Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/255

On the Modification of the Equations of Electromotive Force when the Axes to which they are referred are moving in Space.

600.] Let x', y', z' be the coordinates of a point referred to a system of rectangular axes moving in space, and let x, y, z be the coordinates of the same point referred to fixed axes.

Let the components of the velocity of the origin of the moving system be u, v, w, and those of its angular velocity ω₁, ω₂, ω₃, referred to the fixed system of axes, and let us choose the fixed axes so as to coincide at the given instant with the moving ones, then the only quantities which will be different for the two systems of axes will be those differentiated with respect to the time. If ${\frac {\delta x}{\delta t}}$ denotes a component velocity of a point moving in rigid connexion with the moving axes, and ${\frac {dx}{dt}}$ and ${\frac {dx'}{dt'}}$ that of any moving point, having the same instantaneous position, referred to the fixed and the moving axes respectively, then

${\frac {dx}{dt}}={\frac {\delta x}{\delta t}}+{\frac {dx'}{dt}},$

(l)

with similar equations for the other components.

By the theory of the motion of a body of invariable form,

$\left.{\begin{aligned}{\frac {\delta x}{\delta t}}=u+\omega _{2}z-\omega _{3}y,\\{\frac {\delta y}{\delta t}}=v+\omega _{3}z-\omega _{1}z,\\{\frac {\delta z}{\delta t}}=w+\omega _{1}z-\omega _{2}x.\end{aligned}}\right\}$

(2)

Since F is a component of a directed quantity parallel to x, if ${\frac {dF'}{dt}}$ be be the value of ${\frac {dF}{dt}}$ referred to the moving axes,

${\frac {dF'}{dt}}={\frac {dF}{dx}}{\frac {\delta x}{\delta t}}+{\frac {dF}{dy}}{\frac {\delta y}{\delta t}}+{\frac {dF}{dz}}{\frac {\delta z}{\delta t}}+G\omega _{3}-H\omega _{2}+{\frac {dF}{dt}}.$

(3)

Substituting for ${\frac {dF}{dy}}$ and ${\frac {dF}{dz}}$ their values as deduced from the equations (A) of magnetic induction, and remembering that, by (2),

${\frac {d}{dx}}{\frac {\delta x}{\delta t}}=0,\quad {\frac {d}{dx}}{\frac {\delta y}{\delta t}}=\omega _{3},\quad {\frac {d}{dx}}{\frac {\delta z}{\delta t}}=-\omega _{2},$

(4)

${\begin{aligned}{\frac {dF'}{dt}}={\frac {dF}{dx}}{\frac {\delta x}{\delta t}}+F{\frac {d}{dx}}{\frac {\delta x}{\delta t}}+{\frac {dG}{dx}}{\frac {\delta y}{\delta t}}+G{\frac {d}{dy}}{\frac {\delta y}{\delta t}}+{\frac {dH}{dx}}{\frac {\delta z}{\delta t}}+H{\frac {d}{dx}}{\frac {\delta z}{\delta t}}\\-c{\frac {\delta y}{\delta t}}+b{\frac {\delta z}{\delta t}}+{\frac {dF}{dt}}.\end{aligned}}$

(5)