456 J. W. Gibbs—Equilibrium of Heterogeneous Substances. 
a new surface, are capable of a very simple expression. If the 
surfaces A-B, B-C, C-D, D-A, separating the masses A, B, QO, 
D, meet along a line, it is necessary for equilibrium that their 
tensions and directions at any point of the line should be such 
that a quadrilateral a, 8, 7, d may be formed with sides repre- 
senting in direction and length the normals and tensions of 
the successive surfaces. For the stability of the system with 
reference to the possible formation of surfaces between A and 
C, or between B and D, it is farther necessary that the tensions 
Gxc and ogp should be greater than the diagonals ay and f 
respectively. The conditions of stability are entirely analo- 
gous in the case of a greater number of surfaces. For the 
conditions of stability relating to the formation of a new 
phase at a line in which three surfaces of discontinuity meet, 
or at a point where four different phases meet, the reader is 
referred to the original paper. : 
Liquid films.—When a fluid exists in the form of a very 
thin film between other fluids, the great inequality of its exten- 
sion in different directions will give rise to certain peculiar 
properties, even when its thickness is sufficient for its interior 
The elasticity of a film (i. e., the increase of its tension when 
extended,) is easily accounted for. It follows from the general 
