78 



Mr maxwell, ON FARADAY'S LINES OF FORCE. 



the sphere, disturbing the lines of force, will be the same as that due to three small magnets at 

 the centre, with their axes parallel to a>, y, and ar, and their moments equal to 



k-k' 

 2A; + k 



, a?a. 



k-k' 



Zk + k 



-,a'(i. 



k-k' 

 Sk + k' 



a^y. 



The actual distribution of potential within and without the sphere may be conceived as the 

 result of a distribution of imaginary magnetic matter on the surface of the sphere ; but since 

 the external effect of this superficial magnetism is exactly the same as that of the three small 

 magnets at the centre, the mechanical effect of external attractions will be the same as if the 

 three magnets really existed. 



Now let three small magnets whose lengths are Z, l^ I3, and strengths m^ nii m^ exist at the 

 point xy% with their axes parallel to the axes of wyx; then, resolving the forces on the three 

 magnets in the direction of J^, we have 



— X = till 



da 4" 

 rft/2 



da la 



+ m. 



a + 



da I3 

 dz2 



-«! + 



da , da , 



aw ay 



dy2) 

 da 



dak 



dz2 



Substituting the values of the moments of the imaginary magnets 



~2k + k' \dx '^ dx '^ dx) 2k + &' 2 dx ^ 



The force impelling the sphere in the direction of x is therefore dependent on the variation 

 of the square of the intensity or (a^ + /3' + 7*), as we move along the direction of x, and the 

 same is true for y and z, so that the law is, that the force acting on diamagnetic spheres is 

 from places of greater to places of less intensity of magnetic force, and that in similar distri- 

 butions of magnetic force it varies as the mass of the sphere and the square of the intensity. 



It is easy by means of Laplace's Coefficients to extend the approximation to the value of 

 the potential as far as we please, and to calculate the attraction. For instance, if a north or 

 south magnetic pole whose strength is M, be placed at a distance b from a diamagnetic sphere, 

 radius a, the repulsion will be 



^-'<*-*')?G-^F- 



3.2 





4.3 



Sk + 2k' W 4& + 3k' b* 



+ &c. 



When — is small, the first term gives a sufficient approximation. The repulsion is then as 

 b 



the square of the strength of the pole and the mass of the sphere directly and the fifth power 



of the distance inversely, considering the pole as a point. 



