Then, according to Mariotte's law, the volume varies inversely as the absolute pressure; or, v0 v'; = p p0 (1)
Next, add (or subtract) heat, keeping the pressure p constant, until the volume has changed to v. The temperature is now T and, by Gay-Lussac's law, the volume is directly proportional to the absolute temperature; or, v' v = T0 T (2)
These changes have taken place by two independent successive steps, whose result has been to change the gas from the original volume, pressure, and temperature v0, p0 and T0, respectively, to the new volume, pressure, and temperature v, p, and T respectively. Now, by multiplying together the left-hand members of formulas 1 and 2, v0 v' v' v = v0 v' v' v
But, since v' is in both numerator and denominator, it may be canceled out, and the fraction becomes v0 v Next, by multiplying together the right-hand members of formulas 1 and 2, p p0 T0 T = pT0 p0T
The product of the left-hand members is equal to the product of the right-hand members, so that v0 v = pT0 p0T
By multiplying both members of this equation by p0v T0 and canceling, it is found that p0v0 T0 = pv T(3)
Hence, for any addition or subtraction of heat, and any change in volume the ratio pv T is equal to p0v0 T0, and hence is constant. If, now, the volume does not change while the gas is heated or cooled, v0 = v, and p0T0 = p T showing that,
