482 ON PHYSICAL LINES OF FORCE. 



and that in the absence of free magnetism 



we find 



l(dQ_d_R\ d d_x_ d_dz 



dy dx da. dz da. dy dft dx _ da _ , v 



*"dz~d~t~dz'di~fydi + dydi~di = 



puttin g a= K^"W) (74)> 



aa 1 /a'Cr d 1 JI\ C7z\ 



and -7- = - I , ., j 1. 1 v/ 3 J> 



dt fj. \dz dt dy dt] 



where F, G, and // are the values of the electrotonic components for a fixed 

 point of space, our equation becomes 



d f n dx dz dG\ d / <!;/ fi dx dlf\ 



The expressions for the variations of $ and y give us two other equations 

 which may be written down from symmetry. The complete solution of the three 

 equations is 



p- ty.- fi^ ^_ 

 ~ W ~dt~ PP dt dt dx 



dz dx . dG dV , ^ 



O dx di/ dll dV 

 = --. a -- + -- 







The first and second terms of each equation indicate the effect of the motion 

 of any body in the magnetic field, the third term refers to changes in the 

 electrotonic state produced by alterations of position or intensity of magnets 

 or currents in the field, and V is a function of x, y, z, and t, which is inde- 

 terminate as far . as regards the solution of the original equations, but which 

 may always be determined in any given case from the circumstances of the 

 problem. The physical interpretation of is, that it is the electric tension at 

 each point of space. 



