Page:Scientific Papers of Josiah Willard Gibbs.djvu/152

Comparing the two methods, we observe that in one

${\displaystyle v=x,\eta =y,\epsilon =z,}$

(187)

${\displaystyle p=-{\frac {dz}{dx}},t={\frac {dz}{dy}},\mu =\zeta =z-{\frac {dz}{dx}}x-{\frac {dz}{dy}}y;}$

(188)

and in the other

${\displaystyle t=x,p=y,\mu =\zeta =z,}$

(189)

${\displaystyle \eta =-{\frac {dz}{dx}},v={\frac {dz}{dy}},\epsilon =z-{\frac {dz}{dx}}x-{\frac {dz}{dy}}y.}$

(190)

Now ${\displaystyle {\frac {dz}{dx}}}$ and ${\displaystyle {\frac {dz}{dy}}}$ are evidently determined by the inclination of the tangent plane, and ${\displaystyle z-{\frac {dz}{dx}}x-{\frac {dz}{dy}}y}$ is the segment which it cuts off on the axis of ${\displaystyle Z}$. The two methods, therefore, have this reciprocal relation, that the quantities represented in one by the position of a point in a surface are represented in the other by the position of a tangent plane.

The surfaces defined by equations (187) and (189) may be distinguished as the ${\displaystyle v-\eta -\epsilon }$ surface, and the ${\displaystyle t-p-\zeta }$ surface, of the substance to which they relate.

In the ${\displaystyle t-p-\zeta }$ surface a line in which one part of the surface cuts another represents a series of pairs of coexistent states. A point through which pass three different parts of the surface represents a triad of coexistent states. Through such a point will evidently pass the three lines formed by the intersection of these sheets taken two by two. The perpendicular projection of these lines upon the ${\displaystyle p-t-\zeta }$ plane will give the curves which have recently been discussed by Professor J. Thomson.^[1] These curves divide the space about the projection of the triple point into six parts which may be distinguished as follows: Let ${\displaystyle \zeta ^{(V)},\zeta ^{(L)},\zeta {(S)}}$ denote the three ordinates determined for the same values of ${\displaystyle p}$ and ${\displaystyle t}$ by the three sheets passing through the triple point, then in one of the six spaces

${\displaystyle \zeta ^{(V)}<\zeta ^{(L)}<\zeta ^{(S)},}$

(191)

in the next space, separated from the former by the line for which ${\displaystyle \zeta ^{(L)}=\zeta ^{(S)},}$

${\displaystyle \zeta ^{(V)}<\zeta ^{(S)}<\zeta ^{(L)},}$

(192)

in the third space, separated from the last by the line for which ${\displaystyle \zeta ^{(V)}=\zeta ^{(S)},}$

${\displaystyle \zeta ^{(S)}<\zeta ^{(V)}<\zeta ^{(L)},}$

(193)

in the fourth ${\displaystyle \zeta ^{(S)}<\zeta ^{(L)}<\zeta ^{(V)},}$

(194)

in the fifth ${\displaystyle \zeta ^{(L)}<\zeta ^{(S)}<\zeta ^{(V)},}$

(195)

in the sixth ${\displaystyle \zeta ^{(L)}<\zeta ^{(V)}<\zeta ^{(S)}.}$

(196)

1. ↑ See the Reports of the British Association for 1871 and 1872; and Philosophical Magazine, vol. xlvii. (1874), p. 447.