The existence of similar charged particle trajectories in th
The existence of similar charged particle trajectories in these fields due to the property of homogeneity allows to create analyzers with multi-channel registration of charged particles with respect to energy (electric spectrographs) or mass number (magnetic spectrographs). In the last 10 years, a number of analytical methods have been offered [3–8] within the framework of the uniform ideology for creating spectrographic devices, and different sphingosine-1-phosphate of fields have been built based on them. The properties of some configurations of these families are investigated; the specific schemes of spectrographs are proposed for different applications of energy and mass analysis [9–14].
Our previous article  offered for the first time an analytical theory of 3D harmonic potentials homogeneous in Euler\'s sense, which are presented as polynomials of limited power with respect to one of the Cartesian variables. These potentials are called quasi-polynomials. In this study, there are methods and techniques that give new structures for a family of quasi-polynomial potentials with ideal corpuscular optics characteristics.
Natural degree of homogeneity (special case) In Ref.  mentioned above, we have proposed a new algorithm for synthesis of 3D potentials in order to create effective electric and magnetic spectrographs on their basis. This method can be used for generating potential structures with functions of arbitrary degree of homogeneity k, homogeneous in Euler\'s sense. However, it turns out to be impossible to obtain structures with homogeneous functions with the integer degree of homogeneity k by this algorithm. The statement on a particular chosen solution needs to be changed at one of the steps of this synthesis algorithm. That is why below we present an algorithm to find the 3D potential structures needed. First stage. We build the 3D potential as a polynomial of finite power of 2n or (2n – 1) with respect to the coordinate y with coefficients which are homogeneous functions of the corresponding order with respect to two other coordinates x and z. Here respiratory system is possible to build the potential in odd form as well as in even one. They are divided into two unmixed families:
Second stage. Substitution of decomposition (3) or (4) into 3D Laplace equation and combination of the terms with the same powers of variables y gives, instead of a single equation, a chain of equations which are presented as Poisson equations for the corresponding coefficient functions. An exception is the equality corresponding to the coefficient of the highest power of the polynomial. As a result we get a number of equations for the decomposition (3)and also for another decomposition yielding an odd polynomial (4) Third stage. We solve the Laplace equation that is the last in the chain given, and choose a harmonic function, homogeneous in Euler\'s sense, with the degree of homogeneity p=k – 2n (in a case of the construction of an even polynomial) or s=m – 2n – 1 (in a case of an odd polynomial) as the generating function or where , ; c0(γ) is an unknown function that is even or odd with respect to the argument γ (instead of c0cos pγ or c0sin sγ, i.e., instead of the functions used in Ref. ). Fourth and subsequent stages. Then we search the rest of the multipliers of the lower powers of y, solving the corresponding Poisson equations with the right parts that were found at the previous stage, and satisfying the condition of being symmetrical or anti-symmetrical with respect to the coordinate z. We should note that we shall search for uncertain coefficient functions as a particular solution presented in the following form
The procedure described is repeated until a chain of recurrent equations is stopped at the first term of decomposition (3) or (4). Besides, there are variants if the coefficient multiplying the higher power of y is a harmonic and homogeneous function with zero degree of homogeneity, and therefore it will be defined as