Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/465

element $ds$ is in motion relatively to $ds'$, and that the currents in both elements vary with the time. The expressions thus found will contain terms involving $v^{2}$, $vv'$, $v^{\prime 2}$, $v$, $v'$, and terms not involving $v$ or $v'$, all of which are multiplied by $ee'$. Examining, as we did before, the four values of each term, and considering first the mechanical force which arises from the sum of the four values, we find that the only term which we must take into account is that involving the product $vv'ee'$.

If we then consider the force tending to produce a current in the second element, arising from the difference of the action of the first element on the positive and the negative electricity of the second element, we find that the only term which we have to examine is that which involves $vee'$. We may write the four terms included in $\Sigma (vee')$, thus

$e'(ve+v_{1}e_{1})\;{\text{and}}\;e'_{1}(ve+v_{1}e_{1}).$

Since $e'+e'_{1}=0$, the mechanical force arising from these terms is zero, but the electromotive force acting on the positive electricity $e'$ is $(ve+v_{1}e_{1})$ and that acting on the negative electricity $e'_{1}$ is equal and opposite to this.

858.] Let us now suppose that the first element $ds$ is moving relatively to $ds'$ with velocity $V$ in a certain direction, and let us denote by $\angle (V,ds)$ and $\angle (V,ds')$, the angle between the direction of $V$ and that of $ds$ and of $ds'$ respectively, then the square of the relative velocity, $u$, of two electric particles is

$u^{2}=v^{2}+v^{\prime 2}+V^{2}-2vv'\cos \epsilon +2Vv\cos \angle (V,ds)-2Vv'\cos \angle (V,ds').$

(25)

The term in $vv'$ is the same as in equation (3). That in $v$, on which the electromotive force depends, is

$2Vv\cos \angle (V,ds).$

We have also for the value of the time-variation of $r$ in this case

${\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}=v{\frac {dr}{ds}}+v'{\frac {dr}{ds'}}+{\frac {dr}{dt}},$

(26)

where ${\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}$ refers to the motion of the electric particles, and ${\frac {dr}{dt}}$ to that of the material conductor. If we form the square of this quantity, the term involving $vv'$ , on which the mechanical force depends, is the same as before, in equation (5), and that involving $v$, on which the electromotive force depends, is

$2v{\frac {dr}{ds}}{\frac {dr}{dt}}.$