which, by substitution integration and division, gives

2

4

( 1 + c ) (x2 + y2) + ( 1 − c ) 2 − 2 (N − 1) = = const.,

5

2 6

-

or (1+-esin'?) - 2 (N-1) == const. (7= latitude).

5 5

Supposera (1 + e' sin2 7), which can evidently be made to satisfy this equation, is the equation to the level surface through the equator. By substituting for r and equating the constant parts and also the parts depending on sin3 l, the latter leads to

which gives the form of the level surface.

PROP. To explain how the Tides can be used as a test of the degree of Rigidity of the Earth's mass.

119. It is necessary to premise, that, in an elaborate and difficult investigation on the deformation of a spheroidal elastic mass acted upon by external forces, Professor Thomson has deduced the following formula (Phil. Trans. 1863, p. 574),

h

19 n 1+

2 gwr

where h' and h are the polar elevations of a prolate spheroid drawn out from a spherical figure by an external force (resembling the combined action of the sun and moon in raising the tidal wave), the spheroid being considered a homogeneous mass of incompressible fluid in the first instance, and of incompressible solid matter in the second: in both cases the total mass is equal to the mass of the earth; w denotes the mass of a unit of volume, that is, the density; r the

RIGIDITY OF THE EARTH.

115

radius of the globe, and n the rigidity of the solid substance, that is, n. k is the force necessary to draw a unit of section of the solid through an extremely small space k.

Now suppose the earth an absolutely rigid sphere, and H the polar elevation of the prolate spheroid, or level surface, which is everywhere perpendicular to the resultant action of the sun and moon, which in the actual ocean, produces the tidal wave. Then H is the height of high above low water, in the equilibrium theory of the tides, of an ocean of infinitely small density covering a rigid earth.

Suppose instead of the above that the earth is covered with an ocean, the earth still being a perfectly rigid sphere, and its mean density N times that of the ocean. Let H' be the polar elevation of the prolate spheroid into which the sun and moon draw the ocean. Then by Art. 118, the terrestrial gravitation level would be disturbed by this cause, from the spherical surface to a spheroidal surface of which the polar elevation is (3÷ 5N) H', by the attraction of the ocean in its altered form. The polar elevation in the level surface, before noticed as produced by the direct action of the sun and moon, must be added to this to give the polar elevation of the actual equilibrium level of the free surface of the ocean.

Hence

It will be observed that h' and H'

3

5N

are similar quantities,

but for oceans of different density: when N=1, h'=H',

We have thus far been considering the earth to be absɔlutely rigid. If its solid mass is drawn up from a spherical form to a polar elevation h by the sun and moon, the attraction of the protuberant mass will change the gravitation level from a sphere to a prolate spheroid of polar elevation.

(35) h: and this as before should be added to H to find the whole effect of the sun and moon in changing the gravitation level. The sum will be the absolute tidal elevation above the sphere, of an ocean of infinitely small density covering the elastic globe. By subtracting h, the tidal elevation of the solid itself, we have the difference between high

If the earth be perfectly rigid, n is infinite, and this expression becomes H, as it ought to do. For iron or steel

n=501 × 10,

the unit of mass being 1lb., the unit of space 1 foot. This makes the above expression for the height of the tide equal to 0.59H or about H. For glass

3

n = 2160000 × 144 × 32·2,

which makes the tide 2 H.

"Imperfect," Professor Thomson remarks, "as the comparison between theory and observation as to the absolute height of the tides has been hitherto, it is scarcely possible to believe that the height is in reality only two-ninths of what it would be if, as has hitherto been universally assumed in tidal investigations, the earth were perfectly rigid. It seems therefore nearly certain, with no other evidence than is afforded by the tides, that the tidal effective rigidity of the earth must be greater than that of glass."

PROP. To explain how Precession can be used as a test of the Earth's Rigidity.

120. Conceive the earth to be a fluid mass revolving in a

RIGIDITY OF THE EARTH.

117

day about its axis, and drawn by centrifugal force into a spheroidal figure having the same amount of protuberant matter as the earth actually has. Suppose also that the combined action of the sun and moon produces a tidal wave on each side of the earth which is superimposed upon the spheroid of revolution. As the earth in this case would have no rigidity whatever, it would therefore have no precession. Conceive things to remain the same, except that the fluid becomes an elastic solid, yielding as the fluid to the varying influence of the sun and moon, so as to produce a tidal wave superinduced as before on the spheroid of revolution. This mass would still have no precession. As the mass is now

solid, though elastic, the sun and moon, by attracting the nearer equatorial parts more, and the further equatorial parts less, than the centre, would have a tendency to cause it to rotate round an axis in the equator and produce precession. But as, in the case supposed, no precession takes place, this tendency of the sun and moon to produce precession must be exactly counterbalanced by some opposite tendency: that tendency is the effect of the centrifugal force on the protuberant parts of the tidal wave drawn up on the solid by the sun and moon. This effect, therefore, of the solid tidal wave would in amount exactly equal the actual precessional motion of the earth, on the hypothesis of the earth's mass being perfectly rigid, though it would act in the opposite direction.

Now by the last Article it appears, that if the earth have the rigidity of steel or glass the tides would be reduced to 3-5ths or 2-9ths of their amount on the supposition that the earth's rigidity is perfect; that is, the deformation of the solid parts beneath the ocean would be 2-5ths or 7-9ths of that amount. The result would be that 2-5ths or 7-9ths of the precession caused in a rigid earth would be balanced if the mass have only the rigidity of steel or glass. As no such effect is detected by observation it must be presumed that the rigidity of the earth is decidedly very great.

"The close agreement," Professor Thomson remarks, “between the observed amounts of precession and nutation and the results of theory on the hypothesis of perfect rigidity, renders it impossible to believe that there is enough of.

elastic yielding to influence the phenomena to any considerable extent. It is worthy of remark, however, that in general the theoretical estimates of the amount of precession have been somewhat above the true amount demonstrated by observation. It seems not altogether improbable that this discrepancy is genuine, and is to be explained by some small amount of deformation experienced by the solid parts of the earth, under lunar and solar influence."