Fundamental Concepts of Electrical Theory. 419 



components of the electric and magnetic intensities in an 

 electromagnetic field in the aether * : — 



z "" c -d(x, t) + c A *> T) 3(#, c A/5 ' T/ d(^ 3 



where cr has heen written for 2V. To find the lines of 

 electric and magnetic force in this type of field we must 

 obtain identities of tbe form 



d(a, £)-/(*, tM«, t) +/rft r)d(0, t) = <*(£, ifr), 



d(«r, t) +/(*, t)<*(«, r) +/(£, r) t ?(/3, t) = dfa r). 



the moving lines of magnetic force are given by the equations 



X = (j + #(a, t)+<7(/3, t) = const., t = const. 



The lines of electric force are given by the equations 



</>(a, /?, t) = const., ^(«5 /5, t) = const., 



where these equations give a pair of solutions of the 

 differential equations 



Let us now write t = T as before and consider in the plane 

 of X, Y a two-dimensionjil irrotational motion of an incom- 

 pressible fluid in which the velocity potential <I> and current 

 function ^f are given by the equation 



<f> + iV=g(X + iY, T)=#0, t). 



* The function f(x,y) is supposed to "be real when x and ?/ are real, 

 so that f(a, r) and f(fi,r) are conjugate complex quantities. 



2 G2 



