ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 125 



whence 



S = i (K^ 2 )- 1 sin 1 '* 6 (no + )' sec 1 '' a sec 1 '* a, ( ?- - V'' 



\Zo COS o 2 COS a 2, COS a,/ 



gl (2* 008 ,,-* C0 --, COS a,). 



that is 



S 41 = IK sin a a, sin 1 ' 2 (z t sin ai)~ v * sec 1 ' 1 a sec 1 '- o^ ( ) 



\Zo cos 2 cos a Zi cos a,/ 



,,t (3r cos a^ j cos a Zj cos a]) 



which after some reduction can be put in the form 



O "11 



41 = 



_ 

 QP OxQ/ \acosoo QP 



and this represents the effect at P of the waves reflected from the sphere at Q.* 



From the diagram it appears that the points P for which the condition 

 6 = 2^ a ^ can be satisfied are the points outside the tangent cone drawn 

 from Q! to the sphere (that is the points outside the geometrical shadow), and 

 as P approaches the surface of the cone the value of n a + % approaches KO,; hence 

 the above result holds for all points outside the tangent cone and not close to its 

 boundary. 



Omitting for the present the discussion of the terms of the series S 3 and S 4 for 



* This result can be verified by elementary methods as follows : If M is the amplitude of the magnetic 

 force at Q, MI the amplitude of the magnetic force at P in the waves reflected from Q, the intensity 

 at Q in a beam of rays from Oi of small cross section is 



M 2 o 2 sin (oo - i) cos 00 d (oo - a t ) d<f>, 

 and the intensity in the same beam at P is 



MiVsin 6cosaded<i>, 

 and these intensities are equal, therefore 



M 2 a 2 sin (ao - aj) cos a d(m > -xi) = Mi 2 /' 2 sin 6 cos a dd ; 

 now 



6 = 2a<, - a, - 0.1, 



whence 



ZD cos sto duo = z cos a. da. = ^i COS 04 rfzj, 

 r t cos ai - a cos ao = OiQ = a sin (ao - ai)/sin aj, 



i 2 ?- 2 sin 2 ^ = Ma 2 sin 2 (an - a^ sin sec a sec y.\ I z\ sin a! / \ 



I \*o cos ao 2 cos a ! cos sti/ 



The amplitude of the <f> of the waves from Oi at Q is Ma sin (ao - ^i), and of the reflected waves 

 at P is Mir sin 0, therefore the ^ of the reflected waves at P is 



- IK sin 2 ! sin 1 ' 1 sec 1 '* a. sec 1 ' 2 a.\ (z\ sin ai)" 1 '* ( ) * , 



\3o cos o z cos a zi cos ai/ 



which is the result obtained from the series. 



