ELLIPTICITY FROM PENDULUM EXPERIMENTS.

89

The ratio of increase of gravity from the equator to the pole deduced from these observations is 0.0051828, (see Cosmos, p. 468.)

PROP. To find the ellipticity by Clairaut's Theorem and Pendulum experiments.

93. By Clairaut's Theorem and the last Article

5

€ m-0.0051828 = 0·0086355-0·0051828

2

The investigations of Professor Stokes referred to in Art. 91, show that this should be a little smaller.

1

The actual measurement of the earth, as we shall show in the next Chapter, makes the ellipticity Hence pendulum experiments bear a strong testimony in favour of the fluid theory.

294

But besides the value of the ellipticity deduced from pendulum experiments, the law of variation of gravity on the surface as deduced above from the fluid theory, viz. that it changes as the square of the latitude, also agrees remarkably with those experiments, and bears strong testimony to the truth of that theory.

94. In addition to the general test of the truth of the fluid arrangement thus afforded by pendulum experiments we will add an investigation which still further tests the truth of the theory, by finding what effect certain hypothetical redistributions of the mass, differing from the fluid arrangement, would have upon the motion of the pendulum. If the effect would be sensible we have a further argument in favour of the fluid arrangement being the actual arrangement of the earth's mass.

PROP. To find the effect on the pendulum of certain hypothetical changes in the distribution of the materials of the earth's mass.

95. We will suppose the earth's mass divided into four shells and a nucleus, the radius of the nucleus and the thickness of each shell being equal to one-fifth of the earth's radius, or about 800 miles. We shall make three separate hypotheses:

(1) That the masses of the second and third shells are both altered, each in a different proportion, so as to preserve the whole mass the same and not to alter the form of the strata.

(2) That the form of the strata in one of the shells is altered without affecting the mass.

(3) That the earth consists of a homogeneous mass of the same density as the surface, with the remainder of the mass distributed according to any law in spherical shells.

96. First re-arrangement. Let EEEEE, be the masses of the earth and of the portions of it lying within the inner surfaces of the four successive shells: VVVVV, the corresponding potentials for a point at distance r from the centre and in latitude of which the sine is μ. Then, by Art. 85,

1

E
Ꮴ =- +

1 2

4

2

2

and VV... have corresponding values, e... mm,... being similar quantities to e and m. Then V-V and V- V, are the potentials of the second and third shells. Also E- E2, E-E, are the masses of those shells. Suppose the first of these masses is altered in the ratio a : 1, and the second in the ratio : 1; then as the total mass is unaltered, by hypothesis,

In consequence of this change the potentials of the shells become a (VV) and B (V-V). Hence if U be the potential of the whole earth thus altered in the arrangement of its materials

PENDULUM EXPERIMENTS.

U=V+ (a− 1) (V1 − V2) + (B − 1) (V2 — V2)

E Ea2 m

1

2

2

91

+

203

As m is the ratio of the centrifugal force to gravity at the

equator of the spheroid,

By substituting the values found in future Articles (Art. 107, 8), according to the fluid-hypothesis, we have

=

2

+

3125 2237 3125 2237

0.3892 -0.2569 +0.0220

294

= (α — 1) (0·0005316 – 0·0002623) = (a− 1) × 0·0002693,

the value of e here used being taken from the British Ordnance Survey.

d V

Now gravity

dr

centrifugal force, at the surface.

Hence the ratio of gravity as altered by this change to gravity as it is

dU

=(--centrifugal force) +(-dr-centrifugal force)

= 1 +31 (

and the increase in passing from the equator to the pole

The table in Art. 92 shows that between the equator and Spitzbergen in about 80° north latitude (the highest place north where pendulum experiments have been made) 214 vibrations are lost in 24 hours by a seconds' pendulum. Hence the number which would be lost from the re-arrangement of the mass now under consideration would equal

Suppose that a difference of 5 beats of the pendulum at the equator and at Spitzbergen is easily detected, then if

nearly, that is, if the density of the second shell be increased about 1-7th and that of the third be diminished by about 1-5th, the deranging effect on the pendulum would be capable of detection on the earth's surface. Under these circumstances it may be said, that the near approach to conformity between the observed number of vibrations of the pendulum in

PENDULUM EXPERIMENTS.

93

various places and the same computed on the fluid-arrangement of the mass affords some argument in favour of that arrangement representing the actual condition of the earth, whether it be now in part fluid or not.

97. Second re-arrangement. Suppose that all the strata in one of the shells lose their ellipticity and become spherical, the parts about the poles of the upper surface of the shell swelling up and penetrating the mass of the shell above it, and the parts about the equator of the lower surface of the shell penetrating the shell within; so that the original spheroidal shell may become a spherical shell of the same mass as before, and its mass still co-existing with the other shells and nucleus. This change amounts simply to this: the density of the mass is doubled through a thin space of the form of two hemispherical meniscuses, the rims of which are of no thickness and touch each other at the equator of the upper surface of the shell in question, the thickness of the meniscuses being greatest at the poles and equalling the compression of that surface; and the density is also doubled through a space at the lower surface of the shell generated by the revolution of a crescent round the earth's axis of which the width at the middle equals the distance of the equator of that lower surface from the inscribed sphere. The matter causing the doubling of the density through these thin spaces is drawn from the original shell itself.

We will take the second shell for our example, and will apply the result to the other shells.

1

The potential of this shell is V, V,; this must now be replaced by (E-E)÷r, the potential of a shell of spherical strata.

E

Hence

E

2