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  IX. 
  — 
  On 
  the 
  Power 
  and 
  Economy 
  of 
  Single-Acting 
  Expansive 
  Steam-Engines, 
  

   being 
  a 
  Supplement 
  to 
  the 
  Fourth 
  Section 
  of 
  a 
  Paper 
  On 
  the 
  Mechanical 
  Action 
  

   of 
  Heat. 
  By 
  William 
  John 
  Macquorn 
  Rankine, 
  Civil 
  Engineer, 
  F.R.S.E., 
  

   F.R.S.S.A., 
  &c. 
  

  

  (Read 
  21st 
  April 
  1851.) 
  

  

  (34.) 
  The 
  objects 
  of 
  this 
  paper 
  are 
  twofold 
  : 
  First, 
  To 
  compare 
  the 
  results 
  of 
  

   the 
  formulae 
  and 
  tables 
  relative 
  to 
  the 
  power 
  of 
  the 
  steam-engine, 
  which 
  have 
  

   been 
  deduced 
  from 
  the 
  Dynamical 
  Theory 
  of 
  Heat, 
  with 
  those 
  of 
  experiments 
  on 
  

   the 
  actual 
  duty 
  of 
  a 
  large 
  Cornish 
  engine 
  at 
  various 
  rates 
  of 
  expansion 
  ; 
  and, 
  

   Secondly, 
  To 
  investigate 
  and 
  explain 
  the 
  method 
  of 
  determining 
  the 
  rate 
  of 
  expan- 
  

   sion, 
  and, 
  consequently, 
  the 
  dimensions 
  and 
  proportions 
  of 
  a 
  Cornish 
  engine, 
  

   which, 
  with 
  a 
  given 
  maximum 
  pressure 
  of 
  steam 
  in 
  the 
  cylinder, 
  at 
  a 
  given 
  

   velocity, 
  shall 
  perform 
  a 
  given 
  amount 
  of 
  work 
  at 
  the 
  least 
  possible 
  pecuniary 
  

   cost, 
  taking 
  into 
  account 
  the 
  expense 
  of 
  fuel, 
  and 
  the 
  interest 
  of 
  the 
  capital 
  re- 
  

   quired 
  for 
  the 
  construction 
  of 
  the 
  engine. 
  

  

  This 
  problem 
  is 
  solved 
  with 
  the 
  aid 
  of 
  the 
  tables 
  already 
  printed, 
  by 
  drawing 
  

   two 
  straight 
  lines 
  on 
  a 
  diagram 
  annexed 
  to 
  this 
  paper. 
  

  

  The 
  merit 
  of 
  first 
  proposing 
  the 
  question 
  of 
  the 
  economy 
  of 
  expansive 
  en- 
  

   gines 
  in 
  this 
  definite 
  shape, 
  belongs, 
  I 
  believe, 
  to 
  the 
  Artizan 
  Club, 
  who 
  have 
  

   offered 
  premiums 
  for 
  its 
  solution; 
  having 
  done 
  so 
  (to 
  use 
  their 
  own 
  words), 
  

   " 
  with 
  a 
  view 
  to 
  enable 
  those 
  who, 
  from 
  their 
  position, 
  cannot 
  take 
  part 
  in 
  the 
  

   discussions 
  of 
  the 
  various 
  scientific 
  Societies 
  to 
  give 
  the 
  profession 
  the 
  benefit 
  of 
  

   their 
  studies 
  and 
  experience." 
  The 
  5th 
  of 
  April 
  is 
  the 
  latest 
  day 
  fixed 
  by 
  them 
  

   for 
  receiving 
  papers 
  ; 
  and 
  as 
  this 
  communication 
  cannot 
  possibly 
  be 
  read 
  to 
  a 
  

   meeting 
  before 
  the 
  7th 
  April, 
  nor 
  published 
  until 
  some 
  months 
  afterwards, 
  I 
  trust 
  

   I 
  may 
  feel 
  confident 
  that 
  it 
  will 
  not 
  be 
  considered 
  as 
  interfering 
  with 
  their 
  design. 
  

  

  Formula? 
  applicable 
  to 
  the 
  Cornish 
  Engine. 
  

  

  (35.) 
  The 
  equations 
  of 
  motion 
  of 
  the 
  steam-engine, 
  in 
  this 
  and 
  the 
  original 
  

   paper, 
  are 
  the 
  same 
  in 
  their 
  general 
  form 
  with 
  those 
  of 
  M. 
  de 
  Pambour. 
  The 
  

   differences 
  consist 
  in 
  the 
  expression^ 
  for 
  the 
  pressure 
  and 
  volume 
  of 
  steam, 
  and 
  

   for 
  the 
  mechanical 
  effect 
  of 
  its 
  expansion 
  ; 
  the 
  former 
  of 
  which 
  were 
  deduced 
  

   from 
  a 
  formula 
  suggested 
  by 
  peculiar 
  hypothetical 
  views, 
  and 
  the 
  latter 
  from 
  the 
  

   dynamical 
  theory 
  of 
  heat. 
  

  

  Those 
  equations 
  are 
  Nos. 
  (50) 
  and 
  (51) 
  of 
  the 
  original 
  paper. 
  I 
  shall 
  now 
  

  

  VOL. 
  XX. 
  PART 
  II. 
  3 
  F 
  

  

  