If white light falls on a soap bubble or on a film of oil on a pool of water, the reflected light is very often coloured, due to interference at the surface of the film. Within the last few years fresh interest has been added to the phenomenon by the work of John Strong, Katherine B. Blodgett and others, who have shown that interference films can be deposited on glass and that these films reduce the reflection from the surface and increase the transmission through it, a matter of great importance in optical instruments. In many of the writings on the subject, the intensity of the reflected beam only has been considered, the transmitted beam being dismissed with the statement that, by the Conservation of Energy, light which is not reflected is transmitted. In some of the writings, only perpendicular incidence has been considered. In the discussion below expressions are obtained for the reflected and transmitted beams without the use of the Conservation of Energy for all angles of incidence.

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.

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.