A joint paper by Tranter and the present author concerning the heat flow in an infinite medium heated by a sphere has necessitated a proof that no complex zeros of the equation

The elementary theorem that, if the triangles ΩAB, ΩPQ are directly similar, so also are the triangles ΩAP, ΩBQ, can be elevated into a general principle:

Given a point Ω and any number of points P1, P2,.. lying on a curve Γ, let Q1, Q2, … be points such that the triangles ΩP1Q1, ΩP2Q2, .. are all directly similar. Then Q1, Q2, … lie on a curve similar to Γ, obtainable by rotating Γ through the angle PrΩQr and applying the scale-factor ΩQr/ΩPr.

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.”