## 2 Materials and Methods
Then, from (8), taking into account (7), for charged particles in constant electric and magnetic fields without taking into account the relativistic effect, that is, under the condition \(m=m_{0}\), we get  
\[m\frac{\overset{\rightarrow}{dV}}{dt}=-egrad\phi-\frac{e}{c}\left[\overset{ \rightarrow}{V},grad\omega\right]. \tag{9}\]Next, we examine the properties of the emission and reflective corpuscular-optical systems based on the analysis of Equation (9). Let us introduce a system of generalized curvilinear orthogonal coordinates \(q_{1}\), \(q_{2}\), \(q_{3}\). This coordinate system is characterized by unit vectors (orths) \(i_{1}\), \(i_{2}\), \(i_{3}\) and Lame coefficients \(h_{1}\), \(h_{2}\), \(h_{3}\). In the introduced coordinate system from Equation (9), we obtain the following system of equations  
\[ \begin{array}{c}m\frac{d}{dt}\big{(}h_{1}\dot{q}_{1}\big{)}+mh_{2}\dot{q}_{2 }\left(\frac{\dot{q}_{1}}{h_{2}}\frac{\partial h_{1}}{\partial q_{2}}-\frac{ \dot{q}_{2}}{h_{1}}\frac{\partial h_{2}}{\partial q_{1}}\right)+mh_{3}\dot{q}_ {3}\left(\frac{\dot{q}_{1}}{h_{3}}\frac{\partial h_{1}}{\partial q_{3}}-\frac {\dot{q}_{3}}{h_{1}}\frac{\partial h_{3}}{\partial q_{1}}\right)=\\ =eE_{1}+\frac{e}{c}\left(h_{2}\dot{q}_{2}H_{3}-h_{3}\dot{q}_{3}H_{2}\right), \\ m\frac{d}{dt}\big{(}h_{2}\dot{q}_{2}\big{)}+mh_{1}\dot{q}_{1}\left(\frac{\dot {q}_{2}}{h_{3}}\frac{\partial h_{2}}{\partial q_{1}}-\frac{\dot{q}_{1}}{h_{2 }}\frac{\partial h_{1}}{\partial q_{2}}\right)+mh_{3}\dot{q}_{3}\left(\frac{ \dot{q}_{2}}{h_{3}}\frac{\partial h_{2}}{\partial q_{3}}-\frac{\dot{q}_{3}}{h _{2}}\frac{\dot{q}_{3}}{\partial q_{2}}\right)=\\ =eE_{2}+\frac{e}{c}\left(h_{3}\dot{q}_{3}H_{1}-h_{1}\dot{q}_{1}H_{3}\right), \\ h_{1}^{2}\dot{q}_{1}^{2}+h_{2}^{2}\dot{q}_{2}^{2}+h_{3}^{2}\dot{q}_{3}^{2}=- \frac{2e}{m}[\phi(q_{1},q_{2},q_{3})+\varepsilon],\end{array}  \tag{10}\]  
where \(ee\)--initial energy of the particle, the dot above the symbol denotes differentiation with respect to time. Equations (10)-(12) represent the general equations of motion, and Equation (12) is the equation of the conservation of energy. We study the motion of an arbitrary particle relative to the motion of the main particle, which has zero initial energy and moves along the coordinate (line) \(h_{3}q_{3}\), that is (in the general case), along the curvilinear-optical axis. In an equal period of time, the main particle flies a distance equal to \(h_{3}q_{30}\); the projection of the trajectory of an arbitrary particle will be equal to \(h_{3}q_{32}=h_{3}q_{3}\). The difference between these values \(h_{3}q_{\delta}\) is the longitudinal aberration displacement of an arbitrary particle along the main trajectory  
\[h_{3}q_{3}=h_{3}q_{30}+h_{3}q_{\delta} \tag{13}\]  
From (12) for the main trajectory we have  
\[h_{3}^{2}\dot{q}_{30}^{2}=-\frac{2e}{m}\phi(0,0,q_{30}) \tag{14}\]  
where  
\[h_{3}\dot{q}_{30}=\sigma\sqrt{-\frac{2e}{m}\phi(0,0,q_{30})}=\sigma\sqrt{- \frac{2e}{m}\phi_{0}}. \tag{15}\]Taking into account the last relations and using the notations  
\[ \begin{array}{c}q_{i}{}^{\prime}=\frac{h_{i}q_{i}}{h_{3}q_{3}q_{3}}=\frac{h_{i}dq _{i}}{h_{3}dq_{3}},\\ q_{i}{}^{\prime\prime}=\frac{1}{h_{3}}\frac{d}{dq_{3}}\left(\frac{h_{i}}{h_{3} }\frac{q_{i}}{dq_{3}}\right),\ \phi_{0}{}^{\prime}=\frac{1}{h_{3}}\frac{d\phi_{0}}{dq_{3}},\end{array}  \tag{18}\]  
we obtain the following relations  
\[\frac{d}{dt}\big{(}h_{i}\dot{q}_{i}\big{)}=-\frac{e}{m}\big{(}2\phi_{0}q_{i}{}^ {\prime\prime}+\phi_{0}{}^{\prime}q_{i}{}^{\prime}\big{)}, \tag{19}\]  
\[h_{1}\dot{q}_{1}^{2}+h_{2}\dot{q}_{2}^{2}+h_{3}\dot{q}_{3}^{2}=-\frac{2e}{m} \phi_{0}\Big{[}q_{1}^{\prime}{}^{2}+q_{2}^{\prime}{}^{2}+\big{(}1+q_{\delta}^{ \prime}\big{)}^{2}\Big{]}. \tag{20}\]  
In the last expressions and further, unless otherwise specified, primes denote differentiation with respect to the variable \(q_{30}\). Distributions of electrostatic potential, as is known, can be represented in the form of the following power series  
\[