  Given a triple covering $X$ of genus $g$ of a general (in the sense of
Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free
pencils of degree $d$ which are not composed with the triple covering for any
$d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and
computations. We also discuss about the sharpness of our main result and the
so-called Castelnuovo-Severi bound byexhibiting some examples.
