  We show that the possible drop in multiplicity in an analytic family $F(z,t)$
of complex analytic hypersurface singularities with constant Milnor number is
controlled by the powers of $t$. We prove equimultiplicity of $\mu$-constant
families of the form $f + tg + t^2h$ if the singular set of the tangent cone of
$\{f = 0}$ is not contained in the tangent cone of $\{h = 0}$.
