  A method of measuring the relative phase of the energy gap in a
high-temperature superconductor is suggested for electron energy loss
spectroscopy. Energy-resolved measurements of off-specular scattering should
show a feature similar to the specular feature associated with the gap. Unlike
the specular feature, which reflects an average of the gap over the (normal)
Fermi surface, the energy loss of the off-specular feature depends on the
superconducting energy gap at only two locations on the Fermi surface. The
onset of the feature reflects the relative phase between these two points. This
result is independent of surface characteristics. Such characteristics affect
the {\it magnitude} of the off-specular feature, not its location or onset. The
size of the feature is estimated for a simple surface model. Implications of
specific measurements on $\rm Bi_2Sr_2CaCu_2O_8$ are discussed.
