Effects produced by the Motion of Electrified Bodies. 241 

 The part depending on p in the second integral 

 fiep CCCr^ d 1 Qd 1 



i.By^-j-^Wa dy dz, 



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



_ P e P w 

 - 2"*i. 



Adding this to the term fnepF\ already obtained, we get 



~^y- F\ 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 



-g •(F li > + G'tf + H' l r). 



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

 the axis of x 



= ^T_^^T 

 dx dt dx 

 pe ( d¥\ , dG\ , dR\ dF\ \ 



= YV ? -dx~ + ^^ +r -o^-—drf 



fief dF' ti dG / ll dR\ d¥\ dF\ dF'Al 



= TV y iix- + vijy +r iix--p^-^- r iir)j 



_pef jdG\ d¥\\ f dW, dH\\\ 

 ~2\ q [dx dy J \dz dx J I 



Similarly, the force parallel to the axis of y 



= yK-i^i); (5) 



the force parallel to the axis of z 



= j(ph-q<*i), 



where a 1} b h c x 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 compo- 

 nents are ^, ^-, ^~ t The resultant force is perpendi- 



