103] NEWTONIAN POTENTIAL FUNCTION. 195 



and inserting the values of Q 2 , Q 1} 



where of course all the functions under the integral are to be 

 expressed in terms of q l) q 2 , X. This principle will be made use of 

 in the treatment of the flow of electric currents in thin curved 

 surfaces. 



The theorem becomes very simple in two particular appli- 

 cations. First let q lt q 2 , q s be rectangular coordinates x, y, z, and 

 let V be independent of z, that is, the problem is uniplanar, or 

 the lines of force lie in planes all parallel to the Z- plane. Then 

 X = z const, is one integral and the other is 



From this we obtain 



, a/*, dp, dV, dV . 

 da = f dx + ay = -^ dx - du, 

 dx dy ' dy dx 



djt^dV d^ = _dV 



dx dy ' dy~ dx ' 



and the function //, is the function conjugate to the potential 

 function V, as found in 42. Since by 36 the flux of the 

 vector R across any cross-section of a vector tube defined by four 

 surfaces X, X + d\, //,, //, + dfju is d\dp, the function //, represents 

 the flux through a tube bounded by two parallel planes z = Q, 

 z= 1, by the surface /* = 0, and by the surface p = const. If the 

 vector R represent the velocity of a fluid motion, //, is called 

 Earnshaw's current function, and the amount of fluid crossing unit 

 height perpendicular to the ^-plane of any cylindrical surface 

 projected into a curve on the ^-plane is given by the difference in 

 the values of JJL at the two ends of the curve. We may call the 

 function p for any vector the flux-function. 



In the second case let q 1 , q 2 , q 3 be cylindrical coordinates p, &, z, 

 and let V be independent of o>, so that the lines of force are in 

 planes intersecting in the ^-axis. The figure is then symmetrical 

 around this axis, and we have a problem of revolution. We then 

 have an integral X = <o = const, and for the second, 



(10) IJL = Ip \dp ~ dz\ = const. 



132 



