﻿298 
  Mr. 
  II. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  have 
  to 
  consider 
  the 
  branches 
  in 
  the 
  double 
  point 
  are 
  always 
  

   rectangular. 
  In 
  this 
  case 
  

  

  (20) 
  passes 
  into 
  

  

  7(1 
  — 
  J) 
  sin 
  (7^ 
  — 
  <£) 
  sin 
  (£ 
  — 
  <£)—£ 
  sin 
  7^ 
  sin 
  t 
  

  

  ^/(f) 
  sin 
  (7*-*) 
  sin*.. 
  (22) 
  

  

  -VN 
  

  

  § 
  32. 
  We 
  now 
  shall 
  investigate 
  whether 
  the 
  envelope 
  

   possesses 
  points 
  where 
  the 
  tangential 
  line 
  is 
  parallel 
  to 
  the 
  

   X-axis 
  or 
  to 
  the 
  Y-axis. 
  We 
  suppose 
  /i(£) 
  and 
  / 
  2 
  (f), 
  the 
  

   amplitudes 
  of 
  the 
  vibrations, 
  not 
  to 
  become 
  infinite 
  ; 
  how- 
  

   ever, 
  /i'(f) 
  and/ 
  2 
  '(f) 
  may 
  become 
  infinite. 
  

   From 
  

  

  dy 
  yf 
  2 
  (Z) 
  sin 
  (yt 
  — 
  <f>) 
  

   dx 
  ~ 
  /i(£j 
  sin 
  t 
  

  

  it 
  follows 
  that 
  4 
  = 
  0, 
  when 
  / 
  2 
  (f) 
  = 
  0, 
  or 
  sin 
  (7* 
  — 
  <£) 
  = 
  0, 
  

  

  whilst 
  /i(f)#0 
  and 
  sin 
  ^0. 
  If 
  / 
  2 
  (f)=0j 
  then 
  in 
  general 
  

  

  //(?) 
  or 
  ^'(?) 
  w 
  iH 
  n 
  °t 
  at 
  the 
  same 
  time 
  be 
  infinite, 
  and 
  

  

  fi(£)if2 
  f 
  (£) 
  or 
  sin<£ 
  will 
  not 
  in 
  general 
  be 
  zero 
  at 
  the 
  same 
  

  

  time. 
  Therefore 
  we 
  may 
  deduce 
  from 
  (20, 
  § 
  30) 
  that 
  for 
  

  

  y 
  2 
  (f) 
  = 
  cos 
  (7* 
  — 
  </>) 
  = 
  0. 
  If 
  sin 
  (7*— 
  </>) 
  = 
  0, 
  then 
  either 
  

  

  /iU) 
  = 
  ^,or^ 
  / 
  (0 
  = 
  ^,or/ 
  1 
  (r) 
  = 
  0,or//(p 
  = 
  0,orsin(/> 
  = 
  

  

  (according 
  to 
  20). 
  The 
  last 
  condition, 
  if 
  combined 
  with 
  

  

  sin 
  (7* 
  — 
  ^>) 
  = 
  0, 
  gives 
  the 
  summits 
  of 
  the 
  double 
  curves. 
  

  

  dxi 
  

   For 
  yi(f)=0 
  -p 
  takes 
  the 
  indefinite 
  form; 
  this 
  condition 
  

  

  gives 
  in 
  goneral 
  multiple 
  points. 
  So 
  we 
  may 
  say 
  : 
  the 
  

   tangential 
  line 
  to 
  the 
  envelope 
  is 
  parallel 
  to 
  the 
  X-axis 
  

  

  (1) 
  at 
  the 
  summits 
  of 
  the 
  double 
  curves, 
  and 
  

  

  (2) 
  at 
  the 
  points 
  of 
  the 
  X-axis 
  given 
  by 
  / 
  2 
  (f) 
  = 
  0, 
  

   cos 
  (yt 
  — 
  <f>) 
  =0. 
  

  

  From 
  dy 
  __ 
  yf 
  2 
  ( 
  f 
  ) 
  sin 
  (yt 
  — 
  <f>) 
  

  

  dx~~ 
  /if^sint 
  

  

  it 
  follows 
  that^ 
  =00 
  , 
  when 
  /i(f) 
  = 
  or 
  sin£ 
  = 
  0, 
  whilst 
  

  

  / 
  2 
  (?)#0 
  and 
  sin(7^-<^)#0. 
  If 
  /i(5) 
  = 
  0, 
  then 
  in 
  general 
  

  

  cos£ 
  = 
  (according 
  to 
  20). 
  If 
  sin£ 
  = 
  0, 
  then 
  either 
  

  

  .,/(?) 
  = 
  *>, 
  or 
  ^'(f) 
  = 
  oo, 
  or 
  /,(£) 
  = 
  0, 
  or 
  //(£) 
  = 
  (), 
  or 
  

  

  sin<£ 
  = 
  0. 
  The 
  last 
  condition 
  in 
  connexion 
  with 
  sin 
  £ 
  = 
  

  

  