320 Prof. A. Schuster on Magnetic Precession. 



As <J> only depends on <£ in so far as it contains terms 

 which have cos cr<f> or sin <t$ as factors, the effect of dif- 

 ferentiating with respect to <jf> is the same as a multiplication 



by a and a change of <r<\> to cr(f> + - . Hence the terms 

 depending on <E> will be proportional to the original current- 

 intensities if crcf) is replaced by <r<$> + i) . In other words the 



currents which the forces ^g and ^e tend to produce are of 

 the same type as the original currents, but turned through 



an angle 3- round the axis of rotation, in a direction opposite 



to that of the angular velocity of the body. If the original 

 current function has been proportional to cos a<p or to sin &<$>, 

 inertia will tend to produce currents of the same type but 

 proportional to —sin a<j> or to cos crcf) respectively. The 

 final effect of these will be a rotation of the system of currents. 

 6. The terms depending on Q in (9) and (10) will not 

 produce any permanent currents, but an electrification 

 having — ^iiwQa/t for potential. We obtain Q from (8), by 

 substituting K = ^Q/sin 0d<f). After integration with respect 

 to <f>, it is thus found that 



Q = <l>cos0— /csmfl-y^ . 

 do 



This may be put into the standard form of tesseral har- 

 monics, if we write 



3> = T£C0S(7<£ 



and 



d°V 



T^-siiv^ 

 ± w _sin u dx<r , 



where X = cos# and V n stands for the zonal harmonic o£ 

 degree n. 



By differentiation we obtain 



sin ^ T£ = <7 cos TJ- sin TJ+ 1 . 



We have also the following general equations : — 

 (2rc+l)sin0T; +1 = 



(n + a + 1) (n + <r)T£_ L - (n - a + 1) (n - a) T n+1 \ 

 (2n + l) f iTZ=(n-a+l)T: +1 + (n + c7)T:. 1 . 

 Combining these equations we obtain 

 n.n+1.2n + 1Q = (w-o- + l)n 2 T,T+i cos<7</> 



+ (n + l) 2 (n + a)Tl__ Y cos er<£. 



