Condition that a Triangle self-conjugate with regard to one Conic should be inscribed or circumscribed about another Six vertices of two self-conjugate Triangles lie on a Conic Circle circumscribing self-conjugate Triangle cuts the director circle orthogonally 341 Centre of Circle inscribed in self-conjugate Triangle of equilateral Hyperbola Envelope of Base of Triangle inscribed in one Conic, two of whose sides touch Locus of free Vertex of a Polygon all whose sides touch one Conic, and all Condition that Lines joining to opposite vertices, Points where Conic meets Triangle of reference should form two sets of three meeting in a Point Every line through an imaginary Circular Point, perpendicular to itself Condition for Equilateral Hyperbola and for Parabola in Trilinear Coordinates General Tangential Equation of two Circular Points at infinity To draw a Conic having double contact with a given one, and touching three Four Conics having double contact with S, and passing through three Points, or To draw a Conic through four Points to touch a given Conic Jacobian of three Conics having two Points common, or one of which reduces Area of common conjugate Triangle of two Conics Mixed Concomitants Condition that a line should be cut in involution by three Conics Condition that they should have a common Point Condition that XU+uV+vW can in any case be a perfect Square Direction of Tangents of Conics Determination of Areas of Conics Tangent to any Conic cuts off constant Area from similar and concentric Conic . 874 874 Excess of sum of two Tangents over included Arc, constant when Vertex moves Difference of Arc and Tangent, constant from any Point on Confocal Hyperbola 377 ANALYTIC GEOMETRY. CHAPTER I. THE POINT. 1. THE following method of determining the position of any point on a plane was introduced by Des Cartes in his Géométrie, 1637, and has been generally used by succeeding geometers. We e are supposed to be given the position of two fixed right lines XX', YY' intersecting in the point 0. Now, if through any point P we draw PM, PN parallel to YY' and XX', it is plain that, if we knew the position of the point P, we should know the lengths of the parallels PM, PN; or, vice versâ, that if we knew the lengths of PM, PN, we should know the position of the point P. Suppose, for example, that we are given PN=a, PM = b, Y we need only measure OM=a and ON=b, and draw the parallels PM, PN, which will intersect in the point required. It is usual to denote PM parallel to OY by the letter y, and PN parallel to OX by the letter x, and the point P is said to be determined by the two equations x = a, y = b. 2. The parallels PM, PN are called the coordinates of the point P. PM is often called the ordinate of the point P; while PN, which is equal to OM the intercept cut off by the ordinate, is called the abscissa. B The fixed lines XX and YY' are termed the axes of coordinates, and the point O, in which they intersect, is called the origin. The axes are said to be rectangular or oblique, according as the angle at which they intersect is a right angle or oblique. It will readily be seen that the coordinates of the point M on the preceding figure are xa, y = 0; that those of the point Nare x = 0, y=b; and of the origin itself are x = 0, y = 0. 3. In order that the equations xa, y=b should only be satisfied by one point, it is necessary to pay attention, not only to the magnitudes, but also to the signs of the coordinates. If we paid no attention to the signs of the coordinates, we might measure OM=a and ON=b, on either side of the origin, and any of the four points P, P, P, P, would satisfy the equations x=a, y=b. It is possible, however, to P N P X M M sidered as negative. It is, of course, arbitrary in which direction we measure positive lines, but it is customary to consider OM (measured to the right hand) and ON (measured upwards) as positive, and OM', ON' (measured in the opposite directions) as negative lines. Introducing these conventions, the four points P, P, P1⁄2 P ̧ are easily distinguished. Their co-ordinates are, respectively, x = + a x = − a) x=+a) x=− a) y = + bf' y = + b) ' y = − b] 'y = - b) • |