422 Prof. Rowland on Spherical Waves of Light 



we shall have the following real forms, since 



C n =A n —iB n , 



c'„=a w +;b„, 



c = a — ibj 

 c f = a + ib t 



N 1= - 2 ^®^6^-^{A w cos5(p-Y0-B n sin5(p-Y0^ 

 P 



© 2 = 2 -^^ e ^-y^ [a..,-^] oos J(p-V0 



-[B^^-^Jsin^-Vo}, 

 P 2 =- 2w(n 2 +1) Q re 6^- v ^A n co S 5( / >-Y0-B B sin6(p-V0f. 



r 



And the other terms are readily obtained from these as 

 follows:- — 



N_,= 2 f*®^2 <&- yt) { [(a 2 ~b 2 )K + 2aMBJ cosft^-VO 

 -[(a 2 -5 2 )B w ~-2a6AJ sin6(/>-V0}, 

 Ng= 231500^(0-^ 



-[(a 2 -Z> 2 )B„ + 2a5AJ «B.&0>-Y*)}. 



As these quantities constantly return again to the same form 

 with only a change in the constants, there will be only two 

 cases, when the vector potential, or its analogy in the elastic- 

 solid theory, is made equal to the even or odd vectors. Let 

 us take the first case, making the components of the vector 

 potential equal to © and P . The magnetic induction will 

 then be N 1; and the components of the electric current 



©2 , P, 



and 



47T/X, ^TTfJb 



Hence, in this case, the magnetic induction is in circles around 

 the axis, and the electric currents in planes passing through 

 the axis. At a great distance from the origin the electric 

 currents are on the sphere, but the normal component must 

 still exist in order that the equation of continuity may be 

 satisfied. The case of plane waves is the only one where the 

 normal component entirely vanishes. 



When we make the vector potential equal to N x the electric 



