  We study the problem of gravitational lensing by binary galaxies, idealized
as two isothermal spheres. In a wide binary, each galaxy possesses individual
tangential, nearly astroidal, caustics and roundish radial caustics. As the
separation of the binary is made smaller, the caustics undergo a sequence of
metamorphoses. The first metamorphosis occurs when the tangential caustics
merge to form a single six-cusped caustic, lying interior to the radial
caustics. At still smaller separations, the six-cusped caustic undergoes the
second metamorphosis and splits into a four-cusped caustic and two three-cusped
caustics, which shrink to zero size (an elliptic umbilic catastrophe) before
they enlarge again and move away from the origin perpendicular to the binary
axis. Finally, a third metamorphosis occurs as the three-cusp caustics join the
radial caustics, leaving an inner distorted astroid caustic enclosed by two
outer caustics. The maximum number of images possible is 7. Classifying the
multiple imaging according to critical isochrones, there are only 8
possibilities: 2 three-image cases, 3 five-image cases, and 3 seven-image
cases. When the isothermal spheres are singular, the core images vanish into
the central singularity. The number of images may then be 1, 2, 3, 4 or 5,
depending on the source location, and the separation and masses of the pair of
lensing galaxies. The locations of metamorphoses, and the onset of threefold
and fivefold multiple imaging, can be worked out analytically in this case.
