Winding  Ropes  in  Mines. 
111 
After  this, 
en 
that 
is  when  t  >  -^  and  so  long 
as 
*< 
z 
-  3 
Hence  a  wave  of  extension  sets  out  from  the  upper  end.  When 
the  wave  reaches  a  point  the  point  comes   to    rest  and  the 
extension  of  the  rope  at  the  point  is 
a 
l  +  z 
This  state  continues  so  long  as  t  <  — -  ,  that  is  until  the 
°  a 
eturn  wave  from  the  cage  reaches  the  point. 
When  ,.*  +  *  u  ,  ^-z 
t  > but  < 
a  a 
at—z>l,  and     at\z<Zl, 
¥'(at  +  z)  = 
V 
~~  2a 
/(at 
-*)  = 
V 
2a 
2V 
—  e 
a 
-l_(at—z-i 
ml 
.T{. 
-2e 
(at  —  z- 
1 
2V   - 
:  — e 
a 
_(at~z- 
ml 
-I) 
These  results  show  a  wave  running  back  from  the  cage. 
Every  point  as  the  wave  reaches  it  starts  to  move  downwards 
with  the  velocity  V,  but  its  velocity  rapidly  diminishes. 
The  extension  of  the  rope  at  a  point  when  the  wave  reaches 
2V 
it  becomes  — ,  but  this  also  rapidly  diminishes. 
In  this  manner  we  may  interpret  our  results  from  the 
values  of/'(y)  and  F'(y). 
