A false balance is such that the beam remains horizontal when the pans are empty. An object weighs ${\mathrm{W}}_1$ when placed in one pen and ${\mathrm{W}}_2$ when  placed in the other pan. The true weight of the object is

When pans are empty false balance is as shown in fig

Let ${\mathrm{p}}_1 \mathrm{and} {\mathrm{p}}_2$ be the weight of pans.

Balance is in rotational equilibrium, ${\mathrm{p}}_1{\mathrm{L}}_1={\mathrm{p}}_2{\mathrm{L}}_2$

Now, an object of true weight W is kept on left pan,

$\left(\mathrm{W}+{\mathrm{p}}_1\right){\mathrm{L}}_1=\left({\mathrm{W}}_1+{\mathrm{p}}_2\right){\mathrm{L}}_2$

$\Rightarrow {\mathrm{WL}}_1={\mathrm{W}}_1{\mathrm{L}}_2$.............(1)

When object is kept on right pan, 

$\left({\mathrm{W}}_2+{\mathrm{p}}_1\right){\mathrm{L}}_1=\left(\mathrm{W}+{\mathrm{p}}_2\right){\mathrm{L}}_2$

$\Rightarrow {\mathrm{W}}_2{\mathrm{L}}_1={\mathrm{WL}}_2$...........(2)

Dividing both equation gives,

$\mathrm{W}=\sqrt{{\mathrm{W}}_1{ \mathrm{W}}_2}$