The general (unrestricted) three-body problem is investigated in the case when the force of mutual attraction between the bodies is proportional to the nth power of their distance, where n is an arbitrary real number. A new description is given of the plane problem, based on the introduction of the following Lagrange variables: r--the square root of half the polar moment of inertia, ~--the angle between the two sides of the triangle, and y--the natural logarithm of the quotient of those two sides. The first variable characterizes the size of the triangle, and the other two, its configuration. Routh's equations are derived, in which the variable β’ is 'almost separated' fromy and ~t; the system of equations is reversible. In the special case of the restricted problem, i.e. when the mass of one of the bodies tends to zero, the variables are completely separable, so that the problem describes only the change in the configuration of the triangle.
It is shown that the qualitative results, known for Newtonian interaction (n = -2), are valid throughout the range -3 < n < -1. In particular, for these values of n 'elementary' methods of analysis are used to solve the problems of Hill stability for a pair of bodies, the existence of final motions relating to hyperbolic-elliptic motions is established for n = -2, and a local analysis is carried out of the neighbourhoods of the classical liberation points.
Local analysis show,~d that in the neighbourhood of collinear points two families of Lyapunov periodic motions exist, into which the family of two-dimensional "whiskered" tori degenerates. In the linear approximation, the problems of the stability of triangular points in the restricted and unrestricted formulations are equivalent to one another. Hence the triangular elliptical solutions of the unrestricted problem are stable throughout the domain constructed by Danby for the restricted problem. Allowance for the small non-zero mass of one of the bodies may make the other two bodies leave the unperturbed circular orbit; there is no such effect in the restricted problem.