AND THE HYPERBOLIC LAW OF ELASTICITY. 



[181] 



examination of the experiments in the Report above referred to, that as w is increased by 



w 

 equal successive increments up to the breaking weight, the ratio — decreases in such a manner 



e 



that it may be nearly expressed by the formula 



/3», 



w 



— = a 



e 



(3) 



a and /3 being empirical constants. 



Comparing this with the formula (2) above, we observe that the only difference is in the 

 second term on the right hand side of the equation which is here made proportional to the 

 stretching weight instead of as before to the extension. 



It may be here remarked that the comparison of the formulae (2) and (3) for the longitu- 

 dinal extension of iron applies also to the similar formulae for the deflection of bars bv trans- 

 verse pressure at their centre. The formulae for extension are identical with those for 

 deflection, except that in the latter, d the deflection is substituted for e and w represents the 

 transverse pressure. 



From (3) may be obtained 



ae , to 



w = 



and e = 



1 +/3e a-(Zw' 



These expressions give w in terms of e, and e in terms 

 of w respectively. They may also be put in the form, 



which is the equation to a rectangular hyperbola 

 of which e and w are the co-ordinates. Let CA, CB be 

 the asymptotes of the curve; then referring to the last 

 equation e and w will be measured parallel to the axes 

 Oe, Ow respectively; the origin being above one 

 asymptote and to the left of the other. 



The proposed formula, therefore, exhibits the Hyperbolic Law of Elasticity according 

 to the nomenclature of James Bernouilli who (loc. cit.) represents the relation of the tension 

 to the extension by a curve which he calls the linea tensionum. 



Similarly the formula w = Ae - Be" may be termed the 

 parabolic formula, for it is the equation to a parabola of which 

 w and e are the co-ordinates. Let A be the vertex, AB the 

 axis of the curve, then Oe, Ow are the axes of e and w 

 respectively. O the origin being a point in the curve below b~ 

 the axis. 



Now it has been already observed that in comparison with 

 a series of experimental results the parabolic formula gave 

 the weight too small at the commencement of the series, too 

 large towards the middle of it, and subsequently too small again. This amounts to the same 



