ON PHYSICAL LINES OF FORCE. 465 



Let us first take the case of a single magnetic pole, that is, one end of 

 a long magnet, so long that its other end is too far off to have a perceptible 

 influence on the part of the field we are considering. The conditions then are, 

 that equation (18) must be fulfilled at the magnetic pole, and (19) everywhere 

 else. The only solution under these conditions is 



<j>=- 1 - .................................... (20), 



p, r 



where r is the distance from the pole, and m the strength of the pole. 

 The repulsion at any point on a unit pole of the same kind is 



^ 

 dr 







771 



In the standard medium yn = 1 ; so that the repulsion is simply in that 



medium, as has been shewn by Coulomb. 



In a medium having a greater value of /n, (such as oxygen, solutions of 

 salts of iron, &c.) the attraction, on our theory, ought to be less than in air, 

 and in diamagnetic media (such as water, melted bismuth, &c.) the attraction 

 between the same magnetic poles ought to be greater than in air. 



The experiments necessary to demonstrate the difference of attraction of two 

 magnets according to the magnetic or diamagnetic character of the medium in 

 which they are placed, would require great precision, on account of the limited 

 range of magnetic capacity in the fluid media known to us, and the small 

 amount of the difference sought for as compared with the whole attraction. 



Let us next take the case of an electric current whose quantity is C, 

 flowing through a cylindrical conductor whose radius is R, and whose length is 

 infinite as compared with the size of the field of force considered. 



Let the axis of the cylinder be that of z, and the direction of the current 

 positive, then within the conductor the quantity of current per unit of area is 



_C_ 1 (dft_da\ . } . 



?9"rV& Ty)~ 

 so that within the conductor 



a=-2~y, P = 2~x, y = ..................... (23). 



VOL. I. 59 



