The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.

The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.

The stability of a general toroidal MHD equilibrium with a continuous pressure profile is investigated for the case where the fluid is surrounded by vacuum. It is found that a disturbance which is localized near the free boundary grows exponentially in time unless a certain necessary criterion is satisfied. Because of the weaker boundary condition, this criterion imposes a more stringent restriction on the configuration than does Mercier's criterion near the boundary. For the cylindrically symmetric case the criterion requires a decrease of the rotational transform and yields a critical relation between the shear and the pressure gradient.

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Taylor’s theory of relaxed toroidal plasmas (states of lowest energy with fixed total magnetic helicity) is extended to include a vacuum between the plasma and the wall. In the extended variational problem, one prescribes, in addition to the helicity and the magnetic fluxes whose conservation follows from the perfect conductivity of the wall, the fluxes whose conservation follows from the assumption that the plasma-vacuum interface is also perfectly conducting (if the wall is a magnetic surface, then one has the toroidal and the poloidal flux in the vacuum). Vanishing of the first energy variation implies a pressureless free-boundary magnetohydrostatic equilibrium with a Beltrami magnetic field in the plasma, and in general with a surface current in the interface. Positivity of the second variation implies that the equilibrium is stable according to ideal magnetohydrodynamics, that it is a relaxed state according to Taylor’s theory if the interface is replaced by a wall, and that the surface current is nonzero (at least if there are no closed magnetic field lines in the interface). The plane slab, with suitable boundary conditions to simulate a genuine torus, is investigated in detail. The relaxed state has the same double symmetry as the vessel if, and only if, the prescribed helicity is in an interval that depends on the prescribed fluxes. This interval is determined in the limit of a thin slab.

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M onodentate, Semicarbazone, Thiosem icarbazone, But-2-yne, Tungsten(II) Reaction of [W I(C O )(N C M e)(dppm )(772 -M eC2M e)][BF4] {dppm = Ph2 P(C H 2 )PPh2} with an equim olar quantity of L {L = R R 'C N N H C O N H 2 (R = R ' = Me; R = H, R ' = Ph)} or L {L = R R 'C N N H C S N H 2 (R = R ' = Me, Et; R = Me, R ' = E t, Pr", Bu', Ph; R = H, R ' = Ph)} gives the m onodentately coordinated sem icarbazone or thiosem icarbazone products [W I(CO)L(dppm)(?7 2 -M eC2M e)][BF4] (1 -9 ) . 13C N M R spectroscopy suggests that the but-2 -yne ligand is donating four electrons to the tungsten in [W I(CO){H(Ph)CNN H CSN H 2 }(dppm)(?7 2 -M eC2M e)][BF4].

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