13. In a right angled plane triangle, whose base is equal to b, altitude to a, and the angle opposite a equal to a determine the small error (= d a) committed in calculating a by means of the formula a = b Tan a. From the result, show that the height of a tower may be most accurately determined when the observation of its angular altitude is taken at a distance from its base, as nearly as possible equal to its height...... 14. Prove the formula of Demoivre m m (Cos +1 Sin 0) Cos±√ Sin n and show that the second member of this equation contains n values, as well as the first..... 15. Find an expression for the in terms of the angles-and if E area of a spherical triangle be the spherical excess and E 30 35 cot prove that Cot = 2 2 (Sec 0) Cot C....... 40 COS C Conic Sections. 16. In the Parabola, if QV be parallel to the tangent at P and PV parallel to the axis, QV2 — 4 SP. PV. 17. Draw a tangent to the ellipse at any point; and show that the focus of the perpendicular from the focus on the tangent, is the circumscribing circle,..................... 18. If y2+(1—e2) x2—. m x2 (1+e) equation to all the conic sections, the curve will o be the general be an ellipse, an hyperbola, or a parabola, according as e is less than, greater than, or equal to unity. Determine also in each case the magnitude of the axes or parameter. ..... 19. Find the polar equation to the ellipse when the focus S is the pole and by means of it show that if PS be produced through S to another point P1 in the curve, the rectangle SP SP1 the rectangle contained by the latus rectum and PP1,....................... Statics. Natural Philosophy. 1. What is meant by a line representing a force? Also, by the resultant of forces: and what is the condition that any number of forces acting upon a particle in the same straight line must satisfy, in order that they may be in equilibrium......................... 2. If any number of forces whose directions lie in one plane act upon a point, the sum of their moments about any point in the plane, is equal to the moment of their resultant about the same point. 3. Find how the requisites of a good balance may be satis fied and shew how to graduate the common steel-yard............. 4. Distinguish between stable and unstable equilibrium, and investigate the conditions of stable equilibrium in the case of a paraboloid resting upon a sphere.................... : Dynamics. 5. Explain what is meant by accelerating force, momentum, and moment of inertia, and distinguish between the statical and dynamical measure of force. By what arguments and experiments does it appear that gravity near the earth's surface is an uniform force, and the same in all bodies, whatever be their material or magnitude...... 6. Find the number of seconds gained or lost in a day by a second's pendulum when the force of gravity is slightly altered. Apply this to find the height of a mountain, by observing the loss in the number of oscillations of a second's pendulum at the top of the mountain........................... ............ 7. Determine the equation to the path of a projectile (in vacuo) and state distinctly at what points of the investigation, the first and second laws of motion are applied. Hydrostatics. ...................... 8. Find the resultant of the pressure of a fluid on the surface of a solid immersed in it: and when a solid floats in a fluid, state the necessary conditions to be fulfilled.... 9. Explain how the pressure of the atmosphere may be measured : 15 20 35 30 30 40 35 25 If specific gravity of Mercury = 13.6, and specific gravity of Sulphuric Acid = 1.84, and the mean height of the Mercury Baro 320 meter is 30 inches: find the mean height of an Acid Barometer. 30 Describe the syphon and explain its action. 10. Optics. 11. When a pencil of rays is reflected successively by two plane mirrors, show that it is made to deviate from its original direction, by an angle double of that at which the mirrors are inclined to each other, and explain how this property is made use of in Hadley's sextant. ... 25 12. Determine the modification which a pencil of rays will undergo in passing through a triangular prism, and the relation between the angle of the prism, and the deviation, when the angles of incidence and emergence are equal. Explain also the following terms which apply to the passage of light through a prism: Secondary spectrum-Dispersive power-Irrationality-and state how total reflexion may take place at the second surface....... 40 13. Describe the Gregorian reflecting Telescope, and find its magnifying power. Astronomy. 14. Give the general reasons for supposing that: (1.) The earth moves round its own axis. (2.) The earth moves in an elliptical orbit round the sun. Also explain in a general manner the change of seasons throughout the year, and the way in which the sun's position may affect the climate of different places.................... 15. Explain what is meant by the equation of time, and show that it vanishes four times in the year........ 16. Show how to find the longitude of a place by the observed distance of the moon from the sun, (introducing the corrections for parallax and refraction) the necessary observations being taken by three persons at the same moment. Why is this method of so much service at sea? 30 3360 45 50 Total Value, ... 500 |