Page:Philosophical magazine 21 series 4.djvu/192

element will be urged in the direction of -$x$, transversely to the direction of the current and of the lines of force; that is, an ascending current in a field of force magnetized towards the north would tend to move west.

To illustrate the action of the molecular vortices, let $sn$ be the direction of magnetic force in the field, and let C be the section of an ascending magnetic current perpendicular to the paper. The lines of force due to this current will be circles drawn in the opposite direction from that of the hands of a watch; that is, in the direction $nwse$. At $e$ the lines of force will be the sum of those of the field and of the current, and at $w$ they will be the difference of the two sets of lines; so that the vortices on the east side of the current will be more powerful than those on the west side. Both sets of vortices have their equatorial parts turned towards C, so that they tend to expand towards C, but those on the east side have the greatest effect, so that the resultant effect on the current is to urge it towards the west.

The fourth term,

$+\mu \gamma {\frac {1}{4\pi }}\left({\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right),\ \mathrm {or} \ +\mu \gamma q,$

(10)

may be interpreted in the same way, and indicates that a current $q$ in the direction of $y$, that is, to the north, placed in a magnetic field in which the lines are vertically upwards in the direction of $z$, will be urged towards the east.

The fifth term,

$-{\frac {dp_{1}}{dx}},$

(11)

merely implies that the element will be urged in the direction in which the hydrostatic pressure $p_{1}$ diminishes.

We may now write down the expressions for the components of the resultant force on an element of the medium per unit of volume, thus:

$X=\alpha m+{\frac {1}{8\pi }}\mu {\frac {d}{dx}}\left(v^{2}\right)-\mu \beta r+\mu \gamma q-{\frac {dp_{1}}{dx}},$

(12)

$Y=\beta m+{\frac {1}{8\pi }}\mu {\frac {d}{dy}}\left(v^{2}\right)-\mu \gamma p+\mu \alpha r-{\frac {dp_{1}}{dy}},$

(13)

$X=\gamma m+{\frac {1}{8\pi }}\mu {\frac {d}{dz}}\left(v^{2}\right)-\mu \alpha q+\mu \beta p-{\frac {dp_{1}}{dz}},$

(14)

The first term of each expression refers to the force acting on magnetic poles.