SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 177 



Here (X, Y, Z) denotes the electric force, (a, ft, y) the magnetic force, K is the 

 dielectric constant, /* is the magnetic permeability, and the axes of reference are a 

 Cartesian right-handed system. The units adopted are those of the electromagnetic 

 system, and c is the fundamental constant generally identified with the velocity of 

 radiation in free space ; the equations (E) are those derived from AMPERE'S law, .and 

 the equations (M) are similarly derived from FARADAY'S law, the two together 

 constituting the circuital relations of the electromagnetic field. 



It has proved possible to obtain a solution of a very general type, by assuming that 



(I'l) X = xO Y = yQ Z = zQ 



dx dy dz 



then equations (M) yield 



/, .n\ ^ _ 9Q _ 3Q 3/3 _ 8Q 3Q 3_j/ _ 3Q 3Q 



dt dz dy dt dx dz dt dy dx 



Substitute from equations (l'2) in the first equation (E) and we obtain 



(,,, < = +2 



c 2 or dx\ ax dy dz/ 



3 3 ?f ?t 2 



where A 3 denotes LAPLACE'S operator . + -^-. + -. . 



dx* dy" Sz^ 



On comparing equations (I'l) and (l'3) they will be seen to be consistent provided 

 that 



and that 

 (1-5) 



Thus Q must satisfy the fundamental wave-equation, which is satisfied by any 

 component of the electric or magnetic forces (X, Y, Z) or (a, /3, y). 



For our purpose it is more convenient to express the above solutions in terms of 

 spherical polar co-ordinates r, 0, <j> ; these are supposed to form a right-handed system, 

 when taken in this order, so as to avoid changes of sign in introducing the new 

 co-ordinates. We write here* (R,, K 3 , B 3 ) for the components of electric force in the 

 directions of r, 6, $ respectively ; and (H,, H 2 , H 3 ) for the components of magnetic 

 force. 



Equations (I'l) then become 



/ 1M1 v P 8P 19P T? 1 aP 



(111) -tti = -5 -- ?y> -tta = ~ ^77' its - r ~~a^I 



or r d9 . r sin 9 a$ 



* This is done to avoid confusion with the Cartesian components used in equations (E) and (M) ; but 

 in the subsequent sections we shall use (X, Y, Z) and (, /?, y) for the spherical polar components 

 here denoted by (Ri, Rj, R 3 ) and (Hi, H 2 , H 3 ) respectively. 



2 C 2 



