1. The problem has been proposed by Steiner* of finding the envelope of a system of similar conics circumscribed about a given triangle, and of finding the loci of the centres and foci of the conics of the system. He states that the envelope is a curve of the fourth order having three double points, and gives some of its properties. The problem has been treated by P. H. Schoute in a paper entitled Application de la transformation par droites symétriques à un problème de Steiner.† In this paper the author discusses the problem of the envelope in detail by a geometrical method, and gives the order of the locus of centres, of the locus of foci, and of the locus of vertices, and the class of the envelope of asymptotes, of the envelope of axes, and of the envelope of directrices. But in every case (except that of the locus of centres), he gives the degree or class twice as great as it should be. In a paper published in the Annals of Mathematics‡ I have found explicit equations for the locus of centres and the envelope of asymptotes, showing that each asymptote envelopes a three-eusped hypocycloid; and in a paper in the Transactions of the American Mathematical Society,§ I have obtained an equation for the envelope of the axes, showing that each axis also envelopes a three-cusped hypocycloid. I have also obtained the equation of the envelope of the directrices (a curve of the fourth class) and the equation of the locus of the vertices (a curve of the eighth degree); but the results have not been published. In the solution of a problem in the Educational Times (proposed by himself), Cayley* proved that the locus of the foci of the parabolas which pass through three fixed points is a unicursal quintic passing through the two circular points at infinity, by showing that the co-ordinates of the focus of any such parabola may be explicitly expressed as rational functions of a parameter of the fifth degree. As the locus of the foci of a system of similar conies passing through three fixed points is not in general a unicursal curve, it is hardly likely that Cayley's method could be extended to the general case.