8a Mb maxwell, ON FARADAY'S LINES OF FORCE. 



Returning to equations (l), we get 



l [^ _ ^\ - ^ _ ^ = 

 \dz dyl dm dy 



. (dc^ da^\ dy, da^ 1 I . 

 Kdic dss I dx dz 4nr x 

 (du^ _ dh^\ ^ da^ _ ^^' ^ p 



Idu^ db^X da^ d^^ 

 \dy dx) dy dx 



From which with equation (2) we find 



]_ J_ 7 



A 47r 4 



62 = 0, 



a, = - — T- T sin Ouiss, 



11/.. 



C2 = sin dwx, 



k 4iir 4! 



P2 = — — /w{ (.r* + y^) cos 6 - xz sin 6], 



lOTT 



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 



'Tips 



of a small magnet whose moment is wl sin 6, with its direction along the axis of y, 



^8irk 



so that the magnetism of the field would tend to turn it back to the axis of x*. 



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 



and producing an external effect equal to that of a magnet whose moment is TB?. 



The magnetic inductive components within the shell are 



Ii sin — 9,1' cos (j) in a, 



— 9,1' sin (p in y, 



/i 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 distribution, 



T ■ 



(/i sin 0-2/ cos (p) in m produces 2 w (/, sin - 2/* cos (pi) in y, 



487rft 



T 



(- 9.1' sin 0) in y produces 2 a {9,1' sin 0) in a?; 



487rA 



/, cos 6 in X produces no currents. 



* The expression for ^2 indicates a variable electric tension in the shell, so that currents might be collected by wires touching 

 it at the equator and poles. 



