﻿710 Prof. S. R. Milner on 



we have for the angle of rotation 





Hence 



_ -H^dJiy-Kdey) _ E(ErfA y -H<fe y ) 



Cy '~ E 2 + H 2 ' y '~ EHH 2 



We now further rotate the axes in the plane of zl. (this 

 rotation does not affect directions in the plane xy) through 

 the angle 



lt v < .'Fidhy—'H-dej/ 



d0zi= i 



E 2 + H 2 



The axes are now oriented so that e and h are collinear, 

 and x lies along the electromagnetic line. 



Variations de- z and dh z are similarly transformed by rota- 

 tions in the planes of xz and yl through angles 



E dez + H dli z ,^ i E dh z — H de z 



~~E 2 + H 2 ' yl ~ E 2 + H 2 



Variations de x and dh x do not involve any rotations of the 

 axes, but we have simply 



<iE — de x , dH = dh x . 



Solving these six equations for de X} de y , etc., we obtain 



de x = dE, dI> x =dR, ~\ 



de y = E d6 xy + iK d6 zh dh y = H d6 zy -iE d6 zh I (7) 



de z =. E dOxz — z'H dOyu dh z = H dQ xz 4- z'E dd y i.) 



These form a set of transformation equations by which we 

 can express the infinitesimal variations of e and h between 

 the origin and a neighbouring point in terms of the collinear 

 invariants E and H and of the infinitesimal angles through 

 which the S} r stem of axes has to be rotated in order to lay 

 the axes at the neighbouring point along the appropriate 

 directions for observing the cpllmearity. 



Let the values (7) be substituted in the electromagnetic 

 equations (2) ; if we multiply the resulting equations by E 

 and H, add or subtract them in suitable pairs, and observe 

 that the sign of 6 is reversed by changing the order of its 

 suffixes, we get the following set of eight equations which 



