To avoid the inherent singularity of the spherical coordinates at their poles, a new quasi-spherical coordinate system is developed. In this coordinate system, a finite element procedure is proposed to determining the eigensolutions at three-dimensional vertices in which the field variables are proportional to the (lambda+1)-th power of the distance from the vertices. The resulting global equation is a second order characteristic matrix equation. Several demonstrating problems are investigated. It can be seen that the formulation yield satisfactory results.