ELECTRIC WAVES ROUND A PERFECTLY REFLECTING OBSTACLE. 123 



where 



S, = (2ir)- v *(sin 



5 2 = (2ir)- 1/ '(sin 



5 3 = (2ir)- v *(8in 



5 4 = (27r)-''-' (sin 



Since rOj, the formulae (ii.) of the Appendix can be used for the terms of Si and 

 S 2 , for which n + % is less than z and z(n+%) is of the same or higher order than z' /3 , 

 and the corresponding parts of S! and S 2 are given by 



S u = (2ir)- v - (sin e) >! > (fW 1 a )- 1 "s >/ '(n+|)' / sec 1 /' sec 1 - a^co..-,, 

 S 21 = (27r)-'- (sin 0)'<' (jcr/)- 1 * sTV+i)* sec 1 '' a sec 1 - aie *-i 



where 2 sin a = z t sin a t = n + ^- and A is of the order of unity. 

 Now 



-j- {z cos a 2, cos i 



and a e^ + cannot be a small quantity for >a 1; since '!> 

 of the order Zj" 1 ^, and therefore negligible in comparison with 

 Again 



T {2 cos a Zj cos a 1 



Cu'fli 



and a.a. l will vanish for a value of w between 1 and 2 As 1 / 3 , 

 provided the point P (r, 6} lies inside the sphere described on OO, 

 as diameter, but not close to its boundary. If a and ! now 

 correspond to the value of n for which 0=a. a l , a is the angle 

 OPT and a t the angle OOiP in the figure,* and the principal 

 part of S 21 is given by 



S n = (2ir)- % sin 1 /' (/en 2 )- 1 (z, sin a^ sec 1 /'-' a sec v ' a 1 e'(-' 



hence S u is at most 



where /oi is large, that is 



sn 



sn ai ' sec - a sec ^ 



z cos 



t --- L_p. 



os a 2, cos i J 



The corresponding value of n is given by n + J = up, where _p is the perpendicular from on OiP. 



R 2 



