SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 181 



The general solution of equation (2 '6) is well known, and it is given by* 



(27) F..(r, t) = 



where the functions / and g are arbitrary. 



In the special case of divergent wares, the function g can be omitted in (27) ; and, 

 if the region considered includes the origin, then gfat+r) = f(cj. + r), so as to 

 make F B (r, t) continuous at r = 0. 



It will be convenient to notice that in consequence of equation (2'6) the radial 

 components of force can be written in the simpler forms 



(2'8) X or c = 2 



It will be noticed that we can at once determine the form (2 '5) for U or V 

 when the radial forces have been expressed in the form (2'8) ; this agrees with the 

 general conclusion stated at the end of 1, that (in spherical polar co-ordinates) the 

 remaining components of force are completely determined when the two radial 

 components are known. 



3. SPECIAL, CASE OF SIMPLE HAHMONIC WAVES AND THE APPROPRIATE 



FUNCTIONS. 



We assume in future that the waves are simple harmonic, of wave-length ZTT/K in 

 free space ; we can then suppose the time to occur only in the form of a time-factor 

 e l * c \ with the usual convention that finally only the real (or the imaginary) parts of 

 the formulae will be used. 



The functions /, g occurring in equation (27) above are then exponentials of the 

 types 



g-faf-r) an( J gui.faj + r^ 



where K, is given by 



KG, Ol' (Cj = K 



Thus (if we now suppress the time-factor e"" ct ) the functions given by (27) are 

 of the types 



/ 1 9Ve-'" r / 1 3\"o + "" r 



/ Q * ~I \ M J_ 1 / I " W -I- 1 / 



\a i) 



\ r "or) r \ r 'or I r 



We shall be concerned with two special types only : (i.) divergent waves ; 

 (ii.) waves which are continuous at r 0. The former of these corresponds to the 



* See, for instance, LAMB'S 'Hydrodynamics,' 1906, art. 295; an alternative method of solution 

 given in 3 of my paper in the ' Philosophical Magazine,' quoted on p. 179 above ; compare also A. E. H. 

 LOVE (' Phil, Trans. Roy. Soc.,' A, vol. 197, 1901, pp. 9, 10). 



