SURFACES OF DISCONTINUITY 647 



of its sides, the curvature being reckoned positive for a side if 

 it is convex outwards, negative if concave. On account of this 

 convention of signs it will be seen that the excess may be posi- 

 tive, negative or zero, showing that it is possible for a curvilinear 

 triangle to be like a rectilinear in having the sum of its angles 

 equal to two right angles.) If now a mass of the phase D can 

 exist in equilibrium there is an equilibrium for each of the three 

 triads of tensions at each of the new lines of discontinuity; there 

 is also an equilibrium for the triad of tensions at the original 

 line of discontinuity whose section by the paper is d. We 

 construct a rectilinear triangle whose sides represent the mag- 

 nitudes asc, (TcA, (Tab. Its angles must then be the supplements 

 of the angles between the tangents (or normals) at c^; so we can 



Fig. 8 



set it in such an orientation that its sides are parallel to the 

 normals at d. This is the triangle ajSy of Figures 15 and 16 of 

 Gibbs. On ^y we can construct a triangle ^y8' whose sides 

 represent the magnitudes <tbc, <tcd, (^db', its angles must be the 

 supplements of the angles between the tangents or normals at a. 

 (The sides of this triangle are not parallel to the normals to the 

 surfaces at a unless da is a straight line.) Similarly we can 

 construct triangles 7Q! 5" and a^8"'. There are various ways in 

 which the lines a8", ab'", etc. can fall. If the lines da, dh, dc 

 are straight and abc a curvilinear triangle convex inwards, 

 they fall as in Gibbs' Figure 16; if convex outwards they fall as in 

 Figure 8 of this text. Another case is shown later in Figure 9. 

 Only in special cases when the angles of the triangle abc are 



