204 SCIENCE PROGRESS. 



3. If the spheres have sensible diameter, collisions will 

 occur among them, and we require a law of distribution of 

 velocities. It will be the well-known exponential or 

 Maxwellian law, £ -'"""- being the chance of velocity u 

 in given direction for a sphere of mass m. And if the 

 spheres have room enough the chances are all independent. 

 The permanence of this distribution, once established, has 

 been proved by several writers. 



I shall make some use of the following notation : — 

 Let V be the velocity of the common centre of gravity of 

 a pair of spheres, p their relative velocity. Then if collisions 

 occur at haphazard, it is a property of the sphere that all direc- 

 tions of p after collision are equally probable. It is a property 

 of the above distribution of velocities that for given magnitude 

 and direction of V and given magnitude of p, whether the 

 pair of spheres collide or not, all directions of p are equally 

 probable. This proves the permanence of the distribution 

 if any further proof were wanted. If this law is obeyed, the 

 system is " undisturbed," or, in Professor Tait's language, 

 is in the "special state". If it be not obeyed there is a 

 "disturbance," and the effect of collisions is to remove 

 the disturbance and reduce the system to the " special 

 state '"'. 



4. Boltzmann was the first to show that the "special 

 state" is, in the absence of external forces tending to pro- 

 duce disturbance, not only a sufficient, but a necessary, con- 

 dition for permanence. For that a certain function B is 

 always diminishing with the time, and then only becomes 

 constant when the special state is attained. 



5. If all the spheres in a certain volume have given to 

 them, in addition to their molecular velocities in the special 

 or undisturbed state, any common velocity ?/ in any direc- 

 tion, whether 71 be great or small, the law of uniform distri- 

 bution in direction of the relative velocities is unaffected. 

 So a gas in this state is undisturbed. Professor Tait calls 

 this '' mass motion " of the gas. 



6. One of the most useful forms in which to express the 

 permanent character of the motion is the virial equation — 



\pv = ^\imf + ^SSRr 

 in which R denotes the repulsive force, r the distance, be- 



