110 Prof. R. Clausius on the Theorem of the Mean Ergal, 



plicated motion of the points between the stationary molecules by 

 another motion which has the same mean ergal and the same mean 

 vis viva, and at the same time is represented by periodic variations 

 of the coordinates* 



§ 13. We now go further, to consider an infinitely extended 

 space filled with molecules which are all in motion and strike 

 against one another. 



It has already been hinted that the number of shocks under- 

 gone by a single molecule under these circumstances can be ob- 

 tained if we ascribe to the molecule in question its mean velocity 

 relative to all the other molecules as its proper velocity, assuming 

 that it moves among stationary molecules. From this we derive 

 for the number equation (74), namely 



wherein N denotes the number of the molecules which are simul- 

 taneously present in a volume V*. In order, further, to deduce 

 from this the total number of collisions which take place in the 

 volume V during unit time, the last formula must be applied to 

 all the N molecules. At the same time it must be recollected 

 that each collision concerns not merely one molecule, but two, 

 and thus occurs twice among those which all the individual mo- 

 lecules suffer. In order, therefore, to express the number of 

 collisions which take place, we must not simply multiply the 

 formula by N, but further divide the product by 2, and conse- 

 quently obtain 



NP_ NVpV 

 2 ~2(V — Nfwp*)" 



We get the same number of collisions when we suppose that 

 in a vessel of space-content V — N|-7rp 3 , and superficies N47rp 2 ; 

 AN points are present and moving in all possible directions with 

 velocities the mean value of which is r, and, without reciprocal 

 action on one another, rebound from the sides of the vessel. 



As regards the force of the collisions, this depends on the 

 angles of meeting and on the velocities. The various possible 

 angles occur in the same manner with the collisions of the mo- 

 lecules as with the points striking against the sides of the vessel ; 

 and hence we can leave them out of view. With respect to the 

 velocities, we remark, first, that with two molecules which strike 



* Strictly speaking, if N molecules are present in the selected volume 

 V, each of the N molecules may strike against N— 1 others: hence N — 1 

 must be put in the above formula in place of N. But when V is a volume 

 containing a ponderable quantity (e. g. a unit of weight) of the gas, N is a 

 number so immensely great that 1 may be without hesitation neglected, and 

 N put instead of N— 1. 



