﻿312 
  Miv 
  H. 
  J. 
  E. 
  Beth 
  on 
  the 
  Oscillations 
  

  

  We 
  therefore 
  shall 
  have 
  to 
  determine 
  m, 
  n, 
  and 
  yjr 
  from 
  the 
  

   system 
  

  

  — 
  m$m2ilr=l, 
  

  

  1 
  — 
  m 
  cos 
  2y(r 
  = 
  q, 
  

  

  — 
  n 
  — 
  in 
  sin 
  2 
  yjr 
  = 
  r. 
  

  

  From 
  the 
  first 
  two 
  equations 
  we 
  deduce 
  : 
  

  

  tan 
  2-slr 
  = 
  T 
  , 
  

  

  7 
  q 
  — 
  1 
  

  

  m 
  2 
  = 
  Z 
  2 
  + 
  (?-l) 
  2 
  . 
  

  

  n 
  is 
  to 
  be 
  found 
  from 
  

  

  n 
  = 
  — 
  r 
  — 
  m 
  sin 
  2 
  t^. 
  

  

  So 
  it 
  is 
  always 
  possible 
  to 
  find 
  values 
  of 
  t/r, 
  m, 
  and 
  n 
  ; 
  the 
  

   relation 
  between 
  f 
  and 
  <f>' 
  may 
  therefore 
  be 
  written 
  in 
  this 
  

   form 
  

  

  f'(l_£'jsin 
  2 
  c£' 
  = 
  ?< 
  + 
  rc. 
  

  

  There 
  are 
  two 
  values 
  of 
  yjr, 
  m, 
  and 
  n 
  : 
  so 
  there 
  are 
  two 
  

   different 
  systems 
  of 
  ellipses, 
  each 
  of 
  them 
  having 
  its 
  own 
  

   envelope. 
  

  

  § 
  46. 
  We 
  suppose 
  the 
  turn 
  of 
  the 
  axes 
  to 
  have 
  taken 
  place. 
  

   Therefore 
  we 
  have 
  to 
  take 
  the 
  relation 
  between 
  f 
  and 
  cf> 
  in 
  

   this 
  form 
  : 
  

  

  f(l 
  — 
  ?)sin 
  2 
  = 
  7w?+ra 
  (35) 
  

  

  The 
  equation 
  of 
  the 
  envelope 
  runs 
  : 
  

  

  (1-Osin 
  2 
  (*-<£;-?sin 
  a 
  * 
  = 
  2V?(l-?)/'(f) 
  sin 
  £ 
  sin 
  (£-<£), 
  

   where 
  

  

  /(£> 
  = 
  n/«i- 
  ?)-(»»?+»), 
  

  

  7 
  W 
  2^(1 
  - 
  £) 
  - 
  O? 
  + 
  n) 
  V«l 
  - 
  J) 
  c° 
  s 
  * 
  ' 
  

  

  Therefore 
  

   (1 
  — 
  £) 
  sin 
  2 
  (£ 
  — 
  (£)cosc/> 
  — 
  fsin 
  2 
  ^cos^=(l-2f-?7i)sin^sin(^ 
  — 
  <£). 
  (36) 
  

  

  After 
  reduction 
  : 
  

   (1 
  — 
  f) 
  sin 
  (t 
  — 
  <f>) 
  cos 
  (*~~0) 
  + 
  £cos£ 
  sin 
  £ 
  = 
  \. 
  ; 
  sin 
  (t—<f>) 
  sin 
  £ 
  sin 
  <j>, 
  

  

  mf(l 
  — 
  f) 
  sin 
  (£ 
  — 
  (/>) 
  cos 
  2 
  £ 
  cos 
  (t 
  - 
  <p) 
  -f 
  w(l 
  — 
  f) 
  sin 
  (t 
  — 
  (p) 
  cos 
  (£-— 
  <£) 
  

  

  -J-w^cosisinf 
  + 
  ^fcos 
  £sin 
  i 
  + 
  ?«f(l 
  — 
  f) 
  sin 
  2 
  (£ 
  — 
  <£) 
  sin 
  £ 
  cos 
  £ 
  = 
  0. 
  

  

  