The Lagrangian equations of motion of a general mechanical system ofndegrees of freedom that is executing small motions about a position of equilibrium are subjected to a Laplacian transformation. The transformed equations are then subjected to a similarity transformation. By means of this transformation, a separation of the transformed equations is effected provided the system has the special type of symmetry under consideration. The solution of the usual determinantal equation is avoided. The methods of the Laplacian transformation and matrix algebra as presented in (1) and (2) are assumed in the discussion.