10 Mr. J. J. Thomson on the Magnetic Effects 



taking the real part 



d cos (^pt— X(r — a)) 

 dy r 



Similarly the magnetic force parallel to the axis of x 



d cos (pt — \(r — a)) 



dx r 



and the magnetic force parallel to z vanishes. Thus the 

 magnetic force is the same as that which would be produced 

 by a current-element coe cos pt or ev, v being the velocity of 

 the sphere (see Proc. Math. Soc. xv. p. 214). 



The magnetic force inside the sphere parallel to x equals 



3De^J-S (M 



= 3De^S (V)-- 

 dr v y r 



Substituting the value for D given by equation (7), this 

 equals 



coe 

 a 



— piKyeW; 

 a l 



or, taking the real part and writing X 2 for Kp 2 , 



*^ Qf a 2 Ji/ cos pt. 

 The component parallel to y is 



r QJ a 2 ) x cos pt, 



and the ^-component vanishes. Thus the maximum magnetic 

 force inside the sphere is 



— 2 cos ptiXay. 



If \a is very small, this is very small compared with the force 

 outside the sphere. If the velocity is uniform, p, and there- 

 fore \ = 0, and the magnetic force inside the sphere vanishes. 

 When there is no magnetic force inside the sphere its energy 

 and the force acting upon it have the values assigned to them 

 by Mr. Heaviside. 



Let us next take the case where \a is small and \a large: 

 in this case C and D have the same values as before, so that 

 the magnetic force due to the moving sphere is the same. 



