A TREATISE ON CONIC SECTIONS: 31 CONTAINING AN ACCOUNT OF SOME OF THE MOST IMPORTANT MODERN BY GEORGE SALMON, D.D., D.C.L., F.R.S., REGIUS PROFESSOR OF DIVINITY IN THE UNIVERSITY OF DUBLIN, Fifth Edition. LONDON: LONGMANS, GREEN, READER, AND DYER. 1239 1869. [Junior readers will find all essential parts of the theory of Analytical Geometry included Co-ordinates of Point cutting that Distance in a given Ratio. Transformation of Co-ordinates does not change Degree of an Equation Two Equations represent Points. A single Equation represents a Locus 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 makes with Axes Equation of Line joining two given Points Condition that three Points shall be on one Right Line Co-ordinates of Intersection of two Right Lines Middle Points of Diagonals of a Quadrilateral are in a Right Line (see also p. 62) 26 Equation of Perpendicular on a given Line 27 25 Length of Perpendicular from a Point on a Line Equations of Bisectors of Angles between two given Right Lines Area of any Polygon Condition that three Lines may meet in a Point (see also p. 34) · Equation of Line through the Intersection of two given Lines EXAMPLES ON THE RIGHT LINE. Investigation of rectilinear Loci . of Loci leading to Equations of Higher Degree Problems where it is proved that a Moveable Line always passes through a Centre of Mean Position of a series of Points Right Line passes through a Fixed Point if Constants in its Equation be Test that three Equations may represent Right Lines meeting in a point Connexion between Ratios in which Sides of a Triangle are cut by any Transversal 35 by Lines through the Vertices which meet in a point Meaning of Constant k in Equation a = Bisectors of Angles, Bisectors of Sides, &c. of a Triangle meet in a point Equations of a pair of Lines equally inclined to a, B Theorem of Anharmonic Section proved Expression of Equation of any Right Line in terms of three given ones Harmonic Properties of a Quadrilateral proved (see also p. 305) Homologous triangles: Centre and Axis of Homology Condition that two Lines should be mutually perpendicular Length of Perpendicular on a Line • Trilinear Co-ordinates Trilinear Equation of Parallel to a given Line of Line joining two Points Proof that middle Points of Diagonals of Quadrilateral lie in a Right Line Intersections of Perpendiculars, of Bisectors of Sides, and of Perpendiculars at 66 Angle between two Lines given by a single Equation Equation of Bisectors of Angles between these Lines Condition that Equation of second Degree should represent Right Lines (see also pp. 144, 148, 150, 255) Number of conditions that higher Equations may represent Right Lines Condition that two Circles may be concentric that a Curve should pass through the origin Co-ordinates of Points where a given Line meets a given Circle Line cut harmonically by a Circle, Point, and its Polar Equation of pair of Tangents fram a given Point to a Circle Circle through three Points (see also p. 128) Condition that four Points should lie on a Circle, and its Geometrical meaning If a Point A lie on the polar of B, B lies on the polar of A Conjugate and self-conjugate Triangles Conjugate Triangles Homologous If two Chords meet in a Point, Lines joining their extremities transversely meet Expression of Co-ordinates of Point on Circle by auxiliary Angle |