642 RICE 



ART, L 



50. The Possible Growth of a New Surface at a Point of Meeting 

 of a Number of Lines of Discontinuity 



We might have a system in which there is more than one Hne 

 of discontinuity, these hnes meeting at a point. The latter 

 half of page 289 has a very concise statement about the stability 

 of such a system as regards the development of fresh surfaces 

 at the point. Any reader who is not trained in solid geometry 

 or lacks the power to visualize diagrams in space may require 

 some assistance here. Let us begin with the simplest case of 

 four different fluid masses. In this case there will be six 

 surfaces of discontinuity, and four lines of discontinuity. The 

 easiest way to realize this is to drive three nails into a drawing 

 board, calling them X, Y, Z. Attach three threads to them 

 which can be drawn tight and knotted at a point above the 

 board. A fourth thread, tied to the other three at 0, is stretched 

 tight and tied to another nail U, in a support above 0. One 

 can then see that we can have one mass of fluid in the pyramid 

 OYZU, one in OZXU, one in OX YU and one in OXYZ. Let 

 us call these masses A, B, C, D, respectively. The surface 

 between B and C is OXU; between C and A, OYU; between A 

 and B, OZU; between A and D,OYZ; between B and D, OZX; 

 between C and D, OXY. There are four lines of discontinuity 

 OX, OY, OZ, OU. Since the surfaces OXY, OXZ, OXU 

 meeting in the line OX are in equilibrium, three forces having 

 magnitudes proportional to (tcd, ctbd and (Tbc, and directions 

 normal to these surfaces, are in equilibrium, and can be repre- 

 sented by the sides of a triangle whose corners we shall name 

 /3, 7, 5, the side yb representing aco, 5/3 representing aoB, &y 

 representing gbc In the same way, if the surfaces OYX, 

 OYU, OYZ meeting in OF are in equilibrium, three forces 

 odc, (tca, (Tad normal to these surfaces can be represented by the 

 sides of a triangle 8ya, where a is a fourth point not in the plane 

 of I3y8. The figure a^y8 is a tetrahedron, and it will now be 

 easy for the reader to see that the equilibrium of the other two 

 triads of ^rfaces, viz., OZX, OZY, OZU and OUX, OUY,OUZ 

 is related in a similar way to the triangles fiba and a^y. In 

 short, the tetrahedron a^yb is a geometrical representation of 



