if Xom, Yom, Zom, be the components of the frees etme c a particle èm of fluid at the point

Hence for the equilibrium of Pg. 1 necessary ondition:

Proceeding to the limit when dr. and therefore definitely diminished, p will be he iensity at 2.ada

from which by differentiating, multiplying the equations respectively by Z, X, and Y, and adding, we obtain

a necessary although not a sufficient condition of equilibrium.

18. If the forces tend to fixed centres and are functions of the distances from those centres, we have

when (abc) are co-ordinates of the centre to which the force (r) tends.

it is obvious that the equation (y) is always satisfied, but it is not to be inferred that the equilibrium of a heterogeneous fluid is always possible with such a system of forces.

When the density is constant, the equations (B) become

which are in this case always satisfied, and therefore the equilibrium of a homogeneous fluid under the action of such forces is always possible.

19. If the fluid be elastic, an additional condition is introduced, for if the temperature be constant,

When the forces tend to fixed centres and are functions of the distances, Art. (18), this equation takes the form.

If the temperature be variable, the relation between the pressure density and temperature is found to be

p = kp (1+ at),

where t is the temperature, and a =·003665.

and therefore t must be a function of x, y, and z.

20. From equation (a), if the fluid be homogeneous and Xdx +Y dy+Zid=dV,

p=pV+ C.

If the fluid be heterogeneous and p a function of x, y, and z, such that

p(Xd+Ydy + Zdz) = dU,
p=U+C,

and from (8) if there be equilibrium

In any of these cases, if the pressure at any particular point be given the constant can be determined.

In the last case if the mass of fluid, and the space within which it is contained be given, the constant is determined.

21. In all cases, in which the equilibrium of the fluid is possible we obtain by integration

p = (x, y, z)

If p be constant and equal to p',

$ (x, y, z) = p'....

.....(A),

is the equation to the surface at all points of which the pressure is constant, and by giving different values to p' we obtain a series of surfaces of equal pressure, and the external surface, or free surface, is obtained by making p' equal to the pressure external to the fluid.

If the external pressure be zero the free surface is therefore

which are proportional to the direction cosines of the normal at the point (x, y, z) of the surface A, are equal to

dp, dp dp

dx' dy' dz'

respectively, i.e. to pX, pY, pZ, and are therefore proportional to X, Y, Z.

Hence the resultant force at any point is in direction of the normal to the surface of equal pressure passing through the point*.

*This last result may also be obtained in the following manner:

Consider two consecutive surfaces of equal pressure, containing between them a stratum of fluid, and let a small circle be described about a point P in one surface, and a portion of the fluid cut out by normals through the circumference. The portion of fluid so cut out may be considered rigid, and kept at rest by the impressed force, and the pressures on its ends and on its circumference. Being very nearly a small cylinder, and the pressures on all points of its circumference equal, the difference of pressures on its two faces must be due to the force, which must therefore act in the same direction as these pressures, i. e. in direction of the normal at P.

23. In the particular cases in which Xdx + Ydy + Zdz is a perfect differential dV, p must be a function of V.

For dp = pd V

and dp being a perfect differential, p must be a function of V;

let p=f' (V)
=ƒ'

then dp=f' (V) dV,

p=ƒ(V) + C.

Hence V, and therefore p is a function of p, and surfaces of equal pressure are also surfaces of equal density.

If the fluid be elastic and the temperature variable

Hence by a similar process of reasoning t is a function of p, and surfaces of equal pressure are also surfaces of equal temperature.

24. If however Xdx+ Ydy + Zdz be not a perfect differential these surfaces will not in general coincide.

1st. Let the fluid be heterogeneous and incompressible; then the surfaces of equal pressure and of equal density are given respectively by the equations

These then are the differential equations of surfaces which by their intersections determine curves of equal pressure and density.