rotation about the axis of z, taken as the fixed axis; w.x and w2.y will be the centrifugal force at the point (xyz). Then the differential equation to the surface and the strata becomes

Let r be the distance of the point (xyz) from the origin, and the angler makes with the axis of z, and cos 0=μ: then x2+ y2= sin2 = (1 μ) r2. Also let m be the ratio of the centrifugal force at the equator to gravity at the

equator (or 219) 239); let a' be the mean radius of the stratum

.

through (xyz); a the radius of the equator; then

the strata being considered spherical because of the smallness of the numerator in the value of m;

this arrangement being made, because the second and third terms as they now stand, are Laplace's Functions of the order 0 and 2. (See Art. 39, Ex. 1.)

Now since the mass is supposed to be fluid and the external surface nearly spherical, it follows that as the heavier parts,

which are all free to move, will sink through the lighter, and lie in layers, these layers will also be nearly spherical, otherwise there will be a greater pressure on one part than on another, and the equilibrium of the layer or stratum will not exist. We may therefore assume as a consequence of the fluidity and the form of the surface that the strata also are nearly spherical.

r =a (1 + ¥‚+ ... Y, + ...) and ["p'a”da' = 4 (a),

0

as before. Then substitute this value of V in the equation to the strata and equate terms of the order i. (See Art. 35.)

except when i 2, in which case the second side is

By this equation Y, is to be calculated, and then the form

of the stratum of which the mean radius is a is known by the formula

r = a (1 + Y1 + Y2+ ... + Y + ...).

1

2

PROP. To prove that Y= 0, excepting the case of i=2.

74. Since Y, and p are functions of a, they may be expanded into ascending series of the form

Y1 = Wa' + ..., p =D + D'a" + ...,

where D is the density at the centre of the earth, and is as well as W and D' independent of a: s, n... must not be negative, otherwise Y, and p would be infinite at the centre.

Now when these and the corresponding series obtained by putting a for a, are substituted in the equation of the strata in the last Article, and the first side arranged in powers of a, the various coefficients ought to vanish; excepting when i=2, because then the second side is not zero. We shall therefore substitute these series, and search for values of W and s which satisfy the condition.

After two easy integrations the equation of the strata becomes

No value of s will cause these terms to vanish. The only apparent case is when i=1, for then by putting si-2 the part in the brackets vanishes: but in this particular case s=-1, and is negative and therefore inadmissible.

Hence the only way of satisfying the condition is by putting W=0; this shows that Y, has no first term, that is, that it has no term at all and is therefore zero.

PROP. To find the value of Y, and to prove that the strata are all spheroidal, concentric and with a common axis.

75. The equation for calculating Y, is, by Art. 73,

2

2

Suppose Y, (and similarly Y) is expanded in a series of powers of u2 with indeterminate coefficients to be ascertained by the condition, that they shall satisfy the above equation. These coefficients will be functions of a only, as it is clear from the right-hand side of the equation that a does not enter into the value of Y.; and Y, consists of only one term, that involving the simple power of -μ. Let it be ɛ (1⁄3 — μ2), ɛ being a small quantity of the order of m. Hence

2

r = a {1+ε (} − μ3)}, μ= sin (latitude) = sin l

= a (1 − e) (1 + ɛ cos2 7), since ɛ is small.

This is the equation to a spheroid from the centre, e being the ellipticity. The axis-minor coincides with the axis of revolution of the whole mass. Hence the strata are concentric spheroids, the minor-axes of which coincide with the axis of revolution of the whole mass.

PROP. To obtain an approximate law of the density of the

strata.

76. By Art. 73 we have the following equation for calculating the pressure on the stratum of which the radius is a, neglecting the small term,

Laplace has integrated this equation on the supposition that the change in pressure in descending through the strata varies as the change in the square of the density (Mémoires de l'Institut, Tom. III. p. 496). This law of compression differs from that of elastic fluids, in which the change in pressure varies as the change in density. The law used by

Laplace is à priori more probably true than the law of compression of elastic fluids, for the greater the density of tenacious and semi-fluid masses the greater must be the increase of pressure to produce a given increase of density. See also some remarks on this subject by Professor Challis in the Phil. Mag. Vol. XXXVIII. The approximate truth of this law is, however, shown by the accuracy of the results to which it leads us.

Putting, then, dp = 1⁄2kd.p", k being a constant,

[*1, dp, da' = k (p + constant);

da'

.. ka (p+ const.) = 4π ["p'a”da' + 4πa ["p'a'da',

a

since & (a) = 3 3 [["p'a'da'. Differentiate* with respect to a ;

. . 7 (d,pa + const.) = 4πpа" +4π["p'a'da'—4πpa2=4π[" p'a'da';

k

α

* In order to explain how to differentiate a definite integral with respect to a quantity involved in the limits, let f(x) = F(x) + const.;

da

• 2 S° f (x) dx=dF (a) = f (a); a Sa ƒ(x) dx == dF(b)

α

db

== - ƒ (b).