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 a_{1}, b_{1}, c_{1} 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