Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/459

where the symbol ${\mathfrak {d}}$ indicates that, in the quantity differentiated, the coordinates of the particles are to be expressed in terms of the time.

It appears, therefore, that the terms involving the product $vv'$ in the equations (3), (5), and (6) contain the quantities occurring in (1) and (2) which we have to interpret. We therefore endeavour to express (1) and (2) in terms of $u^{2}$, $\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}$ and ${\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}$. But in order to do so we must get rid of the first and third terms of each of these expressions, for they involve quantities which do not appear in the formula of Ampère. Hence we cannot explain the electric current as a transfer of electricity in one direction only, but we must combine two opposite streams in each current, so that the combined effect of the terms involving $v^{2}$ and $v'^{2}$ may be zero.

Fechner's Hypothesis.

848.] Let us therefore suppose that in the first element, $ds$, we have one electric particle, $e$, moving with velocity $v$ and another, $e_{1}$ and moving with velocity $v_{1}$ and in the same way two particles, $e'$ and $e'_{1}$, in $ds'$ moving with velocities $v'$ and $v'_{1}$ respectively.

The term involving $v^{2}$ for the combined action of these particles is

$\Sigma (v^{2}ee')=(v^{2}e+v_{1}^{2}e_{1})(e'+e'_{1}).$

(7)

Similarly

$\Sigma (v^{\prime 2}ee')=(v^{\prime 2}e+v_{1}^{\prime 2}e'_{1})(e+e_{1});$

(8)

and

$\Sigma (vv'ee')=(ve+v_{1}e_{1})(v'e'+v'_{1}e'_{1}).$

(9)

In order that $\Sigma (v^{2}ee')$ may be zero, we must have either

$e'+e'_{1}=0,\;\;{\text{or}}\;\;v^{2}e+v_{1}^{2}e_{1}=0.$

(10)

According to Fechner's hypothesis, the electric current consists of a current of positive electricity in the positive direction, combined with a current of negative electricity in the negative direction, the two currents being exactly equal in numerical magnitude, both as respects the quantity of electricity in motion and the velocity with which it is moving. Hence both the conditions of (10) are satisfied by Fechner's hypothesis.

But it is sufficient for our purpose to assume, either―

That the quantity of positive electricity in each element is numerically equal to the quantity of negative electricity; or―

That the quantities of the two kinds of electricity are inversely as the squares of their velocities.

Now we know that by charging the second conducting wire as a whole, we can make $e'+e'_{1}$ either positive or negative. Such a charged wire, even without a current, according to this formula, would act on the first wire carrying a current in which $v^{2}e+v_{1}^{2}e_{1}$