Page:Philosophical magazine 21 series 4.djvu/188

We have in general, for the force in the direction of $x$ per unit of volume by the law of equilibrium of stresses^[1],

$X={\frac {d}{dx}}p_{xx}+{\frac {d}{dy}}p_{xy}+{\frac {d}{dz}}p_{xz}$

(3)

In this case the expression may be written

${\begin{array}{l}X={\dfrac {1}{4\pi }}\left\{{\dfrac {d(\mu \alpha )}{dx}}\alpha +\mu \alpha {\dfrac {d\alpha }{dx}}-4\pi {\dfrac {dp_{1}}{dx}}+{\dfrac {d(\mu \beta )}{dy}}\alpha +\mu \beta {\dfrac {d\alpha }{dy}}\right.\\\\\qquad \left.+{\dfrac {d(\mu \gamma )}{dz}}\alpha +\mu \gamma {\dfrac {d\alpha }{dz}}\right\}\end{array}}$

(4)

Remembering that $\alpha {\frac {d\alpha }{dx}}+\beta {\frac {d\beta }{dx}}+\gamma {\frac {d\gamma }{dx}}={\frac {1}{2}}{\frac {d}{dx}}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)$, this becomes

${\begin{array}{l}X=\alpha {\dfrac {1}{4\pi }}\left({\dfrac {d}{dx}}(\mu \alpha )+{\dfrac {d}{dy}}(\mu \beta )+{\dfrac {d}{dz}}(\mu \gamma )\right)+{\dfrac {1}{8\pi }}\mu {\dfrac {d}{dx}}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\\\\\qquad -\mu \beta {\dfrac {1}{4\pi }}\left({\dfrac {d\beta }{dx}}-{\dfrac {d\alpha }{dy}}\right)+\mu \gamma {\dfrac {1}{4\pi }}\left({\dfrac {d\alpha }{dz}}-{\dfrac {d\gamma }{dx}}\right)-{\dfrac {dp_{1}}{dx}}.\end{array}}$

(5)

The expressions for the forces parallel to the axes of $y$ and $z$ may be written down from analogy.

We have now to interpret the meaning of each term of this expression.

We suppose $\alpha ,\beta ,\gamma$ to be the components of the force which would act upon that end of a unit magnetic bar which points to the north.

$\mu$ represents the magnetic inductive capacity of the medium at any point referred to air as a standard. $\mu \alpha ,\mu \beta ,\mu \gamma$ represent the quantity of magnetic induction through unit of area perpendicular to the three axes of $x,y,z$ respectively.

The total amount of magnetic induction through a closed surface surrounding the pole of a magnet, depends entirely on the strength of that pole; so that if dx dy dz be an element, then

$\left({\frac {d}{dx}}\mu \alpha +{\frac {d}{dy}}\mu \beta +{\frac {d}{dz}}\mu \gamma \right)dx\ dy\ dz=4\pi m\ dx\ dy\ dz,$

(6)

which represents the total amount of magnetic induction outwards through the surface of the element dx dy dz, represents the amount of "imaginary magnetic matter" within the element, of the kind which points north.

The first term of the value of $X$, therefore,

$\alpha {\frac {1}{4\pi }}\left({\frac {d}{dx}}\mu \alpha +{\frac {d}{dy}}\mu \beta +{\frac {d}{dz}}\mu \gamma \right),$

(7)

may be written

$\alpha m,\,$

(8)

1. Rankine's 'Applied Mechanics,' art. 116.