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is a complete differential of a function , of the three variable quantities x, y, z; so that
sdx+udy + vdz = dp.
259. If in the equation (57) the variations which are arbitrary, be made equal to the differentials of the same quantities; and if, as in nature, the accelerating forces X, Y, Z be functions of the distance, then Xdr + Ydy + Zdz
will be a complete differential, and may be expressed by dV, so that the equation in question becomes
But the function gives the velocities of the fluid mass in the direc
By the substitution of these values in equation (62), ds, du, dv, and
will be obtained in functions of 4, by which the preceding equation
The constant quantity introduced by integration is included in the function . By the same substitution, the equation of continuity
The two last equations determine the motion of the fluid mass in the
case under consideration.
260. It is impossible to know all the cases in which the function sdx+udy+vdz is an exact differential, but it may be proved that
if it be so at any one instant, it will be an exact differential during the whole motion of a fluid. Demonstration.-Suppose that at any one instant it is a complete differential, it will then be integrable, and may be expressed by do; in the following instant it will become
Now the latter quantity being equal to d. d,
equation (67) gives
And if the density p be a function of p the pressure, the second member of this equation will be an exact differential, consequently the first member will be one also, and thus the function
sdx + udy + vdz
is a complete differential in the second instant, if it be one in the first; it will therefore be a complete differential during the whole motion of the fluid.
Theory of small Undulations of Fluids.
261. If the oscillations of a fluid be very small, the squares and products of the velocities s, u, v, may be neglected: then the preeeding equation becomes
If p be a function of p, the first member will be a complete differential, therefore the second member, and consequently
sdx+udy + vdz
is one also, so that the equation is capable of integration; and as
its last member is equal to d. the integral is
v - ƒ
αφ p dt
This equation, together with equation (68) of continuity, contain
the whole theory of the small undulations of fluids.
262. An idea may be formed of these undulations by the effect
of a stone dropped into still water; a series of small concentric circular waves will appear, extending from the point where the stone fell. If another stone be let fall very near the point where the first fell, a second series of concentric circular waves will be produced; but when the two series of undulations meet, they will cross, each series continuing its course independently of the other, the circles cutting each other in opposite points. An infinite number of such undulations may exist without disturbing the progress of one another. In sound, which is occasioned by undulations in the air, a similar effect is produced: in a chorus, the melody of one voice may be distinguished from the general harmony. Coexisting vibrations may also be excited in solid bodies, each undulation having its perfect effect, independently of the others. If the directions of the undulations coincide, their joint motions will be the sum or the difference of the separate motions, according as similar or dissimilar parts of the undulations are coincident. In undulations of equal frequency, when two series exactly coincide in point of time, the united velocity of the particular motions will be the greatest or least ;-and if the undulations are of equal strength, they will totally destroy each other, when the time of the greatest direct motion of one undulation coincides with that of the greatest retrograde motion of the other.
The general principle of Interferences was first shown by Dr. Young to be applicable to all vibratory motions, which he illustrated beautifully by the remarkable phenomena of two rays of light producing darkness, and the concurrence of two musical sounds producing silence. The first may be seen by looking at the flame of a candle through two extremely narrow parallel slits in a card; and the latter is rendered evident by what are termed beats in music.
The same principle serves to explain why neither flood nor ebb tides take place at Batsham in Tonquin on the day following the moon's passage across the equator; the flood tide arrives by one channel at the same instant that the ebb arrives by another, so that the interfering waves destroy each other.
Co-existing vibrations show the comprehensive nature and elegance of analytical formula. The general equation of small undulations is the sum of an infinite number of equations, each of which gives a single series of undulations, like the surface of water in a shower, which at once contains an infinite number of undulations, and yet exhibits each independently of the rest.
Rotation of a homogeneous Fluid.
263. If a fluid mass rotates uniformly about an axis, its component velocity in the axis of rotation is zero; the velocities in the other two axes are angular velocities-independent of the time, the motion being uniform: indeed, the motion is the same with that of a solid body revolving about a fixed axis. If the mass revolves about the axis z, and if w be the angular velocity at the distance of unity from that axis, the component velocities will be
S = - wy, u = wx, v = 0;
and from equations (63) it will be easily found that
The equation (65) of continuity will be satisfied, since
264. This motion of a fluid mass is therefore possible, although it
is a case in which the function
is not an exact differential; for by the substitution of the preceding values of the velocities, it becomes
sdx + udy + vdz = w (xdy — ydx),
an expression that cannot be integrated. Therefore, in the theory of the tides caused by the disturbing action of the sun and moon on the ocean, the function
sdx+udy + vdx
must not be regarded as an exact differential, since it cannot be integrated even when there is no disturbance in its rotatory motion. 265. Thus a fluid mass or a fluid covering a solid of any form whatever, will rotate about an axis without alteration in the relative position of its particles. This would be the state of the ocean were the earth a solitary body, moving in space; but the attractions of the sun and moon not only trouble the ocean, but also cause commotions in the atmosphere, indicated by the periodic
variations in the heights of the mercury in the barometer. From the vast distance of the sun and moon, their action upon the fluid particles of the ocean and atmosphere, is very small in comparison of that produced by the velocity of the earth's rotation, and by its
Determination of the Oscillations of a homogeneous Fluid covering a Spheroid, the whole in rotation about an axis; supposing the Auid to be slightly deranged from its state of equilibrium by the action of very small forces.
266. If the earth be supposed to rotate about its axis, uninfluenced by foreign forces, the fluids on its surface would assume a spheroidal form, from the centrifugal force induced by rotation; and a particle in the interior of the fluid would be subject to the action of gravitation and the pressure of the surrounding fluid only. But although the fluids would be moving with great velocity, yet to us they would seem at rest. When in this state the atmosphere and ocean are said to be in equilibrio.
Action of the Sun and Moon.
267. The action of the sun and moon troubles this equilibrium, and occasions tides in both fluids. The whole of this theory is perfectly general, but for the sake of illustration it will be considered with regard to the ocean. If the moon attracted the centre of gravity of the earth and all its particles with equal and parallel forces, the whole system of the earth and the waters that cover it, would yield to these forces with a common motion, and the equilibrium of the seas would remain undisturbed. The difference of the intensity and direction of the forces alone, trouble the equilibrium; for, since the attraction of the moon is inversely as the square of the distance, a molecule at m, under the moon M, is as much more attracted than the centre of gravity of the earth, as the square of EM is greater than the square of mM : hence the particle has a tendency to leave the earth, but is retained by its gravitation, which this tendency diminishes. Twelve hours after, the particle is brought to m' by the rotation of the earth, and is then in