Fundamental Concepts of Electrical Theory. 409 



electric and magnetic lines of force are connected in such 

 a way that 



E = e and H = h. 



A method of determining a set of functions X, Y, Z, Y, T 

 which lead to these relations is given in a paper which has 

 been offered to the ' Messenger of Mathematics.' 



§ 3. Geometrical representation of the various Vectors*. — 

 Consider a sphere of radius c whose centre is at the origin 

 and a force R acting along a line L which cuts the sphere in 

 real points. Let (E z , E y , E z ) be the components of this 

 force and (cH x , cH y , cH z ) the moments of the force about 

 the axes of coordinates, then it follows from the relation 

 cR — [vE] that v x , %, v 2 are the coordinates of a point U on 

 the line L. Since the velocity v is generally supposed to be 

 less than c the velocity of light, the point CJ will be within 

 the sphere, and this is why the line L was made to cut the 

 sphere in real points. 



If 

 BY . BY BY 1BY 



& = k " b B^ = %J ' T* = kzi > c-B7 = -^ 



the line L also passes through the point with coordinates 

 (x X} ?/!, z-j). Similarly it passes through a point derived 

 from the function T. 



It follows then that we may write 



^BY^ BT XBY /.BT 



qx OX c ot c ot 



In Sir Joseph Thomson's relation cH = [vE], v was 

 regarded originally as the velocity of the lines of electric 

 force. If we regard v as defined by this relation it is 

 evidently indeterminate, for (v x , v y , vj) may be taken to be 

 the coordinates of any point on the line L. 



The velocity of a line of force when defined in this way 

 may be either greater than, equal to, or less than c the 

 velocity of light. 



If we define the velocity of a line of magnetic force in a 

 similar way by the relation cE+ [wH] =0, the components 

 of this velocity w may be regarded as the coordinates of a 

 point on a line L' which is the polar line of L with regard 

 to the sphere. Since L cuts the sphere in real points, 

 L' does not; and so it follows that the velocity w can never 

 be less than the velocity of light, it can be equal to the 



* Cf. ' Messenger of Mathematics/ Nov. 1915. 



