  We present some properties of a Cauchy family of distributions on the sphere,
which is a spherical extension of the wrapped Cauchy family on the circle. The
spherical Cauchy family is closed under the M\"obius transformation on the
sphere and there is a similar induced transformation on the parameter space.
Stereographic projection transforms the the spherical Cauchy family into a
multivariate $t$-family with a certain degree of freedom on Euclidean space.
Many tractable properties of the spherical Cauchy are derived using the
M\"obius transformation and stereographic projection. A method of moments
estimator and an asymptotically efficient estimator are expressed in closed
form. The maximum likelihood estimation is also straightforward.
