where g is the value of gravity at the surface. The corresponding strains consist of
(1) uniform contraction of all lines of the body of amount
1⁄30 k−1gρa (3 − σ) / (1 − σ),
(2) radial extension of amount 1⁄10 k−1gρ (r²/a) (1 + σ) / (1 − σ),
(3) extension in any direction at right angles to the radius vector of amount
1⁄30 k−1gρ (r²/a) (1 + σ) / (1 − σ),
where k is the modulus of compression. The volume is diminished by the fraction gρa/5k of itself. The parts of the radii vectors within the sphere r = a {(3 − σ) / (3 + 3σ)}1/2 are contracted, and the parts without this sphere are extended. The application of the above results to the state of the interior of the earth involves a neglect of the caution emphasized in § 40, viz. that the strain determined by the solution must be small if the solution is to be accepted. In a body of the size and mass of the earth, and having a resistance to compression and a rigidity equal to those of steel, the radial contraction at the centre, as given by the above solution, would be nearly 1⁄3, and the radial extension at the surface nearly 1⁄6, and these fractions can by no means be regarded as “small.”
76. In a spherical shell of homogeneous isotropic material, of internal radius r1 and external radius r0, subjected to pressure p0 on the outer surface, and p1 on the inner surface, the stress at any point distant r from the centre consists of
(1) uniform tension in all directions of amount
| p1r1³ − p0r0³ | , |
| r0³ − r1³ |