Attraction of a point by two fixed centres

Abstract

We now consider the motion of a point attracted by two fixed centres. Let us restrict ourselves first to the case in which the motion takes place in a plane, which is always the case when the direction of the initial velocity lies in the same plane as the line joining the fixed centres. Let this connecting line be the axis x₂, and the axis x₁ is at right angles to it at the mid point between the centres at a distance 2f from each other. If we now express x₁ and x₂ in terms of λ₁ and λ₂ and choose the constants a₁ and a₂ of the substitution in such a way that the two centres fall at the foci of the confocal system, then the differential equation to be integrated is

$$\begin{gathered} \frac{{\left( {{a_1} + {\lambda _1}} \right)\left( {{a_2} + {\lambda _1}} \right)}} {{{\lambda _1} - {\lambda _2}}}{\left( {\frac{{\partial W}} {{\partial {\lambda _1}}}} \right)^2} + \frac{{\left( {{a_1} + {\lambda _2}} \right)\left( {{a_2} + {\lambda _2}} \right)}} {{{\lambda _2} - {\lambda _1}}}{\left( {\frac{{\partial W}} {{\partial {\lambda _2}}}} \right)^2} \hfill \\ = \frac{1} {2}U + \frac{1} {2}h \hfill \\ \end{gathered}$$

((29.1))

when U likewise must be expressed in terms of λ₁ and λ₂.