Thin lens equation and microscope

Introduction

The thin lens equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac1i + \frac 1o = \frac1f} to compound microscope with two lenses. The lens that is closer to the object is called objective and the the one closer to the eye is called ocular. The distance between the lenses is $L$.

Theory

We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac1{i_1} = \frac1{f_1} - \frac1{o_1} } . The distance between the lenses is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_1 + o_2 = L} which gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o_2 = L-i_1} . Thus we have for the image distance of the second lens

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac1{i_2} &= \frac1{f_2} - \frac1{o_2} \\ &= \frac1{f_2} - \frac1{L-i_1} \\ &= \frac1{f_2} - \frac1{L-\frac{f_1 o_1}{o_1 - f_1}} \\ &= \frac1{f_2} - \frac1{ \frac{L(o_1-f_1)}{o_1-f_1}-\frac{f_1 o_1}{o_1 - f_1}} \\ &= \frac1{f_2} - \frac{o_1-f_1}{ \frac{L(o_1-f_1)- f_1 o_1} \\ \end{align}}

Magnification (for the thin lens) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = - \frac io = - \frac{i_2}{o_1}} .