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 = fno♪ { (r + s) sin (9 + u)}a + (♪V) – (`p),

where (d), (dp), are the values of SV and Sp suited to that state. But we may suppose that in a state of motion,

whence

SV = (SV) + SV', and dp = (dp) + dp';
(SV) = SV — SV', (dp) = dp — dp',

Ꮴ

and n{(r+8) sin (0+u)}2 = SV' —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 80, dw, 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 sin 0,

sin2 0, and put 2n cos 0 = a,

and there will be found the linear equation

be a function of y',

is of the order s or

and as y' is a function of y and V', each of which

is of

same order. If then equation (77), be multiplied by dr its

285. Since this equation has been integrated with regard to r only, A must be a function of 0, w, and t, independent of r, according to the theory of partial equations. And as the function in r is of the order 75 it may be omitted; and then

286. But as d 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, Sa, 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

could be found, would be inadequate to determine the oscillations of the ocean retarded or accelerated by the continents, islands, and innumerable other causes, beyond the reach of analysis. No attempt is therefore made to integrate the equations; but the theory of the tides is determined by comparing the general relations which subsist between the observed phenomena and the causes which produce them.

289. In order to integrate the equation of continuity, it was assumed that if the angles Pob, mPb, or rather

du dv
น, '9 v,
dt dt'

be the same for every particle situate on the same radius throughout the whole depth of the sea at the beginning of the motion, they will always continue to be the same for that set of particles during their motion, therefore all the fluid particles that are at the same instant on any one radius, will continue very nearly on that radius during the oscillations of the fluid. Were this rigorously true, the horizontal flow of the tides would be isochronous, like the oscillations of a pendulum, and their velocity would be inversely as their depth, provided the particles had no motion in latitude; and it may be nearly so in the Pacific, whose mean depth is about four miles, and where the tides only rise to about five feet; but it is very far from being the case in shallow seas, and on the coasts where the tides are high; because the condition of isochronism depends on the omission of quantities of the order of the ratio of the height of the tides to the depth of the sea.

290. The reaction of the sea on the terrestrial spheroid is so small that it is omitted. The common centre of gravity of the spheroid and sea is not changed by this reaction, and therefore the ratio of the action of the sea on the spheroid, is to the reaction of the spheroid on the sea, as the mass of the sea to the solid mass; that is, as the depth of the sea to the radius of the earth, or at most as 1 to 1000, assuming the mean depth of the sea to be four miles. For that reason u, v, express the true velocity of the tides in longitude and latitude, as they were assumed to be.

On the Atmosphere.

291. Experience shows the atmosphere to be an elastic fluid, whose density increases in proportion to the pressure. It is subject to changes of density from the variation of temperature in different latitudes, at different heights, and from various other causes; but in this investigation the temperature is assumed to be constant.

292. Since the air resists compression equally in all directions, the height of the atmosphere must be unlimited if its atoms be infinitely divisible. Some considerations, however, induced Dr. Wollaston to suppose that the earth's atmosphere is of finite extent, limited by the weight of ultimate atoms of definite magnitude, no longer divisible by repulsion of their parts. But whether the particles of the atmosphere be infinitely divisible or not, all phenomena concur in proving its density to be quite insensible at the height of about fifty miles.

Density of the Atmosphere.

293. The law by which the density of the air diminishes as the height above the surface of the sea increases, will appear by considering p, p' p', to be the densities of three contiguous strata of air, the thickness of each being so small that the density may be assumed uniform throughout each stratum. Let p be the pressure of the superincumbent air on the lowest stratum, p' the pressure on the next, and p" the pressure on the third; and let m be a coefficient, such that pap. Then, because the densities are as the pressures, p' = ap', and p" ap".

=

Hence, p — p' = a (p − p') and p − p = a (p' — p'').

But p p-p' is equal to the weight of the first of these strata, and p'p" is equal to that of the second: hence

consequently

p-p' : p'-p' :: p : p' ;

pp" = p2.

The density of the middle stratum is therefore a mean proportional between the densities of the other two; and whatever be the number of equidistant strata, their densities are in continual proportion.

294. If the heights therefore, from the surface of the sea, be taken in an increasing arithmetical progression, the densities of the strata

of air will decrease in geometrical progression, a property that logarithms possess relatively to their numbers.

295. All the circumstances both of the equilibrium and motion of the atmosphere may be determined from equation (70), if the quantities it contains be supposed relative to that compressible fluid instead of to the ocean.

Equilibrium of the Atmosphere.

296. When the atmosphere is in equilibrio v, u, and s are zero, which reduces equation (70) to

Suppose the atmosphere to be every where of the same density as at the surface of the sea, let h be the height of that atmosphere which is very small, not exceeding 5 miles, and let g be the force of gravity at the equator; then as the pressure is proportional to the den sity, ph.g. P,

At the surface of the sea, V is the same for a particle of air, and for the particle of the ocean adjacent to it; but when the sea is in equilibrio

no V + 2

p? sin constant,

therefore p is constant, and consequently the stratum of air contiguous to the sea is every where of the same density.

297. Since the earth is very nearly spherical, it may be assumed that the distance of a particle of air from its centre is equal to R+r', R being the terrestrial radius extending to the surface of the sea, and r' the height of the particle above that surface. V, which relates to the surface of the sea, becomes at the height r';

by Taylor's theorem, consequently the substitution of R+r' for r in the value of hg log. p gives