for a System of Correlated Variables. 25 



positions which a material system may have in statical 

 equilibrium, that one will be chosen for which % is minimum. 

 In fact, by making h infinite, the first theorem is reduced to 

 the statical theorem. If this theorem be not accepted in all 

 its generality, still it must be accepted for a gas, for which in 

 fact Boltzmann proved it. It amounts merely to saying that 

 the system chooses, for given total energy, the motion of 

 least resistance. 



42. It is evident that, as v diminishes, and therefore the 

 coefficients of products in Q increase in absolute magnitude, 

 Q generally diminishes. For the only terms in Q that can 

 be negative are the terms containing products. And such 

 terms can always be made negative by making the correlation 

 coefficient, e. g. b pq , negative if u p and u q have the same sign, 

 or by making it positive if they have opposite signs. As we 

 have already seen, there are, if the system represents a gas, 

 physical reasons why when b pq is not inappreciable, u p and u a 

 should in general have the same sign, and therefore why, if 

 Q is to be as small as possible, b pq should be negative. 



But, at all events when v is very great, and therefore the 



correlation coefficients very small, -|y-j which is equal to 



itpiiq, has the opposite sign to b pq . For D pq = —b pq multiplied 

 by a coaxial minor of D which is positive, plus terms con- 

 taining products of more than one h, which, v being great 

 enough, are negligible compared with the term containing 

 only b pq . Therefore D pq is generally of the opposite sign 

 to r p r g , that is to b pq . On the other hand, /3 pq of (20) is 

 equal to Upliq, and is positive. 



Since, further, Q diminishes as r diminishes, and Q = X»E, 

 it follows that X, diminishes as v diminishes, or if X be the 



least root of B, -=- is positive. Also, since as v diminishes 



the correlation terms in Q assume relatively more importance, 



. generally increases as v diminishes, and — - is negative. 



43. Having thus explained what I consider to be the 

 general form of motion of the dense gas, I now assume that 

 it is being compressed with constant temperature, that is r 

 diminishes, E remaining constant. 



I make an hypothesis, namely this : — That " the critical 

 volume " V at which a gas under compression undergoes 

 liquefaction, is the volume at which D = 0, and that in such 

 liquefaction r, the volume, changes discontinuous!}'. And 



