Effects produced by the Motion of Electrified Bodies. 247 



by the preceding work, that the coefficient of uu' is zero; the 

 coefficient ofri/, 2>git\ and the coefficient ofiuiu^ 5ott. Adding, 

 we get the whole kinetic energy due to the vector-potential 

 arising from e and the electric displacement arising from e' 



= Z 7 ^ (8uu' + (5 + 3 V + (5 + 3)W) 

 zJti 



4-77(7 , , , , x 



yuu + vv + ww ). 



K 



We can get that part of the kinetic energy due to the vector- 

 potential arising from e' and the electric displacement from e 

 by writing e' for e, and i/, v f , w' for u, v, to respectively. 

 Hence, that part of the kinetic energy which is multiplied by ee' 



OTTO" / , , ,\ 



= ~^-{uu + vv + ww') ; 

 or, substituting for a its value, 



= -=Pr- ( uu/ + »t/ + WW?'). 

 OJLV 



Or if ^ and </ be the velocities of the spheres, and e the angle 

 between their directions of motion, this part of the kinetic 

 energy 



fiee' . 

 = 3R 9q ° S6 > 

 and the whole kinetic energy due to the electrification 



/ 2 e 2 q 2 2 e l2 q' 2 ee f \ 



= Kl5 a + l5^r + 8R^' COSe }- • < 6 > 



If x, y, z be the coordinates of the centre of one sphere, 

 x', y', z' those of the other, we may write the last part of the 

 kinetic energy in the form 



fiee f (dx dx' dy dy' dz dz'\ 

 3R\di ~di + dt~dt + di~dtJ* 



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

 acting on the first sphere 



_ C H _ £ ( dT } 

 ~ dx dt \ 7 dx J 



d 'dt 



- ^£{[ d JL ££ . d v d rf dz ££] £\_ d _ ( dx '\ X 



o \\dt dt + di~dt H U 'dt dtJdiB dt\_dt_) j ' 



T 2 



