﻿about 
  a 
  Position 
  of 
  Equilibrium. 
  299 
  

  

  would 
  give 
  the 
  ends 
  of 
  the 
  double 
  curves. 
  For/ 
  2 
  (f)=0 
  -jr- 
  

  

  takes 
  the 
  indefinite 
  form 
  ; 
  this 
  condition 
  gives 
  in 
  general 
  

   multiple 
  points. 
  

  

  So 
  we 
  have 
  found 
  : 
  the 
  only 
  points 
  where 
  the 
  tangential 
  

   line 
  to 
  the 
  envelope 
  is 
  parallel 
  to 
  the 
  Y-axis 
  are 
  given 
  by 
  

   / 
  1 
  (?) 
  = 
  0,cos*=0. 
  

  

  § 
  33. 
  In 
  several 
  simple 
  cases 
  that 
  will 
  be 
  discussed 
  after- 
  

   wards, 
  we 
  may 
  observe 
  that 
  cusps 
  may 
  appear 
  in 
  the 
  envelope. 
  

   In 
  order 
  to 
  examine 
  the 
  appearance 
  of 
  these 
  points 
  we 
  must 
  

  

  put-y^ 
  =00 
  . 
  We 
  write 
  (21, 
  § 
  31) 
  in 
  this 
  form 
  : 
  

   fi(X) 
  sin* 
  J 
  -<y/i(S) 
  sin 
  (yt-<j>) 
  =0. 
  

   Differentiation 
  according 
  to 
  t 
  gives 
  : 
  

   /.(O 
  - 
  t 
  g 
  . 
  J 
  +./VU) 
  - 
  1 
  § 
  d 
  £ 
  +m 
  cos 
  t 
  % 
  - 
  7 
  //(?) 
  § 
  sin 
  (*-« 
  

  

  -T^OD 
  COS 
  (yt-<f>) 
  + 
  7 
  /Kr) 
  sin 
  (yt-f) 
  J 
  =0. 
  

   "We 
  have 
  successively 
  to 
  discuss 
  these 
  conditions 
  : 
  

  

  tfn«-o,/ 
  l( 
  o-a, 
  J-o, 
  J— 
  ./^(O— 
  ./.'« 
  — 
  • 
  J— 
  »#— 
  

  

  Sinf 
  = 
  gives, 
  as 
  we 
  have 
  found 
  §32, 
  either 
  f 
  2 
  '(£) 
  = 
  °o, 
  or 
  

   +'(?) 
  = 
  *>, 
  <>* 
  / 
  2 
  (£)=0, 
  or 
  //(f) 
  =0, 
  or 
  sin 
  = 
  0. 
  For 
  

  

  sin£ 
  = 
  0, 
  combined 
  with 
  / 
  2 
  (f) 
  = 
  0, 
  ~ 
  takes 
  the 
  indefinite 
  

  

  form 
  ; 
  this 
  gives 
  in 
  general 
  multiple 
  points. 
  Sin 
  </> 
  = 
  with 
  

   sin£ 
  = 
  would 
  give 
  the 
  ends 
  of 
  the 
  double 
  curves. 
  There- 
  

   fore 
  we 
  have 
  to 
  combine 
  sin£=0 
  with 
  one 
  of 
  the 
  conditions 
  

  

  /s'(f) 
  = 
  », 
  +1' 
  CO-*, 
  «/i'(0-a 
  

  

  In 
  § 
  32 
  we 
  have 
  seen 
  that 
  tor/i(f) 
  —0 
  cos£ 
  = 
  in 
  general. 
  

   The 
  condition 
  j- 
  =0 
  may 
  be 
  written 
  in 
  this 
  form 
  : 
  

  

  2_ML 
  tan 
  , 
  

  

  -~ 
  = 
  x> 
  involves 
  either 
  /i(f) 
  =0 
  or 
  sin 
  £ 
  = 
  ; 
  these 
  conditions 
  

  

  have 
  been 
  discussed 
  already. 
  

  

  For 
  f 
  l 
  '(^) 
  = 
  -Jo 
  we 
  have 
  in 
  general, 
  according 
  to 
  (20), 
  

   cos£ 
  = 
  0, 
  or 
  sin 
  (yt 
  — 
  (f>)=0. 
  

  

  For 
  f 
  2 
  '(£) 
  = 
  x> 
  we 
  have 
  in 
  general, 
  according 
  to 
  (20), 
  

   sin£ 
  = 
  0, 
  or 
  cos 
  {yt 
  — 
  (p) 
  = 
  0. 
  

  

  X 
  2 
  

  

  