EEV. T. P. KIEKMAN ON AIJTOPOLAE POLTEDEA. 
205 
or 
4-_2G+2G~2=r+3+, 
whence 
-1 =r+, 
which is still more absurd. 
Therefore S' is always made when t is odd from two different generators, and has 
been twice constructed among the (r+3)-edra S. 
Let now r be even. If feUow-generators can be drawn in (Q)' from e and ^+1, we have 
or 
whence 
so that 
Gi“G+l+^, 
2-^G=G+l+ir, 
ir=2G~l, 
r=:4m--2. 
If feUow-generators can be drawn fi-om e and ^—2 in (Q)", we have (VIII.), 
or 
so that 
Gi=G--2+-|r,. 
2-G=G-2+^, 
r=4m 
and 
e being the opposite end of the diameter through \r. 
It is thus proved that a single S' having two leading systems is made by fellow- 
generators only, when r is even, and has been constructed and counted only once among 
the (r-f-3)-edra S. Every other S, having two leading systems, has been twice con- 
structed from the even-angled Q. 
XXIX. The preceding reasoning from XXII. proceeds on the hypothesis that r>5. 
This restriction was made in order the more readily to prove that one summit of the 
triangle a|3y must be o. The same necessity is proved for r=5 thus. 
In this triangle a|3y there is an evanescible edge, which is not ojj, but equal in the 
united rank of its summits with on- The only summits not triaces are the 4-aces e and % 
and either two 4-aces o and o', or the pentace o. 
The evanescible edge must either be oe or en- Now when en is an edge ?s=e+l ; that 
is H, the face in which is =o+l. We see from the signatures 
that either 
or else 
5^4,3 34:2^1^ 
H=3 and o=2, or H=2 and 0=1, 
11=2 and o=3, or H=1 and o=2. 
in all which cases O is a pentagon and O' a triangle. Now no pentagon but O exists in 
the figure ; therefore no triangle aj3y of which o is not a summit has an edge ay, the 
