Proper coordinate systems are constructed in hydromagnetic equilibria and their properties are studied. First, the contra-gradient components of magnetic field and of current density are surface quantities. Second, the equi-pressure surfaces which have no singularity within a finite volume must be topologically torus-shaped. Third, a general condition of no charge separation is deduced as follows: There must be a simple closed loop on every equi-pressure surface having B the property that the integral dl/H is constant for the variable A. Here, the integral is carried A out along a magnetic line from a point A on the loop to the returning point B on the same loop, and dl is the magnetic line element. An expression of the shape of current lines is obtained from the condition in the cases of unclosed field and of twisted field. A method of determining the magnetic surfaces which coincide with equi-pressure surfaces is obtained in the case of closed field. We examine the successive approximation method developed by M. Kruskal et al. with the help of these methods. It fails in the first approximation in almost all cases of twisted fields. It can be used in the case of unclosed field if the rotational transform ratio is one of the continued fractions constructed in this paper. It can be used in the case of the closed field with mirror symmetry when the plasma'pressure gradient is not too steep. An effect of the closed magnetic lines in a twisted field is considered. The diffusing velocity of plasma is infinite in the neighbourhood of magnetic surfaces which are made of closed lines not satisfying the condition of no charge separation. The ratio of the measure of the highly diffusing region to the measure of the whole system is estimated in an easy case. The result suggests that the confinement time of plasma may be considerably shorter than that of plasma in the field compatible with the condition of no charge separation.

A proper coordinate system and a suitable form of equilibrium equations for a successive approximation method are reported in this letter. By these means we can solve equilibrium equations in the case of a rotational transformed field. 1 )2) But we could not apply these methods to the case in which lines of force are closed and have no symmetric property.As is well known, lines of electric current and lines of magnetic force lie on equi-pressure surfaces in the M. H. equilibria. We have proved 3 ) that it is possible to construct a global coordinate system on each equi-pressure surface by taking these two kinds of lines as coordinate lines and defining coordinate Xi on these lines according toIn Eq. (1), Xl and dSl are the magnetic field strength and its line-element, X2 and ds2, the current density and its line-element, respectively.We have studied the structure of these coordinate systems and got three main results. S ) The first is that any closed equi-pressure surface without singular points is topologically torus, as was said by many other authors.2) The second is that there must exist simple closed loops which lie on each equipressure surface with non-vanishing pressure-gradient and have the following property, that is,where the integrals are carried out along lines of force from an arbitrary starting point A on the loop to the final point B on it. Generally, there -are one or more loops passing through any given point on the surface, turning arbitrary given times along and around the magnetic axis in an equilibrium configuration.The third result IS that by taking these loops belonging to two different homology classes as coordinate lines and defining coordinates ~. on these lines in proportion to magnetic flux through an infinitesimally narrow ribbon between the surface and its neighbouring one, limited by ~i=O and ~. itself, we can construct a many valued periodic coordinate system on each equi-pressure surface, and in this system, lines of current and lines of force are represented as parallel straight lines, respectively.

Field reversed configurations with a racetrack shaped separatrix of variable length and constant radius r s are studied by assuming time-dependent uniform temperature and resistivity 17. Only the most useful solutions, namely those with separated radial and time variables, are investigated. The decay times of all relevant quantities are related to those of the magnetic field, te, the mass inventory, t^, and the length, tL. Tjte and r)t^[ are shown to be constant in time, t^ is related to the rate of heat loss, t^. The configuration inside the separatrix is characterized by two parameters, 0 < a < °° and 0 < 0 S < 1, where a = t B /t M and j3 s is the value of beta on the separatrix. Global parameters such as the average beta and the fraction of trapped flux as well as the radial profiles of a number of quantities are shown graphically as functions of oc and j3 s . It is pointed out that te and tM can assume wide ranges of values for fixed 17 and r s . Therefore, the data of tg, t^ and r s are necessary to assign a value of 17 to an experiment. Three conditions for maintaining an internal configuration characterized by (a,/3 s ) throughout an experiment are proposed and evaluated numerically. These can be used for the prediction and control of an FRC. The external configuration is studied with a tentative model and the non-existence of a plasmavacuum interface is shown. For practical purposes, however, the depth of the external plasma layer can be defined and calculated as a function of a and |3 S . All parameters of the model can be determined for any experiment from the data of measurable global quantities, so that the inner configuration can be inferred. The resistivities assigned to current experiments range from one to ten times the classical value.

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The principle of Project MACETIE and the preliminary experiment are described. The basis of our plan is the thought that the goal of controlled thermonuclear fusion may be accessible by the multistage alternating processes of adiabatic compression and irreversible expansion of plasma. In the latter process, the plasma compressed by a strong magnetic field is expanded into a region of weaker magnetic field. A static consideration of this expansion suggests that plasma temperature will not change appreciably in this process, and a second adiabatic compression of plasma in the weak field may further increase its temperature. Preliminary experiments are being carried out to examine the foregoing principle of irreversible expansion.