EEV. T. P. KIEKMAN ON AIJTOPOLAE POLYEDEA. 
189 
odd. When r is odd, the line which bisects the nodal diagonal at right angles will be 
known to us as the nodal axis, or the axis of symmetry. It passes through one summit 
of the base, and bisects the side opposite to that summit. 
When r is even, there is no method of autopolar signature of the pyramid, which shall 
not exhibit a nodal diagonal. This pyramid is only nodally autopolar. But when r is 
odd, the signatures may be enodal, if every summit e is made the pole of the face E 
opposite to it. This odd-angled pyramid is either nodally or enodally autopolar. The 
autopolarity of the even-angled pyramid is purely nodal. 
If, now, any two non-contiguous summits numbered e andyof this base be joined by 
the diagonal ef and the two portions O and O' of be considered as faces about ef, the 
summits e and f are tessaraces in the edge 00'. The vertex a must, therefore, if auto- 
polarity is to be kept, become two summits o and o', and the triangles numbered EF will 
be quadrilaterals intersecting in the edge oo'. E and F, be it remembered, are the two 
numbers e andy. 
The two edges, thus added to 11, are evidently a gamic pair, of which OO' is evanes- 
cible, and oo' convanescible, and the result P is an autopolar (r-i-2)-edron. 
VII. The question to be answered is, how many such results P can be made by 
drawing a gmexator 00' between two summits of FI, such that no two P shall be either 
identical, or one the reflected image of the other % 
If ^and df be two generators 00' of the same P, it is evident thatjf— o= + (y'— o'), 
since O is dhided ahke by both ef and ef. And the figure made by the points of the 
base will be exactly like that presented by the points or its 
reflected image. 
For the only difierence possible between the two results P, P', generated by ef and 
df, will consist in the mutual arrangement of the introduced 4-gons and 4-aces ; and 
any difierence herein will prevent P from being the same with P'. 
The four following equations, along with 
f-e=±{f-d), 
comprise all the possible conditions of similarity between P and P' : — 
E; — o=E^ — o'+ 
E,-e=d-K± 
e,-o=e;-/± 
E; — e=d — F^+.' 
The first afiirms that the distance from the left-hand summit of E to o is the same as 
from the left of E' to d, and measured in the same direction. 
The second afiirms the said distance is that from the right summit of E' to d, measured 
in the opposite direction. In the first case looking at the configuration in Cl, the system 
(df) is a direct repetition of the system (ef). In the other it is its reflected image. 
The configuration of the third condition is that of the first, and of the fourth that of 
