636     Messrs.  Hawthorne  and  Morton  on  Deflexions  caused 
XT 
Since  >  1,  we  take  the  upper  sign  and  so  get 
Xf—\ 
T        xr     _a-\-  ^/a2  —  4^ 
=  oo  Xr—l 
Thus  at  a  distance  from  an  anchor-pole  the  successive 
deflexions  approach  a  geometric  progression.  It  will  appear 
in  the  subsequent  part  of  the  work  that  in  our  actual  case 
this  state  of  affairs  is  closely  approached  after  seven  or  eight 
poles  are  passed. 
The  existence  of  this  limiting  ratio,  taken  in  conjunction 
with  the  last  equation  of  the  set,  leads  to  the  conse- 
quence that  the  maximum  deflexion,  which  of  course  occurs 
in  the  pole  next  to  the  break,  and  which  becomes  greater  as 
this  pole  is  further  removed  from  the  anchor-pole,  approaches 
a  limiting  value.  If  this  limit  is  below  the  allowable  safe 
deflexion,  it  does  not  matter  how  widely  the  anchor-poles  are 
distributed. 
To  find  this  limiting  deflexion  we  have 
xn  \  (a  —  1) ^i  \  =h, 
b 
Xn  — 
(a-l)-~— 
Xn 
Giving  the  ratio  in  the  denominator  its  limiting  value,  we 
find  that  the  deflexion  cannot  exceed 
2b 
a-2+Va2-± 
If  the  original  horizontal  tension  acted  on  one  side  of  the 
T  b 
pole  next  the  break,  it  would  produce  a  deflexion  -r  =  - £. 
cp      a  —  z 
It  will  be  seen  that  the  actual  maximum  deflexion  is  less  than 
this,  on  account  of  the  increased  sag  of  the  remaining  wires, 
in  the  ratio  . 
V    a  — 9. 
2:1 
It  is  easy  to  extend  the  analysis  to  the  case  where  some, 
only,  of  a  number  of  similar  wires  are  broken.  We  shall 
give  the  equations  arrived  at  for  the  deflexions. 
Let  the  break  occur  between  the  nth.  pole  counting  from  one 
anchor-pole,  and  the  ?nth  counting  in  the  opposite  direction 
