  Generalized Chow forms were introduced by Cayley for the case of 3-space,
their zero set on the Grassmannian G(1,3) is either the set Z of lines touching
a given space curve (the case of a `honest' Cayley form), or the set of lines
tangent to a surface. Cayley gave some equations for F to be a generalized
Cayley form, which should hold modulo the ideal generated by F and by the
quadratic equation Q for G(1,3). Our main result is that F is a Cayley form if
and only if Z = G(1,3) \cap {F=0} is equal to its dual variety. We also show
that the variety of generalized Cayley forms is defined by quadratic equations,
since there is a unique representative F_0 + Q F_1 of F, with F_0, F_1
harmonic, such that the harmonic projection of the Cayley equation is
identically zero. We also give new equations for honest Cayley forms, but show
with some calculations that the variety of honest Cayley forms does not seem to
be defined by quadratic and cubic equations.
