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XVI. — On Cases of Instability in Open Structures. By E. Sang, LL.D. 



(Read February 7, 1887.) 



In the course of some remarks on the scheme proposed for the Forth Bridge, 

 which remarks are published in the eleventh volume of the Transactions of the 

 Royal Scottish Society of Arts, I was led to enunciate, among other theorems, 

 one of a somewhat unexpected character, to the effect that any symmetric 

 structure built on a rectangular basis, having no redundant parts, and depend- 

 ing on longitudinal strain alone, is necessarily unstable. This theorem was 

 established by arguments restricted to the single matter under consideration ; 

 it is one of an extensive class, and I now propose to discuss the subject from a 

 general abstract point of view. 



The whole subject is evolved in the working out of two inverse geometrical 

 problems and their corresponding mechanical applications. The relative posi- 

 tions of a number of points being prescribed, we may have to secure these by 

 linear connections ; or, the lengths of these connections being given, we may 

 seek to discover the relative positions of the points. And we may have to 

 compute the strengths needed to enable these connections to resist strains 

 applied at the various points. 



The relative position of two points is determined by the length of the 

 straight line joining them, and the material connection can only serve as the 

 medium for the equipoise of equal and opposite strains applied at its two ends ; 

 it can offer no resistance to stresses directed obliquely to it. The opposing 

 pressures may be directed inwardly so as to cause compression, or outwardly 

 so as to cause distension ; the former is an example of unstable, the latter an 

 example of stable equilibrium. 



The instability in the case of compression is familiarly exemplified by an 

 attempt to balance a load on the top of a walking-stick, or by the buckling of 

 a long, thin rod ; stability can be obtained only by the use of something aside 

 of the straight line. In the case of distension we have to observe that no 

 member of a structure acts upon a contiguous member except by compression ; 

 we do not pull an object toward us, we always push it ; each link of a chain 

 pushes the other link ; the pulling is internal to the links themselves. Every 

 case of stretching necessarily implies at each end compression changed first 

 into transverse strain and then into distension. This phenomenon, which, from 

 habit, we regard as simple, is indeed a most complex one, whose intimate 

 nature as yet surpasses our understanding. Hence it is that, beyond the 



VOL. XXXIII. PART II. 3 B 



