  In math.AG/0108089 we gave sufficient conditions for the irreducibility of
the family V of irreducible curves in the linear system |D| with precisely r
singular points of topological respectively analytical types S1,...,Sr on
several classes of smooth projective surfaces S. The conditions where of the
form (tau'(S1)+2)^2 +...+ (tau'(Sr)+2)^2 < c*(D-K)^2, where tau' is some
invariant of singularity types, K is the canonical divisor of S and c is some
constant. In the present paper we improve this condition, that is the constant
c, by a factor 9.
