or instead of formulas in the case of polynomials of

or instead of formulas in the case of polynomials of arbitrary degree of homogeneity k or m[15]. The final results are given below; there are configurations that differ greatly compared with the common formulas presented in Ref. [15]. Potentials which are symmetrical with respect to z and with even power of y Potentials which are symmetrical ith respect to z and with odd power of y Potentials which are anti-symmetrical with respect to z and with even power of y Potentials which are symmetrical with respect to z and with odd power of y This number of potentials which are homogeneous in Euler\'s sense with natural orders of homogeneity k are an addition to a family of quasi-polynomial 3D potentials homogeneous with any other orders of homogeneity [15]. But this class can be expanded if we explore the transformations of such type when functions remain harmonic and homogeneous in Euler\'s sense, but the degree of homogeneity can differ from the initial one.
Rotation, scaling and parallel shifting of coordinate system It is well known that the Laplace equation keeps its form under transformations such as scaling, rotation and shifting of the Cartesian coordinate system. Besides, if scaling and rotation are used, a function remains homogeneous in Euler\'s sense. That is why if we substitute the variables such as the 3D rotation of the common type [16], it is possible to get new analytical expressions for homogeneous harmonic potentials based on existing analytical formulae. However, as it is easy to see, the rotation in the plane xz yields a linear combination of symmetric and anti-symmetric quasi-polynomials that have already been obtained.
Thomson (Kelvin) formula Treatise [16] established that if U(x, y, z) is an arbitrary harmonic function, then the function where , will be harmonic as well. Substitution of variables is an inversion in a sphere, and the function remains harmonic and homogeneous. The given transformation can be used for synthesizing adrenergic receptors and ion optical systems, Refs. [3,4] are examples of this application. The fact that the 3D function U*(x, y, z) remains harmonic can be verified by substituting formula (12) to 3D Laplace Eq. (5). Besides, it should be noted that formula (12) is not the only one of this type. For example, if we employ inversion and then implement transformation of 3D rotation with respect to the origin of coordinates, allowing for parameterization using three independent parameters in the general form [16], we can obtain formula (12) where the numerators of the arguments of the U function will be linear combinations of variables x, y, z with constant coefficients. The unique property of the transformation expressed by Thomson formula (12) is that if U(x, y, z) is a function homogeneous in Euler\'s sense with the degree of homogeneity k, then the function U*(x, y, z) will be homogeneous in Euler\'s sense as well, but with the degree of homogeneity [3]. If we apply the transformation (12) again, then a reverse transformation will make the transition from the function U*(x, y, z) to the function U(x, y, z), and the degree of homogeneity restores its previous value
Applying the transformation (12) we get a new harmonic function homogeneous in Euler\'s sense, U*(x, y, z). As the function U*(x, y, z) is homogeneous in Euler\'s sense with the degree of homogeneity k, and identity (1) is satisfied, this function can be presented in the following form
Then the new function U*(x, y, z) will be written as which guarantees that this function is homogeneous in Euler\'s sense with the order (–k – 1). Here we take into account the fact that the function U(x, y, z) itself is homogeneous in Euler\'s sense with order k (the details can be found in the first volume of treatise [17] in Appendix of chapter 1 dedicated to spherical harmonic functions). It is easy to verify that transformation (12) retains the properties of an even or odd potential function with respect to the z as well as y coordinate. So, if we choose quasi-polynomial potentials (8)–(11) or (9)–(12) from Ref. [15] as a base with the degree of homogeneity k*= –k – 1, then, by applying Thomson formula (12) to them, we can construct new potentials presented in analytical form with the symmetry we need, expressed by functions homogeneous in Euler\'s sense with the necessary degree of homogeneity k.