INSTABILITY IN OPEN STRUCTURES. 



325 



the five distances DA, AB, BC, CD, DB would suffice to secure the straight - 

 ness of ABC ; but on consideration we perceive that the two triangles ABD, 

 CBD are merely hinged upon the common line DB. 



In the cases of two, three, and four points, we have seen that the length 

 of every line joining them in pairs is needed for fixing the relative positions; 

 this rule does not hold for higher numbers. Thus, if a fifth point E be con- 

 nected with three of the four corners of the tetrahedron ABCD, its relative 

 position is determined, provided always that E be not in the plane of the three 

 points with which it is joined ; so that nine lines suffice for five points. The 

 line joining E with the fourth point of the tetrahedron would be redundant, 



In all such structures, three of the points, as A, B, C in fig. 7, must each 

 have four concurring lines, and the remaining two, D and E, only three ; and 

 if no four of the five points be in one straight line, the system is self-rigid. 



This rigidity will subsist although the two triple points D, E be in the 

 same plane with any one pair of the quadruple ones, as in fig. 8, which is 

 intended to show D, B, E, C as in one plane. The scheme then takes the 



Fig. 7. 



Fig. 8. 



Fig. 9. 



appearance of a pyramid, having A for its apex, and the quadrangle DBEC for 

 its base. Thus the flatness of a tetragon may be secured by connecting each 

 of its corners with a fifth point not in the same plane. 



Moreover, the system still remains rigid although the points D and E be 

 both in one plane with AB also. In this case DBE, the meeting of two planes, 

 must be a straight line, as shown in fig. 9. Thus we see that, for the establish- 

 ment of three points in a straight line, two auxiliary points must be introduced, 

 with seven additional linear members. 



We have now got a possible straight lever DBE. In order to examine the 

 law of the balancing of pressures applied at B, D, E, we must trace out the 

 strains on the various members, and their equilibriums at the five junctions, 

 subject to the condition that there be no external pressure at A or at C. The 

 result of this examination is, that the strains on the members are eliminated ; 

 that the directions of the applied pressures must all pass through one point ; 



