## 3 Results  
It was noted above that the problems of minimizing aberrations can be solved by using emission units with multipole focusing fields. In this connection, we note that the Equations of motion in parametric form for an electrostatic cathode lens with two planes of symmetry can be derived as a special case from Equations (23)-(25). In the Cartesian coordinate system, we take \(q_{1}=x,q_{2}=y,q_{3}=z,h_{1}=1,h_{2}=1,h_{3}=1\). The potential distribution from the form (21), taking into account the properties of the field symmetry under consideration, is reduced to the form  
\[ \begin{array}{c}2\[PERSON]\frac{d\tau}{dz_{\rm eff}^{2}}+\[PERSON]^{\prime}\frac{dx}{ dz_{\rm em}}+\left(\frac{\[PERSON]^{\prime\prime}}{2}-2f_{KB}\right)x=\\ =-\left(\frac{\[PERSON]^{\prime\prime}}{2}-2f_{KB}\right)^{\prime}\eta x-\left( \frac{\[PERSON]^{\prime\prime}}{4}-f_{KB}\right)^{\prime\prime}\eta^{2}x+\left( \frac{\[PERSON]^{\prime\prime}}{16}-\frac{f_{KB}^{\prime\prime}}{3}+4f_{\rm OK} \right)x^{3}+\\ +\left(\frac{\[PERSON]^{\prime\prime}}{16}-12f_{\rm OK}\right)xy^{2},\end{array}  \tag{26}\]  
\[ \begin{array}{c}2\[PERSON]\frac{d\eta}{dz_{\rm em}^{2}}+\[PERSON]^{\prime}\frac{dy}{dz _{\rm em}}+\left(\frac{\[PERSON]^{\prime\prime}}{2}+2f_{KB}\right)y=\\ =-\left(\frac{\[PERSON]^{\prime\prime}}{2}+2f_{KB}\right)^{\prime}\eta y-\left( \frac{\[PERSON]^{\prime\prime}}{4}+f_{KB}\right)^{\prime\prime}\eta^{2}y+\left( \frac{\[PERSON]^{\prime\prime}}{16}+\frac{f_{KB}^{\prime\prime}}{3}+4f_{\rm OK} \right)y^{3}+\\ +\left(\frac{\[PERSON]^{\prime\prime}}{16}-12f_{GK}\right)x^{2}y\\ 2\[PERSON]\frac{d\eta}{dz_{\rm em}}-\[PERSON]^{\prime}\eta=\\ =-\[PERSON] x^{\prime 2}-\[PERSON] y^{\prime 2}-\[PERSON]\eta^{\prime 2}+\frac{\[PERSON]^{ \prime\prime}}{2}\eta^{2}-\left(\frac{\[PERSON]^{\prime\prime}}{4}-f_{KB}\right)x^{ 2}-\left(\frac{\[PERSON]^{\prime\prime}}{4}+f_{KB}\right)y^{2}+\varepsilon.\end{array}  \tag{27}\]  
where \(f_{KB}\)--quadrupole component of focusing fields, \(f_{OK}\)--octupole component of focusing fields. In Equations (26)-(28), the arguments of all the functions are \(z_{on}\), and the primes denote differentiation with respect to the parameter \(z_{on}\). Let us recall that to replace the time value with a spatial variable (i.e., the coordinate of the main optical axis) in the initial equations of motion, Equations of the form (13), (15), and (16) were used. In this case, for electron lenses with a straight optical axis, as noted above, the Lame coefficients \(h_{1}\), \(h_{2}\), \(h_{3}\) are taken to be equal to one. It should be noted that in [16], a corrected method for studying cathode lenses with two planes of symmetry was presented. In this work, equations were also obtained that completely coincide with (26)-(28), which confirms the correctness of the above-proposed method for studying emission systems with a curvilinear axis. Let us now present the results of numerical studies of focusing parameters in doubly symmetrical and axisymmetric cathode lenses, based on the formulas obtained in [16]. The calculations carried out and a comparative analysis of the values of the aberration coefficients of three-electrode axially symmetrical and box-shaped (double-symmetrical) immersion lenses confirms the assumption that multipole cathode lenses can be used to construct sources of charged particles with improved focusing in at least one of the directions of space. Note that in the rotational coordinate system \(r\), \(z\), the equations of the trajectories of charged particles up to the third order of smallness have the form  
\[ \begin{array}{c}re^{i(\varphi+\theta)}=r_{k}u+b_{1}\overline{v}+r_{k}\sqrt{ \varepsilon_{z}}B_{21}+\\ +b_{1}\sqrt{\varepsilon_{z}}B_{22}+r_{k}{}^{3}B_{31}+r_{k}{}^{2}\overline{b}_{ 1}B_{32}+r_{k}{}^{2}b_{1}B_{33}+\\ +r_{k}b_{1}{}^{2}B_{34}+r_{k}b_{1}\overline{b}_{1}B_{35}+b_{1}{}^{2}\overline{ b}_{1}B_{36}+r_{k}\varepsilon_{2}B_{37}+b_{1}\varepsilon_{2}B_{38}\end{array}