The following two propositions are necessary for what is to follow, and are very easy to prove.

I. If A, B, C be three points such that B and C are conjugate with respect to the Polar Conic of A, then are C and A conjugate points with respect to the Polar Conic of B, and similarly with regard to the third vertex.

A triangle possessing the property defined above, viz. that each pair of vertices are conjugate points with respect to the Polar Conic of the third vertex, is called an Apolar Triangle.

This is a note on the theory of continued fractions, in which the chief feature is the use made of the successive remainders or divisors which occur in the reduction of any given ratio to a continued fraction.

The treatment of the Pellian equation also differs from that which is generally given.

The following method of establishing the existence and properties of the Focal Circles of a Circular Cubic is, as far as I know, new, and it has the advantage of dispensing, almost entirely, explicitly with analysis, while many of the properties can be proved without using the complicated method of generating the curve given in Salmon. The results which I have arrived at in connexion with the Nodal Cubic and Cuspidal Cubic are not given in Salmon.

Consider the equation f(ξ η, x, y) = 0. If definite values of ξ and η be taken, and x and y be current co-ordinates, f(ξ η, x, y) = 0 is represented by a curve in the plane of xy. If all the possible values of ξ and η be taken in turn, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the xy plane. Similarly if all possible definite values of x and y be taken in turn and ξ and η be regarded as current co-ordinates, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the ξη plane. Since the first and the second doubly infinite families of curves represent the same equation, it may be expected that some geometrical correspondence exists between them. It is the object of the present paper to investigate this correspondence.

The problem was proposed by Steiner of finding certain loci and envelopes connected with a system of similar conics through three points. In previous papers I have found the locus of centres and the envelope of asymptotes of such a system of conics, also the envelope of axes, and the locus of foci. I now propose to discuss the envelope of the directrices.

The fundamental equation in this paper,

,

can be derived, easily and without using trilinears, from the following property of conics.

A proof of Dupin's theorem with some simple illustrations of the method employed.

Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.

The surfaces here considered were first discussed by Monge as surfaces whose normals are tangents to given developables. Under the name general surfaces moulures, Darboux treats them as the surfaces traced out by a fixed curve on a plane which rolls on a developable. When the developable is a cylinder he gives the general coordinates in his Leçons sur la Theorie Generale des Surfaces (Volume I., page 105). This led me to take up the more general case, but I later found that Darboux had also considered this in an ingenious and elegant manner in his Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes (Tome 1, pages 26–34). Perhaps this quite difFerent discussion will present some interesting points in analysis.

Permit me the liberty of presenting some remarks on an article by Professor Naraniengar inserted in Vol. XXVIII., p. 73, of the Proceedings of the Edinburgh Mathematical Society, 1909–1910.

The method here described applies to the successive integrations of any function whose graph consists of segments of straight lines.

The relation between the eccentric and focal angles of P (Fig. 1) is

or, more compactly,

where

The following direct method of attack establishes the fundamental properties of the Foci and Bi-Tangents of a Bi-Circular Quartic very simply, using only the properties of a homographic correspondence. The method was suggested to me by my previous paper on the focal properties of Circular Cubics.

I am indebted to Mr Peter Fraser, University of Bristol, for his criticism.