1. The following is an account of a theorem whose origin has been traced to Prof. Morley of Johns Hopkins University.

In the course of certain vector analysis, some 14 years ago, Prof. Morley found that if a variable cardioide touch the sides of a triangle the locus of its centre, that is, the centre of the circle on which the equal circle rolls, is a set of 9 lines which are three by three parallel, the directions being those of the sides of an equilateral triangle. The meets of these lines correspond to double tangents; they are also the meets of certain pairs of trisectors of the angles, internal and external, of the first triangle. This result was never published, and it was only the particular case of the internal trisectors that reached the present writers, the existence of the enveloping cardioides and the set of 9 lines being quite unknown to them.