KEY. T. P. KIEKMAJ^ ON AIJTOPOLAE POLYEDEA. 
193 
which give e=f; and unless 
4(?=l+y, 
or 
4^=1 -f-3r, 
which give 
f=e-\-l. 
en e—f, the nodal parallel is simply the point e, an evanescent parallel. When 
it is an edge of Q. From every other point e of Q a nodal parallel can be 
drawn, which has no fellow-generator, and the number of these is — 3). Therefore 
the number of diagonals of Q when r is odd, all different generators, is 
the number of autopolars (P), when r is odd, constructed. 
XIII. But it is next to be determined, how many times P, thus generated by a dia- 
gonal ef, is identical with P', generated by a diagonal hk different from ef*. 
As no two operations on the base Q have been alike, it is certain that P' cannot be 
reduced to a pyramid on the exact base Q by the operation whereby P is so reduced ; 
but it is possible that P' may be reducible to an r-gonal based pyramid, by the union of 
some two faces which are not O and O'. If so, P' will have a convanescible edge differ- 
ent from oo'. This can be none else than OH or OK, or else O'H or O'K, as H and K 
are the only faces not triangles distinct fr-om O and O', which have not a common conva- 
nescible, but an evanescible edge. , 
If OH be convanescible, O is not a triangle ; therefore y > 4, and the two summits of 
OH, when united, must give a summit at least a pentace. Greater than a pentace it 
cannot .be, since there can be no summit of P, neither o nor o', ampler than a tessarace, 
and this must adjoin a triace, because no diagonal hk in Q can make two collateral 
tessaraces. The edge OH must then both be convanescible and have a tessarace, and be 
an edge of the base of a 6-edral pyramid. It is easily seen, either by trial or by inspec- 
tion of 
15248342515 
that the only diagonals fulfilling these' conditions are 13 and 24 (or its fellow 25). The 
first gives a 7-edron P with three convanescible edges, 00 ', 15, 34, which reduces to a pyra- 
mid either on the base 12oo'5, or on 234o'o. The second gives a 7-edron P' having two 
convanescible edges, od and 45, reducing to a pyramid on the base 123o'o. And P, P', 
are plainly not repetitions of each other. Therefore no deduction is to be made from 
the number of (r-j-2)-edra (P) constructed from the pyramid. If H, be this number, 
n,=i{(^’^-3r)4,-f(r^-3r+2}4,_,+(r^-2r-3).2,_.}; 
the number of (r4-2)-edra(P) generable from the (r-]-l)-edral pyramid, where the circu- 
lator 4,.= ! or 0, as r is or is not 4m. 
* It is clearly impossible that any of these P can be either a ])yramid or reducible to a pyramid of higher 
rank than the (r + l)-edral fl. 
2 D 
MDCCCLVII. 
