110  Prof.  J.  Perry  on 
Using  the  value  for  f(y  — 21)  found  in  (5)  we  have 
To  determine  the  constant  C  Ave  notice  that  the  velocity  of 
the  cage  does  not  alter  suddenly. 
•*• 
y=l-o                         y=i+o 
.'. 
_v=v+c^-i 
•'• 
C= em 
a 
•'• 
fM  =  l-T-'**-> 
and  from  (1) 
*<>>-£- 
•'• 
Ky<Zl 
V      2V        i             ^ 
(7) 
By  a  similar  method  we  may  proceed  to  find  the  value  of 
the  functions  /'(#)  and  F'(j/)  for  the  interval  ?>l<y<bl  by 
using  (6)  and  the  values  for  j-'(y  — 21)  andf//(y  —  2l)  from  (7), 
and  assigning  the  constant  of  integration  from  the  fact  that 
the  velocity  of  the  cage  does  not  suddenly  alter. 
We  notice  that  the  functions  f(y)  have  a  discontinuity  of 
V 
amount   —  — ,  and  the  functions    F'(y)    a  discontinuity    of 
amount  —  as  we  pass  through  the  values  of  y  which  are  odd 
multiples  of  I. 
To  study  the  motion,  in  detail,  at  a  point  z  in  the  rope,  we 
have  when  t< 
a 
and  this  state  holds  until  the  wave  reaches  the  point. 
