190 
EEV. T. P. KIEKMA^ ON AIJTOPOLAE POLTEDEA. 
the second, with the difference that the signatures e and f have changed places in e'f 
in the third, and E' and F have changed places about E'F in the fourth. 
VIII. First, let, along with e— /= + (e'— /'), 
E^— e=E;— e'+, i. e. 
2-2e=2-2e'+. 
This, since e=ef and /=/' is contrary to hypothesis (for we are seeking a diagonal e'f 
different from ef), means 
2-2e=2-2e'+7', 
or 
e'=e+-|r ; 
that is, r is even, and e' is the opposite extremity to e of the diameter of fl through e. 
Hence f is opposite /’in the diameter through /*. 
It thus appears that when t is even, for every diagonal ef (not a diameter), there can 
be drawn a diagonal Sf in O, that generates the same (r+2)-edron P which is generated 
by ef‘, and that e' and/*' are diametrically opposite to e and/*. When ^ is a diameter, 
it does not hence appear whether there is any ef a fellow-generator with ef. 
IX. Secondly, let 
E; — e—e' — E^+, i. e. 
2-2e=e^-{l-e')± 
2e+2e'=3+r+. 
We gather that r is odd ; for no multiple of r even can here be added to make this 
equation true. Hence 
e'=i(y+3-2e+) 
and 
/=i(r+3-2/+). 
The diagonal thus found is different from ef unless e'=f and/'=e, i. e. unless 
2/+2e=3+r+, 
the additional + denoting either zero or 2r. When this relation between f and e holds, 
it does not thus far appear whether a pair of generators, ^f and its gamic, can be drawm 
to produce the same P with ef and its gamic. This relation is othei'wise thus ; 
or else 
showing that f is at the same distance measured backwards from the nodal summit ^(?'+l) 
which is also i(3r+l)=r+(|r+l), that e is measm-ed forwards from the nodal 
summit I. The hne ef is either the nodal diagonal through 1 and ^(r-f-l), or it is 
parallel thereto. We may call all these diagonals the nodal par alleh. It is evident that 
when ef and ef are fellow-generators, ee' and ff are nodal parallels ; since 
2e+2e'=2/+2/=r+3+. 
