EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 121 



measured toward the top of the page from Pj^, / m l toward the left 

 from P 2 Q 2 , and ra 2 toward the right from P x Q r It is supposed 

 that P 1 P 2 = 1. Portions of the curves to which these points belong 

 are seen in the figure, and will be denoted by the symbols (A), (B), 

 (C). We may, for convenience, speak of these as separate curves, 

 without implying anything in regard to their possible continuity in 

 parts of the diagram remote from their common tangent AC. The 

 line of dissipated energy includes the straight line AC and portions 

 of the primitive curves (A) and (C). Let us first consider how the 

 diagram will be altered, if the temperature is varied while the 

 pressure remains constant. If the temperature receives the incre- 

 ment dt, an ordinate of which the position is fixed will receive 



the increment (-77) dt, or ydi. (The reader will easily convince 



\C16 / p fn 



himself that this is true of the ordinates for the secondary line AC, 

 as well as of the ordinates for the 

 primitive curves.) Now if we denote 

 by r\ the entropy of the phase repre- 

 sented by the point B considered as 

 belonging to the curve (B), and by rf 

 the entropy of the composite state of 

 the same matter represented by the 

 point B considered as belonging to 

 the tangent to the curves (A) and (C), 

 t(r( r(') will denote the heat yielded by a unit of matter in passing 

 from the first to the second of these states. If this quantity is 

 positive, an elevation of temperature will evidently cause a part of 

 the curve (B) to protrude below the tangent to (A) and (C), which 

 will no longer form a part of the line of dissipated energy. This 

 line will then include portions of the three curves (A), (B), and (C)j 

 and of the tangents to (A) and (B) and to (B) and (C). On the 

 other hand, a lowering of the temperature will cause the curve (B) 

 to lie entirely above the tangent "to (A) and (C), so that all the 

 phases of the sort represented by (B) will be unstable. If t(rf rj") 

 is negative, these effects will be produced by the opposite changes 

 of temperature. 



The effect of a change of pressure while the temperature remains 

 constant may be found in a manner entirely analogous. The varia- 



P, 



b 



PT 



tion of any ordinate will be -r dp or vdp. Therefore, if the 



\U>P't, m 



volume of the homogeneous phase represented by the point B is 

 greater than the volume of the same matter divided between the 

 ^phases represented by A and C, an increase of pressure will give a 

 diagram indicating that all phases of the sort represented by curve 



