SCATTEEING OF PLANE ELECTRIC WAVES BY SPHERES. 205 



Thus, when we regard a/r as small, we may take 



R = 1 a(l sin cos <j> + m sin sin <p + n cos 8), 

 (7-65) 



CJJ-b j 7 ' /\ ' n n\ 



75 = \f sin ft cos <j) + m sin sin <t> + n cos 0). 



Cv 



Now, in applying (7'13), occurs only in the coefficients and not in the exponential 



index : thus we can get the first approximation by putting 6 = -IT in ; this gives 



ov 

 the value 



(7'66) ?5 = n. 



Thus, to our degree of accuracy 



1 / dv dw\ v I 9R a 



4:TT \ Of CV/ 4-7T \ CV CV 



V I CIV 



= IKHW + 

 47T\ 3l/ 



We can now substitute for w and - the values given by (7'63) and (7'64) : it will 



CV 



be seen that the components parallel to y, z give zero (to this order), and that the 



component parallel to x gives 



ucn e" (n "- K> 



2-7T r 

 Accordingly the reflected wave is given by 



(7'67) X = 



27T?' 



where R is. found from (7'65) and the integral extends over the positive hemisphere. 

 For the purpose of integration we write 



I = sin tf cos <j)', m = sin O 1 sin $', n = cos tf. 

 Then in (7 '6 7) we have 



cm R = r + a {(l+cos 0)cos fl' + sin 0sin tf cos (<f> $')}. 



The integration of (7'67) with respect to 0' can be carried out at once, because 

 (7'68) p e ..sinsm'oo8(*-f) ^/ _ 2 T J C ( m sin sin tf). 



