Abstract
The problem of determining the conditions, as regards elasticity, in which a body of planetary dimensions can be gravitationally unstable, was attacked by J. H. Jeans. To turn the difficulty arising from the fact that such a body is necessarily in a state of “ initial stress,” by which gravitation is balanced throughout its interior, he introduced an artificial external field of force to balance gravitation in the undisturbed body, and he thus dispensed with initial stress. A more direct method of treating the problem has been advocated by Lord Rayleigh. In this method the stress at any point of the body, when disturbed from its equilibrium state, is taken to be compounded of two stress-systems ; an initial stress of the nature of hydrostatic pressure, balancing gravitation in the undisturbed body, and an additional stress connected with the strain (reckoned from the initial state) by the same formulæ as connect stress with strain in a homogeneous isotropic elastic solid body which is slightly strained from an initial state of zero stress. In the first part of the present paper the equations of vibratory motion of the planetary body, assumed to be initially spherical, are formed in accordance with this method, and they are solved by means of spherical harmonics. The frequency equation is obtained, and the conditions that it may be satisfied by a zero value of the frequency are found. When these conditions are fulfilled the body is in a state of gravitational instability.
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