opposition to the moon, which attracts it more feebly than it attracts the centre of the earth, in the ratio of the square of EM to the square of m'M. The surface of the earth has then a tendency to leave the particle, but the gravitation of the particle retains it; and gravitation is also in this case diminished by the action of the moon. Hence, when the particle is at m, the moon draws the particle from the earth; and when it is at m', it draws the earth from the particle in both instances producing an elevation of the particle above the surface of equilibrium of nearly the same height, for the diminution of the gravitation in each position is almost the same on account of the distance of the moon being great in comparison of the radius of the earth. The action of the moon on a particle at n, 90° distant from m, may be resolved into two forces-one in the direction of the radius nE, and the other tangent to the surface. The latter force alone attracts the particle towards the moon, and makes it slide along the surface; so that there is a depression of the water in n and n', at the same time that it is high water at m and m'. It is evident that, after half a day, the particle, when at n', will be acted on by the same force it experienced at n.

268. Were the earth entirely covered by the sea, the water thus attracted by the moon would assume the form of an oblong spheroid, whose greater axis would point towards the moon; since the column of water under the moon, and the direction diametrically opposite to her, would be rendered lighter in consequence of the diminution of their gravitation: and in order to preserve the equilibrium, the axis 90° distant would be shortened. The elevation, on account of the

smaller space to which it is confined, is twice as great as the depression, because the contents of the spheroid always remain the same. If the waters were capable of instantly assuming the form of a spheroid, its summit would always be directed towards the moon, notwithstanding the earth's rotation; but on account of their resistance, the rapid motion of rotation prevents them from assuming at every instant the form which the equilibrium of the forces acting on them requires, so that they are constantly approaching to, and receding from that figure, which is therefore called the momentary equilibrium of the fluid. It is evident that the action, and consequently the position of the sun modifies these circumstances, but the action of that body is incomparably less than that of the

moon.

Determination of the general Equation of the Oscillations of all parts of the Fluids covering the Earth.

269. Let pEPQ, fig. 60, be the terrestrial spheroid, Eo the equatorial radius, Pp the axis of rotation. Suppose the spheroid to be entirely covered with the fluid-the

fig. 60.

E

C

n

P D

ocean, for example; and let peP, or pEP, represent the bottom of the sea, CD its surface, PD its depth; also let o be the centre of the spheroid and origin of the co-ordinates, and om the radius. Imagine m to be a fluid particle at any point below the surface of the It is evident that this particle, moved by rotation alone, would be carried to B without changing its distance from the centre of the spheroid, or from the axis of rotation; so that the arcs Pm, PB, are equal to each other, as also the radii om, oB. If peP be assumed as a given meridian, the origin of the time, and y the first point of Aries, then yPB is the longitude of the particle when arrived at B, and EoB is its latitude.

fluid-at the bottom, for example.

Now, if the disturbing forces were to act on the particle during its rotation from m to B, they would cause it to move to b, some point not far from B. By the disturbing forces alone, the longitude of the particle at B would be increased by the very small angle BPb; the latitude would be diminished by the very small angle Bob, and its distance from the centre of the spheroid increased by fb. The angle PB is the rotation of the earth, and any may be represented by nt + w, since it is proportional to the time, (by Article 213;) but in the time t, the disturbing forces bring the particle to b: therefore the angle nt must be increased by BPb or v. Hence

7Pb = nt + w + v.

Again, if be the complement of the latitude EoB, and u, its very small increment, Bob, the angle

POB + U.

In the same manner, if s be the increment of the radius r, then

obr+s.

Hence the co-ordinates of the particle at b are,

x = (r+s) cos (0 + u),

y = (r + s) sin (0 + u) cos (nt + w + v),

z (r+s) sin (0+ u) sin (nt + w + v).

270. v and u very nearly represent the motion of the particle in longitude and latitude estimated from the terrestrial meridian PEp. These are so small, compared with nt the rotatory motion of the earth, that their squares may be omitted. But although the lateral motions v, u of the particle be very small, they are much greater than s, the increase in the length of the radius.

271. If these values of x, y, z, be substituted in (57) the general equation of the motion of fluids; and if to abridge

will determine the oscillations of a particle in the interior of the fluid when troubled by the action of the sun and moon. This equation, however, requires modification for a particle at the surface.

Equation at the Surface.

272. If DH, fig. 60, be the surface of the sea undisturbed in its rotation, the particle at B will only be affected by gravitation and the pressure of the surrounding fluid; but when by the action of the sun and moon the tide rises to y, and the particle under consideration is brought to b, the forces which there act upon it will be gravitation, the pressure of the surrounding fluid, the action of the sun and moon, and the pressure of the small column of water between H and y. But the gravitation acting on the particle at b is also different from that which affects it when at B, for b being farther from the centre of gravity of the system of the earth and its fluids, the gravity of the particle at b must be less than at B, consequently the centrifugal force

K

must be greater: the direction of gravitation is also different at the points B and b.

273. In order to obtain an equation for the motion of a particle at the surface of the fluid, suppose it to be in a state of momentary equilibrium, then as the differentials of v, u, and s express the oscillations of the fluid about that state, they must be zero, which reduces the preceding equation to

n2

12 ♪{ (r + s) sin (0 + u)}a + (8D) = 0;

(71)

for as the pressure at the surface is zero, dp = 0, and (♪V) represents the value of V corresponding to that state. Thus in a state of momentary equilibrium, the forces (V), and the centrifugal force balance each other.

274. Now is the sum of all the forces acting on the particle when troubled in its rotation into the elements of their directions, it must therefore be equal to (SV), the same sum suited to a state of momentary equilibrium, together with those forces which urge the particle when it oscillates about that state, into the elements of their directions. But these are evidently the variation in the weight of the little column of water Hy, and a quantity which may be represented by V', depending on the difference in the direction and intensity of gravity at the two points B and b, caused by the change in the situation of the attracting mass in the state of motion, and by the attraction of the sun and moon.

275. The force of gravity at y is so nearly the same with that at the surface of the earth, that the difference may be neglected; and if y be the height of the little column of fluid Hy, its weight will be gy when the sea is in a state of momentary equilibrium; when it oscillates about that state, the variation in its weight will be gdy, g being the force of gravity; but as the pressure of this small column is directed towards the origin of the co-ordinates and tends to lessen them, it must have a negative sign. Hence in a state of motion,

whence

SV = (SV) + SV' — gdy,

(SV) = SV — SV' + gồy.

276. When the fluid is in momentary equilibrio, the centrifugal force is

but it must vary with dy, the elevation of the particle above the surface of momentary equilibrium. The direction Hy does not coincide with that of the terrestrial radius, except at the equator and pole, on account of the spheroidal form of the earth; but as the compression of the earth is very small, these directions may be esteemed the same in the present case without sensible error; therefore r+s- y may be regarded as the value of the radius at y. Consequently

-dy.rn sin

is the variation of the centrifugal force corresponding to the increased height of the particle; and when compared with - gdy the gravity

of this little column, it is of the order

ner

19

g

the same with the ratio of

the centrifugal force to gravity at the equator, or to

fore may be omitted; hence equation (71) becomes

no 2

dV − ¿V' + gdy + 122 ♪ { (r + 8) sin (0 + u) }2 = 0.

277. As the surface of the sea differs very little from that of a sphere, dr may be omitted; consequently

if

n2

♪{(r+0) sin (0 + u)}2

2

be eliminated from equation (70), the result will be

which is the equation of the motion of a particle at the surface of the sea. The variations dy, V' correspond to the two variables 0 and ☎. 278. To complete the theory of the motions of the atmosphere and ocean, the equation of the continuity of the fluid must now be found.

Continuity of Fluids.

Suppose m'h, fig. 61, to be an indefinitely small rectangular portion of the fluid mass, situate at B, fig. 60, and suppose the solid to be formed by the imaginary rotation of the area Bnhh' about the axis oz; the centre