﻿182 
  Mr. 
  J, 
  Robinson 
  on 
  Konig^s 
  Theory 
  of 
  the 
  

  

  The 
  force 
  exerted 
  by 
  one 
  particle 
  at 
  C 
  on 
  the 
  particle 
  at 
  

   = 
  — 
  ^ 
  2 
  .cos^.(3 
  — 
  5cos^6'). 
  

  

  Now 
  in 
  the 
  length 
  dx 
  o£ 
  the 
  ripple 
  there 
  are 
  n 
  dx 
  particles, 
  

   and 
  the 
  force 
  exerted 
  by 
  these 
  on 
  

  

  = 
  — 
  ^ 
  cos 
  ^(3 
  — 
  5 
  cos^ 
  6) 
  dx. 
  

  

  Fio-. 
  3. 
  

  

  

  

  A 
  ^ 
  

  

  1 
  

  

  / 
  

  

  1 
  

  

  1 
  ) 
  

  

  

  ^0/ 
  

  

  1 
  1 
  

   1 
  ^3 
  

  

  

  

  /^ 
  

  

  

  

  

  1 
  

   1 
  

   1 
  

   1 
  

   1 
  

  

  1 
  

   1 
  

  

  

  

  \ 
  

   1 
  

  

  1 
  

  

  B 
  ^^ 
  

  

  Therefore 
  the 
  total 
  force 
  exerted 
  by 
  the 
  whole 
  ripple 
  

   on 
  0, 
  

  

  = 
  F=— 
  -^ 
  ^- 
  — 
  .cos^ 
  (3 
  — 
  ocos^t/jrf.i; 
  

  

  J 
  -b 
  ^'0 
  

  

  ,r 
  ^os^(3-5co 
  s^6') 
  

   Jo 
  

  

  = 
  — 
  StittctR^z^;^ 
  

  

  dx. 
  

  

  Now 
  

  

  rQ=^a 
  sec 
  6^ 
  . 
  

   x=^a 
  tan 
  ^, 
  

   dx 
  = 
  a 
  sec^ 
  .d6 
  ; 
  

  

  and 
  therefore 
  

  

  '^cos^(3-5cos^6>) 
  

  

  

  c?.:^ 
  

  

  V 
  

  

  -atan" 
  

  

  cos^(9(3-5cos2^) 
  

  

  ^(9 
  

  

  ..tan 
  - 
  

  

  cos^^-5cos^^)f^^. 
  

  

  