210 
EEV. T. P. ETE EMAX ON ATJTOPOLAE POLTEDEA. 
perpendicular to the axis of symmetry, except that r — Tc shall be exchanged fory); and 
let o' be exchanged for k. The result will be a nodaUy autopolar (r+3)-edron S, having 
the triangles and triaces 
0—{T—k—\,d,T—k\ K=(^,/, ^+1), 
k =(KH-1, K, FJ, o' =(0', E-K, E-K-1), 
in which o' and k are evidently nodal summits. 
When r= 4/^-1- 3, the enodal pyramid reads thus: 
...E (E-1) ...(E-K) (E-K-1) ... 
2k-\-l 2k... : k-\-l k... 
The axis of symmetry being di’awn through r, let the generator {k, be drawn from k 
to the mid-point of (^+1, ^-f-2). We have now the 4-laterals K and E— K, and the 
triaces k and r—k. The figure has the triangles and triaces 
a={k, /J:+l,/), K-fl=:(o', r-k, r-k-1), 
o' =(K, K-f 1, F,), k+1 =(0', E-K, E-K-1). 
If we now exchange upon the diagonals perpendicular to the axis of sjunmetry r—e for 
0 , except only that r—k—1 is to be exchanged for/j, and at the same time exchange 
o' and ^4-1, we have again a nodally autopolar (S), having the triangles and triaces 
0'={r—k, o', r—k—1), K+l=(^-f 1, k,f), 
o' =(E-K, O', E-K-1), k+1 =(K+1, K, Fj ; 
in which o' and A;-|-l are the nodal summits. 
Since all summits of the enodal Q are alike, as origins of generators, no generators 
can be di’awn to make an r-gonal O and a triangular O' different from these ; therefore 
no purely enodal (r-f-3)-edron can be constructed to have such an O and O'. 
If 0'=0, the generator {e,f-\-\) is the axis of symmetry through o, and aU the sum- 
mits are triaces, except the pentace e. In this case the resulting (;'+3)-edi’on is 
instantly made nodally autopolar, by the exchange of signatures on each side of the axis, 
which becomes the nodal axis. The gentle reader ■svill kindly assist my demonstrations 
in this new and somewhat intricate subject, by draiving a 7-gon and a 9-gon, and joming 
the angles in the requfred way to two included summits o and o'. 
XXXV. But when O' is not a triangle, nor of equal rank -svith O, it becomes impos- 
sible to give to the S generated by the hne {e,f-\-\) a nodal arrangement; for there is 
no summit with which o' can be exchanged. If/) is a point of the face H, and n ai-e 
tessaraces which may take each other’s place, but o', whether tessarace or m-nce, stands 
alone and immoveable. Consequently, the resulting (r+3)-edron is oi purely enodal 
autopolarity, and, with all the other enodal (S) having no triangular O', is to be added 
to our enumeration. The number of different generators {y',f-\-^) is ■!(/— 5),yrecefring 
every value from y*!:: 2 to f=^\{r—2>)'. this -1(^—5), the number of pui’ely enodal auto- 
