*- 



392 MATHEMATICAL APPENDICES 20* 



If therefore these equations are integrated with the condition 

 that there is no translatory motion but only heat-motions, the same 

 expression 



(ir- l km)*e- km(u * + v * + wZ) du dv dw 



results for every gas to represent the probability that its com- 

 ponents of velocity have the values u, v, w. In this expression 

 the corresponding value of the molecular mass m has to be taken 

 for each kind, while the constant k has the same value for all. 



If, as in 18*, polar coordinates are employed in place of 

 Cartesian, the proposition just shown may be expressed thus, that 

 the probability of a given value w of the speed is 



for each kind of molecules. Since this expression l contains the 

 magnitudes m and w only in the combination raw 2 , the simple 

 meaning of the above theorem is that the probability of a given 

 value of the kinetic energy of a molecule is exactly the same for one 

 component gas as for another. 



From this it follows that the mean value of the kinetic energy 

 of a molecule has the same value for each kind of the molecules, 



or 



3 



Therefore in a mixture of different gases a state of equilibrium 

 results such that a molecule of each kind possesses on the average 

 the same amount of energy ; and the varying values of the energy 

 are distributed among the individual molecules of each kind 

 according to one and the same law of probability, that, namely, 

 whiph regulates the distribution in unmixed gases also. 



, therefore, two gases, the molecular motions of which are of 

 mean energy, are mixed together, no change in the distribu- 

 tion of the energy occurs. The importance of this theoretical 

 proposition stands out when we compare it with a law obtained 

 by experiment, viz. if two gases of the same temperature are 

 mixed together no change in their temperature occurs. Both 

 laws are identical if, in accordance with the fundamental ideas of 

 this theory, the kinetic energy of the molecular motion is taken as 

 the mechanical measure of the temperature. We are therefore 

 justified, on the ground of this agreement with experiment, in con- 



1 [For it is of the form 47i $r z e- r *dr, if r = fo;ia> 2 . TK.] 



