Simple U - tube Manometer

Equating the pressure at the level XX'(pressure at the same level in a continuous body of fluid is equal),

For the left hand side:

P_x = P₁ + rg(a+h)

For the right hand side:

P_x' = P₂ + rga + r_mgh

Since P_x = P_x'

P₁ + rg(a+h) = P₂ + rga + r_mgh

P₁ - P₂ = r_mgh - rgh

i.e. P₁ - P₂ = (r_m - r)gh.
The maximum value of P₁ - P₂ is limited by the height of the manometer. To measure larger pressure differences we can choose a manometer with heigher density, and to measure smaller pressure differences with accuracy we can choose a manometer fluid which is having a density closer to the fluid density.