In the preceding paper of the present subject we have generally formulated the representations of the infinitesimal rotation in the multi-dimensional space, whose fundamental form is real-definite. The second part of the present subject is composed of the representation by polynomials and an application of the general theory to the problem in regard to the angular momentum in the four- and five-dimensional degenerate vibrations. In addition the formulation of the Schrödinger solution of the multi-dimensional harmonic oscillator and the structure of the group of the Schrödinger equation of the system are given in appendices.