for a System of Correlated Variables. 27 



probabil 



43. If P A denote the probability of event A, P B tbat of 

 event B, we have 



P A =e 



P E = 



/A— A' _ rfA\ w 



and P B /P A = ^- y ^^ " 

 Event B is the more probable when X — \ f is greater than 



, /x dX 



N ™ V=X-(,-,')§ +i (.— ') 2 g + &C 



and by making v — 1/ very small we may reject all higher 

 powers of it. And since , 2 is negative (art. 42) — , ^ -,— 



— that is P B > Pa> Further, when v' = Y, A/ = 0, because A/ 

 is the least root of (B), and must therefore vanish when J), 

 the product of the roots, becomes zero. Therefore ultimately 

 _X_ d\ 

 v — v' dv 



4G. It thus appears that, consistently with the theory of 

 probabilities, inequalities of density may arise at any time. 

 But so long as the system remains gas, they will be dispersed 

 by diffusion as fast as they arise. The uniformity of density 

 is therefore stable. Bnt as r approaches V. the portion of 

 gas which assumes the greater density becomes liquid, and is 

 not dispersed by diffusion. Liquefaction will therefore take 

 place of separate portions of gas successively, other portions 

 remaining as gas unaffected, until, with continuing com- 

 pression, they undergo liquefaction in their turn. And the 

 proportion of gas liquefied for a diminution, ~dc, of the volume 



of gas for the time being is . 



47. It is true that we cannot define a priori what particular 



v 

 part of v will as ^ bear the whole loss of volume "ftr. In 



the same way, when a gas liquefies under pressure, we know 

 that as the diminishing volume approaches the critical volume, 

 some portion, though we cannot define what specific portion, 

 will be liquefied, the rest remaining as gas unaffected. In 



