Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/466

Differentiating (26) with respect to $t$, we find

${\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}=v^{2}{\frac {d^{2}r}{ds^{2}}}+2vv'{\frac {d^{2}r}{ds\,ds'}}+v^{\prime 2}{\frac {d^{2}r}{ds^{\prime 2}}}+{\frac {dv}{dt}}{\frac {dr}{ds}}+{\frac {dv'}{dt}}{\frac {dr}{ds'}}+v{\frac {dv}{ds}}{\frac {dr}{ds}}+v'{\frac {dv'}{ds}}{\frac {dr}{ds'}}+{\frac {d^{2}r}{dt^{2}}}.$

(27)

We find that the term involving $vv'$ is the same as before in (6).

The term whose sign alters with that of $v$ is ${\frac {dv}{dt}}{\frac {dr}{ds}}$.

859.] If we now calculate by the formula of Gauss (equation (18)), the resultant electrical force in the direction of the second element $ds'$ arising from the action of the first element $ds$, we obtain

${\frac {1}{r^{2}}}dsds'iV(2\cos \angle (V,ds)-3\cos \angle (V,r)\cos \angle (r,ds))\cos \angle (r,ds').$

(28)

As in this expression there is no term involving the rate of variation of the current $i$, and since we know that the variation of the primary current produces an inductive action on the secondary circuit, we cannot accept the formula of Gauss as a true expression of the action between electric particles.

860.] If, however, we employ the formula of Weber, (19), we obtain

${\frac {1}{r^{2}}}dsds'(r{\frac {dr}{ds}}{\frac {di}{dt}}-i{\frac {dr}{ds}}{\frac {dr}{dt}}){\frac {dr}{ds'}},$

(29)

or

${\frac {dr}{ds}}{\frac {dr}{ds'}}{\frac {d}{dt}}\left({\frac {i}{r}}\right)ds\,ds'.$

(30)

If we integrate this expression with respect to $s$ and $s'$, we obtain for the electromotive force on the second circuit

${\frac {d}{dt}}\int \int {\frac {1}{r}}{\frac {dr}{ds}}{\frac {dr}{ds'}}dsds'.$

(31)

Now, when the first circuit is closed,

$\int {\frac {d^{2}r}{ds\,ds'}}ds=0.$

Hence

$\int {\frac {1}{r}}{\frac {dr}{ds}}{\frac {dr}{ds'}}ds=\int \left({\frac {1}{r}}{\frac {dr}{ds}}{\frac {dr}{ds'}}+{\frac {d^{2}r}{ds\,ds'}}\right)ds=-\int {\frac {\cos \epsilon }{r}}ds.$

(32)

But

$\int \int {\frac {\cos \epsilon }{r}}ds\,ds'=M,\;{\text{by Arts. 423, 524.}}$

(33)

Hence we may write the electromotive force on the second circuit

$-{\frac {d}{dt}}\left(iM\right),$

(34)

which agrees with what we have already established by experiment; Art. 539.