408 J. W. Gibbs — JEqiiilibriu7n of Heterogeneous Substances. 



When r is large, this will be a close approximation for any values of 

 /',, unless ;k,' is very small. The two special conditions (531) and 

 (533) might be derived from very elementary considerations. 



Similar conditions of stability may be found when there are more 

 substances than one in the inner mass or the surface of discontin- 

 uity, which are not components of the enveloping mass. In this case, 

 we have instead of (526) a condition of the form 



{r y,' -f 2 rj |i-^ + (r y,' + 2 i ' J f_- + etc. <p"--p\ (534) 



from which -j--, -=— ^, etc. may be eliminated by means of equations 



derived from the conditions that 



y^' v' + l\s, y.^ v' + r^ '% etc. 

 must be constant. 



Nearly the same method may be applied to the following problem. 

 Two diiferent homogeneous fluids are separated by a diaphragm hav- 

 ing a circular orifice, their volumes being invariable except by the 

 motion of the surface of discontinuity, which adheres to the edge of 

 the orifice: — to determine the stability or instability of this surface 

 when in equilibrium. 



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



written 



dip' — p") ^ da , , ... dr , ^^\ 



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

 sion 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" find a will 

 be constant, and the condition reduces to 



<^^ ^ ^ 



The equilibi-ium 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' when the surface of tension is plane, we shall 

 have the geometrical relations 



R^ z=. 'l r X — x^ ., 



and V :=V' -\-%TtT^x^\n R^ (r - x) 



= V' + TT r ,i'2 —Irrx^. 



