458 J. W. Gibbs — Equilibriwn of Heterogeneous Substances. 



by the following construction, which will also indicate some important 

 pi'operties. If we form a triangle a (3 y (figure 15 or 16) having sides 

 equal to Cab, ^bc, ^ca, the angles of the triangle will be supplements 

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

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

 then construct (as in figure 16) a triangle p y S' having sides equal 

 to (Tbc, ^dc? <5'db5 upon ;/ a a triangle ;/ a 6" having sides equal to 

 O'cA, o'da, (^dc, and upon a /3 a triangle a fid'" having sides equal to 

 <3'ab, <5'db, o'da. These triangles are to be on the same sides of the lines 

 fd y., y <:i', a /5, respectively, as the triangle (x />' y. The angles of 

 these triangles will be supplements of the angles of the surfaces of 

 disco'ntinuity at a, b, and c. Thus fi y d'=.d a b, and a y 6"=.d b a. 

 Now if 6' and 6" fall together in a single point 6 within the triangle 

 afty, 6'" will fall in the same point, as in figure 15. In this 

 case we shall have fi y S + a y S=a y (3, and the three angles of the 

 curvilinear triangle a d b will be together equal to two right angles. 

 The same will be true of the three angles of each of the triangles 

 b d c, cda, and hence of the three angles of the triangle abc. But 

 if 6\ 6", 6'" do not fall together in the same point within the triangle 

 a p y, it is either possible to bring these points to coincide within 

 the triangle by increasing some or all of the tensions Cda, Cdb, <^t>c, 

 or to efiect the same result by diminishing some or all of these ten- 

 sions. (This will easily appear when one of the points S\ 6", cV" 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 d\ 6", d'" fall without the triangle oc ft y^ we may suppose the 

 greatest of the tensions Cda, o'db? ^\ic — the two greatest, when these 

 are equal, and all three when they all are equal — to diminish until 

 one of the points 6\ 6", 6'" is brought within the triangle a ft y.) 

 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 repi-esented 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, bde, cda is less than two right angles 

 (compare figures 14 and 16): in the second case, each pair of the 

 triangles a fid'", fJyd", y a 6" will overlap, at least when the ten- 

 sions CTpA, ^DB, Cdc are only a little too great to be represented as in 

 figure 1 5, and the sum of the angles of each of the triangles adb, 

 bdc, cda will be greater than two right angles. 



