428 



HIKKKLAND. THK NOUWKGIAN AURORA POLAKIS KXPKD1TION, IQO2 1903. 



As it is well known the expression for this quantity involve an additive constant, that may be 

 chosen at pleasure. 



We may therefore, for instance choose such a constant, that the value of the potential at the centn- 

 of the sphere will be zero. Under this supposition we may write the potential, V, as 



ft 



Po. 

 jo 



as the term on the right is an expression for the work done against the field when a positive magnetic 

 pole of unit strength passes from the centre of the sphere to a certain point on its surface. 



Po is only due to the curved portion of the current. We find by equations (3), (8) and (9) 



t'-2 



cos (a* it) sin 6 cos sin L" cos 6 

 = //.- sin i. | s-, - (tii. 



f'\ 



(U- 



cos 



We further have, as will be easily seen 



._ . ,. cos cos tf cos 6 



cos (u /() sin 6 cos t - - sin L cos 6 = - 5 



sin L 



By introducing this expression and by integration with respect to Q, we find, pag. 101, Part I 



","-2 



, 7 . (cos 9 cos cos ft) (o L cos ft) . (cos cos cos d) cos ft , 



I ' = i I j=- da -f- / I ~~2~j "!' 



sin 2 ft yp- -)- L- 2(> L cos [i sm P 



J ,"i ./ ,"i 



or if preferred 



r/'a 



(r*r\c fl . r*rc rric ti\ f/i (J /J\ r-rvc !^\ 



du. 



. , I (cos ft cos cos /?) [Q (L d) cos 



r = I \ . o ,, 7 



sm* ft . d 

 ."i 



(28! 



where d stands for the square root. 



As will be seen, V may also be expressed as elliptic integrals. 



For numerical calculations I think however that the above form is the most practical one. 



By derivation of this expression we find the force-components of the whole current-system. This 

 we have done to control the correctnes of our calculations. 



In our calculations we have imagined the current to lie at a height of about 400 kilometres 

 (L = 1,063 Rl> tne average height of currents, as we found by our calculations in Chapter IV of Part I. 



In the tables, we have employed y as the unit for forces; and / = io 5 [i. e. io fl amperes]. 



TABLE LIX. 

 Values of Po for the horizontal portion of the current. 





