A dielectric plate and a point charge: the problem with series

[reterty]

The problem of the interaction of a point charge with a dielectric plate of finite thickness implies the existence of an infinite series of image charges (see http://www.lorentzcenter.nl/lc/web/2011/466/problems/2/Sometani00.pdf). I introduce notations identical to those used in this work. The original point charge $ q $ is in a medium with a dielectric constant $ \ epsilon_1 $ . At a distance of $ d $ from it is a plate with a thickness of $ c $, made of a material with a dielectric constant $ \ epsilon_2 $. It is necessary to determine the magnitude of the image charges and the distances from them to the original charge $ q $. The first two images charges did not cause me any questions. They are, respectively, $ - \ beta q / \ epsilon_1 $ (here $ \ beta = \ frac {\ epsilon_2- \ epsilon_1} {\ epsilon_2 + \ epsilon_1} $) and $ \ beta q / \ epsilon_1 $. The distances from them to the original charge are $ 2d $ and $ 2 (c + d) $. However, the author does not further use mirror images as such (but I would like to understand it using this language). He writes about corrections to surface density from the first (a) and second (b) plate surfaces.
The first adjustment from (a) to (b) gives image charge- $ - \ beta ^ 2 q / \ epsilon_1 $ (Im confused by the sign) at a distance of $ 2 (c + d) $ (why?). It turns out that we cannot get the image charges simply by reflecting the charge relative to certain planes and multiplying it by certain factors? Please help me to understand.

 

[Cryo]

Please use ## instead of $ to open and close latex commands. The text is very difficult to read

 

[reterty]

Ok. I repeat my message with corrections.
The problem of the interaction of a point charge with a dielectric plate of finite thickness implies the existence of an infinite series of image charges (see http://www.lorentzcenter.nl/lc/web/2011/466/problems/2/Sometani00.pdf). I introduce notations identical to those used in this work. The original point charge $q$ is in a medium with a dielectric constant $\epsilon_1$ . At a distance of $d$ from it is a plate with a thickness of $c$, made of a material with a dielectric constant $\epsilon_2$. It is necessary to determine the magnitude of the image charges and the distances from them to the original charge $q$. The first two images charges did not cause me any questions. They are, respectively, $- \beta q / \epsilon_1$ (here $\beta = \frac {\epsilon_2- \epsilon_1} {\epsilon_2 + \epsilon_1}$) and $\beta q / \epsilon_1$. The distances from them to the original charge are $2d$ and $2 (c + d)$. However, the author does not further use mirror images as such (but I would like to understand it using this language). He writes about corrections to surface density from the first (a) and second (b) plate surfaces.
The first adjustment from (a) to (b) gives image charge $- \beta ^ 2 q / \epsilon_1$ (Im confused by the sign) at a distance of $2 (c + d)$ (why?). It turns out that we cannot get the image charges simply by reflecting the charge relative to certain planes and multiplying it by certain factors? Please help me to understand.