454 J. W. Gibbs — Equilibrium of Heterogeneous /Substances, 



faces cutting the line L at right angles at a point O, In an infini- 

 tesimal variation of the state of the system, we may conceive of a 

 small surface being formed between A and C (to be called A-C), 

 so that the section of the surfaces of discontinuity by the same 

 plane takes the form indicated in figure 12. Let us suppose that 



Fig. 11. 



Fig. 12. 



the condition of equilibrium (615) is satisfied both for the line L in 

 which the surfaces of discontinuity meet in the original state of the 

 system, and for the two such lines (which we may call L' and L") in the 

 varied state of the system, at least at the points O' and O" where they 

 are. cut by the plane of section. We may therefore form a quadri- 

 lateral of which the sides aft, fty, yd, da ai-e equal in numerical 

 value to the tensions of the several surfaces A-B, B-C, CD, D-A, 

 and are parallel to the normals to these surfaces at the point O 

 in the original state of the system. In like manner, for the varied 

 state of the system we can construct two triangles having similar 

 relations to the surfaces of discontinuity meeting at O' and O". 

 But the directions of the normals to the surfaces A-B and B-C 

 at O' and to C-D and D-A at O" in the varied state of the system 

 differ infinitely little from the directions of the corresponding nor- 

 mals at O in the initial state. We may therefore regard afi, fiy 

 as two sides of the triangle representing the surfaces meeting at O', 

 and yd, 6a as two sides of the triangle representing the surfaces 

 meetino- at ()". Therefore, if we join ay, this line will represent the 

 direction of the normal to the surface A-C, and the value of its ten- 

 sion. If the tension of a surface between such masses as A and C had 

 been greater than that represented by ay, it is evident that the initial 

 state of the system of surfaces (represented in figure 1 1 ) would have 

 been stable with respect to the possible formation of any such sur- 

 face. If the tension had been less, the state of the system would 

 have been at least practically unstable. To determine whether it is 

 unstal)le in the strict sense of the term, or whether or not it is prop- 



