238 
PROFESSOR CLERK MAXWELL OX STRESSES IX RARIFIED 
the number of molecules of the kind M x which at a given instant are within the 
element of volume dxdydz, and whose velocity-components lie between the limits 
pi \dr) ^i k/£, Boltzmann has shown that the function f x must satisfy the 
equation 
+ \\\dkd v . 2 d^db\d4??(f 1 f s -f 1 %') = <> .(8) 
where denote what f becomes when in place of the velocity-components of 
before the encounter we put those of M 3 before the encounter, and those of M x and 
M. : after the encounter, respectively, and the integration is extended to all values of (f> 
and b and of r/o, £. : , the velocity-components of the second molecule M 3 . 
It is impossible, in general, to perform this integration without a knowledge, not 
only of the law of force between the molecules, but of the form of the functions f x , f 2 , 
fi'tfd, which have themselves to be found by means of the equation. 
It is only for particular cases, therefore, that the equation has hitherto been solved. 
If the medium is surrounded by a surface through which no communication of 
energy can take place, then one solution of the equation is given by the conditions 
and 
which give 
flfl — *h 
jdfi ( lf , r d fj _I _ _ a 
kz + X 7( + Y Jn + Z 7r °- 
/j=A _ 
( 9 ) 
where xfj 1 is the potential of the force whose components are X l5 Y 1} Z x , and A x is a 
constant which may be different for each kind of molecules in the medium, but h is 
the same for all kinds of molecules. 
This is the complete solution of this problem, and is independent of any hypothesis 
as to the manner in which the molecules act on each other during an encounter. The 
quantity h which occurs in this expression may be determined by finding the mean 
value of £ 2 , which is ^. N ow in the kinetic theory of gases, 
pg 2 =p=UpO .( 10 ) 
where p is the pressure, p the density, 6 the absolute temperature, and R a constant 
for a given gas. Hence 
( 11 ) 
We shall suppose, however, with Boltzmann, that in a medium in which there are 
inequalities of temperature and of velocity 
