Here we put the argument of Section 2.2 into mathematical form. ^[1] It is convenient to think of the three objects, labeled by i=1, 2, 3, as fixed quantities of a compressible fluid, each of whose states can be defined by measuring the pressure p _i and volume V _i. If two objects are both surrounded by insulating walls, p and V can be varied at will for each. On the other hand, if they are separated by a diathermal wall so that equilibrium between them is established, it is found that p and V can no longer be varied arbitrarily; specification of any three of the values fixes the fourth. We can express this fact mathematically by saying that when objects 1 and 2 are in equilibrium, there exists a functional relation of the form

where F ₁₂ is some function which we can determine by measurement.

For example, if object 1 is a mercury thermometer and object 2 is air, F ₁₂ is found empirically to be

where A and B are empirical constants which depend on the properties of air and of mercury (in this particular example, p ₁ does not enter because mercury is virtually incompressible). It is convenient to use the length of the mercury column as a measure of V ₁.

Similarly, if objects 1 and 3 are in equilibrium, there is another relationship

The zero th law...