Effects produced by the Motion of Electrified Bodies. 239 



which these electromotive forces diminish with the distance, 

 we will take the case of a particle stopped by a screen at a 

 distance of t Jq of a millimetre from the glass, and compare 

 the electromotive force at the glass with the electromotive 

 force which would be produced at the glass if there were no 

 screen. By substituting in the formula giving the electromotive 

 force, we find that the electromotive force at the glass when 

 the screen is present is only about yo o o o °^ wna ^ it is when 

 the screen is away ; and as the intensity of the phosphorescence 

 will vary as the square of the electromotive force, we see that 

 when the screen is present the phosphorescence is quite im- 

 perceptible. This explains an experiment of Goldstein's, in 

 which he coated the glass with a layer of collodion whose 

 thickness he estimated at a few hundredths of a millimetre, 

 and found the glass behind quite black. 



§ 5. To find the effect produced by a magnet on a moving 

 electrified sphere. To do this we shall calculate the kinetic 

 energy of the system ; we can then, by means of Lagrange's 

 equations, calculate the force on the sphere. 



Let a, /3, 7 be the components of magnetic induction, a l9 

 fti, 71 the components of magnetic force ; if A ; B, C be the 

 components of magnetization, 



« = « 1 + 47rA, /8 = £ 1 + 47rB, 7 = 7i + 47rC. 



The kinetic energy of the system 



= g^: : \ I ] ( au i + /% + yyi)dx dy dz. 



To get the force on the sphere due to the magnet, we only 

 want that part of the kinetic energy which involves both the 

 coordinates of the sphere and the coordinates of the magnet. 

 We may write the kinetic energy as 



^ )J ]( a2 + ^ 2 + ^ 2 - 47raA - 47r/3B - 47r 7 C >^^^- 



Let F', G 7 , H/ be the components of the vector-potential 

 due to the magnet alone ; then, by equation (4), 



/ d 1 d 1\ dW dG f 



" = ^\ r TyTl-2dznrili~'liz-> 

 o ( d 1 d \\ dW dW 



dx ' 

 dy RJ ' dx dy 



( d 1 d 1\ d(¥ d¥ 



