ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 381 



But the directions of the coordinates are perfectly arbitrary, and therefore this 

 number must depend on the distance from the origin alone, that is 



Solving this functional equation, we find 



If we make A positive, the number of particles will increase with the 

 velocity, and we should find the whole number of particles infinite. We there- 

 fore make A negative and equal to j , so that the number between x and 



x + dx is 



NCe~-'dx. 



Integrating from x = oo to a; = + o> , we find the whole number of particles, 



1 



- 



fix) is therefore 7=e a . 



aVTT 



Whence we may draw the following conclusions : 



1st. The number of particles whose velocity, resolved in a certain direction, 

 lies between x and x + dx is 



dx ................................. (1). 



avir 

 2nd. The number whose actual velocity lies between v and v + dv is 



e*dv ................................. (2). 



3rd. To find the mean value of v, add the velocities of all the particles 

 together and divide by the number of particles ; the result is 



mean velocity = -r= ................................. (3). 



V7T 



4th. To find the mean value of v\ add all the values together and 

 divide by N, 



mean value of zr s = |a 2 .............................. (4). 



This is greater than the square of the mean velocity, as it ought to be. 



