hypotheti

density

earth.

824. To apply the results of the preceding investigation to Laplace's the determination of the law of ellipticity of the layers of cal law of equal density within the earth, on the hypothesis of its within the original fluidity, it is absolutely essential that we commence with some assumption (in default of information) as to the law which connects the density with the distance from the earth's centre. For we have seen (§ 821) how widely different are the results obtained when we take two extreme suppositions, viz., that the mass is homogeneous; and that the density is infinitely great at the centre. In few measurements hitherto made of the Compressibility of Liquids (see Vol. II., Properties of Matter) has the pressure applied been great enough to produce condensation to the extent of one-half per cent. The small condensations thus experimented on have been found, as might be expected, to be very approximately in simple proportion to the pressures in each case; but experiment has not hitherto given any indication of the law of compressibility for any liquid under pressures sufficient to produce considerable condensations. In default of knowledge, Laplace assumed, as an hypothesis, the law of compressibility of the matter of which, before its solidification, the earth consisted, to be that the increase of the square of the density is proportional to the in- Assumed crease of pressure. This leads, by the ordinary equation of tween denhydrostatic equilibrium, to a very simple expression for the law pressure. of density, which is still further simplified if we assume that the density is everywhere finite.

Neglecting the disturbing forces, we have ($$ 822, 752)

But, by the hypothesis of Laplace, as above stated, k being some

constant

.(2).

relation be.

sity and

dp = kpdp

or, by § 822 (5),

Απ

μ

= 4= ['r'p'dr' + + = ['r2p'dr' + "

Multiplying by r, and differentiating, we get

།

Laplace's

hypothetical law of density

within the earth.

Law of density.

If we write 4π/k=1/x2, the integral may be thus expressed—

TP=F'sin (+G).

If we suppose the whole mass to be liquid, i.e., if there be no solid core, or, at all events, the same law of density to hold from surface to centre, G must vanish, else the density at the centre would be infinite. Hence, in what follows, we shall take

Determina-
tion of ellip-
ticities of
surfaces
of equal
density.

We are now prepared to find the value of u, in § 823, upon which depends the ellipticity of the strata. For (15) of § 822 becomes, by (23) of that section and the late equation (4),

where μ' is the mass of fluid, following the density law (3), which is displaced by the core μ, and q is the surface density. In the terrestrial problem we may assume μ μ, and of course a = For simplicity put

0.

Multiply by r, and again differentiate; the result is

..(6),

...............(7).

quences ellipticities

so that u, is known from (6). Now we have already proved that Conseu, increases from the centre outwards, so that we must have as regards C'=0, for otherwise u, would be infinite at the centre. Thus, of surfaces of equal dropping the suffix, to the symbol e for brevity, we have density.

The constants are, of course, to be determined by the known values of the ellipticity of the surface and of the angular velocity of the mass.

Now (5) becomes, at the surface,

We may next eliminate p, dp/dr, and q, being the surface value of p, by means of (3) (4), (6), and (8), and substitute everywhere re for u. Also, if m be the ratio (2) of centrifugal force to gravity at the equator, w is to be eliminated by means of the equation

from which p is to be removed by (3). By the help of these substitutions (10) becomes transformed as follows:

If we put tan r/κ=t, and r/× = 0, so that 0 is the surface value of 9, the integrated expression, divided by Fek2 cos 0/t, with

Ellipticity of internal stratum.

simpler, and may be written

5m 01- 3202 +≈202

2

The mean density of the sphere comprised within the radius

Let p be the mean density of the sphere comprised within this
Po

radius r, and P,

the radius r. It
values of

Then,

as before, the density at the stratum defined by may be noted in passing that 9% and 9 are the

Po and p corresponding to r = r.

If we put ƒ for the ratio of the mean density of this sphere to the density at its bounding surface, we have

Since 3C/F is constant, it follows that (er2)/(1 − 1/f) is the same for all the strata of equal density. If therefore f be the surface

value off, that is to say the ratio q/q of the mean density to the Laplace's surface density of the whole earth,-a quantity which may be cal law of determined by experiment,

This formula gives the ellipticity of any internal stratum accord-
ing to the Laplacian theory.

It may be also reduced to another form which is perhaps
rather curious than important, as follows:-

Differentiate (12) logarithmically with regard to , and we

hypothetidensity within the earth.

Ellipticity of internal stratum.

Thus we may state verbally that the ellipticity of any internal
stratum varies as the rate of decrease, per unit increase of area
of the stratum, of the logarithm of the mean density of the
sphere comprised within that stratum*.

The formula (12) for 5m/2e may now be more simply ex-
pressed. Attributing to ƒ and their surface values f and 0,
we have from (12′′)

From this equation may be found by approximation, and then
(12) gives e in terms of known quantities. In fact, it becomes

* This and the preceding mode of expressing the ellipticity of an internal stratum are taken (with changed notation) from a paper by Mr G. H. Darwin in the Messenger of Mathematics (Vol. vi.), 1877, p. 109.

density.