Differential Equations and Differential Relationships of Geodesic Lines in New Form of Geodetic Coordinate System

Based on the theory of the differential geometry and geodesy,the first and the second order differential equation and 13 order differential relationship based on the new geodetic coordinates are derived for the new form of geodetic coordinate system on the ellipsoidal surface.The given necessary condition expression for geodesic line is the same as that one by the Clairaut theorem in geodetic coordinate system.The first order differential relationship can also be derived from the Liouville formula on the differential geometry.It is useful for the direct and inverse solution of geodetic problem and(applying) to ellipsoidal surface and modeling for three dimensional GIS.