TRANSACTIONS OF THE SECTIONS. 173 
extending or compressing forces. It appears however from experiment, that for 
equal increments of extension the increase of tensile force is continually less and less. 
The decreasing ratio of the elastic force to the corresponding extension or compres- 
sion, indicating what is sometimes termed defect of elasticity, was noticed by Leib- 
nitz, James Bernoulli and others* very soon after Doctor Hooke announced the first- 
mentioned law, which is known in England by his name. 
If the extending weight () be considered such a function of the extension (e) as 
to be capable of being expressed by a convergent series ascending by integral powers 
of e, Dr. Hooke’s law, by which a=Ae, where A is a constant (called the “‘ modu- 
lus of elasticity ””), may be considered as stopping at the first term of such a series. 
To correct the errors of Dr. Hooke’s formula, the step which suggests itself most ob- 
viously is to add on another term of the series; so that, 
a= Ae De aluoilaresina Belted ἀιηθέσγα, gees ἢ 
where Bis another constant ; or dividing by e, 
Be Ass νοῆι δὴ ied Scala οὐ tborsoth soon ἴθ.) 
And as the ratio of to e decreases as e increases, the second term on the second 
side of the equation is necessarily negative. 
Such a formula is compared with experiment in a valuable paper by Eaton Hodg- 
kinson, Esq., Appendix to the Report (1849) of the Royal Commission “‘ appointed 
to inquire into the application of Iron to Railway Structures.”? But on examination 
it will be found that the differences between the results of the formula and experi- 
ments are not + and — promiscuously, but themselves follow a certain order. 
Eight formule are given for extension of different kinds of iron; and we observe 
in all, without exception, that one-half, and generally more, of the results come 
together in the middle of each series of experiments with errors having the same 
sign, and are preceded and followed by errors affected by the contrary sign. The 
formula cannot, however, be considered satisfactory until the errors be affected by 
the + and — signs promiscuously and without regular sequence. From the general 
character of these errors, it may be inferred by sufficient analytical reasoning that a 
formula involving the first three powers of the extension would possess much greater 
accuracy than one inyolving only two powers, if the constant coefficients were 
properly chosen. 
All these formule, however, lead to excessively complicated results when applied to 
investigations respecting the deflection of beams. They have moreover the serious 
inconvenience, that, from the expression for the weight in terms of the extension to 
find conversely the extension in terms of the weight, it is necessary to solve quadratic 
and cubic equations involving very large numerical coefficients. 
The formula about to be proposed is far more accurate than the formula (1,), and 
has greatly the advantage of simplicity of computation from it. It can be applied 
with great facility to obtain the extension from the extending force, or the latter from 
the former. It has also this advantage, that, when applied to the theory of beams, 
it leads to an expression similar in form to itself, from which, with the utmost readi- 
ness, the deflection can be computed from the transverse pressure, or the latter from 
the former. 
If α and β be empirical constants, it will be found that the relation of @ to e, for 
a rod of a unit of length and a unit of sectional area, may be nearly expressed by 
th ti 
e equation ΕΞ ξαϑόβάνδην 6a vine! ψον υρβάτιφῳροῦ Minty Tiga 
This is the equation to a rectangular hyperbola of which e and are the co-ordi- 
nates parallel to the asymptotes, and referred to an origin at a point in the curve. 
The proposed formula therefore exhibits the Hypersotic Law or Evasricity 
according to the nomenclature of James Bernoulli, who (loc. cit.) represents the re- 
lation of the tension to the extension by a curve which he calls the Linea tensionum. 
Similarly, the formula (1.) may be termed the parabolic formula, for itis the equa- 
* Acta Eruditorum of Leipsic, 1694. 
+ A formula involving four powers is given in the Report in one case; but its accuracy 
is little greater than that of the formule of two powers, as the constant coefficients have not 
been chosen by a method which gives the minimum error. 
