Page:Scientific Papers of Josiah Willard Gibbs.djvu/151

or by (98), if we regard $t,\mu _{1},...\mu _{n}$ as the independent variables,

$\left({\frac {d^{2}p}{d\mu _{n}^{2}}}\right)=\infty .$

(185)

In like manner we may obtain

${\frac {d^{2}p}{dt^{2}}}=\infty ,{\frac {d^{2}p}{d\mu _{1}^{2}}}=\infty ,...{\frac {d^{2}p}{d\mu _{n}^{2}}}=\infty .$

(186)

Any one of these equations, (185), (186), may be regarded, in general, as the equation of the limit of stability. We may be certain that at every phase at that limit one at least of these equations will hold true.

Geometrical Illustrations.

Surfaces in which the Composition of the Body represented is Constant.

In the second paper of this volume (pp. 33–54) a method is described of representing the thermodynamic properties of substances of invariable composition by means of surfaces. The volume, entropy, and energy of a constant quantity of a substance are represented by rectangular co-ordinates. This method corresponds to the first kind of fundamental equation described on pages 85–89. Any other kind of fundamental equation for a substance of invariable composition will suggest an analogous geometrical method. Thus, if we make $m$ constant, the variables in any one of the sets (99)–(103) are reduced to three, which may be represented by rectangular co-ordinates. This will, however, afford but four different methods, for, as has already (page 94) been observed, the two last sets are essentially equivalent when $n=1$.

The first of the above mentioned methods has certain advantages, especially for the purposes of theoretical discussion, but it may often be more advantageous to select a method in which the properties represented by two of the co-ordinates shall be such as best serve to identify and describe the different states of the substance. This condition is satisfied by temperature and pressure as well, perhaps, as by any other properties. We may represent these by two of the co-ordinates and the potential by the third. (See page 88.) It will not be overlooked that there is the closest analogy between these three quantities in respect to their parts in the general theory of equilibrium. (A similar analogy exists between volume, entropy, and energy.) If we give m the constant value unity, the third co-ordinate will also represent $\zeta$, which then becomes equal to $\mu$.