Page:Scientific Papers of Josiah Willard Gibbs.djvu/163

Rh $$DTE, FTG, HTA.$$ Now we may make either sheet of the primitive surface sink relatively to the other by the proper 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 $$ETC, DTE,$$ and also $$FTG, HTA.$$ (The lines $$CTD, BTE, ATF, HTG$$ 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 these parts will be merged in one in the surface of dissipated energy, as will be the developable surfaces $$ETC, ATH,$$ and also $$DTE, FTG.$$

It is evident that this is not a case of maximum or minimum temperature for coexistent phases under constant pressure, or of maximum or minimum pressure for coexistent phases at constant temperature.

Another case of interest 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 surface (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 developable surface joining them, and the single point of the sheet $$(C)$$ in which it meets this developable surface. Now, if the temperature or pressure is varied so as to make the sheet $$(C)$$ rise above the developable surface formed on the sheets $$(A)$$ and $$(B)$$ the surface of dissipated energy will be altered in its general features only by the removal of the single point of the sheet $$(C).$$ But if the temperature or pressure is altered so as to make a part of the sheet $$(C)$$ protrude through the developable surface formed on $$(A)$$ and $$(B),$$ the surface of dissipated energy will have the form indicated in figure 8. It will include two plane triangles $$ABC$$ and $$A'B'C',$$ a part of each of the sheets $$(A)$$ and $$(B),$$ represented in the figure by the spaces on the left of the line aAAV and on the right of the line $$bBB'b'$$ a small part $$CC'$$ of the sheet $$(C)$$, and developable surfaces formed upon these sheets taken by pairs $$ACC'A', BCC'B',$$