to the Longitudinal Expansion in Rods of Spring Steel. 29 



different directions, is expanded in a longitudinal direction by 

 a strain, its lateral dimensions undergo a contraction. From 

 Poisson's theoretical considerations, the relation of the lateral 

 contraction to the longitudinal expansion should always be \ ; 

 Wertheim concluded from his experiments that it is \. Accord- 

 ing to a view which has been repeatedly expressed, it has neither 

 one value nor the other, and differs in different substances. In 

 most bodies, in which the same elasticity can be assumed in 

 different directions, the experimental determination of this rela- 

 tion is hindered by the circumstance that, even with very slight 

 changes of form, permanent expansion and elastic secondary 

 effects take place in them to a considerable extent. This is the 

 case with ignited metal wires and glass rods. With hard-drawn 

 metal wires a permanent expansion and an elastic secondary 

 action are much less perceptible ; but in them the elasticity is 

 certainly different in different directions. In the case of 

 hardened steel rods, on the contrary, it may with probability be 

 assumed that there is equal elasticity in different directions ; and 

 since, moreover, these more closely resemble an ideal elastic body 

 than hard-drawn wires do, they appear especially fitted for 

 determining the value of that relation. I have made such expe- 

 riments on several round rods of spring steel, about 2*85 millims. 

 in diameter, and about 300 millims. in length, and will here 

 describe them. 



In fig. 1, Plate I., let A A ; represent an elastic rod which 

 is fixed horizontally at A ; let A' B' be a horizontal arm fixed on 

 A' at right angles to the longitudinal direction of the rod. If a 

 weight P is suspended at B', it will produce both a flexion and a 

 torsion. On the free end of the rod let a mirror C ; be so 

 fastened that its surface is almost horizontal; let a telescope be 

 directed from above upon the mirror, and a scale consisting of 

 two systems of lines cutting each other at right angles at equal 

 distances, be so fixed horizontally that its image appears in the 

 telescope. The flexion and the torsion may then be simultane- 

 ously observed. Assuming that the section of the bar is a circle, 

 the coefficient of elasticity may be calculated from its radius, the 

 weight P, the length A A', and from the observed flexion ; the 

 coefficient of torsion may be calculated from the radius of the 

 rod, the weight P, the length A A', and the torsion observed. 

 Assuming that the elasticity is equal in different directions, the 

 desired relation of the lateral contraction to the longitudinal ex- 

 pansion may be readily calculated from these two coefficients. 

 In determining this relation a knowledge of the radius of the 

 rod is unnecessary, since it occurs in the same form in the ex- 

 pressions for the coefficients of elasticity and of torsion. 



The rods which I subjected to these experiments were pieces 



