  Using the Thue-Siegel method, we obtain effective improvements on Liouville's
irrationality measure for certain one-parameter families of algebraic numbers,
defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some
corresponding Diophantine equations. We obtain bounds for the size of
solutions, which depend polynomially on $a$, and bounds for the number of these
solutions, which are independent of $a$ and in some cases even independent of
the degree of the equation.
