XIV. The Deflection of imperfectly Elastic Beams and the Hyperbolic Law of 

 Elasticity. By Homersham Cox, B.A., Jesus College, Cambridge. 



[Read in part March 11, 1850, and in part October 1850.] 



The equation to the curve of an elastic deflected beam is usually deduced from the assump- 

 tions: 1, that the longitudinal compression, or extension, of an elastic filament is proportional 

 to the compressing or extending force ; 2, that for equal extension and compression the com- 

 pressing and extending forces are equal to each other. 



It appears, however, from experiment that these hypotheses are not exact. All substances 

 appear to be subject to a defect of elasticity, i. e. their elastic forces of restitution increase in a 

 somewhat less degree than in proportion to the extension or compression. The first of the 

 assumptions above mentioned bears in England the name of " Dr Hooke's Law," and its 

 inexactness was noticed very soon after he proposed it, by Leibnitz, James Bernouilli, and 

 others. In the Acta Eruditorum of Leipsic for 1694, Bernouilli gives certain investigations 

 respecting the Elastic Curve: 1, generally when the elastic forces follow any law whatever; 

 2, when they vary as any power of the extension ; 3, when they are directly proportional 

 to the extension. He states that it is worth while to examine the results of the latter 

 hypothesis which had been employed by Leibnitz in his treatise de Resistentia Solidorum, 

 but deems it certain that the law is different in different bodies. " Id quod experimenta, 

 turn nostra turn aliorum, abunde confirmare videntur, quorum plurima praelaudatus auctor 

 [Franciscus Tertius de Lanis] industrius magisterii naturce et artis loco cit. recenset." 



The real law of elasticity of any material can be known only by direct experiments on the 

 material itself, and it seems nearly certain that even for two different specimens of the same 

 metal, the laws would be in some measure different. All therefore that can be done by formulae 

 is to represent approximately the results of experiments. 



If e be a fraction expressing the extension of a rod, and w the direct force producing that 

 extension, w may be put equal to the sum of a series of terms involving constant coefficients 

 and ascending integral powers of e. Such a series will be convergent. But to represent 

 exactly by such means the results of a single set of experiments very complicated formula? 

 would be required. If, for instance, 20 experiments were made with different weights stretching 

 a rod of iron, the formula must generally contain 20 terms ; for the experimental results would 

 probably give equations determining 20 different independent coefficients of such terms. 



The law of Dr Hooke stops at the first term of such a series of ascending algebraical 

 powers of e. The idea readily suggests itself, that by extending the series one term further, 

 its errors may be corrected. The formula so modified would become 



w = Ae-Be*. . . (1) 



Vol. IX. Part II. 47 



