Suppose that initial pressure and volume in a McLeod Gage are given by,
where V is the reservoir volume and A is the cross-sectional area of the sealed tube, as shown schematically below.
Suppose that the final compressed pressure and volume are given by,
According to Boyle's law, we have,
For a typical manometer, . The unknown pressure Pi can be reduced to a function of the height difference h,
Furthermore, the volume of the reservoir is usually much larger than the tube,
V » A·(h0-h)
This allows us to drop the area term, resulting in a simple quadratic function for the pressure,