228 ON FARADAY'S LINES OF FORCE. 



These expressions would determine completely the motion of electricity in 

 a revolving sphere if we neglect the action of these currents on themselves. 

 They express a system of circular currents about the axis of y, the quantity 

 of current at any point being proportional to the distance from that axis. 

 The external magnetic effect will be that of a small magnet whose moment 



TIP 

 is TQ-]. /sin 6, with its direction along the axis of y, so that the magnetism of 



the field would tend to turn it back to the axis of z*. 



The existence of these currents will of course alter the distribution of 

 the electro-tonic functions, and so they will react on themselves. Let the 

 final result of this action be a system of currents about an axis in the plane 

 of xy inclined to the axis of x at an angle <f> and producing an external effect 

 equal to that of a magnet whose moment is TIP. 



The magnetic inductive components within the shell are 



7, sin 6 2f cos $ in x, 

 21' sin <f> in y, 

 /! cos in z. 



Each of these would produce its own system of currents when the sphere 

 is in motion, and these would give rise to new distributions of magnetism, 

 which, when the velocity is uniform, must be the same as the original distri- 

 bution, 



T 



(I, sin 6 2/' cos <f>) in x produces 2 j^ r a (7, sin 6 2/' cos <f>) in y, 



T 

 (-2F sin <f>) in y produces 2 J-T a> (2/' sin <f>) in x ; 



7, eos 6 in z produces no currents. 



We must therefore have the following equations, since the state of the shell 

 is the same at every instant, 



T 



T 



- 2f sin $ = - T <a (/, sin 6- 2f cos <), 



* The expression for />, indicates a variable electric tension in the shell, so that currents might 

 be collected by wires touching it at the equator and poles. 



