  We investigate three dimensional anharmonic phonons in tetrahedral symmetry
and superconductivity mediated by these phonons. Three dimensional anharmonic
phonon spectra are calculated directly by solving Schr\"odinger equation and
the superconducting transition temperature is determined by using the theory of
strong coupling superconductivity assuming an isotropic gap function. With
increasing the third order anharmonicity $b$ of the tetrahedral ion potential,
we find a crossover in the energy spectrum to a quantum tunneling regime. We
obtain strongly enhanced transition temperatures around the crossover point. We
also investigate the anharmonic effects on the Debye-Waller factor, the phonon
spectral functions and the density profile, as a function of the anharmonicity
$b$ and temperature. The isomorphic first order transition observed in KOs_2O_6
is discussed in terms of the first excited state energy $\Delta$, and the
coupling constant $\lambda$ in the strong coupling theory of superconductivity.
Our results suggest the decrease of $\lambda$ and increase of $\Delta$ below
the first order transition temperature. We point out that the change in the
oscillation amplitude <x^2> and <xyz> characterizes this isomorphic transition.
The chemical trends of the superconducting transition temperature, $\lambda$,
and $\Delta$ in the $\beta$-pyrochlore compounds are also discussed.
