Page:Thomson1881.djvu/13

The part depending on p in the second integral

$=-{\frac {\mu ep}{2}}\int \int \int \left(B{\frac {d}{dz}}{\frac {1}{R}}-{\frac {Cd}{dy}}{\frac {1}{R}}\right)dx\ dy\ dz$,

or (see Maxwell's 'Electricity and Magnetism,' § 405)

$=-{\frac {\mu ep}{2}}F'_{1}$.

Adding this to the term $epF'_{1}$ already obtained, we get ${\tfrac {\mu ep}{2}}F'_{1}$ as the part of the kinetic energy depending on p. We have evidently similar expressions for the parts of the kinetic energy depending on q and r. Hence the part of the kinetic energy with which we are concerned will

$={\frac {\mu e}{2}}\cdot \left(F'_{1}p+G'_{1}q+H'_{1}r\right)$.

By Lagrange's equations, the force on the sphere parallel to the axis of x

$={\frac {dT}{dx}}-{\frac {d}{dt}}{\frac {dT}{dx}}$ $={\frac {\mu e}{2}}\left\{p{\frac {dF'_{1}}{dx}}+q{\frac {dG'_{1}}{dx}}+r{\frac {dH'_{1}}{dx}}-{\frac {dF'_{1}}{dt}}\right\}$ $={\frac {\mu e}{2}}\left\{p{\frac {dF'_{1}}{dx}}+q{\frac {dG'_{1}}{dx}}+r{\frac {dH'_{1}}{dx}}-p{\frac {dF'_{1}}{dx}}-q{\frac {dF'_{1}}{dy}}-r{\frac {dF'_{1}}{dz}}\right\}$ $={\frac {\mu e}{2}}\left\{q\left({\frac {dG'_{1}}{dx}}-{\frac {dF'_{1}}{dy}}\right)-r\left({\frac {dF'_{1}}{dz}}-{\frac {dH'_{1}}{dx}}\right)\right\}$ $={\frac {\mu e}{2}}\left(qc_{1}-rb_{1}\right)$.

Similarly, the force parallel to the axis of y

$={\frac {\mu e}{2}}\left(ra_{1}-pc_{1}\right)$,

(5)

the force parallel to the axis of z

$={\frac {\mu e}{2}}\left(pb_{1}-qa_{1}\right)$

where a₁, b₁, c₁ are the components of magnetic induction at the centre of the sphere due to the external magnet. These forces are the same as would act on unit length of a conductor at the centre of the sphere carrying a current whose components are ${\tfrac {\mu ep}{2}}$, ${\tfrac {\mu eq}{2}}$, ${\tfrac {\mu er}{2}}$ . The resultant force is perpendicular