ELECTRIC INDUCTION. 291 



would be reduced to a single positive image, and the sheet would exert a repul- 



ffiYi^ 



give force : - on the pole, whether the pole be in motion or at rest. 



4z 



I need not say that this case does not occur in nature as we know it. 

 Something of the kind is supposed to exist in the interior of molecules in 

 Weber's Theory of Diamagnetism. 



Mathematical Investigation. 



18. Let the conducting sheet coincide with the plane of xy, and let its 

 thickness be so small that we may neglect the variation of magnetic force at 

 different points of the same normal within its substance, and that, for the same 

 reason, the only currents which can produce sensible effects are those which are 

 parallel to the surface of the sheet. 



Current-function. 



19. We shall define the currents in the sheet by means of the current- 

 function <f>. This function expresses the quantity of electricity which, in unit 

 of time, crosses from right to left a curve drawn from a point at infinity to 

 the point P. 



This quantity will be the same for any two curves drawn from this point 

 to P, provided no electricity enters or leaves the sheet at any point between 

 these curves. Hence ^ is a single-valued function of the position of the point P. 



The quantity which crosses the element ds of any curve from right to left is 



*+A. 



1 ' ' o. 



as 



By drawing ds first perpendicular to the axis of x, and then perpendicular 

 to the axis of y, we obtain for the components of the electric current in the 

 directions of x and of y respectively 



u = f, v=-^.. (1). 



dy dx 



The curves for which <f> is constant are called current lines. 



20. The annular portion of the sheet included between the current lines 

 $ and (f> + S(j) is a conducting circuit round which an electric current of strength 



372 



