and its Application to the Molecular Motions of Gases. 109 



vessel let us imagine N' material points which move with the 

 common velocity v in all possible directions. Let the space- 

 content of the vessel be called W, and its superficies S ; then we 

 have, for the mean number of collisions of the single points 

 during unit time, equation (68), viz. 



" 4W' 



and accordingly the total number of collisions taking place in 

 the vessel during unit time will be represented by the product 



' r 4W 



In order, therefore, to get the same number of collisions in this 

 case as in the preceding, we have only to choose the vessel so 

 that W and S are equal or proportional to the quantities 

 V-N|7r /3 3 andN47r / o 2 . 



Besides, the strokes against the surface of the sides of the 

 vessel have the same multifariousness in respect of their direc- 

 tions as those against the surfaces of the spheres of action. If, 

 finally, we assume that the force exerted on the points by the 

 surface of the sides of the vessel is represented by the same 

 function of the distance as the force which the surface of a 

 sphere of action of a molecule exerts upon the points, then the 

 augmentation and diminution of the ergal take place with the 

 strokes against the sides just as with those against the spheres 

 of action of the molecules. We must thus in both cases get the 

 same mean ergal. 



It can, however, be assumed that the velocities of the points 

 moving among the stationary molecules vary according to some 

 law, which law can be mathematically expressed by saying, " If 

 one of the points be taken arbitrarily, the probability that its 

 velocity lies between a given value v and the infinitesimally dif- 

 ferent value v + dv is equal to f(v)dv," where / denotes a given 

 function. In this case we must ascribe to the points which are 

 in the vessel velocities corresponding to the same law ; and then 

 the same mean vis viva and the same mean ergal are again ob- 

 tained for the points in the vessel as for those which move 

 between the stationary molecules. 



As the above conditions which the space-content and the su- 

 perficies of the vessel must satisfy do not determine its form, 

 that of a rectangular parallelepiped can be chosen. If we then 

 determine the positions of the points by rectangular coordi- 

 nates parallel to the edges of the parallelepiped, we shall have, 

 in these coordinates, variables which alter periodically. 



We have accordingly arrived thus far — to replace the very com- 



