Solution of the same problem by introducing the distances of the planet from two fixed points

Abstract

A remarkable relation exists between two radius vectors of the planetary orbit and the arc connecting their end-points. One is led to it if one proceeds from the ordinary differential equations of elliptic motion, but through complicated calculations. We shall derive these relations without difficulty from the partial differential equation and for this we have only to assume that W can be represented by the heliocentric radius vector r and the distance ρ of the planet from one other point M,. The correctness of this hypothesis is not a priori evident¹ without further consideration. However it will be confirmed by calculations.