Electromagnetic Waves found the Earth. 375 



Macdonald's formula * can be written with the present 

 notation 



and this is seen to be equal to (7) for those small values of 

 6 for which cos \6 may be replaced by 1, and 2 sin \Q by 

 sin 6, which is just the limited region of validity of ( 14) as 

 shown by Watson f. 



Love J, who further calculated numerically the values of 

 H for distances = 6°, 9°, 12°, 15°, and 18°, found results 

 in close agreement with Macdonald's formula (14). From a 

 comparison between (7, 8) and (14) or from direct calcu- 

 lations it is obvious that Love's numerical results are in 

 agreement just as well with (7, 8), and as the latter formula 

 has a much wider range of validity it will further be used in 

 the derivation of an expression suitable for comparison with 

 expeiimental results. 



So far all formulae discussed here have been derived for 

 a boundary condition $jq = Q at r = a, i. e. for a sphere of in- 

 finite conductivity. Approximations have also been obtained 

 for the magnetic force at various distances by Love §, 

 Macdonald ||, and Watson ^f for a sphere having a conduc- 

 tivity of <r = l . 10" 11 (Love, Macdonald) and <7 = 3'77 . 10" 11 

 (Watson). 



All three investigations arrive at the same result, viz. that 

 a finite conductivity of the order of <r— LO" 11 such as sea- 

 water actually possesses, only has a small influence on the 

 wave-amplitude, increasing it a few per cent, in comparison 

 with the case of infinite conductivity. The same result is 

 found by Watson for a conductivity of dry earth. When 

 discrepancies therefore occur between experimental values of 

 wave-amplitude and theoretical formulae the cause is not to 

 be sought in the finite conductivity of the earth's crust. 



We now turn to the practical side of the matter. 

 The formulae (7, 8) derived above give the wave-amplitude 

 to be expected due to an oscillating dipol of unit moment. 



* See paper quoted. 



f Proc. Roy. Soc. (ser. A) vol. xcv. p. 84 (1918). 



j Roy. Soc'. Phil. Trans, (ser. A) vol. 215. p. 105. 



§ See paper quoted. 



[| Proc. Roy. Soc. (ser. A) vol. xcii. p. 493 (1916). 



^[ See paper quoted. 



