From the first law Thomson obtained another fundamental equation. For every substance there is a relation connecting the pressure p (or more generally the stress of some type), the volume v (or the configuration according to the specified stress), and the temperature. We may therefore take arbitrary changes of any two of these quantities: the relation referred to will give the corresponding change of the third. Thomson chose v and t as the quantities to be varied, and supposed them to sustain arbitrary small changes dv and dt in consequence of the passage of heat to the substance from without. The amount of heat taken in is Mdv + Ndt, where Mdv and Ndt are heats required for the changes taken separately. But the substance expanding through dv does external work pdv. Thus the net amount of energy given to the substance from without is Mdv + Ndt － pdv or (M － p)dv + Ndt; and if the substance is made to pass through a cycle of changes so that it returns to the physical state from which it started, the whole energy received in the cycle must be zero. From this it follows that the rate of variation of M － p when the temperature but not the volume varies, is equal to the rate of variation of N when the volume but not the temperature varies. To see that this relation holds, the reader unacquainted with the properties of perfect differentials may proceed thus. Let the substance be subjected to the infinitesimal closed cycle of changes defined by (1) a variation consisting of the simultaneous changes dv, dt of volume and temperature, (2) a variation －dv of volume only, (3) a variation －dt of temperature only. M － p and N vary so as to have definite values for the beginning and end of each step, and the proper mean values can be written down for each step at once, and therefore the value of (M － p)dv + Ndt obtained. Adding together these values for the three steps we get the integral for the cycle. The condition that this should vanish is at once seen to be the relation stated above. This result combined with the equation A derived from the second law, gives an important expression for Carnot's function. We shall not pursue this discussion further: so much is given to make clear how certain results as to the physical properties of substances were obtained, and to explain Thomson's scale of absolute thermodynamic temperature, which is by far the most important discovery within the range of theoretical thermodynamics.