on the whole area of the cycloid with that on a circular area of which the axis of the cycloid is a diameter. If P denote the pressure on the cycloidal and Q on the circular

P: Q:: 7 : 2.

area, then

:

a

25. An elliptic area is placed with its major axis vertical, and its vertex is the surface of a fluid; to compare the pressure of the fluid on the area included by the evolute of the ellipse with

pressure on the whole elliptic area. If e be the eccentricity of the ellipse, the ratio of the pressure

е on the area of the evolute to that on the area of the ellipse is equal to

3

e

8

1

26. To find the whole pressure on the surface of a solid cone, including its circular base, when it is immersed in a fluid with its axis vertical and its vertex just at the surface.

If r be the radius of the base, h the length of the axis, and l the distance of each point in the periphery of the base from the vertex, the required pressure will be equal to

žagphr (21 + 3r). 27. A solid hemisphere is immersed in a fluid with its axis inclined at an angle @ to the vertical, the surface of the fluid being a tangent plane to the hemisphere; to find the whole pressure on the convex surface of the hemisphere.

If a denote the radius of the hemisphere, the required pressure will be equal to

agpa(2 - cos 0).

παρα (2 28. To divide a hollow sphere just filled with fluid by a circle parallel to the horizon into two parts which shall be equally pressed

Let 0 represent the inclination of a radius of the sphere passing through the circumference of the required circle to a line drawn vertically upwards from the centre of the sphere; then

a

a

29. A hollow segment of a sphere rests on its base on an inclined plane; supposing it to be just filled with fluid, to find the pressure on the spherical surface.

If a denote the inclination of the plane to the horizon, ß the angle subtended at the centre of the sphere by a diameter of the base of the segment, and r the radius of the sphere, the required pressure will be equal to

B

?

2

4mgpr" sino (1 - cos a cos'

cos)

3).

4

4

30. The axis of a given hollow cone filled with fluid is inclined at a given angle to the horizon; to find how much of the fluid will flow out, and to determine the pressure exerted by the remainder upon the conical surface.

Let 2a = the vertical angle of the cone, ß = the inclination of its axis to the horizon, a = the radius of its base; then the volume of the fluid discharged will be equal to

and the required pressure will be equal to

(sin (B – a) žagpa'.

sina sin ( + a)

COS a.

sin ß

SECTION II.

Normal Pressure of Heterogeneous Incompressible Fluids on the Surfaces

of Immersed Solids.

Let K denote any indefinitely small element of the surface of a solid immersed in a heterogeneous fluid, and let p denote the unit of pressure of the fluid at the area K; then pK will represent the pressure of the fluid on K, and the magnitude of the pressure on the whole surface of the solid will be equal to

Σ (Κρ),

where & denotes the summation of a series of terms, of which Kp is the general type, the limits of the summation being defined by the boundary of the immersed surface. The value of p will be given by the formula

p = (gp dx) = g(pdx), p denoting the density of the fluid at a depth x below the surface of the fluid, the summation being performed with respect to x from x

0 to X = X, the value of x in the second limit being the depth of K below the surface. Supposing p to be invariable, then

p = g(pdx) = gp (dx) = gpx, and

Σ(Κρ) = 9ρ Σ (Κα), and the formulæ here given will degenerate into those of the preceding section for the pressure of homogeneous fluids. If p be not continuous, we must divide the depth of K below

P the surface into a series of parts, such that p may

be continuous for each part, and must then take the sum of the values of g(pdx) for the several strata as constituting the value of p.

1. A cylinder, the axis of which is vertical, is filled with fluid, the density of which varies directly as the depth; to find the whole pressure on the concave surface of the cylinder.

Let h denote the length of the cylinder, c the circumference of a circular section, p the density at the lowest points of the fluid and p at any depth x.

Then

др
dp = gp'dx = x dx,

h
де

gpx?
р
h

2h Hence the whole pressure required is equal to

9px?

.cdx = gpch”.

2h

Encycl. Metrop. Mixed Sciences, vol. 1. p. 176. 2. A circular area is immersed vertically in fluid, its highest point just touching the surface of the fluid ; to determine the

=

h

X d.x

0

S

0

whole pressure on the area, the density of the fluid varying directly as the depth.

Let ß represent the density of the fluid at the centre of the circle; the density, at a depth x below the surface, will there

Box fore be where a is the radius of the circle. Then

Let r denote the distance of any point of the circular area from its centre, and @ the inclination of this distance to the vertical. Then, the pressure on an elemental area rdo dr being

the whole pressure on the circular area will be equal to

gß

27T

0

S.T. *r(co – 2ar cos 0 + r* cos* 6) do dr }gaB) () - } cos 0 + { cos* 6) de

27

0

ga°B (m + 1) = 8 mga*3. 3. Equal masses of n different fluids, the densities of which, beginning with the highest fluid, are Pue Pa, Pza...... Pme being placed in a cylindrical vessel the axis of which is vertical, to compare the pressures which they exert upon the side of the vessel.

Let a,, Ag, Agg... Am, be the portions of the axis of the cylinder which are occupied by the fluids of densities P1, P2, P3s. .....Pne respectively. Let 2 denote the depth of an annular strip 2ordz of the cylinder, r being its radius, below the upper surface of the wth fluid. Then, p being the unit of pressure at the annulus,

p= g(e, a, + P, Q, + Pg az + + Pa-100-1) + gpt

2:

3

3

+

1

or,

2

5

but, the masses of the several fluids being equal, Pia, = P2 a, = Pz az

• Px-1 ani hence

p = g (x - 1) P, Q, +9Pa2, and therefore, P, denoting the pressure on the ath portion of the cylinder, we have

2ar.dz.g {(x - 1) P, Q, + paz}

2tr.g {(x - 1)p, a, a, + } p. a*};
since
Pe Piazy

Pe = agrp, a, {(2x – 2) a, + ac }
= ngrp, a, (2x – 1) a.

а

2x 1 - agr. pia.

P The expression for P. shews that the pressures on the first, second, third, &c. portions of the cylinder are proportional to

1 3

7 Pi P

PA 4. To find the whole pressure on the horizontal base of a vessel full of fluid, supposing the density of the fluid to vary as the altitude above the base. If

p denote the density at the surface, h the height of the cylinder, and A the area of its base, the required pressure will be equal to

3gphA. The same result will

express

the

pressure on the base of the density be supposed to vary as the depth, and p to denote the density at the base.

Encycl. Metrop. vol. 1. Mixed Sc. p. 176.

Bossut: Traité d'Hydrodynamique, p. 39. 5. Masses of n different fluids, which do not mix, the densities of which are pu par Pg. ...pm, being placed in a vessel, to deter

P3, • mine the pressure on the base of the vessel, which is horizontal.

If A represent the area of the base, and h,, h2, h3,. ...hn, the depths of the strata, the required pressure is equal to gA (ph, + pak, + pzh, + .... tp.hu).

+ Bossut: Traité d'Hydrodynamique, tom. I. p. 37.

n

С