This report contains new results on the application of periodic surfaces to the problem of determining the (drag free) orbits of satellites around an oblate earth. The basic mathematical theory is in the NAVORD report (PB-148 042) 'Periodic Surfaces and Satellite Orbits' by S.P. Diliberto. The theory is characterized by the fact that it splits the description of the solutions (in the phase space) into two essentially distinct parts; one element is a description of a surface on which solutions travel, the other is a description of the rotation on such a surface. The theory includes an expansion procedure for finding such surfaces, a simple prerequisite condition which coordinate systems must satisfy so as to allow the possibility of these expansions, and the determination of at least one coordinate system for the satellite problem which satisfies the prerequisite condition. The results include a new method for finding expansions of the solutions on the surfaces (in standard terminology this can be described as an 'expansion for the secular terms'), and the use of energy considerations to find the first term in the surface expansion. In particular, it is shown that the energy integral expressed in these coordinates takes on a fairly simple form.