A dasymeter was meant initially as a device to demonstrate the buoyant effect of gases like air (as shown in the adjacent pictures).
A dasymeter which allows weighing acts as a densimeter used to measure the density of gases.
The Principle of Archimedes permits to derive a formula which does not rely on any information of volume: a sample, the big sphere in the adjacent images, of known mass-density is weighed in vacuum and then immersed into the gas and weighed again.
From the known mass density of the sample (sphere) and its two weight-values, the mass-density of the gas can be calculated as: It consists of a thin sphere made of glass, ideally with an average density close to that of the gas to be investigated.
A dasymeter can be seen as a variant of that pair of scales, only immersed into gas.