or 



Equipotential Curves and Surfaces, fyc. 13 



If we refer this circle to rectangular coordinates with the axis of 

 y equidistant from A and B, then its equation is 



{x-af + y* = ?{(x + aY + if), 



r „ 1 + c 1 — c 

 If V = o, x = a. — !_ora. , 



J 1—C 1+6 



giving the points where any equipotential curve cuts the axis. 



The point B is the electrical image of A with regard to each of these 

 circles ; for if be the centre of one of the circles, 



OA.OB = (J^L) 2 . 



The lines of flow cut the equipotential curves at right angles, or the 

 radius of one of these circles is the tangent to a line of flow ; hence it 

 follows (Euclid, Book III. Prop. 8) that the lines of flow are arcs of 

 circles passing through the points A and B, the positions of the elec- 

 trodes. This result may be arrived at by a method which is also appli- 

 cable to the case of unequal charges at several points in a plane. 



Let us take the equation to the equipotential curves in the form 

 {x — cvf-\-y 2 =c 2 {{x-\-af-\-y 2 }. Differentiating and eliminating c we get 



(oc-af+y* (a+af+y 2 



The lines of flow cut the equipotential curves at right angles ; hence for 

 a line of flow 



(x-af + if (x + af + y 2 



Integrating this equation, we get 



tan -1 — y— — tan -1 ^ = constant, 

 x — a x -\- a 



showing that the lines of flow are circles passing through the positions 

 of the two electrodes. Comparing this result with the curves drawn by 

 experiment in Plate 1. fig. 1, we see that for all parts of the disk which 

 may be considered to be beyond the influence of the edge of the disk, 

 there is a very close agreement between theory and experiment ; and the 

 influence of the edge of the disk will be such that the equipotential 

 curves will cut the edge of the disk at right angles. 



When the conducting-sheet is not of infinite extent, but of very con- 

 siderable extent in every direction, the expression C— A (log r — log r x ) will 

 still be a close approximation to the potential at any point, provided the 

 electrodes are not far from the centre of the sheet. Hence on such a 



