The n-dimensional Helmholtz equation may be separated in ellipsoidal co- ordinates. In this way we find for \(n=2\) the Mathieu function, for \(n=3\) the Lamé function, for \(n=4\) the so called multi-dimensional Lamé function, being considered here. The related eigenvalue problem is to choose the n-1 constants of separation in such a way that the multi- dimensional Lamé wave function - as a solution of a second order ordinary differential equation with coefficients singular in \(a_ 1,a_ 2,...,a_ n\)- has the form \(\prod_{i}(z-a_ i)^{k_ i}u(z)\). It stands to reason that the properties are equivalent to those of the known Lamé functions \((n=3)\) (discreteness and characterization of the eigenvalues, orthogonality of the eigenfunctions, conditions u(z) being a polynomial).