MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
399 
It is obvious that the degrees ot freedom which have been rejected do not in any 
way influence the motion of the sphere, whereas if they are counted as separate 
degrees of freedom, the series of states through which the sphere is made to pass by 
varying all the co-ordinates, is no longer such as to satisfy Maxwell’s condition 
of “Continuity of Path.” (See § 11 of the present paper.) 
PAPvT I. 
The Distribution of Energy in a Gas of which the Molecules are 
Loaded Spheres. 
The Transfer of Energy caused hy Collisions. 
§ 3. We may begin with the consideration of a gas of which the molecules are 
loaded spheres of the kind mentioned in the last section. These spheres are to be 
perfectly elastic, each of radius a, and the centre of gravity of each is to be at a 
small distance r from the geometrical centre. 
We require to find equations giving the transfer of energy between the various 
degrees of freedom in such a gas. If we know the law of distribution of the various 
co-ordinates of the molecules, we shall he able, upon making the usual assumptions of 
the kinetic theory, to calculate the number of collisions which are such that the 
values of the variables, which are required to completely specify a collision, lie 
within certain specified small ranges of value. At each of these collisions the 
transfer of energy is the same, a function of the variables which specify the collision ; 
so that by multiplying this quantity by the number of collisions of the kind under 
consideration which occur during the interval dt, and integrating over all possible 
values of the variables which specify the collision, we shall obtain an expression for 
the transfer of energy during the time dt. 
Let us denote the mean energy of translation of all the molecules at any specified 
instant by K, the mean energy of rotation by H. If we regard the number of mole¬ 
cules in the gas as infinite, we may regard the quantities H and K as varying 
continuously with the time, and the expressions which have just been found for the 
transfer of energy will enable us to calculate and ddLjdt corresponding to any 
state of the gas. The values of dTLjdt, dTE/dt will in general depend on r, a, p (the 
density of the gas) and the coefficients which occur in the various laws of dis¬ 
tribution of co-ordinates. 
If we consider the case in which r = 0, we find that no transfer of energy is 
possible, so that dJd/dt, diK/dt must vanish with r. We further notice that these 
dififerential coefficients must remain unaltered, if — r be written for r, so that, 
assuming for the moment that they can be expanded in ascending powers 
of r, we see that the lowest power of r which can occur is We shall suppose 
r to be so small that terms in C may be^ neglected in comparison with terms 
containing r^. 
