The infinitesimal Kobayashi metric of an ellipsoid of the form
$$
E_m=\{(z_1,z_2)\in \C^2:|z_1|^2+|z_2|^{2m}<1\}
$$
is calculated explicitly, modulo a parameter that is determined by
solving a transcendental equation. Using this result, we show
that the metric is $C^1$ on the tangent bundle away from the zero
section.
We also describe software that will calculate, using a Monte Carlo
method, the infinitesimal Kobayashi metric on a domain of the form
$$
\Omega_\rho=\{(z_1,z_2)\in\C^2:\rho(z_1,z_2)<0\},
$$
where $\rho$ is a real-valued polynomial. We compare results
of computer calculations with those obtained from the
explicit formula for the Kobayashi metric.