SCATTEEING OF PLANE ELECTRIC WAVES BY SPHERES. 191 



On substituting these results (47) becomes 



and 



where, for brevity, we have written 



KO, = p, (*!)* Kp 2 . 

 On reducing these equations, the results are 



, - 



(5 . 4) ^ 



1 and 



To determine A 2 , the series for S 2 (z) and E 2 (z) will be required to the first terms 

 only ; these are 



and on substituting in (4 7), we find 



(5'5) A 2 = 



15 



The field of the scattered waves is then given by 



f A- e~ M ' 

 Y = = +cy = (fd |Ai cos + | A 2 cos 29) cos <j>, 



(5-6) 



Z = -c/3 = ^-^(f A!- id cos 0-|A 2 cos 0) sin 0. 



field (5 '6) is accordingly zero (to the same degree of approximation) in the 

 direction given by 



= 0, cos e = (0,-f 



This conclusion is apparently new ; but it confirms an approximate result due to 

 Lord RAYLEIGH,* according to which the scattered wave is zero in the direction 



given by 



= 0, cos 6 = s (K- 1) ( K a)\ 



when (K-l) is treated as small. But on the other hand, our result contradicts 



* ' Scientific Papers,' vol. 1, p. 531, formula (61). 

 VOL. CCXX. A. 2 E 



