Equilibrium of Vis Viva. 161 



the state referred to, the second law of thermodynamics 

 appears from it to be purely a law of probability. 



It appears to me to be a matter of interest {vide Motto) to 

 examine this latter proof for the case of the molecule provided 

 with elastic springs as imagined by Lord Kelvin, and which, 

 following his example, we shall call a " doublet." By a 

 doublet let us understand the combination of two material 

 points with masses m and m" , which attract each other with 

 a force proportional to their distance apart, m" (the nucleus) 

 is never acted upon by any other force. The masses m (shells) 

 of any two doublets impinge on each other like elastic 

 spheres whenever they come within a distance D" of each 

 other. Besides these, suppose simple atoms of mass mf to be 

 present, which collide with each other if they approach nearer 

 than distance D', and with the shells if they come within dis- 

 tance D from them. We shall always write, for the sake of 

 brevity, " shell " instead of centre of the shell, and " nucleus " 

 instead of centre of the nucleus. Let x, y, z be the coordi- 

 nates of the nucleus of a doublet referred to a system of rect- 

 angular coordinates the origin of which is at the centre of the 

 shell and the axes of which have fixed directions ; let n /f , v /f , w" 

 be the absolute components of the velocity of the nucleus, and 

 It, V, it) the same components taken relatively to the shell ; 

 let #, h, k be the components of the velocity of the centre of 

 gravity of the doublet, u, v, iv those of the shell, and u 1} v u iv 1 

 those of a single atom. Finally, suppose %{x^ y, z, a, V, W, 

 (/, h, k) dx . . . dk to be the number of doublets per unit of 

 volume for values of the variables x . . . k at the time t lying 

 between the limits x and x + dx...k and k + dk, and let 

 /( M ij v ii w \) du i dv 1 div 1 be the number of single atoms in unit 

 of volume whose velocity-components u 1} v 1} w x lie between 

 the limits 



u x and u 1 + du 1 , i\ and v x -f dv ly w 1 and Wi + dw v 



Then the expression the value of which can only be dimi- 

 nished by impacts, and which for brevity we call the entropy, 

 is 



E = 1% lo g X dx - • • dk + j/ lo g.M<i dvi dw r , 



in which the integration extends over all possible values of 

 the variables. The first term of the expression E can be 

 obtained in the following manner. We write down the value 

 of log ^ for every doublet contained in the unit of volume; 

 i. e. we insert in log % for all the variables those values which 

 it takes for the doublet under consideration. Then we add 

 together all the values of log% so obtained. In order to 



