INSTABILITY IN OPEN STRUCTURES. 



331 



Pl&TV. 



Fig. 15 



Among open structures built on a rectangular base, instability is not con- 

 fined to those with rhomboidal tops ; for if, as in fig. 15, the triangle HAE be 

 set up equal to GCF and EBF to HDC, so that 

 AC may be parallel to EF and BD parallel to 

 FG, the structure is movable. The diagonals AC 

 and BD may be on one level and so cross each 

 other, or the one may pass above the other at a 

 distance on the plumb line OP. Since the 

 three lines, AO, OP, PB, are mutually perpen- 

 dicular — 



AB 2 =A0 2 +OP 2 +PB 2 

 and CD 2 = CO 2 + OP 2 + PD 2 ; wherefore 



AB 2 + CD 2 = A0 2 + OC 2 +BP 2 +PD 2 +2.0P 2 . 



But the sum of BC 2 and AD 2 is equal exactly to the same quantity, and con- 

 sequently 



AB2+CD 2 = BC 2 + DA 2 , 



so that one of these four is deducible from the remaining three ; there are then 

 only eleven data in this structure, instead of the twelve needed for rigidity. 

 But it is to be observed that a dislocation must change the horizontality of 

 AC and BD, so that the mutability may be only instantaneous, as in the case 

 of maximum or minimum. 



Passing now to the case of five supported points, we may remark that, by 

 the introduction of a fifth point, a symmetric rigid 

 structure may be built on a rectangular base. 



Thus, if we place, as in fig. 16, the rhombus 

 ABCD vertically over the rectangle EFGH, and 

 complete the construction as in fig. 11, there 

 results a symmetric structure, which, like all those 

 of the same class, is changeable. On assuming, 

 however, a point Z in the vertical axis of the 

 system, and connecting it with each of the points 

 A, B, C, D, we get a fabric both symmetric and 

 rigid. The rigidity is confirmed thus : — If, suppos- 

 ing Z and its connections to be away, the points A 

 and C be brought nearer, B and D would move 

 apart ; now, in virtue of the connections AZ, ZC, 

 the shortening of AB would cause Z to rise, while, in virtue of the connections, 

 BZ, ZD, the widening of BD would bring Z down ; the opposition of these two 

 tendencies keeps Z in its place. 



Fig. 16. 



