  A metric more general than the Euclidean Schwarzschild-Tangherlini metric is
considered. The cosmological constant is not necessarily Zero, and the
hypersphere is replaced by an Einstein variety. A differential equation that
derives from the equation of eigen-values of the Laplace operator in this
metric is studied. Two cases are considered, (i) the cosmological constant is
not Zero, and (ii) the cosmological constant is Zero, and the set of
eigen-values is discrete. Infinity is a singular point of this differential
equation. In these cases only, it is regular. Series of powers solve the
differential equation. With the eigenvalues from the discrete set, the
determinant of the Laplacian is calculated. This is equal to the period of the
imaginary time coordinate. Finally, the stability of the Euclidean
Schwarzschild metric is investigated at the classical level.
