  We report the first X-ray detection of the radio pulsar PSR B1757-24 using
the Chandra X-ray Observatory. We detect point-source emission at the pulsar
position plus a faint tail extending nearly 20" east of the pulsar, in the same
direction and with comparable morphology to the radio tail. Assuming the
point-source X-ray emission is magnetospheric, the observed X-ray tail
represents only ~0.01% of the pulsar's spin-down luminosity. This is
significantly lower than the analogous efficiencies of most known X-ray nebulae
surrounding rotation-powered pulsars. Assuming a non-thermal spectrum for the
tail photons, we show that the tail is unlikely to be emission left behind
following the passage of the pulsar, but rather is probably from
synchrotron-emitting pulsar wind particles having flow velocity ~7000 km/s. We
also show that there must be a significant break in the tail synchrotron
spectrum between the radio and X-ray bands that is intrinsic to the particle
spectrum. No emission is detected from the shell supernova remnant G5.4-1.2.
The upper limits on remnant emission are unconstraining.
