an Arbitrary Electro -magnetic Disturbance, 425 



of finding the disturbance at any point due to any arbitrary 

 electrostatic disturbance throughout the medium. And it is 

 to be noted that these equations are rigidly exact for all dis- 

 tances from the disturbance, and only have to be integrated 

 to give the effect of any disturbance. 



Had the disturbance been magnetic, we should have had for 

 the magnetic induction, 



©" = —^ (2C + G 2 )e- ib ^- vt \ 



p//= 2_cos^ c ^_. 6(p _ W) ^ 



and for the electric displacement, 



N / = _ iYKb Sin Q e -iKp-Vt) 



krrp x 



Hence, taking a small sphere as before, the magnetic induc- 

 tion within it must be 



6 ~ B?\ bR) 



Whence, as before, 



07TI 



And so we have in this case for the magnetic induction, 

 ~„ X"ib 2 sin 2C + C 2 



^ lf 3X f/ ib cos 9G 1 „ MJ 

 Airp 2 C 

 and for the electric displacement, 



S^KX^sinfl C, ra 



32tt 2 p "0 6 



These equations give the complete solution of the disturbance 

 throughout the medium due to an arbitrary magnetic dis- 

 turbance at any point. 



Although the disturbance is harmonic, yet we know by 

 Fourier's theorem that any disturbance can be represented by 

 a series of harmonic terms with the proper coefficients ; and, 

 indeed, we can replace the harmonic term by any function of 

 p — Yt or p + Yt. Should the disturbance not be parallel to 

 the axis of X, we merely have to divide up the disturbance 

 into its components and compute the effect of each, and then 

 add the components of the computed disturbance. In this 



