./ W. Gihbs — Equilibrium, of Ileterogeneoiis Substances. 185 



at the point of contact, the wliole primitive surface as seen from 

 below will in general present four re-entrant furrows, radiating from 

 the point of contact, for each of which a developable surface may he 

 formed by a rolling double tangent plane. The diiferent parts of the 

 surface of dissipated energy in the vicinity of the })oint of contact are 

 represented in figure 7. ATB, ETF are parts of one sheet ot the 

 primitive surface, and CTD, GTH are parts of the other. These are 

 united by the developable surfaces BTC, DTE, FTG, HTA. Now 

 we may make either sheet of the primitive surface sink relatively to 

 the other by the pi'oper variation of temperature or pressure. If the 

 sheet to which ATB, ETF belong is that which sinks relatively, these 



parts of the surface of dissipated energy will 



be merged in one, as well as the developable 



surfaces BTC, DTE, and also FTG, HTA. 



.(The lines CTD, BTE, ATE, HTG will 



separate from one another at T, each forming 



a continuous curve.) But if the sheet of the 



primitive surface which sinks relatively is 



that to which CTD and GTH belong, then 



Fig. 7. these parts will be merged in one in the sur- 



fiice of dissipated energy, as will be the developable surfaces BTC, 



ATH, and also DTE, FTG. 



It is evident that this is not a case of maximum or minimum tem- 

 perature for coexistent phases under constant pressure, or of maximum 

 or minimum pressure for coexistent phases at constant temperature. 



Another case of intei*est is when the composition of one of three 

 coexistent phases is such as can be produced by combining the other 

 two. In this case, the primitive surface must touch the same plane 

 in three points in the same straight line. Let us distinguish the parts 

 of the primitive surface to which these points belong as the sheets (A), 

 (B), and (C), (C) denoting that which is intermediate in position. 

 The sheet (C) is evidently tangent to the developable surface formed 

 upon (A) and (B). It may or it may not intersect it at the point of 

 contact. If it does not, it must lie above the developable sur- 

 face, (unless it represents states which are unstable in regard 

 to continuous changes,) and the surface of dissipated energy 

 will include parts of the primitive sheets (A) and (B), the develop- 

 able surface joining them, and the single point of the sheet (C) 

 in which it meets this developable surface. Now, if the tempera- 

 ture or pressure is varied so as to make the sheet (C) rise above the 

 Tkans. Conn. Acad., Vol. III. 24 February, 1876. 



