Euler’s Equation and (p, r) Coordinates

Extract

1. In a seismological context, it was recently noticed that the use of (p, r) coordinates in a variation problem where polar coordinates had previously been used presented some unusual features. The method followed is one of some generality, and the purpose of the present note is to illustrate it in a suitable elementary problem: “A heavy uniform inelastic string has its ends fastened to two assigned points on the surface of a smooth right circular cone fixed with its axis vertical and vertex O uppermost, and rests in equilibrium on the surface of the cone. Prove that if this surface is developed into a plane, the curve on which the string lay is given by the form p (a + br) = 1.”