by  a  Break  in  an  Overhead  Wire  carried  on  Poles.      633 
The  theoretical  investigation,  given  below,  presents  no 
special  difficulty  and  leads  to  results  of  some  interest.  It  is 
followed  by  a  numerical  evaluation  of  the  deflexions  for 
a  particular  case,  in  which  we  have  used  values  for  the 
constants  derived  from  a  case  occurring  in  actual  engineering 
practice. 
We  shall  suppose  the  wire  to  be  fastened  at  its  ends  to 
massive  "  anchor-poles "  whose  yielding  is  negligible,  and 
to  be  supported  by  equally-spaced  flexible  poles.  The  relation 
between  force  and  deflexion  for  a  horizontal  pull  applied  to 
tho  top  of  a  pole  is  supposed  known.  If  this  be  expressed  by 
the  equation 
then  in  the  region  of  safe  deflexions  the  function  is  linear, 
and  <$  may  be  regarded  as  a  numerical  constant. 
The  other  relation  which  enters  into  the  problem  is  that 
giving  the  alteration  in  the  horizontal  tension  of  a  flat  catenary 
for  a  small  horizontal  displacement  of  its  ends. 
The  arc  of  a  catenary  is  connected  with  the  abscissa  of  its 
end  and  the  parameter  by  the  equation 
*  i* 
s  =  c  sinn— . 
c 
Differentiating  this,  keeping  s  constant, 
dc  \  sinh—  —  —cosh—  I  +  dx  cosh  —  =0, 
\         c       c  c  /  c         * 
dx  =  dc{ tanh  -  ). 
If  <f>  is  the  inclination  of  the  tangent  at  the  end  of  the 
catenary, 
x 
e  & 
=  log  (sec  (/>  +  tan  tf)  ; 
so  that  if  cf>  is  small,  —  is  small  of  the  same  order,  and 
c 
x           .  x  _  xd  t 
tanli o~s  > 
G  C  OC° 
,\    dc= —o-dtf. 
x» 
Phil.  Mag.  S.  6.  Vol.  11.  No.  65.  May  1906.  2  T 
