" of Equipotential Curves and Surfaces. 549 



nected with the poles of a battery, as there are equal currents 

 flowing at those two points, one into and the other out of the 

 sheet, the potential at any point of the sheet is proportional to the 

 difference of the logarithms of its distances from the two points 

 or electrodes where the current enters and leaves the sheet. 



The potential is constant for a series of points if the difference 

 of the logarithms of the distances of each of those points from 

 the electrodes remains ^constant, i. e. if the ratio of the distances of 

 each of those points from the electrodes remains constant. 



The curve joining this series of points is an equipotential curve. 



If r and r, are the distances of any point in the curve from the 

 two electrodes, and c a constant, then 



Hence the equipotential curves are circles with their centres on 

 the line joining the two electrodes ; and the lines of force which 

 cut the equipotential curves at right angles are also arcs of circles 

 passing through the two electrodes. 



The lines of force may be regarded as distinct from one another, 

 but as filling up all the space on the conductor between the two 

 electrodes ; and the distribution would not be altered if we con- 

 ceive of them as divided from one another like separate wires con- 

 ducting currents side b} r side. By taking out any space bounded by 

 lines of force, we shall increase the quantity flowing along the other 

 lines of force, but shall not alter the distribution of the current 

 among them. Hence we may cut out a disk from an unlimited 

 sheet without altering the form of the lines of force, if the boun- 

 dary of the disk be arcs of circles passing through the two elec- 

 trodes ; so that for a circular disk with the electrodes on the edge 

 of it, the equipotential curves are circles having their centres on the 

 straight line joining the electrodes. 



The forms of the equipotential curves may be traced out expe- 

 rimentally by attaching two battery-electrodes to a disk of tinfoil, 

 and having two similar electrodes attached to a delicate galvano- 

 meter ; one of these electrodes being fixed at a point through which 

 the equipotential curve is to be drawn, the other may be moved 

 from point to point to trace out the successive points, so that no 

 current may pass through the galvanometer. A comparison of the 

 experimental results with the theory shows a complete agreement. 



In a large square sheet 310 millims. in diameter, with the elec- 

 trodes 126 millims. apart, the curves in the centre and near the 

 electrodes, which are drawn by pricking fine holes through the 

 tinfoil on a sheet of paper below, are very accurately circular, 

 and mostly coincide with circles, until the points are so far from 

 the centre that the form of the equipotential curves is affected by 

 the edge of the disk. In a circular disk with the electrodes on the 

 edge subtending 60° at the centre, the experimental curves are 

 shown to be accurately arcs of circles, with their centres on the line 

 joining the electrodes. 



In an unlimited sheet, when there are several electrodes by 



