Page:Philosophical magazine 21 series 4.djvu/309

applied to Electric Currents. 285

${\begin{array}{rl}u=&{\frac {1}{2}}{\frac {d\gamma }{dx}}\left(m_{1}\left(x-x_{1}\right)+m_{2}\left(x-x_{2}\right)\right)\\\\+&{\frac {1}{2}}{\frac {d\gamma }{dy}}\left(m_{1}\left(y-y_{1}\right)+m_{2}\left(y-y_{2}\right)\right)+{\frac {1}{2}}{\frac {d\gamma }{dz}}\left(m_{1}\left(z-z_{1}\right)+m_{2}\left(z-z_{2}\right)\right)\\\\-&{\frac {1}{2}}{\frac {d\beta }{dx}}\left(n_{1}\left(x-x_{1}\right)+n_{2}\left(x-x_{2}\right)\right)-{\frac {1}{2}}{\frac {d\beta }{dy}}\left(n_{1}\left(y-y_{1}\right)+n_{2}\left(y-y_{2}\right)\right)\\\\-&{\frac {1}{2}}{\frac {d\beta }{dz}}\left(n_{1}\left(z-z_{1}\right)+n_{2}\left(z-z_{2}\right)\right)\end{array}}$

(31)

In effecting the summation of $\Sigma u\rho dS$, we must remember that round any closed surface $\Sigma ldS$ and all similar terms vanish; also that terms of the form $\Sigma lydS$, where $l$ and $y$ are measured in different directions, also vanish; but that terms of the form $\Sigma lxdS$, where $l$ and $x$ refer to the same axis of coordinates, do not vanish, but are equal to the volume enclosed by the surface. The result is

${\overline {V}}p={\frac {1}{2}}\rho \left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right)\left(V_{1}+V_{2}+etc.\right);$

(32)

or dividing by ${\overline {V}}=V_{1}+V_{2}+etc.$,

$p={\frac {1}{2}}\rho \left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right)$

(33)

If we make

$\rho ={\frac {1}{2\pi }}$

(34)

then equation (33) will be identical with the first of equations (9), which give the relation between the quantity of an electric current and the intensity of the lines of force surrounding it.

It appears therefore that, according to our hypothesis, an electric current is represented by the transference of the moveable particles interposed between the neighbouring vortices. We may conceive that these particles are very small compared with the size of a vortex, and that the mass of all the particles together is inappreciable compared with that of the vortices, and that a great many vortices, with their surrounding particles, are contained in a single complete molecule of the medium. The particles must be conceived to roll without sliding between the vortices which they separate, and not to touch each other, so that, as long as they remain within the same complete molecule, there is no loss of energy by resistance. When, however, there is a general transference of particles in one direction, they must pass from one molecule to another, and in doing so, may experience