= Solution
For positive $\lambda,\alpha$ and $0<s<1$, the constant term of the <quadratic equation> is negative. Its roots are therefore real with opposite signs, so one has positive growth. At $s>1$, both the coefficient of $\beta$ and the constant term are positive. If the roots are real they are both negative; if complex they have negative real parts. Thus \b[the growing band remains $\boxed{0<k^2a^2<1}$]. At $s=0$ or $1$ there is a neutral root. If $\alpha=0$, the result reduces continuously to the clean-film capillary branch with a passive zero root.
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