398 MATHEMATICAL APPENDICES 21* 



whence it is proved that Maxwell's law holds good for composite 

 as weH as for simple molecules. 



There finally remains the constant k to determine ; as in the 

 special case investigated before, it depends in simple wise on the 

 mean value of the energy. 



Since Maxwell's law holds good for polyatomic molecules 

 just as for isolated atoms, the new formulae also lead to the known 

 value 



E = 1 + \m(o? + 6 2 + c 2 ) 



for the mean kinetic energy of the centroid of a molecule, and 

 thus, in the special case of a gas in equilibrium and at rest as a 



whole, to the value 



3 



22*. Atomic Motions 



But another less simple law, which is given by the function 

 (lt, t), IT), determines the distribution of the internal motions among 

 the atoms composing the molecule. To determine the constants 

 which in the expression of this function have been so far left un- 

 determined, we may in the first place remark that the atoms of one 

 kind have no special motion of translation besides the general 

 motion of the molecules. The mean values, therefore, of the 

 components u, tt, tt>, calculated from the probability-function 



must all be equal to zero when all the molecules of the whole 

 system are taken into account, and we therefore have 



a = 0, b = 0, c = 0; 



for in the integrations with respect to dtt, cfo, d\v there are as many 

 atoms with negative components U, tt, tt) as with positive com- 

 ponents of the same absolute magnitude, and these atoms are at 

 the same time endowed with the same values ^ of chemical energy ; 

 the integral can therefore vanish only if a, b, C also vanish. We 

 have then more simply 



x = m (u 2 + & + n> 2 ) + 20, 



or x is double the total energy, i.e. double the sum of the kinetic 

 and potential energies of the atom-complex of a molecule. 



