550 Royal Society .-—-Prof. W. G. Adams on the Forms 

 which currents enter and leave the sheet, the potential at any point 



a log ( ^/;- v 



where r, r\ r" . . . are the distances to the electrodes of one kind, 

 and r v r 4 *, r," are the distances to the electrodes of the other kind. 

 Taking the case of one positive electrode at the centre and four 

 negative electrodes round it at the corners of a square, the curves 

 are traced and are seen to be the same as the curves at the corner 

 of a square sheet with a positive electrode at the corner and two 

 negative electrodes on the edges ; the curves are also the same 

 for a square sheet with a positive electrode at the corner, and one 

 negative electrode along the diagonal. 



The equation for these equipotential curves is 

 r 4 = cr lV * 3 r 4 , 



and is derived, in the case of the limited sheets, by considering 

 that, to every electrode on the limited sheet, there corresponds an 

 equal and like electrode at each of the electrical images of that 

 electrode formed by the edges of the sheet. If we trace the curves 

 for this arrangement of electrodes in the unlimited sheet, the edges 

 of the limited sheet will be some of the hues of force ; and so we 

 may divide the sheet along these edges, without altering the form 

 of the equipotential curves. Where an electrode and its images 

 coincide in position, the index of r is equal to one more than the 

 number of images. 



When there are four electrodes, two of each kind on an un- 

 limited sheet, an equipotential curve is given by the equation 



rr' = cr 1 r,'. 

 If the four points lie on a circle, and the complete quadrilateral be 

 drawn through them, the circles which have their centres at the 

 intersections of opposite sides of the quadrilateral, and which cut 

 the first circle at right angles, will also cut one another at right 

 angles. One of these circles is shown to be an equipotential curve 

 for the four electrodes, and the other is a line of force. 



Hence, if we cut the unlimited sheet along the edge of this 

 latter circle, we shall not alter the forms of the equipotential curves ; 

 and within it we shall have one electrode of each kind, the others 

 being their electric images, the product of the distances of an elec- 

 trode and its image from the centre being equal to the square of 

 the radius of the disk. If an electrode is at the edge of the disk, 

 then the electrode and its image coincide, and the equation to the 

 equipotential curve is 



When one pole is at the edge and the other is at the centre 

 of a circular disk, since the electric image of the centre is at an 

 infinite distance, the equation to the equipotential curves is 



