Abstract: In recent years, using the great circle routes theorem found first by Longuet-Higgins and the energy dispersion of the spherical twodimensional Rossby waves, Hoskins explained both the teleconnections gained with the data analysis and the tele-response calculated in the numerical models, and made important. advance. But the great circle routes theorem is found only when the basic flow angular velocity is constant, however the analytic solutions are difficutly found for the real basic flow on the sphere. On the basis of Fermat's principle, using the variational method, the general ray equation on the sphere is directly found.When the angular velocity of the basic flow is constant, the great circle routes theorem, visibly in the physics, is gained. For a few usual basic flow patterns, the analytical solutions can be found also. In addition, using "the average variational principle", the conservation equation of the wave action is directly found.