In the third part of his paper Maxwell considers the consequences of supposing the particles not to be spherical. In this case the impacts would tend to set up a motion of rotation in the particles. The direction of the force acting on any particle at impact would not necessarily pass through its centre; thus by impact the velocity of its centre would be changed, and in addition the particles would be made to spin. Some part, therefore, of the energy of the particles will appear in the form of the translational energy of their centres, while the rest will take the form of rotational energy of each particle about its centre.
It follows from Maxwell’s work that for each particle the average value of these two portions of energy would be equal. The total energy will be half translational and half rotational.
This theorem, in a more general form which was afterwards given to it, has led to much discussion, and will be again considered later. For the present we will assume it to be true. Clausius had already called attention to the fact that some of the energy must be rotational unless the molecules be smooth spheres, and had given some reasons for supposing that the ratio of the whole energy to the energy of translation is in a steady state a constant. Maxwell shows that for rigid bodies this constant is 2. Let us denote it for the present by the symbol β. Thus, if the translational energy of a molecule is ½ m v², its whole energy is ½ β m v².
The temperature is still measured by the translational energy, or ½ m v²; the heat depends on the whole energy. Hence if H represent the amount of heat—measured as energy—contained by a single molecule, and T its temperature, we have—
H = βT
From this it can be shewn[50] that if γ represent the ratio of the specific heat of a gas at constant pressure to the specific heat at constant volume, then—
β = ⅔ 1/(γ-1)
For air and some other gases the value of γ has been shown to be 1·408. From this it follows that β = 1·634. Now, Maxwell’s theory required that for smooth hard particles, approximately spherical in shape, β should be 2, and hence he concludes “we have shown that a system of such particles could not possibly satisfy the known relation between the two specific heats of all gases.”
Since this statement was made many more experiments on the value of γ have been undertaken; it is not equal to 1·408 for all gases. Hence the value of β is different for various gases.
It is of some importance to notice that the value of β just found for air is very approximately 1·66 or 5/3.