﻿714 Prof. S. R. Milner on 



With these substitutions (8 c) and (8 d) become 



^(V1 ¥ +H 2 .M) = 0, .... (9c) 

 -^(s/W+W.xJ^O (dd) 



It thus appears that a theorem expressing the constancy 

 of the flux over the cross-section of the tube is derivable for 

 each one of the four electromagnetic tubes which can be 

 constructed with any point of the field as origin. The cross- 

 section over which the flux is reckoned is determined by the 

 particular hyperplane which contains the .infinitesimal portion 

 of the tube concerned. The x- and /-tubes lie initially in the 

 hyperplanes xyz, lyz respectively, and have the same cross- 

 section y 1 z 1 ; the y- and 2-tubes lie in yxl, zxl, and have the 

 same cross-section aj^ The same quantity \^E 2 + H 2 

 appears in each case as the function whose flux is constant ; 

 it will be convenient to represent it by a single symbol R. 

 In terms of the invariants of the field as usually expressed 

 we have by (6) 



R = VE 2 + H 2 = {(E 2 -IF) 2 + 4E 2 H 2 }* 



= {(e 2 -h 2 ) 2 + 4(eh) 2 p. . . (10) 



In the special case when e and h are everywhere at right 

 angles to each other (eh) = and 



R = Ve 2 -h 2 . 



This is a result which has already been given in Prof. Whit- 

 taker's paper. The Lorentz transformation also shows that 

 when e and h are perpendicular, H == and R = E, hence 

 when they are viewed at any point in the appropriate hyper- 

 plane the electromagnetic <a?-lines are lines of pure electric 

 force, they may however differ from ordinary electrostatic 

 lines by their whole lengths not being containable in a single 

 hyperplane. 



§ 4. A Theorem, complementary to the preceding, relating 

 to the Twist of each Tube. 



The expression of the constancy of the flux of R for the 

 four tubes only accounts for half the information derivable 

 from the eight electromagnetic equations. The second four 

 of the equations in (8) are concerned with what we may call 



