Measure of the density at any point of a heterogeneous fluid.

Let m be the mass of a volume v of fluid enclosing a given point, and suppose p the density of a homogeneous fluid such that the mass of a volume v is equal to m, or such that

m = pv ;

then p may be defined as the mean density of the portion v of the heterogeneous fluid, and the ultimate value of p when v is indefinitely diminished, supposing it always to enclose the point, is the density of the fluid at that point.

14. To find the work done in compressing a gas.

Letv be the volume of a gas at the pressure p, ds an element of the surface of the vessel containing it, and dn an element of the normal to ds drawn inwards.

Then the work done in a small compression
=pΣdsdn = pdv,

and the work done in compressing from V to V

- [pdv = - [Cdv, if pv = C,

(In these Examples g is taken to be 32, when a foot and a second are units.)

1. ABCD is a rectangular area subject to fluid pressure; AB is a fixed line, and the pressure on the area is a given function (P) of the length BC (x); prove that the pressure at any point of CD is where a = AB.

dP adx

If A be a fixed point, and AB, AD fixed in direction, and

if AB = x and AD=y, the pressure at C=

dxdy

2. In the equation W=gp V, if the unit of force be 100 lbs. weight, the unit of length 2 feet, and the unit of time 4th of a second, find the density of water.

3. If a minute be the unit of time, and a yard the unit of space, and if 15 cubic inches of the standard substance contain 25 oz., determine the unit of force.

4. In the equation, W=gp V, the number of seconds in the unit of time is equal to the number of feet in the unit of length, the unit of force is 750 lbs. weight, and a cubic foot of the standard substance contains 13500 ounces; find the unit of time.

5. A velocity of 4 feet per second is the unit of velocity; water is the standard substance and the unit of force is 125 lbs. weight; find the units of time and length.

6. The number expressing the weight of a cubic foot of water is th of that expressing its volume, th of that expressing its mass, andth of the number expressing the work done in lifting it I foot. Find the units of length, mass, and time.

7. If the pressure of the atmosphere be the unit of pressure, the velocity of sound the unit of velocity, and the acceleration due to gravity the unit of acceleration, find roughly the unit of force.

8. If a feet and b seconds be the units of space and time, and the density of water the standard density, find the relation between a and b in order that the equation, W=gpV, may give the weight of a substance in pounds.

9. A velocity of 8 feet per second is the unit of velocity, the unit of acceleration is that of a falling body, and the unit of mass is a ton; find the density of water.

10. The density at any point of a liquid, contained in a cone having its axis vertical and vertex downwards, is greater than the density at the surface by a quantity varying as the depth of the point. Shew that the density of the liquid when mixed up so as to be uniform will be that of the liquid originally at the depth of one-fourth of the axis of the cone.

11. From a vessel full of liquid of density p is removed 1/nth of the contents, and it is filled up with liquid of density If this operation be repeated m times, find the resulting density in the vessel.

σ.

Deduce the density in a vessel of volume V, originally filled with liquid of density p, after a volume U of liquid of density o has dripped into it by infinitesimal drops.

12. The density of a fluid varies from point to point; considering directions proceeding from a given point, prove that the density varies most rapidly along the normal to the surface of equal density containing the point; and of directions in the tangent plane to this surface, the tangents to its principal sections are those in which the rate of variation of density is greatest and least.

CHAPTER II.

THE CONDITIONS OF THE EQUILIBRIUM OF FLUIDS.

15. TAKING the most general case, suppose a mass of fluid, elastic or non-elastic, homogeneous or heterogeneous, to be at rest under the action of given forces, and let it be required to determine the conditions of equilibrium, and the pressure at any point.

Let x, y, z be the co-ordinates referred to rectangular axes, of any point P in the fluid, and let Q be a point near it, so taken that PQ is parallel to the axis of x.

Take x + dx, y, z, as the co-ordinates of Q; about PQ describe a small prism or cylinder bounded by planes perpendicular to PQ.

Let a be the area of the section of the cylinder perpendicular to its axis, p the pressure at P, and p+ dp the pressure at Q.

Then, a being very small, the pressure at any point of the plane P will be very nearly equal to p, and the pressure upon it will therefore be

(p + y) a,

where y vanishes in comparison with p when a is indefinitely diminished.

We can therefore consider a so small that y may be neglected in comparison with p, and the pressure on the end P of the cylinder may be taken equal to pa, and similarly the pressure on the end Q equal to

(p + Sp) a.

If ρ be the mean density of the cylinder PQ, its mass = ραδι, and Χραδα will represent the force on PQ parallel

to its axis, if Xôm, Yồm, Zồm be the components of the forces acting on a particle dm of fluid at the point xyz.

or

Hence, for the equilibrium of PQ,

(p + Sp) a − pa= Xpadx,

δρ = ρΧδα.

Proceeding to the limit when da, and therefore Sp, is indefinitely diminished, p will be the density at P, and we

dp dx+
dx dy

dp

dp dy

dz;

+

dz

..(a),

dp=p(Xd + Ydy+Zdz).....

the equation which determines the pressure.

16. The pressure is clearly a function of the independent variables x, y, and z, and we know that

dp d2p d2p d2p d2p d2p

=

=

dydz dzdy' dzdx dxdz' dxdy ̄ ̄ dydx Hence we obtain from the preceding equations,

* In the above proof, a is taken so small that its linear dimensions may be neglected in comparison with dx; that is, the change in p, corresponding to a change dx in x, is considered, undisturbed by any alterations in y and z.