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



Hence, multiplying by y and integrating as before, we get 

 cc-\-a x—ax+b oc — b^^ 



r x r r 3 r 2 



or 



COS d — COS (p + cos Q x — cos <p x = c, 

 where 6, 0i, (p, <pi are the angles which the radii make with the line on 

 which the electrodes lie. 

 If we examine the equation 



cos 6+ cos d 1 — cos 0— cos <P{=c, 



which gives the lines of flow where four electrodes are arranged symme- 

 trically along a straight line, we see that when c is greater than 2, the 

 curve does not cut either of the two planes given by the equations 



cos 0-pcos ^ = 0, or cos + cos ^ = 0, 



but lies entirely between them, and that it cuts the axis at the two inte- 

 rior electrodes. The curve 



cos 0+cos 1 — cos p— COS0 X = 2 



meets the plane cos + cos 6 l —0 where it cuts the axis ; for cos $= — 1, 

 and cos cj) l = — 1 for this point. 



When cos 0+cos^ — cos^ — cos^ is less than 2, the curve cuts the 

 planes cos + cos 1 =O and cos + cos ^ = at some distance from the 

 axis, and cuts the axis at the point from which X is measured, i. e. it 

 cuts the axis at the two exterior electrodes. 



The curve cos 0-f-cos 6 1 — cos 0— cos X =2 has very close contact 

 with the plane cos + cos X =O, since the distance between them depends 

 on the differences of [small quantities of the second order. Hence in a 

 rectangular box, when the two electrodes are near to, but not in contact 

 with the ends, the lines of flow and equipotential surfaces should very 

 closely coincide with the lines of flow and equipotential surfaces in space, 

 supposing that two additional like equal electrodes are placed at the 

 electric images of the electrodes. (See Cases 7 and 8, on pp. 9 and 10.) 



The value of the constant, c, in the above equation will . determine 

 directly the cosine of the angle at which the curve cuts the axis. 



For at the axis cos 0= — 1, cos X = — 1, and either cos 6= — 1 when c 

 is less than 2, in which case cos 1 =o— 1, and the curve cuts the axis at 

 the exterior electrodes at an angle cos -1 (c— 1), 

 or cos X =1 when c is greater than 2, in which case cos 6=c— 3, 

 and the curve cuts the axis at the interior eleetrodes at an angle 



cos- 1 (c-3). 

 As in the case of two electrodes, if any two planes, both intersecting 

 along the line of the electrodes, be taken as bounding surfaces of the 

 conducting liquid, there will be no change in the forms of the lines of 

 flow and the equipotential surfaces. The resistance of the liquid will be 

 inversely proportional to the angle between two such bounding planes. 



