  In this paper, we show how to apply a theorem by L\^e D.T. and the author
about linear families of curves on normal surface singularities to get new
results in this area. The main concept used is a specific definition of {\em
general elements} of an ideal in the local ring of the surface. We make
explicit the connection between this notion and the elementary notion of
general element of a linear pencil, through the use of {\em reduction}. This
allows us to prove the invariance of the generic Milnor number (resp. of the
multiplicity of the discriminant), between two pencils generating two ideals
with the same integral closure (resp. the projections associated). We also show
that our theorem, applied in two special cases, on the one hand completes a
theorem by Snoussi on the limits of tangent hyperplanes, and on the other hand
gives an algebraic $\mu$-constant theorem in linear families of planes curves.
