258 On the Lorentz Transformation. 



We have then, finally, 



fE x '=E x 



< 



B.J = H x 



or in real form 



P' =XP+tXtt/c Q 



(20) 



fE x ' = E x 



Ey' =X(EJ f -M/ C Er) 



Hy , =X(H y +t*/c E,) 



B/^ + u/cH,) 

 ^H/=X(H,-«/cE y ), 

 which is the familiar Lorentz-Einstein transformation, 



The transformation of a density of charge having' a velocity-component 

 v parallel to x in A's system is readily found when we have postulated 

 that the two observers agree as to the measure of a charge. For the 

 transformation (8) the Jacobian 



d(#', y>, z', t') _ 



whence 

 and 



dx . dy . dz . dt = dx' , dy' . dz . dt' 



dr _df 

 dr' ~ dt ' 



dr, dr' denoting the measures of the element of volume. 



Now since both observers have the same measure of a charge 



Thus 



p' dr' = p dr. 

 dr dt' 



p _ dr _ dt' / uv\ 



p ~Tt'" di~ K V~^)' 



(21). 



This completes the equations of transformation for the 

 electromagnetic variables. It is now easy to verify by direct 

 substitution that Maxwell's Equations for the general c;ise 

 where there is a moving density of charge in the field pass 

 over into another set of the same form. 



In conclusion I wish to express my thanks to Prof. 

 Eddington for valuable suggestions in connexion with this- 

 paper. 



