160 Prof. Ludwig Boltzmann on the 



But we have just seen that this time is 



CdrdsdQ, 



from which it follows that T$=Cg. "N is therefore quite 

 independent of the angle 6, and not only Lord Kelvin's 

 coefficient A l5 but also A 2 and all the coefficients following 

 vanish. 



I can hardly doubt that Lord Kelvin will be satisfied with 

 this result of his test- case. 



§ 3. On the Distribution of Kinetic Energy among 

 Kelvin's Doublets. 



The other cases suggested by Lord Kelvin relate to a 

 theorem which is closely connected with the problem just 

 considered, but not by any means identical with it, nor even 

 a special case of it ; namely, the question of the equilibrium 

 of heat in the case of polyatomic gaseous molecules. It can 

 easily be shown that in this problem there exists one particular 

 distribution of vis viva which is not altered either by internal 

 motion of the molecules or by impacts. 



Let p v p 2 , . . :p n he the generalized coordinates by which 

 the positions of all constituents of a molecule are determined 

 relatively to its centre of mass, the rotation of the molecule 

 being included. Let q l7 g 2 , . . . q n be the corresponding mo- 

 menta, T the total vis viva of the molecule, Y the potential- 

 function of its internal forces, u, v, to the components of the 

 velocity of its centre of mass, and finally A and k constants ; 

 then the distribution of vis viva referred to is tbat in which 

 the number of molecules per unit volume, for values of the 

 variables u, v, %e, p i} p 2 , . . .jt? n , q l9 q 2 , . . . q n between the limits 

 u x and ui + du^ ...q n and q„ + dq n , is equal to 

 A e -K(v+T) du t . m dqnt 



If there are several kinds of molecules present the constant k, 

 but not A, must have the same value for each. 



The verification of this result in the special case imagined 

 by Lord Kelvin is so simple that I shall not consider it here. 

 But the other proof, that the distribution of kinetic energy 

 given by the above formula is the only possible one, can only 

 be established indirectly by the assumption that a certain 

 particular function must, if altered, be necessarily increased by 

 collisions. As this function is very closely connected on the 

 one hand with the quantity designated by Clausius as entropy*, 

 and on the other hand with the probability of occurrence of 



* Wien. Sitz.-Ber. Bd. lxxvi. (1877), Bd. lxxviii. (1878). 



