SURFACES OF DISCONTINUITY 649 



quantity (and therefore also Ws, since Ws = 2Wv). If Wv is 

 negative the system is stable. (One can hardly say that this is 

 "shown." It can be inferred from the proposition that Wv°^Ws^, 

 proved on page 293, but the inference is not an obvious one; and 

 on the face of it there appears to be a puzzling contradiction 

 between this proposition and [633]. The contradiction, of 

 course, is only apparent; but the reader is asked to defer these 

 difficult points until later and to proceed along the general 

 line of argument.) 



The third step shows how these ideas are to be applied to any 

 given set of circumstances. If the pressures and tensions are 

 known, the figure ahcd can be constructed for the appropriate 

 configuration of equilibrium, if it exists. For since the 

 relative magnitudes of the tensions determine all the angles 

 round the points a, h, c, d we can find the angles of each 

 of the curvilinear triangles hcd, cad, abd, abc. Also since 

 Pd — Pa = ^ffAo/rAD, . . ■, Pb — Pc = 2cTBc/rBc, . . . , we can calcu- 

 late the six radii of the curvilinear sides. The angles and radii 

 are sufficient data to construct the various triangles, if they are 

 consistent with the possibility of a construction ; if they are not, 

 of course no such equilibrium configuration exists, and the 

 problem of stability does not arise. If the construction is 

 possible it shows us that the relative magnitudes of the quan- 

 tities Vd, Va, Vb, Vc (which are the areas of the curvilinear 

 triangles abc, hcd, cad, abd) i.e., the volumes of the mass D per 

 unit length normal to the plane of the paper, and the parts 

 into which it is divided by the surfaces B-C, C-A, A-B) can 

 be determined. These can therefore be taken as known in 

 terms of the tensions and pressures. An inspection is now 

 made of the quantity 



VaVa + VbPb + VcPc 

 Vd 



If the pressure po is greater than this it is obvious that Wv as 

 defined in [627] is positive, and from the second step the equilib- 

 rium of the mass D is unstable so that a disturbance producing 

 a small increase in it would result in a tendency for it to increase 

 still further. If it so happened that this volume Vd were small 



