Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/281

It will be seen that Kirchhoff's electrical researches were greatly influenced by those of Weber. The latter investigations, however, did not enjoy unquestioned authority; for there was still a question as to whether the expressions given by Weber for the mutual energy of two current elements, and for the mutual energy of two electrons, were to be preferred to the rival formulae of Neumann and Riemann. The matter was examined in 1870 by Helmholtz, in a series of memoirs^[1] to which reference has already been made.^[2] Helmholtz remarked that, for two elements ds, ds′, carrying currents i, i′, the electrodynamic energy is

${\frac {ii^{\prime }(\mathbf {ds.ds^{\prime }} )}{r}}$,

according to Neumann, and

${\frac {ii^{\prime }}{r^{3}}}(\mathbf {r.ds} )(\mathbf {r.ds^{\prime }} )$,

according to Weber; and that these expressions differ from each other only by the quantity

${\frac {ii^{\prime }dsds^{\prime }}{r}}\{-\cos(ds.ds^{\prime })+\cos(r.ds)\cos(r.ds^{\prime })\}$,

or $ii^{\prime }dsds^{\prime }{\frac {d^{2}r}{ds\ ds^{\prime }}}$; since this vanishes when integrated round either circuit, the two formulae give the same result when applied to entire currents. A general formula including both that of Neumann and that of Weber is evidently

${\frac {ii^{\prime }(\mathbf {ds.ds^{\prime }} )}{r}}+kii^{\prime }{\frac {d^{2}r}{ds\ ds^{\prime }}}\ ds\ ds^{\prime }$,

where k denotes an arbitrary constant.^[3]

Helmholtz's result suggested to Clausius^[4] a new form for the law of force between electrons; namely, that which is

1. Journal für Math., lxxii (1870), p. 57: lxxv (1873), p. 35: lxxviii (1874), p. 273.
2. Cf. p. 229.
3. Cf. H, Lamb, Proc. Lond. Math. Soc., xiv (1883), p. 301.
4. Journal für Math. lxxxii (1877), p. 85: Phil. Mag., x (1880), p. 255.