ON FARADAY'S LINES OF FORCE. 



227 



The expressions for a,,, $,, % due to the magnetism of the field are 



a,=A <> + -ycoB0 > 



P = B + -(zBin.d x cos 6], 



7o= <?<,-- y sin 0, 



A , B , C being constants ; and the velocities of the particles of the revolving 

 sphere are 



dx dy dz 



j~ = WW, Y~ = OKT, -j- = 0. 



dt dt dt 



We have therefore for the electro-motive forces 



1 da, II 

 * = -^-di = -&2 C 80(aX > 



B= ._1.^ = ^1 



I dy. 1 / . 



y 2 = -j -- f- = - sin 

 47r rft 4ir 2 



Returning to equations (1), we get 



(^L _ ^1} - *?A _ ^y. = 



dc, da t \ dy, da, I I 

 -~----l = -p--^ = - - 

 dx dz 4?r2 



db,\ da, d& 



J - I = -= --- } - 



rfx/ d dx 



From which with equation (2) we find 



1 1 / 



1 1 / 



p,= r^- Io {(x 1 4- y 1 ) coB0 xz sin 6}. 



292 



