Page:Scientific Papers of Josiah Willard Gibbs.djvu/164

128 $$aABb, a'A'B'b',$$ the last two being different portions of the same developable surface.

But if, when the primitive surface is constructed for such a temperature and pressure that it has three points of contact with the same plane in the same straight line, the sheet $$(C)$$ (which has the middle position) at its point of contact with the triple tangent plane intersects the developable surface formed upon the other sheets $$(A)$$ and $$(B)$$, the surface of dissipated energy will not include this developable surface, but will consist of portions of the three primitive sheets with two developable surfaces formed on $$(A)$$ and $$(B)$$ and on $$(B)$$ and $$(C).$$ These developable surfaces meet one another at the point of contact of $$(C)$$ with the triple tangent plane, dividing the portion of this sheet which belongs to the surface of dissipated energy into two parts. If now the temperature or pressure are varied so as to make the sheet $$(C)$$ sink relatively to the developable surface formed on $$(A)$$ and $$(B)$$ the only alteration in the general features of the surface of dissipated energy will be that the developable surfaces formed on $$(A)$$ and $$(B)$$ and on $$(B)$$ and $$(C)$$ will separate from one another, and the two parts of the sheet $$(C)$$ will be merged in one. But a contrary variation of temperature or pressure will give a surface of dissipated energy such as is represented in figure (9), containing two plane triangles $$ABC, A'B'C'$$ belonging to triple tangent planes, a portion of the sheet $$(A)$$ on the left of the line $$aAA'a'$$, a portion of the sheet $$(B)$$ on the right of the line $$bBB'b'$$, two separate portions $$cCy$$ and $$c'C'y'$$ of the sheet $$(C)$$, two separate portions $$aACc$$ and $$a'A'C'c'$$ of the developable surface formed on $$(A)$$ and $$(C)$$, two separate portions $$bBC \gamma$$ and $$b'B'C' \gamma '$$ of the developable surface formed on $$(B)$$ and $$(C)$$, and the portion $$A'ABB'$$ of the developable surface formed on $$(A)$$ and $$(B)$$.

From these geometrical relations it appears that (in general) the temperature of three coexistent phases is a maximum or minimum for constant pressure, and the pressure of three coexistent phases a maximum or minimum for constant temperature, when the composition of the three coexistent phases is such that one can be formed by combining the other two. This result has been obtained analytically on page 99.

The preceding examples are amply sufficient to illustrate the use of the $$m\!-\! \zeta$$ surfaces and curves. The physical properties indicated by the nature of the surface of dissipated energy have been only occasionally mentioned, as they are often far more distinctly indicated by the