EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 291 



equilibrium. But it is evident from the points of application and 

 directions of these forces that they cannot have a zero resultant unless 

 each force is zero. We may therefore introduce a fourth mass D 

 without disturbing the parts which remain of the surfaces A-B, B-C, 

 C-D. 



It will be observed that all the angles at a, b, c, and d in figure 14 

 are entirely determined by the six surface-tensions <TAB> O"BO O"CA> O"DA> 

 <TDB> O"DC- (See (615).) The angles may be derived from the tensions 

 by the following construction, which will also indicate some important 

 properties. If we form a triangle afiy (figure 15 or 16) having sides 

 equal to O- A B> O"BO> <*"OA> ^ ne angles of the triangle will be supplements 

 of the angles at d. To fix our ideas, we may suppose the sides of the 

 triangle to be perpendicular to the surfaces at d. Upon /3y we may 

 then construct (as in figure 16) a triangle f3y$ having sides equal 

 to (7 B c> 0"DC> 0"DB upon ya a triangle yaS" having sides equal to 

 0"CA> O"DA> O"DC> an d upon a/3 a triangle a/3S'" having sides equal to 

 O"AB> O"DB> O"DA- These triangles are to be on the same sides of the lines 

 /Sy, ya, aft, respectively, as the triangle a/3y. The angles of these 

 triangles will be supplements of the angles of the surfaces of discon- 

 tinuity at a, 6, and c. Thus fiyft = dab, and ayS" = dba. Now if $ 

 and 8' fall together in a single point S within the triangle a/3y, ft" 

 will fall in the same point, as in figure 15. In this case we shall have 

 /8y<S -f- ay<$ = ay ft, and the three angles of the curvilinear triangle adb 

 will be together equal to two right angles. The same will be true of 

 the three angles of each of the triangles bdc, cda, and hence of the 

 three angles of the triangle abc. But if S', S", 8" do not fall together 

 in the same point within the triangle a/3y, it is either possible to 

 bring these points to coincide within the triangle by increasing some 

 or all of the tensions o- DA , o- DB > 0"DC> or t effect the same result by 

 diminishing some or all of these tensions. (This will easily appear 

 when one of the points &, <T, 8" falls within the triangle, if we let the 

 two tensions which determine this point remain constant, and the 

 third tension vary. When all the points S', 8", S"' fall without 

 the triangle a/3y, we may suppose the greatest of the tensions 

 O"DA> o"D B > 0"Dc t ne fc wo greatest, when these are equal, and all three 

 when they all are equal to diminish until one of the points <T, <T, "' 

 is brought within the triangle a/3y.) In the first case we may say 

 that the tensions of the new surfaces are too small to be represented 

 by the distances of an internal point from the vertices of the triangle 

 representing the tensions of the original surfaces (or, for brevity, 

 that they are too small to be represented as in figure 15); in the 

 second case we may say that they are too great to be thus represented. 

 In the first case, the sum of the angles in each of the triangles adb, 

 bdc, cda is less than two right angles (compare figures 14 and 16) ; 



