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



For -■ = cos (j>, and -— = cos (tt — 0); 



-A.J3 AJhJ 



, AJf.Bfi AH „ AE 



and AB + AB = AB = 2 -AB ; 



or cos — cos = 2 cos a. 



We may regard the conducting-mass as divided into two parts by a 

 plane passing through the axis of revolution of the lines of flow. The 

 equipotential surfaces will all cut every such plane at right angles. 



Hence we may suppose the conducting-mass on one side of such a 

 plane to be removed without changing the form of the equipotential 

 surfaces. The resistance of the conductor between the two electrodes 

 will then be doubled. 



Hence in a liquid the form of the lines of flow and equipotential sur- 

 faces will be the same when the electrodes are both in the surface of the 

 liquid as when they are immersed to infinitely great depths in it. For 

 the same reason, if the liquid be contained in a very large rectangular 

 vessel with two plane sides parallel to the line joining the electrodes, the 

 forms of the equipotential surfaces will be the same when the electrodes 

 are on the line where the surface of the liquid meets one side of the box, 

 provided they are not near the ends of the box, and provided the oppo- 

 site side of the box is too far off to disturb the forms of the equipoten- 

 tial surfaces. 



A particular solution of the equation to the lines of flow is 



cos d— cos = 1. 

 At one of the electrodes cos <£=— 1, and hence = ^; and at the other 



cos = 1, and hence </> = 4 y; so that the lines of flow cut the axis at right 



angles, i. e. the lines of flow given by this equation touch the two planes 

 drawn through the electrodes at right angles to the axis. 



In order that the lines of flow in a vessel containing liquid may coin- 

 cide accurately with the theoretical lines of flow in a liquid of infinite 

 extent in all directions, it is necessary that every section of the boundary 

 of the vessel through the electrodes should be a line of flow. 



From the above considerations, it appears that near the electrodes the 

 ends of .the vessel may be plane and at right angles to the axis, without 

 causing much change of form in the equipotential surfaces. 



When there are charges of electricity at several points, then the 

 potential at any other point is the algebraic sum of the potentials at that 

 point due to the several charges separately. 



This will also be the case in a liquid conductor of unlimited extent in 

 every direction when currents enter and leave the liquid at several points 

 at a great depth within it. 



