Quasi-polynomial 3D electric and magnetic potentials homogeneous in Euler's sense

Electric and magnetic fields homogeneous in Euler's sense are a useful instrument for designing the systems of charge particle optics. The similarity principle for the charged particle trajectories in these fields was applied by Golikov for the first time to create spectrographic charge particle optical systems in a more systematic and intelligence way when using the fields being homogeneous in Euler's sense. This paper studies the Laplace potentials homogeneous in Euler's sense. The coefficients of the polynomials are functions of the two rest coordinates; they are presented not by the polynomial but ought to be the functions harmonic and homogeneous in Euler's sense. We have solved a finite chain of Poisson equations starting from the highest coefficients. By means of the proposed procedure we obtained new classes of potentials which provided a base for electric and magnetic spectrograph systems.