J. W. Gibhs—Equilibrlaui of Ileterofjeneous SuhsUmres. 175 



representing tlie two states. This will in genorul cut tlie double line 

 formed by the two sheets to which the symbols [L) and ( T^) refer. 

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

 double point thus determined with the i)oints representino- the 

 initial and linal states of the process, and thus form a reversible path 

 for the body between those states. 



The geometrical relations which indicate tlie stability of any state 

 may be easily obtained by applying the principles stated on pp. 156 ff. 

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

 sion (133) as a test of stability will reduce to 



e -t'T/-\-p'v - /.I'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 con- 

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

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



^-l''+(«-0v-(7^-/>V, (198) 



which evidently denotes the distance of the point {t',p', t') below the 

 tangent plane for the point {t, p, t), measured parallel to the axis of 'Q. 

 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 rejjresenting 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- 

 160, 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 jo= will be unstable only when there is a point in the surface 

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

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

 upwards (in the direction in which 'C is measured) the states repre- 

 sented will be stable in respect to adjacent states. This also appears 

 directly from (162). But where the surface is concave upwards in 

 either of its principal curvatures the states represented will be unsta- 

 ble in respect to adjacent states. 



When the number of component substances is greater than unity, 

 it is not possible to represent the fundamental equation by a single 

 surface. We have therefore to consider how it may be represented 

 by an infinite number of surfaces. A natural extension of either of 

 the methods already described will give us a series of surfaces in 



