﻿874 
  Dr. 
  C. 
  Y. 
  Burton 
  on 
  the 
  Apparent 
  

  

  (lateral) 
  displacement 
  of 
  the 
  7^th 
  load. 
  The 
  force 
  on 
  this- 
  

   load 
  is 
  

  

  which 
  is 
  therefore 
  equal 
  to 
  md^yrldt'^. 
  Assume 
  now 
  that 
  the 
  

   motion 
  is 
  periodic, 
  with 
  period 
  27r/'p, 
  and 
  let 
  the 
  motion 
  of 
  

   the 
  7'th 
  load 
  be 
  given 
  by 
  

  

  ijr^Ke'P' 
  (1) 
  

  

  Solutions 
  can 
  be 
  obtained 
  corresponding 
  to 
  the 
  further 
  

   assumptions 
  that 
  the 
  motion 
  of 
  each 
  of 
  the 
  remaining 
  loads 
  

   is 
  periodic, 
  and 
  is 
  represented 
  by 
  an 
  equation 
  of 
  the 
  same 
  

   form 
  as 
  (1), 
  while 
  

  

  A,+i/A,=A,/A,,_i= 
  ... 
  =/^. 
  ... 
  (2) 
  

  

  Then, 
  if 
  for 
  brevity 
  we 
  write 
  

  

  a'^p'bm/^, 
  (3) 
  

  

  it 
  follows 
  that 
  

  

  (a2-2)/^ 
  + 
  l 
  + 
  P 
  = 
  0; 
  (4) 
  

  

  whence 
  

  

  ^ 
  = 
  l_ia2 
  + 
  za(l-l«2>)i 
  (5) 
  

  

  8. 
  It 
  is 
  here 
  snpposed 
  that 
  the 
  wave-length 
  of 
  the 
  dis- 
  

   turbance 
  in 
  question 
  is 
  sufficiently 
  great 
  to 
  make 
  p^bm/^ 
  

   less 
  than 
  4. 
  If, 
  then, 
  we 
  put 
  

  

  Z: 
  = 
  B(cos/e±2sin/3) 
  (6) 
  

  

  ►) 
  , 
  B 
  and 
  /3 
  being 
  real 
  ; 
  we 
  obtain 
  on 
  

   1 
  aginary 
  parts 
  

  

  B 
  = 
  l 
  and 
  tanye=±a(l-ia2y(-jL_.i^2). 
  . 
  (7) 
  

  

  so 
  that 
  

  

  smil3=±iu=±^p^(bm/B). 
  ... 
  (8) 
  

  

  9. 
  Rejecting 
  now 
  the 
  imaginary 
  terms 
  in 
  (1) 
  and 
  in 
  the 
  

   solution, 
  there 
  remain 
  the 
  given 
  motion 
  

  

  7/r 
  = 
  Ar 
  cos 
  pt 
  (9) 
  

  

  and 
  the 
  solutions 
  

  

  yr^q 
  — 
  Kco%{pt±qP)', 
  .... 
  (10) 
  

  

  where 
  q 
  is 
  any 
  integer 
  positive 
  or 
  negative, 
  and 
  /3 
  is 
  an 
  

   acute 
  angle 
  given 
  by 
  (8) 
  . 
  The 
  lower 
  sign 
  corresponds 
  to 
  a 
  

   wave-train 
  propagated 
  in 
  the 
  direction 
  of 
  ^-increasing, 
  the 
  

   upper 
  sign 
  to 
  a 
  train 
  in 
  the 
  contrary 
  direction. 
  In 
  either 
  

   case 
  the 
  amplitude 
  is 
  invariable 
  throughout 
  the 
  train 
  ; 
  that 
  

   is 
  to 
  say, 
  there 
  is 
  no 
  absorption. 
  On 
  the 
  other 
  hand 
  (as 
  

  

  as 
  equivalent 
  to 
  (5), 
  B 
  and 
  /3 
  being 
  real; 
  we 
  obtain 
  on 
  

   equating 
  real 
  and 
  imaginary 
  parts 
  

  

  