758 



BIRKELAND. THE NORWEGIAN AURORA POLARIS EXPEDITION, IQO2 1903. 



What we shall thus have to do is to study the currents that are induced in a sphere by variatic 

 in an external magnetic field. 



This problem has been studied by a number of scientists, some of whom have looked at it nu 

 from a general, others more from a special, point of view. 



The investigations of LORBERG ('), NIVEN (-), and LAMB ( 3 ) are of great interest. If we start with t 

 assumption that the specific resistance of the earth is constant and equals y., we may directly empluv t 

 formulae previously developed by them. 



We assume that we can write the magnetic potential of the inductive current-system in the U 



V 2 S 2 H HS e a '?* ' 



where n may run through all whole positive values from o to oo , and the summation with ret;. 

 extends over a series of p s , which in the special cases are to be determined. 



HS is a solid harmonic of positive degree n, t is the time, * = \ i, and /> s is a constan: 

 employ LAMB'S formulae, and the same system of coordinates as before (cf. fig. 177, p. 424!, 

 then express the currents induced in the following manner. (LAMB has employed the symmetrical 



k- i 



r. 



h 



where u, v, w, are the components of the electric current. Further, 



i 



r) 



and 



- ... = ( !)" 3- 5- ...(2+l) 



By these formulae the induction-currents can always be determined, but the above form is not particular!} 

 well adapted to practical calculation. 

 As, however, 



xu + yv -\- zw = o , 



the currents will run in concentric spherical shells, and these may be more simply expressed by the 

 of a current-function, />. This current-function we will define in the following manner: If, in the spherical 

 shell with radius Q, we move a little way ds, and y>, on this piece, increases from (// to ifi + < 

 then the component of the current at right angles to the direction of this element from left to rig 

 when the observer is imagined to be standing on the spherical shell at the point in quest: 

 looking in the direction of the motion, equals 



dif> 

 4s ' 



(1) Crelle, Vol. 71, p. 53. 

 (3) Phil. Trans. 1881, p 307. 

 ( 3 ) Phi!. Trans. 1883, p. 519. 



