Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/109

ponderomotive forces are exerted on such currents by magnets. To the science which deals with the mutual action of currents he gave the name electro-dynamics;^[1] and he showed that the action obeys the following laws:—

(1) The effect of a current is reversed when the direction of the current is reversed.

(2) The effect of a current flowing in a circuit twisted into small sinuosities is the same as if the circuit were smoothed out.

(3) The force exerted by a closed circuit on an element of another circuit is at right angles to the latter.

(4) The force between two elements of circuits is unaffected when all linear dimensions are increased proportionately, the current-strengths remaining unaltered.

From these data, together with his assumption that the force between two elements of circuits acts along the line joining them, Ampère obtained an expression of this force: the deduction nay be made in the following way:—

Let ds, ds′ be the elements, r the line joining them, and i, i′ the current-strengths. From (2) we see that the effect of ds on ds′ is the vector sum of the effects of dx, dy, dz on ds′, where these are the three components of ds: so the required force must be of the form—

Γ x a scalar quantity which is linear and homogeneous in ds; and it must similarly be linear and homogeneous in ds′; so using (1), we see that the force must be of the form

$\mathbf {F} =ii\prime \mathbf {r} \left\{(\mathbf {ds} .\mathbf {ds\prime } )\phi (r)+(\mathbf {ds} .\mathbf {r} )(\mathbf {ds\prime } .\mathbf {r} )\psi (r)\right\}$

where φ and ψ denote undetermined functions of r.

From (4) it follows that when ds, ds′', r are all multiplied by the same number, F is unaffected: this shows that

$\phi (r)={\frac {A}{r^{3}}}$ and $\psi (r)={\frac {B}{r^{5}}}$

where A and B denote constants. Thus we have

$\mathbf {F} =ii\prime \mathbf {r} \left\{{\frac {A(\mathbf {ds} .\mathbf {ds\prime } )}{r^{3}}}+{\frac {B(\mathbf {ds} .\mathbf {r} )(\mathbf {ds\prime } .\mathbf {r} )}{r^{5}}}\right\}$

1. Loc. cit., p. 298.