Then, $h_{2} – h_{1}$ = h 
If $P_{1}$ and $P_{2}$ are the liquid pressures at depths $h_{1}$ and $h_{2}$ respectively and $\rho$ is its density, then according to the equation 
$P_{1}$    =    $\rho g h_{1}$
$P_{2}$     =    $\rho g h_{2}$

Let the force $F_{1}$ be exerted at the cylinder top by the liquid due to pressure $P_{1}$ and the force $F_{2}$ is exerted at the bottom of the cylinder by the liquid due to $P_{2}$.

$F_{1}$ = $P_{1}$ A    =$\rho gh_{1}A$

And,$F_{2}$ = $P_{2}$ A = $\rho g$ $h_{2}$A
$F_{1}$ and $F_{2}$ are acting on the opposite faces of the cylinder.

Therefore, the net force F will be $F_{2}$ – $F_{1}$ in the direction of $F_{2}$. This net force F on the cylinder is called the upthrust of the liquid.
$F_{2} – F_{1}$ = $\rho g A (h_{2}- h_{1})$

Or, Up thrust of liquid = $\rho g (A h)$
(Ah = V)

Or, = $\rho g V$
= $ (  \rho V ) g$ $( \rho V = m )$
= mg         ……………..(i)                                    

Here Ah is the volume V of the cylinder and is equal to the volume of the liquid displaced by the cylinder. Therefore,  $\rho gV$ is the weight of the liquid displaced. Equation (i) shows that an upthrust acts on the body immersed in a liquid and is equal to the weight of the liquid displaced, which is the Archimedes principle.

DO YOU KNOW?

Hydrometer:

A hydrometer is a glass tube with a scale marked on its stem and heavyweight in the bottom. It is partially immersed in a fluid, the density of which is to be measured. One type of hydrometer is used to measure the concentration of acid in a battery. It is called acid meter