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

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

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

The term whose sign alters with that of $$v$$ is $$\frac{d v}{d t}\frac{d r}{d s}$$.

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

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

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

Now, when the first circuit is closed,

Hence we may write the electromotive force on the second circuit

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