658 RICE ART. L 



viz. E-A, E-B, E-C, E-D, D-A, D-B, D-C, C-B, C-A, 

 B-A, and when we construct all the triangle-of-force diagrams 

 for the various triads of equilibrating tensions we can fit them 

 together as follows. The original system of A, B, C, D being in 

 equilibrium round a point we can construct a tetrahedron of 

 forces for this equilibrium, as pointed out earlier, and call it 

 a^yd. (It is of course rectilinear.) Now in the new system we 

 have, for instance, at the point a of the system a similar equilib- 

 rium existing for the surfaces E-B, E-C, E-D, B-C, B-D, C-D. 

 Hence we can construct a rectilinear tetrahedron of forces for it, 

 and we can arrange three sides of it to coincide with ^yS, with 

 the fourth vertex at a point e'. Similarly a tetrahedron 

 €"y8a can be constructed to represent the tensions of the 

 surfaces E-C, E-D, E-A, C-D, C-A, D-A, and one t"'ba^ to 

 represent the tensions of the surfaces E-D, E-A, E-B, etc., and 

 finally e""a^y to represent the tensions of E-A, E-B, E-C, etc. 

 In the special case when all the surfaces in the system are 

 plane, the four points e', e", t'" , t"" coincide at one point c 

 inside a^yb, and the tetrahedron a^yb can be oriented into a 

 position in which its six edges and the four lines ea, e)3, ty, c5 are 

 normal to the surfaces in the system. 



As before, we construct an expression Zo-pSp — S(r„ Sn, where 

 Sp stands for a new surface which has been formed in developing 

 the system with the phase E from the original system without 

 E, and s„ stands for a portion of one of the original surfaces 

 which has disappeared. We call this expression Ws- As before, 



Wr = PbVb - VaVa — VbVb — VcVc ~ PdVd, 



where Vb is the volume of the phase E, and Va, etc. the volumes 

 of the parts of it originally occupied by the phases A, etc. We 

 can now prove that Ws = I Wv\ for in this case the preservation 

 of similarity of shape in a conceptually growing phase E would 

 require the tensions to vary with linear dimensions of the 

 figure E (the pressures not changing) while the surfaces Sp, Sn 

 vary as the square of the linear dimensions. The argument 

 proceeds in the now familiar way. If we are considering the 

 stability of the system without the phase E, we need only 

 consider the conditions relating to the system when the amount 



