344 Steady Currents in continuous Media [CH. x 



In the TF-plane, the potential difference of the two electrodes is PS, or f , 



1 L r 



while the current is - PQ, or 7- . The equivalent resistance of the plate 



is accordingly rL'/L, so that the quantity we are trying to determine is L'/L. 



Let the coordinates of P, Q, R, S in the 2-plane be z 1} z 2 , z 3 , z. In the 

 f-plane the coordinates of these points are p, q, r, s. Hence from equations 

 (321), we have 



_ a (b d) b (a d) sn 2 mz l (mod K) 

 . (6 d) (a d) sn 2 mzj (mod /c) 



and similar equations for q, r, s. The ratio L'/L of which we are in search 

 is now given by 



L' _ (q r) ( p s) _ (sn 2 mz z sn 2 mz s ) (sn 2 mz l sn 2 mz^) 

 L (p r)(q s) (so 2 mz l sn 2 mz s ) (sn 2 mz 2 sn 2 mz 4 ) ' 



the whole being to modulus K. The values of sn mz can be obtained from 

 Legendre's Tables. 



Moulton has calculated the resistance of a square sheet with electrodes 

 each of length equal to one-fifth of a side, in the following four cases : 



(1) Electrodes at middle of two opposite sides, Resistance = 1*745.22, 



(2) Electrodes at ends of two opposite sides and facing one another, 



Resistance = 2-40&R, 



(3) Electrodes at ends of two opposite sides and not facing one 



another, Resistance = 2*589^, 



(4) Electrodes bent equally round two opposite corners of square, 



Resistance = 3'027^ 3 



where R is the resistance of the square when the whole of two opposite sides 

 form the electrodes. A comparison of the results in cases (2) and (3) shews 

 how large a part of the resistance is due to the crowding in of the lines of 

 force near the electrode, and how small a part arises from the uncrowded 

 part of the path. 



Limits to the Resistance of a Conductor. 



392. The result obtained in 386 enables us to assign an upper and 

 a lower limit to the resistance of a conductor, when this resistance cannot be 

 calculated accurately. For if any parts of the conductor are made into 

 perfect conductors, the resistance of the whole will be lessened, and it may 

 be possible to change parts of the conductor into perfect conductors in such 

 a way that the resistance of the new conductor can be calculated. This 

 resistance will then be a lower limit to the resistance of the original con- 

 ductor. 



