  We derived an analytic formula for T violation by using the perturbation
theory for small quantities, $\Delta m_{21}^2L/2E$ and $\delta a(x)L/2E$, where
$\delta a(x)$ represents symmetric and asymmetric matter fluctuations, i.e.,
deviations from the average density. We analyzed the effect of matter
fluctuations to T violation, by assuming PREM profile of earth matter density.
We found that matter fluctuations do not give any viable contribution for
$L<$6000km, while the fluctuation effect becomes large due to resonances for
$L>$7000km. For 7000km$<L<$8000km, matter fluctuations contribute destructively
to the average density term and the net result is small, while for $L>$8000km,
the contribution from matter fluctuations becomes large but contributes
constructively.
