Page:Philosophical magazine 21 series 4.djvu/365

applied to Electric Currents. 341 along three axes properly chosen, x', y', z', the nine direction-cosines of these axes with their six connecting equations, which are equivalent to three independent quantities, and the three rotations $$\theta_{1},\theta_{2},\theta_{3}$$ about the axes of x, y, z.

Let the direction-cosines of $$x'$$ with respect to x, y, z be $$l_{1},l_{1},l_{1}$$, those of $$y',l_{2},l_{2},l_{2}$$ and those of $$z',l_{3},l_{3},l_{3}$$; then we find

{{MathForm2|(64)|$$\left.\begin{array}{l} \frac{d}{dx}\delta x=l_{1}^{2}\frac{\delta x'}{x'}+l_{2}^{2}\frac{\delta y'}{y'}+l_{3}^{2}\frac{\delta z'}{z'},\\ \\\frac{d}{dy}\delta x=l_{1}m_{1}\frac{\delta x'}{x'}+l_{2}m_{2}\frac{\delta y'}{y'}+l_{3}m_{3}\frac{\delta z'}{z'}-\theta_{3},\\ \\\frac{d}{dz}\delta x=l_{1}n_{1}\frac{\delta x'}{x'}+l_{2}n_{2}\frac{\delta y'}{y'}+l_{3}n_{3}\frac{\delta z'}{z'}+\theta_{2},\end{array}\right\} $$}}

with similar equations for quantities involving $$\delta y$$ and $$\delta z$$.

Let $$\alpha',\beta',\gamma'$$ be the values of $$\alpha,\beta,\gamma$$ referred to the axes of $$x', y', z'$$; then

{{MathForm2|(65)|$$\left.\begin{array}{l} \alpha'=l_{1}\alpha+m_{1}\beta+n_{1}\gamma,\\ \beta'=l_{2}\alpha+m_{2}\beta+n_{2}\gamma,\\ \gamma'=l_{3}\alpha+m_{3}\beta+n_{3}\gamma.\end{array}\right\} $$}}

We shall then have

By substituting the values of $$\alpha',\beta',\gamma'$$, and comparing with equations (64), we find

as the variation of $$\alpha$$ due to the change of form and position of the element. The variations of $$\beta$$ and $$\gamma$$ have similar expressions.

Prop. XI. — To find the electromotive forces in a moving body.

The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element. The whole variation of $$\alpha$$ is therefore

But since a is a function of x, y, z and t, the variation of $$\alpha$$ may be also written