466 Lord Rayleigh on the 



It follows from this that if three fluids can rest in contact, 

 the potential must have the same constant value on all the 

 three intersecting interfaces. But this is clearly impossible, 

 the potential on each being proportional to the sum of the 

 densities of the two contiguous fluids, as we see by considering 

 places sufficiently removed from the point of intersection. 



According to Laplace's hypothesis, then, three fluids cannot 

 rest in contact ; but the case is altered if one of the bodies be 

 solid. It is necessary, however, that the quality of solidity attach 

 to the body of intermediate density. For suppose, for example 

 (fig. 9), that the body of greatest 

 density, a v is solid, and that fluids of 

 densities cr 2 , <r z touch it and one 

 another. It is now no longer neces- 

 sary that the potential be constant 



along the interfaces (1, 2), (1,3) ; but only along the interface 

 (3, 2) . The potential at a distant point of this interface may 

 be represented by cr 2 + cr 3 . But at the point of intersection 

 the potential cannot be so low as this, being at least equal to 

 °"i + °"3> even if the angle formed by the two faces of (2) be 

 evanescent. By this and similar reasoning it follows that the 

 conditions of equilibrium cannot be satisfied, unless the solid 

 be the body of intermediate density <r 2 . 



One case where equilibrium is possible admits of very 

 simple treatment. It occurs when <r 2 — \ {<r x + <r 3 ) , and the con- 

 ditions are satisfied by supposing (fig. 10) that the fluid interface 

 is plane and perpendicular to the solid Yig. 10. 



wall. At a distance from the 

 potential is represented by cr 1 + cr B ; 

 and the same value obtains at a point 

 P, near 0, where the sphere of in- 

 fluence cuts into (2). For the areas of 

 spherical surface lost by (1) and (3) 

 are equal, and are replaced by equal 

 areas of (2) ; so that if the above con- 

 dition between the densities holds 

 good, the potential is constant all the way up to 0. The 

 sub-case, where cr 3 = 0, a 2 = \<r, was given by Clairaut. 



If the intermediate densities differ from the mean of the 

 other two, the problem is less simple ; but the general 

 tendency is easily recognized. If, for example, cr 2 > ^(a l + cr 3 ), 

 it is evident that along a perpendicular interface the potential 

 would increase as is approached. To compensate this the 

 interface must be inclined, so that, as is approached, c^ 

 loses its importance relatively to a s . In this case therefore 

 the angle between the two faces of (1) must be acute. 



