218 Tite Electrical Method of Soil Moisture Determination 



Now if the angle BAP -= 6 we have in three dimensions 



px = py cc ^ Jp^, = ^Acle^er "" '"'" ^' 



if AC remains constant. 



Hence PZ - (PX + PY) cos 6 oc cos^ 6. 



Precisely the same result is obtained from a consideration of the 

 equipotential surfaces. Since the electrification of the electrodes at any 

 moment will be equal and opposite the median plane is the equipotential 

 surface V = 0. 



On either side of this will be equipotential surfaces V = + dV and 

 V = — dV and the current across the median plane therefore will be 

 inversely proportional to the distance x along a line of force between 

 these two surfaces^, since this is a measure of the resistance assuming 

 the medium isotropic. It makes no difference in this consideration 

 whether we take the momentary state or put V in virtual volts and the 

 current in virtual amperes. We have then that the current across the 



median plane at the point P, is equal to K . 



Now considering spheres round the electrodes V = j^ where r is 

 the radius of the sphere, whence 



V 



and since x is inclined at an angle = 6 to the direction of r 



X = dr sec d 



= — ,- r^ sec ddV 



= _ ^^ysec^edV. 



Hence when the potential difference at the ends of x is constant = 2dV 

 we have 



X oc sec* 6, 



whence current across the median plane at P x cos' 6 as before. 



Thus the fraction of the total current passing outside any tore^ 



' This may be considered a straight line. 



- The lines of force are known to be circular arcs vide Masoart and Joubert, Electricity 

 and Magnetism, 1, p. 205. 



