I have had a recent opportunity to recall an early article (1884) which I wrote on the three-cusped hypocycloid. My starting point was the property that the asymptotes of any pencil of equilateral hyperbolas envelop such a hypocycloid. I proved this analytically in the aforesaid article ; perhaps there is some interest in finding geometrical reasons for it.

Principles on pencils of conies are well known. According to these principles :

• (1) The polars of any point a with respect to the various conies of the pencil are concurrent at one and the same point a, which we shall call the corresponding point of a.

• (2) If a describes a straight line D, then a. describes a certain conic C.

• (3) This conic C is also the locus of the poles of D with respect to the conies of the pencil, a consequence being:

• (4) If m, a point of C, is the pole of D with respect to one of the conies H of the pencil and a a point of D with the corresponding point α, then the polar line of a with respect to H is mα.

The Theorem that, if y3 + 3xy + 2x3 = 0, then has been the topic of four recent notes in the Gazette, namely 1542 (not 1546) by E. H. N., 1610 by Mr. F. Bowman, 1684 by Mr. F. M. Goldner, and 1801 by Mr. H. V, Lowry. It seems worth while to supplement these notes by some comments which were for the most part suggested by the concluding lines of the note by E. H. N.

The writers of the four notes have looked at the theorem from the point of view of a candidate in an examination ; if it were set in a Higher Certificate Distinction Paper, I have little doubt that a fair proportion of the candidates would recognise that the cubic is a unicursal cubic, they would write x/y = u (or make some equivalent substitution), they would express x and y as rational functions of u, and they would realise that the question must inevitably come out by calculating accurately the differential coefficients of y with respect to x and then substituting in the left-hand side of the differential equation. In fact, they would reproduce the substance of the work of E. H. N. with the omission of his refinement of taking the reciprocal of y as an auxiliary variable to abbreviate the analysis.

1. We deal with the equation

where ψ(ωt)is continuous, periodic in t with period 2π/ω a, k2 are real and positive, and

A stable (bounded) solution was given by Erdélyi in 1934 employing a complicated transformation and integral equations.

We shall obtain this solution by an elementary method, thereby reducing the analysis to one-third of its former length.