in Conducting Sheets and Solid Bodies. 1 7 



whose surface-density D is given by 



The couple exerted by the system I2 to retard the rotation 

 is therefore 



JJ asmdd<j) 

 = -AA (2w 4 + 1)g . a 2 "+% f ' <T*) 2 . di 



r^CTi) 2 .^ 



-A 2 **»+* ™* - " 2 (" + «) ! 



~ A ° ,a * o- (n + l)(2n + 3) 2n + l (»-«)! 



(Ferrers's ' Spherical Harmonics,' p. 86), on taking 



d 



Y* = sin s 6 



w) 



where /a = cos 6, and P re is the zonal harmonic of order n. 



And by combining such conjugate harmonics in the proper 

 manner, we obtain the general solution in the form 



a =2 Qtjca; cos s <f> + b* sin *</>), 



27Tffl 2 ~ n* 2 (» + *)! .....pA 



<r (>i + l)(2w+l)(2n + 3) , (^-s)!*^ " + n) 



(13) 



and the rate of expenditure of power required to keep up the 

 motion is Q COt 



In a uniform field w=l, s = l; therefore if the magnetic 

 force is F in the direction 0= — , <£ = ; we have F= — A ; 



^i= F • T 2 5 • — • pr am 5 sin i|r, 

 G=^.^a 4 F 2 . 



We see that, still neglecting squares of the small quantity , 



the rotating sphere behaves as a magnet of moment 



Pftfl. 1%. S. 5. Vol. 17. No. 103. Jan. 1884. C 



