  We investigate the effect of surface tension on the linear Rayleigh--Taylor
(RT) instability in stratified incompressible viscous fluids with or without
(interface) surface tension. The existence of linear RT instability solutions
with largest growth rate $\Lambda$ is proved under the instability condition
(i.e., the surface tension coefficient $\vartheta$ is less than a threshold
$\vartheta_{\rm c}$) by modified variational method of PDEs. Moreover we find a
new upper bound for $\Lambda$. In particular, we observe from the upper bound
that $\Lambda$ decreasingly converges to zero, as $\vartheta$ goes from zero to
the threshold $\vartheta_{\rm c}$. The convergence behavior of $\Lambda$
mathematically verifies the classical RT instability experiment that the
instability growth is limited by surface tension during the linear stage.
