239 
GASES ARISING FROM INEQUALITIES OE TEMPERATURE. 
dN=TS{l + F(UQ)f 0 (l v ,Qd£d7M .( 12 ) 
where F is a rational function of f 77, £, which we shall suppose not to contain terms 
of more than three dimensions, and f Q is the same function as in equation (9). 
Now consider two groups of molecules, each defined by the velocity-components, 
and let the two groups be distinguished by the suffixes ( : ) and ( 3 ). We have to 
estimate the number of encounters of a given kind between these two groups in a 
unit of volume in the time St, those encounters only being considered for which the 
limits of b and <f> are bfi\db and <f>^o^d<j>. 
Let us first suppose that both groups consist of mere geometrical points which do 
not interfere with each other’s motion. The group dN 1 is moving through the group 
dN 3 with the relative velocity V, and we have to find how many molecules of the 
first group approach a molecule of the second group in a manner which would, if the 
molecules acted on each other, produce an encounter of the given kind. This will be 
the case for every molecule of the first group which passes through the area bdbd<f> in 
the time St. The number of such molecules is d'Nffbdbdfft for every molecule of 
the second group, so that the whole number of pairs which pass each other within the 
given limits is 
Ybdbd<f>dN l dN 2 St, 
and if we take the time St small enough, this will be the number of encounters of the 
real molecules in the time St. 
(3.) Effect of the Encounters. 
We have next to estimate the effect of these encounters on the average values of 
different functions of the velocity-components. The effect of an individual encounter 
on these functions for the pair of molecules concerned is given in equations (3), (4), (5), 
(6), (7), each of which is of the form 
SP=Q sin 2 6 cos 2 6 .(13) 
where P and Q are functions of the velocity-components of the two molecules, and if' 
we write P for the average value of P for the N molecules in unit of volume, then 
taking the sum of the effects of the encounters— 
2SP=NSP.(14) 
We thus find 
^ = Njj|~j |~ j j |Q sin 2 6 cos 2 6Ybdhd.fffffffffrjffifff.ffrjfff, . . . (15) 
Now, since 6 is a function of b and V, the definite integral 
TStt Too 
V 6 sin 2 6 cos 2 6dbcl(f)= B.(16) 
J o J o 
will be a function of V only. 
