S A TREATISE ON CONIC SECTIONS: CONTAINING AN ACCOUNT OF SOME OF THE MOST IMPORTANT MODERN ALGEBRAIC AND GEOMETRIC METHODS. BY GEORGE SALMON, D.D., FELLOW AND TUTOR, TRINITY COLLEGE, DUBLIN. Fourth Edition. LONDON: LONGMAN, BROWN, GREEN, LONGMAN, AND ROBERTS. 1803. PREFACE. Greater pro This new edition has been carefully revised, and in great part re-written; alterations having been made in some places for the sake of brevity, in others for the sake of greater clearness, in others in order to bring the methods more closely up to the present state of Geometrical Science. Several new examples have been added; as well as a new Chapter on the Applications of the Modern Algebra to the Theory of Conic Sections. minence has been given to the principle of duality; and it has been attempted to show that without the introduction of any new system of co-ordinates, the reciprocity between theorems concerning lines, and theorems concerning points, can be sufficiently manifested. The change most likely to be objected to is the alteration I have made in the mode of writing the equation of the second degree, the letters being now used not in alphabetical order, but in the order suggested by the symmetry of the equation. I believe that the advantage of having uniform notation through the volume, will be found to be so great as to compensate for some temporary inconvenience caused to those who are already familiar with formulæ in the older notation. I have to acknowledge the courtesy of several correspondents who sent me lists of the errata of the former edition, which, if I had now been contented with a simple reprint, would have enabled me to make one nearly free from error. I shall be thankful to any of my readers who may furnish me with similar lists of the errors from which I fear this edition is not exempt, notwithstanding that most of the sheets have been looked over either by Dr. Hart or Mr. Gray or Mr. James MoDowell, who have at various times kindly assisted me in correcting the press. I beg to thank Mr. Burnside I for several notes of which I have in different places made use; and I have derived considerable assistance from the notes and additions in Dr. Fiedler's German translation of this work. TRINITY COLLEGE, DUBLIN, October, 1863. CONTENTS. [It is hoped that the cross references which have been added will enable the Table of Contents to serve as an Index to readers in search of information on any particular theorem. Des Cartes' Method of Co-ordinates Co-ordinates of Point cutting that Distance in a given Ratio Transformation of Co-ordinates . CIIAPTER II. Two Equations represent Points A single Equation represents a Locus Equation of a Right Line parallel to an Axis in any Position Meaning of the Constants in Equation of a Right Line Equation of a Right Line in terms of its Intercepts on the Axes in terms of the Perpendicular on it from Origin, and the Angles it Expression for the Angles a line makes with Axes Equation of Line joining two given Points also p. 64) of line making a given Angle with a given Line . Length of Perpendicular from a Point on a Line . Equations of Bisectors of Angles between two given Right Lines Area of Triangle in terms of Co-ordinates of its Vertices Condition that three Lines may meet in a Point (see also p. 31) a |