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EEV. T. P. KIEKMAJ7 OX AUTOPOLAE POLYEDEA. 
It is evident that P by the vanishing of AP, becomes P' a y?-edral — l)-acron ; or by 
the vanishing of ah becomes P' a (y? — l)-edral ^-cron. Q. E. D. 
If P' be not a pyramid, it can further be reduced by the evanescence or convanescence 
of an edge, until the result of the reduction is a pyramid 11. If there be no jjyramid of 
higher rank than 11, to which P can be thus reduced, P will be said to be geraerahle from 
n, and can evidently be generahle from none but 11. 
When P is autopolar, it has both a convanescible and its evanescible gamic ; and thus, 
from being a ^-edral y?-acron, it reduces to P', a (y? — l)-edral (y>— l)-acron, which will 
also be autopolar ; and P is generahle only from one pyramid 11. 
VI. The problem of the enumeration of N-edra is thus reduced to this ; To find how 
many '^-edra are generahle from the'K.-edral jgyramid H. The solution of this question is 
to be founded on the consideration that if P is generahle from 11, it can be constructed by 
the introduction into IT of new convanescible and evanescible edges; whether P be 
autopolar or heteropolar. I shall first consider the auto^olars, and aftei-wards the hetero- 
polars generahle from the pyramid. 
Problem. To find the number of nodally autopolar (r+2)-e(Zra generahle from the 
{j-\-\yedral pyramid. (See below the definition of nodally autopolar.) 
Let the r triangles and r triaces of the pyramid 11 be numbered thus, 
12 3 ...(P-3) (P-2) (R-1) E , 
r (r-1) (r-2)... 4 3 2 1 
the upper line denoting the faces as they are read by an eye within the pyramid, and the 
lower the summits. 
If and E^ signify the two summits on the right and left of the ^th triangle, and 
e^ ei be the faces on either side of E, 
E^=2 — e+ 
E,.=l — e+, 
where e and E are the same number, as are r and R. 
The + in this and all functions of these signatures, denotes that such a multiple of r 
is to be added or subtracted as shall cause any signature to have an integer value 
between 1 and r inclusive. 
This arrangement is autopolar ; for the vertex a is the pole of the base fl, and the 
%th triangle is placed with respect to the adjoining faces and summits and their signatures, 
as the Tzth summit with respect to the adjoining summits and faces. 
If two gamic edges meet in the ^th face at the ^th summit, we must have either 
k—2—k^ or ^=1—^+ ; 
e. k=l, or A;=^(r+2), or ^=^(r+l). 
This shows that there are only two nodal summits, which have the signatures 1 and 
\{r-\-2) when r is even, and 1 and ^(r+1) when r is odd. 
It will be convenient to call the line joining the nodal summits, nodal diagonal or 
nodal line. When r is even, the nodal diagonal is a diameter; but not when r is 
