The voyage would take 2800÷21=133 hours; but if 5 hours is deducted for longitude difference on the westward voyage, when running before the sun, the apparent time is 128 hours, so that the apparent speed is raised to 21.82 knots.

Examples.

(1) A bucket of water in a balance discharges 4 lb of water per minute through an orifice in its base at 45° to the vertical, and is kept constantly full by a vertical stream which issues from an orifice 8 ft above the surface with velocity 30 f/s.

Prove that the bucket must be counterpoised by about 0.066 lb more than its weight.

(2) The bucket valve in fig. 80 (p. 362) has a small leak, one-800th of the cross section of the barrel, and the height of the water barometer is taken as 32 ft, the height 40 as 16 ft, and the specific volume of the air 800 times that of water.

Prove that the pump will not suck unless the bucket is moved with a velocity greater than /2=1·13 f/s;

but that afterwards water will be lifted if the velocity is greater than 0·04 f/s.

(3) Prove that a hydraulic engine (fig. 78), in which water under pressure is admitted through small orifices to actuate the piston, will do most work when the speed is 3/3 of the unloaded speed, and the load is of the maximum load, and that the efficiency is then 3.

(4) Discuss the influence of inertia and of fluid friction in the pipe, when the Hydraulic Press (§ 12) is actuated by the Accumulator (§ 15). (Cotterill, Applied Mechanics, § 256.)

(5) Prove that the H.P. of the feed pump of a boiler, which evaporates W lb/min of water at a gauge pressure p lb/in2, must exceed

144 Wp÷33000.

(6) Prove that if the jet of § 371 delivers Q ft3/sec, and the hose is 7 ft long, the pumping H.P. of the fire

(7) If water is scooped up from a trough between the rails into a locomotive tender to a height of h ft, determine the minimum velocity required, and the delivery at a given extra speed, taking the frictional losses as represented by a given fraction of the head.

(8) Show how liquid may be raised through a siphon tube, made to revolve about its longer branch,

which is held vertical; and determine the delivery and the mechanical efficiency for a given angular velocity.

(9) Show how to determine the elements of a cyclone from observations at three points.

What is the direction of rotation in the N. and

S. hemispheres ?

CHAPTER XII.

GENERAL EQUATIONS OF EQUILIBRIUM.

387. It was proved in Chapter I., SS 19, 20, that the surfaces of equal pressure and the free surface of a liquid at rest under gravity are horizontal planes; but this assumes that gravity acts in parallel vertical lines.

When we examine more closely the surface of a large sheet of water like the open sea, we find it uniformly curved, so that the surface is spherical; showing that the lines of force of gravity converge to the centre of the Earth; and Archimedes in his diagrams of floating bodies represents them immersed in a spherical ocean.

If three posts are set up, a mile apart in a straight canal, to the same vertical height out of the water, the visual line joining the two extreme posts will, in the absence of curvature by refraction, cut the middle post. 8 ins lower; hence it is inferred that the diameter of the Earth in miles is the number of 8 ins in 1 mile, or 7920.

If miles apart, the visual line cuts at a depth 872 ins; for instance, the Channel tunnel 20 miles long, if made level, would rise in the middle 800 ins from the straight chord; but if made straight, it would have a gradient of about one in 400 at the ends, and water reaching to the ends would have a head of 800 inches in the middle.

486

LEVEL SURFACES OF EQUILIBRIUM.

388. Careful measurements of a degree of the meridian in different latitudes reveal that the mean level surface of the Ocean is not exactly spherical, but slightly ellipsoidal and bulging at the Equator; an effect attributable to the Earth's rotation, and investigated in the theory of the Figure of the Earth.

Lastly, the imperceptible deflections of the plumb line, due to the perturbative attraction of the Moon and Sun, are rendered very manifest by the phenomena of the Tides, due to the same cause of perturbation.

389. All these manifestations are examples of the general principle, enunciated in § 24 as

"Liquids tend to maintain their Level,"

but now the level surface must be taken to mean the surface which is everywhere perpendicular to the resultant force of gravity at the point, as indicated by the plumb line.

To prove the theorem that

"The surfaces of equal pressure in a fluid at rest under given forces are at every point perpendicular to the line of resultant force";

draw two consecutive surfaces of equal pressure PP', QQ', on which the pressures are p and p+Ap suppose; and consider the equilibrium of a cylindrical element of cross section a, the axis PR of which is normal to the surfaces of equal pressure.

The resultant thrust on the curved side of the cylinder of the surrounding liquid, of uniform pressure in planes perpendicular to the axis, being zero, the resultant thrust on the ends must be balanced by the resultant impressed force, which must therefore act along the normal PR to the surface of equal pressure.

SURFACES OF EQUAL PRESSURE.

487

Denoting this force per unit volume by F, and the element PR of the normal by Av,

or the resultant force per unit volume is the space variation, or gradient, of the pressure p in its direction.

This is true also for any other direction PQ, making an angle with the normal PR; for if it meets the consecutive surface QQ' in Q, and PQ=As,

so that "the component force in any direction is the space variation of the pressure in that direction; and the resultant force is the greatest space variation, and therefore normal to the surface of equal pressure."

Thus if the force F has components X, Y, Z parallel to three fixed rectangular axes Ox, Oy, Oz,

and the lines of force must be capable of being cut orthogonally by a system of surfaces, of equal pressure.

390. The impressed forces of gravity and inertia (but not of electricity or magnetism) are proportional at any point to the density p; so that it is usual to multiply F by p, and thus measure F per unit of mass, lb or g, instead of per unit volume, ft3 or cm3; and now we write dp=p(Xd+Ydy+Zdz).

(4)