ASYMPTOTIC DIPOLE RADIATION FORMULAS 665 



/ and m must have their real parts positive and so in taking the 

 square roots of 5- - ^r = - k{^ -^ {ko^ + ^1^) and 5^ — ^2^ = — k2'^ 

 -^ (ki^ + ki~) one halves the smallest angle with the positive real axis. 

 If they both lie on the same side of the real axis P is zero. In order 

 that they may lie on opposite sides of the real axis it is necessary that 



arg ki* < arg {kr + ^2") < arg ^2^. . 



Writing ^r = a -\- il3 and ^2" = •'V" + iy this means that 



y > -T, -^, X + — ^. 



a~ — p~ (x- — p" 



The goal of the paper being asymptotic formulas for the sky waves 

 of vertical and horizontal dipoles the ground wave, P, will hereafter 

 be ignored. This is possible because at the high frequencies for which 

 dipoles are useful the ground wave is very highly damped. 



Sommerfeld gets an asymptotic expression for Qi by noting that if 

 we are at a great distance from the source most of the value of the 

 integral comes from that portion of the path of integration very close 

 to ki. The solution he arrives at is 



-2(i2 + S22 + S2^+ + •••)^ where ^ = 7^^==^ |: •' (7) 



Neglecting higher powers of l/R than the first, equation (7) sums 

 up into 



^ 2^2- cos e, e^ _ ,gv 



^ kJ cos d, + ki-yjk-^ - ki' -^ 



But in getting equation (7) Sommerfeld has replaced -St'^ — ko- by 

 •V^i^ — k2~. This is a needless approximation which ruins the sym- 

 metry, damages the utility and tends to hide the physical meaning of 

 the final result. To get the true asymptotic formula for Qi it is 

 necessary to confine the approximations to the purely operational 

 variety, i.e. make no approximations of substitution before integrating 

 but let the approximation reside wholly in the manner of integrating, 

 as follows 



8 Annalen der Physik, Band 28, 1909, page 705. 



