106 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



remaining case, in which the phase can be varied without altering the 

 value of (133) can hardly be expected to occur. The phase concerned 

 would in such a case have coexistent adjacent phases. It will be 

 sufficient to discuss the condition of stability (in respect to continuous 

 changes) without coexistent adjacent phases. 



This condition, which for brevity's sake we will call the condition 

 of stability, may be written in the form 



e"-tftf'+pV'-fr'm l ''...-fjL n 'm n " > 0, (142) 



in which the quantities relating to the phase of which the stability is 

 in question are distinguished by single accents, and those relating 

 to the other phase by double accents. This condition is by (93) 

 equivalent to 



w ' > 0, (143) 



and to 



-t'n" +P'V" - pW . ..-//>" 



" > 0. (144) 



The condition (143) may be expressed more briefly in the form 



Ae>A?7 pAv+fj. l ^m l ... + /z w Am n , (145) 



if we use the character A to signify that the condition, although 

 relating to infinitesimal differences, is not to be interpreted in accord- 

 ance with the usual convention in respect to differential equations 

 with neglect of infinitesimals of higher orders than the first, but is 

 to be interpreted strictly, like an equation between finite differences. 

 In fact, when a condition like (145) (interpreted strictly) is satisfied 

 for infinitesimal differences, it must be possible to assign limits within 

 which it shall hold true of finite differences. But it is to be remem- 

 bered that the condition is not to be applied to any arbitrary values 

 of A^, Av, Am 1} . . . Am n , but only to such as are determined by a 

 change of phase. (If only the quantity of the body which determines 

 the value of the variables should vary and not its phase, the value of 

 the first member of (145) would evidently be zero.) We may free 

 ourselves from this limitation by making v constant, which will cause 

 the term pAv to disappear. If we then divide by the constant v, 

 the condition will become 



, (146) 



v v v v 



in which form it will not be necessary to regard v as constant. As 

 we may obtain from (86) 



V V V V 



