396 MATHEMATICAL APPENDICES 21* 



the formula defining the state of equilibrium of the medium for 

 which we are looking is 



= 38. fc + fc + to + su + a, + 



f 8m) 



dn> / 



Since the variations here contained must satisfy the above 

 conditions, we obtain by the method of elimination already used 

 the following differential equations for the determination of the 

 function F : 



= 1 + 2km(u-a) 



Here k, a, /3, y are the same constant magnitudes as before ; but 

 a, b, C are to be considered constant only so far that for all atoms' 

 in one and the same molecule they have the same value ; they 

 might possibly have different values for different molecules, 

 and, so far at least, should be taken as dependent on the state of 

 the molecule, that is, as functions of u t v, w. 



We easily see, however, that the above equations would 

 contradict each other if a, b, C changed with u, v, w from molecule 

 to molecule. 



By differentiating, for example, the first and fourth of the 

 above equations, we obtain inconsistent values of the second 

 differential coefficient with respect to the two variables u, u, which 

 are independent of each other, thus : 



T == d_(ldF\ 



dudu du\Fdu) 



