. Sections i-6 give the solution of the main problem for a spheroidal earth with the potential limited to the principal term and the second harmonic which contains the small factor k2. The solution is developed in powers of k2 in canonical variables by a method which is basically the same as that used in treating a different problem by von Zeipel (1916). The periodic terms are divided in two classes: the short-period terms contain the mean anomaly in their argu- ments; the arguments of the long-period terms are multiples of the mean argument of the perigee. The periodic terms, both of short and long period, are developed to 0(k2); the secular motions are obtained to 0(k22). The results are obtained in closed form; no series developments in eccentriciLy or inclination arise. The solution does not apply to orbits near the critical inclination, 63?4, but is otherwise valid for any eccentricity I and any inclination. Section 7 gives the long-period terms and the additions to the secular motions caused by the fourth harmonic in the potential; section 8 gives the contt-ibutions by the third and fifth harmonics; section 9 contains formulas for computation.