  Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly
homotopic generic immersions, then it has been shown in [N] that all generic
regular homotopies between i and i' have the same number mod 2 of quadruple
points. We denote this number by Q(i,i') \in Z/2. We show that for any generic
immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h
are regularly homotopic,
  Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map
induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to
whether h is orientation preserving or reversing, respectively.
