MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENEROY. 
415 
energy is independent of the translational energy, and hence we may suppose that 
the number of molecules per unit volume which lie within a raiige d]^ dq dr ds du is 
* {u) F [q, r, s) f{p) dq) dq dr ds du, 
or, as it will be frequently written, 
(hF/fZp c^P du, 
r/P being written for dq dr ds. 
We shall suppose that the gas has adjusted itself so that the distribution of the 
energy of translation is the permanent distribution. Thus at a point at which the 
potential energy of a molecule is rp we shall have 
As we are going to admit the existence of intermolecular forces, the potential of a 
molecule at a point will depend on the co-ordinates of the molecules as well as the 
position of the point. Thus x(j will in general he a function of x, y, z, p and r. 
Let molecules of which the co-ordinates lie within limits dp c/P du be called 
molecules of class a ; if the limits are dqj c/P' du , let the molecules be described as 
molecules of class Each of these classes will consist of a number of molecules 
wliich is indefinitely small in comparison with the total numljer of molecules 
present. 
At any moment, let us imagine all the molecules placed in position, except those 
belonging to one or other of these two classes. Let them produce a field of force such 
that if a molecule of class a is placed with its centre at the point x, y, z, then 
the potential of this molecule will be U. 
Then the probability that a molecule of class a will be found with its centre within 
an element dx dy dz at x y 2 is 
g-;i(mc2+2n)pjp 
Hence the total number of such molecules to be found in the whole unit volume 
may be obtained by integrating this expression over the whole volume, and may be 
written as 
N, = e-^‘^'^‘'^~^^FfdpdFdu .(i.), 
where 
Q-zh'i _ clx dy dz .(ii.). 
The integral is taken over the whole unit volume, since the integrand is supposed 
to vanish if the point x, y, z is such that the centre of a molecule of class a cannot 
be found there. 
The tpiantity F will be called the mean intermolecular potential for a molecule of 
class a. It is clearly a function of all the coefficients which occur in the law of 
distribution as well as of the co-ordinates of molecules of class a. If we remember 
