﻿72 
  Prof. 
  Barton 
  and 
  Miss 
  Browning 
  on 
  

  

  Thus, 
  we 
  may 
  write 
  : 
  

  

  For 
  t 
  = 
  0, 
  z=f, 
  ] 
  

  

  when 
  it 
  follows 
  statically 
  that, 
  y— 
  9 
  p 
  , 
  

  

  and 
  we 
  have 
  also 
  -f- 
  = 
  0, 
  ^- 
  = 
  0. 
  

  

  dt 
  dt 
  

  

  (71) 
  

  

  Differentiating 
  (69) 
  and 
  (70) 
  with 
  respect 
  to 
  the 
  time, 
  

   and 
  introducing 
  (71) 
  gives 
  equations 
  which 
  are 
  satisfied 
  by 
  

  

  e=|and 
  0=5 
  (72) 
  

  

  E 
  = 
  (2 
  + 
  y 
  g)(-JL4- 
  ;? 
  + 
  A)-^ 
  /; 
  \ 
  

  

  Then, 
  introducing 
  (71) 
  and 
  (72) 
  in 
  (69) 
  and 
  (70) 
  we 
  find 
  

   ( 
  — 
  14-77 
  + 
  

  

  K 
  J 
  \. 
  . 
  . 
  (73) 
  

  

  (2 
  + 
  ff)(l-, 
  y 
  + 
  A) 
  + 
  ff 
  

  

  2(2 
  + 
  /3)A 
  '" 
  J 
  

  

  Finally, 
  (72) 
  and 
  (73) 
  in 
  (69) 
  and 
  (70) 
  give 
  as 
  the 
  re- 
  

   quired 
  special 
  solution 
  

  

  and 
  

  

  l 
  + 
  /3 
  + 
  iy- 
  A 
  

  

  y- 
  — 
  t(2+ffA 
  ^ 
  /C0S 
  ^ 
  

  

  + 
  2C2 
  + 
  /3)A 
  ^ 
  CW,gt 
  - 
  ••■■•■ 
  (75) 
  

  

  IV. 
  Relations 
  among 
  Variables. 
  

  

  It 
  is 
  instructive 
  to 
  plot 
  graphs 
  with 
  the 
  values 
  of 
  the 
  

   coupling 
  7 
  as 
  ordinates, 
  the 
  abscissae 
  being 
  the 
  corresponding 
  

   values 
  of 
  ft 
  (ratio 
  of 
  droop 
  of 
  bridle 
  to 
  pendulum 
  length). 
  

   A 
  different 
  graph 
  is 
  needed 
  for 
  each 
  value 
  of 
  77 
  and 
  p 
  (which 
  

   are 
  respectively 
  the 
  ratios 
  of 
  pendulum 
  lengths 
  and 
  masses 
  

   of 
  bobs). 
  

  

  The 
  data 
  for 
  these 
  graphs 
  are 
  derived 
  from 
  the 
  equations 
  

   and 
  are 
  given 
  in 
  Tables 
  I., 
  II. 
  , 
  and 
  III. 
  (and 
  Table 
  I. 
  p. 
  265 
  

   of 
  October 
  paper). 
  

  

  