trigonometrical operations meets all these conditions. It is found by measurements in widely separated countries, that an ellipse of the kind described can be drawn in the plane of the meridian of any place, cutting the plumb-lines at all the stations where it is examined at right angles; and the ellipticity of this ellipse is almost exactly equal to . There are local deviations from this law, arising from local causes, which are produced by the variations of the surface of the Earth and probably of the interior of the solid crust also. But the average line is this ellipse. Since the variations of the Earth's surface, in mountains and valleys and extensive oceans, are palpable, and must have arisen since the Earth ceased to be fluid and assumed its general form, the fact that deviations from this ellipse are found in the level-curve while the average curve is still this ellipse, is rather confirmatory of the theory of original fluidity than otherwise.

The probability of the truth of the law of density made use of in the previous calculations is strengthened by the value of Precession which it leads to.

PROP. To test the law of density used above by the amount of Precession of the Equinoxes which it leads to.

82. The Annual Precession

=

=

inclination of

=

I obliquity of the ecliptic23° 28′ 18", i Moon's orbit to ecliptic 5° 8' 50", n and n' are the mean motions of the Earth round its axis and round the Sun, and their ratio = 365.26, n" the mean motion of the Moon round the Earth 27.32 days, v = ratio of masses of Earth and Moon 75. (See Mechanical Philosophy, Second Edition, Art. 470: also, changing the notation, Airy's Tracts, Fourth Edition, p. 213, Arts. 36, 38.) Substituting the above quantities,

Annual Precession = 16225".6

C- A

C

where A and C are the principal moments of inertia of the mass, the latter about the axis of revolution. To find these let xyz be the co-ordinates to any element of the mass, row be the polar co-ordinates to the same.

Then the mass of this

element pr2dudwdr, p = cos 0. Also

The terms are here arranged as Laplace's Functions. (See Art. 39, Ex. 4.)

Now r = radius of any stratum = a { 1 + ε ( − μ3) } (Art. 75);

d.ae (1

=

(3

= σ (1) + ↓ (1) ( − 42) suppose;

0 y —

6

8π

Also C= σ (a), neglecting the small term (a).

3

=

5

= a2

૨૨

m

Now (a)-pada-a' 4 (a) ( − 1 )

=

୧

And putting p=

0

(e

sin qa (e–m) z, by Arts. 77, 78.

sin qa, and integrating by parts,

☛ (a) = ["pa'da = Q ["a' sin qada

Substituting for qa, z, e and m their values, this is found

The value generally assigned to the Precession, from observation, is 50"1. The almost complete coincidence of the

result of the theory with this observed value is a remarkable evidence in favour of the law of density we have adopted.

83. Mr Hopkins has endeavoured to ascertain how far the interior of the Earth may at present be fluid, by calculating the value of the Precession upon the supposition of the mass being a spheroidal shell of heterogeneous matter, enclosing a heterogeneous fluid mass, consisting of strata increasing according to the law we have used. In three memoirs in the Philosophical Transactions of 1839, 1840, and 1842, he enters upon a complete investigation of this subject. We will give the evidence upon which he rests his conclusion that the crust is very thick.

PROP. To trace the argument drawn from Precession to show that the crust is of considerable thickness.

84. Mr Hopkins has deduced the following formula (in which we have changed the notation to suit the present treatise),

d.a' (ε' — ε)
da'

=

€

2ea3 [" p'a'da' + €

a

a

ρ' da'

da'

where P is the precession of the equinoxes of a homogeneous spheroid of ellipticity e, which by calculation = 57" nearly if e = 30; P is the precession of the heterogeneous shell, the outer and inner ellipticities being e and e: this = 50"-1 by observation.

The success of the calculation depends upon a remarkable result at which he has arrived, that the precession caused by the disturbing forces in a homogeneous shell filled with homogeneous fluid, in which the ellipticities of the inner and outer surfaces are the same, is the same whatever the thickness of the shell. It is therefore the same for a spheroid solid to the centre. The formula above given is the relation of the amounts of precession in two shells, one heterogeneous and the other homogeneous; and, as the thickness is the quantity sought, neither of these amounts could be calculated, and therefore the relation expressed in the above formula would

be of no avail. Bu in vnserenes if the property that the person of the thell when and the find a homogeneous, is the same as that of the spherd as fiffelly is overcome; and Fan be miniated without knowing the thickness, and themire P will be ov

We have shown At dat de stata fecrease in ellipticity in passing downwards: hence - is never negative, and the faction in the mens hand in the above formula is never negative, and is never so large as mity: let it = = B. Hence

-- 3. or g is less than ţe:

=

and therefore, because the ellipticity decreases in descending, the thickness must be greater than would correspond with an ellipticity of the inner surface of the shell equal to 7-8ths of that of the outer surface.

If solidification took place solely from pressure, the surfaces of equal density would be surfaces of equal degrees of solidity. If we use the formula for finding in Art. 78. and make qa = 150, and the mean density = 24225 times the superficial density (the second of the values in Art. 80, then if ɛ = -e in

7

8

3 the formula of Art. 78, we have, after reduction, aa, or 4

the thickness equal to one fourth of the radius, or 1000 miles. If a smaller ratio of densities is used than 2·4225, the thickness is greater. (Mr Hopkins shows also that a ratio a little larger than 3 makes the thickness 1-5th of the radius: but this ratio is too large. The ratio generally used is about 2·2).

But solidification depends upon temperature, as well as upon pressure. In his third memoir (Phil. Trans. 1842), Mr Hopkins shows that the isothermal surfaces increase in ellipticity in passing downwards. If temperature alone regulated the solidification, these surfaces would be the surfaces of equal solidity. But since both pressure and temperature have their effects, the ellipticities of the surfaces of equal solidity must lie between those of the isothermal and the equi-dense surfaces. Hence the surface of equal solidity at