Page:Scientific Papers of Josiah Willard Gibbs.djvu/137

negative for any homogeneous part of the mass, its value for the whole mass cannot be negative; and if its value cannot be zero for any homogeneous part which is not identical in phase with the mass in its given condition, its value cannot be zero for the whole except when the whole is in the given condition. Therefore, in the case supposed, the value of this expression for any other than the given condition of the mass is positive. (That this conclusion cannot be invalidated by the fact that it is not entirely correct to regard a composite mass as made up of homogeneous parts having the same properties in respect to energy, entropy, etc., as if they were parts of larger homogeneous masses, will easily appear from considerations similar to those adduced on pages 77–78.) If, then, the value of the expression (133) for the mass considered is less when it is in the given condition than when it is in any other, the energy of the mass in its given condition must be less than in any other condition in which it has the same entropy and volume. The given condition is therefore stable. (See page 57.)

Again, if it is possible to assign such values to the constants in (133) that the value of the expression shall be zero for the given fluid mass, and shall not be negative for any phase of the same components, the given condition will be evidently not unstable. (See page 57.) It will be stable unless it is possible for the given matter in the given volume and with the given entropy to consist of homogeneous parts for all of which the value of the expression (133) is zero, but which are not all identical in phase with the mass in its given condition. (A mass consisting of such parts would be in equilibrium, as we have already seen on pages 78, 79.) In this case, if we disregard the quantities connected with the surfaces which divide the homogeneous parts, we must regard the given condition as one of neutral equilibrium. But in regard to these homogeneous parts, which we may evidently consider to be all different phases, the following conditions must be satisfied. (The accents distinguish the letters referring to the different parts, and the unaccented letters refer to the whole mass.)

$\eta '+\eta ''+{\text{etc.}}=\eta ,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (134)
$v'+v''+{\text{etc.}}=v;$
$m'_{1}+m''_{1}+{\text{etc.}}=m_{1},$
$m'_{2}+m''_{2}+{\text{etc.}}=m_{2},$
${\text{etc.}}$

Now the values of $\eta ,v,m_{1},m_{2},$ etc., are determined by the whole fluid mass in its given state, and the values of ${\frac {\eta '}{v'}},{\frac {\eta ''}{v''}},$ etc., ${\frac {m'_{1}}{v'}},{\frac {m''_{1}}{v''}},$ etc., ${\frac {m'_{2}}{v'}},{\frac {m''_{2}}{v''}},$ etc., etc., are determined by the phases of the various