EEV. T. P. EIEKMAN ON AUTOPOLAE POLTEDEA. 
209 
reducible to the nodally-signed pyramid: it must therefore be a different (r+3)-edron 
from all the nodal (Q) before constructed. We may call \i 'purely enodal. 
We have then to add to the nodally autopolars (Q) all those enodally autopolars 
generable by a pair of diagonals of the enodal Q, which are neither a parallel nor an 
equal pair-. And we have to enumerate these pairs, taking care that no one is a repe- 
tition or a reflexion of another in the enodal Q. 
As any of the summits of Q may be r, the number we are seeking cannot be greater 
than ^th of aU the possible pairs of not-crossing diagonals of Q ; that is, the function of 
r, required, is not of a degree higher than the third. And it is evident that it must 
vanish, for r=3 and r=5, and must be 
V,=(y— 3)(r— 5)(ar-|-5). 
By trial we find readily that 
V7=4.2.(7«-1-^)=2, 
and 
Vg=6.4.(9a+^)=8, 
V 7 being the two pairs (72, 73) and (72, 74); and Vg being (92, 93), (92, 94), (92, 95), 
(92, 96), (92, 84), (92, 85), (93, 94), (93, 95). Hence comes 
W=^.(r-3)(r-5)(r-l). 
And this is the number oi purely enodal (r-|-3)-edra (Q), which are generable from the 
(r-)-l)-edral pyramid (when r is odd only). That is, we have to join to our previous 
enumeration the number 
(Q)"=^.(r-3)(r-5)(r-l).2,_„ 
to be added to of XVIII., XXIX., XXX. 
XXXIV. We have yet to determine whether any purely enodal (f-f-3)-edra S can be 
generated by a line drawn from any summit to the mid-point of any edge of Q. 
When r=4^-}-l, the enodal signatures read thus: 
...E (E-1), ... (E-K) (R-K-1) 
... 2 ^ ( 2 ^- 1 )... (^+ 1 ) k : ^- 1 ... 
I.et an axis of symmetry be drawn through r; and let the generator (^-{- 1 ,/)) be drawn 
from (^+1) to the mid-point of (^, ^— 1 ). ^E— (K-f-l)) and (K+l) are now quadri- 
laterals, and {r — (^+l)j and {k-\-\) are tessaraces, if the gamic operations be completed; 
and these tessaraces are upon the diagonal perpendicular to the axis of symmetry, being 
at equal distances from r. We have the triangles and triaces 
0'=(^+l, yl,/), K=(o', r-k, r-k-1), 
d =(K-f 1, K, FJ, k =(0', E-K, E-K-1), 
being the triangle polar to/, introduced between K and K— 1. Leaving now the 
signatures of the faces undisturbed, let r—x be exchanged for upon the diagonals 
MDCCCLVII. 2 F 
