ELECTRIC WAVES OVER THE SURFACE OF THE EARTH. 115 



and we find 



+. (ka) {a-% (to)} -E B (In) - {a- V (*a)} 



and 



fa"'E.(fa) = 



Hence we have at r = a, 



311 _ ie** Ct 



. . (35) 



(36) 



, I \ . n.O p t(*.-.. )T) - iWl _ 2\ n 



' " M) <^ 



n e 2 -(l-M 3 ) ) . . (37) 



where . the first line of the right-hand member represents the efiect of curvature 

 without resistance, and the second line represents the correction for resistance. In 

 practically interesting cases, z is a large number of the order 10 4 or higher, arid the 

 factor r ~ 2 may be replaced by a~ 2 . 



Approximate expressions for R n , <, x n > valid in different ranges of values of n, have 

 been obtained after L. LORENZ* by H. M. MACDONALD( ? ) and J. W. NICHOLSON^'). 

 It appears that, if r a is very small compared with a, </>_ differs very little from </> 

 throughout the whole range, and R n /R n is very nearly unity until n exceeds z by 

 a large number, but when n z is very great this fraction becomes very small. As 

 n increases from 1, R n increases from a number which differs very little from 1, and 

 ultimately becomes very great, while x increases continually from a very small value 

 to ^-TT, passing from a value slightly less than %TT to one slightly greater as n passes 

 through the integer nearest to z^r. For large values of n, provided 8 is not near to 

 or TT, we may use the approximate formulae 



sin 9 



(38) 



so that this factor is the product of a large number of the order ^/n and a simple 

 harmonic function of n. For values of n which are comparable with z, R n is quite 

 moderate. These considerations suggest that the most important terms of the series 



* L. LORENZ, ' CEuvres Scientifiques,' vol. 1 (Copenhagen, 1898), p. 405, 



Q 2 



