SCATTEKING OF PLANE ELECTRIC WAVES BY SPHERES. 



179 



Substituting from (l'6) in (I'll) and (l'2l) we obtain the final expression for the 

 field in terms of U : 



(1-8) 



S 2 U 



c 3 at 2 



i a 2 u 



r 30 3r 



30 3r 



cH, = 



1 3 /K3U\ a 



= + -3T7 --5r)' P = rsinfl, 



In like manner we obtain another set of solutions by making an assumption similar to 

 (l"l) for the components of magnetic force (a, /3, y). This gives the field : 



(1-9) 



R 1 = 



c at 



a* 



H ^ 



1 " " 



3 - 



= r sn 



where V is a second solution of equation (17). 



It can be proved* that (l'8) gives the most general field in which the radial 

 magnetic force (Hi) is zero, while (l'9) gives the most general field in which the 

 radial electric force (RJ is zero. It can also be shown that the field is uniquely 

 determined by the value of R : and HI ; and accordingly the most general solution 

 can be obtained by the superposition of (l'8) and (l'9). 



2. FURTHER SPECIALIZATION OF THE SOLUTION OF 1. 



If we superpose the fields (l'8), (l'9), and now utilize (X, Y, Z), (a, /3, y) to denote 

 the spherical polar components of the field, we have the general solution t : 



(2-1) 



X = 8 2 U MK 3 2 U 

 " " 



- 

 r 30 3r p 30 \c dt 



z = 1 aau . f I . 



3 3r r 30 \c 



(* -sr ) P = r sin e > 



* See a paper in the ' Philosophical Magazine,' July, 1919 (6th ser., vol. 38), p. 143. 

 t Originally worked out in 1899, and first published as a question in Part II. of the ' Mathematical 

 Tripos,' 1910. 



