INSTABILITY IN OPEN STRUCTURES. 329 



in the same vertical line, in which case the opposite members are of equal 

 lengths, AE to CG ; EB to GD, and so on. Such a structure is clearly 

 instable. 



Since the lengths HA, AE are fixed, the point A must be on the circum- 

 ference of a circle, having HE for its axis of rotation ; and similarly for the 

 points B, C, D. If now we suppose the point A to be pushed inwards, the 

 members AB, AD will push the points D and B outwards, and, consequently, 

 C will move inwards by exactly as much as A ; the structure will adapt itself 

 perfectly to its new position. In truth, we have here not twelve, we have only 

 eleven data ; for, if one of the connections, say CD, were removed, and the 

 structure thus made obviously mobile, the distance CD would yet remain 

 always equal to AB ; that distance cannot be reckoned among the data. 



So much for the geometrical mobility ; let us examine the strains. Any 

 horizontal pressure at A is decomposable into two, — one in the direction AB, 

 the other in the direction AD. The stress AB transferred to the point B may 

 again be decomposed into two ; the one of these in the plane EBF parallel to 

 EF is completely resisted at E and F by the stresses EB, EF, but the other, 

 perpendicular to EF, meets with no resisting obstacle ; it and the corresponding 

 pressure at D may be counteracted by extraneous pressure there, or by a single 

 pressure applied at C, equal to the pressure at A, and in the same direction 

 with it. Thus the distortion of the fabric by an eastward pressure at A is 

 prevented by a like pressure applied at C, not westward, but eastward also ; 

 in respect, however, to the strains on EB, BF, HD, DG, the effects of these 

 counteracting pressures are cumulative. 



These considerations would seem to warrant the conclusion that all struc- 

 tures of this class are necessarily unstable ; however, before venturing to accept 

 of this conclusion, it may be prudent for us to inquire whether the arguments 

 on which it is founded be strong enough to bear such a weighty superstructure. 

 Now the chief argument was that the longi- 

 tudinal stress on AB, acting at B, tends to E Plan. 



turn the triangle EBF on EF as an axis ; but 

 this tendency exists only so long as AB is out 

 of the plane EBF, and ceases whenever AB 

 comes to be in that plane ; in other words, when- 

 ever AB is parallel to EF. Hence it follows 

 that structures, represented in plan by fig. 12, Flg< 12 " 



having the two rhomboids ABCD and EFGH placed conformably, are rigid. 

 The conclusion was not absolutely general. 



Though every rhomboid be not a rectangle, every rectangle is a rhomboid, 

 and we might hastily thence conclude that these remarks concerning rhom- 

 boidal structures may be at once extended to rectangular ones. But we have 



VOL. XXXIII. PART II. 3 C 



