But, from (1), p denoting the atmospheric pressure, there is, at the surface, u being neglected as small,

p = p({ w2y2 - gx) + C,

or, by (2),

--

p pgz + C:

and, v denoting the velocity of efflux, we have, putting x = 0, y = 0,

p = − 1 pv2 + C:

from the two last equations we see that

v2 = 2gz.

Hence, putting the value of v here given in the equation (3),

and therefore, integrating from za to z = 0, we obtain for the whole time T of efflux,

Again, the time 7" of efflux, when the fluid is devoid of rotation,

2. A given quantity of fluid in a vertical cylinder revolves about the axis of the cylinder with a given angular velocity in a form of equilibrium: supposing a small orifice to be made in

the side of the vessel, to determine the interval which will elapse before the lowest point of the revolving fluid descends to the level of the orifice.

Let a denote the altitude of any molecule of the fluid above the orifice, and y the distance of this molecule from the axis of the cylinder. Then, u denoting the component of the velocity of this molecule which is not due to the rotation, and w the angular velocity of the revolution of the fluid,

p = p({ w3y2 - gx) - } pu2 + C. . . . . . . . . .(1).

At the surface, u may be regarded as of inconsiderable magnitude, and accordingly, II denoting the atmospheric pressure, II = p (} w3y2 – gx) + C.

If z be the value of x at the lowest point of the surface of the fluid after any time t from the commencement of the efflux,

and therefore we have for the equation to the generating parabola of the surface of the fluid,

If v denote the velocity of the issuing fluid, we have, by (1), a denoting the radius of the cylinder,

If A denote the volume of the fluid in the vessel at the time t, above the horizontal plane through the orifice, then, observing that the volume of a paraboloid of revolution is equal to half that of its circumscribing cylinder,

z' denoting the value of x in the equation (3) when y = a; and therefore, by (3),

and accordingly, z' denoting the initial value of z, the required time will be equal to

Let h denote the length of the portion of the cylinder of which the volume is equal to that of the initial quantity of the fluid.

and therefore the expression for the required time is equal to

πα

kg

{(2gh + { w3a3)* − wa}.

Moseley's Hydrostatics, p. 154.

3. To find the time in which a vertical prismatic vessel, full of fluid, will empty itself through a small orifice in its base into a vacuum, its upper surface being exposed to the pressure of the atmosphere.

Let K represent the area of a transverse section of the vessel, h the altitude of the vessel, and h' the altitude of a column of the fluid the weight of which is equal to the atmospheric pressure on an area equal to the base of the column.

Then, x denoting the altitude of any horizontal section of the fluid after any time t above the base of the vessel, and u the velocity of the section, we have

Hence, denoting the velocity at the surface, we have, putting p = gph', and taking x to represent the depth of the fluid,

gph' =-gpx-pV2 + C:

at the orifice, v representing the velocity of the issuing fluid,

and therefore the whole time of efflux will be equal to

Finite Horizontal Orifices. Uniform Motion.

If a vessel, with a finite horizontal aperture, be continually supplied with fluid, so that its surface remains stationary, the velocity of the fluid will be constantly the same at each point of the vessel, and, accordingly, the equation of steady motion will be applicable.

1. Fluid is supplied with a given uniform velocity to a vessel in the form of a truncated paraboloid with its axis vertical: to determine the position of the surface of the fluid that it may remain stationary.

If a denote the distance of the aperture, and x of the surface, from the vertex of the paraboloid, I the volume of the influx of fluid in a unit of time, and 7 the latus rectum,

2. To determine where a semi-ellipsoidal vessel must be truncated by a horizontal plane parallel to the area of its rim, a principal section of the ellipsoid, that, the vessel being kept constantly full, the efflux in a given time may be the greatest possible.

If c denote the vertical semi-axis of the ellipsoid, and z the distance of the required section from the plane of the rim of the vessel;

2a - 3c2x2 - 2ca:

=

0.

Moseley's Hydrostatics, p. 165.

3. To determine, under the circumstances of the preceding problem, the position of the section that the velocity of efflux may be the greatest possible.

The value of z is given by the equation

z2 = 3 c2.

Moseley's Hydrostatics, p. 166.

SECTION IV.

Finite Vertical Orifices. Uniform Motion.

Suppose a vessel, in a side of which there is a vertical aperture of finite dimensions, to be supplied with fluid at such a rate that its surface may remain stationary. Let A represent the area of an element of the aperture, v the velocity of the fluid issuing through this element, K the area of the surface,