Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/111

contains ds and ds′ linearly and homogeneously, as it should. We can also add any terms of the form

$d\{\mathbf {r} .(\mathbf {ds\prime .r} ).\chi (r)\}$,

where χ(r) denotes any arbitrary function of r, and d denotes differentiation along the arc s, keeping ds′ fixed (so that dr = - ds); this differential may be written

$-\mathbf {ds} .\mathbf {ds\prime .r} .\chi (r)-\mathbf {r} \chi (r)(\mathbf {ds\prime .ds} )-{\frac {1}{r}}\chi \prime (r)\mathbf {r} (\mathbf {ds.r} )(\mathbf {ds\prime .r} )$.

In order that the law of Action and Reaction may not be violated, we must combine this with the former additional term so as to obtain an expression symmetrical in ds and ds′: and hence we see finally that the general value of F is given by the equation

$\mathbf {F} =-ii\prime \mathbf {r} \left\{{\frac {2}{r^{3}}}(\mathbf {ds.ds\prime } )-{\frac {3}{r^{3}}}(\mathbf {ds.r} )(\mathbf {ds.r} )\right\}$

$+\chi (r)(\mathbf {ds\prime .r} )\mathbf {ds} +\chi (r)(\mathbf {ds.r} )\mathbf {ds\prime } +\chi (r)(\mathbf {ds.ds\prime } )\mathbf {r}$

$+{\frac {1}{r}}\chi \prime (r)(\mathbf {ds.r} )(\mathbf {ds\prime .r} )\mathbf {r}$.

The simplest form of this expression is obtained by taking

$\chi (r)={\frac {ii\prime }{r^{3}}}$,

when we obtain

$\mathbf {F} ={\frac {ii\prime }{r^{3}}}\{(\mathbf {ds.r} ).\mathbf {ds\prime } +(\mathbf {ds\prime .r} )\mathbf {ds} +(\mathbf {ds.ds\prime } )\mathbf {r} \}$.

The comparatively simple expression in brackets is vector part of the quaternion product of the three vectors ds, r, ds′. ^[1]

From any of these values of F we can find the ponderomotive force exerted by the whole circuit s on the element ds': it is, in fact, from the last expression,

$ii\prime \int {\frac {1}{r^{5}}}(\mathbf {ds\prime .r} ).\mathbf {ds} -(\mathbf {ds.ds\prime } )\mathbf {r}$,

1. The simpler form of F given in the text is, if the term in ds′ be omitted, the form given by Grassmann, Ann. d. Phys. lxiv (1845), p. 1. For further work on this subject cf. Tait, Proc. R. S. Edio. viii (1873), p. 220, and Korteweg, Journal für Math. xe (1881), p. 45.