194 
EEV. T. P. KIEKMAJs" ON ATJTOPOLAE POLTEDEA. 
In all these (P) there is but one leading system, i. e. system of vanescible gamic pair's 
whereby P reduces to a (r-|-l)-edral pyramid ; except only the two just found, of which 
one has two, and the other three, leading systems. 
These Hi (7’+2)-edra (P), it is to be kept in mind, are all nodally autcrpolar. TMiether 
there be any other enodally autopolar (r+2)-edra generable from the same pyramid, 
when r is odd, remains to be hereafter determined. 
XIV. Problem. To determine the number of nodally aiitopolar (r+3)-f(Zm gerneraUe 
from the {x-\-l)-edral pyramid, by introduction of gamic pairs. 
For the solution of this problem it is necessary to determine the number of pairs of 
diagonals not crossing each other that can be drawn in an x-gon. 
This number is less than that of pau’s of diagonals ; and of diagonals there are 
3r). Therefore P,. 2 the function of r reqrdred is of an order not higher than the 
fourth. It is evident that the function is of the form 
and by trial, 
and 
2 =r . {r— 3)(r— 4)(ar-j- b), 
R5^2=5.2 .l.(5a+J) = 5, 
Rg 2=6.3.2.(6a+^)=21, 
viz. three pairs at every angle, and three parallel pairs ; therefore 
a=b=f2, 
and 
2 = — 3) • (r — 4)(r-i- 1 ) ; 
which can be generalized into 
jy (r— .3)*!-^ 
[Fr2 ’ 
the number of ways in which h diagonals, none crossing each other, can be di’a\^'n in an 
r-gon. 
Let ef and hh be one of these R^ pairs of diagonals of Cl which do not ci'oss each 
other. If we draw them and consider them as new edges. Cl is di\ided into tlmee faces 
OO'O", and the summits efhk become tessaraces if they are all different, or if e=h, e is 
a pentace and f and k are tessaraces. The summit a of the pju'amid, if the result is to 
be autopolar, is broken into three summits odd’, and the faces EFHK are either four 
4-laterals, or a pentagon E and two 4-laterals F and K. The result Q is an autopolar 
(r-l-3)-edron. 
XV. We have to determine how many different (r-l-3)-edra Q can thus be generated 
by a pair of diagonals ef and hk of fl. 
Of two diagonals not crossing each other in an even-angled base, it is plain that both 
cannot be diameters. Let ef and hk not cross, and let not these be fellow-generators ; 
then if neither be a diameter, there is another pair e'f and h’k, the fellow-generators of 
the former two (VIII.), such that the points e'f'h'k' are diametrically opposite to the 
points efhk. And if one of the former two ^ be a diameter, and be the fellow of 
