SCATTERING OF PLANE ELECTRIC WAVES BY SPHERIiS. 
181 
The general solution of equation (2'6) is well known, and it is given by 
* 
(27) 
F„(r, = 
where the functions f and g are arbitrary. 
In the special case of divergent waves, tlie function g can be omitted in (27); and, 
if the region considered includes the origin, then g{cd + 'r') = — J {.cd + r), so as to 
make F„ (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 ca = S F, (r, t) Y, («, 4 .). 
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 splierical polar co-ordinates) the 
remaining components of force are completely determined when the two radial 
components are known. 
3. Special Case of Simple Harmonic Waves and the Appropriate 
Functions. 
We assume in future that the waves are simple harmonic, of wave-length ^ttIk in 
free space; we can then ^suppose the time to occur only in the form of a time-factor 
gi/tcj, usual convention that finally only the real (or the imaginary) parts of 
the formulae will be used. 
The functions f, g occurring in equation (27) above are then exponentials of the 
types 
where «■, is given by 
/CjCi = KC, or /Cl = /C -y/(^K). 
Thus (if we now suppress the time-factor c”"*) the functions given by (27) are 
of the types 
(3-1) 
1 
1 8 Y CU"'’’ 
r dr) r 
^.n+i 
1_8 
r dr) 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. 29.5; 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). 
