EEV. T. P. KIEKMAN ON AUTOPOLAE POLYEDEA. 
195 
M, the points feh!k' are diametrically opposite to efhk. Consequently all distances 
measm’ed on the summits of Cl between the signatures efhk, E^E^F^, &c. will correspond 
exactly with distances between the signature ef’h!k'^...ox else between the signatures 
fehlKYr ; for if 05 — j3 be one of these distances, (a — (|8 + 2^) is the corresponding 
distance. Consequently Q generated by ef and hk will not differ from Q' generated by 
df and h!k!, when ef and hk are neither of them diameters, nor from Q' generated by fe 
and h!]d, when (^is a diameter of Cl. Let, when r is even, 
E|. being the number of pairs ef and hk which are not, and E" that of the pairs ef and 
df which are, fellow-generators. It is evident that with the pairs E^ we cannot obtain 
more than ^E^ autopolars Q. Nor fewer. For let df’ and hJ'Jd' give the same Q with 
the pairs ef and hk, and df and h!K. Q" from the first has the same faces OO'O" with Q 
from the second ; and if hk and its gamic be erased in Q, and h!'k" and its gamic be erased 
in Q", the results P and P" will be identical ; i. e. ef mil be the fellow of df”. In like 
manner hk is proved to be the fellow of A" A/' ; but df and h!Jd are the fellows of ef and 
hk ; wherefore df and h!Jd are not different from df’ and A" A" ; contrary to hypothesis, 
which is absurd. Therefore |^Ey is exactly the number of (rd-3)-edra Q generable by 
the pairs Ey. 
Next, let ^ and df be one of the pairs E", the number of which is easily seen to be 
^r.{r — 4), since r — 4 lines not diameters can be drawn from e. 
There is no pair hk and h!k’ of these E" that can give the same Q with ef and df, else 
would be proved the fellow either of hk or A' A'. Consequently the number of different 
Q obtainable from these E" pairs is E". It follows that the number 11^, of (r-t-3)-edra 
Q generable when r is even from all the E^ pairs is 
n„=-i(R;+2K:)=i(E.+i(»^-4r))=*(.^.(»— 4)(»— 2)). 
But yet this number is subject to a doubt, which will be discussed presently. 
XYI. When r is odd and > 3, we have seen (IX.) that if ef and hk be any two 
diagonals whose fellow-generators are df and h!k', ed, ff, hh!, kJd are nodal parallels. If 
then ^and hk do not meet each other, neither will df and A' A' meet each other. Both these 
pairs have therefore been counted in the E^ pairs of not-crossing diagonals. And these 
pairs give the same result Q; for if B — A be any distance measured on the summits of 
Cl in one between summits affected by the first pair, ^A — i(?^+3)j — (B — ^(^+3)^ is the 
coiTesponding distance in the summits affected by the second pair. 
Next, let ^and df be fellow-generators, which do not meet each other in the base Cl. 
They are opposite sides of a quadrilateral ed ff ; and the resulting Q cannot be identical 
with that made by any other pair gh ij, for this would require that gh should be fellow 
to one and ij to the other of ef and df. 
The number of pairs ef df comprises, first, that of the quadrilaterals whose opposite 
sides are non-contiguous nodal parallels ed and^', of which nodal parallels there are 
\{r— 3) (XII.) ; and secondly, every pair of lines ef, ef drawn from the point {e—f), 
2 D 2 
