J. TK Glhbs — Equilihrimn of Heterogeneous Substances. 495 



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 Ave apply the method of page 685 

 to a systeni containing such a line, it is evident that we shall olitain 

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

 to the surfaces, a term of the form 



f:£{G6T)Dl 



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

 definition of a has 1)een modified in its extension to solid masses. In 

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



will also soon be made good ; nor will the frequency of these occurrences be such as 

 greatly to affect the general smoothness of the surfaces, except near the edges where 

 the surfaces fall off somewliat, as before described. Now a continued growth on any 

 side of a crystal is impossible unless new layers can be formed. This will require a 

 value of //," which may exceed that given by equation (665) by a finite quantity. 

 Since the difBculty in the formation of a new layer is at or near the commencement 

 of the formation, the necessary value of // , " may be independent of the area of the 

 side, except when the side is very small. The value of // , " which is necessary for the 

 growth of the crystal will however be different for different kinds of surfaces, and 

 probably will generally be greatest for the surfaces for whicli a is least. 



On the whole, it seems not improbable that the form of very minute crystals in 

 equilibrium with solvents is principally determined by equation (665), [i. e., by the 

 condition that 2(<t s) shall be a minimum for the volume of the crystal except so far as 

 the case is modified by gravity or the contact of other bodies,) but as they grow 

 larger (in a solvent no more supersaturated than is necessary to make them grow at 

 all), the deposition of new matter on the different surfaces will be determined more by 

 the nature (orientation) of the surfaces and less by their size and relations to the 

 surrounding surfaces. As a final result, a large crystal, thus formed, will generally 

 be bounded by those surfaces alone on which the deposit of new matter takes place 

 least readily, with small, perhaps insensible truncations. If one kind of surfaces 

 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 a 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 2((ts) 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 // , " 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 a 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. 



Trans. Conn. Acad., Vol. III. 6H June, 1878. 



