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m

m € -

2

shown by Burg and others to equal – 8” sin (nt +e); also h=0·0040217 n. Hence after all reductions

1 = 0·0015474,

= 0·0017476;
2 578

1
.. e=0.0032950 = nearly.

303 This method, as was the case with the pendulum experiments, only determines the ellipticity, but gives no evidence on the subject of the original fluidity of the Earth. The near agreement is remarkable.

93. The spheroidal form of the Earth's surface and the circumstance of its being a surface of equilibrium afford us more information, as Professor Stokes has shown, regarding the distribution of matter in the interior of the Earth's

mass. PROP. To show that the centre of the Earth's mass coincides with the centre of its volume, and that the axis of rotation is one of the principal ases of the mass, as a consequence of the form of the surface being a spheroid of equilibrium.

94. By Art. 18 we have the potential of the mass with reference to an external point (fgh), or

pdx dydz V

(f-x)? + (9- y)? + (hz)}}} the integrals extending throughout the whole interior of the Earth. Let pds dydz=dim, and », je, v the direction-cosines of r the radius vector to (fgh); then f=ar, g=ur, h=vr; and expanding the radical according to the inverse powers of r, we get

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E denoting the sum of the three expressions necessary to form a symmetrical function.

Comparing this with the value of Vin Art. 91,

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....

.....

1

m Ea? 2

(2),

3 and other equations from the succeeding terms.

Equations (1) show that the centre of gravity of the mass is at the centre of the spheroid of revolution, or the centre of the volume.

With regard to equation (2) we may observe that duv are tied by the relation 12 + Me + vë= 1. If then we insert this for 1 in the equation, so as to make it symmetrical with regard to X, M, and v, and equate the several coefficients, we shall obtain

1 1
-

xy

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Equations (3) show that the co-ordinate axes are principal axes; and that therefore the axis of revolution is a principal axis.

The last three equations (4) show, first, that

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or, as might be anticipated, the moments of inertia about the axes of x and y, in the plane of the equator, are the same. They give also if, as before, A and °C are the principal moments,

0-1'

=ffsvdm-js=dm=} (e-m

m Ea.

Were we able to deduce C also, without making any hypothesis regarding the internal condition of the earth, the Precession might be obtained under the same circumstances. In that case, the agreement between the amount of precession calculated upon the assumed law of internal density and the observed value (pointed out in Art. 82) would furnish no proof of the truth of that law. But as the precession cannot be thus independently calculated, the result in Art. 82 affords a strong argument for the correctness of the law of internal density which we have adopted.

CHAPTER III.

THE FIGURE OF THE EARTH, DETERMINED BY GEODETIC

OPERATIONS.

95. In a previous Chapter we have shown that if the Earth be considered a fluid mass the form of the surface will be an oblate spheroid of small ellipticity, its axis coinciding with the axis of revolution, and the surface being everywhere at right angles to the direction of gravity; and further, that upon assuming that the density of the strata varies according to a certain very probable law, the ellipticity = 3oo nearly.

In this Chapter we propose to submit this to the test of measurement, by inquiring whether an ellipse can be found with its axis coinciding with the axis of the Earth and cutting the plumb-line at stations along it at right angles; and whether the ellipticity of that ellipse is so

The method of doing this is as follows. A base-line, about 5 or 6 miles in length, is measured with extreme accuracy, near the meridian, the curvature of which we are to find. By a series of triangles this base is connected with a number of stations in succession lying near the meridian, the angles and sides of which are calculated or observed, as the case may be. Thus a connexion is established between the original base and a second base at the termination of the chain of triangles, and the length of this second base obtained by calculation. It is then measured, as the first was, and by a comparison of the calculated and measured results the correctness or not of the operations is tested. This having been satisfactorily performed, the projections of the sides of the triangles on the

*7

P. A.

meridian are found, and their sum gives the length of the meridian arc between its two extremities. The latitudes of these extremities are then observed with great care, and from these data the form of the ellipse, of which the arc is a part, is found by the principles of conic sections, as we shall now show.

PROP. To find the length of an arc of meridian in terms of the amplitude, the semi-axis major, the ellipticity (the ellipticity being small), and the middle latitude.

96. Let I and I be the latitudes of the extremities of the arc, m the mean of these or the middle latitude; 4 the amplitude of the arc or the difference between the latitudes; a, b, and e the semi-axes and ellipticity; 8 the length of the arc, r the radius vector, and 0 the angle r makes with the major axis. Then

1 cos' sine
+
tan /

tan 0;
a 72

7

1

a” cosa 1 + 72 sino ?
a cos* 7+ b*sinol, putting bra (1 – €),
p=a (1 – e sinol), neglecting e'...

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