158 BUTLER 



ART. D 



volume. Secondly, when the composition of the body remains 

 unchanged, (222) becomes 



[A^ + vLv]t, „. > 0, (223) [160] 



and since, by (45), {dxp/clv)t,„, = ~p, this implies that 

 {d^/dv^)t^rn > or dp/dv must be negative. The conditions 

 (220) and (223) thus express the conditions of thermal and 

 mechanical stability of the body. 



The meaning of condition (222), as applied to the \p-v-m 

 diagram for constant temperature, easily follows from considera- 

 tions similar to those used in connection with (211). 



Again, by (15) and (50), (209) becomes 



(f" - n + v"{t" - n - v"ip" - p') 



- Hi (mi" - mi') ... - Hn'imn" - m/) > 0, (224) [161] 

 from which we may obtain the conditions 



[Af + vM - vApU < 0, (225) [162] 



and 



[Ar - /xiAmi ... - M»Aw„],.p > 0. (226) [163] 



In order to show the meaning of this condition, we will 

 consider the f-composition diagram, for constant temperature 

 and pressure, of a two component system.* It is convenient in 

 graphical representations (as in Fig. 7), to use as the variables 

 expressing composition the fractional weights of the com- 

 ponents. If we limit ourselves to phases for which Wi -{- W2 = 1, 

 the quantities mi and rrh become equal to the fractional 

 weights. Then for any change of phase. Ami = — Am2. The 

 curve AB (Fig. 7) represents the f-values of homogeneous 

 phases, at constant temperature and pressure, when m2 is varied 

 from to 1. Let the coordinates of the point D he i;, nh and 

 the coordinates of an adjacent point E he ^ -\- A^, nii -{- Arrh. 

 Let ST he the tangent to the curve AB, at the point D. The 

 slope of this tangent is given by d^/drrh = M2 — Mi, so that if E' 

 is its point of intersection with the vertical through E, the 



* Compare also Article H of this volume. 



