164 



Prof. J. C. Maxwell on Electric Induction. [Feb. 15, 



the sheet, and the induced currents differ little from those which arise from 

 the direct action of the external system (see § 1). 



1 7. If the conductivity of the sheet were infinite, or its resistance zero, 

 R would be zero. The images, once formed, would remain stationary, and 

 all except the last formed positive image would be neutralized. Hence 

 the trail would be reduced to a single positive image, and the sheet would 



2 



exert a repulsive force — , on the pole, whether the pole be in motion or 



As? r 



at rest. 



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. 



Mathema tical Investigation . 



18. Let the conducting sheet coincide with the plane of ccy, 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 (p. This function expresses the quantity of electricity which, in 

 unit of time, crosses from right to left a curve drawn from a point at in- 

 finity 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 q> 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 



Ait. 



ds 



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

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

 in the directions of cc and of y respectively 



• H£ r-4 « 



The curves for which is constant are called current lines. 



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

 and + c0 is a conducting circuit round which an electric current of 

 strength C(p is flowing in the positive direction, that is, from x towards y. 

 Such a circuit is equivalent in its magnetic effects to a magnetic shell of 

 strength 2^>, having the circuit for its edge*. 



* W. Th omsoru " Mathematical Theory of Magnetism." Phil. Trans. 1850. 



