by  a  Break  in  an  Overhead  Wire  carried  on  Poles.       635 
The  first  (n—1)  equations  determine  the  ratios  of  the 
n  deflexions,  in  terms  of  the  single  constant  a.  It  will  be 
seen  that  the  relative  magnitudes  of  the  yieldings  in  the  suc- 
cessive poles  do  not  depend  on  the  position  of  the  break.  The 
latter  fixes  the  absolute  scale,  in  accordance  with  the  last 
equation. 
The    ofs    form   a   recurring    series,    whose    "  generating 
1-ay+f 
i.  e.,  xr  is  the  coefficient  of  yr  in  the  expansion  of  this  fraction 
in  ascending  powers.     Expressed  explicitly  we  have 
xr  .      r-2        _      (r_4)(r-3)         - 
—  =ar~1 — - ar~3+  ± -Li- L  flr-5 
x1  1  1.2 
0— 6)(r-5)Q— 4)  , 
1.2.3         ~a       T-"- 
Consider   now   the    ratios  of  consecutive  deflexions.      We 
have 
x2 
—  =a, 
xx 
X2  .I'j  1 
—  =a =a , 
x2  x2  a 
X±  X2  JL 
a —  — 
a 
Xr 
....(r-l)a's. 
Xr-l  1 
a 
a  — 
Now,  from  the  nature  of  the  case,  a  >  2 ;  and  it  can  be 
shown  that  the  continued  fraction  then  approaches  a  finite 
limit  as  r  increases  *.  In  fact,  if  z  represents  this  limiting 
value,  we  have 
or  z  satisfies 
z2-az+l  =  0. 
The  roots  of  this  are  i{a  +  \/a2— 4},  which  are  real  if  a>2. 
*  Of.  OhiystaTs  { Algebra,'  vol.  ii.  p.  482. 
2  T  2 
