﻿930 
  Mr. 
  A. 
  Ferguson 
  on 
  the 
  Forces 
  acting 
  on 
  a 
  

  

  Remembering 
  that 
  

  

  v. 
  r' 
  

  

  sin 
  4> 
  = 
  U, 
  and 
  therefore 
  sin 
  ^ 
  = 
  yt, 
  

  

  and 
  putting 
  

  

  a? 
  = 
  — 
  , 
  

   99 
  

   (xii.) 
  becomes 
  

  

  m 
  = 
  27ra 
  2 
  p 
  ^ 
  + 
  2tt 
  P 
  ]-^ 
  + 
  v 
  — 
  g 
  — 
  -g-J 
  . 
  . 
  (xm.) 
  

  

  Let 
  d 
  be 
  the 
  value 
  of 
  y 
  at 
  the 
  point 
  (/>! 
  ; 
  then 
  

   r 
  /2 
  = 
  d(m-d), 
  

   and 
  (xiii.) 
  becomes, 
  after 
  a 
  few 
  reductions, 
  

  

  m 
  = 
  2Trp<-j—t-2a°d— 
  ^-- 
  — 
  g- 
  J-. 
  . 
  . 
  (xiv.) 
  

  

  "Returning 
  now 
  to 
  equation 
  (vii.), 
  if 
  we 
  remember 
  that 
  in 
  

   the 
  present 
  problem 
  p 
  = 
  tan0i 
  when 
  y 
  — 
  d, 
  we 
  have, 
  sub- 
  

   stituting 
  these 
  values 
  in 
  equation 
  (vii.), 
  

  

  * 
  = 
  2a\l 
  + 
  cos 
  fr) 
  + 
  ^ 
  (4a*-*)*-g 
  . 
  ^i 
  - 
  2 
  ^. 
  (xv.) 
  

  

  Putting 
  <i 
  2 
  = 
  2a 
  2 
  (l 
  + 
  cos(/>i) 
  in 
  the 
  small 
  terms 
  of 
  (xv.), 
  and 
  

   remembering 
  that 
  cosc^^ 
  — 
  tv~ 
  j 
  we 
  obtain 
  

  

  -"(^D+^Cf.- 
  1 
  ) 
  <" 
  L) 
  

  

  remembering 
  that, 
  since 
  R 
  is 
  large 
  compared 
  with 
  d, 
  we 
  may 
  

   put 
  the 
  approximate 
  value 
  d 
  2 
  = 
  £a 
  2 
  in 
  the 
  second 
  and 
  third 
  

   terms 
  of 
  (xv. 
  a). 
  

  

  (Note 
  that 
  if 
  we 
  put 
  R 
  = 
  oo 
  in 
  equation 
  (xvi.), 
  we 
  recover, 
  

   as 
  we 
  should 
  do, 
  equation 
  (ix. 
  a) 
  of 
  the 
  flat-disk 
  problem.) 
  

  

  To 
  a 
  first 
  approximation 
  we 
  may 
  put 
  r=r' 
  (fig. 
  2) 
  ; 
  if 
  

   desired, 
  a 
  closer 
  approximation 
  may 
  be 
  found 
  by 
  following 
  

   the 
  reasoning 
  which 
  results 
  in 
  equation 
  (xi.). 
  Putting, 
  

   therefore, 
  

  

  _ 
  m 
  — 
  

  

  = 
  d{2B,-d) 
  = 
  2Ud, 
  

  

  