18 Prof. W. G. Adams on the Forms of 



curve for the four equal electrodes A, A v B, B x ; also that the potential 

 at any point of this circle is the same as the potential at the centre O 

 would be if there were only a positive electrode at A and a negative 

 electrode at ~B V which is the same as one half the potential at the centre, 

 O, due to the four electrodes. 



If we take two of these electrodes, B, B x , and through C draw another 

 chord through two other electrodes on the circle, and draw the equi- 

 potential circle for this set of four electrodes, we shall find that all such 

 circles cut both the circles AA X B and EE X mutually at right angles, and 

 that the circle EE X is a line of flow for all such systems of four elec- 

 trodes on the circle AA X , B X B. 



The circles DD 1? EE X , and AA X B will iutersect mutually at right 

 angles. 



This will appear if it is shown that AA X and BB X intersect on the 

 common chord of the circles DD 1 and AA X B ; for in that case the tan- 

 gent to DD X drawn from C is equal to the radius of EE X . 



Let the circle AA X B be referred to two tangents through the point C 

 and their common chord (y=0) ; then its equation is a/3 — y 2 =0. 



Let /x 2 a— /3=0 and X 2 a— /3 = be the equations to the two lines CAA X 

 and CBB 1? 



We may call the points A, A v B, B x , the points /u, — /u, X, — X. 

 The equation to the chord joining fx and X is 



X/ia-(\ + /0 y + /3 = 0, 



and the equation to the chord joining — fx and — X is 



\na + (\ + fx) y + /3 = 0. 



These intersect on the line y=0 where it meets X/ia-f /3 = 0. 

 The equations to the tangents through this point are 



X i ua-2Vx^Ty+/3=0 

 and X/xrt + 2 VX/T.y + /3 = 0, 



which touch the circles at the points V Xjjl and — V\fj.. 

 The equation to the chord joining these points is 



Xfjia — /3 = 0, 



which passes through the point (a, j3). 



The circle HEK is a line of flow for any such system of four equal 

 electrodes as A, A 1} B, B 15 lying on a circle which cuts the circle HEK 

 at right angles. 



A particular case of this arises when two of the electrodes of the same 

 kind as A and A x approach to and ultimately coincide with K ; then the 

 current at K is double each of the currents at B and B x . The two lines 

 of flow, HEK and KBB,, intersect at right angles at the point K, and 

 the equipotential circle DD X diminishes to a point. 



Since the circle HEK is a line of flow, we may cut the conducting 

 sheet along the arc of this circle without changing the form of the equi- 



