204 
EEV. T. P. KTRKMAj^ ON AUTOPOLAE POLTEDEA. 
(^, o'), and (O) has all the summits of S' except o, d and e — 1. By the vanishing 
of one of the leading systems of this S', S' becomes the pyramid, and (O) becomes (Q) ; 
losing the triace and receiving instead the triace [e — 1). By the vanishing of the other 
leading system, the same S' becomes a pyramid on the r-gonal base (Q)', 
..[e+2)(e+l)e.fl,e-2)...(g+iy/{g-l)(g-^) (Q/ 
losing the triace g, and receiving instead the triace o'. Observe that 5^+1 is a tessarace 
and g a triace, as is evident from their polars G and G-j-l, by inspection of 
G-1, G, G+1... 
1 6 0"~1 I 
where G is (o e{e—l)') and G+1 is (o(o— 2)). 
When O is (O)^ (O)^ becomes (Q) by losing the triace and receiving in its stead the 
triace (e+1). Or it may become a pyramid on the r-gonal base 
..(0+2)/ e(e-l)(e-2)...(g+l)g dig-2) (Q)" 
by the vanishing of its second leading system, receiving the triace d and losing the 
triace (^—1). 
It is evident that the pyramids whose bases are Q, (Q)' and (Q)" are identical, having 
the same r signatures with two shght changes of name, and the same nodal line through 
the summit 1. Now S' having the triangle {e{e — 1]/') is generated by drawing in the 
base (Q)' either the generator o(o — 1) from o to a point between / and (o — 2), or the 
generator (^+1)^ from g-\-\ to a point between d and g — 1. And S' having the triangle 
(o(o+l)/) is obtained by drawing in (Q)" either o(o+l) from o to a point between / 
and (o+2), or the generator (^— 2)(^— 1) from ig—2) to a point between d and g. 
XXVIII. Whenever these two generators drawn in (Q)', or are not feUows, we 
have constructed S' twice and twice counted it; but when they are fellows, we have 
constructed and counted it only once ; for (XIX.) we have never used two feUow-gene- 
rators in forming the (r+3)-edra S. 
To examine this, let first r be odd. If fellow-generators can be drawn from e and 
(^+1) in (Q)', we must have, since e is G, in 
(G-l)G G(+l), 
...(,+1) e (,_i) ; 
2G,+2(G+l)=r+3+ (VI.) (X.), 
or 
whence 
4 — 2G+2G+2=f+3+, 
3 =r, 
or the pyramid is a tetraedron, contrary to hypothesis. And if fellow-generators can be 
drawn from e and 2 in (C2)", we have, since e is G^ in 
..(G-2) (G-l)G.. 
: (^+ 1 ) e . ., 
2G,+2(G-2)=r+3,+ 
