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A solution of the equilibrium equation for plasma has been found up to any degree of nonlinearity in form of double series. Applications are presented for the zero- and first-order approximation.

New solutions are found for the Laplacian and Grad-Shafranov operators in toroidal coordinates suitable for the study of gravitational and plasma problems.

An analytic solution of the force-free field equation has been found for a three-dimensional torus.

It is found that charged particles of positive energiesE, when constrained on axisymmetric isoflux surfaces ?, execute sinusoidal motions with typical frequencies O=(2E/m?)1/2). In general, it was found that under equilibrium condition ?p=J ^B/cthe particles develop a non-ambipolar drift velocityvd=(cµ/eb)[1+q2 +2(q/?)2]p.

It is proved that (1) electromagnetic fields with electric and magnetic components parallel to one another are solutions of Maxwell's equations; (2) the equationB?(??B)=0 (B is the magnetic field) is gauge and relativistically-invariant for systems of reference moving with velocityv/c=E?B(1+v2/c2)/(E2+B2).

The equations of motion are derived for a differentially rotating system, and the paper demonstrates the existence of a force term due to coupling of the angular momentum with the gradient of the angular frequency. Translation of the system of reference to the Local Standard of Rest (LSR) shows that the LSR executes a radial oscillatory motion as a necessary consequence of the coupling mentioned above. This mechanism could well provide the theoretical background for expanding galaxies without resorting to a violent explosion in their centre — a mechanism for which the evidence is rather scanty.

An analytical solution has been found for the non-linear equation governing the equilibrium of the plasma torus.

The concept of a static index of refraction is being used to demonstrate that the quantities describing the vibrational ellipse vary because of the torsion induced by the index of refraction. Further, the idea is exploited that in relativistic objects such variations can be induced by the non-Euclidean character of the external geometry. Expressions are given for the induced curvature and torsion and, finally the Hamilton-Jacobi equation is used to derive the effective index of refraction for Kerr's geometry.

The dispersion relation has been found for a galaxy, without the assumption that the centrifugal force is balanced by the gravitational force. It has been shown that such a system (1) can be gravitationally unstable under appropriate conditions, and (2) that there is no resonance at ?=2O (O=angular velocity of the Galaxy).

The dispersion relation has been derived for density waves propagating at an arbitrary angle. The analysis has shown the existence of a resonance which for a two-arm galaxy can be stable, neutral or unstable as $$\omega /2\Omega \tfrac{< }{ > }2$$ , respectively.

A connection is shown to exist between the Gaussian curvature of the associated manifold and the ergodic or non-ergodic behaviour of certain dynamical systems of astronomical and astrophysical importance.

Fermat's principle has been used to derive expressions for the curvature and torsion of the path of the electromagnetic radiation in a medium of refractive indexn (= function of space coordinates). Levi-Civita's notion of parallelism further, in conjunction with Brill's results (cf. reference in text) have enabled the derivation (1) of Einstein's formula for the deflection of light, and (2) an expression for the rotation of the plane of polarization, an entirely general relativistic effect, unaccountable in Newtonian physics. Finally, the idea is put forward that the observed rotation of polarization in pulsars might be a purely general relativistic effect due to the non-Minkowskian geometry around them.

The behavioyr of a force-free field has been studied in a Reissner-Nordström metric. An expansion in tensor harmonics of even-odd parity reduced the radial equations in a differential equation of the Sturm-Liouville system which was solved asymptotically in a conveniently defined space coordinate. Further, it has been possible to regularize the singular behaviour of the Reissner-Nordström metric at the event horizon and the modified metric to be given explicitly.

The effect of the differential rotation on the statistical behaviour of a dynamical system has been studied in the presence of a scalar field. The existence of neutrally stable regimes in a disk-like system has been shown which formally correspond to the Landau levels of a charged particle whose motion is quantized in a magnetic field. The existence also of the so-called anomalous Doppler effect has been pointed out, and a modified form is given to include the differential rotation. Finally, quantitative results are given concerning the instability of the system.

The behaviour of a magnetic line has been studied under continuous injection of plasma. Under increasing density the existence of three zones of possible concentration has been shown. For values greater than a critical valueD=0, withD suitably defined, one zone disappears, whilst the other two degenerate into one.