﻿Electromagnetic Lines and lubes. 



707 



byperspg.ee with Euclidian geometry in place o£ the difficult 

 hyperbolic geometry connected with real time. In it e and h 

 are vectors which obey at each point (where there is no 

 charge) the formal equivalent of the Maxwellian equa- 

 tions * : 



oi oy o~ 



;B^ B^y Mv_ n 



b/ "" B* + B# " ' 



a/ + By a* ~ u > 



.gAy Big B^__ A 



* B/ + 3c 3* ~ ; 



• B/>~~ , B<?« B^ 



B/ 



fl^=0. 



d.v By b; 



(2) 



From the point of view of a super-observer surveying the 

 Four-dimensional field, or for that matter of a person of 

 ordinary mentality who attempts to form a conception of the 

 underlying reality which shows itself to observers in different 

 hyperplanes as electric and magnetic forces of varying 

 strengths and directions, the hyperplane xyz in which the 

 values of e and h are originally specified must be looked 

 npon as an arbitrary one ; the /-axis perpendicular to it is 

 therefore also in an arbitrary direction. By means of the 

 Lorentz transformation the hyperplane may be readily 

 changed. If the observer of e and h at the point is in 

 motion relatively to it with the velocity v along the axis 

 of z, the observed constitution of the field is changed, and 

 the new electric and magnetic forces are given by the well- 

 known equations 



el = /3(e £ -vhy), hj = j3(h x + ve,j), } 



— e, 



h'= K 



(3) 



where {3 = (1 — v 2 )~i. 



On the Minkowski representation this transformation is 



* The mathematical results can always be re-expressed in terms of 



real time by substituting it for I, —i^. for ~. 



o£ o* 



2 Z 2 



