406 
MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
We shall further suppose for the present that the internal energies are only 
slightly altered by collision, as was the case in the former problem. 
The law of distribution of internal energy will now be independent of the velocity 
of translation, so that the mean value of the internal energy of any specified 
mode, Avill be the same whether the average is taken over all collisions or over all 
molecides. 
The same notation as before will be used in connection with the motion of trans¬ 
lation. Tlie energies of the various internal modes will be denoted by e^, Co . . . . 
and their mean values taken over all molecules by E^, . . . . If ty are the 
potential and kinetic energies of which the sum is Ci then the mean value of will 
be Pi. 
AVe begin by calculating the increase at any single collision in (c~ c'-), 
(Ci “b e\), &c. Each of these quantities will be a quadratic function of the velocities 
concerned, and is symmetrical as reyards the two molecules. We next assume the 
law of distribution of tran.slational velocities to be 
(f) (u, V, H’) = 
and average the values we have found over all collisions, the procedure being 
exactly identical with that already followed in the former problem. 
Eor given velocities we arrive at an equation similar to equation (i.), p. 400. It is 
to be })articularly noticed that the translational ^■elocities can only enter through 
the term V~. We now continue in the manner of § 6. The factor A" again occurs 
multiplying every integrand, and giving rise to the term -^/K in the final result. 
By this means we arrive at equations similar to equations (viii.) and (ix.), p, 403, 
giving the rate of change of K, Ej, E,,, . . . arising from collisions. By what has 
been already said, these must be of the form— 
c/Ej/c/^ = pyP{cqjE| -f- cq^E^ + O^K} 
dYjJdt = Pv/K{«.,^E^ -b cqoEo -b . . . + ?>oK}, 
and similar equations, together with 
= py^Kjc'^E^ -b c^Eo -b . . . fi- cK} .... (xviii.). 
It is here assumed tliat a s})ecified value of any internal velocity is just as 
probable as the equal negative velocity, otherwise the mean value of products of 
difterent velocities could not be siqiposed to vanish. The equations determining the 
steady state are 
"b “b . . . + bji. = 0, 
ci^^i ”b ^D^E^ *b . . . “b = 0, 
t-jE^ -b tqE^ “b . . . . “b riv — 0.(xix.). 
