242 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 tempera- 

 ture 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 sur- 

 rounded 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 



9 win' f}"\ ^ P 99 > \ 



~cr / \T: /^ /' \<ji 



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 



2da- = (p f -p") dr+r dp' 



shall be satisfied, when the relations between da-, dp', arid dr are 

 determined from the fundamental equations on the supposition that 

 the conditions of equilibrium relating to temperature and the poten- 

 tials remain satisfied. (The differential coefficients in the equations 

 which follow are to be determined on this supposition.) Moreover, if 



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 



< 524 > 



the equilibrium is unstable. In the remaining case, when 



farther conditions are of course necessary to determine absolutely 

 whether the equilibrium is stable or unstable, but in general the 



