298 Dr. R. 0. Tolraan on the Derivation from the Principle 



H '-^»( H ' + » • • • • (17) 



H'=^(H,-^E,) (18) 



Thus at a given point in space, we may distinguish 

 between the electric vector E as measured by a stationary 

 observer and the vector E' as measured in units of his own 

 system by an observer who is moving past the stationary 

 system with the velocity v in the X direction. If eE is the 

 force acting on a small stationary test charge of magnitude 

 e, then eE' will be the force acting on the same test charge 

 or electron when it is moving through the point in question 

 with the velocity v, the force eE 7 being measured in units of 

 the moving system*. 



We are more particularly interested, however, in the 

 vector F which determines the force eF that acts on the 

 moving charge but which is measured in " stationary units/' 

 thus determining the equations of motion of the test charge 

 e with respect to stationary coordinates. Since, however, it 

 is possible to obtain relations between the units of force used 

 by stationary and moving observers, a method is presented 

 of calculating F from the values of E' already given by the 

 transformation equations (13-18). As a matter of fact the 

 expression for F which can thus be obtained is identical with 

 the fifth fundamental equation of the Maxwell-Lorentz theory. 



Relation between the Units of Force used in Moving 

 and Stationary Systems. 

 Consider a body having the mass m when at rest and 

 moving with the same velocity v as a system of coordinates S'. 

 Evidently its acceleration with respect to those coordinates 

 is determined by Newton's laws of motion, and its acceleration 

 with respect to stationary coordinates can be found by making 

 the proper substitutions, giving us 



F/ = m ux' = m _^* .... (10) 



^;='W='^j (20) 



F z ' = m uJ = m - r ^ ) (21) 



* It should be noticed that according to the first postulate of rela- 

 tivity, if the charge of a stationary electron, for example a hydrogen 

 ion, is €, then when the electron is in motion it must still appear to have 

 the charge e to an observer who is moving along with it, otherwise the 

 possibility would be presented of distinguishing between relative and 

 absolute motion. This justifies us in taking eE' as the force acting on 

 the moving electron and measured in the moving system. 



