XX. First, let neither of them be o or o'; i. e. let g be an edge of fl. The greatest 
of j: and 3 / 5 ; for ef^ being drawn in s to the mid-point of a face H, E becomes a 
4-lateral containing 0 and o', and ri the pole of H becomes a 4-ace containing y]o and o', 
unless E = H, in which case it is a 5-gon, containing 0, o', f, and tAVO summits of CL. As 
no summit or face, except e and E, is affected by ef^ in this case, all the faces but E about 
Cl remain triangles, and therefore there is no convanescent edge g. 'Sllierefore x 4, 
and y is either 4 or 3. First, let ;r=4 and ?/=4 ; whence r=5. We are to look for an 
edge convanescible and between two 4-aces in s made from the pentagonal Cl. The only 
4-aces in s, neither 0 nor o', are e and n- Therefore eri is the edge g, and e=rif:l. We 
see from 
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that in order to have e and ?? contiguous 4-aces, we must either draAv the genei*ator 
