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

Comparing the two methods, we observe that in one

$v=x,\eta =y,\epsilon =z,$

(187)

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

(188)

and in the other

$t=x,p=y,\mu =\zeta =z,$

(189)

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

(190)

Now ${\frac {dz}{dx}}$ and ${\frac {dz}{dy}}$ are evidently determined by the inclination of the tangent plane, and $z-{\frac {dz}{dx}}x-{\frac {dz}{dy}}y$ is the segment which it cuts off on the axis of $Z$. 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 $v-\eta -\epsilon$ surface, and the $t-p-\zeta$ surface, of the substance to which they relate.

In the $t-p-\zeta$ 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 $p-t-\zeta$ 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 $\zeta ^{(V)},\zeta ^{(L)},\zeta {(S)}$ denote the three ordinates determined for the same values of $p$ and $t$ by the three sheets passing through the triple point, then in one of the six spaces

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

(191)

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

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

(192)

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

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

(193)

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

(194)

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

(195)

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

(196)

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