298 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



is therefore sufficient to consider the stability of these lines and sur- 

 faces. We shall suppose that the relations mentioned are satisfied. 



If we denote by W y the work gained in forming the mass E (of 

 such size and form as to be in equilibrium) in place of the portions 

 of the other masses which are suppressed, and by W 8 the work ex- 

 pended in forming the new surfaces in place of the old, it may easily 

 be shown by a method similar to that used on page 292 that 



F s = fTF v , (637) 



whence TF 8 - F v = iTT v ; (638) 



also, that when the volume E is small, the equilibrium of E will be 

 stable or unstable according as W 8 and W v are negative or positive. 



A critical relation for the tensions is that which makes equilibrium 

 possible for the system of the five masses A, B, C, D, E, when all 

 the surfaces are plane. The ten tensions may then be represented in 

 magnitude and direction by the ten distances of five points in space 

 a, /3, y, 8, e, viz., the tension of A-B and the direction of its normal 

 by the line a/5, etc. The point e will lie within the tetrahedron 

 formed by the other points. If we write V E for the volume of E, and 

 V A , V B , v c , V-D for the volumes of the parts of the other masses which 

 are suppressed to make room for E, we have evidently 



W y =p E v E -p&y -p E v B -p v G -PDVD . (639) 



Hence, when all the surfaces are plane, TF V = 0, and TFg = 0. Now 

 equilibrium is always possible for a given small value of V E with any 

 given values of the tensions and of p, p B , p 0) p^. When the tensions 

 satisfy the critical relation, TT S = 0, if p A =p s =p G p I) . But when 

 t E is small and constant, the value of W s must be independent of 

 PA> PE> Pc> Pv> since the angles of the surfaces are determined by the 

 tensions and their curvatures may be neglected. Hence, TFg = 0, and 

 Wy = 0, when the critical relation is satisfied and V E small. This gives 



= VAPA + VBPB + VcPc + v^Py ( 640 ) 



^E 



In calculating the ratios of i> A , i> B , V G , V D , i> E , we may suppose all the 

 surfaces to be plane. Then E will have the form of a tetrahedron, 

 the vertices of which may be called a, b, c, d (each vertex being 

 named after the mass which is not found there), and V A , V E , v c> V-Q will 

 be the volumes of the tetrahedra into which it may be divided 

 by planes passing through its edges and an interior point e. The 

 volumes of these tetrahedra are proportional to those of the five 

 tetrahedra of the figure afiySe, as will easily appear if we recollect 

 that the line ab is common to the surfaces C-D, D-E, E-C, and there- 

 fore perpendicular to the surface common to the lines yS, Se, ey, i.e. 

 to the surface y<$e, and so in other cases (it will be observed that 

 -y, S, and e are the letters which do not correspond to a or b) ; also 



