Let V={(x,y,z):f=z

-npz+(n-1)q=0 for n .geq. 3} be a compled analytic subvariety of a polydisc in

where p=p(x,y) and q=q(x,y) are holomorphic near (x,y)=(0,0) and f is an irreducible Weierstrass polynomial in z of multiplicity n. Suppose that V has an isolated singular point at the origin. Recall that the z-discriminant of f is D(f)=c(p

-q

) for some number c. Suppose that D(f) is square-free. then we prove that by Theorem 2.1 .mu.(p

-q

)=.mu.(f)-(n-1)+n(n-2)I(p,q)+1 where .mu.(f), .mu. p

-q

are the corresponding Milnor numbers of f, p

-q

, respectively and I(p,q) is the intersection number of p and q at the origin. By one of applications suppose that W

={(x,y,z):g

=z

-np

z+(n-1)q

=0} is a smooth family of complex analytic varieties near t=0 each of which has an isolated singularity at the origin, satisfying that the z-discriminant of g

, that is, D(g

) is square-free. If .mu.(g

) are constant near t=0, then we prove that the family of plane curves, D(g

) are equisingular and also D(f

) are equisingular near t=0 where f

=z

-np

z+(n-1)q

=0.}$ =0.