and its Application to the Molecular Motions of Gases. 107 



In order, further, to obtain the mean length of path, we must 

 divide the velocity of the molecule by P ; but as in this case its 

 velocity does not remain invariable, but in general takes another 

 value at each collision, we must employ its mean velocity v. 

 That gives, if the mean length of path is called /, the equation 



V — N47T/0 3 v 



fa J**?P -Z (75) 



The value of the fraction = here occurring depends on the law 



admitted in reference to the velocities of the molecules. 



If the space in which the motion of the molecules takes place 

 is bounded and surrounded by a firm envelope, and if, notwith- 

 standing their small number, we will take into consideration the 

 collisions of the molecule in question against the envelope, we 

 must also take into account the circumstance that this molecule 

 has not the same relative velocity to the envelope as to the other 

 molecules : its mean relative velocity to the former is simply 

 equal to its mean absolute velocity, consequently equal to v. 

 Accordingly equations (72) and (73), for the case of motion of 

 all the molecules, change into 



_ N47rpV + ^ ,„„. 



^^V-NjTrp 3 )' l '° j 



4(V-NfrrpP)5 

 N4firp*r + sv ••■••••*"•* 



§ 12. We must now take up the question whether, and in 

 what manner, the theorem of the mean ergal can be applied to 

 the above-discussed molecular motions. 



This proposition is applicable if the variables which determine 

 the positions of the points change periodically, or if, at least for 

 the single variables, time-intervals can be specified which, when 

 used in forming the variations in the same manner as the periods 

 of periodic variations, conduct to variations that satisfy the equa- 

 tion of condition occurring in the theorem of the mean ergal. 



When from this point of view we contemplate the irregular 

 motions of the molecules, we immediately perceive that no pe- 

 riodicity of the variables which determine the positions of the 

 molecules is present. Even the fixing of the before-mentioned 

 time-intervals appears, on consideration of the actual motion, to 

 present considerable difficulties. I will therefore content myself 

 for the present with stating another mode of treatment — less 

 direct, it is true, but yet leading to reliable results. It is 

 founded on the fact that the motion which really takes place can 

 be replaced by another, in which all the quantities essential to 

 our investigation have the same values as in the former, and 



