Winding  Ropes  in  Mines.  109 
Hence  «,  *  V  A      ^ 
fd/)=-YJ      2/<0    I 
v  $> 
F(y)  =  -^       y<+i) 
Also  since  the  top  end   remains  at  rest   for  all  values  of 
.-.    f(at-l)  +  F(at  +  l)  =  0  t>0, 
i.e.,     F(y)  =  -f(y-2l)  y>L     .     .     (4) 
Consider  the  Interval  0<i/<l. 
From  (3)  we  see  that  in  this  interval 
F'(2/)=-J         and       F"(y)  =  0. 
.'.  equation  (2)  becomes 
To  determine  the  constant  C  we  notice  that  the  velocity 
of  the  cage  does  not  alter  suddenly. 
Now  when  t  <  0  velocity  of  cage  =  —  V.  Since  our  positive 
direction  is  upwards,  and 
•■■  -Y=«{/r'(y)+F'(y)} 
y=0 
Y  =  a(~^4 
I      2a 
>       ml  _  _—    V 
y  =  0 
■-  0=0. 
anc 
.'•/%)  =  - 
V 
'2a 
TT 
V 
"la 
0<y<l}. 
J 
Consider  the  Interval 
1<!I 
<3i. 
From 
(4) 
we  have 
F'(y)  = 
-/'(</- 
-20, 
and  (2) 
may 
be  written 
-f"(y 
-2/), 
(5) 
/"(.y)  +  ^/'0/)=/"0/- 20 -„'/./"(// --'').    .    (6) 
