kept at the same height, the descending stream is bounded by the surface generated by the revolution of the curve y'x = const., about the axis of x.

9. Water is flowing steadily into a large reservoir through a straight tube of small section, inclined at a given angle to the vertical; having given the length of the tube, the depth of its lower extremity below the surface of the water in the reservoir, and the sections of both ends, find the rate at which water is flowing into the reservoir.

10. A vessel containing ink has a small hole pierced in one side, and is placed in a vessel of water; compare the velocity with which the ink will escape into the water, with that which it would have if it were flowing out into the air.

11. A closed cylindrical vessel one foot in height is half full of water, the other half being occupied by atmospheric air; if two small apertures be made, one at the base of the cylinder and the other five inches above it, shew that the density of the

air in the vessel will decrease till it is 1 (1 - 12) times its original

value approximately, and then increase again, h being the height of a water-barometer in feet.

12. A vertical cylinder is supplied with fluid at the top and loses it by an orifice at the bottom: assuming the motion to be by horizontal sections, and supposing the cylinder to be initially empty, g the accelerating force of gravity, v the constant velocity of the entering stream, and m the ratio of its transverse section to that of the cylinder, find the velocity of the issuing stream at any time t, and explain the result when

13. A vessel in the form of a surface of revolution, the axis of which is vertical, has a small orifice at its vertex, and is filled with fluid; determine its form in order that the quantity of fluid which flows out in any time may vary as the square root of the

time.

14. A vertical cylindrical vessel full of fluid has a fine crack extending along a generating line of the cylinder; find the time

of emptying a given portion of the cylinder. Examine the case in which the time of emptying the whole cylinder is required.

15. A right cone is filled with fluid and placed with a generating line horizontal, and uppermost, and a small orifice is made at the lowest point; find the time in which it will be emptied.

16. The surface of a vertical cylinder is pierced by a series of small holes in the form of a helix, the highest hole being at the top of the cylinder, and vertically above the lowest, and no other two holes being in the same vertical line. Determine the equation to the curve traced by the issuing fluid upon the horizontal plane passing through the lowest hole, the cylinder being kept constantly full.

Shew that the mean range is to the height of the cylinder as π 4, and that the area included between the base of the cylinder and the curve above mentioned is

where a is the inclination of the line of holes to the horizon, and h the height of the cylinder..

17. A cubical vessel, having one side horizontal, is divided into two equal parts by a vertical partition, and one of the compartments is filled with fluid. If a small orifice be bored through the partition at a distance below the surface greater than half the depth of fluid, find the time which elapses before the fluid stands at the same height in both compartments.

18. A filament of fluid oscillates in a thin cycloidal tube of uniform bore, the axis of the cycloid being vertical and vertex downwards. Supposing the filament to be placed initially with its lower end at the lowest point of the tube, find the pressure at any point of the filament at any time.

Shew that the pressure is a maximum, during the whole motion, at the middle point of the filament.

19. A filament of fluid oscillates in a thin hypocycloidal tube of uniform bore under the action of a force tending to the centre of the fixed circle, and varying as the distance: supposing

the filament to be placed initially with one end at the vertex of the hypocycloid, find the pressure at any point of the filament at any time.

20. A small orifice of area κ is opened in the base of a vertical cylinder initially full of fluid. The fluid is forced through the orifice by a piston fitting the cylinder, to which is applied an uniform pressure P equal in amount to n times the weight of the fluid which the cylinder can contain. Shew that

1

m

th of the fluid will be evacuated in a time expressed by

where h is the height of the cylinder and A the area of its transverse section.

21. If the orifice of a conical vessel containing water be a section of the cone, perpendicular to its axis and at a distance 8 from its vertex, and v be the velocity with which the water discharges itself, when its surface is at a distance z from the cone's vertex, prove that

22. Two points are connected by a tube of small uniform bore through which heavy fluid is flowing steadily: the axis of the tube being in the vertical plane through the two points and its length being given, find its form when the whole pressure on the tube is a minimum.

CHAPTER X.

FURTHER APPLICATIONS OF THE EQUATIONS OF MOTION.

146. THE following proof of an important theorem is taken, with slight variations, from a paper, by Professor Stokes, in the eighth volume of the Cambridge Philosophical Transactions.

THEOREM. Let the accelerating forces X, Y, Z, which act on the fluid, be such that Xdx + Ydy + Zdz is the exact differential dV of some function of the co-ordinates. Then, if for the whole, or a certain portion of the fluid mass, the motion is at any one instant such that udx + vdy+wdz is an exact differential, that expression will always be an exact differential, for the whole mass, or for the portion of fluid for which it was so at first.

Suppose p a function of p, and let

Differentiate the first of these equations with respect to y, and the second with respect to x, and subtract; then, putting

and observing that, since Xdx + Ydy + Zdz is an exact dif

and it will be observed that, on account of the continuity of the

motion, the differential coefficients finite.

du

dx

cannot become in

Suppose that when t=0, either there is no motion, or the motion is such that udx + vdy+wdz is a perfect differential. This may be the case for the whole or for any portion of the fluid mass.

Then initially,

w' = 0, w" =0, @"" = 0.

Now let L be a superior limit to the numerical values of the coefficients of w' w" w""; then, whether w' w" w"" are, any or all