241. It appears from the preceding investigation, that a homogeneous liquid will remain in equilibrio, if all its particles act on each other, and are attracted towards any number of fixed centres; but in that case, the resulting force must be perpendicular to the surface of the liquid, and must tend to its interior. If there be but one force or attraction directed to a fixed point, the mass would become a sphere, having that point in its centre, whatever the law of the force might be.

242. When the centre of the attractive force is at an infinite distance, its direction becomes parallel throughout the whole extent of the fluid mass; and the surface, when in equilibrio, is a plane perpendicular to the direction of the force. The surface of a small extent of stagnant water may be estimated plane, but when it is of great extent, its surface exhibits the curvature of the earth.

243. A fluid mass that is not homogeneous but free at its surface will be in equilibrio, if the density be uniform throughout each indefinitely small layer or stratum of the mass, and if the resultant of all the accelerating forces acting on the surface be perpendicular to it, and tending towards the interior. If the upper strata of the fluid be most dense, the equilibrium will be unstable; if the heaviest is undermost, it will be stable.

244. If a fixed solid of any form be covered by fluid as the earth is by the atmosphere, it is requisite for the equilibrium of the fluid that the intensity of the attractive forces should depend on their distances from fixed centres, and that the resulting force of all the forces which act at the exterior surface should be perpendicular to it, and directed towards the interior.

245. If the surface of an elastic fluid be free, the pressure cannot be zero till the density be zero; hence an elastic fluid cannot be in equilibrio unless it be either shut up in a close vessel, or, like the atmosphere, it extend in space till its density becomes insensible.

Equilibrium of Fluids in Rotation.

246. Hitherto the fluid mass has been considered to be at rest; but suppose it to have a uniform motion of rotation about a fixed axis, as for example the axis oz. Let w be the velocity of rotation common to all the particles of the fluid, and r the distance of a par

ticle dm from the axis of rotation, the co-ordinates of dm being x, y, z. Then wr will be the velocity of dm, and its centrifugal force resulting from rotation, will be wr, which must therefore be added to the accelerating forces which urge the particle; hence equation (53) will become

dp = Xdr + Ydy + Zdz + w2rdr.

And the differential equation of the strata, and of the free surface of the fluid, will be

Xdx + Ydy + Zdz + w2. rdr = 0.

(55)

from

The centrifugal force, therefore, does not prevent the function being an exact differential, consequently equilibrium will be possible, provided the condition of article 238 be fulfilled.

247. The regularity of gravitation at the surface of the earth; the increase of density towards its centre; and, above all, the correspondence of the form of the earth and planets with that of a fluid mass in rotation, have led to the supposition that these bodies may have been originally fluid, and that their parts, in consolidating, have retained nearly the form they would have acquired from their mutual attractions, together with the centrifugal force induced by rotation when fluid. In this case, the laws expressed by the preceding equations must have regulated their formation.

CHAPTER VII.

MOTION OF FLUIDS.

General Equation of the Motion of Fluids.

248. THE mass of a fluid particle being p dr dy dz, its momentum in the axis x arising from the accelerating forces is, by article 144,

Consequently the equation of the motion of a fluid mass in the axis or, when free, is

dp

= 0.

dx

In the same manner its motions in the axes y and z are

(56)

And by the principle of virtual velocities the general equation of fluids

This equation is not rigorously true, because it is formed in the hypothesis of the pressures being equal on all sides of a particle in motion, which Poisson has proved not to be the case; but, as far as concerns the following analysis, the effect of the inequality of pressure is insensible.

249. The preceding equation, however, does not express all the circumstances of the motion of a fluid. Another equation is requisite. A solid always preserves the same form whatever its motion may be, which is by no means the case with fluids; for a mass

ABCD, fig. 57, formed of particles possessing perfect mobility, changes its form by the action of the forces, so that it always continues to fit into the intervals of the surrounding molecules without leaving any void. In this consists the continuity of fluids, a property which furnishes the other equation necessary for the determination of their motions.

Equation of Continuity.

250. Suppose at any given time the form of a very small fluid mass to be that of a rectangular parallelopiped A B C D, fig. 57. The action of the forces will change it into an oblique angled figure N E F K, during the indefinitely small time that it moves from its first to its second position. Now NE FG may differ from A B C D both in form and density, but not in mass; for if the density depends on the pressure, the same forces that change the form may also produce a fig. 57.

Ꭰ

N

F

G H

K

change in the pressure, and, consequently, in the density; but it is evident that the mass must always remain the same, for the number of molecules in ABCD can neither be increased

nor diminished by the action of the

forces; hence the volume of A B C D into its primitive density must still be equal to volume of N E F G into the new density; hence, if

p dx dy dz,
P

be the mass of A B C D, the equation of continuity will be

251. This equation, together with equations (56), will determine the four unknown quantities x, y, z, and p, in functions of the time, and consequently the motion of the fluid.

Developement of the Equation of Continuity.

252. The sides of the small parallelopiped, after the time dt, become drd.dx, dy + d.dy, dz + d.dz;

or, observing that the variation of dx only arises from the increase of x, the co-ordinates y and z remaining the same, and that the variations of dy, dz, arise only from the similar increments of y and z ;

If the angles GNF and FNE, fig. 58, be represented by andy, the volume of the parallelopiped NK will be NE. NG sin 0. NF sin ;

F

Ꭱ

K

Fa, Nb being at right angles to NE and RG;

but as 0 and were right angles in the primitive volume, they could only vary by indefinitely small arcs in the time dt; hence in the new volume

and the volume becomes NE. NG. NF; substituting for the three edges their preceding values, and omitting indefinitely small quantities of the fifth order, the volume after the time dt is

The density varies both with the time and space; hence p, the primi

consequently, the mass, being the product of the volume and density,

is, after the time dt, equal to

dp_dz

dm=p. dx dy dz (1+ d dt + dL dx + d dy + dp dz

dt

dx

dy d2z

dy

dz