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The rules to write out any one of the linearly independent functionsbelonging to the infinite set of those in polynomial form that satisfy thesourceless Grad-Shafranov equation as stated in the toroidal-polar coordinatesystem are established. It is found that a polynomial solution even in thepoloidal angle is given by the product of an integral power of the radialcoordinate variable by a complete polynomial of equal degree in this samevariable with angular-dependent coefficient functions that are linearcombinations of a finite number of Chebyshev polynomials in the cosine of thepoloidal angle, the numerical coefficients of these being expressed in terms ofthe binomial numbers of Pascal's arithmetic triangle. Tables of the tenpolynomial solutions of the lowest degrees are provided in variables of thetoroidal-polar and of the cylindrical coordinate systems.

Theory of Multipole Solutions to the Sourceless Grad-Shafranov Equation in Plasma Physics