If W₀ be the weight of a cubic meter of air at the temperature 0°C and a barometric pressure of 0.ᵐ76, the weight of this same volume at pressure P and temperature ϴ would be
a being the coefficient of dilatation of air which is here taken at .004 in consequence of the constant presence of watery vapor.
This expresses the weight at the surface of the globe. If transferred to the height z, the weight would be diminished in the ratio of the squares of the distances from the center of the earth. We should then have
Substituting in equation 1, dividing by P and integrating between 0 and z, we get, by calling the pressure at the lower station P₀
the logarithm being Napierian.
From this we obtain