ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 



113 



The boundary conditions now yield, for the determination of the constants C n , C',,, 

 the equations 



. . . (27) 



In particular we find 



0.1 *<*) 



. (ka)} - E. (ta) {a- V. (k'a)} 



. - - (28) 



7. We wish to evaluate the components of electric and magnetic force at points in 

 the dielectric which are close to the sphere. We see that at such points we may put 



- 7T> 



= - 2 (2n+l) P'+Vo'-'E,, (kr u ) a+' {+ (^) + C re E n (ka)} (l-V) ~= ^, (29) 



n = 1 tl/X 



where 



, n (Aa) {a-^E. (*a)} -E, (ka) { 



1-. 



(30) 



If the material of the sphere were perfectly conducting the second factor on the 

 right would be replaced by unity. For any good conductor and high frequency j k' is 

 large compared with k, and we may approximate to the value of the second factor. 



Put 



jfe" = - im 2 , m 2 = 4TT<rkC, (31) 



then we may take 



k'a ^ma */2 (li), ik'a ^ma y/2 (!'+*'), 



and, if a is the radius of the earth, ma is a large number, so that the most important 

 part of sin k'a is ^ie^ k>a . Hence the most important part of ^ n (k'a) is the most 



important term of 



[ 1 d } n e tk ' a 



(k'a d(k'a)\ Zik'a' 

 and this is 



i n e ik ' a 



VOL. CCXV. A. 



2t (k'a) n+l 

 Q 



