A new solution of Hill's lunar problem

Abstract

The two second order differential equations of Hill's lunar problem for rectangular co-ordinates in the rotating system are transformed into a single equation of the fourth order for the geocentric radius vector r, which can be reduced, using Jacobi's integral, to a differential equation of the third order. A discussion of this equation leads to a new exhibition of Hill's “variation orbits”. The first terms of Hill's power series, which represent these periodic solutions of the problem, are exactly confirmed by using a very simple approximative assumption about the mathematical character of the solution.