21

3ra Ag

2a

× {√/2ar — lxa × (arx — — ~ + 2a ;)

x2 3

See Hirsch's Integral Tables, pp. 130, and 122.

; then the time of emptying the whole cone is

701.

Let the sluice be (as sluices usually are,) rectangular, two of its sides, (a) being parallel to the surface of the fluid. Also, let H, and h be the distances of these sides from the surface of the fluid, and supposing this finite orifice a x (H− h) to be composed of innumerable indefinitely small ones, a x dx, distant from the surface by x, the quantity of fluid run out in the time t will be denoted by

on the supposition that the velocity through each small orifice is due to half its depth in the fluid.

Now the reservoir being supplied at a given rate, suppose the supply to be the given quantity Q in the given time t; then H being also known by the question, we have

702.

Let a be the axis of the paraboloid, ẞ the radius of its base, r the distance of any horizontal section from the vertex; then

X = πι = πρα

p being the parameter of the generating parabola. Hence (see 696)

703.

3

1

1:1

Let a be the common altitude of the cylinder and cone, and r the radius of their common base; then since (696)

704.

Let a be the altitude of the cylinder, 2r the diameter of its base, and A the area of the aperture. Also let a be any variable height of the surface of the water; then the quantity discharged in 1" is

Agx

and, if the constant supply in 1" be Agm then the rate of eva

Also by the evacuation the surface descends with the velocity

and by means of the supply it ascends at the same time with the velocity

so that upon the whole, the descending velocity is

2πr2

A g

× {√ x + √ ml. (√ x − √ m) + C }

and putting xa and x = m, and taking the difference between the results, we have the time in which the influx is equal to the efflux, viz.

which, being infinite, shews that there can never exist an equality between the efflux and influx.

705. Let ry be any circular section of the vessel whose altitude is x; and A the area of the orifice, then the velocity at the orifice is

the equation to a parabola of the fourth order.

If a be the given altitude, then the radius of the base is

信

√ga }

Сп

which gives the required dimensions of the vessel.

706.

Let r be the radius of the sphere, and x the altitude of any common section parallel to the horizon of the sphere and cylinder; then (see 696)

X = r2, and X' = xу2 = π (2гx — x2)

707.

Here the area of the descending surface is (2y being

a side of the generating square when at a distance x)

from the equation to the semicubical parabola. Hence (see 696)

708.

Let r be the radius of the sphere, and x the altitude

of any horizontal section made by the water; then (696)

and X' y = n(r2 — x′2)

according as the vertex or base is downwards.

:. t ; t' :: f(2r — x)dx :

::

4r

3

2

Let xx2r; then

tt: S: 3.

709.

Let a be the axis, and 26 the base of either parabola, and supposing them to consist of an indefinite number of orifices 2yx dx, (y being the ordinate, and a the abscissa of the parabola measured from the surface of the fluid,) the quantity of water discharged in the time t, when the vertex is upwards is Q = Szydx x x velocity at the orifice 2ydx

Again, the quantity discharged, when the vertex is downwards in the same time is