EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 241 



may easily be shown (as in a similar case on pages 77, 78) that 

 when the values of 



t, P> t*a> ^ 6 , etc., p gt // A , etc., 



are regarded as fixed, being determined by the surface of discon- 

 tinuity in question, and the values of 



e, r\, m a , m b) etc., m g> m h , etc., 



are variable and may be determined by any body having the given 

 volume v, the first member of this equation cannot have an infinite 

 negative value, and must therefore have a least possible value, which 

 will be negative, if any value is negative, that is, if <r is negative. 



The body determining e, 77, etc. which will give this least value 

 to this expression will evidently be sensibly homogeneous. With 

 respect to the formation of such a body, the system consisting of the 

 two homogeneous masses and the surface of discontinuity with the 

 negative tension is by (53) (see also page 79) at least practically 

 unstable, if the surface of discontinuity is very large, so that it can 

 afford the requisite material without sensible alteration of the values 

 of the potentials. (This limitation disappears, if all the component 

 substances are found in the homogeneous masses.) Therefore, in a 

 system satisfying the conditions of practical stability with respect to 

 the possible formation of all kinds of homogeneous masses, negative 

 tensions of the surfaces of discontinuity are necessarily excluded. 



Let us now consider the condition which we obtain by applying 

 (516) to infinitesimal changes. The expression may be expanded as 

 before to the form (519), and then reduced by equation (502) to the 

 form ' 



That the value of this expression shall be positive when the quanti- 

 ties are determined by two films which differ infinitely little is a 

 necessary condition of the stability of the film to which the single 

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

 so too, and the distinction between the films with respect to stability 

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

 values of /m ff , /ji h , 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 infinitesimally different films 

 within the same limits. For such collective determinations of stability 

 the condition may be written 



sAo- m^A/Zj, ml&/uL h etc.>0, 

 or 



Ao-<-r ff A^-r ft A// A -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 



G.I. 



