The two-parameter Cauchy maximum-likelihood estimator T ( y ) = ( T 1 ( y ), T 2 ( y )) is known to be unique for samples of size n ≽ 3 (J. Copas, Biometrika 62, 701-704 (1975)). In this paper we exploit equivariance under the real fractional linear group to show that the joint density of T has the form p n ( X )/(4 πt 2 2 ) where X = | t ─ θ | 2 / (4 t 2 θ 2 ). Explicit expressions are given for p 3 ( X ) and p 4 ( X ) and the asymptotic large-sample limit. All such densities are shown to have the remarkable property that E ( u(>T )) = u ( θ ) if u(⋅) is harmonic and the expectation is finite. In particular, both components of the maximum -likelihood estimator are unbiased for ≽ 3, E log{ T 1 2 + T 2 2 ) = log ( θ 1 2 + θ 2 2 ), E ( T 1 2 - T 2 2 ) = θ 1 2 - θ 2 2 , E { T 1 T 2 ) = θ 1 θ 2 for n ≽ 4, and so on.