Rigidity of the earth.

tion; of which the first, given by (6) and (7) of § 733, is as follows:

with symmetrical formulæ for ` and `y; which [§ 733 (6)],

which, reduced to harmonics by the proper formula [§ 737 (36)],

becomes

Homogeneous elastic solid globe free at surface; deformed by bodily har

monic force.

་

` Fr=

{[m+ (i + 1) n] r2.
i+1);

dWi+1 dx

(2i+5) m-n d
2i+5
dx

(2i+3) (m + n)
This and the symmetrical formulæ for `Gr and `Hr, with r taken
equal to a, express the components of the force per unit area
which would have to be balanced by the application from without
of surface traction to the bounding surface of the globe, if the
strain through the interior were exactly that expressed by (1).
Hence, still according to the directions of § 732, we must now
find (a, ẞ, y) the state of interior strain which with no force
from without acting bodily through the interior, would result
from surface traction equal and opposite to that (4). Of this
part of the problem we have the solution in § 737 (52), the par-
ticular data being now

with symmetrical terms for B, C., and B.,, C.; but none of other orders than i and i+2. Hence for the auxiliary functions of § 737 (50)

the earth.

Now (52), with the proper terms for i + 2 instead of i added, is Rigidity of to be used to give us a,; and through the vanishing of 1, and

To this we must add 'a, given by (1), to obtain, according to § 732, the explicit solution, a, of our problem. Thus, after somewhat tedious algebraic reductions in which m+n, appearing as a factor in the numerator and denominator of each fraction, is removed, we find a remarkably simple expression for a. This, and the symmetrical formulas for ẞ and y, are as follows:

The infinitely great value of € for the case i=0 depends on the circumstance that the bodily force for this case, being uniform and in parallel lines through the whole mass, is not self equilibrating, and therefore surface stress would be required for equilibrium.

The formulas (8) are susceptible of considerable simplification if we complete the differentiations in their last terms. We shall at the same time separate the formulas into two parts, of which one has for coefficient the bulk-modulus, and the other the rigidity-modulus.

Rigidity of the earth.

Separation of the two moduluses of elasticity.

If k be the bulk-modulus, or modulus of resistance to compression, we have by § 698 (5),

Case of incompressible elastic solid.

Then, on substituting in (9) for m from (9'), carrying the results into (9) and separating the parts depending on k and n, we have

(21n) a = k {[(

i+1

dᎳ .
(i + 4) (a2 — r2) + 3a2 ] dW1+1 + 2(i + 1)æW,

–

dx

+] –

+ }n(i + 1) {(a2 — ro)

and symmetrical formulæ for ẞ and y.

¿... (9iv).

dW 1+1

+ 2x W

dx

In the elastic solids of which we have experimental knowledge [§ 684] the bulk-modulus is larger than the modulus of rigidity, and therefore k is considerably larger than ; thus the terms written in the first line of (9iv) are practically much more important than those in the second. In the ideal case of an absolutely incompressible elastic solid, the terms in the second line of (91v) vanish, and I/k becomes simply 2 (i + 2)2 + 1, and thus we have

The case of i = 1 is that with which we are concerned in the tidal problem. In it (7) and (9) give us

To prepare for terrestrial applications we may conveniently reduce to polar co-ordinates (distance from the centre, r; latitude, ; longitude, A) such that

x = r cos l cos λ, y = r cos l sin λ, z = r sin l ......(11);

the earth.

and denote by p, p, v, the corresponding components of displace- Rigidity of The expressions for these will be precisely the same as

ment.

those for a, ẞ, y, except that instead of

d

dx'

as it appears

in the

d

d dr

expression for a, we have in the expression for p; in that Case of

rdl

Also in transforming from a

top we must put x=r, and in transforming from ẞ and γ to
μ and
v, y and z must be put zero.

Thus if we put

so that S., may denote the surface harmonic, or the harmonic function of directional angular co-ordinates, A, corresponding to W.,, we have from (9iv)

incompres sible elastic solid.

+ 1 ) ( i + 3 )

(i

a 2

In the case of elastic solids, such as we know them experimentally, the terms in k are much more important than those

in n.

Now it is easy to show that, in as far as p depends on the term in k, it reaches a maximum value when ra/1-1/ (i+2)2; and in as far as it depends on the term in n it would algebraically reach a maximum when ra √ 1 + 2 / {(i+ 2) (i − 1)}. But this latter point being outside of the sphere it follows that the term in n increases from the centre to the surface. We thus see that p increases from zero at the centre, to a maximum value near the surface, and then diminishes again.

In a similar manner it appears that p/r reaches a maximum, as far as concerns the term in k, when ra√1 - 3/{i (i + 2)}; and as far as concerns the term in n, when r=a.

When i=1, which corresponds to the case of the tidal problem, we have from (13)

It is obvious that p/r diminishes from the centre outwards to the surface; and its extreme values are

Cases:centrifugal force :

tide-generating force.

(19k+ n) n

P 5k+n

=

ጥ (19k + n) n

When the disturbing action is the centrifugal force of uniform rotation with angular velocity w, we have as found above (§ 794) for the whole potential

W ̧=w{}w2r2 + }w3r2 (} - cos3 0)} ................................(16), where w denotes the mass of the solid per unit volume. The effect of the term ww is merely a drawing outwards of the solid from the centre symmetrically all round; which we may consider in detail later in illustrating properties of matter in our second volume. The remainder of the expression gives us according to our present notation

where

W ̧ = }τ (x2 + y2 – 2≈2); or S, = WT (}} - Cos2 6) ..(17),

2

T = {w3...

......

..(18).

For tide-generating force the same formulæ (14) and (15) hold if (SS 804, 808, 813) we take

and alter signs so as to make the strain-spheroids prolate instead of oblate. The deformed figure of each of the concentric spherical surfaces of the sphere is of course an ellipsoid of revolution; and from (15) we find for the extremes :

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(20).