288 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 0" in the varied state of the system differ infinitely 

 little from the directions of the corresponding normals at O in the 

 initial state. We may therefore regard a/3, /3y as two sides of the 

 triangle representing the surfaces meeting at 0', and yS, Sa as two 

 sides of the triangle representing the surfaces meeting at O". There- 

 fore, if we join ay, this line will represent the direction of the normal 

 to the surface A-C, and the value of its tension. 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 11) would have been stable with 

 respect to the possible formation of any such surface. If the tension 

 had been less, the state of the system would have been at least 

 practically unstable. To determine whether it is unstable in the 

 strict sense of the term, or whether or not it is properly to be 

 regarded as in equilibrium, would require a more refined analysis 

 than we have used.* 



The result which we have obtained may be generalized as follows. 

 When more than three surfaces of discontinuity in a fluid system 

 meet in equilibrium along a line, with respect to the surfaces and 

 masses immediately adjacent to any point of this line, we may form 

 a polygon of which the angular points shall correspond in order to 

 the different masses separated by the surfaces of discontinuity, and 



* We may here remark that a nearer approximation in the theory of equilibrium and 

 stability might be attained by taking special account, in our general equations, of the 

 lines in which surfaces of discontinuity meet. These lines might be treated in a 

 manner entirely analogous to that in which we have treated surfaces of discontinuity. 

 We might recognize linear densities of energy, of entropy, and of the several sub- 

 stances which occur about the line, also a certain linear tension. With respect to 

 these quantities and the temperature and potentials, relations would hold analogous to 

 those which have been demonstrated for surfaces of discontinuity. (See pp. 229-231.) 

 If the sum of the tensions of the lines L' and L", mentioned above, is greater than the 

 tension of the line L, this line will be in strictness stable (although practically unstable) 

 with respect to the formation of a surface between A and C, when the tension of such 

 a surface is a little less than that represented by the diagonal ay. 



The different use of the term practically unstable in different parts of this paper need 

 not create confusion, since the general meaning of the term is in all cases the same. 

 A system is called practically unstable when a very small (not necessarily indefinitely 

 small) disturbance or variation in its condition will produce a considerable change. 

 In the former part of this paper, in which the influence of surfaces of discontinuity 

 was neglected, a system was regarded as practically unstable when such a result 

 would be produced by a disturbance of the same order of magnitude as the quantities 

 relating to surfaces of discontinuity which were neglected. But where surfaces of 

 discontinuity are considered, a system is not regarded as practically unstable, unless 

 the disturbance which will produce such a result is very small compared with the 

 quantities relating to surfaces of discontinuity of any appreciable magnitude. 



