﻿1875.] 
  Exact 
  Rectilinear 
  Motion 
  by 
  Linkwork. 
  569 
  

  

  and 
  

  

  Thus 
  NF=BD'-BN 
  + 
  D'N'* 
  

  

  =(a— 
  kd)— 
  ^{2ab 
  cos 
  B— 
  2 
  cd 
  cos 
  D}, 
  

  

  by 
  (i) 
  

  

  =(«-^)-^{(« 
  2 
  + 
  ^ 
  2 
  )-(^+^)}, 
  (2) 
  

  

  a 
  constant. 
  On 
  the 
  other 
  hand, 
  P 
  N— 
  F 
  N' 
  is 
  in 
  general 
  variable. 
  

  

  The 
  linkage 
  in 
  fig. 
  2, 
  which 
  will 
  assume 
  innumerable 
  forms 
  by 
  giving 
  

   different 
  values 
  to 
  a, 
  b, 
  c, 
  d, 
  and 
  k, 
  is 
  the 
  fundamental 
  linkage 
  upon 
  which 
  

   the 
  various 
  linkworks 
  here 
  discussed 
  depend. 
  As 
  the 
  same 
  lettering 
  will 
  

   be 
  preserved 
  throughout 
  the 
  diagrams, 
  the 
  fundamental 
  linkage 
  may 
  be 
  

   at 
  once 
  recognized 
  in 
  each 
  figure 
  showing 
  its 
  various 
  adaptations. 
  

  

  For 
  clearness 
  the 
  bars 
  are 
  denoted 
  by 
  thick 
  lines, 
  the 
  joints 
  by 
  round 
  

   spots 
  ; 
  when 
  a 
  bar 
  becomes 
  fixed 
  so 
  that 
  its 
  joints 
  are 
  fixed 
  pivots, 
  the 
  

   bar 
  is 
  denoted 
  by 
  a 
  broken 
  line 
  and 
  the 
  pivots 
  by 
  circles 
  round 
  the 
  

   spots. 
  When 
  points 
  in 
  general 
  separate 
  are 
  made 
  coincident, 
  the 
  letters 
  

   denoting 
  all 
  the 
  coincident 
  points 
  are 
  bracketed 
  together. 
  It 
  is 
  found 
  con- 
  

   venient 
  to 
  collect 
  the 
  different 
  linkworks 
  into 
  four 
  groups, 
  a 
  separate 
  

   section, 
  numbered 
  to 
  correspond 
  with 
  the 
  figure, 
  being 
  devoted 
  to 
  each 
  

   separate 
  linkwork 
  described. 
  

  

  I. 
  § 
  3. 
  Take 
  

  

  (a—kd)ab 
  

  

  ~ 
  (a 
  2 
  + 
  b 
  2 
  )-(c 
  2 
  + 
  d 
  2 
  ) 
  

  

  bo 
  thatNN'=?^'. 
  

  

  Fig. 
  3. 
  

  

  Then 
  if 
  the 
  bar 
  A 
  B 
  be 
  fixed 
  and 
  two 
  bars 
  PO, 
  P'O 
  be 
  added, 
  

   PO=PB, 
  P'0=P'D'; 
  

   O 
  clearly 
  lies 
  on 
  A 
  B 
  however 
  the 
  linkwork 
  be 
  deformed, 
  and 
  its 
  locus 
  is 
  

   therefore 
  the 
  straight 
  line 
  A 
  B. 
  

  

  * 
  In 
  the 
  figure 
  D' 
  N' 
  is 
  negative. 
  

  

  2 
  u* 
  2 
  

  

  