| Oct. 23, 1873 | 
ON THE FINAL STATE OF A SYSTEM OF 
MOLECULES IN MOTION SUB¥ECT TO 
FORCES OF ANY KIND 
; LET perfectly elastic molecules of different kinds be in motion 
+ within a vessel with perfectly elastic sides, and let each kind of 
nolecules be acted on by forces which have a potential, the form 
of which may be different for different kinds of molecules. 
Let x, y, , be the the coordinates of a molecule, J/, and é, , ¢ 
_ the components of its velocity, and let it be required to determine 
the number of molecules of a given kind which, on an average, 
have their coordinates between x and. x + dx, yandy + dy, z and 
2+ dz, andalso their component velocities between andi +dé, 
nm andy +dyand ¢and ¢+d¢ This number must depend 
on the coordinates and the components of velocities and on the 
limits of these quantities. We may therefore write it 
aN=f(x,7,%, 81, ()dxdydzdtdnd¢ (1) 
We shall begin by. investigating the manner in which this 
quantity depends on the components of velocity, before we pro- 
_ ceed to determine in what way it depends on the coordinates. 
If we distinguish by suffixes the quantities corresponding to 
different kinds of molecules, the whole number of molecules of 
the first and second kind within a given space which have velo- 
cities within given limits may be written 
A (fy a» G) 2h, day, dG = my (2) 
and Se (Ea) Nz S) @ by Anz Uy = Ny (3) 
The number of pairs which can be formed by taking one 
molecule of each kind is 77, 7. 
Let a pair of molecules encounter each other, and after the 
encounter let their component velocities be &/, 7,', 6’ and 
&', no', G'. The nature of the encounter is completely defined 
when we know f)—&, 1) — 7, (2—G the velocity of the second 
molecule relative to the first before the encounter, and +,— 2, 
Vg—J'xs 2-2, the position of the centre of the second molecule 
relative to the first at the instant of the encounter. When these 
quantities are given, &,' — §', m’ — m’ and ('- G’, the compo- 
nents of the relative velocity after the encounter, are determin- 
able. 
Hence, putting a, B, y for these relative velocities, and a, 4, ¢ 
_ for the relative positions, we find for the number of molecules of 
_ the first kind having velocities between the limits , and & + v, 
_ &c., which encounter molecules of the second kind having velo- 
cities between the limits é and & +, &c., in such a way that 
the relative velocities lie between a and a + da, &c., and the re- 
lative positions between a and a + da, &e. 
4 Eqny ty) d dn dt. fe (E2,n2,F2) a dn dt. (abe aBy) da dbdcda dB dy (4 
and after the encounter the velocity of J/, will be between the 
limits £,' and ¢,' + dé, &c., and that of JZ, between the limits =,’ 
and é,' + dé, &c. 
The differences of the limits of velocity are equal for both 
kinds of molecules, and both before and after the encounter. 
When the state of motion of the system is in its permanent 
condition, as many pairs of molecules change their velocities 
from V,, V, to Vy’, V,' as from V,', V,' to Vy, Vo, and the 
circumstances of the encounter in the one case are precisely simi- 
Jar to those in the second. Hence, omitting for the sake of 
brevity the quantities df, &c., and , which are of the same 
value in the two cases, we find— 
Sz (Ess nvbt) Fe (Ea, nas So) = Sz (Ex, ma Or) So (Eo'y 2's So’) (5) 
writing— 
ll ad ee ee Ee ea ee 
log /(é, 1, ¢) = F(A, fm, x) (6) 
where 7, m, # are the direction cosines of the velocity, V, of the 
molecule JZ. 
Taking the logarithm of both sides of equation (5)— 
e (4, V Zhan) + Fy (MqV.7lom tte) = Fy (MV 7h, my ny) + 
Fy (MV i7lemene) (7) 
The only necessary relation between the variables before and 
after the encounter is— 
MV? + M, Vz? = M, VV," + MV? (8) 
If the righthand side of the equations (7) and (8) are constant, 
the lefthand sides will also be constant ; and since /; m, m, are 
independent of Z, 7. #7, we must have— 
Ff, =A M,V,? and F, = AM, Ve? (9) 
where 4 is a quantity independent of the components of velo- 
city, or— 
NATURE 537 
AM, V2 
A1 (Ey Mv» G) = Cre 7 (10) 
AMyV2 
Fy (Eos Nas (3) = Cye (11) 
This result as to the distribution of the velocities of the mole- 
cules at a given place is independent of the action of finite forces 
on the molecules during their encounter, for such forces do not 
affect the velocities during the infinitely short time of the en- 
counter. 
We may therefore write equation (1) 
2 ‘2 2 
¢aGe | aed did eee 
where C is a function of « yz which may be different for different 
kinds of molecules, while 4 is the same for every kind of mole- 
cule, though it may, for aught we know as yet, vary from one 
place to another. 
Let us now suppose that the kind of molecules under conside- 
ration are acted on by a force whose potential is y. The varia- 
tions of x, y, arising from the motion of the molecules during a 
time 5/ are 
bx=54, 8y = 784, 82 = (dz (13) 
and those of &, 7, ¢ in the same time due to the action of the 
force, are 
sE= _ 25, 67 = ae bg = - 4 5, (14) 
ax dy dz 
If we make 
c=logC (15 
log ites pt BOM» Hondo, 
dtdnd(dxdydz 
The variation of this quantity due to the variations 6x, 54, 52, 
df, 5m, 5G is d Z d 
c e ce 
(s+ ers Cx )ae 
d d d 
— 2am (eS* + nFh + coe) (17) 
c+ AM (4+ 24+ C) (16) 
dz 
Since the number of the molecules does not vary during their 
motion, this quantity is zero, whatever the values of &, 7, ¢ 
Hence we have in virtue of the last term— 
dA dA dA 
= == =o — =9O0 1§ 
Biante tod F dz CF 
or A is constant throughout the whole region traversed by the 
molecules. 
Next, comparing the first and second terms, we find 
c= 2AM) + B) (19) 
We thus obtain as the complete form of dV 
(AM, (F,? +7 + OF +2¥14+ BD 
ae dxdydrdtdnd¢ (20) 
when A is an absolute constant, the same for every kind of mole- 
cule in the vessel, but 4, belongs to the first kind only. To 
determine these constants, we must integrate this quantity with 
respect to the six variables, and equate the result to the number 
of molecules of the first kind. We must then, by integrating 
dN, k M, (&2 + m2 + G2 + 24) determine the whole energy 
of the system, and equate it to the original energy. We shall 
thus obtain a sufficient number of equations to determine the 
constant 4, common to all the molecules, and L,, By, &c, those 
belonging to each kind. 
The quantity A is essentially negative. Its value determines 
that of the mean kinetic energy of all the molecules ina given 
+ U(2 + n+ pez + nA + c24)ee 
ax dy 
ae I E A 
lace, which is =e -, and therefore, according to the kinetic 
piace, Da ig 
theory, it also determines the temperature of the medium at that 
place. Hence, since 4, inthe permanent state of the system, 
is the same for every part of the system, it follows that the tem- 
perature is everywhere the same, whatever forces act upon the 
molecules 
The number of molecules of the first kind in the element 
dzdydr. 
m\i AM, (2x + B) 
= AJ e dxdydz (21) 
The effect of the force whose potential is y, is therefore to cause 
the molecules of the first kind to accumulate in greater numbers 
in those parts of the vessel towards which the force acts, and 
