Winding  Ropes  in  Mines.  107 
(i)  For  an  instrument  consisting  of  a  single  coil  for  one- 
coil  system,  and  of  either  two  or  three  coils  for 
the  other  system, 
(ii)  For  a  symmetrical  instrument  containing  three  coils 
in  one  system  and  two  in  the  other, 
e  =  }&. 
(iii)  For  a  mathematically  symmetrical  six-coil  instrument 
0=0, 
where  e  is  the  square  root  of  mean  square  of  the 
fractions  e1;  e2,  e3  by  which  the  three  load-currents 
differ  from  their  mean  value.  It  is  also  very 
approximately  the  arithmetical  mean  of  the  quan- 
tities elf  e2)  e3  or  two-thirds  of  the  greatest  of  these 
three  quantities,  so  that  e2*  is  essentially  the  same 
as  the  greatest  of  the  quantities  e1?  e2,  and  e3. 
V4 
VI.    Winding  Ropes  in  Mines. 
By  Professor  John  Pekry*. 
[Plate  II.] 
IN  Section  Gr  I  have  just  read  a  short  paper  in  which  I 
give  a  practical  formula  for  the  strength  of  a  winding 
rope.  I  point  out  the  difficulty  of  dealing  mathematically 
with  the  real  problem — "  When  a  cage  is  descending  and  the 
upper  end  of  the  rope  is  stopped,  what  occurs  in  the  rope  ?" 
The  problem  taken  up  in  this  paper  assumes  no  internal 
friction  in  the  rope,  and  it  is  only  towards  the  end  that  I 
refer  to  the  effect  of  a  yielding  in  the  attachments  at  its 
upper  or  lower  ends. 
The  solution  of  the  present  problem  was  given  me  by  Dr. 
Love  ;  I  have  gone  over  his  work  ;  one  of  my  students, 
Mr.  A.  P.  Richardson,  has  carried  the  work  a  good  deal  further 
than  was  done  in  Dr.  Love's  hasty  letter  to  me.  Mr.  Richard- 
son has  taken  a  practical  example  and  drawn  curves  to  show 
how  the  pull  alters  at  the  two  ends  of  the  rope  and  how  the 
speed  of  the  cage  alters,  and  on  the  same  sheets  we  have 
what  pull  and  speed  would  be  if  motions  were  simple  har- 
monic, all  parts  of  the  rope  moving  in  the  same  phase.  It 
will  be  seen,  therefore,  that  my  part  of  the  credit  of  producing 
this  paper  is  very  small. 
*  Communicated  by  the  Author.  Read  in  Section  A  at  the  British 
Association  Meeting-,  Johannesburg,  1005. 
