Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/252

To determine the value of E, let us differentiate the quantity under the integral sign with respect to t, remembering that if the secondary circuit is in motion, x, y and z are functions of the time. We obtain

${\begin{aligned}E=&-\int {\left({\frac {dF}{dt}}{\frac {dx}{ds}}+{\frac {dG}{dt}}{\frac {dy}{ds}}+{\frac {dH}{dt}}{\frac {dz}{ds}}\right)ds}\\&-\int {\left({\frac {dF}{dx}}{\frac {dx}{ds}}+{\frac {dG}{dx}}{\frac {dy}{ds}}+{\frac {dH}{dx}}{\frac {dz}{ds}}\right){\frac {dx}{dt}}ds}\\&-\int {\left({\frac {dF}{dy}}{\frac {dx}{ds}}+{\frac {dG}{dy}}{\frac {dy}{ds}}+{\frac {dH}{dy}}{\frac {dz}{ds}}\right){\frac {dy}{dt}}ds}\\&-\int {\left({\frac {dF}{dx}}{\frac {dx}{ds}}+{\frac {dG}{dz}}{\frac {dy}{ds}}+{\frac {dH}{dz}}{\frac {dz}{ds}}\right){\frac {dz}{dt}}ds}\\&-\int {\left(F{\frac {d^{2}x}{dsdt}}+G{\frac {d^{2}y}{dsdt}}+H{\frac {d^{2}z}{dsdt}}\right)ds}.\end{aligned}}$

(2)

Now consider the second term of the integral, and substitute from equations (A), Art. 591, the values of ${\frac {dG}{dx}}$ and ${\frac {dH}{dx}}$. This term then then becomes,

$-\int {\left(c{\frac {dy}{ds}}-b{\frac {dz}{ds}}+{\frac {dF}{dx}}{\frac {dx}{ds}}+{\frac {dF}{dy}}{\frac {dy}{ds}}+{\frac {dF}{dz}}{\frac {dz}{ds}}\right){\frac {dx}{dt}}ds},$

which we may write

$-\int {\left(c{\frac {dy}{ds}}-b{\frac {dz}{ds}}+{\frac {dF}{ds}}\right){\frac {dx}{dt}}ds},$

Treating the third and fourth terms in the same way, and collecting the terms in ${\frac {dx}{ds}}$, ${\frac {dy}{ds}}$, and ${\frac {dz}{ds}}$, remembering that

$\int {\left({\frac {dF}{ds}}{\frac {dx}{dt}}+F{\frac {d^{2}x}{dsdt}}\right)ds}=F{\frac {dx}{dt}},$

(3)

and therefore that the integral, when taken round the closed curve, vanishes,

${\begin{aligned}E&=\int {\left(c{\frac {dy}{dt}}-b{\frac {dz}{dt}}+{\frac {dF}{dt}}\right){\frac {dx}{ds}}ds}\\&+\int {\left(a{\frac {dz}{dt}}-c{\frac {dx}{dt}}+{\frac {dG}{dt}}\right){\frac {dy}{ds}}ds}\\&+\int {\left(b{\frac {dx}{dt}}-a{\frac {dy}{dt}}+{\frac {dH}{dt}}\right){\frac {dz}{ds}}ds}.\end{aligned}}$

(4)