J. W. Gihbs — Equ'dibriimi of Heterogeneous Habstances. 405 



accents relate. But if one film is stable, the other will in general be 

 so too, and the distinction between the films with res])ect to stability 

 is of importance only at the limits of stability. If all films for all 

 values of yu,^, ///,, etc. are stable, or all within certain limits, it is evident 

 that the value of the expression must be positive when the quantities 

 are determined by any two infinitesimaliy different films within the 

 same limits. For such collective determinations of stability the 

 condition may be written 



— SzjIo'— >>i^ JyWj, — rn\Af.i,^ — etc. ^ 0, 

 or 



J(r< — r], /}/.<„ — r,, Ai.t^ - etc. (521) 



On comparison of this formula with (508), it appears that within the 

 limits of stability the second and higher differential coefficients of the 

 tension considered as a function of the potentials for the substances 

 which are found only at the sui-face of discontinuity (the potentials 

 for the substances fomid 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 fii'st 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 {)ositive, 

 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 Avith respect to changes in the value of r, the radius of the 

 sphere, it is evidently necessary that equation (500), which in this 

 may be written 



2o- = 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 



2do-={p' -p")dr -\- rdp' 



shall be satisfied, when the relations between dff, dp', and dr are 

 determined from the fundamental equations on the supposition that 



