EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 245 



The condition of stability derived from (522) may in this case be 

 written 



where the quantities relating to the concave side of the surface of 

 tension are distinguished by a single accent. 



If both the masses are infinitely large, or if one which contains all 

 the components of the system is infinitely large, p' p" and o- will 

 be constant, and the condition reduces to 



dr 



;r-7 

 dv 



The equilibrium will therefore be stable or unstable according as the 

 surface of tension is less or greater than a hemisphere. 



To return to the general problem : if we denote by x the part of 

 the axis of the circular orifice intercepted between the center of the 

 orifice and the surface of tension, by R the radius of the orifice, and 

 by V the value of v f when the surface of tension is plane, we shall 

 have the geometrical relations 



and v'= F' 



By differentiation we obtain 



(r x)dx + x dr 0, 



and dv' = irx 2 dr + (Sirrx TTX Z ) dx ; 



whence (r x)dv f = irrx 2 dr. (536) 



By means of this relation, the condition of stability may be reduced 

 to the form 



^_^1_? *L<(rf-v\ r " x (537) 



dv' dv' rdv'< (P P) -jrrW 



Let us now suppose that the temperature and all the potentials 

 except one, JUL V are to be regarded as constant. This will be the case 

 when one of the homogeneous masses is very large and contains all 

 the components of the system except one, or when both these masses 

 are very large and there is a single substance at the surface of dis- 

 continuity which is not a component of either ; also when the whole 

 system contains but a single component, and is exposed to a constant 

 temperature at its surface. Condition (537) will reduce by (98) and 

 (508) to the form 



(538) 



