Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/110

Now, by (3), the resolved part of F along ds′ must vanish when integrated round the circuit s, i.e. it must be a complete differential when dr is taken to be equal to -ds. That is to say,

${\frac {A(\mathbf {ds.ds\prime } )(\mathbf {r.ds\prime } )}{r^{3}}}+{\frac {B(\mathbf {ds.r} )(\mathbf {ds\prime .r} )^{2}}{r^{5}}}$

must be a complete differential; or

$-{\frac {A}{2r^{3}}}d.(\mathbf {r.ds\prime } )^{2}+{\frac {B}{r^{5}}}(\mathbf {ds.r} )(\mathbf {r.ds\prime } )^{2}$

must be a complete differential; and therefore

$d.{\frac {A}{2r^{2}}}=-{\frac {B}{r^{5}}}(\mathbf {ds.r} )$,

or${\frac {3A}{2r^{4}}}dr={\frac {B}{r^{4}}}dr$,

or$B=-{\frac {3}{2}}A$

Thus finally we have

$\mathbf {F} =\mathrm {Constant} \times ii\prime \mathbf {r} \left\{{\frac {2}{r^{3}}}(\mathbf {ds.ds\prime } )-{\frac {3}{r^{5}}}(\mathbf {ds.r} )(\mathbf {ds\prime .r} )\right\}$

This is Ampère's formula: the multiplicative constant depends of course on the units chosen, and may be taken to be - 1.

The weakness of Ampère's work evidently lies in the assumption that the force is directed along the line joining the two elements: for in the analogous case of the action between two magnetic molecules, we know that the force is not directed along the line joining the molecules. It is therefore of interest to find the form of F when this restriction is removed.

For this purpose we observe that we can add to the expression already found for F any term of the form

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

where φ(r) denotes any arbitrary function of r; for since

$(\mathbf {ds.r} )=-r.ds.{\frac {dr}{ds}}$,

this term vanishes when integrated round the circuit s; and it