Of all the purely geometrical methods which have been used to exhibit or to demonstrate the fundamental properties of the plane conic sections, perhaps the most elegant is that which consists in the use of a sphere inscribed in a right cone of which the plane conic is a section. This method was indicated, though not quite fully stated, 130 years ago, by Hugh Hamilton, a distinguished Irish Geometer. It has been developed, and its history given in Mr Charles Taylor's 'Geometry of Conics' (Cambridge, 1881). It will therefore be unnecessary for me to do more than briefly to describe it, and to mention some of the results to which it leads, so as to enable the reader to enter more readily upon the consideration of what follows. This combination of the right cone, the inscribed sphere, and the plane, will be found to suggest geometrical properties and trains of mathematical investigation which, I believe, have not yet been distinctly stated. I propose in this Paper to show how this method not only brings into view the focal circles of the plane conic sections, but adapts itself to the proof of propositions relating to similar circles connected with the spherical conics, and indeed of many other fundamental properties of those curves.
2. Final Remarks
A Geometer who has read no new Mathematics for more than twenty years, and has forgotten a great part of what he once knew, need not be surprised if his exercises, jotted down in a desultory way to beguile hours of pain and weakness, in a foreign country, without books of any kind to refer to, or any help from friends, are found to contain nothing that has not been anticipated. Even if that may be said of the part of this Paper which relates to the focals of the plane conics, I retain a hope that some novelty may be recognized in the section which treats of the sphero-conics and their focals. In connection with that part of my subject I have traced, and mean to discuss elsewhere, some of the consequences flowing from a proposition which holds, in the theory of these curves, a place analogous to that which belongs to the definition of a focal mentioned in the first paragraph of this Paper. I mean that I have succeeded in formulating and solving the question which, in the theory of the sphero-conics, is exactly similar to that which has been proposed and answered for the plane conics: viz., By what conditions would a circle be limited, if the length of a tangent drawn to it from any point upon the plane conic is always a rational function of the abscissa?
I gladly acknowledge that in the preparation of this Article for Hermathena I have obtained valuable assistance from Mr Thomas Preston, who has extracted the materials for it from my note-books, and put them in order.
June 18, 1888.