58 
ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OE GASES. 
lie between 
& and g x -M§i> a, and tu+dth, £ x and £ x +f??i» 
and let the number of these molecules be dNj. The velocities of these molecules will 
be very nearly equal and parallel. 
On account of the mutual actions of the molecules, the number of molecules which at 
a given instant have velocities within given limits will be definite, so that 
( 2 ) 
We shall consider the form of this function afterwards. 
Let the number of molecules of the second kind in unit of volume be N 2 , and let 
of these have velocities between £ 2 and fi 2 and t. 2 and where 
The velocity of any of the <£N, molecules of the first system relative to the <ZN 2 mole- 
cules of the second system is V, and each molecule M x will in the time c$£ describe a rela- 
tive path Yht among the molecules of the second system. Conceive a space bounded by 
the following surfaces. Let two cylindrical surfaces have the common axis V ht and 
radii b and b-\-db. Let two planes be drawn through the extremities of the line Y It 
perpendicular to it. Finally, let two planes be drawn through Y&# making angles <p and 
<p dip with a plane through Y parallel to the axis of x. Then the volume included 
between the four planes and the two cylindric surfaces will be Y Idbdtyht. 
If this-volume includes one of the molecules M 2 , then during the time ht there will be 
an encounter between M, and M 2 , in which b is between b and b-\-db, and <p between <p 
and <p-\-d<p. 
Since there are (ZN 1 molecules similar to M t and ^N 2 similar to M 2 in unit of volume, 
the whole number of encounters of the given kind between the two systems will be 
Ybdbd&tdNJ N 2 . 
Now let Q be any property of the molecule M ls such as its velocity in a given direction, 
the square or cube of that velocity or any other property of the molecule which is altered 
in a known manner by an encounter of the given kind, so that Q becomes Q' after the 
encounter, then during the time It a certain number of the molecules of the first kind 
have Q changed to Q', while the remainder retain the original value of Q, so that 
dQdN! = (Q' - Q)Vb dbdqtitdNi dN 2 , 
h -~^=(Q! -Q)Vbdbd<pdN l d'N 2 (3) 
refers to the alteration in the sum of the values of Q for the ^N, molecules, 
due to their encounters of the given kind with the <2N 2 molecules of the second sort. 
In order to determine the value of -It- 1 , the rate of alteration of Q among all the 
Here 
SQrfN, 
molecules of the first kind, we must perform the following integrations : — 
