  The BL Lac object H1426+428 ($z\equiv 0.129$) is an established source of TeV
$\gamma$-rays and detections of these photons from this object also have
important implications for estimating the Extragalactic Background Light (EBL)
in addition to the understanding of the particle acceleration and $\gamma$-ray
production mechanisms in the AGN jets. We have observed this source for about
244h in 2004, 2006 and 2007 with the TACTIC $\gamma$-ray telescope located at
Mt. Abu, India. Detailed analysis of these data do not indicate the presence of
any statistically significant TeV $\gamma$-ray signal from the source
direction. Accordingly, we have placed an upper limit of
$\leq1.18\times10^{-12}$ $photons$ $cm^{-2}$ $s^{-1}$ on the integrated
$\gamma$-ray flux at 3$\sigma$ significance level.
