26 Miv S. H. Burbury on the Lmv of Probability 



that this is the discontinuous change in v at which the 

 .equation <f>(i\ . . r n ) — Ce~~ Q ceases to . be true, expressed 

 above analytically as the failure of Taylor's Theorem. It will 

 appear that this hypothesis leads to results agreeing with 

 experiment in the liquefaction of gases under pressure. 



44. On possible variations of density between different parts 

 of the system, as a consequence of the diminution of v. 



If for the whole gas, or for any separate portion of it, 

 v varies continuously, X the least real root of equation (B), 

 as applied to that portion, will in general vary continuously 

 as a function of v. 



Let us suppose the volume v of the gas to be divided into 



N parts, each equal to ^ . Let v' be a volume less than v 

 divided into N parts each equal to ^ , and let X' and -=— 



denote the values of those functions for ?/. Suppose that 

 by the compression v is reduced to v — "dv. This may 



v — v 1 

 happen in either of two ways. Suppose ~dv= ^ . Then, 



event A, as v diminishes by ~dv, each of the N parts into 

 which v is divided diminishes in the same ratio, so that 

 the density, however varying with the time, remains con- 

 stant in space throughout the whole volume v. Or, event B, 

 as v is diminished by ^v, some one of the N parts into 

 which v is divided, is diminished by "dv, and therefore, since 



cH'= — ^ — , is reduced to ^ , the other parts remaining 



unchanged in volume. 



In event A, X varies continuously throughout, and becomes 



dX 

 for each diminution "ftv of v. X — 7 - "dv. The chance of this 



dv 



happening is therefore, for each dv, represented by 



e 



(*-£*) 



uE. 



In event B, the chance is the product of two chances, 

 (1) that for ~ , X shall become V, and (2) that for vl 1— ^ ) , 

 X shall be unchanged. The chance is therefore 



