10. P is any point in an ellipse, AA' its axis major, NP an ordinate to the point P; to any point Q in the curve draw AQ, A'Q meeting NP in R and S; shew that NR. NS NP2. = 11. PSP is a focal chord of a parabola, RDr the directrix, meeting the axis in D; Q is any point in the curve; prove that if QP, Qp produced meet the directrix in R, r, half the latus rectum will be a mean proportional between DR, Dr. 12. Two bodies acted upon by gravity are projected obliquely from two given points, in given directions and with given velocities; determine their position when their distance is the least possible. 13. A railway train is going smoothly along a curve of 500 yards radius at the rate of 30 miles an hour; find at what angle a plumb-line hanging in one of the carriages will be inclined to the vertical. 14. If a body be projected from a given point in a given direction with a given velocity, and be acted on by a force tending to S and varying as 1 (dist.)2' › prove that if PSP be any focal chord of the body's path the sum of the squares of the velocities at P and p will be constant. A number of balls of given elasticity A, B, C... are placed in a line; A is projected with a given velocity so as to impinge on B; B then impinges on C, and so on; find the masses of the balls B, C... in order that each of the balls A, B, C... may be at rest after impinging on the next; and find the velocity of the nth ball after its impact with the (n − 1)th. 16. An imperfectly elastic ball is projected in a given direction within a fixed horizontal hoop, so as to go on rebounding from the surface of the hoop; find the limit to which the velocity of the ball will approach; and shew that it will attain this limit at the end of a finite time. 17. If Q, q be two points in the radius of a spherical refracting surface whose centre is E, such that EQ Eq :: the sine of incidence: the sine of refraction, determine geometrically the position of the point P so that a ray proceeding from Q and incident upon the surface at P may after refraction proceed from q. 18. If a ray of light after being reflected any number of times in one plane at any number of plane surfaces return on its former course, prove that the same will be true of any ray parallel to the former which is reflected at the same surfaces in the same order, provided the number of reflections be even. 19. An inverted vessel formed of a substance which is heavier than water contains enough of air to make it float; prove that if it be pushed down through a certain space, it will be in a position of unstable equilibrium; and determine the space in question. 20. A uniform piston, terminated by a plane of area A perpendicular to its side, is inserted into an orifice in a vessel containing fluid; prove that the work done in gently pushing in the piston through a small space s is ultimately equal to the work done in lifting a portion of the fluid of volume As through a height equal to the depth of the centre of gravity of the plane below the surface of the fluid. 21. Two equal slender rods AB, AC moveable about a hinge at A and connected by a string BC rest with the angle A immersed in a given fluid; determine the tension of the string BC. 22. If a rectangular court be enclosed within a wall of given height, and one of its sides be inclined at an angle of 30° to the meridian, determine the breadths of the shadows of the walls on a given day at noon, and the portions of the courts and walls which will be enveloped in the shadow, the latitude being 52° 30′ N., and the Sun's declination on the given day 7° 30′ N. 1849. MODERATORS. WILLIAM BONNER HOPKINS, M.A., St Catharine's Hall. EXAMINERS. GEORGE GABRIEL STOKES, M.A., Pembroke College. 1. THURSDAY, Jan. 4. 9...12. DESCRIBE an equilateral triangle upon a given finite straight line. By a method similar to that used in this problem, describe on a given finite straight line an isosceles triangle, the sides of which shall be each equal to twice the base. 2. If a side of any triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. Can you give Legendre's method of demonstrating this proposition, which depends upon the necessary homogeneity of algebraical equations, or any demonstration other than Euclid's? 3. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. Shew that in Euclid's figure four other lines, beside the given line, are divided in the required manner. 4. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle. Give a direct demonstration of this proposition by the method of limits. 5. Inscribe a circle in a given triangle. How may a circle be described touching one side and the produced parts of the other two? 6. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents. What restriction is here implied as to the species of the magnitudes? 7. The sides about the equal angles of equiangular triangles are proportionals, and those sides which are opposite to the equal angles are homologous. Apply this proposition to prove that the rectangle contained by the segments of any chord passing through a given point within a circle is constant. 8. Define compound ratio; and prove that equiangular parallelograms have to each other the ratio which is compounded of the ratios of their sides. Of what use is this proposition in the application of Algebra to Geometry? 9. Draw a straight line perpendicular to a plane from a given point without it. Prove that equal right lines drawn from a given point to a given plane are equally inclined to the plane. 10. In the parabola, the rectangle under the latus rectum and an abscissa of the axis is equal to the square of the semi-ordinate. 11. The normal at any point of an ellipse bisects the angle between the focal distances. Can you deduce the proof of this proposition from mechanical considerations? 12. The perpendiculars from the foci on the tangent to an ellipse intersect the tangent in the circumference of a circle having the axis major as diameter. Deduce from this an analogous proposition for the parabola. 13. In the ellipse, if the conjugate diameter meet either focal distance in E, PE will be equal to AC. 14. Define the circle of curvature; and prove that in the ellipse the diameter, the conjugate diameter, and the chord of curvature passing through the centre, are in continued proportion. 15. If a tangent be drawn to a hyperbola, and be terminated by the asymptotes, it will be bisected in the point of contact. Apply this proposition to prove directly that the area of the triangle contained by the tangent and the asymptotes is constant. If SVS, TVt be two tangents cutting one asymptote in the points S, T, and the other in s, t, prove that 16. VS: Vs Vt: VT. The section of a right cone by a plane parallel to a line in its surface, and perpendicular to the plane containing that line and the axis, is a parabola. The foci of all parabolic sections which can be cut from a given right cone lie upon the surface of another cone. 1. Shew that the value of a product does not depend upon the order of its factors, the factors being commensurable. Extend your proof to the case of incommensurable factors. 2. A person rents a piece of land for £120 a year. He lays out £625 in buying 50 bullocks. At the end of the year he sells them, having expended £12. 108. in labour. How much per head must he gain by them, in order to realize his rent and expenses, and 10 per cent. upon his original outlay? |