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W is evidently equal to the mass, because as r becomes infinitely great the second term vanishes with reference to the first, and we know that in that case the value of the potential must be the mass divided by the distance. Let WE. Also put m = w2. a3÷ E, as in Art. 75;

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the same formula as before. It must always be borne in mind. (what has been implied in the enunciation of the Prop.) that this formula is true only on the hypothesis of the fluid arrangement of the earth's mass being the real state of the earth.

§ 3.

Tests of the truth of the Fluid Theory of the Earth. 88. There are four means of testing the truth of the Fluid Theory of the Earth, which we will now proceed to consider.

First Test. The Law of Gravity upon the surface to which the fluid theory leads; this law can be ascertained with great exactness by means of pendulum experiments.

Second Test. The amount which it assigns to the perturbation of the Moon's motion in Latitude caused by the distribution of the earth's mass in layers which are not spherical; the true amount has been found by astronomical observations.

Third Test. The Ellipticity of the Surface which the theory gives; this is determined by geodesy, as will be set forth in the last Chapter.

Fourth Test. The amount of Precession of the Equinoxes to which the theory leads; the true amount is known by observation.

The investigation on the fluid theory has proceeded thus far without making a single hypothesis-except, of course, the fundamental one which it is our object to test, viz. that the earth was once a fluid mass. The further investigation on which we now enter proceeds also without any hypothesis being made in applying the theory to the first two of these four tests, viz. the law of gravity and the moon's perturba

FOUR TESTS OF THE FLUID THEORY.

85

tions. It is not till we come to the last two, viz. the ellipticity and the precession, that an hypothesis has to be introduced, viz. one regarding the Law of Density of the fluid We therefore take these two tests last of the four.

mass.

All four of these tests bear important and independent testimony to the truth of the hypothesis that the earth was once a fluid mass.

First Test. THE LAW OF GRAVITY AT THE EARTH'S SURFACE.

89. Upon the hypothesis of the earth being a fluid mass it was shown by Clairaut, in his celebrated work, Figure de la Terre, published in 1743, that the increase of gravity in passing from the equator to the poles varies as the square of the sine of the latitude, and that a certain relation must necessarily subsist between the ellipticity and the amount of gravity, a relation which has been ever since known as Clairaut's Theorem. Laplace demonstrated the same, on the simpler hypothesis of the surface only being a surface of equilibrium, and the interior being solid or fluid, but consisting of strata nearly spherical. This we have shown in Art. 74 is a necessary condition of the earth.

Clairaut's Theorem is valuable as it enables us to determine the ellipticity by means of pendulum oscillations, the times of which measure the force of gravity at the several stations where experiments are made, and the result serves as a test of the correctness of the ellipticity deduced by the fluid theory.

PROP. To find the law of gravity at the surface of a spheroid of equilibrium and of small ellipticity; and to prove Clairaut's Theorem.

90. The potential of the earth for an external point is

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Let g be gravity. Then since the angle between the radius vector and the normal varies as the ellipticity and therefore its cosine must be taken 1, the value of gravity

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dv

dr

=

the part of the centrifugal force resolved along r

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Substitute for r and omit small quantities of the second order,

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E

3

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=(1+m+m-e) sin'. latitude}

=

2

- G {1 + ( m − c ) sin' 1},

where G is gravity at the equator.

Hence the increase of gravity in passing from the equator to the poles varies as the square of the sine of the latitude.

This expression immediately leads to the following property, called Clairaut's Theorem after its first discoverer.

Polar gravity equatorial gravity
equatorial gravity

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=

5

+ ellipticity

× ratio of centrifugal force at equator to gravity.

91. When the formula of gravity deduced in the last Article is applied to the several stations where pendulum experiments have been made, discrepancies are brought to light which evidently arise, not from any mistake in the theory, but from the irregularities of the surface of the earth. Mr Airy, in his article on the Figure of the Earth, written in 1830, (Ency. Met.) discussed all the data which had been obtained in various places, and came to the following results: (1) that

CLAIRAUT'S THEOREM.

87

gravity appears to be greater on islands than on continents; (2) that gravity is greater in high north latitudes, less in midàle latitudes than the formula gives it, but is pretty nearly the same about the equator; (3) that gravity does not appear to vary with the longitude alone, nor the hemisphere.

Professor Stokes, in a paper in the Cambridge Philosophical Transactions for 1849, has fully discussed the various causes of disturbance, and has satisfactorily explained the anomalies (1) and (2) deduced by Mr Airy, twenty years earlier, from the experiments. He has also shown, that when allowance is made for the irregularities of the earth's surface, the ellipticity comes out somewhat smaller than it otherwise would.

The chief sources of error are the following. The elevation of the station above the sea-level, and the excess or defect of matter in table-lands or the sea.

The value of gravity obtained by pendulum experiments must be reduced to the standard of the sea-level, and corrected for that level in the way explained in Art. 53. But the sea-level, owing to local attraction, is higher in continents than at sea, as will appear in the next Chapter. Hence gravity obtained from continental experiments will be too small, because it is corrected for a surface too distant from the centre of the earth. This well explains why gravity appears to be less on continents than on islands. The same explanation meets the second anomaly pointed out by Mr Airy. In, the middle latitudes the places where experiments were made are all continental. If this is corrected for, no doubt the deduced ellipticity will come out somewhat smaller, and therefore gravity in high latitudes, as deduced from the formula, no longer be in excess.

Mr Stokes remarks, that if the 49 stations where pendulum experiments have been made are divided into two groups, an equatorial group containing the stations lying between latitudes 35° N. and 35° S., and a polar group containing the rest, it will be found that most if not all of the oceanic stations are contained in the former group, while the stations belonging to the latter are of a more continental character. Hence the observations will make gravity appear too great about the equator and too small about the poles, that is, they will on the whole make

gravity vary too little from the equator to the poles; and

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be best satisfied by a value of e which is too great. This is, in fact, precisely the result of the discussion; the value of e which Mr Airy has obtained from pendulum experiments (0-003535) being, as stated, greater than that which he finds from the discussion of geodetic measures (0.003352).

92. A large collection of the results of pendulum experiments is to be found in Major-General Sabine's work entitled, Account of Experiments to determine the Figure of the Earth by means of the Pendulum Vibrating Seconds in different Latitudes, 1825.

The following abstract is taken from his translation of the Cosmos, Vol. IV. Part I. The column of "computed " vibrations assumes that the change of gravity varies as the change in the square of the sine of the latitude.

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86263,60
86263,60 86269,32
86264,30 86259,77
86267,86 86273,04
86268,48 86268,33
86271,24 86267,27
86274,90 86273,16
86284,80 86285,12
86358,66 86357,73
86390,20 86388,48
86397,06 86396,54
86400,34 86400,59
86400,48 86400,00
86403,12 86403,31
86407,80 86407,23
86408,10 86408,94
86417,02 86417,89
86423,10 86424,60
86433,64 86435,56
86442,24 86438,77

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Greenland

74 32 19 N.

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Spitzbergen 79 49 54 N.

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