  Strings propagating along surfaces with Dirichlet boundaries are studied in
this paper. Such strings were originally proposed as a possible candidate for
the QCD string. Our approach is different from previous ones and is simple and
general enough, with which basic problems can be easily addressed. The Green
function on a surface with Dirichlet boundaries is obtained through the Neumann
Green function on the same surface, by employing a simple approach to Dirichlet
conditions. An easy consequence of the simple calculation of the Green function
is that in the simplest model, namely the bosonic Dirichlet string, the
critical dimension is still 26, and the tachyon is still present in the
spectrum, while the scattering amplitudes differ dramatically from those in the
usual string theory. We discuss the high energy, fixed angle behavior of the
four point scattering amplitudes on the disk and the annulus. We argue for
general power-like behavior of arbitrary high energy, fixed angle scattering
amplitudes. We also discuss the high temperature property of the finite
temperature partition function on an arbitrary surface, and give an explicit
formula of the one on the annulus.
