EEV. T. P. KIEKMAN ON AUTOPOLAE POLTEDEA. 
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ef, can be drawn such that OE and not O'E shall be an edge of S, and another ef^ such 
that OE and not OE shall be an edge. And when OE is no edge, o(e-jrl) is an edge 
of a quadrilateral. And (e+1) is a triace in O'. 
Thus whenever e is nodal in (e, e+l,/]) and O'E is no edge, S may be a repetition of 
some other of the (S) above constructed, having the triangle (o, e+1, e) which contains 
a second leading system. 
Let now e+1 be nodal in the triangle [e, e+1,/]), and let ^+1=G — 2. Here e-\-l 
is a tessarace, because G — 2 is a quadrilateral (e+l,/), e+2, o'); for o'(o+l) is an edge 
because 0'(G — 2) is. In this case therefore O' has two tessaraces, and there is no second 
leading system in S. 
Let then o+l be nodal and o+l = G — 1, or o=G— 2. Here o is a pentace; for E 
has the pointy^. In this case oe is ori, and there is no second leading system. 
In like manner it can be proved that if (e—1) is nodal in the triangle (o, o— 1,/j), 
either e — 1 is a tessarace, as well as e, in O', or that e is the pentace tj. 
Consequently S is never repeated when O' has a nodal summit, except when e is nodal, 
and O'E is no edge of S. 
Thus we are tempted to conclude that S is always twice counted among those of the 
^{r.(r— 2)— 2+2,._i} which have a triangular O', except when (e+1) in O' is nodal, or 
when e is nodal and O'E is an edge. 
XXVI. There are two triangles O' in which ef^ is dra-wn from a nodal summit e to 
make an edge O'E, i. e. e may be either of the nodal summits of the pyramid, but as ef^ 
drawn thus from one nodal summit is the feUow-generator in Cl of the other, only one 
of the resulting S was enumerated in XIX. And this is the s rejected in XXI. Also 
there are two lines efi which can be drawn about either nodal summit to cut off a triangle 
(e, e+l,fi) having (e+1) nodal; but the two lines so drawn about one nodal summit of 
n are fellow-generators of those so drawn about the other. We have therefore among 
the counted in XIX., three of those which have a triangular O', which 
have not two leading systems : and we have not more than three. 
XXVII. We have now to determine how many of these (S) which have a triangle O' 
and two leading systems are repetitions of each other. 
Let S' be one of these. Its base O is r-gonal, having all the summits of Cl except 
(eil), and instead of this the summit/]. By the vanishing of on and oo] the pyramid is 
restored, having the r-gonal base 
...(,+2)(,+I)<,_I)(,-2)....(^+I)y(^-I)(^-2) . . . (O) 
If O' is (e, e— I,/]), the r-gon O is 
...{e-\-2){e-Yl)e.ff,e-2)...{g+\)g{g-l){g-2) (O) 
If O' is {e, the r-gon O is 
(O), 
When O is (O), S' has the triace (e—I), which is (G, G+I, O'), and has also the triangle 
2 E 2 
