On  Testing  the  Metal-resistance  of  Telegraph-wires.     187 
From  equation  (1)  we  can  tell  the  ratio  between  these  forces 
when  the  resistance  of  C  is  known  ;  and  from  equation  (2)  the 
E' 
resistance  of  C  is  found  when  the  ratio  ^  is  known :  but  as 
h 
these  equations  are  derived  one  from  the  other,  we  cannot,  with 
these  equations  alone,  eliminate  either  of  the  unknowns.  The 
author  has  omitted  to  point  out  that  if  we  take  two  tests,  one 
with  copper  to  line  and  the  other  with  zinc  to  line,  and  the  re- 
sults obtained  be  called  D  and  d,  then 
E'_  BC-AD 
-E~™A(D  +  R)  +  B(A  +  R)'    •     •     •     •     W 
_  E'  _  BC-A^ 
+  E  ~A(d+R)+B(A  +  R)'     •■••••     (4) 
E' 
the  signs  before  the  ratio  -^  being  opposite  in  the  two  equa- 
tions, but  the  decision  as  to  which  is  plus  and  which  minus  de- 
pending upon  the  direction  of  the  foreign  electromotive  force, 
which  we  shall  henceforth  call  the  earth-current. 
It  follows  from  this  that  if  we  add  the  two  equations  together 
E' 
we  at  once  eliminate  ^,  and  get 
BC-AD  BC-A^ 
A(D  +  R)+B(A  +  R)"+"A(^+R)+B(A  +  R)~  '     [  j 
from  which  C  is  easily  calculated,  especially  when  (as  is  usually 
the  case  in  testing  line-resistance)  A=B.  The  equation  then 
becomes 
C-D       ,       C-d 
D  +  2R  +  A~t~d+2R  +  A  •     •     ■     W 
Again,  if  we  omit  R  (the  resistance  of  the  testing  battery),  we 
get 
C-D      C-^      .  _ 
BTa  +  ^Ta=0-  ■•..-.    (7) 
As,  however,  the  battery-resistance  is  seldom  so  small  as  to 
be  neglected,  it  is  generally  better  to  use  equation  (6). 
In  Clark  and  Sabine's  '  Electrical  Tables  and  Formulae/  and 
in  a  pamphlet  issued  to  the  Indian  Government  Telegraph  De- 
partment, is  a  formula  by  Schwendler*  which,  while  really  iden- 
*  The  equation  as  given  by  Schwendler  is  as  follows  : — 
_&/(q+6)(W,+W,,)+62(aW'+2W'W/,  +  aWM) 
a6(W'+W")+2<//(a  +  &)-r-2a26 
