SCATTEEING OF PLANE ELECTKIC WAVES BY SPHEEES. 



185 



the wave-train has unit amplitude ; then, in terms of the Cartesian specification, 

 the incident wave is defined by* 



= e 



eft = + e <"<+*>, 



the remaining components of force being zero. 



We must first express this wave in the standard forms of (2'l) and (2'2); we 

 therefore introduce polar co-ordinates, and then proceed to find the radial components 

 of force, which will suffice to determine the functions U, V. 



These radial components are given by 



(4-1) 



X = -sin cos <j> e l * rms \ ca = +sin sin <j> f." rc " s ", 



where the time-factor e" ct is now omitted. 

 Now from (3 '9) we have the formula 



Kt' 



So, differentiating with regard to 6, we find that 



(4'2) sin fle"" 8 " = - 2 (2/t+ 1 ) i"- 1 S " 'y P' n (cos ft) sin fl. 



Accordingly, on substituting (4'2) in (4'l), we find that in the incident wave 



(4-3) 



and 



sine Bin* i _2 



by comparing the two formulae (2'5) and (2'8). 



The corresponding waves in the interior of the sphere will be given by the two 

 functions 



(4-4) 



where 



and 



BinflcoB? 





1) 



p, 



and K, /* are the fundamental constants of the spherical obstacle. 



* It is assumed that in the incident wave we may take /* = 1, K = 1, i = e, 



2 D 2 



