  We study the Euler characteristic of the Milnor fibre of a hypersurface
singularity. This invariant is given in terms of the Euler characteristic of a
fibre in between the original singularity and its Milnor fibre and in terms of
the Euler characteristics associated to strata of the in-between fibre.
  From this we can deduce a result of Massey and Siersma regarding
singularities with a one-dimensional critical locus. The result is also applied
to the study of equisingularity. The famous Brian\c{c}on-Speder-Teissier result
states that a family of isolated hypersurface singularities is equisingular if
and only if its $\mu ^*$-sequence is constant. We show that if a similar
sequence for a family of corank 1 complex analytic mappings from n-space to
(n+1)-space is constant, then the image of the family of mappings is
equisingular. For families of corank 1 maps from 3-space to 4-space we show
that the converse is true also.
