EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 117 



The sheet which gives the least values of f is in each case that which 

 represents the stable states of the substance. From this it is evident 

 that in passing around the projection of the triple point we pass 

 through lines representing alternately coexistent stable and coexistent 

 unstable states. But the states represented by the intermediate 

 values of f may be called stable relatively to the states represented 

 by the highest. The differences Q L) Q V \ etc. represent the amount 

 of work obtained in bringing the substance by a reversible process 

 from one to the other of the states to which these quantities relate, 

 in a medium having the temperature and pressure common to the 

 two states. To illustrate such a process, we may suppose a plane 

 perpendicular to the axis of temperature to pass through the points 

 representing the two states. This will in general cut the double line 

 formed by the two sheets to which the symbols (L) and (V) refer. 

 The intersections of the plane with the two sheets will connect the 

 double point thus determined with the points representing the initial 

 and final states of the process, and thus form a reversible path for the 

 body between those states. 



The geometrical relations which indicate the stability of any state 

 may be easily obtained by applying the principles stated on pp. 100 ff. 

 to the case in which there is but a single component. The expression 



(133) as a test of stability will reduce to 







e-t'q+p'v-fJL'm, (197) 



the accented letters referring to the state of which the stability is in 

 question, and the unaccented letters to any other state. If we consider 

 the quantity of matter in each state to be unity, this expression may 

 be reduced by equations (91) and (96) to the form 



-'+(t-t')q-(p-p')v, (198) 



which evidently denotes the distance of the point (', p', f ') below the 

 tangent plane for the point (t, p, f), measured parallel to the axis of 

 Hence if the tangent plane for every other state passes above the 

 point representing any given state, the latter will be stable. If any 

 of the tangent planes pass below the point representing the given 

 state, that state will be unstable. Yet it is not always necessary to 

 consider these tangent planes. For, as has been observed on page 103, 

 we may assume that (in the case of any real substance) there will 

 be at least one not unstable state for any given temperature and 

 pressure, except when the latter is negative. Therefore the state 

 represented by a point in the surface on the positive side of the 

 plane p = will be unstable only when there is a point in the surface 

 for which t and p have the same values and f a less value. It follows 

 from what has been stated, that where the surface is doubly convex 



