Page:Philosophical magazine 21 series 4.djvu/366

Equating the two values of $\delta \alpha$ and dividing by $\delta t$, and remembering that in the motion of an incompressible medium

${\frac {d}{dx}}{\frac {dx}{dt}}+{\frac {d}{dy}}{\frac {dy}{dt}}+{\frac {d}{dz}}{\frac {dz}{dt}}=0,$

(71)

and that in the absence of free magnetism

${\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}=0,$

(72)

we find

${\begin{array}{c}{\dfrac {1}{\mu }}\left({\dfrac {dQ}{dz}}-{\dfrac {dR}{dy}}\right)+\gamma {\dfrac {d}{dz}}{\dfrac {dx}{dt}}-\alpha {\dfrac {d}{dz}}{\dfrac {dz}{dt}}-\alpha {\dfrac {d}{dy}}{\dfrac {dy}{dt}}+\beta {\dfrac {d}{dy}}{\dfrac {dx}{dt}}\\\\+{\dfrac {d\gamma }{dz}}{\dfrac {dx}{dt}}-{\dfrac {d\alpha }{dz}}{\dfrac {dz}{dt}}-{\dfrac {d\alpha }{dy}}{\dfrac {dy}{dt}}+{\dfrac {d\beta }{dy}}{\dfrac {dx}{dt}}-{\dfrac {d\alpha }{dt}}=0.\end{array}}$

(73)

Putting

$\alpha ={\frac {1}{\mu }}\left({\frac {dG}{dz}}-{\frac {dH}{dy}}\right)$

(74)

and

${\frac {d\alpha }{dt}}={\frac {1}{\mu }}\left({\frac {d^{2}G}{dz\ dt}}-{\frac {d^{2}H}{dy\ dt}}\right),$

(75)

where $F,G,$ and$H$ are the values of the electrotonic components for a fixed point of space, our equation becomes

${\frac {d}{dz}}\left(Q+\mu \gamma {\frac {dx}{dt}}-\mu \alpha {\frac {dz}{dt}}-{\frac {dG}{dt}}\right)-{\frac {d}{dy}}\left(R+\mu \alpha {\frac {dy}{dt}}-\mu \beta {\frac {dx}{dt}}-{\frac {dH}{dt}}\right)=0.$

(76)

The expressions for the variations of $\beta$ and $\gamma$ give us two other equations which may be written down from symmetry. The complete solution of the three equations is

$\left.{\begin{array}{l}P=\mu \gamma {\dfrac {dy}{dt}}-\mu \beta {\dfrac {dz}{dt}}+{\dfrac {dF}{dt}}-{\dfrac {d\Psi }{dx}},\\\\Q=\mu \alpha {\dfrac {dz}{dt}}-\mu \gamma {\dfrac {dx}{dt}}+{\dfrac {dG}{dt}}-{\dfrac {d\Psi }{dy}},\\\\R=\mu \beta {\dfrac {dx}{dt}}-\mu \alpha {\dfrac {dy}{dt}}+{\dfrac {dH}{dt}}-{\dfrac {d\Psi }{dz}}.\end{array}}\right\}$

(77)

The first and second terms of each equation indicate the effect of the motion of any body in the magnetic field, the third term refers to changes in the electrotonic state produced by alterations of position or intensity of magnets or currents in the field, and $\Psi$ is a function of $x,y,z,$ and $t$, which is indeterminate as far as regards the solution of the original equations, but which may always be determined in any given case from the circumstances of the problem. The physical interpretation of $\Psi$ is, that it is the electric tension at each point of space.