  The purpose of this paper is to extend the analysis of gravitational
instability of the Linet - Tian solution to the case $\Lambda > 0$. A
fundamental difference brought about by $\Lambda >0$, as compared to
$\Lambda<0$ is in the structure of the resulting space time. Associated with
each of the two commuting Killing vectors $\partial_{\phi}$, and
$\partial_{z}$, there is a curvature singularity that has the same
characteristics as that associated to $\partial_{\phi}$ in the Levi - Civita
metric, and we show that there is an isometry relating these singularities that
reduces the effective parameter space of the metrics. In attempting to set up
and solve the linearized perturbation equations we are confronted with the
problem of a gauge ambiguity that leads to the introduction of a gauge
invariant function, $W_1$, that is shown to be also a {\em master function},
that satisfies a second order ODE, and in terms of which one can express all
the perturbation functions. Unfortunately, the equation satisfied by $W_1$
contains singular coefficients, and, although {\em all} its solutions are
regular, because of the presence of these singularities one cannot, as in the
case of negative $\Lambda$, set up an associated self adjoint problem that
provides a complete set of solutions for $W_1$. We are thus restricted to
solving numerically the perturbation equations, and using those solutions for
constructing $W_1$, for particular values of the parameters. In all the cases
analyzed we find unstable modes, which strongly suggests that all the Linet -
Tian space times with $\Lambda > 0$ are linearly unstable under gravitational
perturbations. The problem of determining the time evolution of arbitrary
initial data in terms of the $W_1$, or something equivalent, remains open.
