﻿Electromagnetic Lines and Tubes. 719 



infinitesimal rotations in the planes ,v>/, x'z } yV, zV alone. 

 For the corresponding .f'-tnbe ^ve shall have 



= (cos 6 r , . |- + sin 0,i . *-\ ( SW+W . fcfe) 



= by (9 a, 6). 



Hence the flux theorem applies to a tube starting from the 

 origin in any direction in the plane of xl, the properties of 

 the x- and /-tubes are in fact symmetrical in this plane. 

 Since 9 x i is arbitrary, we may choose it so as to satisfy any 

 stated condition, for example, the condition that 



= cos x i^- +sm x j ^-f = 0. 

 00}' OJ ol 



It is clear that in this way an .v'-line may be drawn such 

 that there is no change in the composition of R along it, and 

 it is now also uniquely laid down in hyperspace. When 



— - = 0, (15) shows that J^f, 2 = 0, which means that there 

 o# O t 



is no twist round the V direction perpendicular to x' . Hence 

 the " twist of the xl plane," i. e. the twist round its axis of 

 any infinitely thin tube drawn in it, is a maximum round the 

 particular dZ-line which does not vary in composition along 

 its length, w T hile the /'-line at right angles to x' is charac- 

 terized by no twist round it, and along it a maximum rate 

 of change of composition. In fact, in the xl plane (and all 

 these conclusions apply also to the plane yz) the vectors 

 representing the twist and the gradient of « are in mutually 

 perpendicular directions *. 



* Some sort of a visualization of the effects of twist can be got by 

 picturing an .r-line as being something like a ribbon instead of being the 

 same all round like an ordinar} r line. We can suppose it shows E on 

 the face and H on the edge of the ribbon. The Lorentz transformation 

 enables one to alter the view point, and with untwisted ribbons to 

 change the h-aspect of the field completely into e ; when they are 

 twisted however it is impossible to find any view point from which the 

 e- or the h-aspect alone may be seen ; they must both show simul- 

 taneously. This agrees with what has been deduced : when e and h. are 

 perpendicular, one of them may be transformed away, when they are not 

 perpendicular the lines are twisted and neither can be transformed away 

 completely. It also enables us to visualize how twisted tubes might 

 produce space-time variations in the ratio of H to E. It is, however, 

 only a crude analogy and must not be pushed too far. As a fact the 

 composition of the lines changes in a direction perpendicular to the axis 

 of twist, and not along it as the analogy would suggest. 



