Page:Scientific Papers of Josiah Willard Gibbs.djvu/278

tension considered as a function of the potentials for the substances which are found only at the surface of discontinuity (the potentials for the substances found in the homogeneous masses and the temperature being regarded as constant) satisfy the conditions which would make the tension a maximum if the necessary conditions relative to the first differential coefficients were fulfilled.

In the foregoing discussion of stability, the surface of discontinuity is supposed plane. In this case, as the tension is supposed positive, there can be no tendency to a change of form of the surface. We now pass to the consideration of changes consisting in or connected with motion and change of form of the surface of tension, which we shall at first suppose to be and to remain spherical and uniform throughout.

In order that the equilibrium of a spherical mass entirely surrounded by an indefinitely large mass of different nature shall be neutral with respect to changes in the value of r, the radius of the sphere, it is evidently necessary that equation (500), which in this case may be written

$2\sigma =r(p'-p''),$

(522)

as well as the other conditions of equilibrium, shall continue to hold true for varying values of $r$. Hence, for a state of equilibrium which is on the limit between stability and instability, it is necessary that the equation

$2d\sigma =(p'-p'')dr+rdp'$

shall be satisfied, when the relations between $d\sigma ,dp'$, and $dr$ are determined from the fundamental equations on the supposition that the conditions of equilibrium relating to temperature and the potentials remain satisfied. (The differential coefficients in the equations which follow are to be determined on this supposition.) Moreover, if

$r{\frac {dp'}{dr}}<2{\frac {d\sigma }{dr}}-p'+p'',$

(523)

i.e., if the pressure of the interior mass increases less rapidly (or decreases more rapidly) with increasing radius than is necessary to preserve neutral equilibrium, the equilibrium is stable. But if

$r{\frac {dp'}{dr}}>2{\frac {d\sigma }{dr}}-p'+p'',$

(524)

the equilibrium is unstable. In the remaining case, when

$r{\frac {dp'}{dr}}=2{\frac {d\sigma }{dr}}-p'+p'',$

(525)

farther conditions are of course necessary to determine absolutely whether the equilibrium is stable or unstable, but in general the