300 Fifth Fundamental Equation of Maxwell-Lorentz Theory. 

 or in the case where u x =v, %=w*=0: — 



V — m <> _ du* , 1 ??? o w o ;, 



H? 



and by the further substitution of equations (19-20-21) we 

 obtain 



F X =F Z ' (22) 



F f = Vl^ 2 F/ . . (23) 



F z = V1-/3 2 ¥ s ' } (24) 



which are the desired relations connecting measurements of 

 force in the two systems. 



n 



The Fifth Equation. 



Returning now to the consideration of an electron which 

 is moving with the same velocity as the system S', we see 

 that the transformation equations (13-141-15) together with 

 the above equations lead to the relation 



F Z =F X ' = E/ = E X 



F y = </l=p*F,'= •l^E/ = (E y -~ H*) 



f,= vi^ 2 f/= vi^^e;=(e, + ^ H,y 



which is the desired equation : 



F = E+ - vxH. 

 c 



This result agrees with that obtained by Einstein in his 

 second treatment (JahrbuchderRadioaktivitat,iv. p. 411, 1907) , 

 where instead of defining force as equal to mass times acce- 

 leration, he defined it by the equations 



T? _ A. m ° Ux V — ^ m o u y V — ^ m ° lfz 

 x ~ dt Vl^>' 1y ~ dt x/l-(3* ~ It s/l=J* 



which agree with our definition of force as equal to the rate 

 of increase of momentum. 



