From these results, (8) to (20), we conclude that

Elastic solid tides.

835. The bounding surface and concentric interior spherical surfaces of a homogeneous elastic solid sphere strained slightly by balancing attractions from without, become deformed into harmonic spheroids of the same order and type as the solid harmonic expressing the potential function of these forces, when they are so expressible: and the direction of the component displacement perpendicular to the radius at any point is the same as that of the component of the attracting force perpendicular to the radius. These concentric harmonic spheroids Homogenealthough of the same type are not similar. When they are of solid globe the second degree (that is when the force potential is a solid face; deharmonic of the second degree), the proportions of the ellipti- bodily har cities in the three normal sections of each of them are the same in all but in any one section the ellipticities of the concentric ellipsoids increase from the outermost one inwards to the centre, in the ratio of 5k+ 3n to 8k + 3n, or

ous elastic

free at sur

formed by

monic force.

Ifn/k be small, as is in general the case, the ratio is Case of approximately + 31⁄2n/k: 1.

second degree gives elliptic deformation,

from centre

higher de

greatest pro

For harmonic disturbances of higher orders the amount of de- diminishing viation from sphericity, reckoned of course in proportion to the outwards: radius, increases from the surface inwards to a certain distance, grees give and then decreases to the centre. The explanation of this re- portionate markable conclusion is easily given without analysis, but we from spheri shall confine ourselves to doing so for the case of ellipsoidal at centre disturbances.

deviation

city neither

nor surface.

proof of

ellipticity

for defor

second

836. Let the bodily disturbing force cease to act, and let Synthetic the surface be held to the same ellipsoidal shape by such a maximum distribution of surface traction (§§ 693, 662) as shall maintain at centre, a homogeneous strain throughout the interior. The interior mation of ellipsoidal surfaces of deformation will now become similar order. concentric ellipsoids: and the inner ones must clearly be less elliptic than they were when the same figure of outer boundary was maintained by forces acting throughout all the interior;

VOL. II.

28

the earth.

Rigidity of and, therefore, they must have been greater for the inner surface. And we may reason similarly for the portion of the whole solid within any one of the ellipsoids of deformation, by supposing all cohesive and tangential force between it and the Synthetic solid surrounding it to be dissolved; and its ellipsoidal figure to maximum be maintained by proper surface traction to give homogeneous

proof of

ellipticity

at centre, for defor

mation of second order.

strain throughout the interior when the bodily force ceases to act. We conclude that throughout the solid from surface to centre, when disturbed by bodily force without surface traction, the ellipticities of the concentric ellipsoids increase inwards.

837. When the disturbing action is centrifugal force, or tide generating force (as that of the sun or moon on the earth), the potential is, as we have seen, a harmonic of the second degree, symmetrical round an axis. In one case the spheroids of deformation are concentric oblate ellipsoids of revolution; in the other case prolate. In each case the ellipticity increases from the surface inwards, according to the same law [§ 834 (15)] which is, of course, independent of the radius of the sphere. Oblateness For spheres of different dimensions and similar substances the homogene ellipticities produced by equal angular velocities of rotation solid globe, are as the squares of the radii. Or, if the equatorial surface velocity (V) be the same in rotating elastic spheres of different dimensions but similar substance, the ellipticities are equal. The values of the surface and central ellipticities are respectively

induced in

ous elastic

by rotation.

for solids fulfilling Poisson's hypothesis (§ 685), according to which m= 2n, or k=n.

If the solid be absolutely incompressible these ellipticities are by § 834 (15)

Now since

5 V2w

and

19 2n

8 V2w 19 2n

.(22).

=4211, we see that the compressibility of the elastic solid exercises very little influence on the result.

1

tides.

For steel or iron the values of n and m are respectively Elastic solid 780 x 10 and about 1600 x 106 grammes weight per square centimetre, or 770 × 10° and about 1600 × 10° gramme-centimetre-seconds, absolute units (§ 223), and the specific gravity (w) is about 78. Hence a ball of steel of any radius rotating with an equatorial velocity of 10,000 centimetres per second will be flattened to an ellipticity (§ 801) of 7220. For a specimen of flint glass of specific gravity 2.94 Everett finds n = 244 × 10° grammes weight per square centimetre and very approximately m = 2n. Hence for this substance n/w=83 × 106 [being the length of the modulus of rigidity (§ 678) in centimetres]. But the numbers used above for steel give n/w= 100 x 106 centimetres; Numerical and therefore (§ 838) the flattening of a glass globe is 1/83, or iron and 11⁄2 times that of a steel globe with equal velocities.

results for

glass.

or tidal

but little

by compres

globes of

838. For rotating or tidally deformed globes of glass or Rotational metals, the amount of deformation is but little influenced by ellipticities compressibility, as we see from the numerical comparison given in influenced § 837. For any substance for which 3k5n the surface ellipti- sibility, in city is diminished by three per cent. or by less than three cent., and the centre ellipticity by per cent., or less than per gelatinous cent. if we suppose the rigidity to remain in any case unchanged, but the substance to become absolutely incompressible. For the surface ellipticity, § 834 (22) gives on this supposition

per metallic,

vitreous, or

elastic

solid.

or with

n = 770 × 10° as for steel (§ 837),

a = 640 × 10°, the earth's radius in centimetres,

surface

for globe

839. If now we consider a globe as large as the earth, and Value of of incompressible homogeneous material, of density equal to ellipticities the earth's mean density, but of the same rigidity as steel or same size glass; and if, in the first place, we suppose the matter of such earth, of a globe to be deprived of the property of mutual gravitation tating

and mass a

non-gravi.

material, homogene

ous, incom

pressible,

and same

ri zidity as steel.

Comparison between

maintaining

gravitation

for large

homogene.

ous solid globes.

between its parts: the ellipticities induced by rotation, or by tide generating force, will be those given by the preceding formulæ [S834 (20)], with the same values of n as before; with n/k=0; with 640 × 10° for a, the earth's radius in centimetres; and with 55 for w instead of the actual specific gravities of glass and steel.

But without rigidity at all, and mutual gravitation between the parts alone opposing deviation from the spherical figure, we found before (§ 819) for the ellipticity

840. Hence of these two influences which we have conspheroidal sidered separately:-on the one hand, elasticity of figure, even powers of with so great a rigidity as that of steel; and, on the other hand, and rigidity, mutual gravitation between the parts: the latter is considerably more powerful than the former, in a globe of such dimensions as the earth. When, as in nature, the two resistances against change of form act jointly, the actual ellipticity of form will be the reciprocal of the sum of the reciprocals of the ellipticities that would be produced in the separate cases of one or other of the resistances acting alone. For we may imagine the disturbing influence divided into two parts: one of which alone would maintain the actual ellipticity of the solid, without mutual gravitation; and the other alone the same ellipticity if the substance had no rigidity but experienced mutual gravitation between its parts. Let be the disturbing influence as g, denote measured by § 834 (20), (21); and let T/r and 7/g be the elliptito deforma cities of the spheroidal figure into which the globe becomes spectively altered on the two suppositions of rigidity without gravity and

resistances

tion due re

to gravity and to rigidity.

gravity without rigidity, respectively. Let e be the actual ellipticity and let be divided into ' and 7" proportional to the two parts into which we imagine the disturbing influence to be divided in maintaining that ellipticity. We have T = T'+T", and eT/TT"/g.

e=

T

=

Whence =r+g, or

e

It gives

the earth:

where n/g is the rigidity in grammes weight per square centi- Rigidity of metre. For steel and glass as above (§§ 837, 839) the values of r/g are respectively 2.1 and 66.

σι

introduc

effects of

840'. Mr G. H. Darwin has shown how the introduction of Analytical the effects of the mutual gravitation of the parts of the spheroid tion of may be also carried out analytically instead of synthetically. gravitation. The sphere being in a state of strain is distorted into a spheroid (say ra+o, where σ, is a surface harmonic). Then the state of internal stress and strain in the spheroid is due to three causes, (i) the external disturbing potential W,, (ii) the attraction of the harmonic inequality of which the potential is 3gwr'o¡/(2i + 1) a', (iii) the weight (positive in parts and negative in others) of the inequality o. This last is equivalent to a normal traction per unit area applied to the surface of the sphere equal to -gwo. It is not possible to arrive at the results due to the last cause without a modification of the analysis of § 834, because we have to introduce the effects of surface tractions.

But Mr Darwin shows (p. 9 loc. cit.) that "if W, be the potential of the external disturbing influence, the effective potential per unit volume at a point within the sphere, now free of surface action and of mutual gravitation, is

W1- 2gw (i − 1) r'o;/(2i + 1) a' = roT, suppose.”

The case considered by him is that of an incompressible viscous spheroid, and he goes on to find the height and retardation of tide in such a spheroid. The analysis is, however, almost literatim applicable to the case of an elastic incompressible spheroid.

Suppose now that the external disturbing potential is

* "On the Bodily Tides of Viscous and Semi-elastic Spheroids, &c." Phil. Trans. Part 1. 1879, p. 1.