and its Application to the Molecular Motions of Gases. Ill 



against each other the force of the shock is the same as if one of 

 them were stationary and the other had the relative velocity r of 

 the two molcules as its own velocity. Hence also a material 

 point of the same mass as the molecule, if it strikes with the 

 velocity r against a solid partition, will receive a shock of the 

 same strength. 



The relative velocities of the various occurring combinations 

 of the molecules in twos necessarily differ ; and their ratio can 

 be mathematically expressed by saying that if out of all those 

 combinations one be taken arbitrarily, the probability that the 

 relative velocity of the two molecules belonging to it lies between 

 a value?* and the infinitesimally different value r-\-dr is equal to 

 F(r)^?*, in which F denotes a function dependent on the law that 

 holds for the velocities of the individual molecules. Assuming 

 this function as given, we ascribe also to the ^N points in the 

 vessel such velocities that, if one of the points be taken arbi- 

 trarily, the probability is equal to ~F(r)dr that its velocity lies 

 between r and r -f- dr. 



In this case, not only is the total number of the shocks expe- 

 rienced by the points in the vessel equal to the total number of 

 the collisions which take place between the molecules of the 

 volume V, but there also occur exactly as many collisions of 

 each degree of force in the vessel as among the molecules. 

 Accordingly the JN points in the vessel have the same mean 

 ergal as the N molecules in the volume V. 



Besides, it can be proved that also the vis viva of the JN ma- 

 terial points is equal to that of the N molecules. I have shown, 

 in a recent memoir*, that for any system of mass-points m v m^ 

 7W3, &c. in motion the following equation is valid — 



~2m v m^=^mv*-Mv 2 c , .... (78) 



where m v and m^ are any two given masses, and the summation 

 on the left side refers to all combinations of the given masses in 

 twos, while that on the right-hand side is to be referred simply 

 to all the masses. By M is understood the sum of all the masses, 

 and by v c the velocity of the centre of gravity of the entire system. 

 When we apply this equation to the N molecules simultaneously 

 present in the volume V, we have to put v c = 0, and to assume 

 that all the masses are equal to one another. From the latter 

 circumstance it follows that we may put 



M = Nm, 



and, if v 2 denotes the arithmetic mean of all the values of v 2 



* "On various Forms of the Viral/' Pogg. Ann. Jubelbarul, p. 411; 

 Phil. Mag. S. 4. vol. xlviii. p. 1. 



