Calendar: 1874-1875 Page 543

GENERAL LITERATURE AND SCIENCE 543 If PSp be any focal cliord of parabola then SP χ Sp SP Sp 10 In the ellipse the normal at any point bisects the angle between the focal distances of the point 11 If series of ellipses be described having the same major axis the tangents at the ends of their latera recta will all meet the minor axis in the same point 12 Prove the sum of the squares of two conjugate semi- diameters is constant the area of the parallelogram which touches the ellipse at the ends of conjugate diameters is constant 13 Find the equation to the ellipse referred to axes passing through the ends of the minor axis and meeting at one end of the axis major 14 From the equation to the ellipse deduce that to the parabola 15 Normals at Ρ and the ends of two semi-conjugate diameters of an ellipse meet at Κ find the equation to KG and show thatiTO is perpendicular to PD 16 Two given unequal circles touch externally show that the locus of the centre of the circle which always touches the other two is hyperbola Find the axes and eccentricity and show what the locus becomes when the given circles are equal 17 straight line drawn through one of the vertices of hyperbola and terminated by two straight lines drawn through the other vertex parallel to the asymptotes will be bisected at the other point in which it cuts the hyperbola 18 If there be two triangles such that the straight lines joining the corresponding angles meet at point then the intersections of the corresponding sides lie on straight line II tonwtrg Any one side of triangle is less than the sum and greater than the difference of the two remaining sides If from the ends of the side of triangle there be drawn two straight lines to point within the triangle these shall be less than the other two sides of the triangle but shall contain greater angle If from point within triangle lines be drawn to the angular points of the triangle the sum of these straight lines will he less