  Let $X$ be a smooth algebraic curve. Suppose that there exists a triple
covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we
investigate the existence of morphisms from $X$ to the projective line
$\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose,
we generalize the classical results of Maroni concerning base-point-free
pencils on trigonal curves to the case of triple covers of arbitrary smooth
irrational curves.
