326 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Let us now examine the special condition of equilibrium which 

 relates to a line at which three different masses meet, when one or 

 more of these masses is solid. If we apply the method of pages 316, 

 317 to a system containing such a line, it is evident that we shall 

 obtain in the expression corresponding to (660), beside the integral 

 relating to the surfaces, a term of the form 



to be interpreted as the similar term in (611), except so far as the 

 definition of cr has been modified in its extension to solid masses. In 

 order that this term shall be incapable of a negative value it is 

 necessary that at every point of the line 



2(<rT)^0 (671) 



for any possible displacement of the line. Those displacements are to 

 be regarded as possible which are not prevented by the solidity of the 

 masses, when the interior of every solid mass is regarded as incapable 

 of motion. At the surfaces between solid and fluid masses, the pro- 

 cesses of solidification and dissolution will be possible in some cases, 

 and impossible in others. 



The simplest case is when two masses are fluid and the third is 

 solid and insoluble. Let us denote the solid by S, the fluids by 

 A and B, and the angles filled by these fluids by a and /3 respec- 

 tively. If the surface of the solid is continuous at the line where it 

 meets the two fluids, the condition of equilibrium reduces to 



<r AB cos a = <TBS ~ ^AS (672) 



If the line where these masses meet is at an edge of the solid, the 

 condition of equilibrium is that 



OAB cos a ^ o- BS - <r A8 ,\ 



and <7 AB cos /3 ^ <r A8 - (r B8 ;/ 



which reduces to the preceding when a + /3 = 7r. Since the displace- 

 ment of the line can take place by a purely mechanical process, this 



satisfying this condition cannot form a closed figure, the crystal will be bounded by 

 two or three kinds of surfaces determined by the same condition. The kinds of 

 surface thus determined will probably generally be those for which <r has the least 

 values. But the relative development of the different kinds of sides, even if unmodi- 

 fied by gravity or the contact of other bodies, will not be such as to make S(<rs) a 

 minimum. The growth of the crystal will finally be confined to sides of a single kind. 



It does not appear that any part of the operation of removing a layer of molecules 

 presents any especial difficulty so marked as that of commencing a new layer ; yet 

 the values of fj^" which will just allow the different stages of the process to go on 

 must be slightly different, and therefore, for the continued dissolving of the crystal 

 the value of /*/' must be less (by a finite quantity) than that given by equation (665). 

 It seems probable that this would be especially true of those sides for which cr has 

 the least values. The effect of dissolving a crystal (even when it is done as slowly 

 as possible) is therefore to produce a form which probably differs from that of 

 theoretical equilibrium in a direction opposite to that of a growing crystal. 



