MK. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
411) 
§ 22. In addition to the gain or loss which has just been calculated, Ng will experi¬ 
ence a further g-ain or loss on account of the gradual changes in the co-ordinates of 
the various molecules. 
Let i be a typical co-ordinate, and and -f the limits of the values of ^ for 
> • • • ^ 
molecules counted in Ng. After a time dt, f will have increased to ^ + dt, so that 
id 
some molecules will enter within the above limits, and others will pass out, in course 
of the time dt. 
The molecules Avhich enter within the limits will be those for which, at the begin- 
ning of the interval, ^ had a value which was between |^g — ( —) dt and ^g. The 
number of such molecules is accordingly 
/ ^0 
Similarly the number of molecules which esca^je from between these limits is 
(No f) rf'. 
so that the resvdtant gain to Ng on account of changes in the ^ co-ordinates, the other 
co-ordinates being su|»posed to remain constant, will be 
-al(Nof)'''- 
There is no limit to the smallness of dt, so that if all the co-ordinates vary simul¬ 
taneously, the gain to Ng will be 
— (7^ N — (n — 
or 
— dt N dt^—i*^— 
in which the summation extends to all tlie co-ordinates q, r, and s. 
The value of d^^ dt must be found from the equations of motion of the molecule 
when under the action of no external forces. If -q be any co-ordinate of position, and 
E be written for the total energy T -f- V, then these equations will be of the form 
d /0E\ _ 0E 0G 
dl \c~^) ~ crj ~ dq ■ 
This leads to the following scheme of values for the various time-differentials 
dt 
= 
d?! 
— 0 
di\ 
dt 
— o. 
dt 
ds^ 
A,. _ i 
dl ' d, 
(7Gr 
du.^ 
= 0. 
dt 
~dt 
3 
H 2 
U’ 
