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159. The attraction of the particles of a fluid to each other and to glass being supposed sensible only at very small distances, find the relation between the diameter of a capillary tube, and the height to which the fluid will rise.

160. When a fluid moves, acted on by any forces, determine the effective accelerating force in the direction of its motion at any point. Employ the result to find the velocity with which an incompressible fluid issues through an indefinitely small orifice in a vessel containing it.

161. Define the centre of pressure of a plane surface immersed in a fluid. Why is its position independent of the inclination, and lower than the centre of gravity of the surface?

162. What is the nature of the transmission of fluid pres- 1835 sure? Apply the principle to explain Bramah's press.

163. Compare the specific gravities of a solid and fluid, the specific gravity of the solid being less than that of the fluid.

164. Having given the graduations of two thermometers for freezing and boiling water, determine the graduation of one, when the other marks to.

165. Find the magnitude and direction of the resultant of the pressure of a fluid on the surface of a solid immersed in it.

166. Describe the common barometer, and find the correction to be applied when a small quantity of air remains in the upper part of the tube. What other circumstances affect the accuracy of the apparent altitude?

167. Compare the duties of an atmospheric steam engine for one oscillation of the piston, (1) when the cylinder as usual is open at the top, (2) when hermetically closed.

168. The pressure on a quarter of the surface of a hemispheroidal bowl filled with fluid (bounded by two planes passing through the axis c which is vertical) will be equivalent to a single force acting in the straight line whose equations are Cz 3 a2 c2

x = y

1=3 +

πα 16 α

169. Assuming the differential expression for the pressure

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at any point of a fluid mass in equilibrium when acted on by known forces, shew that the resultant of the forces at a point in the surface is in the direction of the normal; and that the equilibrium is possible when the forces arise from the attraction to a fixed centre, or to every particle of a solid or fluid mass.

170. Find the centre of pressure of the area of a quadrant of a circle, one side of the quadrant coinciding with the surface of the fluid.

171. In what respect does the effect produced by the application of pressure to a fluid differ from that produced by the application of the same pressure to a solid body? what is termed the hydrostatical paradox.

Explain

172. Investigate the conditions to be fulfilled that a body may float in equilibrium on a fluid.

173. Explain the principle and action of the syphon; and thence the phenomena of tide-wells or reciprocating springs.

174. Compare the specific gravities of two fluids by weighing them in the same vessel.

175. Describe the construction of the common mercurial thermometer. Why can it not be depended on for temperatures above 212°? What methods are used to measure intense heat and cold? Explain the construction and use of the register thermometer.

176. Investigate the position of the metacentre of a floating vessel, partly filled with fluid. What effect is produced upon the stability of a ship by throwing out the ballast and leaking an equal weight of sea water?

177. A cubical vessel, full of fluid, is held in a given position, compare the pressures on its several sides.

178. Determine the velocity of the wind, blowing horizontally, when it is just able to overturn a given cylinder standing on a horizontal plane, and prevented from sliding.

179. The times of emptying a segment of a sphere through equal small orifices in its vertex and base are as In, the base being horizontal in both cases; compare the volume of the segment with that of the whole sphere.

180. When a fluid of uniform density is acted on by any forces, state the conditions under which there will be equilibrium; and shew that they are satisfied, if the forces tend to fixed centres and are functions of the respective distances. When the fluid is elastic, shew how the pressure and density at any point are determined. Why cannot the atmosphere of the earth continue in equilibrium?

181. Determine the velocity with which fluid issues through a small orifice in a vessel kept constantly full.

182. Determine the difference of altitude of two stations by means of a barometer and thermometer, the force of gravity being constant.

183. Describe the diving bell, and find the tension of the rope by which it is suspended.

THEORY OF SOUND.

1. Prove that a column of air in a cylindrical pipe of length 1831 l, closed at one end, may be made to vibrate so that at distances 2X, 4X .... 2kλ from the closed end, the air will be stationary, and at distances λ, 3λ, 5λ,. . . . (2k + 1) λ, the last of which = 7, the density will be constant.

2. Prove that the vibrations propagated from any point of disturbance in a cylindrical column of air, are such that the velocity of the particles is proportional to the condensation, and the condensed particles move in the direction of propagation, the rarefied in the contrary direction.

3. Determine the velocity of sound, and shew that it is 1832 independent of its loudness.

How is the difference between the theoretical and experimental determinations accounted for?

4. In consequence of a small disturbance communicated to the particles of an elastic fluid mass at rest, a point whose coordinates are x, y, z is transferred to a position whose coordinates are + λu, y + λv, z + Aw, where is a small

constant positive quantity; prove that the density at that point is increased or diminished according as

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5. Explain the phenomenon of musical beats and the formation of harmonics. If the original notes make m and n vibrations in a given time, find the number of vibrations of the resultant note.

6. The error arising in the computation of the velocity of sound in any gas from neglecting the change of temperature, is corrected by the introduction of a factor √K, explain the physical meaning of K; and shew that if its value for atmospheric air be known, its corresponding value for any known gas may be found by comparing with a monochord the fundamental notes in two equal tubes of the air and gas under the same pressure and at the same temperature.

7. Investigate the condition for the single propagation of a pulse of sound in a thin straight tube filled with uniform elastic fluid. If the pulse meet a second medium, shew that it will be divided into two, running in opposite directions.

SECTION XIX.

QUESTIONS IN OPTICS AND THE THEORY OF LIGHT.

1. Shew that the image of a straight line placed between 1821 the centre and principal focus of a concave mirror is a hyperbola.

2. Shew how to find the focal length of a double concave lens by experiment.

3. If the air near the surface of the ground be less dense than at small altitudes above it, there will be observed inverted images of distant horizontal objects.

4. An object appears brighter, cæteris paribus, when seen through a convex lens, than when seen through a concave lens.

5. Prove that objects appear erect in Galileo's telescope.

6. By what experiments is it proved, that light consists of rays differing in colour and refrangibility?

7. Shew that the image of an indefinite straight line perpendicular to the axis of a convex lens, and nearer to its centre than the principal focus of parallel rays incident in the opposite direction, forms the arcs of two opposite hyperbolas; and find the semi-axes.

8. A person wishes to see distinctly when under water. What kind of glasses must he use, and of what focal length?

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