For the bitangent Cj must lie in the complex determined
by a x a 2 or in that determined by a x b 2 . There is no loss of
generality in supposing it to be in the former ; for the given
complex determined by 04 & x is not altered by interchange of
a 2 and 6 2 . Suppose then that a x a 2 , c x c 2 are pairs of a complex,
and consider the three complexes with pairs

<X-yOj 2 , O-y 2 J C 1 C 2 , ... 5

ttjCj, a 2 c 2 , ... , ... ;

These form a complex-triple and therefore contain all the
bitangents, and in particular a 3 and b 3 . Neither of them is in
the first complex of the triple. They cannot both be in the
second (or both in the third) complex. For they are not a pair
of this complex, and three bitangents of the complex no*two
of which are a pair are asyzygetic, whereas a v a 3 , b 3 are
syzygetic. Since a 3 and b 3 are interchangeable in the given
complex, we may without" loss of generality suppose a 3 to be
in the second (and b 3 in the third) complex of the triple. Let
then c^Cj. a 2 c 2 , a 3 c 3 be three pairs of the second complex.
