Prof. A. Schuster on Magnetic Precession. 323 



per unit linear cross-section, and we may apply equations (5) 

 and (6), leaving t out of the denominator. The investigation 

 of § 7 may be replaced by a simple application of a result 

 given in Prof. Lamb's paper on electric currents in a sphere*. 

 Jf electric currents are once started in a solid they will 

 gradually die out owing to electric resistance, and if these 

 currents are represented by a current- function, as assumed T 

 the time-factor will be of the form g _AT , where the value of 

 X is given by Prof. Lamb for some of the simpler types of 

 currents. The forces per unit volume which act on the 

 currents under these circumstances are — pi, where i is the 

 current-density and p the resistivity ; the corresponding rate 

 of diminution of current will be Xi. 



Hence if <£> represents the current-function and <£' the 

 corresponding force-function, we may put <&'=— pi and 



—— = — X?', and derive 

 dr 



j- = ^ • 



(IT p 



This shows that in equation (12) we must replace — — - 



^ OTTCU 



by -• The angular velocity of the rotating currents then 

 becomes 



O _ /JLO) X 



a n . n + 1 per ' 



A 2 



For n = l, we have for the simplest case \ _1 = - } so that 



9. We may now apply the results obtained to the case of 

 electric currents which we may imagine to circulate in the 

 earth. If terrestrial magnetism is due to such currents, we 

 may represent them by a superposition of different systems, 

 each system producing magnetic forces on the outside, the 

 potential of which is represented by a spherical harmonic. As 

 regards the currents which give rise to zonal harmonics, they 

 must flow in circles at right angles to the axis of rotation and 

 the revolution of the earth cannot affect them. The tesseral 

 harmonics will revolve relatively to the earth in a direction 

 opposite to that of its own rotation. The effects are therefore 

 such as are actually observed. But the calculated angular 

 velocity is much too small to explain the secular variation. 



Taking the case of a solid sphere first. If p = l, which is 

 * Phil. Trans. 174, p. 519 (1888). 



