  Let $\{f_t\}$ be a family of complex polynomial functions with line
singularities. We show that if $\{f_t\}$ has a uniform stable radius (for the
corresponding Milnor fibrations), then the L\^e numbers of the functions $f_t$
are independent of $t$ for all small $t$. In the case of isolated singularities
--- a case for which the only non-zero L\^e number coincides with the Milnor
number --- a similar assertion was proved by M. Oka and D. O'Shea.
  By combining our result with a theorem of J. Fern\'andez de Bobadilla ---
which says that families of line singularities in $\mathbb{C}^n$, $n\geq 5$,
with constant L\^e numbers are topologically trivial --- it follows that a
family of line singularities in $\mathbb{C}^n$, $n\geq 5$, is topologically
trivial if it has a uniform stable radius.
  As an important example, we show that families of weighted homogeneous line
singularities have a uniform stable radius if the nearby fibres
$f_t^{-1}(\eta)$, $\eta\not=0$, are "uniformly" non-singular with respect to
the deformation parameter $t$.
