The nearly commensurable case of the planar restricted three-body problem is treated by application of the two variable expansion procedure. The polar angle of the infinitesimal body, rather than the time, is taken as the independent variable. A set of four coupled first order differential equations, which govern the long-period behavior of the orbital elements, is obtained by imposing the requirement that the assumed form of the expansions must be self-consistent. The independent variable in these equations is the "slow variable". It is then found that the short-period perturbations of the motion of the infinitesimal body do not contain small divisors or secular terms.Approximate solutions for the orbital elements are given, for two different cases. Both libratory and non-libratory solutions are found, depending upon the initial conditions. Numerical results are calculated from these solutions, and are compared to numerical computations recently reported in the literature.
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