Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/460

has a value differing from zero. Such an action has never been observed.

Therefore, since the quantity $e'+e'_{1}$ may be shewn experimentally not to be always zero, and since the quantity $v^{2}e+v_{1}^{2}e_{1}$ is not capable of being experimentally tested, it is better for these speculations to assume that it is the latter quantity which invariably vanishes.

849.] Whatever hypothesis we adopt, there can be no doubt that the total transfer of electricity, reckoned algebraically, along the first circuit, is represented by

$ve+v_{1}e_{1}=c\,i\,ds;$

where $c$ is the number of units of statical electricity which are transmitted by the unit electric current in the unit of time, so that we may write equation (9)

$\Sigma (vv'ee')=c^{2}ii'dsds'.$

(11)

Hence the sums of the four values of (3), (5), and (6) become

$\sum (ee'u^{2})=-2c^{2}ii'dsds'\cos \epsilon ;$

(12)

$\sum \left(ee'\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right)=-c^{2}ii'dsds'{\frac {dr}{ds}}{\frac {dr}{ds'}},$

(13)

$\sum \left(ee'{\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}\right)=-c^{2}ii'dsds'{\frac {d^{r}}{ds\,ds'}},$

(14)

and we may write the two expressions (1) and (2) for the attraction between $ds$ and $ds'$

$-{\frac {1}{c^{2}}}\sum \left[{\frac {ee'}{r^{2}}}\left(u^{2}-{\frac {3}{2}}\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right)\right],$

(15)

and

${\frac {1}{c^{2}}}\sum \left[{\frac {ee'}{r^{2}}}\left(r{\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}-{\frac {1}{2}}\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right)\right].$

(16)

Action At A Distance.

850.] The ordinary expression, in the theory of statical electricity, for the repulsion of two electrical particles $e$ and $e'$ is ${\frac {ee'}{r^{2}}}$ and

$\sum \left({\frac {ee'}{r^{2}}}\right)={\frac {(e+e_{1})(e'+e'_{1})}{r^{2}}},$

(17)

which gives the electrostatic repulsion between the two elements if they are charged as wholes.

Hence, if we assume for the repulsion of the two particles either of the modified expressions

${\frac {ee'}{r^{2}}}\left[1+{\frac {1}{c^{2}}}\left(u^{2}-{\frac {3}{2}}\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right)\right],$

(18)

or

${\frac {ee'}{r^{2}}}\left[1+{\frac {1}{c^{2}}}\left(r{\frac {{\mathfrak {d}}^{2}r}{{\mathfrak {d}}t^{2}}}-{\frac {1}{2}}\left({\frac {{\mathfrak {d}}r}{{\mathfrak {d}}t}}\right)^{2}\right)\right],$

(19)

we may deduce from them both the ordinary electrostatic forces, and the forces acting between currents as determined by Ampère.