﻿566 
  Mr. 
  A. 
  B. 
  Kempe 
  on 
  the 
  production 
  of 
  [June 
  17, 
  

  

  M. 
  Peaucellier's 
  apparatus 
  is 
  shown 
  in 
  fig. 
  10. 
  PO, 
  OK. 
  KD,DP 
  

   are 
  four 
  equal 
  bars 
  jointed 
  together 
  at 
  their 
  extremities 
  ; 
  PB, 
  KB 
  are 
  

   two 
  bars 
  also 
  equal, 
  but 
  unequal 
  to 
  the 
  four 
  others 
  ; 
  they 
  are 
  jointed 
  to 
  

   the 
  others 
  at 
  P 
  and 
  K 
  and 
  to 
  a 
  fixed 
  pivot 
  at 
  B. 
  It 
  is 
  then 
  easily 
  seen 
  

   that, 
  however 
  this 
  linkage* 
  is 
  deformed, 
  B, 
  0, 
  D 
  remain 
  in 
  a 
  straight 
  

   line, 
  and 
  the 
  product 
  BO, 
  BD 
  is 
  constant. 
  Thus 
  if 
  D 
  be 
  made, 
  by 
  

   means 
  of 
  the 
  bar 
  A 
  D 
  jointed 
  to 
  the 
  fixed 
  point 
  A, 
  whose 
  distance 
  from 
  

   B 
  equals 
  A 
  D, 
  to 
  describe 
  a 
  circle 
  through 
  B, 
  the 
  point 
  will 
  describe 
  

   the 
  inverse 
  of 
  the 
  circle 
  — 
  that 
  is, 
  the 
  straight 
  line 
  OL 
  perpendicular 
  

   toBA. 
  

  

  Mr. 
  Hart's 
  apparatus 
  is 
  shown 
  in 
  fig. 
  15. 
  For 
  the 
  six 
  bars 
  BP, 
  BK, 
  

   O 
  P, 
  P 
  D, 
  D 
  K, 
  K 
  O 
  of 
  M. 
  Peaucellier 
  in 
  fig. 
  10 
  he 
  substitutes 
  the 
  

   four 
  bars 
  

  

  BC=B'C, 
  CD 
  = 
  C'D', 
  

  

  and 
  takes 
  three 
  points, 
  P, 
  0, 
  Y, 
  on 
  a 
  Hue 
  parallel 
  to 
  C 
  C 
  ; 
  these 
  points, 
  

   however 
  the 
  linkage 
  be 
  deformed, 
  lie 
  in 
  a 
  straight 
  line, 
  and 
  the 
  product 
  

   P 
  Y, 
  P 
  is 
  always 
  constant. 
  

  

  Thus 
  Y 
  being 
  made, 
  by 
  the 
  bar 
  Y 
  IT 
  equal 
  to 
  P 
  IT 
  and 
  pivoted 
  at 
  U, 
  to 
  

   describe 
  a 
  circle 
  passing 
  through 
  the 
  fixed 
  point 
  P, 
  as 
  in 
  the 
  case 
  of 
  

   M. 
  Peaucellier's 
  linkwork, 
  O 
  describes 
  the 
  straight 
  line 
  L 
  perpendi- 
  

   cular 
  to 
  P 
  U. 
  

  

  A 
  passage 
  in 
  a 
  lecture 
  on 
  M. 
  Peaucellier's 
  discovery 
  delivered 
  by 
  Pro- 
  

   fessor 
  Sylvester 
  at 
  the 
  Boyal 
  Institution, 
  in 
  which 
  he 
  pointed 
  out 
  that 
  

   there 
  might 
  be'other 
  solutions, 
  led 
  me 
  to 
  investigate 
  the 
  subject 
  further 
  ; 
  

   and 
  I 
  succeeded 
  in 
  obtaining 
  certain 
  7-bar 
  linkworks 
  producing 
  recti- 
  

   linear 
  motion, 
  depending 
  on 
  two 
  bars 
  being 
  made 
  to 
  make 
  equal 
  variable 
  

   augles 
  in 
  opposite 
  directions 
  with 
  a 
  third 
  bar. 
  These 
  results 
  were 
  de- 
  

   scribed 
  in 
  a 
  paper 
  published 
  in 
  the 
  ' 
  Messenger 
  of 
  Mathematics 
  ' 
  of 
  De- 
  

   cember 
  1874; 
  they 
  are 
  shown 
  in 
  figs. 
  6, 
  12, 
  13, 
  14 
  of 
  this 
  paper, 
  and 
  

   will 
  be 
  further 
  referred 
  to. 
  

  

  Further 
  investigation 
  led 
  me 
  to 
  the 
  discovery 
  that 
  all 
  these 
  linkworks 
  

   depended 
  for 
  their 
  production 
  of 
  straight 
  lines 
  on 
  an 
  exceedingly 
  simple 
  

   and 
  obvious 
  property 
  of 
  any 
  quadrilateral 
  whose 
  sides 
  are 
  of 
  constant 
  

   length. 
  The 
  observation 
  of 
  this 
  property 
  at 
  once 
  led 
  to 
  the 
  discovery 
  of 
  

   a 
  large 
  number 
  of 
  new 
  7-bar 
  linkworks, 
  of 
  which 
  M. 
  Peaucellier's, 
  Mr. 
  

   Hart's, 
  and 
  those 
  previously 
  discovered 
  by 
  myself 
  proved 
  to 
  be 
  particular 
  

   cases, 
  the 
  inversion 
  property 
  of 
  the 
  two 
  former 
  being, 
  so 
  to 
  say, 
  acci- 
  

   dental. 
  

  

  It 
  is 
  the 
  object 
  of 
  this 
  paper 
  to 
  point 
  out 
  this 
  property, 
  and 
  how 
  it 
  

   may 
  be 
  taken 
  advantage 
  of 
  in 
  the 
  construction 
  of 
  a 
  number 
  of 
  7-bar 
  

   straight-line-producing 
  linkworks 
  . 
  

  

  * 
  Professor 
  Sylvester 
  has 
  employed 
  this 
  term 
  to 
  mean 
  a 
  network 
  composed 
  of 
  an 
  

   even 
  number 
  of 
  bars. 
  When 
  one 
  bar 
  is 
  fixed, 
  so 
  that 
  its 
  joints 
  become 
  fixed 
  pivots, 
  

   the 
  system 
  is 
  termed 
  a 
  linkwork. 
  

  

  