MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY 
425 
Solution for Loiv Temper atU7'es. 
§ 27. Ill the approximately steady state which was found to be possible at low 
temperatures, for the previous system of molecules, it was found that the various mean 
energies varied very slowly with the time. But if such a state were possible for the 
present system, it would not follow that y would only vary slowly with the time, so 
that an approximate solution of cZy/cZ^ = 0, even if it could be found, would be useless. 
Let us, however, examine under what circumstances we could have dxidt equal to 
zero, without approximation. From the remarks at the beginning of the last section, 
it is clear that the coefficients in y can be so chosen as to make dy/di equal to zero, 
but that it is only in the event of G being absent, that these values will also make 
dhjdt = 0. 
Let us suppose that by some external agency h is caused to increase uniformly 
throughout the whole gas, at a rate exactly equal to the rate at which it decreases 
in consequence of the value of dh/dt, found in the manner described at the beginning 
of § 26, being different from zero. Then a completely steady state will have become 
possible, and this is because the imaginary'agency introduces exactly sufficient energy 
to compensate that lost by radiation. In a state such as that which, in the first 
part, was described as approximately steady, the radiation was very small. If a 
similar state can be shown to be possible in the present case, the radiation will be 
very small, and therefore the energy introduced from outside will be very small. 
Hence it will be legitimate to describe the state which would be arrived at by 
checking the external flow of energy as approximately steady. 
The equation which leads to such a state is dy/d^ = 0, or 
€ + (L + M) + 
7\0r ^ 3% 1 G 0Si 3q 
(xxi.). 
The last term in ecjuation (xxi.) is of the same degree as y, so that just as in the 
last section it may be shown that y cannot contain terms of a degree higher than 
the second. And for the same reason as before, y cannot contain terms linear in any 
of the coefficients, so that we may assume 
X — + Kj. 
Now at low temperatures all the terms in equation (xxi.) are small, except terms 
of the form 
^ . _ b ^ 
0q ^ G 35j ^ ‘ 
These terms must therefore vanish approximately for all values of the variable, and 
this requires the relation 
Ti/^i — Cj/dj. 
3 I 
VOL. CXCVI. — A. 
