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It is shown that the new modes introduced by the inclusion of the Hall effect in the gravitational stability problem of a plasma are stable. However, an old mode (which is stable in the absence of a Hall current) becomes unstable. We conclude that the Hall current has a destabilizing effect and that the Jeans criterion for stability is adversely affected by its presence. However, the growth rate of the Hall instability is found to be negligible for any astrophysical system.

The effect of finite ion Larmor radius on the gravitational instability of two superposed fluids in uniform rotation is investigated for interchange perturbations, using the macroscopic equations of motion, where the finite ion Larmor radius effect is incorporated through off-diagonal terms in the pressure tensor. It is found that the region of stable wavelengths is enhanced due to the simultaneous presence of finite Larmor radius and a uniform rotation. A similar conclusion is also arrived at for the situation when a vortex sheet is present between the two superposed fluids.

Jeans's criterion for instability is found to be changed only for the case of perturbations perpendicular to the magnetic field and rotation parallel to it, when rotation and finite Larmor radius effects tend to stabilize the plasma.

An examination has been made of the gravitational instability of an infinite, perfectly-conducting cylinder, maintained by its intrinsic gravitational force, with an axial magnetic field, and surrounded.by fluid of a different density. The magnetic field will suppress any axisymmetric instability due to axial flow if the Alfvén velocity is greater than the streaming velocity.

This paper discusses the gravitational instability of a non-uniformly rotating, infinite, axisymmetric cylinder of a homogeneous, heat conducting medium. It is found that the Bel and Schatzman criterion of gravitational instability remains unaffected, even when the medium is heat conducting.

The stability of finite amplitude internal waves in a weak shear flow in geostrophic balance is investigated theoretically, using a simple model of constant stratification. It is shown that the gravitational stability problem is very important and that, except for waves of small amplitude, a gravitational instability will develop before shear instability, in agreement with some recent theories.

The gravitational instability of a horizontal fluid layer with a univariant phase transition is considered. It is found that the layer can be unstable even when the less dense phase lies above the dense phase and can be stable in the opposite case. Applications of the theory to convection with phase transitions in astrophysical and geophysical problems are briefly discussed.

It is shown that Jeans's criterion for the gravitational instability of an infinite homogeneous medium remains unaffected even if the system is partaking in rotation and Coriolis forces are acting.

PROCESSES originating in the interior of the Earth, such as thermal convection and gravitational instability, are not the only ones whereby energy is dissipated at a rate greater than that required to move lithospheric plates. The rate of loss of kinetic energy resulting from the deceleration of the Earth's rotation has been estimated at 7 × 1026 erg yr−1 (ref. 1); this is probably larger than the drag power on the plates.

A short review of the theory of the formation of galaxies and clusters of galaxies is given within the framework of the non-linear theory of gravitational instability. Probable initial conditions are discussed as well as processes which bring about the origin of galaxies and clusters of galaxies. Possible observational tests are discussed.

[The text or this Chapter contains the first part of the Dicussion on Thursday, September 18, 1969; the second part is given in Chapter 26. I added a long contribution on cosmogonical problems, presented by Dr. Ozernoi on September 10.]

We examine here the problem of the gravitational instability of a medium in nonuniform rotation. We seek therefore to resolve a problem differing somewhat from that examined by Chandrasekhar and Fricke. We imagine a gaseous medium in rotation in a potential field, ɸ, and consider only the plane problem. In the unperturbed motion, the particles describe circular orbits in uniform motion. We examine the stability against perturbations.

The stability of an infinite system of mass points interacting under a Newtonian force is examined through a microscopic analysis. (C.E.S.)

In all conventional methods for particle-size distribution analysis, measuring gear i s introduced or permitted to be present in a suspension of settling particles. This inevitably causes inhomogeneity and gravitational instability in the suspension with the result that the most interesting and relevant features of the particle-size distribution curve are wiped out. Here an apparatus is described where only external measurements (buoyancy) have to be taken on a column of settling particles, and stability is rigorously maintained by the employment of an additional liquid. The design is simple enough to make this method not only suitable for reference purposes but also equipment- and effort-saving in routine determinations. It was combined with a bead-recorder, necessitating no photographic work and almost no mechanical parts.

The problem of stability of infinitesimal disturbances within a Boussinesq fluid confined between horizontal plates, the lower of which is maintained warmer than the upper, is examined for the case of small, constant shear and for several different Prandtl numbers. The results indicate the stabilizing effect of increasing Reynolds and Prandtl numbers for disturbances in a vertical plane aligned in the direction of mean flow. For disturbances in a vertical plane transverse to the mean flow, the stability criterion with shear remains the same as that for pure convection.

CHANDRASEKHAR has derived in a very elegant manner1, from his theory of density fluctuations in an isotropic homogeneous turbulence field2, the generalization to a turbulent medium of Jeans's criterion for the stability against inhomogeneity of an infinite quiescent homogeneous medium. The expression which he obtains may be written: where λ is the critical length for stability, c is the speed of sound in the medium, θ the temperature, m the mass of a gas molecule, and ū2 the mean square velocity of the turbulence field.

It is shown that Jeans's criterion for the gravitational instability of an infinite homogeneous medium is unaffected by the presence of a magnetic field, even if the system is partaking in rotation and Coriolis force is operative.

A study of the stability of a self‐gravitating fluid of variable density with finite resistivity is described. A variational technique is established, and it is proved that the system cannot be overstable. It is also shown that the system is unstable for all wavenumbers and finite resistivity.