Illustration of the Theory of Gases. 439 



pass a given place with velocities included within the pre- 

 scribed range. It will therefore not represent the distribu- 

 tion of velocities in a given space ; for the projectiles, passing 

 in unit time, which move with the higher velocities cover 

 correspondingly greater spaces. If therefore we wish to 

 investigate the effect of a Maxwellian distribution of velocities 

 among the projectiles, we are to take, not v = Be~ kv2 , but 



v=Bve- kv2 (45) 



In this case, by (18) , 



h = k/q; (46) 



and, as we saw, the mean energy of a mass in the steady state 

 is equal to the mean energy of the projectiles which at any 

 moment of time occupy a given space. From (43), 



t f = Bq 2 k~H (47) 



Pendulums in place of Free Masses. 



We will now introduce a new element into the question by 

 supposing that the masses are no longer free to wander in- 

 definitely, but are moored to fixed points by similar elastic 

 attachments. And for the moment we will assume that the 

 stationary state is such that no change would occur in it were 

 the bombardment at any time suspended. To satisfy this con- 

 dition it is requisite that the phases of vibrations of a given 

 amplitude should have a certain distribution, dependent upon 

 the law of force. For example, in the simplest case of a force 

 proportional to displacement, where the velocity u is connected 

 with the amplitude (of velocity) r and with the phase 6 by 

 the relation u = r cos 0, the distribution must be uniform with 

 respect to 0, so that the number of vibrations in phases between 

 6 and 6 + d9 must be ddfeir of the whole number whose am- 

 plitude is r. Thus if r be given, the proportional number with 

 velocities between u and u + du is 



du f .~ 



27rV(r 2 -u 2 ) ^ 



And, in general, if r be some quantity by which the ampli- 

 tude is measured, the proportional number will be of the form 



<f>(r,u)du, (49) 



where <f> is a determinate function of r and u, dependent upon 

 the law of vibration. If now %(r) dr denote the number of 

 vibrations for which r lies between r and r + dr, we have 

 altogether for the distribution of velocities u, 



/i«)=5x.w*fr.«)* ( 5 °) 



