192 
EEV. T. P. ETE E MAN ON AIJTOPOLAE POLTEDEA. 
Now as neither e noxf (^— ^)= + l, and 
e=f±^, 
showing that ef is a diameter of an even-angled base, as is also €’f\ one case of ef. 
Putting we deduce, from k=\, A/ =2, 
2e-[-2/'— 3 -j-r 
2e-j-2e' — |r=3-f-2r 
2^+2/=3+|r; 
proving that r=4A-j-2, and 
2e4-2e'=3-f-|r. 
These equations give e' and/' ditferent from e and/ except when e'=/=e+ir, or ^=e; 
e. when 
. 6 + 3?- 
4e_ 2 , 
and 
«=f(2+/; 
or 
4e=i(6+5r), 
and 
e=i(6+5?^)- 
Hereby we learn that when r=8A— 2, or r=8A-h2, aU the diameters of H pair them- 
selves into generators, ^and e'f, of the same (r+2)-edron P, except only that drawn in 
the first case through e=f(2-fr), and that drawn in the second through e=|-(6-l-5r). 
XII. We can now easily enumerate the autopolar (r-|-2)-edra generable from the 
pyramidal (r+Ij-edron (H). 
When r=4A, all the diagonals of Cl pair themselves into fellow-generators ^and e'f. 
The number of diagonals is ^ — 3r). Hence the number of autopolars 
(P) required is 
When 7’=4A-J-2, all the diagonals but one, namely one of the diameters (XI.), pair 
themselves ; consequently, the number of autopolars (P) sought is 
+ 3 >’+ 2 ). 
The question left unsettled in (IX.), as to whether the nodal parallels pair themselves 
into fellow-generators, is decided in the negative by silence of our formulse in X. and 
XI. on the subject of r=2A-hI. The diagonals of Cl for r odd parr themselves, unless 
when 
2/+2e=3-l-r, 
or 
2/+2e=3+3r. 
For every value of e, whether r=4^+I, or 4^—1, these equations give/ on a nodal 
and unpaired parallel, except when they give e=f, or e=f-\-l ; i. e. unless 
4e=3+r, 
or 
4e=3-j-3r, 
