PART 111. EARTH CURRENTS AND EARTH MAGNETISM. CHAP. i. 759 



ir the current-components /^ and i,,, along respectively meridians (w = constant) and parallels 

 = constant), we then obtain the following expression, changing to polar coordinates by the aid of 

 uations (6) on p. 425: 



)W 



/ = it cos cos to -j- v cos sin u w sin 6 



fra = M Sl'n It) -\- V COS W 



o = ii sin cos lit -J- i> sin sin ti -f- 01 cos 6* , 



icnce we find 



"' = _J__v^ (P 



9 sin 9 si~ * - - 



osinfl 3# " 



' ic expression for the current-function will therefore be 



The numerical calculation according to the above formula will be rather troublesome for an ordi- 

 itry case in which the serial developments are not particularly simple, more especially if the series 

 en verge only slowly. This will be the case with the field of the polar storms, as the acting current- 

 sstems come comparatively near to the earth. 



The formulae can, however, be simplified and put into a better form in the two extreme cases, 



(1) where | k \ . R is very small, and 



(2) where k \ . R is very great, 

 e will especially consider these two extreme cases. 



(i) k . R is assumed to be very small. 

 We may then put, cf. (4) 



we may also assume that this equation holds good for = o . We then obtain 



It now we can write 



/ 2 T 3 



4^ "IT ' a7 



d thus 



