br Assuming that we obtain and consequently br

Assuming that we obtain and, consequently,
Expression (3) yields the most general form of a two-dimensional harmonic function homogeneous in Euler\'s sense. Let us choose two forms for our further investigation: the symmetric and the antisymmetric. The first one is a structure of the form assuming that . The second antisymmetric one takes the form assuming that . The role of electric potentials of the form (2a) as two-dimensional mirrors for an electrostatic smo inhibitor analyzer operating as a spectrograph was explored in Refs. [4,9,14,15] for the case when the potential does not depend on the z coordinate, and the principal motion of charged particles occurs in the OXY plane.
Quasi-polynomial fields (the general case) Let us construct three-dimensional potentials in the form of a polynomial of finite degree 2n or 2n – 1 with respect to one of the Cartesian coordinates (e.g., y) with coefficients that are homogeneous functions of the corresponding order with respect to the other two coordinates: x and z. The main task is in finding the form of these coefficient functions, assuming that their analytical form must be either symmetric, as expression (4), or anti-symmetric, as (5). These potentials fall into two non-overlapping families: the polynomials in even degrees and the polynomials in odd degrees:
As a slight aside from the central subject of our discussion, we should note the following. If a harmonic function (which is a polynomial of finite degree with respect to the y coordinate) is divided into a sum of two polynomials in even and odd degrees of y, it is easy to verify by direct substitution into the three-dimensional Laplace equation that each of the two polynomials separately will also be a harmonic function. This result follows from the fact that the recurrence relations for the factors multiplying different degrees of y, which must be satisfied in order for the polynomial as a whole to be a harmonic function, do not intersect for even and odd degrees of y. Furthermore, if we select the symmetric functions with respect to the argument z with the degree of homogeneity k–j or m–j, respectively, as or functions, as a result we can construct a variation of the electrical potential. If the functions are antisymmetric, we obtain the configuration of the magnetic potential. Let us first consider expansion (6) in even degrees for a symmetric potential, using an even function as a basis. Let us substitute expansion (6) into the three-dimensional Laplace equation and group together the terms multiplying the same degrees of y. Since the whole expression is equal to zero, the coefficients multiplying the different degrees of y must become zero; we then obtain the following sequence of equalities:
Let us take a harmonic function homogeneous in Euler\'s sense with a degree of homogeneity p=k – 2n as the generating one, and substitute it as the coefficient multiplying the highest y2 degree:
The factor multiplying the y2–2 degree can be then determined by solving Poisson\'s equation with the right-hand side U(x, z) and the condition to be symmetric with respect to the z coordinate. It is easy to verify that such a function will have the form where U1 is the free constant, and c1 is chosen so that the result satisfies Poisson\'s equation with a function of the form U0cos pγ on the right-hand side. The factor multiplying the y2–2 degree (8) will be obtained in the form where U1 will be a free constant. Constant d2 is selected so as to coincide with the term in the right-hand side of Poisson\'s equation, and the constant d1 to coincide with the term in the right-hand side of Poisson\'s equation. The described procedure continues until the sequence of recurrent calculations stops at the first term of expansion (8). The formulae for the odd degrees of y and for the antisymmetric potentials are constructed in a similar manner.