584 



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 



then F, G, H, and will be functions of w; and if we put F f , G', H', V 

 for the second differentials of these quantities with respect to w, the equations 

 will be 



' + G ' + a ^ H'-IVV = 



(85). 



X- | .. "- ' \ / 



v 



?' , c ' nm { 



M * + 



If we now put 



If , V 



oW/P m' n'\ (86) ' 



we shall find 



f W \J "^ V3L V \J ^~ V ............ ..P. .......... \O/ / 



\ /* 



with two similar equations for G' and //'. Hence either 



F = (88), 



U=0 (89), 



or 



VF' = lf r , VG'^m^ and VH' = n^ (90). 



The third supposition indicates that the resultant of F', G', H' is in the 

 direction normal to the plane of the wave ; but the equations do not indicate 

 that such a disturbance, if possible, could be propagated, as we have no other 

 relation between ^ and F', G', H'. 



The solution F=0 refers to a case in which there is no propagation. 



* The solution U= gives two values for F' corresponding to values of F', 

 G', H', which are given by the equations 



a' *"b* <? "' '' 



f>*7ji* 



-a'X) + ^,(a'X-& = (92). 



* [Although it is not expressly stated in the text it should be noticed that in finding equations 

 (91) and (92) the quantity *' is put equal to zero. See 98 and also the corresponding treat- 

 ment of this subject in the Electricity and Magnetism il 796. It may be observed that the 



