ON FAKADAYS LINES OF FORCE. 219 



The whole couple about the axis of x is therefore 



OK (; 3 K a j ~- a 3 



(2k s + k')(2k t + k') U 



tending to turn the sphere round from the axis of y towards that of z. Sup- 

 pose the sphere to be suspended so that the axis of x is vertical, and let / 

 be horizontal, then if 6 be the angle which the axis of y makes with the 

 direction of /, m = cos 6, n = sin 0, and the expression for the moment becomes 



k'(k,-k 3 ) , . - 

 ' ' J - 



tending to increase 0. The axis of least resistance therefore sets axially, but 

 with either end indifferently towards the north. 



Since in all bodies, except iron, the values of k are nearly the same as in 

 a vacuum, the coefficient of this quantity can be but little .altered by changing 

 the value of k' to k, the value in space. The expression then becomes 



T * 7V sin 20, 

 independent of the external medium*. 



VII. Permanent magnetism in a spherical shell. 



The case of a homogeneous shell of a diamagnetic or paramagnetic substance 

 presents no difficulty. The intensity within the shell is less than what it would 

 have been if the shell were away, whether the substance of the shell be dia- 

 magnetic or paramagnetic. When the resistance of the shell is infinite, and when 

 it vanishes, the intensity within the shell is zero. 



In the case of no resistance the entire effect of the shell on any point, 

 internal or external, may be represented by supposing a superficial stratum of 



* Taking the more general case of magnetic induction referred to in Art. (28), we find, in the 

 expression for the moment of the magnetic forces, a constant term depending on T, besides those 

 terms which depend on sines and cosines of 0. The result is, that in every complete revolution in 

 the negative direction round the axis of T, a certain positive amount of work is gained; but, since 

 no inexhaustible source of work can exist in nature, we must admit that T=0 in all substances, 

 'with respect to magnetic induction. This argument does not hold in the case of electric conduction, 

 or in the case of a body through which heat or electricity is passing, for such states are main- 

 tained by the continual expenditure of work. See Prof. Thomson, Phil. Mag. March, 1851, p. 186. 



282 



