ON FARADAY'S LINES OP FORCE. 221 



are electric currents, the magnetic effects cannot be represented by a potential. 

 Let p', p l be the external and internal potentials, 



p' = / -5 cos 0, p l = Ar cos #, 

 and since there is no permanent magnetism, -4- = -~ , when r = a, 



If we draw any closed curve cutting the shell at the equator, and at some 

 other point for which is known, then the total magnetic intensity round this 

 curve will be 3/a cos 0, and as this is a measure of the total electric current which 

 flows through it, the quantity of the current at any point may be found by 

 differentiation. The quantity which flows through the element tdd is 3/a sin 6dd, 

 so that the quantity of the current referred to unit of area of section is 



- 31 ? sin 0. 



t 



If the shell be composed of a wire coiled round the sphere so that the number 

 of coils to the inch varies as the sine of 6, then the external effect will be 

 nearly the same as if the shell had been made of a uniform conducting sub- 

 stance, and the currents had been distributed according to the law we have just 

 given. 



If a wire conducting a current of strength /, be wound round a sphere 

 of radius a so that the distance between successive coils measured along the 



2(t 

 axis of x is - - , then there will be n coils altogether, and the value of I I for 



7* 



the resulting electro-magnet will be 



I--I 

 /l 0a * 



The potentials, external and internal, will be 



, T n a* nT n r a 



P =1, Q ^ cos0, p,= -2/ 3 g -cos 0. 



The interior of the shell is therefore a uniform magnetic field. 



