of gravity of Bnhh' will describe an arc, which on account of the small 0 nt +, The colatitude of the point B or Aom = 0, the longitude of B is then the indefinitely small increments of these angles are m'oK' de, m'oB do, for as the figure is independent of the time, nt is constant. Hence if the radii oB, on, be represented by r' and r, the sectors Boh', noh, are r'de and ride; hence But in consequence of the disturbing forces, r, 0, and w, become r + s, 0 + u, ☎ + v, after the time t, and dr, de, dæ, also the density is changed to p + p'. preceding expression for the solid dm, it becomes after the time t If these values be put in the but this must be equal to the original mass; hence ds dv (p+p') (r+s°) (1 + (1+ du) (1+de) sin (0+1) = pr sin 0. do for as u is very small, the arc may be put for the sine, and unity for the cosine, the equation of the continuity of the fluid is 279. The equations (70), (72), and (73), are perfectly general; and therefore will answer either for the oscillations of the ocean or atmosphere. Oscillations of the Ocean. 280. The density of the sea is constant, therefore p' = 0; hence the equation of continuity becomes In order to find the integral of this equation with regard to r only, it may be assumed, that all the particles that are on any one radius at the origin of the time, will remain on the same radius during the motion; therefore r, v, and u will be nearly the same on the small part of the terrestrial radius between the bottom and surface of the sea; hence, the integral will be (r's) is the value of rs at the surface of the sea, but the change in the radius of the earth between the bottom and surface of the sea is so small, that r(s) may be put for (r's); then dividing the whole by r2, and neglecting the terms 27(s), which is the ratio of the depth of the sea to the terrestrial radius, and therefore very small, the mean depth even of the Pacific ocean being only about four miles, whereas the mean radius of the earth is nearly 4000 miles; the preceding equation becomes Now y + s (74) (s) is the whole depth of the sea from the bottom to the highest point to which the tides rise at its surface of momentary equilibrium; and y varies with the angles surface of equilibrium, it becomes and 0; hence at the and as y is the height of a particle above the surface of equilibrium, or 8 − (8) = − y + u dy + v Whence the equation of continuity becomes dy dw 281. In order to apply the other equations to the motion of the sea, it must be observed that a fluid particle at the bottom of the sea would in its rotation from m to B always touch the spheroid, which is nearly a sphere; therefore the value of s would be very small for that particle, and would be to v, u, of the order of the eccentricity of the spheroid, to its mean radius taken as unity; consequently at the bottom of the sea, s may be omitted in comparison of u, v. But it appears from equations (74), that s (s) is a function of u and v independent of r, when the very small quantity 27($) is omitted: hence s is the same throughout every part of the radius r, as it is at the bottom, and may therefore be omitted throughout the whole depth, when compared with u and v, so that equation (72) of the surface of the fluid is reduced to 282. When the fluid mass is in momentary equilibrium, the equation for the motion of a particle in the interior of the fluid becomes 0 = £nod { (r + s) sin (~ + u)}2 + (8V) − (dp), P ያ where (d), (dp), are the values of SV and dp suited to that state. But we may suppose that in a state of motion, whence SV = (SV) + SV', and Sp = (Sp) + Sp'; (SV) = SV — SV', (dp) = dp — dp', - and {(r+s) sin (0+u)}2 = ♪V'−SV + 283. If the first member of this expression be eliminated from equation (70), with regard to the independent variation of r alone, it cients of de, da, be each made zero in equation (76), it will give r add the differential of the last equation relative to t, to the first equa and let the second member of this equation be represented by then divide by y'.r2 sin2 0, 2 sino 0, and put 2n cos 0=a, and there will be found the linear equation be a function of y', and as y' is a function of y and V', each of which is of the same order. If then equation (77), be multiplied by dr its integral will be 285. Since this equation has been integrated with regard to r only, A must be a function of e, w, and t, independent of r, according to the theory of partial equations. And as the function in r is of the 286. But as ♪ does not contain r, s, or y, it is independent of the depth of the particle; hence this equation is the same for a particle at the surface, or in its neighbourhood, consequently it must coincide with equation (76); and therefore 287. Thus it appears, that the whole theory of the tides would be determined if integrals of the equations could be found, for the horizontal flow might be obtained from the first, by making the co-efficients of the independent quantities do, So, separately zero, then the height to which they rise would be found from the second. This has not yet been done, as none of the known methods of analysis have hitherto succeeded. 288. These equations have been formed on the hypothesis of the earth being entirely covered by the sea; hence the integrals, if they |
