﻿718 Prof. S. R. Milner 



on 



laying down in hyperspace x f - and Z'-lines . in a different 

 direction from before, although still in the unaltered xl 

 plane. Similar considerations apply to the y~, and z axes ; 

 indeed, when collinearity of e and h has been obtained it is 

 evident that there is nothing to distinguish their actual 

 directions in the fixed plane yz. 



It thus appears that what is uniquely fixed in hyperspace 

 is not the directions of the individual axes, but the orienta- 

 tions at the point of the two coordinate planes xl and yz, in 

 which planes the axes themselves may be drawn arbitrarily. 

 The entity that we have really to deal with in a four-dimen- 

 sional theory of the electromagnetic field is of the type which 

 is best represented not by lines but by surfaces ; in other 

 words, it is not of the four-vector, but of the six-vector type, 

 being a function of position the properties of which at any 

 point are associated with two absolutely orthogonal planes 

 {xl and yz) which cut each other only at that point *. The 

 six-vector in question is of a restricted type, characterized by 

 the equality of its two parts ; it consists of R = \/E 2 + H 2 

 associated with the yz plane by " acting " in any direction per- 

 pendicular to it, i. <?., along any line in the plane xl, combined 

 with the equal (but imaginary) quantity iR similarly asso- 

 ciated with the xl plane. Although a six- vector, since its 

 two parts are equal, it . only requires five independent 

 quantities to specify it : the four quantities required to fix 

 the orientations of the planes, along with the magnitude 

 of R. We will call it consequently a " five-vector," 

 thereby distinguishing it from the electromagnetic six- 

 vector (h — ze). The six quantities required to specify com- 

 pletely the field at a point are known when a is given in 

 addition. 



§ 6. The Construction of Unique Sets of Tubes, 



These considerations enable us to explain the specialsym- 

 metry which the tubes show in the planes xl and ?/^ and at 

 the same time to derive a set of tubes which are really 

 uniquely laid down in hyperspace. Having obtained from 

 any initial hyperplane the axes oriented so that x lies along 

 the collinear E and H, let the axes be further rotated through 

 the angle 6 x i in the plane xl. This will not affect E, H, or 

 the axes y, z. Starting from the origin along the new direc- 

 tion x' of x, we can construct an #'-line just as before by 



= * The plane xl comprises all points of hyperspace for which ?/ = Q, 

 z=0; yz all points for which .r~0 ; /=0 ; the two planes thus cut only 

 at the origin. 



