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If we take the mean density = 2.4225 x the density of the surface = 2:4225 x 2.75 = 6.66, which accords very nearly with Mr Airy's determination from the Harton Experiment (Phil, Trans. for 1856, p. 355, where it is 6.565), the equation for finding qa is

qa

1 -0.8075q*a?' which is satisfied by qa = 2:618 = 150°. Then

tan qa =-0.57735, 2 = 5.5345, q'a' = 6.8539,

qʻa+ 2 = 1.2384, z = q*a? = 0.8075.

tan qa

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81. In the course of the last eight Articles we have developed the following conditions, which must be satisfied if the Earth has derived its present general form from being in a fluid state. (1) The direction of gravity must everywhere be perpendicular to the surface. (2) The form of the surface must be an oblate spheroid, with its axis coincident with the axis of revolution. * (3) An additional test, though not absolutely infallible yet invested with a large degree of probability, is that furnished by the result of Art. 80, by assuming a law of density of the strata which is of itself à priori very probably true, that the value of the ellipticity is not very different from We shall see in a future Chapter that the actual measurement of the form of the Earth by means of 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 300 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,

1 300

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.

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I = obliquity of the ecliptic = 23° 28' 18", i = inclination of 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,

C-A
Annual Precession = 16225":6

c

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where A and C are the principal moments of inertia of the mass, the latter about the axis of revolution. To find these let syz 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 pr* dudwdr, u = cos 0. Also

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

(See

Now r=
radius of any stratum = a {1+€

pa d.go

• da

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P. A.

6

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87 Also C= o (a), neglecting the small term f (a).

3

Now

*(w) / dodatno da = a*$(a) (en)

(e-m) 2, by Arts. 77, 78.

5 Qa?
sin

qa

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Substituting for qa, z, e and m their values, this is found to 0.00313593.

.. Annual Precession = 16225":6 x 0:00313593

= 50'.8.

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.

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sa" ] "p'a'da'+e

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" (e' – )
p

da
P- P'
E

da
1
P

d.a's
2eao

da

da' where P is the precession of the equinoxes of a homogeneous spheroid of ellipticity €, which by calculation = 57' nearly if e=300; P is the precession of the heterogeneous shell

, the outer and inner ellipticities being e and g: 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

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