﻿708 Prof. S. R. Milner on 



equivalent to rotating the whole system of axes in the plane 

 of zl through an angle 9 given by 



tan 6 = iv. 



If now starting from an arbitrary hyperplane and with the 

 axes of x y z at a given point as origin oriented as in (1), we 

 rotate the axes through the angle zh where 



l>y 

 tan 0,1=*-, (4) 



ex 



we find for the field as observed in the new hyperplane 



ej = E, V = e z ' = 0, h x l = H, h y ' = hj = 0, . (5) 



where B and H are related to the usual invariants of the 

 field by the evident relations 



E 2 -H 2 =e 2 -h 2 , EH-(eh). ... (6) 



E and H are thus themselves also invariants, and the field at 

 the point has been simplified by the orientation of the axes 

 into a combined electric and magnetic force acting in the 

 same direction along the (unchanged) axis of x *. 



At an infinitesimal distance along the t ?;-axis from the 

 origin e and h will no longer be colliuear, but they can be 

 made so again by a suitable orientation of the axes through 

 infinitesimal angles ; and it is evident that in this way a 

 continuous line, at every point characterized by the col- 

 linearity along it of the transformed electric and magnetic 

 forces, may be constructed in the four-dimensional space. 

 This procedure is precisely analogous to the method of 



* When h y >e x the transformation (4) involves a velocity greater than 

 that of light for the observer. To get over the difficulty we might use 

 in this case the strictly legitimate transformation 



tan©ri=*-=r, (4«) 



collinearity is again produced, but now along the y-axis. There is. how- 

 ever, no need to deal with it separately, it can be included with the 

 other in the transformation (4) by imagining that the Lorentz equations 

 are valid for values of v greater than 1. It may be noted here that if 

 we apply transformations (4) and (4 a) to the same field, while (4) gives 

 collinearity along x with e x ' = E, h; C ' = II, (4«) gives collinearity along y 

 Avith ey — — iH, hy'=-^iE. It will be evident in § 5 that these are the 

 yz and xl components of the electromagnetic five-vector (R, i'R), and 

 that the two cases consequently form merely different aspects of the 

 same field. 



