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113. Find the specific gravity of a body lighter than the fluid in which it is weighed.

114. Find the conditions of the stable and unstable equilibrium of a floating body; and if it revolve about a horizontal axis, shew that it passes alternately through positions of stable and unstable equilibrium.

115. Having given the quantity of air in the tube of a barometer, determine the depression below the standard altitude.

116. If given volumes of two fluids of known specific gravities be mixed together, find the specific gravity of the compound, supposing its volume to be equal to the sum of the volumes of

1831

the parts.

n+ 2

117. If the particles of an elastic fuid repel each other with forces varying inversely as the nth power of their distances ; prove that the compressing force varies as (density) 3

118. Find the density of air in the air-pump after n turns.

119. Determine the pressure exercised by an incompressible fluid of uniform density against the surface of the vessel containing it.

120. If A and a be the areas of horizontal sections of a waterfall at heights H and h above the horizontal, find the height of the fall, the (velocity)? of the water at any point being as the height from which it has fallen.

121. Find the times of emptying a given prismatic vessel filled with water, by a cycloidal syphon of small bore placed with its base horizontal; the vertex of the syphon resting on the edge of the vessel.

122. Water retained at a constant elevation, issues from one vessel into another through a cylindrical pipe of radius r, and thence into the air through another cylindrical pipe of radius gud; having given the depth H of the mouth of the latter below the constant surface, find the velocity and pressure at a given depth h in the other pipe.

123. If the particles of a fluid mass, revolving with a given angular velocity, be attracted to a fixed centre by a force,

which is any function of the distance from the centre, the ellipticity, supposed small, will be half the ratio of the centrifugal force to the attraction at the equator.

124. Find the general equation of the equilibrium of incompressible fluids. If the fluid be heterogeneous, shew that all surfaces of equal pressure are of uniform density.

125. Let any particle of a mass of incompressible fluid, the coordinates of which at any time t are x, y, z, be acted upon by the forces X, Y, Z, in the directions of the axes of coordinates; and let u, v, w, be its velocities in the same directions; then if udx + vdy + wdz do, a complete differential, the pressure and motion are determined by the equations

d¢ , ¢ , ¢
dx2 + dy2 + dza

do 1 (do2 do2 dp2 p= S(Xdx + Ydy + Zdx)

dt

2

Zarzt dyz + d22) 126. If the force of gravity be considered constant, and altitudes from the earth's surface be taken in arithmetical progression, the corresponding densities of the air will decrease in geometrical progression, 127. Explain Bramah's hydrostatic press.

What is the limit to its practical application ?

128. When a body floats on a fluid, the weight of the fluid 1832 displaced is equal to the weight of the body; and the centres of gravity of the body and the fluid displaced are in the same vertical line.

129. There is a constant east wind under the equator, and a constant north-east wind near it on the north side, and a constant south-east wind near it on the south side; explain the causes of these phenomena.

130. Describe the barometer, and mention the purposes to which it is applied. Also calculate the correction to be made in any observed altitude of the mercury when the change of level of the mercury in the basin is taken into account.

131. There are three bodies of equal bulk, but of different given specific gravities ; it is required to suspend them from

three points in a straight lever, so that in whatever fluid they be wholly immersed, they may balance about its fulcrum.

132. A cubical vessel is filled with fluid and covered with a lid moveable about one of its edges as a hinge; if the vessel be moved into such a position that three edges meeting in one point are inclined at given angles to the vertical, determine the force which must be applied at a given point of the lid to keep it at rest.

133. Shew that the vertical pressure against the sides and bottom of any vessel is the weight of the Auid contained in it. How does it appear that the horizontal pressure will have no tendency to communicate any lateral motion to the vessel ?

134. If the pressure of the air vary as the density, and the

1
force of gravity vary as find the density at a given

(dist.)2'
height; and shew that it is finite at an infinite height.

135. If a body be let fall into a fluid of the same specific gravity as itself, it will sink to the bottom whatever be the depth of the fluid, the resistance varying as the square of the velocity.

136. Find the velocity of issuing through a small orifice in a vessel in which the water is retained at a given altitude.

137. State the sole condition necessary for the equilibrium of any fluid homogeneous or heterogeneous; and prove that when it is fulfilled, the force at every point of the external surface is perpendicular to the surface.

138. A fluid mass revolves round an axis, and is attracted towards a point in that axis by a force varying as the distance, required its form; and having given the mass of the fluid, determine the axes.

139. A body weighs (a) lbs. in vacuo, and (b) lbs. in water ; another body weighs (a') Ibs. in vacuo, and (b') lbs. in water; compare the specific gravities of the two bodies.

140. Find the pressure on a plane immersed in a heavy fluid. 141. Explain the process of filling the tube of a mercurial

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thermometer, and find what degree on Reaumur's thermometer will correspond to (n)o on Fahrenheit's.

142. Shew how to determine the specific gravities of bodies insoluble in water, by their weights in air, and in water.

143. From the principle that when a mass of fluid is in equilibrium, the state of rest is not altered by supposing any portion of the mass to become solid, deduce the equality of the pressure of fluid in all directions.

144. Explain the action of Watts's steam engine. On what principle is the efficiency of the engine, when employed in raising water, determined from the work done? What is the distinction between high pressure and low pressure engines, and in what cases are the former requisite?

145. The axis of a cylindrical vessel containing a known quantity of fluid is inclined at a given angle to the horizon ; determine the centre of pressure of its base.

146. Between two planes which meet the horizontal plane in the same line, and each of which is inclined to it at an angle whose tangent is ž, a hollow sphere first filled with a homogeneous fluid is placed, shew that the hemispheres on each side of a plane through their intersection may be supposed disunited without disturbing the equilibrium.

147. When a sphere acted on by gravity falls from rest in a medium resisting as the square of the velocity, find the velocity and resistance in terms of the space described. Hence explain the fall of the barometer in rainy weather.

148. State the experiments by which it appears that in all cases the pressure of air at a given temperature varies inversely as the space occupied.

149. Find the resultant of the pressure of a fluid on the 1834 surface of a solid, wholly or partly immersed in it. When the body floats in equilibrium, compare its volume with the volume of the part immersed.

150. Having given the form of the diving bell, find the space occupied by the air in it at any depth below the surface. If the bell be prismatic, solve the resulting equation, and explain the roots.

151, Compare the specific gravities of a solid and a fluid, by weighing the solid in air and in the fluid.

152. How is the pressure at a given point of a mass of fluid measured ? Prove that, when an elastic fluid of uniform temperature acted on by gravity is at rest, its density is the same at all points in the same horizontal plane. What considerations lead us to the conclusion that the height of our atmosphere is finite?

153. Describe Smeaton's air-pump; find the density of the air in the receiver after any number of strokes of the piston, and explain a contrivance by which it may be practically measured.

154. A barometer, consisting of two equal tubes connected by another of larger section into the ends of which they are inserted, has a quantity of lighter fluid above the mercury; shew the greater sensibility of this instrument, by comparing the variation of altitude of the compound column, owing to a given change of atmospheric pressure, with that of the mer. curial column in a common barometer, the common surface of the two fluids always falling within the larger tube.

155. A solid is generated by the revolution, through a given angle, of a right-angled triangle about one of the sides containing the right angle; determine the moment of a couple which, acting in the plane of symmetry, will support the solid with its axis vertical in a given fluid.

156. A given quantity of incompressible fluid, each particle of which is attracted towards a fixed centre by a force which varies as the distance, is separated from it by a given fixed plane; determine the pressure which the fluid exerts upon the plane.

157. Describe and explain the hydraulic ram. Account fully for the downward motion of the valve through which the waste takes place.

158. Find the relation between the density, temperature, and pressure of an elastic fluid, referring to the necessary experiments and assuming the results,

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