the Structure of the Electric Field. 



303 



the same as those possessed by a tube forming part of a con- 

 tinuous electric field. 



(1) That the product of the electric force due to the charge 

 and the cross-section of the tube is constant and equal to iir 

 times the charge at the end of the tube. 



(2) That when the tube is moving relatively to axes which 

 are fixed with reference to the instrument used to measure 

 the physical quantities, there is a magnetic force at each point 

 in the tube, the direction of the magnetic force being at right 

 angles to the plane containing the electric force, and the 

 direction of motion of the tube at the point, the magnitude 

 of the magnetic force being the product of the electric force, 

 and the component of the velocity of the tube at right angles 

 to this force. 



(3) That there is momentum with reference to these axes 

 throughout the tube equal per unit volume to the vector 

 product of the electric and magnetic forces divided by the 

 square of the velocity of light. 



(4) That if R is the resultant electric and H the resultant 

 magnetic force at a point in the tube measured in electro- 

 static units, c the velocity of light, the energy per unit volume 



Fi<r. l of the tube is 



From these principles we shall calculate the 

 values of certain quantities which are required 

 in the discussion of the properties of these tubes. 



The energy in a tube at rest. 



Let the corpuscle be regarded as a sphere of 

 radius «, the tube of force being the portion 

 outside the corpuscle of a double cone whose 

 solid angle is o> with its vertex at the centre 

 of the corpuscle. 



Let A B, A' B' be two adjacent cross-sections 

 of the cone, S the area of AB, r the distance of 

 AB from the vertex of the cone, and dr the 

 distance between AB and A' B' ; then if R is 

 the electric force at AB, the energy per unit 

 volume is R 2 /87r, so that the energy included 

 between AB and A'B' is equal to 



S dr R 2 



Sir • 



If e is the charge on the corpuscle the charge at the base of 

 each half of the cone is \e y thus RS = 2ire and the energy 



