22 Mr. S. H. Burbmy on tlie Law of Probability 



fail if confined to those values which make Q minimum. 

 But E is constant for all possible values of u x . . u n , and there- 

 fore for the values that make Q minimum. 



It appears then that the law of equipartition of energy is 

 not necessarily dependent on Maxwell's law of the distribution 

 of velocities, e~ h ^ mui , but depends on the conditions (1) that 

 every variable s has zero for its mean, and (2) that 2nEi has 

 the form Xmu 2 . 



Application of tlie Method to Gases. 



38. I assume now that the system represents a gas, or two 

 or more gases uniformly mixed, and that u Y . . u n are the 

 vector velocities, m 1 . . m n the masses, of the molecules. For 

 the dimensions of u x . . u n and the other variables in this case 

 see art. 25. 



I assume, further, that each of the gases forming the 

 mixture, or the single gas if there be only one, is homogeneous 

 as regards the constitution of its molecules, and that the 

 mixture is homogeneous as regards the proportions in which 

 different gases are mixed. Also that E represents its tem- 

 perature, and is uniform throughout. And that v represents 

 the volume containing a given number of molecules, and has, 

 up to a certain point hereafter to be defined, the same value 

 at all points. The definitions of temperature at a point and 

 density at a point present no difficulty in a system of this 

 homogeneous character. 



Since the number of molecules in volume v is propor- 

 tional to v, %mv 2 for the molecules in volume v is proportional 

 to rE ; and instead of Q = \»E it is convenient to write 

 Q = \vE, using v to express so much of the space occupied 

 by the gas as contains n molecules. 



I proceed to prove that the velocities of any two molecules 

 if sufficiently near to each other, say at distance less than p 

 from each other, will be appreciably correlated, if at suffi- 

 ciently great distance inappreciably. And I assume — b pq to 

 be, for any distance p, proportional to the force R which at 

 that distance the two molecules denoted by p and q exert on 

 each other, and that for very small values of p such force is 

 repulsive, and — b vq negative. This is consistent with art. 8. 



39. In Glausius' Yirial equation, let P denote pressure and 

 v volume, u the vector velocity, m the mass of a molecule, 

 R the force which acts between two molecules at distance p 

 from each other, R being positive when the force is repulsive.. 

 Then Glausius' equation is 



