520 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



if we take 

 (4) 



The quantities X, Y, Z, L, M, N accordingly have the same set of 

 values for all points for which 



(5 ) Ix + my + nz at = const. 



But this is the equation of a plane whose normal has the 

 direction cosines I, m, n, and whose distance from the origin is 

 at + const. The plane is accordingly travelling in the direction of 

 its normal with the velocity a= 1/J. V//,e. The disturbance is 

 accordingly a plane electromagnetic wave, whatever the nature of 

 the function <. 



The six functions <f> are not independent. For let 

 (6) X = fc, F=0 2 , =</> 3 , Z=fi, Jf=^, ^=^ 3 . 



Then inserting these in the equations (A) and (B) we have 

 - y ^ & =mi/r 3 ' - n^r/, 



(7) -*'-#/- 



\/ 



a system of linear equations to determine the ratios of ^> x 's 

 and i/r"s. 



Multiplying the equations (7) or (8) in order by I, m, n, and 

 adding either set, we obtain 



(9) l# + mfe + nfc = 0, %' + m^r/ + n^ = 0. 



These are two differential equations with regard to the 

 variable s, integrating which gives 



Ifa + m<f> 2 + n$ 3 = C lt Ifa + m^ 2 + nfa = C 2) 

 that is 



(10) lX + m7 + nZ=C lt IL + mM + nN = C 2 . 



This shows that the component of either field resolved parallel 

 to the direction of propagation is constant as we travel in that 

 direction as well as in the plane of the wave, and is therefore 



