116  Winding  Ropes  in  Mines. 
The  values  of /'(#)  and  FY?/)  have  been  calculated  so  far 
as  the  interval  7l<y<yi,  and  from  them  the  values  of  the 
stress  at  the  upper  end  of  the  rope,  the  velocity  of  the  cage, 
and  the  stress  at  the  cage,  have  been  tabulated. 
The  discontinuities  in  the  values  for/'(y)  and  'F'(y)  can  be 
removed  by  considering  a  yielding  at  the  upper  end. 
If  we  consider  a  yielding  at  the  cage  and  at  the  upper  end, 
we  find  that  the  stresses  are  greatly  reduced. 
Consider  the  following:  case  : — 
Weight  of  Head  Gear  =  4600  lbs. 
At   the    upper  end  the  spring  is   such  that  400,000  lbs. 
deflects  it  1  foot.     At  the  lower  end  the  spring  is  such  that 
160,000  lbs.   deflects  it   1   foot,  and  Z=1000  feet.      (These 
round  numbers  are  taken  for  convenience  of  calculation.) 
Then   as  far  t  =  ~  we  find  that  instead  of  the  maximum 
a  V  V 
stress  at  the  upper  end  being  3-    it  is  only  1*4-  .      And  as 
,5/  a  a         .         V 
tar  as  £  =  —  the   maximum    stress  at  the  lower  end  is  1*6 — 
a  a 
V 
instead  of  2  -  . 
a 
Mr.  Richardson's  example  was  this : — Weight  of  cage 
4  tons,  cross  section  of  rope  5*33  sq.  inches,  weight  of  rope 
per  cubic  inch  0*14  lb.  Young's  modulus  of  the  material 
.14'5  x  10G  lb.  per  sq.  inch*.  He  gives  three  cases  in  which 
ihe  lengths  of  rope  out  were  respectively  107  feet,  1000  feet, 
and  2000  feet. 
Explanation  of  the  Diagrams  (PI.  II.) . 
The  scale  of  time  in  each  case  is  such  that  the  distance 
marked  OT  is  at/l,  or  the  time  required  for  a  pulse  to  travel 
from  one  end  of  the  rope  to  the  other.  Thus  in  tig.  1,  where 
the  length  of  rope  is  only  107  feet,  it  is  1/156  of  a  second  ; 
in  fig.  2,  where  /=1()00  feet,  it  is  0-06  second  ;  and  in  tig.  3, 
where  I  —  2000  feet,  it  is  0*12  second. 
The  full  line  curve  V  shows  how  the  speed  of  the  cage 
alters  as  time  goes  on  ;  the  height  OV  represents  40  feet  per 
second.  The  dotted  curve  Y  shows  the  same  thing  on  the 
assumption  that  after  imagining  one-third  of  the  mass  of 
the  rope  to  be  added  to  that  of  the  cage,  the  rope  is  regarded 
as  massless.     This  I  shall  call  the  approximate  solution. 
*  From  some  rough  measurements  upon  wire  rope  since  the  cal- 
culations were  made  I  am  inclined  to  think  that  what  is  equivalent 
to  the  Young's  modulus  in  the  rope  ought  to  be  taken  as  about  one- 
twentieth  of  that  of  the  steel  itself,  or  about  one-tenth  of  what 
Mr.  Richardson  assumed  ;  also  that  the  internal  friction  on  the  rope  must 
be  very  large  indeed. 
