240 EQTJILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



To make this incapable of a negative value, we must have 



fJLg = fJ.' g , unless mf = 0, 

 /*;=/4, unless mf = 0. 



In virtue of these relations and by equation (502), the expression 

 (519), i.e., (516), will reduce to 



a-" s or' s, 

 which will be positive or negative according as 



<r"-<r' (520) 



is positive or negative. 



That is, if the tension of the film is less than that of any other film 

 of the same components which can exist between the same homo- 

 geneous masses (which has therefore the same values of t, /* a , ju. b , etc.), 

 and which moreover has the same values of the potentials /m g , fa, etc., 

 so far as it contains the substances to which these relate, then the 

 first film will be stable. But the film will be practically unstable, 

 if any other such film has a less tension. (Compare the expression 

 (141), by which the practical stability of homogeneous masses is 

 tested.) 



It is, however, evidently necessary for the stability of the surface 

 of discontinuity with respect to deformation, that the value of the 

 superficial tension should be positive. Moreover, since we have by 

 (502) for the surface of discontinuity 



e s - trf - jUL a m* - fji b mf - etc. - /*,mj - // 7t m^ - etc. = a-s, 4 

 and by (93) for the two homogeneous masses 



e' - trf +pv' - fi a m a ' - jui b m b - etc. = 0, 

 e" - tif +pv" - n a m a " - fji b m b rf - etc. = 0, 

 if we denote by 



e, ??, v, m a , m b) etc., m gt m h , etc., 



the total energy, etc. of a composite mass consisting of two such 

 homogeneous masses divided by such a surface of discontinuity, we 

 shall have by addition of these equations 



tr\ +pv fj. a m a /UL b m b etc. fjL g m ff jUL h m h etc. = crs. 



Now if the value of a- is negative, the value of the first member of 

 this equation will decrease as s increases, and may therefore be 

 decreased by making the mass to consist of thin alternate strata of 

 the two kinds of homogeneous masses which we are considering. 

 There will be no limit to the decrease which is thus possible with a 

 given value of v, so long as the equation is applicable, i.e., so long 

 as the strata have the properties of similar bodies in mass. But it 



