  Given a surface F, we are interested in Z/2 valued invariants of immersions
of F into R^3, which are constant on each connected component of the complement
of the quadruple point discriminant in Imm(F,R^3). Such invariants will be
called ``q-invariants.'' Given a regular homotopy class A in Imm(F,R^3), we
denote by V_n(A) the space of all q-invariants on A of order <= n.
  We show that if F is orientable, then for each regular homotopy class A and
each n, dim(V_n(A) / V_{n-1}(A)) <= 1.
