EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



287 



be shown (compare page 252) that this condition is always sufficient 

 for stability with reference to motion of surfaces of discontinuity, 

 even when the variations of t, M 1> M 2 , etc. cannot be neglected in the 

 determination of the necessary condition of stability with respect to 

 such changes. 



On the Possibility of the Formation of a New Surface of Discon- 

 tinuity where several Surfaces of Discontinuity meet. 



When more than three surfaces of discontinuity between homo- 

 geneous masses meet along a line, we may conceive of a new surface 

 being formed between any two of the masses which do not meet in a 

 surface in the original state of the system. The condition of stability 

 with respect to the formation of such a surface may be easily obtained 

 by the consideration of the limit between stability and instability, as 

 exemplified by a system which is in equilibrium when a very small 

 surface of the kind is formed. 



To fix our ideas, let us suppose that there are four homogeneous 

 masses A, B, C, and D, which meet one another in four surfaces, 

 which we may call A-B, B-C, C-D, and D-A, these surfaces all meeting 

 along a line L. This is indicated in figure 11 by a section of the 



Fig. 11. 



Fig. 12. 



Fig. 13. 



surfaces cutting the line L at right angles at a point 0. In an 

 infinitesimal 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 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 0' and O" where 

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

 quadrilateral of which the sides a/3, /3y, yS, Sa are equal in numerical 

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



