MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
407 
These equations are not Independent; when they are added together the resulting 
equation vanishes identically in virtue of the fact that the mean total energy is, 
under all circumstances, unchanged by collisions. 
Hence the equations can all be satisfied when the variables E^, Eo . . . K are in a 
definite ratio. The distribution of energy indicated by this ratio will therefore be 
permanent, and since the equations which determine it are linear, it will be unique. 
This is the distribution discovered by Boltzmaxx, in which the energy is equally 
divided between the various degrees of freedom. 
Continuity of Path. 
§ 11. Anv of the coefficients in the above system of equations may vanish ; so tliat 
it will be possible for the equations to fall into two groups, in such a way that no 
variable occurs in both groups. The motion will in this case be steady provided all 
the variables of the first groiqi are in a given ratio, and all the variables of the 
second group are in a given ratio, but there need be no fixed ratio between the two 
groups. 
Thus the total energy of the first group will be divided according to Boltzmann’s 
Law, and the same applies to the second group, but the distribution between the two 
groups will not follow this law. 
This is the analytical expression of Maxwell’s condition as to “ Continuity of 
Path.”=^^ 
The Two Kinds of Internal Co-ordinates. 
§ 12, Let us suppose, as before, that certain velocities are subject to a retardation 
proportional to the velocities. The mean energies arising from these degrees of 
freedom will be denoted by F^, F^, . . . , the letters E^, E^. . . . being reserved for 
those energies which are not dissipated by friction. 
The system of equations (xvlli., p. 406) must now be replaced ijy 
d.Y.Jdt = pfK {Scq.E, 4- 2p„F, + 6iK}, 
dFfdt = Pv/lx {S(^ijE^ + SrjjFj — ^iFi, 
d^idt = p^/K\tcJI,■Yto:Y,feKd^ .(xx.). 
If we supjjose that at a collision only a small amount of energy can be exchanged 
between the F modes and the remaining modes, then all the coefficients p, q, V, and 
c will be small. 
It is immediately obvious that equations (xx.) may be treated exactly as equations 
* Maxwell, ‘Camb, Phil. Soc. Trans.,’ vol. 12, p. 548 ; or ‘Collected tVorks,’ vol. 2, p. 714. 
