162 Prof. Ludwig Boltzmann on the 



express this symbolically we may denote the sum by 2 log x> 

 In a similar manner the second term gives us % log/, the 

 summation being extended to all the single atoms in a unit of 

 volume. 



In order to prove that E can only diminish, let us first find 

 the change produced in the value of Xlog% simply by the 

 relative motion of nucleus and shell in the doublets, assuming 

 no collisions to take place. In this case </, h, and k would 

 obviously remain constant. On the other hand, we should 

 have at any time t, 



x = Asm (at + B), l\ = Aa cos (at + B), 



and at time t = 0, 



x =A sin B, u = Aa cos B. 



If we consider all doublets for which A and B are included 

 within specified infinitely close limits, then 



dx du = dx du = Aa d A d B ; . . . . (4) 



and similarly for the y- and c-axes, 



dydv = dy dv , dzd)V = dz o d\V ... (5) 



It can easily be shown that the equation 



dx dy dz du dv d\V — dx dy dz du dv dw 



also holds good if the nucleus and shell have any other central 

 motion whatever. (Cf. my investigation of the equilibrium 

 of heat in the case of polyatomic gaseous molecules, previously 

 referred to.) If no collisions were to take place the doublets, 

 whose variables at time £ = lay between the limits x and 

 x + dx . . . k and k + dk , would have their variables at time t 

 between the limits x and x + dx...k and k-\-dk. Let us 

 therefore consider for a moment the variable t as explicitly 

 expressed in the function ^, in order not to exclude the pos- 

 sibility of a variation of ^ with time. We have then as the 

 number of the former doublets, 



X(x . . \V , g, h, k, 0)dx . . dk = Xo ^o • • dk 

 and of the latter 



X(x . .k,t)dx . . dk — xdx . . dk. 

 Hence we have 



Xodx . . dk = xdx..dk, 



and from (4) and (5) 



Xo=X> 



from which it follows that 



Xo log Xo» dx o • • dk = x lo g X dx • • dk. 





