PART III. KARTII CURRENTS AND EARTH MAGNETISM. CHAP. I. 761 



It appears from equation (9) that tp, and with it the strength of the current, diminishes very rapidly 

 as one moves inwards into the sphere. The currents are thus concentrated in the outermost layers ot 

 Lhe sphere, and in this case we may imagine, as LAMB has already shown ('), that all the currents are 

 replaced by the currents in a spherical shell with radius R. If i//! stands for the current-function for 

 :he currents in this spherical shell, we shall have 



R 



= J if>de , 



A 



vvhere p ' s a value of Q, where the strength of the current is insignificant. 

 Now 



271(1 + i)"\lL(f-R) 



X 



-*(v* } 



R l ) 



(II) 



27 C (I + 



ind thus 



y v 2 n ~t~ I **'"' (ffi anip,t _ 1 

 n -\- i 4 7C 4 n 



Thus no serial development is necessary for the determination of this current-system. 



Our next important task is to determine the magnetic effect of the induction-currents. From LAMB'S 

 xpression for the magnetic components in space, we can easily omit the expression for the potential 

 >f the induced currents. We find, if we call this V it that 



* 



/ ' 



f k . R is very small, we may write 



ind if k ! . R is very great, we may put 



Special interest attaches to the value of the potential at the surface. There, too, we can condense, so 

 is to avoid serial developments. 



In the first extreme case then, we have, for p = R , 



, - r-T 

 -f-i (a-f i)(a 



n+ i (zn + i) (2 -f 3) 

 Jut now 



(13) 



(!) loc. cit. p. 537. 



