  This paper gives a detailed derivation of the surface of a tri-axial
ellipsoid. The general result is in terms of the elliptic integrals of the
first and second kind. It is in checked for all special cases included and the
corresponding simplified formulae are given. Next, expressions for the mean and
Gaussian curvature are derived. These curvatures depend only on: a) the three
axes, and from the point under consideration, b) its distance to the centre and
c) the length of the perpendicular from centre to the tangent plane of that
point. Finally, it is noticed that at the four umbilic points of an ellipsoid,
where the cicular sections degenerate to a point, the two principal curvatures
are equal and have the simple expression (a c / b^3)where a>b>c are the
half-axes.
