J. W. Gibhs — Equilihriam of Heterogeneous Substances. 405 



If the system can be in equilibrium when the vertical point has 

 been replaced by a mass E against which the four masses A, B, C, D 

 abut, being truncated at their vertices, it is evident that E will have 

 four vertices, at each of which six surfaces of discontinuity meet. 

 (Thus at one vertex there will be the surfaces formed by A, B, C, 

 and E.) The tensions of each set of six surfaces (like those of the 

 six surfaces formed by A, B, C, and D) must therefore be such that 

 they can be represented by the six edges of a tetrahedron. When 

 the tensions do not satisfy these relations, there will be no particular 

 condition of stability for the point about which A, B, C, and D meet, 

 since if a mass E should be formed, it would distribute itself along 

 some of the lines or surfaces which meet at the vertical point, and it 

 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 TF^ 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 TFs 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 459 that 



TFs=|Trv, (63V) 



whence TT, - TTyz^^ TFy ; (638) 



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

 stable or unstable according as TFg and Wy are negative or positive. 

 A critical relation for the tensions is that which makes equilibrium 

 possible for the system of the live 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, /i, ;/, d', e, viz., the tension of A-B and the direction of its normal 

 by the line a (i, etc. The point e will lie within the tetrahedron 

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

 '^A? ^B) ^C5 ^'d for the volumes of the parts of the other masses which 

 are suppressed to make room for E, we have evidently 



Ww =Pe v^—Pk "a— P^ Vs-pc Vc~2:>D «D- (639) 



Hence, when all the surfaces are plane, Tf^z^O, and TT^r=0. Now 

 equilibrium is always possible for a given small value of v^ with any 

 given values of the tensions and of />a, ^^b, 2^c, />d- When the tensions 

 satisfy the critical relation, Ws = 0, if p^ =2)b =i»c =/>d- But when 

 ?Je is small and constant, the value of Wg must be independent of p^, 

 Pb, Pc-, P-D-> si"ce the angles of the sui-faces are determined by the 

 tensions and their curvatures may be neglected. Hence, Tl^nrO and 



