Waves along a Line of Negligible Leakage. 303 



resistance of the bridge at that value. It was adjusted before 

 each occasion of its use and tested afterwards. The difference 

 was usually of the order 2 or 3 ohms ; but on one occasion 

 it reached 10 ohms. However, to take an extreme case, let it 

 be supposed that the critical value 5G0 is not the correct one, 

 and that on some occasion of its use the bridge-resistance, 

 though differing little from 560 ohms, differed by as much as 

 45 ohms from the value required for complete absorption of the 

 electric waves incident upon it. Then the reflexion-coefficient 

 of the bridge is given by * 



, R,— Lv 45 45 , ,_ x 



p=ivru = 2uTi5 = iT65' nearl y'- • • (7) 



where R x is the resistance of the bridge and Lv the instan- 

 taneous impedance of the line. 



> And it may easily be seen that the electrometer-throw when 

 the absorbing bridge only is in use is, in consequence of its 

 imperfect absorption, affected by a factor rather less than 



(l + p' 2 )/(l- P V 2 ) = l + f /3 ' 2 , nearly; . . (8) 

 since p 2 is of the order one-half, p being the reflexion-coefficient 

 of the oscillator. 



Thus, for the supposed error of 45 ohms, which is an extreme 

 case, we have p'=0'04 and the electrometer-throw increased 

 in consequence by 0*24 per cent. But this is within the 

 limits of the probable error of the determination of the ratios 

 of the throws, and is therefore almost negligible. 



Again, let us now examine the error consequent upon in- 

 complete reflexion from the bridge used to reflect all. The 

 resistance of this bridge should be zero. It consists of a 

 copper wire put across the line, both it and the line were 

 cleaned and polished at the places of contact, and the bridge 

 made so as to spring on tightly into position. The bridge 

 in question, put on with much less care than usual, showed a 

 resistance of 0*04 ohm on testing with a P.O. box. But suppose 

 by any mischance that the resistance of this bridge were half 

 an ohm, then its reflexion-coefficient is given by 

 „ Ro — Lv /, 2R 2 \ „ N 



? = tTLv = -{ 1 -w=- ( - 1 -x>>™y> • ^ 



where R 2 is the resistance of the bridge and Lo the instan- 

 taneous impedance of the line as before. Then the electro- 

 meter-throw when this bridge is in use is, in consequence of its 

 imperfect reflexion, affected by a factor rather nearer unity than 

 1 + pW !-,» _ 1+(1-2 X ) l-p» 



2 l-py» I 1_^(1_2 X )- 1 *%• nea "y- t 10 ' 



* Heaviside's ' Electrical Papers,' vol. ii. pp. 132-133. 



