464 ON PHYSICAL LINES OP FORCE. 



Before going further in the general investigation, we shall consider equations 

 (12, 13, 14), in particular cases, corresponding to those simplified cases of the 

 actual phenomena which we seek to obtain in order to determine their laws by 

 experiment. 



We have found that the quantities p, q, and r represent the resolved parts 

 of an electric current in the three co-ordinate directions. Let us suppose in the 

 first instance that there is no electric current, or that p, q, and r vanish. We 

 have then by (9), 



whence we learn that ajdx + ftdy + ydz = d<l> .............................. (16), 



is an exact differential of <f>, so that 



_d$ d<f> _d<j> 



~dx' P-dy' y ~dz- 



p is proportional to the density of the vortices, and represents the " capacity 

 for magnetic induction" in the medium. It is equal to 1 in air, or in whatever 

 medium the experiments were made which determined the powers of the magnets, 

 the strengths of the electric currents, &c. 



Let us suppose /t constant, then 

 d d 



represents the amount of imaginary magnetic matter in unit of volume. That 

 there may be no resultant force on that unit of volume arising from the action 

 represented by the first term of equations (12, 13, 14), we must have m = 0, or 



Now it may be shewn that equation (19), if true within a given space, 

 implies that the forces acting within that space are such as would result from 

 a distribution of centres of force beyond that space, attracting or repelling 

 inversely as the square of the distance. 



Hence the lines of force in a part of space where p is uniform, and where 

 there are no electric currents, must be such as would result from the theory 

 of "imaginary matter" acting at a distance. The assumptions of that theory 

 are unlike those of ours, but the results are identical 



