J. W. Gihbs — Equilibrium of Heterogeneous Substa?ices. 455 



erly 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 the sides to 

 these surfaces, each side being perpendicular to the corresponding 

 surface, and equal to its tension. With respect to the formation of 

 new surfaces of discontinuity in the vicinity of the point especially 

 considered, the system is stable, if every diagonal of the polygon is 

 less, and practically unstable, if an}^ diagonal is greater, than the 

 tension which woidd belong to the surface of discontinuity between 

 the corresponding masses. In the limiting case, when the diagonal 

 is exactly equal to the tension of the corresponding surface, the sys- 

 tem may often be determined to be unstable by the application of 

 the principle enunciated to an adjacent point of the line in which the 

 surfaces of discontinuity meet. But when, in the polygons con- 

 structed for all points of the line, no diagonal is in any case greater 



* 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. 391-393.) 

 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 difEerent 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. 



Trans. Conn. Acad., Vol. III. 58 March, 1878. 



