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3fofm  Castman  Clarke  Htfirarp 

GIVEN  TO 

BOSTON    UNIVERSITY 

COLLEGE   OF    LIBERAL   ARTS 

BY 

MRS.  JOHN   EASTMAN   CLARKE 

MEMORY  OF  HER   HUSBAND 
JULY,  19,4 


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-*«•*  : 


MISCELLANEOUS 
SCIENTIFIC   PAPERS. 


WORKS  BY  PROFESSOR  RANKINE. 


In  crown  8uo,  cloth. 

I.  A  MANUAL  OF  APPLIED  MECHANICS.     Ninth  Edition,  L2s.  6d. 

II.  A  MANUAL  OF  CIVIL  ENGINEERING.     Thirteenth  Edition,  16a. 

III.  A    MANUAL    OF    MACHINERY    AND    MILLWORK.       Fourth 

Edition,  L2s.  6d. 

IV.  A  MANUAL  oF  THE  STEAM-ENGINE  AND  OTHEE    PRIME 

MOVERS.     Ninth  EdMm,  12s.  6d. 

Y.  USEFUL    RULES    AND    TABLES    for    Engineers,    Surveyors,    and 
Others.     Fifth  Edition 

VI.  A  MECHANICAL  TEXT-BOOK;   or,  Introduction  to  the  Study  of 
Mechanics.      By  Pkop.  RanKINE  and   E.  F.  Bambeb,  C.E.      Second 

Edition,  9s. 


Chables  Griffin  and  Company,  London. 


College  of  Liberal  Arts 

Boston  University 

MISCELLANEOUS 

SCIENTIFIC   PAPERS: 


W.  .1.  MACQUORN  EANKINE,  (J.E..  LL.I)..  F.R.S.,V 

LATE    REGIUS    PROFESSOR   OF    CIVIL    ENGINEERING    AND    MECHANICS    IN    THE 
UNIVERSITY   OF   GLASGOW. 


FROM  THE  TRANSACTIONS  AND  PROCEEDINGS  OP  THE  ROYAL  AND  OTHER  SCIENTIFIC 
AND  PHILOSOPHICAL  SOCIETIES,  AND  THE  SCIENTIFIC  JOURNALS. 


WITH 

-A.     nVCEJMIOXK,     OIF1     THE     ATJTHOB 

By    P.    G.    TAIT,    M.A.,  **' 

PROFESSOR  OF  NATURAL   PHILOSOPHY   IN   THE   UNIVERSITY   OF   EDINBURGH. 
EDITED    BY 

W.    J.    MILLAR,    C.E., 

SECRETARY   TO   THE  INST.    OF    ENGINEERS   AND    SHIPBUILDERS    IN    SCOTLAND. 


titfe  |0thwit,  f  h\Us,  attbf  iiapams. 


LONDON: 
CHARLES    GRIFFIN    AND    COMPANY, 

stationers'    hall    court. 

MDCCCLXXXI. 


Sc\*f\C*   ^      ^SAV\S 


GLASGOW 
PRINTED    BY    BELL    AND    BAIN,    41    MITCHELL    STKEET 


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PREFACE. 


The  republication  of  a  Selection  of  the  Papers  of  the  late  Professor 
Macquorn  Rankine  was  originally  projected  by  several  of  his  personal 
friends  shortly  after  his  decease,  the  object  being  to  combine,  in  a 
suitable  volume-form,  papers  which  were  to  be  found  only  in  the  Records 
of  Scientific  Societies,  and  in  the  Scientific  and  Engineering  Journals,  and 
thus  to  present  to  the  many  admirers  of  the  talented  author  a  memorial 
of  his  great  worth  and  ability.  \ 

Introductory  to  the  Selection  of  Papers  now  published  is  an  exhaustive 
Memoir  by  Professor  Tait,  who  kindly  consented  to  embody  \n  this 
form  the  main  features  of  Rankine's  life,  together  with  his  recollections 
of  one  with  whom  he  had  been  intimately  associated  as  a  fellow- worker. 
There  remains,  therefore,  for  the  Editor  merely  to  point  out,  briefly,  the 
principle  that  has  guided  him  in  making  the  selection  from  the  papers 
placed  in  his  hands  for  the  purpose  of  republication. 

The  object,  then,  kept  in  view  has  been  the  preservation  of  such  papers 
as  are  most  characteristic  of  their  author  in  his  capacity  of  a  scientific 
and  mathematical  inquirer.  Professor  Rankine  was  not,  nor  did  he  claim 
to  be,  a  popular  writer;  his  command  of  mathematical  expression  was 
such,  that  he  naturally  embodied  his  reasoning  and  conclusions  in 
symbols.  All  his  writings,  however,  are  marked  by  a  power  of  state- 
ment so  clear  and  logical,  that  the  reader,  even  should  he  fail  entirely 
to  follow  the  demonstrations,  cannot  but  be  benefited  in  the  attempt  to 
master  them. 

Besides  the  papers  in  the  present  Collection,  Professor  Rankine  con- 
tributed many  others  to  Scientific  and  Engineering  Societies  and  Journals. 
A  number  of  these,  from  their  nature,  (descriptions  of  works  and  of 
machines,)  were  but  of  passing  interest.     The  papers  now  published  are 

b 

BOSTON  UNIVERSITY  LIBRARIES 


vi  PREFACE. 

of  permanent  value,  and  deal  rather  with  general  scientific  principles  and 
their  applications  to  practice. 

The  leading  characteristics  of  Professor  Rankine's  writings  are  too 
well-known  to  require  comment  here.  One  special  feature,  however,  in 
his  method  of  treatment,  may  be  pointed  out — viz.,  the  carefully  arranged 
division  of  the  subject  into  sections.  Starting  with  a  general  statement  of 
the  object  of  the  paper,  he  advances  by  degrees  in  the  argument,  giving  full 
reference  to  the  subordinate  parts  of  the  paper,  which  arc  marked  by 
numerals  or  letters  to  distinguish  them,  thus  progressing  with  logical  pre- 
cision until  the  conclusion  is  reached.  Another  noteworthy  feature  may 
also  be  referred  to — viz.,  that  of  the  introduction  of  new  and  suitable  words 
or  phrases,  proposed  by  the  author,  to  convey  more  dearly  his  meaning. 
Lastly,  we  observe  throughout  all  Professor  Rankine's  writings  the  most 
minute  accuracy  of  statement,  and  the  mosl  scrupulously  honourable  care 
to  give  to  all  fellow-workers  in  the  same  held  with  himself  their  just  due, 
whilst  pointing  out  what  he  considers  to  be  original  on  his  own  part. 

The  principal  papers  in  the  volume  are  those  relating  to  Thermo- 
dynamics and  to  Hydrodynamics,  where  such  subjects  as  the  Action  of 
ileat  in  the  Steam-Engine,  and  the  Forms  of  Waves  and  "Water-Lines  of 
Ships,  arc  discussed  at  length — the  scientific  and  mathematical  investi- 
gation of  these  questions  being  perhaps  most  eminently  characteristic  of 
sor  Rankine. 

The  papers  have  been  grouped  into  three  Divisions,  so  as  to  bring  as 
nearly  as  possible  kindred  subjects  together;  and  in  every  case  the 
name  of  the  Society  or  Journal  through  which  the  paper  was  originally 
brought  forward,  together  with  the  date  of  its  publication,  has  been 
given. 

It  need  hardly  be  added,  that  the  papers  appear  without  change  of  any 
kind.  In  their  present  form  they  stand  precisely  as  they  finally  left  their 
author's  pen;  and  no  pains  have  been  spared  to  ensure  perfect  accuracy 
in  the  reproduction. 

The  Editor,  like  many  others,  has  a  grateful  remembrance  of  Professor 
Rankine,  having  enjoyed  the  great  privilege  of  being  one  of  his  students 
in  the  old  College  of  Glasgow,  and  having  afterwards  had  the  advantage 
of  his  friendship  in  many  ways.  Rankine's  lectures,  although  simpler  than 
his  text-books,  were  marked  by  the  same  clearness  of  arrangement,  and 
were  enforced  by  his  distinct  and  vigorous  enunciation,  and  admirably 
illustrated  by  carefully  prepared  diagrams.  As  chairman  of  the  meetings 
of  the  societies  of  which  he  was  president,  his  methodical  habits  and 
business  qualifications  were  of  marked  service.  These  qualities  were  also 
evidenced  in  the  drawing  up  of  reports  in  committee;  one  of  the  last 
services  which  he  rendered  to  engineering  science  being  in  connection 
with  an  experimental  inquiry  on  safety  valves  on  behalf  of  the  Institu- 


PREFACE.  Vli 

tion  of  Engineers  and  Shipbuilders  in  Scotland — his  decease,  unfortunately, 
occurring  before  the  completion  of  the  experiments. 

The  Papers  classed  under  Part  I.  relate  more  or  less  to  Temperature 
and  Elasticity.  The  First,  originally  published  in  1849,  gives  an 
approximate  equation  for  the  elasticity  of  vapour  in  contact  with  its 
liquid,  (this  equation  appears  to  have  been  one  of  the  results  obtained 
by  Professor  Eankine  whilst  investigating  the  molecular  constitution  of 
matter,)  and  concludes  with  the  statement  of  a  proposition  borne  out  by 
experiment,  which  "  may  be  safely  and  usefully  applied  in  practice." 

In  the  Second  Paper,  published  in  the  same  year,  is  given  a  formula 
for  calculating  the  expansion  of  liquids  by  heat.  This  formula,  the 
author  states,  he  had  found  useful  whilst  considering  the  comparative 
volumes  of  liquids  at  various  temperatures. 

The  Third  Paper,  on  "  The  Centrifugal  Theoiy  of  Elasticity,  as  applied 
to  Gases  and  Vapours,"  published  in  1851,  shows  how  the  laws  of  the 
pressure  and  expansion  of  gases  may  be  deduced  from  the  hypothesis  of 
molecular  vortices.  The  investigation  was  begun  in  1842,  but  was  laid  aside 
on  account  of  the  want  of  experimental  data  ;  it  was  again  resumed  after  the 
publication  of  the  results  of  M.  Kegnault's  experiments  on  gases  and 
vapours,  and  laid  before  the  Royal  Society  of  Edinburgh  in  February,  1850. 
The  hypothesis  is  defined  to  be  "  that  which  assumes  that  each  atom  of 
matter  consists  of  a  nucleus  or  central  point  enveloped  by  an  clastic  atmosphere, 
which  is  retained  in  its  position  by  attractive  forces  ;  and  that  the  elasticity  due 
to  heat  arises  from  the  centrifugal  force  of  these  atmospheres  revolving  or 
oscillating  about  their  nuclei  or  central  points."  After  showing  that  some- 
what similar  ideas  had  been  entertained  by  philosophers  at  different 
times,  there  follows  a  supposition  which,  he  says,  "  so  far  as  I  am  aware, 
is  peculiar  to  my  own  researches.  It  is  this,  that  the  vibration  which, 
according  to  the,  undulatory  hypothesis,  constitutes  radiant  light  and  heat,  is  a 
motion  of  the  atomic  nuclei  or  centres,  and  is  propagated  by  means  of  their 
mutual  attractions  and  repulsions.'"  Tables  are  given  showing  the  closeness 
of  agreement  between  the  formula?  made  use  of,  and  the  experimental 
results. 

In  the  Fourth  Paper  we  have  an  extension  of  the  preceding  one ;  by 
means  of  a  fresh  investigation  the  complete  applicability  of  the  hypothesis 
of  molecular  vortices  to  all  substances  in  all  conditions  is  shown,  and 
this  demonstration  is  followed  by  the  deduction  from  that  hypothesis  of 
the  law  of  the  equivalence  of  heat  and  power. 

The  Fifth  Paper  deals  with  the  Laws  of  Elasticity  in  reference  to 
the  strength  of  structures — the  relations  between  pressures  and  strains ; 
it  shows  how  the  Laws  of  Elasticity  are  simplified  by  adopting  the 
supposition    of   atomic  centres   of   force,  but  also  points  out  that  this 


nil  PREFACE. 

supposition   requires  modification,  and   by  means  of   the   hypothesis  of 
molecular  \  orl  ici    Bimplifie    I  he  i  ion  of  the 

In  the  Sixth  Paper  the  distinction  between  Strain  and  Stress  is  made 
clear,  and  a  nomenclature  adopted  (from  Greek  equivalents)  di  icriptive  of 
their  relations.     Cr   talline    fo]  a  idered,  to  pith   their 

ad  ion  on  li; 

The  Seventh  Paper  treats  of  "The  Vibrations  of  Plane-Polari  ed  Light." 
The  principles  laid  down    in   the   paper  are  shown   to   be   incompatible 
with  the  "idea  of  a  luminiferous  ether  enveloping  ponderable  particles;" 
much  as  thai  "the  Luminiferous  medium  is  a  system  of  atomic  nuclei 
or  centres  of  force,  whose  •  give  form  to  matter;    while  the 

atmo        n      by  which   they  are  surrounded  give  of  themselves  merelj 
extension." 

In  the  Eighth   Paper  an  atl   mpl   is  made  to  diminish  the  difficul 
attending    the    undulatory   theory    of    light,  hy   proposing   a   theory   of 
oscillations  round  axes,  in  tead  of  the  theory  of  vibratii 

In  ill"  Ninth  Paper,  we  have  :i  mathematical  investigation  into  the 
relations  existing  between  the  velocity- of  sound  in  tances  and 

;  he  elastic]      of  the  m 

Pari    II.  relates  principally  to  Energy    and  the   Mechanical  Action  of 

The  first  Paper  of  this  division   |  entitli  d  "  The  Etecon- 

ition  of  the    Mechanical    :  Universe"     a  remarkable 

speculation,  laid  before  the  Briti  iation  al    Belfast,  and  published 

in    1852,  alluded   to  by    Professor   Jevons    hi    bis    Principles  of  Science, 

and    acknowledi I    by  Sir  Wm.  Thomson   f  ;.   characteristic  of 

Rankine.      This    paper,  after   referring  to  all  experimental    evidenci 
being   in    favour   of  the   doctrine   of  the   mutual  convertibility  of   the 
different  kind  i  of  I  he  pi  of  the  universe,  ami  of  the  tendency 

irds  a  uniform  temj  of  matte]  a  to  point  out  how  it  is 

eivable  that  ultimately    the    diffused    energy  may    he  gathered    into 
foci,  and  renewed  chemical  power  produced  from  the  now  inert 

compounds  when  passing  through  Hie  intense  heat  of  these  foci. 

Tie'  next  Paper  deals  with  the  Law  of  the  Transformation  of  Enei 
and   the  ratio  <>f  Work   done  {,,  Energy  expended  in  various  forms  of 
engines;  and  is  followed  by  a  comprehensive  paper  entitled  " Outlines  of 
Science  of  Energetics,"  treating  of  the  laws  of  physical  phenomena — 

the  Sri, 'lice    of    Mechanics  ("  the  Onlj    example    yet    existing   of  a    complete 

physical  theory") — the  use  of  hypotheses— definitions  of  Energy,  Work,&c, 
and  Efficiency  of  Engines. 

The  Thirteenth  Paper  is  mainly  descriptive  of  the  use  of  the  term  Energy, 
and  gives  Professor  Rankine's  reason  for  his  introduction  of  the  phra  c 
"  I'ui,  ntial  Energy." 


PREFACE.  ix 

Xo.  XIV.,  with  its  supplements  XV.-XIX.,  on  "The  Mechanical 
A.ction  of  Heat,"  is  an  important  contribution  of  considerable  length 
to  the  Royal  Society  of  Edinburgh,  extending  over  the  years  1850-1853. 
It  is  based  upon  the  hypothesis  of  molecular  vortices,  and  relates  chiefly 
to  the  "  mutual  conversion  of  heat  and  mechanical  power,  by  means  of 
the  expansion  and  contraction  of  gases  and  vapours." 

In  the  Twentieth  Paper  the  various  conditions  existing  in  a  heated 
substance  are  shown  geometrically  by  curves,  a  method  adopted  first,  it 
is  stated,  by  James  "Watt  in  his  Steam-engine  Indicator.  The  efficiency 
of  thermodynamic  engines  is  considered,  and  Stirling  and  Ericsson's  hot- 
air  engines  described. 

The  next  Paper  is  on  "Formulae  for  the  Maximum  Pressure  and  Latent 
I  [eat  of  Vapours,"  followed  by  one  on  "  The  Density  of  Steam,"  in  which 
the  general  equation  of  thermodynamics,  stated  in  paper  Xo.  XIV.,  is 
again  given,  to  show  the  connection  existing  between  it  and  the  law  of  the 
density  of  steam. 

In  Paper  Xo.  XXIII.  the  Two  Laws  of  Thermodynamics  are  stated,  and 
it  is  shown  that  the  derivation  of  the  Second  Law  from  steady  molecular 
motion  (e.g.,  in  circular  streams  or  in  circulating  streams  of  any  figure), 
as  given  in  previous  papers,  may  lie  more  simply  effected  than  by  the 
methods  adopted  in  these  papers. 

The  following  Paper,  published  in  The  Engineer  in  1867,  refers  to 
the  want  of  popular  illustrations  of  The  Second  Law  of  Thermodynamics, 
and  explains  the  nature  of  the  two  laws,  with  the  particular  questions  to 
which  they  are  respectively  applicable. 

"The  First  Law  informs  us  that  when  mechanical  work  is  done  by 
means  of  heat,  a  quantity  of  heat  disappears.  ...  To  calculate  this 
disappearance  of  heat,  the  work  done  must  be  sensibly  external,  and  subject 
to  direct  measurement." 

"  The  Second  Law  informs  us  how  to  deduce  the  whole  amount  of  work 
done,  internal  and  external,  from  the  knowledge  which  we  have  of  the 
external  work." 

An  illustration  is  given  by  the  expansion  of  a  perfect  or  sensibly  perfect 
gas  ;  but  it  is  pointed  out  that  it  is  different  when  we  have  to  deal  with 
fluids  in  the  act  of  evaporating,  instanced  by  a  case  showing  where  the 
second  law  is  applicable. 

Paper  Xo.  XXVI.  is  on  "The  Working  of  Steam  in  Compound 
Engines,"  defines  such  engines,  and  states  their  advantages,  with  rules  for 
the  construction  of  indicator-diagrams. 

This  is  followed  by  Papers  on  "The  Theory  of  Explosive  Gas-Engines," 
and  on  "  The  Explosive  Energy  of  Heated  Liquids."  In  the  first  of  these 
it  is  shown  that,  in  calculations  respecting  the  practical  use  of  heat- 
engines,  it    is    convenient    to    use    pressures    and    volumes   rather   than 


X  PREFACE. 

temperatures.     The  mixtures  of  gas  and  air  most  suitable  for  gas-engines 

are  also  given. 

The  Papers  in  Part  III.  relate  to  Wave-Forms,  Propulsion  of  Vessels, 
Stability  of  Structures,  &c. 

The  first  of  the  series,  No.  XXIX.,  which  is  entitled  "  On  the  Exact 
Form  of  Waves  near  the  Surface  of  Deep  Water,"  shows  that  the  form 
of  such  waves  is  trochoidal,  and  states  that  this  form  was  first  pointed 
out  by  Mr.  Scott  Eussell. 

The  next  Paper,  "On  Plane  Water-Lines,"  investigates  the  curves 
suitable  for  the  water-lines  of  a  ship.  Water-line  curves  are  designated 
Neoids,  Cyclogenous  Neoids  (or  water-line  curves  generated  by  circles), 
Oogenous  Neoids  (or  those  generated  by  oval  bodies),  and  Lissoneoids,  or 
waterdines  of  smoothest  gliding.  It  is  noticed  that  although,  from 
lengthened  practice  in  the  art  of  shipbuilding,  the  forms  of  water-lines 
have  attained  a  high  degree  of  excellence,  jet  that  this  is  due  rather 
to  empirical  means,  than  to  a  knowledge  of  general  principles.  The 
system  of  Chapman  is  shown  to  be  wholly  empirical,  consisting  of 
parabolic  forms  ;  and  Mr.  Scott  Russell's  is  instanced  as  the  first  useful 
theory  of  ships'  water-lines,  being  based  on  wave  figures.  The  various 
forms  of  water-lines  are  then  considered  in  reference  to  their  fitness  for 
different  classes  of  vessels. 

Other  two  Papers  follow,  the  first  of  which  is  intended  to  assist  those 
who  are  not  familiar  with  the  higher  mathematics  in  understanding  the 
subject  of  Stream-Lines.  A  stream-line  is  defined  as  the  line  traced  by  a 
particle  in  a  current  of  fluid,  and  an  elementary  method  is  given  for 
determining  circular  stream-lines,  a  subject  mathematically  investigated  in 
the  preceding  paper.  The  other  Paper  is  an  investigation  "  to  determine 
the  relations  which  must  exist  between  the  laws  of  the  elasticity  of  any 
substance,  whether  gaseous,  liquid,  or  solid,  and  those  of  the  wave-like 
propagation  of  a  finite  longitudinal  disturbance  in  that  substance."  A 
Paper  on  "  The  Theoretical  Limit  of  the  Efficiency  of  Pro} tellers  "  follows, 
showing  the  theoretical  limit  of  efficiency  which  improvements  in  pro- 
pellers may  attain ;  states  the  formulae  for  reaction  and  effective  power, 
shows  at  what  relative  velocities  the  propeller  is  most  effective,  and 
compares  the  advantages  of  various  forms  of  propeller,  with  numerical 
examples. 

Paper  No.  XXXIV.,  "  On  the  Design  and  Construction  of  Masonry 
Dams,"  originally  consisted  of  a  report  to  the  municipality  of  Bombay, 
made  in  1870,  in  reference  to  proposed  extensions  of  the  Water- Works 
there,  and  was  afterwards  published  in  The  Engineer. 

This  Paper  enters  into  the  question  of  stability  of  structures,  showing 
the  best  and  most  economical  form  which  a  high  masonry  reservoir  wall 


PREFACE.  Xi 

should  have,  and  is  of  value  to  the  civil  engineer  when  proposing  to 
adopt  a  masonry  or  concrete  wall,  instead  of  an  ordinary  embankment,  for 
reservoir  purposes. 

Papers  Xos.  XXXV.  and  XXXVI.  are  extensions  of  methods  adopted 
by  Professor  Eankine  in  his  Manual  of  Applied  Mechanics,  in  connection 
with  the  stability  of  structures  of  various  figures,  such  as  Frames  and 
Arches ;  and  the  last  paper  of  the  series,  Xo.  XXXVII.,  is  a  mathematical 
demonstration  of  a  property  of  certain  curves,  bearing  on  the  forms  of  the 
slopes  of  wave-crests. 

In  conclusion,  the  Editor  desires  to  acknowledge  the  courtesy  shown 
by  the  executive  officers  of  the  various  Societies  and  Journals  to  which 
the  papers,  thus  brought  together,  were  originally  contributed:  and  to 
express  his  thanks — not  only  for  the  permission,  readily  accorded,  to 
republish — but  also  for  the  kindness  which  supplied,  in  many  instances, 
copies  of  the  Papers  selected. 

W.  J.  MILLAP. 


Glasgow, 

October,  18S0. 


CONTENTS. 


PART     I. 


PAPERS    RELATING     TO     TEMPERATURE,    ELASTICITY,    AND 
EXPANSION  OF  VAPOURS,  LIQUIDS,  AND  SOLIDS. 

I'AGE 

I.  ON  AN  EQUATION  BETWEEN  THE  TEMPERATURE  AND 
THE  MAXIMUM  ELASTICITY  OF  STEAM  AND  OTHEE 
VAPOURS. 

Table  of  Vapour  of  Water— Formulae  for  calculating    Temperature  and 
Pressure — Values  of  Constants— Tables  of  Vapour  of  Alcohol,  Ether, 
Turpentine,  Petroleum,  and  Mercury, 1 — I'- 
ll. ON  A  FORMULA  FOR  CALCULATING  THE  EXPANSION  OF 
LIQUIDS  BY  HEAT. 

Table  of  Constants — Tables  of  Expansion  of  Water,  Mercury,  Alcohol,  and 
Sulphuret  of  Carbon,    ...........        13 — 15 

III.  ON    THE    CENTRIFUGAL    THEORY    OF    ELASTICITY,    AS 

APPLIED  TO  GASES  AND  VAPOURS. 

Hypothesis  of  Molecular  Vortices— Relations  between  the  Heat  and 
Elasticity  of  a  Gaseous  Body— Temperature— Specific  Heat— Coefficients 
of  Elasticity  and  Dilatation  of  Gases — Table  of  Coefficients  of  Dilatation- 
Elasticity  of  Vapour,  and  Tables  of  Constants  for  the  same— Mixtures 
of  Gases  and  Vapours, 16— 4S 

IV.  ON  THE  CENTRIFUGAL  THEORY  OF  ELASTICITY  AND  ITS 

CONNECTION  WITH  THE  THEORY  OF  HEAT. 

Relations  between  Heat  and  Expansive  Pressure— Hypothesis  of  Molecular 
Vortices— Elastic  Pressures— Temperature  and  Specific  Heat— Heat  and 
Expansive  Power— Latent  Heat—  Joule's  Law— Carnot's  Law,         .        .        49—66 

V.  LAWS  OF  THE  ELASTICITY  OF  SOLID  BODIES. 

Science  of  Elasticity  as  applied  to  Strength  of  Structures— Strains  and 
Molecular  Pressures— Homogeneous  Bodies— Eigidity— Hypothesis  of 
Atomic  Centres— Hypothesis  of  Molecular  Vortices— Coefficients  of  Pli- 
ability, Extensibility,  and  Compressibility— Modulus  of  Elasticity- 
Tables  of  Coefficients  of  Elasticity— Rigidity,  Extensibility,  and  Com- 
pressibility—Application of  Method  of  Virtual  Velocities  to  the  Theory 
of  Elasticity— Proof  of  the  Laws  of  Elasticity  by  the  Method  of  Virtual 
Velocities, 67-118 


XIV  CONTENTS. 

VI.  ON  AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS. 

PAGE 

Definition   of    Axes    of    Elasticity — Strain — Stress   -Potential    Energy — 

Coefficients  of  Elasticity—  Symmetry— Polarised  J  ,ight,      ....     110     149 


VII.  ON  THE  VIBRATIONS  OE  PLANE-POLARISED  LIGHT. 

Theories — Hypothesis  of  Molecular  Vortices    Experiments  on  Light,  .     150—155 

VIII.  GENERAL   VIEW    OE    AN    OSCILLATORY    THEORY    OE 
LIGHT. 

Various  Hypotheses — Polarised  Light— Hypothesis  of  Oscillations — Diffrac- 
tion— Wave-Surface  in  Crystalline  Bodies  Reflexion — Refraction — 
Dispersion, 150 — 107 

IX.  ON   THE    VELO(  [TY    OF    SOUND    IX    LIQUID    AXI)    SOLID 

BODIES  OF  LIMITED  DIMENSIONS,  ESPECIALLY  ALONG 

PRISMATIC  MASSES  OF  LIQUID. 

Application  to    Elasticity  of  Materials— Vibratory  Movement  in  Homo- 
ueous  Bodies    Transmission  of  a  Definite  Musical  Tone— Propagation 
of  Sound, 168—199 


PART     II. 


PAPERS    RELATING    TO    ENERGY  AND    ITS    TRANSFORMATIONS, 

THERMODYNAMICS,  MECHANICAL  ACTION  OF  HEAT 

TN  THE  STEAM-ENGINE,  .\r. 

X.    ON     THE     RECONOENTRATION     OF     THE     MECHANICAL 
ENERGY  OF  THE  UNIVERSE. 

Convertibility  of  the  different  kinds  of  Physical  Energy  —  Interstellar 
Medium— Diffusion,  an  1  probable  Reconcentration  of  Radiant  Heat,        .    200—202 

XL  ON  THE  GENERAL  LAW  OF  THE  TRANSFORMATION  OF 
ENERGY. 

Actual  or  Sensible  Energy— Potential  or  Latent  Energy— Conservation  of 
Energy— Transformation  of  Energy, 203—208 

XII.  OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS. 

Physical  Theory—  Science  of  Mechanics— Mechanical  Hypotheses— Science 
of  Energetics— Substance— Property— Mass— Accident— Effort— Work- 
Equivalence  of  Energy  and  Work— Equality  of  Action  and  Reaction- 
Potential  Equilibrium— Transformation  of  Energy — Rate  of  Transforma- 
tion— Metabatic  Function — Carnot's  Function— Efficiency  of  Engines- 
Diffusion  of  Actual  Energy—  "Frictional  Phenomena  "—Measurement  of 
Time, 209—228 


CONTENTS.  XV 

XIII.  ON  THE  PHRASE   "  POTENTIAL  ENERGY,"   AND  ON  THE 

DEFINITIONS  OF  PHYSICAL  QUANTITIES. 

PAGE 

Potential  Energy— Sir  John  HerschePs  Views— Newton's  Method— Force — 
The  word  "Energy"  substituted  by  Dr.  Thomas  Young  for  Vis-viva— 
Extension  of  Application  by  Sir  William  Thomson— Qualification  of  the 
Noun  "Energy"  by  the  Adjectives  "Actual"  and  "Potential,"  pro- 
posed by  Eankine,  and  adopted  by  others— Carnot's  "Force  Vivi 
Virtudlc  "—Definitions  of  the  Measurement  of  Time,  Force,  and  Mass,     .     229—233 

XIV.  ON  THE   MECHANICAL  ACTION  OF  HEAT,  ESPECIALLY 

IN  GASES  AND  VAPOURS. 

Hypothesis  of  Molecular  Vortices— Mutual  Conversion  of  Heat  and  Expan- 
sive Power— Joule's  Experiments— Latent  Heat— Conservation  of  Vis- 
viva—  Carnot's  Theory  of  Heat— Specific  Heat— Thermal  Unit— Total 
Heat  of  Evaporation— Power  of  the  Steam-Engine— Pambour's  Principle 
— Numerical  Examples  from  Cornish  Engines — Efficiency  of  Engines- 
Tables  of  Pressure  and  Volume  of  Steam,     234-284 

XV.  NOTE   AS   TO   THE  DYNAMICAL   EQUIVALENT   OF    TEM- 

PERATURE   IN    LIQUID    WATER,    AND    THE    SPECIFIC 
HEAT  OF  ATMOSPHERIC  AIR  AND  STEAM  :  Being  a  Supple- 
ment to  a  Paper  On  the  Mechanical  Action  of  Heat. 
Joule's  Experiments— Specific  Heat  of  Air  and  Steam,  ....     285—287 

XVI.  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. 
Cornish  Engines — Wicksteed's  Experiments, 288—  299 

XVII.  ON  THE  ECONOMY  OF  HEAT  IN  EXPANSIVE  MACHINES. 

Carnot's  Law— Clausius  and  Thomson's  Investigations— Action  of  Steam  in 
Engine, 300—306 

XVIII.  ON    THE    ABSOLUTE    ZERO    OF    THE    PERFECT    GAS 
THERMOMETER. 

Measurement  of  Temperature— Absolute  Zero  of  Temperature,     .        .        .     307-309 

XIX.  ON  THE  MECHANICAL  ACTION  OF  HEAT. 

Properties  of  Expansive  Heat— Heat  Potential— Properties  of  Temperature 
—Hypothesis  of  Molecular  Vortices— Sir  Humphry  Davy's  Researches- 
Thermic  Phenomena  of  Currents  of  Elastic  Fluids, 310—338 

XX.  ON    THE     GEOMETRICAL     REPRESENTATION     OF     THE 

EXPANSIVE    ACTION    OF    HEAT,    AND    THE    THEORY 
OF  THERMODYNAMIC  ENGINES. 

Introduction  of  Indicator-Diagram  and  Diagrams  of  Energy— Isothermal 
Curves— Curves  of  No  Transmission  of  Heat— Total  Actual  Heat— Latent 
Heat  of  Expansion— Law  of  the  Transformation  of  Energy— Thermo- 
dynamic Functions— Curves  of  Free  Expansion— Thomson  and  Joule's 
Experiments— Velocity  of  Sound— Efficiency  of  Thermodynamic  Engines 


XVI  CONTENTS. 

PAGB 

— Use  of  Economiser  or  Regenerator— Stirling  and  Ericsson's  Air-Engines 
Hypothesis  of  Molecular  "Vortices— Relation  between  Temperature  and 
Actual  Heat — Efficiency  of  Air-Engines-  Liquefaction  of  Vapour  by  Ex- 
pansion under  Pressure — Efficiency  of  a  Vapour-Engine — Heat  in  a 
I  'ound  of  Coal — Wire-drawn  Steam — Composite  Vapour-Engines-  <  iurves 
of  Eree  Expansion  for  Nascent  Vapour-  -Total  Heat  of  Evaporation — 
Approximate  Law  for  a  Vapour  which  is  a  Perfect  Gas,    ....    339—409 

XXI.  ON    FORMULAE    FOB    THE    MAXIMUM    PRESSURE    AND 

LATENT  HEAT  OF  VAPOURS. 

Elasticity  of  Vapours— Regnault's  Experiments— Comparison  of  Formulae 
with  Experiments, 410—416 

XXII.  ON  THE  DENSITY  OF  STEAM. 

General  Equation  of  Thermodynamics,  and  its  Application  to  the  Latent 
Heat  and  Density  of  Steam     Fairbairn  and  Tate's  Experiments,      .        .    417—426 

XXIII.  ON  THE  SECOND  LAW  OF  THERMODYNAMICS. 

First  Law  Defined  Second  Law  Defined  Absolute  Temperature — Steady 
Motion — Metamorphic  Function, 427 — 431 

XXIV.  ON  THE  WANT  OF  POPULAB  [LLUSTRATIONS  OF  THE 
SECOND  LAW   OF  THERMODYNAMICS. 

First  and  Second  Laws— Exl   rnaland  Internal  Wori  rofaPer- 

fect  Heat  Engine— Hypothe    i  of  Molecular  Vortices,       ....    432—438 

XXV.  EXAMPLES  OF  THE  APPLICATION  OF  THE  SECOND  LAW 
OF  THERMODYNAMICS  TO  A  PERFECT  STEAM-ENGINE 
AND  A   PERFECT  AIR-ENGINE. 

Definition  of  a   "Perfect  Engin         B       aerator— Examples  from  Steam- 
Engine  and  Air-Engine     Economical  Action  of  Regenerator  in  anAir- 
— ( iomparison  of  Economy  of  Steam  and  Air-En  in<       I  [eat  due  to 
Combustion  of  a  Pound  of  Coal, 139 — 153 

XXVI.  OX  THE  WORKING  OF  STEAM  IN  COMPOUND  ENGINES. 

Principal  Kinds  of  Compound  Engines,  and  their  Advantages— Number  of 
Cylinders  used— Diagrams  of  the  High  and  Low  Pressure  Cylinders- 
Rates  of  Expansion— Theoretical  Expansion  Diagram— The  Common 
Hyperbola  a  good  Approximation  to  the  true  Expansion  Curve — Calcula- 
tion of  Mean  Absolute  Pressure  and  of  Indicated  Work — Theoretical 
Diagrams  of  Compound  Engines  with  and  without  Reservoirs,  .        .     ;."J4  —  4G3 

XXVII.  OX  THE  THEORY  OE  EXPLOSIVE  GAS-ENGINES. 

Thermodynamical  Propositions — Rules  as  to  Heat  and  Expansion— Indicator 
Diagram  of  an  Explosive  Engine — Total  and  Available  Heat  of  Explosion — 
Mixture  of  Air  and  ( J-as— Adiabatic  Curves— Indicated  Work— Efficiency,     4G4— 470 

XXVIII.  OX  THE  EXPLOSIVE  EXERGY  OF  HEATED  LIQUIDS. 

General  Formulas  for  all  Fluids — Efficiency  of  Projection— Full-pressure 
Steam  Gun— Expenditure  of  Water  and  Heat-Efficiency— Full- pressure 
Dry  Steam  Gun, •  471-480 


CONTENTS. 


PART     III. 

PAPERS  RELATING   TO  WAVE  FOP  MS,  PROPULSION  OF  VESSELS, 
STABILITY  OF  STRUCTURES,  ;\r. 

XXIX.  OX  THE  EXACT  FORM  OF  WAVES  NEAR  THE  SURFACE 
OF  DEEP  WATER. 

PAGE 

Trochoidal  Form  of  Waves  first  stated  by  Mr.  Scott  Russell— Demonstra- 
tion of  Trochoidal  Form— Height  of  Crests—  Cycloidal  Waves— Friction 
between  a  Wave  and  a  Wave-shaped  Solid— Mr.  Stokes'  Investigations,   .     481—494 

XXX.  ON  PLANE  WATER-LINES  IN  TWO  DIMENSIONS. 

Plane  Water-Lines  Defined— Flow  of  Liquid  past  a  Solid— ^Water-Line 
Functions— Water- Line  Curves  generated  by  Circles  and  Ovals— Velocities 
of  Gliding— Lines  of  Smoothest  Gliding—  Orbits  of  the  Particles  of  Water 
—Trajectory  of  Transverse  Displacement— General  Problem  of  the  Water- 
Line  of  Least  Friction— Previous  Systems  of  Water-Lines— Chapman's 
System— Scott  Russell's  System— Preferable  Figures  of  Water-Lines— 
Lissoneo'ids  compared  with  Trochoids— Combination  of  Bow  and  Stern- 
Provisional  Formula  for  Resistance •        •    495— 5_1 

XXXI.  ELEMENTARY     DEMONSTRATIONS     OF     PRINCIPLES 
RELATING  TO  STREAM-LINES. 

Stream-Lines  Explained— Condition  of  Perfect  Fluidity— Circular  Stream- 
Lines— Straight  Stream-Lines— Compound  Stream-Lines,  .         .         .     522—529 

XXXII.  ON   THE   THERMODYNAMIC   THEORY   OF   WAVES   OF 
FINITE  LONGITUDINAL  DISTURBANCE. 

Mass- Velocity— Waves  of  Sudden  Disturbance— Thermodynamic  Conditii  his 
— Adiabatic  State— Absolute  Temperature— Previous  Investigations,         .     530  -  543 

XXXIII.  ON   THE   THEORETICAL   LIMIT  OF  THE   EFFICIENCY 
OF  PROPELLERS. 

Slip,  Reaction,  and  Efficiency  of  a  Propeller— Energy  Expended— Useful  and 
Lost  Work— Forms  of  Screw  Propellers— Examples  from  Practice,  . 


544-54'.) 


XXXIV.  REPORT   ON   THE   DESIGN"   AND   CONSTRUCTION    OF 
MASONRY  DAMS. 

Material  and  Mode  of  Building— Form  and  Limits  of  Pressure— Mathemati- 
cal Principles  of  the  Profile  Curves, 550    501 

XXXV.  ON   THE   APPLICATION   OF   BARYCEXTRIC   PERSPEC- 
TIVE TO  THE  TRANSFORMATION  OF  STRUCTURES. 

Parallel  Projection — Application  to  Skew  Arches,  &c,  ....     562—563 

XXXVI.  PRINCIPLE  OF  THE  EQUILIBRIUM  OF  POLYHEDRAL 
FRAMES, ~lM 

XXXVII.  ON   A    PROPERTY    OF    CURVES    FULFILLING    THE 
CONDITION^  +  5^  =  0, 5G5-n6T 


MEMOIE. 


The  life  of  a  genuine  scientific  man  is,  from  the  common  point  of  view, 
almost  always  uneventful.  Engrossed  with  the  paramount  claims  of 
inquiries  raised  high  above  the  domain  of  mere  human  passions,  he  is 
with  difficulty  tempted  to  come  forward  in  political  discussions,  even  when 
they  are  of  national  importance ;  and  he  regards  with  surprise,  if  not 
with  contempt,  the  petty  municipal  squabbles  in  which  local  notoriety 
is  so  eagerly  sought.  To  him  the  discovery  of  a  new  law  of  nature,  or 
even  of  a  new  experimental  fact,  or  the  invention  of  a  novel  mathematical 
method,  no  matter  who  has  been  the  first  to  reach  it,  is  an  event  of  an 
order  altogether  different  from,  and  higher  than,  those  which  are  so 
profusely  chronicled  in  the  newspapers.  It  is  something  true  and  good 
for  ever,  not  a  mere  temporary  outcome  of  craft  or  expediency.  "With 
few  exceptions,  such  men  pass  through  life  unnoticed  by,  almost  unknown 
to,  the  mass  of  even  their  educated  countrymen.  Yet  it  is  they  who,  far 
more  than  any  autocrats  or  statesmen,  are  really  moulding  the  history  of 
the  times  to  come.  Man  has  been  left  entirely  to  himself  in  the  struggle 
for  creature  comforts,  as  well  as  for  the  higher  appliances  which  advance 
civilisation ;  and  it  is  to  science,  and  not  to  so-called  statecraft,  that  he 
must  look  for  such  things.  Science  can  and  does  provide  the  means; 
statecraft  can  but  more  or  less  judiciously  promote,  regulate,  or  forbid 
their  use  or  abuse.  One  is  the  lavish  and  utterly  unselfish  furnisher  of 
material  good,  the  other  the  too  often  churlish  and  ignorant  dispenser  of 
it.  In  the  moral  world  their  analogues  are  charity  and  the  relieving 
officer!  So  much  it  is  necessary  to  say  for  the  sake  of  the  general 
reader;  to  the  world  of  science  no  apology  need  be  made.  In  it 
Rankine's  was  and  is  a  well-known  name. 

It  is  high  eulogy,  but  strictly  correct,  to  say  that  Eankine  holds  a 
prominent  place  among  the  chief  scientific  men  of  the  last  half  century. 
He  was  one  of  the  little  group  of  thinkers  to  whom,  after  the  wondrous 
Sadi  Carnot,  the  world  is  indebted  for  the  pure  science  of  modern  thernio- 


xx  MEMOIR. 

dynamics.      "Were   this    all,   it  would   be   undoubtedly  much.      But  his 

services  to  applied  science  -were  relatively  even  greater.  By  his  admirable 
teaching,  his  excellent  text  books,  and  his  original  memoirs,  he  has  done 
more  than  any  other  man  of  recent  times  for  the  advancement  of  British 
Scientific  Engineering.  He  did  not,  indeed,  himself  design  or  construct 
gigantic  structures,  but  he  taught,  or  was  the  means  of  teaching,  that 
invaluable  class  of  men  to  whom  the  projectors  of  such  works  entrust 
the  calculations  on  which  their  safety  as  well  as  their  efficiency  mainly 
depend.  For  behind  the  great  architect  or  engineer,  and  concealed  by 
his  portentous  form,  there  is  the  real  worker,  without  whom  failure 
would  be  certain.  The  public  knows  but  little  of  such  men.  Not  every 
von  Moltke  has  his  services  publicly  acknowledged  and  rewarded  by  his 
imperial  employer!  But  he  who  makes  possible  the  existence  of  such 
men  confers  lasting  benefit  on  his  country.  And  it  is  quite  certain  that 
Rankine  achieved  the  task. 

William  John  Macquorn  Rankine  was  born  in  Edinburgh  on  the 
nth  July,  1820.  lie  was  a  Scot  of  Scots.  His  father  was  descended 
from  the  Rankines  of  Carrick  and  the  Cochranes  of  Dundonald.  His 
maternal  grandfather  was  Grahame  of  Drumquhassle,  a  descendant  of  tin; 
Grahams  of  Dougalston.  In  Rankine's  .MSS.  there  is  to  be  found  a 
tracing  of  the  various  steps  of  his  pedigree  from  Robert  the  Bruce.  His 
father,  David  Rankine,  was  in  youth  Lieutenant  in  the  21st  Regimenl  (RiHc 
Brigade) ;  but,  as  will  be  seen  by  what  follows,  was  also  a  man  of  great 
general  information,  especially  in  practical  matters.  He  was  employed  in 
later  life  in  constructing  railways,  ami  afterwards  became  Secretary  to  the 
Caledonian  Railway  Company.  Rankine  repeatedly  notes  in  his  journal 
the  hints  and  instruction  he  had  received  from  his  father.  He  was 
profoundly  attached  to  his  parents;  and  one  of  the  most  touching  notes 
in  his  journal  is  the  brief  record  of  his  lasting  obligations  to  them  for 
early  instruction  in  the  fundamental  principles  of  the  Christian  religion 
and  the  character  of  its  Founder. 

It  will  be  convenient  to  give,  in  the  first  place,  a  brief  sketch  of 
Rankine's  career,  and  to  reserve  for  a  time  such  comments  upon  his  more 
important  investigations  and  treatises  as  would  materially  interfere  with 
the  continuity  of  the  sketch. 

From  Rankine's  private  journal  it  appears  that  his  first  introduction  to 
arithmetic  and  elementary  mathematics,  mechanics,  and  physics  was 
obtained  mainly  from  his  father.  He  attended  Ayr  Academy  in  1828-9, 
and  Glasgow  High  School  in  1830.  After  this  he  seems  to  have  been  for 
some  years  privately  instructed  in  Edinburgh,  his  state  of  health  preventing 
his  being  sent  to  a  public  school.    In  December,  1831:,  his  uncle,  Archibald 


: 


MEMOIR.  XX  i 

Grahame,  presented  him  with  a  copy  of  Newton's  Frincijiia,  which  he  read 
carefully.  He  remarks — "  This  was  the  foundation  of  my  knowledge  of  the 
higher  mathematics,  dynamics,  and  physics.  My  knowledge  of  the  higher 
mathematics  was  obtained  chiefly  by  private  study."  About  this  period 
he  paid  much  attention  to  the  theory  of  music.  In  183G  he  studied 
Practical  and  Theoretical  Chemistry  under  David  Boswell  Eeid ;  and  in 
November  of  the  same  year  entered  Edinburgh  University.  He  there 
attended  the  Natural  Philosophy  course  under  Professor  Forbes,  and  gained 
(before  completing  his  seventeenth  year)  the  Gold  Medal  for  an  essay  on 
the  "Undulatory  Theory  of  Light."  In  the  summer  of  1837,  he  studied 
Natural  History  under  Professor  Jameson,  and  Botany  under  Professor 
Graham.  He  attended  the  Natural  Philosophy  Class  a  second  time  in 
1S37-8,  and  obtained  an  extra  prize  for  an  essay  on  "  Methods  in  Physical 
Investigation."  He  records  in  his  journal,  that  in  1836-8,  during 
leisure,  he  read  much  metaphysics,  chiefly  Aristotle,  Locke,  Hume,  Stewart, 
and  Degerando.  I  have  learned  from  himself  that  about  this  period  he 
"  wasted  "  a  great  deal  of  time  in  the  fascinating  but  too  often  delusive 
pursuit  of  "  Theory  of  Numbers." 

In  1837-8  he  made  his  first  accjuaintance  with  the  practice  of  engineer- 
ing, having  assisted  his  father  in  superintending  the  works  of  the  Leith 
branch  of  the  Edinburgh  and  Dalkeith  Eailway.  In  the  latter  year  he 
became  a  pupil  of  the  late  Sir  John  Macneill,  C.E.,  having  among  his 
fellow-pupils  many  who  have  since  risen  to  eminence.  His  journal  records 
the  names  of  G.  W.  Hemans,  J.  W.  Bazalgette,  W.  E.  Le  Fanu,  Matthew 
Blackiston,  John  Moffat,  and  Jonas  S.  Stawell. 

During  the  succeeding  four  years  he  was  employed  by  Macneill  on 
various  surveys,  and  schemes  for  river  improvements,  water-works,  and 
harbours  for  Ireland.  Also,  for  some  time,  on  the  Dublin  and  Drogheda 
Railway.  "While  engaged  on  this  railway  in  1841,  he  contrived  the  method 
of  setting  out  curves  "  by  chaining  and  angles  at  circumference  combined," 
which  has  since  been  known  as  "  Eankine's  method." 

In  1842  appeared  his  first  published  work,  a  pamphlet  entitled  An 
Experimental  Inquiry  into  the  advantage  of  Cylindrical  Wheels  on  Railways. 
This  was  based  upon  experiments  suggested  to  him  by  his  father,  and 
carried  out  by  them  together. 

In  the  same  year  Queen  Victoria  visited  Edinburgh  for  the  first  time, 
and  Eankine  was  charged  with  the  superintendence  of  the  erection  of  the 
huge  bonfire  which  blazed  on  the  top  of  Arthur's  Seat.  He  constructed 
it  with  radiating  air  passages  under  the  fuel,  and  succeeded,  as  he  com- 
placently records,  in  partially  vitrifying  the  rock ! 

During  this  and  the  succeeding  year  he  sent  several  papers  to  the 
Institute  of  Civil  Engineers,  for  some  of  which  prizes  were  awarded  to 
him.     He  records  that  most  of  them  were  based  on  suggestions  by  his 


XX11  MEMOIR. 

tather,  especially  that  on  the  "  Fracture  of  Axles."  He  showed  that  such 
/'iclures  arose  through  gradual  deterioration  or  fatigue,  involving  the 
gradual  extension  inwards  of  a  crack  originating  at  a  square-cut  shoulder. 
In  this  paper  the  importance  of  continuity  of  form  and  fibre  was  first 
shown,  and  the  hypothesis  of  spontaneous  crystallisation  Avas  disproved. 

In  1844-5  he  was  employed  under  Locke  and  Errington  on  the  Clydes- 
dale Junction  Railway  project;  and  afterwards,  till  1S48,  on  various 
schemes  promoted  by  the  Caledonian  Railway  Company.  In  1845-6 
he  engineered  a  project  for  Edinburgh  and  Leith  Water-works,  which  was 
defeated  by  the  opposition  of  the  Edinburgh  AVater  Company. 

About  1848  he  seems  to  have  commenced  that  extensive  series  of 
researches  on  molecular  physics  which  occupied  him  at  intervals  during 
the  rest  of  his  life,  and  which  constitutes  bis  chief  claim  to  distinction  in 
the  domain  of  pure  science.  The  first  paper  he  published  on  the  subject, 
with  the  title  "Elasticity  of  Steam,"  appeared  in  the  Edinburgh  New 
Philosophical  Journal  in  July,  1849;  and  at  the  end  of  that  year  he  sent 
to  the  Royal  Society  of  Edinburgh  his  great  paper  on  the  "Mechanical 
Action  of  Heat,"  It  was  not,  however,  read  to  the  Society  till  February, 
1850.  On  the  contents  of  this,  and  his  subsequent  papers  dealing  with 
similar  subjects,  some  remarks  will  lie  made  below.  In  duly,  1850,  he 
read  to  the  British  Association  at  Edinburgh  another  paper  on  a  closely 
connected  subject,  "Elastic  Solids." 

In  1852  the  Loch  Katrine  Water-works  scheme  for  the  supply  of 
Glasgow  was  revived  by  Rankine  and  John  Thomson.  This  scheme,  now 
successfully  carried  out,  was  first  proposed  by  Lewis  Cordon  and  Laurence 
Hill,  Junior. 

In  1853,  one  of  Rankine's  most  characteristic  papers  in  pure  science, 
••  On  the  General  Law  of  Transformation  of  Energy,"  was  read  by  him  to 
the  Glasgow  Philosophical  Society.  In  the  same  year,  along  with  the  late 
J.  R.  Xapier,  he  projected  and  patented  a  new  form  of  air-engine.  The 
patent  was  afterwards  abandoned. 

He  was  now  elected  to  the  Fellowship  of  the  Royal  Society,  and  sent 
to  that  body  his  next  great  paper  on  Thermodynamics — viz.,  "  On  the 
Geometrical  Representation  of  the  Expansive  Action  of  Heat,"  which  was 
printed  in  the  Philosophical  Transactions. 

From  January  to  April,  1855,  he  acted  in  Glasgow  University  as  substi- 
tute for  Professor  Lewis  Gordon,  on  whose  resignation  he  was  appointed 
to  the  Chair  of  Engineering,  which  he  held  till  his  death.  His  inaugural 
discourse,  delivered  on  Dec.  10  of  the  last-mentioned  year,  bore  the  title 
"  De  concordld  inter  Scientiarum  Machinalium  Contem/plationem  et  Usum."  In 
this  year  he  wrote,  among  several  contributions  to  Kichol's  Cyclopcedia,  an 
article  on  "  Heat,  Mechanical  Action  of,"  the  earliest  formal  treatise  on 
Thermodynamics  in  the  English  language.     In  1856,  the  preparation  of 


memoir.  xxiii 

his  course  of  lectures  led  him  to  the  invention  of  some  remarkable 
methods  connected  with  Transformation  of  Structures.  These  are  based 
on  the  discovery  of  "  reciprocal  diagrams "  of  frames  and  forces,  since 
greatly  extended  and  simplified  by  Clerk-Maxwell.  The  remarkable 
storm  which  occurred  on  February  7  of  this  year,  directed  his  inquiries  to 
the  "  Stability  of  Chimneys,"  on  which  he  has  published  a  valuable  article. 

In  1857,  he  resigned  the  associateship  of  the  Institute  of  Civil 
Engineers ;  and  shortly  afterwards,  on  the  establishment  of  the  Institute 
of  Engineers  in  Scotland,  he  delivered  the  opening  address  as  first 
President.  At  this  time  he  was  busily  engaged  on  a  Treatise  on  Ship- 
building, his  Manual  of  Applied  Mechanics,  an  article  on  the  same 
subject  for  the  Encyclopedia  Britannica,  and  an  investigation  (based  on 
J.  R.  Napier's  experiments)  of  the  theory  of  skin-resistance  of  ships.  He 
also  sent  to  the  French  Academy  of  Sciences  a  memoir, — "  Be  TE<£uililre 
intdrieure  d'un  corps  solide,  elastigue,  ct  homogene." 

In  July,  1859,  an  offer  of  service  was  sent  to  the  Lieutenancy  by  the 
"  Glasgow  University  Rifle  Volunteers."  It  was  accepted  in  October, 
and  Rankine  received  his  commission  as  Captain.  He  spent  the  greater 
part  of  November  at  the  Hythe  School  of  Musketry,  and,  on  his  return, 
instructed  the  officers  and  sergeants  of  his  corps.  In  the  same  year 
appeared  his  valuable  Ma  mud  of  Ihc  Steam-Engine  and  other  Prime  Movers. 
In  1860,  he  was  made  Senior  Major,  and  commanded  the  second 
battalion  of  his  regiment  at  the  memorable  Volunteer  Review,  held  by  the 
Queen  in  the  Queen's  Park,  Edinburgh. 

In  1861,  he  finished  his  Manual  of  Civil  Engineering,  which  was  pub- 
lished early  in  the  following  year.  At  the  International  Exhibition  in 
1862,  at  London,  he  was  a  Juror  in  Class  VIII,  "  Machinery  in  General." 

In  1864,  he  resigned  his  commission  in  the  Volunteers,  "finding  it 
impossible  to  attend  at  once  to  duties  as  field-officer  and  as  professor,  to 
engineering  business,  and  to  literary  work,  especially  Treatise  on  Ship- 
building." 

In  1865,  he  was  appointed  Consulting  Engineer  to  the  Highland  and 
Agricultural  Society  of  Scotland  ;  and  became  a  regular  contributor  to  the 
Engineer,  in  which  many  excellent  articles  of  his  appeared. 

In  1866,  was  published  his  Treatise  em  Shipbuilding,  Theoretical  and 
Practiced.  Though  four  names  were  announced  on  the  title  page  as  joint 
authors,  by  far  the  greater  part  of  the  work  was  written  by  Rankine; 
but  the  proofs  were  revised  by  all  four. 

1869  produced  Machinery  and  Millwork,  the  fourth  of  Rankine's  great 
engineering  treatises.  The  other  three  had  then  reached  their  eighth, 
sixth,  and  fifth  editions  respectively.  In  his  journal  for  this  year  occurs 
the  following  note  : — 

"  Sept.  16.     Thomas  Graham,  Master  of  the  Mint,  died  [son  of  a  cousin 


xxiv  MEMOIR. 

of  my  mother's  father],  I  applied  for  vacancy  to  Chancellor  of  Exchequer 
(Lowe).  Application  well  supported  by  friends,  and  civilly  received  ;  but 
the  office  was  virtually  abolished,  being  conjoined  with  the  Chancellorship 
of  the  Exchequer." 

Eankine  lost  his  father  in  1870,  and  his  mother  in  the  following  year. 
Both  had  passed  the  age  of  seventy.  The  loss  of  his  parents,  to  whom  he 
was  so  fondly  attached,  seems  to  have  accelerated  the  development  of  the 
illness  which  had  for  some  j-ears  been  growing  upon  him.  He  was  well 
enough,  in  1871,  to  contribute  most  valuable  matter  to  the  proceedings 
of  the  t;  Committee  on  Designs  for  Ships  of  War,"  which  was  appointed 
shortly  after  the  loss  of  the  "  ( laptain."  He  investigated  for  that  Committee 
the  "Stability  of  Unmasted  Ships  of  Low  Freeboard,"  and  the  "Stability 
of  Ships  under  Canvas." 

In  February,  1872,  Rankine  completed  his  memoir  of  his  friend,  John 
Elder,  and  in  July  reported  on  the  cause  of  the  explosion  of  the  Tradeston 
Flour  Mill.  In  May  the  increase  of  the  endowment  of  his  chair,  which 
he  had  in  vain  sought  from  Government,  was  given  by  Mr.  Elder's  widow; 
and  the  income  of  Rankine's  post  was  at  last  made  sufficient  to  maintain 
its  occupant.  But  by  this  time  his  energy  was  fast  failing,  the  simplest 
work  fatigued  him;  and  he  died  on  December  24,  of  a  general  decline 
rather  than  of  any  special  disease.  He  had  been  for  some  years  liable  to 
violent  headaches,  and  towards  the  close  of  his  life  these  affected  his  sight. 
They  were  probably  symptoms  of  heart  disease,  which  ultimately  developed 
paralysis.  The  gradual  decay  of  his  physical  powers  is  painfully  evident 
in  the  last  pages  of  his  journal,  where,  though  the  substance  is  correct 
and  to  the  point,  the  handwriting  becomes  more  and  more  irregular  at  each 
succeeding  entry. 

Such  are  the  more  prominent  events  in  the  life  of  this  great  and  good 
man.  Even  now,  after  the  lapse  of  eight  years,  it  is  difficult  to  realise  the 
fact  of  his  death.  He  was  so  many-sided,  and  yet  so  complete  in  himself, 
that  the  mental  image  of  him  formed  by  each  of  his  friends  remains 
almost  as  clear  and  distinct  as  if  it  had  been  formed  but  a  few  days 
ago. 

Of  the  man  himself  it  is  not  easy  to  speak  in  terms  which,  to  a  stranger, 
would  appear  unexaggerated.  His  appearance  was  striking  and  prepos- 
sessing in  the  extreme,  and  his  courtesy  resembled  almost  that  of  a  gentle- 
man of  the  old  school.  His  musical  taste  had  been  highly  cultivated,  and 
it  was  always  exceedingly  pleasant  to  see  him  take  his  seat  at  the  piano 
to  accompany  himself  as  he  sang  some  humorous  or  grotesquely  plaintive 
song — words  and  music  alike  being  generally  of  his  own  composition. 
Some  of  the  best  of  these  songs  have  been  collected  in  a  small  volume, 
Songs  and  Fables  (Second   Edition;    Glasgow,  Maclehose,   1874).      We 


MEMOIR.  XXV 

extract  one  which  give-,  in  a  very  telling  form,  one  point  of  view  of  a 
much-debated  semi-scientific  question  : — 


THE  THEEE-FOOT  EULE. 

When  I  was  hound  apprentice,  and  learned  to  use  my  hands, 
Folk  never  talked  of  measures  that  came  from  foreign  lands  : 
Xow  I  'm  a  British  workman,  too  old  to  go  to  school ; 
So  whether  the  chisel  or  file  I  hold,  I  '11  stick  to  my  three-foot  rule. 

Some  talk  of  millimetres,  and  some  of  kilogrammes, 

And  some  of  decilitres,  to  measure  beer  and  drams ; 

But  I  'm  a  British  workman,  too  old  to  go  to  school ; 

So  by  pounds  I  '11  eat,  and  by  quarts  I  '11  drink,  and  I  '11  work  by  my 

three-foot  rule. 

A  party  of  astronomers  went  measuring  of  the  earth, 
And  forty  million  metres  they  took  to  be  its  girth ; 
Five  hundred  million  inches,  though,  go  through  from  pole  to  pole ; 
So  let 's  stick  to  inches,  feet,  and  yards,  and  the  good  old  three-foot 
rule. 

The  great  Egyptian  pyramid 's  a  thousand  yards  about ; 
And  when  the  masons  finished  it,  they  raised  a  joyful  shout ; 
The  chap  that  planned  that  building,  I  'm  bound  he  was  no  fool ; 
And  now  'tis  proved  beyond  a  doubt,  he  used  a  three-foot  rule. 

Here  's  a  health  to  every  learned  man  that  goes  by  common  sense, 
And  would  not  plague  the  workman  on  any  vain  pretence ; 
But  as  for  those  philanthropists,  who  'd  send  us  back  to  school, 
Oh,  bless  their  eyes,  if  ever  they  tries  to  put  down  the  three-foot  rule. 

When  the  "Red  Lions"  met  during  the  British  Association  week  of  1871, 
in  Edinburgh,  Eankine  was  hailed  with  universal  acclaim  as  the  Lion-King. 
His  versatility  in  that  singular  post  was  very  much  akin  to  that  of  Pro- 
fessor Edward  Forbes;  though  their  paths  in  science  were  widely  different. 
His  conversation  was  always  interesting,  and  embraced  with  equal  seeming 
ease  all  topics,  however  various.  He  had  the  still  rarer  qualification  of 
being  a  good  listener  also.  The  evident  interest  which  he  took  in  all  that 
was  said  to  him  had  a  most  reassuring  effect  on  the  speaker;  and  he  could 
turn  without  apparent  mental  effort  from  the  prattle  of  young  children  to 
the  most  formidable  statement  of  new  results  in  mathematical  or  physical 
science.     Then  his  note-book  was  at  once  produced,  and  in  a  few  lines  he 


xxvi  _  MEMOIR. 

jotted  down  the  essence  of  the  statement,  to  be  pondered  over  at  leisure, 
provided  it  did  not  ut  once  appear  to  him  how  it  was  to  be  verified.  The 
questions  which  he  asked  on  such  occasions  were  always  almost  startlingly 
to  the  point,  and  showed  a  rapidity  of  thought  not  often  met  with  in 
minds  of  such  calibre  as  his,  where  the  mental  inertia,  which  enables  them 
tn  overcome  obstacles  often  prevents  their  being  quickly  set  in  motion. 
His  kindness,  shown  in  the  readiness  with  which  he  undertook  to  read 
proof  sheets  tor  a  friend,  or  even  to  contribute  a  portion  of  a  chapter 
(where  the  subject  was  one  to  which  he  had  paid  special  attention),  was. 
for  a  man  so  constantly  at  work,  absolutely  astonishing.  The  writer  of 
this  brief  notice  has  several  times  availed  himself  of  such  assistance.  It 
was  given  almost  as  soon  as  asked,  and  it  was  invariably  of  sterling  value. 
Nothing  is  more  precious  to  a  writer  on  scientific  subjects  (especially  when 
questions  of  priority  are  involved )  than  the  assistance  of  a  friendly  -though, 
if  necessary,  seven — critic,  such  as  was  Rankine. 

AVe  must  not  refrain  from  pointing  out,  in  connection  with  his  scientific 
merits,  how  very  good  and  how  exemplary  for  scientific  writers  and 
investigators  his  character  was.  I  Ie  was  ambitious  ;  that  is  obvious  from 
the  number  and  variety  of  his  books  and  papers,  and  the  quite  unnecessary 
display  of  symbols  in  several  of  his  less  popular  writings.  But  he  was  tie' 
very  soul  of  honour  in  respect  to  giving  all  credit  to  others,  ami  in  never 
attempting  in  anything,  small  or  great,  to  go  a  hairbreadth  beyond  the 
line  of  right  as  to  his  own  claims.  He  showed  a  particularly  good  and 
generous  temper  in  cases  of  difference  on  scientific  questions — a  temper 
which  proved  the  true  metal,  unalloyed  by  any  mean  quality. 

Rankine  was,  in  many  subject-.,  an  almost  self-taughl  man,  and  the 
direction  of  his  earlier  scientific  wort  seems  not  to  have  been  a  very  pro- 
fitable one.  But,  once  on  the  right  tack,  his  progress  became  very  rapid. 
Every  mathematician  worthy  of  the  name  has  made  himself:  some,  as 
Rowan  Hamilton,  by  attacking  at  an  early  age  the  grander  works  of 
Lagrange  and  Laplace;  others  by  attempting  original  flights  without  the 
assistance  of  books.  Rankine  published  only  one  or  two  papers  on  sub- 
jects of  pure  mathematics;  ami  even  these,  though  not  containing  any 
direct  allusion  to  physics,  were  connected  somewhat  closely  with  kine- 
matical  or  physical  investigations,  such  as  the  deformation  of  an  elastic 
solid. 

The  number  of  Rankine's  scientific  papers  seems  absolutely  enormous, 
when  we  consider  the  minute  and  scrupulous  care  with  which  he  attended 
to  every  point  of  detail  in  the  writing  and  printing  of  them.  How  he 
managed,  in  addition  to  these,  to  find  time  for  the  composition  of  his 
many  massive  (not  heavy)  and  elaborate  volumes — all  marked  with  the 
most  striking  stamp  of  originality — for  his  memoirs,  and  his  almost  weekly 
communications   to   The   Engineer  and    other    professional    papers,    must 


memoir.  xxvii 

always  remain  matter  for  conjecture.  In  the  Royal  Society's  splendid 
Catalogue  of  Scientific  Paper*  we  find  that  from  1843  to  1872  (both 
inclusive)  he  published,  in  recognised  scientific  journals  alone,  upwards  of 
a  hundred  and  fifty  papers— many  of  these  being  exhaustive  essays  on 
mathematical  or  physical  questions,  and  all,  save  one  or  two,  contain- 
ing genuine  contributions  to  the  advance  of  science.  Leaving  out  of 
account  the  more  strictly  professional  of  these  papers,  we  find  among 
the  titles  of  the  rest  such  heads  as  the  following : — Molecular  Vortices, 
Elasticity  of  Solids,  Isorrhopic  Axes,  Compressibility  of  Water  at  Different 
Temperatures,  Centrifugal  Theory  of  Elasticity,  Oscillatory  Theory  of  Light, 
(!.  neral  Law  of  Transformation  of  Energy,  Plane  Water-Lines,  Oogenous 
Neo'ids.  To  indicate  even  briefly  the  nature  and  importance  of  the 
varied  contents  of  these  papers  alone,  would  require  vastly  more  time 
and  space  than  are  at  present  at  our  disposal.  The  more  important  of 
them  are  included  in  the  present  volume;  others  of  less  importance,  or  of 
less  characterised  originality,  may  be  consulted  by  the  reader  in  the 
scientific  publications  where  they  originally  appeared. 

Unquestionably  the  greatest  pure  scientific  work  of  Rankine'sis  contained 
in  his  numerous  papers  bearing  on  the  Dynamical  Theory  of  Heat,  and  on 
Energy  generally.  As  Sir  "William  Thomson  has  remarked,  even  the  mere 
title  of  his  earliest  paper  on  this  subject,  "Molecular  Vortices,"  is  an 
important  contribution  to  physical  science.  The  mode  in  which  Rankine, 
in  1819,  attacked  the  true  theory  of  heat,  which  had  just  been  recalled  to 
the  attention  of  scientific  men  by  the  admirable  experiments  and  numerical 
determinations  of  Joule,  was  quite  different  from  that  adopted  by  any 
one  of  his  concurrents;  and  though  objections  may  fairly  be  raised  to 
certain  parts,  even  his  first  paper  constituted  a  remarkable  contribu- 
tion to  our  physical  knowledge.  The  essential  characteristic  of  his 
method  is  the  introduction  of  a  hypothesis  as  to  the  nature  of  the  motions 
and  displacements  (of  the  ultimate  parts  of  bodies)  upon  which  temper- 
ature depends,  and  in  which  heat,  whether  latent  or  sensible,  consists. 
He  thought  it  necessary  to  defend  this  mode  of  investigation,  and  did  so 
in  a  remarkable  address  to  the  Philosophical  Society  of  Glasgow,  from 
which  we  extract  the  following  passages,  which  are  valuable  not  alone 
from  their  intention,  but  also  from  the  insight^they  give  us  into  the 
character  of  the  man  : — 

"  In  order  to  establish  that  degree  of  probability  which  warrants  the 
reception  of  a  hypothesis  into  science,  it  is  not  sufficient  that  there  should 
be  a  mere  loose  and  general-  agreement  between  its  results  and  those  of 
experiment.  Any  ingenious  and  imaginative  person  can  frame  such 
hypotheses  by  the  dozen.  The  agreement  should  be  mathematically 
exact  to  that  degree  of  precision  which  the  uncertainty  of  experimental 
data  renders  possible,  and  should  be  tested  in  particular  cases  by  numerical 


XXVlll  MEMOIR. 

calculation.  The  highest  degree  of  probability  is  attained  when  a  hypo- 
thesis leads  to  the  prediction  of  laws,  phenomena,  and  numerical  results 
which  are  afterwards  verified  by  experiment"  as  when  the  wave-theory 
of  light  led  to  the  prediction  of  the  true  velocity  of  light  in  refracting 
media,  of  the  circular  polarisation  of  light  by  reflexion,  and  of  the 
previously  unknown  phenomena  of  conical  and  cylindrical  refraction;  and 
as  when  the  hypothesis  of  atoms  in  chemistry  led  to  the  prediction  of 
the  exact  proportions  of  the  constituents  of  innumerable  compounds. 
I  think  I  am  justified  in  claiming  for  the  hypothesis  of  mole- 
cular  vortices,  as  a  means  of  advancing  the  theory  of  the  mechanical 
action  of  heat,  the  merit  of  having  fulfilled  the  proper  purposes  of  a 
Mechanical  hypothesis  in  physical  science,  which  are  to  connect  the  laws 
of  molecular  phenomena  by  analogy  with  the  laws  of  motion,  and  to 
suggest  principles  such  as  the  second  law  of  thermodynamics  and  the 
laws  of  the  elasticity  of  imperfect  gases,  whoso  conformity  to  fact  may  after- 
wards be  tested  by  direct  experiment.  And  I  make  that  claim  the  more 
confidently,  that  1  conceive  the  hypothesis  in  question  to  be  in  a  great 
measure  the  development,  and  the  reduction  to  a  precise  form,  of  ideas 
concerning  the  molecular  condition  which  constitutes  heat,  that  have  been 
entertained  from  a  remote  period  by  the  leading  minds  in  physical 
science  .  .  .  I  wish  it,  however,  to  be  clearly  understood,  that  although 
I  attach  great  value  and  importance  to  sound  mechanical  hypotheses  as 
means  of  advancing  physical  science,  1  firmly  hold  that  they  can  never 
attain  the  certainty  of  observed  facts;  and  accordingly,  I  have  laboured 
assiduously  to  show  that  the  two  laws  of  thermodynamics  are  demon- 
strable as  facts,  independently  of  any  hypothesis;  and  in  treating  of  the 
[aactical  application  of  those  laws,  I  have  avoided  all  reference  to 
hypothesis  whatsoever." 

The  application  of  the  doctrine,  that  heat  "ml  work  an  ble,  to  the 

discovery  of  new  relations  among  the  properties  of  bodies,  was  made 
about  the  same  time  by  three  scientific  men — W.  Thomson,  Rankine,  and 
Clausius. 

Of  these,  Thomson  cleared  the  way  for  the  new  theory  by  his  account 
of  the  almost  forgotten  work  of  Carnot  on  the  "  Motive  Power  of  Heat." 
This  excessively  important  investigation  was  published  in  182-f,  when  the 
world  of  science  was  not  prepared  for  its  reception,  and  had  been  allowed 
to  drop  out  of  notice.  Thomson  gave  a  very  full  abstract  of  its  contents 
in  the  Transacts  ras  of  the  Royal  Society  of  Edinburgh,  1849,  and  pointed  out 
that  they  would  require  modification  if  the  new  theory  were  adopted, 
as  Carnot  had  throughout  assumed  that  heat  is  a  substance,  and  therefore 
indestructible.  He  showed,  besides,  that  Carnot's  method  was  capable 
of  giving  an  absolute  definition  of  temperature ;  independent,  that  is, 
of  the  properties  of  any  particular  substance.      He  also  experimentally 


MEMOIR.  xxix 

verified  a  deduction  made  by  his  brother,  James  Thomson  (from  Carnot's 
theory),  as  to  the  alteration  of  the  freezing  point  of  water  by  pressure. 

Eankine  (late  in  1849)  and  Clausius  (early  in  1850)  took  the  first  step 
towards  the  formation  of  a  true  theory  of  the  action  of  heat  on  bodies,  by 
showing  (by  perfectly  different  modes  of  attacking  the  question)  the  nature 
of  the  modifications  which  Carnot's  theory  required.  The  recent  publica- 
tion of  Carnot's  MSS.  proves  that  that  verjr  remarkable  man  had  himself 
recognised  the  necessity  for  such  modifications  (and  had  all  but  succeeded 
in  making  them)  before  his  premature  death.  Thomson,  in  1851,  put  the 
foundations  of  the  theory  in  the  form  they  have  since  retained. 

In  RanMne's  paper  of  1849,  he  applied  the  theory  to  the  determination 
of  the  relation  between  the  latent  heat  of  steam  and  its  density,  and 
made  a  very  noteworthy  prediction  of  the  true  value  of  the  specific  heat 
of  air,  at  a  time  when  the  experimental  results  which  were  considered 
the  best  were  far  from  the  truth.  [Rankine's  results  were  soon  after 
verified  by  the  experimental  researches  of  Joule  and  Eegnault.]  He  also 
showed  that  saturated  steam,  pressing  out  a  piston  in  a  vessel  impervious 
to  heat,  must  cool  so  as  to  keep  constantly  at  the  temperature  of  satura- 
tion ;  and  that,  besides,  a  portion  of  it  licmefies. 

A  very  excellent  statement  of  the  claims  of  Eankine  in  thermodynamics 
is  given  in  the  following  cmotation  from  an  article  by  Clerk-Maxwell 
{Nature,  1878,  Vol.  XVIL,  p.  257)  :— 

"  Of  the  three  founders  of  theoretical  thermodynamics,  Eankine  availed 
himself  to  the  greatest  extent  of  the  scientific  use  of  the  imagination. 
His  imagination,  however,  though  amply  luxuriant,  was  strictly  scientific. 
Whatever  he  imagined  about  molecular  vortices,  with  their  nuclei  and 
atmospheres,  was  so  clearly  imaged  in  his  mind's  eye,  that  he,  as  a  practical 
engineer,  could  see  how  it  would  work. 

"However  intricate,  therefore,  the  machinery  might  be  which  he 
imagined  to  exist  in  the  minute  parts  of  bodies,  there  was  no  danger  of 
his  going  on  to  explain  natural  phenomena  by  any  mode  of  action  of  this 
machinery  which  was  not  consistent  with  the  general  laws  of  mechanism. 
Hence,  though  the  construction  and  distribution  of  his  vortices  may  seem 
to  us  as  complicated  and  arbitrary  as  the  Cartesian  system,  his  final 
deductions  are  simple,  necessary,  and  consistent  with  facts. 

"  Certain  phenomena  were  to  be  explained.  Eankine  set  himself  to 
imagine  the  mechanism  by  which  they  might  be  produced.  Being  an 
accomplished  engineer,  he  succeeded  in  specifying  a  particular  arrangement 
of  mechanism  competent  to  do  the  work,  and  also  in  predicting  other 
properties  of  the  mechanism  which  were  afterwards  found  to  be  consistent 
with  observed  facts. 

"As  long  as  the  training  of  the  naturalist  enables  him  to  trace  the  action 
only   of  particular  material  systems,   without  giving  him  the  power  of 


XXX  MEMOIR. 

dealing  with  the  general  properties  of  all  such  systems,  he  must  proceed 
by  the  method  so  often  described  in  histories  of  science — he  must  imagine 
model  after  model  of  hypothetical  apparatus,  till  he  finds  one  which  will 
do  the  required  work.  If  this  apparatus  should  afterwards  be  found 
capable  of  accounting  for  many  of  the  known  phenomena,  and  not  demon- 
strably inconsistent  with  any  of  them,  he  is  strongly  tempted  to  conclude 
that  his  hypothesis  is  a  fact,  at  least  until  an  equally  good  rival  hypothesis 
has  been  invented.  Thus  Elankine,*  long  after  an  explanation  of  the 
properties  of  gases  had  been  founded  on  the  theory  of  the  collisions  of 
molecules,  published  what  he  supposed  to  be  a  proof  that  the  phenomena 
of  heat  were  invariably  due  to  steady  closed  streams  of  continuous  fluid 
matter. 

"The  scientific  career  of  Rankine  was  marked  by  the  gradual  develop- 
ment of  a  singular  power  of  bringing  the  most  difficult  investigations 
within  the  range  of  elementary  methods.  In  his  earlier  papers,  indeed, 
1m-  appears  as  if  battling  with  chaos,  as  he  swims,  or  sinks,  or  wades,  or 
creeps,  or  flies, 

'  Ami  through  the  palpable  obscure  finds  out 
I  Ik  uncouth  way  •.' 

but  he  soon  begins  to  pave  a  broad  and  beaten  way  over  the  dark  abyss; 
and  his  Latest  writings  -how  such  a  power  of  bridging  over  the  diiliculties 
oi  science,  that  his  premature  death  must  have  been  almost  as  great  a  loss 
to  the  diflusion  of  Bcience  as  it  was  to  its  advancement. 

"The  chapter  on  thermodynamics  in  his  book  on  the  steam-engine  was 
the  first  published  treatise  on  the  subject,  and  is  the  only  expression  of 
his  views  addressed  directly  to  students. 

"  In  (hi-  boot  he  has  disencumbered  himself  to  a  great  extent  of  the 
hypothesis  of  molecular  vortices,  and  builds  principally  on  observed  facts, 
though  he,  in  common  with  Clausius,  makes  several  assumptions,  some 
expressed  as  axioms,  others  implied  in  definitions,  which  seem  to  us 
anything  but  self-evident.  A-  an  example  of  Kankine's  best  style  Ave 
may  .take  the  following  definition  : — 

"'A  PERFECT  Gas  is  a  substance  in  such  a  condition  that  the  total 
pressure  exerted  by  any  number  oi'  portions  of  it,  at  a  given  temperature, 
against  the  sides  of  a  vessel  in  which  they  are  enclosed,  is  the  sum  of  the 
pressures  which  each  portion  would  exert  if  enclosed  in  the  vessel  separately 
at  the  same  temperature.' 

"  Here  Ave  can  form  a  distinct  conception  of  every  clause  of  the  definition; 
but  when  we  come  to  Eankine's  Second  Law  of  Thermodynamics  we  find 

*  "On  the  Second  Law  of  Thermodynamics,"  Phil.  Mo;/.,  Oct.,  1865,  §  12,  p.  244  ; 
but  in  his  paper  "On  the  Thermal  Energy  of  Molecular  Vortices,"  Trans.  U.S.  Edin., 
XX \".,  p.  557  (1S(50),  he  admits  that  the  explanation  of  gaseous  pressure  by  the  impacts 
of  molecules  has  beeu  proved  to  be  possible. 


MEMOIE.  XXXI 

that  though,  as  to  literary  form,  it  seems  cast  in  the  same  mould,  its 
actual  meaning  is  inscrutable. 

"  'The  Second  Law  of  Thermodynamics. — If  the  total  actual  heat  of 
a  homogeneous  and  uniformly  hot  substance  be  conceived  to  be  divided 
into  any  number  of  equal  parts,  the  effects  of  those  parts  in  causing  work 
to  be  performed  are  equal.' 

"  We  find  it  difficult  enough,  even  in  1878,  to  attach  any  distinct  meaning 
to  the  total  actual  heat  of  a  body,  and  still  more  to  conceive  this  heat 
divided  into  equal  parts,  and  to  study  the  action  of  each  of  these  parts; 
but  as  if  our  powers  of  deglutition  were  not  yet  sufficiently  strained, 
Eankine  follows  this  up  with  another  statement  of  the  same  law,  in  which 
we  have  to  assert  our  intuitive  belief  that 

"  'If  the  absolute  temperature  of  any  uniform^  hot  substance  be  divided 
into  any  number  of  equal  parts,  the  effects  of  those  parts  in  causing  work 
to  be  performed  are  equal.' 

"  The  student  who  thinks  that  he  can  form  any  idea  of  the  meaning  of 
this  sentence  is  quite  capable  of  explaining,  on  thermodynamical  principles, 
what  Mr.  Tennyson  says  of  the  great  Duke — 

'  Whose  eighty  wiuters  freeze  with  one  rebuke 
All  great  self-seekers  trampling  on  the  right.'  " 

Ivankinc's  researches  on  heat  were  for  the  most  part  connected,  as  we 
have  already  said,  with  a  theory  of  the  constitution  of  bodies,  and  a  specula- 
tion as  to  the  physical  nature  of  a  hot  body,  to  which  he  gave  the  name  of 
Theory  of  Molecular  Vortices.  In  this  theory,  the  invisibly  small  parts 
of  bodies  apparently  at  rest  are  supposed  to  be  in  a  state  of  motion,  the 
rapidity  of  which  may  be  compared  with  that  of  a  cannon  ball.  It  was 
distinguished  from  other  theories,  which  attribute  motion  to  bodies 
apparently  at  rest,  by  the  further  assumption  that  this  motion  is  like  that  of 
very  small  vortices,  each  whirling  about  its  own  axis,  and  that  the  centri- 
fugal force  of  this  motion  contributes  to  the  elasticity  of  the  body.  A 
theory  of  a  similar  kind  has  since  been  applied  by  Clerk-Maxwell  to  the 
explanation  of  magnetic  phenomena ;  and  Sir  W.  Thomson  has  made  the 
rigorous  investigation  of  vortices  possible  by  his  paper  on  "  Vortex 
Motion,"  and  has  also  contributed  to  the  philosophy  of  speculation  by  his 
theory  of  "  Vortex  Atoms." 

Eankine's  researches  on  the  general  theory  of  elastic  bodies  are  charac- 
terised by  the  fact  that  while,  in  laying  the  foundation  of  the  theory,  he 
confines  himself  to  the  use  of  rigorous  methods,  and  does  not  shrink  from 
any  mechanical  difficulty  in  their  application,  he  always  prepares  the  way 
for  the  application  of  the  results  to  practice,  by  making  the  definitions  so 
clear,  the  methods   so  simple,  the  results  so   definite,  that  they  can  be 


XXXU  MEMOIR. 

mastered  by  the  exercise  of  a  little  thought,  without  special  mathematical 
training.  This  quality  is  prominent  also  in  his  researches  on  fluid  motion, 
three  of  which  are  of  special  importance. 

I.  The  theory  of  the  propagation  of  waves,  such  as  those  of  sound  in 
elastic  media,  is  generally  supposed  to  belong  to  the  most  abstruse  depart- 
ments of  mathematical  science.  Even  Newton  made  some  oversights  in 
liis  investigation,  and  it  required  more  than  a  century  of  hard  mathematical 
development  before  the  theory  reached  its  present  state — which  is  still  very 
imperfect.  Rankine,  by  the  introduction  of  a  few  new  conceptions  in  the 
elementary  part  of  the  investigations,  has  rendered  it  possible  for  any  one 
acquainted  with  elementary  dynamics  to  follow  the  theory  up  to  the  point 
at  which  it  was  left  by  Laplace  ami  almost  as  much  further  as  it  has  yet 
been  carried 

II.  The  tlnory  of  waves  on  the  surface  of  water,  when  their  height 
is  not  regarded  as  infinitely  small,  is  still  more  difficult  than  that  of 
sound  waves.  Stokes  has,  indeed,  in  a  masterly  series  of  investigations, 
arrived  at  a  second,  and  for  some  purposes  a  third,  approximation.     An 

solution,  however,  but  of  a  particular  ease  only,  was  arrived  at  by 
Rankine.  He  was  not  aware  that  it  had  been  given  by  Gerstner  in  1802, 
having  been  deduced  from  an  assumption,  generally  erroneous,  but  true  for 
this  special  case.  Unfortunately,  as  this  theory  essentially  involves  rota- 
tion of  fluid  elements,  it  is  not  a  solution  of  the  usual  problem  of  waves  at 
the  surface  of  a  perfect  liquid,  for  it  implies  a  kind  of  motion  which  could 
not  be  produced  in  such  a  substance  if  originally  in  a  state  of  rest. 

III.  Rankine's  third  investigation  is  that  of  lines  of  motion  of  water 
flowing  past  a  ship.  lie  begins  with  the  mathematical  theory  of  such 
lines,  but  soon  applies  his  results  to  the  determination  of  good  forms  of 
the  "line-;  of  a  ship,  and  the  investigation  of  the  principal  causes  of  the 
resistance  to  the  motion  of  the  ship,  and  the  means  of  diminishing  that 

nice. 

Xo  other  person  has  done  so  much  as  either  Rankine  or  William 
Froude  to  promote  naval  dynamics,  and  the  application  of  science  to  the 
shaping  of  ships,  and  to  the  estimation  of  their  performances. 

To  Rankine,  tin'  Scientific  Sub-Committee  of  the  late  Admiralty  Com- 
mittee on  Designs  owed  most  of  its  reports,  and  a  very  large  proportion 
of  their  effectiveness.  Even  those  most  disposed  to  disparage  that 
Committee  and  its  work,  have  made  exception  as  regards  the  Reports  of 
tlie  Scientific  Sub-Committee,  and,  in  particular,  Eankine's  contributions. 
(It  seems  to  us  that  only  ignorance  or  unfavourable  bias  could  attempt  to 
disparage  the  committee  at  all  ;  for  it  undoubtedly  did.  though  in  an 
unostentatious  manner,  very  good  sendee  indeed.) 

Eankine's  works  on  Applied  Mechanics,  on  the  Steam-Engine,  and  on 
Engineering,  contain  many  valuable  and  original  methods ;  and  while  the 


MEMOIK.  xxxiii 

publication  of  any  one  of  them  would  have  established  the  fame  of  one  of 
our  average  scientific  men,  that  on  the  steam-engine  could  not  have  been 
produced  by  any  but  an  original  discoverer  of  a  high  order.  Some  of  the 
investigations  contained  in  this  series  of  volumes  are  as  remarkable  for 
the  material  aid  they  afford  to  the  man  of  practice  as  for  the  light  they 
throw  upon  his  work. 

The  following  gives  a  striking  instance  of  Eankine's  tact  under  a  novel 
and  somewhat  puzzling  combination  of  circumstances.  In  August,  1858, 
he  wrote  to  the  Philosophical  Magazine  the  annexed  short  letter,  which 
was  printed  in  the  September  number  of  that  journal : — 

"  In  the  course  of  last  year  there  were  communicated  to  me,  in  con- 
fidence, the  results  of  a  great  body  of  experiments  on  the  engine  power 
required  to  propel  steamships  of  various  sizes  and  figures,  at  various 
speeds.  From  those  results  I  deduced  a  general  formula  for  the  resistance 
of  ships,  having  such  figures  as  usually  occur  in  steamers,  which,  on 
the  23rd  of  December,  1857,  I  communicated  to  the  owner  of  the 
experimental  data,  and  he  has  since  applied  it  to  practice  with  complete 
success. 

"  As  the  experimental  data  were  given  to  me  in  confidence,  I  am  for  the 
present  bound  in  honour  not  to  disclose  the  formula  which  I  deduced  from 
them ;  but  as  I  am  desirous  not  to  delay  longer  the  placing  it  upon  record, 
I  have  recourse  to  the  old  fashion  of  sending  it  to  you  in  the  form  of 
an  anagram,  in  which  the  letters  that  occur  in  its  verbal  statement  are 
arranged  in  alphabetical  order,  and  the  number  of  times  that  each  letter 
occurs  is  expressed  by  figures:— 20  A.  4  B.  6  C.  9  D.  33  E.  8  F.  4  G.  1G  H. 
10 1.  5  L.  3  M.  15  N.  14  0.  4  P.  3  Q.  14  R.  13  S.  25  T.  4  U.  2  V.  2  W.  1  X. 
4  Y.  (219  letters  in  all).  I  hope  I  may  soon  be  released  from  my  present 
obligation  to  secrecy." 

There  could  be  no  doubt  that  this  refers  to  a  remarkable  investigation 
which  Eankine  carried  out  for  his  friend  James  E.  Xapier,  who  had  asked 
him  to  estimate  the  horse-power  necessary  to  propel  at  a  given  rate  a 
vessel  which  he  was  about  to  construct.  Guided  by  this  consideration,  I 
found,  in  1872,  the  following  sentence  of  Eankine's  in  The  Civil  Engineer 
and  Architects  Journal  (October  1,  18G1),  but  without  any  reference  what- 
ever to  the  anagram  or  to  the  Plulosopldcal  Magazine : — 

"  The  resistance  of  a  sharp-ended  ship  exceeds  the  resistance  of  a 
current  of  water  of  the  same  velocity  in  a  channel  of  the  same  length  and 
mean  girth,  by  a  quantity  proportional  to  the  square  of  the  greatest 
breadth,  divided  by  the  square  of  the  length  of  the  bow  and  stern." 

Curiously  enough,  Eankine  seems  to  have  made  an  arithmetical  mistake, 
or  a  mis-spelling,  because  this  sentence  exactly  fits  all  of  the  above 
numerical  data,  with  the  exception  that  it  contains  just  one  E  too  much, 
and  has,  therefore,  220  letters  in  all.     Eankine's  private  MSS.,  to  which  I 


XXXIV  MEMOIR. 

have  recently  had  access,  show  that  my  guess  was  correct,  but  Jo  not 
enable  me  to  find  how  the  numerical  error  just  noticed  arose. 

Mr.  Napier  informed  me  that  in  all  his  business  relations  with  Rankine, 
nothing  had  so  much  impressed  him  as  the  rapid  and  keen  insight  with 
which  he  seemed  at  once  to  seize  upon  the  most  essential  points  in  the 
solution  of  a  practical  question,  though  stated  to  him  for  the  first  time; 
how  he  first  shook  himself  free  from  the  petty  complications,  and  gave 
almost  immediately  an  approximate  estimate  embracing  all  the  larger 
bearings  of  the  question;  and  then,  much  more  formally  and  deliberately, 
and  with  the  minute  accuracy  and  system  of  a  man  of  business,  proceeded 
to  work  the  question  with  the  desired  exactness.  Mr.  Napier  said  that 
on  the  occasion  of  his  first  consultation  with  Rankine  on  the  matter 
referred  to  in  the  anagram  above,  Rankin  e's  very  first  words  pointed  out 
to  him  what  a  large  proportion  of  the  resistance  to  a  vessel's  motion  is  due 
to  friction,  and  how  ill-considered  was  the  then  growing  demand  for  long 
and  narrow  ships. 

Kankine's  text-books  on  engineering  subjects  are  in  many  respects  the 
most  satisfactory  that  have  been  published  in  any  country.  At  the  time 
of  their  publication  they  have  always  been  in  advance  of  the  professional 
knowledge  of  the  day,  but  they  possess  much  greater  merits  than  that  of 
mere  novelty.  Rankine  was  peculiarly  happy  in  discriminating  between 
those;  branches  of  engineering  knowledge  which  grow  from  daily  experi- 
ence, and  those  which  depend  on  unchangeable  scientific  principles.  In 
his  books  he  dealt  almost  exclusively  with  the  latter,  which  may,  and 
certainly  will,  be  greatly  extended,  but  so  far  as  they  have  been  established 
can  never  change.  Hence  his  books  are  a  mine  which  smaller  men  may 
work  for  many  years,  rendering  his  knowledge  more  generally  available 
by  giving  it  a  popular  setting  of  their  own.  By  the  bulk  of  the  engineer- 
ing profession  the  books  are  considered  hard  reading,  but  as  engineering 
education  improves  they  will  more  and  more  be  recognised  as  both 
wonderfully  complete  and  essentially  simple.  Rankine,  by  his  education 
as  a  practical  engineer,  was  eminently  qualified  to  recognise  the  problems 
of  which  the  solution  is  required  in  practice;  but  the  large  scope  of  his  mind 
would  not  allow  him  to  be  content  with  giving  merely  the  solution  of 
those  particular  cases  which  most  frequently  occur  in  engineering  as  we 
now  know  it.  His  method  invariably  is  to  state  the  problem  in  a  very 
general  form,  find  the  solution,  and  then  apply  this  solution  to  special 
cases.  This  method  does  not  make  the  books  easy  reading  for  students, 
nor  does  it  give  the  most  convenient  work  of  reference  for  the  practical 
man ;  but  it  has  produced  writings  the  value  of  which  is  permanent, 
instead  of  being  ephemeral. 

In  his  Applied  Mechanics  we  have  the  best  existing  work  on  the  applica- 
tion of  the  doctrines  of  pure  mechanics  to  general  engineering  problems. 


MEMOIR.  XXXV 

No  specious  reasoning  has  been  detected  in  this  great  work,  a  fact  which 
should  for  ever  dispel  the  old  and  false  antithesis  between  theory  and 
practice — a  contrast  drawn  by  practical  men  who  never  understood  fully 
any  theory,  and  assented  to  by  scientific  men  who  were  not  candid 
enough  to  point  out  where  their  theories  were  incomplete.  In  the 
A\'ork  named  Civil  Engineering,  Rankine  applied  the  general  doctrines  of 
applied  mechanics  to  the  special  problems  which  the  civil  engineer  of 
to-day  meets  with  in  his  practice,  and  his  volume  contains  much  valuable 
statistical  information.  In  his  work  on  Prime  Movers  we  have  a  most 
thoroughly  original  statement  of  the  thermodynamic  theory,  so  far  as  it 
bears  on  the  design  and  use  of  the  steam-engine.  This  work  especially 
shows  Rankine's  clear  discrimination  of  what  is  permanent  and  can  be 
taught,  from  that  which  must  vary  from  day  to  day,  and  can  only  be 
acquired  by  personal  experience;  the  distinction  between  the  science  and  the 
art  of  the  engineer. 

His  treatise  on  Machinery  and  Millvorl  gives  the  mechanical  engineer 
instruction  of  a  kind  analogous  to  that  which  the  civil  engineer  may  derive 
from  the  book  called  Civil  Engineering.  The  problems  stated  generally  in 
the  Applied  Mechanics  are  in  it  applied  to  the  special  cases  which  arise  in  the 
design  of  machinery.  Several  of  these  .Manuals  have  been  recently 
translated  ;  Prime  Movers  into  French,  Civil  Engineering  into  German,  &c.  ; 
and  Machinery  and  Millwork  will  soon  appear  in  Italian. 

Most  of  the  common  treatises  on  engineering  subjects  are  mere  rechauffes 
or  compilations  ;  and  no  library  becomes  sooner  worthless  than  that  of  the 
engineer,  the  practice  of  this  year  being  wholly  different  from  that  of 
five  years  since.  Really  original  papers  and  monographs  rapidly  lose  their 
interest  and  importance,  except  as  historical  landmarks,  but  Eankine's 
works  will  retain  their  value  after  this  generation  has  passed  away. 

In  concluding  the  scientific  part  of  this  brief  notice  of  a  true  man,  we 
need  scarcely  point  out  to  the  reader  how  much  of  Rankine's  usefulness 
was  due  to  steady  and  honest  work.  The  unscientific  are  prone  to  imagine 
that  talent  (especially  when,  as  in  Rankine's  case,  it  rises  to  the  level  of 
genius)  is  necessarily  rapid  and  off-hand  in  producing  its  fruits.  No  greater 
mistake  could  be  made.  The  most  powerful  intellects  work  slowly  and 
patiently  at  a  new  subject.  Such  was  the  case  with  Newton,  and  so  it  is 
still.  Rapid  they  may  be,  and  in  general  are,  in  new  applications  of 
processes  long  since  mastered ;  but  it  is  only  your  pseudo-scientific  man 
who  forms  his  opinion  at  once  on  a  new  subject.  This  truth  was  pro- 
minently realised  in  Rankine,  who  was  prompt  in  reply  when  his  know- 
ledge was  sufficient,  but  patient  and  reticent  when  he  felt  that  more 
knowledge  was  necessary.  With  him  thought  was  never  divorced  from 
work — both  were  good  of  their  kind — the  thought  profound  and  thorough, 


XXXVI  MEMOIR. 

the  work  a  workman-like  expression  of  the  thought.  Few.  if  any, 
practical  engineers  have  contributed  so  much  to  abstract  science,  and  in 
no  case  has  scientific  study  been  applied  with  more  effect  to  practical 
engineering.  RanMne's  name  will  ever  hold  a  high  place  in  the  history 
of  science,  and  will  worthily  be  associated  with  those  of  the  great  men  we 
have  recently  lost.  And,  when  we  think  who  these  were,  how  strangely 
does  such  a  list  —  including  the  names  of  Babbage,  Boole,  Brewster, 
Leslie  Ellis,  Faraday,  Forbes,  Herschel,  Rowan  Hamilton,  Clerk-Maxwell, 
Rankine,  and  others,  though  confined  to  physical  or  mathematical  science 
alone— contrast  with  tfa  inning  utterance  of  the  Prime  Minister  of 

Cieat  Britain  and  Ireland,  to  the  effect  that  the  present  is  by  no  means  an 
age  abounding  in  minds  of  the  first  order !  Tin  such  men  lost  by  this 
little  country  within  the  last  dozen  years  or  so — any  one  of  whom 
would  have  made  himself  an  enduring  name  had  he  lived  in  any  pre- 
ceding it  that  of  Hooke  and  Newton,  or  that  of  Cavendish  and 
Watt:  Nay  more,  even  such  losses  as  these  have  not  extinguished  the 
hopes  of  science  amongst  i  .  E  ry  one  of  these  great  men  has,  by 
some  mysterious  influence  of  hi  .  kindled  the  sacred  thirst  for 
new  knowledge  in  yonngei  but  kindred  spirits,  many  of  whom  will 
certainly  rival,  some  even  may  excel,  their  teachers  ! 

For  the  dates  and  statements  of  fact  in  this  Memoir,  I  am  indebted 
mainly  to  RanMne's  private  MSS.,  access  to  which  has  been  given  me  by 
his  relative-.  Some  special  details  I  have  had  from  his  own  lips.  I  have 
also  to  acknowledge  my  obligations  to  Sir  William  Thomson,  to  Professor 
Jcnkin,  and  specially  to  the  late  Mr.  J.  R.  Napier,  who  was  one  of 
Kankine's  most  enthusiastic  admirers,  lie  furnished  me  with  much  of 
the  more  technical  part  of  the  materials  for  a  notice  of  Kankine's  scientific 
work,  which  I  wrote  immediately  after  his  death  for  the  Glasgow  U'.niJJ. 
December  28,  1872,  and  of  which  I  have  made  considerable  use  in  what 
precedes. 

Of  Kankine's  purely  scientific  work  I  have  spoken  from  actual  acquaint- 
ance with  his  writings ;  but  I  have  found  it  necessary  to  apply  for 
assistance  while  attempting  to  discuss  the  merits  of  his  more  practical 
investigations. 

P.  G.  TAIT. 


College,  Edinbetegh, 
October,  1880. 


PART    I. 

PAPERS  RELATING  TO  TEMPERATURE,  ELASTICITY,  AND 
EXPANSION  OF  VAPOURS,  LIQUIDS,  AND  SOLIDS. 


PAET    I. 

PAPERS  RELATING   TO  TEMPERATURE,  ELASTICITY,  AND 
EXPANSION  OF  VAPOURS,  LIQUIDS,  AND  SOLIDS. 


I.— OX  AN  EQUATION  BETWEEN  THE  TEMPERATURE  AND 
THE  MAXIMUM  ELASTICITY  OF  STEAM  AND  OTHER 
VAPOURS.     (See  Plate  I.)* 

In  the  course  of  a  series  of  investigations  founded  on  a  peculiar  hypothesis 
respecting  the  molecular  constitution  of  matter,  I  have  obtained,  among 
other  results,  an  equation  giving  a  very  close  approximation  to  the  maxi- 
mum elasticity  of  vapour  in  contact  with  its  liquid  at  all  temperatures 
that  usually  occur. 

As  this  equation  is  easy  and  expeditious  in  calculation,  gives  accurate 
numerical  results,  and  is  likely  to  be  practically  useful,  I  proceed  at  once 
to  make  it  known,  without  waiting  until  I  have  reduced  the  theoretical 
researches,  of  which  it  is  a  consequence,  to  a  form  fit  for  publication. 

The  equation  is  as  follows  : — 

Log.P  =  a-£-J2,  .  .  .      (1.) 

where  P  represents  the  maximum  pressure  of  a  vapour  in  contact  with 
its  liquid ; 

/,  the  temperature,  measured  on  the  air-thermometer,  from  a  point  which 
may  be  called  the  absolute  zero,  and  which  is — 

2740-6  of  the  Centigrade  scale  below  the  freezing  point  of  water  ; 
462°-28  of  Fahrenheit's  scale  below  the  ordinary  zero  of  that  scale, 

*  Originally  published  in  the  Edinburgh  New  Phibsophical  Journal  for  July,  1849. 

A 


2 


ON  THE  ELASTICITY  OF  VAPOURS. 


supposing  the  boiling  point  to  have  been  adjusted  under  a  pressure 
of  29-922  inches  of  mercury,  so  that  180°  of   Fahrenheit  may 
be  exactly  equal  to  100  Centigrade  degrees  j 
461°-93    below  the  ordinary  zero  of  Fahrenheit's   scale,  when  the 
boiling  point  has  been  adjusted  under  a  pressure  of  30  inches  of 
mercury,  180°  of  Fahrenheit  being  then  equal  to  100o-0735  of  the 
Centigrade  scale. 
The  form  of  the  equation  has  been  given  by  theory;  but  three  constants, 
represented  by  a,  [3,  and  y,  have  to  be  determined  for  each  fluid  by 
experiment. 

The  inverse  formula,  for  finding  the  temperature  from  the  pressure,  is 
of  course 


M 


-l0£ 


P+0* 


47- 


/3 

-V 


(2.) 


It  is  obvious  that  for  the  determination  of  the  three  constants,  it  is 
sufficient  to  know  accurately  the  pressures  corresponding  to  three  tem- 
peratures ;  and  that  the  calculation  will  be  facilitated  if  the  reciprocals  of 
those  temperatures,  as  measured  from  the  absolute  zero,  are  in  arithmetical 
progression. 

In  order  to  calculate  the  values  of  the  three  constants  for  the  vapour  of 
water,  the  following  data  have  been  taken  from  3L  Regnault's  experi- 
ments : — 


Temperatures  in  Cen- 
tigrade Degrees. 

Common 
Logarithms  of 
the  Pressure  in 
Millimetres  of 

Mercury. 

Remarks. 

Above  the 

Freezing 

Point. 

Above  the 

Absolute 

Zero. 

o 

220- 

100- 
26-86 

o 

494-6 

374-6 

301-46 

o 
4-2403 

2-S80S136 
1-419S 

(  Measured  by  M.   Eegnault   on  his  curve, 
<      representing  the  mean  results  of  his  ex- 
(      periments. 

Logarithm  of  760  millimetres. 

1  Calculated  by  interpolation  from  M.  Eeg- 
{      nault's  general  table. 

These  data  give  the  following  results  for  the  vapour  of  water,  the 
pressures  being  expressed  in  millimetres  of  mercury,  and  the  temperatures 
in  Centierade  decrees  of  the  air-thermometer  : — 


Log.  7  =  5-0827176  Log.  (5- 

a=7'831247 


3-1851091 


ON  THE  ELASTICITY  OF  VAPOURS. 


111 

<d  >  a 

PiO'B 

a-s  o 

§  d  P 

O  OO  O  O  O  OOOO  OOOO     _J 

Ma-  «; 

BOP 

oortTfcowaTfiMH-iwoo 
00090090090000    —. 

0 6 000 0 0000 6 6 00 0 0 

I  ++  -l  +  +  +  +  I  +  +       1     C- 

Differences  be- 
tween Calcula- 
tion and  Experi- 
ment iu  Loga- 
rithms. 

ONflLmN-jdMHHWOil 

00000000000000 
00000000000000    -~-~ 
oooopopoopoooo    ^ 

b bb b b b bo b b b b bb   — ' 
+  l    l  +  1    1    l    1  +  l    l       + 

rithms  of  the 

Millimetres  j 
ng  to 

M.  Roguault's 
Experiments. 

OOMNM«N1QN-*NOOO      ■— » 
QOMNOMOtSNl^NOO-ttl      «! 
KOh«^i"CN00OOh«M     -ii 

cicococococococococo-*'*"*-* 

Common  Loga 

Pressures  in 

accord 

the  Formula. 

0 

CCN^HO-ttOMOWtCOOM 
^h  0  0  0 r-  CO  IO  O  O  ~TI  0  0  0  co 
CO  CO  01  O  O  CO  CO  0  O  -t*  01 1>-  co  O     ■ — - 

OHWt^nMOTtn^^NOOH    -• 

OCWNOWnONt^NOlO-fCl      i_- 

00  O  -h  co  -*  O  O  r-  co  O  O  r-i  Ol  co 

dcoeocococococococo-^-^-'*-* 

&0a-      ■ 

a  o  g 

IP 

8  a  •- 

©   ©  £* 

Hta  a 
S-  o 
D  =  H 

01 10  O  CO  CO  0  0  — 1  10  0  101  •■+  CO  >-<  O  CO  01  01  01  -t<  CO  O  t~-  O  O  CI  0  CO 

■*  01  pcp-Hpoppppppppppppppppppppp    ,_, 

00666006606000066660666060600      CO 

I+  +  +  +  +      lllllll       1    1  +  I    1  +  +  +  1    1    1       1    "-' 

Difference  be- 
tween Calcu- 
lation and  Ex- 
periment in 
Millimetres. 

H«-J«^e0M^»00C150IM 
OOO'-irtOOOHeiNK^C'lOrtHOeOia^ODMiS                             _ 

ooooooooooooooo^^-hocococo— <  0  >n  r~»  0  0    tei 

+  111  11    +  +  +  +  +  +  -+-    ++•++'  <7+++    +  "" 

[illimetres  of 
cording  to 

M.  Regnault's 
Experiments. 

«*-hcOOOOOO-hCOOOtHiOOO 

noooHMti  10  00  r-  ©  0  •**<  ©  t—  00000000 

OOtl^eiN'JSHiiHOJMTllOOMOOKflNOnWOMOlO     -7 
HNMOO'l'nLOtlONiK'NHN-t'O-r'M'OOOH      _u 

h  ci  n  lo  t>  0  ■#  0  n  «  3  c.  >o  -;<  0  co  co  0    c^ 

1-1  1— 1  01  01  CO  -*  10  t-  O  — 1  -P  t^  0 

HHr-lCl 

Pressures  in  1 
Mercury,  ac 

the  Formula. 

«oi-.nw«oi>  10  onanooci 
cocpp^pcpq<lippo^7H^pi^pcp-7Hpcp>pp01t>>0 

6  6  Ol  ■*  b  l--  b  ^h  b  01  b  co  10  b  6  <*  6  co  co  0  co  ^-1  eo  00  »o  10  0  0       • 

Hcmoo"f  mioci'onoci- u>-#w«on — 0  -p    jo 

rH  Ol  CO  O  C-  O  Tfl  0  t-  l-O  0  Ci  l.O  -*  0  CO  co  0     ^ 
1-1  1— lOlOlCO^Ot-O^-^r-O 

i— 1  r— 1  1— 1  01 

es  in  Centi- 
reos  of  the 
motor  from 

the  Abso- 
lute Zero. 

O 

OOOOOO-^OOOOOOOOOOOOOOOOOOOO©     ^_, 
^OOM»C!OOHiMW#OOM»OOHCin-)<lOOI->tOOO      ^~- 

oioioio^o-ic^coeocococococococococoT]HTj<-*TjH-^iTi(T|H-^-*Triin 

Temperatur 
grade  Deg 

Air-Thormo 

the  Freez- 
ing Point. 

0 

CO 

0000000  b  000000000000000000000    ^ 

CO  01  --(         HCKIM^OOt-OOOO-i^M^OOIXBOO-'JlM         • 

4  ON  THE  ELASTICITY  OF  VAPOURS. 

Table  I.  exhibits  a  comparison  between  the  results  of  the  formula  and 
those  of  M.  Eegnault's  experiments,  for  every  tenth  degree  of  the  Centi- 
grade air-thermometer,  from  30°  below  the  freezing  point  to  230°  above 
it,  being  within  one  or  two  degrees  of  the  whole  range  of  the  experi- 
ments. 

M.  Eegnault's  values  are  given,  as  measured  by  himself,  on  the  curves 
representing  the  mean  results  of  his  experiments,  with  the  exception 
of  the  pressures  at  26°*86,  one  of  the  data  already  mentioned,  and  that 
at  -  30°,  which  I  have  calculated  by  interpolation  from  his  table, 
series  h. 

Each  of  the  three  data  used  in  determining  the  constants  is  marked 
with  an  asterisk  *. 

In  the  columns  of  differences  between  the  results  of  the  formula  and 
those  of  experiment,  the  sign  +  indicates  that  the  former  exceed  the  latter, 
and  the  sign  —  the  reverse. 

Beside  each  such  column  of  differences  is  placed  a  column  of  the  cor- 
responding differences  of  temperature,  which  would  result  in  calculating 
the  temperature  from  the  pressure  by  the  inverse  formula.  These  are 
found    by    multiplying    each    number    in    the    preceding    columns    by 

dt       .         -dt 

— z=  or  by  -7i =  as  the  case  may  require. 

dP         J  d  log.  P  J      n 

In  comparing  the  results  of  the  formula  with  those  of  experiment,  as 
exhibited  in  Table  I.,  the  following  circumstances  are  to  be  taken  into 
consideration  : — 

First,  That  the  uncertainty  of  barometric  observations  amounts  in 
general  to  at  least  one-tenth  of  a  millimetre. 

Secondly,  That  the  uncertainty  of  thermometric  observations  is  from 
one-twentieth  to  one-tenth  of  a  degree  under  ordinary  circumstances,  and 
at  high  temperatures  amounts  to  more. 

Thirdly,  That,  in  experiments  of  the  kind  referred  to  in  the  table,  those 
two  sorts  of  uncertainty  are  combined. 

The  fifth  column  of  the  table  shows  that,  from  30°  below  the  freezing 
point  to  20°  above  it,  where  the  minuteness  of  the  pressures  makes  the 
barometric  errors  of  most  importance,  the  greatest  difference  between 
experiment  and  calculation  is  y1^  of  a  millimetre,  or  giro  °f  an  mcn  °^ 
mercury,  a  very  small  quantity  in  itself,  although,  from  the  slowness 
with  which  the  pressure  varies  at  low  temperatures,  the  corresponding 
difference  of  temperature  amounts  to  -££$  of  a  degree. 

The  sixth  and  tenth  columns  show  that,  from  20°  to  230°  above  the 
freezing  point,  the  greatest  of  the  discrepancies  between  experiment  and 
observation  corresponds  to  a  difference  of  temperature  of  only  jfa  of  a 
degree,  and  that  very  few  of  those  discrepancies  exceed  the  amount 
corresponding  to  -^  of  a  degree. 


ON  THE  ELASTICITY   OF  VAPOURS.  5 

A  comparison  between  the  sixth  and  tenth  columns  shows  that,  for 
four  of  the  temperatures  given— viz.,  120°,  150°,  200°,  and  210°— the 
pressures  deduced  from  M.  Eegnault's  curve  of  actual  elasticities,  and 
from  his  logarithmic  curve  respectively,  differ  from  the  pressures  given 
by  the  formula  in  opposite  directions. 

If  the  curves  represented  by  the  formula  were  laid  down  on  M. 
Regnault's  diagram,  they  would  be  almost  undistinguishable  from  those 
which  he  has  himself  drawn,  except  near  the  freezing  point,  where  the 
scale  of  pressures  is  very  large,  the  heights  of  the  mercurial  column  being 
magnified  eight-fold  on  the  plate.  In  the  case  of  the  curves  of  logarithms 
of  pressures  above  one  atmosphere,  the  coincidence  would  be  almost 
perfect. 

The  formula  may,  therefore,  be  considered  as  accurately  representing 
the  results  of  all  M.  Regnault's  experiments  throughout  a  range  of 
temperatures  from  30°  of  the  Centigrade  scale  below  the  freezing  point  to 
230°  above  it,  and  of  pressures  from  o^ott  °f  an  atmosphere  up  to  28 
atmospheres. 

It  will  be  observed  that  equation  (1.)  bears  some  resemblance  to  the 
formula  proposed  by  Professor  Roche  in  1828,  viz. : — 

■p 
W.P=A 


T  +  C 


where  T  represents  the  temperature  measured  from  the  ordinary  zero 
point,  and  A,  B,  and  C,  constants,  which  have  to  be  determined  from 
three  experimental  data.  It  has  been  shown,  however,  by  M.  Regnault, 
as  well  as  by  others,  that  though  this  formula  agrees  very  nearly  with 
observation  throughout  a  limited  range  of  temperature,  it  errs  widely 
when  the  range  is  extensive.  I  have  been  unable  to  find  Professor 
Roche's  memoir,  and  I  do  not  know  the  reasoning  from  which  he  has 
deduced  his  formula. 

The  use  in  computation  of  the  equations  I  have  given,  whether  to 
calculate  the  pressure  from  the  temperature,  or  the  temperature  from  the 
pressure,  is  rapid  and  easy.  In  Table  II.  they  are  recapitulated,  and  the 
values  of  the  constants  for  different  measures  of  pressure  and  temperature 
are  stated. 

In  calculating  the  values  of  a,  the  specific  gravity  of  mercury  has  been 
taken  as  13'596. 

Temperatures  measured  by  mercurial  thermometers  are  in  all  cases  to 
be  reduced  to  the  corresponding  temperatures  on  the  air-thermometer, 
which  may  be  done  by  means  of  the  table  given  by  M.  Regnault  in  his 
memoir  on  that  subject. 


ON  THE  ELASTICITY   OF  VAPOURS. 


Table  II. — Vapour  of  Water. 

Formula  for  calculating  the  Maximum  Elasticity  of  Steam  (P),  from 
the  Temperature  on  the  Air-Thermometer,  measured  from  the  Absolute 
Zero  (0  :— 

Inverse  Formula  for  calculating  the  Temperature  from  the  Maximum 
Elasticity  of  Steam  : — 

1  /q-log.P       ffl"  _  g_ 

t        >        7  472       2y 

Values  of  the  Constants  depending  on  the  Thermometric  Scale : 

For  the  Centigrade  scale  : — 

Absolute  zero  274°'6  below  the  freezing  point  of  water. 

Log.  /3  =  3-1851091  Log.  7  =  5-0827176 

£-  =  0-00G3L".)  i  -ft-  =  0-00004006 

For  Fahrenheit's  scale;  boiling  point  adjusted  at  29-922  inches: — 
Absolute  zero  462°*28  below  ordinary  zero. 
Log.  |3  -  3-4403816  Log.  y  =  5-5932626 

-£=0-0035163  -ft=  0-000012364 

-  y  4y- 

For  Fahrenheit's  scale ;  boiling  point  adjusted  at  30  inches  : — 
Absolute  zero  461°*93  below  ordinary  zero. 
Log.  /3  =  3-4400625  Log.  y  =  5-5926244 

ft=  0-0035189  -ft  =  0-000012383 

2-y  47^ 

Values  of  the  Constant  a  depending  on  the  Measure  of  Elasticity  : 

For  millimetres  of  mercury,         ....        a  =7-831247 
English  inches  of  mercury,    ....  6-426421 


ON   THE  ELASTICITY  OF  VAPOURS. 

Atmospheres   of    760    mil.  =  29*922  inches^ 
=  14-7  lbs.  on  the  square  inch=l-0333  kil.  v      . 
on  the  centimetre2, .         .         .         .         . ) 

Atmospheres  of  30  inches  =  761  mil.  "99  -\ 
=  14'74  lbs.  on  the  square  inch=r  T036  kil.  V 
on  the  centimetre2, .  .         .         .  .J 

Kilogrammes  on  the  square  centimetre, 

Pounds  avoirdupois  on  the  square  inch, 

N.B. — All  the  Constants  are  for  common  logarithms. 


4-950433 


4-949300 

4-964658 
6-117817 


I  have  applied  similar  formula?  to  the  vapours  of  alcohol  and  ether, 
making  use  of  the  experiments  of  Dr.  Ure. 

In  order  to  calculate  the  constants,  the  following  experimental  data 
have  been  taken,  assuming  that,  on  Dr.  Ure's  thermometers,  180°  were 
equal  to  100  Centigrade  degrees. 


For     Alcohol    of     the 
specific  gravity  O'Slo, 


For  Ether,  boiling  at  ) 
105°  F.,  under  30  [ 
inches  of  pressure,        ) 

For  Ether,  boiling  at  ) 
104°  F.,  under  30  [ 
inches  of  pressure,       ) 


Temperatures  on 
Fahrenheit's  Scale 

from 
the  Ordi-   the  Abso- 
nary  Zero,  iuto  Zero. 


250 
173 
111-02 


200 

14S-8 

105 

104 

G6-7 
34 


7123 
635-3 
573  32 


662-3 

611-1 

507-3 

566-3 
529-0 
496-3 


Pressures 
in  Inches 
of  Mer- 
cury. 


132-30 

30  00 

6-30 


142-8 
66-24 
30-00 

30-00 

1376 

6  20 


From  Dr.  Ure's  Table. 

Do. 
Interpolated    in    the    same 
Table. 

From  Dr.  Ure's  Table. 

Interpolated. 
From  the  Table. 

From  Dr.  Ure's  Table. 
Interpolated. 
From  the  Table. 


The  values  of  the  constants  in  equation  (1.),  calculated  from  these  data, 
are  as  folloAvs,  for  inches  of  mercury  and  Fahrenheit's  scale  : — 


Alcohol,  specific  gravity  0-813, 
Ether,  boiling  point  105°F., 
Ether,  boiling  point  104°F., 


6-16620 
5-33590 
5-44580 


Log.  £. 


3-3165220 
3-20S4573 
3-2571312 


Absolute  zero  462°  '3  below  ordinary  zero. 


Log.  y. 


5-7602709 
5-5119893 
5-3962460 


8  ON   THE  ELASTICITY   OF  VAPOURS. 

The  curves  represented  by  the  formulae  for  those  three  fluids  are  laid 
down  on  the  diagram  which  accompanies  this  memoir  (Plate  I.),  and 
which  has  been  reduced  to  one-fourth  of  the  original  scale.  The  horizontal 
divisions  represent  the  scale  of  Fahrenheit's  thermometer,  numbered  from 
the  ordinary  zero ;  the  vertical  divisions,  pressures  of  vapour,  according 
to  the  scales  specified  on  the  respective  curves.  The  points  corresponding 
to  the  experimental  data  are  surrounded  by  small  circles. 

The  curve  for  alcohol  extends  from  32°  to  264°  of  Fahrenheit.  It  is 
divided  into  two  portions,  having  different  vertical  scales,  suitable  to 
high  and  low  pressures  respectively. 

The  curve  for  the  less  volatile  ether  extends  from  105°  to  210°;  that 
for  the  more  volatile  ether,  from  34°  to  104°. 

The  results  of  Dr.  Ure's  experiments  are  marked  by  small  crosses. 

The  irregular  and  sinuous  manner  in  which  those  crosses  are  dis- 
tributed, indicates  that  the  errors  of  observation,  especially  at  high 
temperatures,  must  have  been  considerable.  This  does  not  appear 
surprising,  when  we  recollect  how  many  causes  of  uncertainty  affect  all 
the  measurements  required  in  such  experiments,  especially  the  ther- 
mometric  observations,  and  how  little  those  causes  have  been  understood 
until  very  recently.  The  data  from  which  the  constants  have  been 
calculated  arc,  of  course,  affected  by  the  general  uncertainty  of  the 
experiments. 

When  those  circumstances  are  taken  into  account,  it  is  obvious,  from 
inspection  of  the  diagram,  that  the  curves  representing  the  formulae  agree 
with  the  points  representing  the  experiments  as  nearly  as  the  irregularity 
of  the  latter  and  the  uncertainty  of  the  data  permit ;  and  that  there  is 
good  reason  to  anticipate  that,  when  experiments  shall  have  been  made 
on  the  vapours  of  alcohol  and  ether  with  a  degree  of  precision  equal  to 
that  attained  by  M.  Kegnault  in  the  case  of  the  vapour  of  water,  the 
equation  will  be  found  to  give  the  elasticities  of  those  two  vapours  as 
accurately  as  it  does  that  of  steam. 

Although  the  diagram  affords  the  best  means  of  judging  of  the 
agreement  between  calculation  and  experiment,  three  tables  (III.,  IV., 
and  V.)  are  annexed,  in  order  to  show  the  numerical  amount  of  the 
discrepancies  at  certain  temperatures.  The  data,  as  before,  are  marked 
with  asterisks. 

It  is  worthy  of  remark,  in  the  case  of  alcohol,  that  although  the  lowest 
of  the  experimental  data  is  at  the  temperature  of  lllo-02,  the  formula 
agrees  extremely  well  with  the  experiments  throughout  the  entire  range 
of  79  degrees  Mow  that  point. 


y 

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Degrees  ofJFafavnheits  TIhti 


neiev.^Scaie.  Ihr  Alcohol  ,uui  Etlwt- 


BOSTON  U 

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LIBR 


I 


ON  THE  ELASTICITY   OF  VAPOURS. 


Table  III. — Vapour  of  Alcohol,  of  the  Specific  Gravity  0-813. 


Temperatures  in 

Pressures  in  Inches  of  Mercury 

Differences  be- 

Degrees of 

Fahrenheit  from 

the  Ordinary 

Zero. 

accord 
the  Formula. 

ing  to 

Dr.  Ure's 
Experiments. 

tween  Calculation 

and  Experiment  in 

Inches. 

Corresponding 
Differences  of 
Temperature. 

32 

0  41 

0  40 

+  0  01 

-0-5 

40 

0  57 

0  56 

+  001 

-0-4 

50 

0-84 

0S6 

-0-02 

+  0-7 

60 

1-22 

1-23 

-001 

+  02 

70 

1-74 

1-76 

-0  02 

+  0-3 

80 

2  43 

2-45 

-0  02 

+  0-2 

90 

3  36 

3  40 

-004 

+  04 

100 

456 

4  50 

+  006 

-0-5 

110 

6-12 

6  00 

+  0-12 

-0-7 

*lll-02 

6  30 

6  30 

0  00 

0  0 

120 

8-10 

8-10 

000 

00 

130 

1061 

10  60 

+  0  01 

-00 

140 

1373 

13  90 

-0-17 

+  05 

150 

17-60 

18-00 

-0-40 

+  0-9 

160 

22-32 

22-60 

-0-28 

+  0-5 

170 

2S-06 

28-30 

-0  24 

+  04 

*173 

30-00 

30-00 

0  00 

0  0 

180 

34  96 

34  73 

+0-23 

-03 

190 

43-21 

43-20 

+  0  01 

-0  0 

200 

52-96 

5300 

-0  04 

+o-o 

210 

64  47 

65  00 

-0  53 

+  0-5 

220 

77  92 

78-50 

-0-58 

+  0-4 

230 

93-54 

9410 

-0-56 

+  04 

240 

111-58 

111-24 

+  034 

-0-2 

*250 

132  30 

132-30 

o-oo 

0  0 

260 

155-98 

155-20 

+  0-78 

-0-3 

264 

165 -5S 

166-10 

-0  52 

+02 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

Table  IV. — Vapour  of  Ether;  Boiling  Point,  105°F. 


Temperatures  in 

Pressures  in  Inches  of  Mercury 

Differences  be- 

Degrees of 

according  to 

tween  Calculation 

Corresponding 

Fahrenheit  above 

and  Experiment 

Differences  of 

the  Ordinary 
Zero. 

the  Formula. 

Dr.  Ure's 
Experiments. 

in  Inches  of 
Mercury. 

Temperature. 

*105 

30O0 

30  00 

o-oo 

00 

no 

33 -OS 

32  54 

+  0  54 

-09 

120 

39-98 

39  47 

+  0-51 

-0-7 

125 

43-83 

43-24 

+  0-59 

-OS 

130 

47-95 

47-14 

+  0-S1 

-10 

140 

57-10 

56  90 

+  0-20 

-02 

♦148-8 

66  24 

66-24 

0  00 

0  0 

150 

67  53 

67  60 

-0  07 

+  0-1 

160 

79  35 

SO -30 

-0  95 

+  09 

170 

92-68 

92  80 

-012 

+  0-1 

175 

99-94 

99-10 

+  0-84 

-0-6 

180 

107-62 

108-30 

-0  68 

+  04 

190 

124-29 

124  80 

-051 

+  03 

*200 

142-80 

142-80 

0  00 

o-o 

205 

152  -78 

151-30 

+  1-48 

-0-7 

210 

163-27                   166  00 

l 
1 

-2-73 

+  11 

10  ON  THE   ELASTICITY  OF   VAPOURS. 

Table  V. — Vapour  of  Ether;  Boiling  Point,  104°F. 


Temperatures  in 

Pressures  in  Inches  of  Mercury 

Differences  be- 

Degrees of 

according  to 

tween  Calculation 

Corresponding 

Fahrenheit  above 

and  Experiment 

Differences  of 

the  Ordinary 
Zero. 

the  Formula. 

Dr.  Dre's 
Experiments. 

in  Inches  of 
Mercury. 

Temperature. 

*34 

6-20 

6  20 

o-oo 

0  0 

44 

S-02 

s-io 

-0-08 

+  0-4 

54 

10-24 

10  30 

-0  06 

+  0-2 

64 

12-94 

13  00 

-0-06 

+0-2 

*66-7 

13  76 

13-76 

0  00 

0  0 

74 

16-19 

16-10- 

+  0  09 

-02 

84 

20  06 

20  00 

+  0-06 

-01 

94 

24-64 

24-70 

-0  06 

+  01 

*104 

30-00 

30-00 

0  00 

00 

(1.). 

(2.) 

(3.) 

(4.) 

(5.) 

The  results  of  Dr.  Ure's  experiments  on  the  vapours  of  turpentine  and 
petroleum  are  so  irregular  (as  the  diagram  shows),  and  the  range  of 
temperatures  and  pressures  through  which  they  extend  so  limited,  that 
the  value  of  the  constant  y  cannot  be  determined  from  them  with 
precision.  I  have,  therefore,  endeavoured  to  represent  the  elasticities 
of  those  two  vapours  approximately  by  the  first  two  terms  of  the  formula 
only,  calculating  the  constants  from  two  experimental  data  for  each  fluid. 
The  equation  thus  obtained 


Los.  P 


a  — 


is  similar  in  form  to  that  of  Professor  Roche. 

The  data  and  the  values  of  the  constants  are  as  follows  : — 


Temperatures  ( 
Scalo 

the  Ordinary 
Zero. 

m  Fahrenheit's 
from 

the  Absolute 
Zero. 

Pressures  in 
Inches  of 
Mercury. 

Values  of  the  Constants  for 

Fahrenheit's  Scale  and 

Inches  of  Mercury. 

360 
304 

370 
316 

S22-3 
766-3 

S32-3 

778-3 

Turpentine. 
60 -SO 
30-00 

Petroleum. 
60-70 
30-00 

*  =  5-981S7 
Log.  |3  =  3  -5380701 

*=6-19451 
Log.  |3  =  3  -5648490 

Although  the  temperatures  are  much  higher  than  the  boiling  point  of 
water,  I  have  not  endeavoured  to  reduce  them  to  the  scale  of  the  air- 
thermometer,  as  it  is  impossible  to  do  so  correctly  without  knowing  the 
nature  of  the  glass  of  which  the  mercurial  thermometer  was  made. 


ON  THE   ELASTICITY   OF  VAPOURS. 


11 


The  diagram  shows  that  the  formula  agrees  with  the  experiments  as 
well  as  their  irregularity  entitles  us  to  expect. 

The  followinc;  tables  crive  some  of  the  numerical  results  : — 


Table  VI. — Vapour  of  Turpentine. 


Temperatures  in 

Degrees  of 

Fahrenheit  from 

the  Ordinary 

Zero. 

Pressures  in  In 
accord 

the  Formula 
(of  two  terms). 

ehes  of  Mercury 
ing  to 

Dr.  lire's 
Experiments. 

Differences  be- 
tween Calculation 
and  Experiment 
in  Inches  of 
Mercury. 

Corresponding 
Differences  of 
Temperature. 

*304 

30-00 

30-00 

o-oo 

o 

0  0 

310 

32  52 

33-50 

-0-9S 

+  2-3 

320 

37  09 

37-06 

+0  03 

-o-o 

330 

42-16 

42-10 

+  0-06 

-o-i 

340 

47-78 

47-30 

+  0-48 

-0-9 

350 

53  98 

53-80 

+  0-18 

-0-3 

*360 

60-80 

60 -SO 

o-oo 

o-o 

362 

62-24 

62-40 

-0-16 

+o-o 

Table  VII. — Vapour  of  Petroleum. 


Temperatures  in 

Pressures  in  Inches  of  Mercury 

Differences  be- 

Degrees of 

according  to 

tween  Calculation 

Corresponding 

Fahrenheit,  from 

and  Experiment 

Differences  of 

the  Ordinary 

the  Formula 

Dr.  Ure's 

in  Inches  of 

Temperature. 

Zero. 

(of  two  terms). 

Experiments. 

Mercury. 

*316 

30  00 

30-00 

o-oo 

o 

o-o 

320 

31*71 

31-70 

+001 

-00 

330 

36-35 

36-40 

-0-05 

+0T 

340 

41-52 

41-60 

-0-08 

+  0-2 

350 

47-27 

46-86 

+  0-41 

-0-7 

360 

53-65 

53-30 

+  0-35 

-0-5 

*370 

60-70 

60-70 

o-oo 

0  0 

375 

64-50 

64-00 

+  0-50 

-0-7 

(1.) 

(2.) 

(3.) 

(±.) 

(5.) 

I  have  also  endeavoured,  by  means  of  the  first  two  terms  of  the 
formula,  to  approximate  to  the  elasticity  of  the  vapour  of  mercury,  as 
given  by  the  experiments  of  M.  Kegnault.  The  data  and  the  constants 
are  as  follows  : — 


Temperatures 
Degree 

the  Freezing 
Point. 

in  Centigrade 
s  from 

the  Absolute 
Zero. 

Pressures  in 

Millimetres  of 

Mercury. 

Values  of  the  Constants  in  the 
Formula 

Log.  P  =  i-f 

358 
177-9 

632-6 
452-5 

760-00 
10-72 

a.  for  millimetres            =  7  "5305 
,,  for  English  inches          6T259 
Log.  /3  Centigrade  scale            3  -4685511 
,,     Fahrenheit's      scale,  \ 

*£SF  S*ii(»w" 

inches,     .     .     .     .  ) 

12 


ON    TOE    ELASTICITY    OF    VArOURS. 


The  following  table  exhibits  the  comparative  results  of  observation  and 
experiment  : — 

Tu'.i.r.  VIII. — Vapour  of  Mercury. 


TatnjiiTiiturrs  in 
Centigrade  Degreei 

i  i      area  In  Millimetres  or  Mercury 
ding  to 

DIfferenoea  iiotwoou 
Oaloulatlon  and 

from  Hi"  l''i  I 

Point 

tin>  Formula 
(.if  two  tenna), 

M.  ltognaull's 
Experiments. 

i1  raerlment  in 
Millimetres. 

7274 

0-115 

0-183 

-0  068 

10011 

0-480 

((•107 

+0-073 

100  6 

0-49 

one, 

-0-07 

146-3 

3-49 

3-46 

+  0  03 

•1779 

10*72 

10-72 

o-oo 

200-5 

2 1  -85 

22-01 

-016 

»358  0 

760  00 

760  00 

0  00 

The  discrepancies  are  obviously  of  the  order  of  errors  of  observation, 
and  the  formula  may  be  considered  correct  for  all  temperatures  below 
200°C,  and  for  a  short  range  above  that  point  From  its  wanting  the 
third  term,  however,  it  will  probably  be  found  to  deviate  slightly  from 
the  truth  between  200°  and  358°;  while  above  the  latter  point  it  must 
not  be  relied  oil. 

I  have  not  carried  the  comparison  below  72°,  because  in  that  part 
of  the  scale  the  whole  pressure!  becomes  of  the  order  of  errors  of 
observation. 

In  conclusion,  it  appears  to  me  that  the  following  proposition,  to  which 
1  have  been  led  by  the  theoretical  researches  referred  to  at  the  commence- 
ment of  this  paper,  is  borne  out  by  all  the  experiments  I  have  quoted, 
especially  by  those  of  greatest  accuracy,  and  may  be  safely  and  usefully 
applied  to  practice. 

If  the  maximum  elasticity  of  any  vapour  in  contact  with  its  liquid  be 
tained  for  three  points  on  the  scale  of  the  air-thermometer,  then  the  constants 
of  an  equation  of  the  form 

may  be  determined,  which  equation  will  give,  for  that  vapour,  with  an  accuracy 
limited  only  by  the  errors  of  observation,  the  relation  between  the  temperature  (t), 
measured  from  the  absolute  zero  (27 4'6  Centigrade  degrees  beloiu  the  freezing 
point  of  ivatcr),  and  the  maximum  elasticity  (P),  at  all  temperatures  between 
those  three  points,  and  for  a  considerable  range  beyond  them. 


THE   EXP„  ?  LIQUIDS   EY  HI  13 


EL— OS   A   FORMULA  FOB    CALCULATING  THE  EXPAJ 

:    LIQUIDS   BY  HEAT 

Hating  been  lately  much  engaged  in  researches  involving  the  comparative 
volumes  of  liquids  at  various  temperatures,  I  have  found  the  following 
formula  very  useful : 

Log.  Y  represents  the  common  logarithm  of  the  volar.  .--.ass 

of  liquid,  as  compared  with  its  volume  at  a  certain  standard  temper; 
which,  for  wa:  temperature  of  its  maximum  density,  or  4**1 

i  ie,  and  for  other  liqui  1  rigrade. 

?  is  the  temperature  measured  from  the  absolute  zero  mentioned  in 
paper  on  the  of  Vapours,  in  * 

oi  for  July.   184        as  preat&mg  Vapct  .  ari  u  found  by  a/1  - 

.•  "    : 
B.  and  C,  are  three  constants,  depending  on  the  natur : 
liquid,  whose  values  for  the  Centigrade  scale,  corresponding  to  water, 


Watg*, 

a:-..-/--.\    . 


Log.B_ 

: 

1    ■ 

I  ■    :7"4 

'         - 

(HB29130 

'       .- 

' 

.  15033 

'  - 

1-289X86 

-■ 

'  - 

1-2192054 

HaHstrom :  for  mercury,  from  those  of  Begnault :  and  for  alcohol  and 
n.~:.-:~;  .:  :i:  .-.  :r;n  :._:.-  .:  .--.;-- :'.-..'  As  t:.^  :.~;  :.v-.\ :  : 
1L  Gay-Lossae  give  only  the  apparent  expansion  of  the  liquids  in  glass,  I 
have  assumed,  in  order  to  calrnlate  the  true  expansion,  that  the  dilatation 

"    '.  :..  .l:  .  -I  -  --  EL.  %-.:—.':   ':■        .  •  -    :'  -".:..-.  ':-■' 


14 


THE   EXPANSION   OF   LIQUIDS  BY  HEAT. 


of  the  glass  used  by  him  was  -0000258  of  its  volume  for  each  Centigrade 
degree.  This  is  very  nearly  the  mean  dilatation  of  the  different  kinds  of 
glass.  M.  Kegnault  has  shown  that,  according  to  the  composition  and 
treatment  of  glass,  the  coefficient  varies  between  the  limits  "000022  and 
•000028. 

Annexed  are  given  tables  of  comparison  between  the  results  of  the 
formula  and  those  of  experiment.  The  data  from  which  the  constants 
were  calculated  are  marked  with  asterisks. 

The  table  for  water  shows,  that  between  0°  and  30°  Centigrade,  the 
formula  agrees  closely  with  the  experiments  of  Hallstrom,  and  that  from 
30°  to  100°  its  results  lie  betAveen  those  of  the  experiments  of  Gay-Lussac 
and  Deluc. 

The  experiments  of  Gay-Lussac  originally  gave  the  apparent  volume  of 
water  in  glass  as  compared  with  that  at  100°.  They  have  been  reduced 
to  the  unit  of  minimum  volume  by  means  of  Hallstrijm's  value  of  the 
expansion  between  40,1  and  30°,  and  the  coefficient  of  expansion  of  glass 
already  mentioned. 

In  the  fifth  column  of  the  table  of  comparison  for  mercury,  it  is  stated 
which  of  the  experimental  results  were  taken  from  M.  Eegnault's  own 
measurements  on  the  curve,  representing  the  mean  results  of  his  experi- 
ments, and  which  from  his  tables  of  actual  experiments,  distinguishing  the 
series. 

In  the  experimental  results  for  alcohol  and  sulphuret  of  carbon,  the 
respective  units  of  volume  are  the  volumes  of  those  liquids  at  their  boiling 
points,  and  the  volumes  given  by  the  formula  have  been  reduced  to  the 
same  units. 

Expansion  of  Water. 


Temperature  on 

Volume  as  compared  with  that  at 

Difference  between 

Authorities  for 

the  Centigrade 

4"1  C.  ac( 

oraing  to 

Calculation  and 

the 

Scale. 

the  Formula. 

the  Experiments. 

Experiment. 

Experiments. 

o 

0 

1-0001120 

1 -00010S2 

+ -000003S 

HallstrUm. 

*4'1 

1-0000000 

1-0000000 

0000000 

Do. 

10 

1 -0002234 

1-0002200 

+ 

0000034 

Do. 

20 

1  0015668 

1-0015490 

+ 

00001 7S 

Do. 

*30 

1  0040245 

1-0040245 

0000000 

Do. 

1  -0041489 

- 

0001244 

Deluc. 

40 

100750 

1-00748 

+ 

00002 

Gay-Lussac. 

1-00774 

_ 

00024 

Deluc. 

60 

1  -01718 

1  -01670 

+ 

00048 

Gay-Lussac. 

... 

1-01773 

_ 

00055 

Deluc. 

80 

1-03007 

1-02865 

+ 

00142 

Gay-Lussac. 

1-03092 

_ 

00085 

Deluc. 

100 

1-04579 

1-04290 

+ 

00289 

Gay-Lussac. 

... 

104664 

-  -00085 

Deluc. 

THE   EXPANSION   OF  LIQUIDS  BY  HEAT. 


15 


Expansion  of  Mercury. 


Volume  as  compared  with  that  at 

Temperature  on 
the  Centigrade 

0°  C.  ace 

ording  to 

M.  Regnault's 
Experiments. 

Difference  between 
Calculation  and 

Remarks. 

Scale. 

the  Formula. 

Experiment. 

o 

*0 

1-000000 

1-000000 

■oooooo 

Curve. 

90-22 

1-016333 

1-016361 

-  -000028 

Series  I. 

100  00 

1-018134 

1-018153 

- -000019  • 

Curve. 

100-52 

1-018230 

1-018267 

-  -000037 

Series  I. 

*150-00 

1-027419 

1027419 

•oooooo 

Curve. 

198-79 

1-036597 

1-036468 

+  -000129 

Series  11. 

205  07 

1-037786 

1-037805 

-  -000019 

Series  IV. 

205-57 

1  -037905 

1-037910 

-  -000005 

Series  III. 

*300-00 

1-055973 

1  055973 

•oooooo 

Curve. 

Expansion  of  Alcohol. 


Temperature  on  the 

Volume  as  compared  with  that  at  78°-41  C. 
according  to 

Difference  between 
Calculation  and 

Centigrade  Scale. 

the  Formula. 

m.  way-l-iussac  s 
Experiments. 

Experiment. 

o 

3  41 

•91795 

•91796 

-  -ooooi 

*1S-41 

•93269 

•93269 

•ooooo 

33-41 

•94803 

•94799 

+  -00004 

*48-41 

•96449 

•96449 

•ooooo 

63-41 

•9S183 

•98210 

-  -00027 

*7S-41 

1-00000 

1-00000 

•ooooo 

Expansion  of  Sulphuret  of  Carbon. 


Temperature  on  the 
Centigrade  Scale. 

Volume  as  compared 
accorc 

the  Formula. 

with  that  at  4Go-60C. 
ing  to 

M.  Gay-Lussac's 
Experiments. 

Difference  between 

Calculation  and 

Experiment. 

*  - 13-40 

+  1-60 

*16-60 

31-60 

*46-60 

•93224 
•9476S 
•96417 
•98163 
1-00000 

•93224 
•94776 
•96417 
•98163 
1-00000 

•ooooo 

-  -00008 

•ooooo 
•ooooo 
•ooooo 

1G  ELASTICITY  OF  GASES  AND   VAPOURS. 


HI.— ON    THE    CENTRIFUGAL    THEORY    OF    ELASTICITY, 
AS  APPLIED  TO  GASES  AND  VAPOURS.* 

1.  The  following  paper  is  an  attempt  to  show  how  the  laws  of  the 
pressure  and  expansion  of  gaseous  substances  may  be  deduced  from  that 
which  may  be  called  the  hypothesis  of  molecular  vortices,  being  a  peculiar 
mode  of  conceiving  that  theory  which  ascribes  the  elasticity  connected 
with  heat  to  the  centrifugal  force  of  small  revolutions  of  the  particles  of 

bodies. 

The  fundamental  equations  of  this  theory  were  obtained  in  the  year 
1842.  After  having  been  laid  aside  for  nearly  seven  years,  from  the  want 
of  experimental  data,  its  investigation  was  resumed  in  consequence  of  the 
publication  of  the  experiments  of  M.  Regnault  on  gases  and  vapours.  Its 
results  having  been  explained  to  the  Royal  Society  of  Edinburgh  in 
February,  1850,  a  summary  of  them  was  printed  as  an  introduction  to  a 
paper  on  the  Mechanical  Action  of  Heat  in  the  twentieth  volume  of  the 
Transactions  of  that  body.  I  now  publish  the  investigation  in  detail  in 
its  original  form,  with  the  exception  of  some  intermediate  steps  of  the 
analysis  in  the  second  and  third  sections,  which  have  been  modified  in 
order  to  meet  the  objections  of  Professor  William  Thomson,  of  Glasgow, 
to  whom  the  paper  was  submitted  after  it  had  been  read,  and  to  whom  I 
feel  much  indebted  for  his  friendly  criticism. 

This  paper  treats  exclusively  of  the  relations  between  the  density,  heat, 
temperature,  and  pressure  of  gaseous  bodies  in  a  statical  condition,  or 
when  those  quantities  are  constant.  The  laws  of  their  variation  belong 
to  the  theory  of  the  mechanical  action  of  heat,  and  are  investigated  in  the 
other  paper  already  referred  to. 

The  present  paper  consists  of  six  sections. 

The  first  section  explains  the  hypothesis. 

The  second  contains  the  algebraical  investigation  of  the  statical  relations 
"between  the  heat  and  the  elasticity  of  a  gas. 

The  third  relates  to  temperature  and  real  specific  heat. 

The  fourth  treats  of  the  coefficients  of  elasticity  and  dilatation  of  gases, 
and  compares  the  results  of  the  theory  with  those  of  M.  Regnault's 
experiments. 

*  Eead  before  the  Royal  Society  of  Edinburgh,  February  4,  1850,  and  published  in 
the  Philosophical  Magazine  for  December,  1851. 


ELASTICITY  OF   GASES  AND  VAPOURS.  17 

The  fifth  treats  of  the  laws  of  the  pressure  of  vapours  at  saturation. 

The  sixth  relates  to  the  properties  of  mixtures  of  gases  of  different 
kinds. 

I  have  endeavoured  throughout  this  paper  to  proceed  as  directly  as 
possible  to  results  capable  of  being  compared  with  experiment,  and  to  carry 
theoretical  researches  no  further  than  is  necessary  in  order  to  obtain  such 
results  with  a  degree  of  approximation  sufficient  for  the  purpose  of  that 
comparison. 


Section  I. — On  the  Hypothesis  of  Molecular  Vortices. 

2.  The  hypothesis  of  molecular  vortices  may  be  defined  to  be  that 
which  assumes — that  each  atom  of  matter  consists  of  a  nucleus  or  central  point 
enveloped  by  an  elastic  atmosphere,  which  is  retained  in  its  position  by  attractive 
forces,  and  that  the  elasticity  due  to  heat  arises  from  the  centrifugal  force  of  those 
atmospheres,  revolving  or  oscillating  about  tlmr  nuclei  or  central  points. 

According  to  this  hypothesis,  quantity  of  heat  is  the  vis  viva  of  the 
molecular  revolutions  or  oscillations. 

Ideas  resembling  this  have  been  entertained  by  many  natural  philoso- 
phers from  a  very  remote  period;  but,  so  far  as  I  know,  Sir  Humphry 
Davy  was  the  first  to  state  the  hypothesis  I  have  described  in  an  intelli- 
gible form.  It  appears  since  then  to  have  attracted  little  attention,  until 
Mr.  Joule,  in  one  of  his  valuable  papers  on  the  Production  of  Heat  by 
Friction,  published  in  the  London  and  Edinburgh  Philosophical  Magazine  for 
May,  1845,  stated  it  in  more  distinct  terms  than  Sir  Humphry  Davy  had 
done.  I  am  not  aware,  however,  that  any  one  has  hitherto  applied 
mathematical  analysis  to  its  development. 

3.  In  the  present  stage  of  my  researches,  there  are  certain  questions 
connected  with  the  hypothesis  as  to  which  I  have  not  found  it  necessary 
to  make  any  definite  supposition,  and  which  I  have  therefore  left  indeter- 
minate.    Those  questions  are  the  following : — 

First,  Whether  the  elastic  molecular  atmospheres  are  continuous,  or 
consist  of  discrete  particles.  This  may  be  considered  as  including  the 
question,  whether  elasticity  is  to  a  certain  extent  a  primary  quality  of 
matter,  or  is  wholly  the  result  of  the  repulsions  of  discrete  particles. 

Secondly,  Whether  at  the  centre  of  each  molecule  there  is  a  real  nucleus 
having  a  nature  distinct  from  that  of  the  atmosphere,  or  a  portion  of  the 
atmosphere  in  a  highly  condensed  state,  or  merely  a  centre  of  condensation 
of  the  atmosphere,  and  of  resultant  attractive  and  repulsive  forces.  There- 
fore, although  the  word  nucleus  properly  signifies  a  small  central  body,  I 
shall  use  it  in  this  paper,  for  want  of  a  better  term,  to  signify  an  atomic 
centre,  whether  a  real  nucleus  or  a  centre  of  condensation  and  force.     I 

B 


18  ELASTICITY   OF   GASES    AND   VAPOURS. 

assume,  however,  that  the  volume  of  the  nucleus,  if  any.  is  inappree: . 
small  as  compared  with  that  of  the  atmosphere. 

■4.  I  have  now  :  .  supposition,  which,  so  far  as  I  am  aware,  is 

peculiar  to  my  own  researches.  It  is  this : — that  the  vibration  ichich, 
according  to  the  undulatory  hypothesis,  t  radiant  light  and  heat,  is  a 

motion  of  the  atomic  nuclei  or  centres,  and  by  means  of  then 

mutual  attractions  and  repulsions. 

It  will  be  perceived  at  once,  that  from  the  combination  of  this  sup- 
position with  the  hypothesis  of  molecular  vortices,  it  follows  that  the 
absorption  of  light  and  of  radiant  heat  consists  in  the  transference  of 
motion  from  the  nuclei  to  their  atmospheres;  and  conversely,  that  the 
emission  of  light  and  of  radiant  heat  is  the  b  ace  of  motion  from 

the  atmospheres  to  the  nuclei 

It  appears  to  me  that  the  supposition  I  have  stated 

advantages  over  the  ordinary  hypothesis  of  a  luminiferous  ether  pervad- 
ing the  spaces  between  ponderable  particles,  especially  in  the  following 
respects  : — 

.  ue  propagation  of  transverse  vibrations  requires  the  operation 
of  forces,  which,  if  not  altogether  attractive,  are  of  a  very  different 
nature  from  those  capable  of  producing  gaseous  and  which  it 

is  difficult  to  ascribe  to  such  a  substance  as  the  ether  is  suppo; 
whii  between  the  atomic  centres  are  perfectly 

with  their  "being  kept  asunder  by  the  ela~"  Leir  atm 

nd.  The  immense  velocity  of  light  and  radiant  heat  is  a  natural 
consequence  of  th:  ition,  according  to  which  the  vibrating  m 

be  extremely  small  as  compared  with  the  forces  exerted  by  them. 

rding  to  the  most  probable  view  of  the  theory  of  dispersion, 
the  unequal  refrangibility  of  undulations  of  different  lengths  is  a  c 
quence  of  the  distances  between  the  pa:  the  vibrating  medium 

having  an  appreciable   magnitude  as  compared  with  tl  _  v.s  of  the 

undulations.  This  is  scarcely  conceivable  of  the  ether,  but  easily  con- 
ceivable of  the  atomic  n 

The  manner  in  which  the  propagation  of  light  and  of  radiant 
heat  is  affected  by  the  molecular  arrangement  of  crystalline  bodi-- 
I  much  more  intelligible  if  the  vibrations  are  supposed  to  be  t 
of  the  atomic  nuclei,  on  whose  mutual  forces  and  positions  the  form  of 
.llisation  must  depend. 

— The  consequences  of  this  supposition,  in  the  theory  of  double 

:tion  and  polarisation,  are  pointed  out  and  shown  to  be  corroborated 

by  Pi  .    5l  kest  experiments  on  diffraction,  in  a  paper  read  to  the 

of  Edinburgh  on  the  2nd  of  December,  1850,  and  publ. 
in  the  1         fkkai  Magazine  for  June,  1851.] 


ELASTICITY   OF  GASES  AND  VAPOURS.  19 


Section    II— Investigation   of  the  General   Equations   between 
the  Heat  and  the  Elasticity  of  a  Gas. 

5.  I  now  proceed  to  investigate  the  statical  relations  between  the  heat 
and  the  elasticity  of  a  gaseous  body ;  that  is  to  say,  their  relations  when 
both  are  invariable.  The  dynamical  relations  between  those  phenomena 
which  involve  the  principles  of  the  mutual  conversion  of  heat  and 
mechanical  power  by  means  of  elastic  fluids,  and  of  the  latent  heat  of 
expansion  and  evaporation,  form  the  subject  of  another  paper. 

6.  It  is  obvious  that,  in  the  condition  of  perfect  fluidity,  the  forces 
resulting  from  attractions  and  repulsions  of  the  atomic  centres  or  nuclei 
upon  their  atmospheres  and  upon  each  other,  must  be  considered  as  being 
sensibly  functions  merely  of  the  general  density  of  the  body,  and  as  being 
either  wholly  independent  of  the  relative  positions  of  the  particles,  or 
equal  for  so  many  different  positions  as  to  be  sensibly  independent  of 
them ;  for  otherwise  a  certain  degree  of  viscosity  would  arise,  and  con- 
stitute an  approach  to  the  solid  state.  For  the  same  reason,  in  the  state 
of  perfect  fluidity  each  atomic  atmosphere  must  be  considered  as  being 
sensibly  of  uniform  density  in  each  spherical  layer  described  round  the 
nucleus  with  a  given  radius,  and  the  total  attractive  or  repulsive  force  on 
each  indefinitely  small  portion  of  an  atmosphere  must  be  considered  as 
acting  in  a  line  passing  through  its  nucleus ;  that  force,  as  well  as  the 
density,  being  either  independent  of  the  direction  of  that  line,  or  equal  for 
so  many  different  and  symmetrical  directions  as  to  be  sensibly  independent 
of  the  direction. 

7.  An  indefinite  number  of  equal  and  similar  atoms,  under  such  con- 
ditions, will  arrange  themselves  so  that  the  form  of  their  bounding 
surfaces  will  be  the  rhombic  dodecahedron,  that  being  the  nearest  to  a 
sphere  of  all  figures  which  can  be  built  together  in  indefinite  numbers. 

8.  I  may  here  explain  that  by  the  term  bounding  surfaces  of  the  atoms, 
I  understand  a  series  of  imaginary  surfaces  lying  between  and  enveloping 
the  atomic  centres,  and  so  placed  that  at  every  point  in  these  surfaces  the 
resultant  of  the  joint  actions  of  all  the  atomic  centres  is  null.  To  secure 
the  permanent  existence  of  each  atom,  it  must  be  supposed  that  the  force 
acting  on  each  particle  of  atomic  atmosphere  is  centripetal  towards  the 
nearest  nucleus  or  centre. 

The  variation  of  that  force  in  the  state  of  perfect  fluidity  must  be  so 
extremely  small  in  the  neighbourhood  of  those  surfaces,  that  no  appreci- 
able error  can  arise,  if,  for  the  purpose  of  facilitating  the  calculation  of 
the  elasticity  of  the  atmosphere  of  an  atom  at  its  bounding  surface,  the 
form  of  that  surface  is  treated  as  if  it  were  a  sphere,  of  a  capacity  equal 
to  that  of  the  rhombic  dodecahedron. 

9.  If  the  several  atoms  exercised  no  mutual  attractions  nor  repulsions, 


20  ELASTICITY  OF  GASES  AND   VAPOURS. 

the  total  elasticity  of  a  body  would  be  equal  to  the  elasticity  of  the  atomic 
atmospheres  at  their  bounding  surfaces.  Supposing  such  attractions  and 
repulsions  to  exist,  they  will  produce  an  effect,  which,  in  the  state  of 
perfect  fluidity,  will  be  a  function  of  the  mean  density  of  the  body ;  and 
which,  for  the  gaseous  state,  will  be  very  small  as  compared  with  the 
total  elasticity.  Therefore,  if  p  be  taken  to  represent  the  superficial 
elasticity  of  the  atomic  atmospheres,  P  the  actual  or  total  elasticity  of  the 
fluid,  and  D  its  general  density, 

P-P+/(D),  ....     (1.) 

where  /(D)  is  a  function  of  the  density,  which  may  be  positive  or 
negative  according  to  the  nature  of  the  forces  operating  between  distinct 
atoms. 

10.  The  following  relations  must  subsist  between  the  masses  of  the 
atmosphere  and  nucleus,  and  the  density  and  volume  of  each  atom  : 

Let  R  represent  the  radius  of  the  sphere  already  mentioned,  whose 
capacity  is  equal  to  the  volume  of  an  atom,  that  volume  being  equal  to 

tes- 

Let  /i  denote  the  mass  of  the  atmosphere  of  an  atom,  m  that  of  the 
nucleus,  and  M  =  fi  +  m  the  whole  mass  of  the  atom  (so  that  if  there  is  no 
real  nucleus,  but  merely  a  centre  of  condensation,  m  =  0,  and  M  =  /u). 

Then  D  being  the  general   density  of  the  body,  ^-D   is    the   mean 

density  of  the  atomic  atmosphere,  and  M=— E3D. 

o 

If  ull  be  taken  to  denote  the  distance  of  any  spherical  layer  of  the 
atmosphere  from  the  nucleus,  the  density  of  the  layer  may  be  repre- 
sented by 

and  the  function  ifm  will  be  subject  to  this  equation  of  condition, 


ix=\      du\ 

J  u-0      V 


M 

which  is  equivalent  to 


iTrW^Du^u), 


f  du(u*rpu) (2.) 


11.  So  far  as  our  experimental  knowledge  goes,  the  more  substances 
are  rarefied — that  is  to  say,  the  more  the  forces  which  interfere  with  the 
operation  of  the  elasticity  of  the  atomic  atmospheres  are  weakened — the 
more  nearly  do  they  approach  to  a  condition  called  that  of  perfect  gas,  in 


ELASTICITY  OF   GASES  AND  VAPOURS.  21 

which  the  elasticity  is  simply  proportional  to  the  density.  I  therefore 
assume  the  elasticity  of  the  atomic  atmosphere  at  any  given  point  to  be 
represented  by  multiplying  its  density  at  that  point  by  a  constant 
coefficient  b,  which  may  vary  for  different  substances,  but,  as  I  have 
already  stated,  without  deciding  whether  that  elasticity  is  a  primary 
quality  or  the  result  of  the  repulsion  of  particles.  Consequently,  the 
superficial  atomic  elasticity 

P^bp)m,  ....  (3.) 

^(1)  being  the  value  of  \pu,  which  corresponds  to  the  bounding  surface  of 
the  atom,  where  u  =  1. 

12.  Let  an  oscillatory  movement  have  been  propagated  from  the 
nuclei  to  every  part  of  their  atmospheres,  the  size  of  the  orbits  of 
oscillation  being  everywhere  very  small  as  compared  with  the  radii  of 
the  atoms,  and  let  this  movement  have  attained  a  permanent  state,  which 
will  be  the  case  when  every  part  of  each  atmosphere,  as  well  as  each 
nucleus,  moves  with  the  same  mean  velocity,  v — mean  velocity  signifying 
that  part  of  the  velocity  which  is  independent  of  periodic  changes.  It  is 
necessary  to  suppose  that  the  propagation  of  this  movement  to  all  parts 
of  a  molecular  atmosphere  is  so  rapid  as  to  be  practically  instantaneous. 

We  shall  conceive  all  the  masses  and  densities  referred  to,  to  be 
measured  by  weight.  Then  taking  g  to  represent  the  velocity  generated 
by  the  force  of  gravity  at  the  earth's  surface  in  unit  of  time,  the  whole 
mechanical  power  to  which  the  oscillatory  movement  in  question  is 
equivalent  in  one  atom  will  be  represented  in  terms  of  gravity  by 

V=2; {L) 

that  is  to  say,  the  weight  of  the  atom,  M,  falling  through  the  height  — 

due  to  the  velocity  v ;  and  this  is  the  mechanical  measure  of  the  quantity 
of  heat  in  one  atom  in  terms  of  gravity. 

13.  Any  such  motion  of  the  particles  of  a  portion  of  matter  confined 
in  a  limited  space,  will  in  general  give  rise  to  a  centrifugal  tendency  with 
respect  to  that  space.  In  order  to  obtain  definite  results  with  respect  to 
that  centrifugal  tendency  in  the  case  now  under  consideration,  it  is 
necessary  to  define,  to  a  certain  extent,  the  general  character  of  the 
supposed  movement. 

In  the  first  place,  it  is  periodical ;  secondly,  it  is  similar  with  respect  to 
so  large  a  number  of  radii  drawn  in  symmetrical  directions  from  the 
atomic  centre,  as  to  be  sensibly  similar  in  its  effects  with  respect  to  all 
directions  round  that  centre.  Thi3  symmetry  exists  in  the  densities  of 
the  different  particles  of  the  atomic  atmosphere  in  a  gas,  and  in  the 


22  ELASTICITY  OF  GASES  AND   VAPOURS. 

forces  which  act  upon  them ;  and  we  are  therefore  justified  in  assuming 
it  to  exist  in  their  motions. 

Two  kinds  of  motion  possess  these  characteristics  : 

First,  Radial  oscillation,  by  which  a  portion  of  a  spherical  stratum  of 
atmosphere  surrounding  an  atomic  centre,  being  in  equilibrio  at  a  certain 
distance  from  that  centre,  oscillates  periodically  to  a  greater  and  a  less 
distance.  This  forms  part  of  the  vis  viva  of  the  molecular  movements ; 
but  it  can  only  affect  the  superficial  atomic  elasticity  by  periodic  small 
variations,  having  no  perceptible  effect  on  the  external  elasticity. 

Second,  Small  rotations  and  revolutions  of  particles  of  the  atomic 
atmosphere  round  axes  in  the  direction  of  radii  from  the  atomic  centre, 
by  which  each  spherical  layer  is  made  to  contain  a  great  number  of  equal 
and  similar  vortices,  or  equal  and  similar  groups  of  vortices  having  their 
axes  at  right  angles  to  the  layer,  and  similarly  situated  with  respect  to  a 
great  many  symmetrical  directions  round  the  atomic  centre. 

Let  us  now  consider  the  condition,  as  to  elasticity,  of  a  small  vortex 
of  an  atmosphere  whose  elasticity  is  proportional  to  its  density,  inclosed 
within  a  cylindrical  space  of  finite  length,  and  not  affected  by  any  force 
at  right  angles  to  the  axis  except  its  own  elasticity.  Let  Z  denote  the 
external  radius  of  the  cylinder,  px  its  external  density,  p  its  mean  density, 
p  the  density  at  any  distance  z  from  the  axis  (all  the  densities  being 
measured  by  weight),  w  the  uniform  velocity  of  motion  of  its  parts.  The 
condition  of  equilibrium  of  any  cylindrical  layer  is,  that  the  difference  of 
the  pressures  on  its  two  sides  shall  balance  the  centrifugal  force ;  con- 
sequently (b  being  the  coefficient  of  elasticity) 

(jz  dz  ' 

The  integral  of  this  equation  is 

,>■'■ 
p'  =  azb'S 

The  coefficient  a  is  determined  by  the  following  relation,  analogous  to 
that  of  equation  (2),  between  the  densities  : 

Z2 


p-=J  dz(pz)= ^—Z» 


whence 


V+2 


■ti+I 


And  the  general  value  of  the  density  is 


,,2 


£+i)®*  '  (5° 


ELASTICITY   OF  GASES  AND  VAPOURS.  23 

Making  z  =  Z,  and  multiplying  by  the  coefficient  of  elasticity  b,  we 
obtain  for  the  elasticity  of  the  atmosphere,  at  the  cylindrical  surface  of 
the  vortex, 

bPi  =  bP  +  -l£->  •         •         •     W 

which  exceeds  the  mean  elasticity  bp  by  a  quantity  equivalent  to  the 
weight  of  a  column  of  the  mean  density  p,  and  of  the  height  due  to  the 
velocity  u;  and  independent  of  the  radius  of  the  vortex. 

Supposing   a   spherical   layer,   therefore,    to    contain   any   number   of 
vortices  of  any  diameter,   in  which   the   mean  density  is  equal,  it  is  . 
necessary  to  a  permanent  condition  of  that  layer  that  the  velocities  in 
all  these  vortices  should  be  equal,  in  order  that  their  lateral  elasticities 
may  be  equal. 

Although  the  mean  elasticity  at  the  plane  end,  or  any  plane  section  at 
right  angles  to  the  axis  of  a  vortex,  is  simply  =  bp,  being  the  same  as  if 
there  were  no  motion,  yet  the  elasticity  is  variable  from  point  to  point, 
and  the  law  of  variation  depends  on  the  velocity.  Therefore,  if  two 
vortices  are  placed  end  to  end,  it  is  necessary  to  a  stable  condition  of 
the  fluid,  not  only  that  their  terminal  planes  should  coincide,  and  that 
their  mean  elasticities  should  be  in  cquilibrio,  but  also  that  their  velocities 
should  be  equal,  or  subject  only  to  periodical  deviations  from  a  state  of 
equality. 

Therefore,  the  mean  velocity  of  vortical  motion,  independent  of  small 
periodic  variations,  is  the  same  throughout  the  whole  atomic  atmosphere; 
and  the  mean  total  velocity,  independent  of  small  periodic  variations, 
being  uniformly  distributed  also,  the  vis  viva  of  the  former  may  be 
expressed  as  a  constant  fraction  of  that  of  the  latter,  so  that 

v?  =  ^,         .         .         .         .     (5b.) 

-  being  the  mean  value  of  a  coefficient  which  is  subject  to  small  periodical 
k 

variations  only  * 

*  As  it  has  been  represented  to  me  that  I  have,  without  stating  sufficient  grounds, 
assumed  the  velocity  of  revolution  w  to  be  constant  throughout  each  individual 
vortex,  I  add  this  note  to  assign  reasons  for  that  supposition. 

First,  Unless  w,  the  velocity  of  revolution  of  a  particle,  is  independent  of  z,  its 
radius  vector,  the  atomic  atmosphere  cannot  be  in  a  permanent  condition. 

For  if  w  is  a  function  of  z,  the  external  elasticity  of  a  vortex  will  be  a  function  of 
its  diameter.  If  the  whole  atmosphere  is  in  motion,  vortices  of  different  diameters 
must  exist  in  the  same  spherical  layer  ;  and  if  their  external  elasticities  are  different, 
their  condition  cannot  be  permanent. 

,Second,  Whatsoever  may  be  the  nature  of  the  forces  by  which  velocity  is  communi- 
cated throughout  the  atmosphere,  the  tendency  of  those  forces  must  be  to  equalise 
that  velocity,  and  thus  to  bring  about  a  permanent  condition. 


24  ELASTICITY  OF  GASES  AND  VAPOURS. 

This  coefficient,  being  the  ratio  of  the  vis  viva  of  motion  of  a  peculiar 
kind  to  the  whole  vis  viva  impressed  on  the  atomic  atmospheres  by  the 
action  of  their  nuclei,  may  be  conjectured  to  have  a  specific  value  for  each 
substance,  depending,  in  a  manner  "as  yet  unknown,  on  some  circumstance 
in  the  constitution  of  its  atoms.  It  will  afterwards  be  seen  that  this 
circumstance  is  the  chemical  constitution. 

Let  the  entire  atmosphere  of  an  atom  be  conceived  to  be  divided  into 
a  great  number  of  very  acute  pyramids  meeting  at  the  centre,  and  having 
even  numbers  of  faces,  equal  and  opposite  in  pairs ;  and  let  one  of  these 
pyramids,  intersecting  a  spherical  layer  whose  distance  from  the  nucleus 
is  R?i  and  thickness  Tulu,  cut  out  a  frustum,  containing  and  surrounded 
by  vortices.  Consider  one  pair  of  the  faces  of  that  frustum ;  their  length 
being  TMu,  let  their  breadth  be  h,  and  their  distance  asunder/.  Then 
they  make  with  each  other  the  angle  at  the  apex  of  the  pyramid 

o  s;n-i ;    J     . 

their  common  area  is  KRdu ;  and  the  sum  of  the  volumes  of  the  two 
triangular  frusta  of  the  spherical  layer,  included  by  diagonal  planes  drawn 
between  their  radial  edges,  is 

fKRflll 

2     ' 

the  sum  of  all  such  triangular  frusta  being  the  whole  volume  of  the 
spherical  layer. 

The  additional  pressure  due  to  the  centrifugal  force  of  vortices,  viz. — 

2gV 

acts  on  the  two  lateral  faces,  its  total  amount  for  each  being 

2gk 

The  transverse  components  of  this  pair  of  forces  balance  each  other. 
Their  radial  components,  amounting  to 


/        v2p1-rt7       iPpfhdu 


Hu      2gk  2gku 

constitute  a  centrifugal  force  relatively  to  the  atomic  centre,  acting  on  the 
pah*  of  triangular  frusta  whose  mass  is 

pfJiRdu 

2     ' 

The  condition  of  permanent,  or  periodical,  equilibrium  of  this  pair  of 
frusta,  requires  that  this  centrifugal  force  shall  be  balanced  by  the  varia- 
tion of  the  mean  elasticity  of  the  atmosphere  at  the  two  surfaces  of  the 


ELASTICITY  OF   GASES  AND  VAPOURS.  25 

spherical  layer,  combined  with  the  attraction  of  the  nucleus.  The  action 
of  the  former  of  these  forces  is  represented  by 

7dpy        ill 
—  b^au  X  — . 
cm  2 

Let  the  accelerating  force  of  attraction  towards  the  nucleus  be  repre- 
sented by 

<f>(Rn) 
E,    ' 

^  being  a  function,  which,  by  the  definition  of  an  atomic  bounding 
surface  in  article  8,  is  null  at  that  surface,  or  when  u  =  1.  Then  the 
attraction  on  the  pair  of  frusta  is 

pfhdu(p(Ru) 
2         " 

Add  these  three  forces  together;  let  the  sum  be  divided  by 

■^pfhdw, 
and  let  the  density  p  be  denoted,  as  in  article  10,  by 

then  the  folloAving  differential  equation  is  obtained  as  the  condition  of  a 
permanent  state  of  the  atomic  atmosphere  : 

-T-77T  --r^-0  =  O.   .         .         .     (5c.) 
gku      \p(u)      du        r 

This  equation  will  be  realised  for  each  layer  at  its  mean  position,  on 
each  side  of  which  its  radial  oscillations  are  performed. 

The  variation  of  this  expression  being  of  opposite  sign  to  the  variation 

of  ,  shows  that  any  small  disturbance  of  the  density  produces  a  force 

tending  to  restore  that  distribution  to  the  state  corresponding  to  the 
position  of  equilibrium  of  the  layers,  and  therefore  that  the  state  indicated 
by  equation  (5  c)  is  stable. 

1 4.  The  integral  of  equation  (5c)  is 

xPu=u^ea~"fU1du-<P.  .         .         .     (6.) 

The  arbitrary  constant  a  is  determined  from  the  equation  of  condition 
(2)  in  the  following  manner  : — 


2G  ELASTICITY   OF   GASES   AND   VAPOURS. 

Substituting  for  $11  in  equation  (2)  its  value  as  given  above,  we  obtain 


or 


r-  r w 

-•=s/*(«3B+V*/1*,*)J    .         .         .     (7.) 


which  integration  having  been  effected,  we  shall  obtain  for  the  value  of 
the  superficial  elasticity  of  the  atomic  atmospheres, 


*=OWl)  =  6£D* 


•     (8.) 


M     rv  J       M 

To  obtain  an  infinite  series  for  approximating  to  the  value  of  the 
integral  in  equation  (7),  let  the  following  substitutions  be  made  : — 


Lo"   u=X 


1 


gkb 


+  3  =  30 


J    l  — (O, 


(9.) 


and  let  the  values  of  the  successive  differential  co- 
efficients of  10  with  respect  to  A,  when  A  =  0,  $  =  0, 
and  w=r  1,  be  denoted  by 

(a/),  (to"),  (w'"),  &c. 
Then 

e-a=3J     d\.esexu>. 

-    -so 

The  value  of  which  (when  the  function  (p  is  such  as  to  admit  of  its  having 
a  finite  value)  is 

6     -0V      36  +  90-        270s +  &C'/' 
whence  i 


1  +  W  +  ^90^ 


+ 


27  03 


+  &c.) 

Now,  because  (a/)  =  —  t0  (m  =  d  =  0, 

^(l)  =  e«  =  e(l-^]+&c.)j' 
which  may  be  represented  by 

Gi+l)(l-r<D^)'   ' 


(10.) 


(10a.) 


ELASTICITY  OF   GASES  AND  VAPOURS.  27 

F(D,0)  being  a  quantity  which  becomes  continually  less  as  the  density 
becomes  less  and  the  heat  greater.  The  complete  expression  for  the 
elasticity  of  a  gas  is  therefore,  according  to  equations  (1),  (8),  and  (10a), 

P^+/(D)-^D(^  +  j)(l-F(D,e))+/(D);.    (11.) 

when  each  atom  contains  a  quantity  of  heat  measured  by  the  mechanical 
power  corresponding  to  the  velocity  v  in  the  weight  M,  or 


Mr2 


according  to  equation  (4). 


Section  III. — Of  Temperature,  and  of  Real  Specific  Heat. 

15.  The  definition  of  temperature  consists  of  two  parts: — First,  the 
definition  of  that  condition  of  two  portions  of  matter  when  they  are  said 
to  be  at  the  same  temperature ;  and"  second,  the  definition  of  the  measure  of 
differences  of  temperature. 

Two  bodies  are  said  to  be  at  the  same  temperature  when  there  is  no 
tendency  for  one  to  become  hotter  by  abstracting  heat  from  the  other ; 
that  is  to  say  (calling  the  two  bodies  A  and  B),  when  there  is  either  no 
tendency  to  transmission  of  heat  between  them,  or  when  A  transmits  as 
much  heat  to  B  as  B  does  to  A.  Now  it  is  known  by  experiment,  that 
any  surface  or  other  thing  which  affects  the  transmission  of  heat  being 
placed  between  B  and  A,  has  exactly  the  same  influence  upon  the 
same  quantity  of  heat  passing  in  either  direction;  therefore,  to  produce 
equilibrium  of  temperature  between  A  and  B,  the  powers  of  their  atoms 
to  communicate  heat  must  be  equal. 

15a.  If  we  apply  to  vortices  at  the  surface  of  contact  of  the  atmo- 
spheres of  two  atoms  of  the  same  or  different  kinds,  the  conditions  of 
permanency  laid  down  in  article  1 3  for  vortices  in  the  same  atmosphere, 
these  conditions  take  the  folloAving  form : — 

First,  The  superficial  atomic  mean  elasticities  must  be  the  same ;  in 
other  words,  the  superficial  atomic  mean  densities  must  be  inversely  as 
the  coefficients  of  elasticity  of  the  atmospheres.  This  is  the  condition  of 
equilibrium  of  pressure. 

Second,  The  law  of  variation  of  the  elasticity  from  the  centre  to  the 
circumference  of  a  vortex,  as  expressed  in  equation  (5),  must  be  the  same 

.2  2 

for  both  atoms;  and  this  law  depends  on  the  quantity  ~  =yh;  therefore 

the  condition  of  equilibrium  of  heat  is,  that  the  square  of  the  velocity  of 
vortical  motion,  divided  by  the  coefficient  of  atmospheric  elasticity,  shall 


28  ELASTICITY  OF  GASES  AND   VAPOURS. 

1)0  the  same  for  each  atom.     Of  this  quantity,  therefore,  and  of  constants 
common  to  all  substances,  temperature  must  be  a  function. 

Taking  the  characteristics   (A)  and  (B)  to  distinguish  the  quantities 
proper  to  the  two  atoms,  we  have  the  following  equation : — 

jgD*(l)(A>=8|D*(l)(B)' 

>=>),[  •        •     (12.) 

temperature  =  0  f  — , universal  constants ) . 


16.  In  sx  perfect  gas,  equation  (11)  is  reduced  to 


the  pressure  being  simply  proportional  to  the  mean  elasticity  of  the  atmo- 
spheric part  of  the  gas  multiplied  by  a  function  of  the  heat,  which,  as 
equation  (12)  shows,  is  a  function  of  the  temperature,  from  its  involving 

only  —  and  universal  constants. 
kb 

Therefore,  in  two  perfect  gases  at  the  same  pressure  and  temperature, 
the  mean  elasticities  of  the  atmospheric  parts  arc  the  same,  and 
consequently — 

The  mean  specific  gravities  (if  the  atmospheric  parts  of  all  perfect  gases  are 
inversely  proportional  to  the  coefficients  of  atmospheric  elasticity. 

Let  n  therefore  represent  the  number  of  atoms  of  a  perfect  gas  which 
fill  unity  of  volume  under  unity  of  pressure  at  the  temperature  of  melting 
ice,  so  that  nM  is  the  total  specific  gravity  of  the  gas,  and  njx  that  of  its 
atmospheric  part ;  then 

bn/n  =  constant  for  all  gases,        .         .          (12b.) 
and  consequently 

kb  k {        '} 

Therefore, 

Temtperatwre  is  a  function  of  universal  constants,  and  of  the  vortical  vis  viva 
of  the  atomic  atmospheres  of  so  much  of  the  substance  as  would,  in  the  condition 
of  perfect  gas,  fill  unity  of  volume  under  unity  of  pressure  at  some  standard 
temperature. 

The  equation  (12a)  further  shows,  that  in  any  two  perfect  gases  the 
respective  values  of  the  quotient  of  the  pressure  by  the  density  corresponding  to 
the  same  temperature,  bear  to  each  other  a  constant  ratio  for  all  temperatures, 

being  that  of  the  values  of  the  coefficient  b  ~. 
°  M 


ELASTICITY   OF   GASES  AND  VAPOURS.  29 

Therefore  the  pressure  of  a  perfect  gas  at  a  given  density,  or  its  volume 
under  a  given  pressure,  is  the  most  convenient  measure  of  temperature. 

Let  P0  represent  the  elasticity  of  a  perfect  gas  of  the  density  D  at  the 
temperature  of  melting  ice,  P  that  of  the  same  gas  at  the  same  density,  at 
a  temperature  distant  T  degrees  of  the  thermometric  scale  from  that  of 
melting  ice,  and  C  a  constant  coefficient  depending  on  the  scale  employed; 
then  the  value  of  T  is  given  by  the  equation 

P-Pol 


T=C 


Po 


(13.) 


or  r  • 

T-fC=C~j 

The  value  of  the  constant  C  is  found  experimentally  as  follows: — Let 
Pj  represent  the  elasticity  of  the  gas  at  the  temperature  of  water  boiling 
under  the  mean  atmospheric  pressure,  T1  the  number  of  degrees,  on  the 
scale  adopted,  between  the  freezing  and  boiling  points  of  water;  then 

P  — P   1 

rp    p  rl         X0 

1  15 

and  1-  (14.) 

c=t'pFpJ 

C  is  in  fact  the  reciprocal  of  the  coefficient  of  increase  of  elasticity  with 
temperature,  or  the  reciprocal  of  the  coefficient  of  dilatation,  of  a  perfect 
gas  at  the  temperature  of  melting  ice. 

17.  As  it  is  impossible  in  practice  to  obtain  gases  in  the  theoretical 
condition  referred  to,  the  value  of  C  can  only  be  obtained  by  approxima- 
tion. From  a  comparison  of  all  M.  Eegnault's  best  experiments,  I  have 
arrived  at  the  following  values,  which  apply  to  all  gaseous  bodies. 

For  the  Centigrade  scale,  C  =  2740,6,  being  the  reciprocal  of 
0-00364166. 

For  Fahrenheit's  scale,  if  adjusted  so  that  180°  are  equal  to  100° 
Centigrade, — 

C  for  temperatures  measured  from  the  freezing-point  of  water 

=  494°-28. 

C  for  temperatures  measured  from  the  ordinary  zero 

=  494°-28-32°  =  462°-28. 

The  point  C  degrees  below  the  ordinary  zero  of  thermometric  scales 
may  be  called  the  absolute  zero  of  temperature ;  for  temperatures  measured 
from  that  point  are  proportional  to  the  elasticities  of  a  theoretically 
perfect  gas  of  constant  density. 


30 


ELASTICITY   OF   GASES   AND  VAPOURS. 


Temperatures  so  measured  may  be  called  absolute  temperatures.     Through- 
out this  paper  I  shall  represent  them  by  the  Greek  letter  t,  so  that 


T  +  C. 


(15.) 


It  is  to  be  observed,  that  the  absolute  zero  of  temperature  is  not  the 
absolute  zero  of  heat. 

18.  If  we  now  substitute  for  P  in  equation  (13)  its  value  according 
to  equation  (12a),  we  obtain  the  following  result: — 


'=T+C=c£i(£*+J) 


oV:V/''- 

Let  n  represent,  as  before,  the  theoretical  number  of  atoms  in  unity  of 
volume  under  unity  of  pressure,  at  the  temperature  of  melting  ice,  of  the 
gas  in  question,  supposing  the  disturbing  forces,  represented  by  -  F(D,0) 
and /(D),  to  be  inappreciable;  then  nM  is  the  weight  of  unity  of  volume 
under  those  circumstances,  and  it  is  evident  that 


Consequently 


D  AT 

1  0 


*{jfk+h 


(16.) 


being  the  complete  expression  for  that  function  of  heat  called  temperature. 
It  follows  that  the  function  0,  which  enters  into  the  expressions  for  the 
elasticity  of  gases,  is  given  in  terms  of  temperature  by  the  equation 


oglcb 


Cnjub' 


(16a.) 


If,  according  to  the  expression  4,  for  the  quantity  of  heat  in  one  atom 
we  substitute  -^-  for  v2  in  equation  (16),  we  obtain  the  following  equa- 
tions : 


Mv2     3£M/  r        A 


(17.) 


and  if  Q  represent  the  quantity  of  heat  in  unity 
of  weight, 

19.  The  real  specific  heat  of  a  given  substance  is  found  by  taking  the 
differential  coefficient  of  the  quantity  of  heat  with  respect  to  the  tempera- 


ELASTICITY   OF   GASES   AND  VAPOURS.  31 

ture.     Hence  it  is  expressed  in  various  forms  by  the  following  equations, 

in  which  the  coefficient  j  is  supposed  not  to  vary  sensibly  with  the  tem- 

perature. 

Eeal  specific  heat  of  one  atom, 

dq  _  3&M 
dr  ~2CV 

real  specific  heat  of  unity  of  weight, 

dQ__U_ 

dT~2CV  <18-> 

real  specific  heat  of  so  much  of  a  perfect  gas  as 
occupies  unity  of  volume  under  unity  of  pressure 
at  the  temperature  of  melting  ice, 

dq  _  3&M 
ndr~~2Cjx- 

The  coefficient  - — ,  representing  the  ratio  of  the  total  vis  viva  of  the 

motions  of  the  molecular  atmospheres  to  the  portion  of  vis  viva  which  pro- 
duces elasticity,  multiplied  by  the  ratio  of  the  total  mass  of  the  atom  to 
that  of  its  atmospheric  part,  is  the  specific  factor  in  the  capacity  of  an 
atom  for  heat.  The  view  which  I  have  stated  as  probable  in  article  1 3 
— that  the  first  factor  of  this  coefficient  is,  like  the  second,  a  function  of 
some  permanent  peculiarity  in  the  nature  of  the  atom — is  confirmed  by 
the  laws  discovered  by  Dulong  :  that  the  specific  heats  of  all  simple  atoms 
bear  to  each  other  very  simple  ratios,  and  generally  that  of  equality  ;  that 
the  same  property  is  possessed  by  the  specific  heats  of  certain  groups  of 
similarly  constituted  compound  atoms ;  and  that  the  specific  heats  of  equal 
volumes  of  all  simple  gases,  at  the  same  temperature  and  pressure,  are 
equal. 

The  coefficient  —  varies  in  many  instances  to  a  great  extent  for  the 
A* 
same  substance  in  the  solid,  liquid,  and  gaseous  states.  So  far  as  experi- 
ment has  as  yet  shown,  it  appears  not  to  vary,  or  not  sensibly  to  vary, 
with  the  temperature  j  and  this  I  consider  probable  a  priori,  except  at  or 
near  the  points  of  fusion  of  solid  substances. 

Apparent  specific  heat  differs  from  real  in  consequence  of  the  con- 
sumption and  production  of  certain  quantities  of  heat  by  change  of 
volume  and  of  molecular  arrangement,  which  accompany  changes  of 
temperature. 

This  subject  belongs  to  the  theory  of  the  mechanical  action  of 
heat. 


32  ELASTICITY   OF   GASES  AND   VAPOURS. 

Section  IV. — Of  the  Coefficients  of  Elasticity  and  Dilatation 

of  Gases. 

1)  T 

20.  If  in  equation  (11)  we  substitute  f or  — -?  +  b  its  value  ~—  we 
obtain  the  following  value  for  the  elasticity  of  a  gas, 
D 


^1_F(D,0))+/(D);      .         .     (19.) 


P  ~~  nM    C 

in  which  -     denotes  the  ratio  of  the  actual  weight  of  unity  of  volume  to 
wM 

the  weight  of  unity  of  volume  under  unity  of  pressure,  at  the  absolute 
temperature  C,  in  the  theoretical  state  of  perfect  gas ; 

t  is  the  absolute  temperature ; 

-  F(D,  6)  is  a  function  of  the  temperature  and  density,  representing  the 
effect  of  the  attraction  of  the  atomic  nucleus  or  centre  in  diminishing  the 
superficial  elasticity  of  its  atmosphere ; 

And  /(D)  is  a  function  of  the  density  only,  representing  the  effect  of  the 
mutual  attractions  and  repulsions  of  the  atoms  upon  the  whole  elasticity 
of  the  body. 

From  this  equation  are  now  to  be  determined,  so  far  as  the  experiments 
of  M.  Eegnault  furnish  the  requisite  data,  the  laws  of  the  deviation  of 
gases  from  that  theoretical  state  in  which  the  elasticity  is  proportional  to 
the  density  multiplied  by  the  absolute  temperature. 

21.  The  value  of  -F(D,  0)  is  given  by  the  infinite  series  of  equations 

T 

(10),  (10fl),  substituting  in  which  for  0  its  value  » — =  we  obtain  the 
following  result : — 

A.,  A2,  A3,  &c,  being  a  series  of  functions,  the  value  of  which  is  given  by 
the  following  equation  : — 

~K= 31^ ®l+mi  '  ■       (20-) 

351+7n  being  the  coefficient  of  z1+m  in  the  development  of  the  reciprocal  of 

the  series 

1  -  (w')x  +  (a/>2-  (a/ ")a?  +  &c, 

when  (d)')  &c.  have  the  values  given  in  equation  (9). 
Equation  (19)  is  thus  transformed  into 

p=a(e-v-?-&c-W<D>--     •  ^ 

The  series  in  terms  of  the  negative  powers  of  the  absolute  temperature 


ELASTICITY   OF  GASES   AND   VAPOURS.  S3 

converges  so  rapidly,  that  I  have  found  it  sufficient,  in  all  the  calculations 
I  have  hitherto  made  respecting  the  elasticity  of  gaseous  bodies,  to  use  the 

first  term  only, . 

T 

22.  Instead  of  making  any  assumption  respecting  the  laws  of  the 
attractions  and  repulsions  which  determine  the  functions  A  and  /(D),  I 
have  endeavoured  to  represent  those  functions  by  empirical  formulae, 
deduced  respectively  from  the  experiments  of  M.  Kegnault  on  what  he 
terms  the  coefficient  of  dilatation  of  gases  at  constant  volume,  which  ought 
rather  to  be  called  the  coefficient  of  increase  of  elasticity  with  temperature,  and 
from  his  experiments  on  the  compressibility  of  elastic  fluids  at  constant 
temperature. 

From  the  data  thus  obtained  I  have  calculated,  by  means  of  the  theory, 
the  coefficients  of  dilatation  of  gases  under  constant  pressure,  which,  as  a  test  of 
the  accuracy  of  the  theory,  I  have  compared  with  those  deduced  by 
M.  Kegnault  from  experiment. 

23.  The  mean  coefficient  of  increase  of  elasticity  with  temperature  at 
constant  volume  between  0°  and  100°  of  the  Centigrade  thermometer  is 
found  by  dividing  the  difference  of  the  elasticities  at  those  two  tempera- 
tures by  the  elasticity  at  0°,  and  by  100°,  the  difference  of  temperature. 
It  is  therefore  represented  by 

P  —  P 

E  —  -^ — °-  (22/) 

ioo°p0'  •  K     } 

where  E  represents  the  coefficient  in  question,  and  P0  and  Px  the  elasti- 
cities at  0°  and  100°  Centigrade  respectively. 

Now  by  equation  (21),  neglecting  powers  of  -  higher  than  the  first,  we 
have 

^=30 -£)+'<« 

D_/100°  +  C_        A      \ 

1_»M\       C  100°  +  C/+M    h 

whence 

D     /l  A  \  ,      . 

_7iMP0VC+C(C  +  100o)/•      '         '         '     t     ■' 

Supposing  the  value  of     ...    ■  to  be  known,  this  equation  affords  the 
11        °  wMP0  '  * 

means  of  calculating  the  values  of  the  function  A  corresponding  to  various 

densities,  from  those  of  the  coefficient  E  as  given  by  experiment. 

C 


34, 


ELASTICITY   OF   GASES   AND  VAPOURS. 


As  a  gas  is  rarefied       _    approximates  to  unity,  A  diminishes  without 

limit,  and  the  value  of  E  consequently  approximates  to   ~,  the  reciprocal 

of  the  absolute  temperature  at  0°  Centigrade.     This  conclusion  is  verified 
by  experiment ;  and  by  means  of  it  I  have  determined  the  values  already 

given— viz.,  C  =  274°6  Centigrade,  and  ^  =  -003641 6G  for  the  Centigrade 


scale. 


C 


24.  In    order   to    calculate  the  values  of 


D 

//-Ml',, 


1  have  made  use  of 


empirical   formulae,  deduced   from  those  given  by  M.   Regnault  in  his 

memoir  on  the  Compressibility  of   Elastic    Fluids.      In  M.   Regnault's 

formulae,  the  unit  of  pressure  is  one  metre  of  mercury,  and  the  unit  of 

density  the  actual  density  corresponding  to  that  pressure.     In  the  formula? 

which  I  am  about  to  state,  the  unit  of  pressure  is  an  atmosphere  of  760 

millimetres  of  mercury,  or  29-922  inches;  and  the  unity  of  density,  the 

theoretical  density  in  the  perfectly  gaseous  state  at  0°  Centigrade,  under  a 

pressure  of  one  atmosphere,  which  has  been  found  from  M.  Regnault's 

M?iP 
formula'  by  making  the  pressure  =  0  in  the  value  of     .  — °.    M.  Regnault's 

experiments  were  made  at  temperatures  slightly  above  the  freezing  point, 

but  not  sufficiently  so  to  render  the  formula;  inaccurate  for  the  purpose  of 

,    ,    .       ,         .    .  .         D 

calculating  the  ratio  in  question,    ^p  . 

The  formula;  are  as  follows  : — 


Supposing  —  given, 


(Vdi* 
rD 


L>         a 


D\2 
M/  : 


which,  when  T  is  small,  or  -  nearly  =  C,  gives  an 
approximate  value  of  -         . 


Supposing  P0  given, 


-I) 


CwMP 


=  l+7P0-f£P02; 


which,  when  T  is  small,  gives  an   approximate 
value  of 


wMP0' 


, 


-4. 


The  values  of  the  constants  a,  /3,  7,  e,  and  of  their  logarithms,  are  giverj, 
together  with  the  mean  temperatures  above  the  freezing  point  at  which 


ELASTICITY  OF  GASES  AND   VAPOUES. 


35 


M.  Eegnault's  experiments  were  made,  for  atmospheric  air,  carbonic  acid 
gas,  and  hydrogen. 


ft 


Atmospheric  Air. 

Constants. 
-  7  =  -  -000860978 
+  -000011182 
-  -000009700 


4°-75. 
Logarithms. 
4-9349920 
5-0485140 
6-9867717 


Carbonic  Acid  Gas.     T  =  3°-27. 

a  =  -  y  =  -  -00641836  3-8074242 

/3  =  _  -0000041727        6-6204126 

£  =  +  -0000865535       5-9372846 


Hydrogen.  T  =  4°-75. 


0  = 


+  -000403324 
+  -0000048634 
_  -0000044981 


4-6056546 
6-6869401 
6-6530291 


The  three  substances  above-mentioned  are  the  only  gases  on  which 
experiments  have  yet  been  made,  under  circumstances  sufficiently  varied 
to  enable  me  to  put  the  theory  to  the  test  I  have  described  in  article 
22. 

25.  M.  Regnault  has  determined  the  values  of  the  coefficient  of 
elasticity  E  for  carbonic  acid  at  four  different  densities,  and  for  atmospheric 
air  at  ten.  By  applying  equations  (23)  and  (24)  to  those  data,  I  have 
ascertained  that  the  function  A  for  these  two  gases  may  be  represented 
empirically,  for  densities  not  exceeding  that  corresponding  to  five 
atmospheres,  by  the  formula?  given  below,  which  lead  to  formulae  for  the 
coefficient  E. 


For  Carbonic  Acid, 


A  =  a 


D 


where  log.  a  =  0-3344538,  and  consequently 

«MP0   CV   "rC+100°   mM 
a 


lo 


C  +  1001 


=  3-7608860. 


(25.) 


3G 


ELASTICITY  OF   GASES   AND   VAPOURS. 


For  Atmospheric  Air, 


A  =  a[  —T7 


r, 
10  . 


where  log.  «=0-3176168,  and  consequently 


E  = 


»MP0'C 


S0+ 


l0£ 


"C  +  100° 

1 


c  +  ioo° 

3-7440490 


*) 


(26.) 


The  value  of  log.  ~  is  3-5012995. 

The  following  table  shows  that  those  empirical  formula;  accurately 
represent  the  experiments,  the  greatest  differences  being  less  than  one- 
half  of  -00001 3 C,  which  M.  Kegnault,  in  the  seventy-first  page  of  his 
memoir,  assigns  as  the  limit  of  the  errors  of  observation  due  to  barometric 
measurements  alone. 

As  the  coefficient  E  for  hydrogen  has  been  determined  for  one  density 
only,  it  is  impossible  to  obtain  an  empirical  formula  for  that  gas.  The 
single  ascertained  value  of  E  is  nevertheless  inserted  in  the  table. 

Table  of  Coefficients  of  Increase  of  Elasticity  with 
Temperature  at  Constant  Volu.m  i 


Pressure  at 

Density 

Coefficient  E 

Coefficient  E 

Difference  be- 

Atmospheres 

=  £„. 

D 
nM" 

according  to  the 

Formula. 

according  to 
Experiment. 

Formula;  and 
Experiment. 

CARBONIC  ACID. 

I. 

0  9980 

1-00448 

•003G865 

•0036856 

+  -0000009 

II. 

1  -1857 

1-19487 

•0036951 

•0036943 

+•0000008 

III. 

2  2931 

2-327S8 

•0037465 

•0037523 

-  -0000058 

IV. 

4-7225 

4-87475 

•003S647 

•0038598 

+  -0000049 

ATMOSPHERIC  AIR. 

I. 

01444 

0-1444 

•0036484 

•0036482 

+  '0000002 

II. 

0-22'J4 

0-2294 

•0036507 

•0036513 

-  -0000006 

III. 

0-3501 

0-3502 

•0036535 

•0036542 

-  -0000007 

IV. 

0*4930 

0-4932 

•0036564 

•0036587 

-  -0000023 

V. 

0-4937 

0-4939 

•0036564 

•0036572 

-  -oooooos 

VI. 

1-0000 

1-000S5 

•0036652 

•0036650 

+  -0000002 

VII. 

2-2084 

2-2125 

•0036810 

•0036760 

+  -0000050 

VIII. 

2-2270 

2-2312 

•0036812 

•0036800 

+  -0000012 

IX. 

2-8213 

2S279 

•0036880 

•0036894 

-  -0000014 

X. 

4-8100 

4-8289 

•0037081 

•0037091 

-  -oooooio 

] 

iYDROGEN. 

1-0000 

0-9996 

No  formula. 

•0036678 

1 

ELASTICITY   OF   GASES   AND  VAPOURS. 


37 


26.  The  empirical  formula?  (24),  representing  the  experiments  of  M. 
Eegnault  on  the  compressibility  of  carbonic  acid  gas,  atmospheric  air, 
and  hydrogen  at  certain  temperatures,  give  for  these  temperatures  the 
values  of  a  function  which  is  theoretically  expressed  by 


CraMP  CA       CwM/(D) 

rD  Tf  tD      * 


(27.) 


-r        ■  1  •  C»MP      p  .        ■         .  ,  ., 

It  is  evident,  that  supposing  the  value  of  — =—  tor  any  given  density 

TXJ 

to  be  known  by  experiment,  and  that  of  A  to  be  calculated  from  the 
value  of  the  coefficient  E,  or  from  the  empirical  formulae  (25)  and  (26), 

nMf(D) 


the  corresponding  value  of  the  function 


1) 


may  be  determined  by 


means  of  equation  (27). 

By  this  method  I  have  obtained  the  following  empirical  formulae  for 
calculating  the  values  of  that  function  : — 


For  Carbonic  Acid, 


«M/(D)  _      D 


D 


mM' 


where  log.  A  =3*1083932. 

Far  Atmospheric  Air, 


"M/(D)       _  /  D 


D 
where  log.  h=  3-8181545. 

nMf(D) 


* 


»il 


(28.) 


As  only  one  value  of 


I) 


for  hydrogen  can  at  present  be  ascer- 


tained, it  is  impossible  to  determine  a  formula  for  that  gas.     The  single 
value  in  question  is — 

For  P0  =  1  atmosphere,  wM^D)  =  -01059.        .     (29.) 

27.  I  now  proceed  to  determine  theoretically,  from  the  data  which 
have  already  been  obtained,  the  mean  coefficients  of  dilatation  at  constant 
pressure,  between  0°  and  100°  of  the  Centigrade  scale,  for  the  three  gases 
under  consideration,  at  various  pressures. 

Let  E'  represent  the  coefficient  required:    S0  and  Sx  the  respective 

%M 

values  of  -=r  for  0°  and  100°  under  the  pressure  P,  that  is  to  say,  the 

volumes  occupied  by  the  weight  nM  at  those  temperatures ;  A0  and  Av 


38 


ELASTICITY   OF   GASES  AND  VAPOURS. 


/0  and  fv  the  corresponding  values  of  A  and  /(D).     Then  from  equation 
(21)  we  deduce  the  following  results: — 


So  =  f(l-^Q+S0/o) 

s  -V14-1000-     A<     +S  f  V 
•  '  ~i'V    r   C        C  +  lOO0  '  '  lJl  )' 


and  consequently 


E'  = 


si_  ,s,> 

100  s0 

1  /l 


+ 


S0P\C  '  100C       100(C+100) 


100 


)•  •     (30.) 


In  applying  the  empirical  formuhe  (25),  (26),  and  (28),  to  determine 
the  values  of  Aj   and   S,  /,   in  the   above   equation,  it  will  produce  no 

appreciable  error  to  use  TTTfJ^o  as  an  approximate  value  of  Dx  for 

that  purpose  only.      By  making  the  necessary  substitutions,  the  following 
formula?  arc  obtained  : — 


For  Carbonic  Acid, 


wMPVC"1"      »M/J 
where  log.  0=  5*5189349. 

For  Atmospheric  Air, 

L",»Ml'  Vc+aV»M/     ~P'\nW  ) 

where  log.  a  =5*4717265 
log.  /3  =  6*9759738 


J 


(31.) 


28.  The  following  table  exhibits  a  comparison  between  the  results 
of  the  formulae  and  those  of  M.  Eegnault's  experiments.  It  is  not,  like 
the  preceding  table  (article  25),  the  verification  of  empirical  formulae, 
but  is  a  test  of  the  soundness  of  the  theoretical  reasoning  from  which 
equations  (30)  and  (31)  have  been  deduced. 

It  is  impossible,  from  the  want  of  a  sufficient  number  of  experimental 
data,  to  give  a  formula  similar  to  (31)  for  hydrogen.  I  have  calculated, 
however,  the  value  of  the  coefficient  E'  for  that  gas,  corresponding  to  the 
pressure  of  one  atmosphere,  on  the  assumption  that  at  that  pressure  a 
formula  similar  to  that  for  carbonic  acid  gas  is  applicable  without  sensible 
error. 


ELASTICITY   OF   GASES   AND   VAPOURS. 


39 


The  table  shows  only  one  instance  in  which  the  difference  between  the 
result  of  the  theory  and  that  of  experiment  exceeds  '0000136;  the  limit, 
according  to  M.  Eegnault,  of  the  errors  of  observation  capable  of  arising 
from  one  cause  alone, — the  uncertainty  of  barometric  measurements. 
That  discrepancy  takes  place  in  one  of  the  determinations  of  the  coefficient 
E'  for  carbonic  acid  gas  under  the  pressure  of  one  atmosphere.  In  the 
other  determination  the  discrepancy  is  less  than  the  limit. 

The  agreement  between  theory  and  experiment  is  most  close  for  the 
highest  pressures;  and  M.  Eegnault  has  shown  that  the  higher  the  pressure 
the  less  is  the  effect  of  a  given  error  of  observation  in  producing  an  error 
in  the  value  of  the  coefficient. 

The  theory  is  therefore  successful  in  calculating  the  coefficients  of 
dilatation  of  gases,  so  far  as  the  means  at  present  exist  of  putting  it  to 
the  test. 


Table  of  Coefficients  of  Dilatation  under  Constant  Pressure, 
showing  a  Comparison  between  Theory  and  Experiment. 


Pressure  in 

Atmospheres. 

Coefficient  E'  according 
to  the  Theory. 

Coefficient  E'  according 

to  M.  Kegnault's 

Experiments. 

Difference  between 
Theory  and 
Experiment. 

CARBONIC  ACID  GAS. 

1-000 
3-316 

•00369S8 
■0038430 

First  Memoir. 
•0037099 

Second  Memoir. 
•003719 

First  Memoir. 
•0038450 

-  0000111 

-  -0000202 

-  -0000020 

ATMOSPHERIC    AIR. 

1-0000 

3-3224 

3-4474 

•0036650 

■0036955 
•0036969 

First  Memoir. 

•0036706 

Second  Memoir. 
003663 
•003667 

First  Memoir. 
•0036944 
•0036965 

-  -0000056 

+  -0000020 

-  -0000020 

+  •0000011 
+  -0000004 

HYDROGEN. 

1-0000 

•0036598 

•0036613 

-  -0000015 

40  ELASTICITY   OF   GASES  AND   VAPOURS. 

Section  V. — Of  the  Elasticity  of  Vapour  in  Contact  with  the 
same  Substance  in  the  Liquid  or  Solid  State. 

29.  As  the  most  important  phenomena  of  evaporation  take  place 
from  the  liquid  state,  I  shall  generally  use  the  word  liquid  alone  through- 
out this  section  in  speaking  of  the  condition  opposed  to  the  gaseous  state  ; 
but  all  the  reasonings  are  equally  applicable  to  those  cases  in  which  a 
substance  evaporates  from  the  solid  state. 

30.  In  considering  the  state  of  a  limited  space  entirely  occupied  by 
a  portion  of  a  substance  in  the  liquid  form,  and  by  another  portion  of  the 
same  substance  in  the  form  of  vapour,  both  being  at  rest,  the  most  obvious 
condition  of  equilibrium  is,  that  the  total  elasticity  of  the  substance  in  each 
of  the  two  states  must  be  the  same;  that  is  to  say, 

P=^4V(D0)=iJi+/(D1),    •  •  •     (32.) 

where  p0  represents  the  superficial  atomic  elasticity  in  the  liquid  state,  px 
that  in  the  gaseous  state,  and  /(D0),  /(DJ,  the  corresponding  values  of  the 
pressures,  positive  or  negative,  due  to  mutual  actions  of  distinct  atoms. 

31.  A  second  condition  of  equilibrium  is,  that  the  superficial  elastici- 
ties of  two  contiguous  atoms  must  be  equal  at  their  surface  of  contact. 
Hence,  although  there  may  be  an  abrupt  change  of  density  at  the  bounding 
surface  between  the  liquid  and  the  vapour,  there  must  be  no  change  of 
superficial  atomic  elasticity  except  by  inappreciable  degrees;  and  at  that 
bounding  surface,  if  there  is  an  abrupt  change  of  density  (as  the  reflexion 
of  light  renders  probable),  there  must  be  two  densities  corresponding  to 
the  same  superficial  atomic  elasticity. 

32.  A  third  condition  of  equilibrium  is  to  be  deduced  from  the  mutual 
attractions  and  repulsions  of  the  atoms  of  liquid  and  of  vapour.  In  a  gas 
of  uniform  density,  those  forces,  acting  on  each  individual  particle  at  an 
appreciable  distance  from  the  bounding  surface,  balance  each  other,  and 
have  accordingly  been  treated  as  merely  affecting  the  total  elasticity  of  the 
body  by  an  amount  denoted  by  /(D) ;  but  near  the  bounding  surface  of  a 
liquid  and  its  vapour,  it  is  obvious  that  the  action  of  the  liquid  upon  any 
atom  must  be  greater  than  that  of  the  vapour.  A  force  is  thus  produced 
which  acts  on  each  particle  in  a  line  perpendicular  to  that  bounding 
surface,  and  which  is  probably  attractive  towards  the  liquid,  very  intense 
close  to  the  bounding  surface,  but  inappreciable  at  all  distances  from  it 
perceptible  to  our  senses.  Such  a  force  can  be  balanced  only  by  a  gradual 
increase  of  superficial  atomic  elasticity  in  a  direction  towards  the  liquid. 
Hence,  although  at  perceptible  distances  from  the  surface  of  the  liquid  the 
density  of  vapour  is  sensibly  uniform,  the  layers  close  to  that  surface  are 
probably  in  a  state  of  condensation  by  attraction,  analogous  to  that  of  the 
earth's  atmosphere  under  the  influence  of  gravity. 


ELASTICITY   OF   GASES  AND   VAPOURS.  41 

Professor  Faraday  has  expressed  an  opinion,  founded  on  his  own  experi- 
ments and  those  of  MM.  Dulong  and  Thenard,  that  a  state  of  condensation 
exactly  resembling  that  which  I  have  described  is  produced  in  gases  by 
the  superficial  attraction  of  various  substances,  especially  platinum,  and 
gives  rise  to  chemical  actions  which  have  been  called  catalytic. 

To  express  this  third  condition  algebraically,  let  the  boundary  between 
the  liquid  and  the  vapour  be  conceived  to  be  a  plane  of  indefinite  extent, 
perpendicular  to  the  axis  of  x ;  and  let  positive  distances  be  measured  in 
a  direction  from  the  liquid  towards  the  vapour. 

Let  x,  x  +  dx  represent  the  positions  of  two  planes,  perpendicular  to  the 
axis  of  x,  bounding  a  layer  whose  thickness  dx  is  very  great  as  compared 
with  the  distance  between  two  atomic  centres,  but  very  small  as  compared 
with  any  perceptible  distance,  and  let  a  portion  of  the  layer  be  considered 
whose  transverse  area  is  unity. 

Let  p  represent  the  mean  density  of  the  layer.     Then  it  is  acted  upon 

by  a  force 

-pXrfx, 

the  resultant  of  the  actions  of  all  the  neighbouring  atoms,  which  has  the 
negative  sign,  because  it  is  attractive  towards  the  liquid,  X  being  a  func- 
tion of  the  position  of  the  layer  in  question,  and  of  the  densities  and 
positions  of  all  the  neighbouring  layers. 

The  superficial  atomic  elasticity  behind  the  layer  being  p,  and  in  front 

of  it  p  +  -j-  dx,  it  is  also  acted  on  by  the  force 

-to6*'* 

hence  its  condition  of  equilibrium  is 

!+»x=° (33-> 

In  order  to  integrate  this  equation,  so  as  to  give  a  relation  applicable  at 
perceptible  distances  from  the  surface,  let  x0,  xx  represent  the  positions  of 
two  planes  perpendicular  to  the  axis  of  x,  the  former  situated  in  the 
liquid,  the  latter  in  the  vapour,  and  so  far  asunder  that  the  densities 
beyond  them  are  sensibly  uniform,  and  equal  respectively  to  D0  for  the 
liquid  and  D1  for  the  vapour,  the  corresponding  superficial  atomic  elastici- 
ties being  p0  and  pv  Then  dividing  equation  (33)  by  p,  and  integrating 
between  the  limits  xQ  and  xv  the  result  obtained  is 

o   lJ  o 

Had  we  a  complete  knowledge  of  the  laws  of  molecular  forces  in  the 
solid,  liquid,  and  gaseous  states,  this  equation,  taken  in  conjunction  with  the 


42  ELASTICITY  OF   GASES  AND   VAPOURS. 

two  conditions  previously  stated,  would  be  sufficient  to  determine  formulae 
for  calculating  the  total  elasticity  and  the  respective  densities  of  a  liquid 
and  its  vapour  when  in  contact  in  a  limited  space,  at  all  temperatures. 

33.  In  the  absence  of  that  knowledge,  I  have  used  equation  (34), 
so  as  to  indicate  the  form  of  an  approximate  equation  suitable  Re- 
calculating the  elasticity  of  vapour  in  contact  with  its  liquid  at  all 
ordinary  temperatures,  the  coefficients  of  which  I  have  determined 
empirically, — for  water  and  mercury,  from  the  experiments  of  M.  licgnault, 
and  for  alcohol,  ether,  turpentine,  and  petroleum,  from  those  of  Dr.  Ure. 

It  has  been  shown  (equation  19)  that  the  superficial  atomic  elasticity 
is  expressible  approximately  in  terms  of  the  density  and  temperature 
for  gases  by  f      ,  /         r     \\ 

where  the  function  F  is  a  very  rapidly  converging  series,  in  terms  of  the 
negative  powers  of  the  absolute  temperature,  the  coefficients  being 
functions  of  the  density.  It  is  probable  that  a  similar  formula  is 
applicable  to  liquids,  the  series  being  less  convergent. 

It  follows  that  the  density  is  expressible  approximately  in  terms  of 
the  superficial  atomic  elasticity  by 

P=i,^(i+*(P,^)), 

the  function  4>  being  also  a  converging  scries  in  terms  of  the  negative 

powers  of  the  absolute  temperature,  and  the  coefficients  being  functions  of  p. 

Making   this    substitution    in   the    first    side   of   equation    (34),    and 

abbreviating  <p  (j),     T    J  into  <£,  we  obtain  the  following  result : — 

?1   j      1        _r_     |»,  _J_ 

]po   1K p       CnMJpQ  ^'j»(l+$) 

=  cii(l0-  *  "  los-^o  -//  *£(ifl)) 

=  -(Vx; (35.) 

o 
from  which,  making 

log-  tPo+l1  dP  ■    n\*\  =  *>  and  CnM  I  *  ch  ■  X  =  Q' 

8c/0    Jj;  x    p(i+*)  )\ 

the  following  value  results  for  the  hyperbolic  logarithm  of  the  super- 
ficial atomic  elasticity  of  the  vapour  at  sensible  distances  from  the 
surface  of  the  liquid  : —  ~ 

log.,  ^  =  ¥  - -.  .  •  •     (36.) 


ELASTICITY  OF   GASES  AND  VAPOURS.  43 

In  the  cases  which  occur  in  practice,  the  density  of  the  vapour  is  very 
small  as  compared  with  that  of  the  liquid.  Hence  it  follows  that  in 
such  cases  the  value  of  ^F  depends  chiefly  on  the  superficial  atomic  elasticity 
of  the  liquid,  and  that  of  Q  on  its  density.  The  density  is  known  to 
diminish  with  the  temperature,  but  slowly.  The  superficial  atomic 
elasticity,  according  to  equation  (32),  is  expressed  by 

Po  =  Pi  +f(Pi)  -/(*U 

where  px  and  f(Dx)  are  obviously  small  as  compared  with  /(D0),  a  function 
of  the  density  of  the  liquid,  so  that  the  variations  of  p0  and  of  M*  with 
the  temperature  are  comparatively  slow  also. 

Therefore,  when  the  density  of  the  vapour  is  small  as  compared  with 
that  of  the  liquid,  the  principal  variable  part  of  the  logarithm  of  its 
superficial  atomic  elasticity,  and  consequently  of  its  whole  pressure, 
is  negative,  and  inversely  proportional  to  the  absolute  temperature;  and 

ft 

a 

(a  and  /3  being  constants)  may  be  regarded  as  the  first  two  terms  of  an 
approximate  formula  for  the  logarithm  of  the  pressure. 

A  formula  of  two  terms,  similar  to  this,  was  proposed  about  1828 
by  Professor  Eoche.  I  have  not  been  able  to  find  his  memoir,  and 
do  not  know  the  nature  of  the  reasoning  from  which  he  deduced  his 
formula.  It  has  since  been  shown,  by  M.  Eegnault  and  others,  to  be 
accurate  for  a  limited  range  of  temperature  only.  The  quantity  corre- 
sponding in  it  to  t  is  reckoned  from  a  point  determined  empirically, 
and  very  different  from  the  absolute  zero. 

Thus  far  the  investigation  has  been  theoretical.  The  next  step  is  to 
determine  empirically  what  other  terms  are  requisite  in  order  to  approxi- 
mate to  the  effect  of  the  function  /(D),  and  of  the  variation  of  the 
functions  ^P  and  Q. 

The  analogy  of  the  formula  for  the  dilatation  of  gases,  the  obvious 
convenience  in  calculation,  and  the  fact  that  the  deviations  of  the  results 
of  the  first  two  terms  from  those  of  experiment  are  greatest  at  low 
temperatures,  naturally  induced  me  to  try,  in  the  first  place,  the  effect 
of  a  third  term  inversely  proportional  to  the  square  of  the  absolute 
temperature,  making  the  entire  formula  for  the  logarithm  of  the  pressure 
of  vapour  in  contact  with  its  liquid 


a  — 

r       7" 


and  the  inverse  formula,  for  calculating  the  abso- 
lute temperature  from  the  pressure, 

1  _       ja  -  log.  P       g2  _  (3_ 
y  4y2       2y 


(37.) 


44  ELASTICITY   OF  GASES  AND   VAPOURS. 

the  values  of  the  constants  a,  /3,  7,  being  determined  by  the  ordinary 
methods  from  three  experimental  data  for  each  substance. 

34.  The  agreement  of  those  formulae  with  the  results  of  experiment 
proved  so  remarkable,  that,  as  they  are  calculated  to  be  practically  useful, 
I  thought  it  my  duty  not  to  delay  their  publication  until  I  should  have 
an  opportunity  of  submitting  my  theoretical  researches  to  the  Royal 
Society  of  Edinburgh.  I  therefore  communicated  the  formulas  to  the 
Edinburgh  New  Philosophical  Journal  for  July,  1849,  together  with  the 
full  details  of  their  comparison,  graphic  and  tabular,  with  the  experiments 
of  M.  Regnault  upon  water  and  mercury,  and  with  those  of  Dr.  Ure  upon 
alcohol,  ether,  turpentine,  and  petroleum,  but  without  giving  any  account 
of  the  reasoning  by  which  1  had  been  led  to  them. 

Without  repeating  those  details  here,  I  may  state  that  the  agreement 
between  the  results  of  the  formula'  and  those  of  observation  is  in  every 
case  as  close  as  the  precision  of  the  experiments  renders  possible.  This 
is  remarkable,  especially  with  respect  to  the  experiments  of  M.  Regnault 
on  the  elasticity  of  steam,  which  extend  throughout  a  range  of  tempera- 
tures from  30°  below  zero  of  the  Centigrade  scale  to  230°  above  it,  and 
of  pressures  from  ytgv  °f  an  atmosphere  to  28  atmospheres,  and  which, 
from  the  methods  of  observation  adopted,  especially  those  of  measuring 
temperature,  necessarily  surpass  by  far  in  precision  all  other  experiments 
•  if  the  same  kind.  From  20°  to  230°  Cent,  the  greatest  discrepancy 
between  calculation  and  experiment  corresponds  to  a  difference  of  jfa  of 
a  Centigrade  degree,  and  very  few  of  the  other  differences  amount  to  so 
much  as  -^  of  a  degree.  Below  20°,  where  the  pressure  varies  so  slowly 
with  the  temperature  that  its  actual  value  is  the  proper  test  of  the 
formula,  the  greatest  discrepancy  is  -j3^  of  a  millimetre  of  mercury,  or 
2-jjxy  of  an  inch.  If  the  curves  representing  the  formula?  were  laid  down 
on  M.  Regnault's  diagram,  they  would  be  scarcely  distinguishable  from 
those  which  he  has  himself  drawn  to  exhibit  the  mean  results  of  his 
experiments. 

Annexed  is  a  table  of  the  values  of  the  constants  a,  log.  /3,  log.  y, 

ly    Ay- 

existing  experiments  on  mercury,  turpentine,  and  petroleum,  are  not 
sufficiently  extensive  to  indicate  any  precise  value  for  the  coefficient  y 
(which  requires  a  great  range  of  temperatures  to  evince  its  effect),  I  have 
used  for  these  fluids,  as  an  approximation,  the  first  two  terms  of  the 

formula  only,  a  —  — . 

T 

For   different   measures  of   pressure,  the  contact    a   evidently   varies 
equally  with  the  complement  of  the  logarithm  of  the  unit  of  pressure. 
For  different  thermometric  scales  /3  varies  inversely  as  the  length  of  a 


„  for  the  fluids  for  which   they  have  been  calculated.     As  the 

41  -v'     4.-V2'  J 


ELASTICITY    OF   GASES   AND   VAPOURS. 


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8 
degree,  y  inversely  as  the  square  of  that  length,  '     directly  as  the  length 

of  a  degree,  and  -  ----,-  directly  as  the  square  of  that  length. 

For  all  the  fluids  except  water,  it  will  probably  he  found  necessary  to 
correct  more  or  less  the  values  of  the  constants,  when  more  precise  and 
extensive  experiments  have  been  made,  especially  those  for  the  more 
volatile  ether,  and  for  turpentine,  petroleum,  and  mercury,  which  have  all 
been  determined  from  data  embracing  but  a  small  range  of  pressures. 

In  reducing  the  constants  for  the  Centigrade  scale  to  those  for  Fahren- 
heit's scale,  180°  of  the  latter  have  been  assumed  to  be  equal  to  100°  of 
the  former.  In  order  that  this  may  be  the  case,  the  boiling  point  of 
Fahrenheit's  scale  must  be  adjusted  under  a  barometric  pressure  of  760 
millimetres,  or  29"922  inches,  of  mercury,  whose  temperature  is  0° 
Centigrade. 

In  the  ninth  and  tenth  columns  of  tin-  table  are  given  the  limits  on  the 
scales  of  temperature  and  pressure  between  which  the  formulae  have  been 
compared  with  experiment.  It  is  almost  certain  that  the  formula  for  the 
pressure  of  steam  may  be  employed  without  material  error  for  a  consider- 
able range  beyond,  and  probably  also  that  for  the  pressure  of  vapour  of 
alcohol;  but  none  of  the  formula'  are  to  be  regarded  as  more  than 
approximations  to  the  exact  physical  law  of  the  elasticity  of  vapours,  for 
the  determination  of  which  many  data  are  still  wanting,  that  can  only  be 
supplied  by  extensive  series  of  experiments. 

The  following  are  some  additional  values  of  the  constant  a  for  steam, 
corresponding  to  various  units  of  pressure  used  in  practice  : — 

I'nits  of  Pressure.  Values  of  a. 

Atmospheres  of  760  millimetres  of  mercury — 
=  29-922  inches  of  mercury 
=  14*7  lbs.  on  the  square  inch 
=  1-0333  kilog.  on  the  square  centim.,        4-950433 

Atmospheres  of  30  inches  of  mercury — 

=  761-99  millimetres 

=  14-74  lbs.  on  the  square  inch 

=  1-036  kilog.  on  the  square  centim.,         4-949300 

Kilogrammes  on  the  square  centimetre,  .  .         4-964658 

Kilogrammes  on  the  circular  centimetre,  .         4 "859 748 

Pounds  avoirdupois  on  the  square  inch,  .  .         6-117662 

Pounds  avoirdupois  on  the  circular  inch,         .  6-012752 

Pounds  avoirdupois  on  the  square  foot,  .  .         8-276025 

All  the  numerical  values  of  the  constants  are  for  common  logarithms. 


ELASTICITY   OF   GASES   AND   VAPOURS.  47 

35.  According  to  the  principles  which  form  the  basis  of  calculation 
in  this  section,  every  substance,  in  the  solid  or  liquid  state,  is  surrounded 
by  an  atmosphere  of  vapour,  adhering  to  its  surface  by  molecular  attrac- 
tion ;  and  even  when  the  presence  of  vapour  is  imperceptible  at  all  visible 
distances  from  the  body's  surface,  the  elasticity  of  the  strata  close  to  that 
surface  may  be  considerable,  and  sufficient  to  oppose  that  resistance  to 
being  brought  into  absolute  contact,  which  is  well  known  to  be  very  great 
in  solid  bodies,  and  perceptible  even  in  drops  of  liquid.  It  is  possible 
that  this  may  be  the  only  cause  which  prevents  all  solid  bodies  from 
cohering  when  brought  together. 

The  action  of  an  atmosphere  of  vapour,  so  highly  dense  and  elastic  as 
to  operate  at  visible  distances,  may  assist  in  producing  the  spheroidal  state 
of  liquids. 

If  the  particles  of  clouds  are  small  vesicles  or  bubbles  (which  is  doubt- 
ful), the  vapour  within  them  may,  according  to  these  principles,  be 
considerably  more  dense  than  that  which  pervades  the  external  air,  and 
may  thus  enable  them  to  preserve  their  shape. 

Section  VX — Of  Mixtures  of  Gases  and  Vapours  of 
Different  Kinds. 

36.  The  principle  stated  in  Section  II.  article  11,  that  the  elasticity  of 
the  atomic  atmosphere  is  proportional  to  its  density,  might  be  otherwise 
expressed  by  saying,  that  the  elasticity  of  any  number  of  portions  of  atomic 
atmosphere,  compressed  into  a  given  space,  is  equal  to  the  sum  of  the  elasticities 
which  such  portions  would  respectively  have,  if  they  occupied  the  same  space 
separately. 

If  the  same  principle  here  laid  down  for  portions  of  atomic  atmosphere 
of  any  one  kind  of  substance,  be  considered  as  true  also  of  portions  of 
atomic  atmosphere  of  substances  of  different  kinds  mixed,  and  if  it  be 
supposed  that  when  two  or  more  gases  are  mixed  there  is  no  mutual  force 
exerted  between  atoms  of  different  kinds,  except  the  elastic  pressure  of 
the  atomic  atmospheres,  it  will  then  evidently  follow. — 

First,  that  the  mixed  gases  will  only  be  in  equilibria  when  the  particles 
of  each  of  them  are  diffused  throughout  the  whole  space  which  contains 
them. 

Secondly,  that  the  particles  of  each  gas  taken  separately  will  be  in  the 
same  condition  as  to  density,  elasticity,  arrangement,  and  mutual  action, 
and  also  as  to  gravitation,  or  any  other  action  of  an  external  body,  as  if 
that  gas  occupied  the  space  alone. 

Thirdly,  that  the  joint  elasticity  of  the  mixed  atomic  atmospheres  at  any 
given  point,  will  be  the  sum  of  the  elasticities  which  they  would  respec- 
tively have  had  at  that  point  if  each  gas  had  occupied  the  space  alone. 


48  ELASTICITY  OF   GASES   AND   VAPOURS. 

Fourthly,  that  the  value  of  the  elasticity,  positive  or  negative,  resulting 
from  the  attractions  and  repulsions  of  separate  atoms,  will  be  the  sum  of 
the  values  it  would  have  had  if  each  gas  had  occupied  the  space  alone ; 
and, 

Fifthly,  that  the  total  elasticity  of  the  mixed  gases  will  be  the  sum  of 
the  elasticities  which  each  would  have  had  separately  in  the  same  space. 

If  there  are  any  mutual  actions  between  the  particles  of  different  gases 
except  the  elasticity  of  the  molecular  atmospheres,  these  conclusions  will 
no  longer  be  rigidly  true  •  but  they  will  still  be  approximately  true  if  the 
forces  so  operating  are  very  small.  This  is  probably  the  actual  condition 
of  mixed  gases. 

37.  On  applying  the  same  principle  to  the  case  of  a  gas  mixed  with  a 
vapour  in  contact  with  its  liquid,  it  is  obvious  that  if  the  attractions  and 
repulsions  of  the  particles  of  the  gas  upon  those  of  the  vapour  are  null  or 
inappreciable,  the  direct  effect  of  the  presence  of  the  gas  upon  the  elasticity 
assumed  by  the  vapour  at  a  given  temperature  will  also  be  null  or 
inappreciable. 

The  gas,  however,  may  have  a  slight  indirect  influence,  by  compressing 
the  liquid,  and  consequently  increasing  its  superficial  atomic  elasticity  and 
its4  attractive  power,  on  which  the  functions  ¥  and  £2  in  equation  (3G) 
depend.  The  probable  effect  of  this  will  be  to  make  the  elasticity  of  the 
vapour  somewhat  less  than  if  no  gas  were  present.  There  appear  to  be 
some  indications  of  such  an  effect ;  but  they  are  not  sufficient  to  form  a 
basis  for  calculation. 

Supposing  the  gas,  on  the  contrary,  to  exercise  an  appreciable  attraction 
on  the  particles  of  vapour,  the  elasticity  of  the  latter  will  be  increased. 
Traces  of  an  effect  of  this  kind  are  perceptible  in  M.  Regnault's  experi- 
ments on  the  vapour  of  mercury  in  which  air  was  present. 

38.  I  have  already  referred  to  the  property  ascribed  by  Professor 
Faraday  to  various  substances,  of  attracting,  and  retaining  at  their 
surfaces,  layers  of  gas  and  vapour  in  a  high  state  of  condensation.  Sup- 
posing a  solid  body  to  acquire,  in  this  manner,  a  mixed  atmosphere, 
consisting  partly  of  its  own  vapour  and  partly  of  foreign  substances,  the 
total  elasticity  of  that  atmosphere  at  any  point  will  be  equal,  or  nearly 
equal,  to  the  sum  of  the  elasticities  which  each  ingredient  would  have  had 
separately;  and  thus  solid  metals,  glass,  charcoal,  earthy  matters,  and 
other  substances,  may  acquire  a  great  power  of  resisting  cohesion,  although 
producing  no  perceptible  vapours  of  their  own  at  ordinary  temperatures. 


THE   CENTRIFUGAL   THEORY   OF   ELASTICITY.  49 


IV.— ON  THE  CENTRIFUGAL  THEORY  OF  ELASTICITY  AND 
ITS  CONNECTION  WITH  THE  THEORY  OF  HEAT.* 

Section  I. — Relations  between  Heat  and  Expansive  Pressure. 

1.  In  February,  1850,  I  laid  before  the  Royal  Society  of  Edinburgh 
a  paper  in  which  the  laws  of  the  pressure  and  expansion  of  gases  and 
vapours  were  deduced  from  the  supposition  that  that  part  of  the  elasticity 
of  bodies  which  depends  upon  heat  arises  from  the  centrifugal  force  of  the 
revolutions  of  the  particles  of  elastic  atmospheres  surrounding  nuclei  or 
atomic  centres.  A  summary  of  the  results  of  this  supposition,  which  I 
called  the  Hypothesis  of  Molecular  Vortices,  was  printed  in  the  Transactions 
of  this  Society,  Vol.  XX.,  as  an  introduction  to  a  series  of  papers  on  the 
Mechanical  Action  of  Heat;  and  the  original  paper  has  since  appeared  in 
detail  in  the  Philosophical  Magazine. 

In  that  paper  the  bounding  surfaces  of  atoms  were  defined  to  be  imaginary 
surfaces,  situated  between  and  enveloping  the  atomic  nuclei,  and  sym- 
metrically placed  with  respect  to  them,  and  having  this  property — that  at 
these  surfaces  the  attractive  and  repulsive  actions  of  the  atomic  nuclei  and 
atmospheres  upon  each  particle  of  atomic  atmosphere  balance  each  other. 
The  pressure  of  the  atomic  atmospheres  at  those  imaginary  boundaries  is 
the  part  of  the  total  expansive  pressure  of  the  body  which  varies  with 
heat,  the  effect  of  the  centrifugal  force  of  molecular  vortices  being  to 
increase  it. 

In  the  subsequent  investigation  it  was  assumed  that,  owing  to  the  sym- 
metrical action  of  the  particles  of  gases  in  all  directions,  and  the  small 
amount  of  those  attractive  and  repulsive  forces  which  interfere  with  the 
elasticity  of  their  atmospheres,  no  appreciable  error  would  arise  from 
treating  the  boundary  of  the  atmosphere  of  a  single  atom,  in  calculation, 
as  if  it  were  spherical,  an  assumption  which  very  much  simplified  the 
analysis. 

An  effect,  however,  of  this  assumption  was  to  make  it  doubtful  whether 
the  conclusions  deduced  from  the  hypothesis  were  applicable  to  any  sub- 
stances except  those  nearly  in  the  state  of  perfect  gas.     I  have,  therefore, 

*  Read  before   the   Royal  Society  of  Edinburgh  on  December   15,   1S51,   and 
published  in  Vol.  XX. ,  Part  iii. ,  of  the  Transactions  of  that  Society. 

D 


50  THE  CENTRIFUGAL  THEORY   OF   ELASTICITY. 

in  the  present  paper,  investigated  the  subject  anew,  without  making  any 
assumption  as  to  the  arrangement  of  the  atomic  centres,  or  the  form  of 
the  boundaries  of  their  atmospheres.  The  equations  deduced  from  the 
hypothesis  between  expansive  pressure  and  heat,  are  therefore  applicable 
to  all  substances  in  all  conditions;  and  it  will  be  seen  that  they  are 
identical  with  those  in  the  original  paper,  showing  that  the  assumption 
that  the  atomic  atmospheres  might  be  treated,  in  calculation,  as  if  spherical, 
did  not  give  rise  to  any  error. 

By  the  aid  of  certain  transformations  in  those  equations  I  have  been 
enabled,  in  investigating  the  principles  of  the  mutual  transformation  of 
heat  and  expansive  power,  to  deduce  Joules  law  of  the  equivalence  of  heat 
and  mechanical  power  directly  from  them,  instead  of  taking  it  (as  I  did  in 
my  previous  papers)  as  a  consequence  of  the  principle  of  vis  viva.  Carnofs 
law  is  also  deduced  directly  from  the  hypothesis,  as  in  one  of  the  previous 
papers. 

2.  Classification  of  Elastic  Pressures. — The  pressures  considered  in  the 
present  paper  are  those  only  which  depend  on  the  volume  occupied  by  a 
given  weight  of  the  substance,  not  those  which  resist  change  of  figure  ill 
solids  and  viscous  liquids.  Certain  mathematical  relations  exist  between 
those  two  classes  of  pressures ;  but  they  do  not  affect  the  present 
investigation. 

To  illustrate  this  symbolically,  let  V  represent  the  volume  occupied  by 

unity  of  weight  of  the  substance,  so  that       is  the  mean  density;  Q,  the 

V 

quantity  of  heat  in  unity  of  weight,  that  is  to  say,  the  vis  viva  of  the  mole- 
cular revolutions,  which,  according  to  the  hypothesis,  give  rise  to  the 
expansive  pressure  depending  on  heat ;  and  let  P  denote  the  total  expan- 
sive pressure.     Then, 

P  =  F(V,Q)+/(V),.         .         .         .     (1.) 

In  this  equation,  F(V,  Q)  is  the  pressure  of  the  atomic  atmospheres  at 
the  surfaces  called  their  boundaries,  which  varies  with  the  centrifugal 
force  of  the  molecular  vortices  as  well  as  with  the  mean  density ;  and 
/( V)  is  a  portion  of  pressure  due  to  the  mutual  attractions  and  repulsions 
of  distinct  atoms,  and  varying  with  the  number  of  atoms  in  a  given 
volume  only.  If  the  above  equation  be  differentiated  with  respect  to  the 
hyperbolic  logarithm  of  the  density,  we  obtain  the  coefficient  of  elasticity 
of  volume 

V  V  V 

where  ft  denotes  the  cubic  compressibility. 


THE   CENTRIFUGAL  THEORY   OF  ELASTICITY.  51 

The  latter  portion  of  this  coefficient,  —  -==f(V),  consists  of  two  parts, 

T 

one  of  which  is  capable  of  being  resolved  into  forces,  acting  along  the 
lines  joining  the  atomic  centres,  and  gives  rise  to  rigidity  or  elasticity  of 
figure  as  well  as  to  elasticity  of  volume,  while  the  other,  which  is  not  capable 
of  being  so  resolved,  gives  rise  to  elasticity  of  volume  only.  The  ratio  of 
each  of  those  parts  to  their  sum  must  be  a  function  of  the  heat,  the  former 
part  being  greater  and  the  latter  less,  as  the  atomic  atmosphere  is  more 
concentrated  round  the  nucleus ;  that  is  to  say,  as  the  heat  is  less ;  but 
their  sum,  so  far  as  elasticity  of  volume  is  concerned,  is  a  function  of  the 
density  only. 

That  is  to  say,  as  in  equation  (12)  of  my  paper  on  the  Laws  of  the 
Elasticity  of  Solids  {Cambridge  and  Dublin  Mathematical  Journal,  February, 
1851),  let  the  total  coefficient  of  elasticity  of  volume  be  denoted  thus — 

~  =  J  +  cp(CvC2,Cs),       .  •     (IB.) 

C1?  C2,  C3,  being  the  coefficients  of  rigidity  round  the  three  axes  of  elas- 
ticity, and  J  a  coefficient  of  fluid  elasticity ;  then 


J  =  -^F(V,Q)-^(V,Q,)-^/(V) 

T  V 

d 


y  •  (ic.) 


^(C1;C2,C3)  =  -(l-^V,Q))-^/(V) 

V  j 

For  the  present,  we  have  to  take  into  consideration  that  portion  only 
of  the  expansive  pressure  which  depends  on  density  and  heat  jointly,  and 
is  the  means  of  mutually  converting  heat  and  expansive  power ;  that  is 
to  say,  the  pressure  at  the  boundaries  of  the  atomic  atmospheres,  which 
I  shall  denote  by 

Pressures  throughout  this  paper  are  supposed  to  be  measured  by  units 
of  weight  upon  unity  of  area ;  densities  by  the  weight  of  unity  of  volume. 

3.  Determination  of  the  External  Pressure  of  an  Atomic  Atmosphere. — 
Let  a  body  be  composed  of  equal  and  similar  atomic  nuclei,  arranged  in 
any  symmetrical  manner,  and  enveloped  by  an  atmosphere,  the  parts  of 
which  are  subject  to  attractive  and  repulsive  forces  exercised  by  each 
other  and  by  the  nuclei.  Let  it  further  be  supposed  that  this  atmosphere, 
at  each  point,  has  an  elastic  pressure  proportional  to  the  density  at  that 
point,  multiplied  by  a  specific   coefficient  depending  on  the  nature  of  the 

BOSTON  UNIVERSITY 

C.ni  i  FGF  or  LIBERAL  ARTS 


52  THE   CENTRIFUGAL  THEORY  OF   ELASTICITY. 

substance,  which  I  shall  denote  by  h.  (This  coefficient  was  denoted  by  b 
in  previous  papers.) 

Let  p  and  p  denote  the  density  and  pressure  of  the  atomic  atmosphere 
at  any  point ;  then 

/  =  hP 

be  the  accelerative  forces  operating  on  a  particle  of  atomic  atmosphere  in 
virtue  of  the  molecular  attractions  and  repulsions  which  I  have  made 
explicitly  negative,  attractions  being  supposed  to  predominate.  The 
property  of  the  surfaces  called  the  boundaries  of  the  atoms  is  this 


(^=0,(1)^0,(^=0, 


the  suffix  ,  being  used  to  distinguish  the  value  of  quantities  at  those 
surfaces.  Hence  $x  is  a  maximum  or  minimum.  Those  surfaces  are 
symmetrical  in  form  round  each  nucleus,  and  equidistant  between  pairs  of 
adjacent  nuclei.     Their  equation  is 

$  -  $!  =  0. 

Let  M  denote  the  total  weight  of  an  atom ;  /x  that  of  its  atmospheric 
part,  and  M  —  fi  that  of  its  nucleus;  tl  en 
MV  is  the  volume  of  the  atom, — 

the  mean  density  of  the  atmospheric  part,  measured  by  weight,  the 

nucleus  being  supposed  to  be  of  insensible  magnitude; — 
and  we  have  the  following  equations 


MV  =  //./(1)'/'''/ 

M  =  m v j  1  i  Qd  x '' y ''  "■ =  1 1 1  d)p  dxdy(l  -■ 


(2.) 


J 


The  suffix  (j)  denoting  that  the  integration  is  to  be  extended  to  all 
points  within  the  surface 

($-$1  =  0). 

According  to  the  hypothesis  now  under  consideration,  Meat  consists  in 
a  revolving  motion  of  the  particles  of  the  atomic  atmosphere,  com- 
municated to  them  by  the  nuclei.  Let  v  be  the  common  mean  velocity 
possessed  by  the  nucleus  of  an  atom  and  the  atmospheric  particles,  when 
the  distribution  of  this  motion  has  been  equalised.  I  use  the  term  mean 
velocity  to  denote  that  the  velocity  of  each  particle  may  undergo  small 
periodic  changes,  which  it  is  unnecessary  to  consider  in  this  investigation. 


THE   CENTRIFUGAL   THEORY   OF   ELASTICITY. 


53 


Then  the  quantity  of  heat  in  unity  of  weight  is 


2g 

being  equal  to  the  mechanical  power  of  unity  of  weight  falling  through 

the  height  — .     The  quantity  of  heat  in  one  atom  is  of  course  MQ,  and  in 

the  atmospheric  part  of  an  atom,  pQ. 

I  shall  leave  the  form  of  the  paths  described  by  the  atmospheric  par- 
ticles indeterminate,  except  that  they  must  be  closed  curves  of  permanent 
figure,  and  included  within  the  surface  (<!>  -c&^O).  Let  the  nucleus  be 
taken  as  the  origin  of  co-ordinates,  and  let  a,  )3,  y,  be  the  direction- 
cosines  of  the  motion  of  the  particles  at  any  point  (x,  y,  z).  Then  the 
equations  of  a  permanent  condition  of  motion  at  that  point,  are 


P   dx      ax  \  dx     '    ay        dz/ 


1    dp 

P 

1    dp'  d$      2 

p'  dy  dy 

1    dp'  d& 

p'  dz  dz 


1 


di 


■f.+pi+^ 


■     (3.) 


rt  /    d      ~.  d         d\        „ 


Let  r  be  the  radius  of  curvature  of  the  path  of  the  particles  through 
(x,  y,  z),  and  o'  /3'  7',  its  direction-cosines;  then  the  above  equations 
obviously  become 


1  dp_ 
p  dx 
1  dp' 
p  dy 
1  rf_p' 
p   dz 


d$ 
dx 

d$> 
dy 

d$ 
dz 


a 
r 


0 


-  2Q 
_2q£-=C 


2Q^ 

r 


0 


(3A.) 


If  these  equations  are  integrable, 

—  dx  +  $-  dy  +^-dz 
r  r  r 

must  be  an  exact  differential.  Let  —  <p  be  its  primitive  function ;  the 
negative  sign  being  used,  because  a,  /3',  y'  must  be  generally  negative. 
Then  the  integral  of  the  equations  (3)  is 


loSvP  = 


dp'       1  , 

-£-  =  -=-(2  Q0  —  <P)  +  constant: 


h 


54 


THE  CENTRIFUGAL  THEORY  OF   ELASTICITY. 


or  taking  f>y  to  denote  the   pressure  at   the  bounding   surface   of  the 
atom : — 


P  =  Pie 


■        (4.) 


Our  present  object  is  to  determine  the  superficial-atomic  density,  pv 
and  thence  the  pressure  2)=hpv  ^u  terms  °f  tne  mean  density        and 

heat  Q.     For  this  purpose  we  must  introduce  the  above  value  of  p  into 
equation  (2),  giving 

^t^p^  J  J  f/>fi  dxdydz 

whence 

\ ( f  h  dxdydz.    (5.) 

Let  the  volume  of  the  atom  be  conceived  to  be  divided  into  layers,  in 
each  of  which  <p  has  a  constant  value.  Then  we  may  make  the  following 
transformations. 


p  =  hPl=hfi+J  j  |( 


f  f  Id  x  d  y  d  z=kMvfe  ~  d<J> 


—  1  (*_*1) 


h(<p-<Pi) 


>     (C) 


dxdydz=kMYje  ^d<p 


k  being  a  specific  constant,  and  xp  and  w  functions  of  cf>,  and  of  the  nature 
and  density  of  the  substance. 

The  lower  limit  of  integration  of  $  must  be  made  —  cc  ,  that  it  may 
include  orbits  of  indefinitely  small  magnitude  described  round  the  atomic 
centre. 

The  nature  of  the  function  \p  is  limited  by  the  following  condition, 


l=k\Vl     e  -T-d<h 

J    —  GO  Wt 


(7.) 


20 
Let^+1  = 

hie 


Then  these  transformations  give  the  following  result  for  the  pressure  at 
the  bounding  surface  of  an  atom  : — 


p=hPl 


Jl/UL 

MV 


+£ 


h<px     fl*(0-0i) 


kdcp 


hfx 
MV 


0Wl 


w\    i    w  l        w    l    ,    p 


►  .     (8.) 


e  ■  e2 


THE   CENTRIFUGAL   THEORY  OF  ELASTICITY.  55 

to'v  &c,  being  the  successive  differential  coefficients  of  w  with  respect  to 
k([>,  when  <p  =  (pv 

4.  The  following  transformation  will  be  found  useful  in  the  sequel. 

Let  X  be  the  indefinite  value  of  log.e  V,  and  \  its  actual  value  in  the 
case  under  consideration.  Let  G  be  the  same  function  of  A  which  w  is 
of  k<p,  and  let  G',  G",  &c,  be  its  successive  differential  coefficients  with 
respect  to  A. 

Let 

„     f\     *(*-*!)„  ,.    gx    g;  ,  g;'     , 

Then 

p-uv^ {J-> 

The  function  H  has  the  following  properties,  which  will  be  afterwards 
referred  to : — 


<^  +  6li1-G1=0 


\    mix=-d^ 

—  oo  a  v 


(10.) 


5.  Case  of  a  Perfect  Gas.  —  As  a  substance  is  rarefied,  it  gradually 
approaches  a  condition  in  which  the  pressure,  under  like  circumstances  as 
to  heat,  varies  proportionally  to  the  density.  This  is  because  the  effect 
of  the  molecular  attractions  and  repulsions  on  the  pressure  diminishes 
with  the  density,  so  that  <P,  w,  and  G  approximate  to  constant  quantities. 
In  the  limiting  or  perfectly  gaseous  condition,  therefore, 

G, 


Hi=7T 


and 


hnQ       hfi  /2Q        \ 
P=MY=MvKhk+1)'   *  •     (1L 


6.  Equilibrium  of  Heat :  Nature  of  Temperature  and  Real  Specific 
Heat. — When  the  atmospheres  of  atoms  of  two  different  substances  are 
in  contact  at  their  common  bounding  surface,  it  is  necessary  to  a  per- 
manent condition  that  the  pressure  in  passing  that  surface  should  vary 
continuously. 

Let  (a)  and  (b)  be  taken  as  characteristics,  to  distinguish  the  specific 
quantities  peculiar  to  the  two  media  respectively.  Let  dm  denote  the 
volume  of  an  indefinitely  thin  layer,  close  to  the  bounding  surface. 
Then  the  following  equations  must  be  fulfilled  to  ensure  a  permanent 
condition  : — 

i;(a)^;(6);^(a)=^(J)when/^    *  *     (12° 


56  THE   CENTRIFUGAL   THEORY   OF   ELASTICITY. 

By  making  the  proper  substitutions  in  equation  (4),  it  appears  that 
j/=pe  7- 


Hence 


*•? 


£p=^=^-dM    +    dm 


dm 
Now,  j)  is  the  same  for  both  media  : 


&  •  _   --*<•-•>> 


/  =  c 


is  either  a  maximum  or  a  minimum,  so  that  its  differential  is  null ;  and 
dm  is  a  continuous  function  of  lc<p,  so  that 

dm  v  dm  v 

There  remains  only  the  function  of  heat : 

20 

Therefore  the  condition  of  a  permanent  state  of  molecular  motion,  that 
is  to  say,  the  condition  of  equttibrivm  of  heat,  is  that  this  function  shall  be 
the  same  for  the  two  substances ;  or  that 


?%=|a as.) 


Hence,  temperature  depends  on  the  above  function  only ;  for  the 
definition  of  temperature  is,  that  bodies  at  the  same  temperature  are  in 
a  permanent  condition  as  to  heat,  so  far  as  their  mutual  action  is 
concerned. 

The  ratio  of  the  real  specific  heat  of  (a)  to  that  of  (b)  is  obviously 

KK  ■■  *A-     •  •      •    (u-> 

7.  Measure  of  Temperature  ami  Specific  Heat.  —  The  function  6  is 
proportional  to  the  pressure  of  a  perfect  gas  at  a  constant  density.  That 
pressure,  therefore,  is  the  most  convenient  measure  of  temperature. 

Let  t  denote  absolute  temperature,  as  measured  by  the  pressure  of  a 
perfect  gas  at  constant  density,  and  reckoned  from  a  certain  absolute  zero, 
2740-G  Centigrade,  or  494°-28  Fahrenheit  below  the  temperature  of 
melting  ice.  Let  k  be  a  constant  which  depends  on  the  length  of  a 
degree  on  the  thermometric  scale,  and  is  the  same  for  all  substances  in 
nature.     Then 


2^,      1 
M 

hi 


+  K 

Q  =  (r-«) 


C    I 


'2«  J 


THE   CENTRIFUGAL  THEORY  OF   ELASTICITY.  57 

and  the  real  specific  heat  of  the  substance,  that  is  to  say,  the  depth  of 
fall,  under  the  influence  of  gravity,  which  is  equivalent  to  a  rise  of  one 
degree  of  temperature  in  the  body,  is  represented  by 

fe=P (1G.) 

The  pressure  of  a  perfect  gas  is  represented  in  terms  of  temperature  by 

h  U.  T  ,-.  „  \ 

It  may  also  be  expressed  thus  :  let  r0  denote  the  absolute  temperature 
of  meltiDg  ice  in  degrees  of  the  scale  employed,  and  V0  the  volume  of 
unity  of  weight  of  the  substance  in  the  theoretical  state  of  perfect  gas,  at 
the  temperature  of  melting  ice  and  pressure  unity  : — then 

*-%P     ■  ■  <18-> 

v     To 
On  comparing  this  with  equation  (17)  we  see  that 

kfx  _  V0_  1 


h  flT0        h  JUL  K 


(19.) 


MV0'MV0      r0         J 
Xow  h  is  the  specific  elasticity  of  the  atomic  atmosphere  of  the  substance ; 

=-£=-  is  the  mean  specific  gravity  of  that  atmosphere,  when  the  body  is  in 
M  V0 

the  theoretical  state  of  perfect  gas;  and  k  and  r0  are  the  same  for  all 

substances  in  nature.      Therefore,  for  every  substance  in  nature,  the  mean 

specific  gravity  of  the  atomic  atmosphere  in  the  theoretical  state  of  perfect  gas  is 

inversely  proportional  to  the  specific  elasticity  of  that  atmosphere. 

Real  specific  heat  may  also  be  thus  expressed : — 

k==yBi*M (20) 

V  1  -,  ^M,    3£M 

in  which  -°-  corresponds  to  t^-tv  m  my  former  papers,  and  — —  to  -r— - 

1 

or  =r. 

N 

The  latter  factor  appears  to  depend  on  the  chemical  constitution  of  the 
substance,  being  the  same  for  all  simple  gases. 

8.  Total  Pressure  of  Substances  in  general,  expressed  in  terms  of  Temperature. 


In  equation  (9)  let  -  be  put  for  9  :  then 


58  THE   CENTRIFUGAL  THEORY   OF   ELASTICITY. 

v=P  +/(v)  =/(V)  +  %  £»,+  {«.-  K^+^':  -  M 

where 

Ax  =  —  — ;  A2  =  —  ^  2(G/2  —  G/') 

a3  =  -  j£3(<V3 - 20/6/'+ G/");  &c. 

This  formula  is  identical  with  that  which  I  employed  in  my  former 
paper  to  represent  the  pressure  of  an  imperfect  gas,  and  which  I  found  to 
agree  with  M.  Regnault's  experiments,  when  the  coefficients  A  and  the 
function /(V)  had  been  calculated  empirically. 


Section  II. — Eelations  between  Heat  and  Expansive  Power. 

9.  Variations  of  Sensible  and  Latent  Heat :  Fundamental  Equation  of  the 
Theory. — If  the  forms,  positions,  and  magnitudes  of  the  paths  described  by 
the  revolving  particles  of  the  atomic  atmospheres  be  changed,  whether  by 
a  variation  of  mean  density,  or  by  a  variation  of  temperature,  an  increase 
or  diminution  of  the  vis  lira  of  their  motion,  that  is  to  say,  of  the  heat  of 
the  body,  will  take  place  in  virtue  of  that  change  of  the  paths  of  motion — 
an  increase  when  they  are  contracted,  and  a  diminution  when  they  are 
dilated. 

Let  S.Q  represent,  when  positive,  the  indefinitely  small  quantity  of 
heat  which  must  be  communicated  to  unity  of  weight  of  a  substance,  and 
when  negative,  that  which  must  be  abstracted  from  it,  in  order  to  produce 
the  indefinitely  small  variation  of  temperature  S  t  simultaneously  with  the 
indefinitely  small  variation  of  volume  2V.  Let  S.Q  be  divided  into  two 
parts 

SQ  +  SQ'=S.Q, 

of  which  SQ,  being  directly  employed  in  varying  the  velocity  of  the  particles, 
is  the  variation  of  the  actual  or  sensible  heat  possessed  by  the  body ;  while 
SQ',  being  employed  in  varying  their  orbits,  represents  the  amount  of  the 
mutual  transformation  of  heat  with  expansive  power  and  molecular  action, 
or  the  variation  of  what  is  called  the  latent  heat;  that  is  to  say,  of  a 
molecular  condition  constituting  a  source  of  power,  out  of  which  heat  may 
be  developed.  (§Q'  in  this  paper  corresponds  to  —  2Q'  in  my  former 
papers.) 


THE   CENTRIFUGAL   THEORY   OF   ELASTICITY.  59 

The  variation  of  sensible  heat  has  evidently  this  value, 

ZQ  =  \\St (22.) 

Let  dx,  By,  $z,  be  the  displacements  of  the  orbit  of  the  particles  of  atomic 
atmosphere  at  the  point  (x,  y,  z) .  A  molecule  p  dxclydz  is  acted  upon 
by  the  accelerative  forces  (see  equation  3A). 

parallel  to  the  three  axes  respectively. 

The  sum  of  the  actions  of  those  forces  on  the  molecule  pdxdydz 
during  the  change  of  temperature  and  volume,  is 


<>+^+^-V-^--< 


dy 

=  —  2Q<$<ppdxd  yd  :. 

The  sum  of  such  actions  upon  all  the  particles  in  unity  of  weight  is 
equal  in  amount  and  opposite  in  sign  to  the  variation  of  latent  heat ;  that 
is  to  say, 

W  =  ^fff{i)p$<pdxdydz.        .         .     (23.) 

To  determine  the  value  of  the  variation  Bcp,  let  it  be  divided  into  two 
parts,  thus  : — 

t(p  —  ccpl  +  oAcp, 
where  A<j>  =  <p  —  (pr 

First,  With  respect  to  $<f>v  it  is  obvious  that  because,  according  to 
equations  (6,  7),  • 

,-0i       ks<p  j. 
MY  =  IMY  e      -f-dd>, 

J  -co         \px    r 

we  must  have 

BV=:kYB(p1  and  $<Pi  =  t^- 
Hence  the  first  part  of  the  integral  (23)  is 

=&-4-   •   •   •  <23A-> 

To  determine  the  second  part  of  the  integral  we  have  the  condition  that 
the  quantity  of  atomic  atmosphere  inclosed  within  each  surface  at  which 
A(f>  has  some  given  value  is  invariable;  that  is  to  say, 


GO  THE   CENTRIFUGAL  THEORY   OF   ELASTICITY. 

Hence 

ftp.  M  Ve        — , 

The  value  of  the  second  part  of  the  integral  (23)  is  now  found  to  be: — 


l-  e±$ 


M 


[j  f  fQPSA<pdxdydz=  ^  kPl  M  v/*  x  .  '     ^  SA0^ 

In  the  double  integral,  let  X  =  log.,,  V  be  put  for  fc0,  G  for  w,  and  H  for 
the  single  integral,  as  in  equation  (9).     Then  the  double  integral  becomes 

G,   dr' 

Also,because  PlMV  =  M  ' :  by  Eq.  (9),  and  ^  =  -  (t-  k), the  second  part 

of  the  integral  (23)  is  found  to  be 


hfx 
M 


('-o(»'c+,v^)©-  •     •  (33R) 


Hence,  adding  together  (2 3 A)  and  (23B)  we  find  for  the  total  variation 
of  latent  heat 

so'-  hfL  (r      *\  i  £r  ^loS>Hi   ■  SV  ■  (1    4--10-'  HA  1     (24  ) 
8Q-^(r-K)|Sr.— ^-     +6V    ^y  +    dTdy)$-W 

To  express  this  in  terms  of  quantities  which  may  be  known  directly  by 
experiment,  we  have,  by  equations  (10)  and  (9) — 

^  +  0-^  =  0,  that  is  to  say, 

d\og.,TL1_    Gj    _  r        M    _    t_ 
dV      ~HXV     K\~hfxP      kY' 
and,  therefore, 

TYT   f  — 

log.e  Hx  =  —  h?fZ  V-  -logvV+/(r)  +  constant. 

Il  [A  J  K 

/(r)  is  easily  found  to  be  =  —  log.er  for  a  perfect  gas,  and,  being  indepen- 
dent of  the  density,  is  the  same  for  all  substances  in  all  conditions ;  hence 
we  find  (the  integrals  being  so  taken  that  for  a  perfect  gas  they  shall  =  0) 


THE   CENTRIFUGAL   THEOEY   OF   ELASTICITY.  61 


dr  J   Vi/mclr      kVJ  t 


<PlogH1=Mf^|   1 
dr2  A^c.'  c?r2  t2' 

d?\og.enx  ■_  M  dp  _  ± 
drdV  h/u.  dr      kV 

and,  therefore, 

W-(r-.){*.(&  +  /glv)  +  «7.4}       .     (3,) 

is  the  variation  of  latent  heat,  expressed  in  terms  of  the  pressure,  volume, 
and  temperature  ;  to  which,  if  the  variation  of  sensible  heat,  3Q  =  feSr,  be 
added,  the  complete  variation  of  heat,  SQ  +  SQ'  =  S.Q,  in  unity  of  weight 
of  the  substance,  corresponding  to  the  variations  SV  and  Sr  of  volume  and 
temperature,  will  be  ascertained. 

It  is  obvious  that  equation  (25),  with  its  consequences,  is  applicable  to 
any  mixture  of  atoms  of  different  substances  in  equilibria  of  pressure  and 

7  72 

temperature;  for  in  that  case  r,  J-,  and  —  are  the  same  for  each  substance. 
We  have  only  to  substitute  for  ~  the  following  expression  : — 

XMX         2M2 

where  nv  «.„  &c,  are  the  proportions  of  the  different  ingredients  in  unity 
of  weight  of  the  mixture,  so  that  n^  +  n2  +  &c.  =  1. 

Equation  (25)  agrees  exactly  with  equation  (6)  in  the  first  section  of  my 
original  paper  on  the  Theory  of  the  Mechanical  Action  of  Heat.  It  is  the 
fundamental  equation  of  that  theory ;  and  I  shall  now  proceed  to  deduce 
the  more  important  consequences  from  it. 

10.  Equivalence  of  Heat  and  Expansive  Power:  Joule's  Law. — From  the 
variation  of  the  heat  communicated  to  the  body,  let  us  subtract  the 
variation  of  the  expansive  power  given  out  by  it,  or 

PSV={j>+/(V)}SV. 

The  result  is  the  variation  of  the  total  power  exercised  upon  or  com- 
municated to  unity  of  weight  of  the  substance,  supposing  that  there  is  no 
chemical,  electrical,  magnetic,  or  other  action  except  heat  and  pressure ; 
and  its  value  is — 

*  This  coefficient  corresponds  to  -  E  in  the  notation  of  my  previous  paper  on  the 
Mechanical  Action  of  Heat.  * 


G2  THE  CENTRIFUGAL  THEORY   OF   ELASTICITY. 

^SQ  +  SQ'-P^Sr.{k+^-^)+(r-K)/0^v} 

+  SV.{(t-k)-^-P-/(V)}.   ■     .         .     (2G.) 

This  expression  is  obviously  an  exact  differential,  and  its  integral  is  the 
following  function  of  the  volume  and  temperature  : — 

¥=k(r-K)+^(log,T+^)  +j{(r-K)^T-P}dY-ff(Y)dV.  (27.) 

Accordingly,  the  total  amount  of  power  which  must  be  exercised  upon 
unity  of  weight  of  a  substance,  to  make  it  pass  from  the  absolute 
temperature  r0  and  volume  V0  to  the  absolute  temperature  rx  and  volume 

Y,,  is 

<P(V1,r1)-¥(Vfllr0). 

This  quantity  consists  partly  of  expansive  or  compressive  power,  and 
partly  of  heat,  in  proportions  depending  on  the  mode  in  which  the 
intermediate  changes  of  temperature  and  volume  take  place;  but  the 
total  amount  is  independent  of  these  changes. 

Hence,  if  a  body  be  made  to  pass  through  a  variety  of  changes  of  temperature 
and  volume,  and  at  length  be  brought  back  to  its  primitive  volume  and  tempera- 
tun',  the  algebraical  sum  of  the  portions  of  power  applied  to  and  evolved  from 
the  body,  whether  in  the  form  of  expansion  and  compression,  or  in  that  of  heat, 
is  equal  to  zero. 

This  is  one  form  of  the  law,  proved  experimentally  by  Mr.  Joule,  of 
the  equivalence  of  heat  and  mechanical  power.  In  my  original  paper  on 
the  Mechanical  Action  of  Heat,  I  used  this  law  as  an  axiom,  to  assist  in 
the  investigation  of  the  equation  of  latent  heat.  I  have  now  deduced 
it  from  the  hypothesis  on  which  my  researches  are  based — not  in  order 
to  prove  the  law,  but  to  verify  the  correctness  of  the  mode  of  investigation 
which  I  have  followed. 

Equations  (26)  and  (27),  like  equation  (23),  are  made  applicable  to 

unity  of  weight  of  a  mixture,  by  putting  2«ft  for  ft,  and  2ra  -^  for  =£. 

The  train  of  reasoning  in  this  article  is  the  converse  of  that  followed 
by  Professor  William  Thomson  of  Glasgow,  in  article  20  of  his  paper  on 
the  Dynamical  Theory  of  Heat,  where  he  proves  from  Joule's  law  that 
the  quantity  corresponding  to  S^  is  an  exact  differential. 

11.  Mutual  Conversion  of  Heat  and  Expansive  Poiver :  Carnot's  Law  of 
the  Action  of  Expansive  Machines. — If  a  body  be  made  to  pass  from  the 
volume  V0  and  absolute  temperature  r0  to  the  volume  Vx  and  absolute 
temperature  tv  and  be  then  brought  back  to  the  original  volume  and 


THE   CENTRIFUGAL   THEORY   OF   ELASTICITY.  Go 

temperature,  the  total  power  exerted  (¥)  will  have,  in  those  two 
operations,  equal  arithmetical  values,  of  opposite  signs.  Each  of  the 
quantities  ¥  consists  partly  of  heat  and  partly  of  expansive  power,  the 
proportion  depending  on  the  mode  of  intermediate  variation  of  the  volume 
and  temperature,  which  is  arbitrary.  If  the  mode  of  variation  be  different 
in  the  two  operations,  the  effect  of  the  double  operation  will  be  to 
transform  a  portion  of  heat  into  expansive  power,  or  vice  versa, 

Let  (a)  denote  the  first  operation,  (b)  the  reverse  of  the  second.     Then 

The  terms  of  ¥  which  involve  functions  of  r  only,  or  of  V  only,  are 
not  affected  by  the  mode  of  intermediate  variation  of  those  quantities. 
The  term  on  which  the  mutual  conversion  of  heat  and  expansive  power 
depends,  is  therefore 

f{(r-K)fT-p}dY(b)  =  f{(T-K)^-p}dY(a), 

/(^)"©=/(^-')"» 

f^dY(a)-f^dY(b)  =jpdY(a)-fpd  Y  (b), 


or, 


Hence, 


which  last  quantity  is  the  amount  of  the  heat  transformed  into  expansive 
power,  or  the  total  latent  heat  of  expansion  in  the  double  operation. 
Let 


J  dr  J  t  —  k   a  \ 

dY=(T-K)dF, 


Then  because 

dV 
we  have 

"V  V  -F  F 

J  jnlY(a)-f  1pdY(b)  =  j   \t-k)  d  F  («)-  f  '(i— *)tfF(&) 

=  fFl  (ra-rb)dF=  f  "LlZH  ^  d Y.        .      (28.) 
•'F0  Jv0ra-K    d\ 

In  which  ra  and  r^  are  the  pair  of  absolute  temperatures,  in  the  two 
operations  respectively,  corresponding  to  equal  values  of  F. 

This  equation  gives  a  relation  between  the  heat  transformed  into 
expansive  power  by  a  given  pair  of  operations  on  a  body,  the  latent  heat 
of  expansion  in  the  first  operation,  and  the  mode  of  variation  of  tempera- 
ture in  the  two  operations.  It  shows  that  the  proportion  of  the  original 
latent  heat  of  expansion  finally  transformed  into  expansive  power,  is  a 


04  THE  CENTRIFUGAL  THEORY  OF   ELASTICITY. 

function  of  the  temperatures  alone,  and  is  therefore  independent  of  the 
nature  of  the  body  employed. 

Equation  (28)  includes  Carnot's  law  as  a  particular  case.      Let  the 
limits  of  variation  of  temperature  and  volume  be  made  indefinitely  small. 

Then 

,    7A.       dr      dq  7,. 


and  dividing  by  drdY 


dp=   _J_    d<$ 

dr~  t—  k'  dV 


This  differential  equation  is  also  an  immediate  consc  luence  of 
equation  (25). 

If  -  be  put  for ,  and  JM  for  (V?T,  it  becomes  identical  with   the 

J  i — k  d  V 

equation  by  which  Professor  William  Thomson  expresses  Carnot's  law, 

as  deduced  by  him  and  by  Mr.  Clausius  from  the  principle,  that  it  is 

impossible  to  transfer  heat  from  a  colder  to  a  hotter  body  without  expenditure  of 

mechanical  pov:  if. 

The  investigation  which  I  have  now  given  is  identical  in  principle 
with  that  in  the  fifth  section  of  my  paper  on  the  Mechanical  Action  of 
Heat ;  but  the  result  is  expressed  in  a  more  comprehensive  form. 

Equation  (28),  like  (25),    (26),  and  (27),  is  applicable  to  a  mixture, 

composed  of  any  number  of  different   substances,   in   any  proportions, 

dp,  '/-/'. 
provided  the  temperature,  the  pressure,  and  the  coefficients  j-  -5   ._,  are 

the  same  throughout  the  mass. 

12.  Apparent  Specific  Heat. — The  general  value  of  apparent  specific 
heat  of  unity  of  weight  is 

„     dQ     dq     dq   dY  •  (  hfi      [cPp      -] 

y    (29.) 

,dY   dp  I 

"*"  dr'dr\  J 

agreeing  with  equation  (13)  of  my  previous  paper. 

The  value  in  each  particular  case  depends  on  the  mode  of  variation 
of  volume  with  temperature.     Specific  heat  at  constant  volume  is 

*,-*+('-«>(&  +  /£«).•     •  w 

When  the  pressure  is  constant,  we  must  have 

dV         .dp 

wclY+-dr  =  0, 


THE   CENTRIFUGAL   THEORY   OF   ELASTICITY. 

and,  consequently, 

dp 

dV_     _thL 

dr~       dF' 

dV 

therefore  specific  heat  at  constant  pressure  is 


65 


Kp  =  KV  +  (t  —  k) 


(ML 

_dY 

dY 


(31.) 


J. 


This  agrees  with  equation  (10)  of  Professor  Thomson's  paper,  if  —  in  his 
notation  —  r  —  k. 

If  the  body  be  a  perfect  gas,  then 


r0\2/x"rr 


K.  =  ^  +  5 


►r      t—k   d Y 
r2  +     V     '  d 


*> 


TQ\Zfl 


y   (32.) 


Kp  =  Kv  + 


_  *) = lg* + 1  _ 


/' 


The   fact   that   the   specific   heats   of   all   simple   gases  for   unity  of 
weight  are  inversely  proportional  to  their  specific  gravities,  shows  that 


is  the  same  for  them  all. 


-> 


13.   Velocity  of  Sound  in  Fluids. — Let  a  denote  the  velocity  of  sound 
in  a  fluid,  and  d .  P  the  total  differential  of  the  pressure.     Then 


«  =  V(,-^)=V{,V<- 


dj 

dY 


dV_  d 
dr'  dY 


;)}■  (33.) 


If  it  were  possible  to  maintain  the  temperature  of  each  particle  of  the 
fluid  invariable  during  the  passage  of  sound,  this  velocity  would  be 
simply 

v(,-5> 

d.y 

But  Ave  have  reason  to  believe  that  there  is  not  time,  during  the 
passage  of  sound,  for  an  appreciable  transfer  of  heat  from  atom  to  atom, 
so  that  for  each  particle 

E 


G6  THE   CENTRIFUGAL   THEORY   OF   ELASTICITY. 

dQ  +  dQ'  =  0;  or,  K  =  0  in  equation  (29). 
To  fulfil  this  condition,  we  must  have 

dr  i —  k    dp 

dV=  ~    Kv   '  It' 
Consequently, 

or,  by  equation  (31), 

,/    d?    KP\  ,nt. 

That  is  to  say,  the  action  of  heat  increases  the  velocity  of  sound  in  a  fluid 
beyond  what  it  ivould  be,  if  heat  did  not  act,  in  the  ratio  of  the  square  root  of 
the  specific  heat  at  constant  pressure,  to  the  square  root  of  the  specific  heat  at 
constant  volume. 

This  is  Laplace's  law  of  the  propagation  of  sound,  which  is  here  shown 
to  be  applicable,  not  only  to  perfect  gases,  but  to  all  fluids  whatsoever. 


LAWS   OF  THE   ELASTICITY   OF  SOLID   BODIES.  67 


Y.— LAWS  OF  THE  ELASTICITY  OF  SOLID  BODIES.* 

Introduction. 

1.  The  science  of  the  elasticity  of  solid  bodies,  considered  with 
reference  to  its  most  important  application,  the  determination  of  the 
strength  of  structures,  consists  of  three  parts  : 

First,  The  investigation  of  what  may  be  specially  termed  the  Laws  of 
Elasticity  ;  that  is  to  say,  the  mutual  relations  which  must  exist  between 
the  elasticities  of  different  kinds  possessed  by  a  given  solid,  and  between 
the  different  values  of  those  elasticities  in  different  directions. 

Secondly.  The  integration  of  the  equations  of  equilibrium  and  motion 
of  the  particles  of  an  elastic  solid.  The  results  of  this  process  enable  us 
to  determine  the  relative  displacements  of  the  particles  from  their  natural 
positions  in  a  solid  body  of  a  given  material  and  figure,  subjected  to  a 
given  combination  of  forces. 

Thirdly.  The  application  of  the  results  derived  from  the  first  two 
branches  of  the  theory  to  our  experimental  knowledge  of  the  pressures  and 
relative  displacements  to  which  the  particles  of  known  materials  may  safely 
be  subjected  in  practice.  This  enables  us  to  compute  the  strength  of 
actual  structures. 

2.  Notwithstanding  the  great  amount  of  attention  which  has  been  paid 
to  the  strength  of  materials,  and  the  numerous  and  elaborate  experiments 
which  have  been  made  respecting  it,  few  examples  exist  of  the  sound 
application  of  physical  and  mathematical  principles  to  practice  in  con- 
nection with  this  subject.  This  has  arisen  chiefly  from  the  fact,  that  the 
first  and  second  branches  of  the  inquiry  have  to  a  great  extent  been 
carried  on  without  reference  to  their  application  to  the  third,  and  the 
third  conducted  without  regard  to  the  principles  of  the  first  and  second. 
The  results  of  investigation,  on  correct  principles,  into  the  theory  of 
elasticity  have  been  limited  in  their  applications,  with  a  few  exceptions, 
to  the  laws  of  the  propagation  of  vibratory  movements ;  and  those  few 
exceptions  relate  almost  exclusively  to  bodies  of  equal  elasticity  in  all 
directions — a  class  which  excludes  many  of  the  most  useful  materials  of 

*  Eead  before  the  British  Association  at  Edinburgh,  on  August  1,  1S50,  and 
published  in  the  Cambridge  and  Dublin  Mathematical  Journal,  May,  1851. 


G8  LAWS   OF  THE   ELASTICITY  OF   SOLID   BODIES. 

construction.  On  the  other  hand,  when  it  has  been  found  necessary  to 
adopt  theoretical  principles,  for  the  purpose  of  reducing  the  results  of 
experiments  on  the  strength  and  elasticity  of  materials  to  a  system, 
assumptions  have  often  been  made,  with  a  view  chiefly  to  simplicity 
in  calculation,  of  a  kind   inconsistent  with    the    real    nature    of  elastic 

bodies. 

3.  The  present  inquiry  relates  to  the  first  part  of  the  theory  of 
elasticity — viz.,  the  laws  of  the  relations  which  must  exist  between  the 
elasticities  of  different  kinds  possessed  by  a  given  substance,  and  between 
their  various  values  in  different  directions. 


Section  I. — Composition  and  Resolution  of  Strains  and 
Molecular  Pressures. 

4.  At  the  outset  of  the  inquiry  two  preliminary  problems  present  them- 
selves :  the  composition  and  resolution  of  relative  molecular  displacements, 
and  the  composition  and  resolution  of  pressures  such  as  the  parts  of  elastic 
bodies  exert  upon  each  other.  The  former  is  a  question  of  pure  geometry ; 
the  latter,  of  pure  statics.  They  are  usually  considered  simultaneously, 
on  account  of  the  analogy  which  exists  between  their  solutions.  This 
is  not  the  result  of  the  physical  connection  between  the  two  classes  of 
phenomena,  and  it  would  still  exist  although  there  were  no  such  physical 
connection ;  it  is  merely  a  consequence  of  the  analogy  between  forces  in 
statics  and  straight  lines  in  geometry. 

Those  two  problems  have  been  so  fully  investigated  by  MM.  Cauchy, 
Lam£,  and  Clapeyron,  as  to  leave  nothing  further  to  be  done.  The 
theorems  and  formulae  which  they  have  obtained  are  many  and  important. 
In  the  present  paper  I  shall  state  those  principles  and  results  only  to 
which  there  will  be  occasion  to  refer  in  the  sequel. 

5.  It  is  desirable  that  some  single  word  should  be  assigned  to  denote 
the  state  of  the  particles  of  a  body  when  displaced  from  their  natural 
relative  positions.  Although  the  word  strain  is  used  in  ordinary  language 
indiscriminately  to  denote  relative  molecular  displacement,  and  the  force 
by  which  it  is  produced,  yet  it  appears  to  me  that  it  is  well  calculated  to 
supply  this  want.  I  shall  therefore  use  it,  throughout  this  paper,  in  the 
restricted  sense  of  relative  displacement  of  particles,  whether  consisting  in 
dilatation,  condensation,  or  distortion ;  while  under  the  term  pressure  I 
shall  include  every  kind  of  force  which  acts  between  elastic  bodies,  or  the 
parts  of  an  elastic  body,  as  the  cause  or  effect  of  a  state  of  strain,  whether 
that  force  is  tensile,  compressive,  or  distorting. 

The  nature  and  magnitude  of  a  simple  and  uniform  strain  are  defined 
by  three  things  : 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  69 

First.  The  direction  of  the  lines  along  which  the  particles  of  the  body 
are  displaced  from  their  natural  position. 

Secondly.  The  direction  along  which  the  rate  of  variation  of  the  displace- 
ment from  point  to  point  is  a  maximum.  This  direction  is  normal  to  a 
series  of  planes  of  equal  displacement,  and  may  be  called  the  strain-normal. 

Thirdly.  The  amount  of  that  rate  of  variation,  being  the  differential 
coefficient  of  the  displacement  with  respect  to  distance  along  the  strain- 
normal. 

C.  A  strain  may  be  resolved  into  three  components,  in  which  the 
directions  of  displacement  shall  be  respectively  parallel  to  three  rectangular 
axes,  while  the  strain-normal  remains  unchanged,  by  multiplying  its 
amount  by  the  direction-cosines  of  the  total  displacement. 

Each  of  these  three  components  may  itself  be  resolved  into  three  com- 
ponents, in  which,  the  direction  of  displacement  remaining  unchanged,  the 
strain-normals  are  respectively  parallel  to  the  three  axes,  by  multiplying 
its  amount  by  the  direction-cosines  of  the  original  strain-normal. 

Thus  every  strain  is  reducible  to  nine  components. 

These  nine  components,  however,  are  equivalent  to  but  six  distinct 
strains.  If  we  consider  the  strains  as  thus  reduced  to  three  rectangular 
axes,  we  shall  find  that  they  are  of  two  kinds  :  longitudinal,  that  is  to  say, 
strains  of  linear  extension  or  condensation,  where  the  displacements  are 
parallel  to  the  strain-normals ;  and  transverse,  or  strains  of  distortion,  when 
these  directions  are  at  right  angles.  Thus,  if  x,  y,  z,  denote  the  three 
rectangular  axes,  and  £,  jj,  £,  small  molecular  displacements  respectively 
parallel  to  them,  then 

dE,     dr\     dZ, 

</ .!•'    dif    d  z' 

are  longitudinal  strains,  which  are  dilatations  when  positive,  and  condensa- 
tions when  negative.  I  shall  denote  them  respectively  by  N^  N2,  N3  ; 
their  sum,  when  positive,  is  the  cubic  dilatation  of  the  particles,  and  when 
negative,  the  cubic  condensation. 

Transverse  strains,  or  distortions,  are  represented  by  the  six  differential 
coefficients  of  the  displacements  with  respect  to  axes  at  right  angles  to 
them — viz., 

dr\     dZ, ,    dZ,     f?s.    d%     dr) 

dz'    dy'    dx'   dz'  dy'  dx' 

Let  the  axis  of  x  be  perpendicular  to  the  plane   of  the  paper.     Let 
.ABCD  be  the  section,  by  the  plane  yz,  of  a  prism  which  in  its  natural 
state  is  square,  and  has  its  faces  normal  to  the  axes  of  y  and  z.     A  distor- 
tion in  the  plane  yz,  relatively  to  these  axes,  is  measured  by  the  deviation 
from  rectangularity  of  this  originally  square  section,  that  deviation  being 


70 


HWS   OF  THE   ELASTICITY   OF   SOLID   BODIES. 


considered  positive  which  makes  the  angles  B  and  D  acute.  Now,  so  far 
as  the  positions  of  the  particles  in  this  prism  relatively  to  each  other  are 
concerned,  it  is  immaterial  whether  that  devia- 
tion from  rectangularity  is  produced  by  keep- 
ing the  sides  AD  and  BC  parallel  to  their 
original  positions,  and  giving  angular  motion 
to    AB    and   DC — a    change    represented  by 


+ 

y 

A 

n 

D 

y 

e 

ch) 

dz 


■  or   by  keeping   AB  and  DC  parallel   to 


their   original    positions,    and    giving    angular 
motion   to   AD    and    BC — a    change    repre- 

7  V 

sented  by  — - ;  or  by  combining  those  two  oper- 
dy 

ations:  so  that  the  total  transverse  strain  in  the  plane  yz  is  represented 

by  the  sum  of  these  two  coefficients, 

dr\       dZ,  _  0  T 
Tz  +  dy~JiLv 

Similar  reasoning  gives,  for  the  total  distortion  in  the  plane  zx, 


and  in  the  plane  x  y, 


dz      dz 

dj,      dn  _  0  T 
dy+ dx      "    a' 


The  factor  2  is  used  in  these  expressions  for  the  sake  of  convenience 
in  the  employment  of  certain  formula?,  to  be  afterwards  quoted. 
The  halved-differences  of  the  pairs  of  differential  coefficients, 

2\dz      dyP  2  \dx      dz)'2\dy      dxF 

represent  rotations  of  the  prism  as  a  whole  round  the  axes  of  x,  y,  z,  respec- 
tively, which  have  no  connection  with  the  positions  of  its  particles 
relatively  to  each  other. 

The  component  strains  into  which  all  others  can  be  resolved  with 
respect  to  a  given  set  of  axes  are  thus  reduced  to  six,  three  longitudinal 
and  three  transverse. 

7.  A  pressure,  like  a  strain,  is  defined  by  three  things : 

1st.  The  direction  of  the  pressure. 

2nd.  The  position  of  the  surface  at  which  the  pressure  is  exerted. 

3rd.  The  amount  of  the  pressure  as  expressed  in  units  of  force  pes 
unit  of  area  of  the  surface  of  action. 

A  pressure  on  a  plane,  in  whatsoever  direction  it  may  act,  may  be 
resolved   into  three  rectangular   components,  one  normal  to    the   plane 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  71 

and  two  tangential.  The  normal  pressure  may  be  compressive  or  tensile: 
when  compressive,  it  is  considered  as  positive;  when  tensile,  negative. 

In  an  elastic  solid  which  is  in  equilibrio,  let  a  cube  be  conceived  to 
exist  with  its  faces  normal  to  the  axes  of  co-ordinates,  and  let  the  pressures 
throughout  its  extent  be  uniform.  This  cube  exerts  on  the  matter  round 
it,  and  is  reacted  on  by  three  pairs  of  normal  pressures,  at  the  faces 
respectively  normal  to  the  axis  of  x,  y,  z,  which  may  be  denoted  by 
Pj,  Pq,  P3,  the  pressures  at  opposite  faces  being  equal. 

Let  A  B  C  D  represent  the  section  of  this  cube  by  the  plane  yz.  On  the 
faces  A  B  and  C  D,  parallel  to  x  z,  let  a  pair  of  tangential  forces  act  in  the 
directions  denoted  by  the  order  of  the  letters,  tending  to  produce  dis- 
tortion by  making  the  angles  B  and  D  acute  and  A  and  C  obtuse.  Let  a 
pair  of  forces  of  similar  tendency  act  on  the  faces  C  B  and  A  D,  parallel 
to  zy.  These  two  pairs  of  forces  are  equal  and  opposite  to  those  which 
the  cube,  in  consequence  of  the  transverse  displacements  of  its  particles, 
exerts  on  the  surrounding  portion  of  the  solid.  No  displacement  of  the 
relative  situations  of  a  system  of  particles  can  give  the  system  a  tendency 
to  revolve  as  a  whole  round  an  axis.  Such  a  tendency  must  exist  in  the 
cube  unless  the  tangential  forces  on  the  faces  A B,  CD  are  equal  to  those 
on  the  faces  C  B,  A  D. 

Therefore,  the  tangential  pressure  parallel  to  z,  on  a  plane  normal  to  y,  is 
equal  to  the  tangential  pressure  parallel  to  y,  on  a  plane  normal  to  z:  a  theorem 
first  proved  by  Cauchy. 

The  common  value  of  those  forces  may  be  denoted  by  Q1,  as  they  are 
both  perpendicular  to  x. 

Similar  reasoning  shows  that  the  two  pairs  of  tangential  forces  perpen- 
dicular to  y  have  one  common  value,  Q2. 

In  like  manner,  those  perpendicular  to  z  may  be  denoted  by  Q3. 

Thus  the  pressures  exerted  by  and  on  the  cube  are  reduced  to  six, 
three  normal  and  three  tangential. 

8.  The  composition  of  pressures  applied  to  different  planes,  and  their 
reduction  to  new  axes,  depends  on  the  following  principle  : — 

Conceive  a  small  triangular  pyramid,  with  its  apex  at  the  origin  of 
rectangular  co-ordinates,  its  sides  being  formed  by  the  three  co-ordinate 
planes,  and  its  base  by  a  plane  in  any  given  direction  intersecting  them. 
Let  pressures  in  one  given  direction  act  on  the  three  sides,  and  be 
balanced  by  a  pressure  in  the  same  direction  on  the  base.  Each  of  the 
three  sides  is  equal  to  the  base  multiplied  by  the  cosine  of  the  angle 
between  the  normal  to  the  base  and  the  normal  to  the  side  in  question. 
Therefore,  the  total  pressure  on  the  base  is  equal  to  the  sum  of  the 
pressures  on  the  sides,  each  multiplied  by  the  cosine  of  the  angle  between 
the  normal  to  the  side  in  question  and  the  normal  to  the  base.  If  the 
normal  to  this  base  is  one  of  three  new  axes  of  rectangular  co-ordinates,  the 


72 


LAWS  OF  THE   ELASTICITY  OF  SOLID   BODIES. 


total  pressure  thus  found  may  be  reduced  to  normal  and  tangential  pressures, 
by  multiplying  it  by  its  direction-cosines  with  respect  to  the  new  axes. 

9.  I  annex,  for  convenience  of  reference,  the  general  formulae  which 
have  been  deduced  from  this  principle. 

Let  x,  y,  g,  be  rectangular  axes  of  co-ordinates,  and  P^  P2,  P3,  Qx,  Q0,  <v>  . 
normal  and  tangential  pressures  which  act  as  shown  in  the  following 
table  : 

Pressuros  parallel  to 
Normals.  Planes.  X  jf  Z 

x  yz  Px  Q8  Q2 

y  z*  Q3         P-        Qi 

z  xy  Q2  Qj  Ps 

Let  Ej,  R2,  R3,  be  the  rectangular  components  of  the  total  pressure  at 
a  plane,  the  direction-cosines  of  whose  normal  are  av  a2,  as. 
Then 

Rx  =  ay  Pj  +  a2Qs  +  «aQ2,->> 

R2  =  a1Q3  +  «2P2  +  «3Q1,  I  (i.) 

E3  =  axQ2  +  a.Xii  +  «3P3,  J 

Let  this  normal  be  taken  as  the  axis  of  x  in  a  new  set  of  rectangular 
axes  x,  y',  z,  which  make  with  the  original  axes  the  angles  whose  cosines 
are  given  in  the  following  table : — 

Original  Axes.  New  Axes. 

x'  y  z' 


h 


c2  VDirection-cosines. 


c,J 


y3  L3- 

Let  P/,  P2',  P3',  Q/,  Q2',  Q3',  be  the  normal  and  tangential  pressures,  un- 
reduced to  the  new  axes :  then 

P/^P^  +  P^  +  p^ 

+  2Q1a2a3  +  2Q2a3aj,  +  2Q^a1a2, 

+  2Q1b2b3  +  2Q.2bsb1  +  2Q.J)1b2, 

P3'=PlCl2  +  P2^  +  P3c32 

+  2QlC2c3  +  2Q2c3c1  +  2Q3c1r2, 
Q/  =  P151c1  +  P2  b2c2  +  P363c3 

+  Qi(^3  +  b3c2)  +  Q2(6gCl  +  b^)  4-  Q3(V2  +  62cj). 

Qo  —  Px^i^x  +  P2C2a2  +  P3C3a3 

+  Qj(c2a3  +  c3a2)  +  Qo(c3a1  +  cx«3)  +  Q3(c1«2  +  c./^), 
Q3'  =  P1a161  +  P2«/;2  +  P3«363 

+  Q^ao&a  +  azb2)  +  Q2(a36x  +  a^3)  +  Q^a-ft.,  +  a.px), 


y  (2.) 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  73 

By  the  substitution  of  N  for  P  and  T  for  Q,  the  formulae  given  above 
are  made  applicable  to  the  reduction  of  strains  to  new  axes  of  co-ordinates. 

I  shall  not  here  recapitulate  the  many  elegant  and  important  theorems 
which  MM.  Cauchy  and  Lame  and  Clapeyron  have  deduced  from  those 
equations,  as  they  do  not  relate  to  the  branch  of  the  theory  of  elasticity 
of  which  this  paper  treats. 

I  may  mention  that  in  their  memoir  in  the  seventh  volume  of  Crelle's 

Journal,  MM.  Lame  and  Clapeyron  have  used  X  and  T  to  denote  pressures, 

d  ? 
and  have  expressed  strains  simply  by  the  differential  coefficients  — ,  &c. 


Section  II.— Physical  Relations  between  Pressures  and  Strains, 

SO  FAR  AS  THEY  ARE  INDEPENDENT  OF  HYPOTHESES  RESPECTING  THE 

Molecular  Constitution  of  Matter. 

10.  In  almost  all  investigations  which  have  hitherto  been  made 
respecting  the  elasticity  of  bodies  which  have  different  degrees  of 
elasticity  in  different  directions,  it  has  been  the  practice  to  take  some 
hypothesis  as  to  the  molecular  constitution  of  solid  bodies  as  the  basis  of 
calculation  from  the  outset  of  the  inquiry.  It  appears  to  me,  however, 
that  the  more  philosophical  course  is,  to  ascertain,  in  the  first  place,  what 
conclusions  can  be  attained  as  to  the  laws  of  elasticity  without  the  aid 
of  any  such  hypothesis,  and  afterwards  to  inquire  how  far  the  theory 
can  be  simplified,  and  what  additional  results  can  be  gained  by  introducing 
suppositions  respecting  the  ultimate  constitution  of  matter. 

For  the  present,  therefore,  I  shall  make  no  assumption  as  to  the 
questions  whether  bodies  are  systems  of  physical  points,  or  of  atoms 
of  definite  bulk  and  figure,  or  are  continuous,  or  have  a  constitution 
intermediate  between  those  three;  and  I  shall  use  the  word  particle  in  its 
literal  sense  of  a  small  part 

11.  I  shall  restrict  the  present  inquiry  to  homogeneous  bodies  possessing 
a  certain  degree  of  symmetry  in  their  molecular  actions,  which  consists 
in  this :  that  the  actions  upon  any  given  particle  of  the  body  of  any  two 
equal  particles  situated  at  equal  distances  from  it  within  the  sphere  of 
molecular  action,  in  opposite  directions,  shall  be  equal  and  opposite. 

Substances  may  possess  higher  degrees  of  molecular  symmetry,  but  this 
is  the  lowest. 

The  statement  that  a  body  is  homogeneous  means,  when  applied  to- 
molecular  action,  that  the  mutual  actiou  of  a  pair  of  particles,  situated  at 
a  given  distance  from  each  other  in  a  given  direction,  shall  be  equal  to 
that  of  any  other  pair  of  particles  equal  to  the  first,  situated  at  an  equal 
distance  from  each  other  in  a  parallel  direction. 

12.  It  is  known  by  observation,  that  strains  and  pressures  are  physically 


74<  LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 

connected.  It  is  also  known  by  observation,  that  the  pressure  with  which 
a  strain  is  connected  consists  in  a  tendency  of  the  body  to  recover  its 
natural  state,  and  is  opposite  or  nearly  opposite  in  direction  to  the  strain  ; 
thus  longitudinal  condensation  is  accompanied  with  positive  normal,  or 
nearly-normal  pressure ;  longitudinal  dilatation,  with  negative  normal,  or 
nearly-normal  pressure ;  and  distortion  in  a  given  plane,  with  tangential 
pressure  in  the  same  plane,  of  opposite  sign. 

It  is  known  by  experiment,  that  when  a  pressure  and  the  strain  with 
which  it  is  connected  are  given  in  direction,  and  when  the  strain  does  not 
exceed  a  certain  limit,  being  in  most  cases  the  utmost  limit  to  which  a 
structure  can  be  strained  without  danger  to  its  permanency,  the  pressure 
and  the  strain  are  sensibly  proportional  to  each  other.  The  quantity  by 
which  a  strain  is  to  be  multiplied  to  give  the  corresponding  pressure  is  a 
coefficient  of  elasticity,  and  is  expressed,  like  a  pressure,  by  a  certain 
number  of  units  of  force  per  unit  of  surface. 

I  have  said  that  a  strain  and  the  corresponding  pressure  referred  to  the 
same  plane  are  opposite  or  nearly  opposite  in  direction ;  for  they  are  not 
of  necessity  exactly  opposite  for  all  directions  of  strain,  except  in  sub- 
stances which  are  possessed  of  the  highest  degree  of  molecular  symmetry ; 
that  is  to  say,  which  are  equally  elastic  in  all  directions.  For  those 
having  lower  degrees  of  symmetry,  the  following  proposition  is  true  : — 

Theorem  I.  In  an  elastic  substance  which  is  homogeneous  and  symmetrical 
with  respect  to  molecular  action,  there  are  three  directions  at  right  angles  to  each 
other  in  which  a  longitudinal  strain  produces  an  exactly  normal  pressure  on  a 
plane  at  right  angles  to  the  direction  of  the  drum. 

Those  three  directions  are  called  Axes  of  Elasticity.  The  proposition  is 
equivalent  to  an  assertion,  that  the  lowed  degree  of  symmetry  of  molecular 

action  necessarily  involves  symmetry  with  respect 
to  three  rectangular  co-ordinate  planes. 

This  theorem  has  been  often  demonstrated 
for  systems  of  atoms.  But  it  is  easily  seen 
that  the  truth  of  these  demonstrations  de- 
pends, not  on  the  special  hypotheses  which 
they  involve,  but  on  the  fundamental  con- 
dition of  symmetry. 

The  following  demonstration  involves  no 
hypothesis. 

Let  a  point  0  in  the  interior  of  a  body 

be    assumed    as   the    origin    of   rectangular 

co-ordinates,    the    axes    being    considered    as   fixed,    and    the    body   as 

movable  angularly  in  all  directions  about  the  origin.     Space  round  0  is 

divided  by  the  co-ordinate  planes  into  eight  similar  indefinitely-extended 


LAWS   OF   THE    ELASTICITY   OF   SOLID   BODIES. 


75 


rectangular  three-sided  pyramids.     Let  those  pyramids  be  designated  as 
follows,  according  to  the  signs  of  the  co-ordinates  comprised  in  them. 

Signs  of  Designation  of 

x  'i I  z  Pyramid. 

.  A 
.  B 
.  C 
.  D 
.  E 
.  F 
.  G 
.     H 

To  express  the  relative  situations  of  these  pyramids,  as  taken  in  pairs, 
let  the  following  terms  be  used  : 

Diametrically  opposite — when  the  pyramids  touch  at  the  apex  only; 
comprising  the  following  pairs, 

A,  G ;  B,  H ;  C,  E ;  D,  F. 

Diagonally  opposite — when  they  touch  at  an  edge ;  comprising  the  pairs 

A,  H  ;  D,  E ;  B,  G ;  C,  F  ; 
A,  F  ;  B,  E  ;  D,  G  ;  C,  H  ; 
A,  C ;   B,  D  ;  E,  G  ;  F,  H. 

Contiguous — when  they  touch  in  a  face ;  comprising  the  pairs 

A,  B  ;  D,  C  ;  H,  G  ;  E,  F  ; 
A,  D  ;  B,  C  ;  F,  G  ;  E,  H  ; 
A,  E ;  B,  F  ;  C,  G ;   D,  H. 

Each  pair  of  contiguous  pyramids  forms  a  rectangular  wedge,  which 
has  an  opposite  wedge  touching  it  along  the  edge,  and  a  contiguous  wedge 
touching  it  at  each  of  its  two  faces. 

The  pairs  of  opposite  wedges  are 

A  B,  G  H  ;  C  D,  E  F  ; 
AD,FG;  BC,EH; 
A  E,  C  G  ;    B  F,  D  H. 

The  pairs  of  contiguous  wedges  are 

AB,  CD;  CD,  GH;  GH,  EF;  E  F,  AB; 

AD,  BC;  BC,  FG;  FG,  EH;  EH,  AD; 

AE,  BF;  BF,  CG;    CG,  DH;  DH,  AE. 


76  LAWS   OF   THE   ELASTICITY    OF   SOLID   BODIES. 

According  to  the  condition  of  symmetry  already  stated,  the  portions  of 
matter  comprised  in  any  pair  of  diametrically  opposite  pyramids  must  be 
symmetrical  in  their  actions  on  a  particle  placed  at  0,  or  on  any  pair  of 
equal  particles  symmetrically  placed  with  respect  to  0,  whatsoever  may 
be  the  angular  position  of  the  body  with  respect  to  the  axes. 

Suppose  the  body  to  receive  a  longitudinal  strain  in  the  direction  of 
the  axis  of  z.  Let  a  small  circular  area  w  be  conceived  to  exist  in  the 
plane  of  xy,  with  its  centre  at  0;  and  let  this  area  be  the  base  of  a 
cylinder  extending  indefinitely  in  a  negative  direction  along  the  axis  of  z, 
and  denoted  by  to  z.  The  pressure  on  the  plane  x  y  is  proportional  and 
parallel  to  the  resultant  of  the  actions  of  the  four  pyramids  A,  B,  C,  D,  on 
the  cylinder  u>  ~,  divided  by  the  area  w.  The  action  of  each  of  those 
pyramids  consists  of  a  normal  component  parallel  to  z,  and  a  tangential 
component  parallel  to  the  plane  xy.  In  order  that  the  total  pressure  may 
be  normal,  those  tangential  actions  must  balance  each  other,  which  can 
only  be  the  case  when  the  tangential  action  of  the  wedge  AB,  parallel  to 
the  axis  of  y,  is  equal  and  opposite  to  that  of  the  contiguous  wedge  CD, 
and  the  tangential  action  of  the  wedge  B  C,  parallel  to  the  axis  of  x,  is 
equal  and  opposite  to  that  of  the  contiguous  wedge  A  D. 

The  pair  of  contiguous  wedges  A  B,  C  D,  touch  in  the  plane  of  xz,  having 
the  axis  of  x  for  their  common  edge.  If  the  actions  of  this  pair  of  wedges 
on  w~,  when  longitudinally  strained  along  .:,  are  unsymmetrical,  this 
cannot  arise  from  the  form,  position,  or  strain  of  these  wedges,  which  are 
exactly  symmetrical  with  respect  to  each  layer  of  particles  in  w:,  but  from 
the  nature  of  the  particles  occupying  tin-  wi  dges.  Now  by  rotating  the 
body  through  a  right  angle  about  the  axis  of  x,  we  can  bring  the  particles 
which  formerly  occupied  C  D  into  A  B,  and  the  particles  which  formerly 
occupied  G  H  (which  consists  of  two  pyramids  diametrically  opposite,  and 
therefore  molecularly  symmetrical  to  A  and  B)  into  C  D.  In  this  new 
situation  of  the  body  with  respect  to  the  axes  of  co-ordinates,  the  resultant 
of  the  tangential  actions,  parallel  to  y,  of  the  wedges  A  B  and  C  D,  on  wz, 
though  not  necessarily  eqml,  will  be  opposite  in  direction  to  the  original 
resultant;  and  this  change  will  have  been  produced,  not  abruptly,  but 
continuously,  so  that  the  value  of  the  resultant  must  have  passed  through 
zero.  Therefore,  whatsoever  may  be  the  situation  of  the  axis  of  x 
amongst  the  particles  of  the  body,  it  is  possible,  by  rotating  the  body 
about  that  axis,  to  find  a  position  in  which  the  tangential  actions  of  the 
wedges  AB  and  CD,  parallel  to  y,  on  the  cylinder  wz,  shall  balance  each 
other.  And  by  similar  reasoning  it  may  be  proved,  that  whatsoever  may 
be  the  situation  of  the  axis  of  y  amongst  the  particles  of  the  body,  it  is 
possible,  by  rotating  the  body  about  that  axis,  to  find  a  position  in  which 
the  tangential  actions  of  the  Avedges  B  C  and  A  D,  parallel  to  x,  on  the 
cylinder  w  z,  shall  balance  each  other. 


LAWS   OF   THE    ELASTICITY   OF   SOLID   BODIES.  77 

Therefore,  by  combining  rotations  about  the  axes  of  x  and  y,  it  is 
possible  to  find  a  position  of  the  solid  with  respect  to  the  axes  of 
co-ordinates  such  that  the  tangential  actions  of  the  four  pyramids  A,  B, 
C,  D,  on  the  cylinder  wz,  arising  from  a  longitudinal  strain  along  ,~  shall 
be  in  equilibrio,  and  that  the  total  pressure  on  x  y  shall  be  normal. 

The  direction,  with  respect  to  the  solid,  which  fulfils  this  condition,  is 
called  an  axis  of  elasticity. 

Let  —  z  O  +  z,  being  now  an  axis  of  elasticity,  be  considered  as  fixed  in 
the  solid. 

From  the  manner  in  which  the  two  pairs  of  wedges  AB,  CD  and  BC, 
AD,  are  composed  of  the  four  pyramids  A,  B,  C,  D,  it  is  clear  that  the 
actions  of  the  pair  of  diagonally  opposite  pyramids  A,  C,  are  symmetrical, 
and  also  those  of  the  diagonally  opposite  pyramids  B,  D.  From  this  and 
the  symmetry  of  the  actions  of  diametrically  opposite  pyramids  it  follows, 
that  the  actions  of  the  four  pairs  of  contiguous  pyramids,  A,  E ;  D,  H ; 
B,  F;  C,  G,  are  symmetrical,  and  also  those  of  the  two  pairs  of  diagonally 
opposite  pyramids,  E,  G ;  F,  H.  This  symmetry  of  action  (subject  to  the 
condition  of  symmetry  of  strain)  is  not  disturbed  by  rotation  about  the 
axis  of  z. 

Let  the  small  circular  area  to  be  now  conceived  to  exist  in  the  plane  y~, 
and  let  the  cylinder  of  which  it  is  the  base  extend  in  a  negative  direction 
along  the  axis  of  x,  and  be  called  the  cylinder  wx.  Let  the  solid  receive 
a  longitudinal  strain  along  the  axis  of  x.  The  action  of  A  on  iox  is  sym- 
'  metrical  to  that  of  E,  and  the  action  of  D  to  that  of  II ;  therefore,  the 
tangential  actions  of  the  wedges  A  D,  E  H,  parallel  to  z,  balance  each  other. 
It  remains  only  to  make  the  tangential  actions  of  the  wedges  A  E,  D  H, 
parallel  to  y,  on  wx,  balance  each  other,  which  is  to  be  done  by  rotation 
about  the  axis  of  :. 

The  solid  is  now  in  such  a  position  that  a-,  as  well  as  z,  is  an  axis  of 
elasticity. 

The  pairs  of  contiguous  pyramids  are  now  all  molecularly  symmetrical 
about  their  common  faces.  Therefore  the  pairs  of  contiguous  wedges  AB, 
E  F ;  A  E,  B  F,  are  symmetrical  in  their  actions  on  a  cylinder  to  y,  when 
longitudinally  strained  along  y. 

Therefore  y  also  is  an  axis  of  elasticity,  and  the  theorem  is  proved. 
It  is  not  necessary  to  the  existence  of  rectangular  axes  of  elasticity  that 
the  body  should  be  homogeneous  (in  the  sense  in  which  I  have  used  the 
word)  throughout  its  whole  extent,  but  only  round  each  point  throughout 
a  space  which  is  large  as  compared  with  the  sphere  of  appreciable  mole- 
cular action  of  each  particle.  Hence  the  rectangular  axes  of  elasticity 
may  vaiy  in  direction  at  different  points  of  the  same  body ;  and  some,  or 
all  of  them,  may  follow  the  course  of  a  system  of  curves,  as  they  do  in  a 
rope,  a  piece  of  bent  timber,  or  a  curved  bar  of  fibrous  metal. 


LXIS. 

Plane. 

X 

yz 

y 

zx 

z 

xy 

78  LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 

13.  The  axes  of  elasticity  are  evidently  those  which  ought  to  be 
selected  as  axes  of  co-ordinates,  for  the  resolution  of  all  pressures  and 
strains,  in  researches  on  the  laws  of  elasticity.  The  strains  and  pressures 
being  so  resolved,  we  shall  have  the  expression  -  Ax  Nx  for  part  of 
the  normal  pressure  on  the  plane  yz :  Ax  being  the  coefficient  of  longitudinal 
elasticity  for  the  axis  of  x.  But  this  is  not  the  whole  of  that  pressure ; 
for  it  is  known  by  observation,  that  the  normal  pressure  on  a  given 
plane  is  augmented  by  condensation,  and  diminished  by  dilatation  of 
the  particles,  in  a  direction  parallel  to  the  given  plane  as  well  as  normal 
to  it.  The  normal  pressure  Px  on  yz,  therefore,  depends  not  only  on  the 
longitudinal  strain  Nx  along  x,  but  also  on  the  longitudinal  strains  N2  and 
N  alon"-  y  and  z.  Applying  similar  reasoning  to  the  other  normal 
pressures,  they  are  found  to  be  represented  as  follows  : 

Px  =  -  AXNX  -  B3N0  -  Bo'Ng  "] 

p^-b^-aX-bX  r      •  (3-) 

P3=-B2N1-B1'N2-A8NSJ 

The  tangential  pressures  are  represented,  in  terms  of  the  distortions,  in 
the  following  manner : 

Tlane. 

yz       .       .       Q1=-  20^ 

zx        .         .         Q„=  -2C.X    y    .  .         .     (4.) 

xy        .         .  Q3=-2C3T3J 

These  six  equations  are  merely  the  representation  of  observed  facts, 
framed  with  regard  to  the  principle  of  axes  of  elasticity. 

They  contain  twelve  coefficients  of  elasticity,  which  may  be  thus 
classified  : 

Ax,  A2,  A3,  are  the  coefficients  of  longitudinal  elasticity  for  the  axes  of 
x,  y,  z,  respectively ; 

B1?  Bx',  are  the  coefficients  of  lateral  elasticity  in  the  plane  of  yz :  the 
former  expressing  the  effect  of  a  strain  along  z  in  producing  normal  pres- 
sure parallel  to  y :  the  latter,  the  effect  of  a  strain  along  y  in  producing 
normal  pressure  parallel  to  z. 

B2,  B2',  are  the  coefficients  of  lateral  elasticity  in  the  plane  of  zx;  and 

B3,  B3',  in  the  plane  of  x  y. 

C1;  C2,  C3,  are  the  coefficients  of  transverse  or  tangential  elasticity,  or  of 
rigidity,  in  the  planes  of  yz,  zx,  and  xy,  respectively.  The  possession  of 
this  species  of  elasticity  is  the  property  which  distinguishes  solids  from 
fluids,  and  is  that  upon  which  the  strength  and  stability  of  solid  structures 
entirely  depend.  When  a  beam,  or  any  other  portion  of  a  solid  structure, 
takes  a  set,  as  it  is  called  (or  undergoes  permanent  alteration  of  figure),  it 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  79 

is  the  rigidity  which  has  been  overstrained  and  has  given  way.  So  far  as 
I  am  aware,  however,  it  has  not  hitherto  been  directly  referred  to  in 
researches  on  the  strength  of  materials,  except  in  those  relative  to 
torsion. 

The  principal  object  of  the  present  inquiry  is  to  determine  what  mutual 
relations  must  necessarily  exist  amongst  those  twelve  coefficients  of 
elasticity  in  each  substance. 

14.  The  three  coefficients  of  rigidity,  so  far  as  we  have  as  yet  seen, 
represent  the  elasticity  called  into  play  by  three  kinds  of  distortion, 
measured  respectively  by  the  alteration  of  the  angles  of  the  three  rectan- 
gular sections  of  a  cube  whose  faces  are  normal  to  the  three  axes  of 
elasticity.  I  shall  now,  however,  prove  that  the  tangential  pressures  pro- 
duced by  equal  distortions  are  equal,  so  long  as  the  plane  in  which  the 
distortion  takes  place  is  unchanged,  and  are  not  altered  by  any  change  of 
the  direction,  in  that  plane,  of  the  sides  of  the  figure  on  which  the  distor- 
tion is  measured ;  that  is  to  say — 

Theorem  II.  The  coefficient  of  rigidity  is  the  same  for  all  directions  of  dis- 
tortion in  a  given  plane. 

Let  ABCD  be  the  section  at  right  angles  to  the  edges  of  a  rhombic 
prism  having  any  angles ;  and  G  E  and  F  H  two 
lines  normal  respectively  to  the  faces  of  the  prism. 
Let  this  prism  undergo  a  small  alteration  in  the        ^ 
angles  of  its  section  ABCD. 

Whether  we  estimate  the  distortion  so  produced 
as  a  transverse  displacement  of  the  particles  in 
lines  parallel  to  AB,  and  varying  along  the  strain- 
normal  G  E,  or  as  a  transverse  displacement  of 
the  particles  in  lines  parallel  to  A  D,  and  vary- 
ing along  the  strain-normal  F  H,  the  result,  so  far 
as  the  relative  transverse  displacements  of  the  par-  G 

tides  are  concerned,  will  be  the  same. 

Also,  the  tangential  pressures  are  the  same  at  the  pair  of  faces  A  B  and 
C  D,  and  at  the  pair  of  faces  B  C  and  A  D ;  for  otherwise  a  relative  dis- 
placement among  the  particles  of  a  body  would  produce  a  force  tending  to 
make  it  revolve  as  a  whole  round  an  axis,  which  is  impossible. 

Therefore,  the  tangential  forces  produced  by  equal  transverse  displace- 
ments relatively  to  two  strain-normals  which  make  any  angle  with  each 
other  are  equal,  provided  the  displacements  are  in  the  same  plane  with  the 
normals ;  therefore  the  coefficient  of  rigidity  is  the  same  for  all  directions 
in  a  given  plane. 

15.  This  theorem  leads  to  another,  which  expresses  the  relations 
between  the  twelve  coefficients  of  elasticity,  as  far  as  it  is  possible  to 


<S0  LAWS   OF  THE   ELASTICITY   OF   SOLID   BODIES. 

determine  them  independently  of  hypotheses  respecting  the  constitution  of 
matter. 

Theorem  III.  In  each  of  the  co-ordinate  planes  of  elasticity  the  coefficient  of 
rigidity  is  equal  to  one-fourth  part  of  the  sum  of  the  two  coefficients  of  longitudinal 
elasticity  for  the  axes  which  lie  in  that  plane,  diminished  by  one-fourth  part  of  the. 
sum  of  the  two  coefficients  of  lateral  elasticity  in  tin'  same  plane. 

For  example,  let  the  plane  be  that  of  yz,  in  which  the  coefficient  of 
rigidity  is  Cv  those  of  longitudinal  elasticity  A2  and  A3,  and  those  of 
lateral  elasticity  Bx  and  B/. 

Let  2  ly  represent  a  distortion  in  the  plane  yz,  relative  to  two  new 

A 

.axes  y, ,:',  in  the  same  plane,  and  let  the  angle  yy'  =  9.  Let  this  distortion 
be  resolved,  with  respect  to  the  original  axes,  according  to  equation  (2). 
Then 

Nx  =  0;  N2  =  -  2  TV  °os  0  sin  6>;  X.,  =  2  T/  cos  6  sin  0; 

2  T,  =  2  T/  (cos-  e  -  sin'2  0);  T2  =  0;  T3  =  0. 

The  corresponding  pressure.--  referred  to  the  original  axes  arc 

T1  =  -  2  T/  (II.  -  IV)  cos  6  sin  0, 

P2=  -  2TX'  (- A.+  15,)  cos  6  sin  0. 

p3  =  -  2  t;  (-  ]y  +  A3)  cos  e  sin  e, 

Q2  =  -  2  C^T/  (cos2  B  -  sin2  0),  Q,  =  0,  Q,  =  0. 

Let  us  now  determine,  according  to  equation  (2),  from  the  above 
pressures,  the  tangential  pressure  Q/  as  referred  to  the  new  axes.     Then 

Q/  =  -  2  T/  {Cx  +  cos2  e  sin2  6  (X,  +  A3  -  B,  -  B,'  -  4  CJ}. 

But  by  the  preceding  theorem  we  have  also 

0'=  -2TT 

for  all  values  of  0,  which  cannot  be  true  unless  the  coefficient  of 
cos2  0  sin2  0  in  the  first  value  of  Q/  is  =  0.     Consequently 

Ci  =  \  (A2  +  A3  -  Bj  -  B/). 


%  applying  similar  reasoning  to   the   planes   of 
sx  and  xy  it  is  also  proved  that 

1 
C2  =  I(A3+A1-B2-B2'), 

C3  =  |(A1  +  A2-B3-B30  , 

being  the  algebraical  statement  of  the  theorem  enunciated. 


(50 


LAWS   OF   THE   ELASTICITY  OF  SOLID   BODIES.  81 

Thus  the  number  of  independent  coefficients  of  elasticity  is  reduced  to 
nine,  of  which  the  other  three  are  functions ;  and  this  is  the  utmost  reduc- 
tion of  their  number  which  can  be  made  without  the  aid  of  suppositions 
as  to  the  constitution  of  matter. 

The  determination  by  experiment  of  nine  constants  for  each  substance 
is  an  undertaking  almost  hopeless ;  it  is  therefore  desirable  to  ascertain 
whether,  by  the  introduction  of  some  probable  hypothesis,  their  number 
can  be  further  reduced. 


Section  III. — Eesults  of  the  Hypothesis  of  Atomic  Centres. 

16.  Almost  all  the  investigations  of  the  laws  of  elasticity  which  have 
hitherto  appeared,  are  founded  on  the  hypothesis  of  Boscovich :  that 
matter  consists  of  physical  points  or  centres  of  force,  or  of  atoms  acting 
as  if  their  masses  were  concentrated  at  their  centres;  which  physical 
points  or  atoms  occupy  space,  and  produce  the  phenomena  of  elasticity, 
because  the  forces  which  act  between  them,  and  which  depend  on  their 
relative  distances  and  positions,  tend  to  make  them  remain  in  certain 
relative  positions,  and  at  certain  distances  apart. 

Although  the  results  of  this  supposition  are  not  verified  by  all  solid 
substances,  still  it  seems  probable  that  its  errors  are  to  be  corrected,  not 
by  rejecting  it,  but  by  combining  it  with  another,  to  which  I  shall 
afterwards  refer. 

I  shall  now,  therefore,  show  to  what  extent  the  laws  of  elasticity  are 
simplified  by  adopting  Boscovich's  supposition  of  atomic  centres  of  force, 
acting  on  each  other  by  attractive  and  repulsive  forces  along  the  lines 
joining  them.  It  will  be  seen,  that  in  consequence  of  the  course  adopted, 
of  determining,  in  the  first  place,  the  necessary  relations  between  the 
coefficients  of  elasticity  which  must  exist  independently  of  all  special 
hypotheses,  this  investigation  is  almost  entirely  freed  from  the  algebraical 
intricacy  in  which  it  would  otherwise  be  involved. 

17.  All  the  consequences  peculiar  to  this  hypothesis  flow  from  the 
following  single  theorem,  in  which  the  term  'perfect  solid  is  used  to  denote 
a  body  whose  elasticity  is  due  entirely  to  the  mutual  attractions  and 
repulsions  of  atomic  centres  of  force. 

Theorem  IV.  In  each  of  the  co-ordinate  planes  of  elasticity  of  a  perfect 
solid,  the  tivo  coefficients  of  lateral  elasticity,  and  the  coefficient  of  rigidity,  are 
all  equal  to  each  other. 

Take,  for  example,  the  plane  of  y  z.  The  proposition  enunciated  is 
equivalent  to  the  assertion,  that  the  tangential  pressure  parallel  to  y  at 

F 


S2  LAWS   OF  THE   ELASTICITY   OF   SOLID   BODIES. 

the  plane  of  xy,  produced  by  a  given  transverse  strain  2TX  =  -5-,  which 

consists  in  a  displacement  of  the  atomic  centres  parallel  to  y  and  varying 
with  z,  is  equal  to  the  normal  pressure  parallel  to  :  at  the  same  plane  xy, 

d 

produced  by  a  longitudinal  strain  N8  =    ,   .  which  consists  in  condensing 

or  dilating  the  atomic  centres  in  a  direction  parallel  to  y.  provided  that 
longitudinal  strain  is  equal  in  amount  to  the  transverse  strain. 

The  pressure  on  a  given  area  of  the  plane  xy,  is  the  effect  of  the  joint 
actions  of  the  atomic  centres  on  the  negative  side  of  that  plane  upon  the 
atomic  centres  on  the  positive  side. 

In  the  natural  or  unstrained  condition  of  the  body,  this  pressure  is 
null,  showing  that  those  forces  neutralise  each  other.  "When  the  body 
is  strained,  therefore,  the  pressure  is  the  resultant  of  the  variations  of  all 
those  forces,  arising  from  the  displacements  of  the  atomic  centres  from 
their  natural  relative  positions.* 

Let  m  and  //.  denote  a  pair  of  atomic  centres,  m  being  situated  on  the 
positive  side  of  the  plane  xy,  and  ju  on  the  negative  side.  The  force 
acting  between  m  and  /j.  is  supposed  to  act  along  the  line  joining  them, 
and  to  be  a  function  of  its  length.  When  the  relative  displacement  of 
the  atoms  is  very  small  as  compared  with  their  distance,  the  variation  of 
this  force  will  be  sensibly  proportional  to  the  variation  of  distance, 
multiplied  by  some  function  of  the  distance.  It  may  therefore  be  denoted 
by  <pr.dr,  where  /•  denotes  the  distance  (jim).  Let  this  line  make  with  the 
axes  the  angles  a.  /3.  y. 

Let  the  strain  to  be  considered,  in  the  first  place,  be  transverse,  the 
displacements  being  parallel  to  y  and  varying  with  .:,  the  rate  of  variation 
being 

dz  ~  *  L» 

and  the  force  to  be  estimated  being  in  the  direction  y.     Then  the  dis- 
placement of  m  relatively  to  /x  is 

A  ?;  =  2  T,  r  cos  y. 

The  variation  of  their  distance  apart  is 

£  r  =  cos  j3  A  ij  =  2  T1  r  cos  /3  cosjy. 

The  variation  of  the  force  acting  between  them  is 

(pr  .^r  =  2  T1r  (pr  .  cos  ft  cos  y. 

*  Small  quantities  of  the  second  order  relatively  to  the  strains  Tj,  kc,  are  here 
nedeeted. 


LAWS   OF   THE  ELASTICITY  OF  SOLID   BODIES.  83 

And  the  component  of  that  variation  parallel  to  y,  which  forms  the  part 
of  the  tangential  pressure  due  to  the  action  of  li  on  m,  is 

cos  /3  <pr  .  S  r  =  2  Tx  r  <p  r  .  cos2  /3  cos  y.     .         .     (a.) 

Next,  let  the  strain  be  longitudinal,  parallel  to  y,  and  denoted  by 

N  =^ 
dy 

Then  the  displacement  of  m  relatively  to  fx  is 

A  tj  =  N2  r  cos  |3. 

The  variation  of  their  distance  apart  is 

8r  =  cosj3ATj  =  N2rcos*j3. 

The  variation  of  the  force  acting  between  them  is 

(f>r  .Sr  =  ~N2r  <pr  cos2  /3. 

And  the  component  of  that  variation  parallel  to  z,  which  forms  the  part 
of  the  normal  pressure  on  the  plane  xy  due  to  the  action  of  fx  on  m,  is 

cos  y  (j)  r .  S  r  =  N2  r  (j>  r  cos2  /3  cos  y.      .         .     (b.) 

On  comparing  the  expressions  (a)  and  (b)  it  will  be  seen  that  the 
quantities  by  which  2l\  and  N2  are  multiplied  are  identical.  Therefore, 
the  tangential  force  in  the  direction  y  on  the  plane  xy  produced  by  a 
distortion  in  the  plane  yz,  and  the  normal  force  in  the  direction  z  produced 
by  a  longitudinal  strain  along  y,  are  equal  when  the  strains  are  equal,  for 
each  pair  of  atomic  centres.  They  are  therefore  equal  for  a  perfect  solid, 
because  its  elasticity  is  wholly  due  to  the  mutual  actions  of  atomic  centres; 
and  the  theorem  is  proved  for  the  plane  yz,  and  may  in  the  same  manner 
be  proved  for  the  other  co-ordinate  planes  of  elasticity.  It  is  expressed 
algebraically  as  follows : 


Plane. 

yz  .  .  .  B^B^G^ 

z  x  .  .  .  B2  =  B2'  =  C2 

xy  .  .  .  B3  =  B3'  =  C3 


(6.) 


18.  The   combination  of  these  equations  with  the   equations  (5)    of 
Theorem  III.  leads  immediately  to  the  following  results : 


84  LAWS   OF  THE  ELASTICITY  OF   SOLID   BODIES. 


p     A2  +  A3  "I 

Li-    6 

p     A3  +  Al 

p     Al  +  A2 

^  — 6~  j 

A2  =  3  (C2  +  Cs  -  < 

3^ 

A2  =  3  (C8  +  Cx  - 

DJ 

A3  =  3  (Cj  +  c2  - 

D8)^ 

•     (7.) 


•     (8.) 


that  is  to  say, 


Theorem  V.    In  each  of  the  three  co-ordinate  planes  of  elasticity  of  a  perfect 
solid,  the  coefficient  of  rigidity  is  equal  to  one-sixth  part  of  the  sum  of  the  two 
coefficients  of  longitudinal  elasticity  ; 
and  consequently, 

For  each  axis  of  elasticity  of  a  perfect  solid,  the  coefficient  of  longitudinal 
elasticity  is  equal  to  three  times  the  sum  of  the  two  coefficients  of  rigidity  for  the 
co-ordinate  planes  which  pass  through  that  axis,  diminished  by  three  times  the 
coefficient  of  rigidity  for  the  plane  normal  to  that  axis. 

We  have  now  arrived  at  the  conclusion,  that  in  a  body  whose  elasticity 
arises  wholly  from  the  mutual  actions  of  atomic  centres,  all  the  coefficients 
of  elasticity  are  functions  of  the  three  coefficients  of  rigidity.  Rigidity 
being  the  distinctive  property  of  solids,  a  body  so  constituted  is  properly 
termed  a  perfect  solid. 

When  the  three  coefficients  of  rigidity  are  equal,  the  body  is  a  perfect 
solid,  equally  elastic  in  all  directions.  The  equations  6  and  8  become 
A  =  3  C ;  B  =  C,  agreeing  with  the  results  deduced  by  various  mathe- 
maticians from  the  hypothesis  of  Boscovich. 


Section  IV. — Results  of  the  Hypothesis  of  Molecular  Vortices. 

19.  The  great  and  obvious  deviations  from  the  laws  of  elasticity,  as 
deduced  from  the  hypothesis  of  atomic  centres,  which  many  substances 
present,  render  some  modification  of  it  essential. 

Supposing  a  body  to  consist  of  a  continuous  fluid,  diffused  through 
space  with  perfect  uniformity  as  to  density  and  all  other  properties,  such 
a  body  must  be  totally  destitute  of  rigidity  or  elasticity  of  figure,  its  parts 
having  no  tendency  to  assume  one  position  as  to  direction  rather  than 
another.     It  may,  indeed,  possess  elasticity  of  volume  to  any  extent,  and 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  85 

display  the  phenomena  of  cohesion  at  its  surface  and  between  its  parts. 
Its  longitudinal  and  lateral  elasticities  will  be  equal  in  every  direction ; 
and  they  must  be  equal  to  each  other  by  equation  (5),  which  becomes 

0  =  A  -  B;  C  =  0. 

If  we  now  suppose  this  fluid  to  be  partially  condensed  round  a  system 
of  centres,  there  will  be  forces  acting  between  those  centres  greater  than 
those  between  other  points  of  the  body.  The  body  will  now  possess  a 
certain  amount  of  rigidity;  but  less,  in  proportion  to  its  longitudinal  and 
lateral  elasticities,  than  the  amount  proper  to  the  condition  of  perfect 
solidity.  Its  elasticity  will,  in  fact,  consist  of  two  parts,  one  of  which, 
arising  from  the  mutual  actions  of  the  centres  of  condensation,  will  follow 
the  laws  of  perfect  solidity ;  while  the  other  will  be  a  mere  elasticity 
of  volume,  resisting  change  of  bulk  equally  in  all  directions. 

In  a  paper  on  the  Mechanical  Action  of  Heat  in  connection  with  the 
Elasticity  of  Gases  and  Vapours  (Trans,  of  the  Royal  Society  of  Edinburgh, 
Vol.  XX.,  Part  I.),  I  have  attempted  to  develop  some  of  the  consequences 
of  a  supposition  of  this  kind,  called  the  hypothesis  of  molecular  vortices."" 
It  assumes  that  each  atom  of  matter  consists  of  a  nucleus  or  central 
physical  point,  enveloped  by  an  elastic  atmosphere,  which  is  retained  in 
its  position  by  forces  attractive  towards  the  atomic  centre,  and  which, 
in  the  absence  of  heat,  would  be  so  much  condensed  round  that  centre 
as  to  produce  the  condition  of  perfect  solidity  in  all  substances;  that  the 
changes  of  condition  and  elasticity  due  to  heat  arise  from  the  centrifugal 
force  of  revolutions  among  the  particles  of  the  atmospheres,  diffusing  them 
to  a  greater  distance  from  their  centres,  and  thus  increasing  the  elasticity 
which  resists  change  of  volume  alone,  at  the  expense  of  that  which  resists 
change  of  figure  also;  and  that  the  medium  which  transmits  light  and 
radiant  heat  consists  of  the  nuclei  of  the  atoms,  of  small  mass,  but 
exerting  intense  forces,  vibrating  independently,  or  almost  independently 
of  their  atmospheres;  absorption  being  the  communication  of  that  motion 
to  the  atmosphere,  so  that  it  is  lost  by  the  nuclei. 

20.  A  body  so  constituted,  in  which  the  rigidity  is  considerable,  may 
be  called,  in  general,  an  imperfect  solid  ;  and  it  is  obvious  that  its  various 
elasticities  may  be  represented  in  the  following  manner : 

Theorem  VI.  In  an  imperfect  solid,  according  to  the  hypothesis  of 
molecular  vortices,  each  of  the  coefficients  of  longitudinal  and  lateral  elasticity 
is  equal  to  the  same  function  of  the  coefficients  of  rigidity  which  would  be  its 
value  in  a  perfect  solid,  added  to  a  coefficient  of  fluid  elasticity  which  is  the 
same  in  all  directions. 

*  An  abstract  of  that  paper  is  published  in  PoggendorfF's  Annalen  for  1850,  No  IX. 


86 


LAWS   OF  THE   ELASTICITY  OF  SOLID   BODIES. 


Denoting  this  fourth  coefficient  by  J,  we  have  the  following  equations, 
giving  the  values  of  the  coefficients  of  longitudinal  and  lateral  elasticity 
in  terms  of  the  coefficient  of  fluid  elasticity,  and  of  the  three  coefficients 
of  rigidity. 


Ax  =  3  (C2  +  C,  -  Cx)  +  J 
A,  =  3  (C3  +  Oj  -  C2)  +  J 
A3  =  3  (Cx  +  02  -  C3)  +  J 
Bx  =  Cx  +  J 
B2  =  C,  +  J 
B3  =  C3  +  J 


(0.) 


The  utmost  number  of  independent  coefficients  is  thus  increased 
to  four. 

If  the  coefficients  of  rigidity  be  progressively  diminished  without  limit, 
as  compared  with  the  coefficient  of  fluid  elasticity,  the  body  will  pass 
through  every  stage  of  the  gelatinous  state;  and  when  the  coefficients  of 
rigidity  vanish,  its  condition  will  be  that  of  a  perfect  fluid,  in  which  the 
longitudinal  and  lateral  elasticities  are  all  equal,  and  represented  by  the 
single  coefficient  J. 

It  is  to  be  observed,  that  in  this  condition  the  independent  actions  of 
the  nuclei  or  physical  points  at  the  atomic  centres  upon  each  other,  which 
are  the  means  of  radiation,  may  be  very  great;  their  sensible  effect  on 
the  elasticity  of  the  body  being  neutralised  by  other  forces,  exerted  by 
the  parts  of  the  atmospheres. 

If  two  of  the  coefficients  of  rigidity  are  equal  (as  C9  =  C3),  the  body  is 
equally  elastic  in  all  directions  round  an  axis,  which  in  this  case  is  that 
of  x ;  and  equations  (9)  become 


Ax  =  6  C,2  -  3  Cx  +  J 

A2  =    A3  =  3  Cx  +  J 

B2=    B3  =  C2  +  J     j 


(9A.) 


When  the  three  coefficients  of  rigidity  are  all  equal,  the  body  is  an 
imperfect  solid  equally  elastic  in  all  directions.  The  results  of  this 
condition  have  been  investigated  by  Professor  Stokes,  M.  Wertheim,  and 
Mr.  Clerk  Maxwell. 

Equations  (9)  in  this  case  become 


C  +  J;  B  =  C  +  J. 


(9B.) 


LAWS   OF  THE   ELASTICITY   OF   SOLID   BODIES. 


87 


Note  respecting  Previous  Investigations. 


(18  a.)  The  investigations  of  Poisson  (Mem.  de  VAcad.  des  Sciences, 
XVIIL),  of  M.  Cauchy  (Exercices  des  MatMmatiques,  passim),  and  of  Mr. 
Haughton  (Trans.  Boy.  Irish  Acad.,  XXL),  respecting  the  elasticity  of 
substances  unequally  elastic  in  different  directions,  are  all  founded  on  the 
hypothesis  of  atomic  centres.  So  far  as  they  relate  to  substances  possessed 
of  rectangular  axes  of  elasticity,  they  agree  in  expressing  the  elasticity  of 
such  bodies  by  means  of  six  coefficients,  corresponding  respectively  to  those 
which  I  have  denoted  by 

Av  A2,  A3,  C15  C2,  C3. 

None  of  those  investigations  indicate  any  relations  amongst  these  six 
coefficients. 

The  researches  of  Mr.  Green  on  the  propagation  of  vibratory  movement 
(Camb.  Trans.,  VII.)  differ  materially  from  those  which  preceded  them, 
inasmuch  as  they  are  applicable,  not  merely  to  systems  of  atomic  centres 
or  physical  points,  but  to  solid  substances  constituted  in  any  manner 
whatsoever.*  So  far  as  they  are  applicable  to  bodies  possessed  of 
axes  of  elasticity,  they  involve  nine  coefficients:  three  of  longitudinal, 
three  of  lateral,  and  three  of  transverse  elasticity.  The  following  table 
exhibits  a  comparison  between  Mr.  Green's  notation  and  that  of  this  paper: 

Coefficients  of  Elasticity. 


In  the  Notation 
of 

Longitudinal. 

Lateral. 

Transverse. 

Mr.  Green,  . 
This  Paper, 

G     H     I 

K    A2    A3 

P     Q    & 
Bx    B2    B3 

L     M     N 
C\    C2    C3 

There  is  nothing,  however,  in  the  researches  of  Mr.  Green  to  indicate 
any  mutual  relations  amongst  those  nine  coefficients;  and  to  establish 
such  relations,  indeed,  it  appears  to  me  that  the  subject  must  be  in- 
vestigated, not  dynamically,  but  statically. 

It  may  here  be  observed,  that  Mr.  Green's  equations  contain  three 
additional  coefficients,  to  represent  the  effect  of  a  strained  condition  of  the 
medium  on  the  propagation  of  vibratory  movement;    but    those   three 

*  A  second  paper  by  Mr.  Haughton  (Trans.  Roy.  Irish  Acad.,  XXII.)  is  equally 
comprehensive. 


88  LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 

quantities,  being  foreign  to  the  subject  of  this  paper,  have  no  expressions 
corresponding  to  them  in  its  notation. 

In  the  equations  of  the  propagation  of  light,  Mr.  Green  effects  an 
apparent  reduction  in  the  number  of  coefficients  by  introducing  the 
supposition  that  the  vibrations  are  of  necessity  wholly  tangential  to  each 
wave-front.  But  this  supposition  is  quite  at  variance  with  the  nature 
of  elastic  solids,  and  is  obviously  intended  by  the  author  as  merely  an 
assumption  for  the  purpose  of  facilitating  calculation,  and  obtaining 
approximately  true  results,  in  the  case  of  luminiferous  undulations. 

Mr.  M'Cullagh's  researches  on  the  propagation  of  light  (Trans.  Roy.  Irish 
Acad.,  XXI.)  involve  a  similar  assumption. 

The  result  peculiar  to  the  investigations  contained  in  the  present 
paper  is  the  establishment  of  certain  mutual  relations  amongst  the  different 
elasticities  of  a  given  substance,  whereby  the  six  coefficients  of  Poisson 
and  Cauchy  are  reduced  to  functions  of  three,  and  the  nine  coefficients 
of  Mr.  Green  to  functions  of  four  ;  the  former  representing  the  condition 
of  a  medium  whose  elasticity  is  wholly  due  to  the  mutual  actions  of  atomic 
centres,  the  latter  that  of  a  su bstance  whose  condition  is  intermediate 
between  those  of  a  system  of  centres  of  force,  and  of  a  continuous  and 
uniformly  diffused  fluid. 

General  equations  of  vibratory  movement,  in  the  particular  case  of 
uncrystallised  media,  agreeing  with  those  of  Mr.  Green,  are  given  by 
Professor  Stokes  in  his  memoir  on  Diffraction  (Camb.  Trans.  IX.)  His  two 
coefficients  of  elasticity  have  the  following  values  in  the  notation  of 
this  paper : — 

Prof.  Stokes.  This  Paper. 


62 


D" 


g  denotes  the  accelerating  force  of  gravity;  and  D,  the  weight  of  unity 
of  volume  of  the  vibrating  medium. 

a  and  b,  in  Professor  Stokes's  paper,  are  the  velocities  of  propagation 
of  normal  and  tangential  vibrations  respectively. 

In  the  researches  of  Poisson,  Navier,  Cauchy,  Lam6,  and  others,  on 
the  elasticity  of  bodies  equally  elastic  in  all  directions,  the  coefficients 
are  often  expressed  in  terms  of  two  quantities,  denoted  by  k  and  K,  in 
the  following  expression  for  a  normal  pressure  on  the  plane  yz : 

P1  =  -^N1-K(N1  +  N2  +  N3): 

k  represents  a  species  of  longitudinal  elasticity,  under  the  condition  that 


LAWS   OF    THE   ELASTICITY   OF   SOLID   BODIES.  89 

the  volume  remains  unchanged;  and  K,  an  elasticity  resisting  change  of 
volume.     Their  values  in  the  notation  of  this  paper  are  as  follows : — 

&  =  A-B  =  2Cj   K  =  B  =  C  +  J. 

It  is  evidently  impracticable  to  apply  an  analogous  notation  to  bodies 
unequally  elastic  in  different  directions. 

M.  Wertheim  has  recently  made  a  most  elaborate  and  valuable  series 
of  experiments  on  the  elasticity  of  brass,  glass,  and  caoutchouc,  according 
to  a  method  suggested  by  M.  Eegnault,  for  the  purpose  of  determining 
the  laws  of  elasticity  of  uncrystallised  substances  (Ami.  de  Chun,  ct  de 
Phys.,  Ser.  III.,  torn.  XXIII.)  He  concludes  that  for  brass  and  glass,  and 
for  caoutchouc  moderately  strained,  the  following  equation  is  nearly  if  not 
exactly  true,  in  the  notation  to  which  I  have  just  referred,  h  =  K,  which, 
in  the  notation  of  this  paper,  is  equivalent  to  the  following  : 

J  =  C;    B  =  2C;    A  =  4  C. 

M.  "Wertheim  has  investigated  the  consequences  which  must  follow  in 
the  solution  of  several  problems  connected  with  elasticity,  if  this  law  be 
universally  true  for  solid  bodies. 

This  supposition  must  be  regarded  as  doubtful ;  and  it  is  not,  indeed, 
advanced  by  M.  Wertheim  as  more  than  a  conjecture.  So  far  as  our 
present  knowledge  goes,  it  seems  more  probable  that  the  relations  between 
C  and  J  may  be  infinitely  varied.  If  the  effect  of  heat  is  to  diminish  C 
and  increase  J,  there  maybe  some  temperature  for  each  substance  at  which 
M.  Wertheim's  equation  is  verified.  In  the  sequel  I  shall  consider  more 
fully  the  consequences  to  be  deduced  from  M.  Wertheim's  experiments  on 
this  subject. 


Section  V. — Coefficients  of  Pliability,  and  of  Extensibility 
and  Compressibility,  Longitudinal,  Lateral,  and  Cubic. 

Examples  of  their  Experimental  Determination. 

21.  Coefficients  of  elasticity  serve  to  determine  pressures  from  the 
corresponding  strains.  We  have  now  to  consider  the  determination  of 
strains  from  pressures. 

To  determine  a  distortion  from  the  corresponding  tangential  pressure, 
it  is  sufficient  to  multiply,  using  the  negative  sign,  by  the  reciprocal  of  the 
proper  coefficient  of  rigidity.  This  reciprocal  may  be  called  a  coefficient 
of  pliability. 

A  similar  process,  however,  cannot  be  applied  to  the  calculation  of 
longitudinal    strains   from   normal   pressures;    because,    as    each   normal 


90 


LAWS   OF  THE   ELASTICITY  OF   SOLID   BODIES. 


pressure  is  a  function  of  all  the  three  longitudinal  strains,  so  each  longi- 
tudinal strain  is  a  function  of  all  the  three  normal  pressures. 

Let  the  longitudinal  strains  be  represented,  in  terms  of  the  normal 
pressures,  by  the  following  equations  : 

N1  =  -;r1P1  +  b;2P,  +  ^P3^ 

N2=       h,^-  BgPss  +  JbrjPa   L     .         .     (10.) 

N3  =       ^Pi  +  ^Ps-KgPj 

Then  the  coefficients  in  these  equations  are  found,  by  a  process  of 
elimination,  to  have  the  following  values  in  terms  of  the  coefficients  of 
elasticity. 

Let 

K  -  24(C22C3  +  C2C32  +  C32C  x  +  C8(V  +  C^C,  +  (W  -C,»  -  C8«  -  C33) 
-  52C1C2C3  +  J{8(C2C3  +  0,0x4-  0,0.,)  -  4(CX2  +  C22  +  C32)}. 

Then 

1 


{80^-9(0,  -03)2  +  40^} 


%=^{8C22-9(C3-C1)2  +  4C2Jj 

a3=g{8C32-9(C1-C2)2  +  4C,.l; 

1 


fr3=^{3(cx+c2--c,)C8-c1c2+2(c1+c2-c,)j} 


(ii.) 


The  above  coefficients  may  be  thus  classified  : 

ttv  cl2,  n3,  are  the  coefficients  of  longitudinal  extensibility  and  compressibility 
parallel  respectively  to  the  three  axes  of  elasticity. 

bv  l)2,  I)3,  are  the  coefficients  of  lateral  extensibility  and  compressibility  for 
the  three  co-ordinate  planes  of  elasticity,  serving  to  determine  the  effect  of 
a  normal  pressure  on  those  dimensions  of  a  body  which  lie  at  right  angles 
to  its  direction. 

From  the  manner  in  which  the  coefficient  J  enters  into  the  common 
denominator  K,  it  is  obvious  that  when  the  coefficients  of  rigidity  diminish 
without  limit  as  compared  with  that  of  fluid  elasticity,  the  six  coefficients 
of  linear  extensibility  and  compressibility  increase  ad  infinitum. 

In  a  body  whose  three  coefficients  of  rigidity  are  different,  the  coefficient 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 


91 


of  cubic  compressibility,  that  is  to  say,  the  quotient  of  the  sum  of  the 
three  longitudinal  strains  by  the  mean  of  the  three  normal  pressures,  with 
the  sign  changed,  has  no  fixed  value  unless  some  arbitrary  relation  be 
fixed  between  those  pressures.  Let  them  be  supposed,  then,  to  be  all 
equal ;  let  their  common  value  be  P,  and  let  the  coefficient  of  cubic  com- 
pressibility in  this  case  be  denoted  by  &  :  then 


h  =  _  *Ti+%  +  3gT.  =  Ri  +  ,  +  ,3  _  2(Irx  +  ,b2  +  *,) 


=  4"  {8(°2C3  +  CaCi  +  CxC2)  -  4(<V  +  C2s  +  C32)} 


y     a2-) 


Hence  i  =  J  +  6(C1+C,  +  C3) 


lDGC^CgC. 


~  8  (C203  +  O^  +  CXC^  -  4(0/  +  C22  +  C3*) 

So  that  this  coefficient  is  the  sum  of  the  three  longitudinal  coefficients 
of  compressibility,  diminished  by  twice  the  sum  of  the  three  lateral 
coefficients.  It  does  not,  like  them,  increase  ad  infinitum  when  the 
rigidity  vanishes ;  its  ultimate  value  in  that  case  being 

1 
J' 

the  reciprocal  of  the  coefficient  of  fluid  elasticity,  as  might  have  been 
expected. 

If  C2  =  C8,  so  that  the  body  is  equally  elastic  in  all  directions  round  the 
axis  of  x,  equations  (11)  and  (12)  take  the  following  forms : 


K  =  4C1{12C1C3-6C12-  a2  +  J(4C2-C1)} 

»i  =g  (8Ci2  +  4ClJ)  =  UQXG,  -  bCV  -  C„2  +  J(4C2  -  i\) 
R2  =  »,«^{8C1«-9(01-Ciy  +  40aJ} 

b±  =  ~  {6CVC,  -30^-a2  +  J(4C2-  2CJ} 

1  (  (  Co  +   J 

&2  =  I>3  =  g(2C1C2  +  2CX  J)  =  jMCi0g-12C1*-308*  +  J(SC2  -  2CX) 

ir  =  s(16C1C2-4C12)  =  ^C^o  -  6CX2  -  C22  +  J(4C2  -  C~) 

1  490  2 

£  =  J  +  12CO  +  6C!- jp     ' 


M12A) 


4Co-Cx 


92  LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 

For  bodies  equally  elastic  in  all  directions  the  coefficients  of  com- 
pressibility and  extensibility  take  the  following  values  : 

2C  +  J  .  C  +  J 

^-^=50^  ••■rJ+|a 

In  substances  of  this  kind  the  coefficient  of  cubic  compressibility  is  the 
same,  whether  the  three  normal  pressures  arc  equal  or  unequal,  being 
equal  to  the  sum  of  the  three  longitudinal  strains  divided  by  the  mean  of 
the  three  normal  pressures  with  the  sign  changed :  that  is  to  say, 

*~  " "  i\  +  p2  +  iv 

One  of  the  most  frequent  errors  in  investigations  respecting  the  elasti- 
city and  strength  of  materials,  and  the  propagation  of  sound,  has  been  to 
confound  the  coefficients  of  longitudinal  elasticity  with  the  reciprocals  of 
the  coefficients  of  longitudinal  compressibility.  The  equations  of  this 
section  show  clearly  how  widely  these  two  classes  of  quantities  may 
differ. 

The  reciprocal  of  the  longitudinal  extensibility,    ,  is  what  is  commonly 

termed  the  Weight  of  the  modulus  of  elasticity. 

22.  The  following  formula  may  be  found  useful  in  the  determination  of 
the  coefficient  J  of  fluid  elasticity  from  experimental  data. 

Let  us  suppose  that  the  three  coefficients  of  rigidity  of  a  substance, 
Cv  C2,  C3,  have  been  determined  by  experiments  on  torsion,  and  that 
some  one  of  the  coefficients  of  compressibility  and  extensibility  in  equa- 
tion (11),  or  those  derived  from  it,  has  also  been  determined  by  experi- 
ment. Let  the  actual  value  of  this  coefficient  be  called  f,  and  the  value 
which  it  would  have  had,  had  J  been  =  0,  f0.  Also  let  K0  denote  the 
value  which  the  denominator  K  would  have  had,  had  J  been  =  0,  and  let  n 
be  the  factor  by  which  J  is  multiplied  in  the  numerator  of  f,  and  in,  in 
the  denominator. 

Then 

J  =  K0.-^LC (13.) 

u    mi  —  n 

When  applied  to  coefficients  of  longitudinal  extensibility,  this  formula 
labours  under  the  disadvantage  that  a  comparatively  slight  error  in  the 
experimental  data  may  cause  a  serious  error  in  the  determination  of  J. 
Let  us  take,  for  example,  an  uncrystallised  substance,  and  make  succes- 
sively the  two  following  suppositions, 

J  =  0,  J  =  C: 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES.  93 

it  will  be  found  that  the  results  are  respectively, 

a  =  -  x  0-4,  n  =  r-  X  0375, 

being  in  the  ratio  of  1 6  : 1 5  ;  so  that  any  uncertainty  in  the  experiments 

is  in  this  case  increased  fifteenfold  in  computing  the  value  of  — .       Hence 

it  appears,  that  without  very  great  precision  in  the  experiments,  the 
coefficient  of  fluid  elasticity  cannot  be  satisfactorily  determined  by  a 
comparison  of  the  effects  of  longitudinal  tension  with  those  of  torsion.  It 
is  especially  desirable  that  the  two  sets  of  experiments  should  be  made 
on  the  same  piece  of  the  material. 

The  best  data  for  calculations  of  this  kind  would  be  experiments  on 
cubic  compressibility,  in  conjunction  with  experiments  on  torsion  ;  for,  as 
equations  (12),  (12 A),  and  (12B)  show,  in  order  to  determine  J  we  have 
simply  to  subtract  a  certain  symmetrical  function  of  the  rigidities  from 
the  reciprocal  of  the  cubic  compressibility.  In  the  process  of  calculation, 
the  errors  in  the  experiments  on  rigidity  are  multiplied,  on  an  average, 
by  ~  only,  while  those  of  the  experiments  on  compressibility  sustain  no 
augmentation  whatsoever. 

Next  after  data  of  this  kind  may  be  ranked  experiments  on  longitudinal 
extensibility,  as  compared  with  the  cubic  extensibility  or  compressibility 
of  the  same  piece  of  material.  Of  this  method,  suggested  by  M.  Eegnault, 
and  carried  into  effect  by  M.  Wertheim,  I  shall  presently  speak  more 
fully. 

Were  it  possible  to  ascertain  the  velocity  of  sound  in  an  unlimited 
mass  of  an  elastic  material  along  each  of  the  axes  of  elasticity,  the 
coefficients  of  longitudinal  elasticity  could  be  determined  with  great 
precision  by  the  formula 

where  v  is  the  velocity  of  sound,  D  the  weight  of  unity  of  volume  of  the 
substance,  and  g  the  accelerating  force  of  gravity.  But  it  is  only  practi- 
cable to  determine  the  velocity  of  sound  along  prismatic  or  cylindrical  rods; 
and,  as  I  shall  show  in  a  subsequent  paper,  it  is  impossible,  in  the  present 
state  of  our  knowledge  of  the  molecular  condition  of  the  superficial 
particles  of  solid  bodies,  to  assign  theoretically  the  ratio  in  which  the 
velocity  of  sound  along  a  rod  is  less  than  its  velocity  in  an  indefinitely 
extended  mass.  That  ratio  is  only  known  empirically  in  a  few  cases, 
having  various  values  lying  between  1  and  ^/f . 

23.  The  experiments  of  M.  Wertheim,  on  longitudinal  and  cubic 
extensibility  {Ann.  de  Chim.  et  de  Phys.,  Ser.  III.,  Tom.  XXIII.)  were  made 


94  LAWS  OF   THE  ELASTICITY   OF   SOLID   BODIES. 

upon  brass  and  crystal,  the  results  being  calculated  on  the  supposition 
that  those  substances  are  homogeneous  and  equally  elastic  in  all  directions. 
There  can  be  no  doubt  of  the  correctness  of  this  supposition  with  respect 
to  well  annealed  crystal ;  and  with  respect  to  brass,  it  is  probably  very 
near  the  truth. 

In  those  experiments,  a  cylindrical  tube  of  the  substance  to  be  examined 
was  strained  by  longitudinal  tension.  The  increase  of  length  was  observed 
directly.  The  increase  of  bulk  was  found  by  observing  the  depression  in 
a  capillary  tube  connected  with  the  summit  of  the  strained  tube,  of  a 
column  of  liquid  with  which  they  were  filled.  Let  R  denote  the  tensile 
force  reduced  to  unity  of  surface  ;  let  L  be  the  original  length  of  a  given 
portion  of  the  tube,  U  its  original  volume,  A  L  and  A  U  the  increase  of 
those  quantities  by  the  tension  R ;  and  let  the  axis  of  x  be  that  of  the 
tube.     Then  we  have 

N   =AL-    X  +N  +N   =^J- 

Pl  =  -R;    P2  =  0;    P3  =  0: 
and  consequently,  for  uncrystallised  substances, 

1     AL    .        3    AU 

a=B,'^r>  *  =  r-  it-     •     •  <14-> 

To  determine  the  coefficients  of  rigidity  and  fluid  elasticity  from  these 
data,  we  have  the  following  formulae  : 


C  = 


3  a  -  £  &      „AL      AU 
6T~  IT 

Vn        /         VL     AU       / 


(14A.) 


J 


The  experiments  of  M.  Wertheim  were  made  on  three  tubes  of 
brass  and  five  of  crystal.  In  the  following  table  those  tubes  are 
designated  as  M.  Wertheim  has  numbered  them.  The  coefficients  a  and 
Jr,  transcribed  or  calculated  from  his  statement  of  the  mean  results  of 
numerous  experiments  on  each  tube,  express  the  fraction  by  which  the 
material  is  elongated,  or  increased  in  bulk,  by  tension  at  the  rate  of  one 
kilogramme  on  the  square  millimetre;  that  is  to  say,  1422-34  pounds 
avoirdupois  on  the  square  inch.  (The  common  logarithm  of  this  number 
is  3-153004.)  The  reciprocals  of  those  coefficients,  and  the  coefficients  of 
elasticity,  as  calculated  by  equation  (14A),  are  given  in  kilogrammes  on 
the  square  millimetre. 


LAWS   OF  THE   ELASTICITY   OF   SOLID  BODIES. 


95 


H 

S3 
55 

H 
X! 

H 

K 

O 


H 

a 

»— t  ■ 

m 
m 
X 


1-1 


5,«j 

o 

Hi 


H 

O 


P5 

■  I 

q  pq 


05 
a 

H 

< 

t3 
O 
iJ 

o 

H 
55 
SI 


E=4 

o 

Eh 
O 

H 


a£ 

<!-  — 3 
lap 

O   01   g 

•S  =  W 

«|J 
a  a 
23 

S-l 


3  "-» 


a  g" 


-|o 


^H  O  ^H 


•-5 

a  a 
©  o 


cS 

os 

m 

>o 

o 

Tfl 

CO 

CO 

T* 

CO 

CO 

C5 

CM 

t— 

lO 

CO 

Ci 

o 

o 

in 

CO 

CO 

CM 

CM 

o 

~H 

CM 

<M 

CM 

<M 

CM 

o 

o 

CO 

O 

o 

o 

o 

O 

o 

o 

o 

o 

o 

o 

Q 

o 

a 

o 

o 

o 

o 

o 

o 

O 

o 

h1 

o 

o 

o 

o 

o 

© 

o 

o 

H 


96 


LAWS   OF   THE   ELASTICITY   OF   SOLID   BODIES. 


The  various  degrees  of  elasticity  of  the  brass  tubes  are  ascribed  by  M. 
Wertheim  to  the  relative  frequency  with  which  they  were  subjected  to 
wire-drawing,  to  reduce  the  thickness  of  metal.  It  may  be  observed,  that 
this  operation  seems  to  increase  the  rigidity  more  than  the  fluid  elasticity, 
a  fact  which  might  naturally  have  been  expected. 

The  means  of  the  three  sets  of  results  for  brass  are  given  in  the 
following  table  : — 


Coefficients  of 

Rigidity, 

Fluid  Elasticity,  . 
Longitudinal  Elasticity, 
Lateral  Elasticity, 


Reciprocals  of  Extensibilities. 

Longitudinal  (or  Weight  of  the  )   1 

Modulus  of  Elasticity),      .    )   a 

1 
Cubic,  .....        t 


Kilogrammes  on 
the  Sq.  Millim. 

3745-3 

4389-0 

15G25-0 

8134-3 


10D.V1--1 


10G31-0 


Lbs.  Avoird.  on  the 
Square  Inch. 

5327100 

6242700 
22224000 
11579000 


14301000 
15121000 


Coefficients  of  Extensibility 

Per  Kilog.  on  the 

Per  lb.  on  the  Square 

and  Compressibility. 

Square  Millim. 

Inch. 

Longitudinal, 

;t 

0-00009946 

0-0000000699 

Cubic, 

b 

0-00009406 

0-0000000661 

Latei'al, 

b 

0-00003405 

0-0000000239 

The  following  result  is  calculated  from  the  experiments  of  M.  Savart 
on  the  torsion  of  brass  wire  (Ann.  dc  Chim.  et  de  Phys.,  August. 
1829):— 


Coefficient  of  Rigidity, 
The  difference  beins 


Kilog.  on  the 
Square  Millim. 

C  3682 

63-3 


Lbs.  on  the 
Square  Inch. 

5237100 
900U0 


Hence  we  see  that  the  rigidity  of  wire-drawn  brass,  as  determined 
directly  by  torsion,  differs  from  that  calculated  from  the  longitudinal  and 
cubic  extensibilities  by  only  one-sixtieth  part,  being  a  very  small  discrepancy 
in  experiments  of  this  kind. 

The  following  table  gives  the  means  of  the  four  sets  of  results,  I.,  III., 
IV.,  V.,  for  crystal : — 


LAWS   OF   THE  ELASTICITY  OF  SOLID   BODIES. 


07 


Kilog.  on  the 

Square  Millim. 

Lbs.  on  the  Square  Inch 

c     . 

1518 

2159100 

J    . 

1438 

2045300 

A     . 

5992 

8522600 

B      . 

2956 

4204400 

1 

4039 

5746100 

l 

3968 

5643800 

Per  Kilog.  on  the 
Square  Millim. 

0-0002476 
0-0002520 
0-0000818 


Per  lb.  on  the  Square 
Inch. 

0-0000001740 
0-0000001772 
0-0000000575 


It  is  obvious  that  the  above  mean  values  for  crystal  are  not  to  be 
relied  upon  as  equally  accurate  with  those  for  brass ;  for  the  wide  dis- 
crepancies between  the  results  of  the  experiments  on  the  five  crystal  tubes 
show  that  this  substance,  like  every  kind  of  glass,  is  subject  to  great 
variations  in  the  physical  properties  of  different  specimens. 

24.  So  far  as  I  am  aware,  there  is  no  substance  whose  elasticity  varies 
in  different  directions,  for  which  experimental  data  as  yet  exist,  adequate 
to  determine  the  three  coefficients  of  rigidity,  and  the  coefficient  of  fluid 
elasticity. 

Supposing  the  three  coefficients  of  rigidity  of  a  substance  of  this  kind 
to  be  known  by  experiments  on  torsion,  the  process  of  MM.  Regnault  and 
AVertheim  would  readily  furnish  data  for  calculating  the  fluid  elasticity. 

For  example :  let  a  tension  E  per  unit  of  area  be  applied  to  the  ends 
of  a  tube  whose  axis  is  one  of  the  axes  of  elasticity,  say  that  of  x.     Let 

—==-  De  the  fraction  by  which  its  volume  is  increased,  as  before.     Then 


AU_      N1  +  N2  +  N3_ 

"\ 

=  1{8C;--2C1(C2  +  C3)-6(C2- 

-  C  V21r 

v3/  /      1 

y  (is.) 


Let  the  above  equation  be  abbreviated  into 

A  U  _      0(C) 
R  U  ~  K0  +  m  J ' 

G 


98  LAWS    OF  THE   ELASTICITY   OF   SOLID   BODIES, 

where  K  =  K0  +  mJ,  as  in  equation  (13).     Then 

The  formula?  corresponding  to  equation  (15)  for  tubes  whose  axes  are 
parallel  to  y  and  z,  are  easily  found  by  permutations  of  the  indices  1,  2, 

3.     The  sum  of  the  three  values  of  ~TT  thus  obtained  is  obviously  =  tf. 

h  U 

[It  may  be  remarked,  with  reference  to  Sect.  1 7  of  the  preceding  paper, 
that  the  effect  of  alterations  of  direction  in  the  lines  joining  pairs  of 
particles  is  not  taken  into  account  in  the  investigation  of  the  elastic  forces 
arising  from  the  states  of  strain  which  are  there  considered.  It  appears 
to  me  that  this  effect,  except  for  particular  laws  of  force,  will  be  of  the 
same  order  as  that  which  depends  on  the  alterations  of  the  mutual 
distances  between  the  particles ;  and  that  if  it  be  taken  into  account,  the 
demonstration  of  Theorem  IV.  fails. 

This  objection  occurred  to  me  after  the  whole  of  the  paper  was  in  type, 
and  I  immediately  suggested  it  to  the  author ;  but,  as  he  was  not  con- 
vinced of  the  correctness  of  my  view,  he  desired  that  the  paper  should  be 
published  as  it  stands,  reserving  additional  explanations  or  modifications, 
if  necessary,  for  the  next  number  of  the  journal. — W.  T.] 


Supplementary  Paper  to  Section  III.,  Article  17. 

In  the  portion  above  referred  to  of  my  paper  on  the  Elasticity  of  Solids, 
published  in  the  Cambridge  and  Dublin  Mathematical  Journal  for  February, 
1851,  the  theorem  is  laid  down,  that  in  a  given  plane  in  an  elastic  solid 
consisting  entirely  of  atoms  acting  on  each  other  by  attractions  and 
repulsions  between  their  centres,  the  coefficients  of  rigidity  and  of  lateral 
elasticity  are  equal. 

The  proof  of  this  proposition  depends  on  the  principle  that  the  elastic 
force  in  such  a  solid,  called  into  play  by  a  strain,  in  which  the  relative  displace- 
ments of  the  atoms  arc  very  small  as  compared  with  tlieir  distances  apart,  is 
sensibly  the  resultant  of  the  variations  of  force  due  to  the  variations  of  distance 
only,  the  variations  of  relative  direction  producing  no  appreciable  effect.  This 
principle  being  granted,  it  is  easily  shown  that  the  portion  of  that 
resultant  for  each  pair  of  atoms  is  the  same  for  a  given  amount  of  strain 
in  a  given  plane,  whether  lateral  or  transverse  with  respect  to  the  plane 
on  which  elastic  pressure  is  estimated. 


LAWS    OF   THE   ELASTICITY  OF  SOLID  BODIES.  99 

In  the  paper  referred  to,  I  assumed  this  principle  without  demonstration. 
The  editor  of  this  journal,  however,  has  since  shown  me,  that  my  having 
done  so  may  be  considered  as  causing  a  defect  in  the  chain  of  reasoning. 
I  shall  now,  therefore,  proceed  to  prove  it. 

Let  it  be  possible  for  a  solid  to  exist  in  an  unstrained  condition, 
consisting  entirely  of  atomic  centres  of  force  acting  on  each  other  along 
the  lines  joining  them,  with  forces  which  are  functions  of  the  lengths 
of  these  lines.  Then  must  the  pressure,  estimated  in  any  direction,  on 
any  portion  of  any  plane  in  that  solid  be  null.  That  pressure  is  the 
resultant,  in  the  direction  assumed,  of  the  mutual  actions  of  all  the  atoms 
whose  lines  of  junction  pass  through  the  given  portion  of  the  given 
plane. 

Let  the  given  portion  be  indefinitely  small,  and  let  it  be  called   w, 
being  situated  in  the  arbitrarily-assumed  plane 
PWjp,  which  divides  the  solid  into  two  por-  +x 

tions,  A  and  B.  Let  —  X  o>  +  X  be  an  arbi- 
trary axis  along  which  pressure  is  to  be  esti- 
mated. The  pressure  exerted  by  the  portion 
A  upon  the  infinitesimal  area  10  of  the  portion 
B,  is  the  resultant,  reduced  to  the  direction 
—  X  to  +  X,  of  all  the  forces  exerted  by  the 
atoms  in  A  on  the  atoms  in  B,  in  lines  passing 
through  (o;  and  the  body  being  unstrained, 
this  resultant  must  be  null. 

Assume  a  new  position,  Vuyp',  for  the  plane  of  separation,  making  an 
equal  angle  P'w  +  X=+XwPon  the  opposite  side  of  the  axis  to  the 
original  position.  The  same  letters  applying  to  the  two  portions  of  the 
solid,  the  pressure  of  A  on  the  area  10  of  B  along  -XwX  must  still 
be  null. 

The  two  planes  divide  the  solid  into  two  pairs  of  opposite  wedges. 
The  action  of  A  on  B  along  X  through  w  in  the  original  position  of  the 
plane,  may  be  divided  into  two  parts,  viz. — 

The  resultant  of  the  actions  of  the  atoms  in  the  wedge  P  top'  on  those 
in  the  opposite  wedge  Ywp ; 

The  resultant  of  the  actions  of  the  atoms  in  the  wedge  p  u>p  on  those 
in  the  wedge  P  w  P'. 

In  the  new  position  of  the  plane,  the  pressure  on  u>  is  made  up  as 
follows : 

The  resultant  of  the  actions  of  the  atoms  in  the  wedge  Pw/on  those 
in  the  wedge  P'wp,  which  is  the  same  as  hi  the  original  position  of 
the  plane; 

The  resultant  of  the  actions  of  the  atoms  in  the  wedge  PwP'  on  those 
in  the  wedge  p'wjp,  being  identical  in  amount  but  opposite  in  direction  to 


100  LAWS  OF  THE  ELASTICITY   OF  SOLID   BODIES. 

that  of  the  atoms  in  p'  up  on  those  in  P to  P',  which  formed  part  of  the 
pressure  in  the  original  position  of  the  plane. 

Now  the  pressures  in  the  two  positions  of  the  plane  of  separation  cannot 
both  be  null,  unless  the  resultant  of  the  mutual  actions  of  the  atoms  in 
each  pair  of  opposite  wedges  is  separately  null;  for  we  see  that  the  action 
of  a  pair  of  wedges  can  be  reversed  in  direction  without  affecting  the 
nullity  of  the  total  resultant.  The  position  of  the  pair  of  opposite 
wedges  is  arbitrary;  so  also  is  their  angular  magnitude,  which  may  be 
indefinitely  small. 

Therefore,  no  mere  change  of  angular  position  of  a  pair  of  opposite 
elementary  wedges  can  produce  a  pressure. 

Every  strain  in  which  the  relative  displacements  of  the  particles  are 
small  as  compared  with  their  relative  distances,  may  be  reduced  to 
angular  displacements  of  pairs  of  opposite  elementary  wedges,  and  varia- 
tions of  the  mutual  distances  of  the  particles  contained  in  them.  The 
angular  displacements  can  produce  no  pressure  of  themselves;  the 
variations  of  distance  are  therefore  the  sole  cause  of  that  portion  of  the 
pressure  which  is  of  the  same  order  of  small  quantities  with  the  strain: 
being  the  principle  to  be  proved. 

The  combination  of  the  angular  displacements  with  the  variations  of 
distance  will  give  rise  to  pressures  of  the  second  and  higher  orders  of 
small  quantities  as  compared  with  the  strain;  but  for  the  small  strains 
to  which  the  present  inquiry  is  limited,  those  are  inappreciable,  and  may 
be  nesdected. 


Note  respecting  Mr.  Clerk  Maxwell's  Paper  <;  On  the 
Equilibrium  of  Elastic  Solids."  (Trans.  Hoy.  Soc.  Edin., 
Vol.  XX.,  Part  I.) 

I  HAVE  already  referred  to  the  researches  of  Mr.  Clerk  Maxwell,  of  the 
general  nature  of  which  only  I  was  aware  at  the  time  of  the  publication 
of  my  paper  on  this  subject  in  the  Cambridge  and  Dublin  Mathematical 
Journal  for  February,  1851. 

Since  then  I  have  had  an  opportunity  of  reading  Mr.  Maxwell's  paper, 
so  as  to  compare  his  notation  with  my  own. 

Mr.  Maxwell's  investigations  relate  to  such  solids  only  as  are  equally 
elastic  in  all  directions.  He  expresses  their  elasticity  by  means  of  two 
coefficients,  fx  and  m,  having  the  following  properties: 


LAWS   OF    ELASTICITY.  101 


M  = 

1    Px  +  P9  +  P, 
3    d£     dv     dV 

dx     dy     d  z 

m  = 

P   —  P             P   - 
d  £      dr\            dr\ 

d  x     dy           dy 

-  P., 
d  z 

dZ, 
dz~ 

-Pi 
_dj 

dx 

From  which  it  is  clear,  that  those  coefficients  have  the  following  values 
in  the  notation  of  the  paper  which  I  have  published. 

ju  =  ^.  =  -JC  +  J  =  reciprocal  of  the  cubic  compressibility, 

m  =  2  C  =  twice  the  rigidity ; 

consequently 

C  =  ^  m,  J  =  fi  —  4  m. 

The  particular  problems  solved  by  Mr.  Clerk  Maxwell  are  of  a  very 
interesting  character,  especially  those  relative  to  the  optical  changes 
produced  in  transparent  bodies  by  straining  them. 


ON  THE  LAWS  OF  ELASTICITY.* 

Section  VI.— Ox  the  Application  of  the  Method  of  Virtual 
Velocities  to  the  Theory  of  Elasticity. 

25.  Lagrange's  method  of  virtual  velocities  having  been  applied  to  the 
problems  of  the  equilibrium  and  motion  of  elastic  media  by  Mr.  Green 
(Camb.  Trans.,  VII.)  and  by  Mr.  Haughton  (Trans.  Royal  Irish  Acad.,  XXL, 
XXIL),  it  is  my  purpose  in  this  and  the  following  section  to  point  out  the 
mutual  correspondence  between  the  coefficients  in  the  formulas  arrived  at 
by  these  gentlemen,  and  the  coefficients  of  elasticity  which  form  the  sub- 
ject of  the  previous  portion  of  this  paper,  and  also  to  show  how  far  the 
laws  of  relation  between  the  nine  coefficients  of  elasticity  of  a  homogene- 
ous body,  which  I  originally  proved  by  a  method  chiefly  geometrical,  are 
capable  of  being  deduced  symbolically  from  equations  found  according  to 
Lagrange's  method. 

26.  The  principle  of  virtual  velocities,  as  applied  to  molecular  action,  it 

*  Originally  published  in  the  Cambridge  and  Dublin  Mathematical  Journal,  Nov., 
1S52. 


102  LAWS    OF   ELASTICITY. 

as  follows  •  Let  X,  Y,  Z,  denote  the  total  accelerativc  forces  applied  to 
any  particle,  whose  mass  is  m,  of  an  elastic  medium,  through  agencies  dis- 
tinct from  molecular  action  (such  as  the  attraction  of  gravitation);  let 
u  v  to  be  the  components  of  the  velocity  of  m;  let  dx,  By,  Bz,  denote 
indefinitely  small  virtual  variations  of  x,  y,  z;  let  S  be  the  total  accelera- 
tive  molecular  force  applied  to  m,  Bs  an  indefinitely  small  virtual  variation 
of  the  line  along  which  it  acts  j  then  the  following  equation 

»[-{(x-^..+(y-50«»+(--^.}]\w 

+  S(mS8s)  =  0  J 

(the  summation  2  being  extended  to  all  the  particles  of  the  medium), 
expresses  at  once  all  the  conditions  of  equilibrium  and  motion  of  every 
particle  of  the  medium. 

27.  In  applying  this  principle  to  the  theory  of  elastic  media,  both  Mr. 
Green  and  Mr.  Haughton  assume  the  following  postulates  : 

First.  That  in  calculation  Ave  may  treat  each  particle  m  as  if  it  were 
a  small  rectangular  space,  dx  dy  dz,  filled  with  matter  of  a  certain 
density  p :  so  that  for  the  symbol  2  m  we  may  substitute  that  of  a  triple 
integration. 

fffpdxdydz. 

Secondly.  That  the  virtual  moment  m  S  B  s  of  the  total  molecular  force 
acting  on  any  particle  m,  is  capable  of  being  expressed  by  the  product 
of  the  small  rectangular  space  dx  dy  dz  into  the  variation,  B  V,  of  a 
certain  function  V  of  the  relative  position  of  m  and  the  other  particles  of 
the  body* 

Equation  (16)  is  thus  transformed  into  the  following: — 


+  (z  -  ~^Bz\dxdydz  +ff/BYdxdydz  =  0 


(17.) 


J 


28.  The  term  elasticity  properly  comprehends  those  molecular  forces 
only  whose  variations  are  produced  by,  and  tend  to  produce,  variations 
in  the  volume  and  figure  of  bodies.  There  are,  therefore,  conceivable 
kinds  of  molecular  force,  which  are  not  included  in  the  term  elasticity. 
For  example,  let  us  take  the  forces  which  Mr.  MacCullagh  ascribed  to 
the  particles  of  the  medium  which  transmits  light. 

*  This  amounts,  in  fact,  to  the  assumption  that  no  part  of  the  power  developed  by  a 
variation  of  the  relative  positions  of  the  particles  is  permanently  converted  into  heat, 
or  any  other  agency:  in  other  words,  that  the  body  is  perfectly  elastic. 


LAWS   OF    ELASTICITY.  103 

Let  £,  17,  £,  denote  displacements  of  a  point  in  the  medium  parallel 
respectively  to  x,  y,  z.  Then  Mr.  MacCullagh  supposes  the  molecular 
forces  to  be  functions  of 

dr\       dZ,      dZ,       d£      d£       dr\ 

dz       dy}    d x      dz '    dy      dx' 

which  are  proportional  to  the  rotations  of  an  element  dxdydz  from  its 
position  of  equilibrium  about  the  three  axes  respectively.  This  amounts 
to  ascribing  to  the  particles  of  the  medium  a  species  of  polarity,  tending 
to  place  three  orthogonal  axes  in  each  particle  parallel  respectively  to 
the  three  corresponding  axes  in  each  of  the  other  particles :  the  rotative 
force  acting  between  the  corresponding  axes  in  each  pair  of  particles 
being  a  function  of  the  projection  of  the  relative  angular  displacement 
of  the  axes  on  the  plane  passing  through  them,  of  the  position  of  that 
plane,  and  of  the  distance  between  the  particles. 

A  portion  of  a  medium  endowed  with  such  molecular  forces  only 
would  transmit  oscillations;  but  it  would  not  tend  to  preserve  any 
definite  bulk  or  figure,  nor  would  it  resist  any  change  of  bulk  or 
figure.  It  would  be  a  medium  or  system,  but  not  a  body.  Molecular 
forces  of  this  kind,  therefore,  are  not  comprehended  under  the  term 
elasticity ;  and  the  limits  of  the  present  investigation  exclude  those  forms 
of  the  function  V  which  represent  the  laws  of  their  action. 

29.  The  inquiry  being  thus  restricted  to  molecular  forces  dependent 
on  the  variations  of  the  bulk  and  figure  of  bodies,  there  is  to  be  intro- 
duced a 

Third  Poshdate:  That  supposing  the  body  to  be  divided  mentally 
into  small  parts,  which,  in  the  undisturbed  state  of  the  body,  are 
rectangular  and  of  equal  size,  those  parts,  in  the  disturbed  state,  continue 
to  be  sensibly  of  equal  bulk  and  similar  figure,  throughout  a  distance 
round  each  point  at  least  equal  to  the  greatest  extent  of  appreciable 
molecular  action. 

This  assumption  has  been  made  in  all  previous  investigations,  except 
those  respecting  the  dispersion  of  light;  and  it  seems,  indeed,  to  be 
perfectly  consistent  with  the  real  state  of  tangible  bodies. 

Its  advantage  in  calculation  is,  that  it  enables  us  to  treat  the  variations 
of  the  molecular  forces  acting  on  a  given  particle,  as  functions  simply  of  the 
variations  of  bulk  and  figure  of  an  originally  rectangular  element  situated 
at  that  particle:  seeing  that  the  adjoining  elements  throughout  the 
extent  of  appreciable  molecular  action  continue  always  to  undergo 
sensibly  the  same  variations  of  bulk  and  figure  as  the  element  under 
consideration. 

Let  x0,  y0,  z0  be  the  co-ordinates  of  any  physical  point  in  a  homogeneous 


104  LAWS    OF    ELASTICITY. 

body  in  equilibrio,  and  whose  particles  are  not  operated  upon  by  any 
extraneous  forces,  X,  Y,  Z.     In  this  condition  it  is  evident  that 

SV  =  0 

at  every  point,  and  that  we  may  also  make 

V  =  0. 

In  the  disturbed  condition,  let  £,  ?j,  Z,,  be  the  displacements  of  the 
point  whose  undisturbed  position  is  (x0,  y0,  z0),  so  that 

x  =  o:0  +  £,  &c. 

Then  all  the  variations  of  bulk  and  figure  which  can  be  undergone 
by  an  originally  rectangular  element,  consistently  with  the  third  postulate, 
may  be  expressed  by  means  of  the  following  six  quantities,  which  I  have 
elsewhere  called  strains : 

d%  _         dt)  _  r.      ell,  _ 

Tz~a>  Ty-pm'  Tz~7' 

dj  ,dX_        d%      &%,  _        cZj;       djj  _      ,. 
dz  +  dy~    '   dx+dz~fM}   dy+ dx~V' 

of  which  a,  ft,  y,  are  longitudinal  extensions  if  positive,  compressions  if 
negative,  and  X,  ju,  v,  are  distortions  in  the  planes  perpendicular  to  x,  y,  z, 
respectively. 

Hence  it  appears  that 

Y  =  <f>(a,ft,7>\,fx,v).       .  .  .      (18.) 

30.  The  first  assumption,  that  we  may  treat  the  body  in  calculation 
as  composed  of  rectangular  elements  p  d  x  d  y  d  z,  involves  the  consequence 
that  we  may  express  all  the  molecular  forces  which  act  on  each  such 
element  by  means  of  pressures,  normal  and  tangential,  exerted  on  its 
six  faces.  Taking  yz,  zx,  xy,  to  denote  the  position  of  the  faces  of  such 
an  element,  P  to  denote  generally  a  normal  pressure  expressed  in  units 
of  force  per  unit  of  area,  and  Q  a  tangential  pressure  similarly  expressed, 
let  the  nine  component  pressures  on  unity  of  area  of  those  faces  be 
thus  denoted : 

*  This  notation  is  substituted  for 

Ni,  N2,  N3,  2Tn,  2Ta  2T3, 
as  being  more  convenient. 


LAWS   OF   ELASTICITY.  105 


Position  of  Face. 

Direction  of  Pressure. 

X 

y 

Z 

y~      Pi 

% 

<¥ 

nx      q; 

p. 

Qi 

xy         Q2 

Qi' 

p3 

By  the  definition  of  elasticity  all  pressures  are  excluded,  except  those 
whose  variations  produce  and  are  produced  by  variations  of  volume  and 
figure  of  the  parts  of  the  body.     Hence  the  pressures 

Qi-Q/;  Q«-Qii  Q3-Q/; 

whose  tendency  is  to  make  the  element  dxclydz  rotate  about  its  three 
axes  respectively,  without  change  of  form,  must  be  null;  and  therefore 

Qi  =  Qi';  Q2  =  Q2';  Q3  =  Q3'-        •       ■    (19-) 

Mr.  Haughton  correctly  remarks  that  this  often  quoted  theorem  of 
Cauchy  is  not  true  for  all  conceivable  media.  It  is  not  true,  for  instance, 
for  a  medium  such  as  that  which  Mr.  MacCullagh  assumed  to  be  the 
means  of  transmitting  light.  It  is  true,  nevertheless,  for  all  molecular 
pressures  which '  properly  fall  under  the  definition  of  elasticity,  if  that 
term  be  confined  to  the  forces  which  preserve  the  figure  and  volume  of 
bodies. 

Let  us  now  express  the  sum  of  the  virtual  moments  of  the  molecular 
forces  acting  on  the  element  d  x  d  y  d  ~,  in  terms  of  the  pressures  P1?  &c. ; 
to  do  which,  we  must  multiply  each  pressure  by  the  virtual  variation  of 
the  effect  which  it  tends  to  produce  in  its  own  direction.  Thus  we  obtain 
the  following  result : — 

SV  =  P1Sa  +  PoS/3  +  P3£7  +  Q1gA  +  Q2Syu  +  Q3Si/.    .    (20.) 

Hence  the  function  V  bears  the  following  relations  to  the  normal  and 
tangential  pressures  at  the  faces  of  a  rectangular  element : 


1_  da'        2~  d(}'        3_  dy 
^~d\>     ^-df,'    ^~Jv) 


(21.) 


31.  A  Fourth  Postulate,  generally  assumed  in  investigations  of  this  kind, 
is  that  the  pressures  are  sensibly  proportional  simply  to  the  strains  with 
which  they  are  connected.  This  assumption  must  be  approximately  true 
of  any  law  of  molecular  action,  when  the  pressures  and  strains  are  suffi- 
ciently small.     It  is  known  to  be  sensibly  true  for  almost  all  bodies,  so 


106  LAWS   OF   ELASTICITY. 

long  as  the  pressures  and  strains  are  not  so  great  as  to  impair  their  power 
of  recovering  their  original  volume  and  figure. 

According  to  this  postulate,  the  pressures  P1?  &c.,  are  algebraic  functions 
of  the  first  order  of  the  strains  a,  &c. ;  and  consequently  V  is  an  algebraic 
function  of  the  second  order  of  those  strains.  The  constant  part  of  V,  as 
we  have  already  seen  (Art.  29),  is  null. 

Following  the  notation  adopted  by  Mr.  Haughton,  let  (a)  denote  the 

2 

coefficient  of  a  in  V,  (a2)  that  of  —  — ,  (/3y)  that  of  —  (5y,  &c.     Then 
V  =  (a)a  +  (/3)/3  +  (7)7  +  W  ^  +  (/*)/*  +  W» 

«2  P2  .,2  \2  ..2  ,,2 

-  («2)  2  -  (02) f-  ~  M  I  -  (X2)  y  "  M  J  -  M>  •> 

—  ($y)  fiy  —  (yet)  ya  —  (a/3)  a/3  —  (fiv)  fxv  —  (vX)  v\  —  (\/ul)  \/jl 
—  (aX)  a\  —  (a/u)  a/j.  —  (av)  av 
-(P\)P\--(M(5f*-((3v)pv 
-(y\)y\  —  (yfi)yfA—(yv)yv.  .  .     (22.) 

The  six  coefficients  of  the  terms  of  the  first  order. in  this  equation 
obviously  represent  the  pressures,  uniform  throughout  the  whole  extent  of 
the  body,  to  which  it  is  subjected  when  its  particles  arc  in  those  positions 
from  which  the  displacements  are  reckoned:  that  is  to  say,  when 

I  =  0;    r,  =  Oj    Z  =  0. 

Let  P1>0  &c,  denote  those  pressures.     Then 

(«)  =  i\,o;   0)-pmj   (7)  =  p,,o| 
W  =  Qi,oJ    (m)=Q2,0;    W^Q^J 

The  twenty-one  coefficients  of  the  terms  of  the  second  order  are  the 
coefficients  of  elasticity  of  the  body,  as  referred  to  the  three  axes  selected. 
The  negative  sign  is  prefixed  to  each,  because  it  is  essential  to  the  stability 
of  a  body  that  molecular  pressures  should  be  opposite  in  direction  to  the 
strains  producing  them. 

The  transformation  of  the  quantities  in  equation  (22)  for  any  set  of 
rectangular  axes,  is  effected  by  means  of  equation  (2)  of  Sect.  I,  Art.  9, 
by  making  the  following  substitutions  : — 

for  PPP20'>O'>O 

x  1'    -""  2'        3'    A  H'ij    *  k'01    -  V3J 

substitute  a,     (3,    y,     X,        p,        v, 

and  make  similar  substitutions  for  the  accented  symbols.     By  multiplying 
the  six  equations  referred  to  together  by  pairs,  twenty-one  equations  are 


LAWS    OF    ELASTICITY.  107 

obtained,  serving  to  transform  the  squares  and  products  of  a,  (5,  7,  X,  jjl,  v. 
Formulae  similar  to  those  which  transform  the  strains  a,  &c.,  and  their 
half-squares  and  products,  serve  also  to  transform  the  respective  coefficients 
of  those  quantities  in  equation  (22).* 

It  is  shown  by  Mr.  Haughton,  that  by  properly  selecting  the  axes  of 
co-ordinates,  the  number  of  independent  coefficients  of  elasticity  may 
always  be  reduced  to  three  less  than  when  the  axes  are  indefinite ;  and 
by  Mr.  Haughton  and  Mr.  Green,  that  when  the  body  has  orthogonal 
axes  of  elasticity  at  each  point,  then,  if  those  axes  be  taken  as  the  axes 
of  co-ordinates,  the  coefficients  of  elasticity  are  reduced  to  the  first  nine. 
The  latter  proposition  is  obvious,  because  if  molecular  action  be  sym- 
metrical about  three  orthogonal  planes,  and  those  be  taken  for  co-ordinate 
planes,  then  the  value  of  that  part  of  the  function  V  which  is  of  the 
second  order  cannot  be  altered  by  a  change  in  the  sign  of  either  of  the  dis- 
tortions A,  ju,  v;  so  that  the  coefficients  of  the  last  twelve  terms  of 
equation  (22)  must  each  be  null. 

The  nine  coefficients  of  elasticity  of  a  body  in  those  circumstances  have 
the  following  values,  in  terms  of  the  notation  of  the  previous  sections  : 

Coefficients  of  Longitudinal  Elasticity. 
(a2)  =  Ai;     m=A2;      (72)=A3 


(24.) 


Coefficients  of  Lateral  Elasticity. 
(/37)  =  Bi;     (7a)  =  B2;     (a/3)=B3. 

Coefficients  of  Eigidity. 

32.  Def.  Let  the  term  Perfect  Fluid  be  used  to  denote  the  state  of  a 
body,  which  under  a  given  uniform  normal  pressure,  and  at  a  given  temperature, 
tends  to  preserve,  and  if  disturbed  to  recover,  a  certain  bulk;  but  offers  no 
resistance  to  change  of  figure. 

In  such  a  body,  if  the  element  whose  original  bulk  was  dxdydz, 
becomes  of  the  bulk  (1  +  0-)  dxdydz  (o-  being  a  small  fraction),  we  shall 
have 

a  =  a  +  /3  +  J, 

and  the  function  V  must  be  of  the  form 

Y  =  Y0„-(o»-)a~  .         .         .     (25.) 

where  P0  is  the  uniform  normal  pressure  when  the  particles  are  not  dis- 
placed, and  (o-2)  a  coefficient  of  elasticity,  whose  value,  in  the  notation  of 
the  previous  sections,  is  : 

*  See  the  Note  at  the  end  of  this  paper. 


108  LAWS    0F   ELASTICITY. 

Coefficient  of  Fluid  Elasticity. 

(«*)  =  (a2)  =  (/32)  =  (72)  =  (/3y)  =  (y«)  =  («0)  =  J-    (26.) 

The  normal  pressure  in  the  disturbed  state,  which  is  the  same  in  all 
directions,  is  obviously 

The  tangential  pressures  are  each  =  0. 


Section  VII.— On  the  Proof  of  the  Laws  of  Elasticity  by  the 
Method  of  Virtual  Velocities. 

33.  Having  thus  followed  very  nearly  the  steps  of  the  researches  of 
Mr.  Haughton  and  Mr.  Green,  so  as  to  compare  their  coefficients  with 
those  used  in  the  previous  part  of  this  paper,  I  shall  now  investigate  how 
far  the  method  ■  of  Lagrange  can  be  used  to  establish  those  relations 
between  the  coefficients  of  elasticity  of  different  kinds  in  homogeneous 
solid  bodies,  which  I  have  elsewhere  deduced  from  geometrical  and 
physical  considerations. 

The  fluid  elasticity  considered  in  the  last  article  cannot  arise  from  the 
mutual  actions  of  centres  of  force;  for  such  actions  would  necessarily 
tend  to  preserve  a  certain  arrangement  amongst  those  centres,  and  would 
therefore  resist  change  of  figure.  Fluid  elasticity  must  arise  either  from 
the  mutual  actions  of  the  parts  of  continuous  matter,  or  from  the  centri- 
fugal force  of  molecular  motions,  or  from  both  those  causes  combined. 

On  the  other  hand,  it  is  only  by  the  mutual  action  of  centres  of  force 
that  resistance  to  change  of  figure  and  molecular  arrangement  can  be 
explained,  that  property  being  inconceivable  of  a  continuous  body.  The 
elasticity  peculiar  to  solid  bodies  is,  therefore,  due  to  the  mutual  action  on 
centres  of  force.  Solid  bodies  may  nevertheless  possess,  in  addition,  a 
portion  of  that  species  of  elasticity  which  belongs  to  fluids. 

The  investigation  is  simplified  by  considering,  in  the  first  place,  the 
elasticity  of  a  solid  body  as  arising  from  the  mutual  action  of  centres  of 
force  only,  and  afterwards  adding  the  proper  portion  of  fluid  elasticity. 

It  is  known  that  solid  bodies  are  capable  of  preserving  bulk  and  figure, 
although  their  surfaces  are  acted  upon  by  no  sensible  pressure,  normal  or 
tangential.  We  may  take  the  positions  of  the  particles  in  this  condition 
as  points  from  which  to  measure  their  displacements.  Thus  we  cause  the 
coefficients  of  all  the  terms  of  the  first  order  in  equation  (22)  to  vanish. 

To  investigate  the  properties  of  the  coefficients  of  elasticity,  the  function 
SV  is  to  be  expressed  in  a  new  form, — viz.,  as  the  sum  of  the  virtual 
moments  of  the  actions  exerted  upon  each  of  the  centres  of  force  in  the 


LAWS   OF   ELASTICITY.  10D 

particle  under  consideration,  by  the  centres  of  force  in  all  the  other 
particles.  Mr.  Haughton,  in  his  first  memoir,  having  performed  this 
process,  shows  by  means  of  its  results,  that  in  a  body  composed  entirely 
of  centres  of  force  acting  along  the  lines  joining  them,  the  number  of 
independent  coefficients  of  elasticity  for  any  system  of  orthogonal  axes  is 
reduced  to  fifteen,  which,  by  properly  selecting  those  axes,  may  be  reduced, 
for  bodies  in  general,  to  twelve,  and  for  those  having  axes  of  elasticity, 
to  six. 

I  shall  now  endeavour  to  prove  by  the  method  of  virtual  velocities, 
what  I  have  in  the  third  section  proved  by  other  modes  of  reasoning, 
that  in  a  homogeneous  body  constituted  of  centres  of  force  only,  the 
independent  coefficients  of  elasticity  are  reducible  to  three,  of  which,  and 
of  the  position  of  the  axes,  the  twenty-one  in  equation  (22)  are  functions. 

A  fourth  independent  coefficient  is  to  be  added  in  solids  possessing  a 
portion  of  fluid  elasticity;  that  is  to  say,  in  all  known  solids. 

34.  It  is  known  that  a  homogeneous  solid  can  exist,  with  its  particles 
in  an  unstrained  condition,  bounded  by  plane  surfaces  in  any  direction. 
In  this  condition,  therefore,  the  total  molecular  action  upon  a  particle 
situated  at  any  bounding  plane  must  be  null.  Conceive  the  bounding 
plane  still  to  pass  through  the  same  particle,  but  to  have  its  position 
shifted  through  any  angle.  The  molecular  action  on  the  particle  will  still 
be  null.  Now  the  effect  of  the  shifting  of  the  bounding  plane  is  to  take 
away  a  wedge  of  matter  from  one  side  of  the  particle,  and  to  substitute  an 
equal  and  similarly  constituted  wedge,  lying  in  a  diametrically  opposite 
direction.  Hence,  in  the  unstrained  condition  of  a  solid  body,  the  action 
exercised  upon  any  particle,  by  a  wedge  of  matter  bounded  by  any  two 
planes  passing  through  the  particle,  is  null. 

This  shows  that  the  action  of  a  wedge  of  solid  matter  on  a  particle 
situated  at  its  edge,  is  not  altered  by  varying  the  angular  position  of  the 
wedge;  and  consequently,  that  the  molecular  actions  which  produce 
elasticity  are  not  directly  functions  of  the  relative  angular  positions  of 
the  centres  of  force  which  act  on  each  other,  but  merely  of  their  distances 
apart,  so  that  if  the  actions  of  the  several  equal  wedges  into  which  a  body 
may  be  conceived  to  be  divided,  round  a  given  particle,  are  different,  this 
does  not  arise  directly  from  the  angular  positions  of  the  wedges,  but  from 
the  different  distribution  of  their  centres  of  force  as  to  distance  from  those 
of  the  particle  operated  upon. 

(I  have  proved  this,  in  a  manner  slightly  different  in  form,  in  a 
supplementary  paper  to  Sect.  III.,  Art.  17.)     See  p.  98. 

This  further  shows,  that  the  mutual  action  of  two  centres  of  force  in  a 
solid  must  be  directed  along  the  line  joining  them;  for  otherwise  it  would 
tend  to  bring  that  line  into  some  definite  angular  position,  and  would  be 
a  function  of  the  direction  of  the  line. 


HO  LAWS   OF    ELASTICITY. 

It  finally  results,  from  what  has  been  stated,  that  the  action  of  an 
indefinitely  slender  pyramid  of  a  solid  body  upon  a  particle  at  its  apex 
must  be  a  direct  attraction  or  repulsion  along  the  axis  of  the  pyramid, 
which  is  a  function  of  the  several  distances  of  the  centres  of  force  in  the 
pyramid  from  those  in  the  particle  at  the  apex,  and  which,  in  the 
unstrained  condition  of  the  body,  must  be  null. 

The  principles  stated  above  have  to  a  greater  or  less  extent  been  taken 
for  granted  in  previous  investigations,  but  have  not  hitherto  been  demon- 
strated. They  may  all  be  regarded  as  the  necessary  consequences  of  the 
following : — 

Def.  Let  the  term  Elastic  Solid  be  used  to  denote  the  condition  of  a  body, 
which,  when  acted  upon  by  any  given  system  of  pressures,  or  by  none,  and  at  a 
given  temperature,  tends  to  preserve,  and  if  disturbed,  to  recover,  a  definite  bulk 
and  figure;  and  such  that,  if  lohile  in  an  unstrained  condition  it  be  cut  into 
parts  of  any  figure,  those  parts,  when  separate,  will  tend  to  preserve  the  same 
bulk  and  figure  as  they  did  when  they  formed  one  body. 

Experience  informs  us  that  bodies  sensibly  agreeing  with  this  definition 
exist;  its  consequences  are,  therefore,  applicable  to  them  in  practice. 

35.  Let  r  denote  the  distance  apart  of  two  centres  of  force  in  an 
unstrained  solid,  and  let  <pr  be  proportional  to  their  mutual  action. 
Then 

dxdydzd2 a  .  2i<pr  —  0 

may  be  taken  to  represent  the  total  action  of  an  indefinitely  slender 
pyramid  which  subtends  the  element  of  angular  space  d2  <o  upon  a  particle 
at  its  apex  d  x  d  y  d  z. 

In  consequence  of  a  strain,  let  each  of  the  distances  r  become 

(1  +*)r, 

£  being  a  very  small  fraction.  Then  the  total  action  of  the  pyramid 
becomes 

d  x  d  y  d  z  d2  o> .  2  (<p  r  +  tr<p'r)  =  dx  dy  dzd2  w .  e  2  r  <p'  r; 

for  by  the  third  postulate,  e  is  uniform  throughout  the  extent  of  appreci- 
able molecular  action. 

The  quantity  which  the  force  acting  between  two  centres  of  force  tends 
to  vary,  is  their  relative  displacement  along  the  line  joining  them,  or  e  r. 
Hence  the  sum  of  the  virtual  moments  of  the  actions  of  all  the  slender 
pyramids  into  which  the  solid  is  conceived  to  be  divided,  that  is  to 
say,  the  total  virtual  moment  of  its  molecular  action  upon  the  particle 
dx  dy  dz,  is 

SV dxdy  dz  =  dx  dy  dzffeSt.S  (r2  <f,' r)~.d2  u, 


LAWS    OF   ELASTICITY. 


Ill 


the  double  integration  extending  to  all  angular  space.     Consequently,  we 
obtain  as  a  new  value  of  the  function  V, 

V  =  ff\  £2  2  (r2  <f>'  r)  .cPw.  .         .     (28.) 

Let  a,  b,  c  be  the  direction-cosines  of  the  axis  of  a  given  slender  pyramid. 
Then  it  is  easily  seen  that  the  strain  e  along  that  axis  has  the  following  value, 
in  terms  of  the  six  strains  as  referred  to  the  axes  of  co-ordinates  : 

e  =  a  a2  +  /3  b2  +  7  c2  +  X  b  c  +  fx  c  a  +  v  a  b, 
and,  consequently,  that 


9  9  /")f> 

£         o      4.    ,    P    Tj    , 


C4  +  y%2C2+    £*rf  +  .£rfP 


+  /3  7  62  c2  +  7  a  c2  ft2  +  a  j3  ft2  &2 

+  /tv(i2J(;+  v\«i2c-|-  \  jul  a  be2 

+  a  \  or  be  +  a  fj.  as  c  +  av  a3b 

+  fi\b3c    +  /3m«^2c+  /3va&3 

+  7  X  k3    +  7  /a  a  c3  +  7  v  a  5  c2. 

If  this  value  of  1 12  be  substituted  in  equation  (28),  and  the  result 
compared  with  equation  (22),  it  is  at  once  obvious  that  the  twenty-one  co- 
efficients of  elasticity  have  the  following  values  (putting  S  (r2  <p'  r)  =  —  R, 
which  is  negative  that  equilibrium  may  be  stable) : 

(a2)=//VRd2W 
(/32)=//S*E^<u 

(72)=yyVRrZ2a, 

(\2)^(/37)  =//-52c2Rrf2(u 

(/M2)  =  (7  a)  =ffc2a2TLd2(o 

(„*)=  (aj3)  =  //a2b2Ud2(o 

(fjiv)  =  (oX)  =ffa2bc~Rd2u> 

(vA)  =  (pV)  =//aS2CRfZ2w 
(X/*)=  (tv)  =  ffabciB,d2<a 
(0A)  =ffVicnd*to 
(yX)  =//ic3EfZ2fc> 

iir).= //***&» 

(a  fx)  =  f/c  a3  ~Rd2  to 
(av)  =ffasb-Rd2to 
C/3v)  =  //*a&3Rtf2w 


112  LAWS    OF   ELASTICITY. 

In  the  above  equations,  which  agree  with  those  given  by  Mr.  Haughton, 
the  number  of  independent  coefficients  is  fifteen. 

36.  Their  reduction  to  a  smaller  number  arises  from  the  nature  of  the 
function 

E  =  -  2  (r2  r/>'  r). 

This  quantity  is  a  function  of  the  distances  of  the  centres  of  force  in  a 
given  indefinitely  slender  pyramid  from  those  in  a  particle  at  its  apex,  and 
can  vary  with  the  direction-cosines  a,  b,  c  of  the  axis  of  the  pyramid,  solely 
because  those  distances  vary  with  them.  Now  in  a  homogeneous  solid, 
that  is,  one  composed  of  a  succession  of  similar  and  regularly  placed 
groups  of  centres  of  force,  those  distances  depend  upon  a  quantity  which 
may  be  called  the  mean  interval  between  the  centres  of  force  jn  a  given 
direction :  a  quantity  of  such  a  nature  that  the  product  of  its  three  values 
for  any  three  orthogonal  directions  is  a  constant  quantity;  being  the  space 
occupied  by  a  centre  of  force,  or  by  a  definite  group  of  such  centres.  To 
have  this  property,  the  mean  interval  must  be  a  quantity  of  this  form: 

•  __  ,f+ga"  +  hh'2  +  Jcc-  +  lbc+mca  +  nab.  (30) 

that  is  to  say,  its  logarithm  must  be  proportional  to  the  reciprocal  of  the 
square  of  the  radius  of  an  ellipsoid,  whose  axes  are  those  of  molecular 
arrangement,  and  therefore  of  molecular  action,  and  of  elasticity. 

Let  the  axes  of  this  ellipsoid  be  taken  as  axes  of  co-ordinates.  Then 
I  =  0,  m  =  0,  n  =  0;  and  the  above  equation  is  reduced  to 

i  =  cf+gat  +  hv  +  i-c^       m         t         m     (3QA.) 

and  because  the  quantity 

R  =  F  (0  =^(f+rj  a?  +  h  &  +  k  c2),  .     (31.) 

does  not  change  its  sign  or  value  by  any  change  of  the  signs  of  the  cosines 
a,  I),  c,  it  follows  that  all  the  coefficients  in  (29)  containing  odd  powers  of 
those  cosines,  that  is  to  say,  all  except  the  first  six,  disappear  when  the 
axes  of  molecular  arrangement  are  taken  for  axes  of  co-ordinates. 

These  six,  for  all  known  homogeneous  substances,  are  reducible  to  three, 
by  the  following  reasoning : 

Let  us  assume  as  a  Fifth  Postulate,  what  experience  shows  to  be  sensibly 
true  of  all  known  homogeneous  substances — viz.,  that  their  elasticity  varies 
very  little  in  different  directions.  Those  substances,  such  as  timber,  whose 
elasticity  in  different  directions  varies  much,  are  not  homogeneous,  but 
composed  of  fibres,  layers,  and  tubes  of  different  substances. 

If  this  be  assumed,  it  follows  that,  in  the  expression  (31),  for  the 
quantity  E,  the  variable  terms 

g  a2  +  h  62  +  k  r, 


LAWS   OF   ELASTICITY. 


113 


are  very  small  compared  with  the  constant  term  /,  and  that  E  may  be 
developed  in  the  form 

R  =  f  (/)+  f.  (/).  (ga*  +  h&  +  Ice")  +  &c. 

If  this  value  of  R  be  introduced  into  equation  (29),  and  if  small  quantities 
of  the  second  order  be  neglected,  it  is  easily  seen,  on  performing  the  inte- 
grations, that  the  following  relations  exist  amongst  the  six  coefficients 
already  specified: 

(/32)  +  (72)  =  6(\2)  =  6(/37) 

(72)  +  («2)  =  6  (M2)  =  8  (r  «) 

(a2)+(/32)=6(v2)  =  6(«/3) 

or,  by  transformation,  S~   .     (32.) 

(a2)  =  3  {(&  +  (v2)  -  (X2)} 
(/32)  =  3  {(v2)  +  (X2)  -  0.2)} 

(r)  =  3  {(x^)  +  (tf)  -  (v2)} 

These  equations  reduce  the  number  of  independent  coefficients  of  elas- 
ticity, arising  from  the  actions  of  centres  of  force,  to  three.  They  are 
identical  with  the  equations  (7)  and  (8),  embodied  in  the  fifth  theorem  in 
Sect.  III.,  although  arrived  at  by  a  different  process. 

37.  Let  us  suppose  the  solid  under  consideration  to  possess  a  portion 
of  fluid  elasticity,  represented  by  the  coefficient  J.  Then  the  coefficients 
of  elasticity  have  evidently  the  following  relations  : 

(a2)  =  3  (Ox2)  +  („2)  -  (X2)}  +  J  1 

(/32)=3{(v2)+(X2)-(M2)}  +  J 

(72)  =  3{(X2)+at2)-(v2)}  +  J 
(/3y)  =  (A2)+J 
(7«)=(,x2)  +  J 
(a/3)  =  (v2)  +  J 

which  are  identical  with  the  six  equations  (9)  comprehended  under  the 
sixth  theorem,  in  Sect.  IV. 

38.  The  laws  of  elasticity  stated  in  this  paper  are  the  necessary  con- 
sequences of  the  definitions  of  elasticity  and  of  fluid  and  solid  bodies, 
given  in  Arts.  28,  32,  and  34,  respectively,  when  taken  in  conjunction 
with  five  postulates  or  assumptions,  which,  however,  may  be  summed  up 
in  two — viz. 

H 


y 


(33.) 


114  LAWS   OF    ELASTICITY. 

First,  That  the  variations  of  molecular  force  concerned  in  producing 
elasticity  are  sufficiently  small  to  be  represented  by  functions  of  the  first 
order  of  the  quantities  on  which  they  depend ;  and, 

Secondly,  That  the  integral  calculus  and  the  calculus  of  variations  are 
applicable  to  the  theory  of  molecular  action.  It  is  thus  apparent  that 
the  science  of  elasticity  is,  to  a  great  extent,  one  of  deduction  a  priori. 

The  functions  of  perceptive  experience  in  connection  with  it  are  two- 
fold :  first,  by  observation,  to  inform  us  of  the  existence  of  substances, 
agreeing  to  a  greater  or  less  degree  of  approximation  with  the  definitions 
and  postulates;  and,  secondly,  by  experiment,  to  ascertain  the  numerical 
value  of  the  coefficients  of  elasticity  of  each  substance. 


Note  to  Sections  VI.  and  VII.  of  Preceding  Paper. 

On  the  Transformation  of  Coefficients  of  Elasticity,  hy  the  aid  of  a  Surface 
of  the  Fourth  Order. 

(The  following  note  contains  no  original  principle,  and  is  designed 
merely  to  put  on  record,  for  the  sake  of  convenient  reference,  a  series  of 
equations  which  will  be  found  useful  in  future  investigations.) 

It  has  been  pointed  out  by  Mr.  Haughton,  in  his  first  paper,  that  if 
we  take  into  consideration  that  part  only  of  the  elasticity  of  a  solid 
which  arises  from  the  mutual  actions  of  centres  of  force,  so  that  the 
function  V  shall  contain  at  most  but  fifteen  unequal  coefficients  (viz., 
those  whose  values  are  given  in  equation  29),  and  if,  with  those  fifteen 
coefficients,  we  construct  a  surface  of  the  fourth  order,  whose  equation  is 
the  following, 

U=  («2K  +  (/32)^+(72)^ 

+  6  (X2)  f$  +  6  Ou2)  z^  +  G  (v2)  x}if 

+  12  («\)  x^yz  +  12  (j3M)  xfz  +  12  (yv)  xyz* 

+  4  (a  i<)  :c"  y  +  4  (a  n)  xz  z 

+  i(fi\)fz+  4:(i3v)y*z 

+  4:(yn)£x+4:(y\)z*y 

=  l  ■         •         •         •■ (A.) 

then  will  U  be  the  same  function  of  the  six  quantities 

a?,     y2,    z\     2yz,     2zx,     2xy, 


LAWS   OF    ELASTICITY. 


115 


that  —  2  V  is  of  the  six  strains 

«>     /3,     7,     A,     ix,     v, 

which  are  known  to  be  transformed  by  the  same  equations  "with  the 
above  functions  of  the  second  order  of  x,  y,  g;  and  consequently,  the 
same  equations  which  serve  to  transform  the  coefficients  of  the  surface 
U  =  1,  into  those  suitable  for  a  new  set  of  rectangular  co-ordinates,  will 
also  serve  to  transform  the  coefficients  of  elasticity  in  the  function  V. 
Now,  it  is  obvious  that  if  the  equation 

^  (»,  y,  z)  =  xp  (x,  y,  z) 

be  true  for  two  sets  of  rectangular  co-ordinates  having  the  same  origin, 
then  must  the  equation 


<1> 


f  d      d      d  \ 
\dx'  dv'  Tz) 


i 


?) 


(B.) 


d         d         d 
dx'  dy'  dz  J  ~~  r  \  d  aj"   d  y"   d 
be  true  also. 

It  follows  that  the  fifteen  coefficients  of  elasticity  (a2),  &c,  which  are 
proportional  to  the  differential  coefficients  of  U  of  the  fourth  order  with 
respect  to  x,  y,  z,  are  transformable  hj  means  of  the  same  equations  which 
serve  to  transform  the  fifteen  algebraical  functions  of  the  fourth  order  of 
x,  y,  z,  by  which  they  are  respectively  multiplied  in  the  value  of  U. 

The  following  is  the  investigation  of  those  fifteen  equations  of  the 
fourth  order,  as  well  as  of  the  six  equations  of  the  second  order,  from 
which  they  are  formed  by  multiplication. 

Let  the  relative  direction-cosines  of  the  two  sets  of  rectangular  axes  be 
expressed  as  follows: 


1 


Original 

Axes. 

N 

ew  Axes. 

V 

z 

X 

ai 

h 

cl 

V 

tt.y 

h 

c2 

z 

az 

h 

c'3 

\-  cosines. 
J 


Let  the  following  notation  be  used  for  functions  of  those  cosines.     (It 
is  the  same  which  is  employed  by  Mr.  Haughton.) 


q1  =  cla1; 


p2  —  b2  c2 ; 
q2  =  c2a2; 
r„  —  a„  k2; 


%  —  H  az> 
r*  =  a,  \\ 


116  LAWS    OF   ELASTICITY. 

l1  =  b2c3  +  b3  c2;  l2  =  b3  cx  +  \  c3;  ls  —  \  c2  +  \  cx; 

m1  —  c2 a3  +  cz  a2;         rn2  =  czax  +  cx  az;         m3  =  cxa2  +  c2 ax; 
nx  =  a2b3  +  a3  \;         n2  =  a3bx  +  ax  b3;         nz  =  axb2  +  a2bx; 

then  the  following  are  the  six  equations  of  transformation  of  the  second 
order  for  the  surface  U  =  1, 

x2  =  x2  a2  +  y2  a22  +  z2  az2  +  2y  z  a2  az  +  2  z  x  azax  +  2  x  y  ax  a2, 

(for  y'2,  z'2,  similar  equations  in  b,  c,  respectively). 

y'z'  =  x2px  +  fv2  +  z2p3  +  yzlx  +  zxl2  +  xyl3, 

(for  z  x,  a  similar  equation  in  q  and  m), 

(for  x  y,  a  similar  equation  in  r  and  n),  .         .         .     (C.) 

Those  equations  are  made  applicable  to  the  transformation  of  strains  by 
the  following  substitutions : 

for  x2,  y2,  z2,   2yz,   2zx,  2xy, 

substitute    a,  (5,   j,      A,        p,        v; 

and  to  that  of  pressures,  by  the  following : 

for  o:2,    f,     z2,    yz,  zx,  xy, 

substitute  P1?  P2,   P3,    Qv   Q2,    Q3, 

and  similar  substitutions  for  the  accented  symbols. 

The  following  are  the  fifteen  equations  of  transformation  of  the  fourth 
order: 

a/4  =  re*  ax*  +  y*  a2*  +  z*  a3*  +  0>y2z2  «22  a32  +  Qz2  x2  a2  a2  +  6  x2  y2  ax2  a* 

+  12  x2 y  z ax2 a2 a3  +  12  x y2 z ax a22 a3  +  12  xy  z2 ax  a2 az2 

+  4  x3  y  ax3  a2  +  4  x3  z  ax3  a3  +  4  y3  z  a23  az  +  Ay3x  a23  ax  +  4  c3  x  az3  al 

+  4  s3  y  az3  a2; 

(for  yri,  zri,  similar  equations  in  b,  c,  respectively). 

y'2z'2  =  a*p*  +  y'P-f  +  s>32  +  y2#  (h2  +  2p2pz)  +  z2x2{l2  +  2psPx) 

+  *2y2(h2  +  2PiPi)  +  2«?yz{lhh  +  hh)  +  %xy2z{2hk  +  hld 

+  2  xyz2  (pz  l3  +  lx  l2)  +  2x3ypxlz  +  2  x3zpx  L 

+  2y3zp2lx  +  2y3xp2l3  +  2z3xpzl2  +  2z3ypz(x; 

(for  z'2  x'2,  a  similar  equation  in  q  and  m)  ; 

(for  x'2  y'2,  a  similar  equation  in  r  and  n) ; 


LAWS   OF   ELASTICITY.  117 

x1  y  z'  =  x*  a2  px  +  y*  a2  p2  +  s4  a32  p3 

+  z2  x2  {a32Pi  +  a\Vz  +  2  a3  ax  ?2) 
+  x2  y2  {a2p2  +  a2px  +  2ata2  l3) 
+  x2  y  z  (ax2  lx  +  2  a2  a3  pt  +  2  as  ax  l3  +  2  ax  a.2  I.) 
+  xy2z  (a.?  1.2  +  2a3  ax  p2  +  2  ax  a.2  lL  +  2  a2  a3  l3) 
+  xy  z2  (a32  l3  +  2  ax  a.2 p3  +  2  a2  a3l2  +  2  a3  ax  lx) 
+  x3y(al2l3  4-  2a1a2pl)  +  x3z(a12l2  +  2a1a3p1) 
4-  2/3  z  (a.22  lx  +  2  a2  a3p2)  +  y3  x  (a.22  l3  +  2  a2  ax  a.2) 
+  z3  x  (a2  l2  +  2a3  ax  p3)  +  &  y  (a.2  lx  +  2  a3  a.2  ps) ; 

(for  x  y'2  z',  a  similar  equation  in  b,  q,  m); 

(for  x  y'  z'2,  a  similar  equation  in  c,  r,  n) ; 
Ay'  -  x^a^rx  +  yia22r2  +  zia32r3 
+  3  y2  z2  a2  a3  ?ix 
+  3  z2  x2  a3  ax  n0 
+  3  x2  y2  ax  a2  n3 
+  3  x2  y  z  (ax2  nx  +  2  a.2  a3  rx) 
+  3xy2z (a2 n2  +  2  a3 ax r2) 
+  3  x  y  z2  (a32  n3  +  2  ata2  r3) 

+  x3  y  (ax2  n3  +  2  ax  a.2  rx)  +  x3  z  (a-,2  n2  +  2  a3  ax  ry) 
+  y3  z  (a22  nx  +  2a2  a3  r2)  +  y3  x  (cc22  n3  +  2ax  a2  r2) 
+  z3  x  (a32  n2  +  2  a3ax  r3)  +  z3y  («32  nx  +  2a2a3  r3) ; 

(for  x'3  z',  a  similar  equation  in  a,  q,  and  m) ; 

(for  y'3  z',  a  similar  equation  in  b,  p,  and  I) ; 

(for  y'3  x',  a  similar  equation  in  b,  r,  and  n); 

(for  z'3  x',  a  similar  equation  in  c,  q,  and  m) ; 

(for  z'3  y',  a  similar  equation  in  c,  p,  and  I);         .       (D.) 

The  above  equations  are  made  applicable  to  the  transformation  of  the 
■coefficients  of  elasticity  arising  from  the  mutual  actions  of  centres  of  force 
only,  by  the  following  substitutions: 

for   %A,      f,       z4,  y2z2,  z2x\  x2y2, 

substitute  (a2),    (/32),    (72),    (j37)  =  (X2),    (y  a)  =  (ft),   (a/3)  =  (v2); 

for  x2yz,  xy2z,  xyz2, 

substitute  (a  X)  =  (//  v),    (/3  jui.)  =  (v  X),    (7  v)  =  (\fi); 


118  LAWS   OF    ELASTICITY. 

for  a3  y,      x3  z,      y3  z,      f  x,      z3  x,      z3  y, 
substitute  (a  v),    (aim),    (/3  A),    (/3  v),    (y  ft),    (7  A); 

and  similar  substitutions  for  the  accented  symbols. 

Should  the  substance  under  consideration  be  endowed  with  a  portion 
of  fluid  elasticity  in  addition  to  that  which  arises  from  the  mutual  action 
of  centres  of  force,  the  coefficient  of  that  fluid  elasticity  J  must  be  sub- 
tracted from  the  coefficients  into  which  it  enters,  viz. — 

(a-),  (/32),  (r),  (/3  7)  =  (X2)  +  J>  (7  «)  =  0"e)  +  J>   (°/3)  =  O2)  +  J> 

before  effecting  the  transformation. 

The  results  of  the  transformation  for  those  six  coefficients,  being  in- 
creased by  the  same,  quantity  J  which  was  previously  subtracted,  will  give 
their  entire  values  for  the  new  axes. 

If  the  original  axes  of  co-ordinates  are  those  of  elasticity,  each  of  the 
fifteen  equations  of  transformation  is  reduced  to  its  first  six  terms,  in 
which  the  following  substitutions  are  to  be  made  for  the  unaccented 
symbols : 

for      x*,  y4,  z4,  fz\  z2  x2,  x2  fy 

substitute  Aj-  J,   A2-J,   A3-J,   Bj-J^C^   B2-J=C2,   B3-J  =  C3. 


AXES   OF  ELASTICITY  AND   CRYSTALLINE   FORMS.  119 


VI.—ON  AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS.* 


Section  1. — General  Definition  of  Axes  of  Elasticity. 

As  originally  understood,  the  term  "  axes  of  elasticity "  was  applied  to 
the  intersections  of  three  orthogonal  planes  at  a  given  point  of  an  elastic 
medium,  with  respect  to  each  of  which  planes  the  molecular  actions  causing 
elasticity  were  conceived  to  be  symmetrical. 

If  the  elasticity  of  solids  arose  either  wholly  from  the  mutual  attractions 
and  repulsions  of  centres  of  force,  such  attractions  and  repulsions  being 
functions  of  the  mutual  distances  of  those  centres,  or  partly  from  such 
mutual  actions  and  partly  from  an  elasticity  like  that  of  a  fluid,  resisting 
change  of  volume  only,  it  is  easy  to  prove  that  there  would  be  three  such 
orthogonal  planes  of  symmetry  of  molecular  action  in  every  homogeneous 
solid. 

But  there  is  now  no  doubt  that  the  elastic  forces  in  solid  bodies  are  not 
such  as  can  be  analysed  into  fluid  elasticity  and  mutual  attractions  between 
centres  simply;  and  though  there  are,  as  will  presently  be  shown, 
orthogonal  planes  of  symmetry  for  certain  kinds  of  elastic  forces,  those 
planes  are  not  necessarily  the  same  for  all  kinds  of  elastic  forces  in  a  given 
solid. 

The  term  "  axes  of  elasticity,"  therefore,  may  now  be  taken  in  a  more 
extended  sense,  to  signify  all  directions  with  respect  to  which  certain  hinds  of 
clastic  forces  are  symmetrical ;  or  speaking  algebraically,  directions  for  which 
certain  functions  of  the  coefficients  of  elasticity  are  null  or  infinite. 

The  theory  of  axes  and  coefficients  of  elasticity  is  specially  connected 
with  that  branch  of  the  calculus  of  forms  which  relates  to  linear  trans- 
formations, and  which  has  recently  been  so  greatly  advanced  by 
the  researches  of  Mr.  Sylvester,  Mr.  Cayley,  and  Mr.  Boole.  In  such 
applications  of  that  calculus  as  occur  in  this  paper,  the  nomenclature  of 
Mr.    Sylvester   is   followed ;  f    and   by   the    adoption    of  the   "  Umbral 

*  Read  before  the  Royal  Society  of  London,  on  June  21,  1855. 
t  See  Cambridge  and  Dublin  Mathematical  Journal,  Vol.  VII.  ;  and  Philosophical 
Transactions,  1853. 


120  AXES   OF  ELASTICITY  AND   CRYSTALLINE  FORMS. 

Notation"  of  that  author  immense  advantages  are  gained  in  conciseness 
and  simplicity.  * 

Section  2. — Strains,  Stresses,  Potential  Energy,  and  Coefficients 

of  Elasticity. 

In  this  paper,  the  word  "  Strain "  will  be  used  to  denote  the  change 
of  volume  and  figure  constituting  the  deviation  of  a  molecule  of  a  solid 
from  that  condition  which  it  preserves  when  free  from  the  action  of 
external  forces ;  and  the  word  "  Stress  "  will  be  used  to  denote  the  force, 
or  combination  of  forces,  which  such  a  molecule  exerts  in  tending  to 
recover  its  free  condition,  and  which,  for  a  state  of  equilibrium,  is  equal 
and  opposite  to  the  combination  of  external  forces  applied  to  it. 

In  framing  a  nomenclature  for  quantities  connected  with  the  theory 
of  elasticity,  OXtyt-Q  is  adopted  to  denote  strain,  and  raaig  to  denote  stress. 

It  is  well  known  that  the  condition  of  strain  at  any  given  point  in  the 
interior  of  a  molecule  may  be  completely  expressed  by  means  of  the 
following  six  elementary  strains,  in  which  £,  )j,  Z,  are  the  components  of  the 
molecular  displacement  parallel  to  three  rectangular  axes,  x,  y,  :. 

PI        ,.  dl£  dr\       r.     dZ, 

Elongations,    .     —  =  a;   -j-=  ft;    -^  =  y; 


dj  = 

dx 

a; 

dr\ 

dy~ 

ft; 

dZ> 
dz 

dy^ 

dz 

=  A; 

dl 

dz 

+  T 

dx 

Distortions,     .     ~  +  j^  =  A;   "p    -  ^  =  ft;  -p  +  —  =  v. 

dy      dz  dz      dx  dx      ay 

It  is  also  well  known  that  the  condition  of  stress  at  a  given  point  may 
be  completely  expressed,  relatively  to  the  three  rectangular  co-ordinate 
planes,  by  means  of  six  elementary  stresses,  viz. — 

Normal  Pressures,  .         .  P1?     P2,     P3, 

Tangential  Pressures,       .         .     Qlt     Q2,     Q3; 

these  quantities  being  estimated  in  units  of  force  per  unit  of  surface. 

Let  each  elementary  stress  be  integrated  with  respect  to  the  elementary 
strain  which  it  tends  directly  to  diminish,  from  the  actual  amount  of  that 
strain,  to  the  condition  of  freedom;  the  sum  of  the  integrals  is  the 
potential  energy  of  elasticity  of  the  molecule  dxdydz,  expressed  in  units 
of  work  per  unit  of  volume ;  viz. — 

+  j>Q1d\  +  jq.2d^+  j°Q3dv.      .         .     (1.) 

*  See  the  Note  at  the  end  of  the  paper. 


AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS.  121 

The  condition  that  the  function  U  shall  have  the  same  value,  in  what 
order  soever  the  variations  of  the  different  elementary  strains  take  place, 
amounts  to  supposing  that  no  transformation  of  energy  of  the  kind  well 
distinguished  by  Professor  Thomson  as  frictlonal  or  irreversible,  takes  place 
during  such  variations ;  in  other  words,  that  the  substance  is  perfectly 
elastic. 

Each  of  the  elementary  stresses  being  sensibly  a  linear  function  of  the 
six  elementary  strains,  the  potential  energy  of  elasticity  is,  as  Mr.  Green 
first  showed,  a  function  of  those  strains  of  the  second  degree,  having 
twenty-one  constant  coefficients,  which  are  the  coefficients  of  elasticity  of 
the  body,  and  will  in  this  paper  be  called  the  Tasinomic  Coefficients; 
that  is  to  say,  adopting  Mr.  Green's  notation  for  such  coefficients, — 

'    U  =  („2)|2+  (/32)f  +  (72)|2+  (X2)^  +  (m2)|3  +  if)t 
+  (/37)/3y+(7«)7«  +  («£W3 

+   (fX  v)  /H  V  +   (v  X)  V  X  +   (XjUi)  XyU 

+  (a  \)a  X  +  (/3/x)  /3/x  +  (7  v)  y  v 
+  (/3X)  /3X  +  (yfx)  yn  +  (a  v)  a  v 
+  (y\)y\+(a[i)an+(Pv)Pv  .  .     (2.) 

From  a  theorem  of  Mr.  Sylvester  it  follows,  that  every  such  function 
as  U  is  reducible  by  linear  transformations  to  the  sum  of  six  positive 
squares,  each  multiplied  by  a  coefficient.  The  nature  and  meaning  of  this 
reduction  have  been  discussed  by  Professor  William  Thomson. 

The  following  classification  of  the  tasinomic  coefficients  will  be  used 
in  the  sequel : — 

Designation  of  Coefficients.  Elasticities.  Symbols. 

C  Euthytatic,      Direct  or  Longitudinal,       (a2)       (82)       (72) 

Orthotatic  -l    Platytatic,        Lateral,     .         .  (/3  7)     (7  a)     (a  /3) 

I  Goniotatic,      Rigidities,  .         .         .        (X2)       (;u2)       (v2) 

Plagiotatic,    .         .         .      Unsymmetrical, .         .       (jut  v),  &c,  &c. 

The  twenty-one  equations  of  transformation  by  which  the  values  of 
these  coefficients,  being  known  for  any  one  set  of  orthogonal  axes,  are 
found  for  any  other,  are  founded  on  the  following  principles. 

It  is  well  known,  that  for  rectangular  transformations,  the  operations 

d        d        d 
dz'    dy'    dz 


122  AXES   OF   ELASTICITY   AND   CRYSTALLINE   FORMS, 

are  respectively  covariant  with 

•'•,  y,  ■"-> 

from  which  it  is  easily  deduced,  that  because  the  displacements 
are  respectively  covariant  with 

&,  y,  z, 

therefore  the  elementary  strains, 

a,    /3,     y,     A,    fx,     v 

the  operations, 

d        d        d  d       0  d  d^ 

da'    dp'    d^'      IX'    ~d~ii'      dv 

and  the  strains 

P       P       P       ^  O       °  0       2  0 

■*•  1'  2'  3'      "     'I'      ~  v*-2>  »-o' 

must  be  respectively  covariant  with  the  squares  and  products, 
/-,    f,    #,    2yz,    2zx,    2xy. 


Section  3. — Thlipsimetric  and  Tasimetric  Surfaces  and 
Invariants. 

Isotropic  functions  of  the  elementary  strains  and  stresses,  which  may 
be  called  respectively  Thlipsimetric  and  Tasimetric  Invariants,  are  easily 
deduced  from  the  principle,  that  the  strains  may  be  represented  by  the 
coefficients  of  the  following  Thlipsimetric  Surface, 

as5  +  fitf  +  7^  +  \yz  +  ftzx  +  vxy=l,        .     (3.) 

and  the  stresses  by  the  coefficients  of  the  Tasimetric  Surface, 

T>l3?  +  J>2y*  +  Y3z*  +  2Cl1yz+2Q2zx+2%xy=l.       (4.) 

These  surfaces,  and  others  deduced  from  them,  have  been  fully  discussed 
by  M.  Cauchy  and  M.  Lame. 

The  invariants  in  question  may  all  be  deduced  from  the  following  pair 
of  contrasredient  matrices  : — 


AXES   OF   ELASTICITY  AND   CRYSTALLINE   FORMS. 

For  Stresses. 

Pi   Q3   Q-2 


123 


For  Strains. 

V         JUL 


(5.)  <! ;  fi .  | 


yU.  X 

9,    9  *y 


Q3    P2     Qj   ^  (5  a.) 


Q*   Qi   p3 


J 


The  following  are  the  primitive  thlipsimetric  invariants,  from  which  an 
indefinite  number  of  others  may  be  deduced  by  involution,  multiplication, 
addition,  and  subtraction  : — 


a  +  /3  +  7  =  91  (the  cubic  dilatation) ; 
/37  +  7«  +  a/3-i(A2  +  /x2  +  i/2)  =  02; 


(6.) 


The   potential   energy  U  is  what    Mr.    Sylvester   calls  a   "  Universal 
Mixed  Concomitant,"  its  value  being 

U  =  -  |  (Pl0  +  P2/3  +  P37  +  QXA  +  Q,fx  +  Q3v).      (7.) 


Section  4. — Tasinomic  Functions,  Surfaces,  and  Umbrae. 

If,  in  any  isotropic  function  of  the  co-ordinates  and  the  elementary 
strains,  there  be  substituted  for  each  square  or  product  of  elementary  strains 
that  tasinomic  coefficient  which  is  covariant  with  it,  the  result  will  be 
an  isotropic  function  of  the  co-ordinates  and  tasinomic  coefficients,  called 
a  Tasinomic  Function. 

The  following  Table  of  Covariants  is  readily  deduced  from  the  prin- 
ciples stated  at  the  end  of  Sect.  2 : — 


r  Squares  of  )     „ 
!o-    J      Strains,     }a'        P~>      ?'       A' 


Co 

variant  1  Tasinomic    "i 
Coefficients  J 


'-(«2)>     m,   ir),    4  (A2),    4(M2),    4(v2); 


Co- 
variant 


Products      ") 

J 

Tasinomic    ") 


y£y,     7a,     a/3,    Mv,         vA,         A//, 
(/3  y),  (7  a),  (a  /3),  4  Gu  v),  4  („  A),  4  (AM), 


MS-) 


.  Coefficients  j 

a  A,        a  /a,        a  v,        /3  A,       /3  /*,       /3  v,       7  A,       7  /z,       y  v, 

2(aA),2(a/x),2(av),2(/3A),2(/3A0;2(/3v),2(7A),2(7it<),2(7r).J 


124  AXES   OF  ELASTICITY  AND   CRYSTALLINE  FORMS. 

Each  tasinomic  function  being  equated  to  a  constant,  forms  the 
equation  of  a  Tasinomic  Surface;  and  on  the  geometrical  properties  of 
such  surfaces  depend  many  of  the  laws  of  coefficients  and  axes  of 
elasticity. 

A  convenient  and  expeditious  mode  of  forming  tasinomic  functions 
is  obtained  by  the  aid  of  an  Umbral  Notation  analogous  to  that  intro- 
duced by  Mr.  Sylvester  in  the  calculus  of  forms. 

Let  each  tasinomic  coefficient  be  regarded  as  compounded  of  two 
Tasinomic  Umbrae,  those  umbra?  being  expressed  by  the  following  notation  : 

(a),     ((3),     (y),     (X),     W,     (v); 

then  the  following  equation,  deduced  from  that  of  the  thlipsimetric 
surface  (3),  by  substituting  uinbraj  for  elementary  strains  according 
to  the  following  Table  of  Covariance, 

Strains,     ...      a,        /3,        y,  X,  f.t,  v, 

Umbrae,    .     .     .     (a),     (/?),     (7),     2  (X;,     2  fa),     2  („), 

is  the  equation  of  the  Tasinomic  Umbral  Ellipsoid,  from  which,  by  elimi- 
nation, multiplication,  involution,  addition,  subtraction,  and  differentia- 
tion, various  tasinomic  functions  may  be  deduced, 

(a)  a2  +  (/3)  f  +  (y)  ,:2  +  2  (X)  yz  +  2  fa)  zx  +  2  (v)  xy  =  (0)  =  1.    (8a.) 

Section  5. — Tasinomic  Invariants  and  Spheres. 

Tasinomic  invariants  are  constant  isotropic  functions  of  the  tasinomic 
coefficients,  which  are  deduced,  either  by  substitution  from  thlipsimetric 
invariants,  or  directly  from  the  Umbral  Matrix, 

(«)      W       fa)  \ 

(v)         08)        (X)   I  .     (9.) 

G»)      (X)      (y)  ) 

The  following  invariant  is  umbral  of  the  first  order : — 

(&-  +  i?  +  £)  •  W  =  W  +  W>  +  M  =  W>  (9«0 

Invariants  of  the  second  order  in  umbra;  are  real  quantities  of  the 
first  order,  viz. — 

(a2)     +  (/32)  +  (y2)    +  2  (/3  y)  +  2  (y  a)  +  2  (a  /3)  =  (6J2  (the  Ctt&ic  <««%) 

(/3y)+(ya)  +  (a/3)-    (X2)  -    (M2)  -    (*2)  =(02) 

(a2)     +  (/32)  +  (y2)  +2(X2)    +  2  fa2)  +2(v2)    =(01)2~2(02).      .      (10). 


AXES  OF  ELASTICITY  AND   CRYSTALLINE   FORMS.  125 

The  equation  of  a  Tasinomic  Sphere  is  formed  by  multiplying  a  tasinomic 
invariant  by 

a?  +  f  +  *\ 
or  any  power  of  that  quantity,  and  equating  the  result  to  a  constant. 


Section  6. — Of  Two  Tasinomic  Ellipsoids,  and  their  Axes, 
Orthotatic  and  Heterotatic. 

The  equations  of  two  ellipsoids  with  tasinomic  coefficients  are  derived 
from  that  of  the  umbral  ellipsoid  (8a.),  in  one  case  by  multiplying 
each  term  by  the  umbral  invariant  (0),  and  in  the  other  by  substituting 
for  each  umbra  in  the  function  (fa),  the  contravariant  component  of  the 
Inverse  to  the  umbral  matrix  (9).     The  results  are  as  follows : — 


Orthotatic  Ellipsoid. 

(ei)x(^  =  {(a2)  +  (a^)  +  (y«)}.r2+{(^)  +  (^)  +  (y«)}!/2 

+  {(y2)+(y«)+(£y)K 

+  2{(aX)  +  (/5X)  +  (yX)}^~+2{(aM)  +  (/3/>c)  +  (y/,)}.rC 

+  2{(«v)  +  (/3v)  +  (yv)}^=l.    .  .  (11). 

Heterotatic  Ellipsoid. 

{(^)-(A2)!r+{(ya)-W}!/2+[M)-(v2)}^ 
+  2{^v)-(a\yjyz+2{(v\)-^^}zx+2{(\^-(yv)}xy=l.       (12.) 

The  three   Orthotatic  Axes   are  three  rectangular  directions  for  which 
the  following  sums  of  plagiotatic  coefficients  are  null : — 


(aA)  +  (/3X)  +  (yX)  =  0;    (apt)  +  (/3  *)  +  (y  /»)  =  0; 

(a  v)  +  (/3  v)  +  (y  v)  =  0. 


}      (13.) 


It  was  proved  by  Mr.  Haughton,  in  a  paper  published  in  the  Trans- 
actions of  the  Royal  Irish  Academy,  Vol.  III.,  Part  2,  that  there  are  three 
rectangular  directions  having  this  property  in  a  solid  whose  elasticity 
arises  solely  from  the  mutual  actions  of  physical  points,  and  which  has 
but  fifteen  independent  coefficients  of  elasticity.  The  present  investi- 
gation shows  that  there  are  three  such  axes  at  each  point  of  every  solid, 


126  AXES  OF   ELASTICITY  AXD  CRYSTALLINE   FORMS. 

independently  of  call  hypothesis.     The   physical  meaning  of  this   result 
is  expressed  by  the  following 

Theorem  as  to  Orthotatic  Axes. 

At  each  point  of  an  clastic  solid  there  is  one  position  in  which  a  cubical 
molecule  may  be  cut  out,  such  that  a  uniform  dilatation  or  condensation  of 
that  molecule  by  equal  elongations  or  equal  compressions  of  its  three  dimensions, 
shall  produce  no  tangential  stress  on  the  faces  of  the  molecule. 

The  properties  of  the  Hetcrotatic  Axes  are  expressed  by  the  following 
equations : — 

(/*v)-(aX)  =  0;  (i/X)-(/3ju)  =  0;  (X^)-(yv)  =  0;       (14.) 

or  by  the  following 

Theorem  as  to  Heterotatic  Axes. 

At  each  point  of  an  elastic  solid  there  is  one  fiosition  in  which  a  cubical 
molecule  may  be  cut  out,  such  that  if  there  he  a  distortion  of  that  molecule 
round  x  (x  being  any  one  of  its  three  axes),  and  an  equal  distoiiion  round  y 
(y  being  either  of  its  other  two  axes),  the  normal  stress  on  the  faces  normal 
to  x  arising  from  the  distortion  round  x  shall  be  equal  to  the  tangential  stress 
round  z  arising  from  the  distortion  round  y. 

The  six  coefficients  of  the  heterotatic  ellipsoid  may  be  called  the 
Heterotatic  Differences.  For  a  solid  whose  elasticity  is  wholly  due  to  the 
mutual  attractions  and  repulsions  of  physical  points,  each  of  those 
differences  is  necessarily  null ;  therefore,  they  represent  a  part  of  the 
elasticity  which  is  necessarily  irreducible  to  such  attractions  and  repul- 
sions. There  is  reason  to  believe  that  part  at  least  of  the  elasticity  of 
every  substance  is  of  this  kind. 

If  this  part  of  the  elasticity  of  a  solid  be,  as  suggested  in  a  series  of 
papers  in  the  Cambridge  and  Dublin  Mathematical  Journal,  for  1851-52, 
a  species  of  fluid  elasticity,  resisting  change  of  volume  only,  the  solid  may 
be  said  to  be  heterotatically  isotropic.  The  equations  (14)  will  be  fulfilled 
for  all  directions  of  axes,  and  also  the  following  equations  : — 

(/3y)-(X2)  =  (ya)-(M2)  =  (a/8)-(v2)j     .  .      (15.) 

that  is  to  say,  the  excess  of  the  platytatic  above  the  goniotatic  coefficient 
will  be  the  same  in  every  plane. 

In  a  substance  orthotatically  isotropic,  the  equations  (13)  are  fulfilled 
for  all  directions,  and  also  the  following : — 

(a2)  +  (a/3)  +  (y  a)  =  (/32)  +  (/3y)  +  (a/3)  =  (y2)  +  (y  a)  +  (/3y),       (1 G.) 


AXES  OF   ELASTICITY  AND   CRYSTALLINE  FORMS.  127 

that  is  to  say,  a  uniform  compression  in  all  directions  produces  a  uniform 
normal  stress  in  all  directions,  and  no  tangential  stress. 

The  equations  (16)  may  be  reduced  to  the  following  form : — 

(a2)  -  (/3  y)  =  (/32)  -  (y  «)  =  (y2)  -  (a  /3).  .     (17.) 

In  a  substance  which  is  at  once  orthotatkaUy  and  hetcrotatically  isotropic, 
there  may  still  be  eleven  independent  quantities  amongst  the  tasinomic 
coefficients,  viz. — 


(18.) 


Three  Euthytatic  Coefficients,  (a2),  (/32),  (y2), 

The  Isotropic  excess,  .  .     (a2)  —  (/3  y), 

The  Isotropic  excess,  .          .     (/?  y)  —  (X2), 

Six  Plagiotatic  Coefficients,  (/3X),  (yX),  (y/x),  (a/ui),  (av),  (/3v),  j 

Such  a  substance  may,  therefore,  be  far  from  being  completely  isotropic 
with  respect  to  elasticity. 

Section  7. — Biquadratic  Tasinomic  Surface;  Homotatic 
Coefficients;  Euthytatic  Axes  Defined. 

If  the  equation  (8a.)  of  the  umbral  ellipsoid  be  squared,  there  is 
obtained  the  following  equation  of  a  Biquadratic  Tasinomic  Surface. 

(#=(«2K  +  (/32)^  +  (y2).:! 
+  2{(/3y)  +  2(X2)}^2+  2{(ya)  +  2(^2)},2^  +  2{(«/3)  +  2(v2)}^2 
+  4{2(/xv)  +  {a\))xhjz  +  4{2(„A)  +  (fti)}atf»  +  4{2(XM)  +  (yv)}x^ 
+  l(PX)fz  +  4(yX)//:3  +  4(7M)r,,:  +  4(a//)^3  +  i(av):>hj  +  4((3v)xf  =1    (19) 

The  fifteen  coefficients  of  this  surface  (which  will  be  called  the  Homo- 
tatic Coefficients)  are  covariant  respectively  with  the  fifteen  biquadratic 
powers  and  products  of  the  co-ordinates,  with  proper  numerical  factors. 

It  is  obvious,  that  when  the  fifteen  homotatic  coefficients,  and  the 
six  heterotatic  differences,  are  known  for  any  set  of  orthogonal  axes, 
the  twenty-one  tasinomic  coefficients  are  completely  determined. 

Mr.  Haughton,  in  the  paper  previously  referred  to,  discovered  the 
biquadratic  surface  for  a  solid  constituted  of  centres  of  force.  It  is  here 
shown  to  exist  for  all  solids,  independently  of  hypotheses. 

Those  diameters  of  the  biquadratic  surface  which  are  normal  to  that 
surface,  are  axes  of  maximum  and  minimum  direct  elasticity,  and  have  also 
this  property,  that  a  direct  elongation  along  one  of  them  produces,  on  a 
plane  perpendicular  to  it,  a  normal  stress,  and  no  tangential  stress;  so 
that  they  may  be  called  Euthytatic  Axes.     Though  such  axes  sometimes 


128  AXES  OF  ELASTICITY  AND   CRYSTALLINE  FORMS. 

form  orthogonal  systems,  their  complete  investigation  requires  the  use 
of  oblique  co-ordinates,  and  is  therefore  deferred  till  after  the  eighteenth 
section  of  this  paper,  which  relates  to  such  co-ordinates. 

Section  8. — Orthogonal  Axes  of  the  Biquadratic  Surface. 
Metatatic  Axes,  Orthogonal  and  Diagonal. 

By  rectangular  linear  transformations,  it  is  always  possible  to  make 
three  of  the  terms  Avith  odd  exponents,  or  three  functions  of  such  terms, 
vanish  from  the  equation  of  the  biquadratic  surface.  Thus  are  ascertained 
sets  of  orthogonal  axes  having  special  properties. 

To  exemplify  this,  let  the  rectangular  transformation  be  such  as  to  make 
the  following  functions  vanish  : — 

(G8X)-(yX)}(^-tf>;  {(yri-MK*-^«i  {(«v)-(Pv)}(?-f)xy. 

A  cubical  molecule  having  its  faces  normal  to  the  axes  fulfilling  this 
condition  has  the  following  property  : — if  there  be  a  linear  elongation  along 
y,  and  an  equal  linear  compression  along  z  (or  vice  versti),  no  tangential  stress 
ivill  result  round  x  on  planes  normal  to  y  and  z  •  and  similarly  of  other 
pairs  of  axes. 

This  set  of  axes  may  be  called  the  Orthogonal  or  Principal  Metatatic 
Axes,  and  their  planes,  Metatatic  Planes. 

Let  the  suffix  1  designate  co-ordinates  and  coefficients  referred  to  these 
axes.  Let  Oy,  0:  be  any  new  pair  of  orthogonal  axes  in  the  plane  yx zv 
Then  since  (|3  X)  —  (7  X)  is  co variant  with  (y2  —  f?)yz,  it  follows  that 

(/3  X)  -  (y  X)  =  {2  (0  y\  +  4  (X2)x  -  C/38),  -  (y\)  •  ^~~     (20.) 

(where  w  =  <  yx  0  y), 

a  quantity  which  is  =  0  for  all  values  of  to  which  are  multiples  of  45°* 
There  are  of  course  similar  equations  for  the  other  metatatic  planes. 
Hence  it  appears  that  in  each  of  the  three  Metatatic  Planes  there  is  a  pair  of 
Diagonal  Metatatic  Axes,  bisecting  the  right  angles  formed  by  the  Principal 
Metatatic  Axes. 

Each  pair  of  diagonal  axes  is  metatatic  for  that  plane  only  in  which  it 
is  situated. 

Thus  there  are  in  all  nine  metatatic  axes,  three  orthogonal  axes,  and 
three  pairs  of  diagonal  axes.  The  diagonal  axes  are  normal  to  the  faces 
of  a  regular  rhombic  dodecahedron. 

Let  Oy,  Oz  be  a  pair  of  rectangular  axes  in  any  plane  ivhatsoever ; 
Oy',  Oz'  any  other  pair  of  rectangular  axes  in  the  same  plane ;  and  let 

<yOy   =  u/ ; 


AXES   OF  ELASTICITY  AND  CRYSTALLINE   FORMS.  129 

then 

(/3  X)'  -  (y\y  =  [2  O  y)  +  4  (X2)  -  m  -  (y2)}  ^^ 

+  {(/3A)-(y\)}cos4w',  .         .     (21.) 

a  quantity  which  is  null  for  eight  values  of  &/,  differing  from  each  other 
by  multiples  of  45°.     Hence,  in  each  plane  in  an  elastic  solid,  there  is  a 
system  of  two  pairs  of  axes  meiatatic  for  that  plane,  and  forming  with  each  other 
eight  equal  angles  of  45°. 
In  equation  (21),  make 

to'   =    —    O) 

(/3\)'-(yX)'  =  (/3X)1-(yX)1  =  0; 
then  from  equations  (20)  and  (21),  it  is  easily  seen  that 
2  (By)  +  4(X2)-OT-(y2) 
=  {2  (£  7)i  +  ^  (A2)x  -  (/32)x  -  (y2X}  •  cos  4  «.   .     (22.) 

The  trigonometrical  factor  cos  4  w  is  +  1  for  all  values  of  to  which  are 
even  multiples  of  45°,  —  1  for  all  odd  multiples  of  45°,  and  =  0  for  all 
odd  multiples  of  22i°.  Hence,  in  every  plane  in  an  elastic  solid,  the 
quantity  (22),  which  may  be  called  the  Metastatic  Difference,  is  a  maximum 
for  one  of  the  two  pairs  of  metatatic  axes,  a  minimum  of  equal  amount  and 
negative  sign  for  the  other,  and  null  for  the  eight  intermediate  directions. 

Section  9.— Of  Metatatic  Isotropy. 

A  solid  is  Metatatically  Isotropic  when,  if  a  cubical  molecule,  cut  out  in 
any  position  whatsoever,  undergoes  simultaneously  an  elongation  along  one 
axis,  and  an  equal  and  opposite  linear  compression  along  another  axis,  no 
tangential  stress  will  result  on  the  faces  of  that  molecule. 

For  such  a  substance,  the  metatatic  differences  must  be  null  for  all  sets 
of  axes,  viz. : — 

2(/3y)  +  4(X2)-(/32)-(r)  =  C>n 

2(ya)  +  40u2)-(y2)-(a2)  =  O;    L  .      (23.) 

2(a/3)  +  4(„2)-(«s)-(/32):=0.   J 

In  a  paper  in  the  Cambridge  and  Dublin  Mathematical  Journal,  Vol.  VI., 
this  theorem  was  alleged  of  all  homogeneous  solids,  it  having  been,  in 
fact,  tacitly  taken  for  granted,  that  homogeneity  involves  metatatic 
isotropy,  as  above  denned. 

I 


130  AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS. 


Section  10. — Of  Orthotatic  Symmetry. 

If  it  be  taken  for  granted  that  symmetrical  action  with  respect  to  a 
certain  set  of  axes,  between  the  parts  of  a  body  under  one  kind  of  strain, 
involves  symmetrical  action  with  respect  to  the  same  axes  under  all  kinds 
of  strains,  then  one  and  the  same  set  of  orthogonal  axes  will  be  at  once 
orthotatic,  heterotatic,  metatatic,  and  euthytatic,  and  for  them  the  whole 
twelve  plagiotatic  coefficients  will  vanish  at  once,  and  the  independent 
tasinomic  coefficients  be  reduced  to  the  nine  orthotatic  coefficients 
enumerated  in  Sect.  2.  As  long  as  the  rigidity  of  solid  bodies  was 
ascribed  wholly  to  mutual  attractions  and  repulsions  between  centres  of 
force,  it  is  difficult  to  sec  how,  with  respect  to  homogeneous  substances, 
the  above  assumption  could  be  avoided.  It  is  probable  that  there  exist 
substances  for  which  it  is  true.  Such  substances  may  be  said  to  be 
Orthotatically  Symmetrical. 

Orthotatic  symmetry  requires  that  the  equation  (1 9)  of  the  biquadratic 
surface  should  be  reducible  by  rectangular  transformations  to  its  first  six 
terms,  and  that  the  axes  so  found  should  also  be  those  of  the  heterotatic 
ellipsoid.  The  conditions  which  must  be  fulfilled  in  order  that  a 
biquadratic  function  of  three  variables  may  be  reducible  by  rectangular 
transformations  to  its  first  six  terms,  have  been  investigated  by  Mr.  Boole.*' 


Section  11. — Of  Cybotatic  Symmetry. 

Let  a  substance  be  conceived  which  is  not  only  orthotatically  symme- 
trical, but  for  which  the  three  kinds  of  orthotatic  coefficients  are  equal  for 
the  three  orthotatic  axes,  viz. — 

(a2)  =  (/32)  =  (y2);    (/3y)  =  (y«)  =  (aft;    (X2)  =  (M2)  =  (v2).     (24.) 

Then,  for  such  a  substance  the  metatatic  difference  may  be  expressed  by 

2(/3y)  +  4(\2)-2(a2);     .         -         •     (25.) 

and  if  the  body  be  not  metatatically  isotropic,  this  difference  will  have 
equal  maxima  or  minima  for  the  three  orthogonal  axes,  normal  to  the 
faces  of  a  cube,  and,  conversely,  equal  minima  or  maxima  for  the  six 
diagonal  axes,  normal  to  the  faces  of  a  regular  rhombic  dodecahedron. 

Symmetry  of  this  kind  may  be  called  Cybotatic,  from  its  analogy  to  that 
of  crystals  of  the  tessular  system. 

*  Cambridge  and  Dublin  Mathematical  Journal,  Vol.  VI. 


AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS.  131 


Section  12. — Of  Pantatic  Isotropy. 

When  a  body  fulfils  the  conditions  of  cybotatic  symmetry,  and  at  the 
same  time  those  of  metatatic  isotropy,  it  is  completely  isotropic  with 
respect  to  elasticity,  or  pantatically  isotropic.  It  has  but  three  tasinomic 
coefficients — viz.,  the  euthytatic,  platytatic,  and  goniotatic  coefficients, 
which  are  equal  for  all  sets  of  axes,  and  are  connected  by  the  following 
equation,  expressing  the  condition  of  metatatic  isotropy  : 

(a2)  =  (/3y)  +  2(A2).         .  .  .     (26.) 

The  properties  of  such  bodies  have  been  fully  investigated  by  various 
authors. 


Section  13. — Of  Thlipsinomic  Coefficients. 

If  the  six  elementary  strains,  a,  &c,  at  a  given  point  in  an  elastic  solid, 
be  expressed  as  linear  functions  of  the  six  elementary  stresses,  Pp  &c, 
these  expressions  will  contain  twenty-one  coefficients  of  compressibility, 
extensibility,  and  pliability,  which  are  the  second  differential  coefficients 
of  the  potential  energy  of  elasticity  with  respect  to  the  six  elementary 
stresses ;  that  energy  being  represented  as  follows : — 

p2  p2  p2  O2  O2  o2 

U  =  («2)f +(52)^  +  (c2)-f +  (Z2)-f +  K)f+(^)f 

+  (5C)P2P3  +  (ca)P3P1+(a&)P1P2+(Ww)Q2Q3+(wOQ3Qi  +  (MQiQ2 
-\-{(al)  P1+(5/)P2+(d)  P3}Q1 

+  {(am)P1  +  (6OT)P2+(cm)P3}Q2 

+  (HP1  +  (k)P2+(«)P3}Q3.        .        .         .     (27.) 

The  twenty-one  coefficients  in  the  above  equation  may  be  comprehended 
under  the  general  term  Thlijisinomic,  and  classified  as  follows : — 

Designations  of  Coefficients.        Properties  expressed  by  them.        Symbols. 

rEuthythliptic,  Longitudinal  Extensibilities,  (a2),    (JP),    (c2) 
Orthothliptic<j  Platythliptic,   Lateral  Extensibilities,  .     .  (be),    (ca),    (ab), 

[Goniothliptic,  Pliabilities, (Z2),    (m2),  (re2), 

Plagiothliptic,      ....     Unsymmetrical  Pliabilities,  (mn),  &c,  &c. 


132  AXES   OF   ELASTICITY  AND   CRYSTALLINE   FORMS. 


Section  14. — Of  Thlipsinomic  Transformations,  Umbrae, 
Surfaces,  and  Invariants. 
» 
The  equations  of  transformation  of  the  thlipsinomic  coefficients  are 
easily  deduced  from  the  principle,  that  the  operations 

d  d  d  d  d  d 

dP^    d¥2'    dP3'    dCfc    JQ2'    dQ3 

are  respectively  covariant  with 

V       *2>  , 3>       *  ^-1'       "  ^-2>       ~"  ^v3» 

and  these  with 

A     f,      r,      2yz,     2zz,     2xy. 

We  may  regard  the  thlipsinomic  coefficients,  like  the  tasinomic  coeffi- 
cients, as  binary  compounds  of  the  following  six  Umbrcr, 

(a),     (b),     (c),     (I),     (m),     (n), 

which  being  respectively  substituted  for 

"l>       *  2»  3'       *  ^1'       *  ^-2>       "  tw3» 

in  the  equation  of  the  tasimetric  surface  (4),  produce  the  following- 
equation  of  the  Umbral  Thlipsinomic  Ellipsoid, 

(a)  :>?  +  (b)  f  +  (c)  r  +  (I)  yz  +  (m) zx  +  {n)xy  =  1,     (28.) 

from  which,  by  involution,  multiplication,  and  other  operations  exactly 
analogous  to  those  performed  on  the  umbral  tasinomic  ellipsoid,  there 
may  be  deduced  the  equations  of  Thlipsinomic  Surfaces  exactly  correspond- 
ing to  the  tasinomic  surfaces  already  described ;  while,  from  the  umbral 
matrix, 

(a)  }(»)        J(m)>| 

too       (»)    *©  y  •     •     •  (29.) 

may  be  formed  Thlipsinomic  Invariants  corresponding  to  the  tasinomic 
invariants. 

Hence  it  appears,  that  every  function  of  the  tasinomic  coefficients  is 
converted  into  a  function  of  the  thlipsinomic  coefficients  with  analogous 


AXES   OF   ELASTICITY  AND  CRYSTALLINE   FORMS. 


133 


properties,   by  the   substitution   of    thlipsinomic   for    tasinomic    umbrae 
according  to  the  following  table  : — 


Tasinomic  Umbra?, 
Thlipsinomic  Umbra?,    . 


(«),  08).  (y),  (X),  W,  (v), 
(«),   (»).   (0»  K0ki(»)»iW- 


Amongst  the  thlipsinomic  invariants  may  be  distinguished  the  Cm6/c 
Compressibility,  which  is  formed  by  squaring  the  umbral  invariant 
(a)  +  (^)  +  (c),  and  has  the  following  value : 

O2)  +  (J2)  +  (r)  +  2  (6  e)  +  2  (c a)  +  2  (a  J). 


Section  15. — Thlipsinomic  and  Tasinomic  Contragredient 

Systems. 

Let  the  following  square  matrices  be  formed  with  the  tasinomic  and 
thlipsinomic  coefficients  respectively  : — 

(a2)  (a/5)  (ya)  (a  X)  (ap)  (av) 

(a/3)  OT  (/3y)  G8X)  (/5m)  (/5.) 

(ya)  (/3y)  (y2)  (yX)  (y/t)  (yv) 

(aX)  (/3X)  (yX)  (X2)  (\M)  (vX) 

(a//)  (/3/x)  (y/*)  (X/x)  (m2)  0*v) 

(av)  Q3v)  (yv)  (v\)  (fiv)  (v2)   , 


> 


(a2) 
(aft) 
(ca) 
(a2) 


(ab) 

(b2) 
(be) 

(bl) 


(ca) 

(be) 

(c2) 

(cl) 


(a  I) 
(bl) 

(cl) 
(I2) 


(a  m)  (a  n) 

(b  m)  (b  n) 

(c  m)  (c  n) 

(Im)  (nl) 


(30.) 


(31.) 


(a  m)     (b  m)     (c  m)     (I  m)     (m2)      (m  n) 
(an)      (bn)      (en)      (nl)      (mn)     (n2) 


Then  will  these  matrices  be  mutually  inverse,  the  two  systems  of 
coefficients  arrayed  in  them,  with  their  respective  systems  of  functions 
mutually  contragredient,  and  each  coefficient  or  function  belonging  to  one 
system  contravariant  to  the  corresponding  coefficient  or  function  belonging 
to  the  other  system. 

The  values  of  the  coefficients  in  either  of  those  matrices  are  expressed 


134 


AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS. 


in  terms  of  those  in  the  other  matrix,  in  Mr.  Sylvester's  umbral  notation, 
by  twenty-one  equations,  of  which  the  following  are  examples  : — 


(«2)  = 
(ab)  = 


03),  (y),  (X),  (a*),  (v) 

0),  (y),  (a),  0*),  to 

(/3),  (y),  (X),  (/*),  W 
(a),  (y),  (X),  GO,  to 


(«),  (/3),  (y),  (X),  W  to 

(«),  O),  (y)>  ^)'  0*).  to 

(«),  (/3),  (y),  (A),  W  to 

(«).  (/3),  (y),  (X),  Gk),  to 


V.    (32.) 


Section  1G. — Of  Thlipsinomic  Axes. 

If,  under  given  conditions,  any  symmetrical  system  or  function  of 
the  constituents  of  one  of  the  above  matrices  be  null,  then  under  the 
same  conditions  will  the  contravariant  system  or  function  of  the  constituents 
of  the  inverse  matrix  be  null  or  infinite.  Therefore,  Systems  of  Thlipsinomic 
Axes  coincide  with  the  corresponding  systems  of  Tasinomic  Axes. 


Section  17. — Platythliptic  Coefficients  are  Negative. 

It  may  be  observed,  as  a  matter  of  fact,  that  in  consequence  of  the 
largeness  of  the  euthytatic  coefficients  (a2),  (/32),  (y2),  as  compared  with 
the  other  tasinomic  coefficients,  the  platythliptic  coefficients  (be),  (ca), 
(a  b),  are  generally,  if  not  always,  negative. 

To  illustrate  this,  the  case  of  pantatic  isotropy  may  be  taken,  for 
which  the  two  matrices  have  the  following  forms : — 


(a2)  (/3y)  (/3y)  0  0       0 

0y)  («2)  (Pv)    o  o     o 

03  y)  (/3y)  (a2)  0  0      0 

0         0         0  (A2)  0       0 

0         0         0  0  (A2)     0 

0         0         0        0  0  (A2) 


(a2)  (be)  {be)  0  0  0' 

(be)  (a2)  (6c)  0  0  0 

(be)  (be)  (a2)  0  0  0 

0        0  0  (P)  0  0 

0       0  0  0  (I2)  0 

0        0  0  0  0  (Z2)  J 


y  (33.) 


from  which  it  is  easily  seen   that  the  sole  platythliptic  coefficient  has 
the  following  value : 

-(fly) 


(be) 


(33A) 


(a2)2  +  (a2)(/3y)-2(/3y)2,  " 
The   denominator   of   this   fraction   is  always  positive   so   long   as    (a2) 


AXES  OF  ELASTICITY  AND   CRYSTALLINE  FORMS.  135 

exceeds  (/3y);  a  condition  invariably  fulfilled  by  solid  bodies,  and,  in 
fact,  necessary  to  their  existence. 


Section  18. — Of  Oblique  Co-ordinates  and  Contra-ordinates. 

As  there  are,  in  the  relations  between  two  systems  of  oblique  co-ordinates, 
or  between  a  system  of  oblique  co-ordinates  and  a  system  of  rectangular 
co-ordinates,  six  independent  constants  of  transformation,  it  is  possible, 
by  referring  the  equation  of  the  biquadratic  surface  (19)  to  oblique 
co-ordinates,  to  make  the  six  terms  vanish  which  contain  the  cubes  of  the 
co-ordinates. 

The  conception  of  the  physical  meaning  of  such  a  transformation  is 
much  facilitated  by  the  employment  of  a  system  of  three  auxiliary 
variables,  which  will  be  designated  as  Contraordinates. 

The  relations  between  co-ordinates  and  contraordinates  are  as  follows : — 

Through  an  origin  0  let  any  three  axes  pass,  right  or  oblique.  Let  E 
be  any  point,  and  let 

0~R  =  r. 

Through  It  draw  three  planes,  parallel  respectively  to  the  three  co-ordinate 
planes,  and  intersecting  the  axes  respectively  in  the  points  X,  Y,  Z. 
Also,  on  OE,  as  a  diameter,  describe  a  sphere,  intersecting  the  axes 
respectively  in  U,  V,  W.     Then  will 

OX  =  <c,   OY  =  y,   OZ  =  z, 

be  the  co-ordinates  of  E,  as  usual,  and 

OU  =  w,   OV  =  v,   OW  =  «e, 

its  contra-ordinates,  being,  in  fact,  the  projections  of  0  E  on  the  three  axes. 

For  rectangular  axes,  co-ordinates  and  contra-ordinates  are  identical. 

Co-ordinates  and  contra-ordinates  are  connected  by  the  following 
equation : — 

r2  =  u  x  +  v  y  +  we.  .         .         .     (34.) 

In  the  language  of  Mr.  Sylvester,  a  system  of  co-ordinates,  and  the 
concomitant  system  of  contra-ordinates,  are  mutually  Contragredient ;  and 
the  square  of  the  radius  vector  is  their  universal  mixed  concomitant. 

Let  the  cosines  of  the  angles  made  by  the  axes  with  each  other  be 
denoted  as  follows: 

cos2/Os  =  q;   coszOx  =  ci;    cos  xOy  =  c3; 


136 


AXES  OF  ELASTICITY  AND   CRYSTALLINE  FORMS. 


then  the  contra-ordinates  of  a  given  point  are  the  following  functions  of 
the  co-ordinates: 


Also  let 


u  =  x     +  cs  y  +  c2  z  } 

v  =  czx  +  y     -f  c1  z  \ 

I 

to  =  c2  x  +  cx  y  +  z     J 


=  l-cf-4-ci+2c1c2c8  =  C; 


(35.) 


then  the  co-ordinates  are  the  following  linear  functions  of  the  contra- 
ordinates  : — 


Also, 


x  =  hx  u  —  k3  v  —  k2  w ; 
y  =  —  k,A  u  +  h2  v  —  kx  w ; 
z  =  —  h2  u  —  l\  v  +  hz  w ;  J 


r  =  x'  -f  y2  +  z-  +  2  cx  y  z  +  2c2zx  +  2c%xy 


(3C) 


(37.) 


=  hx u2  +  /;2 v2  +  hz to2  -ll^vw  -  2k2ivu-  2 fcs m v.   (37a) 

Differentiations   with    respect   to   the   contra-ordinates   are   obviously 
covariant  with  the  co-ordinates,  and  vice  versa;  that  is  to  say, 


,,  ,.  d        d        d        d        d        d 

the  operations      .     — -,     — ,     — ,     — ,     — ,     — - 
dx      cly      dz      du      dv      dw 


are  respectively  ) 
covariant  with  ) 


y  ■ 


(38.) 


*■>      y, 


By  making  substitutions  according  to  the  above  law  of  covariance  in 
the  equations  (34),  (37),  (37a),  three  equivalent  symbols  of  operation  are 
obtained,  which,  being  applied  to  isotropic  functions  of  the  second  degree, 
produce  invariants  of  the  first  degree. 


AXES  OF   ELASTICITY  AND   CRYSTALLINE  FORMS.  137 


Section  19. — Of  Molecular  Displacements  and  Strains  as 
referred  to  oblique  axes. 

If  the  displacement  of  a  particle  from  its  free  position  be  resolved  into 
three  components  g,  r\,  Z,,  parallel  respectively  to  three  oblique  axes,  0  x, 
0  y,  0  z,  those  components  are  evidently  covariant  respectively  with  the 
co-ordinates  x,  y,  z. 

It  is  now  necessary  to  find  a  method  of  expressing  the  strain  at  any 
particle  in  an  elastic  solid  by  a  system  of  six  elementary  strains,  which 
shall  be  covariant  respectively  with  the  squares  and  doubled-products 
of  these  oblique  co-ordinates.  This  condition  is  fulfilled  by  considering 
the  elementary  strains  as  being  constituted  by  the  variations  of  the 
components  of  the  molecular  displacement  with  respect  to  the  distances 
of  the  strained  particle  from  three  planes  passing  through  the  origin, 
and  normal  respectively  to  the  three  axes;  that  is  to  say,  with  respect  to 
the  contra-ordinates  of  the  particle,  as  expressed  in  the  following  equations: — 


Elongations,        .     a  =  ^—  ;   j3  —  ^  )    Y  —  T~  ■> 
°  du}  '       dv3    *      dw' 

n       -tv*     i-          ^       d%,drl             d^.dK.  _dri      d% 

Quasi-Distortions,  A  =  -; — h  t~ ;   ix  =  -3 — h  3-  ,  v  —  5 — (-7- 

dv      dw            die     du  du      dv 


y  (39.) 


The  six  elementary  strains,  as  above  defined,  are  obviously  covariant 
with  the  squares  and  doubled-products  of  the  co-ordinates,  according  to 
the  following  table : 

a,    /3,    y,      X,        /*,        v,     ") 

r     .      .  (40.) 

x2,   y\   z2,    2yz,    2zx,    2xy.  ) 


Section  20. — Of  Stresses,  as  referred  to  Oblique  Axes. 

It  is  next  required  to  express  the  stress  at  any  particle  of  an  elastic 
solid  by  means  of  a  system  of  six  elementary  stresses,  which  shall  be 
contragredient  to  the  system  of  six  elementary  strains  defined  in  the 
preceding  section.     This  is  accomplished  in  the  following  manner. 

It  is  known  that  the  total  stress  at  any  point  may  be  resolved  into 
three  normal  stresses  on  the  three  principal  planes  of  the  tasimetric 
surface.  Let  the  direction  and  sign  of  any  one  of  those  three  principal 
stresses  be  represented  by  those  of  a  line  0  K,  and  its  magnitude,  as 
reduced  to  unity  of  area  of  the  plane  normal  to  that  direction,  by  the 
square  of  that  line. 

OR2  =  r2. 


138 


AXES  OF   ELASTICITY  AND  CRYSTALLINE  FORMS. 


Let  u,  v,  iv  be  the  contraordinates  of  E,  as  referred  to  the  oblique  axes 
0  X,  0  Y,  0  Z.  Then  will  the  stresses  on  unity  of  area  of  planes  normal 
to  those  axes,  in  the  direction  0  E,  be  represented  respectively  by 

u  r,     v  r,    to  r. 

Let  the  Elementary  Stresses  be  defined  to  be,  the  projections  on  the  three 
axes  of  co-ordinates  of  the  total  stresses  on  unity  of  area  of  the  three  pairs  of 
faces  of  a  parallelepiped,  normal  to  the  three  axes  respectively :  then,  if  we 
take  S  to  denote  the  summation  of  three  terms  arising  from  the  three 
principal  stresses,  the  elementary  stresses  will  be  expressed  as  follows : — 


Normal  stresses  on  the  faces  normal  to 

x,  y,  z, 

P1  =  S.tt2;     P2  =  S.*2;     P3  =  S.ir; 

Oblique  stresses  on  the  faces  normal  to 


(41.) 


Iu  the  directions 


"V 


These  expressions  fulfil  the  condition  of  making  the  elementary  stresses 
Plf    P2,    P3,    Qlf    Q2,    Q3 

contravariant  respectively  to  the  elementary  strains 

a,     ft,     y>     \     /">     v> 

so  that  for  oblique  axes,  as  for  rectangular  axes,  the  potential  energy  of 
elasticity  is  represented  by 

♦ 
U  =  - 1  (Pia  +  Po/3  +  P3y  +  QXX  +  Q,fx  +  Q3v), 

the  universal  concomitant;  and  may  be  expressed  either  by  a  homo- 
geneous quadratic  function  of  the  six  elementary  strains  (as  iu  equation  2) 
with  twenty-one  tasinomic  coefficients,  or  by  a  homogeneous  quadratic 
function  of  the  six  elementary  stresses,  as  in  equation  (27),  with  twenty- 
one  thlipsinomic  coefficients,  forming  a  system  contragredient  to  that  of 
the  tasinomic  coefficients. 


axes  of  elasticity  and  crystalline  forms.  139 

Section  21*. — Of  Tasinomic  and  Thlipsinomic  Umbrae  for 
Oblique  Axes. 

The  tasinomic  coefficients  for  oblique  axes  may  be  regarded  as  com- 
pounded of  umbra? 

(a),     08),     (y),     (A),     (/*),     (v), 

contravariant  respectively  to  the  elementary  strains 

«,     (3,     7>     I  A,     I  ft,     hv, 

and  consequently  covariant  with  the  squares  and  products  of  the  contra- 
ordinates 

v?,     v2,     w2,     1)10,     wu,     uv; 

and  the  thlipsinomic   coefficients   for   oblique  axes  may  be  regarded   as 
compounded  of  umbra? 

(a),     (b),     (c),     (/),     (m),     («), 

contravariant  respectively  to  the  stresses 

Plf     P2,     Ps,     2QX,     2Q2,     2Q3, 

and  consequently  covariant  with  the  squares  and  products  of  the  co-ordinaks 

ar,      ij ,     (.  ,      *></~,     &%j>,      *jxy. 

Section  22. — Of  the  Biquadratic  Surface,  and  of  Principal 
euthytatic  axes. 

For  oblique  as  well  as  for  rectangular  axes  of  co-ordinates,  the  char- 
acteristic function  of  the  biquadratic  tasinomic  surface  is  represented 
by  equation  (19);  and  the  fifteen  homotatic  coefficients  are  covariant 
respectively  with  suitable  multiples  of  the  fifteen  biquadratic  powers  and 
products  of  the  contraordinates. 

If  by  linear  transformations  a  system  of  three  axes,  oblique  or  rectan- 
gular, be  found  which  reduces  the  characteristic  function  of  the  biquad- 
ratic surface  to  the  canonical  form,  consisting  of  not  more  than  nine 
terms,  viz. — 

(0)2  =  (a2K+(W+(rK 

+  2{(/3y)  +  2(\2)}^2  +  2{(ya)  +  2(ya2)}^2+2{(a/3)  +  2(v2)}a;Y 
+  4{2(Mv)  +  (aA)}ry:  +  4{2(^)  +  (|3,i)}^22  +  4{2(\M) 

H-(7v)}^2  =  l;      ....     (42.) 


140  AXES  OF  ELASTICITY  AND  CRYSTALLINE   FORMS. 

then,  for  that  system  of  axes,  the  following  six  plagiotatic   coefficients 

are  null, 

(|3X)  =  0;  (yX)  =  0;  (yfi)  =  0;  (afx)  =  0;  (av)  =  0;  ((Sv)  =  0;  (43.) 

and  each  of  those  axes  is  Euthytatic,  according  to  the  definition  in  Sect. 
7,  that  is  to  say,  is  a  direction  of  maximum  or  minimum  direct  elasticity 
(absolute  or  relative),  and  also  a  direction  in  which  a  direct  elongation 
or  compression  produces  a  simply  normal  stress. 

There  are  necessarily  three  euthytatic  axes  at  least  in  every  solid — 
viz.,  the  three  Principal  Euthytatic  Axes,  as  above  described,  which  are 
normal  to  the  faces  of  a  hexahedron,  right  or  oblique,  as  the  case  may 
be;  but  in  special  cases  of  symmetry  there  are  additional  or  secondary 
euthytatic  axes,  of  which  examples  will  now  be  given. 

Section  23. — Of  Rhombic  and  Hexagonal  Symmetry. 

When  a  solid  has  three  oblique  principal  euthytatic  axes  making  equal 
angles  with  each  other  round  an  axis  of  symmetry,  and  having  equal 
systems  of  homotatic  coefficients  corresponding  to  them,  viz. — 

(«2)  =  Q32)  =  (y2);  (|3y)  +  2(^)=:(7a)  +  2(^)  =  (a/3)  +  2(v2)1 

r  (43a0 

2(MV)  +  (aA)  =  2(vA)  +  (/3/0  =  2(X/0  +  (7v)  J 

it  may  be  said  to  possess  rhombic  symmetry,  because  the  three  oblique 
axes  are  normal  to  the  faces  of  one  rhombohedron,  and  to  the  edges 
of  another  belonging  to  the  same  series,  crystallographically  speaking. 
It  is  evident  in  this  case,  that  the  axis  of  symmetry  must  be  a  fourth 
Euthytatic  Axis. 

In  the  limiting  case,  when  the  three  oblique  axes  make  with  each 
other  equal  angles  of  120°,  they  lie  in  the  same  plane,  normal  to  the 
axis  of  symmetry,  and  are  normal  to  the  faces  of  one  hexagonal  prism 
and  the  edges  of  another. 

Let  0  yx  denote  the  longitudinal  axis  of  symmetry  of  the  prism ;  0  zx 
any  one  of  the  three  transverse  axes  perpendicular  to  0  yv  The  equation 
of  a  section  of  the  biquadratic  surface  by  the  Plane  of  Hexagonal  Symmetry 
yx  zv  is  as  follows : — 

(i32)1^  +  (y2)1^  +  2{(/3y)1+2(X2)1}2/i,1=.l.        .     (44.) 

The  equation  of  the  same  section,  referred  to  any  other  pair  of 
orthogonal  axes  0  y,  Oz,  in  the  plane  of  yxzv  is  as  follows  : — 

(j32) . y*  +  (y2) . **+  2 {(/3y)  +  2(\2)}!/V+  4{(/3 A) /  +  (yA)*»Jy*=  1.  (44a.) 


AXES  OF  ELASTICITY  AND   CRYSTALLINE   FORMS.  141 

From  considerations  of  symmetry,  it  is  evident  that  the  coefficient 
(/3»/)  must  be  null  for  every  direction  of  the  axis  0  y,  in  the  plane  of  y^zy ; 
consequently,  every  direction  Oy  in  that  plane,  for  which  (/3\)  =  0,  is 
an  euthytatic  axis. 

To  ascertain  whether,  and  under  what  conditions,  there  are  other 
euthytatic  axes  in  the  planes  of  hexagonal  symmetry  besides  the 
longitudinal  and  transverse  axes,  it  is  to  be  considered,  that  for  rectangular 
co-ordinates  (/3\)  is  covariant  with  yh;  hence,  let 

ZylOy  =  w, 
then 

(£A)  =  sin_2<o  m  j-{,  (/,  y^  +  4  (x%  _  ^^  _  (y2)i}  cog  2  ^ 

-  (P\  +  (r)i]  •    (4r>.) 

The  first  factor  of  the  above  expression  is  null  for  the  longitudinal 
and  transverse  axes  only.  The  condition  of  there  being  additional 
euthytatic  axes  in  the  plane  yxzx  is,  that  the  second  factor  shall  vanish ; 
that  is  to  say,  that 

coso„_  (ff)l  -  (V% (i6) 

and  that  the  value  of  w  which  makes  it  vanish  shall  neither  be  0°  nor  90°; 
that  is  to  say,  that  the  second  member  of  the  above  equation  (46)  shall 
lie  between  + 1  and  —  1 ;  in  which  case  the  equation  is  satisfied  by  equal 
values  of  to  with  opposite  signs.  Hence  are  deduced  the  following 
theorems,  which  are  stated  in  such  a  form  as  to  be  applicable  to  planes 
of  symmetry,  whether  hexagonal  or  otherwise : 

If,  in  any  plane  of  tasinomic  symmetry  containing  a  pair  of  orthogonal 
euthytatic  axes,  the  difference  of  the  euthytatic  coefficients  for  these  axes  be 
equal  to  or  greater  than  the  metatatic  difference,  there  are  no  additional 
euthytatic  axes  in  that  plane. 

If,  on  the  other  hand,  the  difference  of  such  euthytatic  coefficients  be  less 
than  the  metatatic  difference,  there  are,  in  such  plane  of  symmetry,  a  pair  of 
additional  euthytatic  axes  making  with  each  other  a  pair  of  angles  bisected 
by  the  orthogonal  euthytatic  axes. 

2  to  is  the  angle  bisected  by  the  axis  0  yv 

In  the  case  of  hexagonal  symmetry,  the  additional  axes  thus  found 
are  normal  to  the  faces  of  one  pyramidal  dodecahedron  and  the  edges  of 
another. 


142  AXES  OF  ELASTICITY  AND  CRYSTALLINE  FORMS. 


Section  24. — Of  Orthorhombic  Symmetry. 

Let  a  solid  have  one  of  the  three  principal  euthytatic  axes,  0  zv  normal 
to  the  other  two,  Oxv  0^;  let  the  last  two  be  oblique  to  each  other,  and 
have  equal  sets  of  homotatic  coefficients,  viz. — 

(«2)>  =  (/32)i ;  O  y)i  +  2  (A2)!  =  (y  «)x  +  ^  (S\ ; 

2frv)  +  (a\)  =  2(v\)  +  (Pfi),       •  (i7-) 

then,  that  solid  may  be  said  to  have  Orthorhombic  Symmetry,  its  principal 
euthytatic  axes  being  normal  to  the  faces  of  a  right  rhombic  prism. 

The  existence  or  non-existence,  and  the  position,  of  a  pair  of  additional 
euthytatic  axes  in  the  longitudinal  planes  of  yx  r.v  st  o\,  is  to  be  determined 
as  in  the  preceding  section.  "When  such  axes  exist,  they  are  normal  to 
the  faces  of  an  octahedron  with  a  rhombic  base. 


Section  25. — Of  Orthogonal  Symmetry. 

If  the  three  principal  euthytatic  axes  be  orthogonal,  they  are  normal 
to  the  faces  of  a  right  rectangular  or  square  prism,  and  to  the  edges  of  a 
rigid  rhombic  or  square  prism.  The  existence  or  non-existence,  and 
position,  of  a  pair  of  additional  euthytatic  axes  in  each  of  the  principal 
planes  of  such  a  solid,  are  determined  as  in  Sect.  23. 

If  there  be  a  pair  of  such  additional  axes  in  each  of  the  three  principal 
planes,  they  are  normal  to  the  faces  of  an  irregular  rhombic  dodecahedron, 
and  to  the  edges  of  a  rhombic  octahedron. 

If  there  be  a  pair  of  such  additional  axes  in  two  of  the  three  principal 
planes,  those  axes  are  normal  to  the  faces  of  an  octahedron  with  a 
rectangular  or  square  base,  and  to  the  edges  of  an  octahedron  with  a 
rhombic  or  square  base. 

If  there  be  a  pair  of  such  additional  axes  in  one  of  the  planes  of 
orthotatic  symmetry  only,  those  axes  are  normal  to  the  lateral  faces  of  a 
right  rhombic  prism. 


Section  26. — Of  Cyboid  Symmetry. 

The  case  of  Cyboid  Symmetry  is  that  in  which  the  homotatic  coefficients 
are  equal  for  three  orthogonal  axes,  viz. — 

(«2)  -  m  =  (y2);    (73  y)  +  2  (X2)  =  (y  a)  +  2  Ou2)  =  (a  0)  +  2  (t,2); 
2  0uv)+(aX)  =  2(vX)  +  (j3ju)  =  2(Xiu)  +  (7v)  =  O.         (48.) 


AXES  OF  ELASTICITY  AND   CRYSTALLINE  FORMS.  143 

In  this  case,  the  principal  metatatic  axes  coincide  with  the  principal 
euthytatic  axes,  which  are  normal  to  the  faces  of  a  cube;  the  diagonal 
metatatic  axes  normal  to  the  faces  of  a  regular  rhombic  dodecahedron, 
are  euthytatic  also;  and  there  are,  besides,  four  additional  euthytatic 
axes  symmetrically  situated  between  the  first  nine,  and  normal  to  the 
faces  of  a  regular  octahedron,  making  in  all  thirteen  euthytatic  axes. 

Section  27. — Of  Monaxal  Isotropy. 

Monaxal  Isotropy  denotes  the  case  in  which  the  homotatic  coefficients 
are  completely  isotropic  round  one  axis  only.  In  this  case,  the  principal 
euthytatic  axes  are,  the  axis  of  isotropy,  and  every  direction  perpendi- 
cular to  it ;  and  when  there  are  additional  axes,  determined  as  in  the 
preceding  sections,  they  are  normal  to  the  surface  of  a  cone. 

Section  28. — Of  Complete  Isotropy. 

In  the  case  of  complete  isotropy  of  the  homotatic  coefficients,  every 
direction  is  a  euthytatic  axis. 


Section  29. — Probable  Eelations  between  Euthytatic  Axes  and 
Crystalline  Forms. 

In  the  preceding  sections  it  has  been  shown  what  must  be  the  nature 
of  the  relations  between  the  fifteen  homotatic  coefficients,  for  various 
solids,  having  systems  of  euthytatic  axes  normal  to  the  faces  and  edges 
of  the  several  Primitive  Forms  known  in  crystallography. 

It  is  probable  that  the  normals  to  Planes  of  Cleavage  are  euthytatic 
axes  of  minimum  elasticity. 

It  may  also  be  considered  probable,  that  in  some  cases,  especially  in 
the  tessular  system,  which  corresponds  to  cyboi'd  symmetry,  and  in  the 
case  of  the  pyramidal  summits  of  crystals  of  the  rhombohedral  system, 
euthytatic  axes  correspond  to  symmetrical  summits  of  crystalline  forms. 
In  the  icositetrahedral  crystals  of  leucite  and  analcime,  and  the  tetra- 
contaoctahedral  crystals  of  diamond,  there  are  twenty-six  symmetrical 
summits,  one  pair  corresponding  to  each  of  the  thirteen  axes  of  cybo'id 
symmetry. 

The  following  is  a  synoptical  table  of  the  various  possible  systems  of 
euthytatic  axes,  arranged  according  to  their  degrees  and  kinds  of  symmetry, 
and  of  the  crystalline  forms  to  the  faces  and  edges  of  which  such  systems 
of  axes  are  respectively  normal. 


144  AXES   OF   ELASTICITY  AND   CRYSTALLINE  FORMS. 

Systems  of  Euthytatic  Axes.  Crystalline  Forms. 

Faces.  Edges. 

I.  Asymmetry.  Tetarto-prismatic  System. 

1.  Three  unequal  oblique  axes,    .     Oblique  hexahedron. 

II.  Symmetry  about  One  Plane.  Hemiprismatic  System. 

2.  Two  unequal  oblique  axes,  and  )         ,.,,.,,.  „.,.  .    _.. 

\        ,  .  \  Ri<ditrhornboidal  prism,    Oblique  rhombic  prism, 

one  rectangular  axis,  .  )      a  r        '  * 

wo  equa  <    (  Oblique  rhombic  prism,      Right  rhomboidal  prism, 

oblique  axis,  .  )  x  * 

III.  Rhombic  and  Hexagonal  Rhomboiiedral  System. 

Symmetry. 

4.  Three    equi-oblique    principal  j 

axes    round    one    axis    of  >  Rhombohedron,    .         .     Rhombohedron. 
symmetry,  .         .         .         .  ) 

5.  Three    equi-oblique    principal  j 

axes  in  one  plane,   normal  >  Hexagonal  prism,  .     Hexagonal  prism, 

to  axis  of  symmetry,    .         .  ) 

6.  Three  pairs  of  secondary  axes    Pyramidal       dodecahe-     Pyramidal      dodecahe- 

in  planes  of  symmetry,        .         dron,         .         .         .         dron. 

IV.  Orthoriiombic  Symmetry.  Prismatic  and  Pyramidal  Systems. 

7.  Two  equal  oblique  transverse  \ 

axes  normal  to  one  longi-  >  Right  rhombic  prism,  .     Rectangular  prism, 
tudinal  axis,         .         .         .  ) 

8.  Two  pairs  of  secondary  axes  in    Octahedron  with  rhom-     Octahedron  with  rectan- 

longitudinal  planes,     .         .         bic  base,    .         .         .         gular  base. 

V.  Orthogonal  Symmetry. 

9.  Three  orthogonal  axes,  not  all    Rectangular  and  square    Right     rhombic      and 

equal, prisms,      .         .         .         square  prisms. 

!  Octahedron  with  rhom- 
bic base  and  rectan- 
gular prism. 

,,_,.„  I  Octahedron  with  square     Octahedron  with  square 

11.  Two  pairs  of  secondary  axes,  .  j      or  rectangular  base,  .        or  rhombic  base. 

Same  with  7.     One  pair  of  second-  )  .  '       .         . 

arv  axes  \  raSut  rhombic  prism,   .     Rectangular  prism. 

VI.  Cyboid  Symmetry.  Tessular  System. 

12.  Three  equal  orthogonal  axes,  .     Cube. 

10    „.     ,.  ,  I  Regular  rhombic  dode-     Cube  and  regular  octahe- 

13.  Six  diagonal  axes,  .  \         ,    j  n 

°  (      cahedron,  .         .         .         dron. 


14.  Four  symmetrical  intermediate 
axes, 


Regular  octahedron,     .     Rhombic  dodecahedron. 


AXES  OF  ELASTICITY  AND  CRYSTALLINE   FORMS.  145 


VII.   MONAXAL  ISOTKOPY. 


15.  One  axis  of  isotropy,        .        .  Isotropic  lamina?. 

16.  Innumerable  transverse  axes, .  Istropic  fibres. 

17.  Innumerable  equi-oblique  axes,  Conical  cleavage. 

VIII.  Complete  Isotropy. 

18.  Innumerable  axes  of  isotropy, .  Amorphism. 


Section  30. — Mutual  Independence  of  the  Euthytatic  and 
Heterotatic  Axes,  and  of  the  Homotatic  and  Heterotatic 
Coefficients. 

The  fifteen  homotatic  coefficients  of  the  biquadratic  surface,  on  which 
the  euthytatic  axes  depend,  and  the  six  heterotatic  differences,  coefficients 
of  the  heterotatic  ellipsoid,  constitute  twenty-one  independent  quantities; 
so  that  the  euthytatic  axes  may  possess  any  kind  or  degree  of  symmetry 
or  asymmetry,  and  the  heterotatic  axes  any  other  kind  or  degree,  in 
the  same  solid. 

Hence,  if  it  be  true  that  crystalline  form  depends  on  the  arrangement 
of  euthytatic  axes,  it  follows  that  two  substances  may  be  exactly  alike 
in  crystalline  form,  and  yet  differ  materially  in  the  laws  of  their  elasticity, 
owing  to  differences  in  then  respective  heterotatic  coefficients. 

It  may  be  observed,  however,  that  this  complete  independence  of  those 
two  systems  of  axes  and  coefficients  is  mathematical  only,  and  that  their 
physical  dependence  or  independence  is  a  question  for  experiment. 


Section  31. — On  Eeal  and  Alleged  Differences  between  the  Laws 
of  the  Elasticity  of  Solids,  and  those  of  the  Luminiferous 
Force. 

For  every  conceivable  system  of  tasinomic  coefficients  in  a  solid,  the 
plane  of  polarisation  of  a  wave  of  distortion  is  that  which  includes  the 
direction  of  the  molecular  vibration  and  the  direction  of  its  propagation, 
being,  in  fact,  the  plane  of  distortion. 

On  the  other  hand,  it  appears  to  be  impossible  to  avoid  concluding, 
from  the  laws  of  the  diffraction  of  polarised  light,  as  discovered  by 
Professor  Stokes,  and  from  those  of  the  more  minute  phenomena  of  the 
reflexion  of  light,  as  investigated  theoretically  by  M.  Cauchy  and 
experimentally  by  M.  Jamin,  that  in  plane-polarised  light  the  plane  of 
polarisation  is  perpendicular  to  the  direction  of  vibration,  or  rather  (to 
avoid  hypothetical  language)  to  the  direction  of  some  physical  phenomenon 

K 


146  AXES  OF  ELASTICITY  AND  CRYSTALLINE   FORMS. 

whose  laws  of  communication  are  to  a  certain  extent  analogous  to  those 

of  a  vibratory  movement. 

This  constitutes  an  essential  difference  between  the  laws  of  the  elastic 

forces  in  a  solid,  and  those  of  the  luminiferous  force. 

In  order  to  frame,  in  connection  with  the  wave  theory  of  light,  a 

mechanical  hypothesis  which  should  take   that   difference   into   account, 

it  has  been  proposed  to  consider  the  elasticity  of  the  luminiferous  medium 

to  be  the  same  in  all  substances,  and  for  all  directions,  or  Pantatically 
Isotropic  ;  and  to  ascribe  the  various  retardations  of  light  to  variations  in 
the  inertia  of  the  mass  moved  in  luminiferous  waves,  in  different  substances, 

and  for  different  directions  of  motion.* 

Another  essential  difference  between  the  laws  of  solid  elasticity  and 

those  of  the  luminiferous  force  is,  that  under  no  conceivable  system  of 
tasinomic  coefficients  in  a  homogeneous  solid,  would  the  plane  of  distortion 
in  a  wave  be  rotated  continuously  round  the  direction  of  propagation. 

Much  has  been  written,  both  recently  and  in  former  times,  concerning 
an  alleged  difficulty  in  the  theories  of  waves,  both  of  sound  and  of  light, 
arising  from  the  physical  impossibility  of  the  actual  divergence  of  waves 
from,  or  their  convergence  to,  a  mathematical  point.  This  impossibility 
must  be  admitted;  but  the  supposed  difficulty  to  which  it  gives  rise  in 
the  theories  of  waves  is  completely  overcome  in  Mr.  Stokes'  paper  "  On 
the  Dynamical  Theory  of  Diffraction,"!  in  which  that  author  proves,  that 
waves  spreading  from  a  focal  space,  or  origin  of  disturbance,  of  finite 
magnitude,  and  of  any  figure,  sensibly  agree  in  all  respects  with  waves 
spreading  from  an  imaginary  focal  point,  so  soon  as  they  have  attained  a 
distance  from  the  focal  space  which  is  large  as  compared  with  the 
dimensions  of  that  space ;  so  that  the  equations  of  the  propagation  of 
waves  spreading  from  imaginary  focal  points  may  be  applied,  without 
sensible  error,  to  all  those  cases  of  actual  waves  to  which  it  is  usual  to 
apply  them. 

The  physical  impossibility  of  focal  points  applies  to  light  independently 
of  all  hypotheses;  for  at  such  points  the  intensity  would  be  infinite. 
It  appears  to  be  worthy  of  consideration,  whether  this  impossibility  may 
not  be  connected  with  the  appearance  of  spurious  disks  of  fixed  stars 
in  the  foci  of  telescopes. 


Section  32. — On  the  Action  of  Crystals  on  Light. 

If  we  set  aside  those  actions  on  light  to  which  there  is  nothing  analogous 
in  the  phenomena  of  the  elasticity  of  homogeneous  solids,  the  laws  of 

*  Philosophical  Magazine,  June,  1851,  December,  1853. 
+  Cambridge  Transactions,  Vol.  IX.,  Part  1. 


AXES   OF   ELASTICITY  AND   CRYSTALLINE   FORMS.  147 

the  refractive  action  of  a  crystal  on  light  are  in  general  of  a  more  sym- 
metrical kind,  or  depend  on  fewer  quantities  than  those  of  its  elasticity. 

Thus,  the  elasticity  of  a  homogeneous  solid  depends  on  twenty-one 
quantities ;  its  crystalline  form,  on  fifteen  (the  homotatic  coefficients), 
while  its  refractive  action  on  homogeneous  light  in  most  cases  is  expres- 
sible by  means  of  the  magnitudes  and  directions  of  the  orthogonal  axes 
of  Fresnel's  wave-surface,  making  in  all  six  quantities.  Crystals  which 
possess  only  rhombic  or  hexagonal  symmetry  in  their  euthytatic  axes,  are 
usually  monaxally  isotropic  in  their  action  on  light ;  while  crystals  which 
possess  only  cybo'id  symmetry  in  their  euthytatic  axes,  are  completely 
isotropic  in  their  action  on  light. 

From  these  remarks?  however,  there  are  exceptions,  as  in  the  case  of 
the  extraordinary  optical  properties  discovered  by  Sir  David  Brewster 
in  analcime,  which,  in  its  refraction  as  well  as  in  its  form,  is  cyboidally 
symmetrical  without  being  isotropic. 


Note  referred  to  at  page  120,  On  Sylvestrian  Umbrae. 

Without  attempting  to  enter  into  the  abstract  theory  of  the  umbral 
method,  it  may  here  be  useful  to  explain  the  particular  case  of  its  appli- 
cation which  is  employed  in  this  paper. 

Let  U  be  a  quantity  having  an  absolute  value,  constant  or  variable, 
(such,  for  example,  as  any  physical  magnitude),  and  u,  v,  .  .  .  &c. 
a  set  of  quantities,  m  in  number,  such  that  tJ  is  of  them  a  homogeneous 
rational  function  of  the  nth.  degree.  There  are  an  indefinite  number  of 
possible  sets  of  m  quantities  satisfying  this  condition;  and  the  quantities 
of  each  set  are  related  to  those  of  each  other  set  by  m  equations  of  the 
first  degree,  called  equations  of  linear  transformation.     Let 


the 


Up  vv     .     .     . 

be  two  such  sets. 

^t   (U     .     .     . 
development  of 

and  let 

M2>    ^2> 

denote   the   coefficient  of  uavb     . 
(u  +  v+      .     .     .)» 

U  =  2{Ca,6, 

•      •       •      Al,a,6,      •       •      •       KVl        ■ 

•       ■} 

=  2{Ca.&, 

.     .     .     A.,     ,      ...     u-,av?     . 

2.  a,  b,       •       •       •       w2    "2 

•       •}• 

The  two  sets  of  coefficients  Av  A.z,  are  connected  by  linear  equations 


148 


AXES  OF  ELASTICITY  AND   CRYSTALLINE  FORMS. 


of  transformation,  the  investigation  of  which  is  much  facilitated  by  the 
following  process. 

Let  two  sets,  each  of  m  symbols,  av  [3V  &c.    .    .    .    a2,  /32,    .    .    .    &c, 
be  assumed  such  that 


«i  «i  +  &  vx  + 
and  that,  consequently, 


=  a2u2  +  fi2v2  + 


(«1M1+/Vl+  • 

■    .)*~2{€U    • 

.    .    <&&    . 

.    ufvf    . 

•     •} 

=  (a2u2+ /32v2+    ■ 

•  -)n=^K,b,  ■ 

■  <A>&  • 

•     «/^26     • 

•     •}• 

Then,  if  the  m  equations  of  transformation  between  the  two  sets  of 
symbols  av  &  •  •  •  and  a2,  (32  .  .  .  be  formed,  and  if  from 
them  be  deduced  the  equations  between  the  two  sets  of  products  axre  fi^ 


,  and  a2a(32 


,  &c,  and  if   in  the  latter  system  of 


equations,  there  be  substituted  for  each  product  aa /3b  .  .  .  the 
corresponding  coefficient  Aa_  &>  .  .  .  ,  the  result  will  be  the  system  of 
equations  sought.  Also,  if -any  function  of  the  products  a°/S&  .  .  . 
be  invariant  (i  e.,  a  function  whoso  value,  like  that  of  the  original  function 
U,  is  not  altered  by  the  transformation),  the  corresponding  function  of  the 
coefficients  A  will  be  invariant. 

The  symbols  a,  (3,  &c,  with  reference  to  their  relation  to  the  coefficients 
A,  are  called  umbrce  ;  that  is,  factors  of  symbols,  whose  equations  of  transfor- 
mation are  similar  to  those  of  the  coefficients  A.  In  the  umbral  notation, 
umbrae  are  usually  distinguished  from  symbols  denoting  actual  quantities 
by  being  enclosed  in  brackets  thus : 

(a),  08),  &c.     .     .     . 

and  each  coefficient  A  is  represented  by  enclosing  in  brackets  that  product 
of  umbrae  with  which  it  is  covariant;  thus : 


a,  I, 


=  (aa/36 


•)• 


The  umbral  notation  is  applied  to  abbreviate  the  expression  of  deter- 
minants in  a  manner  of  which  the  following  are  examples : — 


a,   (3,   y,   &c. 
a,   /3,    y,    &c. 


denotes 


(«2)      («/3) 

(ay) 

&c. 

(«/3)    (/32) 

(/3y) 

&c. 

(ay)  (/3y) 

(y2) 

&c. 

&c.      &c. 

&c. 

&c. 

AXES   OF  ELASTICITY  AND   CRYSTALLINE   FORMS. 


149 


a,    y,    S,    &c. 

/5,    y,    S,     &C. 


denotes 


(«/3) 

(/3y) 

CSS) 

&c. 

(«y) 

(r) 

(yS) 

&c. 

(«  8) 

(yS) 

(S2) 

&c. 

&c. 

&c. 

&c. 

&c. 

150  THE  VIBRATIONS  OF   PLANE-POLARISED   LIGHT. 


VII— ON  THE  VIBRATIONS  OF  PLANE-POLARISED  LIGHT.* 

1.  The  important  question,  whether  the  direction  of  vibration  in  plane- 
polarised  light  is  normal  or  parallel  to  the  plane  of  polarisation,  is 
equivalent  to  this  : — whether  the  velocity  of  propagation  of  a  rectilinear 
transverse  vibratory  movement  in  the  medium  which  transmits  light  in 
crystallised  bodies  is  a  function  of  the  direction  of  vibration  only,  or  of  the 
position  of  the  plane,  which  includes  the  direction  of  vibration  and  the  direction 
of  transmission. 

The  former  of  these  views  was  adopted  by  Fresnel,  as  being  necessary 
to  explain  the  phenomena  of  polarisation  by  reflexion;  and  in  Mr.  Green's 
investigation  of  the  laws  of  these  phenomena,  in  which  the  conclusions 
of  Fresnel  are  shown  to  agree  either  exactly  or  approximately  with  the 
consequences  of  strictly  mechanical  principles,  the  same  supposition  is 
adopted. 

But  if  we  follow  the  generally  received  theory,  that  the  different 
velocities  of  differently  polarised  rays  in  crystalline  bodies  are  due  solely 
to  the  different  degrees  of  elasticity  possessed  by  the  vibrating  medium 
in  different  directions,  Fresnel's  supposition  must  be  abandoned,  and  the 
opposite  one  adopted.  For  if  there  is  any  proposition  more  certain  than 
others  respecting  the  laws  of  elasticity,  it  is  this  : — that  the  transverse 
elasticity  of  a  medium,  or  the  elasticity  which  resists  distortion  of  the 
particles,  depends  upon  the  position  of  the  plane  of  distortion,  being  the 
same  for  all  directions  of  distortion  in  a  given  plane.  This  law  is  impli- 
citly involved  in  the  researches  of  Poisson,  of  M.  Cauchy,  of  Mr.  Green, 
and  others  on  elasticity;  and  in  a  memoir  read  to  the  British  Association 
at  Edinburgh,  in  1850,  and  published  in  the  Cambridge  and  Dublin 
Mathematical  Journal  for  February,  1851,  I  have  shown  that  it  is  true 
independently  of  all  hypotheses  respecting  the  constitution  of  matter, 
being  a  necessary  consequence  of  the  conception  of  an  elastic  medium. 
Now  a  wave  of  plane-polarised  light  is  a  wave  of  distortion  :  the  plane 
of  distortion  is  the  plane  which  includes  the  direction  of  transmission  and 
the  direction  of  vibration:  the  elasticity  called  into  play  depends  on  the 
position  of  this  plane;  therefore,  if  the  velocity  of  propagation  depends 
upon  elasticity  alone,  the  plane  of  distortion  must  be  the  plane  of  polaris- 
ation; and  if  a  normal  be  drawn  to  that  plane,  the  velocity  of  propagation 

*  Read  before  the  Royal  Society  of  Edinburgh,  on  December  2,  1850,  and  published 
in  the  Philosophical  Magazine,  June,  1851. 


THE   VIBRATIONS   OF  PLANE-POLARISED   LIGHT.  151 

will  be  a  function  of  the  position  of  that  normal,  and  not,  as  supposed 
by  Fresnel,  of  the  direction  of  vibration  itself. 

2.  Up  to  a  very  recent  period,  no  experimental  data  existed  adequate 
to  determine  which  of  these  suppositions  is  supported  by  facts ;  for  the 
phenomena  of  double  refraction  are  consistent  with  either ;  and  the  theory 
of  polarisation  by  reflexion  is  not  regarded  as  sufficiently  certain  to  afford 
the  means  of  deciding  this  question.  At  length,  however,  the  experimentum 
cruris  has  been  made  by  Professor  Stokes,  and  the  result  is  conclusive  in 
favour  of  the  supposition  of  Fresnel. 

In  his  paper  on  Diffraction  (Cambridge  Transactions,  Vol.  IX.,  Part  1), 
Professor  Stokes  has  shown,  that  on  any  conceivable  theory  of  the 
propagation  of  undulations  of  light,  vibrations  normal  to  the  plane  of 
diffraction  must  be  transmitted  round  the  edge  of  an  opaque  body  with  less 
diminution  of  intensity  than  vibrations  in  that  plane.  Therefore,  Avhen 
light,  of  which  the  vibrations  are  oblique  to  the  plane  of  diffraction,  is  so 
transmitted,  the  plane  of  vibration  will  be  more  nearly  perpendicular  to  the 
plane  of  diffraction  in  the  diffracted  ray  than  in  the  incident  ray.  He 
has  found  by  experiment,  that  when  light  of  which  the  plane  of  polarisa- 
tion is  oblique  to  the  plane  of  diffraction  is  transmitted  round  the  edge 
of  an  opaque  body,  the  plane  of  polarisation  is  more  nearly  parallel  to  the 
plane  of  diffraction  in  the  diffracted  than  in  the  incident  ray.  The 
necessary  conclusion  is,  that  the  direction  of  vibration  in  plane-polarised 
light  is  normal  to  the  plane  of  polarisation;  in  other  words,  that  the  velocity 
of  light  in  crystallised  media  depends  on  the  direction  of  vibration,  as  con- 
jectured by  Fresnel. 

This  result  of  experiment  is  at  variance  with  the  necessary  consequences 
of  the  supposition,  that  the  velocity  of  light  depends  on  elasticity  alone; 
therefore,  that  supposition  is  inadequate  to  explain  the  phenomena  of 
polarised  light. 

3.  Having  considered  what  modifications  must  be  introduced  into  our 
hypothetical  conceptions  of  the  nature  of  the  medium  which  transmits 
light,  to  make  them  adequate  to  explain  the  facts  which  have  thus  been 
established,  I  now  offer  the  following  suggestions. 

In  a  paper  read  to  the  Royal  Society  of  Edinburgh,  and  published  in 
their  Transactions,  Vol.  XX.,  Part  1,  I  proposed,  as  a  foundation  for  the 
theory  of  heat,  and  of  the  elasticity  of  gases  and  vapours,  an  hypothesis 
called  that  of  molecular  vortices ;  and  in  a  subsequent  paper,  already 
referred  to,  I  deduced  from  the  same  hypothesis  some  principles  relative 
to  the  elasticity  of  solids.  I  shall  now  show  that  Fresnel's  conjecture 
as  to  the  direction  of  vibration  in  plane-polarised  light  is  a  natural 
consequence  of  that  hypothesis. 

The  fundamental  suppositions  of  the  hypothesis  of  molecular  vortices 
are  the  following: — 


152  THE  VIBRATIONS   OF  PLANE-POLARISED  LIGHT. 

First.  That  each  atom  of  matter  consists  of  a  nucleus  or  central  physical 
point  enveloped  by  an  elastic  atmosphere,  which  is  retained  round  it  by 
attraction;  so  that  the  elasticity  of  bodies  is  made  up  of  two  parts — one 
arising  from  the  diffused  portion  of  the  atmospheres,  and  resisting  change 
of  volume  only ;  the  other  arising  from  the  mutual  actions  of  the  nuclei, 
and  of  the  portions  of  atmosphere  condensed  round  them,  and  resisting 
not  only  change  of  volume,  but  also  change  of  figure. 

Secondly.  That  the  changes  of  elasticity  due  to  heat  arise  from  the 
centrifugal  force  of  revolutions  or  oscillations  among  the  particles  of  the 
atomic  atmospheres,  diffusing  them  to  a  greater  distance  from  their  nuclei, 
and  thus  increasing  the  elasticity  which  resists  change  of  volume  only,  at 
the  expense  of  that  which  resists  change  of  figure  also. 

Thirdly.  That  the  medium  which  transmits  light  and  radiant  heat 
consists  of  the  nuclei  of  the  atoms  vibrating  independently,  or  almost 
independently,  of  their  atmospheres;  absorption  being  the  transference  of 
motion  from  the  nuclei  to  the  atmospheres,  and  emission  its  transference 
from  the  atmospheres  to  the  nuclei. 

This  last  supposition  is  peculiar  to  my  own  researches,  the  first  two 
having  more  or  less  resemblance  to  ideas  previously  entertained  by  others. 

If  an  indefinitely  extended  vibrating  medium,  equally  elastic  in  all 
directions,  consists  of  a  system  of  atomic  nuclei,  tending  to  preserve  a 
certain  configuration  in  consequence  of  their  mutual  attractions  and 
repulsions,  it  is  well  known  that  such  a  medium  is  capable  of  transmitting 
two  sorts  of  vibrations  only,  longitudinal  and  transverse,  the  latter  alone 
being  supposed  to  be  concerned  in  the  phenomena  of  light.  It  is  also 
well  known,  that  the  square  of  the  velocity  of  propagation  of  transverse 
vibrations  is  directly  proportional  to  a  quantity  called  the  transverse 
elasticity  of  the  medium,  arising  from  the  mutual  actions  of  the  nuclei, 
and  inversely  proportional  to  its  density — that  is,  to  the  sum  of  the  masses 
of  all  the  nuclei  contained  in  unity  of  volume.  To  account  for  the 
immense  velocity  of  light,  the  masses  of  the  atomic  nuclei  must  be  sup- 
posed to  be  very  small  as  compared  with  the  mutual  forces  exerted  by 
them. 

In  stating  the  third  supposition  of  the  hypothesis,  the  nuclei  are  said 
to  vibrate  independently,  or  almost  independently,  of  their  atmospheres; 
for  the  absolute  independence  of  their  vibrations  is  probably  an  ideal  case, 
not  realised  in  nature,  though  approached  very  nearly  in  the  celestial 
space,  where  the  atomic  atmospheres  must  be  inconceivably  rarefied. 

As  a  pendulum  is  known  to  be  accompanied  in  its  oscillations  by  a 
portion  of  the  air  in  which  it  swings,  so  the  nuclei  probably  in  all  cases 
carry  along  with  them  in  their  vibrations  a  small  portion  of  their  atmos- 
pheres, which  acts  as  a  load,  increasing  the  vibrating  mass  without  increas- 
ing  in   the   same   proportion  the  elasticity,  and  consequently  retarding 


THE  VIBRATIONS   OF  PLANE-POLARISED   LIGHT.  153 

the  velocity  of  transmission.  The  amount  of  this  load  must  depend  on 
the  density  of  the  atomic  atmosphere;  and,  accordingly,  we  find  that, 
generally  speaking,  the  most  dense  substances  are  those  in  which  the 
velocity  of  light  is  least. 

Now,  if  we  assume,  what  is  extremely  probable,  that  in  crystallised 
media  the  atomic  atmospheres  are  not  similarly  diffused  in  all  directions 
round  their  nuclei,  but  are  more  dense  in  certain  directions  than  in  others, 
we  must  at  once  conclude  that  in  such  media  the  velocity  of  propagation 
of  vibratory  movement  depends  on  the  direction  of  vibration  ;  for  upon  that 
direction  depends  the  load  of  atmosphere  which  each  nucleus  carries  along 
with  it. 

4.  Having  thus  shown  that  the  conjecture  of  Fresnel,  which  has  been 
confirmed  by  the  experiments  of  Professor  Stokes,  is  a  natural  consequence 
of  the  hypothesis  of  molecular  vortices,  I  shall  now  prove  that  that 
hypothesis  leads  to  those  mathematical  laws  of  the  transmission  of  light 
in  crystalline  media  which  Fresnel  discovered. 

Considering  it  desirable  in  this  paper  to  avoid  lengthened  algebraical 
analysis,  I  shall  with  that  view  state,  in  the  first  place,  certain  known 
geometrical  properties  of  the  ellipsoid,  to  which  it  will  be  necessary  for 
me  to  refer. 

I.  If  a  curved  surface  be  described  about  a  centre,  such  that  the  sum 
of  the  reciprocals  of  the  squares  of  any  three  orthogonal  diameters  is  a 
constant  quantity,  that  surface,  if  no  diameter  is  infinite,  is  an  ellipsoid. 

II.  Every  function  of  direction  round  a  centre,  whose  variation  from  a 
given  amount  varies  as  the  reciprocal  of  the  square  of  the  diameter  of 
an  ellipsoid  described  about  that  centre,  is  itself  proportional  to  the 
reciprocal  of  the  square  of  the  diameter  of  another  ellipsoid  described 
about  the  same  centre  with  the  first,  and  having  the  directions  of  its  axes 
the  same. 

III.  It  follows  from  the  last  proposition,  that  if  there  be  a  function  of 
direction  round  a  centre  which  is  proportional  to  the  reciprocal  of  the 
square  of  the  diameter  of  an  ellipsoid  of  small  excentricities,  so  that  the 
range  of  variation  of  the  function  is  small  as  compared  with  its  amount, 
then  any  function  of  that  function,  whose  range  of  variation  is  small  also, 
may  be  represented  approximately  by  the  reciprocal  of  the  square  of  the 
diameter  of  another  ellipsoid,  having  its  centre  and  the  directions  of  its 
axes  the  same  with  those  of  the  first. 

The  square  of  the  velocity  of  propagation  of  transverse  vibrations  is 
proportional  to  the  transverse  elasticity  of  the  medium  divided  by  the 
mean  density;  that  is,  by  the  sum  of  all  the  vibrating  masses  in  unity 
of  volume.  That  sum  is  the  sum  of  the  masses  of  the  nuclei,  added  to 
the  masses  of  atmosphere  with  which  they  are  loaded.  The  atmospheric 
load  of  each  nucleus  depends  on,  or  is  a  function  of,  the  density  of  the 


154-  THE   VIBRATIONS    OF    PLANE-POLARISED   LIGHT. 

atmosphere  adjoining  the  nucleus  along  the  line  in  which  the  latter 
vibrates.  The  mode  of  distribution  of  the  atmospheres  depends  on  the 
attraction  of  the  nuclei  upon  them,  and  therefore  on  the  mode  of  arrange- 
ment of  the  nuclei.  The  mode  of  arrangement  of  the  nuclei,  when  it 
is  symmetrical  and  uniform,  may  be  expressed  by  means  of  their  mean 
intervals. 

The  mean  interval  of  the  nuclei  is  a  function  of  direction,  of  such  a 
nature  that  its  three  values  for  any  three  orthogonal  directions  being 
multiplied  together  give  a  constant  result — viz.,  the  space  occupied  (not 
filled)  by  one  nucleus,  or  the  quotient  of  a  given  space  by  the  number  of 
nuclei  contained  in  it.  Hence  the  sum  of  the  values  of  the  logarithm 
of  the  mean  interval  for  any  three  orthogonal  directions  is  a  constant 
quantity;  and  that  logarithm,  therefore,  is  proportional  to  the  reciprocal 
of  the  square  of  the  diameter  of  an  ellipsoid,  -whose  three  axes  may  be 
called  the  axes  of  atomic  distribution.  Therefore,  the  mean  interval,  the 
atmospheric  load  of  the  nuclei,  and  the  square  of  the  velocity  of  propa- 
gation, for  a  given  direction  of  transverse  vibration,  are  all  functions  of 
the  reciprocal  of  the  square  of  the  diameter  of  an  ellipsoid,  and  have 
maxima  and  minima  corresponding  to  its  three  axes,  which  are  those  of 
atomic  distribution. 

Now,  in  all  known  crystalline  media,  the  range  of  variation  of  these 
quantities  for  different  directions  is  very  small  compared  with  their 
amount.  Therefore,  each  of  them  may  be  approximately  represented  by 
the  reciprocal  of  the  square  of  the  diameter  of  an  ellipsoid  whose  axes  are 
parallel  to  those  of  atomic  distribution. 

If,  then,  the  directions  of  vibration  in  a  given  crystal  which  correspond 
to  the  greatest  and  least  velocities  of  transmission  are  known,  let  these 
directions  (which  are  at  right  angles  to  each  other),  and  a  third  direction 
at  right  angles  to  them  both,  be  taken  for  the  axes  of  an  ellipsoid,  the 
lengths  of  those  axes  being  inversely  proportional  to  the  corresponding 
velocities  of  transmission.  Then  will  the  velocity  of  transmission  of  any 
transverse  vibratory  movement  be  sensibly  proportional  to  the  reciprocal 
of  a  diameter  of  that  ellipsoid,  drawn  parallel  to  the  direction  of  vibration. 
And  if  a  plane  be  drawn  through  the  centre  of  the  ellipsoid  parallel  to  a 
series  of  plane-waves,  the  two  axes  of  the  elliptic  section  so  made  will 
represent,  in  magnitude,  the  reciprocals  of  the  greatest  and  least  normal 
velocities  of  transmission  of  waves  parallel  to  that  plane,  and,  in  direction, 
the  corresponding  directions  of  vibration. 

This  agrees  exactly  with  the  construction  given  by  Fresnel,  on  which 
his  entire  theory  of  double  refraction  is  founded. 

The  degree  of  symmetry  and  uniformity  of  arrangement  of  the  atoms 
which  is  necessary  in  order  that  the  mean  interval  may  have  a  definite 
value,  and  that  three  axes  of  distribution  may  exist,  is  the  same  which 


THE   VIBRATIONS   OF    PLANE-POLARISED   LIGHT.  155 

is  necessary  to  the  existence  of  rectangular  axes  of  elasticity  in  a  solid 
body.  It  must  extend  round  each  point,  throughout  a  space  which  is 
large  as  compared  with  the  sphere  of  appreciable  molecular  action. 

The  experiments  of  Sir  David  Brewster  and  of  Fresnel  on  the  action 
of  compressed  glass  on  polarised  light,  show  that  rays  polarised  in  a  plane 
normal  to  the  direction  of  compression — that  is  to  sa}^  vibrations  parallel 
to  that  direction — are  accelerated.  This  indicates  that  the  atmospheric 
load  on  each  vibrating  nucleus  in  that  direction  is  diminished,  probably 
by  the  displacement  of  a  portion  of  the  atmosphere  out  of  that  line. 

5.  Though  I  have  assumed,  in  the  course  of  this  investigation,  that  the 
luminiferous  medium  is  equally  elastic  in  all  directions,  I  by  no  means 
intend  to  assert  that  it  is  necessarily  so  in  all  substances ;  but  merely 
that,  in  most  known  crystalline  media,  an  atmospheric  load  on  the  vibrating 
nuclei  is  the  predominant  cause  of  variation  in  the  velocity  of  transmission 
with  the  direction  of  transverse  vibration. 

It  is  remarkable  that  Fresnel,  in  his  theory  of  the  intensity  of  reflected 
and  refracted  light,  speaks  of  the  particles  of  the  luminiferous  medium  as 
being  more  or  less  loaded  in  substances  of  greater  or  less  refractive  power. 
He  did  not,  however,  apply  this  idea  to  double  refraction,  although  he 
adopted  a  theory,  which,  as  we  have  seen,  results  from  it. 

The  principles  laid  down  in  this  paper  are  not  compatible  Avith  the 
prevalent  idea  of  a  luminiferous  ether  enveloping  ponderable  particles. 
The  fundamental  idea  from  which  they  spring  is  the  converse:  that  the 
luminiferous  medium  is  a  system  of  atomic  nuclei  or  centres  of  force,  whose 
office  it  is  to  give  form  to  matter ;  while  the  atmospheres  by  which  they 
are  surrounded  give,  of  themselves,  merely  extension. 


156  AN   OSCILLATORY  THEORY   OF  LIGHT. 


VIII— GENERAL  VIEW  OF  AN  OSCILLATORY  THEORY  OF 

LIGHT.* 


Section  I. — Difficulties  of  the  Present  Hypothesis. 

Notwithstanding  the  perfection  to  which  the  geometrical  part  of  the 
undulatory  theory  of  light  has  been  brought,  it  is  admitted  that  great 
difficulty  exists  in  framing,  to  serve  as  a  basis  for  the  theory,  a  physical 
hypothesis  ■which  shall  at  once  be  consistent  with  itself  and  with  the 
known  properties  of  matter. 

The  present  paper  is  a  summary  of  the  results  of  an  attempt  to  diminish 
that  difficulty.  All  the  conclusions  stated  have  been  deduced  by  means 
of  strict  mathematical  analysis ;  and  although  it  is  impossible  to  read  the 
investigations  before  the  British  Association  in  detail,  their  results  can 
easily  be  verified  by  every  mathematician  who  is  familiar  with  the 
undulatory  theory  in  its  present  form. 

It  may  be  considered  as  established,  that  if  we  assume  the  supposition 
that  plane-polarised  light  (out  of  the  varieties  of  which  all  other  light  can 
be  compounded)  consists  in  the  wave-like  transmission  of  a  state  of 
motion,  the  nature  and  magnitude  of  which  are  functions  of  the  direction 
and  length  of  a  line  transverse  to  the  direction  of  propagation,  we  can 
deduce  from  this  supposition,  with  the  aid  of  experimental  data,  and  of 
certain  auxiliary  hypotheses,  the  laws  of  the  phenomena  of  the  inter- 
ference of  light,  of  its  propagation  in  crystalline  and  uncrystalline 
substances,  of  diffraction,  of  single  and  double  refraction,  of  dispersion  by 
refraction,  and  of  partial  and  total  reflexion. 

It  has  hitherto  been  always  assumed,  that  the  kind  of  motion  which 
constitutes  light  is  a  vibration  from  side  to  side,  transmitted  from  particle 
to  particle  of  the  luminiferous  medium,  by  means  of  forces  acting  between 
the  particles.  In  order  to  account  for  the  transmission  of  such  transverse 
vibrations,  the  luminiferous  medium  has  been  supposed  to  possess  a  kind 
of  elasticity  which  resists  distortion  of  its  parts,  like  that  of  an  elastic 
solid ;  and  in  order  to  account  for  the  non-appearance  in  ordinary  cases  of 
effects  which  can  be  ascribed  to  longitudinal  vibrations,  it  has  been  found 

*  Read  before  the  British  Association  at  Hull,  September  10,  1^53,  and  published 
in  the  PhilosojjJdcal  Magazine,  December,  1853. 


AN  OSCILLATORY   THEORY  OF  LIGHT.  157 

necessary  to  suppose  further,  that  this  medium  resists  compression  with  an 
elasticity  immensely  greater  than  that  with  which  it  resists  distortion; 
the  latter  species  of  elasticity  being,  nevertheless,  sufficiently  great  to 
transmit  one  of  the  most  powerful  kinds  of  physical  energy  through 
interstellar  space,  with  a  speed  in  comparison  with  which  that  of  the 
swiftest  planets  of  our  system  in  their  orbits  is  appreciable,  but  no 
"more. 

It  seems  impossible  to  reconcile  these  suppositions  with  the  fact,  that 
the  luminiferous  medium  in  interstellar  space  offers  no  sensible  resistance 
to  the  motions  of  the  heavenly  bodies. 

A  step  towards  the  solution  of  this  difficulty  was  made  by  Mr. 
MacCullagh.  The  equations  which  he  used  to  express  the  laws  of  the 
propagation  of  light,  when  interpreted  physically,  denote  the  condition 
of  a  medium  whose  molecules  tend  to  range  themselves  in  straight  lines, 
and  when  disturbed,  to  return  to  those  lines  with  a  force  depending  on  the 
curvature  of  the  lines  into  which  they  have  been  moved.  But  even  this 
hypothesis  requires  the  assumption  that  the  elasticity  of  the  luminiferous 
medium  to  resist  compression  is  immensely  greater  than  the  elasticity 
which  transmits  transverse  vibrations. 

The  difficulty  just  referred  to  arises  from  a  comparison  of  the  hypo- 
thesis of  transverse  vibrations  with  the  observed  phenomena  of  the  world. 

Another  difficulty  arises  within  the  hypothesis  itself.  Fresnel  originally 
assumed,  that  in  crystalline  media,  where  the  velocity  of  light  varies  with 
the  position  of  the  plane  of  polarisation,  the  direction  of  vibration  is 
perpendicular  to  that  plane.  This  is  equivalent  to  the  supposition,  that 
the  velocity  with  which  a  state  of  rectilinear  transverse  vibration  is 
transmitted  through  such  a  medium,  is  a  function  simply  of  the  direction 
in  which  the  particles  vibrate.  From  this  hypothesis  he  deduced  the 
form  of  that  wave-surface .  which  expresses  completely  the  law  of  the 
propagation  of  plane-polarised  light  through  crystalline  media,  and  he 
obtained  also  a  near  approximation  to  the  laws  of  the  intensity  of  plane- 
polarised  light  reflected  from  singly  refracting  substances. 

But  it  was  afterwards  demonstrated,  that  the  elastic  forces  which 
propagate  a  transverse  movement  in  any  medium  must  necessarily  be 
functions,  not  merely  of  the  direction  of  the  movement,  but  jointly  of  this 
direction  and  the  direction  of  propagation ;  that  is  to  say,  of  the  position 
of  the  plane  containing  these  two  directions.  Consequently,  if  the  various 
velocities  of  variously  polarised  light  in  a  doubly  refracting  medium,  arise 
from  variations  of  elasticity  in  different  directions,  the  direction  of  vibra- 
tion is  in  the  plane  of  polarisation,  contrary  to  the  hypothesis  of  Fresnel. 
Fresnel's  wave-surface,  and  his  approximate  formulas  for  the  intensity  of 
reflected  light,  are  deducible  from  this  supposition  as  readily  as  from  his 
original  hypothesis;  and  Mr.  MacCullagh  obtained  from  it  formula?  for  the 


158  AN   OSCILLATORY   THEORY  OF  LIGHT. 

intensity  of  light  reflected  from  doubly  refracting  substances,   agreeing 
closely  with  the  experiments  of  Sir  David  Brewster. 

On  the  other  hand,  the  formula?  of  M.  Cauchy,  and  those  of  Mr.  Green, 
as  modified  by  Mr.  Haughton,*  expressing  the  effects  of  reflexion  on  the 
intensity  and  phase  of  polarised  light,  all  of  which  are  founded  on  the 
supposition  that  the  direction  of  vibration  is  perpendicular  to  the  plane  of 
polarisation,  have  been  shown  to  be  capable,  by  the  introduction  of  proper 
constants,  of  giving  results  agreeing  closely  with  those  of  the  important 
experiments  of  M.  Jamin  {Annales  de  Chimie  et  de  Physique,  3rd  Series, 
Vol.  XXIX.,  1850);  and  it  is  difficult,  if  not  impossible,  to  see  how 
such  formulae  could  have  been  deduced  from  the  opposite  supposition. 

But  the  true  crucial  experiment  on  this  subject  has  been  furnished  by 
the  researches  of  Professor  Stokes  on  the  Diffraction  of  Polarised  Light 
(Camb.  Trans.,  Vol.  IX.).  Whatsoever  may  be  the  nature  of  the  motion 
that  constitutes  light,  if  it  can  be  expressed  by  a  function  of  the  direction 
and  length  of  a  line  perpendicular  to  the  direction  of  propagation  (which 
may  be  called  a  transversal),  it  is  certain  that  this  motion  will  be  more 
abundantly  communicated  round  the  edge  of  an  obstacle,  when  its  trans- 
versal is  parallel  than  when  it  is  perpendicular  to  that  edge ;  so  that  the 
effect  of  diffraction  is,  to  bring  every  oblique  transversal  into  a  position 
more  nearly  parallel  to  the  diffracting  edge.  But  it  has  been  shown  by 
the  experiments  of  Professor  Stokes,  that  the  effect  of  diffraction  upon 
every  ray  of  light  polarised  in  a  plane  oblique  to  the  diffracting  edge,  is  to 
bring  the  plane  of  polarisation  into  a  position  more  nearly  perpendicular  to 
the  diffracting  edge.  Therefore,  the  transversal  of  a  ray  of  plane-polarised 
light  (which,  if  light  consists  in  linear  vibratory  movement,  is  the  direction 
of  vibration)  is  perpendicular  to  the  plane  of  polarisation. 

Hence  it  follows,  that,  in  a  crystalline  medium,  the  velocity  of  light 
depends  simply  on  the  direction  of  the  transversal  characteristic  of  the 
movement  propagated,  and  not  on  the  direction  of  propagation. 

This  conclusion  is  opposed  to  the  laws  of  the  propagation  of  transverse 
vibrations  through  a  crystalline  elastic  solid,  or  through  any  medium  in 
which  the  velocity  of  propagation  depends  on  elasticity  varying  in  different 
directions.  Therefore,  the  velocity  of  light  depends  on  something  not 
analogous  to  the  variations  of  elasticity  in  such  a  medium. 

To  solve  this  difficulty,  the  author  of  this  paper  some  time  since 
suggested  the  hypothesis,  that  the  luminiferous  medium  consists  of  particles 
forming  the  nuclei  of  atmospheres  of  ordinary  matter ;  that  it  transmits 
transverse  vibrations  by  means  of  an  elasticity  which  is  the  same  in  all 
substances  and  in  all  directions ;  and  that  the  variations  in  the  velocity  of 
the  transmission  of  vibrations  arise  from  variations  in  the  atmospheric  load, 
which  the  luminiferous  particles  carry  along  with  them  in  their  vibrations, 
*  Philosophical  Magazine,  August,  1S53. 


AN   OSCILLATORY   THEORY  OF   LIGHT.  159 

and  which  is  a  function  of  the  nature  of  the  substance,  and,  in  a  crystalline 
body,  of  the  direction  of  vibration. 

But  although  this  hypothesis  removes  the  inconsistency  just  pointed 
out  as  existing  within  the  theory  itself,  it  leaves  undiminished  the 
difficulty  of  conceiving  a  medium  pervading  all  space,  and  possessed  of  an 
elasticity  of  figure,  at  once  so  strong  as  to  transmit  the  powerful  energy 
of  light  -with  its  enormous  velocity,  and  so  feeble  as  to  exercise  no  direct 
appreciable  effect  on  the  motions  of  visible  bodies. 


Section  II — Statement  of  the  Proposed  Hypothesis  of 
Oscillations. 

The  hypothesis  now  to  be  proposed  as  a  groundwork  for  the  undulatory 
theory  of  light,  consists  mainly  in  conceiving  that,  the  luminiferous 
medium  is  constituted  of  detached  atoms  or  nuclei  distributed  throughout 
all  space,  and  endowed  with  a  peculiar  species  of  polarity,  in  virtue  of 
which  three  orthogonal  axes  in  each  atom  tend  to  place  themselves 
parallel  respectively  to  the  corresponding  axes  in  every  other  atom ;  and 
that  plane-polarised  light  consists  in  a  small  oscillatory  movement  of  each 
atom  round  an  axis  transverse  to  the  direction  of  propagation. 

Such  a  movement  would  be  transmitted  through  such  a  medium  with 
a  velocity  proportional,  directly,  to  the .  square  root  of  the  total  rotative 
force  exercised  by  the  luminiferous  atoms  in  a  given  small  space,  upon 
those  in  a  given  adjacent  small  space  lying  in  the  direction  of  propagation, 
in  consequence  of  a  given  amount  of  relative  angular  displacement  round 
the  axis  of  oscillation ;  and  inversely,  to  the  square  root  of  the  sum  of 
the  moments  of  inertia  round  the  axes  of  oscillation  of  the  atoms  contained 
in  a  given  space,  loaded  with  such  portions  of  molecular  atmospheres 
surrounding  them  as  they  may  carry  along  with  them  in  their  oscilla- 
tions. 

Then,  denoting  by 

h,  the  velocity,  in  a  given  direction  of  plane-waves,  of  oscillation  round 
transverse  axes  parallel  to  a  given  line; 

C,  a  coefficient  of  polarity  or  rotative  force  for  the  given  directions  of 
propagation  and  of  axes ; 

M,  a  coefficient  of  moment  of  inertia  for  the  given  direction  of  axes; 
the  above  principle  may  be  represented  by  this  equation, 

M 

The  coefficient  of  polarity  in   question  is  proper  only  to  an  axis  of 


100  AN   OSCILLATORY  THEORY   OF  LIGHT. 

oscillation  transverse  to  the  direction  of  propagation.  To  account  for  the 
stability  of  direction  of  the  axes  of  the  atoms,  and  also  for  the  non-appear- 
ance, in  ordinary  cases,  of  phenomena  capable  of  being  ascribed  to 
oscillations  round  axes  parallel  to  the  direction  of  propagation,  it  is 
necessary  to  suppose  the  corresponding  coefficient  for  the  latter  species  of 
oscillations  to  be  much  greater  than  the  coefficient  for  transverse  axes  of 
oscillation. 

It  is  evident,  that  how  powerful  soever  the  polarity  may  be  which  is 
here  ascribed  to  the  atoms  of  the  luminiferous  medium,  it  is  a  kind  of 
force  which  must  be  absolutely  destitute  of  direct  influence  on  resistance 
to  change  of  volume  or  change  of  figure  in  the  parts  of  that  medium,  or  of 
any  body  of  which  that  medium  may  form  part ;  and  that,  consequently, 
the  difficulty,  which  in  the  hypothesis  of  vibrations  arises  from  the 
necessity  of  ascribing  to  the  luminiferous  medium  properties  like  those  of 
an  clastic  solid,  has  no  existence  in  the  hypothesis  of  oscillations  now 
proposed.  * 

The  luminiferous  atoms  may  now  be  supposed  to  be  diffused  throughout 
all  space,  and,  as  molecular  nuclei,  throughout  all  bodies ;  the  distribution 
and  motion  of  their  centres  being  regulated  by  forces  wholly  independent 
of  that  species  of  polarity  which  is  the  means  of  transmitting  a  state  of 
oscillation  round  those  centres. 


Section  III. — Of  the  Diffraction  of  Plane-polarised  Light,  and  the 
Relation  of  Axes  of  Oscillation  to  Planes  of  Polarisation. 

In  the  diffraction  of  an  oscillatory  movement  round  transverse  axes  past 
the  edge  of  an  obstacle,  a  law  holds  good  exactly  analogous  to  that 
demonstrated  by  Professor  Stokes  for  a  transverse  vibratory  movement, 
substituting  only  the  axis  of  oscillation  for  the  direction  of  vibration — that 
is  to  say : 

The  direction  of  the  axes  of  oscillation  in  the  diffracted  icave  is  the  projection 
of  that  of  the  axes  of  oscillation  in  the  incident  wave  on  a  plane  tangent  to  the 
front  of  the  diffracted  wave. 

Consequently,  oscillations  in  the  incident  wave,  round  axes  oblique  to 
the  diffracting  edge,  give  rise  to  oscillations  in  the  diffracted  wave  round 
axes  more  nearly  parallel  to  the  diffracting  edge. 

But  the  experiments  of  Professor  Stokes  .have  proved,  that  light 
polarised  in  a  plane  oblique  to  the  diffracting  edge,  becomes,  after 
diffraction,  polarised  in  a  plane  more  nearly  perpendicular  to  the  diffracting 
edge. 

Therefore,  the  axes  of  oscillation  in  plane-polarised  light  are  perpendicular 
to  the  plane  of  polarisation. 


AN   OSCILLATORY  THEORY  OF   LIGHT.  161 

Therefore,  the  velocity  of  transmission  of  oscillations  round  transverse 
axes,  through  the  luminiferous  medium  in  a  crystalline  body,  is  a  function 
simply  of  the  direction  of  the  axes  of  oscillation. 

Now,  if  the  variations  of  the  velocity  of  transmission  arose  from 
variations  of  the  coefficient  of  transverse  polarity  (denoted  by  C),  they 
would  depend  on  the  direction  of  propagation  as  well  as  upon  that  of  the 
axes  of  oscillation,  so  that  the  plane  of  polarisation  would  be  that  which 
contains  these  two  directions.  Since  the  velocity  of  transmission  depends 
on  the  direction  of  the  axes  of  oscillation  only,  it  follows  that  its  variations 
in  a  given  crystalline  medium  arise  wholly  from  variations  of  the  moment 
of  inertia  of  the  luminiferous  atoms,  together  with  their  loads  of  extraneous 
matter. 

Consequently,  the  coefficient  of  polarity,  C,  for  transverse  axes  of  oscilla- 
tion is  the  same  for  all  directions  in  a  given  substance. 

To  account  for  the  known  laws  of  the  intensity  and  phase  of  reflected 
and  refracted  light  consistently  with  the  hypothesis  of  oscillations,  it  is 
necessary  to  suppose,  also,  that  this  coefficient  is  the  same  for  all  substances ; 
so  that  the  variations  of  the  velocities  of  light,  and  indices  of  refraction  for 
different  media,  depend  solely  on  those  of  the  moments  of  inertia  of  the 
loaded  luminiferous  atoms. 

There  is  reason  to  anticipate,  that,  upon  further  investigation,  it  will 
appear  that  this  condition  is  necessary  to  the  stability  of  the  luminiferous 
atoms. 


Section  IV. — Of  the  Wave-surface  in  Crystalline  Bodies. 

Let  the  axes  of  co-ordinates  be  those  of  molecular  symmetry  in  a 
crystalline  medium. 

Let  Mv  M9,  M3,  be  coefficients  proportional  to  the  moments  of  inertia  of 
the  luminiferous  atoms  with  their  loads  of  extraneous  matter,  round  axes 
parallel  to  x,  y,  z,  respectively. 

Let  r  be  a  radius  vector  of  the  diverging  wave-surface  in  the  direction 

(«>  A  y)- 

Then  the  equation  of  that  surface  for  polar  co-ordinates  is, 


~  -  -2  •  ~  { (M2  +  M3)  cos2  a  +  (M3  +  Mx)  cos2  /3  +  (Mx  +  M2)  cos2  y  } 
+  -^  {M2  M3  cos2  a  +  M3  Mx  cos2/3  +  Mx  M2  cos2  y }  =  0  ; 


162  AN   OSCILLATORY  THEORY  OF  LIGHT, 

and  for  rectangular  co-ordinates, 

^(x2  +  2/2  +  ^).(M2M3x2  +  M3Ml2/2  +  M1M2.2) 

-i{(M2+M3)x2+(M3+M1)2/2+(M1+M2)x2}=:l. 

The  above  equations  are  exactly  those  of  Fresnel's  wave-surface,  with 
the  following  semi-axes: — 

Directions.  Semi-axes. 

x'      Vm2'     Vm3; 
y'       Vm3'     Vm^ 

the  squares  of  the  semi-axes  of  the  wave-surface  along  each  axis  of 
co-ordinates  being  inversely  proportional  to  the  moments  of  inertia  of  the 
loaded  luminiferous  atoms  in  a  given  space  round  the  other  two  axes  of 
co-ordinates. 

The  plane  of  polarisation  at  each  point  of  the  wave-surface  is  perpen- 
dicular to  the  direction  of  greatest  declivity. 

The  equation  of  the  index-surface,  whose  radius  in  any  direction  is 
inversely  proportional  to  the  normal  velocity  of  the  wave,  is  formed  from 
that  of  the  wave-surface  by  substituting  respectively, 

r     J-     ±      ! 

'     Mx'     M2'     M3' 
for 

c,     M1?     M2,    M3, 

These  equations  are  obtained  on  the  supposition  that  the  coefficient  of 
polarity  for  axes  of  oscillation  parallel  to  the  direction  of  propagation, 
(which  we  may  call  A),  is  either  very  large,  or  very  small,  compared  with 
that  for  transverse  axes.  By  treating  the  ratio  of  these  quantities  as 
finite,  there  is  obtained  an  equation  of  the  sixth  order,  representing  a 
wave-surface  of  three  sheets,  differing  somewhat  from  that  of  the  propaga- 
tion of  vibrations  in  an  elastic  crystalline  solid ;  inasmuch  as  the  former 
has  always  three  circular  sections,  while  the  latter  has  none,  unless  it  is 

symmetrical  all  round  one  axis  at  least.      By  increasing  the  ratio   ~ 


AN  OSCILLATOEY  THEOKY   OF  LIGHT.  163 

without  limit,  this  equation  is  made  to  approximate  indefinitely  to  the 
product  of  the  equation  of  Fresnel's  wave-surface  by  the  following — 

which  represents  a  very  large  ellipsoidal  wave  of  oscillations  round  axes 
parallel  to  the  direction  of  propagation. 


Section  V. — Of  Reflexion  and  Effraction. 

According  to  the  proposed  hypothesis  of  oscillations,  the  laws  of  the 
phase  and  intensity  of  light  reflected  and  refracted  at  the  bounding 
surface  of  two  transparent  substances,  are  to  be  determined  by  conditions 
analogous  to  those  employed  in  the  hypothesis  of  vibrations  by  M.  Cauchy 
and  Mr.  Green.  They  are  the  consequences  of  the  principle,  that  if  we 
have  two  sets  of  formula?  expressing  the  nature  and  magnitude  of  the 
oscillations  in  the  two  substances  respectively,  then  either  of  those 
formulae,  being  applied  to  a  particle  at  the  bounding  surface,  ought  to  give 
the  same  results. 

According  to  this  principle,  the  following  six  quantities  for  a  particle 
at  the  bounding  surface  must  be  the  same  at  every  instant,  when  computed 
by  either  of  the  two  sets  of  formulae : — 

The  three  angular  displacements  round  the  three  axes  of  co- 
ordinates, 

The  three  rotative  forces  round  the  same  three  axes. 

There  is,  generally  speaking,  a  change  of  phase  when  light  undergoes 
refraction  or  reflexion.  It  is  known  that  we  may  express  this  change 
of  phase  by  subdividing  each  reflected  or  refracted  disturbance  into  two, 
of  suitable  intensities  and  signs;  one  synchronous  in  phase  with  the 
corresponding  incident  disturbance,  and  the  other  retarded  by  a  quarter 
of  an  undulation.  There  are  thus  twelve  quantities  to  be  found — viz.,  the 
amplitudes  of  the  six  components  of  the  reflected  disturbance,  and  those 
of  the  six  components  of  the  refracted  disturbance.  To  determine  these 
quantities  there  are  twelve  conditions — viz.,  the  equality  at  every  instant, 
according  to  the  formulas  for  either  medium,  of  the  total  angular  displace- 
ments, and  of  the  total  rotative  forces,  round  each  of  the  three  axes  of 
co-ordinates,  for  the  set  of  waves  composed  of  the  incident  wave  and  those 
synchronous  with  it,  and  for  the  set  of  waves  retarded  by  one  quarter  of 
an  undulation. 


164  AN  OSCILLATORY   THEORY   OF   LIGHT. 

The  results  of  these  conditions  have  been  investigated  in  detail  for 
singly  refracting  substances. 

The  indices  of  refraction  of  such  substances  are  proportional  to  the 
square  roots  of  the  moments  of  inertia  of  the  loaded  luminiferous  atoms 
in  a  given  space.  Thus,  if  the  coefficients  M',  M"  are  proportional  to 
these  moments  in  two  given  substances  respectively,  then  the  index  of 
refraction  of  the  second  substance  relatively  to  the  first  is 


M 


/M" 


In  the  case  of  light  incident  on  a  plane  surface  between  two  such 
media,  the  axes  of  co-ordinates  may  be  assumed  respectively  perpendicular 
to  the  reflecting  surface,  perpendicular  to  the  plane  of  reflexion,  and  along 
the  intersection  of  those  two  planes ;  and  oscillations  round  axes  normal 
and  parallel  to  the  plane  of  reflexion  may  be  considered  separately. 

When  the  axes  of  oscillation  are  normal  to  the  plane  of  reflexion — that 
is  to  say,  when  the  light  is  polarised  in  that  plane — the  formula}  for  the 
intensities  of  the  reflected  and  refracted  light  agree  exactly  with  those  of 
Fresnel.  When  the  reflexion  takes  place  in  the  rarer  medium,  the 
reflected  light  is  retarded  by  half  an  undulation;  when  in  the  denser, 
there  is  no  change  of  phase,  unless  the  reflexion  is  total,  when  there  is  a 
certain  acceleration  of  i  hase  depending  on  the  angle  of  incidence.  In  the 
last  case,  the  disturbance  in  the  second  medium  is  an  evanescent  wave, 
analogous  to  those  introduced  into  the  vibratory  theory  by  M.  Cauchy  and 
Mr.  Green — that  is  to  say,  a  wave  in  which  the  amplitude  of  oscilla- 
tion diminishes  in  proportion  to  an  exponential  function  of  the  distance 
from  the  bounding  surface  (called  by  M.  Cauchy  the  modulus),  and  which 
travels  along  that  surface  with  a  velocity  less  than  the  velocity  of  an 
ordinary  wave ;  the  square  of  the  negative  exponent  of  the  modulus  being 
proportional  to  the  difference  of  the  squares  of  those  velocities,  divided  by 
the  square  of  the  velocity  of  an  ordinary  wave. 

This  is  an  evanescent  wave  of  oscillation  round  transverse  axes. 

How  large  soever  the  coefficient  of  polarity  for  oscillations  round 
longitudinal  axes  may  be,  an  evanescent  wave  of  such  oscillations  may 
travel  along  the  bounding  surface  of  a  medium  with  any  velocity,  however 
slow,  provided  the  negative  exponent  of  the  modulus  is  made  large  enough. 
Consequently,  in  framing  the  formulas  to  represent  oscillations  round  axes 
parallel  to  the  plane  of  incidence,  we  must  introduce  in  each  medium  two 
such  evanescent  waves  of  suitable  exponents  and  indeterminate  amplitudes ; 
one  travelling  along  the  surface  with  the  incident  wave,  and  the  other  a 
quarter  of  an  undulation  behind  it.  The  maximum  amplitudes  of  oscilla- 
tion in  these  evanescent  waves  constitute  four  unknown  quantities ;  the 


AN  OSCILLATORY  THEORY   OF  LIGHT.  1G5 

amplitudes  in  the  two  ordinary  reflected  waves,  and  the  two  ordinary 
refracted  waves,  differing  by  one  quarter  of  an  undulation,  constitute  four 
more  unknown  quantities,  making  eight  in  all :  four  conditions  having 
been  fulfilled  by  the  waves  polarised  in  the  plane  of  incidence,  there 
remain  to  be  fulfilled  eight  conditions — viz.,  the  identity,  as  calculated 
by  the  formulae  for  the  first  and  second  substance  respectively,  of  the 
following  eight  functions  at  the  bounding  surface ;  the  angular  displace- 
ment, and  the  rotative  forces,  round  each  of  the  two  axes  in  the  plane 
of  incidence,  for  the  incident  wave,  and  the  set  of  waves  synchronous 
with  it,  and  for  the  set  of  waves  retarded  by  one  quarter  of  an  undula- 
tion. These  conditions  are  sufficient  to  determine  the  unknown  quantities, 
and  to  complete  the  solution  of  the  problem. 

The  following  is  a  general  statement  of  the  results  of  the  solution  when 
the  second  medium  is  the  denser.  They  agree  with  the  results  of  the 
experiments  of  M.  Jamin,  and  are,  in  every  respect,  analogous  to  those 
deduced  from  the  hypothesis  of  vibrations  by  M.  Cauchy,  Mr.  Green,  and 
Mr.  Haughton. 

Light  polarised  in  a  plane  perpendicular  to  the  plane  of  incidence,  suffers 
by  reflexion  at  a  perpendicular  incidence  no  alteration  of  phase. 

At  a  grazing  incidence  (or  when  the  angle  of  incidence  differs  insensibly 
from  90°),  the  phase,  like  that  of  light  polarised  in  the  plane  of  incidence, 
is  retarded  by  half  an  undulation. 

The  variation  of  phase  with  the  angle  of  incidence  is,  in  fact,  con- 
tinuous; but  it  is,  generally  speaking,  not  appreciable  by  observation, 
except  in  the  immediate  neighbourhood  of  an  angle,  called  by  M.  Jamin 
the  principal  incidence,  where  the  retardation  of  phase  is  a  quarter  of  an 
undulation. 

This  angle  differs  by  a  very  small  amount,  appreciable  only  in  certain 
substances,  from  the  polarising  angle,  at  which  the  intensity  of  light 
polarised  in  a  plane  at  right  angles  to  the  plane  of  incidence  is  a 
minimum. 

The  "  law  of  Brewster,"  that  the  tangent  of  the  polarising  angle  is  equal 
to  the  index  of  refraction,  is,  theoretically,  only  approximately  true;  but 
the  error  is  quite  inappreciable. 

When  the  second  medium  is  the  less  dense,  the  phase  of  the  reflected 
light  is  half  an  undulation  in  advance  of  its  value  when  the  second  medium 
is  the  denser. 

In  either  case,  light  polarised  in  planes  perpendicular  to  the  plane  of 
incidence  is  less  retarded — that  is  to  say,  is  accelerated  in  phase — as 
compared  with  light  polarised  in  that  plane,  according  to  the  following 
table : — 


166  AN  OSCILLATORY   THEORY  OF  LIGHT. 

Angle  of  Relative 

Incidence.  Acceleration. 

o 

Perpendicular  incidence,    .         .       0  \   undulation. 

Principal  incidence, \  undulation. 

Grazing  incidence,     .         .         .90  0 

In  the  case  of  total  reflexion,  light  polarised  in  planes  perpendicular  to 
the  plane  of  incidence,  lias  its  phase  more  accelerated  than  light  polarised 
in  that  plane,  by  an  amount  to  which  the  formulae  of  Fresnel  give  a  close 
approximation. 

The  proposed  hypothesis  has  not  yet  been  applied  to  reflexion  from 
doubly  refracting  crystals ;  but  there  can  be  little  doubt  that  it  will  be 
found  to  represent  the  phenomena  correctly. 


Section  VI. — Of  Circular  and  Elliptic  Polarisation. 

Light  polarised  in  a  plane  oblique  to  the  angle  of  incidence  is,  generally 
speaking,  elliptically  polarised  after  reflexion,  the  plane-polarised  com- 
ponents of  the  disturbance  being  in  different  phases. 

According  to  the  hypothesis  of  oscillations,  circularly  and  elliptically 
polarised  light,  being  compounded  of  oscillations  in  different  phases  round 
two  transverse  axes,  consist  in  a  sort  of  nutation  of  the  longitudinal  axis 
of  each  luminiferous  atom.  The  direction  of  this  nutation,  and  the  form 
of  the  circle  or  ellipse  described  by  the  ends  of  the  longitudinal  axes,  serve 
to  define  the  character  of  the  light.  The  ellipse  of  nutation  has  its  axes 
in  the  same  proportion  with,  but  perpendicular  in  position  to,  those  of  the 
elliptic  orbit  supposed  to  be  described  by  each  atom  according  to  the 
hypothesis  of  vibrations. 

The  molecular  mechanism  by  which  certain  media  transmit  *  right  and 
left-handed  circularly  or  elliptically  polarised  light  with  different  velocities, 
is  still  problematical,  according  to  either  hypothesis.  The  laws  of  the 
phenomena,  however,  may  be  represented  by  means  of  the  assumption^ 
that  in  the  substances  in  question  the  extraneous  load  on  the  luminiferous 
atoms  is  a  function  of  the  direction  of  nutation. 


Section  VII. — Of  Dispersion. 

If  we  assume  the  extent  of  sensible  direct  action  of  the  polarity  of  the 
luminiferous  atoms  to  be  appreciable,  as  compared  with  the  length  of  a 
wave,   the    velocity    of    propagation    (precisely    as   with   the    vibratory 


AN  OSCILLATORY  THEORY  OF  LIGHT.  167 

hypothesis)  is  found  to  consist  of  a  constant  quantity,  diminished  by  the 
sum  of  a  series  in  terms  of  the  reciprocal  of  the  square  of  the  length  of  a 
wave. 

It  may  be  doubted,  however,  whether  this  supposition  is  of  itself 
adequate  to  explain  the  phenomena  of  dispersion,  and  whether  it  may  not 
be  necessary  to  assume,  also,  that  the  load  upon  the  luminiferous  atoms  is 
a  function  of  the  time  of  oscillation,  as  well  as  of  the  nature  of  the 
substance  and  the  position  of  the  axes  of  oscillation. 

In  conclusion,  it  may  be  affirmed,  that,  as  a  mathematical  system,  the 
proposed  theory  of  oscillations  round  axes  represents  the  laws  of  all  the 
phenomena  which  have  hitherto  been  reduced  to  theoretical  principles,  as 
well,  at  least,  as  the  existing  theory  of  vibrations ;  while,  as  a  physical 
hypothesis,  it  is  free  from  the  principal  objections  to  which  the  hypothesis 
of  vibrations  is  liable. 


168  VELOCITY   OF  SOUND   IN  LIQUID  AND  SOLID   BODIES. 


IX.— ON  THE  VELOCITY   OF  SOUND  IN   LIQUID  AND  SOLID 

BODIES    OF    LIMITED    DIMENSIONS,    ESPECIALLY 

ALONG  PEISMATIC  MASSES  OF  LIQUID.* 

Intkoductory  Kemarks. 

1.  The  velocity  of  sound  in  elastic  substances  of  different  kinds,  solid, 
liquid,  and  gaseous,  has  been  made  the  subject  of  numerous  and  careful 
experiments,  most  of  which  are  well  known.  The  object  of  this  investi- 
gation is  to  determine  to  what  extent  our  present  knowledge  of  the 
condition  and  properties  of  elastic  bodies,  and  of  the  laws  of  elasticity, 
enables  us  to  use  those  experiments  as  data  for  calculating  the  elasticity 
of  the  materials ;  and,  also,  to  point  out  circumstances  which,  so  far  as  I 
am  aware,  have  been  insufficiently  attended  to,  if  not  altogether  overlooked, 
in  previous  theoretical  researches,  and  which  must  limit  our  power  of 
drawing  definite  conclusions  from  those  experiments,  until  our  knowledge 
of  molecular  forces  shall  be  in  a  more  advanced  state. 

2.  If  it  were  possible  for  us  to  ascertain  by  experiment  the  velocities 
of  transmission  of  vibratory  movements  along  the  axes  of  elasticity  of  an 
indefinitely  extended  mass  of  any  substance,  we  could  at  once  calculate 
the  coefficients  of  elasticity  of  that  material ;  for  in  such  a  mass  we  can 
assign  the  direction  of  vibratory  movement  corresponding  to  each  given 
direction  of  transmission,  and  consequently  the  nature  of  the  molecular 
forces  which  are  called  into  play,  and  whose  intensity  is  indicated  by  the 
velocity  of  transmission.  In  an  uncrystallised  medium,  for  instance,  the 
direction  of  vibration  must  either  be  exactly  longitudinal  or  exactly 
transverse  with  respect  to  the  direction  of  transmission,  so  that  we  can 
calculate  from  the  velocity  of  transmission  the  longitudinal  or  the  transverse 
elasticity,  as  the  case  may  be.  In  a  crystalline  medium  having  rectangular 
axes  of  elasticity,  the  directions  of  vibration,  though  not  always  exactly 
longitudinal  or  transverse,  unless  the  direction  of  transmission  coincides 
with  an  axis,  have  still  certain  definite  positions. 

3.  It  is  only  in  air  and  water,  however,  that  such  experiments  are 
possible.  For  other  substances,  the  best  experiments  which  it  is  prac- 
ticable for  us  to  make,  are  those  upon  the  transmission  of  nearly  longi- 
tudinal vibrations  along  prismatic  or  cylindrical  bodies.     Were  we  able 

*  Eead  before  the  British  Association  at  Ipswich,  July  3,  1851,  and  published  in  the 
Cambridge  and  Dublin  Mathematical  Journal,  Nov.,  1851. 


VELOCITY   OF   SOUND   IN    LIQUID   AND   SOLID   BODIES.  169 

to  ensure  that  the  vibrations  of  those  prisms  and  cylinders  should  be 
exactly  longitudinal,  we  might  compute  from  their  velocity  of  transmission, 
as  from  that  of  such  vibrations  in  an  unlimited  mass,  the  true  longitudinal 
elasticity.  This  we  can  do  for  gaseous  substances,  as  M.  Wertheim  has 
proved  (Ann.  de  Chim.  et  de  Phys.,  Se>.  III.,  torn,  xxiii.),  by  making  the 
organ-pipes  in  which  they  vibrate  of  proper  construction. 

In  liquid  and  solid  columns,  on  the  other  hand,  it  is  impossible  to 
prevent  a  certain  amount  of  lateral  vibration  of  the  particles,  the  effect 
of  which  is  to  diminish  the  velocity  of  transmission  in  a  ratio  depending 
on  circumstances  in  the  molecular  condition  of  the  superficial  particles, 
which  are  yet  almost  entirely  unknown. 

4.  It  has,  indeed,  been  sometimes  supposed,  that  the  coefficient  of 
elasticity,  as  calculated  from  the  vibrations  of  a  solid  rod,  is  that  called 
the  weight  of  the  modulus  of  elasticity — that  is  to  say,  the  reciprocal  of  the 
fraction  by  which  the  length  of  a  rod  is  increased  by  a  tension  applied 
to  its  ends  of  unity  of  weight  upon  unity  of  area;  that  coefficient  being- 
less  than  the  true  coefficient  of  longitudinal  elasticity,  because  the  lateral 
collapsing  of  the  particles  enables  them  to  yield  more  in  a  longitudinal 
direction  to  a  given  force,  than  if  their  displacements  were  wholly 
longitudinal. 

This  conjecture,  however,  is  inconsistent  with  the  mechanics  of  vibratory 
movement;  and,  accordingly,  experiment  has  shown  that  the  elasticity 
corresponding  to  the  velocity  of  sound  in  a  rod  agrees  neither  with  the 
modulus  of  elasticity,  nor  with  the  true  longitudinal  elasticity;  although 
it  is,  in  some  cases,  nearly  equal  to  the  former  of  those  quantities,  and  in 
others  to  the  latter. 

5.  In  liquids,  it  has  been  shown  by  the  experiments  of  M.  "Wertheim 
(Ann.  de  Chim.  el  de  Phys.,  Se>.  III.,  torn,  xxiii.),  that  the  velocity  of  sound 
in  a  mass  contained  in  a  trough,  and  set  in  motion  through  an  organ-pipe, 
bears  to  that  in  an  unlimited  mass  the  ratio  of  ^2  to  y/3.  This  has  led 
him  to  form  the  conjecture,  that  liquids  possess  a  momentary  rigidity 
for  very  small  molecular  displacements  as  great  in  comparison  with  their 
other  elastic  forces  as  that  of  solids.  This  conjecture,  paradoxical  as  it 
may  seem,  would  indeed  be  necessary  to  account  for  the  facts,  if  the 
supposition  I  have  already  mentioned  were  true,  that  the  velocity  of 
sound  in  a  rod  depends  upon  the  modulus  of  elasticity.  I  shall  show, 
however,  in  the  sequel,  that  if  we  suppose  that  at  the  free  surface  of 
every  mass  of  liquid,  an  atmosphere  of  its  own  vapour  is  retained  by 
molecular  attraction  under  certain  conditions  of  equilibrium,  the  ratio 
mJ2  :  *J3  between  the  velocities  of  sound  in  a  prism  and  an  unlimited 
mass,  is  a  consequence  of  the  equations  of  motion  in  all  cases  in  which 
the  liquid  has  any  rigidity  whatsoever,  even  although  so  small  as  to  be 
insensible  by  any  means  of  observation;  so  that  the  supposition  of  a 


170  VELOCITY  OF  SOUND    IN  LIQUID  AND    SOLID   BODIES. 

rigidity  for  small  displacements  equal  to  that  of  solids  becomes  unneces- 
sary. 

6.  With  respect  to  solids,  all  that  theory  is  yet  adequate  to  show  us 
is,  that  the  velocity  of  sound  along  a  rod  must  be  less  than  in  an  unlimited 
mass,  a  conclusion  in  accordance  with  experiment.  The  precise  ratio 
depends  on  properties  of  the  superficial  particles  yet  unknown. 


General  Equations  of  Vibratory  Movement  in  Homogeneous 

Bodies. 

7.  Having  now  stated  generally  the  objects  of  this  paper,  I  shall 
proceed,  in  the  first  place,  to  the  mathematical  investigation  of  the  integrals 
of  the  general  differential  equations  of  vibratory  movement  in  homo- 
geneous bodies;  because,  although  those  equations  have  already  been 
integrated  by  many  mathematicians,  it  will  be  necessary  in  this  paper  to 
introduce  functions  into  the  integrals  which  have  hitherto  been  almost 
totally  neglected  in  such  researches;  having  been  applied  only  to  the 
theory  of  waves  rolling  by  the  influence  of  gravity,  to  that  of  total 
reflexion,  by  Mr.  Green  (Camb.  Trans.,  Vol.  VI.),  and  by  Professor  Stokes 
to  represent  the  gradual  extinction  of  sound  by  its  conversion  into 
heat. 

8.  Let  g  represent  the  accelerating  force  of  gravity  : 

D  the  weight  of  unity  of  volume  of  a  homogeneous  substance,  having 
orthogonal  axes  of  elasticity  whose  directions  are  the  same  throughout 
its  extent ; 

Ap  A2,  A3,  the  coefficients  of  longitudinal  elasticity  for  the  axes  of  x, 
y,  z,  respectively : 

B1?  B2,  B3,  the  coefficients  of  lateral  elasticity ;  and 

Cv  C2,  C3,  those  of  rigidity  for  the  planes  of  yz,  zx,  xy,  respectively ; 

£,  7),  £,  the  displacements  of  a  particle  parallel  to  x,  y,  z,  respectively. 

Then,  it  is  well  known  that  the  differential  equations  of  small  vibratory 
movements  are  the  following,  when  small  quantities  of  the  second  order 
are  neglected : 

.       /     D    d?  d?    ,   n    d?    ,  n    d?\t 


g    dt2         1dz2         3dy 

d'2  n  d2? 

+  (B3  +  C3)  -£4-  +  (B2  +  C2)  f-±- 

x  •*         6I  dz  dy      v   l         u  dz  dz 

_/     D    d?       «    <?  d*    LP    i2\ 

~  Vlj'W  +  ^da*  +     2dy*  +  Ulrf7V 


(1.) 


VELOCITY  OF  SOUND  IN  LIQUID  AND  SOLID   BODIES.  171 


\      g    df         "dxl         ldy         "dz-/ 

of  which  the  integrals  are 

I  =  ^{L^i^e.t  +  ax  +  (3y  +  yz+  k)} 
n  .=  S  {L2  0(^£  J  +  aa  +  j8y  +  y  2  +  *)} 
£  =  2{L30(Ve.*  +  a^;  +  /3y  +  ys  +  *)}• 


^    •     (I-) 


Y    ■     (2.) 


The  form  of  the  function  <£  being  arbitrary,  subject  to  a  restriction 
to  be  afterwards  referred  to,  and  2  extending  to  any  number  of  terms, 
the  coefficients  of  which  fulfil  the  following  conditions.     Let 


w1  =  Ax  a2  +  C3  /32  +  C2  y2 

w,  =  C3a2  +  A.2J82  +  C1y2 
u,3  =  C2a2  +  C1/32  +  A3y- 

Pl  =  ^1.+  C1)i87 

Pi  —  (B2  +  C2)  7  a 

p3  =  (B3  +  C3)«/3 
9 


(a.) 


E 


Then  the  following  equations  must  be  satisfied  by  the  coefficients  of 
each  set  of  terms  in  equation  (2) : 

0  =  Lx(Wl  -  E)  +  L2p3  +  L3p2  - 

0  =  LlP3  +  L2(a>2-E)  +  L3/3l    [     .         .     (3.) 

0  =  Ll  Pi  +  L2  P\  +  L3  (W3  —  E)-  „ 

By  elimination  we  transform  those  equations  as  follows :  let 

G  =  wx  +  w2  +  (o3, 

H  =  (o.2  w3  +  (osw1-\r  wx  Wo  —  p^  —  p2   —  Pz>  r*   (b-) 

K  =.  wx  Wo  <*)z  +  2  pj  p2  p3  —  wx  p*  —  w2  p2   —  h>z  p32.  J 


172  VELOCITY  OF   SOUND  IN   LIQUID  AND    SOLID  BODIES. 

Then  for  each  set  of  values  of  a,  /3,  y,  E  has  three  values  which  are 
the  roots  of  the  cubic  equation, 

0  =  E3-GE2  +  HE-K;     .         .         .     (4.) 

so  that  ^e  has  six  values,  three  positive  and  three  negative,  of  equal 
arithmetical  amount. 

The  absolute  values  of  Lv  L2,  L3,  are  arbitrary,  but  their  mutual  ratios 
are  fixed  by  the  following  equations  : 

L1{(w1-E)^1-p:,p.i}=L2{(w2-E)p.2-p3p1}    "J 

consequently,  they  have  in  general  three  sets  of  ratios  for  each  set  of 
values  of  a,  ft,  y,  corresponding  to  the  three  values  of  E. 

9.  The  condition  that  the  motions  of  the  particles  of  the  body  must  be 
small  oscillations  restricts  the  variations  of  the  displacements  £,  rj,  £,  within 
certain  limits.  Now,  as  the  time  t  increases  ad  infinitum,  this  can  be 
fulfilled  only  when  each  of  those  quantities  is  either  a  periodical  circular 
function  of  t,  or  a  function  developable  into  a  sum  or  definite  integral 
of  such  functions.  We  may,  therefore,  make  each  of  the  functions  <j>  a 
trigonometrical  function  of  t.  This  being  the  case,  those  functions  must 
be  either  trigonometrical  or  exponential  with  respect  to  x,  y,  and  z,  or 
compounded  of  both,  being  trigonometrical  so  far  as  a,  (3,  y,  are  real, 
and  exponential  so  far  as  they  are  imaginary. 

If  we  suppose  each  of  these  coefficients  to  consist  of  a  real  and  an 
imaginary  part,  then  each  of  their  functions  which  enters  into  the  equations 
of  condition,  will  also  consist  of  a  real  and  an  imaginary  part.  Each  of 
the  equations  of  condition  thus  becomes  divided  into  two,  which  must  be 
separately  satisfied. 

Thus  we  arrive  at  the  following  results : 

For  the  symbol  <£  {  },  put  e2W-l  {  };  so  as  to  make  £,  &c,  trigono- 
metrical with  respect  to  t.     Let  X  be  a  line  of  such  a  length  that 


and  let 


a?  +  l2  +  c2=  1, 


a  =  z-(+a—  a  J  —  1), 
0  =  £(T&-&V-1). 


y  =  <(  +  c~  cV~  !) 


VELOCITY   OF  SOUND   IN   LIQUID   AND  SOLID  BODIES.  173 

also,  let 

Li  =  I  +  l's/  —  1>  L2  —  m  +  m ' s/  —  li  L3  =  w  +  n'*/  —  1 ; 
so  that  the  displacements  become 


-{a'x  +  b'y  +  c'zdz V-l(Ve-  i-ax-Jy-cs)}     | 


rj  =  S  {terms  in  m,  m],    £  =  2  {terms  in  w,  w'} 


>  a) 


J 


Let  the  quantities  in  the  equations  of  condition  be  thus  represented 

«i  -i\  ± iAV-  !>  &c- ;   pi  =  &  ±  2W-  *>  &c- ; 
g  =  g  ±  aV-  i ;  H  =  &  ±  &V-  1 ;  K  =  k  ±  kV-  1- 

The  equations  of  notation  now  become 

Pl  =  Ax  (a2  -  a'2)  +  C3  (62  -  6'2)  +  C2(c2  -  c'2) 
^2  =  C3  (a2  -  a'2)  +  A2(62  -  b'2)  +  Cx  (c2  -  c'2) 
j?3  =  C2  (a2  -  a'2)  +  C^fi2  -  6'2)  +  A3(c2  -  c'2) 
<?1  =  (B1  +  C1)(&c-6V) 

#2  =  (^2  +  ^2)  (ca~c' a') 

q3  =  (B3  +  C3)(ab-a'b') 
Pi  =  2  (Ax  a  a'  +  C3  6  6'  +  Co  c  c') 
p2  =  2  (C3  a  a'  +  A26  6'  +  Cj  c  c') 
j93'  =  2  (C2a a'  +  Cxb b'  +  A3c c') 
?1'  =  (B1  +  C1)(6c'  +  6/c) 
?2'  =  (B2  +  C2)(ca'  +  c'a) 
?3'=(B3  +  C3)(a&'  +  a'£) 

—  =  Las  before,  or  c  =  -^r 
9  D 

q  =Pl+P2+  p3,    q'  =pj  +  2h  +  #$' 

&  =  P2#5  +^1  +  ^l2>2  ~  ^l2  ~  ?22  -  ?32 

-P2P&-P3P1-P1P2  +  9i2  +  <lil  +  9z2 

#  =P2P3  +P2P3  +PzP\  +  #$>i  +  Pi  Pz+PlPi 
-2q1q1,-2q.2q2'-2q2q3' 

H  =plp.2pz+2qlq2qs-p1q12-p2q22-p.3q^ 


Mc) 


He) 


174  VELOCITY  OF   SOUND   IN  LIQUID  AND  SOLID  BODIES. 

-ftft'ft'  -  ft'ft  ft'  "ft'ft'ft 

+Pi  ft'2  +  ft  ft'2  +  ft  ft'2  +  2  (ft.'  ft  ft'  +  ft'ft  ft'  +  ft'  ft  ft') 

K=pMp&  +  ^^'^Is-Pi^-P^-Pil^ 

-  ft'ft  ft  -  ftft'ft  -  Pi  P2P3 

-  2  (fc'ft  ft  +  ft  ft' ft  +  ft  ft  ^s') 

+  Kft2  +  iVft2+Kft2  +  2(/?lftft'  +  ^ftft'+^3ftft')- 

Also,  let 

*!  =  (^  -  E)  qx  -  p{qx'  -  ftft  +  ft'ft' 
*2  =  (ft  -  E)  ft  "  ft'ft'  -  ft  ft  +  ft'ft' 
*3  =  (ft  ~  E)  ft  -  ft'ft'  ~  ft  ft  +  ft'ft' 
%l  =  (^  -  E)  jfc'  +  ^'ft  -  q2  2s  -  ft'ft 
X2'  =  (p2  -  E)  q2'  +  ^2'<72  -  ft  ?/  -  ?3'ft 

*s  =  (ft  -  E)  ft'  +  ft'ft  -  ft  ft'  -  ft'ft  J 

Then  the  equations  of  condition  relative  to  the  coefficients  become  the 
following : — 


0  =  E8-flE2+  |jE-h, 

0  =  g'E2  -  b'E  +  k',       • 
ltx  +  1'X{=  m%2  +  m'X2'  =  nxz  +  w%'  "| 
lx{-  l'Xx  =  mXo-  m'%2  =n%&' -  n'xz.  J 


(8.) 
(9.) 

(10.) 


The  three  original  'equations  of  condition  are  transformed  into  the 
following  six,  to  which  (8),  (9),  (10)  are  equivalent : — 


0  =  I  (px  —  E)  -r  I'pi  +  mqs  +  m'q3'  +  nq2  +  n'q.2 
0  =  l'(Pi  —  E)  —  lpx  +  m'qz  —  mqs'  +  n'q2  —  nq2 
0  =  I  qz  +  l'qz'  +  m  (p2  —  E)  +  m'p2  +  nqx  +  n'qx 
0  =  l'qs  -  lqz'  +  «i'(ft  ~  E)  -  mpz'  +  n'qx  -  nqx 
0  =  lq2  +  l'q2  +  mqx  +  m'q-l  +  n{p3  -  E)  +  ripz 
0  =  l'q2  -  lq2  +  m'qx  —  mqx  +  n'(ps  —  E)  —  npz 


y  (10A.) 


To  give  an  intelligible  result,  the  terms  of  the  series  in  equation  (7) 
must  be  taken  in  pairs,  with  the  imaginary  exponents  in  each  pair  of 
equal  arithmetical  value  and  opposite  signs. 


VELOCITY   OF    SOUND  IN   LIQUID  AND  SOLID  BODIES. 
Hence  equations  (7)  are  equivalent  to  the  following : — 

—  {a'x  +  b'y  +  cz)  f  2  tt  . 

ex  -J  I.  cos-—  (^Z e  .  i  —  ax  —  by  —  cz) 

2-rr  ")  "1 

+  T .  sin  —  (^/e  .t  —  ax  —  by  —  cz)r 

r\  =  S  {terms  in  w,  m'},    £=  2  {terms  in  w,  ft'} 


175 


^  (H-) 


The  above  equations  (11),  together  with  the  equations  of  condition  (8), 
(9),  (10),  or  their  equivalent  (10A),  and  the  equations  of  notation  (c), 
contain  the  complete  representation  of  the  laws  of  small  molecular 
oscillations  in  a  homogeneous  body  of  any  dimensions  and  figure ;  it  being 
understood  that  in  the  symbol  of  summation  2  are  included  as  many 
definite  integrations  as  the  problem  may  require  with  respect  to  inde- 
pendent variables  of  which  the  coefficients  A,  ^  z,a,b,c,a'  ,b'  ,c  ,l,m,n,l'  ,m  ,n , 
are  functions. 

As  there  are  fourteen  coefficients,  connected  by  seven  equations — viz., 
a?-\-b2-{-c2=l,  and  the  six  equations  of  condition,  the  greatest  number  of 
independent  variables  is  limited  to  seven ;  therefore,  in  the  most  general 
case, the  symbol  2  {...}  in  equations  (11)  may  be  replaced  by 

^ffflffl^v  0»  %,  ev  dp  e6,  e7)  { . . }  dov  aev  ae9  de„  ae„  ae&  ae7,  (i  2.) 

0V  &c,  being  variables  of  which  the  coefficients  are  functions,  and  F  an 
arbitrary  function.* 

10.  Let  us  consider  the  physical  meaning  of  a  single  set  of  terms  of 
the  sums  in  equations  (11),  containing  but  one  set  of  values  of  the 
coefficients.  It  represents  a  system  of  plane  waves,  the  wave  surfaces, 
or  planes  of  equal  phase,  in  which  are  normal  to  the  line  whose  direction- 
cosines  are  a,  b,  c.     A  is  the  length  of  a  wave  measured  along  that  line. 


2  7T  I 

-7T-  (*Je  .t  —  ax  —  by  —  cz)-\-  tan-1  -y 

~t-  (s/s  .t  —  ax  —  by  —  cz)  +  tan-1  — > 
A  '  m 


—  (*/s  .t  —  ax  —  by  —  cz)  +  tan-1  -7 


are  the 

phases  of 

vibration 

for 


%, 


*To  make  the  functions  in  equations  (11)  satisfy  the  conditions  of  equilibrium, 
instead  of  those  of  oscillation,  it  is  only  necessary  to  make  s  =  0,  and  to  substitute 
h=0,k'  =  0,  for  equations  (8)  and  (9).  Some  additional  functions,  however,  are  necessary 
in  order  to  complete  the  values  of  £ ,  n,  £. 


176  VELOCITY   OF  SOUND   IN    LIQUID  AND  SOLID  BODIES. 

^/e  =  J(—^- )  is  the  normal  velocity  of  propagation,  —j-  is  the  periodic 
time  of  an  oscillation  of  a  particle 

and  the  corresponding  expressions  in  m  and  n  are  the  semi-amplitudes  of 
vibration  parallel  to  x,  y,  z,  respectively;  a,  b\  c,  are  proportional  to  the 
direction-cosines  of  a  normal  to  a  series  of  planes  of  equal  amplitude  of 
vibration. 

The  trajectory  of  each  particle  affected  by  a  single  series  of  plane 
waves  is  in  general  an   ellipse,  the  position  and  magnitude  of  which  are 

2  7T 

found  as  follows.     Let  (p0  denote  the  value  of        (y/e .  t  —  ax  —  by  —  cz), 

which  makes  the  total  displacement  ^/{^  +  tj2  +  £2)  a  maximum  or 
minimum.     It  is  easily  seen  that 

where 

J2  +  m2  +  n2  _  p  _  m>2  _  nt 

1 1'  +  in  m  +  /i  n' 

The  values  of  £,  tj,  ^,  calculated  from  <f>0  by  equations  (11),  are  the 
co-ordinates  of  the  extremities  of  the  axes  of  the  elliptic  trajectory,  referred 
to  the  natural  position  of  the  particle  as  origin. 

The  processes  of  summation  and  definite  integration  denote  the  repre- 
sentation of  an  arbitrary  manner  of  oscillation  by  the  combination  of  a 
definite  or  indefinite  number  of  systems  of  plane  waves. 


Case  of  an  Indefinitely  Extended  Medium. 

11.  Let  the  medium,  in  the  first  place,  be  supposed  to  be  indefinitely 
extended  in  all  directions.  This  case  having  been  thoroughly  investigated 
by  MM.  Poisson,  Cauchy,  Green,  MacCullagh,  Haughton,  Stokes,  and 
others,  I  shall  give  merely  an  outline  of  the  general  results.  The  condition 
that  the  motion  shall  consist  of  small  oscillations,  here  makes  it  necessary 
that  the  exponential  factor  in  the  displacements  should  in  all  cases  be 
equal  to  unity,  and  therefore  that 

a'  =  0  ;     V  =  0 ;     e  =  0  j 


VELOCITY  OF   SOUND  IN   LIQUID   AND   SOLID   BODIES. 


177 


and,  consequently",  each  of  the  accented  symbols  in  equations  (c)  =  0. 
Equation  (9)  vanishes,  and  the  normal  velocity  of  propagation  for  each 
set  of  direction-cosines  a,  b,  c,  has,  generally  speaking,  three  values,  corre- 
sponding to  the  three  values  of  E,  roots  of  equation  (8).  Equations  (10) 
become 


,  ,,       ,      ,       1     1     1 

I  :  m  :  n  : :  I   :  m   :  n   : :  -  :  -  :  — . 

r,    X.,   r. 


•     (14.) 


consequently,  the  phases  of  £,  rj,  £,  are  simultaneous;  so  that  >v/(^2  +  Z/2), 
x/(m2  +  m'2),  *J(n2  +  «'2),  are  proportional  to  the  direction-cosines  of  a 
rectilinear  vibratory  movement  of  the  semi-amplitude  +J(l2-\-l'2-\-m2  +  m'2 
-\-n2-\-n'2),  which  cosines  have  in  general  three  sets  of  values  corresponding 
to  the  three  values  of  E.  It  is  easily  shown  that  those  three  directions 
are  at  right  angles  to  each  other.  The  number  of  coefficients  being  in 
this  case  reduced  to  eleven,  connected  by  six  equations — viz.,  a2  +  62+c2=  1, 
equation  (8),  and  the  proportional  equation  (14),  which  is  equivalent  to 
four,  the  greatest  number  of  definite  integrations  in  the  operation  (12) 
is  restricted  to  five. 

Thus  it  appears  that  the  velocity  of  transmission  of  vibratory  movement 
through  an  indefinitely  extended  mass,  has  a  set  of  definite  values,  not 
exceeding  three,  for  each  position  of  plane  waves.  When  the  direction 
of  propagation  coincides  with  an  axis  of  elasticity,  we  find  those  values 
to  be : 


For 

vibrations 

parallel 

to 

Velocity  of  propagation  along 

X      . 

X 

m 

% 

m 

V     ■     ■ 

m 

m 

m 

z     .     . 

m 

m 

V(¥) 

(15.) 


When  the  substance  is  equally  elastic  in  all  directions,  we  have  simply, 
Velocities  of  propagation  in  any  direction  for  longitudinal  vibrations 


V(t& 


178  VELOCITY  OF   SOUND   IN   LIQUID  AND   SOLID  BODIES. 

For  transverse  vibrations,  in  any  direction  perpendicular  to   that  of 
propagation, 


m- 


Hence,  experiments  on  the  velocity  of  sound  in  an  indefinitely  extended 
mass,  or  one  so  large  as  to  be  practically  such,  afford  the  means  of  directly 
calculating  the  coefficients  of  elasticity. 


General  Case  of  a  Body  of  Limited  Dimensions. 

12.  It  is  not  so,  however,  in  a  body  of  limited  dimensions;  for  the 
coefficients  a,  V ,  c,  in  the  exponents  of  the  exponential  factors,  are  no 
longer  necessarily  null,  but  have  values  which  must  depend  on  the  mole- 
cular condition  of  the  external  surface  of  the  body,  and  on  the  forces 
applied  to  it.  The  velocity  of  propagation  is  no  longer  a  function  of  the 
direction-cosines  a,  b,  c,  alone,  but  also  of  the  coefficients  a,  b',  c.  It 
has  in  general  but  one  value,  corresponding  to  the  common  root  of  the 
equations  (8)  and  (9).  By  substituting  successively  the  two  roots  of 
equation  (9) — viz., 

6'  +   \(\r-     r 


for  E  in  equation  (8),  the  latter  is  converted  into  two  alternative  equations 
between  the  six  quantities  a,  b,  c,  a',  b',  c\  showing  the  relations  they  must 
have  in  order  that  equations  (8)  and  (9)  may  have  a  common  root. 
In  the  only  particular  problems,  however,  of  which  I  shall  here  give  the 
solutions,  those  relations  are  obvious  without  going  through  that  process, 
for  they  belong  to  a  class  of  cases  in  which  the  three  quantities  g',  ])',  f{', 
have  a  common  factor;  which  being  made  =  0,  the  necessary  conditions  are 
fulfilled. 

It  is  obvious  that  in  all  cases  the  effect  of  the  coefficients  a',  b',  c,  is 
to  diminish  the  velocity  of  propagation. 

1 3.  The  following  are  the  values  of  the  three  components  of  the  velocity 
of  a  particle  : 

2*  1 

d£     JT2ir    ,       y {a'z  +  b'y+e'z)  C  ^ 

^7=2I  -y VE-C  {  -lsm—(^/£.t-ax-by-cz) 

+  V  cos  ~  (Jt .  t-  ax  -  by  -  cz)  }  ]  f  (16-) 
-t.—^  {terms  in  m,  m)  ;      y-  =  2  {terms  in  n,  n). 


VELOCITY   OF   SOUND   IN   LIQUID  AND  SOLID  BODIES.  179 


The  strains,  or  coefficients  of  relative  molecular  displacement,  are  as 
follows : 


Along 


T2 


tr  —{a'x  +  b'y  +  c'z) 


Longitudinal  Strains. 

■j  {la  —I' a)  cos—  (y/s  .t  —  ax— by— cz) 
-\- (la +  1' a')  sin  —(^e.t  — ax—  by  — cz)  r 


(1 11 

-p  =  2  {terms  in  (in  V  —  m  b)}   (m b  +  m' V) } 

7  Y 

-5-  =  2 {terms  in  (nc'—n'c),  (nc-\-n'c)} 


(17.) 


In  the 
plane 

yz 


xy 


z^dy-ZL 


Distortions. 


2tt  —  (a'x  +  b'y  +  c'z) 


dz     di  1 

C  /  27T 

1  (mc'—m'c  -\-nV —  n'b)  cos  —  (^A .  t— ax— by— cz) 

+  (wc  +m'c'-f-7i5  +?''^/)sin  —  (^/s.t— ax— by— cz)  r 

dZ     d£ 

-j- +  -7- =2  {terms  in  (na'  —  na  -\-lc—l'c), 

(na  -{-n'a  -\-lc-\-l'c')} 

jy         7 

j- -+— =  2 {terms  in  (Ib'  —  l'b  -\-ma  —rn'a), 

Lb  if  Cv  JO 

(lb  -\-l'b'-\-ma  +mV)} 


J,  (17 A.) 


The  pressures  on  the  co-ordinate  planes,  arising  from  those  strains 
(using  the  notation  of  my  paper  on  the  Laws  of  Elasticity,  Cambridge  and 
Dublin  Mathematical  Journal,  February,  1851),  are  the  following: — 


On 


yz 


Normal. 


n 


*>  —  **f.-*- 


— -  —  B„  - 
dy         2  d 


JZ 


180  VELOCITY  OF   SOUND   IN   LIQUID   AND   SOLID   BODIES. 


L2tt  —(a'x  +  b'y  +  c'z)  (/.,,,       r  , 
~ex  |  [A^la  -la) 


\  2  7T 

+  B3(m V  -  m'6)  +  B2(« c'  -  n'en  .  cos  -zr-^e.t  —  ax  —  by-c .:) 
+  (Ax(/a  +  Va)  +  B3(m6  +  m  V)  +  B2(» c  +  »' c')) 


sin  -  -  (Je  .t  —  a  x  —  ly  —  e  .: 
A 


)}] 


(18.) 


The  other  normal  pressures  are  found  by  substituting  symbols  accord- 
ing to  the  following  table  : — 

Plane.  Pressure.  Coefficients. 

yz     .     .     .     Px     .     .     .     Alt     B3,     B2, 

zx    .     .     .    P2    .     .     .     B3,     A2,     Bv 


-■'  y 


B2,     Bp     A3. 


Tangential  I'r- 


Plane  of 
Distortion      Along      On  the  plane 


"- 


xy 


fa 


y*1 

y*J 


*=-*®+fi 


Case  of  an  Uncrystallised  Medium. 

14.  I  shall  now  take  the  particular  case  of  an  uncrystallised  medium, 
in  which  the  coefficients  of  elasticity  are  the  same  for  all  axes,  and  may 
be  represented  thus  : — 

rigidity  =  C ;    fluid  elasticity  =  J ; 
longitudinal  elasticity  A  =  3  C  +  J, 

lateral  elasticity  B  =  C  +  J  =  A—  2  C. 


VELOCITY   OF   SOUND  IN   LIQUID  AND  SOLID   BODIES.  181 

The  position  of  the  axes  being  in  this  case  arbitrary,  I  shall  take  the 
direction  of  propagation  as  the  axis  of  x,  so  as  to  make 

o=l,    b  =  0,   c  =  0. 

To  fulfil  the  condition  that  equations  (8)  and  (9)  shall  have  common 
roots,  we  must  make 

a  =  0, 

being  in  this  case  a  common  factor  of  g',  |j',  fe'. 
The  equations  of  notation  (c)  now  become 

i72  =  -A6'2  +  C(l-c'2) 
^3  =  -Ac'2  +  C(l  -V1) 
2l  =  -(A-C)6V;  q2  =  0;  q,  =  0 

g  =  (A  +  2C)(l-Z/2-c'2) 
ft  =  (2AC  +  C2)(1  -b'2-c'-2)2 
fe  =AC2(1  -6'2-c'2)3 
g'  =0;  jj'=0;   fe'=0 

'1  =  (Pi  —  E)?i  +  fe'&'i  r2  =  °;  rs  =  °; 

Xi=0;  x:=(p2  -  e)?2'-  &'&;  rs,=  (#$-  E)&'-  &&'•  ^ 

Hence  it  appears,  that  for  an  uncrystallised  medium,  equation  (8)  has 
three  roots,  viz., — 


r-  (d.) 


one  root         .         .     E  =  A 
two  equal  roots,  each  E  =  C 


i(i  -  &,a  -  n,  \ 

)  (1    _  tf*  _  C'2).    J 


(19.) 


So  that  the  velocity  of  propagation  is  less  than  that  in  an  unlimited  mass, 
in  the  ratio  ^/(l  —  &'2  —  c'2)  :  1.     Equation  (9)  disappears. 
Equations  (10)  become 


or 


lx1  =  m'x%  =  w'r3' 

— Vx1  =  ror2'  =  nx3' 
I  :  m   :  n'  : :  —  I'  :  m  :  n 


(20.) 


i.i.I  ! 

ti    fa    r'3J 


(20A.) 


182  VELOCITY  OF  SOUND   IN   LIQUID   AND   SOLID   BODIES. 

Equations  (10A)  become 

0  =  I  {px  —  E)  +  m'jg  +  riq2' 
0  =  r(jpj  -  E)  -  m  jg'  -  n  j,' 
0  =  Z'gs'  +  m  (jp2  —  E)  +  n  qx 
0  =  —  lq^  +  m  {p,  —  E)  +  »'& 
0  =  l'q2'  +  m  q1  +  n  (p3  -  E) 
0  =  -  lq2'  +  m'q1  +  n'(jp3  -  E)  ^ 

15.  It  may  be  shown  that  the  vibrations  corresponding  to  the  roots  C 
(1_Z/2_C'2)  cannot  take  place  in  a  body  of  which  the  surface  is  free, 
unless  V  =  0,  c  =  0,  in  which  case  they  are  reduced  to  ordinary  transverse 
vibrations.     (See  Appendix,  No.  II.) 


Nearly-longitudinal  Vibrations  in  an  Uncrystallised  Medium. 

16.  For  the  present,  therefore,   I   shall   confine    the  investigation    to 

the  root 

E  =  A  (1  -  V-  -  c'2), 

corresponding  to  the  velocity  of  propagation 

V—V{^<1-'* -*">}•    •     •  (21-> 

The  vibrations  to  which  this  root  is  applicable  may  be  called  nearly-longi- 
tudinal;   because  in  them  the  longitudinal  component  predominates,  and 
their  velocity  of  transmission  is  a  function  of  the  longitudinal  elasticity  A. 
This  value  being  substituted  for  E  in  the  expressions  for  Xv  &c,  gives 


'  =  —  V I ;    n  =  —  c  I  } 
.  =  V  X ;       ri  =  c  X.     J 


.     (22.) 


Which  values  being  substituted  in  equations  (11),  (16),  (17),  (18),  give 
the  following  results  : — 
For  brevity's  sake,  let 


also,  let 


$  =  2  {  — e^  (Z'  cos  $  -  I  sin  0)  J- . 


VELOCITY  OF   SOUND   IN   LIQUID   AND   SOLID  BODIES.  183 


Then  the  displacements  are 


£  =  2  {e*  (Z  cos  (f>  +  V  sin  <p)}  = 
n  =  2  {6'  e^  (I'  cos  0  —  Z  sin  <£)}  = 
£  =  2  {c'  e^  (/'  cos  0  -  J  sin  0)}  = 


da 
<Zs 


(23.) 


The  velocities  of  the  particles  are  : 

^5  =  2  |  ^  ^  •  «*  (*'  cos  0  -  Z  sin  0)  } 

^=-2J^r/v/^&'^Gcos0  +  rsin0)}     ["      (24.) 


dZ 


=  -  ?  \  ^  +/e .  c  e*  (I  cos  $  +  I'  sin  0)  J- 


The  longitudinal  strains  : 


dj 

dx 


=  -  2  j  ^e^  (Z'  cos  0  -  Z  sin  0)  J- 

^  =  2  |  ~6'2  e*  (Z'  cos  0  -  Z  sin  0)  j 

^  =  2  j  ^  c"2  ^  (r  cos  0  -  Z  sin  0)  } 
a  z  (.    A  J 

The  total  change  of  volume  : 
.xdvdz  IX  j 


y  (25.) 


The  distortions : 


^  +  ^  =  22  j  ^-h'c'e*  (V  cos  rf>  -  Z  sin  0)  } 
<Z,s       tfy  l  A  J 

^  +  ^  =  22J^c'^(Zcosri  +  Z'sin9f>)}        V     (25.) 
dx       dz  I    A  J 

^  +  ^  =  22  (  ~  b'e+{l  cos  0  +  r  sin  0)  } 
dy       dx  IX  J       J 


>  (26.) 


184  VELOCITY  OF  SOUND  IN  LIQUID  AND  SOLID  BODIES. 

The  pressures  due  to  the  displacements  are  as  follows : 

Normal  Pressures. 
Pj  =  2r^r^{A(l-P-c/2)  +  2C(&'2  +  c'2)}(r  cos  0-Z  sin  tf>)J 

P2  =  2r^e^{A(l-&'2-r'2)-2C(l-c'2)}(rcos</>-?sin0)] 

P3  =  2r~7re^{A(l-i/2-c'2)-2C(l-^2)}(rcos«/,-^sin^ 

The  tangential  pressures  %  Q2,  Q3,  are  found  by  multiplying  the 
distortions  by  —  C. 

Let  Rj,  R2,  R3,  be  the  three  components  of  the  pressure  exerted  by 
the  particles  of  the  body,  in  consequence  of  the  molecular  displacements, 
at  any  part  of  its  external  surface,  the  normal  to  which  makes  with  the 
axes  the  angles  a,  /3,  y     Then 

Rx  =  Px  cos  a  +  Q3  cos  /3  +  Q2  cos  7 
R2  =  Q3  cos  a  +  P2  cos  /3  +  Qx  cos  y 
R3  =  Q2  cos  a  +  Qi  cos  /3  +  P3  cos  y  ^ 

Should  there  be  any  surface  along  which  the  particles  are  constrained 
to  slide,  it  is  obvious  that  at  that  surface  the  following  condition  must 
be  fulfilled : 

0  =  £  cos  o  +  i?  cos  /3  +  Z  cos  y ; 

or  if  zx  =  /  (x,  y)  be  the  equation  of  the  surface, 


f 


(27/) 


(28.) 


J2.  d  zx 

dx  dy 


Were  we  acquainted  with  the  laws  which  determine  the  superficial 
pressures  in  vibrating  bodies,  equations  (27)  would  enable  us  to  determine 
the  values  which  b'  and  c  must  have,  in  virtue  of  those  laws,  during  the 
transmission  of  sound  in  a  limited  mass  of  an  uncrystallised  material, 
and  thence  the  ratio  ^/(l  —  Z>'2— c'2)  :  1,  in  which  the  velocity  of  sound  in 
such  a  body  is  less  than  in  an  unlimited  mass  of  the  same  material. 
Those  laws,  however,  are  as  yet  a  matter  of  conjecture  only. 


Transmission  of  a  Definite  Musical  Tone. 

17.  When  the  body  transmits  one  or  more  definite  musical  tones  (which 
is  the  case  in  all  experiments  capable  of   yielding  useful  results),  the 


VELOCITY  OF   SOUND  IN  LIQUID  AND   SOLID  BODIES. 


18/ 


50 


velocity  of  propagation  must  be  the  same  for  all  the  elementary  vibrations 
into  which  the  motion  may  be  resolved:  that  is  to  say,  1  —  b"2  —  c'2  must 
have  the  same  value  in  all  the  terms  of  the  sums  2.  This  affords  the 
means  of  simplifying  the  equations.     Let 

6'2  +  c'-2  =  tf .    i>  -  ]h  cos  q  .    c'  =  hsin  6; 

h  being  the  same  for  all  the  terms  in  the  sums  2.  Then  the  velocity  of 
propagation  is 

'   ^  =  a/{tt(1-*2)}  •     •     •  (29° 

and  this  factor  may  be  removed  outside  the  sign  of  summation. 

2  7T 

When  but  one  musical  tone  is  transmitted,  the  factor  -y-  also  may  be 

removed  outside  that  sign,  and  for  2  {  }  may  be  substituted  a  definite 
integration, 

2/F0  {  .  .  .  }d0, 
F  6  being  arbitrary. 
We  have  also 


r 


e*  =  ex 


h  (y  cos  6  +  z  sin  6) 


(30.) 


2tt 


in  which  :~-  h,  y,  and  z,  are  independent  of  9,  and  may  be  treated  as 
A 

constants  in  the  definite  integration. 

Introducing  these  modifications  into  equations  (23),  &c,  we  find 


$  =  A  (l'  cos  <j>  -  I  sin  <f)  2/«*  FQdO 

Displacements. 
l  =  {l  cos  <p  +1'  sin  <j>)  2/^  F  OdO 
r,  =  {V  cos  $  -  I  sin  tfh.'S/cos  6e*F  9d0 
I  =  (lf  cos  <j>  -  l  sin  0)  h.  2/ sin  0e*F0d0 


>  (31.) 


Velocities  of  the  Particles. 

~  =  ^V£-  {V  cos  0  -  I  sin  0)2/^  F0d0 
ttf  A 

^  =  -  ^  ^e.  (Z  cos  0  +  r  sin  0)  A.  2/cos  0^  F  0c7  0 

(it  A 


1SG  VELOCITY   OF  SOUND   IN   LIQUID  AND    SOLID   BODIES. 


j^=  -^Js.Qcos  <p  +  r  sin  0)  ft.  2/ sin  Oc^FOdO 


dt 


dx 


Longitudinal  Strain*. 
27r  (/'  cos  0  -  I  sin  <j>)  2/e^  F6d0 


p  =  ?£(T  cos<j>  -l8m$)h2.'2fcoB20e*F6d0 

^?  =  ^ (7  cos  0  -  Z  sin  <f)  h2 .  2/ sin2  Oe^FBdO. 
clz         A 


CuJ/c  Dilatation, 


d%      dri      dZ 

dx       dy      d : 


*T  ,v 


{t  cosf-l  sin  <£)  ( 1  -  /r)  2/e*  F  0  d  0. 


Distortions. 


7  7X  A 

-,-"  +  ',    =    J  (I'  cos  d>  -  I  sin  <h)  h-  2 /cos  0  sin  0e*F0d0 
dz       dy        A 

^?  +  ^  =  1^(Z  costf,  +  r  sin  0)A2/Sin  0e*F0<Z0 
J~  +  P=  y  ('  costf,  +  Z'  sin  0)&2/cos  de^FOdO. 

Which,  being  multiplied  by  —  C,  give  the  tangential  pressures 
Qi»  Q»  Q3,  on  the  co-ordinate  planes. 


Normal  Pressures  on  the  Co-ordinate  Planes,  due  to  the 
Displacements. 

Px  =  -=-  (/'  cos  $  -  I  sin  <f)  {A  (1  -  /j2)  +  2  C/r}  2/e^  F  0  d  0 

p2  =  ^  (*'  cos  *  -  *  sin  0)  [{A  (1  -  !r)  -  2  C}  2/e^  F  6  d  0 

+  2C/r2/sin20^F0</0] 

2tt 
P3  =  —  (T  cos<j>-l  sin  0)  [  {A  (1  -  F)  -  2  C}  2/e^  F  0rf  0 

+  2C/?2/cos20<^F0<2  0].  J 


(31.) 


VELOCITY   OF  SOUND   IN  LIQUID   AND   SOLID   BODIES. 


187 


Let  Rx,  R2,  R3,  be  the  components  of  the  pressure  exerted  by  the  body, 
in  consequence  of  the  molecular  displacements,  at  a  point  of  its  surface 
normal  to  the  direction  (a,  /3,  y).     Also,  let 


cos  (3  —  sin  a  cos  X, 
cos  y  =  sin  a  sin  x, 


so  as  to  make  x  the  axis  of  polar  co-ordinates,  and  xy  the  plane  from  which 
longitudes  X  are  measured.     Then, 


I>    =  ?£[cosa(rcos0-Zsin0){A(l  -A2)  +  2CA2}  Ife+VBdB 


2  sin  a  (Z  cos  <f>  +  X  sin  <p)  C  /t  2/cos  (0  -  X)  e^  F  0  d  0] 


2tt, 


R2  =  —  [  -  2  cos  a  (Z  cos  0  +  Z'  sin  <p)  C  A  2/cos  0  e^  F  0<Z  0 
+  sina(rcos«/,-/sin^){cosx(A(l-/r)-2C)v^F0(/0 

+  2  C  /r  2/sin  0  sin  (0  -  x)  e^  F  0  d  0}] 

R   =  —  [  -  2  cos  a  (/  cos  0  +  I'  sin  0)  C  h  2/sin  0  e^  F  0  d  0 
A 

+  sin  a  (r  cos  tf>  -  / sin  j,)  {sin  X  (A  (1  -A2)  -  2  c)  2/«* F  0  rf  0 
-  2  C  /r  2/cos  0  sin  (0  _x)  c^  F  0  rf  0}]  , 


[>   (32.) 


Let  P'  represent  the  normal  pressure  at  the  given  point  of  the  surface 
due  to  molecular  displacements  :  then 

P'  =  Rx  cos  a  +  sin  a  (II.,  cos  X  +  R3  sin  X) 

=  Px  cos2  a  +  P2  sin2  a  cos2  X  +  P3  sin2  a  sin2  X 

-f  2  Qx  sin2  a  cos  X  sin  x  +  2  Q2  cos  a  sin  a  sin  X 
+  2  Q3  cos  a  sin  a  cos  x 

2tt  / 

=  —  [(?  cosf—l  sin  0)  {(A  (1  —  lr)  )-(32A) 

+  2  C  (/i2  cos2  a  -  sin2a) )  2/V*  F  0  d  0 

+  2  C  K-  sin2  a  2/sin2  (0  -  X)  c^  F  6d  0} 

—  4(/cos0  +  *'  sin  <p)  C  A  cos  a  sin  a  2/C0S  (0  -  x)e*FedB 


188  VELOCITY   OF   SOUND  IN   LIQUID  AND  SOLID   BODIES. 


Propagation  of  Sound  by  Nearly-longitudinal  Vibrations  along 
a  Horizontal  Prism  of  Liquid  contained  in  a  Rectangular 
Trough,  investigated  according  to  a  Peculiar  Hypothesis. 

18.  I  shall  now  suppose  the  vibrating  body  to  be  a  rectangular 
horizontal  prism  of  liquid  contained  in  a  trough  of  some  substance  so 
dense,  hard,  and  smooth,  that  the  particles  at  the  sides  and  bottom  of  the 
trough  are  constrained  to  slide  along  those  surfaces,  and  that  the  vertical 
ends  of  the  trough  are  capable  of  perfectly  reflecting  a  wave  of  sound 
travelling  horizontally;  so  that  the  propagation  of  that  wave  may  take 
place  as  if  in  a  trough  of  indefinite  length ;  and  I  shall  investigate  the 
velocity  of  such  a  wave  according  to  a  peculiar  hypothetical  view  of  the 
molecular  condition  of  the  upper  surface  of  the  liquid. 

The  axis  of  x  being  the  horizontal  axis  of  the  trough,  and  parallel  to 
the  direction  of  propagation,  let  that  of  y  be  transverse,  and  that  of  2 
vertical.  Let  the  middle  of  the  bottom  of  the  trough  be  the  origin  of 
co-ordinates,  2y1  being  its  breadth,  and  z1  the  depth  of  liquid  in  it. 

The  conditions  to  be  fulfilled  at  the  bottom  are,  when 

z  =  0,    a  =  ^  ir,    and  x  =  ~  i  ~- 
Let 

,  h  y  cos  6  „  .   ,  „ 

2/sin0^F0(Z0  =  2/sin0cA     *         F6dd  =  0 

at  the  sides,  when 

y  =  ±  yv    a  =  l ■  ir,    and  y_  =  0  or  7r. 
Let 

2-r 
t  „    — ^(±2/1  cos0  +  2Sin4) 

2/cos0^F0fZ0  =  2y"cos0e*  F0<Z0  =  O; 

which  conditions  are  fulfilled  by  making 

cos  0  =  0,    sin  0  =  ±  1, 

and  putting  for  2/F  6  d  6  a  summation  of  two  terms  in  which  the  signs 
of  the  exponent  are  respectively  positive  and  negative. 
Thus  we  obtain 


~^1, 


£  =  (I  cos  $  +  V  sin  (p)    \e   A       +  e 

7/    =    0 


Z,  =  (I'  cos  0  —  I  sin  (f)h\c  x       —  e     x     /  > 


y    (33.) 


VELOCITY   OF  SOUND   IX  LIQUID  AND    SOLID   BODIES.  189 

The  trajectory  of  each  particle  is  an  ellipse  in  a  vertical  longitudinal 
plane ;  the  motion  being  direct  in  the  upper  part  of  the  ellipse,  because 

the  sien  of  —   is  the  same  with  that  of  £.     The  axes  are  vertical  and 
°  at 

horizontal  respectively,  and  have  the  following  values  : — 


-  V  , 


-hz  -  —hz 


Horizontal  axis  =  2  J  (I2  +  I"2)    .    \e   x 

Vertical  axis       =  2  J  (I2  +  l'2) .  h.  \e  *    *  -  e     *    7  ; 

so  that  the  motion  is  analogous  to  that  of  waves  propagated  by  gravitation, 
being  entirely  horizontal  at  the  bottom  of  the  trough,  and  elliptical  else- 
where, the  ellipse  being  larger  and  less  eccentric  as  the  height  above  the 
bottom  increases.  The  ratio  of  the  axes,  however,  instead  of  approaching 
equality  as  the  depth  of  the  trough  increases  (which  is  the  case  with  waves 
of  gravitation),  approaches  1  :  h. 

19.  To  determine  this  ratio,  upon  which  the  velocity  of  sound  along 
such  a  mass  of  liquid  must  depend,  I  shall  assume  the  following  hypo- 
thetical principles  respecting  the  state  of  the  particles  at  the  upper  surface : 

First,  That  (as  laid  down  in  a  previous  paper,  Cambridge  and  Dublin 
Mathematical  Journal,  February,  1851)  the  elasticity  of  bodies  is  due 
partly  to  the  mutual  actions  of  atomic  centres  producing  elasticity  both 
of  volume  and  figure,  and  partly  to  a  mere  fluid  elasticity  resisting  change 
of  volume  only,  and  exerted  by  atmospheres  surrounding  those  centres ; 
and  that  the  effect  of  the  mutual  actions  of  the  atomic  centres  in  producing 
pressure  is  very  small  in  liquids,  and  absolutely  inappreciable  in  gases 
and  vapours. 

Secondly,  That  every  liquid  maintains  at  its  surface,  by  molecular 
attraction,  an  atmosphere  of  its  own  vapour,  under  these  conditions — 
that  the  total  pressures  of  the  liquid  and  vapour,  and  also  their  fluid 
pressures,  shall  be  equal  at  the  bounding  surface.  (From  this  hypothesis 
I  have  already  deduced  the  form  of  an  approximate  equation  between  the 
pressure  and  temperature  of  vapour  at  saturation.)  The  total  pressure 
of  the  vapour  on  the  liquid  is  sensibly  equal  to  its  fluid  pressure ;  the 
total  pressure  of  the  liquid  on  the  vapour  consists  of  its  fluid  pressure, 
and  a  pressure  due  to  atomic  centres ;  the  latter  quantity  must,  therefore, 
be  null. 

Thirdly,  That  the  pressure  of  the  vapour  follows  that  of  the  liquid 
throughout  its  variations  during  the  propagation  of  sound;  so  that  the 
portion  of  the  pressure  of  the  liquid  on  the  vapour,  due  to  atomic 
centres,  must  continue  null  throughout  these  variations. 

Let  &  be  the  mutual  pressure  of  the  liquid  and  its  vapour  in  a  state 


190  VELOCITY   OF   SOUND   IN    LIQUID   AND   SOLID   BODIES. 

of  rest,  then  a»  +  P'  is  their  momentary  mutual  pressure  during  the  passage 
of  a  wave  of  sound  horizontally  along  the  trough.     The  portion  of  P' 
depending   on  the    coefficient    of   rigidity   C    being  made  =  0,    we    shall 
obtain  an  equation  from  which  the  value  of  h  may  be  deduced. 
Making  the  proper  substitutions  in  equation  (3 2 A),  viz. — 

cos  a  —  0,    sin  a  =  1,    cos  ^  =  0,    sin  ^  =  1,    \L  =  ±  -st-  h  .. 

cos  6  =  0,   sin  0  =  ±  1,   F  6  =  1,   z  =  zv  &c, 

we  find 

/2r.  _2»      \ 

w  +  r'=w+~(rcos^-/sin0){A(l-/t2)-2C}Vc  x    Zl+e     x     7- 
A 

The  part  of  this  depending  on  mere  fluid  elasticity,  in  which  the  liquid 
is  followed  by  the  vapour,  is 

\dx      dz 

/  2  t  2  * 

2  7T  (  h  ~l  T-nZi 

=  u+T (/'  cos  (p  —  I  sin  0) .  J  (1  —  Ir)\c  x        +  e      * 
A 

which,  being  subtracted,  there  remains  for  the  part  depending  on  atomic 
centres, 

/  2*-  It      \ 


A 

Consequently, 


0  =  ^f  (T  cos  0  -  Z  sin  <j>)  C  (1  -  3  h°-)\e  *        +  e 


1  -  3/i2  =  0,    or  h  =  Jl     .         .         .     (33.) 


is  the  equation  of  condition  sought,  arising  from  the  state  of  the  free 
surface ;  and  this  equation  is  independent  of  the  amount  of  rigidity  of 
the  liquid,  requiring  only  that  it  shall  be  something,  however  small,  while 
that  of  the  vapour  is  null. 

It  follows  from  this  equation,  that  the  velocity  of  propagation  of  sound 
along  a  trough  of  liquid  of  the  density  D,  and  longitudinal  elasticity  A,  is 

+= J  {&-»}=&¥)•     ■  <*> 

or  less  than  the  velocity  in  an  unlimited  mass  in  the  ratio  of  *J2  to  y/3. 

20.  This  is  precisely  the  result  arrived  at  by  M.  Wertheim  from  a 
comparison  of  his  numerous  experiments  on  the  propagation  of  sound  in 
water  at  various  temperatures,  from  15°  to  60°  Centigrade,  in  solutions  of 
various  salts,  in  alcohol,  turpentine,  and  ether  (Ann.  de    Chim,,  Ser.  Ill, 


VELOCITY   OF   SOUND   IN    LIQUID   AND    SOLID   BODIES.  191 

torn,  xxiii.),  with  those  of  M.  Grassi  on  the  compressibility  of  the 
same  substances  (Comptes  Rendus,  XIX.,  p.  153),  and  with  the  experiments 
of  MM.  Colladon  and  Sturm  on  the  velocity  of  sound  in  an  expanse  of 
water. 

M.  Wertheim  having  given  this  comparison  in  detail,  I  shall  quote  one 
example  only. 

The  velocity  of  sound  in  an  unlimited  mass  of  water,  at  the  temperature 
of  16°  Centigrade,  as  ascertained  by  MM.  Colladon  and  Sturm,  was  1435 
metres  per  second. 

That  of  sound  in  water  contained  in  a  trough,  the  vibrations  of  which 
were  regulated  by  an  organ-pipe,  was  found  by  M.  Wertheim,  at  15° 
Centigrade,  to  be  11 734  metres  per  second. 

The  ratio  of  the  squares  of  those  quantities  is  0-6G8G  :  1,  differing 
from  f  by  0-0009  only. 


REMARKS    ON   THE   PROPAGATION    OF    SOUND   ALONG    SOLID   RODS. 

21.  I  refrain  from  giving,  in  the  body  of  this  paper,  detailed  investigations 
of  particular  problems  respecting  the  propagation  of  sound  along  a  solid 
prism  or  cylinder;  for,  in  the  present  state  of  our  knowledge  of  the 
condition  of  the  superficial  particles  of  such  bodies,  the  conclusions  would 
be  almost  entirely  speculative  and  conjectural. 

I  may  mention  briefly,  however,  the  following  general  results.  If 
we  adopt  for  solids  the  same  hypothesis  as  for  liquids,  then  the  ratio  of 
the  velocity  of  sound  in  a  rod  of  an  uncrystallised  material  to  that  in  an 
unlimited  mass  has  the  following  values  : 

For  a  rectangular  prismatic  rod,  the  lateral  vibrations  of  the  particles 
of  which  are  confined  to  planes  parallel  to  one  pair  of  faces  of  the  prism, 
but  are  perfectly  free  in  other  respects,  the  ratio  is  ^2:^/3,  being  the 
same  as  for  a  liquid. 

For  a  cylindrical  rod,  the  surface  being  perfectly  free,  the  ratio  has 
various  values,  from  ^Jl :  ^/f ,  approaching  the  less  value  as  the  diameter 
of  the  rod  diminishes,  and  the  greater  as  it  increases  ad  infinitum.  (See 
Appendix,  No.  I.) 

22.  These  conditions,  however,  cannot  be  realised  in  practice;  and 
the  lateral  vibrations  being  more  or  less  confined  by  the  means  used 
in  fixing  the  rods,  we  find  that  the  ratio  generally  exceeds  ,J'2  :  ^3, 
and  sometimes  approaches  equality. 

The  following  table  illustrates  this  fact.  I  have  selected,  in  the  first 
place,  the  experiments  of  M.  Wertheim  on  tubes  of  crystal  (Ann.  de  Chim., 
Ser.  III.,  torn,  xxiii.),  because  in  them  the  coefficients  of  elasticity  and 
the  velocities  of  sound  were  ascertained  by  experiments  on  the  same 


192 


VELOCITY   OF  SOUND   IN  LIQUID  AND   SOLID  BODIES. 


pieces  of  the  material.  To  these  I  have  added  a  calculation,  founded 
on  a  comparison  of  the  experiments  of  M.  Wertheim  on  the  elasticity  of 
brass,  with  those  of  M.  Savart  on  the  velocity  of  sound  in  it,  as  being 
the  only  other  data  of  the  kind  now  existing  from  which  a  satisfactory 
conclusion  can  be  drawn. 

The  coefficients  of  longitudinal  elasticity,  calculated  by  myself  from 
M.  Wertheim's  experiments,  are  extracted  from  my  paper  on  elasticity 
in  the   Cambridge  and  Dublin  Mathematical  Journal  for  February,  1851. 

The  quantities  -  -  for  crystal  are  given  as  calculated  by  M.  Wertheim. 

For  brass  I  have  used  the  following  data : 

tj  e  —  velocity  of  sound  in  brass  rods ;  mean  of  many  experiments  by 
M.  Savart  =3560  metres  per  second. 

D  =  density,  in  kilogrammes  per  cubic  metre,  8395. 


TABLE. 


Crystal. 

Longitudinal 

Elasticity 

A 

Kilogrammes  per 

square  millimetre. 

jD 

.'/ 

Kilogrammes 
per  square 

millimetre. 

llatio 
1  -  /** 

Tube  No.     L, 

.      5514-2 

5354-0 

0-970, 

„        „    HI, 

.      5611-0 

5476-7 

0-976, 

„       „    IV., 

.      6183-1 

5597-3 

0-905, 

v 

.     6659-9 

5489-8 

0-824, 

Brass, 

.   15625 

10847 

0-694. 

Concluding  Remarks. 


23.  The  chief  positive  results  arrived  at  in  this  paper  may  be  summed 
up  as  follows: — 

(i.)  In  liquid  and  solid  bodies  of  limited  dimensions,  the  freedom  of 
lateral  motion  possessed  by  the  particles  causes  vibrations  to  be  propagated 
less  rapidly  than  in  an  unlimited  mass. 

(u.)  The  symbolical  expressions  for  vibrations  in  limited  bodies  are 
distinguished  by  containing  exponential  functions  of  the  co-ordinates  as 
factors;  and  the  retardation  referred  to  depends  on  the  coefficients  of 
the  co-ordinates  in  the  exponents  of  those  functions,  which  coefficients 
depend  on  the  molecular  condition  of  the  body's  surface — a  condition  yet 
imperfectly  understood. 

(in.)  If  we  adopt  the  hypothetical  principle,  that  at  the  free  surface  of  a 
vibrating  mass  of  liquid  the  normal  pressure  depending  on  the  actions  of  atomic 


VELOCITY  OF  SOUND   IN  LIQUID  AND    SOLID  BODIES.  193 

centres  is  always  null,  then  we  deduce  from  theory  that  the  ratio  of  the 
velocity  of  sound  along  a  mass  of  liquid  contained  in  a  rectangular  trough 
to  that  in  an  unlimited  mass  is  ^f'2  :  ^3,  that  ratio  being  independent 
of  the  specific  rigidity  of  the  liquid — a  conclusion  agreeing  with  our 
present  experimental  knowledge. 

24.  I  do  not  put  forward  the  hypothetical  part  of  these  researches 
as  more  than  a  probable  conjecture;  nor  should  I  be  justified  in  so  doing 
in  the  present  state  of  our  knowledge  of  molecular  forces.  I  think, 
however,  that  these  investigations  are  sufficient  to  prove  that  we  are 
not  warranted  in  concluding  from  M.  Wertheim's  experiments  (as  he  is 
disposed  to  do)  that  liquids  possess  a  momentary  rigidity  as  great  as 
that  of  solids,  seeing  that  any  amount  of  rigidity,  howsoever  small,  will 
account  for  the  phenomena  if  we  adopt  certain  suppositions  as  to  mole- 
cular forces;  and  to  show  that  our  knowledge  of  those  forces  is  not  yet 
sufficiently  advanced  to  enable  us  to  use  experiments  on  sound  as  a  means 
of  determining  the  coefficients  of  elasticity  of  solids. 


Appendix. — No.  I. 


Propagation  of  Sound  by  Nearly-Longitudinal  Vibrations  along 
a  Cylindrical  Uncrystallised  Eod. 

Let  the  vibrating  body  be  cylindrical  round  the  axis  of  x,  and  let  the 
vibrations  of  all  particles  in  a  given  circle  round  that  axis  be  assumed 
to  be  equal  and  simultaneous.    Let  r  represent  the  distance  of  any  particle 

A 

from  the  axis  of  x,  and  ^,  the  angle  y  r. 
Then 


To  make  the  results  of  the  definite  integrations  ^fFQdd  independent 
of  the  angle  ^,  we  must  have  F  $  =  constant,  and  the  limits  of  integra- 
tion 0  and  2  tt. 

The  following  are  the  definite  integrals  which  enter  into  the  solution  of 
this  problem. 

N 


194  VELOCITY  OF  SOUND   IN  LIQUID  AND   SOLID  BODIES, 

Let 

2  7T 

— —  hr  =  k, 


/■2t  fja-.fccos^    ,    p-kcosS 

e=     <f«»iao  =  i\-    =    y    *  to 

Jo  Jo  A 

-^L  +  ila*(r(»+i)),JJ 


9': 


cosOe7ccoatdd 


>        l36.) 


""■2|  22n_! 


pi-i 


] 

!».!>+  1)  i 


O' 


cos2  Oekcos<>dO 


~  d&  ~Jo  l 

r      v  j        (2w+  i)*2n        )  "1 
_  2,r L*  +    1 22n+1  i>  +  i)i>  +  2)  J  J 

the  values  of  n  comprehending  all  integers  from  1  inclusive. 
Those  series  have  the  following  properties  : 

(i.)  The  term  (n)  of  0  =  term  {n  —  1)  X  —  - , ;  therefore,  this  series 

always  becomes  convergent  at  the  term  for  which  n  >  \  k. 

k2 
(n.)  Term  (n)  of  0'  =  term  (n  —  1)  X 

series  becomes  convergent  when  n2  —  n  >  i  W, 


4  (n  —  l)w 


;   therefore,  this 


(in.)   Term   (w)    of    0"  =  term   (w  —  1)  X 


(2n+  \)k2 


4(2»-  !).».(»+  1)' 


w 


2  n  -  1  >  4" 


therefore,  it  begins  to  converge  when  n2  — 
(iv.)  Q'=k(e-G"). 

(v.)  Term  (n)  of  0"=  term  (n)  of  0  x —77 ;  a  ratio  which  is  \  for 

w  '  2  n  +  2 

the  first  term  (w  =  0),  and  approaches  equality  as  n  increases ;  therefore, 

0" 

when  ^  k-  is  an  inappreciably  small  fraction,  — -  =  \  sensibl}'. 


0 


6' 


And  the  larger  k  is,  the  more  nearly  is  —  =1. 


VELOCITY   OF  SOUND   IN  LIQUID  AND  SOLID  BODIES.  195 

The  following  table  of  a  few  numerical  results  illustrates  this  : — 


¥ 

e 

G" 

e" 

4 

2<r 

2* 

e 

0 

1-0000 

0-5000 

0-5000 

1 

1-2661 

0-7010 

0-5537 

i 

3 

1-3622 

0-7741 

0-5683 

1 

1-5661 

09302 

0-5490 

1 

2-2796 

1-4843 

0-6511 

2 

4-2523 

3-0550 

0-7160 

3 

•  7-1590 

5-4238 

0-7576 

4 

11-3019 

9-3620 

0-8284 

The  displacements  in  this  case  are  as  follows : — 

£  =  (I  cos  <p  +  X  sin  0)  G 

rj  =  (I'  cos  (j)  —  I  sin  <f)  h  G'  cos  ^ 

£  =  (7  cos  0  —  Z  sin  ^)/i6'  sin  x'> 


(37.) 


whence  it  appears  that  the  two  transverse  displacements  ij  and  £  compose 
a  radial  displacement, 


p  =  (I'  cos  <j>  —  I  sin  0)A  6'. 


.     (37A.) 


Therefore,  the  trajectory  of  each  particle  is  an  ellipse,  in  a  plane  passing 
through  the  axis  of  the  cylinder;  and  the  axes  of  the  ellipse  are  longi- 
tudinal and  radial,  and  have  the  following  values: 

Longitudinal  axis,  =  2A/(72  +  P)   .   0  ") 

L         .     (38.) 
Eadial  axis,       .      =  2  J{p  +  P) .  h  Q'  j 

If  we  now  adopt  the  same  hypotheses  with  respect  to  the  outer  surface 
of  the  cylinder  that  have  been  used  in  the  problem  respecting  liquids,  we 
shall  have  for  the  mutual  pressure  of  the  solid  and  its  atmosphere  of  vapour 


»+!?=» +^(l' cos  <f>-ltmfi{(A-2V)(l-h*)Q1-2Ch*ei"}> 
A 


196  VELOCITY  OF  SOUND   IN  LIQUID  AND  SOLID  BODIES. 

©1?  ©j"  being  the  values  of  those  integrals  corresponding  to  the  radius  of 
the  cylinder. 

The  portion  of  this  pressure  depending  on  mere  fluid  elasticity  is 

w_j(^+^  +  ^=w  +  ^  _Zs 

\dx     ay     dz/  A  * 

which,  being  subtracted,  leaves 

0  =  ~(r  cos  0  -  J  sin  0)C  {©!  -  h2(S1  +  2  ©/')} ; 


therefore,  according  to  the  hypothesis  adopted, 


h*  = 1-ttt,:.         ■         •         .     (39.) 


1  +  2ex 

and  the  velocity  of  sound  along  the  cylinder  is 


Now,  the  limits  of  the  ratios  in  the  above  formulae  are  the  following : 

•       1, 


T  .    .,      .  2  7r  A  r,        , 

Lrmits  of  —  — l  =  \      .      .      .      0 

A 


©/ J    • 

A V2   •     •   */i» 

„  */(!-]£) Jl  .   .  Jl 

That  is  to  say,  if  the  hypothesis  already  explained  with  reference  to 
liquids  is  applicable  to  a  solid  cylinder  of  an  uncrystallised  material, 
the  velocity  of  sound  along  such  a  cylinder,  when  its  surface  is  perfectly 
free,  will  be  less  than  that  in  an  unlimited  mass  in  some  ratio  between 
^  and  J?6 . 


VELOCITY  OF  SOUND  IN  LIQUID  AND  SOLID   BODIES.  197 

Appendix. — No.  II. 

General  Equations  of  Nearly-Transverse  Vibrations. 

The  two  equal  roots  of  equation  (8)  in  uncrystallised  bodies,  viz. 
E  =  C  (1  -  b'2  -  c'% 

correspond  to  what  may  be  called  nearly-transverse  vibrations,  propagated 
with  the  velocity 


x/£  =  V{^(1~&'2"C'2)}-        '         *     (41>) 


Equations  (20)  in  this  case  give  no  result;  but  equations  (20A)  are 
reducible  to  the  following  two : 

I  =  —  m'b'  —  n'  c'     .     .     .     . ) 

f       •         •     (42); 
I'  —  mb'  +  nc' ) 

the  ratios  m  :  n  and  m' :  n'  are  arbitrary. 

Equations  (11),  (16),  (17),  (18),  become  the  following: 

Displacements. 
£  —  2  [e+  {-  (m! b'  +  ri c')  cos  0  +  (m b'  +  n c)  sin  0  }] 
t)  =  2  {e^  (m  cos  0  +  m  sin  <£) } 
£  =  2  {e^  (n  cos  0  +  n'  sin  cj>)} 

Velocities  of  the  Particles. 
^  =  2^^Je.e*  {(m'b'  +  n'c)sm<p  +  (mb'  +  nc')cos<p}J    y  ^ 

—  =  2  -j  ^—  ^/  g .  e^  (—  m  sin  0  +  m'  cos  0)  >• 


dt 


dj 

dx 


=  2  -j  ^—  a/  e .  e^  (  —  w  sin  0  +  %'  cos  $>)  r 

Longitudinal  Strains. 
=  _  2  I^V  {(»»  &'  +  n  0  cos  0  +  (m'  b'  +  n'  c')  sin  <p}  \ 


198  VELOCITY  OF  SOUND   IN   LIQUID   AND   SOLID   BODIES. 

^  =  2  \~e*  (mV  cos  <h  +  m'V  sin  d>)\ 
ay  (A  ) 


dz 


2  \  -t—  e^  (w  c'  cos  $  -\-  n'  c  sin  </j)  r 


Cufo'c  Dilatation. 

dj,     ,     <*JJ     ,     dt   __   Q 
d  re         rf  y         rZ  « 


Distortions. 


>  (43.) 


g+f!=2[x^{-(n/(i+^+m'&v)co^ 

+  (n  (1  +  0"  +  rub'  A  sin  0}    | 

+  (m  (1  +  Z>'2)  +  nb' A  sin  0}   | 
Which,  being  multiplied  by  —  C,  give  the  tangential  pressures 

%>  ^2'  **3' 

Normal  Pressures  on  the  Co-ordinate  Planes  due  to  the 
Displacements. 

Px  =  2  C .  2 1  ~  e*  {(m  V  +  n c)  cos  0  +  {m  V  +  w' c)  sin  </>}   | 
P2  —  —  2  C .  2 1  -j—e^  {m  V  cos  <f>  -\-  m  b'  sin  0}   I 

P3  —  —  2  C .  2j  ~y-e^  {71  c'  cos  <j>  -\-  n  c  sin  0} 
Px   +   P2   +   P3  =   0. 

The  normal  pressure  due  to  the  displacements  at  any  point  of  the 
surface  of  a  prism  or  cylinder  described  round  x  is 


VELOCITY   OF   SOUND   IN   LIQUID   AND   SOLID   BODIES.  199 

P'  =  —  2  C  .  2  I  --- e$ {(m  b'  cos'-'  ^  +  n  c  sin2  ^  +  (m  <:'  +  n  b') 

cos  ^  sin  ^J  cos  <p 

+  ( wi'  6'  cos"  x  +  w'  c'  sin2  y_  +  (m'  c'  +  n  b')  cos  ^  sin  y)  sin  0}    I  (44.) 

If  this  pressure  is  to  be  null  at  all  points  of  the  surface,  we  must  have 
b'  =  0,  c'=0,  and,  consequently,  Z=0,  l'  =  0;  and  the  motion  is  restricted 
to  common  exactly-transverse  vibrations,  for  which 

E  =  Cand^a  =  >/  Cd)' 

Nearly-transverse  vibrations,  therefore,  cannot  be  transmitted  along  a 
cylindrical  or  prismatic  uncry  stall  ised  body  whose  surface  is  absolutely 
free. 


PART    II. 

PAPERS  RELATING  TO  ENERGY  AND  ITS  TRANSFORMATIONS, 

THERMODYNAMICS,  MECHANICAL  ACTION  OF 

HEAT  IN  THE  STEAM  ENGINE,  &a 


PART    IT. 

PAPERS  RELATING  TO  ENERGY  AND  ITS  TRANSFORMATIONS, 

THERMODYNAMICS,  MECHANICAL  ACTION  OF 

HEAT  IN  THE  STEAM  ENGINE,  &a 


X.— ON  THE  ^CONCENTRATION   OF  THE  MECHANICAL 
ENERGY  OF  THE  UNIVERSE.* 

THE  following  remarks  have  been  suggested  by  a  paper  by  Professor 
William  Thomson f  of  Glasgow,  on  the  tendency  which  exists  in  nature  to 
the  dissipation  or  indefinite  diffusion  of  mechanical  energy  originally 
collected  in  stores  of  power. 

The  experimental  evidence  is  every  day  accumulating,  of  a  law  which 
has  long  been  conjectured  to  exist, — that  all  the  different  kinds  of  physical 
energy  in  the  universe  are  mutually  convertible;  that  the  total  amount 
of  physical  energy,  whether  in  the  form  of  visible  motion  and  mechanical 
power,  or  of  heat,  light,  magnetism,  electricity,  or  chemical  agency,  or  in 
other  forms  not  yet  understood,  is  unchangeable ;  the  transformations  of 
its  different  portions  from  one  of  those  forms  of  power  into  another,  and 
their  transference  from  one  portion  of  matter  to  another,  constituting  the 
phenomena  which  are  the  objects  of  experimental  physics. 

Professor  William  Thomson  has  pointed  out  the  fact,  that  there  exists 
(at  least  in  the  present  state  of  the  known  world),  a  predominating 
tendency  to  the  conversion  of  all  the  other  forms  of  physical  energy  into 
heat,  and  to  the  uniform  diffusion  of  all  heat  throughout  all  matter.  The 
form  in  which  we  generally  find  energy  originally  collected,  is  that  of  a 
store  of  chemical  power,  consisting  of  uncombined  elements.  The  com- 
bination of  these  elements  produces  energy  in  the  form  known  by  the 
name  of  electric  currents,  part  only  of  which  can  be  employed  in  analysing 

*  Read  before  the  British  Association  at  Belfast,  on  September  2,  1S52,  and  published 
in  the  Philosophical  Magazine  for  November,  1852. 
f  ISIow  Sir  William  Thomson. 


MECHANICAL   ENERGY  OF  THE  UNIVERSE.  201 

compounds,  and  thus  reconverted  into  a  store  of  chemical  power;  the 
remainder  is  necessarily  converted  into  heat :  a  part  only  of  this  heat  can 
be  employed  in  analysing  compounds,  or  in  reproducing  electric  currents.  If 
the  remainder  of  the  heat  be  employed  in  expanding  an  elastic  substance, 
it  may  be  entirely  converted  into  visible  motion,  or  into  a  store  of  visible 
mechanical  power  (by  raising  weights,  for  example),  provided  the  elastic 
substance  is  enabled  to  expand  until  its  temperature  falls  to  the  point 
which  corresponds  to  absolute  privation  of  heat ;  but  unless  this  condition 
be  fulfilled,  a  certain  proportion  only  of  the  heat,  depending  upon  the 
range  of  temperature  through  which  the  elastic  body  works,  can  be  con- 
verted, the  rest  remaining  in  the  state  of  heat.  On  the  other  hand,  all 
visible  motion  is  of  necessity  ultimately  converted,  entirely  into  heat  by 
the  agency  of  friction.  There  is  thus,  in  the  present  state  of  the  known 
world,  a  tendency  towards  the  conversion  of  all  physical  energy  into  the 
sole  form  of  heat. 

Heat,  moreover,  tends  to  diffuse  itself  uniformly  by  conduction  and 
radiation,  until  all  matter  shall  have  acquired  the  same  temperature. 

There  is,  consequently,  Professor  Thomson  concludes,  so  far  as  we 
understand  the  present  condition  of  the  universe,  a  tendency  towards  a 
state  in  which  all  physical  energy  will  be  in  the  state  of  heat,  and  that 
heat  so  diffused  that  all  matter  will  be  at  the  same  temperature ;  so  that 
there  will  be  an  end  of  all  physical  phenomena.  ' 

Vast  as  this  speculation  may  seem,  it  appears  to  be  soundly  based  on 
experimental  data,  and  to  represent  truly  the  present  condition  of  the 
universe,  so  far  as  we  know  it. 

My  object  now  is  to  point  out  how  it  is  conceivable  that,  at  some 
indefinitely  distant  period,  an  opposite  condition  of  the  world  may  take 
place,  in  which  the  energy  which  is  now  being  diffused  may  be  recon- 
centrated  into  foci,  and  stores  of  chemical  power  again  produced  from  the 
inert  compounds  which  are  now  being  continually  formed. 

There  must  exist  between  the  atmospheres  of  the  heavenly  bodies  a 
material  medium  capable  of  transmitting  light  and  heat ;  and  it  may  be 
regarded  as  almost  certain  that  this  interstellar  medium  is  perfectly 
transparent  and  diathermanous ;  that  is  to  say,  that  it  is  incapable  of 
converting  heat,  or  light  (which  is  a  species  of  heat),  from  the  radiant  into 
the  fixed  or  conductible  form. 

If  this  be  the  case,  the  interstellar  medium  must'  be  incapable  of 
acquiring  any  temperature  whatsoever ;  and  all  heat  which  arrives  in  the 
conductible  form  at  the  limits  of  the  atmosphere  of  a  star  or  planet,  will 
there  be  totally  converted,  partly  into  ordinary  motion,  by  the  expansion 
of  the  atmosphere,  and  partly  into  the  radiant  form.  The  ordinary 
motion  will  again  be  converted  into  heat,  so  that  radiant  heat  is  the 
ultimate  form  to  which  all  physical  energy  tends ;  and  in  this  form  it  is, 


202  MECHANICAL   ENERGY  OF  THE  UNIVERSE. 

in  the  present  condition  of  the  world,  diffusing  itself  from  the  heavenly 
bodies  through  the  interstellar  medium. 

Let  it  now  be  supposed,  that,  in  all  directions  round  the  visible  world, 
the  interstellar  medium  has  bounds  beyond  which  there  is  empty 
space. 

If  this  conjecture  be  true,  then  on  reaching  those  bounds  the  radiant 
heat  of  the  world  will  be  totally  reflected,  and  will  ultimately  be  recon- 
centrated  into  foci.  At  each  of  these  foci  the  intensity  of  heat  may  be 
expected  to  be  such,  that  should  a  star  (being  at  that  period  an  extinct 
mass  of  inert  compounds)  in  the  course  of  its  motions  arrive  at  that  part 
of  space,  it  will  be  vaporised  and  resolved  into  its  elements ;  a  store  of 
chemical  power  being  thus  reproduced  at  the  expense  of  a  corresponding 
amount  of  radiant  heat. 

Thus  it  appears,  that  although,  from  what  we  can  see  of  the  known 
world,  its  condition  seems  to  tend  continually  towards  the  equable  dif- 
fusion, in  the  form  of  radiant  heat,  of  all  physical  energy,  the  extinction 
of  the  stars,  and  the  cessation  of  all  phenomena ;  yet  the  world,  as  now 
created,  may  possibly  be  provided  within  itself  with  the  means  of  recon- 
centrating  its  physical  energies,  and  renewing  its  activity  and  life. 

For  aught  we  know,  these  opposite  processes  may  go  on  together ;  and 
some  of  the  luminous  objects  which  we  see  in  distant  regions  of  space  may 
be,  not  stars,  but  foci  in  the  interstellar  ether. 


GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY.    203 


XI.— ON  THE  GENERAL  LAW  OF  THE  TRANSFORMATION 

OF  ENERGY.* 

Actual,  or  Sensible  Energy,  is  a  measurable,  transmissible,  and  trans- 
formable condition,  whose  presence  causes  a  substance  to  tend  to  change 
its  state  in  one  or  more  respects.  By  the  occurrence  of  such  changes, 
actual  energy  disappears,  and  is  replaced  by 

Potential  or  Latent  Energy;  which  is  measured  by  the  product  of  a 
change  of  state  into  the  resistance  against  which  that  change  is  made. 

(The  vis  viva  of  matter  in  motion,  thermometric  heat,  radiant  heat,  light, 
chemical  action,  and  electric  currents,  are  forms  of  actual  energy;  amongst 
those  of  potential  energy  are  the  mechanical  powers  of  gravitation, 
elasticity,  chemical  affinity,  statical  electricity,  and  magnetism). 

The  law  of  the  Conservation  of  Energy  is  already  known — viz.,  that  the 
sum  of  all  the  energies  of  the  universe,  actual  and  potential,  is  unchange- 
able. 

The  object  of  the  present  paper  is  to  investigate  the  law  according  to 
which  all  transformations  of  energy,  between  the  actual  and  potential  forms, 
take  place. 

Let  V  be  the  magnitude  of  a  measurable  state  of  a  substance; 

U,  the  species  of  potential  energy  which  is  developed  when  the  state  V 
increases; 

P,  the  common  magnitude  of  the  tendency  of  the  state  V  to  increase, 
and  of  the  equal  and  opposite  resistance  against  which  it  increases ;  so 
that— 

dU  =  FdV;  andP  =  ^.    .         .         .     (A.) 

Let  Q  be  the  quantity  which  the  substance  possesses,  of  a  species  of 
actual  energy  whose  presence  produces  a  tendency  of  the  state  V  to 
increase. 

It  is  required  to  find  how  much  energy  is  transformed  from  the  actual 
form  Q  to  the  potential  form  U,  during  the  increment  d  V;  that  is  to  say, 
the  magnitude  of  the  portion  of  d  U,  the  potential  energy  developed,  which 
is  due  to  the  disappearance  of  an  equivalent  portion  of  actual  energy  of 
the  species  Q. 

The  development  of  this  portion  of  potential  energy  is  the  immediate 

*  Read  before  the  Philosophical  Society  of  Glasgow,  on  January  5,  1S53,  and 
published  in  the  Proceedings  of  that  Society,  Vol.  III.,  No.  V. 


204    GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY. 

effect  of  the  presence  in  the  substance  of  the  total  quantity  Q  of  actual 
energy. 

Let  this  quantity  be  conceived  to  be  divided  into  indefinitely  small 
equal  parts  d  Q.  As  those  parts  are  not  only  equal,  but  altogether  alike 
in  nature  and  similarly  circumstanced,  their  effects  must  be  equal ;  there- 
fore, the  effect  of  the  total  energy  Q  must  be  equal  simply  to  the  effect  of 

Q 

one  of  its  small  parts  d  Q,  multiplied  by  the  ratio  -y-y 

But  the  effect  of  the  indefinitely  small  part  d  Q  in  causing  development 
of  potential  energy  of  the  species  U,  during  the  increment  of  state  d  V,  is 
represented  by — 

whence  it  follows,  that  the  effect  of  the  presence  of  the  total  actual  energy 
Q,  in  causing  transformation  of  energy  from  the  actual  form  Q  to  the 
potential  form  U,  is  expressed  by  the  following  formula : — 

«-^v w 

which  is  the  solution  required,  and  is  the  symbolical  expression  of  the 
General  Law  of  the  Transformation  of  Energy  : — 

The  effect  of  the  whole  actual  energy  present  in  a  substance,  in  causing 
transformation  of  energy,  is  the  sum  of  the  effects  of  all  its  parts. 

The  difference  between  this  quantity  and  the  potential  energy  developed, 
viz : — 


(p-«-;nD«v< 


dW 

represents  a  portion  of  potential  energy,  clue  to  causes  different  from  the 

actual  energy  Q.     This  difference  is  null,  when  the  resistance  ( P  =  -p\j) 

against  which  the  state  V  increases,  is  simply  proportional  to  the  total 
actual  energy  Q. 

It  is  next  proposed  to  find  the  quantity  of  actual  energy  of  the  form  Q, 
which  must  be  transmitted  to  the  substance  from  without,  in  order  that 
its  total  actual  energy  may  receive  the  increment  d  Q,  and  its  state  V  at 
the  same  time  the  increment  d  V. 

This  quantity  is  composed  of  three  parts — viz.,  actual  energy,  which 
preserves  its  form,  dQ,;  actual  energy  which  transforms  itself  to  some 
unknown  form,  in  consequence  of  the  resistance  which  is  offered  to  the 
increase  of  the  total  actual  energy,  L  d  Q ;  actual  energy,  already  deter- 


GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY.    205 

mined,  which   transforms   itself  into  potential   energy  of  the  form  U, 

p 
Q .  -7-^  •  d  V ;  the  sum  of  these  parts  being — 

d.Q  =  (l+L)dQ  +  Q.^|.dV,       .        .     (2.) 

in  which  nothing  remains  to  be  determined  except  the  function  L. 

If  Ave  subtract  from  the  above  formula  the  total  potential  energy 
developed  during  the  increment  d  Y,  viz  : — 

■we  obtain  the  algebraical  sum  of  the  energies,  actual  and  potential,  received 
and  developed  by  the  substance  during  the  changes  d  Q,  d  V;  which  is 
thus  expressed : — 

d¥  =  d.Q-d.TJ  =  (l+L)dQ  +  (Q^-l)-p.dY.    (B.) 

This  quantity  must  be  the  exact  differential  of  a  function  of  Q  and  V; 
for  otherwise  it  would  be  possible,  by  varying  the  order  of  the  increments 
d  Q,  d  V,  to  change  the  sum  of  the  energies  of  the  universe. 

It  follows  that — 

*J±-A-(qA. -\?-q  JLy- 
dV-dQK^dq    )r-H'dQ?r' 

and,  consequently,  that 

l=/(Q)  +  Q.^/.wv, 

where  /'  (Q)  is  a  function  of  Q  and  constants,  the  first  derivative  of/'  (Q). 
We  find  at  length  the  following  equation — 

dV=d.Q-d.\J  =  (l+f®)  +  Q.^fvdv)dQ+(Q±-l) 
=  ^{q+/(Q)  +  (q^-i)/p*v}        •    (3.) 

which  represents  the  algebraical  sum  of  the  energy,  actual  and  potential, 
received  and  developed  by  a  substance,  when  the  total  actual  energy  of 
the  species  Q,  and  the  state  V,  receive  respectively  the  increments  d  Q,  d  V. 
It  is  to  be  observed,  that  in  the  last  equation,  the  symbol  J"P .  d  V 
denotes  a  partial  integral,  taken  in  treating  the  particular  value  of  Q,  to 
which  it  corresponds  as  a  constant  quantity ;  while  d .  U  represents  the 
real  magnitude  of  the  potential  energy  developed. 


206    GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY. 

The  application  of  the  general  law  of  the  transformation  of  energy  may 
be  extended  to  any  number  of  kinds  of  energy,  actual  and  potential,  by 
means  of  the  following  equation  : 

=  2  {(l  +/(«)  +  Q.S^i/p<T>q{  +  2  {(SQ^-  1>*TJ 

=  rf  jsQ  +  2/(Q)  +  2(2.Q^-l)/p<Zv|      .     (4.) 

This  equation  is  the  complete  expression  of  the  general  law  of  the 
transformation  of  energy  of  all  possible  kinds,  known  and  unknown.  It 
affords  the  means,  so  soon  as  the  necessary  experimental  data  have  been 
obtained,  of  analysing  every  development  of  potential  energy,  and  referring 
its  several  portions  to  the  species  of  actual  energy  from  which  they  have 
been  produced. 

Amongst  the  consequences  of  this  law,  the  author  deduces  that  which 
may  be  called  the  general  principle  of  the  maximum  effect  of  engines. 

An  engine  consists  essentially  in  a  substance  whose  changes  of  state, 
and  of  actual  energy,  between  given  limits,  are  so  regulated  as  to  produce 
a  permanent  transformation  of  energy. 

Let  Qx  be  the  given  superior  limit  of  actual  energy ;  Q2,  the  inferior 
limit. 

To  produce  the  maximum  permanent  transformation  of  energy  from 
the  actual  to  the  potential  form,  the  substance  must  undergo  a  cycle  of 
four  operations,  viz  : — 

First  Operation. 

The  substance,  preserving  the  constant  quantity  Ql  of  actual  energy, 
passes  from  the  state  VA  to  the  state  VB,  receiving  from  without  the 
following  quantity  of  actual  energy,  which  is  converted  into  potential 
energy: — 


H-«.-/o/v:p-'n- 


Second  Operation. 


The  substance  passes  from  the  superior  limit  of  actual  energy  Qv  to 
the  inferior  limit  Q2.  Let  Vc  be  the  value  of  the  state  V  at  the  end  of 
this  operation. 


GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY.    207 


Third  Operation. 

The  substance,  preserving  the  constant  quantity  Q2  of  actual  energy,, 
passes  from  the  state  Vc  to  the  state  VD,  transmitting  to  external  sub- 
stances the  following  quantity  of  actual  energy,  produced  by  the  disap- 
pearance of  potential  energy: — 


h*=q4/v:p<*v- 


Fourth  Operation. 


The  substance  is  brought  back  to  its  original  actual  energy  Qls  and 
state  VA,  thus  completing  the  cycle  of  operations. 

In  order  that  the  second  and  fourth  operations  may  be  performed 
without  expenditure  of  energy,  the  following  condition  must  be  fulfilled : — 

hi  kVllY  (for  Q  =  Qa) = m  /£  p  d  v  (fOT  Q  =  *«■ 

This  being  the  case,  the  total  expenditure  of  energy  during  a  cycle  of 
operations  will  be  Hv  being  the  quantity  converted  from  the  actual  to 
the  potential  form  during  the  first  operation;  the  energy  lost  will  be  H2, 
the  quantity  reconverted  to  the  actual  form,  and  transmitted  to  external 
substances,  during  the  third  operation ;  and  the  quantity  of  energy  per- 
manently transformed  from  the  actual  to  the  potential  form,  that  is  to 
say,  the  work  done  by  the  engine,  will  be — 

H1-H2=(Q1-Q2)^|^P^V(forQ  =  Q1)    .     (6.) 


Q 

The  ratio  of  this  work  to  the  total  expenditure  of  energy  is 
Ht  —  H9  _  Q2  —  Q0 


(7.) 


This  principle  is  applicable  to  all  possible  engines,  known  and  unknown. 

In  the  sequel  of  the  paper,  the  author  gives  some  examples  of  the 
application  of  the  general  principles  of  the  transformation  of  energy  to 
the  theory  of  heat,  and  to  that  of  electro-magnetism ;  and  deduces  from 
them,  as  particular  cases,  several  laws  already  known  through  specific 
researches. 

The  details  of  the  application  of  these  principles  to  the  theory  of 


208    GENERAL  LAW  OF  THE  TRANSFORMATION  OF  ENERGY. 

heat  are  contained  in  the  sixth  section  of  a  memoir  read  before  the  Royal 
Society  of  Edinburgh,  "  On  the  Mechanical  Action  of  Heat." 

The  actual  energy  produced  by  an  electric  pile  in  unity  of  timo  is 
expressed  by — 

where  M  is  the  electro-motive  force,  and  u,  the  strength  of  the  current. 
The  actual  energy  of  an  electric  circuit  is  expressed  by — 

~Ru2, 

where  R  is  the  resistance  of  the  circuit.  This  energy  is  immediately  and 
totally  transformed  into  sensible  heat. 

The  proportion  of  the  actual  energy  produced  in  the  pile,  which  is 
transformed  into  mechanical  work  by  an  electro-dynamic  machine,  is 
represented  by — 

Qt-Q2  _  M-Rtt 
Q2  M     ' 

The  strength  of  the  current  is  known  to  be  found  by  means  of  the 
equation — 

M-N 

where  N  is  the  negative  or  inverse  electro-motive  force  of  the  apparatus 
by  means  of  which  electricity  is  transformed  into  mechanical  work. 
Hence, 

Qi  -  Q2  _  n 

Qi       ~M" 

The  above  particular  forms  of  the  general  equation  agree  with  formulae 
already  deduced  from  special  researches  by  Mr.  Joule  and  Professor 
William  Thomson. 


OUTLINES   OF  THE  SCIENCE   OF   ENERGETICS.  209 


XII— OUTLINES   OF   THE   SCIENCE   OF   ENEKGETICS.* 

Section  I. — What  constitutes  a  Physical  Theory. 

An  essential  distinction  exists  between  two  stages  in  the  process  of 
advancing  our  knowledge  of  the  laws  of  physical  phenomena;  the  first 
stage  consists  in  observing  the  relations  of  phenomena,  whether  of  such 
as  occur  in  the  ordinary  course  of  nature,  or  of  such  as  are  artificially 
produced  in  experimental  investigations,  and  in  expressing  the  relations 
so  observed  by  propositions  called  formal  laws.  The  second  stage  consists 
in  reducing  the  formal  laws  of  an  entire  class  of  phenomena  to  the 
form  of  a  science ;  that  is  to  say,  in  discovering  the  most  simple  system 
of  principles,  from  which  all  the  formal  laws  of  the  class  of  phenomena 
can  be  deduced  as  consequences. 

Such  a  system  of  principles,  with  its  consequences  methodically 
deduced,  constitutes  the  physical  theory  of  a  class  of  phenomena. 

A  physical  theory,  like  an  abstract  science,  consists  of  definitions  and 
axioms  as  first  principles,  and  of  propositions,  their  consequences ;  but 
with  these  differences: — First,  That  in  an  abstract  science,  a  definition 
assigns  a  name  to  a  class  of  notions  derived  originally  from  observation, 
but  not  necessarily  corresponding  to  any  existing  objects  of  real  pheno- 
mena; and  an  axiom  states  a  mutual  relation  amongst  such  notions,  or 
the  names  denoting  them :  while  in  a  physical  science,  a  definition  states 
properties  common  to  a  class  of  existing  objects,  or  real  phenomena ;  and 
a  physical  axiom  states  a  general  law  as  to  the  relations  of  phenomena. 
And,  secondly,  That  in  an  abstract  science,  the  propositions  first  discov- 
ered are  the  most  simple ;  whilst  in  a  physical  theory,  the  propositions 
first  discovered  are  in  general  numerous  and  complex,  being  formal  laws, 
the  immediate  results  of  observation  and  experiment,  from  which  the 
definitions  and  axioms  are  subsequently  arrived  at  by  a  process  of  reason- 
ing differing  from  that  whereby  one  proposition  is  deduced  from  another 
in  an  abstract  science,  partly  in  being  more  complex  and  difficult, 
and  partly  in  being,  to  a  certain  extent,  tentative — that  is  to  say, 
involving  the  trial  of  conjectural  principles,  and  their  acceptance  or 
rejection,  according  as  their  consequences  are  found  to  agree  or  disagree 
with  the  formal  laws  deduced  immediately  from  observation  and  experi- 
ment. 

*  Read  before  the  Philosophical  Society  of  Glasgow  on  May  2,  1855,  and  published 
in  the  Proceedings  of  that  Society,  Vol.  III.,  No.  VL 

O 


210  OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS. 


Section  II. — The  Abstractive  Method  of  forming  a  Physical 
Theory  distinguished  from  the  Hypothetical  Method. 

Two  methods  of  framing  a  physical  theory  may  be  distinguished, 
characterised  chiefly  by  the  manner  in  which  classes  of  phenomena  are 
defined.  They  may  be  termed,  respectively,  the  abstractive  and  the 
hypothetical  methods. 

According  to  the  abstractive  method,  a  class  of  objects  or  phenomena 
is  defined  by  describing,  or  otherwise  making  to  be  understood,  and 
assigning  a  name  or  symbol  to,  that  assemblage  of  properties  which  is 
common  to  all  the  objects  or  phenomena  composing  the  class,  as  perceived 
by  the  senses,  without  introducing  anything  hypothetical. 

According  to  the  hypothetical  method,  a  class  of  objects  or  pheno- 
mena is  defined,  according  to  a  conjectural  conception  of  their  nature,  as 
being  constituted,  in  a  manner  not  apparent  to  the  senses,  by  a  modifica- 
tion of  some  other  class  of  objects  or  phenomena  whose  laws  are  already 
known.  Should  the  consequences  of  such  a  hypothetical  definition  be 
found  to  be  in  accordance  with  the  results  of  observation  and  experiment, 
it  serves  as  the  means  of  deducing  the  laws  of  one  class  of  objects  or 
phenomena  from  those  of  another. 

The  conjectural  conceptions  involved  in  the  hypothetical  method  may 
be  distinguished  into  two  classes,  according  as  they  are  adopted  as  a  pro- 
bable representation  of  a  state  of  things  which  may  really  exist,  though 
imperceptible  to  the  senses,  or  merely  as  a  convenient  means  of  expressing 
the  laws  of  phenomena ;  two  kinds  of  hypotheses,  of  which  the  former 
may  be  called  objective,  and  the  latter  subjective.  As  examples  of  objec- 
tive hypotheses  may  be  taken,  that  of  vibrations  or  oscillations  in  the 
theory  of  light,  and  that  of  atoms  in  chemistry;  as  an  example  of  a 
subjective  hypothesis,  that  of  magnetic  fluids. 


Section  III. — The  Science  of  Mechanics  considered  as  an 
Illustration  of  the  Abstractive  Method. 

The  principles  of  the  science  of  mechanics,  the  only  example  yet  exist- 
ing of  a  complete  physical  theory,  are  altogether  formed  from  the  data  of 
experience  by  the  abstractive  method.  The  class  of  objects  to  which  the 
science  of  mechanics  relates — viz.,  material  bodies — are  defined  by 
means  of  those  sensible  properties  which  they  all  possess — viz.,  the  pro- 
perty of  occupying  space,  and  that  of  resisting  change  of  motion.  The 
two  classes  of  phenomena  to  which  the  science  of  mechanics  relates  are 
distinguished   by   two    words,   motion   and  force — motion  being   a   word 


OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS.  211 

denoting  that  which  is  common  to  the  fall  of  heavy  bodies,  the  flow  of 
streams,  the  tides,  the  winds,  the  vibrations  of  sonorous  bodies,  the 
revolutions  of  the  stars,  and,  generally,  to  all  phenomena  involving  change 
of  the  portions  of  space  occupied  by  bodies ;  and  force,  a  word  denoting 
that  which  is  common  to  the  mutual  attractions  and  repulsions  of  bodies, 
distant  or  near,  and  of  the  parts  of  bodies,  the  mutual  pressure  or  stress 
of  bodies  in  contact,  and  of  the  parts  of  bodies,  the  muscular  exertions 
of  animals,  and,  generally,  to  all  phenomena  tending  to  produce  or  to 
prevent  motion. 

The  laws  of  the  composition  and  resolution  of  motions,  and  of  the 
composition  and  resolution  of  forces,  are  expressed  by  propositions  which 
are  the  consequences  of  the  definitions  of  motion  and  force  respectively. 
The  laws  of  the  relations  between  motion  and  force  are  the  consequences 
of  certain  axioms,  being  the  most  simple  and  general  expressions  for  all 
that  has  been  ascertained  by  experience  respecting  those  relations. 


Section  IV. — Mechanical  Hypotheses  in  Various  Branches 

of  Physics. 

The  fact  that  the  theory  of  motions  and  motive  forces  is  the  only 
complete  physical  theory,  has  naturally  led  to  the  adoption  of  mechanical 
hypotheses  in  the  theories  of  other  branches  of  physics;  that  is  to*say, 
hypothetical  definitions,  in  which  classes  of  phenomena  are  defined  con- 
jecturally  as  being  constituted  by  some  kind  of  motion  or  motive  force 
not  obvious  to  the  senses  (called  molecular  motion  or  force),  as  when  light 
and  radiant  heat  are  defined  as  consisting  in  molecular  vibrations,  thermo- 
metric  heat  in  molecular  vortices,  and  the  rigidity  of  solids  in  molecular 
attractions  and  repulsions. 

The  hypothetical  motions  and  forces  are  sometimes  ascribed  to  hypo- 
thetical bodies,  such  as  the  luminiferous  ether;  sometimes  to  hypothetical 
jparts,  whereof  tangible  bodies  are  conjecturally  defined  to  consist,  such  as 
atoms,  atomic  nuclei  with  elastic  atmospheres,  and  the  like. 

A  mechanical  hypothesis  is  held  to  have  fulfilled  its  object,  when,  by 
applying  the  known  axioms  of  mechanics  to  the  hypothetical  motions  and 
forces,  results  are  obtained  agreeing  with  the  observed  laws  of  the  classes 
of  phenomena  under  consideration;  and  when,  by  the  aid  of  such  a  hypo- 
thesis, phenomena  previously  unobserved  are  predicted,  and  laws  antici- 
pated, it  attains  a  high  degree  of  probability. 

A  mechanical  hypothesis  is  the  better  the  more  extensive  the  range 
of  phenomena  whose  laws  it  serves  to  deduce  from  the  axioms  of 
mechanics ;  and  the  perfection  of  such  a  hypothesis  would  be,  if  it  could, 


212  OUTLINES  OF  THE   SCIENCE  OF   ENERGETICS. 

by  means  of  one  connected  system  of  suppositions,  be  made  to  form  a 
basis  for  all  branches  of  molecular  physics. 


Section  V. — Advantages  and  Disadvantages  of  Hypothetical 

Theories. 

It  is  well  known  that  certain  hypothetical  theories,  such  as  the  wave 
theory  of  light,  have  proved  extremely  useful,  by  reducing  the  laws  of  a 
various  and  complicated  class  of  phenomena  to  a  few  simple  principles, 
and  by  anticipating  laws  afterwards  verified  by  observation. 

Such  are  the  results  to  be  expected  from  well-framed  hypotheses  in 
every  branch  of  physics,  when  used  with  judgment,  and  especially  with 
that  caution  which  arises  from  the  consideration,  that  even  those  hypo- 
theses whose  consequences  are  most  fully  confirmed  by  experiment  never 
can,  by  any  amount  of  evidence,  attain  that  degree  of  certainty  which 
belongs  to  observed  facts. 

Of  mechanical  hypotheses  in  particular,  it  is  to  be  observed,  that  their 
tendency  is  to  combine  all  branches  of  physics  into  one  system,  by  making 
the  axioms  of  mechanics  the  first  principles  of  the  laws  of  all  phenomena — 
an  object  for  the  attainment  of  which  an  earnest  wish  was  expressed  by 
Newton.* 

In  the  mechanical  theories  of  elasticity,  light,  heat,  and  electricity, 
considerable  progress  has  been  made  towards  that  end. 

The  neglect  of  the  caution  already  referred  to,  however,  has  caused 
some  hypotheses  to  assume,  in  the  minds  of  the  public  generally,  as  well 
as  in  those  of  many  scientific  men,  that  authority  which  belongs  to  facts 
alone ;  and  a  tendency  has,  consequently,  often  evinced  itself  to  explain 
away,  or  set  aside,  facts  inconsistent  with  these  hypotheses,  which  facts, 
rightly  appreciated,  would  have  formed  the  basis  of  true  theories.  Thus, 
the  fact  of  the  production  of  heat  by  friction,  the  basis  of  the  true  theory 
of  heat,  was  long  neglected,  because  inconsistent  with  the  hypothesis  of 
caloric ;  and  the  fact  of  the  production  of  cold  by  electric  currents,  at 
certain  metallic  junctions,  the  key  (as  Professor  William  Thomson  recently 
showed)  to  the  true  theory  of  the  phenomena  of  thermo-electricity,  was, 
from  inconsistency  with  prevalent  assumptions  respecting  the  so-called 
"  electric  fluid,"  by  some  regarded  as  a  thing  to  be  explained  away,  and 
by  others  as  a  delusion. 

Such  are  the  evils  which  arise  from  the  misuse  of  hypotheses. 

Utinam  caetera  naturae  phenomena  ex  principiis  mechanicis  eodem  argumentandi 
genere  derivare  liceret.— {Phil.  Nat.  Prin.  Math.;  Protf.) 


OUTLINES   OF   THE   SCIENCE  OF   ENERGETICS.  213 


Section  VI. — Advantages  of  an  Extension  of  the  Abstractive 
Method  of  framing  Theories. 

Besides  the  perfecting  of  mechanical  hypotheses,  another  and  an  entirely 
distinct  method  presents  itself  for  combining  the  physical  sciences  into 
one  system;  and  that  is,  by  an  extension  of  the  Abstractive  Process 
in  framing  theories. 

The  abstractive  method  has  already  been  partially  applied,  and  with 
success,  to  special  branches  of  molecular  physics,  such  as  heat,  electricity, 
and  magnetism.  We  are  now  to  consider  in  what  manner  it  is  to  be 
applied  to  physics  generally,  considered  as  one  science. 

Instead  of  supposing  the  various  classes  of  physical  phenomena  to  be 
constituted,  in  an  occult  way,  of  modifications  of  motion  and  force,  let 
us  distinguish  the  properties  which  those  classes  possess  in  common  with 
each  other,  and  so  define  more  extensive  classes  denoted  by  suitable 
terms.  For  axioms,  to  express  the  laws  of  those  more  extensive  classes 
of  phenomena,  let  us  frame  propositions  comprehending  as  particular 
cases  the  laws  of  the  particular  classes  of  phenomena  comprehended 
under  the  more  extensive  classes.  So  shall  we  arrive  at  a  body  of 
principles,  applicable  to  physical  phenomena  in  general,  and  which,  being 
framed  by  induction  from  facts  alone,  will  be  free  from  the  uncertainty 
which  must  always  attach,  even  to  those  mechanical  hypotheses  whose 
consequences  are  most  fully  confirmed  by  experiment. 

This  extension  of  the  abstractive  process  is  not  proposed  in  order  to 
supersede  the  hypothetical  method  of  theorising;  for  in  almost  every 
branch  of  molecular  physics  it  may  be  held,  that  a  hypothetical  theory  is 
necessary,  as  a  preliminary  step,  to  reduce  the  expression  of  the  phenomena 
to  simplicity  and  order,  before  it  is  possible  to  make  any  progress  in 
framing  an  abstractive  theory. 


Section  VII. — Nature  of  the  Science  of  Energetics. 

« 

Energy,  or  the  capacity  to  effect  changes,  is  the  common  characteristic 
of  the  various  states  of  matter  to  which  the  several  branches  of  physics 
relate;  if,  then,  there  be  general  laws  respecting  energy,  such  laws  must 
be  applicable,  mutatis  mutandis,  to  every  branch  of  physics,  and  must 
express  a  body  of  principles  as  to  physical  phenomena  in  general. 

In  a  paper  read  before  the  Philosophical  Society  of  Glasgow,  on  the  5  th 
of  January,  1853  (seep.  £08),  a  first  attempt  was  made  to  investigate  such 
principles  by  defining  actual  energy  and  potential  energy,  and  by  demonstrat- 
ing a  general  law  of  the  mutual  transformations  of  those  kinds  of  energy, 


214  OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS. 

of  which  one  particular  case  is  a  previously  known  law  of  the  mechanical 
action  of  heat  in  elastic  bodies,  and  another,  a  subsequently  demonstrated 
law  which  forms  the  basis  of  Professor  William  Thomson's  theory  of 
thermo-electricity. 

The  object  of  the  present '  paper  is  to  present,  in  a  more  systematic 
form,  both  these  and  some  other  principles,  forming  part  of  a  science 
whose  subjects  are,  material  bodies  and  physical  phenomena  in  general, 
and  which  it  is  proposed  to  call  the  Science  of  Energetics. 


Section  VIII. — Definitions  of  Certain  Terms. 

The  peculiar  terms  which  will  be  used  in  treating  of  the  Science  of 
Energetics  are  purely  abstract ;  that  is  to  say,  they  are  not  the  names  of 
any  particular  object,  nor  of  any  particular  phenomena,  nor  of  any 
particular  notions  of  the  mind,  but  are  names  of  very  comprehensive 
classes  of  objects  and  phenomena.  About  such  classes  it  is  impossible  to 
think  or  to  reason,  except  by  the  aid  of  examples  or  of  symbols.  General 
terms  are  symbols  employed  for  this  purpose. 

Substance. 
The  term  "substance"  will  be  applied  to  all  bodies,  parts  of  bodies, 
and  systems  of  bodies.  The  parts  of  a  substance  may  be  spoken  of  as 
distinct  substances,  and  a  system  of  substances  related  to  each  other  may 
be  spoken  of  as  one  complex  substance.  Strictly  speaking,  the  term 
should  be  "material  substance;"  but  it  is  easily  borne  in  mind,  that  in  this 
essay  none  but  material  substances  are  referred  to. 

Properly. 
The  term  "property"  will  be  restricted  to  invariable  properties;  whether 
such  as  always  belong  to  all  material  substances,  or  such  as  constitute 
the  invariable  distinctions  between  one  kind  of  substance  and  another. 

Mass. 
Mass  means  "  quantity  of  substance."     Masses  of  one  kind  of  substance 
may  be  compared  together  by  ascertaining  the  numbers  of  equal  parts 
which  they  contain ;  masses  of  substances  of  different  kinds  are  compared 
by  means  to  be  afterwards  referred  to. 

Accident. 
The  term  "accident"  will  be  applied  to  every  variable  state  of  substances, 
whether  consisting  in  a  condition  of  each  part  of  a  substance,  how  small 
soever,  (which  may  be  called  an  absolute  accident),  or  in  a  physical  relation 


OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS.  215 

between  parts  of  substances,  (which  may  be  called  a  relative  accident). 
Accidents  to  be  the  subject  of  scientific  inquiry,  must  be  capable  of  being 
measured  and  expressed  by  means  of  quantities.  The  quantity,  even  of 
an  absolute  accident,  can  only  be  expressed  by  means  of  a  mentally- 
conceived  relation. 

The  whole  condition  or  state  of  a  substance,  so  far  as  it  is  variable,  is 
a  complex  accident;  the  independent  quantities  which  are  at  once  necessary 
and  sufficient  to  express  completely  this  complex  accident,  are  independent 
accidents.  To  express  the  same  complex  accident,  different  systems  of 
independent  accidents  may  be  employed;  but  the  number  of  independent 
-accidents  in  each  system  will  be  the  same. 

Examples. — The  variable  thermic  condition  of  an  elastic  fluid  is  a 
complex  accident,  capable  of  being  completely  expressed  by  two  independent 
accidents,  which  may  be  any  two  out  of  these  three  quantities — the 
temperature,  the  densihj,  the  pressure — or  any  two  independent  functions 
of  these  quantities. 

The  condition  of  strain  at  a  point  in  an  elastic  solid,  is  a  complex  acci- 
dent, capable  of  being  completely  expressed  by  six  independent  accidents, 
which  may  be  the  three  elongations  of  the  dimensions  and  the  three 
distortions  of  the  faces  of  a  molecule  originally  cubical,  or  the  lengths  and 
directions  of  the  axes  of  the  ellipsoidal  figure  assumed  by  a  molecule 
originally  spherical ;  or  any  six  independent  functions  of  either  of  those 
systems  of  quantities. 

The  distinction  of  accidents  into  absolute  and  relative  is,  to  a  certain 
extent,  arbitrary;  thus,  the  figure  and  dimensions  of  a  molecule  may  be 
regarded  as  absolute  accidents  when  it  is  considered  as  a  whole,  or  as 
relative  accidents  when  it  is  considered  as  made  up  of  parts.  Most  kinds 
of  accidents  are  necessarily  relative;  but  some  kinds  can  only  be  considered 
as  relative  accidents  when  some  hypothesis  is  adopted  as  to  the  occult 
condition  of  the  substances  which  they  affect,  as  when  heat  is  ascribed 
hypothetically  to  molecular  motions;  and  such  suppositions  are  excluded 
from  the  present  inquiry. 

Accidents  may  be  said  to  be  homogeneous  when  the  quantities  expressing 
them  are  capable  of  being  put  together,  so  that  the  result  of  the  com- 
bination of  the  different  accidents  shall  be  expressed  by  one  quantity. 
The  number  of  heterogeneous  kinds  of  accidents  is  evidently  indefinite. 

Effort,  or  Active  Accident. 

The  term  "effort"  will  be  applied  to  every  cause  which  .varies,  or  tends 
to  vary,  an  accident.  This  term,  therefore,  comprehends  not  merely 
forces  or  pressures,  to  which  it  is  usually  applied,  but  all  causes  of  variation 
in  the  condition  of  substances. 

Efforts  may  be  homogeneous  or  heterogeneous. 


21 G  OUTLINES   OF   THE  SCIENCE   OF  ENERGETICS. 

Homogeneous  efforts  are  compared  by  balancing  them  against  each  other. 

An  effort  being  a  condition  of  the  parts  of  a  substance,  or  a  relation 
between  substances,  is  itself  an  accident,  and  may  be  distinguished  as  an 
"  active  accident." 

With  reference  to  a  given  limited  substance,  internal  efforts  are  those 
which  consist  in  actions  amongst  its  parts;  external  efforts  those  which 
consist  in  actions  between  the  given  substance  and  other  substances. 

Passive  Accident. 

The  condition  which  an  effort  tends  to  vary  may  be  called  a  "passive 
accident"  and  when  the  word  "accident"  is  not  otherwise  qualified, 
''passive  accident"  may  be  understood. 

Radical  Accident. 

If  there  be  a  quantity  such  that  it  expresses  at  once  the  magnitude  of 
the  passive  accident  caused  by  a  given  effort,  and  the  magnitude  of  the 
active  accident  or  effort  itself,  let  the  condition  denoted  by  that  quantity 
be  called  a  "  radical  accident." 

[The  velocity  of  a  given  mass  is  an  example  of  a  radical  accident,  for 
it  is  itself  a  passive  accident,  and  also  the  measure  of  the  kind  of  effort 
called  accelerative  force,  which,  acting  for  unity  of  time,  is  capable  of 
producing  that  passive  accident.] 

[The  strength  of  an  electric  current  is  also  a  radical  accident.] 

Effort  as  a  Measure  of  Mass. 

Masses,  whether  homogeneous  or  heterogeneous,  may  be  compared 
by  means  of  the  efforts  required  to  produce  in  them  variations  of  some 
particular  accident.  The  accident  conventionally  employed  for  this  pur- 
pose is  velocity. 

Work 

u  Work "  is  the  variation  of  an  accident  by  an  effort,  and  is  a  term 
comprehending  all  phenomena  in  which  physical  change  takes  place. 
Quantity  of  work  is  measured  by  the  product  of  the  variation  of  the  passive 
accident  by  the  magnitude  of  the  effort,  when  this  is  constant;  or  by 
the  integral  of  the  effort,  with  respect  to  the  passive  accident,  when  the 
effort  is  variable. 

Let  x  denote  a  passive  accident ; 

X  an  effort  tending  to  vary  it ; 

W  the  work  performed  in  increasing  x  from  x0  to  xx :  then, 


OUTLINES   OF  THE   SCIENCE   OF   ENERGETICS.  217 


W  =  f  X  Xdx,  and 

W  =  X  (xx  —£(,),  if  X  is  constant. 


(10 


Work  is  represented  geometrically  by  the  area  of  a  curve,  whereof  the 
abscissa  represents  the  passive  accident,  and  the  ordinate,  the  effort. 


Energy,  Actual  and  Potential. 

The  term  "energy"  comprehends  every  state  of  a  substance  which 
constitutes  a  capacity  for  performing  work.  Quantities  of  energy  are 
measured  by  the  quantities  of  work  which  they  constitute  the  means  of 
performing. 

" Actual  energy"  comprehends  those  kinds  of  capacity  for  performing 
work  which  consist  in  particular  states  of  each  part  of  a  substance,  how 
small  soever ;  that  is,  in  an  absolute  accident,  such  as  heat,  light,  electric 
current,  vis  viva.     Actual  energy  is  essentially  positive. 

"Potential  energy"  comprehends  those  kinds  of  capacity  for  performing 
work  which  consist  in  relations  between  substances,  or  parts  of  substances; 
that  is,  in  relative  accidents.  To  constitute  potential  energy  there  must  be 
a  passive  accident  capable  of  variation,  and  an  effort  tending  to  produce 
such  variation;  the  integral  of  this  effort,  with  respect  to  the  possible 
variation  of  the  passive  accident,  is  potential  energy,  which  differs  in  work 
from  this — that  in  work  the  change  has  been  effected,  which,  in  potential 
energy,  is  capable  of  being  effected. 

Let  x  denote  an  accident;  xv  its  actual  value;  X,  an  effort  tending 
to  vary  it;  xQ,  the  value  to  which  the  effort  tends  to  bring  the  accident ; 
then 


X  d  x  =  U,  denotes  potential  energy. 


r 

J  x, 


Examples  of  potential  energy  are,  the  chemical  affinity  of  uncombined 
elements ;  the  energy  of  gravitation,  of  magnetism,  of  electrical  attraction 
and  repulsion,  of  electro-motive  force,  of  that  part  of  elasticity  which  arises 
from  actions  between  the  parts  of  a  body,  and,  generally,  of  all  mutual 
actions  of  bodies,  and  parts  of  bodies. 

Potential  energy  may  be  passive  or  negative,  according  as  the  effort  in 
question  is  of  the  same  sign  with  the  variation  of  the  passive  accident,  or 
of  the  opposite  sign ;  that  is,  according  as  X  is  of  the  same  sign  with  dx, 
or  of  the  opposite  sign. 


218  OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS. 

It  is  to  be  observed,  that  the  states  of  substances  comprehended  under 
the  term  actual  energy,  may  possess  the  characteristics  of  potential  energy 
also ;  that  is  to  say,  may  be  accompanied  by  a  tendency  or  effort  to  vary 
relative  accidents ;  as  heat,  in  an  elastic  fluid,  is  accompanied  by  a  ten- 
dency to  expand;  that  is,  an  effort  to  increase  the  volume  of  the  receptacle 
containing  the  fluid. 

The  states  to  which  the  term  potential  energy  is  especially  applied,  aro 
those  which  are  solely  due  to  mutual  actions. 

To  put  a  substance  into  a  state  of  energy,  or  to  increase  its  energy,  is 
obviously  a  hind  of  work. 


Section  IX. — First  Axiom. 
All  kinds  of  Work  and  Energy  are  Homogeneous. 

This  axiom  means,  that  any  kind  of  energy  may  be  made  the  means  of 
performing  any  kind  of  work.  It  is  a  fact  arrived  at  by  induction  from 
experiment  and  observation,  and  its  establishment  is  more  especially  due 
to  the  experiments  of  Mr.  Joule. 

This  axiom  leads,  in  many  respects,  to  the  same  consequences  with  the 
hypothesis  that  all  those  kinds  of  energy  which  are  not  sensibly  the  results 
of  motion  and  motive  force  are  the  results  of  occult  modifications  of  motion 
and  motive  force. 

But  the  axiom  differs  from  the  hypothesis  in  this,  that  the  axiom  is 
simply  the  generalised  allegation  of  the  facts  proved  by  experience,  while 
the  hypothesis  involves  conjectures  as  to  objects  and  phenomena  which 
never  can  be  subjected  to  observation. 

It  is  the  truth  of  this  axiom  which  renders  a  science  of  energetics 
possible. 

The  efforts  and  passive  accidents  to  which  the  branches  of  physics  relate 
are  varied  and  heterogeneous ;  but  they  are  all  connected  with  energy,  a 
uniform  species  of  quantity  which  pervades  every  branch  of  physics. 

This  axiom  is  also  equivalent  to  saying,  that  energy  is  transformable  and 
transferable  (an  allegation  which,  in  the  previous  paper  referred  to,  was 
included  in  the  definition  of  energy) ;  for,  to  transform  energy,  means  to 
employ  energy  depending  on  accidents  of  one  kind  in  putting  a  substance 
into  a  state  of  energy  depending  on  accidents  of  another  kind ;  and  to 
transfer  energy,  means  to  employ  the  energy  of  one  substance  in  putting 
another  substance  into  a  state  of  energy,  both  of  which  are  kinds  of  work, 
and  may,  according  to  the  axiom,  be  performed  by  means  of  any  kind  of 
energy. 


OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS.  219 


Section  X. — Second  Axiom. 

TJie  Total  Energy  of  a  Substance  cannot  he  altered  by  the  Mutual  Actions  of 

its  Parts. 

Of  the  truth  of  this  axiom  there  can  be  no  doubt;  but  some  difference 
of  opinion  may  exist  as  to  the  evidence  on  which  it  rests.  There  is  ample 
experimental  evidence  from  which  it  might  be  proved;  but  independently 
of  such  evidence,  there  is  the  argument,  that  the  law  expressed  by  this 
axiom  is  essential  to  the  stability  of  the  universe,  such  as  it  exists. 

The  special  application  of  this  law  to  mechanics  is  expressed  in  two 
ways,  which  are  virtually  equivalent  to  each  other,  the  principle  of  vis  viva, 
and  that  of  the  equality  of  action  and  reaction.  The  latter  principle  is 
demonstrated  by  Newton,  from  considerations  connected  with  the  stability 
of  the  universe  (Principia,  Scholium  to  the  Laws  of  Motion) ;  for  he  shows, 
that  but  for  the  equality  of  action  and  reaction,  the  earth,  with  a  continually 
accelerated  velocity,  would  fly  away  through  infinite  space. 

It  follows,  from  the  second  axiom,  that  all  work  consists  in  the  transfer 
and  transformation  of  energy  alone; .  for  otherwise  the  total  amount  of 
energy  would  be  altered.  Also,  that  the  energy  of  a  substance  can  be 
varied  by  external  efforts  alone. 


Section  XL — External  Potential  Equilibrium. 

The  entire  condition  of  a  substance,  so  far  as  it  is  variable,  as  explained 
in  Sect.  VIII.,  under  the  head  of  accident,  is  a  complex  accident,  which 
may  be  expressed  in  various  ways  by  means  of  different  systems  of 
quantities  denoting  independent  accidents ;  but  the  number  of  independent 
accidents  in  each  system  must  be  the  same. 

The  quantity  of  work  required  to  produce  any  change  in  the  condition 
of  the  substance,  that  is  to  say,  the  potential  energy  received  by  it  from 
without  during  that  change,  may  in  like  manner  be  expressed  in  different 
ways  by  the  sums  of  different  systems  of  integrals  of  external  efforts,  each 
integrated  with  respect  to  the  independent  accident  which  it  tends  to 
augment ;  but  the  number  of  integrals  in  each  system,  and  the  number  of 
efforts,  like  the  number  of  independent  accidents,  must  be  the  same ;  and 
so  also  must  the  suras  of  the  integrals,  each  sum  representing  the  same 
quantity  of  work  in  a  different  way. 

The  different  systems  of  efforts  which  correspond  to  different  systems  of 
independent  accidents,  each  expressing  the  same  complex  accident,  may 


220  OUTLINES   OF  THE   SCIENCE   OF   ENELIGETICS. 

be  called  equivalent  systems  of  efforts;  and  the  finding  of  a  system  of  efforts 
equivalent  to  another  may  be  called  conversion  of  efforts.  * 

When  the  law  of  variation  of  potential  energy,  by  a  change  of  condition 
of  a  substance,  is  known,  the  system  of  external  efforts  corresponding  to 
any  system  of  independent  accidents  is  found  by  means  of  this  principle  : 

Each  effort  is  equal  to  the  rate  of  variation  of  the  potential  energy  with  respect 
to  the  independent  accident  which  that  effort  tends  to  vary;  or,  symbolically, 

X-£         ....     (,) 

External  Potential  Equilibrium  of  a  substance  takes  place,  when  the 
external  effort  to  vary  each  of  the  independent  accidents  is  null ;  that  is  to  say, 
when  the  rate  of  variation  of  the  potential  energy  of  the  substance  with  the 
variation  of  each  independent  a  null. 

For  a  given  substance  there  are  as  many  conditions  of  equilibrium,  of 
the  form 

-  =  0,  ....     (3.) 

as  there  are  independent  accidents  in  the  expression  of  its  condition. 

The  special  application  of  this  law  to  motion  and  motive  force  consti- 
tutes the  principle  of  virtual  velocities,  from  which  the  whole  science  of 
statics  is  deducible. 


Section  XII. — Internal  Potential  Equilibrium. 

The  internal  potential  equilibrium  of  a  substance  consists  in  the  equili- 
brium of  each  of  its  parts,  considered  separately ;  that  is  to  say,  in  the 
nullity  of  the  rate  of  variation  of  the  potential  energy  of  each  part  with 
respect  to  each  of  the  independent  accidents  on  which  the  condition  of 
such  part  depends. 

Examples  of  particular  cases  of  this  principle  are,  the  laws  of  the 
equilibrium  of  elastic  solids,  and  of  the  distribution  of  statical  electricity. 


Section  XIII. — Third  Axiom. 

The  Effort  to  Perform  Work  of  a  Given  Kind,  caused  by  a  Given  Quantity 
of  Actual  Energy,  is  the  Sum  of  the  Efforts  caused  by  the  Parts,  of  that 
Quantity. 

A  law  equivalent  to  this  axiom,  under  the  name  of  the  "General 

*  The  conversion  of  efforts  in  physics  is  connected  with  the  theory  of  lineal  trans- 
formations in  aliiehra. 


OUTLINES   OF   THE   SCIENCE   OF   ENERGETICS.  221 

Law  of  the  Transformation  of  Energy,"  formed  the  principal  subject 
of  the  previous  paper  already  referred  to.     (See  p.  £03.) 

This  axiom  appears  to  be  a  consequence  of  the  definition  of  actual 
energy,  as  a  capacity  for  performing  woi'k  possessed  by  each  part  of  a 
substance  independently  of  its  relations  to  other  parts,  rather  than  an 
independent  proposition. 

Its  applicability  to  natural  phenomena  arises  from  the  fact,  that  there 
are  states  of  substances  corresponding  to  the  definition  of  actual  energy. 

The  mode  of  applying  this  third  axiom  is  as  follows  : — 

Let  a  homogeneous  substance  possess  a  quantity  Q,  of  a  particular  kind 
of  actual  energy,  uniformly  distributed,  and  let  it  be  required  to  determine 
the  amount  of  the  effort  arising  from  the  actual  energy,  which  tends  to 
perform  a  particular  kind  of  work  "W,  by  the  variation  of  a  particular 
passive  accident  x. 

The  total  effort  to  perform  this  kind  of  work  is  represented  by  the  rate 
of  its  increase  relatively  to  the  passive  accident,  viz., — 

d  X  ' 

Divide  the  quantity  of  actual  energy  Q  into  an  indefinite  number  of 
indefinitely  small  parts  SQ;  the  portion  of  the  effort  X  due  to  each  of 
those  parts  will  be 

and  adding  these  partial  efforts  together,  the  effort  caused  by  the  whole 
quantity  of  actual  energy  will  be 

„dX       _    d*W  ,    x 

If  this  be  equal  to  the  effective  effort  X,  then  that  effort  is  simply 
proportional  to,  and  wholly  caused  by,  the  actual  energy  Q.  This  is  the 
case  of  the  pressure  of  a  perfect  gas.  and  the  centrifugal  force  of  a  moving- 
body. 

If  the  effort  caused  by  the  actual  energy  differs  from  the  effective  effort, 
their  difference  represents,  when  the  former  is  the  less,  an  additional 
effort, 

and  when  the  former  is  the  greater,  a  counter  effort  }■  (5.) 

due  to  some  other  cause  or  causes. 


222  OUTLINES  OF  THE  SCIENCE  OF   ENERGETICS. 


Section  XIV. — Rate  of  Transformation;  Metamorphic  Function. 

The  effort  to  augment  a  given  accident  x,  caused  by  actual  energy  of 
a  given  kind  Q,  may  also  be  called  the  "rate  of  transformation"  of  tho 
given  kind  of  actual  energy,  with  increase  of  the  given  accident ;  for  the 
limit  of  the  amount  of  actual  energy  which  disappears  in  performing  work 
by  an  indefinitely  small  augmentation  dx,  of  the  accident,  is 

dII  =  Qd~dx     ....     (6.) 
a  v^j 

n<?w     ,  aw 

—  QjTTj-    dx  =  Qd- 

dQ,dx  aQ 

The  last  form  of  the  above  expression  is  obviously  applicable  when  the 
work  W  is  the  result  of  the  variation  of  any  number  of  independent 
accidents,  each  by  the  corresponding  effort.  For  example,  let  x,  y,  z,  &c, 
be  any  number  of  independent  accidents,  and  X,  Y,  Z,  &c,  the  efforts  to 
augment  them ;  so  that 

dV?  =  Xdx  +  Ydy  +  Zdz  +  &c. 
Then, 

7TT       ._    f  d  X  7         dY  ,        dZ  ,         .        )  /n. 

'*-*il$*'  +  i$'t+i$"  +  *-     J      '     (7-> 

=  Q  d  -777 ,  as  before. 
d*4 

The  function  of  actual  energy,  efforts,  and  passive  accidents,  denoted  by 

S-i?-'.-  •  •  ■« 

whose  variation,  multiplied  by  the  actual  energy,  gives  the  amount  of 
actual  energy  transformed  in  performing  the  work  d  W,  may  be  called  the 
"  Metamorphic  Function  "  of  the  kind  of  actual  energy  Q,  relatively  to 
the  kind  of  work  W. 

When  this  metamorphic  function  is  known  for  a  given  homogeneous 
substance,  the  quantity  H  of  actual  energy  of  the  kind  Q  transformed  to 
the  kind  of  work  W,  during  a  given  operation,  is  found  by  taking  the 
integral 


H 


JQdF (9.) 


The  transformation  of  actual  energy  into  work  by  the  variation  of 
passive  accidents  is  a  reversible  operation;  that  is  to  say,  if  the  passive 


OUTLINES  OF  THE  SCIENCE  OF  ENEEGETICS.  223 

accidents  be  made  to  vary  to  an  equal  extent  in  an  opposite  direction, 
potential  energy  will  be  exerted  upon  the  substance,  and  transformed 
into  actual  energy:  a  case  represented  by  the  expression  (9)  becoming 
negative. 

The  metamorphic  function  of  heat  relatively  to  expansive  power,  was 
first  employed  in  a  paper  on  the  Economy  of  Heat  in  Expansive  Machines, 
read  before  the  Royal  Society  of  Edinburgh  in  April,  1851.  (Trans.  Boy. 
Soc.  Edin.,  Vol.  XXI.) 

The  metamorphic  function  of  heat  relatively  to  electricity  was  employed 
by  Professor  William  Thomson,  in  a  paper  on  Thermo-Electricity,  read  before 
the  Eoyal  Society  of  Edinburgh  in  May,  1854  (Trans.  Boy.  Soc.  Edin., 
Vol.  XXL),  and  was  the  means  of  anticipating  some  most  remarkable  laws, 
afterwards  confirmed  by  experiment. 


Section  XV. — Equilibrium  of  Actual  Energy  ;   Metabatic 

Function. 

It  is  known  by  experiment,  that  a  state  of  actual  energy  is  directly 
transferable ;  that  is  to  say,  the  actual  energy  of  a  particular  kind  (such 
as  heat),  in  one  substance,  may  be  diminished,  the  sole  work  performed 
being  an  equal  augmentation  of  the  same  kind  of  actual  energy  in  another 
substance. 

Equilibrium  of  actual  energy  of  a  particular  kind  Q  between  substances 
A  and  B,  takes  place  when  the  tendency  of  B  to  transfer  this  kind  of 
energy  to  B  is  equal  to  the  tendency  of  B  to  transfer  the  same  kind  of 
energy  to  A. 

Laws  respecting  the  equilibrium  of  particular  kinds  of  actual  energy 
have  been  ascertained  by  experiment,  and  in  some  cases  anticipated  by 
means  of  mechanical  hypotheses,  according  to  which  all  actual  energy  con- 
sists in  the  vis  viva  of  motion. 

The  following  law  will  now  be  proved,  respecting  the  equilibrium  of 
actual  energy  of  all  possible  kinds  : — 

Theorem. — If  equilibrium  of  actual  energy  of  a  given  kind  take 

PLACE  BETWEEN  A  GIVEN  PAIR  OF  SUBSTANCES,  POSSESSING  RESPECTIVELY 
QUANTITIES  OF  ACTUAL  ENERGY  OF  THAT  KIND  IN  A  GIVEN  RATIO,  THEN 
THAT  EQUILIBRIUM  WILL  SUBSIST  FOR  EVERY  PAIR  OF  QUANTITIES  OF 
ACTUAL  ENERGY  BEARING  TO  EACH  OTHER  THE  SAME  RATIO. 

Demonstration. — The  tendency  of  one  substance  to  transfer  actual  energy 
of  the  kind  Q  to  another,  must  depend  on  some  sort  of  effort,  whose 
nature  and  laws  may  be  known  or  unknown.     Let  YA  be  this  effort  for 


224  OUTLINES  OF  THE  SCIENCE  OF  ENERGETICS. 

the  substance  A,  YB  the  corresponding  effort  for  the  substance  B.     Then 
a  condition  of  equilibrium  of  actual  energy  is 

YA  =  YB (10.) 

The  effort  Y  may  or  may  not  be  proportionate  to  the  actual  energy  Q 
multiplied  by  a  quantity  independent  of  Q, 
Case  first. — If  it  is  so  proportional,  let 

K  being  independent  of  Q ;  then  the  condition  of  equilibrium  becomes 

or 

Qb  _  Kj, 

a  ratio  independent  of  the  absolute  amounts  of  actual  energy. 

Case  second. — If  the  effort  Y  is  not  simply  proportional  to  the  actual 
energy  Q,  the  portion  of  it  caused  by  that  actual  energy,  according  to 
the  principle  of  Sect,  XIII.,  deduced  from  the  third  axiom,  is,  for  each 
substance, 

o'/v 

and  a  second  condition  of  equilibrium  of  actual  energy  is  furnished  by  the 
equation 

*&  =  *&      '     •     '  (11') 

In  order  that  this  condition  may  be  fulfilled  simultaneously  with  the  con- 
dition (10),  it  is  necessary  that 

<ZQA  _  dQB 

that  is  to  say,  that  the  ratio  of  the  quantities  of  actual  energy  in  the  two 
substances  should  be  independent  of  those  quantities  themselves ;  a  con- 
dition expressed,  as  before,  by 

?i  =  ?? (11.) 

Q.E.D. 

This  ratio  is  a  quantity  to  be  ascertained  by  experiment,  and  may  be 


OUTLINES   OF   THE  SCIENCE   OF   ENERGETICS.  225 

called  the  ratio  of  the  specific  actual  energies  of  the  substances  A  and 
B,  for  the  kind  of  energy  under  consideration. 
The  function 

K-K-e,  ....   (i2v 

whose  identity  for  the  two  substances  expresses  the  condition  of  equili 
brium  of  the  actual  energy  Q  between  them,  may  be  called  the  "  meta- 
batic  FUNCTION  "  for  that  kind  of  energy. 

In  the  science  of  thermo-dynamics,  the  metabatic  function  is  absolute 
temperature;  and  the  factor  K  is  real  specific  heat.  The  theorem  stated 
above,  when  applied  to  heat,  amounts  to  this  :  that  the  real  specific  heat  of  a 
substance  is  independent  of  its  temperature. 


Section  XVI. — Use  of  the  Metabatic  Function;  Transformation 
of  Energy  in  an  aggregate. 

From  the  mutual  proportionality  of  the  actual  energy  Q,  and  the  meta- 
batic function  9,  it  follows  that  the  operations 

are  equivalent ;  and  that  the  latter  may  be  substituted  for  the  former  in 
all  the  equations  expressing  the  laws  of  the  transformation  of  energy. 
"We  have  therefore 

dX_     dX_      d?W  , 

qd-q-eTe-0deTx-      '      ■   (W 

for  the  effort  to  transform  actual  energy  of  the  kind  Q  into  work  of  the 
land  W,  when  expressed  in  terms  of  the  metabatic  function ;  and 

dn  =  9d~,  ....    (H.) 

for  the  limit  of  the  indefinitely  small  transformation  produced  bj»  an 
indefinitely  small  variation  of  the  accidents  on  which  the  kind  of  work 
W  depends. 

There  is  also  a  form  of  metamorphic  function, 


♦=£=/"  =  "•  ■         :  w 


22G  OUTLINES  OF  THE  SCIENCE  OF   ENERGETICS. 

suited  for  employment  along  with  the  metabatic  function,  in  order  to  find, 
by  the  integration 


B.  =  fed<t>,      ....     (1G.) 


the  quantity  of  actual  energy  of  a  given  kind  Q  transformed  to  the  kind 
of  work  W  during  any  finite  variation  of  accidents. 

The  advantage  of  the  above  expressions  is-,  that  they  are  applicable  not 
merely  to  a  homogeneous  substance,  but  to  any  heterogeneous  substance  or 
aggregate,  which  is  internally  in  a  state  of  equilibrium  of  actual  and  potential 
energy;  for  throughout  all  the  parts  of  an  aggregate  in  that  condition,  the 
metabatic  function  6  is  the  same,  and  each  of  the  efforts  X,  &c,  is  the 
same,  and  consequently  the  metamorphic  function  <p  is  the  same. 

"  Carnot's  function  "  in  thermo-dynamics  is  proportional  to  the  reciprocal 
of  the  metabatic  function  of  heat. 


Section  XVII. — Efficiency  of  Engines. 

An  engine  is  a  contrivance  for  transforming  energy,  by  means  of  the 
periodical  repetition  of  a  cycle  of  variations  of  the  accidents  of  a  sub- 
stance. 

The  efficiency  of  an  engine  is  the  proportion  which  the  energy  perma- 
nently transformed  to  a  useful  form  by  it,  bears  to  the  whole  energy  com- 
municated to  the  working  substance. 

In  a  perfect  engine  the  cycle  of  variations  is  thus  : — 

I.  The  metabatic  function  is  increased,  say  from  60  to  9V 

II.  The  metamorphic  function  is  increased  by  the  amount  A  <f>. 

III.  The  metabatic  function  is  diminished  from  Qx  back  to  60. 

IV.  The  metamorphic  function  is  diminished  by  the  amount  A  (p. 

During  the  second  operation,  the  energy  received  by  the  working  sub- 
stance, and  transformed  from  the  actual  to  the  potential  form  is  Bx  A  0. 
During  the  fourth  operation  energy  is  transformed  back,  to  the  amount 

0O  A  0.     So  that  the  energy  permanently  transformed  during  each  cycle 

a    a 

is  (91  —  00)  A  <f> ;  and  the  efficiency  of  the  engine  -— — °. 

"i 


Section  XVIII. — Diffusion  of  Actual  Energy;  Irreversible  or 
Frictional  Operations. 

There  is  a  tendency  in  every  substance,  or  system  of  substances,  to  the 
equable  diffusion  of  actual  energy;  that  is  to  say,  to  its  transfer  between  the 


OUTLINES  OF  THE   SCIENCE  OF  ENERGETICS.  227 

parts  of  the  substance  or  system,  until  the  value  of  the  metabatic  function 
becomes  uniform. 

This  process  is  not  directly  reversible;  that  is  to  say,  there  is  no  such 
operation  as  a  direct  concentration  of  actual  energy  through  a  tendency  of 
the  metabatic  function  to  become  unequal  in  different  parts  of  a  substance 
or  system. 

Hence  arises  the  impossibility  of  using  the  energy  reconverted  to  the 
actual  form  at  the  lower  limit"  of  the  metabatic  function  in  an  engine. 

There  is  an  analogy  in  respect  of  this  property  of  irreversibility,  between 
the  diffusion  of  one  kind  of  actual  energy  and  certain  irreversible  trans- 
formations of  one  kind  of  actual  energy  to  another,  called  by  Professor 
William  Thomson,  "  Frictional  Phenomena  " — viz.,  the  production  of  heat 
by  rubbing,  and  agitation,  and  by  electric  currents  in  a  homogeneous  sub- 
stance at  a  uniform  temperature. 

In  fact,  a  conjecture  may  be  hazarded,  that  immediate  diffusion  of  the 
actual  energy  produced  in  frictional  phenomena,  is  the  circumstance  which 
renders  them  irreversible ;  for,  suppose  a  small  part  of  a  substance  to  have 
its  actual  energy  increased  by  the  exertion  of  some  kind  of  work  upon  it, 
then,  if  the  increase  of  actual  energy  so  produced  be  immediately  diffused 
amongst  other  parts,  so  as  to  restore  the  uniformity  of  the  metabatic 
function,  the  whole  process  will  be  irreversible.  This  speculation,  how- 
ever, is,  for  the  present,  partly  hypothetical ;  and,  therefore,  does  not, 
strictly  speaking,  form  part  of  the  science  of  energetics. 


Section  XIX. — Measurement  of  Time. 

The  general  relations  between  energy  and  time  must  form  an  important 
branch  of  the  science  of  energetics;  but  for  the  present,  all  that  I  am 
prepared  to  state  on  this  subject  is  the  following  DEFINITION  OF  EQUAL 
times  : — 

Equal  times  are  the  times  in  which  equal  quantities  of  the  same  kind  of 
work  are  performed  by  equal  and  similar  substances,  under  loholly  similar 
circumstances. 


Section  XX. — Concluding  Eemarks. 

It  is  to  be  observed,  that  the  preceding  articles  are  not  the  results  of  a 
new  and  hitherto  untried  speculation,  but  are  the  generalised  expression 
of  a  method  of  reasoning  which  has  already  been  applied  with  success  to 
special  branches  of  physics. 


£28  OUTLINES  OF  THE  SCIENCE  OF   ENERGETICS. 

In  this  brief  essay,  it  has  not  been  attempted  to  do  more  than  to  give 
an  outline  of  some  of  the  more  obvious  principles  of  the  science  of  ener- 
getics, or  the  abstract  theory  of  physical  phenomena  in  general ;  a  science 
to  which  the  maxim,  true  of  all  science,  is  specially  applicable — that  its 
subjects  are  boundless,  and  that  they  never  can,  by  human  labours,  be 
exhausted,  nor  the  science  brought  to  perfection. 


THE  PHRASE   "POTENTIAL   ENERGY."  229 


XIII.— ON  THE  PHEASE  "POTENTIAL  ENERGY,"  AND  ON 
THE  DEFINITIONS  OF  PHYSICAL  QUANTITIES* 

1.  In  the  course  of  an  essay  by  Sir  John  Herschel  "On  the  Origin  of 
Force,"  which  appeared  some  time  ago  in  the  Fortnightly  Review,  and  has 
lately  been  republished  in  a  volume,  entitled  Familiar  Lectures  on  Scientific 
Subjects,  the  opinion  is  expressed  that  the  phrase  "  Potential  Energy  "  is 
"  unfortunate,  inasmuch  as  it  goes  to  substitute  a  truism  for  the  announce- 
ment of  a  great  dynamical  fact"  (Familiar  Lectures,  page  469). 

2.  There  is  here  no  question  as  to  the  reality  of  the  class  of  relations 
amongst  bodies  to  which  that  phrase  is  applied,  nor  as  to  any  matter  of 
fact  concerning  those  relations,  but  as  to  the  convenient  and  appropriate 
use  of  language.  This  is  a  sort  of  question  in  the  discussion  of  which 
authority  has  much  weight ;  and  when  an  objection  to  the  appropriateness 
of  a  term  is  made  by  an  author  who  is  not  less  eminent  as  a  philosopher 
than  as  a  man  of  science,  and  whose  skill  in  the  art  of  expressing  scientific 
truth  in  clear  language  is  almost  unparalleled,  it  becomes  the  duty  of  those 
who  use  that  term  to  examine  carefully  their  grounds  for  doing  so. 

3.  As  the  phrase  "Potential  Energy,"  now  so  generally  used  by  writers  on 
physical  subjects,  was  first  proposed  by  myself  in  a  paper  "  On  the  General 
Law  of  the  Transformation  of  Energy,"  f  read  before  the  Philosophical 
Society  of  Glasgow,  on  the  5th  of  January,  1853  (seep.  203),  I  feel  that 
the  remark  of  Sir  John  Herschel  makes  it  incumbent  upon  me  to  explain 
the  reasons  which  led  me,  after  much  consideration,  to  adopt  that  phrase 
for  the  purpose  of  denoting  all  those  relations  amongst  bodies,  or  the  parts 
of  bodies,  which  consist  in  a  power  of  doing  work  dependent  on  mutual 
configurations. 

4.  The  kind  of  quantity  now  in  question  forms  part  of  the  subject  of 
the  thirty -ninth  proposition  of  Newton's  Principia;  but  it  is  there  repre- 
sented by  the  area  of  a  figure,  or  by  symbols  only,  and  not  designated 
by  a  name ;  and  such  is  also  the  case  in  many  subsequent  mathematical 
writings. 

5.  The  application  of  the  word  "force"  to  that  kind  of  quantity  is  open 

*  Pvead  before  the  Philosophical  Society  of  Glasgow  on  Jan.  23,  1867,  and  published 
in  the  Proceedings  of  that  Society,  Vol.  VI.,  No.  III. 

t  Viz.,— that  the  effect  of  the  presence  of  a  quantity  of  actual  energy,  in  causing 
transformation  of  energy  between  the  actual  and  the  potential  forms,  is  the  sum  of  the 
effects  of  all  the  parts  of  that  quantity. 


230  THE  PHRASE   "POTENTIAL  ENERGY." 

to  the  objection,  that  when  "force"  is  taken  in  the  sense  in  which  Newton 
defines  "vis  motrix,"  the  power  of  performing  work  is  not  simply  force, 
but  force  multiplied  by  space.  To  make  such  an  application  of  the  word 
"force,"  therefore,  would  have  been  to  designate  a  product  by  the  name 
properly  belonging  to  one  of  its  factors,  and  would  have  added  to 
the  confusion  which  has  already  arisen  from  the  ambiguous  employ- 
ment of  that  word. 

6.  The  word  "  power,"  though  at  first  sight  it  might  seem  very  appro- 
priate, was  already  used  in  mechanics  in  at  least  three  different  senses  : — 
viz.,  first,  the  power  of  an  engine,  meaning  the  rate  at  which  it  performs 
work,  and  being  the  product  of  force  and  space  divided  by  time;  secondly, 
the  power,  in  the  sense  of  effort  or  pressure,  which  drives  a  machine ;  and 
thirdly,  "mechanical  powers,"  meaning  certain  elementary  machines.  Thus 
"'power"  was  open  to  the  same  sort  of  objection  with  "force." 

7.  About  the  beginning  of  the  present  century,  the  word  "energy"  had 
been  substituted  by  Dr.  Thomas  Young  for  "vis  viva,"  to  denote  the 
capacity  for  performing  work  due  to  velocity ;  and  the  application  of  the 
same  word  had  at  a  more  recent  time  been  extended  by  Sir  William 
Thomson  to  capacity  of  any  sort  for  performing  work.  There  can  be  no 
doubt  that  the  word  "energy"  is  specially  suited  for  that  purpose;  for  not 
only  does  the  meaning  to  be  expressed  harmonise  perfectly  with  the 
etymology  of  avipyua,  but  the  word  "energy"  has  never  been  used  in 
precise  scientific  writings  in  a  different  sense;  and  thus  the  risk  of 
ambiguity  is  avoided. 

8.  It  appeared  to  me,  therefore,  that  what  remained  to  be  done,  was  to 
qualify  the  noun  "energy"  by  appropriate  adjectives,  so  as  to  distinguish 
between  energy  of  activity  and  energy  of  configuration.  The  well-known 
pair  of  antithetical  adjectives,  "  actual "  and  "  potential,"  seemed  exactly 
suited  for  that  purpose ;  and  I  accordingly  proposed  the  phrases  "  actual 
energy  "  and  "  potential  energy,"  in  the  paper  to  which  I  have  referred. 

9.  I  was  encouraged  to  persevere  in  the  use  of  those  phrases,  by  the 
fact  of  their  being  immediately  approved  of  and  adopted  by  Sir  William 
Thomson ;  a  fact  to  which  I  am  disposed  to  ascribe,  in  a  great  measure, 
the  rapid  extension  of  their  use  in  the  course  of  a  period  so  short  in 
the  history  of  science  as  fourteen  years.*  I  had  also  the  satisfaction 
of  receiving  a  very  strong  expression  of  approval  from  the  late  Professor 
Baden  Powell. 

10.  Until  some  years  afterwards  I  was  not  aware  of  the  fact,  that  the 
idea  of  a  phrase  equivalent  to  "  potential  energy,"  in  its  purely  mechanical 
sense,  had  been  anticipated  by  Carnot,  who,  in  an  essay  on  machines  in 
general,  employed  the  term  "force  vive  virtuelle,"  of  which  "potential 

*  Sir  William  Thomson  and  Professor  Tait  have  lately  substituted  the  word  "kinetic  " 
for  "actual." 


THE  PHRASE   "POTENTIAL  ENERGY."  231 

energy"  might  be  supposed  to  be  almost  a  literal  translation.  That  coin- 
cidence shows  how  naturally  the  phrase  "  potential  energy,"  or  something 
equivalent,  occurs  to  one  in  search  of  words  appropriate  to  denote  that 
power  of  performing  work  which  is  due  to  configuration,  and  not  to  activity. 

11.  Having  explained  the  reasons  which  led  me  to  propose  the  use  of 
the  phrase  "  potential  energy,"  I  have  next  to  make  some  observations  on 
the  objection  made  by  Sir  John  Herschel  to  that  phrase,  that  "  it  goes  to 
substitute  a  truism  for  a  great  dynamical  fact." 

12.  It  must  be  admitted  that  the  use  of  the  term  "potential  energy" 
tends  to  make  the  statement  of  the  law  of  the  conservation  of  energy  wear, 
to  a  certain  extent,  the  appearance  of  a  truism.  It  seems  to  me,  however, 
that  such  must  always  be  the  effect  of  denoting  physical  relations  by  words 
that  are  specially  adapted  to  express  the  properties  of  those  relations ;  or, 
what  amounts  virtually  to  the  same  thing,  of  drawing  up  precise  and  com- 
plete definitions  of  physical  terms.  Let  A  and  B  denote  certain  conceivable 
relations,  and  let  them  be  precisely  and  completely  defined ;  then,  from  the 
definitions  follows  the  proposition,  that  A  and  B  are  related  to  each  other 
in  a  certain  way;  and  that  proposition  wears  the  appearance  of  a  truism, 
and  is  virtually  comprehended  in  the  definitions.  But  it  is  not  a  bare 
truism ;  for  when  with  the  definitions  are  conjoined  the  two  facts,  ascer- 
tained by  experiment  and  observation,  that  there  are  relations  amongst 
real  bodies  corresponding  to  the  definition  of  A,  and  that  there  are  also 
relations  amongst  real  bodies  corresponding  to  the  definition  of  B,  the  pro- 
position as  to  relation  between  A  and  B  becomes  not  a  bare  truism,  but  a 
physical  fact.  In  the  present  case,  for  example,  "actual  energy"  and 
"potential  energy"  are  defined  in  such  a  way  as  to  make  the  proposition : 
That  what  a  body  or  a  system  of  bodies  gains  in  one  form  of  energy  through 
mutual  actions,  it  loses  in  the  other  form — in  other  words,  that  the  sum  of 
actual  and  potential  energies  is  "conserved" — follow  from  the  definitions, 
so  as  to  sound  like  a  truism ;  but  when  it  is  proved  by  experiment  and 
observation  that  there  are  relations  amongst  real  bodies  agreeing  with  the 
definitions  of  "  actual  energy"  and  "  potential  energy,"  that  which  otherwise 
would  be  a  truism  becomes  a  fact. 

1 3.  A  definition  cannot  be  true  or  false ;  for  it  makes  no  assertion,  but 
says,  "  let  such  a  word  or  phrase  be  used  in  such  a  sense;"  but  it  may 
be  real  or  fantastic,  according  as  the  description  contained  in  it  corresponds, 
or  not,  to  real  objects  and  phenomena;  and  when,  by  the  aid  of  experi- 
ment and  observation,  a  set  of  definitions  have  been  framed  which  possess 
reality,  precision,  and  completeness,  the  investing  of  a  physical  fact  with 
the  appearance  of  a  truism  is  often  an  unavoidable  consequence  of  the 
use  of  the  term  so  defined. 

14.  In  the  case  of  physical  quantities  in  particular,  the  definition  involves 
a  rule  for  measuring  the  quantity ;  and  the  proof  of  the  reality  of  the 


232  THE  PHRASE   "POTENTIAL  ENERGY." 

definition  is  the  fact,  that  the  application  of  the  rule  to  the  same  quantity 
under  different  circumstances  gives  consistent  results,  which  it  would  not 
do  if  the  definition  were  fantastic ;  and  hence  the  definitions  of  a  set  of 
physical  quantities  necessarily  involve  mathematical  relations  amongst 
those  quantities,  which,  when  expressed  as  propositions  and  compared 
with  the  definitions,  wear  the  appearance  of  truisms,  and  are  at  the  same 
time  statements  of  fact. 

15.  In  illustration  of  the  foregoing  principles,  it  may  be  pointed  out 
that  there  is  a  certain  set  of  definitions  of  the  measurement  of  time,  force, 
and  mass,  which  reduce  the  laws  of  motion  to  the  form  of  truisms,  thus — 

I.  Let  "  equal  times "  mean  the  times  in  which  a  moving  body,  under 
the  influence  of  no  force,  describes  equal  spaces.  This  definition  is 
proved  to  be  real  by  the  fact,  that  times  which  are  equal  when  compared 
by  means  of  the  free  motion  of  one  body,  are  equal  when  compared  by 
means  of  the  free  motion  of  any  other  body.  If  the  definition  were 
fantastic,  times  might  be  equal  as  measured  by  the  free  motion  of  one 
body,  and  unequal  as  measured  by  that  of  another. 

II.  Let  "fwce"  mean  a  relation  between  a  pair  of  bodies  such  that 
their  relative  velocity  changes,  or  tends  to  change,  in  magnitude  or 
direction,  or  both ;  and  let  "  equal  forces  "  mean  those  which  act  when 
equal  changes  of  the  relative  velocity  of  a  given  pair  of  bodies  occur  in 
equal  times.  This  definition  is  proved  to  be  real  by  the  fact,  that  the 
comparative  measurements  of  forces  made  in  different  intervals  of  time 
are  consistent  with  each  other,  which  would  not  be  the  case  if  the 
definition  were  fantastic. 

III.  Let  the  "  mass  "  of  a  body  mean  a  quantity  inversely  proportional 
to  the  change  of  velocity  impressed  on  that  body  in  a  given  time  by  a 
given  force.  This  definition  is  proved  to  be  real  by  the  fact,  that  the 
ratio  of  the  masses  of  two  given  bodies  is  found  experimentally  to  be 
always  the  same,  when  those  masses  are  compared  by  means  of  the 
velocities  impressed  on  them  by  different  forces,  and  in  different  times ; 
and  is  also  the  same,  whether  each  of  the  masses  is  measured  as  a  whole 
or  as  the  sum  of  a  set  of  parts. 

Assuming  those  definitions  as  merely  verbal,  without  reference  to 
their  reality,  the  laws  of  motion  take  the  form  of  verbal  truisms;  but 
when  experiment  and  observation  inform  us  that  permanent  relations 
exist  amongst  real  bodies  and  real  events  corresponding  to  the  definitions, 
those  apparent  truisms  become  statements  of  fact. 

16.  One  of  the  chief  objects  of  mathematical  physics  is  to  ascertain, 
by  the  help  of  experiment  and  observation,  what  physical  quantities  or 
functions  are  "conserved."    Such  quantities  or  functions  are,  for  example — 

I.  The  mass  of  every  particle  of  matter,  conserved  at  all  times  and 
under  all  circumstances. 


THE  PHRASE  "  POTENTIAL  ENERGY."  233 

II.  The  resultant  momentum  of  a  body,  or  system  of  bodies,  conserved 
so  long  as  internal  forces  act  alone. 

III.  The  resultant  angular  momentum  of  a  body  or  system  of  bodies, 
conserved  so  long  as  internal  forces  act  alone. 

IV.  The  total  energy  of  a  body,  or  system  of  bodies,  conserved  so  long 
as  internal  forces  act  alone. 

V.  The  tliermo-dynamic  function,  conserved  in  a  body  while  it  neither 
receives  nor  gives  out  heat. 

In  defining  such  physical  quantities  as  those,  it  is  almost,  if  not  quite, 
impossible  to  avoid  making  the  definition  imply  the  property  of  con- 
servation ;  so  that  when  the  fact  of  conservation  is  stated,  it  has  the  form 
of  a  truism. 

1 7.  In  conclusion,  it  appears  to  me  that  the  making  of  a  physical  law 
wear  the  appearance  of  a  truism,  so  far  from  being  a  ground  of  objection 
to  the  definition  of  a  physical  term,  is  rather  a  proof  that  such  definition 
has  been  framed  in  strict  accordance  with  reality. 


234  THE  MECHANICAL  ACTION   OF  HEAT. 


XIV.— ON  THE  MECHANICAL  ACTION  OF  HEAT,  ESPECIALLY 
IN  GASES  AND  VAPOURS.  * 

Introduction — Summary  of  the  Principles  of  the  Hypothesis 

of  Molecular  Vortices,  and  its  Application  to  the  Theory 

of  Temperature,  Elasticity,  and  Real  Specific  Heat. 

The  ensuing  paper  forms  part  of  a  series  of  researches  respecting  the 
consequences  of  an  hypothesis  called  that  of  Molecular  Vortices,  the 
object  of  which  is,  to  deduce  the  laws  of  elasticity,  and  of  heat  as  connected 
with  elasticity,  by  means  of  the  principles  of  mechanics,  from  a  physical 
supposition  consistent  and  connected  with  the  theory  which  deduces  the 
laws  of  radiant  light  and  heat  from  the  hypothesis  of  undulations.  Those 
researches  were  commenced  in  1842,  and  after  having  been  laid  aside 
for  nearly  seven  years,  from  the  want  of  experimental  data,  were  resumed 
in  consequence  of  the  appearance  of  the  experiments  of  M.  Regnault  on 
gases  and  vapours. 

The  investigation  which  I  have  now  to  describe,  relates  to  the  mutual 
conversion  of  heat  and  mechanical  power  by  means  of  the  expansion  and 
contraction  of  gases  and  vapours. 

In  the  introduction,  which  I  here  prefix  to  it,  I  purpose  to  give  such  a 
summary  of  the  principles  of  the  hypothesis  as  is  necessary  to  render  the 
subsequent  investigation  intelligible. 

The  fundamental  suppositions  are  the  following : — 

First,  That  each  atom  of  matter  consists  of  a  nucleus,  or  central  physical 
point,  enveloped  by  an  elastic  atmosphere,  which  is  retained  in  its  position  by 
forces  attractive  towards  the  nucleus  or  centre. 

Suppositions  similar  to  this  have  been  brought  forward  by  Franklin, 
iEpinus,  Mossotti,  and  others.  They  have  in  general,  however,  conceived 
the  atmosphere  of  each  nucleus  to  be  of  variable  mass.  I  have  treated 
it,  on  the  contrary,  as  an  essential  part  of  the  atom.  I  have  left  the 
question  indeterminate,  whether  the  nucleus  is  a  small  body  of  a  character 
distinct  from  that  of  the  atmosphere,  or  merely  a  portion  of  the  atmosphere 
in  a  highly  condensed  state,  owing  to  the  mutual  attraction  of  its  parts. 

According  to   this  first  supposition,  the  boundary  between  two  con- 

*  Read  before  the  Royal  Society  of  Edinburgh  on  Feb.  4,  1850,  and  published  in 
the  Transactions  of  that  Society,  Vol.  XX.,  Part  I.     (See  also  p.  16.) 


THE  MECHANICAL  ACTION  OF   HEAT.  235 

tiguous  atoms  of  a  body  is  an  imaginary  surface  at  which  the  attractions 
of  all  the  atomic  centres  of  the  body  balance  each  other;  and  the  elasticity 
of  the  body  is  made  up  of  two  parts :  First,  the  elasticity  of  the  atomic 
atmospheres  at  the  imaginary  boundaries  of  the  atoms,  which  I  shall  call 
the  superficial-atomic  elasticity;  and,  secondly,  the  force  resulting  from 
the  mutual  actions  of  distinct  atoms.  If  the  atmospheres  are  so  much  con- 
densed round  their  nuclei  or  centres,  that  the  superficial-atomic  elasticity 
is  insensible,  and  that  the  resultants  of  the  mutual  actions  of  all  parts  of 
the  distinct  atoms  are  forces  acting  along  the  lines  joining  the  nuclei  or 
centres,  then  the  body  is  a  perfect  solid,  having  a  tendency  to  preserve 
not  only  a  certain  bulk,  but  a  certain  figure ;  and  the  elasticity  of  figure, 
or  rigidity,  bears  certain  definite  relations  to  the  elasticity  of  volume. 

If  the  atmospheres  are  less  condensed  about  their  centres,  so  that  the 
mutual  actions  of  distinct  atoms  are  not  reducible  to  a  system  of  forces 
acting  along  the  lines  joining  the  atomic  centres,  but  produce  merely  a 
cohesive  force  sufficient  to  balance  the  superficial-atomic  elasticity,  then 
the  condition  is  that  of  a  perfect  liquid ;  and  the  intermediate  conditions 
between  this  and  perfect  solidity  constitute  the  gelatinous,  plastic,  and 
viscous  states. 

When  the  mutual  actions  of  distinct  atoms  are  very  small  as  compared 
with  the  superficial-atomic  elasticity,  the  condition  is  that  of  gets  or  vapour; 
and  when  the  substance  is  so  far  rarefied  that  the  influence  of  the  atomic 
nuclei  or  centres  in  modifying  the  superficial  elasticity  of  their  atmos- 
pheres is  insensible,  it  is  then  in  the  state  of  perfect  gas. 

So  far  as  our  experimental  knowledge  goes,  the  elasticity  of  a  perfect 
gas,  at  a  given  temperature,  varies  simply  in  proportion  to  its  density. 
I  have  therefore  assumed  this  to  be  the  law  of  the  elasticity  of  the 
atomic  atmospheres,  ascribing  a  specific  coefficient  of  elasticity  to  each 
substance. 

The  second  supposition,  being  that  from  which  the  hypothesis  of  mole- 
cular vortices  derives  its  name,  is  the  following : — That  the  elasticity  due 
to  heat  arises  from  the  centrifugal  force  of  revolutions  or  oscillations  among  the 
particles  of  the  atomic  atmospheres  ;  so  that  quantity  of  heat  is  the  vis  viva  of 
those  revolutions  or  oscillations. 

This  supposition  appears  to  have  been  first  definitely  stated  by  Sir 
Humphry  Davy.  It  has  since  been  supported  by  Mr.  Joule,  whose 
valuable  experiments  to  establish  the  convertibility  of  heat  and  mechanical 
power  are  well  known.  So  far  as  I  am  aware,  however,  its  consequences 
have  not  hitherto  been  mathematically  developed. 

To  connect  this  hypothesis  with  the  undulatory  theory  of  radiation,  I 
have  introduced  a  third  supposition :  That  the  medium  which  transmits  light 
and  radiant  heat  consists  of  the  nuclei  of  the  atoms,  vibrating  independently, 
or  almost  independently,  of  their  atmospheres;    so  that   the   absorption  of 


236  THE   MECHANICAL   ACTION   OF   HEAT. 

light  and  of  radiant  heat,  is  the  transference  of  motion  from  the  nuclei  to 
their  atmospheres;  and  the  emission  of  light  and  of  radiant  heat,  the 
transference  of  motion  from  the  atmospheres  to  their  nuclei. 

Although  in  all  undulations  of  sensible  length  and  amplitude,  such  as 
those  of  sound,  the  nuclei  must  carry  their  atmospheres  along  with  them, 
and  vibrating  thus  loaded,  produce  a  comparatively  slow  velocity  of 
propagation;  yet,  in  all  probability,  the  minute  vibrations  of  light  and 
radiant  heat  may  be  performed  by  the  atomic  nuclei  in  transparent  and 
diathermanous  bodies,  without  moving  the  atmospheres  more  than  by 
that  amount  which  constitutes  absorption;  and  those  vibrations  will 
therefore  be  transmitted  according  to  the  laws  of  the  elasticity  of  perfect 
solids,  and  with  a  rapidity  corresponding  to  the  extreme  smallness  of  the 
masses  set  in  motion,  as  compared  with  the  mutual  forces  exerted  by  them. 

This  supposition  is  peculiar  to  my  own  view  of  the  hypothesis,  and  is, 
in  fact,  the  converse  of  the  idea  hitherto  adopted,  of  an  ether  surrounding 
ponderable  particles. 

The  second  and  third  suppositions  involve  the  assumption,  that  motion 
can  be  communicated  between  the  nuclei  and  their  atmospheres,  and 
between  the  different  parts  of  the  atmospheres;  so  that  there  is  a  tendency 
to  produce  some  permanent  condition  of  motion,  which  constitutes  equili- 
brium of  heat.  It  is  now  to  be  considered  what  kind  of  motion  is  capable 
of  producing  increase  of  elasticity,  and  what  are  the  conditions  of  perma- 
nency of  that  motion. 

It  is  obvious,  that  the  parts  of  the  atomic  atmospheres  may  have 
motions  of  alternate  expansion  and  contraction,  or  of  rectilinear  oscillation 
about  a  position  of  equilibrium,  without  affecting  the  superficial  atomic 
elasticity,  except  by  small  periodical  changes.  Should  they  have  motions, 
however,  of  revolution  about  centres,  so  as  to  form  a  group  of  vortices,  the 
centrifugal  force  will  have  the  effect  of  increasing  the  density  of  the 
atmosphere  at  what  I  have  called  the  bounding  surfaces  of  the  atoms,  and 
thus  of  augmenting  the  elasticity  of  the  body. 

In  this  summary,  I  shall  not  enter  into  the  details  of  mathematical 
analysis,  but  shall  state  results  only.  The  following,  then,  are  the  con- 
ditions which  must  be  fulfilled,  in  order  that  a  group  of  vortices,  of  small 
size  as  compared  with  the  bulk  of  an  atom,  and  of  various  diameters,  may 
permanently  co-exist,  whether  side  by  side,  or  end  to  end,  in  the  atomic 
atmospheres  of  one  substance,  or  of  various  substances  mixed. 

First,  The  mean  elasticity  must  vary  continuously,  which  involves  the 
condition,  that  at  the  surface  of  contact  of  two  vortices  of  different 
substances,  side  by  side,  or  end  to  end,  the  respective  densities  at  each 
point  of  contact  must  be  inversely  proportional  to  the  coefficients  of 
elasticity.  Hence,  the  specific  gravities  of  the  atmospheric  parts  of  all 
substances,  under  -precisely  similar  circumstances  as  to  heat  and  molecular  forces 


THE  MECHANICAL   ACTION   OF   HEAT.  237 

(a  condition  realised  in  perfect  gases  at  the  same  pressure  and  temperature), 
are  inversely  proportional  to  the  coefficients  of  atmospheric  elasticity.  Therefore, 
let  juL  represent  the  mass  of  the  atmosphere  of  one  atom  of  any  substance, 
b  its  coefficient  of  elasticity,  and  n  the  number  of  atoms  which,  in  the 
state  of  perfect  gas,  occupy  unity  of  volume  under  unity  of  pressure  at 
the  temperature  of  melting  ice; — then 

n  fib (I.) 

is  a  constant  quantity  for  all  substances. 

Secondly,  The  superficial  elasticity  of  a  vortex  must  not  be  a  function 
of  its  diameter :  to  fulfil  which  condition,  the  linear  velocity  of  revolution 
must  be  equal  throughout  all  parts  of  each  individual  vortex. 

Thirdly,  In  all  contiguous  vortices  of  the  same  substance,  the  velocities 
of  revolution  must  be  equal;  and  in  contiguous  vortices  of  different 
substances,  the  squares  of  the  velocities  must  be  proportional  to  the 
coefficients  of  elasticity  of  the  molecular  atmospheres. 

The  second  and  third  conditions  are  those  of  equilibrium  of  heat,  and 
are  equivalent  to  this  law  : — 

Temperature  is  a  function  of  the  square  of  the  velocity  of  revolution  in 
the  molecular  vortices,  divided  by  the  coefficient  of  elasticity  of  the  atomic  atmos- 
pheres; or 


Temperature  =  0  ( —J,       .         .         .     (II.) 


where  w  represents  that  velocity. 

The  mean  elasticity  which  a  vortex  exerts  endways  is  not  affected  by 
its  motion,  being  equal  to 

bP, (HI.) 

where  p  is  its  mean  density.  The  superficial  elasticity  at  its  lateral 
surfaces,  however,  is  expressed  by 

%  +  *p (iv.) 

The  additional  elasticity  ■—-*-,  being  that  which  is  due  to  the  motion,  is 

independent  of  the  diameter.  The  divisor  #,(the  force  of  gravity)  is 
introduced,  on  the  supposition  of  the  density  p  being  measured  by 
weight. 

Supposing  the  atmosphere  of  an  atom  to  be  divided  into  concentric 
spherical  layers,  it  may  be  shown  that  the  effect  of  the  co-existence  of  a 
great  number  of  small  vortices  in  one  of  those  layers  whose  radius  is 
r,  and  mean  density  p,  is  to  give  it  a  centrifugal  force,  expressed  by 

^ (V.) 

gr  v 


238  THE  MECHANICAL  ACTION  OF  HEAT. 

which  tends  to  increase  the  density  and  elasticity  of  the  atmosphere  at 
the  surface,  which  I  have  called  the  boundary  of  the  atom.  The  layer 
is  also  acted  upon  by  the  difference  between  the  mean  elasticities  at  its 
two  surfaces,  and  by  the  attraction  towards  the  atomic  centre;  and  these 
three  forces  must  balance  each  other. 

I  have  integrated  the  differential  equation  which  results  from  this 
condition,  for  substances  in  the  gaseous  state,  in  which  the  forces  that 
interfere  with  the  centrifugal  force  and  atmospheric  elasticity  are  com- 
paratively small ;  and  the  result  is 

p  =  JsD(5+1)(1-F)+/(D)-      •  <VL> 

P  is  the  entire  elasticity  of  the  gas,  and  D  its  mean  density.  M  repre- 
sents the  total  mass  of  an  atom,  measured  by  weight,  and  /.i  that  of  its 

atmospheric  part;    so   that        D   is  the  mean   density  of  the   atomic 

atmospheres. 

/  (D)  denotes  the  effect  of  the  mutual  actions  of  separate  atoms. 

The  first  term  represents  the  superficial-atomic  elasticity.  F  denotes 
the  effect  of  the  attraction  of  the  nucleus  in  modifying  that  elasticity, 
and  can  be  represented  approximately  by  a  converging  series,  in  terms  of 

ID2 

the  negative  powers  of  „     r  +  1,  commencing  with  the  inverse  square, 

the  coefficients  being  functions  of  the  density  D. 

By  using  the  first  term  of  such  a  series,  and  determining  its  coefficient 
and  the  quantity  /(D)  empirically,  I  have  obtained  formula?  agreeing 
closely  with  the  results  of  M.  Eegnault's  experiments  on  the  expansion 
of  atmospheric  air,  carbonic  acid,  and  hydrogen. 

In  a  perfect  gas,  the  above  expression  is  reduced  to 


p=iHDW>  +  1>  <VIL> 

Let  n,  as  before,  denote  the  number  of  atoms  of  a  substance  which,  in 
the  state  of  perfect  gas,  occupy  unity  of  volume  under  unity  of  pressure, 
at  the  temperature  of  melting  ice,  so  that  wM  is  its  specific  gravity  in 
that  state :  then, 


p  =  ^J(sT*  +  1)-  •    (VIIL> 


The  factor  by  which  -^-=  is  here  multiplied,  fulfils  the  condition  of 


THE  MECHANICAL  ACTION  OF  HEAT.  239 

being  a  function  of  - --,  and  of  constants  which  are  the  same  for  all 

substances,  and  is,  therefore,  fitted  for  a  measure  of  temperature.  It 
obviously  varies  proportionally  to  the  pressure  of  a  perfect  gas  of  a  given 
density,  or  its  volume  under  a  given  pressure. 

Let  r,  therefore,  denote  temperature,  as  measured  from  an  imaginary  zero, 
C  degrees  of  the  scale  adopted,  below  the  temperature  of  melting  ice, 
at  which 

3gb 
Then,  for  all  substances 

and  in  perfect  gases  •     O^*) 

0    »lff 

r  may  be  termed  absolute  temperature,  and  the  point  from  which  it  is 
measured,  the  absolute  zero  of  temperature.  This,  as  I  have  observed,  is  an 
imaginary  point,  being  lower  than  the  absolute  zero  of  heat  by  the 
quantity  Ciifib,  which  is  the  same  for  all  substances. 

The  value  of  C,  or  the  absolute  temperature  of  melting  ice,  as  determined 
from  M.  Eegnault's  experiments,  is 

274°-6  Centigrade, 
being  the  reciprocal  of 

0-00364166  per  Centigrade  degree, 

the  value  to  which  the  coefficients  of  dilatation  of  gases  at  the  temperature 
of  melting  ice  approximate  as  they  are  rarefied. 

For  Fahrenheit's  scale  C  =  494°-28. 

In  the  sequel  I  shall  represent  temperatures  measured  from  that  of 
melting  ice  by 

T  =  r  -  C. 

We  have  now  to  consider  the  absolute  quantity  of  heat,  or  of  mole- 
cular vis  viva  which  corresponds  to  a  given  temperature  in  a  given  substance. 
It  is  obvious  that 

/AW2 

represents,  in  terms  of  gravity,  the  portion  of  vis  viva,  in  one  atom,  due 


240 


THE  MECHANICAL  ACTION  OF  HEAT. 


to  the  molecular  vortices ;  but  besides  the  vortical  motion,  there  may  be 
oscillations  of  expansion  and  contraction,  or  of  rectilinear  vibration  about  a 
position  of  equilibrium.  The  velocity  with  which  these  additional  motions 
are  performed  will  be  in  a  permanent  condition,  when  the  mean  value 
of  its  square,  independent  of  small  periodic  changes,  is  equal  throughout 
the  atomic  atmosphere.     We  may  therefore  represent  by 


fX  V L  _      fX  w* 

27         "27' 


(X.) 


the  total  vis  viva  of  the  atomic  atmosphere.     To  this  we  have  to  add  that 
of  the  nucleus,  raising  the  quantity  of  heat  in  one  atom  to 


Mr2 
2<7 


=  <Z> 


while  the  quantity  of  heat  in  unity  of  weight  is    y  . 


(XI.) 


2?     H 


The  coefficient  k  (which  enters  into  the  value  of  specific  heat)  being 
the  ratio  of  the  vis  viva  of  the  entire  motion  impressed  on  the  atomic 
atmospheres  by  the  action  of  their  nuclei,  to  the  vis  viva  of  a  peculiar  kind  of 
motion,  may  be  conjectured  to  have  a  specific  value  for  each  substance 
depending,  in  a  manner  as  yet  unknown,  on  some  circumstance  in  the 
constitution  of  its  atoms.  Although  it  varies  in  some  cases  for  the  same 
substance  in  the  solid,  liquid,  and  gaseous  states,  there  is  no  experimental 
evidence  that  it  varies  for  the  same  substance  in  the  same  condition.  In 
the  investigation  which  follows,  therefore,  I  have  treated  it  as  sensibly 
constant. 

The  following,  then,  are  the  expressions  for  quantity  of  heat  in  terms 
of  temperature  :  in  one  atom, — 


v1  __        3fcM  ,        n       .. 


In  unity  of  weight, 


3k 


Q  =  2^  =  2^r-Cw^- 


1 


Y    (xii.) 


J 


Real  specific  heat  is,  consequently,  expressed  by  the   following   equa- 
tions : — 


THE  MECHANICAL   ACTION   OF  HEAT. 


241 


For  one  atom, — 

dq 
dr~ 

3kM 
2Crc// 

For  unity  of  weight, — 

dQ 

'dr  ~ 

3k 

2Cwu 

y  (xiii.) 


For  so  much  of  a  perfect  gas  as  occupies  unity  of  volume 
under  unity  of  pressure  at  the  temperature  of  melting 
ice, — 

dq       3kM 


d< 


2C 


fi 


The  laws  established  experimentally  by  Dulong,  that  the  specific  heats 
of  simple  atoms,  and  of  certain  groups  of  compound  atoms,  bear  to  each 
other  simple  ratios,  generally  that  of  equality,  and  that  the  specific 
heats  of  equal  volumes  of  all  simple  gases  are  equal,  show  that  the  specific 

factor depends  on  the  chemical  constitution  of  the  atom,  and  thus 

M 
confirm  the  conjecture  I  have  stated  respecting  the  coefficient  k. 

As  I  shall  have  occasion,  in  the  investigation  which  follows,  to  refer 
to  and  to  use  the  equation  for  the  elasticity  of  vapours  in  contact  with 
their  liquids,  which  I  published  in  the  Edinburgh  New  Philosophical 
Journal  for  July,  1849,  I  shall  here  state  generally  the  nature  of  the 
reasoning  from  which  it  was  deduced. 

The  equilibrium  of  a  vapour  in  contact  with  its  liquid  depends  on  three 
conditions  : 

First,  The  total  elasticity  of  the  substance  in  the  two  states  must  be 
the  same. 

Secondly,  The  superficial  atomic  elasticity  must  vary  continuously; 
so  that  if  at  the  surface  which  reflects  light  there  is  an  abrupt  change 
of  density  (which  seems  almost  certain),  there  must  there  be  two  densities 
corresponding  to  the  same  superficial-atomic  elasticity. 

Thirdly,  The  two  forces  which  act  on  each  stratum  of  vapour  parallel 
to  the  surface  of  the  liquid — namely,  the  preponderance  of  molecular 
attraction  towards  the  liquid,  and  the  difference  of  the  superficial-atomic 
elasticities  at  the  two  sides  of  the  stratum — must  be  in  equilibrio. 

Close  to  the  surface  of  the  liquid,  therefore,  the  vapour  is  highly 
condensed.  The  density  diminishes  rapidly  as  the  distance  from  the 
liquid  increases,  and  at  all  appreciable  distances  has  a  sensibly  uniform 
value,  which  is  a  function  of  the  temperature  and  of  certain  unknown 
molecular  forces. 

Q 


242 


THE  MECHANICAL  ACTION   OF   HEAT. 


The  integration  of  a  differential  equation  representing  the  third  condition 
of  equilibrium,  indicates  the  form  of  the  approximate  equation, 


P     v 


Log  P  =  a  -  ^  - 

T  T" 


(XIV.) 


the  coefficients  of  which  have  been  determined  empirically  by  three 
experimental  data  for  each  fluid.  For  proofs  of  the  extreme  closeness 
with  which  the  formula?  thus  obtained  agree  with  experiment,  I  refer  to 
the  Journal  in  which  they  first  appeared. 

I  annex  a  table  of  the  coefficients  for  water,  alcohol,  ether,  turpentine, 
petroleum,  and  mercury,  in  the  direct  equation,  and  also  in  the  inverse 
formula, 


J- 


•         A  1 

y  4r 


ay 


(XV.) 


by  which  the  temperature  of  vapour  at  saturation  may  be  calculated  from 
the  pressure.  In  the  ninth  and  tenth  columns  are  stated  the  limits 
between  which  the  formulae  have  been  compared  with  experiment. 

For  turpentine,  petroleum,  and  mercury,  the  formula  consists  of  two 
terms  only. 


LokP 


/3 
a  —  — . 


(XVI.) 


the  small  range  of  the  experiments  rendering  the  determination  of  y 
impossible. 

The  following  are  some  additional  values  of  the  constant  a  for  steam, 
corresponding  to  various  units  of  pressure  used  in  practice. 


Units  of  Pressure. 

Atmospheres  of  7 GO  millimetres  of  mercury, 

=  29*922  inches  of  mercury, 

=  14" 7  lbs.  on  the  square  inch, 

=  1,0333  kilogrammes  on  the  square  centimetre, 
Atmospheres  of  30  inches  of  mercury, 

=  761 -99  millimetres, 

=  14-74  lbs.  on  the  square  inch, 


=  T036  kilogrammes  on  the  sqv 
Kilogrammes  on  the  square  centimetre, 
Kilogrammes  on  the  circular  centimetre, 
Pounds  avoirdupois  on  the  square  inch, 
Pounds  avoirdupois  on  the  circular  inch, 
Pounds  avoirdupois  on  the  square  foot, 


Values  of  a. 


4-950433 


are  centimetre,  4-949300 
4-964658 
4-859748 
6-117662 
6-012752 
8-276025 


All  the  numerical  values  of  the  constants  are  for  common  logarithms. 


THE  MECHANICAL  ACTION  OF  HEAT. 


243 


H 

a 

A 

o 
O 


m 
W 

5 

o 

PL, 
< 

o 

m 
W 
»— i 
H 

a 


S  o 


;«5       2 


9  33 


o 


S  + 


•s  + 


OO        fe(M 


W 

a 
W 
H 

« 

O 

P 

a 

O 
PR 

W 

W 
H 


02 
H 

H 

O 

O 

a 

W 
H 

O 

PQ 


-H  CO 


!>.  —I 


°3 

"3  ^ 


rd  r^  ,£3 


Cm        Cm 


&H 


fe  CM 


-d  A 


Eq 


.O        .S 


.    c3 

.    c3 

.3 

a 

tJD 

bD 

p 

^.3 

c.g 

"3 

O  --H 

« ."^ 

Oh 

£    is  3 


-d'o      -d'o 


K' 


a1 


H     Cm 


244  THE  MECHANICAL  ACTION   OF   HEAT. 


Section  I. — Of  the  Mutual  Conversion  of  Heat  and  Expansive 

Power. 

1.  The  quantity  of  heat  in  a  given  mass  of  matter,  according  to  the 

hypothesis  of  molecular  vortices,  as  well  as  every  other  hypothesis  which 

ascribes  the  phenomena  of  heat  to  motion,  is  measured  by  the  mechanical 

power  to  which   that  motion  is  equivalent,  that  being  a  quantity  the 

total  amount  of  which,  in  a  given  system  of  bodies,  cannot  be  altered  by 

their  mutual  actions,  although  its  distribution  and  form  may  be  altered. 

This  is  expressed  in  equation  XII.  of  the  introduction,  where  the  quantity 

v2 
of  heat  in  unity  of  weight,  Q,  is  represented  by  the  height  -  ,  from  which 

a  body  must  fall  in  order  to  acquire  the  velocity  of  the  molecular  oscilla- 
tions. This  height,  being  multiplied  by  the  weight  of  a  body,  gives  the 
mechanical  power  to  which  the  oscillations  constituting  its  heat  are 
equivalent.  The  real  specific  heat  of  unity  of  weight,  as  given  in  equa- 
tion (XIII.)  of  the  introduction, 

d  Q  3  h 


dr        2Cn 


/•- 


represents  the  dijdli  of  fall,  which  is  equivalent  to  one  degree  of  rise  of 
temperature  in  any  given  weight  of  the  substance  under  consideration. 

We  know,  to  a  greater  or  less  degree  of  precision,  the  ratios  of  the 
specific  heats  of  many  substances  to  each  other,  and  they  are  commonly 
expressed  by  taking  that  of  water  at  the  temperature  of  melting  ice  as 
unity ;  but  their  actual  mechanical  values  have  as  yet  been  very  imper- 
fectly ascertained,  and,  in  fact,  the  data  necessary  for  their  determination 
are  incomplete. 

2.  Mr.  Joule,  indeed,  has  made  several  very  interesting  series  of 
experiments,  in  order  to  ascertain  the  quantity  of  heat  developed  in 
various  substances  by  mechanical  power  employed  in  different  ways — viz., 
by  electric  currents  excited  by  the  rotation  of  a  magnet,  by  the  forcing 
of  water  through  narrow  tubes,  by  the  agitation  of  water  and  oil  with  a 
paddle,  by  the  compression  of  air,  and  by  the  friction  of  air  rushing 
through  a  narrow  orifice.  The  value  of  the  depth  of  fall  equivalent 
to  a  rise  of  one  degree  of  Fahrenheit's  scale  in  the  temperature  of  a  mass 
of  water,  as  determined  by  that  gentleman,  varies,  in  the  different  series 
of  experiments,  between  the  limits  of  760  feet  and  890  feet,  the  value 
in  which  Mr.  Joule  appears  to  place  the  greatest  confidence  being  about 
780  feet. 

Although  the  smallness  of  the  differences  of  temperature  measured  in 
those   experiments   renders   the   numerical   results   somewhat   uncertain, 


THE  MECHANICAL  ACTION   OF  HEAT.  245 

it  appears  to  me  that,  as  evidence  of  the  convertibility  of  heat  and 
mechanical  power,  they  are  unexceptionable.  Nevertheless,  there  is 
reason  to  believe  that  the  true  mechanical  equivalent  of  heat  is  consider- 
ably less  than  any  of  the  values  deduced  from  Mr.  Joule's  experiments; 
for  in  all  of  them  there  are  causes  of  loss  of  power  the  effect  of  which 
it  is  impossible  to  calculate.  In  all  machinery,  a  portion  of  the  power 
which  disappears  is  carried  off  by  waves  of  condensation  and  expansion, 
along  the  supports  of  the  machine,  and  through  the  surrounding  air : 
this  portion  cannot  be  estimated,  and  is,  of  course,  not  operative  in 
producing  heat  within  the  machine.  It  is  also  impossible  to  calculate, 
where  friction  is  employed  to  produce  heat,  what  amount  of  it  has  been 
lost  in  the  production  of  electricity,  a  power  which  is,  no  doubt,  conver- 
tible into  heat,  but  which,  in  such  experiments,  probably  escapes  without 
undergoing  that  conversion.  To  make  the  determination  of  the  mechanical 
equivalent  of  heat  by  electro-magnetic  experiments  correct,  it  is  necessary 
that  the  whole  of  the  mechanical  power  should  be  converted  into  magnetic 
power,  the  whole  of  the  magnetic  power  into  what  are  called  electric 
currents,  and  the  whole  of  the  power  of  the  electric  currents  into  heat, 
not  one  of  Avhich  conditions  is  likely  to  be  exactly  fulfilled.  Even  in 
producing  heat  by  the  compression  of  air,  it  must  not  be  assumed  that 
the  whole  of  the  mechanical  power  is  expended  in  raising  the  tempera- 
ture. 

3.  The  best  means  of  determining  the  mechanical  equivalent  of 
heat  are  furnished  by  those  experiments  in  which  no  machinery  is 
employed.  Of  this  kind  are  experiments  on  the  velocity  of  sound  in  air 
and  other  gases,  which,  according  to  the  received  and  well-known  theory 
of  Laplace,  is  accelerated  by  the  heat  developed  by  the  compression  of 
the  medium. 

The  accuracy  of  this  theory  has  lately  been  called  in  question.  There 
can  be  no  doubt  that  it  deviates  from  absolute  exactness,  in  so  far  that 
the  magnitude  of  the  displacements  of  the  particles  of  air  is  neglected 
in  comparison  with  the  length  of  a  wave.  It  appears  to  me,  however, 
that  the  Astronomer-Eoyal,  in  his  remarks  on  the  subject  in  the  London 
and  Edinburgh  Philosophical  Magazine  for  July,  1849,  has  shown,  in  a 
satisfactory  manner,  that  although  the  effect  of  the  appreciable  magnitude 
of  those  displacements,  as  compared  with  the  length  of  a  wave  of  sound, 
is  to  alter  slowly  the  form  of  the  function  representing  the  wave,  still 
that  effect  is  not  sufficiently  great  to  make  Laplace's  theory  practically 
erroneous.  I  have,  therefore,  in  the  sequel,  adhered  to  the  experiments 
of  Dulong,  and  to  those  emoted  by  Poisson,  on  the  velocity  of  sound,  as 
the  best  data  for  determining  the  mechanical  equivalent  of  heat. 

4.  The  expression  already  given  for  the  real  specific  heat  of  unity  of 
weight  of  a  given  substance  may  be  resolved  into  two  factors,  thus  : — 


246  THE  MECHANICAL  ACTION   OF  HEAT. 

dQ, 1_       3&M  (l) 

dr        C«M  2/i  '         '  k  '' 

The  first  factor,  ~ — r>,  may  be  considered  in  general  as  a  known  quantity; 
CwM 

for  C  represents,  as  already  stated,  274"6  Centigrade  degrees,  the  absolute 
temperature  of  melting  ice,  and  n  M  the  theoretical  weight,  in  the  per- 
fectly gaseous  state,  of  unity  of  volume  of  the  substance,  under  unity  of 

pressure,  at  that  temperature;  or  what  is  the  same  thing,  — ^  is  the 

height  of  an  imaginary  column  of  the  substance,  of  uniform  density,  and 
at  the  temperature  of  melting  ice,  whose  pressure  by  weight  upon  a  given 
area  of  base  is  equal  to  its  pressure  by  elasticity,  supposing  it  to  be 

.    3&M  . 

perfectly  gaseous.     The  determination  of  the  ratio  — —  is  necessary  to 

complete  the  solution  of  the  problem. 

5.  The  relation  now  to  be  investigated  between  heat  and  mechanical 
power,  is  that  which  exists  between  the  power  expended  in  compressing 
a  body  into  a  smaller  volume,  and  the  increase  of  heat  in  consequence  of 
such  a  compression ;  and,  conversely,  between  the  heat  which  disappears, 
or,  as  it  is  said,  becomes  latent  during  the  expansion  of  a  body  to  a  greater 
volume,  and  the  mechanical  power  gained  or  developed  by  that  expansion. 
Those  phenomena,  according  to  the  hypothesis  now  under  consideration, 
as  well  as  every  hypothesis  which  ascribes  heat  to  motion,  are  simply  the 
transformation  of  mechanical  power  from  one  shape  into  another. 

It  is  obvious,  in  the  first  place,  without  the  aid  of  algebraical  symbols, 
that  the  general  effect  of  the  compression  of  an  oscillating  atomic  atmos- 
phere, or  molecular  vortex,  must  be  to  accelerate  its  motion,  and  of  its 
dilatation,  to  retard  its  motion;  for  every  portion  of  such  an  atmosphere 
is  urged  toward  the  nucleus  or  atomic  centre  by  a  centripetal  force  equal 
to  the  centrifugal  force  arising  from  the  oscillation;  so  that  when,  by 
compression,  each  portion  of  the  atmosphere  is  made  to  approach  the  centre 
by  a  given  distance,  the  vis  viva  of  its  motion  will  be  increased  by  the 
amount  corresponding  to  the  centripetal  force  acting  through  that  dis- 
tance ;  and,  conversely,  when  by  expansion  each  portion  of  the  atmosphere 
is  made  to  retreat  from  the  centre,  the  vis  viva  of  its  motion  will  be 
diminished  by  a  similar  amount. 

It  is  not,  however,  to  be  taken  for  granted,  that  all  the  power  expended 
in  compressing  a  body  appears  in  the  form  of  heat.  More  or  less  power 
may  be  consumed  or  developed  by  changes  of  molecular  arrangement, 
or  of  the  internal  distribution  of  the  density  of  the  atomic  atmospheres; 
and  changes  of  molecular  arrangement  or  distribution  may  develop  or 
consume  heat,  independently  of  changes  of  volume. 

6.  We  shall  now  investigate,  according  to   the   hypothesis   of   mole- 


THE  MECHANICAL  ACTION   OF   HEAT.  247 

cular  vortices,  the  amount  of  heat  produced  by  an  indefinitely  small 
compression  of  one  atom  of  a  body  in  that  state  of  perfect  fluidity  which 
admits  of  the  bounding  surface  of  the  atom  being  treated  as  if  it  were 
spherical :  its  radius  being  denoted  by  R,  and  the  radius  of  any  internal 
spherical  layer  of  the  atmosphere  by  multiplying  R  by  a  fraction  u. 

I  shall  denote  by  the  ordinary  symbol  of  differentiation  d,  such  changes 
as  depend  on  the  various  positions  of  portions  of  the  atomic  atmosphere 
relatively  to  each  other,  when  changes  of  volume  and  temperature  are  not 
taken  into  consideration;  while  by  the  symbol  8  of  the  calculus  of 
variations,  I  shall  represent  such  changes  as  arise  from  the  variations  of 
volume  and  temperature. 

Let  us  consider  the  case  of  an  indefinitely  thin  spherical  layer  of  the 
atomic  atmosphere,  whose  distance  from  the  nucleus  is  Rw,  its  thickness 

R  d  u,  its  area  4  tt  R2  u2,  and  its  density  ^ :  D  \p  (u,  D,  t). 

The  weight,  then,  of  this  layer  is 

4 ttR3^  Dm2 ?/,(«,  D,  T)du. 

Its  velocity  of  oscillation  is  v,  and  having,  in  virtue  of  that  velocity,  a 
mean  centrifugal  force,  as  explained  in  the  introduction  (Equation  V.), 
equal  to 

/      t'2  2Q  \ 

its  weight  X  (      ?  p     =  y~—  ) 

°  \  g k R u       kliu/ 

it  is  kept  in  equilibrio  by  an  equal  and  opposite  centripetal  force,  arising 
from  attraction  and  elastic  pressure,  which  is  consequently  represented  by 

4ttR2£^-Dmi/,(w,  D>  T)du 
M  gk 

=  8  7t  R2  r^  Q  D  u  $  («,  D,  r)  d  u. 

Let  the  mean  density  of  the  atom  now  be  increased  by  the  indefinitely 
small  quantity  §  D.  Then  the  layer  will  approach  the  nucleus  through 
the  distance  —  S(Rw)=— wSR  — RSm,  and  being  acted  upon  through 
that  distance  by  the  centripetal  force  already  stated,  the  vis  viva  of 
oscillation  will  be  increased  by  a  quantity  corresponding  to  the  mechanical 
power  (that  is  to  say,  the  heat),  represented  by  the  product  of  that 
distance  by  that  force,  or  by 

-8ttR2t^|QDm^  («,  J),T)duxd(Ru) 


248  THE  MECHANICAL  ACTION   OF  HEAT. 

=  -8»^QD^D,r)^  +  ^. 
which,  because 

SR  1    SD       ,  47rE3D      __  . 

—  =  —  -  .  -jp  and =  M,  is  equal  to 

+  QM.^frD,r)tf(^-8^)*iL 

"We  must  suppose  that  the  velocity  of  oscillation  is  equalised  throughout 
the  atomic  atmosphere,  by  a  propagation  of  motion  so  rapid  as  to  be 
practically  instantaneous. 

Then,  if  the  above  expression  be  integrated  with  respect  to  du,  from 
u  =  0  to  u=l,  the  result  will  give  the  whole  increase  of  heat  in  the 
atom  arising  from  the  condensation  SD;  and  dividing  that  integral  by 
the  atomic  weight  M,  we  shall  obtain  the  corresponding  development  of 
heat  in  unity  of  weight.     This  is  expressed  by  the  following  equation : — 

8Q'  =  *<*&{irfodu  •  **(*  D' r) 

-3  (ldu.vBu\l,(ut'D,  r)|    .     (2.) 

The  letter  Q'  is  here  introduced  to  denote,  when  negative,  that  heat 
which  is  consumed  in  producing  changes  of  volume  and  of  molecular 
arrangement ;  and  when  positive,  as  in  the  above  equation,  the  heat  which 
is  produced  by  such  changes. 

The  following  substitutions  have  to  be  made  in  equation  (1)  of  this 
Section  : 

For  Q  is  to  be  substituted  its  value,  according  to  equation  XII.  of  the 
introduction;  or  abbreviating  Cn/nb  into  k, — 

«=rab<'— >      •     •     •  <3) 

The  value  of  the  first  integral  in  equation  (2)  of  this  Section  is 

/•i  1 

I   du  .  u2\p (u,  D,  t)  =  x. 

JO  "J 

The  value  of  the  second  integral, 

—  31    du  .  uBu-ip (u,  D,  t) 
J  o 

remains  to  be  investigated.     The  first  step  in  this  inquiry  is  given  by 


THE  MECHANICAL  ACTION   OF  HEAT. 


2-19 


the  condition,  that  whatsoever  changes  of  magnitude  a  given  spherical 
layer  undegoes,  the  portion  of  atmosphere  between  it  and  the  nucleus  is 
invariable.     This  condition  is  expressed  by  the  equation 


0 


„  A  +  8  r  4-  +  §  D  ^)fd  u  .  M2  +  (n,  D,  r),       (4.) 


du  '  dr 

from  which  it  follows  that 


d~D, 


Su  =  - 


1T—  (8  r  f  +  S  D  -A  )  f\*  »  .  ^  (w,  D,  r), 

u,  D, t)  \       dr  a D/J  o 


u2  \p  (m,  D,  r) 
and,  consequently,  that 

—  3  I   du  .  u$uip (u,  D,  r)  = 
•Jo 

Hence,  making 

9  I1—  I"  du.  u2  xL  (u,  D,  r)  =  U.         .         .     (5.) 

J  0  U  J  0 

The  second  integral  in  equation  (2)  is  transformed  into 


By  means  of  those  substitutions  we  obtain  for  the  mechanical  value  of 
the  heat  developed  in  unity  of  weight  of  a  fluid  by  any  indefinitely  small 
change  of  volume  or  of  molecular  distribution — 


SQ'  = 


r  —  k 
CnM 


(SD(1+^WC^ 


dW 


or  taking  V  =  ==r  to  denote  the  volume  of  unity  of  weight 
of  the  substance, 

C»M\        \V       d  V/  d  t 


(6.) 


Of  this  expression,  the  portion 


-k     SD 


T-.C         SV 

CnM  '   Y~  repre" 


CnM      D 
sents  the  variation  of  heat  arising  from  mere  change  of  volume. 

t-k  gyfg  =  JLZJLgr)45,  denotes  the  variation  of  heat  pro- 
CnM         dY       CnM         dD 

duced  by  change  of  molecular  distribution  dependent  on  change  of  volume. 


250  THE  MECHANICAL  ACTION   OF  HEAT. 

— — — -  8  t  —, —  expresses  the  variation  of  heat  due  to  change  of  mole- 
(jnM  dr 

cular  distribution  dependent  on  change  of  temperature. 

7.  The  function  U  is  one  depending  on  molecular  forces,  the  nature 
of  which  is  as  yet  unknown.  The  only  case  in  which  it  can  be  calculated 
directly  is  that  of  a  perfect  gas.  Without  giving  the  details  of  the  integra- 
tion, it  may  be  sufficient  to  state,  that  in  this  case 

1 


y  .      .      .   (7.) 


T 

and,  therefore,  that 

dU  _        K      clU  _ 

dr   ~        t25     dV~       J 

In  all  other  cases,  however,  the  value  of  this  function  can  be  determined 
indirectly,  by  introducing  into  the  investigation  the  principle  of  the 
conservation  of  vis  viva. 

Suppose  a  portion  of  any  substance,  of  the  weight  unity,  to  pass  through 
a  variety  of  changes  of  temperature  and  volume,  and  at  length  to  be 
brought  back  to  its  primitive  volume  and  temperature.  Then  the  absolute 
quantity  of  heat  in  the  substance,  and  the  molecular  arrangement  and 
distribution,  being  the  same  as  at  first,  the  effect  of  their  changes  is 
eliminated ;  and  the  algebraical  sum  of  the  vis  viva  expanded  and  produced, 
whether  in  the  shape  of  expansion  and  compression,  or  in  that  of  heat,  must  be 
equal  to  zero:  that  is  to  say,  if,  on  the  whole,  any  mechanical  power  has 
appeared,  and  been  given  out  from  the  body,  in  the  form  of  expansion, 
an  equal  amount  must  have  been  communicated  to  the  body,  and  must 
have  disappeared  in  the  form  of  heat ;  and  if  any  mechanical  power 
has  appeared  and  been  given  out  from  the  body  in  the  form  of  heat,  an 
equal  amount  must  have  been  communicated  to  the  body,  and  must 
have  disappeared  in  the  form  of  compression.  This  principle  expressed 
symbolically  is 

A  n  +  A  Q'  =  0,     .         .         .         .     (8.) 

where  n,  when  positive,  represents  expansive  power  given  out,  when 
negative,  compressive  power  absorbed;  and  Q'  represents,  when  positive, 
heat  given  out,  when  negative,  heat  absorbed. 

To  take  the  simplest  case  possible,  let  the  changes  of  temperature  and 
of  volume  be  supposed  to  be  indefinitely  small,  and  to  occur  during 
distinct  intervals  of  time,  so  that  t  and  V  are  independent  variables; 
let  the  initial  absolute  temperature  be  t,  the  initial  volume  V,  and  the 
initial  total  elasticity  P ;  and  let  the  substance  go  through  the  following 
four  changes : 


THE  MECHANICAL  ACTION   OF  HEAT.  251 

First,  Let  its  temperature  be  raised  from  r  to  r  +  8  r,  the  volume 
remaining  unchanged.     Then  the  quantity  of  heat  absorbed  is 

-     (d®       T  ~  K  dJJ 
dT\dr        CmM    dr 

and  there  is  no  expansion  nor  compression. 

Secondly,  Let  the  body  expand,  without  change  of  temperature,  from 
the  volume  V  to  the  volume  V  +  S  V.  Then  the  quantity  of  heat 
absorbed  is 

+  Sr-ic/l  d      nT    ,    d\J 


while  the  power  given  out  by  expansion  is 
SY(P  +  ^Sr). 

Thirdly,  Let  the  temperature  fall  from  r  +  8  r,  to  its  original  value  r, 
the  volume  V  +  S  V  continuing  unchanged ;   then  the  heat  given  out  is 

+  «'(£-wh  £<"  +  £">> 

and  there  is  no  expansion  nor  compression. 

Fourthly,  Let  the  body  be  compressed,  without  change  of  temperature,  to 
its  original  volume  V ;  then  the  heat  given  out  is 

+  6VC7iMVV       dYJ' 

while  the  power  absorbed  in  compression  is  —  8  V .  P.  The  body  being 
now  restored  in  all  respects  to  its  primitive  state,  the  sum  of  the  two 
portions  of  power  connected  with  change  of  volume,  must,  in  virtue  of  the 
principle  of  vis  viva,  be  equal  to  the  sum  of  the  four  quantities  of  heat 
with  their  signs  reversed.  Those  additions  being  made,  and  the  sums 
divided  by  the  common  factor  S  V  8  t,  the  following  equation  is  obtained : 

dV _      1     (1        dU\ 

dr       C»M\V        dVJ'     '         '         '     K  'J 

The  integral  of  this  partial  differential  equation  is 

U  =  0.r  +  fdV^-CnM^).    .          .     (10.) 


252 


THE   MECHANICAL   ACTION   OF   HEAT. 


Now  <f).r  being  the  same  for  all  densities,  is  the  value  of  U  for  the 
perfectly  gaseous  state,  or  -;  for  in  that  state,  the  integral  =  0.  The 
values  of  the  partial  differential  coefficients  are  accordingly 


dV 

dY  ~ 

1 
V 

— 

c 

dr 

dJJ 
dr  ~ 

- 

K 

— 

CnM  fdY 

d2~P 
dr* 

i 
J 

and  they  can,  therefore,  be  determined  in  all  cases  in  which  the  quantity 
ic  =  Cii{xb,  and  the  law  of  variation  of  the  total  elasticity  with  the  volume 
and  temperature  are  known,  so  as  to  complete  the  data  required  in  order 
to  apply  equation  6  of  this  section  to  the  calculation  of  the  mechanical 
value  of  the  variations  of  heat,  due  to  changes  of  volume  and  molecular 
arrangement. 

The  total  elasticity  of  an  imperfect  gas,  according  to  equations  VI.  and 
XII.  of  the  introduction,  being 

p-uiavO-FM)+'<w 

its  first  and  second  partial  differential  coefficients  with  respect  to  the 
temperature  arc 


dV 

dr 

d?V 

dr2 


i 


C  n  M  V 
1 


0- 


1+T 

a  t- 


d  t         d  r 


C  n  M  V 
Consequently,  for  the  imperfectly  gaseous  state, 


u 

du 

dY 


'M^l 


)j<ivXT)>0 


l 

A' 


n-r>fc 


y  (I2-) 


—  =    --i 

a  r  r" 


*£+'£)' 


*)/< 


rt 


8.  It  is  to  be  observed  that  the  process  followed  in  ascertaining  the 
nature  of  the  function  U  is  analogous  to  that  employed  by  M.  Carnot 
in  his  theory  of  the  motive  power  of  heat,  although  founded  on  contrary 


THE  MECHANICAL  ACTION  OF  HEAT.  253 

principles,  and  leading  to  different  results.  Carnot,  in  fact,  considers 
heat  to  be  something  of  a  peculiar  kind,  whether  a  condition  or  a  substance, 
the  total  amount  of  which  in  nature  is  incapable  of  increase  or  of 
diminution.  It  is  not,  therefore,  according  to  his  theory,  convertible  into 
mechanical  power ;  but  is  capable,  by  its  transmission  through  substances 
under  parti cidar  circumstances,  of  causing  mechanical  power  to  be  developed. 

He  supposes  a  body  to  go  through  certain  changes  of  temperature 
and  volume,  and  to  return  at  last  to  its  primitive  volume  and  temperature, 
and  conceives,  in  accordance  with  his  view  of  the  nature  of  heat,  that  it 
must  have  given  out  exactly  the  same  quantity  of  heat  that  it  has  absorbed. 
The  transmission  of  this  heat  he  regards  as  the  cause  of  the  production 
of  an  amount  of  mechanical  power  depending  on  the  quantity  of  heat 
transmitted,  and  on  the  temperature  at  which  the  transmission  has  taken 
place.  According  to  these  principles,  a  body,  having  received  a  certain 
quantity  of  heat,  is  capable  of  giving  out,  not  only  all  the  heat  it  has 
received,  but  also  a  quantity  of  mechanical  power  which  did  not  before  exist. 

According  to  the  theory  of  this  essay,  on  the  contrary,  and  to  every 
conceivable  theory  which  regards  heat  as  a  modification  of  motion,  no 
mechanical  power  can  be  given  out  in  the  shape  of  expansion,  unless  the 
quantity  of  heat  emitted  by  the  body  in  returning  to  its  primitive  tem- 
perature and  volume  is  less  than  the  quantity  of  heat  originally  received ; 
the  excess  of  the  latter  quantity  above  the  former  disappearing  as  heat, 
to  appear  as  expansive  power,  so  that  the  sum  of  the  vis  viva  in  these  two 
forms  continues  unchanged. 


Section  II. — Of  Eeal  and  Apparent  Specific  Heat,  especially  in 
the  State  of  Perfect  Gas. 

9.  The  apparent  specific  heat  of  a  given  substance  is  found  by  adding 
to  the  real  specific  heat  (or  the  heat  which  retains  its  form  in  producing 
an  elevation  of  one  degree  of  temperature  in  unity  of  weight)  that 
additional  heat  which  disappears  in  producing  changes  of  volume  and  of 
molecular  arrangement,  and  which  is  determined  by  reversing  the  sign 
of  Q1  in  equation  6  of  Sect.  I  (so  as  to  transform  it  from  heat  evolved 
to  heat  absorbed),  and  taking  its  total  differential  coefficient  with  respect 
to  the  temperature.     Hence,  denoting  total  apparent  specific  heat  by  K, 

K  =  dQ_  d.  Q1  _dQ      dQ1      dQ1    dV 


dr  dr  dr         dr         dV  '  d 


r 


1      (31M,  ,         ,/tiV/l       dV\      dV\)     ,„„. 


251  THE  MECHANICAL  ACTION   OF  HEAT. 

Another  mode  of  expressing  this  coefficient  is  the  following  : 
Denote  the  ratio     .  M  by  N,  and  the  real  specific  heat  by 

ft  =  ClM'     ....     (H.) 
then 

K  -i  {1  +N(r  -  k)  (7z7(y  -  <nr)  -  7/  J  }.     (15.) 

The  value  of  ■=—  is  to  be  determined  from  the  conditions  of  each  par- 
ticular case,  so  that  each  substance  may  have  a  variety  of  apparent  specific 
heats,  according  to  the  manner  in  which  the  volume  varies  with  the 
temperature. 

dV 
If  the  volume  is  not  permitted  to  vary,  so  that  — -  =  0,  there   is 

a  t 

obtained  the  following  result,  being  the  apparent  specific  heat  at  constant 

volume : — 

*'-o^(i-fr-'»£)-*(l-*<'-")i7>    (10-» 


10.  Then  the  substance   under   consideration  is  a  perfect  gas,   it  has 
already  been  stated,  equation  (7),  that 


d]J  k  dV  _ 

dr   ==        r2'  d  V  ~     ' 


and  because  the  volume  of  unity  of  weight  is  directly  as  the  absolute 
temperature  and  inversely  as  the  pressure. 


1  cTV  _    1        1  dP_ 

V  dr    ~  r         P  dr         '  "  "      (    '^ 


Hence,  the  following  are  the  values  of  the  apparent  specific  heats  of 
unity  of  weight  of  a  theoretically  perfect  gas  under  different  circum- 
stances : — 

General  value  of  the  total  apparent  specific  heat : 


THE  MECHANICAL  ACTION   OF   HEAT. 


255 


K-- M1-  + 

c»m  In  t 


*-*&+4h)} 


=  chi  { s  + (r  _  k)  ( ?  +  r "  ^7)  1 

Apparent  specific  heat  at  constant  volume  :  y  (18.) 

*-jot{s+!-S-»0-+»(-:-s)) 

Apparent  specific  heat  under  constant  pressure  : 

^  =  o^m(s  +  1-?)  =  s{1  +  n(1-?)}  , 

The  ratio  of  the  apparent  specific  heat  under  constant  pressure  to  the 
apparent  specific  heat  at  constant  volume,  is  the  following : — 


1  +N 


(i-a 


1  - 


l+N  -- 


=  1  +N 


1  +N 


-;-5) 


(19.) 


The  value  of  k  is  unknown,  and,  as  yet,  no  experimental  data  exist 
from  which  it  can  be  determined.  I  have  found,  however,  that  practically, 
results  of  sufficient  accuracy  are  obtained  by  regarding  k  as  so  small  in 


comparison   with 


that   -,    and    ic  fortiori    —,,   may   be    neglected   in 

T  T~ 


calcvdation. 

Thus  are  obtained  the  following  approximate  results  for  perfect  gases, 
and  gases  which  may  without  material  error  be  treated  as  perfect. 

General  value  of  the  total  apparent  specific  heat : — 


_    _L  M  r   dY 

CnM\W  +  Y    dr 


=  fe+P 


dY 


1 


-1  _  r   dV 

N   +  P    dr 


CwM 

Apparent  specific  heat  at  constant  volume 
1 


K,  = 


% 


y  (20.) 


CnMN 
being  equal  to  the  real  specific  heat. 

Apparent  specific  heat  under  constant  pressure  :- 


K„  = 


cin(-i  +  i)=ft(i+N). 


256  THE  MECHANICAL  ACTION   OF  HEAT. 

Eatio  of  those  two  specific  heats  : — 

|-P=1  +  N (21.) 

This  ratio  is  the  quantity  called  by  Poisson  y,  in  his  researches  on  the 
propagation  of  sound. 

11.  It  is  unnecessary  to  do  more  than  to  refer  to  the  researches 
of  Poisson,  and  to  those  of  Laplace,  for  the  proof  that  the  effect  of  the 
production  of  heat  by  the  compression  of  air  is  the  same  as  if  the  elasticity 
varied  in  proportion  to  that  power  of  the  density  whose  index  is  the 
ratio  of  the  two  specific  heats ;  so  that  the  actual'  velocity  of  sound  is 
greater  than  that  which  it  would  have  if  there  were  no  such  development 
of  heat,  in  the  proportion  of  the  square  root  of  that  ratio. 

The  following  is  the  value  of  the  velocity  of  sound  in  a  gas,  as  given 
by  Poisson  in  the  second  volume  of  his  TraiU  de  M6caniqu& : — 


B=^.y.(l+ET)^,  .  .  .     (22.) 


where  a  denotes  the  velocity  of  sound,  g  the  velocity  generated  by  gravity 
in  unity  of  time,  E  the  coefficient  of  increase  of  elasticity  with  temperature, 
at  the  freezing  point  of  water,  T  the  temperature  measured  from  that 
point,  m  the  specific  gravity  of  mercury,  A  that  of  the  gas  at  the  tem- 
perature of  melting  ice,  and  pressure  corresponding  to  a  column  of 
mercury  of  the  height  h.  It  follows  that  the  ratio  y  is  given  by  the 
formula 

Y  =  l+Nneariy  =  gm/((^ET).    .        .    (23.) 

Calculations  have  been  made  to  determine  the  ratio  y  from  the  velocity 
of  sound ;  but  as  many  of  them  involve  erroneous  values  of  the  coefficient 
of  elasticity  E,  the  experiments  have  to  be  reduced  anew. 

The  following  calculation  is  founded  on  an  experiment  quoted  by 
Poisson  on  the  velocity  of  sound  in  atmospheric  air,  the  values  of  E,  w, 
and  A  being  taken  from  the  experiments  of  M.  Eegnault. 

a  =  340-89  metres  per  second. 

g  =  9m-S0896.  h  =  0m-76.  T  =  150<9  Centigrade. 

in. 
E  =  0-003665;  ~  =  10513. 

A 

Consequently,  for  atmospheric  air, 

y  =  1-401. 


THE   MECHANICAL  ACTION   OF  HEAT. 


257 


The  results  of  a  reduction,  according  to  correct  data,  of  the  experiments 
of  Dulong  upon  the  velocity  of  sound  in  atmospheric  air,  oxygen,  and 
hydrogen,  are  as  follows  : — 


Atmospheric  air, 
Oxygen,    . 
Hydrogen, 


7- 


1-410 
1-426 
1-426 


Thus  it  appears,  that  for  the  simple  substances,  oxygen  and  hydrogen, 
the  ratio  N  is  the  same,  while  for  atmospheric  air  it  is  somewhat  smaller* 

12.  The  ordinary  mode  of  expressing  the  specific  heats  of  gases  is 
to  state  their  ratios  to  that  of  an  equal  volume  of  atmospheric  air  at  the 
same  pressure  and  temperature. 

When  -  is  a  very  small  fraction,  specific  heats  of  unity  of  volume  of  a 


perfect  gas  are  given  by  the  equations — 


M  Kn  = 


n  M  Kt 


1 
CN 

1  /l 


SGr  +  0 


(24.) 


That  is  to  say,  the  specific  heat  of  unity  of  volume  at  constant  volume 
is  inversely  proportional  to  the  fraction  by  which  the  ratio  of  the  two 
specific  heats  exceeds  unity;  a  conclusion  already  deduced  from  experiment 
by  Dulong. 

The  following  is  a  comparison  of  the  ratios  of  the  apparent  specific 
heats  under  constant  pressure,  of  unity  of  volume  of  oxygen  and  hydrogen 

*  The  following  are  some  additional  determinations  of  the  value  of  y  for  atmospheric 
air,  founded  upon  experiments  on  the  velocity  of  sound  : — 


Observers. 
Bravals  and  Martins :  mean  of  several  experi- 
ments at  temperatures  varying  from  5°  to 
11°  Centigrade,  reduced  to  0°   (Comptes 
JRendus,  xix.)    ...... 

Moll  and  Van  Beeh :  reduced  to 

Stampfer  and  Myrbach:  reduced  to  0°  (not 
corrected  for  moisture)     .... 

Academic  des  Sciences,  173S :   (not  corrected  ) 
for  moisture)    .         .         .         .         .         .  S 

A  variation  of  one  metre  per  second  in  the  velocity  of  sound  at  0°  corresponds  to  a 
variation  of  '0085  in  the  value  of  y. 

K 


T 

Centigrade. 

a 

Metres  per  second. 

y 

0° 

332-37 

1-40955 

0° 

332-25 

1-40853 

1        0' 

332-96 

1-41456 

G°l 


337-10 


1-41S 


258  THE  MECHANICAL  ACTION  OF  HEAT. 

respectively,  to  that  of  atmospheric  air,  as  deduced  from  equation  (24), 
with  those  determined  experimentally  by  De  la  Roche  and  Berard: — 

_  A.         wMKp(Gas) 
Ratio  ■ 


n  M  Kr(Atmos.  air) 
Gas.  By  Theory.  By  Experiment. 

Oxygen,  .         .         .         0-973  0*9705 

Hydrogen,       .        .         .         0-973  0-9033 

This  comparison  exhibits  a  much  more  close  agreement  between  theory 
and  experiment  than  has  been  hitherto  supposed  to  exist,  the  errors  in 
the  constants  employed  having  had  the  effect  of  making  the  ratio  1+N 
seem  greater  for  atmospheric  air  than  for  oxygen  and  hydrogen,  while  in 
fact  it  is  smaller. 

To  treat  the  other  substances  on  which  both  M.  Dulong  and  MM.  De 
la  Roche  and  Berard  made  experiments  as  perfect  gases,  would  lead  to 
sensible  errors.  I  have,  therefore,  confined  my  calculations  for  the 
present  to  oxygen,  hydrogen,  and  atmospheric  air. 

13.  The  heat  produced  by  compressing  so  much  of  a  perfect  gas  as 
would  occupy  unity  of  volume  under  the  pressure  unity,  at  the  temperature 

0°  Centigrade,  from  its  actual  volume  nMY1  =  p-p>  into  a  volume  which 

is  less  in  a  given  ratio  ^  (when  k  is  neglected  as  compared  with  r),  is 
expressed  by  the  following  motion  : — 

sV  s 

«MQ'  =  -i(      1dY.^  =  -nM\1t   Vds,  .     (25.) 

v  1 

being,  in  fact,  equal  to  the  mechanical  power  used  in  the  compression. 
When  the  temperature  is  maintained  constant,  this  becomes 

»MQ'(T)  =  £loge.i        .  .     (26.) 

which  is  obviously  independent  of  the  nature  of  the  gas. 

Hence,  equal  volumes  of  all  substances  in  the  state  of  perfect  gas,  at  the  same 
pressure,  and  at  equal  and  constant  temperatures,  being  compressed  by  the  same 
amount,  disengage  equal  quantities  of  heat;  a  law  already  deduced  from 
experiment  by  Dulong. 

14.  The  determination  of  the  fraction  N  affords  the  means  of 
calculating  the  mechanical  or  absolute  value  of  specific  heat,  as  defined  by 
equation  (1),  section  first.  The  data  for  atmospheric  air  being  taken  as 
follows : — 

N  =:  0-4,  C  =  274°-6  Centigrade, 


THE  MECHANICAL  ACTION  OF  HEAT.  259 

—=-=  :=  height  of  an  imaginary  column  of  air  of  uniform  density,  at  the 
n  M 

temperature  0°  Cent.,  whose  pressure  by  weight  on  a  given  base  is  equal  to 
its  pressure  by  elasticity, 

=  7990  metres,     =26214  feet  :— 

the  real  specific  heat  of  atmospheric  air,  or  the  depth  of  fall  equivalent 
to  1  Centigrade  degree  of  temperature  in  that  gas,  is  found  to  be 


C»H 


=  72-74  metres  =  238*60  feet.       .     (27.) 


The  apparent  specific  heat  of  atmospheric  air,  under  constant  pressure, 
according  to  De  la  Eoche  and  Berard,  is  equal  to  that  of  liquid  water  at 
0°  Centigrade  x  0-2669.  The  ratio  of  its  real  specific  heat  to  the  apparent 
specific  heat  of  water  at  0°  Centigrade  is,  therefore, 

•2669  x  ~=  -1906. 
1"4 

And,  consequently,  the  mechanical  value  of  the  apparent  specific  heat  of 
liquid  water,  at  the  temperature  of  melting  ice,  is 

;n'7  =  38T64  metres  =  1252  feet  per  Centigrade  degree,   (       .      x 
•1906  >-      [to.) 

or  695*6  feet  per  degree  of  Fahrenheit's  scale.         ) 

This  quantity  we  shall  denote  by  Kw.  It  is  the  mechanical  equivalent 
of  the  ordinary  thermal  unit 

I  have  already  pointed  out  (in  article  2  of  the  first  section)  the  causes 
which  tend  to  make  the  apparent  value  of  the  mechanical  equivalent  of 
heat,  in  Mr.  Joule's  experiments,  greater  than  the  true  value.  The 
differences  between  the  result  I  have  just  stated,  and  those  at  which  he 
has  arrived,  do  not  seem  greater  than  those  causes  are  capable  of  producing 
when  combined  with  the  uncertainty  of  experiments,  like  those  of  Mr. 
Joule,  on  extremely  small  variations  of  temperature. 

15.  Besides  the  conditions  of  constant  volume  and  constant  pressure, 
there  is  a  third  condition  in  which  it  is  of  importance  to  know  the 
apparent  specific  heat  of  an  elastic  fluid — namely,  the  condition  of  vapour 
at  saturation,  or  in  contact  with  its  liquid. 

The  apparent  specific  heat  of  a  vapour  at  saturation  is  the  quantity  of 
heat  which  unity  of  weight  of  that  vapour  receives  or  gives  out,  while 
its  temperature  is  increased  by  one  degree,  its  volume  being  at  the  same 
time  compressed  so  as  to  bring  it  to  the  maximum  pressure  corresponding 
to  the  increased  temperature. 


2G0  THE  MECHANICAL   ACTION   OF  HEAT. 

It  has  been  usually  taken  for  granted,  that  this  quantity  is  the  same, 
with  the  variation  for  one  degree  of  temperature,  of  what  is  called  the 
total  heat  of  evaporation.  Such  is,  indeed,  the  case  according  to  the  theory 
of  Carnot;  but  I  shall  show  that,  according  to  the  mechanical  theory  of 
heat,  these  two  quantities  are  not  only  distinct,  but  in  general  of  con- 
trary signs. 

I  shall,  for  the  present,  consider  such  vapours  only  as  may  be  treated 
in  practice  as  perfect  gases,  so  as  to  make  the  first  of  the  equations  (20) 
applicable. 

It  has  been  shown  that  the  logarithm  of  the  maximum  elasticity  of  a 
vapour  in  contact  with  its  liquid  may  be  represented  by  the  expression 

Log.  P  =  a  -  £  -  -<. 

T  T- 

The  coefficients  a,  /3,  y,  being  those  adapted  for  calculating  the  common 
logarithm  of  the  pressure,  I  shall  use  the  accented  letters  a,  /3',  y,  to 
denote  those  suited  to  calculate  the  hyperbolic  logarithm,  being  equal 
respectively  to  the  former  coefficients  X  2,3025851. 

Then  for  vapour  at  saturation, 

11.  _  £  +  Ix  (29 , 

Prfr_r-+    J*'  ■         ■  ■     (->) 

Making  this  substitution  in  the  general  equation  (21,)  wo  find  the 
following  value  for  the  apparent  specific  heat  of  perfectly  gaseous  vapour 
at  saturation  : 

Ks^h  +  P(-  =  h(l+N.?-) 

.     (30.) 


>{i+*(i-f£)} 

C  n  M  \N  ^  T        r2  / 


16.  For  the  vapours  of  which  the  properties  are  known,  the  negative 
terms  t>f  this  expression  exceed  the  positive  at  all  ordinary  temperatures, 
so  that  the  kind  of  apparent  specific  heat  now  under  consideration  is  a 
negative  quantity : — that  is  to  say,  that  if  a  given  weight  of  vapour  at 
saturation  is  increased  in  temperature,  and  at  the  same  time  maintained 
by  compression  at  the  maximum  elasticity,  the  heat  generated  by  the 
compression  is  greater  than  that  which  is  required  to  produce  the  elevation 
of  temperature,  and  a  surplus  of  heat  is  given  out;  and  on  the  other 
hand,  if  vapour  at  saturation  is  allowed  to  expand,  and  at  the  same  time 


THE  MECHANICAL  ACTION   OF  HEAT.  261 

maintained  at  the  temperature  of  saturation,  the  heat  which  disappears 
in  producing  the  expansion  is  greater  than  that  set  free  by  the  fall  of 
temperature ;  and  the  deficiency  of  heat  must  be  supplied  from  without, 
otherwise  a  portion  of  the  vapour  will  be  liquefied  in  order  to  supply  the  heat 
necessary  for  the  expansion  of  the  rest. 

This  circumstance  is  obviously  of  great  importance  in  meteorology,  and 
in  the  theory  of  the  steam-engine.  There  is  as  yet  no  experimental 
proof  of  it.  It  is  true  that,  in  the  working  of  non-condensing  engines, 
it  has  been  found  that  the  steam  which  escapes  is  always  at  the  tem- 
perature of  saturation  corresponding  to  its  pressure,  and  carries  along  with 
it  a  portion  of  water  in  the  liquid  state ;  but  it  is  impossible  to  distinguish 
1  ictween  the  water  which  has  been  liquefied  by  the  expansion  of  the  steam, 
and  that  which  has  been  carried  over  mechanically  from  the  boiler. 

The  calculation  of  the  proportion  of  vapour  liquefied  by  a  given 
expansion,  requires  the  knowledge  of  the  latent  heat  of  evaporation,  which 
forms  the  subject  of  the  next  section. 


Section  III— Of  the  Latent  and  Total  Heat  of  Evaporation, 

ESPECIALLY  FOR  WATER. 

1 7.  The  latent  heat  of  evaporation  of  a  given  substance  at  a  given 
temperature,  is  the  amount  of  heat  which  disappears  in  transforming 
unity  of  weight  of  the  substance  from  the  liquid  state,  to  that  of  vapour 
of  the  maximum  density  for  the  given  temperature,  being  consumed  in 
producing  an  increase  of  volume,  and  an  unknown  change  of  molecular 
arrangement. 

It  is  obvious,  that  if  the  vapour  thus  produced  is  reconverted  into  the 
liquid  state  at  the  same  temperature,  the  heat  given  out  during  the  lique- 
faction must  be  equal  to  that  consumed  during  the  evaporation;  for  as 
the  sum  of  the  expansive  and  compressive  powers,  and  of  those  dependent 
on  molecular  arrangement  during  the  whole  process,  is  equal  to  zero, 
so  must  the  sum  of  the  quantities  of  heat  absorbed  and  evolved. 

The  heat  of  liquefaction,  at  a  given  temperature,  is  therefore  equal 
to  that  of  evaporation,  with  the  sign  reversed. 

18.  If  to  the  latent  heat  of  evaporation  at  a  given  temperature,  is 
added  the  quantity  of  heat  necessary  to  raise  unity  of  weight  of  the 
liquid  from  a  certain  fixed  temperature,  (usually  that  of  melting  ice),  to 
the  temperature  at  which  the  evaporation  takes  place,  the  result  is 
called  the  total  heat  of  evaporation  from  the  fixed  temperature  chosen. 

According  to  the  theory  of  Carnot,  this  quantity  is  called  the  constituent 
heat  of  vapour ;  and  it  is  conceived,  that  if  liquid  at*  the  temperature  oi 
melting  ice  be  raised  to   any  temperature  and  evaporated,   and  finally 


262  THE  MECHANICAL  ACTION  OF  HEAT. 

brought  in  the  state  of  vapour  to  a  certain  given  temperature,  the  whole 
heat  expended  will  be  equal  to  the  constituent  heat  corresponding  to  that 
given  temperature,  and  will  be  the  same,  whatsoever  may  have  been  the 
intermediate  changes  of  volume,  or  the  temperature  of  actual  evaporation. 

According  to  the  mechanical  theory  of  heat,  on  the  other  hand,  the 
quantity  of  heat  expended  must  vary  with  the  intermediate  circumstances; 
for  otherwise  no  power  could  be  gained  by  the  alternate  evaporation  and 
liquefaction  of  a  fluid  at  different  temperatures. 

19.  The  law  of  the  latent  and  total  heat  of  evaporation  is  immediately 
dcducible  from  the  principle  of  the  constancy  of  the  total  vis  viva  in  the 
two  forms  of  heat  and  expansive  power,  when  the  body  has  returned  to 
its  primitive  density  and  temperature,  as  already  laid  down  in  article  7. 

That  principle,  when  applied  to  evaporation  and  liquefaction,  may  be 
stated  as  follows  : — 

Let  a  portion  of  fluid  in  the  liquid  state  be  raised  from  a  certain 
temperature  to  a  higher  temperature :  let  it  be  evaporated  at  the  higher 
temperature  :  let  the  vapour  then  be  allowed  to  expand,  being  maintained 
always  at  the  temperature  of  saturation  for  its  density,  until  it  is  restored 
to  the  original  temperature,  at  which  temperature  let  it  be  liquefied : — 
then,  the  excess  of  the  heat  absorbed  by  the  fluid  above  the  heat  given  out,  will 
he  equal  to  the  expansive  powt  v  generated. 

To  represent  those  operations  algebraically, — let  the  lower  absolute 
temperature  be  t0;  the  volume  of  unity  of  weight  of  liquid  at  that 
temperature  r0,  and  that  of  vapour  at  saturation  V0:  let  the  pressure  of 
that  vapour  be  P0;  the  latent  heat  of  evaporation  of  unity  of  weight  L0; 
and  let  the  corresponding  quantities  for  the  higher  absolute  temperature 
tv  be  vv  Vv  Px,  Lr  Let  KL  represent  the  mean  apparent  specific  heat 
of  the  substance  in  the  liquid  form  between  the  temperatures  r0  and  rr 
Then,— 

First,  Unity  of  weight  of  liquid  being  raised  from  the  temperature  r0 
to  the  temperature  rv  absorbs  the  heat, 

Kl(ti  -  ro)> 
and  produces  the  expansive  power, 


J     dv.V. 


Secondly,  It  is  evaporated  at  the  temperature  rv  absorbing  the  heat 
and  producing  the  expansive  power, 


THE  MECHANICAL  ACTION   OF   HEAT.  263 

Px  (Vx  -  vj. 

Thirdly,  The  vapour  expands,  at  saturation,  until  it  is  restored  to  the 
original  temperature  r.     In  this  process  it  absorbs  the  heat, 


-r 


dr  .  K3, 


and  produces  the  expansive  power 


I      dV.F. 

Fourthly,  It  is  liquefied  at  the  original  temperature,  giving  out  the  heat 


\y 


and  consuming  the  compressive  power, 

Po(V0-*'o)- 

The  equation  between  the  heat  which  has  disappeared,  and  the  expansive 
power  which  has  been  produced,  is  as  follows  : — 


Lx  -  L0  +  Kt  (rx  -r0)  -        dr  .  Ks 
=  Px  (Vx  -  vx)  -  P0  (Y0  -  v0)  + /\ *  .  P  +/ '  °  <Z  V  .  P. 


y  (3i.) 


j 


■V0 

If  the  vapour  be  such  that  it  can  be  regarded  as  a  perfect  gas  without 

^forKs,andof^ 


d  V  r 

sensible  error,  the  substitution  of  ft  +  P  -r-  for  Ks,  and  of  n     M  =  HNr 


for  P  V,  transforms  the  above  to 

Lx  -  L0  +  {KL  -  k  (1  +  N)}  (r,  -  r0)  ^j 

fix  /-Po  \      (32.) 

=  -P1t;1  +  P0»0+J     dv.?  =  -j      dP.v   I 

In  almost  all  cases  which  occur  in  practice,  v  is  so  small  as  compared 
with  V,  that  —  [  d  P .  v  may  be  considered  as  sensibly  =  0 ;  and,  there- 
fore, (sensibly) 

L1  +  Kt(r1-r0)  =  Lto  +  fc(l+N)(r1-r0).      .     (33.) 


2G4  THE  MECHANICAL  ACTION   OF   HEAT. 

Now  this  quantity,  which  I  shall  denote  by  H,  is  the  total  heat  required 
to  raise  unity  of  weight  of  liquid  from  r0  to  rj  of  absolute  temperature, 
and  to  evaporate  it  at  the  latter  temperature.  Therefore,  the  total  heat  of 
evaporation,  where  the  vapour  may  be  treated  as  a  perfect  gas,  increases  sensibly 
at  an  unifwm  rate  icith  the  temper  attire  of  evaporation;  and  the  coefficient  of 
Us  increase  with  temperature  is  equal  to  the  apparent  specific  heat  of  the  vapour 
at  constant  pressure,  lt(l  +  X). 

20.  There  have  never  been  any  experiments  from  which  the  apparent 
specific  heat  of  steam  under  constant  pressure  can  be  deduced  in  the 
manner  in  which  that  of  permanent  gases  has  been  ascertained. 

The  experiments  of  M.  Eegnault,  however,  prove  that  the  total  heat 
of  evaporation  of  water  increases  uniformly  with  the  temperature  from 
0°  to  200°  Centigrade,  and  thus  far  fully  confirm  the  results  of  this  theory. 

The  coefficient  of  increase  is  equal  to 


Kw  x  0-305. 

Its  mechanical  value  is,  consequently, 

1 16*4  metres  =  382  feet  per  Centigrade  degree,  or 
212  feet  per  degree  of  Fahrenheit. 


(34.) 


Although  the  principle  of  the  conservation  of  vis  viva  has  thus  enabled 
us  to  ascertain  the  law  <>f  increase  of  the  total  heat  of  evaporation,  it  does 
not  enable  us  to  calculate  a  priori  the  constant  L0  of  the  formula,  being 
the  latent  heat  of  evaporation  at  the  fixed  temperature  from  which 
the  total  heat  is  measured ;  for  the  changes  of  molecular  arrangement 
which  constitute  evaporation  are  unknown. 

"When  the  fixed  temperature  is  that  of  melting  ice,  M.  Eegnault's 
experiments  give  606  5  Centigrade  degrees,  applied  to  liquid  water  as  the 
value  of  this  constant :  so  that 


H  =  Kw(606°-5  +  -30:)T°), 

for  the  Centigrade  scale, 

H  =  Kw(l091°-7  +  -305  (T°  -  32°)\ 
for  Fahrenheit's  scale, 


(35.) 


is  the  complete  expression  for  the  heat  required  to  raise  unity  of  weight 
of  water  from  the  temperature  of  melting  ice  to  T°  above  the  ordinary 
zero,  and  to  evaporate  it  at  the  latter  temperature.  This  formula  has- 
been  given  by  M.  Eegnault  as  merely  empirical ;  but  we  have  seen  that 
it  closely  represents  the  physical  law  when  quantities  depending  on  the 
expansion  of  water  are  neglected. 


THE  MECHANICAL  ACTION   OF  HEAT.  2G5 

It  must  be  remarked,  that  the  unit  of  heat  in  M.  Regnault's  tables  is 
not  precisely  the  specific  heat  of  water  at  0°  Centigrade,  but  its  mean 
specific  heat  between  the  initial  and  final  temperatures  of  the  water  in  the 
calorimeter.  The  utmost  error,  however,  which  can  arise  from  this 
circumstance,  is  less  than  y^-  of  the  total  heat  of  evaporation,  so  that  it 
may  safely  be  neglected. 

The  coefficient  -305  Kw  =  382  feet  per  Centigrade  degree,  is  the 
apparent  specific  heat  of  steam  at  constant  pressure ;  that  is  to  say,  for 
steam, — 

f{  +  — — m  =  382  feet  per  Centigrade  degree,  but  ~ — ^  =  153  ft. 


1 

Therefore,  the  real  specific  heat  of  steam  is  K  =  7. — ^r^p  =229 
1  C 11 JV1  -N 

feet  per  Centigrade  degree  =  127*4  feet  per  deg.  of  Fahrenheit 

153  _  2 
229  ~  3' 


>  (36.) 


153       ° 
=  KWX  -183,andX  =  — = 


The  quantity  —   I      d  P .  v  has  been  neglected,  as  already  explained,  in 


these  calculations,  on   account  of  its  smallness.      When  r0  =  C,  or  the 
fixed  point  is  0°  Centigrade,  this  integral  is  nearly  ecpial  to 


which  for  steam,  is  equal  to 

—  T\      x   -122   V-  t 

v  1 

For  a  pressure  of  eight  atmospheres,  ~=  =  -—7   nearly,  ra  =  4450,5 

(T  =  170°-9  Cent.);  consequently,  -  v  P2  =  -  Kw  X  0°-22  Cent.,  a 
quantity  much  less  than  the  limit  of  errors  of  observation  in  experiments 
on  latent  heat.  This  shows  that  in  practice  we  are  justified  in  overlook- 
ing the  influence  of  the  volume  of  the  liquid  water  on  the  heat  of 
evaporation. 


Section  IV. — Or  the  Mechanical  Action  of  Steam  treated  as  a 
Perfect  Gas,  and  the  Power  of  the  Steam  Engine. 

21.  In  the  present  limited  state  of  our  experimental  knowledge  of  the 
density  of  steam  at  pressures  differing  much  from  that  of  the  atmosphere. 


200  THE  MECHANICAL  ACTION   OF  HEAT. 

it  is  desirable  to  ascertain  whether  any  material  error  is  likely  to  arise 
from  treating  it  as  a  perfect  gas.  For  this  purpose  the  ratio  of  the 
volume  of  steam  at  100°  Centigrade  under  the  pressure  of  one  atmosphere, 
to  that  of  the  water  which  produces  it  at  40,1  Centigrade,  as  calculated 
theoretically  on  the  supposition  of  steam  being  a  perfect  gas,  is  to  be 
compared  with  the  actual  ratio. 

The  weight  of  one  volume  of  water  at  40-l  Centigrade  being  taken 
as  unity,  that  of  half  a  volume  of  oxygen  at  0°  Centigrade,  under 
the  pressure  of  one  atmosphere,  according   to   the    experiments    of  M. 

Eegnault,  is 0*000714900 

That  of  one  volume  of  hydrogen,  ....  0-000089578 


The  sum  being, 0-000804478 

The  reciprocal  of  this  sum  being  multiplied  by  -»,„  =  1'3 64 16 6,  the 

ratio  of  dilatation  of  a  perfect  gas  from  0°  to  100°  Centigrade,  the  result 
gives  for  the  volume  of  steam  of  saturation  at  100°  Centigrade,  as  com- 
pared with  that  of  water, 

At  4°-l 1095-72 

And  for  its  density,      ....  0-00058972 

The  agreement  of  those  results  with  the  known  volume  and  density  of 
steam  is  sufficiently  close  to  show,  that  at  pressures  less  than  one  atmos- 
phere, it  may  be  regarded  as  a  gas  sensibly  perfect ;  from  which  it  may  be 
concluded,  that,  in  the  absence  of  more  precise  data,  the  errors  arising 
from  treating  it  as  a  perfect  gas  at  such  higher  pressures  as  occur  in 
practice,  will  not  be  of  much  importance. 

Representing,  then,  by  v  the  volume  of  unity  of  weight  of  water  at 
4  "1  Centigrade,  that  of  unity  of  weight  of  steam  at  any  pressure  and 
temperature  will  be  given  by  the  formula 

_  1696e<o     r 
\   -  — ^y-  .p.  •         •         •     (38.) 

(o  representing  the  number  of  units  of  weight  per  unit  of  area  in  the 
pressure  of  one  atmosphere,  and  (r)  the  absolute  temperature  at  which  the 
pressure  of  saturation  is  one  atmosphere ;  being  for  the  Centigrade  scale 
374°-6,  and  for  Fahrenheit's  scale  6740>28. 

The  mechanical  action  of  unity  of  weight  of  steam  at  the  temperature  r 
and  pressure  P,  during  its  entrance  into  a  cylinder,  before  it  is  permitted 
to  expand,  is  represented  by  the  product  of  its  pressure  and  volume, 
or  by 


THE  MECHANICAL  ACTION   OF   HEAT.  207 

PV,16^.,         .         .         .     (39.) 
The  coefficient  — y-r —  represents  a  certain  depth  of  fall  per  degree  of 

absolute  temperature,  and  is  the  same  with  the  coefficient  ~ — ^.,  already 

referred  to. 

By  taking  the  following  values  of  the  factors: — v  =  0*016  cubic  foot 
per  pound  avoirdupois,  w  =  2117  pounds  avoirdupois  per  square  foot,  we 
find  this  coefficient  to  be 


153*35  feet  =  46*74  metres  per  Centigrade  degree, 
85*19  feet  per  degree  of  Fahrenheit. 


(40.) 


This  determination  may  be  considered  correct  to  about  1210o  part. 
When  French  measures  are  used  in  the  calculation,  the  following  is  the 
result : — 

v  =  1  cubic  centimetre  per  gramme, 

to  =  1033*3  grammes  per  square  centimetre. 

1 


,,  =  46*78  metres  per  Centigrade  degree, 
C»M 

153*48  feet,  or  85*27  feet  per  degree  of  Fahrenheit. 


(41.) 


The  difference,  which  is  of  no  practical  importance  in  calculating  the 
power  of  the  steam-engine,  arises  in  the  estimation  of  the  density  of 
liquid  water. 

22.  Unit  of  weight  of  steam  at  saturation,  of  the  elasticity  Px  and 
volume  V19  corresponding  to  the  absolute  temperature  rv  being  cut  off 
from  external  sources  of  heat,  it  is  now  to  be  investigated  what  amount 
of  power  it  will  produce  in  expanding  to  a  lower  pressure  P2  and 
temperature  r2. 

It  has  already  been  shown,  at  the  end  of  the  second  section,  that  if 
vapour  at  saturation  is  allowed  to  expand,  it  requires  a  supply  of  heat 
from  without  to  maintain  it  at  the  temperature  of  saturation,  otherwise  a 
portion  of  it  must  be  liquefied  to  supply  the  heat  required  to  expand  the 
rest.  Hence,  when  unity  of  weight  of  steam  at  saturation,  at  the  pressure 
Pj  and  volume  Vv  expands  to  a  lower  pressure  P,  being  cut  off  from 
external  sources  of  heat,  it  will  not  occupy  the  entire  volume  V  corre- 
sponding to  that  pressure,  according  to  equation  (38),  but  a  less  volume, 
S  =  mV,  where  m  represents  the  weight  of  water  remaining  in  the 
gaseous  state,  the  portion  1  —  m  having  been  liquefied  during  the  expan- 


2GS 


THE  MECHANICAL  ACTION   OF   HEAT. 


sion  of  the  remainder.     The  expansive  action  of  the  steam  will,  therefore, 
be  represented  by 

S 
%ZS.P (42.) 


/: 


The  law  of  variation  of  the  fraction  m  flows  from  the  following  con- 
siderations : — Let  d  m  represent  the  indefinitely  small  variation  of  m 
corresponding  to  the  indefinitely  small  change  of  temperature  S  r ;  L, 
the  latent  heat  of  evaporation  of  unity  of  weight ;  Ks,  as  in  equation  (30), 
the  specific  heat  of  vapour  at  saturation,  which  is  a  negative  coefficient 
varying  with  the  temperature ;  then  we  must  have 

—  L  o  m  =  m  EL  o  r,  or  —  =  —   T  -  S  t, 
in  L 


in  order  that  the  heat  produced  by  the  liquefaction  of  $  m  may  be  equal 
to  the  heat  required  to  expand  m.  Hence  making,  according  to 
equation  (30) — 


KsSr  =  1i(Sr  +  N^Sv), 


and  Sr  =  -y8V-^ 


1 


+ 


2  V 


we  obtain 


m    "  IA         ft' 


27 


-  +  -I-  -  1 


)  v' 


and  denoting  the  coefficient  of 


(43.) 


SV 


by  -  v, 


d  log.  m 


d  log. 

V 

and  because 

d  log. 

V 

d  log. 

P 

d  log. 

m 

d  log. 

P 

d  log. 

S 

V 

cl  log.  S 
V  '  d  log.  V 


P...I   2V 

T    +     T2 


+   V     1   - 


1 


/3' 


+ 


2y 


d  log.  P 


=  -0-v)     1- 


1 


^4 

T  T~ 


2y 


^       (44.) 


THE   MECHANICAL  ACTION   OF  HEAT.  269 

As  the  mean  temperature  of  the  liquid  thus  produced,  more  or  less 
exceeds  that  of  the  remaining  vapour,  a  small  fraction  of  it  will  be 
reconverted  into  vapour,  if  the  expansion  is  carried  on  slowly  enough ; 
but  its  amount  is  so  small,  that  to  take  it  into  account  would  needlessly 
complicate  the  calculation,  without  making  it  to  any  material  extent  more 
accurate. 

23.  The  extreme  complexity  of  the  exponent  a,  considered  as  a  function 
of  the  pressure  P,  would  render  a  general  formula  for  the  expansive  action 

/PfZS  very  cumbrous  in  its  application.      For  practical  purposes,  it  is 

sufficient  to  consider  the  exponent  o-  as  constant  during  the  expansion 
which  takes  place  in  any  given  engine,  assigning  it  an  average  value 
suitable  to  the  part  of  the  scale  of  pressures  in  which  the  expansion  takes 
p>lace.  For  engines  in  which  the  steam  is  introduced  at  pressures  not 
exceeding  four  atmospheres,  I  conceive  that  it  will  be  sufficiently  accurate 

to  make   tr  =  —  -}    while  for  engines  in  which  the  initial   pressure  lies 

5 
between  four  and  eight  atmospheres,  the  suitable  value  is  a  =  -. 

The  utmost  error  which  can  arise  from  using  these  exponents  is 
about  yijj  of  the  whole  power  of  the  engine,  and  that  only  in  extreme 
cases.     Making,  therefore, 

*  =  *>$ 

we  obtain  for  the  value  of  the  expansive  action  of  unity  of  weight  of 
steam, 


V, 

1 
P,V 


y       (45.) 


^iUl-'l's) 


s 

s  being  used  to  denote  ==^,  or  the  ratio  of  the  volumes  occupied  by  steam 

at  the  end  and  at  the  beginning  of  the  expansion  respectively. 

A  table  to  facilitate  the  computation  is  given  in  the  Appendix. 

The  gross  mechanical  action  of  unity  of  weight  of  steam  on  one  side  of 
the  piston  is  found  by  adding  to  the  above  quantity  the  action  of  the 
steam  before  it  begins  to  expand,  or  Px  Vv  and  is  therefore, 


(46.) 


270  THE  MECHANICAL   ACTION   OF   HEAT. 

The  values  of  the  coefficients  and  exponent  being 


1 

a 

!_1 

1  -  a 

1   -  <7 

x  — -      . 
a 

For  initial  pressures  between 

1  and  4  atmospheres, 

.      7 

6 

1 
6' 

4  and  8  atmospheres, 

.     6 

5 

1 
5' 

24.  The  following  deductions  have  to  be  made  from  the  gross  action, 
in  order  to  obtain  the  action  effective  in  overcoming  resistance. 

First,  For  loss  of  power  owing  to  a  portion  of  the  steam  being  employed 
in  filling  steam-passages,  and  the  space  called  the  clearance  of  the  cylinder 
at  one  end.  Let  the  bulk  of  steam  so  employed  be  the  fraction  c  S2  of 
the  space  filled  by  steam  at  the  end  of  the  expansion ;  then  the  loss  of 
power  from  this  cause  is 

PjCSo  =  csP1V1. 

Secondly,  For  the  pressure  on  the  opposite  side  of  the  piston,  of  the  steam 
which  escapes  into  the  condenser,  or  into  the  atmosphere,  as  the  case  may 
be.  Let  P3  be  the  pressure  of  this  steam ;  the  deduction  to  be  made  for 
its  action  is 

P3S.,(1  -c)=  P8Vt(l  -  c)s. 

These  deductions  having  been  made,  there  is  obtained  for  the  effect  of 
unity  of  weight  of  water  evaporated, 

VijP,  (y^  -  ^-J  ~  »-  cs)  -  P,(l  -  c)s)     (47.) 

25.  The  effect  of  the  engine  in  unity  of  time  is  found  by  multiplying 
the  above  quantity  by  the  number  of  units  of  weight  of  water  evaporated 
in  unity  of  time. 

If  this  number  be  denoted  by  W, 

"W  S2 (1  -  c)  =  W V1  (1  -  c) s  =  Au,  .         .     (48.) 

will  represent  the  cubical  space  traversed  by  the  piston  in  unity  of  time, 
A  denoting  the  area  of  the  piston,  and  u  its  mean  velocity. 

Now,  let  the  whole  resistance  to  be  overcome  by  the  engine  be  reduced, 
by  the  principles  of  statics,  to  a  certain  equivalent  pressure  per  unit  of  area 
of  piston,  and  let  this  pressure  be  denoted  by  P.     Then, 


THE  MECHANICAL  ACTION  OF  HEAT.  271 

E  A  u  =  E  W  Vj  (1  -  c)  5,    .         .         .     (49.) 

expresses  the  effect  of  the  engine  in  terms  of  the  gross  resistance. 

We  have  now  the  means  of  calculating  the  circumstances  attending  the 
"working  of  a  steam-engine,  according  to  the  principle  of  the  conservation 
of  vis  viva,  or,  in  other  words,  of  the  equality  of  power  and  effect,  which 
regulates  the  action  of  all  machines  that  move  with  an  uniform  or 
periodical  velocity. 

This  principle  was  first  applied  to  the  steam-engine  by  the  Count  de 
Pambour ;  and,  accordingly,  the  formulae  which  I  am  about  to  give  only 
differ  from  those  of  his  work  in  the  expressions  for  the  maximum  pressure 
at  a  given  temperature,  and  for  the  expansive  action  of  the  steam,  which 
are  results  peculiar  to  the  theory  of  this  essay. 

In  the  first  place,  the  effect  as  expressed  in  terms  of  the  pressure,  is  to  be 
equated  to  the  effect  as  expressed  in  terms  of  the  resistance,  as  follows: — 

ea^kwv^i  -c)s  =  wVijp^j-^-  yzt^s     a~cs) 

-P3(l-c)s}    .         .  .     (50.) 

This  is  the  fundamental  equation  of  the  action  of  the  steam-engine,  and 
corresponds  with  equation  A.  of  M.  de  Pambour's  theory. 

26.  Dividing  both  sides  of  equation  (50)  by  the  space  traversed  by 
the  piston  in  unity  of  time,  "W  Vx  (1  —  c)  s,  and  transferring  the  pressure 
of  the  waste  steam,  P3,  to  the  first  side,  we  obtain  this  equation  : — 


i-L 


cs 


R+p3=p11  '  ll-'). ■   ■  («•) 

which  gives  the  means  of  determining  the  pressure  Px  at  which  the  steam 
must  enter  the  cylinder,  in  order  to  overcome  a  given  resistance  and 
counter-pressure  with  a  given  expansion ;  or,  supposing  the  expansion  s  to 
be  variable  at  pleasure,  and  the  initial  pressure  Px  fixed,  the  equation 
gives  the  means  of  finding,  by  approximation,  the  expansion  best  adapted 
to  overcome  a  given  resistance  and  counter-pressure. 

The  next  step  is  to  determine,  from  equations  (XV.)  of  the  introduction 
and  (38)  of  this  section,  the  volume  V1  of  unity  of  weight  of  steam 
corresponding  to  the  maximum  pressure  Pr  Then  equation  (48)  gives 
the  space  traversed  by  the  piston  in  unity  of  time,  which,  being  multiplied 
by  the  resistance  E  per  unit  of  area  of  piston,  gives  the  gross  effect  of 
the  engine. 


272  THE  MECHANICAL  ACTION   OF   HEAT. 

27.  If,  on  the  other  hand,  the  space  traversed  by  the  piston  in  unity 
of  time  is  fixed,  equation  (48)  gives  the  means  of  determining,  from  the 
evaporating  power  of  the  boiler  W,  either  the  volume  Y1  of  unity  of 
weight  of  steam  required  to  work  the  engine  at  a  given  velocity  with 
a  given  expansion,  or  the  expansion  s  proper  to  enable  steam  of  a  given 
initial  density  to  work  the  engine  at  the  given  velocity.  The  initial 
pressure  Px  being  then  determined  from  the  volume  Yv  the  resistance 
which  the  engine  is  capable  of  overcoming  with  the  given  velocity  is  to  be 
calculated  by  means  of  equation  (51). 

28.  This  calculation  involves  the  determination  of  the  pressure  P2 
from  the  volume  V1  of  unity  of  weight  of  steam  at  saturation,  which  can 
only  be  done  by  approximation.  The  following  formula  will  be  found 
useful  for  this  purpose  : — 

12 


F-i^Hv/1     •         •         •         •     (52-) 

where  to  represents  the  pressure  of  one  atmosphere,  V0  the  volume  of 
steam  of  saturation  at  that  pressure  (being  1G96  times  the  volume  of 
water  at  4°'l  Cent.,  or  27*130  cubic  feet  per  pound  avoirdupois),  and  V, 
the  volume  of  steam  of  saturation  at  the  pressure  Pr  This  formula  is 
only  applicable  between  the  pressures  of  one  and  eight  atmospheres  :  that 
is  to  say,  when  the  volume  of  steam  is  not  greater  than  27  cubic  feet  per 
pound,  nor  less  than  4,  and  the  temperature  not  lower  than  100°  Centigrade, 
nor  higher  than  171°  Centigrade  (which  correspond  to  212°  and  340° 
Fahrenheit). 

The  greatest  error  in  computing  the  pressure  by  means  of  this  formula 
is  about  -Jg-  of  an  atmosphere,  and  occurs  at  the  pressure  of  four  atmos- 
pheres, so  that  it  is  pj  of  the  whole  pressure.  This  is  sufficiently 
accurate  for  practice,  in  calculating  the  power  of  steam-engines ;  but 
should  a  more  accurate  result  be  required,  the  approximate  value  of  the 
pressure  may  be  used  to  calculate  the  temperature  by  means  of  equation 
(XV.)  ;  and  the  temperature  thus  determined,  (which  will  be  correct  to  ~  of 
a  Centigrade  degree),  may  then  be  used  in  conjunction  with  the  volume  to 
compute  a  corrected  value  of  the  pressure,  according  to  equation  (38). 
The  pressure,  as  thus  ascertained,  will  be  correct  to  ^wo  of  its  amount, 
which  may  be  considered  the  greatest  degree  of  accuracy  attainable. 

The  most  convenient  and  expeditious  mode,  however,  of  computing  the 
pressure  from  the  volume,  or  vice  versd,  is  by  interpolation  from  the  table 
given  in  the  Appendix  to  this  paper. 

29.  The  resistance  denoted  by  R  may  be  divided  into  two  parts ;  that 
which  arises  from  the  useful  work  performed,  and  that  which  is  independent 
of  it,  being,  in  fact,  the  resistance  of  the  engine  when  unloaded.  Now  it 
is  evident,  that  the  maximum  usrful  effect  of  the  steam  has  been  attained, 


THE  MECHANICAL  ACTION  OF   HEAT.  273 

as  soon  as  it  has  expanded  to  a  pressure  which  is  in  equilibrio  with  the 
pressure  of  the  waste  steam  added  to  the  resistance  of  the  engine  when 
unloaded ;  for  any  further  expansion,  though  increasing  the  total  effect, 
diminishes  the  useful  effect.     Therefore,  if  we  make 

R  =  K'+/, 

R'  being  the  resistance  arising  from  the  useful  work,  and  /  the  resistance 
of  the  engine  when  unloaded,  both  expressed  in  the  form  of  pressure 
on  the  piston,  the  expansion  corresponding  to  the  maximum  of  useful 
effect  will  take  place  when 

P2  =  P3+/> 
the  corresponding  ratio  of  expansion  being  !  ,_    . 


\p.+// 


The  maximum  useful  effect  with  a  given  pressure  on  the  safety-valve 
has  been  so  fully  discussed  by  M.  de  Pambour,  that  it  is  unnecessary  to 
do  more  than  to  state  that  it  takes  place  when  the  initial  pressure 
in  the  cylinder  is  equal  to  that  at  the  safety-valve  :  that  is  to  say,  when 
it  and  the  useful  resistance  are  the  greatest  that  the  safety-valve  will 
permit. 

30.  Annexed  is  a  table  of  the  values  of  some  of  the  quantities  which 
enter  into  the  preceding  equations  in  the  notation  of  the  Count  de 
Pambour's  works  : — 

Expression  in  the  Notation  Equivalent  Expression  in 

of  this  Paper.  M.  tie  Pambour's  Notation. 

R  =  R'+/ (1  +  8)r+/ 

Aw av 

W  .  .         .  S  x  weight  of  one  cubic 

foot  of  water. 

P3  .  .  p 

I  +  c 
V  +  c 


I'  +  c 

31.  As  an  illustration,  I  shall  calculate  the  maximum  useful  effect  of 
one  pound,  and  of  one  cubic  foot  of  water,  in  a  Cornish  double-acting 

S 


27-i  THE  MECHANICAL  ACTION   OF   HEAT. 

engine,  in  the  circumstances  taken  by  M.  de  Pambour  as  an  example  for 
that  kind  of  engine,  that  is  to  say, — 

Clearance  one-twentieth  of  the  stroke,  or  c  =  — 

Eesistance  not  depending  on  the  useful  load,         /  =      72  lbs.  per  sq.  ft. 
Pressure  of  condensation,    .         .         .         .        P3  =    576  lbs.     „      „ 
Consequently,  to  give  the  maximum  useful  effect, 

P2  =  PS+/  =    048  lbs.     „      „ 

Total  pressure  of  the  steam  when  first  admitted,  V1  —  7200  lbs.     „      „ 
Volume  of  1  lb.  of  steam  Vx  =  8*7825  cubic  feet. 
Therefore  Px  Vj  =  G3234  lbs.  raised  one  foot. 

P         7200 

— 1  =  — — ;  and,  consequently, 

P2  04o 

Expansion  to  produce  the  maximum  useful  effect  s  =  (rr  )  '  —  7'877. 

Space  traversed  by  the  piston  during  the  action  of  one  pound  of  steam, 

=  Vx  (1  -  c)  s  =  G5-8SG  cubic  feet. 

Gross  effect  of  one  pound  of  steam,  in  pounds  raised  one  foot  high, 

1 
=  PjV,  (7 -6s'6-  £)  -  P3V1(1  -  c)s     =  112001 

Deduct  for  resistance  of  engine  when  unloaded  f\1  (1  —  c)s      =      4744 


Effect  of  one  pound  of  steam  in  overcoming  resistance  depend-"!      io7°60 
ing  on  useful  load,       .         .         .         .         .         .         -J 

This  being  multiplied  by  G2£,  gives  for  the  effect  of  one  cubic 

foot  of  water  evaporated,  in  pounds  raised  one  foot,        .      G,703,750 

It  is  here  necessary  to  observe,  that  M.  de  Pambour  distinguishes  the 
useful  resistance  into  two  parts,  the  resistance  of  the  useful  load  indepen- 
dently of  the  engine,  and  the  increase  in  the  resistance  of  the  engine 
arising  from  the  former  resistance,  and  found  by  multiplying  it  by  a 
constant  fraction,  which  he  calls  $.  In  calculating  the  net  useful  effect,  he 
takes  into  account  the  former  portion  of  the  resistance  only ;  consequently  : 
Net  useful  effect  as  defined  by  M.  de  Pambour 


THE   MECHANICAL  ACTION   OF   HEAT.  275 

Gross  effect  —  f  Y,  (1  —  c)  s  /t, .  N 

= i  +  g  •     •  <"•> 

The  value  of  S,  for  double-acting  steam-engines  generally,  is  considered 
by  M.  de  Pambour  to  be  y ;  consequently,  to  reduce  the  effect  of  one 
cubic  foot  of  water,  as  calculated  above,  to  that  which  corresponds  with  his 
definition,  we  must  deduct  £,  which  leaves, 

5,865,781  lbs.  raised  one  foot. 

M.  de  Pambour's  own  calculation  gives, 

6,277,560, 

being  too  large  by  about  one-fifteenth. 

32.  In  order  to  show  the  limit  of  the  effect  which  may  be  expected 
from  the  expenditure  of  a  given  quantity  of  heat  in  evaporating  water, 
and  also  to  verify  the  approximate  method  employed  in  calculating  the 
expansive  action  of  the  steam,  I  shall  now  investigate  the  maximum  gross 
cjfed,  including  resistance  of  all  kinds,  producible  by  evaporating  unity  of 
weight  of  Avater  at  a  higher  temperature  and  liquefying  it  at  a  lower,  and 
compare,  in  two  examples,  the  power  produced  with  the  heat  which 
disappears  during  the  action  of  the  steam,  as  calculated  directly. 

To  obtain  the  maximum  gross  effect,  the  steam  must  continue  to  act 
expansively  until  it  reaches  the  pressure  of  condensation,  so  that  P2  =  P3. 
The  clearance  must  also  be  null,  or  c  =  0.  Making  those  substitutions  in 
the  formula  (47),  we  find,  for  the  maximum  gross  effect  of  unity  of  weight 
of  water,  evaporated  under  the  pressure  Px,  and  liquefied  under  the 
pressure  POJ 

,   (     -M         ^ST" 
p^r-V1-8    ''  -  P'Y'     i - .        (55) 

In  order  to  calculate  directly  the  heat  which  is  converted  into  power  in 
this  operation,  let  rv  r2,  respectively  represent  the  absolute  temperatures 
of  evaporation  and  liquefaction,  and  L2  the  latent  heat  of  evaporation  at 
the  lower  temperature  r2;  then  the  total  heat  of  evaporation  at  tv  starting 
from  t0  as  the  fixed  point,  by  equation  (33),  is 

H2,  ,  =  L2  +  -305  Kw  (r,  -  r2). 

This  is  the  heat  communicated  to  the  water  in  raising  it  from  r2,  to  tx  and 
evaporating  it.  Now  a  weight  1  —  m  of  the  steam  is  liquefied  during  the 
expansion  at  temperatures  varying  from  ra  to  r2,  so  that  it  may  be  looked 
upon  as   forming  a   mass  of  liquid  water  approximately  at  the   mean 


276  THE  MECHANICAL  ACTION  OF  HEAT. 

temperature  ^-— — -,  and  from  which  a  quantity  of  heat,  approximately 

a 

represented  by 

Kw(i-m)^p, 

must  be  abstracted,  to  reduce  it  to  the  primitive  temperature  r2. 

Finally,  the  weight  of  steam  remaining,  m,  has  to  be  liquefied  at  the 
temperature  r2,  by  the  abstraction  of  the  heat 

mL2. 

The  difference  between  the  heat  given   to  the  water,  and  the  heat 
abstracted  from  it,  or 


Hjs,,  -  Kw(l  -  m)Tl      T*  -  mL2 
=  (1  -  m)U  +  Kw(-305  -  1-J=-^)(r1  -  r2) 


>      (560 


is  the  heat  which  has  disappeared,  and  ought  to  agree  with  the  expression 
(55)  for  the  power  produced,  if  the  calculation  has  been  conducted 
correctly. 

As  a  first  example,  I  shall  suppose  unity  of  weight  of  water  to  be 
evaporated  under  the  pressure  of  four  atmospheres,  and  liquefied  under 
that  of  half  an  atmosphere ;  so  that  the  proper  values  of  the  coefficients 
and  exponent  are 

1  7       1  l 

=  7,     1  —  a  = 


1  -a  '  T 

The  data,  in  this  case,  for  calculating  the  power  are, 

Px  =  84G8  lbs.  per  square  foot. 
Vj  =  7 -58 4  cubic  feet  for  1  lb.  of  steam. 
Px  Vx  =  64221  lbs.  raised  one  foot. 

P        1  " 

—  =  -,  whence  s  =  87  =  5 '9 44. 
J-i       b 

Maximum  possible  effect  of  one  pound  of  water, 

=  P1V1x7H-  Qm  =  115600  lbs.  raised  one  foot. 

Being  the  mechanical  equivalent  of  9 2° '3  Centigrade  applied  to  one 
pound  of  liquid  water  at  0°  C;  or, 


THE  MECHANICAL  ACTION  OF  HEAT.  277 

92°-3  Kw. 

Maximum  possible  effect  of  one  cubic  foot  of  water,  7,225,000  lbs.  raised 
one  foot. 

In  order  to  calculate  directly  the  heat  converted  into  power,  we  have, 

Tj  =  C  +  144°-1  Cent.  r2  =  C  +  81°'7. 
L2  =  549°-7  Kw. 
H2n  =  5  6  80- 7  Kw  =  heat  expended  in  the  boiler. 
1  —  m  =  '14  nearly  =  proportion  of  steam  liquefied  during  the  expansion. 

The  heat  converted  into  mechanical  power,  as  calculated  from  these 
data,  is  found  to  be, 

01°'6KW, 

differing  by  only  0o,7  from  the  amount  as  calculated  from  the  power 
produced. 

The  direct  method,  however,  is  much  less  precise  than  the  other,  and  is 
to  be  regarded  as  only  a  verification  of  the  general  principle  of  calculation. 

92'3 

The  heat  rendered  effective,  in  the  above  example,  is  ,  or  less  than 

one-sixth  of  that  expended  in  the  boiler. 

As  a  second  example,  I  shall  suppose  the  steam  to  be  produced  at  a 
pressure  of  eight  atmospheres,  and  to  expand  to  that  of  one  atmosphere. 
In  this  case, 

Pj  =  16936  lbs.  per  square  foot. 

Vj  =  4'03  cubic  feet  per  lb.  of  steam. 

VlY1  =  68252  lbs.  raised  one  foot. 

PI  6 

£?=  *  -.s  =  5-657  =  8*. 

Maximum  possible  effect  of  one  pound  of  water, 

=  P1V1xen-  (q)0  =  119'942  lbs- raised  one  foot' 

Being  the  equivalent  of  950-8  Kw  (Centigrade). 

Maximum  possible  effect  of  one  cubic  foot  of  water  =  7,496,375  lbs. 
raised  one  foot. 

The  data  for  calculating  directly  the  heat  rendered  effective  are, 


278  THE  MECHANICAL  ACTION   OF  HEAT. 

Tl  =  C  +  170o,9  Cent,  r,  =  C  +  100°. 
L2  =  537°  Kw. 
Hojj  =  5580,6  Kw  =  heat  expended  in  the  boiler. 
1  —  m  =  '148  nearly  =  steam  liquefied  during  the  expansion. 
Whence,  the  heat  converted  in  power,  as  calculated  directly,  is 

95°-S  Kw, 

agreeing  with  the  calculation  from  the  power  produced. 

95"8 
In  this  example,  the  heat  rendered  effective  is         -,  or  somewhat  more 

than  one-sixth  of  that  expended  in  the  boiler. 

33.  The  results  of  the  calculations  of  maximum  possible  effect,  of 
which  examples  have  just  been  given,  are  limits  which  may  be  approached 
in  practice  by  Cornish  and  similar  engines,  but  which  cannot  be  fully 
realised ;  and  yet  it  has  been  shown,  that  in  those  theoretical  cases  only 
about  one-sixth  of  the  heat  expended  in  the  boiler  is  rendered  effective. 
In  practice,  of  course,  the  proportion  of  heat  rendered  effective  must  be 
still  smaller;  and,  in  fact,  in  some  unexpansivc  engines,  it  amounts  to 
only  onc-t ve a f {/-fourth,  or  even  less. 

Dr.  Lyon  Playfair,  in  a  memoir  on  the  Evaporating  Power  of  Fuel,  has 
taken  notice  of  the  great  disproportion  between  the  heat  expended  in  the 
steam-engine  and  the  work  performed.  It  has  now  been  shown  that  this 
waste  of  heat  is,  to  a  great  extent,  a  necessary  consequence  of  the  nature 
of  the  machine.  It  can  only  be  reduced  by  increasing  the  initial  pressure 
of  the  steam,  and  the  extent  of  the  expansive  action;  and  to  both  of  those 
resources  there  are  practical  limits,  which  have  already,  in  some  instances, 
been  nearly  attained. 


APPENDIX  TO  THE  FOURTH  SECTION. 

Containing  Tables  to  be  used  in  Calculating  the  Pressure, 

Volume,  and  Mechanical  Action  of  Steam, 

Treated  as  a  Perfect  Gas. 

The  object  of  the  First  of  the  annexed  Tables  is  to  facilitate  the  calcula- 
tion of  the  volume  of  steam  of  saturation  at  a  given  pressure,  of  the 
pressure  of  steam  of  saturation  at  a  given  volume,  and  of  its  mechanical 
action  at  full  pressure. 


THE  MECHANICAL   ACTION   OF  HEAT.  279 

The  pressures  are  expressed  in  pounds  avoirdupois  per  square  foot,  and 
the  volumes  by  the  number  of  cubic  feet  occupied  by  one  pound  avoirdu- 
pois of  steam,  when  considered  as  a  perfect  gas ;  those  denominations 
being  the  most  convenient  for  mechanical  calculations  in  this  country. 

The  columns  to  be  used  in  determining  the  pressure  from  the  volume, 
and  vice  versd,  are  the  third,  fourth,  sixth,  and  seventh. 

The  third  column  contains  the  common  logarithms  of  the  pressures  of 
steam  of  saturation  for  every  fifth  degree  of  the  Centigrade  thermometer, 
from  —  30°  to  +  260°:  that  is  to  say,  for  every  ninth  degree  of 
Fahrenheit's  thermometer,  from  —  22°  to  +  500°. 

The  fourth  column  gives  the  differences  of  the  successive  terms  of  the 
third  column. 

The  sixth  column  contains  the  common  logarithms  of  the  volume  of 
one  pound  of  steam  of  saturation  corresponding  to  the  same  temperatures. 

The  seventh  column  contains  the  differences  of  the  successive  terms  of 
the  sixth  column,  which  are  negative ;  for  the  volumes  diminish  as  the 
pressures  increase. 

By  the  ordinary  method  of  taking  proportional  parts  of  the  differences, 
the  logarithms  of  the  volumes  corresponding  to  intermediate  pressures, 
or  the  logarithms  of  the  pressures  corresponding  to  intermediate  volumes, 
can  be  calculated  with  great  precision.  Thus,  let  X  +  h  be  the  logarithm 
of  a  pressure  not  found  in  the  table,  X  being  the  next  less  logarithm 
which  is  found  in  the  table ;  let  Y  be  the  logarithm  of  the  volume  cor- 
responding to  X,  and  Y  —  h  the  logarithm  of  the  volume  corresponding  to 
X  +  h;  let  H  be  the  difference  between  X  and  the  next  greater  logarithm 
in  the  table,  as  given  in  the  fourth  column,  and  K  the  corresponding 
difference  in  the  seventh  column ;  then  by  the  proportion 

H  :  K  : :  h  :  i 

either  Y  —  Jc  may  be  found  from  X  -f  h,  or  X  +  h  from  Y  -  k 

In  the  fifth  and  eighth  columns  respectively,  are  given  the  actual 
pressures  and  volumes  corresponding  to  the  logarithms  in  the  third  and 
sixth  columns,  to  five  places  of  figures. 

In  the  ninth  column  are  given  the  values  of  the  quantity  denoted  by 
P  Vj  in  the  formulae,  which  represents  the  mechanical  action  of  unity  of 
weight  of  steam  at  full  pressure,  or  before  it  has  begun  to  expand,  in 
raising  an  equal  weight.  Those  values  are  expressed  in  feet,  being  the 
products  of  the  pressures  in  the  fifth  column  by  the  volumes  in  the 
eighth,  and  have  been  found  by  multiplying  the  absolute  temperature  in 
Centigrade  degrees  by  153*48  feet.  Intermediate  terms  in  this  column, 
for  a  given  pressure  or  a  given  volume,  may  be  approximated  to  by  the 
method  of  differences,  the  constant  difference  for  5°  Centigrade  being  767'4 


280  THE  MECHANICAL  ACTION   OF  HEAT. 

feet ;  but  it  is  more  accurate  to  calculate  them  by  taking  the  product  of 
the  pressure  and  volume. 

When  the  pressure  is  given  in  other  denominations,  the  following 
logarithms  are  to  be  added  to  its  logarithm,  in  order  to  reduce  it  to 
pounds  avoirdupois  per  square  foot : — 

For  Millimetres  of  mercury,          .         .         .  0*44477 

„  Inches  of  mercury,        ....  1 '84960 

„  Atmospheres  of  7 GO  millimetres,    .         .  3*32559 

„  Atmospheres  of  30  inches,    .         .         .  33 2 G 72 

.  Kilogrammes  on  the  square  centimetre,  .  3-31136 

„  Kilogrammes  on  the  circular  centimetre,  3*41  G 27 

„  Kilogrammes  on  the  square  metre,          .  1*31136 

„  Pounds  avoirdupois  on  the  square  inch, .  2*15836 

„  Pounds  avoirdupois  on  the  circular  inch,  2*26327 

To  reduce  the  logarithm  of  the  number  of  cubic  metres  occupied  by  one 
kilogramme  to  that  of  the  number  of  cubic  feet  occupied  by  one  pound 
avoirdupois,  add  1*20463. 

The  logarithms  are  given  to  five  places  of  decimals  only,  as  a  greater 
degree  of  precision  is  not  attainable  in  calculations  of  this  kind. 

The  Second  Table  is  for  the  purpose  of  calculating  the  mechanical  action 
of  steam  in  expansive  engines. 

The  first  column  contains  values  of  the  fraction  of  the  entire  capacity 
of  the  cylinder  which  is  filled  with  steam  before  the  expansion  commences 

(being  the  quantity  -  of  the  formulas),  for  every  hundredth  part,  from 

1*00,  or  the  whole  cylinder,  down  to  0*10,  or  one-tenth. 

If  I  be  the  entire  length  of  stroke,  V  the  portion  performed  at  full 
pressure,  and  c  the  fraction  of  the  entire  capacity  of  the  cylinder  allowed 
for  clearance,  then 

1  _ 
I'       s      C  ,  1        „         ,  V 

i=Tzre  and-=(i -,)-  +  ,. 

The  entire  capacity  of  the  cylinder  is  to  be  understood  to  include  clearance 
at  one  end  only. 

The  second  column  gives  the  reciprocals  of  the  quantities  in  the  first, 
or  the  values  of  the  ratio  of  expansion  s. 


THE  MECHANICAL  ACTION  OF  HEAT.  281 

The  third   and   fourth   columns,  headed   Z,  give   the   values  of  the 
quantity 

1  C7  1~i 


1  -  a-         1  -  a 

of  article  23,  which  represents  the  ratio  of  the  entire  gross  action  of  the 
steam  to  its  action  at  full  pressure,  without  allowing  for  clearance.  The 
third  column  is  to  be  used  for  initial  pressures  of  from  one  to  four 
atmospheres ;  and  the  fourth  for  initial  pressures  of  from  four  to  eight 
atmospheres. 

The  deduction  to  be  made  from  the  quantity  Z  for  clearance  is  c  s,  or 
the  product  of  the  fraction  of  the  cylinder  allowed  for  clearance  by  the 
ratio  of  expansion.  Hence,  to  calculate  from  the  tables  the  net  mechanical 
action  of  unity  of  weight  of  steam,  allowing  for  the  counter-pressure  of  the 
waste  steam  P3,  as  well  as  for  clearance,  we  have  the  formula 

p1y1(z-cs)-p3v1(i-c),, 

being  equivalent  to  the  formula  (47)  of  this  paper. 


282 


THE  MECHANICAL   ACTION   OF   HEAT. 


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THE  MECHANICAL  ACTION   OF  HEAT.  283 


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284 


THE  MECHANICAL  ACTION   OF  HEAT. 


TABLE  II. — Expansive  Action  of  Steam. 


(1.) 

(2.) 

(3.) 

(4.) 

(1.) 

(2.) 

(3.) 

(4.) 

Fraction  of 

Coefficient  of  Gross 

Fraction  of 

Coefficient  of  Cros3 

Cylinder 

Action  =  Z. 

i  rider 

Action  =  Z. 

filled  with 

Steam  at  full 

Pressuro 

Patio  of 

Expansion 

Cnitj  il  Pres- 
sure One 

Initial  Pres- 
sure Four 

filled  with 

Steam  at  full 

Pressure 

Ratio  of 
Expansion 

Initial  Pres- 
sure One 

Initial  Pros- 
sure  Four 

1 

' 

to  Four 

to  Eight 

1 

=  •'• 

to  Four 

to  Eight 

Atmos- 

Atmos- 

Atmos- 

— s' 

pheres. 

ph(  l 

s 

pheres. 

pheros. 

1-00 

1-000 

1-000 

1-000 

•54 

1-852 

1-586 

1 -5S0 

■99 

1010 

1010 

1010 

•53 

1-887 

1G02 

1-596 

•98 

1-020 

1020 

1  020 

•52 

1  -923 

1  -620 

1-613 

•97 

1031 

1030 

1  -030 

•51 

1-961 

1-637 

1-G30 

•9G 

1  -042 

1041 

1041 

•50 

2  000 

1  -655 

1  -647 

•95 

1  053 

1051 

1-051 

•49 

2  041 

1-G73 

L-665 

•94 

1-064 

1-0G2 

1-0G2 

•48 

2-083 

1-691 

1-683 

•93 

1075 

1072 

L-072 

•47 

2-12S 

1-709 

1-701 

•92 

1-087 

1-083 

10S3 

•46 

2174 

1-728 

1-719 

•91 

1  -099 

1-094 

1  093 

•45 

2*222 

1-748 

1-738 

•90 

1-111 

1-104 

1-104 

•44 

,    2-273 

1-767 

1-757 

•89 

1-124 

1115 

1-115 

•43 

2-326 

1-787 

1777 

•88 

1-136 

1-126 

1-126 

•42 

2  381 

1-803 

1 -79G 

•87 

1-149 

1-138 

1-137 

•11 

2-439 

1-S29 

1-817 

•86 

1-1G3 

1-149 

1-149 

•40 

2-500 

1-850 

1-837 

•85 

1-176 

1-1G0 

1-1G0 

•39 

2-564 

1-871 

1-858 

•84 

1-190 

1-172 

1-171 

•38 

2-632 

1-894 

1-880 

•83 

1-205 

1183 

1-1S3 

•37 

2-703 

1-916 

1-902 

•82 

1  -220 

1-195 

1-195 

•36 

2-778 

1-939 

1-924 

•81 

1  -235 

1-207 

1  206 

•35 

2-857 

1-963 

1-947 

•80 

1-250 

1-219 

1-218 

•34 

2-941 

1-987 

1-970 

•79 

1-266 

1-231 

1-230 

•33 

3  030 

2012 

1-994 

•78 

1-282 

1-243 

1-242 

•32 

3125 

2-038 

2  019 

•77 

1  -209 

1  -256 

1  -255 

•31 

3-225 

2  064 

2  044 

•76 

1-316 

1-268 

1-2G7 

•30 

3-333 

2091 

2  070 

•75 

1  -333 

1-281 

1-280 

•29 

3-44S 

2119 

2  097 

•74 

1-351 

1-294 

1  -292 

•28 

3  571 

2-147 

2-124 

•73 

1-370 

1  -307 

1-305 

•27 

3-704 

2176 

2152 

•72 

1-389 

1-320 

1-318 

•26 

3-S46 

2-207 

2-181 

•71 

1-408 

1-333 

1-331 

•25 

4-000 

2-238 

2-211 

•70 

1-429 

1-346 

1-344 

•24 

4-167 

2-270 

2-242 

•69 

1-449 

1-3G0 

1-358 

•23 

4-34S 

2  304 

2-273 

•6S 

1-471 

1  -374 

1-371 

•22 

4-545 

2-338 

2-306 

•67 

1-493 

1-387 

1-385 

•21 

4-762 

2-374 

2-341 

•66 

1-515 

1-401 

1-399 

•20 

5  000 

2412 

2-376 

•65 

1-538 

1-416 

1-413 

•19 

5-203 

2-451 

2-413 

•64 

1-563 

1-430 

1-427 

•18 

5-556 

2-492 

2-452 

•63 

1-587 

1-445 

1-441 

•17 

5-8S2 

2-534 

2-492 

•62 

1-613 

1-459 

1  -456 

•16 

6-250 

2-579 

2-434 

•61 

1-640 

1-474 

1-471 

•15 

6-667 

2-626 

2-579 

•60 

1-667 

1-490 

1-486 

•14 

7143 

2-676 

2-626 

•59 

1-695 

1-505 

1-501 

•13 

7-692 

2-730 

2-675 

•58 

1-724 

1-521 

1-516 

•12 

8-333 

2-786 

2-728 

•57 

1-754 

1-537 

1-532 

•11 

9  091 

2-847 

2-784 

•56 

1-786 

1-553 

1-547 

•10 

10  000 

2-912 

2-845 

•55 

1-818 

1-569 

1-563 

THE  MECHANICAL  ACTION  OF  HEAT.  285 


XV.—  NOTE    AS    TO     THE    DYNAMICAL     EQUIVALENT     OF 

TEMPERATURE  IN  LIQUID  WATER,  AND  THE  SPECIFIC 

HEAT    OF    ATMOSPHERIC    AIR    AND    STEAM: 

Being   a   Supplement   to   a   Paper   On   the   Mechanical   Action 

of  Heat.* 

33.  In  my  paper  on  the  Mechanical  Action  of  Heat  (see  p.  *23Jt),  published 
in  the  first  part  of  the  twentieth  volume  of  the  Transactions  of  Hie  Royal 
Society  of  Edinburgh,  some  of  the  numerical  results  depend  upon  the 
dynamical  equivalent  of  a  degree  of  temperature  in  liquid  -water.  The 
value  of  that  quantity  which  I  then  used,  was  calculated  from  the  experi- 
ments of  De  la  Roche  and  Be>ard  on  the  apparent  specific  heat  of 
atmospheric  air  under  constant  pressure,  as  compared  with  liquid  water. 

The  experiments  of  Mr.  Joule  on  the  production  of  heat  by  friction 
give,  for  the  specific  heat  of  liquid  water,  an  equivalent  about  one-ninth 
part  greater  than  that  which  is  determined  from  those  of  De  la  Roche 
and  B6rard.  I  was  formerly  disposed  to  ascribe  this  discrepancy,  in  a 
great  measure,  to  the  smallness  of  the  differences  of  temperature  measured 
by  Mr.  Joule,  and  to  unknown  causes  of  loss  of  power  in  his  apparatus, 
such  as  the  production  of  sound  and  of  electricity;  but,  subsequently 
to  the  publication  of  my  paper,  I  have  seen  the  detailed  account  of 
Mr.  Joule's  last  experiments  in  the  Philosophical  Transactions  for  1850, 
which  has  convinced  me,  that  the  uncertainty  arising  from  the  smallness 
of  the  elevations  of  temperature,  is  removed  by  the  multitude  of  experi- 
ments (being  forty  on  water,  fifty  on  mercury,  and  twenty  on  cast  iron) ; 
that  the  agreement  amongst  the  results  from  substances  so  different, 
shows  that  the  error  by  unknown  losses  of  power  is  insensible,  or  nearly 
so;  and  that  the  necessary  conclusion  is,  that  the  dynamical  value 
assigned  by  Mr.  Joule  to  the  specific  heat  of  liquid  water — viz.,  772  feet 
per  degree  of  Fahrenheit — does  not  err  by  more  than  two,  or,  at  the 
utmost,  three  feet;  and  therefore,  that  the  discrepancy  originates  chiefly 
in  the  experiments  of  De  la  Roche  and  Berard. 

I  therefore  take  the  earliest  opportunity  of  correcting  such  of  my 
calculations  as  require  it,  so  as  to  correspond  with  Mr.  Joule's  equivalent. 

*  Eead  before  the  Eoyal  Society  of  Edinburgh  on  December  2,  1850,  and  published 
in  the  Transactions  of  that  Society,  Vol.  XX.,  Part  II. 


286  THE  MECHANICAL  ACTION   OF  HEAT. 

They  relate  to  the  specific  heat  of  atmospheric  air  as  compared  with 
liquid  water,  and  to  that  of  steam,  and  are  contained  in  the  second  and 
third  sections  of  my  paper,  Articles  14  and  20 ;  equations  28,  34,  and  36. 


Specific  Heat  of  Atmospheric  Air  as  Compared  with  liquid 
Water. — (Section  II.,  Article  14.) 

The  dynamical  values  of  the  specific  heat  of  atmospheric  air  are 
calculated  independently  from  the  velocity  of  sound,  without  reference 
to  the  specific  heat  of  liquid  water;  and  from  the  closeness  of  the  agree- 
ment of  the  experiments  of  MM.  Bravais  and  Martins,  Moll  and  Van 
Beek,  Stampfcr  and  Myrbach,  Wertheim  and  others,  it  is  clear  that 
the  limits  of  error  are  about  ^  for  the  velocity  of  sound,  j^  for  the 
ratio,  and  from  -4T0-  to  -5V  for  the  dynamical  values  of  the  specific  heat  of  air, 
at  constant  volume  and  constant  pressure.  Those  values,  as  given  by 
equation  (27),  are — 

Real  specific  heat, — 

ft  =  23S'GG  feet  =  7274  metres  per  Centigrade  degree. 
=  132-G  feet  per  degree  of  Fahrenheit. 
Apparent  specific  heat  under  constant  pressure, — 

Kp  =  334  fcet=  101-8  metres  per  Centigrade  degree, 
=  185*6  feet  per  degree  of  Fahrenheit. 

The  ratio  of  these  two  quantities  being  taken  as 

^-p=l  +N=  1-4. 
R 

The  dynamical  equivalent  of  the  specific  heat  of  liquid  water,  as 
determined  by  Mr.  Joule,  is 

Kw=  1389-6  feet  =  423-54  metres  per  Centigrade  degree, 
=  772  feet  per  degree  of  Fahrenheit. 

The  specific  heat  of  air,  that  of  liquid  water  being  taken  as  unity,  has 
therefore  the  following  values : — 
Real  specific  heat, — 

It        1326 

IF-—       r-nn      =    0   1<  1  I. 

Kw        772 


THE  MECHANICAL  ACTION   OF   HEAT.  287 

Apparent  specific  heat  under  constant  pressure, — 

KP.      185-6       nnlnt 

—  -  = =  0,o404 

Kw        772        V~Wi- 

This  last  quantity,  according  to  De  la  Eoche  and  Berard,  is 

0-2669 


The  discrepancy  being        .         .         .         .     0'0265 
or  one-ninth  of  the  value,  according  to  Mr.  Joule's  equivalent. 

Specific  Heat  of  Steam. — (Section  III,  Art.  20.) 

The  apparent  specific  heat  of  steam  (equations  34  and  36)  as  a  gas 
under  constant  pressure,  is  equal  to  that  of  liquid  water  x0'305.  Its 
dynamical  value  is,  therefore, 

KP  =  fc  +  ?1     Vr  =  1389-6  x  0-30.3 
C  n  M 

=  422-83  feet  =  129-18  metres  per  Cent,  degree. 
But 

— — —r  =  153*48  feet  =    46-78  metres  per  Cent,  decree. 
Therefore,  the  real  specific  heat  is 

ft  =  269*35  feet  =    82-40  metres  per  Cent,  degree. 
Or,  that  of  liquid  water  being  taken  as  unity, 

A  =  2^  =  0-194 
Kw       1389-6       U1J*- 

The  ratio  of  these  two  values  of  the  specific  heat  of  steam  is 

l+N=l-57. 

Their  dynamical  equivalents  for  Fahrenheit's  scale  are, 

ft  =  149-64  feet,        .         .         .     KP  =  235'46  feet. 

Neither  the  formula?  in  the  fourth  Section,  respecting  the  working  of  the 
steam-engine,  nor  the  tables  at  the  end  of  the  paper,  require  any  alteration ; 
for  the  action  of  steam  at  full  pressure  being  calculated  from  data 
independent  of  its  specific  heat,  is  not  at  all  affected  by  the  discrepancy  I 
have  mentioned ;  and  the  expansive  action  is  not  affected  to  an  extent 
appreciable  in  practice. 


2SS  THE  MECHANICAL  ACTION  OF  HEAT. 


XVI.— 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.* 

34.  The  objects  of  this"  paper  are  twofold :  First,  To  compare  the 
results  of  the  formula?  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,  Second!;/,  To  investigate  and  explain 
the  method  of  determining  the  rate  of  expansion,  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 
required  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 
engines  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 

*  Eead  before  the  Royal  Society  of  Edinburgh  on  April  21,  1851,  and  published  in 
the  Transactions  of  that  Society,  Vol.  XX.,  Part  II. 


SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES.  289 

Pambour.  The  differences  consist  in  the  expressions  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.  (Seep. 
971.)  I  shall  now  express  them  in  a  form  more  convenient  for  practical 
use,  the  notation  being  as  follows  : — 

Let  A  be  the  area  of  the  piston  ; 

/,  the  length  of  stroke ; 

v,  the  number  of  double  strokes  in  unity  of  time ; 

c,  the  fraction  of  the  total  bulk  of  steam  above  the  piston  when  down, 
allowed  for  clearance,  and  for  filling  steam-passages;  so  that  the  total 
bulk  of  steam  at  the  end  of  the  effective  stroke  is 

...  .  .      .     (a.) 

1  —  c 

I',  the  length  of  the  portion  of  the  stroke  performed  when  the  steam  is 
cut  off. 

s,  the  ratio  of  expansion  of  the  steam,  so  that 


(b.) 


Let  W  be  the  weight  of  steam  expended  in  unity  of  time. 

Pj,  the  pressure  at  which  it  enters  the  cylinder. 

A"  ,  the  volume  of  unity  of  weight  of  steam  at  saturation  at  the  pressure 
Pxj  which  may  be  found  from  Table  I.  of  the  Appendix  to  the  original 
paper.     (See  p.  282) 

F,  the  sum  of  all  the  resistances  not  depending  on  the  useful  load, 
reduced  to  a  pressure  per  unit  of  area  of  piston;  whether  arising  from 
imperfect  vacuum  in  the  condenser,  resistance  of  the  air-pump,  feed-pump, 
and  cold-water  pump,  friction,  or  any  other  cause. 

E,  the  resistance  arising  from  the  useful  load,  reduced  to  a  pressure  per 
unit  of  area  of  piston. 

Z,  the  ratio  of  the  total  action  of  steam  Avorking  at  the  expansion  s, 
to  its  action  without  expansion.  Values  of  this  ratio  are  given  in  the 
second  table  of  the  Appendix  to  the  original  paper. 

Then  the  following  are  the  two  fundamental  equations  of  the  motion  of 
the  steam-engine  as  comprehended  in  equation  (50)  of  the  original  paper.  . 

First,  Equality  of  power  and  effect, — 

T 


1 

J' 

-\ 

s 

=  (!■ 

-;h 

+ 

; 

1 

-  c 

f 

V 

s 

I 

=  T^ 

-  c 

J 

290  SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES. 

R  A I  a  =  W  V,  {P2  (Z  -  c  s)  -  F  (1  -  c)  *}.  .     (c.) 

Secondly,  Equality  of  two  expressions  for  the  weight  of  steam  expended 
in  unity  of  time, — 

w  =  v  ,f*n. (d.) 

\j(l  -  c)s 

From  these  two  equations  is   deduced  the   following,  expressing  the 
ratio  of  the  mean  load  on  the  piston  to  the  initial  pressure  of  the  steam : — 

li  +  F  _     Z  —  cs  .  . 

being  equivalent  to  equation  (51). 

In  computing  the  effect  of  Cornish  engines  these  formulae  require  to  be 
modified,  owing  to  the  following  circumstances  : — 

The  terms  depending  on  the  clearance  c  have  been  introduced  into 
equations  (c),  (//),  on  the  supposition  that  the  steam  employed  in  filling 
the  space  above  the  piston  at  the  top  of  its  stroke  is  lost,  being  allowed 
to  escape  into  the  condenser,  without  having  effected  any  work;  so 
that  a  weight  of  steam  Wcs  is  wasted,  and  an  amount  of  power 
WVj  (Pj  —  F)ox  lost,  in  unity  of  time.  But  in  Cornish  engines  this  is 
not  the  case;  for  by  closing  the  equilibrium-valve  at  the  proper  point  of 
the  up  or  out-door  stroke,  nearly  the  whole  quantity  of  steam  necessary 
to  fill  the  clearance  and  valve-boxes  may  be  kept  imprisoned  above  the 
piston,  so  as  to  make  the  loss  of  power  depending  on  it  insensible  in 
practice.  This  portion  of  steam  is  called  a  cushion,  from  its  preventing 
a  shock  at  the  end  of  the  upstroke;  and,  as  Mr.  Pole  in  his  valuable 
work  on  the  Cornish  engine  has  observed,  its  alternate  compression  and 
expansion  compensate  each  other,  and  have  no  effect  on  the  duty  of  the 
engine.  The  proper  moment  of  closing  the  equilibrium-valve  is  fixed  by 
trial,  which  is,  perhaps,  the  best  way;  but  if  it  is  to  be  fixed  by  theory, 
the  following  is  the  proper  formula :  let  I"  be  the  length  of  the  portion 
of  the  upstroke  remaining  to  be  performed  after  the  equilibrium-valve 
has  been  closed :  then — 

1"   _C(8-1)  m 

I  '       I  -  c  '     '        '  *    u'; 

A  slight  deviation  from  this  adjustment  will  produce  little  effect  in 
practice,  if  the  fraction  c  is  small. 

In  forming  the  equations  of  motion,  therefore,  of  the  Cornish  engine, 
we  may,  without  material  error,  in  practice  omit  the  terms  denoting  a 
waste  of  steam  and  loss  of  power  due  to  clearance  and  filling  of  steam- 
passages  ;  and  the  results  are  the  following : — 

Equation  of  effect  and  power  in  unity  of  time : — 


SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES.  291 

Useful  effect  E  =  E  AZ»  =  W  V,  {Px  Z  -  F}.     .     (57.) 
"Weight  of  steam  expended  in  unity  of  time : — 

w  =  4^ (58o 

From  those  two  fundamental  equations  the  following  are  deduced: — 
Ratio  of  mean  load  on  piston  to  maximum  pressure, — 


R  +  F  _  Z 

Pi         =   » 


(59.) 


Duty  of  unity  of  weight  of  steam, — 


W-V^Z-F),       •  •     (6Q.) 

which,   being   multiplied    by  the  number  of  units  of  weight  of  steam 
produced  by  a  given  weight  of  fuel,  gives  the  duty  of  that  weight  of  fuel. 
Weight  of  steam  expended  per  stroke, — 

^  =  ^ (61.) 

In  fact,  it  is  clear  that  if  any  five  quantities  out  of  the  following  seven 
be  given,  the  other  two  may  be  determined  by  means  of  the  equations  : 

R  +  F,  the  mean  load  on  unit  of  area  of  piston. 

Pp  the  maximum  pressure  of  steam  in  the  cylinder. 

s,  the  ratio  of  expansion. 

W,  the  weight  of  steam  produced  in  unity  of  time. 

A,  the  area  of  the  piston. 

/,  the  length  of  stroke. 

n,  the  number  of  strokes  in  unity  of  time. 

The  other  quantities,  E,  Yv  Z,  are  functions  of  those  seven. 

Comparison  of  the  Theory  with  Mr.  Wicksteed's  Experiments. 

36.  In  order  to  test  the  practical  value  of  this  theory,  I  shall  compare 
its  results  with  those  of  the  experiments  which  were  made  by  Mr. 
Wicksteed  on  the  large  Cornish  pumping  engine,  built  under  the  direction 
of  that  eminent  engineer,  by  Messrs.  Harvey  and  West,  for  the  East  London 


292  SINGLE-ACTING    EXPANSIVE   STEAM-ENGINES. 

Water-Works- at  Old  Ford,  and  which  were  published  in  1841.  The 
dimensions  and  structure  of  the  engine,  and  the  details  of  the  experiments, 
are  stated  with  such  minuteness  and  precision,  that  there  is  none  of  that 
uncertainty  respecting  the  circumstances  of  particular  cases,  which  is  the 
most  frecpient  cause  of  failure  in  the  attempt  to  apply  theoretical  principles 

to  practice. 

The  eno-ine  was  worked  under  a  uniform  load  at  five  different  rates  of 
expansion  successively.  The  number  of  strokes,  and  the  consumption 
of  steam  during  each  trial,  having  been  accurately  registered,  Mr.  ^Wieksteed 
gives  a  tabic  showing  the  weight  of  steam  consumed  per  stroke  for  each 
of  the  five  rates  of  expansion.  I  shall  now  compute  the  weight  of  steam 
per  stroke  theoretically,  and  compare  the  results. 

Throughout  these  calculations  I  shall  uniformly  use  the  foot  as  the 
unit  of  length,  the  avoirdupois  pound  as  that  of  weight,  and  the  hour  as 
that  of  time.  Pressures  are  consequently  expressed  in  pounds  per  square 
foot  for  the  purpose  of  calculation;  although  in  the  table  of  experimeuts 
I  have  reduced  them  to  pounds  per  square  inch,  as  being  the  more 
familiar  denomination. 

The  data  respecting  the  dimensions  and  load  of  the  engine,  which  are 
constant  throughout  the  experiments,  are  the  following : — 

Area  of  piston,        .  ....     A  =:  34'854  square  feet. 

Stroke,  .  ...      7  =  10  feet. 

Cubic   space  traversed    by   piston   during    one 

down  stroke,      .  .         .         .        =  A I  —  3  48  '5  4  cubic  feet 

Clearance  and  valve-boxes,       ....  18-00         „ 

Sum,  .  300-54 

Therefore,  c  =  0*05 

E  =  useful  load  of  piston,       .         .  •  =  1597-0  lbs. per  sq.ft. 

F  =  additional  resistance,       .         .  •  =200*0  „ 

R  +  F  =  total  mean  pressure  on  piston, .         .         =  1803-0  „ 

The  mode  of  calculation  is  the  following : — 

Mr.  Wicksteed  states  the  fraction  -  of  the  stroke  performed  at  full 

pressure  in  each  experiment.      From  this  the  ratio   of  expansion  s  is 
computed  by  equation  (b),  giving  in  this  case 

-  =  0-95  -7  +  0-05. 

s  I 


SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES.  293 

The  value  of  Z  corresponding  to  s  is  then  found  by  means  of  the  third 
column  of  table  second;  that  column  being  selected  because  the  initial 
pressures  were  all  below  four  atmospheres.  This  affords  the  means  of 
determining  the  initial  pressure  of  the  steam  by  equation  (59),  viz.  — 

Px  =  |(E  +  F)  =  1863-6  |.  , 

By  using  table  first  according  to  the  directions  prefixed  to  it,  the 
volume  of  one  pound  of  steam  at  the  pressure  Pp  in  cubic  feet,  is 
calculated,  and  thence,  by  equation  (60),  the  weight  of  steam  per  stroke, 
according  to  theory,  which  is  compared  with  the  weight  as  ascertained 
by  experiment. 

Further,  to  illustrate  the  subject,  the  useful  effect,  or  duty  of  a  pound 
of  steam,  is  computed  according  to  the  theory  and  the  experiments 
respectively,  and  the  results  compared. 

The  following  table  (See  j>.  ,!UJt)  exhibits  the  results. 

This  comparison  sufficiently  proves  that  the  results  of  the  theory  are 
practically  correct. 

It  is  remarkable,  that  in  every  instance  except  one  (experiment  E),  the 
experimental  results  show  a  somewhat  less  expenditure  of  steam  per 
stroke,  and  a  greater  duty  per  pound  of  steam,  than  theory  indicates. 
This  is  to  be  ascribed  to  the  fact,  that  although  the  action  of  the  steam 
is  computed  theoretically,  on  the  assumption  that  during  the  expansion 
it  is  cut  off  from  external  sources  of  heat,  yet  it  is  not  exactly  so  in 
practice;  for  the  cylinder  is  surrounded  with  a  jacket  or  casing  communi- 
cating with  the  boiler,  in  which  the  temperature  is  much  higher  than 
the  highest  temperature  in  the  cylinder,  the  pressure  in  the  boiler  being 
more  than  double  the  maximum  pressure  of  the  steam  when  working, 
as  columns  (2)  and  (5)  show.  There  is,  therefore,  a  portion  of  steam,  of 
whose  amount  no  computation  can  be  made,  which  circulates  between  the 
boiler  and  the  jacket,  serving  to  convey  heat  to  the  cylinder,  and  thus 
augment  by  a  small  quantity  the  action  of  the  steam  expended ;  and 
hence  the  formulae  almost  always  err  on  the  safe  side. 

Supposing  one  pound  of  the  best  Welsh  coals  to  be  capable  (as  found 
by  Mr.  Wicksteed)  of  evaporating  9'493  lbs.  of  water  at  the  pressure 
in  the  boiler  during  the  experiment  F,  then  the  duty  of  a  Cornish  bushel, 
or  94  lbs.  of  such  coals,  in  the  circumstances  of  that  experiment  would  be — 

By  theory, 88,288,000  ft.  lbs. 

By  experiment, 90,801,000      „ 

Difference,      .         .     +   2,513,000      „ 


294 


SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES. 


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SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES.  295 


Economy  of  Single-Acting  Expansive  Engines. 

37.  By  increasing  the  ratio  of  expansion  in  a  Cornish  engine,  the 
quantity  of  steam  required  to  perform  a  given  duty  is  diminished ;  and 
the  cost  of  fuel,  and  of  the  boilers,  is  lowered.  But  at  the  same  time, 
as  the  cylinders  and  every  part  of  the  engine  must  be  made  larger,  to 
admit  of  a  greater  expansion,  the  cost  of  the  engine  is  increased.  It  thus 
becomes  a  problem  of  maxima  and  minima  to  determine  what  ratio  of 
expansion  ought  to  be  adopted  under  given  circumstances,  in  order  that 
the  sum  of  the  annual  cost  of  fuel,  and  the  interest  of  the  capital 
employed  in  construction,  may  be  the  least  possible,  as  compared  with  the 
work  done. 

That  this  problem  may  admit  of  a  definite  solution,  the  following  five 
quantities  must  be  given  : — 

Pj,  the  initial  pressure  in  the  cylinder. 

F,  the  resistance  not  depending  on  the  useful  load. 

I  n,  the  amount  of  the  length  of  the  effective  strokes  made  in  unity  of 
time. 

k,  the  annual  cost  of  producing  unity  of  weight  of  steam  in  unity  of 
time,  which  consists  of  two  parts — the  price  of  fuel,  and  the  interest  of 
the  cost  of  the  boilers. 

k,  the  interest  of  the  cost  of  the  engine,  per  unit  of  area  of  piston. 

Hence  the  annual  expenditure  to  be  taken  into  consideration,  reduced 
to  unity  of  weight  of  steam,  is  , 

And  the  useful  effect  of  unity  of  weight  of  steam  being 

V^Z-Fs), 

the  problem  is  to  determine  the  ratio  of  expansion  s,  so  that 

V^Z-Fs) 
V  s 
In 
shall  be  a  maximum. 

Dividing  the  numerator  of  this  fraction  by  Vt  P1?  and  the  denominator 

by  -r-1,  both  of  which  are  constants  in  this  problem,  we  find  that  it  will 


296  SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES. 

be  solved  by  making  the  ratio 

7  F 

Z   —   =r-S 

Py  ....      (G2.) 

a  maximum. 

The  algebraical  solution  would  be  extremely  complicated  and  tedious. 
The  graphic  solution,  on  the  other  hand,  is  very  simple  and  rapid,  and 
sufficiently  accurate  for  all  practical  purposes ;  and  1  have  therefore 
adopted  it. 

In  the  diagram  (See  Piatt  II.,  Fig.  I),  the  axis  of  abscissa;  —  XO 
+  X,  is  graduated  from  0  towards  +  X  into  divisions  representing 
ratios  of  expansion,  or  values  of  &  The  divisions  of  the  axis  of  ordinates 
OY  represent  values  of  Z.  The  curve  marked  "locus  of  Z,"  is  laid 
down  from  the  third  column  of  Table  II.  of  the  Appendix  to  the  original 
paper,  being  applicable  to  initial  pressures  not  exceeding  four  atmospheres. 

Through  the  origin  0  draw  a  straight  line  BOA,  at  such  an  inclination 

F 

to    —  X  0  +  X   that   its   ordinates   are  represented  by       s.     Then  the 

-*■  i 

ordinates  measured  from  this  inclined  line  to  the  locus  of  Z  represent  the 

F 
value  of  the  numerator  Z  —       s,  of  the  ratio  (62),  corresponding  to  the 

-*-  i 

various  values  of  .*. 

Take  a  point  at  C  on  the  line  B  0  A,  whose  abscissa,  measured  along 

0  —  X,  represents   —       7.     Then  the  ordinates,  measured  from  BOA, 
k  V  j 

of  any  straight  line  drawn  through  C,  vary  proportionally  to  the  denomi- 
nator — —  +  s  of  the  ratio  (62). 

/,'  V , 

Through  the  point  C,  therefore,  draw  a  straight  line  C  T,  touching  the 
locus  of  Z  :  then  the  ratio  (62)  is  a  maximum  at  the  point  of  contact  T, 
and  the  abscissa  at  that  point  represents  the  ratio  of  expansion  required. 

Example. 

i 
38.  To  exemplify  this  method,  let  us  take  the  following  data: — 

Greatest  pressure  in  the  cylinder  Px  =  20  lbs.  per  square  inch,  =  2880 
lbs.  per  square  foot. 

The  corresponding  value  of  Vx  is  20-248  cubic  feet  per  pound  of  steam. 

To  obtain  this  initial  pressure  in  the  cylinder,  it  will  be  necessary  to 
have  a  pressure  of  about  50  lbs.  per  square  inch  in  the  boiler. 

F,  resistance  not  depending  on  the  useful  load  =  2  lbs.  per  square  inch, 
=  288  lbs.  per  square  foot,  =  TV  Pr 


SINGLE-ACTING   EXPANSIVE   STEAM-ENGINES.  21)7 

In,  amount  of  clown  strokes,  =  4800  feet  per  hour;  being  the  average 
speed  found  to  answer  best  in  practice. 

To  estimate  h,  the  annual  cost  of  producing  one  pound  of  steam  per 
hour,  I  shall  suppose  that  the  engine  works  6000  hours  per  annum;  that 
the  cost  of  fuel  is  one  penny  per  100  lbs.  of  steam;'"  that  the  cost  of 
boiler  for  each  pound  of  steam  per  hour  is  0*016  ton,  at  £27,  =  £0*432; 
and  that  the  interest  of  capital  is  five  per  cent,  per  annum.  Hence  h  is 
thus  made  up — 

Fuel  for  6000  lbs.  of  steam  at  0*01d.,     .          .          .       £0*2500 
Interest  on  £0*432,  at  5  per  cent,,  .         .         .  0*0216 


h  =  £0*2716 

Estimating  the  cost  of  the  engine  at  £250  per  square  foot  of  piston, 
we  find  /;  =  5  per  cent,  per  annum  on  £250  =  £12*5. 

,    h        „  ,  ^-.  ~     h  1  n        _  ,  .  . 
and       =0*0217;  =-=■  =  5*144. 
I;  k  V  j 

The  line  E  O  A,  then,  is  to  be  drawn  so  that  its  ordinates  are 

F  1 

p;5  =  To5- 

The  point  C  is  taken  on  this  line,  at  £=-   =  5*144  divisions  of  the 

axis  of  abscissa?  to  the  left  of  O  Y. 

The  tangent  C  T  being  drawn,  is  found  to  touch  the  locus  of  Z  at  2*800 
divisions  to  the  right  of  O  Y. 

Then  s  =  2*800  is  the  ratio  of  expansion  sought,  corresponding  to  the 
greatest  economy. 

If  we  make  c=r0'05,  as  in  Mr.  Wicksteed's  engine,  then  the  fraction  of 
the  stroke  to  be  performed  at  full  pressure  is 

-  =  0*323, 

I 

being  nearly  the  same  as  in  experiment  F. 

The  mean  resistance  of  the  useful  load  per  square  foot  of  piston  is 

E  =  —  R  -  F  -=1713*6  lbs. 
s      1 

*  This  estimate  is  maile  on  the  supposition  that  coals  capable  of  producing  nine  times 
their  weight  of  steam  are  worth  about  16s.  9d.  per  ton. 


298  SINGLE-ACTING  EXPANSIVE  STEAM-ENGINES. 

The  duty  of  one  square  foot  of  piston  per  liour — 

B,ln,  =  8,225,300  foot-lbs. 

And  one  horse-power  being  1,980,000  foot-lbs.  per  hour,  the  real  horse- 
power of  the  engine  is 

4-154  per  square  foot  of  piston. 

The  duty  of  one  pound  of  steam  is 

RV1s  =  97,154  foot-lbs. 

To  give  an  example  of  a  special  case,  let  the  duty  to  be  performed  be 
198,000,000  foot-pounds  per  hour,  being  equal  to  100  real  horse-power, 
for  G,000  hours  per  annum.  This  being  called  E,  we  find  from  the  above 
data  that  the  area  of  piston  required  is 

i; 

A  =  _  .     =  21-072  square  feet, 
li  /  n 

The  consumption  of  steam  per  hour  is 

W  =,,!':      =  2038  lbs., 
Jtt  V  x  .S' 

which  requires  2038  x  0-01 G  =  32-608  tons  of  boilers. 
The  expenditure  of  steam  per  annum  is 

2038  x  G000  =  12,228,000  lbs. 

Hence  we  have  the  following  estimate  : — 

Cost  of  engine,   24*072    square  feet  of  piston  at 

£250, £6018-000 

Cost  of  boilers,  32-G08  tons  at  £27,     .          .         .  880*416 


Total  capital  expended,         .         .         .  £6898-416 

Interest  at  five  per  cent,  per  annum,     .         .         .  344-921 

Cost  of  fuel  per  annum,  12,228,000  lbs.  of  steam 

at  0-01d., 509-500 


Annual  cost  for  interest  and  fuel,  .         .       £854*421 
I  wish  it  to  be  understood  that  the  rates  I  have  adopted  in  the  fore- 


SINGLE-ACTING  EXPANSIVE  STEAM-ENGINES.  299 

going  calculations,  for  interest,  cost  of  fuel,  and  cost  of  construction,  are 
not  intended  as  estimates  of  their  average  amount,  nor  of  their  amount  in 
any  particular  case ;  but  are  merely  assumed  in  order  to  illustrate,  by  a 
numerical  example,  the  rules  laid  down  in  the  preceding  article.  It  is, 
of  course,  the  business  of  the  engineer  to  ascertain  those  data  with 
reference  to  the  special  situation  and  circumstances  of  the  proposed  work; 
and  having  clone  so,  the  method  explained  in  this  paper  will  enable  him 
to  determine  the  dimensions  and  ratio  of  expansion  which  ought  to  be 
adopted  for  the  engine,  in  order  that  it  may  effect  its  duty  with  the 
greatest  possible  economy. 


300  ECONOMY   OF   HEAT   IN    EXPANSIVE   MACHINES. 


XVII.— ON  THE  ECONOMY  OF  HEAT  IN  EXPANSIVE 
MACHINES  :* 

Forming  the  Fifth  Section  of  a  Paper  On  the 
Mechanical  Action  of  Heat. 

39.  A  MACHINE  working  by  expansive  power  consists  essentially  of  a 
portion  of  some  substance  to  which  heat  is  communicated,  so  as  to  expand 
it,  at  a  higher  temperature,  being  abstracted  from  it,  so  as  to  condense  it 
to  its  original  volume,  at  a  lower  temperature.  The  quantity  of  heat 
given  out  by  the  substance  is  less  than  the  quantity  received;  the 
difference  disappearing  as  heat,  to  appear  in  the  form  of  expansive  power. 

The  heat  originally  received  by  the  working  body  may  act  in  two  ways: 
to  raise  its  temperature,  and  t<<  expand  it.  The  heal  given  out  may  also 
act  in  two  ways:  to  lower  the  temperature,  and  to  contract  the  body. 
Now,  as  the  conversion  of  heat  into  expansive  power  arises  from  changes 
of  volume  only,  and  not  from  changes  of  temperature,  it  is  obvious,  that 
the  proportion  of  the  heat  received  which  is  converted  into  expansive 
power  will  lie  the  greatest  possible,  when  the  reception  of  heat,  and  its 
emission,  each  take  place  at  a  constant  temperature. 

40.  Carnot  was  the  first  to  assert  the  law,  that  the  ratio  of  the  maximum 
mechanical  effect  to  the  vial  heat  expended  in  an  expansive  machine,  is  a 
function  solely  of  tin  fir,,  temperatures  at  which  the  loot  is  respectively  received 
ami  emitted,  and  is  independent  of  (he  nature  of  the  working  substance,  But 
his  investigations  not  being  based  on  the  principle  of  the  dynamical 
convertibility  of  heat,  involve  the  fallacy  that  power  can  be  produced  out 
of  nothing. 

•41.  The  merit  of  combining  Carndt's  Law,  as  it  is  termed,  with  that 
of  the  convertibility  of  heat  and  power,  belongs  to  Mr.  Clausius  and 
Professor  William  Thomson;  and  in  the  shape  into  which  they  have 
brought  it,  it  may  be  stated  thus  : — 

The  maximum  proportion  of  heat  converted  in  la  expansive  power  by  any 
machine,  is  a  function  solely  of  the  temperatures  at  which  heat  is  received  and 
emitted  by  the  working  substance;  which  function  for  each  pair  of  temperatures 
is  the  same  for  all  substances  in  nature. 

*  Bead  before  the  Eoyal  Society  of  Edinburgh  on  April  21,  1851,  and  published  in 
the  Transactions  of  that  Society,  Vol.  XX.,  Part  II. 


in  which  the  negative  sign  denotes  aDsorpuou,  ana  cue  jjustvttg  emission. 


Bn.l. 


ECONOMY   OF   HEAT   IN   EXPANSIVE   MACHINES.  301 

This  law  is  laid  down  by  Mr.  Clausius.  as  it  originally  had  been  by 
Carnot,  as  an  independent  axiom;  and  I  had  at  first  doubts  as  to  the 
soundness  of  the  reasoning  by  which  he  maintained  it.  Having  stated 
those  doubts  to  Professor  Thomson,  I  am  indebted  to  him  for  having 
induced  me  to  investigate  the  subject  thoroughly;  for  although  I  have 
not  yet  seen  his  paper,  nor  become  acquainted  with  the  method  by  which 
he  proves  Carnot's  law,  I  have  received  from  him  a  statement  of  some 
of  his  more  important  results. 

42.  I  have  now  come  to  the  conclusions, — First,  That  Carndt's  Law 
is  not  an  independent  principle  in  the  theory  of  heat;  hut  is  deducible,  as  a 
consequence,  from  the  equations  of  the  mutual  conversion  of  heat  and  expansive 
power,  as  given  in  the  first  section  of  this  paper. 

Secondly,  That  the  function  of  the  temperatures  of  reception  and  emission, 
which  expresses  the  maximum  ratio  of  the  heat  converted  into  power  to  the  total 
heat  received  by  the  working  body,  is  the  ratio  of  the  difference  of  those  temper- 
atures to  the  absolute  temperature  of  reception  diminished  by  the  constant,  which 
I  have  called  tc  =  Cn/j.b  ;  and  which  must,  as  I  have  shown  in  the  intro- 
duction, be  the  same  for  all  substances,  in  order  that  molecular  equilibrium 
may  be  possible. 

43.  Let  abscissa^  parallel  to  0  X  in  the  diagram,  Plate  II,  Fig.  2, 
denote  the  volumes  successively  assumed  by  the  working  body,  and 
ordinates,  parallel  to  0  Y,  the  corresponding  pressures.  Let  rx  be  the 
constant  absolute  temperature  at  which  the  reception  of  heat  by  the  body 
takes  place :  t0,  the  constant  absolute  temperature  at  which  the  emission 
of  heat  takes  place.  Let  A  B  be  a  curve  such  that  its  ordinates  denote 
the  pressures,  at  the  temperature  of  reception  tv  corresponding  to  the 
volumes  denoted  by  abscissa\  Let  D  C  be  a  similar  curve  for  the  temper- 
ature of  emission  r0.  Let  A  D  and  B  C  be  two  curves,  expressing  by 
their  co-ordinates  how  the  pressure  and  volume  must  vary,  in  order 
that  the  body  may  change  its  temperature,  without  receiving  or  emitting 
heat ;  the  former  corresponding  to  the  most  condensed  and  the  latter  to 
the  most  expanded  state  of  the  body,  during  the  working  of  the  machine. 

The  quantity  of  heat  received  or  emitted  during  an  operation  on  the 
body  involving  indefinitely  small  variations  of  volume  and  temperature, 
is  exj>ressed  by  adding  to  equation  (G)  of  section  fourth  the  heat  due  to 
change  of  temperature  only,  in  virtue  of  the  real  specific  heat.  "We  thus 
obtain  the  differential  equation 

—  fe  £  r, 
in  which  the  negative  sign  denotes  absorption,  and  the  positive  emission. 


302  ECONOMY    OF   HEAT   IN   EXPANSIVE   MACHINES. 

If  wo  now  put  for    }  v,     ,    ,  their  values  according  to  equation  (11), 
d  V     (It 

we  find 

8Q'_SQ=  _(T_«)^?.gy 

_  jfc  +      1      («  _  «*)  +  (T  _  .)  ''  j™  ,,  V  \»T.    (63.) 
(.  C«M\t        rv  d  tJ  (It  ) 

The  first  term  represents  the  variation  of  heat  due  to  variation  of 
volume  only;  the  second,  that  due  to  variation  of  temperature.  Let 
us  now  apply  this  equation  to  the  cycle  of  operations  undergone  by  the 
working  body  in  an  expansive  machine,  as  denoted  by  the  diagram. 

First  operation. — The  body,  being  at  first  at  the  volume  VA  and  pressure 
PA,  is  made  to  expand,  by  the  communication  of  heat  at  the  constant 
temperature  tv  until  it  reaches  the  volume  VB  and  pressure  PB,  A  B  being 
the  locus  of  the  pressures. 

Here  S  r  =  0 ;  therefore,  the  total  heat  received  is 


H1  =  -Q'1  =  (r1-K)Jv 


v 

1  dV 


d 


r 


=  0"!-  *){<P(V*>tJ-  *(V*r,)}.J 


(a.) 


Set:  lion. — The  body,  being  prevented  from  receiving  or  emitting 

heat,  exj  lands  until  it  falls  to  the  temperature  r0,  the  locus  of  the  pressures 
being  the  curve  B  C.  During  this  operation  the  following  condition 
must  be  fulfilled, — 

0  =  8  q  -  8  <  >. 

which,  attending  to  the  fact  that  V  is  now  a  function  of  r,  and  trans- 
forming the  integrals  as  before,  gives  the  equation 

0  =  ,!  +  cTm  \r  ~  ?)  +  <r  -  K)  KTr  +  ,77  •  2  v)  *  <*■  T> 

This  equation  shows  that 

0  (V.  rx)  -  96  (Vc,  r0)  =  ^  (rr  r0).         .  .     (b.) 

Third  operation. — The  body,  by  the  abstraction  of  heat,  is  made  to 
contract,  at  the  constant  temperature  r0,  to  the  volume  VD  and  pressure  PDr 
which  are  such  as  to  satisfy  conditions  depending  on  the  fourth  operation. 
C  D  is  the  locus  of  the  pressures.     The  heat  emitted  is  evidently 

H0  =  Q'0  =  (r0  -  k)  {cp  (Vc,  r0)  -  d>  (Vw  r0)}.  .      (&) 


ECONOMY  OF  HEAT  IN   EXPANSIVE  MACHINES.  303 

Fourth  operation. — The  body,  being  prevented  from  receiving  or  emitting 
heat,  is  compressed  until  it  recovers  its  original  temperature  tv  volume  VA, , 
and  pressure  PA;   the   locus  of  the  pressures  being  DA.     During   this 
operation,  the  same  conditions  must  be  fulfilled  as  in  the  second  operation ; 
therefore, 

<P  (Ya,  O  -  </>  (VD,  r0)  =  xP  (tv  r0).         .  .      (d.) 

ib  being  the  same  function  as  in  equation  (b). 

By  comparing  equations  (b)  and  (d),  we  obtain  the  relation  which  must 
subsist  between  the  four  volumes  to  which  the  body  is  successively  brought, 
in  order  that  the  maximum  effect  may  be  obtained  from  the  heat.  It  is 
expressed  by  the  equation 

tiVr,^)   -   c/>(Yx,T1)  =  ci>(Yc,T0)-  <P(\\,T0).       •       (64.) 

From  this  and  equations  (a)  and  (c),  it  appears  that 


Ho  _  To   —   K 
Hl  Tl  -   * 


(65.) 


That  is  to  say  :  ivhen  no  heat  is  employed  in  producing  variations  of  tempera- 
ture, the  ratio  of  the  heat  received  to  the  heat  emitted  by  the  working  body  of  an 
ivi  machine,  is  equal  to  that  of  the  absolute  temperatures  of  reception  and 
emission,  each  diminished  by  the  constant  k,  which  is  the  same  for  all  substances. 

Hence,  let 

n  =  -  q\  -  Q'0  =  hx  -  h0 

denote  the  maximum  amount  of  power  which  can  be  obtained  out  of 
the  total  heat  H^  in  an  expansive  machine  working  between  the  tempera- 
tures tx  and  r0.     Then 

£  =  ^=A     ....     (66.) 
Hi        Ti  ~  K 

being  the  law  which  has  been  enunciated  in  Article  42,  and  which  is 
deduced  entirely  from  the  principles  already  laid  down  in  the  introduction 
and  first  section  of  this  paper. 

The  value  of  the  constant  K  is  unknown;  and  the  nearest  approximation 
to  accuracy  which  we  can  at  present  make,  is  to  neglect  it  in  calculation, 
as  being  very  small  as  compared  with  r. 

44.  This  approximation  having  been  adopted,  I  believe  it  will  be 
found  that  the  formula  (66),  although  very  different  in  appearance  from 
that  arrived  at  by  Professor  Thomson,  gives  nearly  the  same  numerical 
results.  For  example :  let  the  machine  work  between  the  temperatures 
140°  and  30°  Centigrade  :  then 


304  ECONOMY   OF   HEAT   IX   EXPANSIVE   MACHINES. 

r,  =  414°-G,  r0  =  304°'6, 
and 

"   =  0-2653. 

Trofessor  Thomson  has  informed  me,  that  for  the  same  temperatures 
he  finds  this  ratio  to  be  0*271 3.* 

45.  To  make  a  steam-engine  work  according  to  the  conditions  of 
maximum  effect  here  laid  down,  the  steam  must  enter  the  cylinder  frorn^ 
the  boiler  without  diminishing  in  pressure,  and  must  be  worked  expan- 
sively down  to  the  pressure  and  temperature  of  condensation.  It  must 
then  be  so  far  liquefied  by  conduction  alone,  that  on  the  liquefaction 
being  completed  by  compression,  it  may  be  restored  to  the  temperature 
of  the  boiler  by  means  of  that  compression  alone.  These  conditions 
are  unattainable  in  steam-engines  as  at  present  constructed,  and  different 
from  those  which  form  the  basis  of  the  formulas  and  tables  in  the 
fourth  section  of  this  paper;  hence  it  is  found,  both  by  experiment 
and  by  calculation  from  those  formulas,  that  the  proportion  of  the  total 
heat  converted  into  power  in  any  possible  steam-engine  is  less  than  that 
indicated  by  equation  (GG). 

The  annexed  table  illustrates  this. 

The  heat  transformed  into  power,  as  given  in  the  fifth  column,  has 
been  reduced  to  Centigrade  degrees  in  liquid  water,  by  dividing  the  duty 
of  a  pound  of  steam  by  Mr.  Joule's  equivalent,  1389-G  feet  per  Centigrade 
degree.  Hence,  the  first  two  numbers  in  that  column  are  less  than  those 
given  in  Art.  32,  which  were  computed  from  too  small  an  equivalent. 

The  first  two  cases  fulfil  the  conditions  required  by  Canmt's  law  in 
every  respect  except  one  : — viz.,  that  the  steam  remaining  at  the  end 
of  the  stroke,  instead  of  being  partially  liquefied  by  refrigeration,  and 
then  reduced  to  water  at  the  temperature  of  the  boiler  by  compression, 
is  supposed  to  be  entirely  liquefied  by  refrigeration.  This  occasions  the 
loss  of  the  heat  necessary  to  raise  the  water  from  the  temperature  of 
the  condenser  to  that  of  the  boiler;  but  at  the  same  time,  there  is  a 
gain  of  the  power  which  would  be  required  to  liquefy  part  of  the 
steam  by  compression,  and  those  two  quantities  partially  compensate 
for  each  other's  effects  on  the  ratio  of  the  power  to  the  heat  expended, 
so  that  although  it  is  below  the  maximum,  the  difference  is  small. 

*  From  information  which  I  have  received  from  Professor  Thomson  subsequently  to 
the  completion  of  this  paper,  it  appears  that  his  formula  becomes  identical  with  the 

approximate  formula  here  proposed,  on  making  the  function  called  bj'  him  fc  =  — ,  J 

being  Joule's  equivalent. 

Mr.  Joule  also,  some  time  since,  arrived  at  this  approximate  formula  in  the 
particular  case  of  a  perfect  gas. 


ECONOMY   OF   HEAT   IN   EXPANSIVE   MACHINES. 


305 


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306  ECONOMY   OF   HEAT   IN   EXPANSIVE  MACHINES. 

In  the  third  and  fourth  examples,  founded  on  the  calculated  and 
observed  duty  of  Mr.  Wicksteed's  engine  during  experiment  F,  the  actual 
ratio  is  less  than  half  the  maximum.  This  waste  of  heat  is  to  he 
ascribed  to  the  following  causes  : — 

First,  The  mode  of  liquefaction,  which  has  already  been  referred  to. 

Secondly,  The  initial  pressure  in  the  cylinder  is  but  18  "9 3  lbs.  on  the 
square  inch,  while  that  in  the  boiler  is  45-7;  so  that  although  the  steam 
is  produced  at  135°-2  Centigrade,  it  only  begins  to  work  at  107°'2G. 
This  great  fall  of  pressure  is  accounted  for  by  the  fact,  that  the  steam 
for  each  stroke,  which  is  produced  in  the  boiler  in  about  seven  or  eight 
seconds,  escapes  suddenly  into  the  cylinder  in  a  fraction  of  a  second. 

Thirdly,  The  expansive  working  of  the  steam,  instead  of  being  continued 
down  to  30°  Centigrade,  the  temperature  of  the  condenser,  stops  at  a 
much  higher  temperature,  74°-GG.  This  is  the  most  important  cause 
of  loss  of  power. 

If  we  now  take  for  rx  and  r0  the  absolute  temperatures  at  the  beginning 
and  end  of  the  expansive  working,  and  calculate  the  maximum  duty  of 
one  pound  of  steam  by  Carn&t's  Law  between  those  temperatures,  we  find, — 

tx  =  107°-2G  +  274°-G  =  381°-8G 
Tq  =     74°-GG  +  274°-G  =  349  '26 

~  =  0-08f) !  2 

K1=5U0'5i.:Il= 48°'22 

To  this  has  to  be  added  the  duty,  at  full  pressure, 
of  steam  at  r0,  diminished  by  one-third  for  back- 
pressure and  friction,  and  by  OHe-fifteenth  for 
liquefaction  in  the  cylinder,  =    .  .  .  .      230,14 

The  whole  amounting  to      .         .         .     71°#3G 

Which  agrees  very  nearly  with  73°'23,  the  observed  duty,  and  almost 
exactly  with  71°*2,  the  duty  as  calculated  by  the  formulae  and  tables  of 
section  fourth. 

These  examples  show  clearly  the  nature  and  causes  of  the  waste  of  heat 
in  the  steam-engine. 


ABSOLUTE   ZERO   OF   THE   PERFECT   GAS   THERMOMETER.  307 


XVIIL— ON  THE  ABSOLUTE  ZERO  OF  THE  PERFECT  GAS 
THERMOMETER :  * 

Being  a  Note  to  a  Paper  On  the  Mechanical  Action  of  Heat. 

{Seep.  23 If.) 

Temperature  being  measured  by  the  pressure  of  a  perfect  gas  at  constant 
density,  the  absolute  zero  of  temperature  is  that  point  on  the  thermometric 
scale  at  which,  if  it  were  possible  to  maintain  a  perfect  gas  at  so  low  a 
temperature,  the  pressure  would  be  null. 

The  position  of  this  point  is  of  great  importance,  both  theoretically  and 
practically;  for  by  reckoning  temperatures  from  it,  the  laws  of  phenomena 
depending  on  heat  are  reduced  to  a  more  simple  form  than  they  are  when 
any  other  zero  is  adopted. 

As  we  cannot  obtain  any  substance  in  the  perfectly  gaseous  condition 
(that  is  to  say,  entirely  devoid  of  cohesion),  Ave  cannot  determine  the 
position  of  the  absolute  thermometric  zero  by  direct  experiment,  which 
furnishes  us  with  approximate  positions  only.  Those  approximate  posi- 
tions are  always  too  high ;  because  the  effect  of  cohesion  is  to  make  the 
pressure  of  a  gas  diminish  more  rapidly  with  a  diminution  of  temperature, 
than  if  it  were  devoid  of  cohesion. 

As  a  gas  is  rarefied,  the  cohesion  of  its  particles  diminishes,  not  only  in 
absolute  amount,  but  also  in  the  proportion  which  it  bears  to  the  pressure 
due  to  heat.  The  gas,  therefore,  approaches  more  and  more  nearly  to  the 
state  of  a  perfect  gas  as  its  density  diminishes ;  and  from  a  series  of 
experiments  on  the  rate  of  increase  of  its  elasticity  with  temperature,  at 
progressively  diminishing  densities,  may  be  calculated  the  positions  of  a 
series  of  points  on  the  thermometric  scale,  approaching  more  and  more 
nearly  to  the  true  absolute  zero. 

By  observing  the  law  which  those  successive  approximations  follow,  the 
true  position  of  the  absolute  zero  can  be  determined. 

Having  performed  this  operation  by  means  of  a  graphic  process,  soon 
after  the  publication  of  the  experiments  of  M.  Regnault  on  the  elasticity 
and  expansion  of  gases,  I  stated  the  result  in  a  paper  on  the  Elasticity  of 
Vapours  (Edinburgh  New  Philosophical  Journal,  July,  1849),  {See  p.  1).  and 

*  Read  before  the  Royal  Society  of  Edinburgh  on  January  4,  1333,  and  pnblis!  ed 
in  the  Transactions  of  that  Society,  Vol.  XX.  Part  IV. 


80S        ABSOLUTE   ZERO   OF   THE   TERFECT   GAS   THERMOMETER. 

also  in  a  paper  on  the  Mechanical  Action  of  Heat  (Trans.  Royal  Sue.  Edin., 
Vol.  XX.,  Part  I.),  (See  p.  ..'■'.'/' — viz.,  that  the  absolute  zero  is — 

274°-G  Centigrade,  .)  below  ^  tc        rature  of  melting  ice ; 

or  494°-28  Fahrenheit,         J 

or  4G20-28  below  the  ordinary  zero  of  Fahrenheit's  scale. 
To  enable  others  to  judge  of  the  accuracy  of  this  result,  1  shall  now 
explain  the  method  by  which  it  was  obtained. 

Let  E  denote  the  mean  rate  of  increase,  per  degree,  between  the 
freezing  and  boiling  points,  of  the  pressure  of  a  gas  whose  volume  is 

maintained  constant.     Then   the  reciprocal  of  this  coefficient,       ,   is   an 

approximation  to  the  number  of  degrees  below  the  freezing  point,  at 
which  the  absolute  zero  is  situated. 

The  experimental  data  in  the  following  table  were  copied  from  the 
memoirs  of  M.  Regnault  on  the  Expansion  of  Gases.  The  numbers  in 
the  first  column  designate  the  series  of  experiments.  The  second  column 
contains  the  pressures  of  the  gases  at  the  freezing  point.  The  third 
column  contains  the  mean  coefficients  of  increase  of  pressure  per  Centi- 
grade degree,  between  0°  sod  100°  Centigrade.  The  fourth  column 
contains  the  reciprocals  of  those  coefficients,  with  the  negative  Bign,  being 
approximate  positions  of  the  absolute  zero,  in  Centigrade  degrees,  below 
the  temperature  of  melting  ice.  The  gases  employed  were  atmospheric  air 
and  carbonic  acid. 

The  approximate  positions  of  the  absolute  zero  contained  in  this  table 
wen*  laid  down  on  ;i  diagram,  in  which  they  were  marked  by  crosses. 
The  longitudinal  divisions  representing  Centigrade  degrees  divided  into 
tenths;  the  transverse  divisions,  atmospheres  of  pressure  a1  (l  Centigrade, 
also  divided  into  tenths.  The  positions  of  the  crosses  indicating  at 
once  the  pressures  in  the  second  column  of  the  table,  and  the  approximate 
zeros  in  the  fourth  ;  and  the  numbers  affixed  to  them  corresponding  with 
those  in  the  first  column. 

As  the  effect  of  cohesion  is  greater  and  more  easily  eliminated  in  car- 
bonic acid  gas  than  in  atmospheric  air,  the  determination  of  the  true 
absolute  zero  was  made  from  the  experiments  on  the  former  gas.  The 
approximate  positions  of  the  absolute  zero  for  carbonic  acid  lie  nearly  in  a 
straight  line.  A  straight  line  having  been  drawn  so  that  it  should,  as 
nearly  as  possible,  traverse  them,  was  found  to  intersect  the  line  corre- 
sponding to  the  zero  of  pressure,  that  is,  to  the  state  of  perfect  gas,  at  a 
point  on  the  scale  of  temperatures  274-6  Centigrade  degrees  below  the 
temperature  of  melting  ice,  which  point  was  accordingly  taken  as  the  true 
absolute  zero  of  the  perfect  gas  thermometer. 


ABSOLUTE   ZERO   OF   THE   PERFECT   GAS   THERMOMETER. 


309 


No. 

Pressure  at  0°  Centigrade 
in  Atmospheres. 

Coefficient  of  Increase  of 

Elasticity  with  Temperature 

=  E. 

Approximate  Positions 

of  the  Absolute  Zero 

in  Centigrade  Degrees 

1 

~  E" 

CARBONIC  ACID. 

1. 

0-99S0 

0-0036S5G 

-271-33 

o 

1-1857 

0  0036943 

-270-63 

3. 

22931 

0-0037523 

-266-50 

4, 

4-7225 

0-003S59S 

-259 -OS      • 

ATMOSPHERIC  AIR. 

1. 

0-1444 

00036482 

-274-11 

o 

0-2294 

0-003G513 

-273-88 

3. 

0-3501 

0-0036542 

-273-66 

4. 

0-4930 

0  00365S7 

-27332 

5. 

0-4937 

0  0036572 

-273-43 

6. 

1-0000 

0-003G650 

-272-85 

7. 

2-2084 

0-003G700 

-272-03 

8. 

2-2270 

0-0036800 

-27174 

9. 

2-8213 

0  0036S94 

-27105 

10. 

4-S100 

0  0037091 

-269  61 

So  far  as  their  irregularity  permits,  the  experiments  on  atmospheric  air 
confirm  this  result,  for  the  approximate  positions  of  the  absolute  zero 
deduced  from  them  evidently  tend  towards  the  very  same  point  on  the 
diagram  with  those  deduced  from  the  experiments  on  carbonic  acid. 

The  values  of  the  coefficient  of  dilatation  and  of  increase  of  pressure  of 
a  perfect  gas,  per  degree,  in  fractions  of  its  volume  and  pressure,  at  the 
temperature  of  melting  ice,  are,  accordingly,— 


For  the  Centigrade  Scale    7——  =  0-003641  CO 
°  274-0 


For  Fahrenheit's  Scale 


1 

494-L'S 


0-00202314. 


810  THE   MECHANICAL   ACTION    OF   HEAT. 


XIX.— OX  THE  MECHANICAL  ACTION  OF  HEAT.* 

Section  VI. — A  Review  of  the  Fundamental  Principles  of  the 

Mechanical  Theory  of  Heat;   with    Remarks   on  the 

Thermic  Phenomena  of  Currents  of  Elastic 

Fluids,  as  illustrating  those 

Principles. 

4G.  I  have  been  induced  to  write  this  section  in  continuation  of  a 
paper  on  the  Mechanical  Action  of  Heat,  by  the  publication  (in  the 
Philosophical  Magazine  for  December,  1852,  Supplementary  Number)  of  a 
series  of  experiments  by  Mr.  Joule  and  Professor  "William  Thomson,  On 
the  Thermal  Effects  experienced  by  Air  in  rushing  through  Small  Apertures. 
Although  those  authors  express  an  intention  to  continue  the  experiments 
on  a  large  scale,  so  as  to  obtain  more  precise  results;  yet  the  results 
already  obtained  are  sufficient  to  constitute  the  first  step  towards  the 
experimental  determination  of  that  most  important  function  in  the  theory 
of  the  mechanical  action  of  heat,  which  has  received  the  name  of  Carnot's 
function. 

By  the  theoretical  investigations  of  Messrs.  Clausius  and  Thomson, — 
which  are  based  simply  on  the  fact  of  the  convertibility  of  heat  and 
mechanical  power,  the  determination  of  their  relative  value  by  Mr.  Joule, 
and  the  properties  of  the  function  called  temperature,  without  any  definite 
supposition  as  to  the  nature  of  heat, — Carnot's  function  is  left  wholly 
indeterminate. 

By  the  investigations  contained  in  the  previous  sections  of  this  paper, 
and  in  a  paper  on  the  Centrifugal  Theory  of  Elasticity  (See  p.  ^9), — in 
which  the  supposition  is  made,  that  heat  consists  in  the  revolutions  of 
what  are  called  molecular  vortices,  so  that  the  elasticity  arising  from  heat 
is  in  fact  centrifugal  force, — a  form  is  assigned  to  Carnot's  function;  but 
its  numerical  values  are  left  to  be  ascertained  by  experiment. 

The  recent  experiments  of  Messrs.  Joule  and  Thomson  serve  (so  far 
as  the  degree  of  precision  of  their  results  permits)  at  once  to  determine 
numerical  values  of  Carnot's  function  for  use  in  practice,  and  to  test  the 

*  Read  before  the  Eoyal  Society  of  Edinburgh  on  January   17,   1853,   and  pub- 
lished in  the  Transactions  of  that  Society,  Vol.  XX.,  Part  IV. 


THE   MECHANICAL  ACTION   OF  HEAT.  oil 

accuracy  with  which  the  phenomena  of  heat  are  represented  by  the 
consequences  of  the  hypothesis  of  molecular  vortices,  from  which  the 
investigation  in  this  paper  sets  out. 


Sub-Section  1. — Properties  of  Expansive  Heat. 

47.  To  show  more  clearly  the  nature  of  the  questions,  towards  the 
decision  of  which  these  experiments  are  a  step,  I  shall  now  briefly  review 
the  fundamental  principles  of  the  theory  of  heat,  and  the  reasoning  on 
which  they  are  based;  and  the  object  of  this  being  illustration  rather 
than  research,  I  shall  use  algebraical  symbols  no  further  than  is  absolutely 
necessary  to  brevity  and  clearness,  and  shall  follow  an  order  of  investigation 
which,  though  the  same  in  its  results  with  that  pursued  in  the  previous 
sections  of  this  paper,  is  different  in  arrangement. 

By  a  mind  which  admits  as  an  axiom  that,  in  the  present  order  of 
things,  physical  power  cannot  be  annihilated,  nor  produced  out  of  nothing, 
the  law  of  the  mutual  convertibility  of  heat  and  motive  power  must  be 
viewed  as  a  necessary  corollary  from  this  axiom,  and  Mr.  Joule's  experi- 
ments, as  the  means  of  determining  the  relative  numerical  value  of  those 
two  forms  of  power.  By  a  mind  which  does  not  admit  the  necessity  of 
the  axiom,  these  experiments  must  be  viewed  also  as  the  proof  of  the  law. 

This  law  was  virtually,  though  not  expressly,  admitted  by  those  who 
introduced  the  term  latent  heat  into  scientific  language;  for  Avhen  divested 
of  ideas  connected  with  the  hypothesis  of  a  subtle  fluid  of  caloric,  and 
regarded  simply  as  the  expression  of  a  fact,  this  term  denotes  heat  which 
has  disappeared  during  the  appearance  of  expansive  power  in  a  mass  of 
matter,  and  which  may  be  made  to  reappear  by  the  expenditure  of  an 
equal  amount  of  compressive  power. 

48.  Without  for  the  present  framing  any  mechanical  hypothesis  as  to 
the  nature  of  heat,  let  us  conceive  that  unity  of  weight  of  any  substance 
occupying  the  bulk  V  under  the  pressure  P,  and  possessing  the  absolute 
quantity  of  thermometric  heat  whose  mechanical  equivalent  is  Q,  undergoes 
the  indefinitely  small  increase  of  volume  d  V;  and  let  us  investigate  how 
much  heat  becomes  latent,  or  is  converted  into  expansive  power,  during 
this  process;  the  thermometric  heat  being  maintained  constant,  so  that 
the  heat  which  disappears  must  be  supplied  from  some  external  source. 

During  the  expansion  d  V,  the  body,  by  its  elastic  pressure  P,  exerts 
the  mechanical  power  P  d  V.  Part  of  this  power  is  produced  by  mole- 
cular attractions  and  repulsions;  and  although  this  part  may  be  modified 
by  the  influence  of  heat  upon  the  distribution  of  the  particles  of  the  body, 
it  is  not  the  direct  effect  of  heat.  The  remainder  must  be  considered  as 
directly  caused  by  the  heat  possessed  by  the  body,  of  which  the  pressure 


312  THE  MECHANICAL  ACTION   OF   HEAT. 

P  is  a  function;  and  to  this  portion  of  the  power  developed,  the  heat 
which  disappears  during  the  expansion  must  be  equivalent. 

To  determine  the  portion  of  the  mechanical  power  P<ZV  which  is 
the  effect  of  heat,  let  the  total  heat  of  the  body,  Q,  be  now  supposed  to 
vary  by  an  indefinitely  small  quantity  d  Q.  Then  the  mechanical  power 
of  expansion  VdY  will  vary  by  the  indefinitely  small  quantity 

This  is  the  development  of  power  for  the  expansion  dV,  caused  by  each 
indefinitely  small  portion  rf'Q  of  the  total  heat  possessed  by  the  body; 
and,  consequently,  the  whole  mechanical  power  for  the  expansion  dY  due 
to  the  whole  heat  possessed  by  the  body  Q,  is  expressed  as  follows: — 

«ii" (67-» 

and  this  is  the  equivalent  of  the  heat  transformed  into  mechanical  power, 
or  the  latent  heat  of  expansion  of  unity  of  weight,  fur  the  small  increment 
of  volume  d  V,  at  the  volume  V  and  total  heat  Q. 
Now,  a  part  only  of  this  power,  viz. — 

P  d  Y, 

is  visible  mechanical  energy,  expended  in  producing  velocity  in  the 
expanding  body  itself,  or  in  overcoming  the  resistance  of  the  bodies  which 
inclose  it.     The  remainder 


(„;:;;-  .•)..- v.  .  w 


r/O 

is,  therefore,  expended  in  overcoming  molecular  attraction. 

Molecular  attraction  depends  on  the  density  and  distribution  of  the 
particles  of  the  body;  and  is,  consequently,  a  function  of  the  volume  and 
total  heat  of  unity  of  weight.  It  is,  therefore,  possible  to  find  a  potential 
S,  being  a  function  of  V  and  Q,  of  such  a  nature,  that  the  difference 
between  its  two  values 

S2  —  9j, 

corresponding  respectively  to  two  sets  of  values  of  the  volume  and  total 
heat  (V15  Q,  and  V„  Q2),  shall  represent  the  power  which  is  the  equivalent 
of  the  heat  consumed  in  overcoming  molecular  attraction,  during  the 
passage  of  the  body  from  the  volume  Y1  and  heat  Qx  to  the  volume  V2 


THE   MECHANICAL   ACTION   OF   HEAT.  313 

and  heat  Q2.     The  form  of  the  expression  (68)  shows  that  this  potential 
has  the  following  property  : — 

The  integration  of  which  partial  differential  equation  gives  the  following 
value  for  the  potential  of  molecular  action: — ■ 

S  =  /(Q^!-P)eZV  +  tf>(Q),       •         •     (70.) 

<j)  (Q)  being  some  unknoAvn  function  of  the  heat  only,  and  the  integral 
being  taken  as  if  the  heat  Q  were  constant. 

The  heat  which  disappears  in  overcoming  molecular  action,  during  a 
small  increase  of  total  heat  d  Q,  while  the  volume  remains  constant,  is 
expressed  as  follows: — 

the  heat  Q  being  treated  as  a  constant  in  the  integration. 

If  we  now  investigate  the  entire  quantity  of  heat,  both  sensible  and 
latent,  which  is  consumed  by  a  body  during  a  simultaneous  small  change 
of  total  heat  d  Q  and  volume  d  V,  we  find  the  following  results  : — 

Sensible  heat  (which  retains  its  condition),        .      =  d  Q 
Latent  heat,  or  heat  which  disappears  in 

overcoming  molecular  action,         .         .     yf\  ^  ^   •    Tv        " 

Latent  heat  equivalent  to  the  visible  me- 
chanical effect, P  d  V 

The  amount  being 

^  +  ^S  +  P.V=(l+^)iQ+(^  +  p).V  = 


(72.) 


This  formula  expresses  completely  the  relations  between  heat,  mole- 
cular action,  and  expansion,  in  all  those  cases  in  which  the  expansive 
power  developed,  P  d  V,  is  entirely  communicated  to  the  bodies  inclosing 
the  substance  which  expands. 

49.  The  following  coefficients  are  contained  in,  or  deducible  from  it, 


314 


THE   MECHANICAL   ACTION   OF   HEAT. 


The  ratio  of  the  specific  heat  at  constant  volume  to  the  real  specific 
heat : — 

T  =  1+s|  =  1  +  Q/^-dV  +  ^(Q)'     •    (73) 

The  coefficient  of  latent  heat  of  expansion  at  constant  heat  :— 


dS 


d\ 


+  P  =  Q 


dV 


(74.) 


The  ratio  of  the  specific  heat  at  constant  pressure  to  the  real  specific 
heat  is  found  as  follows.     To  have  the  pressure  constant  we  must  have 


dV  lrt       dF  ITT       ,  dV 

a  (^  d\  d  Q 


dV 
dQ 

dly 

dX 


consequently,  the  ratio  in  question  is, 


dV 


dX 


dVY- 


\d  (  ) 


^  (75.) 


50.  In  order  to  investigate  the  laws  according  to  which  heat  is  con- 
verted into  mechanical  power,  in  a  machine  working  by  the  expansion  of 
an  elastic  body,  it  will  be  convenient  to  use  a  function, 

F=  j^dY(Q  =  const.), 

of  such  a  nature  that  the  difference  between  two  of  its  values,  correspond- 
ing to  different  volumes  of  the  body  at  the  same  total  heat,  represents  the 
ratio  of  the  heat  converted  into  power  by  expansion  between  those  volumes, 
to  the  given  constant  total  heat.  I  shall  call  this  function  a  heat-potential. 
Introducing  this  function  into  equation  (72),  we  find  for  the  total  heat 
consumed  by  a  body  during  the  increase  of  total  heat  d  Q,  and  the  expan- 
sion d  V, 

dQ,  +  d  .  S  +  P<7V  =  (l  +  f  .  (Q)  W  +  Qd  .  F         (7G.) 


THE  MECHANICAL   ACTION   OF   HEAT.  315 


( 


d  F  dF 

observing  that  d  .  F  =  T  —  d  Q  +  T,r  d  V 
«  y  a  V 

"OS"- )'«  +  £«•> 

Let  us  now  suppose  that  the  body  changes  its  volume  without  either 
losing  or  gaining  heat  by  conduction.  This  condition  is  expressed  by  the 
equation 

0  =  (1  +$'  .Q)dQ  +  Qd.  F, 

from  which  we  deduce  the  following, 

-i.r-l+^W.iq.       .      .    (77.) 

which  expresses  the  following  theorem  : — 

When  the  quantity  of  heat  in  a  body  is  varied  by  variation  of  volume  only,  the 
variation  of  the  heat-potential  depends  on  the  heat  only,  and  is  independent  of  the 
volume. 

In  order  that  a  machine  working  by  the  expansive  power  of  heat  may 
produce  its  greatest  effect,  all  the  heat  communicated  from  external  bodies 
should  be  employed  in  producing  expansive  power,  and  none  in  producing 
variations  of  the  quantity  of  heat  in  the  body;  for  heat  employed  for  the 
latter  purpose  would  be  wasted,  so  far  as  the  production  of  visible  motion 
is  concerned.  To  effect  this  the  body  must  receive  heat  by  conduction, 
and  convert  it  into  expansive  power,  while  containing  a  certain  constant 
quantity  of  heat  Q:;  give  out  by  conduction  heat  produced  by  compression, 
while  containing  a  smaller  constant  quantity  of  heat  Q9;  and  change 
between  those  two  quantities  of  thermometric  heat  by  means  of  changes 
of  volume  only,  without  conduction.  For  this  purpose  a  cycle  of  opera- 
tions must  be  performed  similar  to  that  described  by  Carnot,  as 
follows  : — 

(I.)  Let  FA  be  the  initial  value  of  the  heat-potential ;  let  the  body 
expand  at  the  constant  heat  Ql  till  the  heat-potential  becomes  FB.  Then 
the  heat  received  and  converted  into  expansive  power  is 

Ha  =  Q,  (FB  -  FA). 

(II.)  Let  the  body  further  expand  without  receiving  or  emitting  heat 
till  the  quantity  of  heat  in  it  falls  to  Q2 ;  the  heat-potential  varying 
according  to  equation  (77),  and  becoming  at  length  Fc.  The  heat  con- 
verted into  expansive  power  in  this  operation  is 

Q,  -  Q,- 


31G  THE   MECHANICAL   ACTION   OF   HEAT. 

(III.)  Let  the  body  be  compressed,  at  the  constant  heat  Q„  till  the 
heat-potential  becomes  FD;  a  quantity  differing  from  the  initial  heat- 
potential  Fv  by  as  much  as  Fc  differs  from  FB.  In  this  operation  the 
following  amount  <>;'  power  is  reconverted  into  heat,  and  given  out  by 
conduction  : — 

IL  =  Q,  (Fc  -  F„). 

(IV.)  Let  the  body  be  further  compressed,  till  the  heat-potential  returns 
to  FA,  its  original  value.  Then,  by  the  power  expended  in  this  com- 
pression alone,  without  the  aid  of  conduction,  the  total  heat  of  the  body 
will  be  restored  to  its  original  amount,  exactly  reversing  the  operation  II. 

At  the  end  of  this  cycle  of  operations,  the  following  quantity  of  heat 
will  have  been  converted  into  mechanical  power: — 

Hx  -  H2  =  Q,  (FB  -  FA)  -  Q2  (Fc  -  FD), 

but  it  is  obvious  that  the  difference  between  the  heat-potentials  is  the 
same  in  the  first  and  third  operations ;  therefore,  the  useful  effect  is 
simply 

11,-11,,  =  ^,-  t>,KFu-FA),    1 

while  the  whole  heat  expended  is,  '    .         .     (78.) 

H^Q^F,-  V  J 

Hence,  the  ratio  of  the  heat  convert*  J  into  mechanical  effect,  in  cm  expansive 
machine,  working  to  the  greatest  advantage,  to  the  whole  heat  expended,  is  the 
same  with  that  which  the  difference  between  the  quantities  of  heat  possessed  by  the 
expansive  body  during  the  i 

to  the  quantity  of  heat  possessed  by  it  during  the  operation  of  receiving  . 
and  is  independent  of  the  nature  and  condition  of  the  body. 

This  theorem  is  thus  expressed  symbolically, — 

H,  -  H.,  Effect  _  Q,  -  Q., 


\li  i  teat  expended  Qx 


(79.) 


51.  When  a  body  expands  without  meeting  with  resistance,  so  that  all 
its  expansive  power  is  expended  in  giving  velocity  to  its  own  particles, 
and  when  that  velocity  is  ultimately  extinguished  by  friction,  then  a 
quantity  of  heat  equivalent  to  the  expansive  power  is  reproduced. 

The  heat  consumed  is  expressed  by  taking  away  the  term  representing 
the  expansive  power,  ~P  dY,  from  the  expression  (72),  the  remainder  of 
which  consists  merely  of  the  variation  of  actual  heat,  and  the  heat  expended 
in  overcoming  molecular  attraction,  viz. : — 

/         d  S\  ,  „       d  S  , ,, 


THE   MECHANICAL   ACTION   OF   HEAT.  317 

This  expression  is  a  complete  differential,  and  may  be  written  thus  : — 
d  (Q  +  S)  =  d  {  Q  +  <p  (Q)  +  (Q  ~  -  l) j  P  rf  v|-        (80.) 

(Q  being  treated  as  a  constant  in  performing  the  integration  I  P  d  X ). 

Its  integral,  Q  +  8,  the  sum  of  the  heat  of  the  body,  and  of  the  potential 
of  its  molecular  actions,  is  the  same  quantity  which  I  have  denoted  by  the 
symbol  ^F  in  the  tenth  article  of  a  paper  on  the  Centrifugal  Theory  of 
Elasticity  (See  p.  G2),  and  whose  differences  are  there  stated  to  repre- 
sent the  total  amount  of  power  which  must  be  exercised  on  a  body, 
whether  in  the  form  of  expansive  or  compressive  power,  or  in  that  of  heat, 
to  make  it  pass  from  one  volume  and  temperature  to  another.  This 
integral  corresponds  also  to  the  function  treated  of  by  Professor  William 
Thomson  in  the  fifth  part  of  his  paper  on  the  Dynamical  Theory  of  Heat, 
under  the  name  of  "  Total  Mechanical  Energy." 

52.  We  have  now  obtained  a  system  of  formula3,  expressing  all  the 
relations  between  heat  and  expansive  power,  analogous  to  those  deduced 
from  a  consideration  of  the  properties  of  temperature,  by  Messrs.  Clausius 
and  Thomson,  and  from  the  hypothesis  of  molecular  vortices  in  the 
previous  sections  of  this  paper;  but,  in  the  present  section,  both  the 
theorems  and  the  investigations  are  distinguished  from  former  researches 
by  this  circumstance — that  they  are  independent,  not  only  of  any  hypo- 
thesis respecting  the  constitution  of  matter,  but  of  the  properties,  and  even 
of  the  existence,  of  such  a  function  as  temperature;  being,  in  fact,  simply 
the  necessary  consequences  of  the  following 

"    Definition  of  Expansive  Heat. 

Let  the  term  Expansive  Heat  he  used  to  denote  a  hind  of  physical  energy 
•convertible  with,  and  measurable  by,  equivalent  quantities  of  mechanical  power, 
and  augmenting  the  expansive  elasticity  of  matter  in  which  it  is  -present. 

52a.  It  is  further  to  be  remarked,  that  the  theorems  and  formulae  in 
the  preceding  articles  of  this  section  are  applicable,  not  only  to  heat  and 
expansive  power,  but  to  any  two  directly  convertible  forms  of  physical 
energy,  one  of  which  is  actual,  and  the  other  potential.  They  are,  in  fact, 
the  principles  of  the  conversion  of  energy  in  the  abstract,  when  interpreted 
according  to  the  following  definitions  of  the  symbols  : — 


318  THE   MECHANICAL   ACTION    OF   HEAT. 

Let  Q  denote  the  quantity  of  a  form  of  actual  physical  energy  present 
in  a  given  body ; 

V,  a  measurable  state,  condition,  or  mode  of  existence  of  the  body, 
whose  tendency  to  increase  is  represented  by 

P,  a  force,  depending  on  the  condition  V,  the  energy  Q,  and  permanent 
properties  of  the  body,  so  that 

P  d  V  is  the  increment  of  a  form  of  potential  energy,  corresponding  to 
a  small  increment  d  V  of  the  condition  V. 

Let  d&  be  the  quantity  Avhcrcby  the  increment  of  potential  energy 
P(7V  falls  short  of  the  quantity  of  actual  energy  of  the  form  Q,  which  is 
converted  into  the  potential  form  by  the  change  of  condition  d  V. 

Then,  as  in  equation  (G9), 

dX  ~HdQ         ' 

an  equation  from  which  all  those  in  the  previous  articles  are  deducible, 
and  which  comprehends  the  whole  theory  of  the  mutual  conversion  of  the 

actual  form  of  energy  Q,  and  the  potential  form  I  "PdV,  whatsoever  those 

forms  may  be,  when  no  other  form  of  energy  interferes.  The  application 
of  these  principles  to  any  form,  or  any  number  of  forms,  of  actual  and 
potential  energy,  is  the  subject  of  a  paper  read  before  the  Philosophical 
►Society  of  Glasgow,  on  the  5th  January,  18.33,  and  published  in  the 
Philosophical  Magazine  for  February,  1853.     (See  p.  203.) 


Si  B-SECTION  2. — PROPERTIES  OF  TEMPERATURE. 

53.  Still  abstaining  from  the  assumption  of  any  mechanical  hypothesis, 
let  us  proceed  a  step  beyond  the  investigation  of  the  foregoing  articles, 
and  introduce  the  consideration  of  temperature — that  is  to  say,  of  an 
arbitrary  function  increasing  with  heat,  and  having  the  following  pro- 
perties : — 

Definition  of  Equ.d  Temperatures. 

Two portion;  of  matter  are  said  to  have  equal  temperatures  ivhen  neither  tends 
to  communicate  heat  to  the  other. 

Corollary. 

All  bodies  absolutely  destitute  of  heat  have  equal  temperatures. 

The  ratio  of  the  real  specific  heats  of  two  substances  is  that  of  the 


THE   MECHANICAL   ACTION   OF   HEAT.  319 

quantities  of  heat  which  equal  weights   of  them    possess  at  the   same 
temperature. 

Theorem. 

The  ratio  of  the  real  specific  heats  of  any  pair  of  substances  is  the  same  at  all 
temperatures. 

For,  suppose  equal  weights  of  a  pair  of  homogeneous  substances  to  be 
in  contact,  containing  heat  in  such  proportions  as  to  be  in  equilibrio. 
Then,  let  additional  portions  of  each  substance,  of  equal  weight,  and 
destitute  of  heat,  be  added  to  the  original  masses  ;  so  that  the  quantities 
of  heat  in  unity  of  weight  may  be  diminished  in  each  substance,  but  may 
continue  to  be  in  the  same  ratio.  Then,  if  the  equality  of  temperature  do 
not  continue,  portions  of  heat  which  were  in  equilibrio  must  have  lost  that 
equilibrium,  merely  by  being  transferred  to  other  particles  of  a  pair  of 
homogeneous  substances,  which  is  absurd.  Therefore,  the  temperatures 
continue  equal. 

It  follows,  that  the  quantity  of  heat  in  unity  of  weight  of  a  substance 
at  a  given  temperature,  may  be  expressed  by  the  product  of  a  quantity 
depending  on  the  nature  of  the  substance,  and  independent  of  the 
temperature,  multiplied  by  a  function  of  the  temperature,  which  is  the 
same  for  all  substances. 

Let  r  denote  the  temperature  of  a  body  according  to  the  scale  adopted ; 
k,  the  position,  on  the  same  scale,  of  the  temperature  corresponding  to 
absolute  privation  of  heat ;  ft,  a  quantity  depending  on  the  nature  of  the 
substance,  and  independent  of  temperature.  Then  the  quantity  of  heat 
in  unity  of  weight  may  be  expressed  as  follows  : — 

Q  =  feO/,.r-^.K).         .  .  .      (81.) 

54.  If  we  introduce  this  notation  into  the  formula  (79),  which  expresses 
the  proportion  of  the  total  heat  expended  which  is  converted  into  useful 
power  by  an  expansive  machine  working  to  the  best  advantage,  the 
quantity  ft,  peculiar  to  the  substance  employed,  disappears,  and  we  obtain 
Carnot's  theorem,  as  modified  by  Messrs.  Clausius  and  Thomson — viz., 
that  this  ratio  is  a  function  solely  of  the  temperatures  at  which  heat  is 
received  and  emitted  respectively,  and  is  independent  of  the  nature  of  the 
substance ;  or  symbolically, 

Effect  _  Qi^Q.2  _  \p  .  Tj  -  }p  .  r,  ,g2  . 

Heat  expended  Qx  \f;  .  rx  —  \p  .  k 

55.  Let  us  now  apply  the  same  notation  to  the  formula  (G7)  for  the 
latent  heat  of  a  small  expansion,  d  V,  at  constant  heat,  viz  : — 


320 


THE   MECHANICAL   ACTION   OF   HEAT. 


we  have  evident lv 


d?        1       dV 

1        dv 

dQ ~ dQ     dr  ~ 

\l  \jj     .    T         ll  7 

dr 

and.  consequently,  the  heat  which  disappears  by  the  expansion  dX  is 

dV 

dr 


T 


d  V, 


dQ  \f/ 

from  which  formula  the  specific  quantity  ft  has  disappeared. 
Now,  in  the  notation  of  Professor  Thomson  we  have 

\p  .  T  —  Tp  .  K  _   «J 

where  J   is  Joule's  equivalent,  and  ju.  a  function  of  the  temperature,  the 
same  for  all  substances,  to  be  determined  empirically ;  and,  consequently, 

hyp.  log.  (i//  .  t  —  i//  .  k)  =  1 J  '  fx  d  -. 

if 


/J- 


ip  .  t  —  ifj .  k  =  e 


and 


(84.) 


sr 


^ 


j 


Q  =  fe  (i/,  .  t  -  i£  .  k)  =  ft 

These  expressions  will  be  re  by  those  who  have  studied  Professor 

Thomson's  papers  on  the  dynamical  theory  of  heat.  By  introducing  the 
value  given  above  of  the  quantity  of  heat  in  unity  of  weight,  into  the 
formula'  of  the  preceding  articles  of  this  section,  they  are  at  once  trans- 
formed to  those  of  Professor  Thomson,  and  in  particular,  the  formulae  (79) 
and  (82)  become  the  following: — 

1     "I       7  1    ~, 

j)      "dr         J, 


*  Effect  of  Machine 
Heat  expended 


M 


=   1   -  £ 


J 

k 


d 


,"- 


(85.) 


*  It  is  to  be  observed,  that  in  Professor  Thomson's  notation,  heat  is  supposed  to  be 
measured  by  an  arbitrary  unit,  whose  ratio  to  a  unit  of  mechanical  power  is  denoted  by 
J ;  while  in  this  paper,  the  same  unit  is  employed  in  expressing  quantities  of  heat  and 
of  mechanical  power. 


THE   MECHANICAL   ACTION   OF   HEAT.  321 


Sub-Section  3. — On  the  Hypothesis  of  Molecular  Vortices. 

56.  The  use  of  a  mechanical  hypothesis  in  the  theory  of  heat,  as  in 
other  branches  of  physics,  is  to  render  it  a  branch  of  mechanics,  the  only 
"complete  physical  science ;  and  to  deduce  its  principles  from  the  laws  of 
force  and  motion,  Avhich  are  better  understood  than  those  of  any  other 
phenomena. 

The  results  of  the  investigations  in  the  £>receding  part  of  this  section 
are  consistent  alike  with  all  conceivable  hypotheses  which  ascribe  the 
phenomena  of  heat  to  invisible  motions  amongst  the  particles  of  bodies. 

Those  investigations,  however,  leave  undetermined  the  relation  between 
temperature  and  quantity  of  heat,  except  in  so  far  as  they  show  that  it 
must  follow  the  same  law  of  variation  in  all  substances. 

By  adopting  a  definite  hypothesis,  we  are  conducted  to  a  definite 
relation  between  temperature  and  quantity  of  heat;  which,  being  intro- 
duced into  the  formulae,  leads  to  specific  results  respecting  the  phenomena 
of  the  mutual  transformation  of  heat  and  visible  mechanical  power  ;  and 
those  results,  being  compared  with  experiment,  furnish  a  test  of  the 
soundness  of  the  hypothesis. 

Thus,  the  hypothesis  of  molecular  vortices,  which  forms  the  basis  of  the 
investigations  in  the  first  five  sections  of  this  paper,  and  in  a  paper  on  the 
centrifugal  theory  of  elasticity,  leads  to  the  conclusion,  that,  if  tempera- 
ture be  measured  by  the  expansion  of  a  perfect  gas,  the  total  quantity  of 
heat  in  a  body  is  simply  proportional  to  the  elevation  of  its  temperature 
above  the  temperature  of  absolute  privation  of  heat ;  or,  in  the  notation 
of  the  preceding  article, 

xp  .  r  =  r,  \f/  .  r  =  1, 
and 

Q  =  fc  (r  -  k),    .  .  .  .     (86.) 

ft  being  the  real  specific  heat  of  the  body. 

If  this  value  be  substituted  for  the  quantity  of  heat  Q,  in  all  the 
formulae,  from  (67)  to  (80)  inclusive,  which  are  founded  simply  on  the 
definition  of  expansive  heat,  it  reproduces  all  the  formula?  which,  in  this 
and  the  other  paper  referred  to,  have  been  deduced  directly  from  the 
hypothesis.  In  the  sequel,  I  shall  apply  one  of  these  formulae  to  the 
calculation,  from  the  experiments  of  Professor  Thomson  and  Mr.  Joule  on 
the  heating  of  currents  of  air  by  friction,  of  approximate  values  of  the 
absolute  temperature  corresponding  to  total  privation  of  heat,  that  the 
mutual  consistency  of  those  values  may  serve  as  a  test  of  the  soundness  of 
the  hypothesis,  and  the  accuracy  of  the  formulae  deduced  from  it. 

57.  Before  proceeding  further,  it  may  be  desirable  to  point  out  how  far 

x 


322  THE  MECHANICAL  ACTION   OF   HEAT. 

this  hypothesis  agrees  with,  and  how  far  it  differs  from,  that  proposed  by 
Mr.  Herapath  and  Mr.  Waterstom,  which  supposes  bodies  to  consist  of 
extremely  small  and  perfectly  clastic  particles,  which  fly  about  in  all 
directions  with  a  velocity  whose  half-square  is  the  mechanical  equivalent 
of  the  heat  possessed  by  unity  of  weight,  and  are  prevented  from  dispersing 
by  their  collisions  with  each  other  and  with  the  particles  of  surrounding 
bodies.     Let  v  be  the  velocity  of  motion,  then 

9 

represents  the  heat  possessed  by  unity  of  weight,  expressed  in  terms  of 
the  force  of  gravity. 

The  expansive  pressure  due  to  such  motions  is  found  by  conceiving  a 
hard,  perfectly  elastic  plane  of  the  area  unity  to  be  opposed  to  the  collision 
of  the  particles,  and  calculating  the  pressure  which  would  be  required  to 
maintain  its  position  against  them.  If  all  the  particles  were  to  strike  and 
rebound  from  such  a  plane  at  right  angles,  the  pressure  would  bi 
represented  thus: 

1_ 

9  '  T 

» 

where  V  is  the  volume  which  contains  so  many  particles  as  amount  to 
unity  of  weight.  But  the  particles  are  supposed  to  tly  in  equal  numbers 
in  all  directions.     Then,  if  9  denote  the  angle  of  incidence  on  the  plane 


/ 


sin  0  <1  9  .     „  ,  » 

=  sin  9  I'  9 
— 

sin  6  d  0 
0 


represents  the  proportion  of  the  whole  particles  which  fly  in  those  direc- 
tions which  make  the  angle  0  with  the  normal  to  the  plane.  Of  this 
proportion,  again,  the  fraction  cos  9  only  strikes  the  plane;  while  the 
force  of  the  blow  also  is  less  than  that  of  a  normal  blow  in  the  ratio 
cos  0:1.     Hence,  the  mean  force  of  collision  is 


/ 


.7T 

2  1 

cos2  9  sin  9  (I  9  =  ^ 

0  6 


of  the  force  of  a  perpendicular  collision ;  so  that  the  expansive  pressure  i& 
represented  by 

1       *2      ^_2  Q 

3   '    cj    '  V  ~  3  '  V 


THE  MECHANICAL  ACTION   OF  HEAT.  323 

Hence,  according  to  this  hypothesis,  we  should  have  for  a  perfect  gas 

P  V  =  |  Q, 

o 

or  the  product  of  the  pressure  and  volume  of  a  mass  of  a  perfect  gas 
equal  to  two-thirds  of  the  mechanical  equivalent  of  its  total  heat. 

It  is  known,  however,  that  the  product  of  the  pressure  and  volume  of  a 
mass  of  sensibly  perfect  gas  is  only  about  four-tenths  of  the  equivalent  of 
its  total  heat.     The  hypothesis,  therefore,  requires  modification. 

By  supposing  the  particles  to  attract  each  other,  or  to  be  of  appreciable 
bulk  compared  with  the  distances  between  them,  the  ratio  in  question  is 
diminished;  but  either  of  these  suppositions  is  inconsistent  with  the 
perfectly  gaseous  condition.  * 

It  appears  to  me,  that,  besides  this  difficulty  connected  with  the  gaseous 
condition,  there  exists  also  great  difficulty  in  conceiving  how  the  hypothesis 
can  be  applied  to  the  solid  condition,  in  which  the  particles  preserve 
definite  arrangements.  The  limited  amount  of  time  and  attention, 
however,  which  I  have  hitherto  bestowed  on  this  hypothesis,  is  not 
sufficient  to  entitle  me  to  pronounce  whether  these  difficulties  admit  of  a 
solution. 

58.  The  idea  of  ascribing  expansive  elasticity  to  the  centrifugal  force  of 
vortices  or  eddies  in  elastic  atmospheres  surrounding  nuclei  of  atoms, 
originated  with  Sir  Humphry  Davy.  The  peculiarity  of  the  view  of  the 
hypothesis  taken  in  this  paper  consists  in  the  function  ascribed  to  the 
nuclei  or  central  physical  points  of  the  atoms,  which,  besides  retaining  the 
atmospheres  round  them  by  their  attraction,  are  supposed,  by  their  actions 
on  each  other,  to  constitute  the  medium  which  transmits  radiant  heat  and 
light ;  so  that  heat  is  radiant  or  thermometric,  according  as  it  affects  the 
nuclei  or  their  atmospheres. 

In  this  form  the  hypothesis  of  molecular  vortices  is  not  a  mere  special 
supposition  to  elucidate  the  theory  of  expansive  heat,  but  becomes 
connected  with  the  theory  of  the  elasticity  of  matter  in  all  conditions, 
from  solid  to  gaseous,  and  with  that  of  the  transmission  of  radiations. 

I  have  already  investigated  mathematically  the  consequences  of  this 
hypothesis  by  two  different  processes,  which  are  necessarily  somewhat 
complicated. 

When  the  question,  however,  is  confined  to  the  relations  between  tem- 
peratures and  quantities  of  heat,  a  more  simple  process  may  be  followed, 
analogous  to  that  which  has  been  applied  in  the  preceding  article  to  the 
hypothesis  of  molecular  collisions. 

If  a  mass  of  elastic  fluid,  so  much  rarefied  that  the  effect  of  molecular 
attraction  is  insensible,  be  entirely  filled  with  vortices,  eddies,  or  circu- 
lating currents  of  any  size  and  figure,  so  that  every  particle  moves  with 


324  THE  MECHANICAL  ACTION   OF  HEAT. 

the  common  velocity  iv,  then,  if  the  planes  of  revolution  of  these  eddies 
be  uniformly  distributed  in  all  possible  positions,  it  follows,  from  reasoning 
precisely  similar  to  that  employed  in  the  preceding  article,  that  the 
pressure  exerted  by  the  fluid  against  a  plane,  in  consequence  of  the  cen- 
trifugal force  of  the  eddies,  has  the  following  value  in  terms  of  gravity  : — 

\-'?-l (870 

3      g      V 

or  two-thirds  of  the  hydrostatic  pressure  due  to  the  velocity  of  the  eddies 
w  j  V  being,  as  before,  the  volume  occupied  by  unity  of  weight. 

It  is,  however,  reasonable  to  suppose,  that  the  motion  of  the  particles  of 
atomic  atmospheres  does  not  consist  merely  in  circulating  currents ;  but 
that  those  currents  afe  accompanied  with  a  certain  proportionate  amount 
of  vibration, — a  kind  of  motion  which  docs  not  produce  centrifugal  force. 
To  these  Ave  have  to  add  the  oscillations  of  the  atomic  nuclei,  in  order  to 
obtain  the  mechanical  equivalent  of  the  whole  molecular  motions ;  which 
is  thus  found  to  be  expressed  for  unity  of  weight  by 

I  f-  •   =  Q,     •          •         ■         •     (88.) 
2g 

2  k 
h  being  a  specific  coefficient.     Hence  it  follows  (denoting  —  by  N),  that 

the  expansive  pressure  due  to  molecular  motions  in  a  perfect  gas  is  equal 
to  the  mechanical  equivalent  of  those  motions  in  unity  of  volume  multi- 
plied by  a  specific  constant 

N  •  | (89.) 

The  coefficient  N  has  to  be  determined  by  experiment ;  its  value  for 
atmospheric  air  is  known  to  be  between  0'4  and  0*41. 

In  order  to  account  for  the  transmission  of  pressure  throughout  the 
molecular  atmospheres,  it  is  necessary  to  suppose  them  possessed  of  a 
certain  amount  of  inherent  elasticity,  however  small,  varying  proportionally 
to  density,  and  independent  of  heat.     Let  this  be  represented  by 

h 
V' 

then 

P=(NQ  +  A)i  .  •     (90.) 

is  the  total  pressure  of  a  perfect  gas. 

Equilibrium  of  heat  and  pressure  between  portions  of  two  different 
perfect  gases  in  contact  requires  that  the  pressures  independent  of  heat 


THE  MECHANICAL  ACTION   OF  HEAT.  325 

and  the  pressures  caused  by  heat,  shall  separately  be  in  equilibria  Let 

the   suffixes   a  and   b  be    used  to  distinguish  quantities  relative  to  two 

different  substances  in  the  perfectly  gaseous  condition.     Then  the  first 
condition  of  equilibrium  is  expressed  as  follows  : — 

h\ ,  N        (h 


(4)  w =(£)<»»  •  (9L) 


that  is  to  say,  the  densities  of  two  perfect  gases  in  equilibrio  are  inversely 
proportional  to  the  coefficients  of  elasticity  of  their  atomic  atmospheres. 
The  second  condition  is  expressed  as  follows : — 


("*)«  =  &*)» 


which,  being  taken  in  connection  with  the  first  condition,  gives 


(!«)«=(!«)»         •  <93-> 


Now,  by  equation  (90),  we  have 

Hence  the  condition  of  equilibrium  of  heat  between  two  perfect  gases  is 

(^)(°>  =  (i>>>  •    •     •  o*> 

consequently,  temperature  may  be  measured  by  the  product  of  the  pressure  and 
volume  of  a  perfect  gas,  divided  by  a  coefficient,  which  is  proportioned  to  the 
volume  of  the  gas  at  a  standard  pressure  and  temperature. 

Temperatures  thus  measured  are  reckoned  from  the  point  known  as  the 
zero  of  gaseous  tension,  or  absolute  zero  of  a  perfect  gas  thermometer,  274°*6 
Centigrade  below  the  temperature  of  melting  ice. 

Let  V0  denote  the  volume  of  unity  of  weight  of  a  perfect  gas,  at  a 
standard  pressure  P0,  and  absolute  temperature  t0  ;  then  any  other  absolute 
temperature  has  the  following  value  : — 


PV 

t  —  r, 


-^-(NQ  +  A),       •         •     (94.) 


op  v  ~p  v 

x0   v0  x0   v0 


while  the  absolute  temperature  of  total  privation  of  heat  is 


K  —  To  p  v  " 

r0   v0 


(94a.) 


320  THE  MECHANICAL   ACTION   OF   HEAT. 

Hence  it  appears  that  quantity  of  heat  in  unity  of  weight  bears  the 
following  relation  to  temperature, — 

Q  =  i(PT-*)  =  ^.(r-.4      •         •     (95.) 

in  which,  if  we  substitute  the  symbol  of  real  specific  heat, 
»  p  V 

ft=iv   •    •    •    •  ^ 

we  obtain  the  formula  already  given  (8*6)  for  the  relation  between  heat 
and  temperature.* 

59.  The  introduction  of  the  value  given  above  of  the  quantity  of  heat 
in  terms  of  temperature,  into  the  formula  (67),  gives  for  the  latent  heat 
of  a  .small  expansion  d  V  at  constant  temperature 

(r-K)7>-<>V.  .  .  .      (97.) 

The  formula'  (79)  and  (82),  for  the  proportion  of  heart  rendered  available 
by  an  expansive  engine  working  to  the  greatest  advantage,  becomes 


rj  -  r2 


(98.) 


or  the  ratio  of  the  difference  between  the  temperatures  of  receiving  and 
emitting  heat,  to  the  elevation  of  the  former  temperature  above  that  of 
total  privation  of  heat.  This  is  the  law  already  arrived  at  by  a  different 
process  in  Section  V  of  this  paper. 

When  the  same  substitution  is  made  in  equation  (80),  which  represents 
the  total  energy,  whether  as  heat  or  as  compressive  power,  which  must  be 
applied  to  unity  of  weight  of  a  substance  to  produce  given  changes  of 
heat  and  volume,  the  following  result  is  obtained  : — 

d  .  ¥  =  dQ  +  d  .  S  =  I  ft  +/'(t)  +  (r  -  k)J~  d  V  I  dr 

=  d  .   {  ft  r  +/(r)  +  ((t  -  k)  f~T  -  l)  /  P  d  V  }     .      (99.) 

As  it  cannot  be  simplified,  it  is  unnecessary  here  to  recapitulate  the 
investigation,  which  leads  to  the  conclusion  that  the  functions  /  (r)  and 
/  (t)  have  the  following  values  : — 

*  See  Appendix,  Note  A,  p.  33G. 


THE  MECHANICAL  ACTION  OF  HEAT.  327 

f{r)  =  *  N  (k  hyp.  log.  r  +  £) ;  /'  (r)  =  ft  N  (^  -  J).       (99  A.) 

We  have  thus  reproduced  equation  (26)  of  the  paper  formerly  referred  to, 
on  the  Centrifugal  Theory  of  Elasticity.     (See  p.  49.) 

The  coefficient  of  the  variation  of  temperature  in  the  first  form  of 
equation  (99)  is  the  specific  heat  of  the  suhstance  at  constant  volume. 
Denoting  this  by  Kv,  the  formula  becomes 

d  .  *  =  Kv  .  d  t  +  |  (r  -  k)  ^f  -  P  |  d  V.     .     (100.) 


Sub-Section  4. — Thermic  Phenomena  of  Currents  of 
Elastic  Fluids. 

60.  When  a  gas  previously  compressed  is  allowed  to  escape  through 
small  apertures,  as  in  the  experiments  of  Mr.  Joule  and  Professor  Thomson, 
and  has  its  velocity  destroyed  entirely  by  the  mutual  friction  of  its  particles, 
without  impediment  from  any  other  substance,  and  without  conduction  of 
heat  to  or  from  any  other  substance ;  then  its  condition  is  expressed  by 
making 

d  ¥  =  0, 
that  is  to  say, 

1       f        (d?       P\  r/P) 


If  we  assume  (as  is  really  the  case  in  the  experiments)  that  the  specific 
heat  of  the  gas  at  constant  volume  does  not  sensibly  vary  within  the 
limits  of  the  experiments  as  to  temperature  and  volume,  so  that  KT  is 
sensibly  constant,  and  also  that  the  variation  of  temperature  is  very 
small  as  compared  with  the  absolute  temperatures,  then  we  have  the 
following  approximate  integral : 

i  i 

which  represents  the  cooling  effect  of  an  expansion  from  the  volume  Vt  to 
the  volume  V2. 

If  it  were  possible  to  obtain  any  substance  in  the  state  of  perfect  gas  to 
be  used  in  experiments  of  this  kind,  the  first  integral  in  the  above  expres- 
sion would  disappear,  because  for  a  perfect  gas 

dr        t 


328  THE  MECHANICAL  ACTION   OF   HEAT. 

and  as  the  other  terra  is  negative,  the  result  would  be  a  slight  heating 

dP    ,  ,    P    , 

effect.     As  no  gas,  however,  is  perfect,  and  as     —  always  exceeds  — ,  the 

(I  T  T 

mode  of  reducing  the  experimental  data  is  to  calculate  the  value  of  the 
first  term,  which  represents  the  effect  of  cohesion,  from  the  known  pro- 
perties of  the  gas,  to  subtract  from  it  the  actual  cooling,  and  from  the 
remainder  to  compute  values  of  k,  the  temperature  of  absolute  privation  of 
heat,  according  to  the  following  formula  : — 


v 
2riP 


(103.) 


r     i     T-dY 
KT  J  v  dr 


When  the  gas  is  nearly  perfect,  as  in  the  case  of  atmospheric  air,  it  is 
unnecessary  to  take  into  consideration  its  deviation  from  the  perfect 
condition  in  computing  the  integral  in  the  denominator,  whose  approxi- 
mate value  is  found  to  be 

P  V  V0  P 

-  ° — -  .  hyp.  log.  ~  =  N  .  hyp.  log.     -  nearly  (t  being  nearly  constant), 
Kvr0  Vj  12 

and  Kv  nearly  =  ft. 

The  value  of  the  integral  in  the  numerator  is  found  as  follows : — 

The  centrifugal  theory  of  elasticity  indicates  that  the  pressure  of  an 

imperfect  gas  may  be  represented  by  the  following  formula  : — 

P  =  P»v{r0  +  A»-A;-r'-&C-}'      '      (1M0 

where  V0  is  the  volume  in  the  perfectly  gaseous  state,  at  a  standard 
pressure  P0,  and  absolute  temperature  r0,  and  A0,  A1}  &c,  are  a  series  of 
functions  of  the  density,  to  be  determined  empirically.  From  this  formula 
it  is  easily  seen  that 

dP       P_pV0f       A        2A  ) 

so  that  the  first  term  in  the  numerator  of  the  expression  (103)  has  the 
following  value  : — 

i  ii 

p  v 

in  which  — ^— °  =  N  r0  nearly. 


THE   MECHANICAL   ACTION   OF   HEAT.  329 

In  order  to  represent  correctly  the  result  of  M.  Regnault's  experiments 
on  the  elasticity  and  expansion  of  gases,  it  was  found  sufficient  to  use,  in 
the  formula  for  the  pressure  (104),  the  first  three  terms;  and  the  functions 
of  the  density  which  occur  in  these  terms,  as  determined  empirically  from 
the  experiments,  were  found  to  have  the  following  values,  in  which  the 
unit  of  volume  is  the  theoretical  volume  of  unity  of  weight  of  air  under 
the  pressure  of  one  atmosphere,  at  the  temperature  of  melting  ice,  *  and 
the  values  of  the  constants  are  given  for  the  Centigrade  scale. 

v=t(v)?;  v1  =  "G)f    •    •  <107-> 

Com.  log.  b  —  3-8181545  ;  Com.  log.  a  =  0-317616S. 

Hence  it  appears  that  the  integrals  in  the  formula  (106)  have  the 
following  values  : — 

/>y=«.A.(^!!/>T=¥.t.?.A.(^)*aMi.) 

i  i 

in  which  the  common  logarithms  of  the  constants  are 

Com.  log.  2  b  =  2-1101845  ;   log.  —  .  -  =  2-4017950  ; 

3      r0 

and  these  values  suit  any  scale  of  temperatures. 

In  calculating,  for  use  in  these  formulae,  the  densities  —  from  the  observed 

pressures,  it  is  sufficiently  near  the  truth,  in  the  case  of  air,  to  use  the 
approximate  equation 

1  T 

—    °  .  P  (in  atmospheres). 

V  T 

The  common  logarithm  of  r0,  the  absolute  temperature  of  melting  ice, 
for  the  Centigrade  scale,  is  2-4387005. 

The  constant  N  for  atmospheric  air  is  0"4  nearly ;  therefore 

Com.  log.  (N  X  hyp.  log.  10)  =  1-9642757. 

The  following,  therefore,  is  the  approximate  value  of  the  formula  (103) 
to  be  used  (with  the  numerical  constants  already  given)  in  reducing  the 
experiments  of  Mr.  Joule  and  Professor  Thomson  on  atmospheric  air,  so 
as  to  obtain  approximate  values  of  the  absolute  temperature  of  total 
privation  of  heat : — 

*  This  unit  of  volume  is  greater  than  the  actual  volume  of  air,  under  the  circum- 
stances described,  in  the  ratio  of  1  00085  to  1. 


330  THE   MECHANICAL   ACTION   OF   HEAT. 

-={^(jg.0)»A.lP*)-«^A.CP»))-X-A^ 

-4-  N  hyp.  log.  10  x  A  .  com.  log.  ,,.  .     (106.) 

In  using  this  formula,  the  mean  absolute  temperature  should  be  taken 
as  the  value  of  r. 

The  following  table  shows  the  values  of  the  quantity  k,  computed  from 
ten  mean  experimental  data,  taken  respectively  from  the  first  ten  series  of 
experiments  described  in  the  recent  paper  of  Messrs.  Joule  and  Thomson, 
in  the  supplementary  number  of  the  Philosophical  Magazine  for  December, 
L852.  The  temperatures  in  the  table,  for  the  sake  of  convenience,  are 
reduced  to  the  Centigrade  scale,  because  that  scale  has  been  used  through- 
out the  previous  sections  of  this  paper. 

The  final  pressure  in  each  case  was  thai  of  the  atmosphere. 

Professor  Thomson  and  Mr.  Joule  have  expressed  the  opinion,  which  is 
undoubtedly  correct,  that  those  experiments  in  which  the  largest  quantities 
of  air  Avere  used  were  the  least  liable  to  error  from  disturbing  causes,  such 
as  the  conduction  of  heat. 

Now,  it  may  1 bserved   in   the   table,   that    the    calculated  values 

of  k  are  generally  greatest,  and  the  discrepancies  amongst  them  least, 
for  the  experiments  in  which  most  air  was  used.  To  illustrate  this,  the 
results  of  the  last  eight  series  are  arranged  below  in  the  order  of  the 
quantities  of  air  i  mployed. 

Cubic  inches  1 1>4         ^         ^       g.fi         &l       ^       n.2       {].., 

per  second,  ) 

Values  of  ic,       1-683     1'762     2'09     2-228      151     2-087     2-345     2'14 

It  is  further  to  be  remarked,  that  the  discrepancy  between  the  highest 
and  the  lowest  of  the  values  of  k  is 

2°-345  -  l°-08  =  r-265  Centigrade: 

a  quantity  which  corresponds  to  a  difference  of  less  than  one  three-hundredth 
part  in  computing  the  proportion  of  heat  converted  into  mechanical  power 
by  any  ordinary  expansive  engine,  according  to  the  formula  (98),  which 
has  been  deduced  from  the  hypothesis  of  molecular  vortices. 

The  experiments,  therefore,  may  1  »e  considered  as  tending  to  prove,  that 
the  formulas  deduced  from  this  hypothesis  are  sufficiently  correct  for 
practical  purposes;  and  also  as  affording  a  strong  probability  that  the 
principles  to  which  it  leads  are  theoretically  exact,  and  that  the  tempera- 
ture of  absolute  privation   of  heat  is  a  real  fixed   point  on   the  scale, 


THE   MECHANICAL  ACTION   OF  HEAT. 


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332  THE   MECHANICAL   ACTION    OF    HEAT. 

somewhat  more  than  two  Centigrade  degrees  above  the  absolute  zero  of  a 
perfect  gas-thermometer  (which  is,  of  course,  an  imaginary  point) ;  that  is 
to  say,  about  272h  Centigrade  degrees,  or  490|  degrees  of  Fahrenheit, 
below  the  temperature  of  melting  ice. 

If  these  conclusions  be  correct,  it  follows,  that  when  the  temperatures 
Tt  and  T.„  between  which  an  expansive  engine  works,  are  measured  from 
the  ordinary  zero  points  of  the  Centigrade  and  of  Fahrenheit's  scales 
respectively,  the  following  are  the  utmost  proportions  of  the  total  heat 
expended  which  it  can  be  made  to  convert  into  mechanical  power : — 


For  the  Centigrade  scale, 
For  Fahrenheit's  scale, 


T,  -  T2 
Tj  +  2721 

Tx  +  458£  J 


(109.) 


In  the  fifth  section  of  this  paper,  where  a  comparison  is  made  between 
the  actual  duty  of  the  Cornish  engine  at  Old  Ford,  as  determined  by  Mr. 
Wicksteed,  and  the  greatest  possible  duty  which  could  be  obtained  from  a 
given  quantity  of  heat  by  a  theoretically  perfect  engine  working  between 
the  same  temperatures,  the  constant  k  is  treated  as  being  so  small  that  it 
may  be  neglected  in  practice.  If  the  value  of  k  is  really  20,1  Centigrade, 
as  computed  above,  the  calculated  maximum  theoretical  duty  in  Section  \ 
is  too  small  by  about  one  one-hundred-and-ninetieth  part  of  its  amount,, — 
a  quantity  of  no  practical  importance  in  such  calculations. 

61.  It  may  be  anticipated,  that  when  Mr.  Joule  and  Professor  Thom- 
son shall  have  performed  experiments  on  the  thermic  phenomena  exhibited 
by  air  in  more  copious  currents,  and  by  gases  of  more  definite  composition, 
and  more  simple  laws  of  elasticity,  much  more  precise  results  will  be 
obtained. 

When  a  gas  deviating  considerably  from  the  perfectly  gaseous  condition, 
or  a  vapour  near  the  point  of  saturation,  is  employed,  it  will  no  longer  be 
sufficiently  accurate  to  treat  the  specific  heat  at  constant  volume  as  a  con- 
stant quantity,  nor  the  cooling  effect  as  very  small.  It  will,  therefore,  be 
necessary  to  employ,  for  the  reduction  of  the  experiments,  the  integral 
form  of  equation  (99) — that  is  to  say, 

0  =  A  ^  =  A  -   h  t  +  fe  X  k-  (hyp.  log.  r  +  -J 

+  (('-^/r-0fp''v} 


THE   MECHANICAL   ACTION   OF  HEAT.  335 

-  k  {  A  (-^  rf  V  -  k  N  (A  .  5  +  A  hjT.  log.  t)  }  .   (110.) 

62.  Preliminary  to  the  application  of  this  equation,  it  is  necessary  to 
determine  the  mechanical  value  of  the  real  specific  heat  fe.  Supposing  the 
law  which  connects  the  pressure,  density,  and  temperature  of  the  gas  to 
be  known,  it  is  sufficient  for  this  purpose  to  have  an  accurate  experimental 
determination,  either  of  the  apparent  specific  heat  at  constant  pressure  for 
a  given  temperature,  or  the  velocity  of  sound  in  the  gas  under  given 
circumstances. 

First,  let  us  suppose  that  the  apparent  specific  heat  at  constant  pressure 
is  known. 

The  value  of  this  coefficient  (Centrifugal  Theory  of  Elasticity,  Art.  1 2)  is 


(dV 

\  dr 


v  dv   ; 


In  order  that  the  lower  limit  of  the  integral  may  correspond  with  the 
condition  of  perfect  gas,  it  is  convenient  to  transform  it  into  one  in  terms 
of  the  density.     Let  D  be  the  weight  of  unity  of  volume,  then 


/£"--#•£<*■     <1I1A-> 


If,  then,  we  have  the  pressure  of  the  gas  undeu  consideration  expressed 
by  the  following  approximate  formula  : — 


B-^U+v*} 


The  following  will  be  the  values  of  the  functions  of  the  pressure  which 
enter  into  the  above  equation  : — 


:p      PoV0  f  1       At  1    ^P  P0V0    A,  _      ,pvA1] 


334 


THE   MECHANICAL   ACTION   OF   HEAT. 


)  dr>         ~       i0D2"  dr**  +"     r-      i0D 

(dF.S 
\dr, 


—  =  p  v 
d  p       °  ° 

,/ v 


1  +  AA» 

rf.  A0  D       1      <Z .  A1 D 


To  illustrate  the  application  of  these  formula?,  let  us  calculate  the 
difference  between  the  real  specific  heat  and  the  apparent  specific  heat,  at 
constant  pressure,  of  carbonic  acid  gas.  at  the  temperature  of  melting  ice, 
and  at  the  density  which,  it'  the  gas  were  perfect,  would  correspond  to  a 
pressure  of  one  atmosphere  al  the  temperature  of  melting  ice.  Let  this 
density  be  denoted  by  D0,  and  its  reciprocal  by  V0.  As  the  constants  have 
been  deduced  from  M.  Regnault's  experiments,  the  calculations  will  be 
made  in  French  measures  and  for  the  latitude  of  Paris 

The  actual  density  of  carbonic  acid  at  0°  Centigrade,  and  under  one 

atmosphere  of  pressure,  exceeds  the  theoretical  density,  in  the  perfectly 

gaseous   state,    in  the   ratio   of  1'0065  to    1   nearly.     Hence,  the  height 

of  a  homogeneous  atmosphere  of  actual  carbonic  acid  at  0°  Centigrade 

being      .........         5225*5  metres, 

the  corresponding  height  in  the  state  of  perfect  gas  is  P0  V0=5259-5 

P  V 
and     "    -  =  19*53  metres  per  Centigrade  i  ---  02*84  !■ 

r 

The  functions  which  express  the  influence  of  density  on  thejdeviation 

of  carbonic  acid  gas  from  the  perfectly  gaseous  state,  have  the  following 

values : — 


b 


D 


A  T) 

A1  =  «.- 


when 


> 


Com.  log.  h  =  3*1083932;  Com.  log.  a  =  03344538 
6  =  0-00128349  a  =2*16: 


Mine.) 


Bn    d.D 


D       d 


J     ndD-J      A1]D.    ^    -«-DflJ  rfD'-^o^ 


=  2b 


D       d 


D0'  rfD' 


A,D 


i> 


For  the  purposes  of  a  first  approximation,  we  may  assume  that  the 


THE   MECHANICAL   ACTION   OF   HEAT.  335 

value  of  k  already  found  is  sufficiently  near  the  truth — viz.,  20,1   Centi- 
grade, so  that,  in  the  present  instance,  t  —  k  =  2  7  2°  5  Centigrade. 
Then  we  find  the  following  results  when  r  =  r0,  and  D=D0: 

Metres.  Feet. 

(T  -  K)  P° -V°  .  -7  =  per  Centigrade  degree,  .         .          .     0*145  0-48 

/'  <T-  P 
(T  —  K\\ — -  d  V  =■  per  Centigrade  degree,  .         .         .     0*150         0-49 

J    tl  t  

Sum  =  Kv  —  ll  =  excess  of  apparent  specific  heat  at  con- 
stant volume  above  real  specific 
heat 0-295  0-97 

,/r 

r 


(j  _  K)         T/     =  difference  between  apparent  specific 


heats  at  constant  volume   and  at 

constant  pressure,  .  .  .   19'565        64-19 


Kr  —  ft  =  excess  of  apparent  specific  heat  at  con- 
stant pressure  above  real  specific 
heat 19-8G0       65-16 

4  of  the  above  quantities  are  of  course  the  corresponding  quantities  for 
Fahrenheit's  scale. 

Secondly,  If  the  velocity  of  sound  in  the  gas  is  given,  let  this  =  u.     Then 
Ave  know  that 

v?  =  q    -P     K''  •  •  •     (112.) 

9    dD    Kv  V        ; 

in  which 


d? 

d 


P       p  y   f  r    ,   d.A0D       ld^AjD)         .        A) 


So  that  from  the  velocity  of  sound  we  can  calculate  the  ratio  of  the  specific- 
heats  at  constant  pressure  and  at  constant  volume.  Let  this  ratio  be 
denoted  by  y,  and  let 


Kv  =  fc+  c;  Kp  =  ft  +  c'; 

then 

fc  +  c 

;  ana  k  =  - 

7 


=  l±^;  andft  =  ^-C    .         .     (H2B.) 

B+C  y  —    1 


i.3G 


THE  MECHANICAL   ACTION   OF   HEAT. 


in  which  c  and  c  are  to  be  calculated  as  above.* 

G3.  In  using  the  formula  (110)  for  a  gas  whose  pressure  is  represented 
by  the  formula 

the  integrals  may  be  transformed  so  as  to  be  taken,  with  respect  to  the 
density,  as  in  the  preceding  article.     Thus  we  obtain 

*J(^*)"--4*v-S-*)»= 

-   fl.^Dn:  -PoV0A(^hyP.log.I)+^/;V;^D)   (113.) 
For  carbonic  acid,  the  first  of  these  formula;  becomes  simply 


+ 


^gQ-^KOw. 


J 


and  the  second,  )■  (113  A.) 

n„   fl,        .       D,         a  /Dt       DA  ) 
+  P0Yu-|.)hvp.log.I)'-Do(^-^)^ 


APPENDIX. 

Note  A.  (to  Article  58). — Since  this  section  was  read,  the  theoretical 
views  relative  to  the  relation  between  heat  and  temperature  contained  in 
it  and  the  previous  sections  of  this  paper,  have  received  a  strong  confir- 
mation by  the  publication  by  M.  Eegnault  of  the  fact,  that  he  has  found 
the  specific  heat  of  air  to  be  sensibly  constant  at  all  temperatures 
from  —  30°  Centigrade  to  +  225°,  and  at  all  pressures  from  one  to  ten 
atmospheres  (Comptes  Rcnclus,  April  18,  1853);  so  that  equal  lengths  on 
the  scale  of  the  air  thermometer  represent  equal  quantities  of  heat. 

Note  B.  (to  Article  62). — Until  very  recently,  there  existed  no  exact 

*  See  Appendix,  Note  B. 


THE   MECHANICAL   ACTION   OF   HEAT. 


337 


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338  THE  MECHANICAL   ACTION   OF   HEAT. 

experimental  determination  of  the  specific  heat  of  any  gas.  The  specific 
heat  of  air  at  constant  pressure,  as  compared  with  that  of  water,  was 
calculated  theoretically  in  the  previous  part  of  this  paper,  from  Joule's 
equivalent  and  the  velocity  of  sound,  and  found  to  be  0'24.  This  value 
has  since  been  confirmed  very  closely  by  Mr.  Joule's  experiments,  whose 
mean  result  was  0'23,  and  still  more  exactly  by  M.  Kegnault's  experiments, 
already  referred  to,  which  give  the  value  0-2379.  The  table  {See p.  337), 
shows  the  results  of  the  application  of  the  formula?  of  this  paper  to  the 
specific  heats  of  five  different  gases  at  constant  pressure,  selected  from  M. 
Kegnault's  table  (Comptes  Rendus,  April  1 8),  as  being  those  in  which  the 
velocity  of  sound  can  be  computed,  and  has  been  determined  experi- 
mentally. The  table  shows  also  a  comparison  of  the  calculated  and 
observed  velocities  of  sound.  This  table  appeared  originally,  in  French 
measures,  in  the  Philosophical  Magazine  for  June,  1853:  the  metres  are 
here  reduced  to  feet.  Kp,  Kv,  and  Kw,  are  expressed  in  feet  of  fall  per 
Centigrade  degree.     Kw  (Joule's  equivalent)  =  1389'G 

The  real  specific  heat  of  carbonic  acid  gas  is  235*5  feet  of  fall  per 
Centigrade  degree.  That  of  the  other  gases  does  not  differ  from  the 
apparent  specific  boat  at  constant  volume  by  an  amount  appreciable  in 
practice. 


ON  THERMODYNAMICS. 


339 


XX.— ON  THE  GEOMETRICAL  REPRESENTATION  OF 

THE  EXPANSIVE  ACTION  OF  HEAT,  AND  THE 

THEORY  OF  THERMODYNAMIC  ENGINES  * 

Section  I. — Introduction  and  General  Theorems. 


1.  The  first  application  of  a  geometrical  diagram  to  represent  the  expan- 
sive action  of  heat  was  made  by  James  Watt,  when  he  contrived  the 
well-known  steam-engine  indicator,  subsequently  altered  and  improved 
by  others  in  various  ways.  As  the  diagram  described  by  Watt's  Indicator 
is  the  type  of  all  diagrams  representing  the  expansive  action  of  heat,  its 
general  nature  is  exhibited  in  Fig.  1. 

Let  abscissas,  measured  along,  or  par- 
allel to,  the  axis  0  X  represent  the  vol- 
umes successively  assumed  by  a  given 
mass  of  an  elastic  substance,  by  whose 
alternate  expansion  and  contraction  heat 
is  made  to  produce  motive  power ;  0  VA 
and  0  VB  being  the  least  and  greatest 
volumes  which  the  substance  is  made  to 
assume,  and  OV  any  intermediate  vol- 
ume. For  brevity's  sake,  these  cmantities 
will  be  denoted  by  VA,  YB,  and  V,  respec- 
tively. Then  VB  —  VA  may  represent  the 
space  traversed  by  the  piston  of  an  engine 
during  a  single  stroke. 

Let  ordinates,  measured  parallel  to  the  axis  0  Y,  and  at  right  angles  to 
0  X,  denote  the  expansive  pressures  successively  exerted  by  the  substance 
at  the  volumes  denoted  by  the  abscissae.  During  the  increase  of  volume 
from  VA  to  VB,  the  pressure,  in  order  that  motive  power  may  be  produced, 
must  be,  on  the  whole,  greater  than  during  the  diminution  of  volume  from 
VB  to  VA;  so  that,  for  instance,  the  ordinates  V  Px  and  V  P2,  or  the  symbols 


*  Read  before  the  Royal  Society  of  London  on  January  19,  1834,  and  published  in 
the  Philosophical  Transactions  for  1854. 


340  ON   THERMODYNAMICS. 

P,  and  P0,  may  represent  the  pressures  corresponding  to  a  given  volume 
V  during  the  expansion  and  contraction  of  the  substance  respectively. 

Then  the  area  of  the  curvilinear  figure,  or  Indicator-diagram,  A  Px  B  P2  A, 
will  represent  the  motive  power,  or  "  potential  energy,"  developed  or  given 
out  during  a  complete  stroke,  or  cycle  of  changes  of  volume  of  the  elastic 
substance.     The  algebraical  expression  for  this  area  is 


P,)  i  V (l.) 


The  practical  use  of  such  diagrams,  in  ascertaining  the  power  and  the 
mode  of  action  of  the  steam  in  steam-engines,  where  the  curve  A  Pt  B  P2  A 
is  described  by  a  pencil  attached  to  a  pressure-gauge  on  a  curd  whose 
motion  corresponds  with  that  of  the  piston,  is  sufficiently  well  known. 

2.  It  appears  that  the  earliest  application  of  diagrams  of  energy  (as  they 
may  be  called)  to  prove  and  illustrate  the  theoretical  principles  of  the 
mechanical  action  of  heat,  was  made  either  by  Carnot,  or  by  M.  Clapeyron 
in  his  account  of  Carnot's  theory ;  but  the  conclusions  of  those  authors 
were  in  a  great  measure  vitiated  by  the  assumption  of  the  substantiality 
of  heat. 

In  the  fifth  section  of  a  paper  on  the  Mechanical  Action  of  Heat, 
published  in  the  Transactions  of  the  Royal  Society  of  Edinburgh  (Seep. 
300),  a  diagram  of  energy  is  employed  to  demonstrate  the  general  law  of 
the  economy  of  heat  in  thermodynamic  engines  according  to  the  correct 
principle  of  the  action  of  such  machines — viz.,  that  the  area  of  the  diagram 
represents  at  once  the  potential  energy  or  motive  power  which  is  de- 
veloped at  each  stroke  and  the  mechanical  equivalent  of  the  actual  energy, 
or  heat,  Avhich  permanently  disappears. 

As  the  principles  of  the  expansive  action  of  heat  are  capable  of  being 
presented  to  the  mind  more  clearly  by  the  aid  of  diagrams  of  energy  than 
by  means  of  words  and  algebraical  symbols  alone,  I  purpose,  in  the  present 
paper,  to  apply  those  diagrams  partly  to  the  illustration  and  demonstration 
of  propositions  already  proved  by  other  means,  but  chiefly  to  the  solution 
of  new  questions,  especially  those  relating  to  the  action  of  heat  in  all 
classes  of  engines,  whether  worked  by  air,  or  by  steam,  or  by  any  other 
material ;  so  as  to  present,  in  a  systematic  form,  those  theoretical  principles 
which  are  applicable  to  all  methods  of  transforming  heat  to  motive  power 
by  means  of  the  changes  of  volume  of  an  elastic  substance. 

Throughout  the  whole  of  this  investigation,  quantities  of  heat,  and 
coefficients  of  specific  heat,  are  expressed,  not  by  units  of  temperature  in 
a  unit  of  weight  of  water,  but  by  equivalent  quantities  of  mechanical 
power,  stated  in  foot-pounds,  according  to  the  ratio  established  by  Mr. 
Joule's  experiments  on  friction  (Phil.  Trans.,  1850);  that  is  to  say, 


ON   THERMODYNAMICS. 


341 


772  foot-pounds  per  degree  of  Fahr.,  or 
1389-6  foot-pounds  per  Centigrade  degree, 

applied  to  one  pound  of  liquid  water  at  atmospheric  temperatures. 

3.   Of  Isothermal  Curves,  and  Curves  of  No  Transmission  of  Heat. 

A  curve  described  on  a  diagram  of  energy,  such  that  its  ordinates 
represent  the  pressures  of  a  homogeneous  substance  corresponding  to 
various  volumes,  while  the  total  sensible  or  actual  heat  present  in  the  body- 
is  maintained  at  a  constant  value,  denoted,  for  example,  by  Q,  may  be 
called  the  isothermal  curve  of  Q  for  the  given  substance  (See  Fig.  2). 
Suppose,  for  instance,  that  the  co-ordinates  of  the  point  A,  VA  and  PA, 
represent  respectively  a  volume  and  a  pressure  of  a  given  substance,  at 
which  the  actual  heat  is  Q ;  and  the  co-ordinates  of  the  point  B — viz.,  VB 
and  PB,  another  volume  and  pressure '  at  which  the  actual  heat  is  the 
same ;  then  are  the  points  A  and  B  situated  on  the  same  isothermal 
curve  Q  Q. 

On  the  other  hand,  let  the  substance  be  allowed  to  expand  from  the 
volume  and  pressure  VA,  PA, 
without  receiving  or  emitting 
heat;  and  when  it  reaches  a 
certain  volume,  Vc,  let  the 
pressure  be  represented  by 
Pc,  which  is  less  than  the 
pressure  would  have  been 
had  the  actual  heat  been 
maintained  constant,  because 
by  expansion  heat  is  made 
to  disappear.  Then  C  will 
be  a  point  on  a  certain  curve 
N  N  passing  through  A, 
which  may  be  called  a  Curve 
of  No  Transmission. 

It  is  to  be  understood,  that  "during  the  process  last  described,  the 
potential  energy  developed  during  the  expansion,  and  which  is  represented 
by  the  area  A  C  Vc  VA,  is  entirely  communicated  to  external  substances ; 
for  if  any  part  of  it  were  expended  in  agitating  the  particles  of  the 
expanding  substance,  a  portion  of  heat  would  be  reproduced  by  friction. 

If  o  o  o  be  a  curve  whose  ordinates  represent  the  pressures  corresponding 
to  various  volumes  when  the  substance  is  absolutely  destitute  of  heat,  then 
this  curve,  which  may  be  called  the  Curve  of  Absolute  Cold,  is  at  once  an 
isothermal  curve  and  a  curve  of  no  transmission. 

So  far  as  we  yet  know,  the  curve  of  absolute  cold  is,  for  all  substances, 
an  asymptote  to  all  the  other  isothermal  curves  and  curves  of  no  trans- 


Fig.  2. 


342  ON  THERMODYNAMICS. 

mission,  which  approach  it  and  each  other  indefinitely  as  the  volume  of 
the  substance  increases  without  limit. 

Note. — The  following  remarks  are  intended  to  render  more  clear  the 
precise  meaning  of  the  term  Total  Actual  Heat. 

The  Total  Actual  Heat  of  a  given  mass  of  a  given  substance  at  a  given 
temperature,  is  the  cpiantity  of  physical  energy  present  in  the  mass  in  the 
form  of  heat  under  the  given  circumstances. 

If,  for  the  purpose  of  illustrating  this  definition,  we  assume  the  hypo- 
thesis that  heat  consists  in  molecular  revolutions  of  a  particular  kind, 
then  the  Total  Actual  Heat  of  a  mass  is  measured  by  the  mechanical 
power  corresponding  to  the  vis  viva  of  those  revolutions,  and  is  repre- 
sented by 

-  2 .  m  v2, 

m  being  the  mass  of  any  circulating  molecule,  and  i>2  the  mean  square  of 
its  velocity. 

But  the  meaning  of  the  term  Total  Actual  Heat  may  also  be  illustrated 
without  the  aid  of  any  hypothesis. 

For  this  purpose,  let  us  take  the  ascertained  fact  of  the  production  of 
heat  by  the  expenditure  of  mechanical  power  in  friction,  according  to  the 
numerical  proportion  determined  by  Mr.  Joule;  and  let  E  denote  the 
quantity  of  mechanical  power  which  must  be  expended  in  friction,  in 
order  to  raise  the  temperature  of  unity  of  weight  of  a  given  substance 
from  that  of  absolute  privation  of  heat  to  a  given  temperature  r. 

During  this  operation,  let  the  several  elements  of  the  external  surface 
of  the  mass  undergo  changes  of  relative  position  expressed  by  the 
variations  of  quantities  denoted  generally  by^,  and  let  the  increase  of  each 
such  quantity  as  p  be  resisted  by  an  externally-applied  force  such  as  P. 

Then,  during  the  elevation  of  temperature  from  absolute  cold  to  r,  the 
energy  converted  to  the  potential  form  in  overcoming  the  external  pressures 
P  will  be 


fvdp. 


Also,  let  the  internal  particles  of  the  mass  undergo  changes  of  relative 
position,  expressed  by  the  variations  of  quantities  denoted  generally  by  r, 
and  let  the  increase  of  each  such  quantity  as  r  be  resisted  by  an  internal 
molecular  force  such  as  R  : 

Then  the  energy  converted  to  the  potential  form  in  overcoming  internal 
molecular  forces  will  be 

2  .  I E  d  r. 


ON  THERMODYNAMICS.  343 

Subtracting  these  quantities  of  energy  converted  to  the  potential  form 
by  means  of  external  pressures  and  internal  forces,  from  the  whole  power 
converted  into  heat  by  friction  in  order  to  raise  the  temperature  of  the 
mass  from  that  of  absolute  privation  of  heat  to  the  given  temperature 
t,  we  find  the  following  result : — 


Q  =  E-2.[p<Zp-2.( 


Hdr 


and  this  remainder  is  the  quantity  of  energy  which  retains  the  form  of  heat 
in  unity  of  weight  of  the  given  substance  at  the  given  temperature ;  that 
is  to  say,  the  Total  Actual  Heat. 

It  is  obvious  that  Total  Actual  Heat  cannot  be  ascertained  directly ; 
first,  because  the  temperature  of  total  privation  of  heat  is  unattainable ; 
and,  secondly,  because  the  molecular  forces  R  are  unknown. 

It  can,  however,  be  determined  indirectly  from  the  latent  heat  of 
expansion  of  the  substance.  For  the  heat  which  disappears  during  the 
expansion  of  unity  of  weight  of  an  elastic  substance  at  constant  actual 
heat  from  the  volume  VA  to  the  volume  VB,  under  the  constant  or  variable 
pressure  P,  is  expressed  (as  will  be  shown  in  the  sequel)  by 


q-/q/,>V; 


so  that  from  a  sufficient  number  of  experiments  on  the  amount  of  heat 
transformed  to  potential  energy  by  the  expansion  of  a  given  substance, 
the  relations,  for  that  substance,  between  pressure,  volume,  and  total 
actual  heat,  may  be  determined. 

4.  Proposition  I. — Theorem.  The  mechanical  equivalent  of  the  heat 
absorbed  or  given  out  by  a  substance  in  passing  from  one  given  state  as  to 
pressure  and  volume  to  another  given  state,  through  a  series  of  states  represented 
by  the  co-ordinates  of  a  given  curve  on  a  diagram  of  energy,  is  represented  by 
the  area  included  between  the  given  curve  and  two  curves  of  no  transmission  of 
heat  drawn  from  its  extremities,  and  indefinitely  prolonged  in  the  direction 
representing  increase  of  volume. 

(Demonstration :  see  Fig.  3.)  Let  the  co-ordinates  of  any  two  points, 
A  and  B,  represent  respectively  the  volumes  and  pressures  of  the  substance 
in  any  two  conditions ;  and  let  a  curve  of  any  figure,  A  C  B,  represent, 
by  the  co-ordinates  of  its  points,  an  arbitrary  succession  of  volumes  and 
pressures  through  which  the  substance  is  made  to  pass,  in  changing  from 
the  condition  A  to  the  condition  B.  From  the  points  A  and  B  respectively 
let  two  curves  of  no  transmission  AM,  BN,  extend  indefinitely  towards  X; 
then  the  area  referred  to  in  the  enunciation  is  that  contained  between 


344 


ON  THERMODYNAMICS. 


the  given  arbitrary  curve  A  C  B  and  the  two  indefinitely  prolonged  curves 
of  no  transmission;  areas  above  the  curve  AM  bciii^-  considered  as 
representing  heat  absorbed  by  the  substance,  and  those  below  heat 
given  out. 

To  fix  the  ideas,  let  us,  in  the  first  place,  suppose  the  area  MACBN 


Re.  3. 


to  be  situated  above  A  M.  After  the  substance  has  reached  the  state  B, 
let  it  be  expanded  according  to  the  curve  of  no  transmission  BN,  until 
its  volume  and  pressure  are  represented  by  the  co-ordinates  of  the  point  D'. 
.Next,  let  the  volume  V„  be  maintained  constant,  while  heat  is  abstracted, 
until  the  pressure  falls  so  as  to  be  represented  by  the  ordinate  of  the 
point  D,  situated  on  the  curve  of  no  transmission  A  M.  Finally,  let  the 
substance  be  compressed,  according  to  this  curve  of  no  transmission,  until 
it  recovers  its  primitive  condition  A.  Then  the  area  AcBD'DA,  which 
represents  the  whole  potential  energy  developed  by  the  substance  during 
one  cycle  of  operations,  represents  also  the  heat  which  disappears,  that  is. 
the  difference  between  the  heat  absorbed  by  the  substance  during  the 
change  from  A  to  1!.  and  emitted  during  the  change  from  D'  to  D;  for 
if  this  were  not  so,  the  cycle  of  operations  would  alter  the  amount  of 
energy  in  the  universe,  which  is  impossible. 

The  farther  the  ordinate  Vr,PD'  is  removed  in  the  direction  of  X.  the 
smaller  does  the  heat  emitted  during  the  change  from  D'  to  D  become; 
and,  consequently,  the  more  nearly  does  the  area  ACBD'DA  approximate 
to  the  equivalent  of  the  heat  absorbed  during  the  change  from  A  to  B; 
to  which,  therefore,  the  area  of  the  indefinitely  prolonged  diagram 
M  A  C  B  N  is  exactly  equal.     Q.  E.  D. 

It  is  easy  to  see  how  a  similar  demonstration  could  have  been  applied, 
■mutatis  mutandis,  had  the  area  lain  below  the  curve  AM.  It  is  evident 
also,  that  when  this  area  lies,  part  above  and  part  below  the  line  A  M. 
the  difference  between  these  two  parts  represents  the  difference  between 


ON   THERMODYNAMICS.  345 

the  heat  absorbed  and  the  heat  emitted  during-  different  parts  of  the 
operation. 

5.  First  Corollary. — Theorem.  The  difference  between  the  whole,  heat 
absorbed,  and  the  whole  expansive  power  developed,  during  the  operation  repre- 
sented by  any  curve,  such  as  A  C  B,  on  a  diagram  of  energy,  depends  on  the 
initial  and  final  conditions  of  the  substance  alone,  and  not  on  the  intermediate 
process. 

(Demonstration.)  In  Fig.  3,  draw  the  ordinates  AVA,  BVB  parallel  to 
0  Y.  Then  the  area  VA  A  C  B  V„  represents  the  expansive  power 
developed  during  the  operation  ACB;  and  it  is  evident  that  the  difference 
between  this  area  and  the  indefinitely-prolonged  area  MAC  B  N,  which 
represents  the  heat  received  by  the  substance,  depends  simply  on  the 
positions  of  the  points  A  and  B,  which  denote  the  initial  and  final 
conditions  of  the  substance  as  to  volume  and  pressure,  and  not  on  the 
form  of  the  curve  ACB,  which  represents  the  intermediate  process.    Q.E.D. 

To  express  this  result  symbolically,  it  is  to  be  considered,  that  the 
excess  of  the  heat  or  actual  energy  received  by  the  .substance  above  the 
expansive  power  or  potential  energy  given  out  and  exerted  on  external 
bodies,  in  passing  from  the  condition  A  to  the  condition  B,  is  equal  to 
the  whole  energy  stored  "jp  in  the  substance  during  this  operation,  which 
consists  of  two  parts,  viz. — 

Actual  energy;  being  the  increase  of  the  actual  or  sensible  heat  of  the 
substance  in  passing  from  the  condition  A  to  the  condition  B,  which  is  to 
be  represented  by  this  expression, 

A.Q  =  QB-QA; 

Potential  energy;  1  icing  the  power  which  is  stored  up  in  producing 
changes  of  molecular  arrangement  during  this  process;  and  which  it  appears 
from  the  theorem  just  proved,  must  be  represented,  like  the  actual  energy, 
by  the  difference  between  a  function  of  the  volume  and  pressure  corre- 
sponding to  A,  and  the  analogous  function  of  the  volume  and  pressure 
corresponding  to  B ;  that  is  to  say,  by  an  expression  of  the  form, 


Let 


AS  =  SB-SA. 


HA>B  =  areaMACBX 


represent    the    heat    received    by    the   substance    during    the    operation 
A  C  B,  and 

P  d  V  =  area  VA  A  C  B  VB 
the  power  or  potential  energy  given  out. 


346 


ON  THERMODYNAMICS. 


Then,  the  theorem  of  this  article  is  expressed  as  follows  : — 

,V„ 


HAlB-  I 


PrfV  =  QB-QA  +  SB-SA  =  AQ  +  A.S,    (2.) 


being  a  form  of  the  general  equation  of  the  expansive  action  of  heat,  in 
which  the  potential  of  molecular  action,  S,  remains  to  be  determined. 

6.  Second  Corollary  (see  Fig.  4). — The  Latent  Heat  of  Expansion  of 
a  substance  from  one  given  volume  VA  to  another  VB,  for  a  given  amount 
of  actual  heat  Q; 


Fig.  4. 

that  is  to  say,  the  heat  which  must  be  absorbed  by  the  substance  in  expand- 
ing from  the  volume  VA  to  the  volume  VB,  in  order  that  the  actual  heat  Q 
may  be  maintained  constant,  is  represented  geometrically  as  follows: 
Let  Q  Q  be  the  isothermal  curve  of  the  given  actual  heat  Q  on  the 
diagram  of  energy;  A,  B  two  points  on  this  curve,  whose  co-ordinates 
represent  the  two  given  volumes  and  the  corresponding  pressures.  Through 
A  and  B  draw  the  two  curves  of  no  transmission  A  M,  B  N,  produced 
indefinitely  in  the  direction  of  X.  Then  the  area  contained  between  the 
portion  of  isothermal  curve  AB,  and  the  indefinitely-produced  curves  AM, 
B  N,  represents  the  mechanical  equivalent  of  the  latent  heat  sought,  Avhose 
symbolical  expression  is  formed  from  equation  (2)  by  making  Q,.  —  QA  =  0, 
and  is  as  follows: — 

HA,  B  (for  Q  =  const.)  =  J  ^  P  tf  V  +  SB  -  SA.        .     (3.) 


Section  II. — Propositions  Belative  to  Homogeneous  Substances. 

7.  Proposition  II.— Theorem.     In  Fig.   5,  let  Ax  A2  M,  Bx  B2  N  he 
any  two  curves  of  no  transmission,  indefinitely  extended  in  the  direction  of  X, 


ON   THERMODYNAMICS. 


347 


intersected  in  the  points  Av  Bv  A2,  B2,  by  two  isothermal  curves  Qx  Ax  Bx  Qx, 
Q0A0B0  Q0,  which  are  indefinitely  near  to  each  other;  that  is  to  say,  which 
correspond  to  two  quantities  of  actual  heat,  Qx  and  Q2,  differing  by  an  indefinitely 
small  quantity  Qx  —  Q2  =  SQ. 

Then   the  elementary  quadrilateral  area,  A1  Bx  B2  A2,  bears  to  the  whole 


indefinitely-prolonged  area  MA1B1N,  the  same  proportion  which  the  indefi- 
nitely small  difference  of  actual  heat  SQ  bears  to  the  ivhole  actual  heat  Q:;  or 

area  Ax  Ba  B2  A2  S  Q 

area  M  Ax  B1  N  ~~    Qx' 

(Demonstration.)  Draw  the  ordinates  A1VAl,  A2VA2,  B^^,  B2VB2. 
Suppose,  in  the  first  place,  that  8  Q  is  an  aliquot  part  of  Q1?  obtained  by 
dividing  the  latter  quantity  by  a  very  large  integer  n,  which  we  are  at 
liberty  to  increase  without  limit. 

The  entire  indefinitely-prolonged  area  M  A1 B1  N  represents  a  quantity 
of  heat  which  is  converted  into  potential  energy  during  the  expansion  of  the 
substance  from  VAl  to  VBl,  in  consequence  of  the  continued  presence  of  the 
total  actual  heat  Qx ;  for  if  no  heat  were  present  no  such  conversion  would 
take  place.  Mutatis  mutandis,  a  similar  statement  may  be  made  respecting 
the  area  MA.,B,K  By  increasing  without  limit  the  number  n  and 
diminishing  8  Q,  we  may  make  the  expansion  from  VA2  to  VB2  as  nearly 
as  we  please  an  identical  phenomenon  with  the  expansion  from  VAl  to  VBl. 
The  quadrilateral  Ax  Bl  B2  A2  represents  the  diminution  of  conversion  of 
heat  to  potential  energy,  which  results  from  the  abstraction  of  any  one 
whatsoever  of  the  n  small  equal  parts  §  Q  into  which  the  actual  heat  Q: 
is  supposed  to  be  divided,  and  it  therefore  represents  the  effect,  in 
conversion  of  heat  to  potential  energy,  of  the  presence  of  any  one 
of  those  small  portions  of  actual  heat.  And  as  all  those  portions 
8  Q,  are  similar  and  similarly  circumstanced,  the  effect  of  the  presence 
of  the  whole  actual  heat  Qx  in  causing  conversion  of  heat  to  potential 


348  ON   THERMODYNAMICS. 

energy,  will  be  simply  the  sum  of  the  effects  of  all  its  small  portions, 

and  will  bear  the  same  ratio  to  the  effect  of  one  of  those  small  portions 

which   the    whole  actual   heat   bears  to  the   small   portion.      Thus,    by 

virtue  of  the  general  law  enunciated  below  and  assumed  as  an  axiom, 

the  theorem  is  proved  when  SQ  is  an  aliquot  part  of  Ql;  but  SQ  is 

either  an  aliquot  part,  or  a  sum  of  aliquot  parts,  or  may  be  indefinitely 

approximated  to  by   a  series  of  aliquot  parts;   so   that  the  theorem  is 

universally  true.     Q.  E.  D. 

The  symbolical  expression   of   this  theorem  is  as  follows :   when    the 

actual  heat  Ql5  at  any  given  volume,  is  varied  by  the  indefinitely  small 

quantity   8  Q,   let  the  pressure  vary  by  the  indefinitely  small   quantity 

(IP 

y-r-  B  Q ;   then   the  area  of  the  quadrilateral  Ax  ~B1  B2  A2  will  be  rcpre- 

(l  v^ 

sented  by 

v 

V  A,  1 

and,  consequently,  that  of  the  whole  figure  MAjBjN",  or  the  latent  heat 
of  expansion  from  VAl  to  VBl,  at  Qu  by 

V 

'    "'  clV 

■V;         .  .  .      (4.) 


^    A   1 


a  result  identical  with  that  expressed   in   the   sixth  section  of   a  paper 
published  in  the  Transactions  of  th  Royal  Society  of  Edinburgh.  (Seep.  310) 
The  demonstration  of  this  theorem  is  an  example  of  a  special  application 
of  the  following 


General  Law  of  the  Transformation  of  Energy. 

The  effect  of  the  presence,  in  a  substance,  of  a  quantify  of  actual  energy, 
in  causing  transformation  of  energy,  is  the  sum  of  the  effects  of  all  its  parts: — 
a  law  first  enunciated  in  a  paper  read  by  me  to  the  Philosophical  Society 
of  Glasgow  on  the  5th  of  January,  1853.     (See  p.  203.) 

8.  General  Equation  of  the  Expansive  Action  of  Heat. 

The  two  expressions  for  the  latent  heat  of  expansion  at  constant 
actual  heat,  given  in  equations  (3)  and  (4)  respectively,  being  equated, 
furnish  the  means  of  determining  the  potential  energy  of  molecular  action 
S,  so  far  as  it  depends  on  volume,  and  thus  of  giving  a  definite  form  to 
the  general  equation  (2). 

The  two  expressions  referred  to  may  be  thus  stated  in  words: — 


ON   THERMODYNAMICS.  349 

I.  The  heat  which  disappears  in  producing  a  given  expansion,  while 
the  actual  heat  present  in  the  substance  is  maintained  constant,  is  equiva- 
lent to  the  sum  of  the  potential  energy  given  out  in  the  form  of  expansive 
power,  and  the  potential  energy  stored  up  by  means  of  molecular  attractions. 

II.  It  is  also  equivalent  to  the  potential  energy  due  to  the  .action  during 

d  P 

the  expansion  of   a  pressure  Q-ttt,  at  each  instant  equal  to  what  the 

pressure  would  be,  if  its  actual  rate  of  variation  with  heat  at  the  instant 
in  question  were  a  constant  coefficient,  expressing  the  ratio  of  the  whole 
pressure  to  the  whole  actual  heat  present. 

The  combination  of  these  principles,  expressed  symbolically,  gives  the 
following  result : 

V  V 

HA,B(for  Q  =  const.)  =  Q  f      ^d\  =  f  ' P dV  +  SB  -  8A; 

*  A  '  A 

whence  Ave  deduce  the  following  general  value  for  the  potential  of  mole- 
cular action : — 

S  =  /(Qff-P)<*V  +  *-Q'  *  •     <5-> 

in  which  <£.Q  denotes  some  function  of  the  total  actual  heat  not  depending 
on  the  density  of  the  substance.  This  value  being  introduced  into 
equation  (2),  produces  the  following  : — 

HA,J      PiV  =  Q.-Q4  +  S,-SA 

=  Qb  -  Qa  +  <j>  .  QB  -  <j> .  QA  +  /  "(q  ~-  p)tf  v  =  ¥B  -  *A.    (6.) 

'   A 

The  symbol  ¥"  =  Q  +  S  is  used  to  denote  the  sum  of  the  actual  energy 
of  heat,  and  the  potential  energy  of  molecular  action,  present  in  the 
substance  in  any  given  condition. 

The  above  is  the  General  Equation  of  the  Expansive  Action  of 
Heat  in  a  Homogeneous  Substance,  and  is  the  symbolical  expression 
of  the  Geometrical  Theorems  I.  and  II.  combined. 

When  the  variations  of  actual  heat  and  of  volume  become  indefinitely 
small,  this  equation  takes  the  following  differential  form : — 

d.¥  =  d.K-FdV  =  dQ  +  d.S 


=  (l  +  0'.Q  +  Q^/P(Jv)iQ  +  (Qf|-P>T) 


otherwise 


y  (<) 


*.H-£.*Q  +  Qg.iV. 


350 


ON   THERMODYNAMICS. 


The  coefficient  of  d  Q  in  the  above  expressions,  viz. — 


(8.) 


is  the  ratio  of  the  apparent  specific  heat  of  the  substance  at  constant 
volume  to  its  real  specific  heat ;  that  is,  the  ratio  of  the  whole  heat 
consumed  in  producing  an  indefinitely  small  increase  of  actual  heat,  to 
the  increase  of  actual  heat  produced. 

These  general  equations  are  here  deduced  independently  of  any  special 
molecular  hypothesis,  as  they  also  have  been,  by  a  method  somewhat 
different,  in  the  sixth  section  of  a  paper  previously  referred  to.  Equations 
equivalent  to  the  above  have  also  been  deduced  from  the  hypothesis 
of  molecular  vortices,  in  the  paper  already  mentioned,  and  in  a  paper  on 
the  Centrifugal  Theory  of  Elasticity.     (See  p.  4$.) 

9.  First  Corollary  from  Proposition  II. — Theorem.  If  a  succession  of 
isothermal  curves  corresponding  to  quantities  of  heal  diminishing  by  equal  small 
differences  $  Q,  be  drawn  across  any  pair  of  curves  of  no  transmission,  they  ivill 
cut  off  a  series  of  equal  small  quadrilaterals. 

Second  Corollary. — Theorem.  In  Fig.  G,  let  ADM,BCN  be  any  two 
curves  of  no  transmission,  indefinitely  prolonged  in  the  direction  of  X,  and  let 


Fig.  6. 

any  two  isothermal  curves  Qt  Q15  Q2  Q2,  corresponding  respectively  to  any  two 
quantities  of  actual  heat  Qx,  Q2,  be  drawn  across  them.  Then  will  the 
indefinitely-prolonged  areas  MABN,  M DCN,  bear  to  each  other  the  simple 
ratio  of  the  quantities  of  actual  heat,  Qv  Q2. 

Or,  denoting  those  areas  respectively  by  ~H_V  H2 — 

H 


Hi  ~  Q/ 


|      ....  (9, 

This  corollary  is  the  geometrical  expression  of  the  law  of  the  maximum 


ON  THERMODYNAMICS.  351 

efficiency  of  a  perfect  thermodynamic  engine,  already  investigated  by 
other  methods.  In  fact,  the  area  MABN  represents  the  whole  heat 
expended,  or  the  latent  heat  of  expansion,  the  actual  heat  at  which  heat 
is  received  being  Qt ;  M  D  C  N,  the  heat  lost,  or  the  latent  heat  of  com- 
pression, which  is  carried  off  by  conduction  at  the  actual  heat  Q2 ;  and 
ABCD  (being  the  indicator-diagram  of  such  an  engine),  the  motive 
power  produced  by  the  permanent  disappearance  of  an  equivalent  quantity 
of  heat ;  and  the  efficiency  of  the  engine  is  expressed  by  the  ratio  of  the 
heat  converted  into  motive  power  to  the  whole  heat  expended,  viz. : — 

ABCD       Ht-Hg,       Qt-Qg  (w) 

MABN  Hx  Qx     "      '  K     '' 

1 0.  Third  Corollary  (of  Thermodynamic  Functions). 

If  the  two  curves  of  no  transmission  in  Fig.  6,  ADM,  BCN,  be 
indefinitely  close  together,  the  ratio  of  the  heat  consumed  in  passing  from 
one  of  those  curves  to  the  other  to  the  actual  heat  present,  will  be  the 
same,  whatever  may  be  the  form  and  position  of  the  curve  indicating  the 
mode  of  variation  of  pressure  and  volume,  provided  it  intersects  the  two 
curves  of  no  transmission  at  a  finite  angle;  because  the  area  contained 
between  this  connecting  curve  and  the  two  indefinitely-prolonged  curves 
of  no  transmission  will  differ  from  an  area  whose  upper  boundary  is  an 
isothermal  curve,  by  an  indefinitely  small  area  of  the  second  order. 

To  express  this  symbolically,  let 

f  =  8F, 

be  the  ratio  in  question,  for  a  given  indefinitely-close  pair  of  curves  of  no 
transmission.  Let  the  change  from  one  of  these  curves  to  the  other  be 
made  by  means  of  any  indefinitely-small  changes  of  actual  heat  and  of 
volume,  SQ,  SV.  Then  by  the  general  equation  (7),  the  following 
quantity — 

+  m-SY  =  m-^  +  '^v'  '    '  (11) 

is  constant  for  a  given  pair  of  indefinitely-close  curves  of  no  transmission, 
and  is,  therefore,  the  complete  variation  of  a  function,  having  a  peculiar 
constant  value  for  each  curve  of  no  transmission,  represented  by  the 
following  equation : — 


352 


ON   THERMODYNAMICS. 


This  function,  which  I  shall  call  a  thermodynamic  function,  has  the 
following  properties  : — 

H=(QdF,     ....     (13.) 

is  equivalent  to  the  general  equation  (6) ; 

<1F  =  0,         .         .         .         .     (14.) 
is  the  equation  common  to  all  curves  of  no  transmission;  and 

F  =  a  given  constant,      .  .  .      (14A.) 

is  the  equation  of  a  particular  curve  of  no  transmission. 

11.  Proposition  111. — Problem.  Let  it  be  supposed  find  for  a  given 
substance,  the  forms  of  all  pc  \ermal  cm  .  hut  of  only  one 

curve  of  no  tra  ;  it  is  required  /■  determination  of 

points,  another  curve  of  no  transmi  sing  through  a  given  point,  situated 

anywhere  out  of  the  known  curve. 

(Solution  :  see  Fig.  7.)  Let  L  M  be  the  known  curve  of  no  trans- 
mission;  B  the  given  point.  Through  B  draw  an  isothermal  curve 
Q1ABQ1,  cutting  LM  in  A.  o,  being  the  quantity  of  heat  to  which  this 
curve  corresponds,  draw,  indefinitely  near  to  it,  the  isothermal  curve  ql  qv 
corresponding  to  the  quantity  of  heat  Q,  —  2  Q,  where  c  Q  is  an  indefinitely 
small  quantity.      Draw  any  other  pair  of  indefinitely  close  isothermal 


Kg.  7. 

curves  Q„  Q2,  q2  q2,  corresponding  to  the  quantities  of  heat  Q2  Q2  —  $  Q;  2  Q 
being  the  same  as  before.  Let  D  be  the  point  where  the  isothermal  curve 
Q2Q2  cuts  the  known  curve  of  no  transmission.  Draw  the  ordinates 
A  VA,  B  VB  parallel  to  0  Y,  enclosing,  with  the  isothermal  curves  of  Qx 
and  Qj  —  S  Q,  the  small  quadrilateral  A  B  b  a.     Draw  the  ordinate  D  VD 


ON   THERMODYNAMICS.  353 

parallel  to  0  Y,  intersecting  the  isothermal  curve  of  Q2  —  §  Q  in  d. 
Lastly,  draw  the  ordinate  C  Vc  in  such  a  position  as  to  cut  off  from  the 
space  between  the  isothermal  curves  of  Q2  and  Q2  —  8  Q  a  quadrilateral 
D  C  c  d,  of  area  equal  to  the  quadrilateral  ABba. 

Then  will  C,  where  the  last  ordinate  intersects  the  isothermal  curve  of 
Q2,  approximate  indefinitely  to  the  position  of  a  point  in  the  curve  of  no 
transmission  passing  through  the  given  point  B,  when  the  variation  of 
actual  heat  S  Q  is  diminished  without  limit.  And  thus  may  be  determined 
to  as  close  an  approximation  as  we  please,  any  number  of  points  in  the 
curve  of  no  transmission  NBE  which  passes  through  any  given  point  B, 
when  any  one  curve  of  no  transmission  L  M  is  known. 

(Demonstration.)  For  when  the  variation  §  Q  diminishes  indefinitely, 
the  curves  qx  qv  q.2  q2,  approach  indefinitely  towards  the  curves  Qx  Q15  Q2  Q2 
respectively;  and  the  small  quadrilaterals  bounded  endways  by  the 
ordinates  approximate  indefinitely  to  the  small  quadrilaterals  bounded 
endways  by  the  curves  of  no  transmission ;  which  latter  pair  of  quadri- 
laterals are  equal,  by  the  first  corollary  of  Proposition  II. 

The  symbolical  expression  of  this  proposition  is  as  follows  : — 

Let  VA,  VB,  Vc,  VD,  be  the  volumes  corresponding  to  the  four  points  of 
intersection  of  a  pair  of  isothermal  curves  with  a  pair  of  curves  of  no 
transmission;  A  and  B  being  on  the  isothermal  curve  of  Qp  C  and  D  on 
that  of  Q2,  A  and  D  on  one  of  the  curves  of  no  transmission,  B  and  C  on 
the  other ;  then 

C  § d  v  (for  Q  =  Ql)  =  -C*  8 '' v  (for  Q  =  Qi>'  1 

'  a  v  i>  y  (is.) 

or 

F   —  F    =  F  —  F 

J-B-LA-1-C-I-D- 

The  second  form  of  this  equation  is  in  the  present  case  identical,  because 

F    =  F   •    F    =  F 

12.  Proposition  IV. — Problem  (see  Fig.  8).  The  forms  of  all 
isothermal  curves  for  a  given  substance  being  given,  let  E  F  be  a  curve  of  any 
form,  representing  an  arbitrarily  assumed  succession  of  pressures  and  volumes. 
It  is  required  to  find,  by  the  determination  of  points,  a  corresponding  curve 
passing  through  a  given  point  B,  such  that  the  quantity  of  heat  absorbed  or 
emitted  by  the  substance,  in  passing  from  any  given  isothermal  curve  to  any  other, 
shall  be  the  same,  ichether  the  pressures  and  volumes  be  regulated  according  to 
the  original  curve  E  F,  or  according  to  the  curve  passing  through  the  point  B. 

(Solution.)     The  process  by  which  the  latter  curve  is  to  be  deduced 

z 


354  ON  THERMODYNAMICS. 

from  the  former  is  precisely  the  same  with  that  by  Avhich  one  curve  of  no 
transmission  is  deduced  from  another  in  the  last  problem. 


«*\ 

<Di^ 

\"~~"~~~><1L 

S^~- Ha 

p 

— -^£__  ^^\ 

^c 

0. 

H      ^^^- 

**"     ~^5j-m 

^Kl 

L 

Fig.  8. 

(Demonstration.)  Let  GBH  be  the  required  curve.  This  curve,  and 
the  curve  EF,  in  their  relation  to  each  other,  may  be  called  Curves  of 
Equal  Transmission.  Through  B  draw  the  isothermal  curve  Qx  Q19  inter- 
secting the  curve  E  F  in  A.  Draw  also  any  other  isothermal  curve  Q2  Q2, 
intersecting  E  F  in  D  and  G  H  in  C.  Through  A,  B,  C,  D,  respectively, 
draw  the  four  indefinitely-prolonged  curves  of  no  transmission,  AK, 
intersecting  Q2  Q2  in  d,  B  L,  intersecting  Q2  Q2  in  c,  C  M,  and  D  N. 
Conceive  the  whole  space  between  the  isothermal  curves  Q,  Qt,  Q2  Q2,  to 
be  divided  by  other  isothermal  curves,  into  a  series  of  indefinitely-narrow 
stripes,  corresponding  to  equal  indefinitely-small  variations  of  actual  heat. 
Then,  by  the  construction  of  the  solution,  the  quadrilaterals  cut  from  those 
stripes  by  the  pair  of  curves  E  F,  G  H  are  all  equal ;  and  so  also  are  the 
quadrilaterals  cut  from  the  stripes  by  the  pair  of  curves  of  no  transmission, 
A  K,  B  L.  Therefore  the  area  A  B  C  D  is  equal  to  the  area  A  B  c  d.  The 
indefinitely-prolonged  areas,  M  C  D  N,  L  c  d  K,  are  evidently  equal ; 
therefore,  adding  this  pair  of  equal  areas  to  the  preceding,  the  pair  of 
indefinitely-prolonged  areas  LEAK.  MCBADN  are  equal.  Subtracting 
from  each  of  these  areas  the  part  common  to  both,  ABR,  and  adding  to 
each  the  indefinitely -prolonged  area  K  K  C  M,  we  find,  finally,  that  the 
indefinitely-prolonged  areas  KADN,  LBCM  are  equal. 

But  the  former  of  those  areas  (by  Prop.  I.)  represents  the  mechanical 
equivalent  of  the  heat  absorbed  by  the  substance  in  passing  from  the 
actual  heat  Q2  to  the  actual  heat  Qx  through  a  series  of  pressures  and 
volumes  represented  by  the  co-ordinates  of  the  curve  E  F ;  and  the  latter, 
the  corresponding  quantity  for  the  curve  G  H ;  therefore  those  curves 
are,  with  respect  to  each  other,  curves  of  equal  transmission,  which  was 
required. 

The  algebraical  expression  of  this  result  is  that  the  equation  (15)  holds 


ON   THERMODYNAMICS. 


>oa 


for  any  pair  of  curves  of  equal  transmission,  as  well  as  for  a  pair  of  curves 
of  no  transmission ;  or,  in  other  terms,  let  FA,  FB,  Fc,  FD  be  the  thermo- 
dynamic functions  for  the  curves  of  no  transmission  passing  through  the 
four  points  where  a  pair  of  isothermal  curves  cut  a  pair  of  curves  of  equal 
transmission :  A,  B  being  on  the  upper  isothermal  curve ;  C,  D  on  the 
lower;  A,  D  on  one  curve  of  equal  transmission,  B,  C  on  the  other:  then, 


F.  =  K  -  Fr 


(16.) 


13.  Proposition  V. — Theorem.  The  difference  between  the  quantities 
of  heat  absorbed  by  a  substance  in  passing  from  one  given  amount  of  actual  heat 
to  another,  at  two  different  constant  volumes,  is  equal  to  the  difference  between 
the  two  latent  heats  of  expansion  in  passing  from  one  of  those  volumes  to  the 
other  at  the  two  different  amounts  of  actual  heat  respectively,  diminished  by  the 
corresponding  difference  between  the  quantities  of  expansive  power  given  out. 

(Demonstration:  see  Fig.  9.)  Let  Qt  Qa  be  the  isothermal  curve  of  the 
higher  amount  of  actual  heat;  Q2Q2  that  of  the  lower.     Let  VA,  VB  be 


Fig.  9. 

the  two  given  volumes.  Draw  the  two  ordinates  VA  a  A,  VB  b  B,  and  the 
four  indefinitely-prolonged  curves  of  no  transmission  A  M,  a  m,  B  1ST,  b  n. 
the  quantities  of  heat  absorbed,  in  passing  from  the  actual  heat  Q2  to  the 
actual  heat  Qv  at  the  volumes  VA  and  VB,  are  represented  respectively 
by  the  indefinitely-prolonged  areas  MAam,  NBbn.  Then  adding  to 
each  of  those  areas  the  indefinitely-prolonged  area  n iBAM  (observing 
that  the  space  below  the  intersection  R  is  to  be  treated  as  negative),  we 
find  for  their  difference 

NBJn-  MAam  =  NBAM-  ubBAam  =(NBAM  -nbam) 
-(VBBAVA-VB5«VA); 

but  NBAM  and  n  b  a  m  represent  the  latent  heats  of  expansion  from 
VA  to  VB,  at  the  actual  heats  Qx  and  Q2  respectively ;  and  VB  B  A  VA  and 


856 


ON   THERMODYNAMICS. 


VB  b  a  VA  represent  the  power  given  out  by  expansion  from  V A  to  V,,  at 
the  actual  heats  Qx  and  Q2  respectively :  therefore,  the  proposition  is 
proved.     Q.  E.  D. 

This  proposition,  expressed  symbolically,  is  as  follows :  A  Q  being  the 
difference  of  actual  heat,  Qx  —  Q2,  let  A  (Q  +  SA)  be  the  heat  absorbed 
in  passing  from  Q2  to  Qt  at  the  volume  VA,  and  A  (Q  +  S„)  the  corre- 
sponding quantity  ;tt  the  volume  VB;  A  SA  and  A  SB  representing  quantities 
of  potential  energy  stored  up  in  altering  molecular  arrangement.     Then 


»H-«-i(^-l)/'MT.  .     (17.) 

A 

14.  Of  Curves  of  Free  Expansion. 

In  all  the  preceding  propositions,  the  whole  motive  power  developed 
by  an  elastic  substance  in  expanding  is  supposed  to  be  communicated 
to  external  bodies;  to  a  piston,  for  example,  which  the  substance  causes 
to  move  and  to  overcome  the  resistance  of  a  machine. 

Let  us  now  suppose  that  as  much  as  possible  of  the  motive  power 
developed  by  the  expansion  is  expended  in  agitating  the  particles  of  the 
expanding  substance  itself,  by  whose  mutual  friction  it  is  finally  recon- 
verted into  heat  (as  when  compressed  air  escapes  freely  from  a  small 
orifice);  and  let  us  examine  the  properties  of  the  curves  which,  on  a 
diagram  of  energy,  represent  the  law  of -expansion  of  the  substance  under 
these  circumstances,  and  which  may  be  called  Curves  of  free  Expansion. 

15.  Proposition  VI. — Theorem.  If  from  two  points  on  a  curve  of  free 
expansion  there  be  drawn  tiro  straight  lines  perpendicular  to  and  terminating 
at  the  axis  of  ordinates,  and  also  tv:o  curves  of  no  transmission,  indefinitely 
prolonged  away  from  the  origin  of  co-ordinates ;  then  the  area  contained  between 

the  curve  of  free  expansion,  the  two  straight 
lines,  and  the  axis  of  ordinates,  will  be 
erpial  to  the  area  contained  between  the 
curve  of  free  expansion  and  the  two 
indefinitely-prolonged  curves  of  no  trans- 
mission. 

(Demonstration.)  Let  FF  (Fig.  10) 
be  a  curve  of  free  expansion ;  G  H 
any  two  points  in  it ;  G  VG,  H  VH 
ordinates ;  G  PG,  H  Pn  lines  per- 
pendicular to  O  Y ;  G  M,  HN 
■*  curves  of  no  transmission,  indefinitely 
Fig.  10.  prolonged    in    the    direction    of    X. 

Then  the  indefinitely-prolonged  area  MGHN  represents  the  heat  which 
would  have  to  be  communicated  to  the   substance,  if  the  motive  power 


ON   THERMODYNAMICS. 


357 


developed  were  entirely  transferred  to  external  bodies,  while  the  area 
Y,.  G  H  VH  represents  that  motive  power.  The  excess  of  the  rectangular 
area  PH  H  VH  0  above  the  area  PG  G  VG  0,  is  the  power  necessarily  given 
out  by  the  elastic  fluid  in  passing  from  a  vessel  in  which  the  pressure  is 
PG  and  volume  VG,  to  a  vessel  in  which  the  pressure  is  PH  and  volume  VH. 
The  remainder  of  the  expansive  power,  represented  by  the  area  P0  G  H  PH, 
by  the  mutual  friction  of  the  particles  of  the  expanding  substance,  is 
entirely  reconverted  into  heat,  and  is  exactly  sufficient  (by  the  definition 
of  the  curve  of  free  expansion)  to  render  the  communication  of  heat  to 
the  substance  unnecessary;  from  which  it  follows,  that  this  area  is  equal 
totheareaMGHX.     Q.  E.  D. 

The  equation  of  a  curve  of  free  expansion  is 


d  (¥  +  P  V)  =  0. 


(17  a.) 


16.  Corollary. — In  Fig.  11,  the  same  letters  being  retained  as  in  the 
last  figure,  through  G  draw  an  isothermal  curve  QL  Qx,  which  the  line  PHH 
produced  cuts  in  h; 


and  from  h  draw  the  indefinitely-prolonged  curve  of  no  transmission,  hn. 
Then  because,  by  the  proposition  just  proved,  the  areas  PG  G  H  PH  and 
M  G  H  N  are  equal,  it  follows  that  the  indefinitely-prolonged  area,  MGhn, 
which  represents  the  latent  heat  of  expansion  at  the  constant  actual 
heat  Q1?  from  the  volume  VG  to  the  volume  Yh,  exceeds  PG  G  h  PH,  by  the 
indefinitely-prolonged  area  NH/j  n,  which  represents  the  heat  which  the 
substance  would  give  out,  in  falling,  at  the  pressure  PH,  from  the  actual 
heat  Qt  to  the  actual  heat  corresponding  to  the  point  H  on  the  curve  of 
free  expansion  passing  through  G.  Subtracting  from  this  area  the  excess 
of  the  rectangle  PH  Yh  above  the  rectangle  PG  VG,  we  obtain  the  excess  of 
the  area  M  G  h  n  above  the  area  VG  G  h  Yh. 

This  conclusion  may  be  thus  expressed : — Let  Q2  be  the  actual  heat  for 


358 


ON   THERMODYNAMICS. 


K, 


the  point  H ;  -~  the  ratio  of  specific  heat  at  the  constant  pressure  PH  to 
real  specific  heat ;  then 


/ 


Qa 


■rfQ-PHVA  +  PaVfl 


(^-1)LApdV(f0rQ  =  Ql);    r-    •     (18.) 


otherwise : — 


Qi  r  Po 

Equation  (18)  may  he  used,  either  to  find  points  in  the  curve  of  free 
expansion  which  passes  through  G,  when  the  isothermal  curves  and  the 
curves  of  no  transmission  are  known;  or  to  deduce  theoretical  results 
from  experiments  on  the  form  of  curves  of  free  expansion,  such  as  those 
which  have  been  for  some  time  carried  on  by  Mr.  Joule  and  Professor 
"William  Thomson. 

Considered  geometrically,  these  experiments  give  values  of  the  area 
XH/ni,     The  area 

,P<; 
PflG/»PH=  VdT 

'  I\, 

is  known  in  each  case  from  previous  experiments  on  the  properties  of 
the  gas  employed ;  and  this  area,  by  Proposition  VI.,  is  equal  to  the  area 
MG/iHN;  to  which,  adding  the  area  N  H  h  n,  ascertained  by  experiment, 
we  obtain  the  area  MG/j n,  that  is,  the  latent  heat  of  expansion  from  the 
volume  V0  to  the  volume  Yh,  at  the  constant  actual  heat  Q1}  denoted 
symbolically  by 

*   G 

Now  the  problem  to  be  solved  is  of  this  kind.  "We  know  the  differences 
of  actual  heat  corresponding  to  a  certain  series  of  isothermal  curves  for 
the  substance  employed ;  and  we  have  to  ascertain  the  absolute  quantities 
of  actual  heat  corresponding  to  those  curves.  Of  the  above  expression 
for  the  area  M  G  h  n,  therefore,  the  factor  Qt  is  to  be  determined,  while 
the   other    factor,    being    the    difference    between    two    thermodynamic 


ON   THERMODYNAMICS. 


359 


functions,  is  known ;  and  the  experiments  of  Messrs.  Thomson  and  Joule, 
by  giving  the  value  of  the  product,  enable  us  to  calculate  that  of  the 
unknown  factor,  and  thence  to  determine  the  point  on  the  thermometric 
scale  corresponding  to  absolute  privation  of  heat. 

1 7.  Proposition  VII. — Problem.  To  determine  the  ratio  of  the  apparent 
specific  heats  of  a  substance  at  constant  volume  and  at  constant  pressure,  for  a 
given  pressure  and  volume;  the  isothermal  curves  and  the  curves  of  no 
transmission  being  known. 

(Solution.)  In  Fig.  12,  let  A  be  the  point  whose  co-ordinates  represent 
the  given  volume  VA  and  pressure  PA ;  Q  A  Q  the  isothermal  curve  passing 


Fis.  12. 


through  A;  qq  another  isothermal  curve,  very  near  to  Q Q.  Through  A 
draw  the  ordinate  VA  A  a  parallel  to  0  Y,  cutting  qq  in  a;  draw  also  A  B 
parallel  to  0  X,  cutting  q  q  in  B.  From  A,  a,  B,  draw  the  three  indefi- 
nitely-prolonged curves  of  no  transmission  A  M,  a  in,  B  N. 

Then  the  heat  absorbed  in  passing  from  the  actual  heat  Q  to  the  actual 
heat  q,  at  the  constant  volume  VA,  is  represented  by  the  indefinitely- 
prolonged  area  MAaro,  while  at  the  constant  pressure  PA  it  is  represented 
by  the  area  MABN.  Let  the  curve  qq  be  supposed  to  approximate 
indefinitely  to  QQ.  Then  will  the  three-sided  area  A«B  diminish 
indefinitely  as  compared  with  the  areas  between  the  curves  of  no  trans- 
mission A  M,  a  m,  B  N ;  and,  consequently,  the  area  MABN  will  approxi- 
mate indefinitely  to  the  sum  of  the  areas  MA  am  and  m«BN;  the 
ultimate  ratio  of  which  sum  to  the  area  M  A  a  m  is  therefore  the  required 
ratio  of  the  specific  heats.  Now  in  a  B  N,  as  qq  approaches  Q  Q,  approxi- 
mates indefinitely  to  the  latent  heat  of  the  small  expansion  VB  —  VA  at 
the  actual  heat  Q,  and  this  small  expansion  bears  ultimately  to  the 
increment  of  pressure  Pa  —  PA,  the  ratio  of  the  subtangent  of  the  isothermal 
curve  Q  Q  to  its  ordinate  at  the  point  A. 

The  symbolical  expression  of  this  proposition  is  as  follows : — Let  o  Q 
denote    the    indefinitely-small    difference   of    actual    heat    between   the 


360 


ON   THERMODYNAMICS. 


isothermal  curves  Q  Q,  qq;  B  V  the  indefinitely-small  variation  of  volume 
VB  —  VA ;  B  P  the  indefinitely  small  variation  of  pressure  Pa  —  PA ; 

the  quantities   of  heat   required   to  produce   the  variation   BQ,  at  the 
constant  volume  VA,  and  at  the  constant  pressure  PA  respectively. 
Then 

d? 


BY  = 


SP 


d? 

dV 


dQ 
d  P 

"  dY 


■  BQ; 


and 


^.8Q=^.8Q  +  Qi£.SV 


consequently. 


dQ 


,/p 


r~  (19.) 


_ ';  ? 


J 


equations  agreeing  with  equation  (31)  of  a  paper  on  the  Centrifugal  Theory 
of  Elasticity  before  referred  to. 

18.  First  Corollary. — As  the  curves  AM,  am,  BN  approximate  indefi- 
nitely towards  parallelism,  and  the  point  a  towards  C,  where  am  intersects 
A B,  the  ratio  of  the  areas  MABN:  MAaih,  approximates  indefinitely 
to  that  of  the  lines  AB:AC,  which  are  ultimately  proportional,  respectively, 
to  the  subtangents  of  the  isothermal  curve  and  the  curve  of  no  transmission 
passing  through  A.     Therefore, 


Kv 


Subtangent  of  Isothermal  Curve 


Subtangent  of  Curve  of  No  Transmission" 


(20.) 


19.  Second  Corollary.— Velocity  of  sound.  The  subtangents  of  different 
curves  at  a  given  point  on  a  diagram  of  energy  being  inversely  proportional 
to  the  increase  of  pressure  produced  by  a  given  diminution  of  volume 
according  to  the  respective  curves,  are  inversely  proportional  to  the  squares 
of  the  respective  velocities  with  which  waves  of  condensation  and  rare- 
faction will  travel  when  the  relations  of  pressure  to  volume  are  expressed 
by  the  different  curves.  Therefore,  if  there  be  no  sensible  transmission 
of  heat  between  the  particles  of  a  fluid  during  the  passage  of  sound,  the- 


ON   THERMODYNAMICS. 


3G1 


square  of  the  velocity  of  sound  must  be  greater  than  it  would  have  been 
had  the  transmission  of  heat  been  instantaneous  in  the  ratio  of  the 
subtangent  of  an  isothermal  curve  to  that  of  a  curve  of  no  transmission 
at  the  same  point,  or  of  the  specific  heat  at  constant  pressure  to  the 
specific  heat  at  constant  volume. 

This  is  a  geometrical  proof  of  Laplace's  law  for  all  possible  fluids.  The 
same  law  is  deduced  from  the  hypothesis  of  molecular  vortices  in  the 
paper  before  referred  to  on  the  Centrifugal  Theory  of  Elasticity. 

20.  Proposition  VIII. — Problem.  The  isothermal  curves  for  a  given 
substance  being  knoum,  and  tin-  quantities  of  heat  required  to  produce  all  variations 
of  actual  heat  at  a  given  constant  volume;  it  is  required  to  find  any  number  of 
points  in  a  curve  of  no  transmission  passing  through  a  given  point  in  the  ordinate 
corresponding  to  that  volume. 

(Solution.)  In  Fig.  13,  let  VAAX  be  the  given  ordinate;  Q1Q1,  A2Q, 
isothermal  curves  meeting  it  in  Alf  A2,  respectively;  and  let  it  be  required. 


Fig.  13. 

for  example,  to  find  the  point  where  the  curve  of  no  transmission  passing 
through  A1  intersects  the  isothermal  curve  A2  Q2.  On  the  line  VA  A2  A1? 
as  an  axis  of  abscissa?,  describe  a  curve  C  C,  whose  ordinates  (such  as 
A2  C2,  aA  cA,  &c.)  are  proportional  to  the  specific  heat  of  the  substance  at 
the  constant  volume  VA,  and  at  the  degrees  of  actual  heat  corresponding 
to  the  points  where  they  are  erected,  divided  by  the  corresponding  rate 
of  increase  of  pressure  with  actual  heat ;  so  that  the  area  of  this  curve 
between  any  two  ordinates  (e.g.,  the  area  a4  c4  c3  a3)  may  represent  the 
mechanical  equivalent  of  the  heat  absorbed  in  augmenting  the  actual 


362  ON   THERMODYNAMICS. 

heat  from  the  amount  corresponding  to  the  lower  ordinate  to  that  corre- 
sponding to  the  higher  (e.g.,  from  the  amount  corresponding  to  «4  to  that 
corresponding  to  a'3). 

Very  near  to  the  isothermal  curve  A2  Q2,  draw  another  isothermal  curve 
a2  #2,  and  let  the  difference  of  actual  heat  corresponding  to  the  interval 
between  these  curves  be  3  Q.  Draw  a  curve  D  D,  such  that  the  part  cut 
off  by  it  from  each  ordinate  of  the  curve  C  C  shall  bear  the  same  proportion 
to  the  whole  ordinate  which  the  difference  8  Q  bears  to  the  whole  actual 
heat  corresponding  to  the  ordinate ;  for  example,  let 

A^q  :  A7Dx  :  :  Qi  =  8  Q 
ATO;  :  A^TI,   :  :  Q2   :  8  Q,  &C. 

Then  draw  an  ordinate  VB  B  b,  parallel  to  0  Y,  cutting  off  from  the 
space  between  the  isothermal  curves  A,  Q2,  a2  q2,  a  quadrilateral  area 
A2  B  b  a2  equal  to  A{  D4  D2  A2,  the  area  of  the  curve  D  D  between  the 
ordinates  at  A,  and  A.. 

Then,  if  the  difference  8  Q  be  indefinitely  diminished,  the  point  B  will 
approximate  indefinitely  to  the  intersection  required  of  the  isothermal 
curve  A2  Q2  with  the  curve  of  no  transmission  passing  through  A4 ;  and 
thus  may  any  number  of  points  in  this  curve  of  no  transmission  be  found. 

(Demonstration.)  Let  A!  M]  be  the  curve  of  no  transmission  required. 
Let  «3  c3,  «4  c4  be  any  two  indefinitely-close  ordinates  of  the  curve  C  C, 
corresponding  to  the  mean  actual  heat  Q3  4.  Let  a3m3,  a4w?4  be  curves  of 
no  transmission,  cutting  the  curves  a.,  q2,  A,  Q2,  so  as  to  enclose  a  small 
quadrilateral  area  e.     Then,  by  the  construction,  and  Proposition  I., 

The  area  a3  c3  c4  «4  =  the  indefinitely-prolonged  area  m3  a3  «4  mA ; 

and  by  the  first  corollary  of  the  second  proposition  and  the  construction, 

the  area  e        8  Q        area  a3  d3  d4  a4 
m3  a3  a4  mi       Q3  4       area  a3  cs  c4  a^ 

Therefore,  the  area  e  =  the  area  a3  d3  </4  a4 ;  but  the  area  Ax  Dx  D2  A2  is 
entirely  made  up  of  such  areas  as  a3  d3  di  a4)  to  each  of  which  there  corre- 
sponds an  equal  area  such  as  e;  and  when  the  difference  3  Q  is  indefinitely 
diminished,  the  area  A2  B  b  a2  approximates  indefinitely  to  the  sum  of  all 
the  areas  such  as  e,  that  is,  to  equality  with  the  area  AjDjD^o.  Q.E.D. 
The  symbolical  expression  for  this  proposition  is  found  as  follows  : — 

rQiK 

The  area  Ax  Dx  D2  A<,  ultimately  =  8  Q  .  /      ^  .  d  Q  (for  V  =  YA) ; 

J  q  KQ, 


ON   THERMODYNAMICS.  363 

V 

The  area  A2  B  b  a.2  ultimately  =  %  Q  .  I       TfidV  (fov  Q  =  Q2) ; 


divide  both  sums  by  S  Q  and  equate  the  results ;  then, 
r*Bf;p  A?  1 


I" % " v (for Q - Qi> " L h " Q (for v - v-)'  (21) 


which  denotes  the  equality  of  two  expressions  for  the  difference,  Fx  —  Fa 
between  the  thermodynamic  functions  for  the  curve  of  no  transmission 
Ax  M,  and  for  that  passing  through  the  point  A2. 

When  the  relations  between  pressure,  volume,  and  heat,  for  a  given 
substance,  are  known,  the  equation  (21)  may  be  transformed  into  one 
giving  the  volume  VB  corresponding  to  the  point  at  which  the  required 
curve  of  no  transmission  cuts  the  isothermal  curve  of  Q.. 

Suppose,  for  instance,  that  for  a  perfect  gas 

P  V  =  N  Q  sensibly  ;  and  \Y  =  1  sensibly;        .     (22.) 

N  being  a  constant  (whose  value  for  simple  gases  and  for  atmospheric  air 
and  carbonic  oxide  is  about  0-41);  then  the  thermodynamic  function  for 
a  perfect  gas  is  sensibly 

F  =  hyp.  log.  Q  +  N  hyp.  log.  V;    .         .     (22a.) 

and  equation  (21)  gives,  for  the  equation  of  a  curve  of  no  transmission, 

X,=  a>".  .         .         .     (23.) 


whence 

P„ 

P 


!  =  (r)""1_Sk-    •  •  (24) 


Equations  (23)  and  (24)  are  forms  of  the  equation  of  a  curve  of  no 
transmission  for  a  perfect  gas,  according  to  the  supposition  of  Mayer ; 
and  are  approximately  true  for  a  perfect  or  nearly  perfect  gas  on  any 
supposition. 

According  to  the  hypothesis  of  molecular  vortices,  the  relations  between 
pressure,  volume,  and  actual  heat,  for  a  perfect  gas,  are  expressed  by  these 
equations : — 

PV  =  NQ  +  /i;I|=l+^^;        .     (25.) 


T)G4  ON    THERMODYNAMICS. 

where  h  is  a  very  small  constant,  which  is  inversely  proportional  to  the 
specific  gravity  of  the  gas,  and  whose  value,  in  the  notation  of  papers  on 
the  hypothesis  in  question,  is 

h  =  Nlr,     .         .         .         .     (25A.) 

k  being  the  height,  on  the  scale  of  a  perfect  gas-thermometer,  of  the  point 
of  absolute  cold  above  the  absolute  zero  of  gaseous  tension.  Hence  we 
find,  for  the  thermodynamic  function  of  a  perfect  gas, 

F  =  hyp.  log.  Q  -  y^rh  +  N  hyp.  log.  V,      .     (26.) 

and  for  the  equation  of  a  curve  of  no  transmission, 

1        \        h  h        ) 

-B  =  (9AN  . e  t »  Q* .+  h     N  Qx  +  A  I  _      /07) 

V  .         \\j.tJ 


For  all  practical  purposes  yet  known,  these  equations  may  be  treated  as 
sensibly  agreeing  with  equation  (23),  owing  to  the  smallncss  of  h  as 
compared  with  N  Q, 


Section    III. — Of    the    Efficiency    of    Thermodynamic    Engines, 

Worked  by  the  Expansion  and  Condensation  of 

Permanent  Gases. 

21.  The  efficiency  of  a  thermodynamic  engine  is  the  proportion  of 
the  whole  heat  expended  which  is  converted  into  motive  power ;  that  is  to 
say,  the  ratio  of  the  motive  power  developed  to  the  mechanical  equivalent 
of  the  whole  heat  consumed. 

To  determine  geometrically  the  efficiency  of  a  thermodynamic  engine, 
it  is  necessary  to  know  its  true  indicator-diagram ;  that  is  to  say,  the 
curve  whose  co-ordinates  represent  the  successive  volumes  and  pressures 
which  the  elastic  substance  working  the  engine  assumes  during  a  complete 
revolution.  This  true  indicator-diagram  is  not  necessarily  identical  in 
figure  with  the  diagram  described  by  the  engine  on  the  indicator-card ; 
for  the  abscissae  representing  volumes  in  the  latter  diagram  include  not 
only  the  volumes  assumed  by  that  portion  of  the  elastic  substance  which 
really  performs  the  work  by  alternately  receiving  heat  while  expanding, 
and  emitting  heat  while  contracting,  in  such  a  manner  as  permanently  to 
transform  heat  into  motive  power,  but  also  the  volumes  assumed  by  that 
portion  of  the  elastic  substance,  if  any,  which  acts  merely  as  a  cushion  for 
transmitting  pressure  to  the  piston,  undergoing,  during  each  revolution,  a 


ON   THERMODYNAMICS. 


3G5 


scries  of  changes  of  pressure  and  volume,  and  then  the  same  series  in  an 
order  exactly  the  reverse  of  the  former  order,  so  as  to  transform  no  heat 
permanently  to  power. 

The  thermodynamic  engines  to  be  considered  in  the  present  section, 
are  those  in  which  the  elastic  substance  undergoes  no  change  of  condition. 
We  shall,  in  the  first  place,  investigate  the  efficiency  of  those  which  work 
without  the  aid  of  the  contrivance  called  an  ;'  economiser  "  or  "  regenerator," 
and  afterwards,  those  which  work  with  the  aid  of  that  piece  of  apparatus. 

22.  Lemma. — Problem.  To  determine  the  true  from  the  apparent 
indicator-diagram  of  a  thermodynamic  engine ;  the  portion  of  the  elastic 
substance  which  acts  as  a  cushion  being  known,  and  the  law  of  its  changes  of 
pressure  and  volume. 


-X 


Fig.  14 

(Solution.)  In  Fig.  14,  let  abed  be  the  apparent  indicator-diagram. 
Parallel  to  0  X  draw  H  a  and  L  c,  touching  this  diagram  in  a  and  c 
respectively ;  then  those  lines  will  be  the  lines  of  maximum  and  minimum 
pressure.  Let  H  E  and  L  G  be  the  volumes  occupied  by  the  cushion  at  the 
maximum  and  minimum  pressures  respectively :  draw  the  curve  E  G, 
such  that  its  co-ordinates  shall  represent  the  changes  of  volume  and 
pressure  undergone  by  the  cushion  during  a  revolution  of  the  engine.  Let 
K  F  d  b  be  any  line  of  equal  pressure,  intersecting  this  curve  and  the 
apparent  indicator-diagram ;  so  that  K  b,  K  d  shall  represent  the  two 
volumes  assumed  by  the  whole  elastic  body  at  the  pressure  0  K,  and  K  F 
the  volume  of  the  cushion  at  the  same  pressure.     On  this  line  take 

JB  =  Iff)  =  ~KF; 

then  it  is  evident  that  B  and  D  will  be  two  points  in  the  true  indicator- 
diagram  ;  and  in  the  same  manner  may  any  number  of  points  be  found. 

The  area  of  the  true  diagram  ABCD  is  obviously  equal  to  that  of  the 
apparent  diagram  abed. 


3G6  ON   THERMODYNAMICS. 

23.  Proposition  IX. — Problem.  The  true  Indicator-diagram  of  a 
thermodynamic  engine  worked  by  the  expansion  and  contraction  of  a  substance 
which  does  not  change  its  condition,  and  without  a  regenerator,  being  given,  it  is 
required  to  determine  the  efficiency  of  the  engine. 

(Solution.)  In  Fig.  15,  let  A  a  a  Bb'  b  A  be  the  given  true  indicator- 
diagram.     Draw  two  curves  of  no  transmission,  A  M,  B  N,  touching  this 

figure  at  A  and  B  respec- 
tively, and  indefinitely  pro- 
duced towards  X.  Then 
during  the  process  denoted 
by  the  portion  Aaa'B  of 
the  diagram  the  elastic  sub- 
stance is  receiving  heat,  and 
the  mechanical  equivalent  of 
the  total  quantity  received 
r  is  represented  by  the  in- 
Firr.  15.  definitely  -  prolonged      area 

MAaa'BN;  during  the 
process  denoted  by  the  portion  Bb'bA  of  the  diagram,  the  substance  is 
giving  out  heat,  and  the  mechanical  equivalent  of  the  total  heat  given  out 
is  represented  by  the  indefinitely-prolonged  area  M  A  b  b'  B  N ;  while  the 
difference  between  those  areas,  that  is,  the  area  of  the  indicator-diagram 
itself,  represents  at  once  the  heat  which  permanently  disappears  and  the 
motive  power  given  out.  The  EFFICIENCY  of  the  engine  is  the  ratio  of 
this  last  quantity  to  the  total  heat  received  by  the  elastic  substance  during 
a  revolution ;  that  is  to  say,  it  is  denoted  by  the  fraction, 

area  A  a  a'  B  b'  b  A 
area  M  A  a  «'  B  N' 

To  express  this  result  symbolically,  find  the  limiting  points  A  and  B  by 
combining  the  equation  of  the  indicator-diagram  Avith  the  general  equation 
of  curves  of  no  transmission,  viz. : — 

d  F  =  0. 

Then  draw  two  indefinitely-close  and  indefinitely-prolonged  curves  of  no 
transmission,  a  b  m,  a  V  m,  through  any  part  of  the  diagram,  cutting  out  of 
it  a  quadrilateral  stripe,  a b V a.  Let  Qx  be  the  mean  actual  heat  corre- 
sponding to  the  upper  end  ad  of  this  quadrilateral  stripe;  Q2,  that 
corresponding  to  the  lower  end,  b  V. 

The  area  of  this  indefinitely-narrow  stripe,  representing  a  portion  of 
the  heat  converted  into  motive  power,  is  found,  according  to  the  principles 
and  notation  of  the  third  corollary  to  Proposition  II.  and  of  Proposition 


ON   THERMODYNAMICS. 


3G7 


III.,  by  multiplying  the  difference  between  the  actual  heats  by  the 
difference  between  the  thermodynamic  functions  for  the  curves  of  no 
transmission  that  bound  the  stripe,  thus  : — 

SE  =  (Q1-Q2)SF: 

while  the  area  of  the  indefinitely-prolonged  stripe,  mactfm',  representing 
part  of  the  total  heat  expended,  is,  according  to  the  same  principles, 

and  that  of  the  indefinitely-prolonged  stripe  m  b  V  m',  representing  part 
of  the  heat  given  out,  is 

o  H2  =  Q2  o  F. 

Integrating  these  expressions  we  find  the  following  results  : — 

Whole  heat  expended, 

FB 
H1  =  f   BQ^F; 


Heat  given  out. 


Motive  power  given  out, 


-F, 


H„  =  (    BQ,dF; 


F 
E  =  H1-H2  =  f   "(Qi-Q^ZF; 


(28.) 


Efficiency, 


J  V 


QtdF 


J 


formulae  agreeing  with  equation  (28)  of  a  paper  on  the  Centrifugal  Theory 
of  Elasticity  (Seep.  63);  it  being  observed  that  the  symbol  F  in  the  last- 
mentioned  paper  denotes,  not  precisely  the  same  quantity  which  is  denoted 
by  it  in  this  paper,  and  called  a  thermodynamic  function,  but  the  pro- 
duct of  the  part  of  that  function  which  depends  on  the  volume  by  the 
real  specific  heat  of  the  substance. 

24.  First  Corollary.    Maximum  efficiency  between  given  limits  of  actual  heat. 

When  the  lushest  and  lowest  limits  of  actual  heat  at  which  the  engine 


3GS 


ON   THERMODYNAMICS. 


can  work  are  fixed,  it  is  evident  that  the  greatest  possible  efficiency  of  an 
engine  without  a  regenerator  will  be  attained  when  the  whole  reception  of 
heat  takes  place  at  the  highest  limit,  and  the  whole  emission  at  the  lowest; 
so  that  the  true  indicator-diagram  is  such  a  quadrilateral  as  is  shown  in 
Fig.  6,  and  referred  to  in  the  second  corollary  of  Proposition  II.;  bounded 
above  and  below  by  the  isothermal  curves  denoting  the  limits  of  actual 
heat,  and.  laterally,  by  any  pair  of  curves  of  no  transmission.  The 
•efficiency  in  this  case,  as  has  been  already  proved  in  various  ways,  is 
represented  by 


E 


(29.) 


being  the  maximum  efficiency  possible  between  the  limits  of  actual  heat, 
Qx  and  Q,. 

25.  Second  Corollary. — Problem.  To  drew  the  diagram  of  greatest 
efficiency  of  a  thermodynamic  engine  without  a  regenerator,  when  the  extent  of 
variation  of  volume  is  limited,  as  well  as  that  of  the  ra  notion  of  actual  heat. 

(Solution.)  In  Fig.  16,  let 
Qi  ( i  i  • ( I ■•  Q-2  ^e  the-  isothermal 
curves  denoting  the  limits  of 
actual  heat;  YA,  Y„  the  limits 
of  volume.  Draw  the  ordinates 
\\  I  >  A.  VB  C  B,  intersecting  the 
isothermal  curves  in  the  points 
A.IU',D.  Through  A  and  C 
respectively  draw  the  curves  of 
no  transmission,  A  M  cutting 
Q2Q2  in  d,  and  ( '  \  cutting  C^Q, 
in  b.  Then  will  AbCd  be  the 
diagram  required  An  anal- 
ogous construction  would  give  the  diagram  of  greatest  efficiency  when  the 
variations  of  pressure  and  of  actual  heat  are  limited ;  as  in  the  air-engine 
proposed  by  Mr.  Joule. 

26.  Of  the  use  of  the  Economizer  or  Regenerator  in  Thermodynamic  Engines. 
As  the  actual  heat  of  the  elastic  substance  which  works  a  thermo- 
dynamic engine  requires  to  be  alternately  raised  and  lowered,  it  is  obvious 
that  unless  these  operations  are  performed  entirely  by  compression  and 
expansion,  without  reception  or  emission  of  heat  (as  in  the  case  of 
maximum  efficiency  described  in  the  first  corollary  of  Proposition  IX.), 
part,  at  least,  of  the  heat  emitted  during  the  lowering  of  the  actual  heat 
may  be  stored  up,  by  being  communicated  to  some  solid  conducting 
substance,  and  used  again  by  being  communicated  back  to  the  elastic 
substance,  when  its  actual  heat  is  being  raised.     The  apparatus  used  for 


ON   THERMODYNAMICS.  3G9 

this  purpose  is  called  an  economiser  or  regenerator,  and  was  first  invented 
about  181 G,  by  the  Rev.  Robert  Stirling.  In  the  air-engine  proposed 
by  him,  it  consisted  of  a  sheet-metal  plunger  surrounded  by  a  wire 
grating  or  network;  in  that  of  Mr.  James  Stirling,  it  is  composed  of  thin 
parallel  plates  of  metal  or  glass  through  which  the  air  passes  longitudinally, 
and  in  the  engine  of  Captain  Ericsson  of  several  sheets  of  wire  gauze. 

A  regenerator  may  be  regarded  as  consisting  of  an  indefinite  number 
of  strata  with  which  the  elastic  substance  is  successively  brought  into 
contact;  each  stratum  serving  to  store  up  and  give  out  the  heat  required 
to  produce  one  particular  indefinitely-small  variation  of  the  actual  heat 
of  the  working  substance. 

A  perfect  regenerator  is  an  ideal  apparatus  of  this  kind,  in  which  the 
mass  of  material  is  so  large,  the  surface  exposed  so  extensive,  and  the 
conducting  powers  so  great  as  to  enable  it  to  receive  and  emit  heat 
instantaneously  without  there  being  any  sensible  difference  of  temperature 
between  any  part  of  the  regenerator  and  the  contiguous  portion  of  the 
working  substance;  and  from  which  no  appreciable  amount  of  heat  is  lost 
by  conduction  or  radiation.  In  theoretical  investigations  it  is  convenient, 
in  the  first  place,  to  determine  the  saving  of  heat  effected  by  a  perfect 
regenerator,  and  afterwards  to  make  allowance  for  the  losses  arising  from 
the  non-fulfiment  of  the  conditions  of  ideally  perfect  action;  losses  which, 
in  the  present  imperfect  state  of  our  knowledge  of  the  laws  of  the 
conduction  of  heat,  can  be  ascertained  by  direct  experiment  only.* 

27.  Proposition  X. — Problem.  The  true  indicator-diagram  of  any 
thermodynamic  engine  being  given,  to  determine  the  amount  of  heat  saved  by  a 
perfect  regenerator. 

(Solution.)  Let  ABCD  (in  Fig.  1 7)  be  the  given  indicator-diagram. 
Across  it  draw  any  two  indefinitely-close  isothermal  curves,  q1  qx  inter- 
secting it  in  a,  b,  and  q2  q2  intersecting  it  in  d,  c  To  the  stripe  between 
those  two  curves,  speaking  generally,  a  certain  layer  or  stratum  of  the 
regenerator  corresponds,  which  receives  heat  from  the  working  substance 
during  the  change  from  b  to  c,  and  restores  the  same  amount  of  heat 
during  the  change  from  d  to  a.  The  amount  of  heat  economised  by  the 
layer  in  question  is  thus  found.  Through  the  four  points  a,  b,  c,  d,  draw 
the  indefinitely-prolonged  curves  of  no  transmission,  ah,  bl,  cm,  dn;  then 
the  smaller  of  the  two  indefinitely-prolonged  areas,  lb  cm,  Icadn,  represents 
the  heat  saved  by  the  layer  of  the  regenerator  corresponding  to  the 
indefinitely-narrow  stripe  between  the  isothermal  curves  qx  q1  and  q2  q2. 

*  It  is  true  that  the  problem  of  the  waste  of  heat  in  the  action  of  the  regenerator 
is  capable  of  a  hypothetical  solution  by  the  methods  of  Fourier  and  Poisson;  and  I 
have  by  these  methods  obtained  formulae  which  are  curious  in  a  mathematical  point  of 
view;  but  owing  to  our  ignorance  of  the  absolute  values  and  laws  of  variation  of  the 
coefficients  of  conductivity  contained  in  these  formulae,  they  are  incapable  of  being 
usefully  applied ;  and  I  therefore  for  the  present  refrain  from  stating  them. 

2  A 


370 


ON   THERMODYNAMICS. 


Draw  two  curves  of  no  transmission,  BL,  DN,  touching  the  diagram; 
and  through  the  points  of  contact,  B  and  I),  draw  the  isothermal  curves, 


-X 


Fig.  17. 

Qj  Q,  cutting  the  diagram  in  A  and  B,  and  Q2Q2  cutting  it  in  C  and  D. 
Then  because,  during  the  whole  of  the  change  from  D  through  A  to  B, 
the  Avorking  substance  is  receiving  heat,  and  during  the  whole  of  the 
change  from  B  through  C  to  D,  emitting  heat,  the  regenerator  can  have 
no  action  above  the  isothermal  curve  QjQp  nor  below  the  isothermal 
curve  Q2Q0. 

The  whole  of  the  diagram  between  these  curves  is  to  be  divided  by 
indefinitely-close  isothermal  curves  into  stripes  like  abed;  and  the  saving 
of  heat  effected  by-  the  layer  of  the  regenerator  corresponding  to  each 
stripe  ascertained  in  the  manner  described,  when  the  whole  saving  may 
be  found  by  summation  or  integration. 

The  symbolical  expression  of  this  result  is  as  follows :  Let  the  points 
of  contact,  B,  D,  which  limit  the  action  of  the  regenerator,  and  the 
corresponding  quantities  of  actual  heat,  Q15  Q.„  be  found,  as  in  Proposition 
IX.,  by  means  of  the  equation  dF  =  0. 

Then,  the  saving  of  heat 


■Qi 


Q, 


=  Lqw:dq  =  L  It  +%iqJ^  (3(X) 


Qs 


Q2 


.7       T? 

care  being  taken,  when     'y-  nas  different  values  for  the  same  value  of  Q, 


ON   THERMODYNAMICS. 


371 


corresponding  respectively  to  the  two  sides  of  the  diagram,  to  choose  the 
smaller  in  performing  the  integration. 

28.  Corollary. — It  is  evident  that  the  regenerator  acts  most  effectually 
when  the  outlines  of  the  indicator-diagram  from  A  to  D,  and  from  B  to  C, 
are  portions  of  a  pair  of  curves  of  equal  transmission  (determined  as  in 
Proposition  IV.) ;  for  then,  if  the  operation  of  the  regenerator  is  perfect, 
the  changes  from  B  to  C  and  from  D  to  A  will  be  effected  without 
expenditure  of  heat;  the.  heat  transmitted  from  the  working  substance 
to  a  given  stratum  of  the  regenerator,  during  any  part,  such  as  be,  of  the 
operation  B  C,  being  exactly  sufficient  for  the  corresponding  part,  da,  of 

d .  F 

the  operation  D  A.     In  this  case  for  each  value  of  Q  between  Q,  and 

d^ 

Q2,  has  the  same  value  at  either  side  of  the  diagram. 

In  fact,  the  effect  of  a  perfect  regenerator  is,  to  confer  upon  any  pair  of 

curves    of   equal   transmission  the  properties   of  a   pair  of   curves    of  no 

transmission. 

29.  Proposition  XL — Theorem.  The  greatest  efficiency  of  a  thermo- 
dynamic engine,  working  between  given  limits  of  actual  heat,  with  a  perfect 
regenerator;  is  equal  to  the  greatest  efficiency  of  a  thermodynamic  engine,  working 
between  the  same  limits  of  actual  heat  without  a  regenerator. 

(Demonstration.)  In  Fig.  1 8,  let  Qx  Qv  Q2  Q.2  be  the  isothermal  curves 
denoting  the  given  limits  of  actual  heat.  Let  AD,  B C  be  a  pair  of 
curves  of  equal  transmission  of  any  form.  Then  by  the  aid  of  a  perfect 
regenerator,  the  whole  of  the  heat  given  out  by  the  elastic  substance  during 
the  operation  B  C  may  be  stored  up,  and  given  out  again  to  that  substance 
in  such  a  manner  as  to  be  exactly  sufficient  for  the  operation  DA;  so 
that  the  whole  consumption  of  heat  in  one  revolution  by  an  engine  whose 


Fig.  18. 


indicator-diagram  is  A  B  C  D,  may  be  reduced  simply  to  the  latent  heat 
of  expansion   during   the   operation   AB,  which  is  represented  by  the 


372  ON   THERMODYNAMICS. 

indefinitely-prolonged  area  M  A  B  N,  A  d  M  and  B  c  N  being  curves  of  no 
transmission.     The  efficiency  of  such  an  engine  is  represented  by 

the  area  A  B  C  D 


the  area  MABY 

Now  the   maximum    efficiency  of   an    engine  without   a    regenerator, 
working  between  the  same  limits  of  actual  heat,  is  represented  by 

the  area  AB  cd        Q,  —  Q2 


the  area  MABN  Q, 

and  from  the  mode  of  construction  of  curves  of  equal  transmission, 
described  in  Proposition  IV.,  it  is  evident  that 

the  area  A B C D  =  the  area  ABn/; 

hence  the  maximum  efficiencies,  working  between  the  given  limits  of  actual 
heat,  Qj  and  {}.»  are  equal,  with  or  without  a  perfect  regenerator.     Q.E.D. 

30.  Advantage  of  a  Regenerator. 

It  appears  from  this  theorem  that  the  advantage  of  a  regenerator  is,  not 
to  increase  the  maximum  efficiency  of  a  thermodynamic  engine  between 
given  limits  of  actual  heat,  but  to  enable  that  amount  of  efficiency  to  be 
attained  with  a  less  amount  of  expansion,  and,  consequently,  with  a 
smaller  engine. 

Suppose,  for  instance,  that  to  represent  the  isothermal  curves,  and  the 
curves  of  no  transmission,  for  a  gaseous  substance,  Ave  adopt  the  approxi- 
mate equations  already  given  in  Article  20,  viz.  : — 

For  the  isothermal  curve  of  Q,  P  V  =  N  Q ; 

]  - 1        I  (31.) 

V         /n  \  -    .        /P  V        f      I 
For  a  curve  of  no  transmission  ~  =  (  -  --  J     ^ '  =  (  ~ ) 1  +  * '  ;  j 

and  let  us  compare  the  forms  of  the  indicator-diagrams  without  and  with 
a  regenerator,  for  a  perfect  air-engine,  working  between  given  limits  as  to 
actual  heat,  defined  by  the  isothermal  curves  Q2  Q1}  ().,  Q.,  in  Fig.  19. 

The  amount  of  expansion  at  the  higher  limit  of  heat  being  arbitrary, 
let  us  suppose  it  to  be  from  the  volume  VA  to  the  volume  VB,  corre- 
sponding respectively  to  the  points  A  and  B,  and  to  be  the  same  in  all 
cases,  whether  with  or  without  a  regenerator. 

The  engine  being  without  a  regenerator,  the  diagram  corresponding  to 
the  maximum  efficiency  has  but  one  form,  viz.,  A B c d,  where  Be,  A d  are 
curves  of  no  transmission.  Hence,  in  this  case,  there  must  be  an  additional 
expansion,  from  the  volume  VB  to  the  volume 


ON   THERMODYNAMICS. 


373 


V.  =  V, 


Qi\n 

Q2/   ' 


(32.) 


for  the  purpose  merely  of  lowering  the  actual  heat  of  the  air  without  loss 
of  heat ;  and  the  engine  must  be  made  large  enough  to  admit  of  this 
expansion,  otherwise  heat  will  lie  wasted. 


Fig.  19. 

On  the  other  hand,  if  the  engine  be  provided  with  a  perfect  regenerator, 
any  pair  of  curves  of  equal  transmission  passing  through  A  and  B  will 
complete  a  diagram  of  maximum  efficiency.  The  property  of  a  pair  of 
these  curves  being,  as  shown  in  Proposition  IV.,  that  the  difference  of  their 
thermodynamic  functions, 


AF 


(  =  I  -r-=-  d  V,  when  Q  is  constant  j, 


is  the  same  for  every  value  of  Q,  it  follows,  that  for  a  gas,  according  to 
the  approximate  equation  (23),  the  property  of  a  pair  of  curves  of  equal 
transmission  is,  that  the  volumes  corresponding  to  the  intersections  of  the 
two  curves  by  the  same  isothermal  curve,  are  in  a  ratio  which  is  the  same 
for  every  isothermal  curve.  Thus,  let  Va,  Yb  be  such  a  pair  of  volumes, 
then  this  equation 


v. 


V.' 


(33.) 


defines  a  pair  of  curves  of  equal   transmission.     From  this  and   from 
equation  (31)  it  follows,  that  for  such  a  pair  of  curves 


P         P 


(34.) 


374  OX    THERMODYNAMICS. 

If  one  of  the  curves,  or  lines,  of  equal  transmission  is  a  straight  line  of 
equal  volumes,  that  is,  an  ordinate  A  D  parallel  to  0  Y,  then  the  other  is 
an  ordinate  B C,  parallel  to  0  Y  also.  Then  ABCI)  is  the  diagram  of 
maximum  efficiency  for  an  air-engine  with  a  perfect  regenerator,  when  the 
air  traverses  the  regenerator  without  alteration  of  volume;  and  by  adopting 
this  diagram,  the  additional  expansion  from  VB  to  Vc  is  dispensed  with. 

If  one  of  the  curves,  or  lines,  of  equal  transmission  is  a  straight  line  of 
equal  pressures  A  D'  parallel  to  0  X,  then  the  other  also  is  a  straight  line 
of  equal  pressures  BC.  The  diagram  thus  formed,  A  BCD',  is  suitable, 
when  the  air,  as  in  Ericsson's  engine,  has  to  traverse  the  regenerator 
without  change  of  pressure. 

It  must  be  observed,  that  no  finite  mass,  or  extent  of  conducting 
surface,  will  enable  a  regenerator  to  act  with  the  ideal  perfection  assumed 
in  Propositions  X.  and  XL,  and  their  corollaries. 

Owing  to  the  want  of  a  general  investigation  of  the  theory  of  the  action 
of  the  regenerator  based  on  true  principles,  those  who  have  hitherto 
written  respecting  it  have  either  exaggerated  its  advantages  or  unduly 
depreciated  them.  From  this  remark,  however,  must  be  excepted  a 
calculation  of  the  expenditure  of  heat  in  Captain  Ericsson's  engine,  by 
Professor  Barnard  of  the  University  of  Alabama.* 

31.  General  Hi  urn  1 1,<  on  the  preceding  Propositions. 

The  eleven  preceding  propositions,  with  their  corollaries,  are  the 
geometrical  representation  of  the  theory  of  the  mutual  transformation  of 
heat  and  motive  power,  by  means  of  the  changes  of  volume  of  a  homo- 
geneous elastic  substance  which  does  not  change  its  condition.  All  these 
propositions  are  virtually  comprehended  in  the  first  two,  of  which,  perhaps, 
the  most  simple  enunciations  are  the  following  : — 

I.  The  mechanical  equivalent  of  the  heat  absorbed  or  given  out  by  a 
substance  in  passing  from  one  given  state  as  to  pressure  and  volume  to 
another  given  state,  through  a  series  of  states  represented  by  the  co- 
ordinates of  a  given  curve  on  a  diagram  of  energy,  is  represented  by  the 
area  included  between  the  given  curve  and  two  curves  of  no  transmission 
of  heat  drawn  from  its  extremities,  and  indefinitely  prolonged  in  the 
direction  representing  increase  of  volume. 

II.  If  across  any  pair  of  curves  of  no  transmission  on  a  diagram  of 
energy  there  be  drawn  any  series  of  isothermal  curves  at  intervals  corre- 
sponding to  equal  differences  of  actual  heat,  the  series  of  quadrilateral 
areas  thus  cut  off  from  the  space  between  the  curves  of  no  transmission 
will  be  all  equal  to  each  other. 

These  two  propositions  are  the  necessary  consequences  of  the  definitions 
of  isothermal  curves  and  curves  of  no  transmission  on  a  diagram  of  energy, 
and  are  the  geometrical  representation  of  the  application  to  the  particular 
*  SUliman's  Journal,  September,  1853. 


ON   THERMODYNAMICS.  375 

case  of  heat  and  expansive  power,  of  two  axioms  respecting  Energy  in  the 
abstract,  viz.: — 

I.  The  sum  of  Energy  in  the  Universe  is  unalterable. 

II.  The  effect,  in  causing  Transformation  of  Energy,  of  the  whole 
quantity  of  Actual  Energy  present  in  a  substance,  is  the  sum  of  the  effects 
of  all  its  parts. 

The  application  of  these  axioms  to  Heat  and  Expansive  Power  virtually 
involves  the  following  definition  of  expansive  heat : — 

Expansive  Heat  is  a  species  of  Actual  Energy,  the  presence  of  iddch  in  a 
substance  affects,  and  in  general  increases,  its  tendency  to  expand. 

And  this  definition,  arrived  at  by  induction  from  experiment  and 
observation,  is  the  foundation  of  the  theory  of  the  expansive  action  of 
heat. 


Section   IV. — Of    Temperature,   the    Mechanical    Hypothesis  of 

Molecular  Vortices,  and  the  Numerical  Computation 

of  the  Efficiency  of  Air-Engines. 

32.  In  order  to  apply  the  propositions  of  the  preceding  articles  to 
existing  substances,  besides  experimental  data  sufficient  for  the  determina- 
tion, direct  or  indirect,  of  the  isothermal  curves  and  curves  of  no 
transmission,  it  is  necessary  also  to  know  the  relation,  for  the  substance  in 
question,  between  the  quantity  of  heat  actually  present  in  it  under  any 
circumstances,  and  its  temperature ;  a  quantity  measured  by  the  product 
of  the  pressure,  volume,  and  specific  gravity  of  a  mass  of  perfect  gas,  when 
in  such  a  condition  that  it  has  no  tendency  to  communicate  heat  to,  or  to 
abstract  heat  from,  the  substance  whose  temperature  is  ascertained. 

The  nature  of  the  relation  between  heat  and  temperature  has  been 
discussed  in  investigations  already  published,  as  a  consequence  deducible 
from  a  hypothesis  respecting  the  molecular  constitution  of  matter,  with 
the  aid  of  data  supplied  by  the  experiments  of  Messrs.  Thomson  and 
Joule  and  of  M.  Eegnault.  Nevertheless,  it  seems  to  me  desirable  to 
add  here  a  few  words  respecting  the  grounds,  independent  of  direct 
experiment,  for  adopting  the  hypothesis  of  molecular  vortices  as  a  probable 
conjecture,  the  extent  to  which,  by  the  aid  of  this  hypothesis,  the  residts 
of  experiment  were  anticipated,  and  its  use,  in  conjunction  with  the  results 
of  experiment,  as  a  means  of  arriving  at  a  knowledge  of  the  true  law  of 
the  relation  between  temperatures  and  total  quantities  of  heat. 

To  introduce  a  hypothesis  into  the  theory  of  a  class  of  phenomena,  is  to 
suppose  that  class  of  phenomena  to  be,  in  some  way  not  obvious  to  the 
senses,  constituted  of  some  other  class  of  phenomena  with  whose  laws  we 
are  more  familiar.     In  thus  framing  a  hypothesis,  we  are  guided  by  some 


376  ON   THERMODYNAMICS. 

analogy  between  the  laws  of  the  two  classes  of  phenomena :  we  conclude, 
from  this  analogy  of  laws,  that  the  phenomena  themselves  are  probably 
alike.  This  act  of  the  mind  is  the  converse  of  the  process  of  ordinary 
physical  reasoning;  in  which,  perceiving  that  phenomena  are  alike,  we 
conclude  that  their  laws  are  analogous.  The  results,  however,  of  the 
latter  process  of  reasoning  may  be  certainly  true,  while  those  of  the  former 
can  never  be  more  than  probable ;  for  how  complete  soever  the  analogy 
between  the  laws  of  two  classes  of  phenomena  may  be,  there  will  always 
remain  a  possibility  of  the  phenomena  themselves  being  unlike.  A 
hypothesis,  therefore,  is  incapable  of  absolute  proof;  but  the  agreement 
of  its  results  with  those  of  experiment  may  give  it  a  high  degree  of 
probability. 

The  laws  of  the  transmission  of  radiant  heat  are  analogous  to  those  of 
the  propagation  of  a  transverse  oscillatory  movement.  The  laws  of 
thermometric  heat  are  analogous  to  those  of  motion,  inasmuch  as  both 
are  convertible  into  mechanical  effect ;  and  motion,  especially  that  of 
eddies  in  liquids  and  gases,  is  directly  convertible  into  heat  by  friction. 
If,  guided  by  these  analogies,  we  assume  as  a  probable  hypothesis  that 
heat  consists  in  some  kind  of  molecular  motion,  we  must  suppose  that 
thermometric  heat  is  such  a  molecular  motion  as  will  cause  bodies  to  tend 
to  expand;  that  is  to  say,  a  motion  productive  of  centrifugal  force.  Thus 
we  are  led  to  the  hypothesis  of  Molecular  Vortices. 

This  hyp i  :hesis,  besides  the  principles  already  enunciated,  of  the 
mutual  tram  formation  of  heat  and  motive  power  in  homogeneous  sub- 
stances, leads  to  the  following  special  conclusion  respecting  the 

Eelation  between  Temperature  and  Actual  Heat: — 
When  the  temperature  of  a  substance,  as  measured  by  a  perfect  gas  ther- 
mometer, rises  by  equal  increments,  the  actual  heat  present  in  the  substance  rises 
also  by  equal  increments — a  principle  expressed  symbolically  by  the  equation 

Q  =  h  (r  -  k),    .  .  .  .     (35.) 

where  Q  is  the  actual  heat  in  unity  of  weight  of  a  substance,  r  its 
temperature,  measured  from  the  absolute  zero  of  gaseous  tension,  k  the 
temperature  of  absolute  cold,  measured  from  the  same  point,  and  f>  the 
real  specific  heat  of  the  substance,  expressed  in  terms  of  motive  power.* 

The  enunciation  of  this  law  was  originally  an  anticipation  of  the  results 
of  experiment;  for  when  it  appeared  no  experimental  data  existed  by 
which  its  soundness  could  be  tested. 

Since  then,  however,  one  confirmation  of  this  law  has  been  afforded 

*  The  hypothesis  of  Mayer  amounts  to  supposing  that  »  =  0,  or  that  the  zero  of 
gaseous  tension  coincides  with  the  point  of  absolute  cold. 


ON   THERMODYNAMICS.  o77 

by  the  experiments  of  M.  Regnault,  showing  that  the  specific  heat  of 
atmospheric  air  is  sensibly  constant  at  all  temperatures  and  at  all  densities 
throughout  a  very  great  range;  and  another,  by  the  experiments  of  Messrs. 
Joule  and  Thomson,  referred  to  in  Proposition  VI.,  on  the  thermic 
phenomena  of  gases  rushing  through  small  apertures,  which  not  only 
verify  the  theoretical  principle,  but  afford  the  means  of  computing  approxi- 
mately the  position  k  of  the  point  of  absolute  cold  on  the  thermometric 
scale. 

According  to  this  relation  between  temperature  and  heat,  every 
isothermal  curve  on  a  diagram  of  energy  is  also  a  curve  of  equal  tempera- 
ture. The  isothermal  curve,  for  example,  corresponding  to  a  constant 
quantity  of  actual  heat,  Q,  corresponds  also  to  a  constant  absolute 
temperature, 

r  =  fH-K (36.) 

The  curve  of  absolute  cold  is  that  of  the  absolute  temperature  k. 
Any  series  of   isothermal    curves  at  intervals  corresponding  to   equal 
differences  of  heat,  correspond  to  a  series  of  equidistant  temperatures. 
Hence  we  deduce 

Proposition  XII. — Theorem.  Everything  that  has  been  predicated,  in 
the  propositions  of  the  preceding  articles,  of  the  mutual  proportions  of  quantities 
of  actual  heat  and  their  differences,  may  be  predicated  also  of  the  mutual 
proportions  of  temperatures  as  measured  from  the  point  of  absolute  cold,  and 
their  differences. 

The  symbolical  expression  of  this  theorem  is,  that  in  all  the  equations 
of  the  preceding  sections,  we  may  make  the  following  substitutions: — 

Qa  _  r2  -  k  _    (A,  g,  or  d)  Q  _  (A,  g,  or  d)  r  ^ 

Ql         TX  —  K  '  Q  r  —  K 

This  theorem  is  not,  like  those  which  have  preceded  it,  the  consequence 
of  a  set  of  definitions.  It  is  a  law  known  by  induction  from  experiment, 
aided  by  a  hypothesis  or  conjecture,  with  the  results  of  which  those  of 
experiment  have  been  found  to  agree. 

It  is  true  that  the  theorem  itself  might  have  been  stated  in  the  form  of 
a  definition  of  degrees  of  temperature;  but  then  induction  from  experiment 
would  still  have  been  required,  to  prove  that  temperature,  as  measured  in 
the  usual  way,  agrees  with  the  definition. 

By  substituting  symbols  according  to  the  above  theorem,  and  making 


378 


ON   THERMODYNAMICS. 


the  general  equation  of  the  expansive  action  of  heat  is  made  to  take  the 
following  form: — 


A  .  ¥  =  A  .  H  -  IPdV  =  AQ+A.S  =  fc.Ar«f  A/ .  - 

which  agrees  with  the  equation  deduced  directly  from  the  hypothesis  of 
molecular  vortices,  if  we  admit  that 


and,  consequently, 


/.<r=ftNie  (hyp.  log.  r  + 


/\r  =  *N 


(37  a.) 


J 


The  differential  form  of  equation  (37)  is 

d  .*  =  d  .H  -  VdX  =  dQ  +  d  .*  =  Kv  o?  r 


(38.) 


where 


Kv  =  k+/'.r  +  (r-K)|//r, 


rfV. 


The  expression  for  the    thermodynamic   function   denoted   by    F    takes 
the  form 


rl +£./'.  r 


(39.) 


but  a  more  convenient  thermodynamic  function,  bearing  the  same  relation 
to  temperature  as  reckoned  from  the  point  of  absolute  cold,  which  the 
function  F  does  to  actual  heat,  is  formed  by  multiplying  the  latter  by  the 
real  specific  heat  fe,  thus:  — 

$  =  kF=fii+/^,r+(''l\,Y,  .      (40.) 

J      r  —  K  Jut 

which,  being  introduced  into  the  general  equation,  transforms  it  to 

.     (10  A.) 


A.¥=  f(r-K)i$-  fvdY 


ON    THERMODYNAMICS.  370 

33.  Of  the  Numerical  Computation  of  the  Efficiency  of  Air-Engines,  with  or 
without  a  perfect  Regenerator. 

The  relation  between  temperature  and  heat  being  known,  the  preceding 
propositions  can  be  applied  to  determine  the  efficiency,  and  other  circum- 
stances relative  to  the  working  of  thermodynamic  engines.  To  exemplify 
this  application  of  the  theory,  let  the  substance  working  the  engine  be 
atmospheric  air,  and  let  the  real  indicator-diagram  be  such  as  to  develop 
the  maximum  efficiency  between  two  given  absolute  temperatures  rx  and  r2, 
being  a  quadrilateral,  as  in  Fig.  1 9,  of  which  two  sides  are  portions  of  the 
isothermal  curves  of  those  temperatures,  and  the  other  two  portions  of  a 
pair  of  curves  of  equal  transmission,  of  such  a  form  as  may  be  best  suited 
to  the  easy  working  of  the  engine.  Should  these  curves  be  curves  of  no 
transmission,  a  regenerator  may  be  dispensed  with.  In  every  other  case  a 
regenerator  is  necessary,  to  prevent  waste  of  heat;  and  for  the  present, 
its  action  will  be  assumed  to  be  perfect,  as  the  loss  which  occurs  from  its 
imperfect  action  cannot  be  ascertained  except  by  direct  experiment. 

In  this  investigation  it  is  unnecessary  to  use  formula?  of  minute 
accuracy ;  and  for  practical  purposes  those  will  be  found  sufficient  which 
treat  air  as  a  perfect  gas,  whose  thermometric  zero  of  pressure  coincides 
with  the  point  of  absolute  cold,  viz. — 

272 J°  Centigrade,  or. 

!  below  melting;  ice  ;  * 
490i°  Fahrenheit,       3 

whose  real  specific  heat  is  equal  to  its  specific  heat  at  constant  volume,  being 

,  2'M'G  feet  of  fall  per  Centigrade  degree,  Gi- 
ft =  Kv  = 

I  130-3  feet  of  fall  per  degree  of  Fahrenheit; 

whose  specific  heat  at  constant  pressure  (as  determined  by  M.  Regnault) 
is  0-23S  x  the  specific  heat  of  liquid  water;  or 

c  3 30 -8  feet  of  fall  per  Centigrade  degree,  or 

KP  =  \ 

i.  183-8  feet  of  fall  per  degree  of  Fahrenheit; 

the  ratio  of  these  two  quantities  being 

|?=  1+X=1-41, 

as  calculated  from  the  velocity  of  sound. 

*  This  estimate  of  the  position  of  the  point  of  absolute  cold  is  to  be  considered  as 
merely  approximate,  recent  experiments  and  calculations  having  shown  that  it  may 
possibly  be  too  high  by  about  1^°  Centigrade.  It  is,  however,  sufficiently  correct  for 
all  practical  purposes. 


380 


ON   THERMODYNAMICS. 


The  volume  occupied  by  an  avoirdupois  pound  of  air,  at  the  temperature 
of  melting  ice,  under  the  pressure  of  one  pound  on  the  square  foot,  as 
calculated  from  the  experiments  of  M.  Regnault,  is 

P0V0=  2G214-4  cubic  feet. 

This  represents  also  the  length  in  feet  of  a  column  of  air  of  uniform 
density  and  sectional  area,  whose  weight  is  equal  to  its  elastic  pressure  on 
the  area  of  its  section  at  the  temperature  of  melting  ice. 

Tt  will  be  found  convenient,  in  expressing  the  temperature,  as  measured 
from  the  point  of  absolute  cold,  to  make  the  following  substitution  : — 


k  =  T  +Tfl 


(41.) 


where  T  represents  the  temperature  as  measured  on  the  ordinary  scale 
from  the  temperature  of  melting  ice,  and  T0  the  height  of  the  temperature 
of  melting  ice  above  the  point  of  absolute  cold,  as  already  stated. 
Then  we  have 

P  V 


IN  B  -      T     • 


(41  A.) 


According  to  these  data,  the  equation  of  the  isothermal  curve  of  air  for 
any  temperature  T  is 

PV  =  P0V0.^l°  =  Nfe(T  +  T0).  .     (42.) 

The  thermodynamic  functions  are — 
For  quantities  of  actual  heat, 

F  =  hyp.  log.  Q  +  N  hyp.  log.  V; 
For  temperatures, 
$  =  hF  +  constant  =  Kv  {hyp.  log.  (T  +  T0)  +  N  hyp.  log.  V] 

=  Kv  hyp.  log.  (T  +  T^  +  P„0  °  •  hyp.  log.  V; 

1o 

consequently,  the  equation  of  any  curve  of  no  transmission  is 

$  =  constant ; 

otherwise 

N  1  +  N 

(T  +  T0)  .  V     =  const.;  orP  .  V  =  const.;  K      (43.) 


y  (42a.) 


or 


(T  +  T0)  .  PI  +  N  -  constant ; 


ON   THERMODYNAMICS. 


381 


111  W 


liicli 


N  =  0-41,  1  +  N  =  1-41, 


N 
1  +  N 


0-2908. 


The  maximum  possible  efficiency  between  any  two  temperatures  Tx  and 
T.,  is  given  by  the  universal  formula, 


Ht  -  H,       T,  -  T, 


Hx 


Tx  +  T0 


(44.) 


The  latent  heat  of  expansion  of  unity  of  weight  of  air  at  a  given 
constant  temperature  Tv  from  the  volume  VA  to  the  volume  VB,  is  sensibly 
equivalent  simply  to  the  expansive  power  developed,  being  given  by  the 
following  formula  : — 


Hx  =  (T,  +  T0) 


(*,-*a)  =  P0V„ 
,Y 
V 


Tx  +  T0 


hyp.  log. 


c 


Pd  v. 


(45.) 


Let  Ya  and  V&  be  the  volumes  corresponding  to  the  points  at  which  any 
isothermal  curve  intersects  a  given  pair  of  curves  of  no  transmission,  or  of 
equal  transmission ;  then  the  ratio  of  these  volumes, 


V' 


(46.) 


is  constant  for  every  such  pair  of  points  on  the  given  pair  of  curves ; 
because  the  difference  of  the  thermodynamic  functions,  which  is  pro- 
portional to  the  logarithm  of  this  ratio,  is  constant. 

Hence,  if  in  Fig.  19  a,  two  isothermal  curves,  T^,  T2T2,  be  the  upper 
and  lower  boundaries  of  an  y 
indicator-diagram  of  maximum 
energy  for  an  air-engine,  AaD 
an  arbitrary  curve  bounding 
the  diagram  at  one  side,  and  B 
the  other  limit  of  the  expan- 
sion at  the  higher  temperature ; 
the  fourth  boundary,  being  a 
curve  of  equal  transmission  to 
AftD,  may  be  described  by 
this  construction;  draw  any 
isothermal  curve  1 1  cutting  A  a  D  in  a,  and  make 

VA:VB::V    ■  V,.  .      (47.) 


Fis-  19  A. 


382  ON    THERMODYNAMICS. 

then  will  b  be  a  point  in  the  curve  sought,  B  b  C. 

Suppose,  for  example,  that  the  form  assumed  for  AaD  is  a  hyperbola, 
concave  towards  0  Y,  and  haviug  the  following  equation  : — 

.         .         .     (47  A.) 


a    /s  -  v; 

v 

in  which  a  and  (3  are  two  arbitrary  constants;  and  let  the  ratio  v"  =  r. 

Then  must  the  curve  B  b  C  be  another  hyperbola  concave  towards  0  Y, 
having  for  its  equation 

p.  =  r^.     ■     •     •  («») 

The  total  expenditure  of  heat,  per  pound  of  air  per  stroke,  in  a  perfect 
air-engine,  is  the  latent  heat  of  expansion  from  VA  to  VB,  given  by 
equation  (45). 

The  heat  to  be  abstracted  by  refrigeration  is  the  latent  heat  of  com- 
pression from  Vc  to  VDj  and  is  found  by  substituting  in  the  same 
equation  the  lower  temperature  T2  for  the  higher  temperature  Tr 

The  indicated  work  per  pound  of  air  per  stroke,  being  the  difference 
between  those  two  quantities,  is  found  by  multiplying  the  range  of 
temperature  by  the  difference  of  the  thermodynamic  functions  ll>  for  the 
curves  AD,  BC,  or  by  multiplying  the  latent  heat  of  expansion  by  the 
efficiency,  and  has  the  following  value  : — 

E^^-H^^-T,).^-^) 


-Po^o.^V^-^P-log.^.       •         •     (48.) 


o  '  o  •        nn         •  'vr  —&■  y 


The  heat  alternately  stored  up  and  given  out  by  the  regenerator 
(supposing  it  to  work  perfectly),  is  to  be  computed  as  follows : — Let  the 
arbitrary  manner  in  which  volume  is  made  to  vary  with  temperature,  on 
either  of  the  curves  D  a  A,  C  b  B,  be  expressed  by  an  equation 

then  the  thermodynamic  function  $  takes  the  form 

$  =  Kv  hyp.  log.  (T  +  T0)  +  P^-°  hyp.  log.  V  .  T  : 
and  the  total  heat  stored  up  and  given  out  per  pound  of  air  per  stroke,  is 


ON   THERMODYNAMICS.  383 

T 

/  1(T  +  T0)^^T  =  KV(T1-T2) 


T 

f 

-0       '    rp 

For  example,  if,  as  before, 


+  gAJ  H? +  t0)V .? dT  (49) 


ft      0  -  Va 
be  the  equation  of  the  curve  D  A,  then 

P  (T  +  T0) 


V„  = 


T  +  To(l+pV)' 

x  x  0    v  0 

and  the  heat  stored  up  per  pound  of  air  per  stroke,  is 

Kv  (T,  -  T2)  +  «  .  hyp.  log.  <{    ) ^    V  (49  a.) 

LT3  +  T0(l+^)j 

33A.  Numerical  Examples?. 

To   illustrate   the   use   of  these   formula*,  let  us  take    the   following; 
example : — 

Temperature  of  receiving  heat, 

Tx  =  343°-3  Centigrade. 
Tl  +  T0  =  615°-8  Centigrade. 

Temperature  of  emitting  heat, 

T2  =     35°-4  Centigrade. 
T2  +  T0  =  307°-9  Centigrade. 

Ratio  of  Effective  Expansion, 

Y.  _  lo  _  Pa  =  ?d  =  3 
VA       VD       PB       Pc       2' 

From  these  data  are  computed  the  following  results  : — 


3S4  OX   THERMODYNAMICS. 

Maximum  Efficiency. — 

307°'9       1 


G15°-8        2 

Heat  expended,  or  latent  heat  of  expansion, — 

G15-S  3 

H,  =  P0\0  x  ^^  x  hyp.  log.  - 

=  24020  foot-pounds  per  pound  of  working  air  per  stroke. 
Heat  abstracted  by  refrigeration, — 

H2  =  P0  V0  X  y^  X  hyp.  log.  - 
=  12010  foot-pounds  per  pound  of  working  air  per  stroke. 
Work  performed, — 

H1-H,  =  P0Vox  |^xhyP-log| 

=  12010  foot-pounds  per  pound  of  working  air  per  stroke. 

To  exemplify  the  computation  of  the  heat  stored  by  a  perfect  regenerator, 
let  it  be  supposed,  in  the  first  place,  that  the  indicator-diagram  resembles 
ABC'D'  in  Fig.  1 9,  where  the  curves  of  equal  transmission  are  represented 
by  a  pair  of  lines  of  constant  pressure.     Then  the  heat  to  be  stored  is 

Kp  (T,  —  l\))  =  101,800  foot-pounds  per  pound  of  working  air  per  stroke. 

(Secondly,  let  the  diagram  resemble  ABCD  in  Fig.  19,  where  the 
curves  of  equal  transmission  are  represented  by  a  pair  of  lines  of  constant 
volume.     Then  the  heat  to  be  stored  is 

KV(T2  —  T2)  =  72,233  foot-pounds  per  pound  of  working  air  per  stroke. 

Thirdly,  let  the  curves  of  equal  transmission,  as  in  a  recent  example,  be 
hyperbolas,  concave  towards  O  Y,  and  let  the  arbitrary  constant  a  have 
the  following  value, — 

a  — .  P0V0  =  26214*4  foot-pounds; 

then  the  heat  to  be  stored,  according  to  equation  (49  a),  is 

72,233  +  26214-4  x  hyp.  log.||^|  =  72,233  +11,157 
=  83,390  foot-pounds  per  pound  of  working  air  per  stroke. 


ON  THERMODYNAMICS.  385 

The  large  proportions  borne  by  these  quantities  to  the  whole  heat 
expended,  show  the  importance  of  efficient  action  in  the  regenerator  to 
economy  of  fuel.  The  quantity  of  heat  to  be  stored,  however,  becomes 
smaller,  as  the  curves  of  equal  transmission  approach  those  of  no  trans- 
mission, for  which  it  is  null.  The  additional  expansion  requisite  in  this 
last  case  is  found  by  the  following  computation : — 


/%  +  T0\s 

"  VT2  +  Tj 

l 
20-41 

=  5-423, 

the  result  of  which  shows  the  great  additional  bulk  of  engine  required,  in 
order  to  obtain  the  maximum  efficiency  without  a  regenerator. 

Supposing  one  pound  of  coal,  by  its  combustion,  to  be  capable  of  com- 
municating heat  to  the  air  working  in  an  engine  corresponding  with  the 
above  example,  to  an  amount  equivalent  to 

6,000,000  foot-pounds 

(an  amount  which  would  evaporate  about  7  lbs.  of  water),  the  maximum 
theoretical  duty  of  one  pound  of  such  coal  in  such  an  engine,  without 
waste  of  heat  or  power,  would  be 

3,000,000  foot-pounds, 
corresponding  to 


3,000,000 


=  249  strokes  of  a  pound  of  working  air,  with  the  effective 


12,010 

.      3 
expansion  -. 

u 

The  deductions  to  be  made  from  this  result  in  practice  must,  of  course, 
be  determined  by  experience. 


Section  V. — Propositions  Relative  to  a  Heterogeneous  Mass,  or 
Aggregate,  especially  in  Vapour-Engines. 

34.  The  heterogeneous  mass  to  which  the  present  investigation  refers, 
is  to  be  understood  to  mean  an  aggregate  of  portions  of  different  in- 
gredients, in  which  each  ingredient  occupies  a  space,  or  a  number  of 
spaces,  of  sensible  magnitude. 

The  results  arrived  at  are  not  applicable  to  mixtures  in  which  there  is  a 
complete  mutual  diffusion  of  the  molecules  of  the  ingredients,  so  that  every 
space  of  appreciable  magnitude  contains  every  ingredient  in  a  fixed  pro- 
portion.    A  mixture  of  this  kind,  when  the  relations  between  its  pressure, 

2b 


38G  ON   THERMODYNAMICS. 

volume,  heat,  and  temperature  are  known,  may  be  treated,  so  far  as  regards 
the  expansive  action  of  heat,  as  a  homogeneous  substance. 

The  ingredients  of  an  aggregate  are  heterogeneous  with  respect  to  the 
expansive  action  of  heat,  when  either  their  specific  heats,  or  their  volumes 
for  unity  of  weight  at  a  given  pressure  and  temperature,  or  both  these 
classes  of  quantities,  arc  different. 

Hence  a  portion  of  a  liquid,  and  a  portion  of  its  vapour,  enclosed  in 
the  same  vessel,  though  chemically  identical  and  mutually  transformable, 
are  heterogeneous,  and  are  to  be  treated  as  an  aggregate,  with  respect  to 
the  expansive  action  of  heat. 

M.  Clausius  and  Professor  William  Thomson  have  applied  their  formulae 
to  the  aggregate  composed  of  a  liquid  and  its  vapour,  and  have  pointed  out 
certain  relations  which  must  exist  between  the  pressure  and  density  of  a 
liquid  and  its  vapour,  and  the  latent  heat  of  evaporation. 

I  shall  now  apply  the  geometrical  method  of  this  paper  to  the  theory 
of  the  expansive  action  of  heat  in  an  aggregate,  especially  that  consisting 
of  a  liquid  and  its  vapour.  The  total  volumes  are,  for  the  present, 
supposed  not  to  be  large  enough  to  exhibit  any  appreciable  differences 
of  pressure  due  to  gravitation. 

35.  Proposition  XIII. — Theorem.  In  an  aggregate  in  equilibria,  the 
pressure  of  each  ingredient  must  be  the  same;  and  the  quantity  of  heat  in  unity 
of  weight  of  each  ingredient  must  fa  inversely  proportional  to  its  real  specific 
heat;  that  is  to  say,  the  temperature  must  be  equal. 

The  following  is  the  symbolical  expression  of  this  theorem,  with  certain 
conclusions  to  which  it  leads : — 

Let  t  —  k  be  the  common  temperature  of  the  ingredients,  as  measured 
from  the  point  of  absolute  cold ; 

P,  their  common  pressure ; 

nv  w2,  ??3,  &c.j  their  proportions  by  weight,  in  unity  of  weight  of  tin- 
aggregate  ; 

vv  v2,  vz,  &c,  the  respective  volumes  of  unity  of  weight  of  the  several 
ingredients. 

V,  the  volume  of  unity  of  weight  of  the  aggregate ; 

qv  q2,  q3,  Sec,  the  respective  quantities  of  actual  heat  in  unity  of  weight 
of  the  several  ingredients; 

fci>  fc-2>  ^3>  c^'c-'  their  respective  real  specific  heats ; 

0,  the  quantity  of  heat  in  unity  of  weight  of  the  aggregate ; 

«1>,  a  thermodynamic  function  for  the  aggregate. 


ON  THERMODYNAMICS.  387 

Then  these  quantities  are  connected  by  the  following  equations: — 

2  .  n  =  1 (50.) 

V-=.2..»«. (51.) 

T  -  I.  -  -  —  -  cVC.     .  .  .       {0^.) 

**1  "2  ^3 

Q  =  S  .  n  J  =  (r  -  k)  .  2  .  n Is.       .         .         .     (53.) 

t.p''»+/,',).i,'+fg.<T.         .     (54.)     ' 

It  is  evident  that  all  these  equations  hold,  whether  the  proportions  of 
the  ingredients  nv  &c,  are  constant,  as  in  an  aggregate  of  chemically 
distinct  substances,  or  variable,  as  in  the  aggregate  of  a  liquid  and  its 
vapour. 

Let  3H  be  the  heat  which  disappears  in  consequence  of  a  small 
expansion  of  aggregate  at  constant  temperature,  represented  by 

SV  =  2.S«,     ....     (55.) 

$  u  representing  any  one  of  the  parts  arising  from  the  changes  undergone 
by  the  different  ingredients,  of  which  the  whole  expansion  of  the  aggregate 
S  V  is  made  up. 
Then 

gH  =  2-[(r-K)^.S«};        .         •     (56.) 

the  same  for  every  ingredient,  as  well  as  the  tei 
perature ;  therefore,  the  factor  (r  -  k)  -j-  is  the  same  for  every  ingredient. 


but  the  pressure  P  is  the  same  for  every  ingredient,  as  well  as  the  tem- 

(V? 

(It 
and,  consequently,  for  the  whole  aggregate;  that  is  to  say, 


SH  =  (r-K)^.SV=(r-K)S$.  .     (57.) 

(It 

This  equation  shows  that  the  relation  of  temperature  to  the  mutual 
transformation  of  heat  and  expansive  power  is  the  same  in  an  aggregate 
as  in  a  homogeneous  substance. 

Consequently,  if  we  define  isothermal  curves  for  an  aggregate  to  be  cm 
of  constant  temperature,  we  arrive  at  the  following  conclusion: — 

Proposition  XIV. — Theorem.  Isothermal  curves  on  the  diagram  of 
energy  of  an  aggregate  have  the  same  properties,  with  reference  to  the  mutual 
transformation  of  heat  and  expansive  power,  with  those  on  the  diagram  of  energy 
of  a  homogeneous  substance. 

It   is   unnecessary   to    enunciate   separately  a  similar   proposition  for 


388  ON  THERMODYNAMICS. 

curves  of  no  transmission;  for  the  demonstration  of  Proposition  I.,  on 
which  all  their  properties  depend,  is  evidently  appli  cable  to  an  aggregate, 
constituted  in  any  manner. 

Hence  it  appears,  that  if  the  isothermal  curves  for  an  aggregate  be  drawn 
according  to  the  above  definition,  all  the  propositions  proved  in  this 
paper  respecting  homogeneous  substances  become  true  of  the  aggregate. 

36.  Proposition  XV. — Theorem.  Every  isothermal  line  for  an  aggre- 
gate of  a  liquid  and  its  vapour,  is  a  straight  line  of  equal  pressure,  from  the  volume 
corresponding  to  complete  liquefaction  to  the  volume  corresponding  to  complete 
evaporation. 

This  is  a  fact  known  by  experiment.  The  theorem  is  equivalent  to 
a  statement,  that  the  pressure  of  a  liquid  and  its  vapour  in  contact  with 
each  other  is  a  function  of  the  temperature  only. 

Corollary. — Theorem.  At  any  given  temperature,  the  volume  of  an  aggre- 
gate of  liquid  and  vapour  is  arbitrary  between  and  up  to  the  limits  of  total 
liquefaction  and  total  evaporation. 

To  express  this  symbolically,  let  P  be  the  pressure  of  an  aggregate  of 
liquid  and  vapour  corresponding  to  the  absolute  temperature  r ;  and 
unity  of  weight  being  the  quantity  of  the  aggregate  under  consideration, 
let  v  be  the  volume  corresponding  to  complete  liquefaction,  v  that 
corresponding  to  complete  evaporation,  and  V  the  actual  volume  at  any 
time;  let  n  be  the  proportion  of  liquid,  and  1  —  n  that  of  vapour,  corre- 
sponding to  the  aggregate  volume  V ;  then 

V  —  nv  +  (1  -  n)v',        .         .         .     (58.) 

and  V  may  have  any  value  not  less  than  v  nor  greater  than  v,  while  P 
and  r  remain  constant;  the  proportion  of  liquid,  n,  being  regulated 
according  to  the  foregoing  equation. 

37.  Proposition  XVI. — Problem.  The  density  of  a  liquid  and  of  Us 
vapour,  when  in  contact  at  a  given  temperature,  being  given,  and  the  isothermal 
lines  of  the  aggregate,  it  is  required  to  determine  the  latent  heat  of  evaporation 
of  unity  of  weight  of  the  fluid. 

(Solution.)  The  densities  of  the  liquid  and  of  its  vapour  are  respectively 
the  reciprocals  of  the  volumes  of  total  liquefaction  and  total  evaporation 
of  unity  of  weight,  above-mentioned.  In  Fig.  20,  let  the  abscissae  Ov,  Ov 
represent  these  volumes,  and  the  equal  ordinates,  vA,  v'B,  the  pressure 
corresponding  to  the  given  temperature;  so  that  AB,  parallel  to  OX,  is 
the  isothermal  line  of  the  aggregate  for  that  temperature.  Suppose  two 
curves  of  no  transmission  AM,  BN,  to  be  drawn  from  A  and  B  respectively, 
and  indefinitely  prolonged  towards  X;  then  the  indefinitely-prolonged 
area  MABN  represents  the  mechanical  equivalent  of  the  latent  heat 
sought,  and  this  area  is  to  be  computed  in  the  following  manner  : — Draw 
a   second   isothermal  line,  a  b,  indefinitely   near  to  A  B,   at   an  interval 


ON  THERMODYNAMICS. 


389 


A                                                        1 

3 

N 

M 

X 

Fv'.  20. 


corresponding  to  the  indefinitely-small  difference  of  temperature  dr;  then, 
ultimately, 


d t  :  t  —  k  :  :  area  A B b a  :  area  MABN; 


or,  symbolically, 
L 


d~P 
latent  heat  of  evaporation  =  (r  —  k)  -=-  (v  —  v).    (59.) 


This  is  simply  the  application  of  Propositions  I.  and  II.  to  the  aggregate 
of  a  liquid  and  its  vapour,  mutatis  mutandis. 

(Remarks?) — The  existence  of  a  necessary  relation  between  the  density, 
pressure,  and  temperature  of  a  vapour  and  its  liquid  in  contact,  and  the 
latent  heat  of  evaporation,  was  first  shown  by  Carnot.  If  for  r  —  k  in 
the  preceding  equation  be  substituted,  according  to  Professor  Thomson's 

notation,  -,  J  being  "Joule's  equivalent"  and  /x  "Carnot's  function,"  the 
,u 

equation    is    transformed    into    that    deduced   by   Messrs.    Clausius    and 

Thomson  from  the  combination  of  Carnot's  theory  with  the  law  of  the 

mechanical  convertibility  of  heat. 

38.  Corollary. — The  volume  occupied  by  unity  of  weight  of  vapour  at 

saturation  may  be  computed  from  its  latent  heat  of  evaporation  by  means 

of  the  inverse  formula, — 

L 


(r  -  k) 


dj>' 

It 


(GO.) 


the  latent  heat,  L,  being,  of  course,  always  stated  in  units  of  motive  power. 
The  want  of  satisfactory  experiments  on  the  density  of  vapours  of  any 
kind,  has  hitherto  prevented  the  use  of  the  direct  formula  (59). 


390  ON  THERMODYNAMICS. 

It  is  otherwise,  however,  with  the  inverse  formula  (60),  at  all  events 
in  the  case  of  steam;  for,  so  far  as  we  are  yet  able  to  judge,  the  experi- 
ments of  M.  Kegnault  have  determined  the  latent  heat  of  evaporation  of 
water  with  accuracy  throughout  a  long  range  of  temperature. 

M.  Clausius,  applying  to  those  experimental  data  a  formula  founded  on 
the  supposition  of  Mayer  (that  is  to  say,  similar  to  the  above,  with  the 
exception  that  k  is  supposed  =  0),  has  calculated  the  densities  of  steam 
at  certain  temperatures,  so  as  to  show  how  much  they  exceed  the  densities 
calculated  from  the  pressures  and  temperatures  on  the  supposition  that 
steam  is  a  perfect  gas.  From  these  calculations  he  concludes,  that 
either  the  supposition  of  Mayer  is  erroneous,  or  steam  deviates  very  much 
at  high  densities  from  the  condition  of  a  perfect  gas. 

In  the  following  table,  the  value  of  k  is  supposed  to  be  20,1  Centigrade, 
and  use  has  been  made  of  the  formula  for  calculating  the  pressure  of  steam 
and  other  vapours  at  saturation,  first  published  in  the  Edinburgh  Neiv 
Philosophical  Journal  for  July,  1849  (Sec  p.  1),  viz. — 

log.P  =  a-£-^.       •         •         •     (61.) 


This  table  exhibits,  side  by  side,  the  volume  in  cubic  feet  occupied  by 
one  pound  avoirdupois  of  steam  at  every  twentieth  Centigrade  degree, 
from  -  20°  to  +  200°  (that  is,  from  -4°  to  +  500'  Fahrenheit)— first,  as 
extracted  from  a  table  for  computing  the  power  of  steam-engines,  in  the 
Transactions  of  the  Royal  Society  of  Edinburgh,  Vol  XX.  (See  p.  2S2),  which 
was  calculated  on  the  supposition  that  steam  is  a  perfect  gas;  and,  secondly, 
as  computed  by  equation  (GO)  from  the  latent  heat  of  steam  as  determined 
by  M.  Regnault  The  excess  of  the  former  quantity  above  the  latter  is 
also  given  in  each  case,  with  its  ratio  to  the  second  value  of  the  volume. 

For  convenience  sake,  a  column  is  added  containing  the  pressures  of 
steam  corresponding  to  the  temperatures  in  the  table  in  pounds  per 
scpiare  foot. 

The  fourth  column  of  this  table  could  easily  be  extended  and  filled  up, 
so  as  to  replace  the  column  of  volumes  of  steam  for  every  fifth  Centigrade 
degree  in  the  table  previously  published;  but  it  would  be  unadvisable  to 
do  so  at  present,  for  the  following  reasons : — 

First,  the  value  of  the  constant  k  is  still  uncertain."" 

Secondly,  the  results  of  M.  Regnault's  direct  experiments,  on  the  density 
of  steam  and  other  vapours,  may  soon  be  expected  to  appear. 

*  It  is  probable  that  x.  may  be  found  to  be  inappreciably  small ;  in  which  case  the 
numbers  in  column  (4)  will  have  to  be  diminished  to  an  extent  varying  from  Tl,-, 
to  TJ  „  of  their  amount. 


ON   THERMODYNAMICS.  391 

Table  of  Computed  Volumes  of  1  lb.  Avoirdupois  of  Steam. 


Temperature. 

Volume  sup- 
posed a  Per- 
fect Gas. 

Volume  com- 
puted from 
Latent  Heat. 

Difference. 

Ratio  of 
Difference  to 
lesser  Value 
of  Volume. 

Pressure. 

Fahrenheit. 

Centigrade. 

Deg. 

Deg. 

Cubic  feet. 

Cubic  feet. 

Cubic  feet. 

lb.  per  square 
foot. 

-   4 

-20 

15757 

15718 

39 

0-0025 

2-4799 

+  32 

0 

3390  4 

3377-2 

13-2 

0-0039 

12-431 

68 

+  20 

936-81 

934-50 

2-31 

0  0025 

48-265 

104 

40 

314-88 

313  56 

1-32 

0  0042 

153-34 

140 

60 

123  65 

122-63 

1-02 

0-0083 

415-33 

176 

80 

55  05 

54-19 

0  86 

0-0158 

988-67 

212 

100 

27-166 

26-47S 

0-68S 

0-0260 

2116-4 

248 

120 

14-596 

14  076 

0-520 

0-0369 

4149-3 

284 

140 

8-420 

8-004 

0416 

0-0502 

7557-0 

320 

160 

5-15S 

4-838 

0-320 

0  0661 

12931 

356 

180 

3-326 

3-071 

0-255 

0  0830 

20979 

392 

200 

2-241 

2  033 

0-20S 

0  1023 

32512 

428 

220 

1-568 

1-396 

0-172 

0T232 

48425 

464 

240 

1134 

0-990 

0144 

0-1455 

69680 

500 

260 

0-843 

0  722 

0  121 

0-1676 

97275 

Col.  (1.) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

Thirdly,  it  is  possible  that  the  values  of  the  latent  heat  of  evaporation 
of  water,  as  deduced  from  M.  Eegnault's  experiments,  may  still  have  to 
undergo  some  correction ;  because,  according  to  the  theoretical  definition 
of  the  latent  heat  of  evaporation,  the  liquid  is  supposed  to  be  under  the 
pressure  of  an  atmosphere  of  its  own  vapour,  which  atmosphere,  as  it 
increases  in  bulk,  performs  work  of  some  kind,  such  as  lifting  a  piston; 
whereas,  in  M.  Regnault's  experiments,  the  water  is  pressed  by  an  atmo- 
sphere of  mingled  steam  and  air,  whose  united  pressure  is  that  corre- 
sponding to  the  tenrperature  of  internal  ebullition  of  the  water;  so  that  the 
pressure  of  the  steam  alone  on  the  surface  of  the  water,  which  regulates 
the  superficial  evaporation,  may  be  less  than  the  maximum  pressure 
corresponding  to  the  temperature  of  ebullition;  and  this  steam,  moreover, 
has  no  mechanical  work  to  perform,  except  to  propel  itself  along  the 
passage  leading  to  the  calorimeter,  and  to  agitate  the  water  in  the  latter 
vessel.  Under  these  circumstances,  it  is  possible,  though  by  no  means 
certain,  that  the  latent  heat  of  evaporation  of  water,  as  deduced  from  M. 
Eegnault's  experiments,  may  be  somewhat  smaller  than  that  which  corre- 
sponds to  the  theoretical  definition,  especially  at  high  pressures;  and  a 
doubt  arises  as  to  the  precise  applicability  of  the  formulas  (59)  and  (60) 
to  those  experimental  results,  which  cannot  be  solved  except  by  direct 
experiments  on  the  density  of  steam. 

Notwithstanding  this  doubt,  however,  the  preceding  table  must  be 
regarded  as  adding  a  reason  to  those  already  known,  for  believing  that 


392 


ON   THERMODYNAMICS. 


saturated  steam  of  high  density  deviates  considerahly  from  the  laws  of 
the  perfectly  gaseous  condition.* 

39.  Proposition  XVII. — Problem.  The  isothermal  lines  for  a  liquid  and 
Us  vapour,  and  the  apparent  specific  heat  of  the  liquid  at  all  temperatures  hcing 
given,  and  the  expansion  of  the  liquid  by  heat  being  treated  as  inappreciably 
small:  to  determine  a  curve  of  no  transmission  for  the  aggregate,  passing 
through  a  given  point  on  the  ordinate  ichose  distance  from  the  origin  approxi- 
mately represents  the  volume  of  the  liquid, 

(Solution.)  In  Fig.  21,  let  Ov  represent  the  volume  of  the  liquid 
assumed  to  be  approximately  constant  for  all  temperatures  under  con- 
sideration;  let   jjA  be   an   ordinate  parallel  to   0  Y,   and  let    the  heat 


Fig.  21. 

consumed  by  the  liquid  in  passing  from  the  temperature  corresponding  to 
any  point  on  this  ordinate  to  that  corresponding  to  any  other  point,  be 
known;  let  the  isothermal  lines  for  the  aggregate  of  liquid  and  vapour, 
all  of  which  are  straight  lines  of  equal  pressure  parallel  to  0  X,  such  as 
AT1?  «BT2,  be  known.  Then  to  draw  a  curve  of  no  transmission  through 
any  point  A  on  the  ordinate  vA,  the  same  process  must  be  followed  as  in 
Proposition  VIII. 

To  apply  to  this  case  the  symbolical  representation  of  Proposition  VIIL, 
viz.,  equation  (21),  let  rx  be  the  absolute  temperature  corresponding  to 
the  point  A  (that  is,  to  the  isothermal  line  A  Tx) ;  r2  that  corresponding 
to  any  lower  isothermal  line  aBT9;  VB  the  volume  of  the  aggregate  of 
liquid  and  vapour  corresponding  to  the  point  B,  where  the  curve  sought, 
AM,  intersects  the  latter  isothermal  line;  KL  the  apparent  specific  heat 


*  Evidence  in  favour  of  this  opinion  is  afforded  by  the  experiments  recorded  by 
Mr.  C.  W.  Siemens  (Civil  Engineer  and  Architect's  Journal).  A  remarkable  cause, 
however,  of  uncertainty  in  all  such  experiments  has  lately  been  investigated  by 
Professor  Magnus  (Poggendorff's  Annalen,  1853,  No.  8),  viz.,  a  power  which  solid 
bodies  have  of  condensing,  by  attraction  on"  their  surfaces,  appreciable  quantities 
of  gases. 


ON  THERMODYNAMICS.  393 

of  the  liquid;    then,  making  the  proper  substitutions  of  the  symbols  of 
temperature  for  those  of  heat,  and  observing  that  the  operation 

/      dV 

-'vA 

is  in  this  case  equivalent  to  multiplication  by  VB  —  v,  we  have 

l^  =  fT(YB-v)(iovr  =  r.1)=f1^Kdr,    .     (62.) 

being  an  equation  between  two  expressions  for  the  difference  between  the 
thermodynamic  functions  $  for  the  curve  A  B,  and  for  that  which  passes 
through  a. 

If  the  specific  heat  of  the  liquid  is  approximately  constant,  this 
equation  becomes 

A  *  =  ~  (VB  -  v)  (for  t  =  t.J  =  Kt  hyp.  log.  *LZ*>     (63.) 

»T  T2   —   K 

40.  Corollary. — Problem.  The  same  data  being  given  as  in  the  preceding 
problem,  and  the  expansion  of  the  liquid  by  heat  neglected,  a  mass  of  liquid, 
having  been  raised  from  the  absolute  temperature  t2  to  the  absolute  temperature  rl7 
is  supposed  to  be  allowed  to  evaporate  partially,  under  pressure,  without  receiving 
or  emitting  heat,  until  its  temperature  falls  again  to  t2,  at  which  temperature  it 
is  liquefied  under  constant  pressure  by  refrigeration :  it  is  required  to  find  the 
power  developed. 

(Solution.)  The  power  developed  is  represented  by  the  area  of  the 
three-sided  diagram  of  energy  in  Fig.  21,  A  B  a  ;  that  is  to  say,  by 

T2  T2 

which,  if  KL  is  nearly  constant,  becomes 

KL  j  *  hyp.  log.  ^^  .dr  =  KL  {  (r,  -  k) 


r9 


-  (r2  -k)  .  (l  +  hyp.  log.  J-^) }•   •         .     (65.) 

41.  Numerical  Example. 

Let  one  pound  avoirdupois  of  water  be  raised,  in  the  liquid  state,  from 
T2  =  40°  Centigrade  to  T\  =  140°  Centigrade.     Then 


394  ON  THERMODYNAMICS. 

Tl  -  K  =  T,  +  T0  =  140°  +  272r  =  4m°  Centigrade. 

r2  -  k  =  T2  +  T0  =     40°  +  272?,°  =  3121°  Centigrade. 

The  mean  apparent  specific  heat  of  liquid  water  between  those  tem- 
peratures is 

KL  =  Kw  (or  Joule's  equivalent)  X  l'OOG  =  1398  feet  per  Cent,  degree; 

consequently,  the  heat  expended  is  equivalent  to  139,800  foot-pounds. 
The  other  numerical  data  are, — 

dP 

-r—  at  40°  Centigrade  =  S'2075  lbs.  per  square  foot  per  Cent,  degree; 

(i  - 

v  =  mean  volume  of  1  lb.  of  liquid  water  =  0-017  cubic  foot  nearly. 

Let  it  be  required  to  find,  in  the  first  place,  VB,  the  volume  to  which 
the  water  must  be  allowed  to  expand  by  partial  evaporation  under 
pressure,  in  order  that  its  temperature  may  fall  to  40°  Centigrade;  and, 
secondly,  how  much  power  will  be  developed  in  all,  after  the  water  has 
been  totally  reliquefied  by  refrigeration  at  constant  pressure,  at  the 
temperature  of  40°. 

First,  by  the  equation  (G3), 

A  $  =  ~  (VB  -  v)  =  1398  x  hyp.  log.  |i||  =  402-G24  ; 

divide  by  |-  =  8-2075 ;  then  YB-v  =  49-055  cubic  feet. 

addv=    0-017 


Aggregate  volume  of  water  and  steam  at  40°,  VB  =  49-072  ., 

As  the  volume  of  one  pound  of  steam  at  40°  Centigrade,  according  to 
the  fourth  column  of  the  table  in  Article  38,  is  313-5G  cubic  feet,  it 
appears  from  this  calculation  that  somewhat  less  than  one-sixth  of  the 
water  will  evaporate. 

Secondly,  it  appears,  from  equation  (G5),  that  after  the  water  has  been 
restored  to  the  liquid  state  by  refrigeration  at  40°  Centigrade,  the  whole 
power  developed — that  is  to  say,  the  area  A  B  a — will  be 


1398  foot-pounds  X   j  412°-5  -  312°-5(l  +  hyp.  log.  <— |— J  j 
=  1398  ft.  lbs.  X  10°  Centigrade  =  13,980  ft.  lbs., 

or  one-tenth  of  the  equivalent  of  the  heat  expended.  The  other  nine- 
tenths  constitute  the  heat  abstracted  during  the  reliquefaction  at  40° 
Centigrade. 


ON   THERMODYNAMICS. 


395 


This  calculation  further  shows,  that  in  order  that  one  pound  of  water 
and  steam  at  40°  C.  may  be  raised  to  140°  C.  solely  by  compressing  it 
into  the  liquid  state,  it  must  occupy  at  the  commencement  of  the  operation 
the  volume  VB  =  4 9 -07 2  cubic  feet;  and  that  the  power  expended  in  the 
compression  will  be  as  follows  : — 

Foot-pounds. 
Area  of  the  curvilinear  triangle  ABa,  Fig.  21,  as  already  calculated,    1 3,9S0 

Area  of  the  rectangle  aBYBv  —  P2  (VB  —  v)  =  .         .         .       7,522 


Total, 


21,502 


42.  Proposition  XVIII. — Problem.  Having  the  same  data  as  in  the 
last  proposition,  it  is  required  to  draw  a  curve  of  no  transmission  through  any 
point  on  the  diagram  of  energy  for  the  aggregate  of  a  liquid  and  its  vapour. 

(Solution.)  In  Fig.  22,  through  the  given  point  B  draw  the  straight 
isothermal  line  AB  corresponding  to  the  absolute  temperature  rv  and 
cutting  the  ordinate  corresponding  to  the  volume  of  total  liquefaction  in  A. 
Through  A,  according  to  the  last  proposition,  draw  the  curve  of  no  trans- 
mission, A  D  M.  Let  E  D  C  be  any  other  isothermal  line,  corresponding 
to  the  absolute  temperature  r2,  and  cutting  the  curve  A  M  in  D.  Draw 
isothermal  lines  a  b,  e  cl  c  at  indefinitely-small  distances  from  A  B,  EDO 


v    A                 B 

a 

c 

fa55^ 

\ 

\d                                                "^^T£- J 

\ 

C                 — jnt 

0 

d 

NM 

ArE     Vn 


-X 


Fk 


respectively,  corresponding  to  the  same  indefinitely  small  difference  of 
temperature  dr.  Draw  the  ordinates  VDdD,  VBbB;  then  draw  the 
ordinate  VccC  at  such  a  distance  from  VDrfD,  that  the  indefinitely-small 
rectangles  DCcd,  ABba  shall  be  equal.  Then,  as  the  difference  8 r  is 
indefinitely  diminished,  C  approximates  indefinitely  to  a  point  on  the 
required  curve  of  no  transmission,  B  N. 

This  is  Proposition  III.  applied  to  aggregates,  mutatis  mutandis. 


39G  ON   THERMODYNAMICS. 

The  symbolical  representation  of  this  proposition  is  as  follows : — let  Fl 
and  P2  be  the  pressures  of  the  aggregate  of  liquid  and  vapour  corresponding 
respectively  to  the  temperatures  rr  and  r2 ;  then  the  following  expressions 
for  the  difference  between  the  thermodynamic  functions  $  of  the  curves 
A  M,  B  N  are  equal, 

A$  =  ^(V0-VD)  =  ^(VB-»).  •     (GG.) 

43.  Corollary.     (Absolute  Maximum  Efficiency  of  Vapour-Engines.) 
If  the  volume  VB  be  that  corresponding  to  complete  evaporation  at  the 
temperature  tu  that  is  to  say,  if 

VB  =  •, 

then  the  curve  B  C  N  will  represent  the  mode  of  expansion  under  pressure, 
of  vapour  of  saturation  in  working  an  engine,  and  will  be  defined  by  the 
equation 

YV{v  ~v) 

V   _  y    =  — .     .         .         .     (G7.) 

(It 

If  in  this  equation  be  substituted  the  value  of  v  —  v,  in  terms  of  the 
latent  heat  of  evaporation  at  the  higher  temperature,  given  by  equation 
(60),  it  becomes 


h. 


-'-up-  •         ■  (°a> 


In  this  case  the  diagram  A  B  C  D,  Fig.  22,  is  evidently  that  of  a  vapour- 
engine  working  with  the  absolute  maximum  of  efficiency  between  the 
absolute  temperatures  rx  and  r2.  The  heat  expended  at  each  single  stroke, 
per  unit  of  weight  of  fluid,  is  the  latent  heat  of  evaporation  at  the  higher 
temperature,  or  Lx;  the  area  of  the  diagram  is  given  by  the  following 
equation : — 


T-,   —  T., 


E  =  (r1-r2)A$  =  ^ 3.1*       .  .      (09.) 

This  is  the  mechanical  power  developed  at  each  single  stroke  by  a  unit 
of  weight  of  the  substance  employed.     The  efficiency  is  represented  by 

r  =  T^Ll2,  ....    (to.) 

Ll  Tl  -  « 

being   the    expression   for   the    maximum    efficiency   of    thermodynamic 
engines  in  general. 


ON  THERMODYNAMICS.  397 

The  conditions  of  obtaining  this  efficiency  are  the  following : — 
First,  That  the  elevation  of  temperature  from  r2  to  tx,  during  the 
operation  represented  by  the  curve  D  A  on  the  diagram,  shall  be  produced 
entirely  by  compression.  The  volume  at  which  this  heating  by  compression 
must  commence  is  given,  according  to  Proposition  XVII.,  by  the  following 
equation  : — 

V.  =  .  +  ^  .  K,  hn,  log.  *=$    .        .     (71.) 

dr 

Secondly,  That  the  expansive  working  of  the  vapour  shall  be  carried  on 
until  the  temperature  falls,  by  expansion  alone,  to  its  lower  limit ;  that  is 
to  say,  until  the  volume  reaches  the  following  value,  obtained  by  adding 
together  equations  (68)  and  (71) : — 

Ve  =  .  +  /p;  •  {  K,  h)T.  log.  ^  +  -h-  } .      (72.) 

dr 

44.  Numerical  Example. 

To  exemplify  this  numerically,  let  the  same  data  be  employed  as  in 
Article  41,  the  substance  working  being  one  pound  avoirdupois  of  water. 
These  data,  with  some  additional  data  deduced  from  them,  are  given  in 
the  folio  win  sj  table  : — 


Temperature  in  Centigrade  Degrees : — 

Above  melting  ice  (T), 

Above  zero  of  gaseous  tension  (t), 

Above  absolute  cold  (r  —  k), 
Pressure  in  pounds  per  square  foot  (P),   . 

„  „      per  square  inch, 

Initial  Volume  of  saturated  steam,  VB  =  v\,  =  8*004  cubic  ft.  per  lb. 

Latent  Heat  of  Evaporation : — 

In  degrees,  applied  to  one  pound  of  liquid  water,  509o,l  Centigrade. 

In  foot-pounds  (Lj),  .         .         .         .         .     707,445 -3G 


At  upper  limit 

At  lower  limit 

of  Actual  Heat. 

.       140° 

40° 

414-6 

314-6 

412-5 

312-5 

.     7557 

153-34 

52-5 

1-065 

From  these  data  are  deduced  the  following  results  : — 
Absolute  Maximum  Efficiency  ;  =  0-2424 


398  ON   THERMODYNAMICS. 

Duty  of  one  pound  of  water;  being  the  area 

of  the  diagram  ABCD,  .         .     1 71,484-75  ft.  lbs. 

Volume   at    which    the    compression    must 

commence;  calculated  as  in  Art.  41,  .     VD  =    49"1  cubic  ft.  per  lb. 

Volume  to  which  the  expansion  must  be  car- 
ried; calculated  by  equation  (72),  .     Vc  =  258*1  cubic  ft.  per  lb. 

Vc       258*1 
Ratio  of  Expansion,        .        .         .         .     =  y-  =  8.qq^  =  32-25. 

45.  Liquefaction  of  Vapour  by  Expansion  under  Pressure. 

In  Fig.  22,  let  the  abscissa?  of  the  curve  BFE  indicate  the  volumes 
corresponding  to  complete  evaporation  at  the  pressures  denoted  by  its 
ordinates.  For  most  known  fluids,  a  curve  of  no  transmission,  B  C  N, 
drawn  from  any  point  B  of  the  curve  of  complete  evaporation  in  the 
direction  of  X,  falls  within  that  curve ;  so  that  by  expansion  of  saturated 
vapour  under  pressure,  a  portion  in  most  cases  will  be  liquefied. 

To  ascertain  whether  this  will  take  place  in  any  particular  case,  and  to 
what  extent,  equation  (60),  which  gives  the  volume  of  unity  of  weight  of 
saturated  vapour  at  the  temperature  t,,  is  to  be  compared  with  equation 
(72),  which  gives  the  volume  at  the  same  temperature  of  unity  of  weight 
of  an  aggregate  of  liquid  and  vapour,  which  lias  expanded  under  pressure 
from  a  state  of  complete  evaporation  at  the  temperature  rv  The  difference 
1  letween  the  volumes  given  by  these  equations  is  as  follows  (neglecting,  as 
usual,  the  expansibility  in  the  liquid  state) : — 

'■'•'  -  v<  =  tV  •  { — —  -  k* •  hyp-  los-  Ty^ }  (73-) 

d  F2  (.  T2  —   K  T1  —   K  T2  —  K  ) 

JV 

That  this  quantity  is  almost  always  positive  appears  from  the  following 
considerations.  The  latent  heat  of  evaporation  L,  is  in  general  capable  of 
being  represented  approximately  by  an  expression  of  this  form  : 

L  =  a-  &(r-  k),  .         .         .         .     (74.) 

(For  water,  a  =  79G°  Centigrade  x  Kw  =  1,106,122  ft.  lbs. ;  b  =  0-695 
X  Kw  =  965-772  ft.  lbs.  per  Centigrade  degree). 
Hence,  the  second  factor  in  equation  (73)  is  nearly  equal  to 

»(r,-rj  KL .  hyp.  logA- '         .     (75.) 

(rx  —  k)  .  (t2  —  k)  r2  —  k 


Now 


hyp.  log.  — < -. 


ON  THERMODYNAMICS.  399 

Therefore,  the  expression  (75)  is  positive  so  long  as 

exceeds  KL,  the  specific  heat  of  the  liquid.  .     (75  A.) 

For  water  this  condition  is  fulfilled  for  all  temperatures  lower  than 
523^°  Centigrade  (at  which  rx  —  k  =  796°  Centigrade);  and  there  is 
reason  to  believe  that  it  is  fulfilled  also  for  other  fluids  at  those  tempera- 
tures at  which  their  vapours  can  be  used  for  any  practical  purpose. 

To  determine  the  proportion  of  the  fluid  which  is  liquefied  by  a  given 

expansion  under  pressure,  we  have  the  following  formula,  deduced  from 

equation  (58)  : — 

v    —  V 
n  =    -] -c (76.) 

v2  —  v 

As  a  numerical  example,  we  may  take  the  case  of  Art.  44,  where 
saturated  steam  at  140°  Centigrade  is  supposed  to  be  expanded  under- 
pressure until  its  temperature  falls  to  40°  Centigrade.  The  volume  of  one 
pound  of  water  and  steam  at  the  end  of  the  expansion  has  already  been 
found  to  be 

Vc  =  258-1  cubic  feet. 

While,  according   to  the  table  in  Art.  38,  the  volume   of   a   pound   of 
steam  at  that  temperature  is 

v'.2  =  313'56  cubic  feet. 

Consequently,  the  fraction  liquefied  by  the  expansion  is 

313-56  -  258-1  55-46 


313-56  -  0-016       313-544 


=  0-177. 


This  conclusion  was  arrived  at  contemporaneously  and  independently, 
by  M.  Clausius  and  myself,  about  four  years  since.  Its  accuracy  was 
subsequently  called  in  question,  chiefly  on  the  ground  of  experiments, 
which  show  that  steam,  after  being  expanded  by  being  "  wire-drawn,"  that 
is  to  say,  by  being  allowed  to  escape  through  a  narrow  orifice,  is  super- 
heated, or  at  a  higher  temperature  than  that  of  liquefaction  at  the  reduced 
pressure.  Soon  afterwards,  however,  Professor  William  Thomson  proved 
that  those  experiments  are  not  relevant  against  the  conclusion  in  question^ 
by  showing  the  difference  between  the  free,  expansion  of  an  elastic  fluid,  in 
which  all  the  power  due  to  the  expansion  is  expended  in  agitating  the 
particles  of  the  fluid,  and  is  reconverted  into  heat,  and  the  expansion  of 
the  same  fluid  under  a  pressure  equal  to  its  own  elasticity,  when  the  power- 
developed  is  all  communicated  to  external  bodies,  such,  for  example,  as  the 
piston  of  an  engine. 


400 


ON   THERMODYNAMICS. 


The  free  expansion  of  a  vapour  will  be  considered  in  the  sequel. 

46.  Efficiency  of  a  Vapour-Engine  without  heating  by  compression. 

The  numerical  example  of  Art.  44  sufficiently  illustrates  the  fact, 
that  the  strict  fulfilment  of  the  condition  specified  in  Art.  43,  as 
necessary  to  the  attainment  of  the  absolute  maximum  of  efficiency  of  a 
vapour-engine,  is  impossible  in  practice. 

Let  us  consider,  in  the  first  place,  the  effect  of  dispensing  with  the 
process  D  A,  during  which  the  fluid  is  supposed  to  have  its  high  tempera- 
ture restored  solely  by  compression. 

The  effect  of  this  modification  is  evidently  to  add  to  the  heat  expended 
that  which  is  necessary  to  elevate  the  temperature  of  the  liquid  from  r2  to 
tv  and  to  add  to  the  power  developed  an  amount  represented  by  the  area 
AD  E  (Fig.  22). 

To  express  this  symbolically,  we  have — 


The  latent  heat  of  evaporation  at  tv  as 
before,  ...... 

The  additional  heat  expended  (Kh  being 
the  mean  specific  heat  of  the  liquid 
between  tx  and  r2), 

Total  heat  expended,     . 
Then,  for  the  power  developed,  we  have 


L, 


K,.(r.-T2) 
I^  +  K^-r,)     (77.) 


The  area  ABCD,  as  in  Art.  43,  =  -* -2 . L,, 

the  area  AD  E,  as  in  Art.  40,  equation  (G5), 

=  Kt  {  (Tl  -  K)  -  (r2  -  k)  (l  +  hyp.  log.  Tf^j  } 

the  sum  of  which  quantities  is  the  total  power  developed,  . 
The  efficiency  may  be  expressed  in  the  following  form : — 


Power  developed  _  rl  —  r0 
Heat  expended         tx  —  k 

\  +  KL  (rx  -  r2) 


(78.) 


y     (79.) 


an  equation  which  shows  at  once  how  far  the  efficiency  falls  short  of  the 
absolute  maximum. 


ON  THERMODYNAMIC'S. 


401 


For  a  numerical  example,  the  same  data  may  be  taken  as  in  Arts. 
41  and  44.  Then  the  heat  expended,  per  pound  of  steam,  is  thus 
made  up : — 


Latent  heat  of  evaporation,  as  in  Art.  44,    . 

Heat  required  to  raise  the  water  100°  C,  as  in  Art.  41, 

Total  heat  expended  per  lb.  of  water, 

The  power  developed  consists  of, — 

The  area  A  B  C  D,  as  in  Art.  4  4, 
The  area  ADE,  as  in  Art.  41,    . 


Foot-pounds. 
707,445-30 

139,800-00 
847,245-30 


Foot-pounds. 
171,484-75 

13,980-00 


Total  power  developed  per  lb.  of  water,      185,464*75 


.  185,484-75 

•Efficiency  847,245-36        ' 

Absolute  maximum  efficiency,  as  in  Art.  44, 


0-2189 
0-2424 


Loss  of  efficiency  by  omitting  the  heating  by  compression,     0-0235 

or  about  one-tenth  part  of  the  absolute  maximum. 

47.  Efficiency  of  a  Vapour-Engine  with  incomplete  expansion. 

It  is  in  general  impossible  in  practice  to  continue  the  expansion  of  the 
vapour  down  to  the  temperature  of  final  liquefaction ;  and  from  this  cause 
a  further  loss  of  efficiency  is  incurred. 

Let  it  be  supposed,  for  example,  that  while  the  pressure  of  evaporation 


/ 

r 

^      Pi         1 

i 

1 

H 

P* 

^\ 

Cr 

3A 

E 

P3 

K 

^~~-^C 

D\ 

-s 

0 

>iyt 

-r 

i 

l 

A 

"> 

c 

Fi<r.  23. 


Pi  corresponds  to  the  line  AB,  in  Fig.  23,  and  the  pressure  of  liquefaction, 
P3,  to  the  line  E  D  C,  the  pressure  at  which  the  expansion  terminates,  P,, 

2c 


402  ON  THERMODYNAMICS. 

corresponds  to  an  intermediate  line  H  L  G.  Let  v  A,  ?•/  B,  as  before,  be 
the  ordinates  corresponding  to  complete  liquefaction  and  to  complete 
evaporation,  at  the  pressure  Pj. 

Draw,  as  before,  the  curves  of  no  transmission  A  M,  B  N,  cutting  H  L  G 
in  L  and  G,  and  EDO  in  D  and  C;  draw  also  the  ordinate  V0KG, 
cutting  E  D  C  in  K. 

Then  the  expansion  terminates  at  the  volume  V(;,  and  ABGKE  is  the 
indicator-diagram  of  the  engine. 

To  find  the  power  represented  by  this  diagram,  the  area  A  L  H  is  to  be 
found  as  in  Art.  40,  the  area  A  B  G  L  as  in  Art.  43,  and  the  rectangle 
HGKE  by  multiplying  its  breadth  VG  —  v  (found  as  in  Art.  43)  by  its 
height  H  E,  which  is  the  excess  of  the  pressure  at  the  end  of  the  expansion 
P2,  above  the  pressure  of  final  liquefaction,  P3. 

Hence,  we  have  the  following  formula  for  the  indicated  power  developed, 
per  unit  of  weight  of  fluid  evaporated  : — 

E  =  area  A  B  G  K  E  =  KL  j  (r,  -  ic)  -  (r,  -  k) 

1  +hvp.  log.  T|  "^l  +  L./1-^ 

n       5  r2  -  J  J  '    rx-K  y       (80.) 


+  (P2-P3)^{^  +  Kthyp.log^ 


r,-«J 

,/■ 

The  heat  expended  is  of  course  L,  +  K,  (rj  —  t3). 

To  illustrate  this  numerically,  let  the  fluid  be  water;  let  the  temperature 
of  evaporation  be  1 40°  Centigrade,  and  that  of  liquefaction  40°,  as  in  the 
previous  examples;  and  let  the  expansion  terminate  when  the  pressure 
has  fallen  to  100°  Centigrade. 

The  numerical  data  in  this  case  are  the  following : — 

1.  2.                        3. 

During  the  At  the  end  of          During  the 

evaporation,  the  expansion,  final  liquefaction. 
Temperature  in  Cent,  degrees  :— 

Above  melting  ice,       .         .      140°  100°  40° 

Above  zero  of  gaseous  ten- 
sion, T  —      . 

Above  absolute  cold,  r  —  k, 

Pressure,  in  lb.  per  square  f oot,  P  =    7557 

Pressure,  in  lb.  per  square  inch,  . 


14-6 

3746 

31 4-6 

-12-5 

372-5 

312-5 

57 

2116-4 

153 -34 

52-5 

14  7 

1-065 

ON   THERMODYNAMICS.  403 

1.  2.  3. 

During  the      At  the  end  of  During  the 

evaporation,     the  expansion,    linal  liquefaction. 

in  lb.   per  square  foot   per 

Centigrade  degree,      .         .       21416  75-617  8-2075 

Initial  Volume  of  steam  in  cubic- 
feet  per  lb.,  .         .         .  8-004 

Latent   heat  of  evaporation,  Lt,  in 

foot-pounds  per  lb.  of  steam,      707,445-36 

Total  heat  expended,  in  foot-pounds 

per  lb.  of  steam,  .         .      847,245*36 

Mean  specific  heat  of  liquid  water — 

Between  40°  and  140°  Cent.,      1398  feet  of  fall. 
Between  100°  and  140°  Cent.,    1409 

Applying  equation  (80)  to  these  data,  we  obtain  the  following  results  : — 

Foot-pounds. 
AreaALH, 2,818 

AreaABGL, 68,601 

Area  HGKE=  (P,  -  P3) .  (V6  -  v)  =  1963  lbs.  per 

square  foot  x  24'58  cubic  feet,  .         .         .  =  48,250 


Total  power  developed  by  1  lb.  of  water  evaporated,     119,669 

Efficiency  =  ^|^| =0-1413 

J        847,245 

Efficiency  computed  in  the  last  article,    .          .  .          0"21S9 

Difference  =  loss  of  efficiency  by  incomplete  expansion,       0-0776 

V         24-60 
Ratio  of  expansion  — y  =  —-—  =  3-07  nearly. 
1  v         8-004  J 

If  the  power  of  the  same  engine  be  now  computed  by  the  tables  and 
formulae  published  in  the  twentieth  volume  of  the  Transactions  of  the  Roijol 
Society  of  Edinburgh  (See  p.  278),  which  were  calculated  on  the  supposition 
that  steam  is  sensibly  a  perfect  gas,  the  following  results  are  obtained  : — 

Ratio  of  expansion,   "  =  2-921  =  s  in  tables. 

8'4204 


404 


ON   THERMODYNAMICS. 


"Action  at  full  pressure  "  (PiV,  in  tables), 

"  Coefficient  of  gross  action  "  (Z  in  tables)  for  the  ex- 
pansion 2  "9  2 1 , 

Gross  action  (P^  Z),       . 
Deduct  for  back-pressure  of  liquefaction,  P3VG  =  1533 4 
X  24-6,  .   ■ 


Foot-pounds. 
63,633 


Power  developed  per  lb.  of  steam,       .     122,221 

This  result  is  too  large  by  about  one  forty-seventh  part,  a  difference  to 
be  ascribed  chiefly  to  the  error  of  treating  steam  as  a  perfect  gas.  This 
difference,  however,  is  not  of  material  consecpience  in  computing  theoreti- 
cally the  power  of  a  steam-engine,  being  less  than  the  amount  of  error 
usually  to  be  expected  in  such  calculations. 

48.  My  object  in  entering  thus  minutely  into  the  theory  of  the  efficiency 
of  vapour-engines  is,  not  so  much  to  provide  new  formulae  for  practical  use, 
as  to  illustrate  the  details  of  the  mechanical  action  of  heat  under  varied 
and  complicated  circumstances,  and  to  show  with  precision  the  nature  and 
influence  of  the  circumstances  which  prevent  the  production,  by  steam- 
engines,  of  the  absolute  maximum  of  efficiency  corresponding  to  the  tem- 
peratures between  which  they  work. 

To  illustrate  the  results  of  these  calculations  with  respect  to  the  con- 
sumption of  coal,  let  it  De  assumed,  as  in  Art.  33,  that  each  pound  of 
coal  consumed  in  the  furnace  communicates  to  the  water,  or  air,  or  other 
elastic  substance  which  performs  the  work,  an  amount  of  heat  equivalent 
to  6,000,000  foot-pounds,  which  corresponds  to  a  power  of  evaporating,  in 


Absolute  Theoretical  Maximum,  being 
the  same  for  every  perfect  thermo- 
dynamic engine  working   between 
the   same   limits   of   temperature, 
140°  -  40° 
L40°+2724°          ' 

Efficiency. 

Effect  per  pound  of  coal 
in  foot-pounds. 

0  2424 

1,454,400 

Deductions: — 

For  raising  the  temperature  of  the 

feed-water  from  40°  to  140°  Cent., 
For  stopping  the  expansive  working 

at  3 "07  times  the  initial  volume 

instead  of  32  times,    . 

Reduced  Efficiency  and  Effect, 

0-0235 
0-0776 

o-ioii 

141,000 
465,600 

606,600 

01413 

847, SOO 

ON   THERMODYNAMICS.  405 

round  numbers,  about  seven  times  its  weight  of  water.  Then  the  following 
calculation  shows  the  theoretical  indicated  duty  of  one  pound  of  such  coal, 
when  the  limits  of  working  temperature  are  140°  and  40°  Centigrade,  at 
the  absolute  maximum  of  theoretical  efficiency,  and  at  the  reduced 
efficiency,  computed  in  the  preceding  article,  on  the  supposition  that  the 
expansive  working  ceases  at  the  atmospheric  pressure. 

The  last  of  these  quantities  corresponds  to  a  consumption  of  about  2*34 
lbs.  of  coal  per  indicated  horse-power  per  hour. 

The  conditions  of  the  preceding  investigations  are  very  nearly  fulfilled 
in  steam-engines  with  valves  and  steam-passages  so  large,  and  a  velocity  of 
piston  so  moderate,  that  the  pressure  in  the  cylinder  during  the  admission 
of  the  steam  is  nearly  the  same  with  that  in  the  boiler. 

In  many  steam-engines,  however,  the  steam  is  more  or  less  "wire- 
drawn ; "  that  is  to  say,  it  has  to  rush  through  the  passages  with  a  velocity, 
to  produce  which  there  is  required  a  considerable  excess  of  pressure  in  the 
boiler  above  that  in  the  cylinder.  The  power  developed  during  the 
expansion  of  the  steam  from  the  pressure  in  the  boiler  to  that  in  the 
cylinder  is  not  altogether  lost;  for,  as  already  stated  in  Art.  45,  it  is 
expended  in  agitating  the  particles  of  the  steam,  and  is  ultimately  con- 
verted into  heat  by  friction,  so  that  the  steam  begins  its  action  on  the 
piston  in  a  superheated  state;  and  both  its  initial  pressure  and  its 
expansive  action  are  greater  than  those  of  steam  of  saturation  of  the  same 
density.  The  numerical  relations  of  the  temperature,  pressure,  and  density 
of  superheated  steam  are  not  yet  known  with  sufficient  precision  to 
constitute  the  groundwork  of  a  system  of  exact  formulae  representing  its 
action.  Some  general  theorems,  however,  will  be  proved  in  the  sequel, 
respecting  superheated  vapours,  which  may  be  found  useful  when  the 
necessary  experimental  data  have  been  obtained. 

Calculation  and  experiment  concur  to  prove  that  in  Cornish  single- 
acting  engines  the  initial  pressure  of  the  steam  in  the  cylinders  is  very 
much  less  than  the  maximum  pressure  in  the  boilers ;  generally,  indeed, 
less  than  one-half.*  It  is  doubtful,  however,  whether  this  arises  altogether 
from  wire-drawing  in  the  steam-passages  and  valves ;  for  when  it  is  con- 
sidered that  in  such  engines,  even  at  their  greatest  speed,  the  steam-valve 
remains  shut  nearly  the  whole  of  each  stroke,  being  opened  during  a  small 
portion  of  the  stroke  only,  it  may  be  regarded  as  probable  that  the  sudden 
opening  of  this  valve  causes  a  temporary  reduction  of  temperature  and 
pressure  in  the  boiler,  itself. 

49.  Composite  Vapour-Engines. 

The  steam-and-ether  engine  of  M.  du  Trembley  is  an  example  of  what 
may  be  called  a  composite  vapour-engine,  in  which   two    fluids  are  era- 

"  See  Mr.  Pole's  work  on  the  Cornish  Engine,  and  Art.  36  of  a  paper  on  the 
Mechanical  Action  of  Heat,  Trans.  Roy.  Soc.  o/Edin.,  Vol.  XX.     {Vide  p.  291.) 


400  ON   THERMODYNAMICS. 

ployed,  a  less  and  a  more  volatile;  the  heat  given  out  during  the  lique- 
faction of  the  less  volatile  fluid  serving  to  evaporate  the  more  volatile 
fluid,  which  works  an  auxiliary  engine,  and  is  liquefied  in  its  turn  °y 
refrigeration. 

Let  the  efficiency  of  the  engine  worked  by  the  less  volatile  fluid  be 
expressed  in  the  form 

»-* 

so  that      is  the  fraction  of  the  whole  heat  expended  which  is  given  out  to 

n 
the  more  volatile  fluid.     Let  the  efficiency  of  the  engine  worked  by  the 
mure  volatile  fluid  be 

i-ii 

n 
then  the  efficiency  of  the  combined  engines  will  be 

1-- (81.) 

nn 

If  both  the  engines  are  perfect  thermodynamic  engines,  let  Tj  be  the 
absolute  temperature  at  which  the  first  fluid  is  evaporated;  t,  that  at 
which  it  is  condensed,  and  the  second  fluid  evaporated ;  and  r3  that  at 
which  the  second  fluid  is  condensed ;  then, 

l-Ul-1:-'-*;   1-1  =  1-^^;  l-±.=  l_Il=^      (81  A.) 

n  t1  —  k  n  To  —  k  nn  tx  —  k 

bein"  equal  to  the  theoretical  maximum  efficiency  of  a  simple  thermo- 
dynamic engine  working  between  the  limits  of  temperature  rt  and  r3. 

Composite  vapour-engines,  therefore,  have  the  same  theoretical  maximum 
efficiency  with  simple  vapour-engines,  and  other  engines  moved  by  heat, 
working  between  the  same  temperatures;  but  they  may,  nevertheless, 
enable  the  same  efficiency  to  be  obtained  with  smaller  engines. 

50.  Curves  of  free  expansion  for  nascent  vapour. 

By  nascent  vapour  is  to  be  understood  that  which  is  in  the  act  of 
risin^  from  a  mass  of  liquid.  If  this  vapour  be  at  once  conducted  to  a 
condenser,  without  performing  any  work,  and  there  liquefied  at  a  tempera- 
ture lower  than  that  at  which  it  was  evaporated,  its  expansion,  from  the 
pressure  of  evaporation  down  to  the  pressure  of  liquefaction,  will  take 
place  according  to  a  law  defined  by  a  curve  analogous,  in  some  respects,  but 
not  in  all,  to  the  curve  of  free  expansion  for  a  homogeneous  substance, 
referred  to  in  Proposition  VI.  To  determine  theoretically  the  form  of  this 
curve,  it  is  necessary  to  know  the  properties  of  the  isothermal  curves  and 
curves  of  no  transmission  for  the  fluid  in  question  in  the  gaseous  state, 


OX   THERMODYNAMICS. 


407 


when  above  the  temperature  of  saturation  for  its  pressure.  Having  these 
data,  we  can  solve  numerically  the  following  problem  : — 

Proposition  XIX. — Problem.  To  draw  the  curve  of  free  expansion  for 
vapour  nascent  under  a  given  pressure. 

(Solution.)  In  Fig.  24,  let  AB,  parallel  to  OX,  be  the  isothermal  line 
of  an  aggregate  of  liquid  and  vapour  at  the  pressure  of  evaporation  Px 
corresponding  to  the  temperature  tx  :  let  A  vv  B  v^  be  ordinates  parallel  to 
OY;  so  that  v1  is  the  volume  of  unity  of  weight  of  the  liquid  at  this 


temperature,  and  i\  that  of  unity  of  weight  of  the  vapour  at  saturation. 
Let  D  F  be  a  line  drawn  parallel  to  0  X,  at  a  distance  representing  any 
lower  pressure  P2  corresponding  to  the  temperature  r0.  It  is  required  to 
find  the  point  where  the  curve  of  free  expansion  drawn  from  B  inter- 
sects DF. 

Let  v.,  be  the  volume  of  unity  of  weight  of  the  liquid  at  the  lower 
pressure  and  temperature,  ?;.,  D  an  ordinate  parallel  to  0  Y,  and  D  A  a 
curve  representing  the  law  of  expansion  of  the  liquid  as  the  pressure  and 
temperature  increase.  Draw  the  curves  of  no  transmission  D  N,  B  L 
indefinitely  prolonged  towards  X;  ascertain  the  indefinitely-prolonged 
area  LBADN;  draw  a  curve  of  no  transmission  MC,  cutting  DF  in  C, 
such  that  the  indefinitely-prolonged  area  MCDN  shall  be  equal  to  the 
indefinitely-prolonged  area  LBADN;  then  will  C  be  the  point  required 
where  the  curve  of  free  expansion  B  C  intersects  the  line  D  F. 

(Demonstration.)  Unity  of  weight  of  the  fluid  being  raised  in  the 
liquid  state  from  the  temperature  r2  and  corresponding  pressure  P2,  to  the 
temperature  tx  and  corresponding  pressure  Px;  then  evaporated  com- 
pletely at  the  latter  pressure  and  temperature;  then  expanded  without 
performing  work,  until  it  falls  to  the  original  pressure  P2;  then  cooled  at 
this  pressure  till  it  returns  to  the  original  temperature  r2,  at  which  it  is 
finally   liquefied ;    the    area    A  B  C  D    represents    the    expansive    power 


408  ON  THERMODYNAMICS. 

developed  during  this  cycle  of  operations,  which,  as  no  work  is  performed, 
must  be  wholly  expended  in  agitating  the  thud,  and  reproducing  by 
friction  the  heat  consumed  by  the  free  expansion  represented  by  the 
curve  B  C,  which  heat  is  measured  by  the  indefinitely-prolonged  area 
MCBL,  which  area  is  therefore  equal  to  the  area  A B C D.  Subtracting 
from  each  of  these  equal  areas  the  common  area  B  U  C,  and  adding  to 
each  of  the  equal  remainders  the  indefinitely-prolonged  area  LUDN. 
we  form  the  areas  MCDN,  LBADN,  which  are  consequently  equal. 
Q.  E.  D. 

51.  Of  the  total  heat  of  evaporation. 

The  symbolical  expression  of  the  preceding  proposition  is  formed  in 
the  following  manner.  The  area  LBADN  represents  the  total  heat  of 
evaporation,  at  the  temperature  rv  from  the  temperature  r2,  and  is  composed 
of  two  parts,  as  follows  : — 


LBADN 


f1Khdr  +  L11  .         •     (82.) 


of  which  the  first  is  the  heat  necessary  to  raise  the  liquid,  whose  specific 
heat  is  KL,  from  t2  to  rv  and  the  second  is  the  latent  heat  of  evaporation 
at  tv 

Let  v£  be  the  volume  of  unity  of  weight  of  the  vapour  at  the  pressure 
P2  and  temperature  of  saturation  t2;  draw  the  ordinate  P2'E,  meeting  DF 
in  E,  through  which  point  draw  the  indefinitely-prolonged  curve  of  no 
transmission  Eli:  then  is  the  area  MCDN  divided  into  two  parts,  as 
follows : — 

MCDN  =  MCER  +  EEDX=         Krdr  +  L2,  .     (83.) 

in  which  equation  rc  denotes  the  temperature  corresponding  to  the  point 
C  on  the  curve  of  free  expansion,  and  Kp  the  specific  heat  of  the  vapour, 
at  the  constant  pressure  P2  when  above  the  temperature  of  saturation; 
so  that  the  first  term  represents  the  heat  abstracted  in  lowering  the 
temperature  of  the  vapour  from  rc  to  the  temperature  of  saturation  r2, 
at  the  constant  pressure  P2;  and  the  second  term,  the  latent  heat  of 
evaporation  at  r2  abstracted  during  the  liquefaction. 

By  equating  the  formula?  (82)  and  (83),  the  following  equation  is 
obtained : — 

f1KLdT  +  L1-L.2  =  fTCKvdr,        .         .     (84.) 

7".,  T.-, 


ON  THERMODYNAMICS.  409 

which  is  the  symbolical  solution  of  Proposition  XIX.,  and  shows  a  relation 
between  the  total  heat  of  evaporation  of  a  fluid,  the  free  expansion  of  its 
vapour,  and  the  specific  heat  of  that  vapour  at  constant  pressure. 

52.  Approximate  Jaw  for  a  vapour  which  is  a  perfect  gas. 

If  the  vapour  of  the  fluid  in  question  be  a  perfect  gas,  and  of  very 
great  volume  as  compared  with  the  fluid  in  the  liquid  state,  the  curve; 
BC  will  be  nearly  a  hyperbola,  and  will  nearly  coincide  with  the 
isothermal  curve  of  the  higher  temperature  rv  to  which,  consequently, 
rc  will  be  nearly  equal;  and  the  following  equation  will  be  approximately 
true : 

I  1Kl4dr  +  LX  -L,=  j  'ly/r,        .         .     (85.) 

which,  when  the  difference  between  the  higher  and  lower  temperatures 
diminishes  indefinitely,  is  reduced  to  the  following: — 

K,  + ^  =  K„,    ....     (86.) 

that  is  to  say: — 

Corollary. — Theorem.  When  a  vapour  is  a  perfect  gas,  and  very  hilly 
as  compared  with  its  liquid,  the  rate  of  increase  of  the  total  heat  of  evaporation 
with  temperature  is  nearly  equal  to  the  specific  heat  of  the  vapour  at  constant 
pressure. 

This  was  demonstrated  by  a  different  process,  in  a  paper  read  to  the 
Royal  Society  of  Edinburgh  in  1850.  It  has  not  yet  been  ascertained, 
however,  whether  any  vapour  at  saturation  approaches  sufficiently  near  to 
the  condition  of  perfect  gas  to  render  the  theorem  applicable. 

53.  Concluding  Remarks. 

In  conclusion,  it  may  be  observed,  that  the  theory  of  the  expansive 
action  of  heat  embodied  in  the  propositions  of  this  paper  contains  but  one 
principle  of  hypothetical  origin — viz.,  Proposition  XII.,  according  to  which 
the  actual  heat  present  in  a  substance  is  simply  proportional  to  its 
temperature  measured  from  a  certain  point  of  absolute  cold,  and  multiplied 
by  a  specific  constant ;  and  that  although  existing  experimental  data  may 
not  be  adequate  to  verify  this  principle  precisely,  they  are  still  sufficient 
to  prove  that  it  is  near  enough  to  the  truth  for  all  purposes  connected 
with  thermodynamic  engines,  and  to  afford  a  strong  probability  that  it  is 
an  exact  physical  law. 


410  MAXIMUM   PRESSURE  AND   LATENT   HEAT   OF   VAPOURS. 


XXL— OX  FORMULAE  FOR  THE  MAXIMUM  PRESSURE 
AXD  LATENT  HEAT  OF  VAPOURS.* 

1.  It  is  natural  to  regard  the  pressure  which  a  liquid  or  solid  and  its 
vapour  maintain  when  in  contact  with  each  other  and  in  equiUbrio,  as  the 
result  of  an  expansive  elasticity  in  the  vapour,  balanced  by  an  attractive 
force  which  tends  to  condense  it  on  the  surface  of  the  liquid  or  solid,  and 
which  is  very  intense  at  that  surface,  but  inappreciable  at  all  sensible 
distances  from  it.  According  to  this  view,  every  solid  or  liquid  substance 
is  enveloped  by  an  atmosphere  of  its  own  vapour,  whose  density  close  to 
the  surface  is  very  great,  and  diminishes  at  first  very  rapidly  in  receding 
from  the  surface ;  but  at  appreciable  distances  from  the  surface  is  sensibly 
uniform,  being  a  function  of  the  temperature  and  of  the  attractive  force  in 
question. 

2.  Many  rears  since  I  investigated  mathematically  the  consequences 
of  this  supposition,  and  arrived  at  the  conclusion,  that  although  it  is 
impossible  to  deduce  from  it,  in  the  existing  condition  of  our  knowledge 
of  the  laws  of  molecular  forces,  the  exact  nature  of  the  relation  between 
the  temperature  and  the  maximum  pressure  of  a  vapour,  yet  that  if  the 
hypothesis  be  true,  it  is  probable  that  an  approximate  formula  for  the 
logarithm  of  that  pressure  for  any  substance  will  be  found,  by  subtracting 
from  a  constant  quantity,  a  converging  series  in  terms  of  the  powers  of 
the  reciprocal  of  the  absolute  temperature,  the  constant  and  the  coefficients 
of  the  series  being  determined  for  each  substance  from  experimental  data. 
Such  a  formula  is  represented  by 

Log.  P  =  A  -  -  -  C,  -  &a, 

T  T- 

where  P  denotes  the  pressure,  r  the  absolute  temperature,  that  is,  the 
temperature  as  measured  from  the  absolute  zero  of  a  perfect  gas-ther- 
mometer, A  the  constant  term,  and  B,  C,  &c,  the  coefficients  of  the 
converging  series. 

3.  On  applying  this  formula  to  M.  Regnault's  experiments  on  the 
pressure  of  steam,  it  was  found  that  the  first  three  terms  were  sufficient  to 

*  Read  before  the  British  Association  at  Liverpool,  September,  1854,  and  published 
in  the  Philosophical  Magazine,  December,  185-4. 


MAXIMUM   PRESSURE  AND   LATENT  HEAT   OF   VAPOURS.  411 

represent  the  results  of  these  experiments  with  minute  accuracy  through- 
out their  whole  extent;  that  is  to  say,  between  the  temperatures  of 

-  30°  and  +  230°  Centigrade 
=  -  22°  and       44G°  Fahrenheit, 

and  between  the  pressures  of  .,.-}0()  of  an  atmosphere,  and  82  atmospheres. 
Formula?  of  three  terms  were  also  found  to  represent  the  results  of 
Dr.  Ure's  experiments  on  the  vapours  of  alcohol  and  ether,  and  formulae 
of  two  terms  those  of  his  experiments  on  the  vapours  of  turpentine  and 
petroleum,  as  closely  as  could  be  expected  from  the  degree  of  precision  of 
the  experiments.  A  formula  of  two  terms  was  found  to  represent  accu- 
rately the  results  of  M.  Eegnault's  experiments  on  the  vapour  of  mercury. 

4.  These  formula?,  with  a  comparison  between  their  results  and  those  of 
the  experiments  referred  to,  were  published  in  the  Edinburgh  New  Philo- 
sophical Journal  for  July,  1849,  in  a  paper  the  substance  of  which  is 
summed  up  at  its  conclusion  in  the  following  proposition  (See  p.  1) : — 

If  the  maximum  elasticity  of  any  vapour  in  contact  with  its  liquid  be  ascer- 
tained for  three  points  on  the  scale  of  the  air-thermometer,  then  the  constants  of 
an  equation  of  the  form 

Log.P=A-B-^ 

T  T 

•may  be  determined,  which  equation  will  give,  for  that  vapour,  with  an  accuracy 
limited  only  by  the  errors  of  observation,  the  relation  behceen  the  temperature  (t), 
measured  from  the  absolute  zero,  and  the  maximum  elasticity  (P),  at  all 
temperatures  between  those  three  points,  and  for  a  considerable  range  beyond 
them. 

5.  In  the  case  of  water  and  mercury,  the  precision  of  the  experimental 
data  left  nothing  to  be  desired.  I  have,  however,  in  the  table  of  constants 
at  the  end  of  this  paper,  so  far  modified  the  coefficients  for  Avater  and 
mercury  as  to  adapt  them  to  a  position  of  the  absolute  zero  (274°  Centi- 
grade, or  493°-2  Fahrenheit  below  the  temperature  of  melting  ice),  which 
is  probably  nearer  the  truth  than  that  employed  in  the  original  paper, 
which  was  six-tenths  of  a  Centigrade  degree  lower.  This  modification, 
however,  produces  no  practically  appreciable  alteration  in  the  numerical 
results  of  the  formula?. 

6.  It  was  otherwise  with  respect  to  the  other  fluids  mentioned,  for 
which  the  experimental  data  were  deficient  in  precision,  so  that  the  values 
of  the  constants  could  only  be  regarded  as  provisional. 

7.  A  summary  published  in  the  Comptes  Bendus  for  the  14th  of  August, 
1854,*  of  the  extensive  and  accurate  experiments  of  M.  Eegnault  on  the 

*  See  Phil.  Mag.,  Series  4,  Vol.  VIII.,  p.  2G9. 


412  MAXIMUM   PRESSURE  AND   LATENT  HEAT  OF  VAPOURS. 

elasticities  of  the  vapours  of  ether,  sulphuret  of  carbon,  alcohol,  chloroform, 
and  essence  of  turpentine,  has  now  supplied  the  means  of  obtaining 
formula?,  founded  on  data  as  precise  as  it  is  at  present  practicable  to 
obtain,  for  the  maximum  pressures  of  these  vapours. 

A  synopsis  of  these  formulae,  and  of  the  constants  contained  in  them,  is 
annexed  to  this  paper.  The  constants,  as  given  in  the  table,  are  suited 
for  millimetres  of  mercury  as  the  measure  of  pressures,  and  for  the  scale 
of  the  Centigrade  thermometer;  but  logarithms  are  given,  by  adding 
which  to  them  they  can  be  easily  adapted  to  other  scales. 

The  limited  time  which  has  elapsed  since  the  publication  of  M.  Regnault's 
experiments  prevents  my  being  yet  able  to  bring  the  details  of  the 
investigation  of  the  formula?,  and  of  the  comparison  of  their  results  with 
those  of  experiment,  into  a  shape  suited  for  publication ;  but  I  shall  here 
add  some  remarks  on  their  degree  of  accuracy  and  the  extent  of  their 
applicability. 

8.  M.  Regnault  explains,  that  his  experiments  were  made  by  two 
methods;  at  low  temperatures,  by  determining  the  pressure  of  the  vapour 
in  vacuo;  at  high  temperatures,  by  determining  the  boiling-point  under 
the  pressure  of  an  artificial  atmosphere.  For  each  fluid  the  pressures 
determined  by  both  those  methyl-  were  compared  throughout  a  certain 
.series  of  intermediate  temperatures. 

For  all  Huids  in  a  state  of  absolute  purity,  the  results  of  those  two 
methods  agreed  exactly  (as  M.  Regnault  had  previously  shown  to  be  the 
case  for  water). 

The  presence,  however,  of  a  very  minute  quantity  of  a  foreign  substance 
in  the  liquid  under  experiment  was  sufficient  to  make  the  pressure  of  the 
vapour  in  vacuo  considerably  greater  than  the  pressure  of  ebullition  at 
a  given  temperature ;  and  it  would  appear,  also,  that  a  slight  degree  of 
impurity  affects  the  accuracy  even  of  the  latter  method  of  observation, 
although  by  far  the  more  accurate  of  the  two  when  they  disagree. 

9.  The  degree  of  precision  with  which  it  has  been  found  possible  to 
represent  the  results  of  the  experiments  by  means  of  the  formula?,  cor- 
responds in  a  remarkable  manner  Avith  the  degree  of  purity  in  which, 
according  to  M.  Regnault,  the  liquid  can  be  obtained. 

Sulphuret  of  Carbon,  M.  Regnault  states,  can  easily  be  obtained  perfectly 
pure.  For  this  fluid,  the  agreement  of  the  pressures  computed  by  the 
formula  with  those  determined  by  experiment  throughout  the  whole  range 
of  temperature  from  —  16°  Centigrade  to  +  136°,  is  almost  as  close  as 
in  the  case  of  steam. 

Ether  and  Alcohol  are  less  easy  to  be  obtained  perfectly  pure.  The 
discrepancies  between  calculation  and  experiment  in  these  cases,  though 
still  small,  are  greater  than  for  sulphuret  of  carbon. 

For  ether  the  formula  may  be  considered  as  practically  correct  through- 


MAXIMUM   PRESSURE  AND   LATENT  HEAT  OF   VAPOURS.  413 

out  the  whole  range  of  the  experiments,  from  —  20°  Centigrade  to 
+  116°;  but  for  alcohol  below  0°  Centigrade,  the  discrepancies,  though 
absolutely  small  quantities,  are  large  relatively  to  the  entire  pressures; 
and  the  formula  can  be  considered  applicable  above  this  temperature  only. 

Essence  of  Turpentine  has  been  discovered  by  M.  Eegnault  to  undergo  a 
molecular  change  by  continued  boiling.  For  this  fluid  the  agreement 
between  the  formula  and  the  experiments  is  satisfactory  above  40° 
Centigrade,  and  up  to  the  limit  of  the  experiments,  222°,  but  not 
below  40°. 

It  is  impossible  to  obtain  Chloroform  free  from  an  admixture  of  foreign 
substances.  Accordingly,  M.  Eegnault  has  found  that  the  two  methods 
of  determining  the  pressure  of  the  vapour  of  this  fluid  give  widely  different 
results,  neither  of  which  can  be  represented  accurately  by  the  formula 
now  proposed  below  the  temperature  of  70°  Cent.  From  this  temperature, 
however,  up  to  130°  Cent.,  the  limit  of  the  experiments,  the  agreement 
is  close. 

10.  In  the  cases  of  alcohol  and  turpentine,  the  discrepancies  between 
the  formulae  and  the  experiments  at  very  low  temperatures  are  such 
as  to  indicate  that  they  might  be  removed  by  introducing  a  fourth  term 
into  the  formulae,  inversely  proportional  to  the  cube  of  the  absolute 
temperature;  but  the  trifling  and  uncertain  advantage  to  be  thus  obtained 
would  be  outweighed  by  the  inconvenience  in  calculation,  and  especially  by 
the  necessity  for  solving  a  cubic  equation  in  computing  the  temperature 
from  the  pressure;  whereas,  with  formula?  of  three  terms,  it  is  only 
necessary  to  extract  a  square  root,  as  the  formula  No.  2  shows. 

1 1 .  Although,  for  the  mere  determination  of  the  maximum  pressure  of 
a  vapour  at  a  given  temperature,  or  its  temperature  at  a  given  pressure, 
a  table,  or  a  curve  drawn  on  a  diagram  may  be  sufficient,  still  there  are 
many  questions  of  thermodynamics  respecting  vapours  for  the  solution 
of  which  a  formula  is  essential. 

Amongst  these  is  the  computation  of  the  latent  heat  of  evaporation, 
which  is  equivalent  to  the  potential  energy  or  work  exerted  by  the 
vapour  in  overcoming  external  pressure,  added  to  that  exerted  in  over- 
coming molecular  attraction.  For  unity  of  iceight  of  a  given  substance, 
this  is  a  function  of  the  pressure,  temperature,  and  density;  but  for  a 
quantity  of  the  substance  such  that  its  volume  when  evaporated  exceeds 
its  volume  in  the  liquid  or  solid  state  by  unity  of  cubic  space,  the  latent 
heat  of  evaporation  is  simply  the  differential  coefficient  of  the  pressure 
with  respect  to  the  hyperbolic  logarithm  of  the  absolute  temperature,  as 
shown  in  the  formula  No.  3  ;  so  that,  although  the  densities  of  the 
vapours  of  the  seven  fluids  referred  to  in  this  paper  are  yet  known  by 
conjecture  only,  and  not  by  direct  experiment,  we  can,  from  the  relation 
between  the  pressure  and  the  temperature,  determine  accurately  how  much 


414  MAXIMUM   PRESSURE   AND   LATENT    HEAT   OF   VAPOURS. 

heat  must  be  expended  in  the  evaporation  of  so  much  of  each  of  them  as 
is  necessary  in  order  to  propel  a  piston  through  a  given  space  under  a 
given  constant  pressure,  and  thus  to  solve  many  problems  connected  with 
engines  driven  by  vapours  of  different  kinds. 

12.  It  is  somewhat  remarkable,  that  the  coefficients  of  the  reciprocal  of 
the  temperature  (B)  in  the  formulae  for  ether,  sulphuret  of  carbon,  and 
alcohol,  are  nearly  equal;  as  also  those  of  the  square  of  the  reciprocal  of 
the  temperature  (C)  for  ether  and  sulphuret  of  carbon. 

In  consequence  of  this,  the  pressure  of  the  vapour  of  ether,  and  its 
latent  heat  for  unity  of  space,  as  above  denned,  at  a  given  temperature, 
exceed  the  corresponding  quantities  for  sulphuret  of  carl  ion  at  the  same 
temperature,  in  a  ratio  which  is  nearly,  though  not  exactly,  constant,  and 
whose  average  value  is  somewhat  less  than  1*5. 


Synopsis  of  the  Formula,  &c. 

Notation. 

t  =  absolute  temperature  =  temp.  Cent.  +  274c  C. 

=  temp.  Fahr.  +  4G1°2  F. 

P  =  maximum  pressure  of  vapour  at  the  absolute  tempera- 
ture T. 

r   —  volume  of  unity  of  weight  of  the  liquid. 

V  =  volume  of  unity  of  weight  of  saturated  vapour. 

L  =  latent  heat  of  evaporation  of  unity  of  weight  of  the  fluid 
expressed  in  units  of  work. 

A,  B,  C,  constants. 

Formula. 

1.  To  find  the  maximum  pressure  from  the  temperatu:c. 

Com.  \or'  P  =  A  —      —    ... 

°  T  T1 

2.  To  find  the  temperature  from  the  maximum  pressi  n 

1  /  f  A  -  com,  log.  P        B^  )  _    B 

r  ~  V  t  ~  C  +  4  C-  i       2  G 


MAXIMUM   PRESSURE   AND   LATENT   HEAT   OF   VAPOURS. 


415 


3.  To  find  the  latent  heat  of  evaporation  (expressed  in  units  of  -work) 
of  so  much  of  the  fluid  that  its  hulk  when  evaporated  exceeds  its  bulk  in 

the  liquid  state  by  an  unit  of  space,  that  is  to  say,  of  the  weight  ^ — 

of  fluid.     In  this  formula  the  pressure  must  be   expressed  in  units   of 
weight  per  square  unit. 

=  r—   =P(-  +     ■:,      X  hyp.  log.  10. 


V  -  o 


d; 


(Hyp.  log.  10  =  2-30258509, 
the  common  logarithm  of  which  is  0-3622157.) 

Units  of  work  are  reduced  to  units  of  heat  (degrees  in  unity  of  weight 
of  liquid  water)  by  dividing  by  Joule's  equivalent  of  the  specific  heat  of 
liquid  water,  which  has  the  following  values,  according  to  the  units  of 
temperature  and  length  employed  : — 

Logarithms. 
Centigrade  scale,  and  metres,     .  .        423-54  2-6268969 

Centigrade  scale,  and  feet,         .         .     1389-6  3'1428898 

Fahrenheit's  scale,  and  feet,       .  .        772'0  2-8876173 


Constants  in  the  Formulae  for  Pressures  in  Millimetres  of 
Mercury,  and  Temperatuf.es  in  Centigrade  Degrees. 


Fluids. 

A. 

Log.  B. 

Log.  C. 

B 
2  0" 

B2 

4C-" 

Ether, 

7  1284 

3-0596504 

4-7065130 

0011275 

0-00012712 

Sulphuret  of  carhon,  . 

G-S990 

3-0520049 

4-7078426 

0-011044 

000012197 

Alcohol  above  0°  C,  . 

7  3259 

3-0570610 

5-2426805 

0  0032610 

0-0000 10634 

Water, 

7-8143 

3-1S11430 

5-0881S57 

0-0061934 

0-000038358 

Essence  of  turpentine  I 
above  40°  C.,.         .  \ 

6  2522 

2  9025209 

5-3712157 

0-0019511 

0  0000038067 

Chloroform  above  70°  C, 

5-S075 

2-4007279 

5-3919420 

0  00051022 

0-00000026032 

Mercury  up  to  35S°  C, 

7-5243 

3-4675637 

To  adapt  the  formula?  to  other  scales  of  pressure,  add  the  following 
logarithms  to  the  constants  A  : — 


41  6  MAXIMUM   PRESSURE  AND   LATENT  HEAT  OF  VAPOURS. 

For  inches  of  mercury,  .  .  .  .  .  •  2  5951/ 
For  kilogrammes  on  the  square  metre,  .  .  .  1*1 3341 
For  pounds  avoirdupois  on  the  square  foot,        .         .     0*44477 

To    adapt    the    formula'    to    the    scale    of    Fahrenheit's    thermometer 
multiply  B  by  1*8,  and  C  by  (1*8)2  =  3-24  ;  that  is  to  say, 

Add  to  log.  B, 0-2552725 

Add  to  log.  C •     0-5105450. 


THE  DENSITY   OF   STEAM.  417 


XXIL— ON   THE  DENSITY   OF   STEAM.* 

1 .  The  object  of  the  present  paper  is  to  draw  a  comparison  between  the 
results  of  the  mechanical  theory  of  heat,  and  those  of  the  recent  experi- 
ments of  Messrs.  Fairbairn  and  Tate  on  the  density  of  steam,  published  in 
the  Philosophical  Transactions,  for  1SG0. 


General  Equation  of  Thermodynamics. 

2.  The  equation  which  expresses  the  general  law  of  the  relations  between 
heat  and  mechanical  energy  in  elastic  substances  was  arrived  at  indepen- 
dently and  contemporaneously  by  Professor  Clausius  and  myself,  having 
been  published  by  him  in  Poggendorff's  Annalen  for  February,  1850,  and 
communicated  by  me  to  the  Royal  Society  of  Edinburgh  in  a  paper  which 
was  received  in  December,  1849,  and  read  on  the  4th  of  February,  1850 
(See  p.  23 Jf).  The  processes  followed  in  the  two  investigations  were  very 
different  in  detail,  though  identical  in  principle  and  in  results ;  Professor 
Clausius  having  deduced  the  law  in  question  from  the  equivalence  of  heat 
and  mechanical  energy  as  proved  experimentally  by  Mayer  and  Joule, 
combined  with  a  principle  which  had  been  previously  applied  to  the  theory 
of  substantial  caloric  by  Sady  Carnot,  while  by  me  the  same  law  was 
deduced  from  the  "  hypothesis  of  molecular  vortices,"  otherwise  called  the 
"  centrifugal  theory  of  elasticity." 

3.  Although,  since  the  appearance  of  the  paper  to  which  I  have  referred, 
the  notation  of  the  general  equation  of  thermodynamics  has  been  improved 
and  simplified  in  my  own  researches,  as  well  as  in  those  of  others,  I  shall 
here  present  it,  in  the  first  place,  precisely  in  the  form  in  which  I  first 
communicated  it  to  this  Society,  in  order  to  show  the  connection  between 
that  equation  in  its  original  form,  and  the  law  of  the  density  of  steam, 
which  has  since  been  verified  by  the  experiments  of  Messrs.  Fairbairn  and 
Tate.  The  equation,  then,  as  it  originally  appeared  in  the  twentieth 
volume  of  the  Transactions  of  the  Royal  Society  of  Edinburgh  (See  p.  24-9) 
is  as  follows  : — 

*  Read  before  the  Eoyal  Society  of  Edinburgh  on  April  2S,  1S62,  and  published  in 
Vol.  XXIII.  of  the  Transactions  of  that  Society. 

2  D 


418  THE   DENSITY   OF   STEAM. 

in  which  the  symbols  have  the  following  meanings  : — 

r,  the  absolute  temperature  of  an  elastic  substance  as  measured  from 
the  zero  of  gaseous  tension,  a  point  which  was  then  estimated  to  be 
at  274C-G  Centigrade  below  that  of  melting  ice,  but  which  is  now 
considered  to  be  more  nearly  at  274°  Centigrade,  or  4930,2  Fahr. 
below  that  temperature ; 

k,  A  constant,  expressing  the  height  on  the  thermometric  scale  of  the 
temperature  of  total  privation  of  heat  above  the  zero  of  gaseous 
tension.  This  constant  was  then  only  known  to  be  very  small ; 
according  to  later  experiments,  it  is  either  null  or  insensible ; 

n  M,  The  ideal  or  theoretical  weight,  in  the  perfectly  gaseous  state,  of 
an  unit  of  volume  of  the  substance,  under  unity  of  pressure,  at  the 
temperature  of  melting  ice ; 

C,  The  absolute  temperature  of  melting  ice,  measured  from  zero  of 
gaseous  tension  (that  is  to  say,  according  to  the  best  existing  data 
C  =  274°  Centigrade,  or  493°-2  Fahr.); 

V,  The  actual  volume  of  unity  of  weight  of  the  substance ; 

8  V,  An  indefinitely  small  increment  of  that  volume; 

<5  r,  An  indefinitely  small  increment  of  temperature ; 

U,  A  certain  function  of  the  molecular  forces  acting  in  the  substance ; 

+  B  Q',  The  quantity  of  heat  which  appears,  or  —  B  Q',  the  quantity 
of  heat  which  disappears,  during  the  changes  denoted  by  o  V  and 
Bt,  through  the  actions  of  molecular  forces,  independently  of  heat 
employed  in  producing  changes  of  temperature ;  such  quantity  of 
heat  being  expressed  in  equivalent  units  of  mechanical  energy. 

The  equation  having  been  given  in  the  above  form,  it  is  next  shown 
(See  p.  252),  that  the  differential  coefficients  of  the  function  U  have  the 
following  values  : — 

-=---C«M/cZV.^.  .     (3.) 

4.  The  physical  law  of  which  the  general  equation  just  cited  is  the 
symbolical  expression,  may  be  thus  stated  in  words : — The  mutual  trans- 


THE   DENSITY  OF  STEAM.  419 

formation  of  heat  and  mechanical  energy  during  any  indefinitely  small  change  in 
the  density  and  temperature  of  an  elastic  substance,  is  equal  to  the  temperature, 
reckoned  from  the  zero  of  absolute  cold,  multiplied  by  the  complete  differential  of 
a  certain  function  of  the  pressure,  density,  and  temperature;  which  function  is 
either  nearly  or  exactly  equal  to  the  rate  of  variation  with  temperature  of  the 
work  performed  by  indefinite  expansion  at  a  constant  temperature. 
5.  It  may  be  remarked  that  the  quantity, — 

cp  =  il  hyp.  log.  r  +  ^-^  (hyp.  log.  V  -  U)  j 

r      ■     (±0 

=  ft  hyp.  log.  r  +  /  —  d  V 

J  a  t  J 

(ft  being  the  real  specific  heat  of  the  substance  in  units  of  mechanical 
energy),  is  what,  in  later  investigations,  I  have  called  the  "  thermo- 
dynamic function;"  and  that  by  its  use,  and  by  making  k  =  0,, 
equation  (1)  is  reduced  to  the  simplified  form, 

-8Q'  =  rS0-feSr;  .  .  .     (5.) 

but  the  following  notation  is  more  convenient :  Let  $  h  denote  the 
whole  heat  absorbed  by  the  substance,  not  in  units  of  mechanical 
energy,  but  in  ordinary  thermal  units,  and  J  the  value  of  an 
ordinary  thermal  unit  in  units  of  mechanical  energy,  commonly 
called  "Joule's  equivalent,"  so  that 

JSh  =  k$T-  SQ'; 

then  the  general  equation  of  thermodynamics  takes  the  form 

J3/i  =  rS<£ (6.) 

G.  For  the  purposes  of  the  present  paper,  the  most  convenient  form  of 
the  thermodynamic  function  is  that  given  in  the  second  line  of  equation 
(i) ;  but  it  may  nevertheless  be  stated,  that  in  a  paper  read  to  this  Society 
in  1855,  and  which  now  lies  unpublished  in  their  archives,  it  was  shown 
that  another  form  of  that  function,  via. : — 

P„V„\,        ,  fdV 


fc+^)hyp.log.r-/^P,   .         .     (7.) 


P  V 
was  useful  in  solving  certain  questions ;   — -    °  denoting  the  same  thing 

u 

with  - — —  in  equation  (1). 
C  n  JM 


420  THE  DENSITY  OF   STEAM. 


Application  of  the  General  Equation  of  Thermodynamics  to 
the  Latent  Heat  and  Density  of  Steam. 

7.  At  the  time  when  the  general  equation  (1)  was  first  published, 
sufficient  experimental  data  did  not  exist  to  warrant  its  application  to  the 
computation  of  the  density  of  a  vapour  from  its  latent  heat.  But  very 
soon  afterwards,  various  points,  which  had  previously  been  doubtful,  were 
settled  by  the  experiments  of  Mr.  Joule  and  Professor  William  Thomson ; 
and  in  particular  Mr.  Joule,  by  his  experiments,  published  in  the  Philoso- 
phical Transactions  for  1850,  finally  determined  the  exact  value  of  the 
mechanical  equivalent  of  a  British  unit  of  heat,  to  which  he  had  been 
gradually  approximating  since  18-43,  viz.: — 

J  =  772  foot-pounds; 

and  Messrs.  Joule  and  Thomson  in  1851, 1852,  and  1853,  made  experiments 
on  the  free  expansion  of  gases,  especially  dry  air  and  carbonic  acid,  which 
established  the  very  near,  if  not  exact,  coincidence  of  the  true  scale  of 
absolute  temperature  with  that  of  the  perfect  gas-thermometer ;  that  is  to 
say,  those  experiments  proved  that  k  in  the  equation  (1)  is  sensibly  =  0. 
When,  with  a  knowledge  of  these  farts,  equation  (1)  is  applied  to  the 
phenomenon  of  the  evaporation  of  a  liquid  under  a  constant  pressure,  and 
at  a  constant  temperature,  it  takes  the  following  form  : — 

Jh  =  r^(Y-v),  .         .         .     (8.) 

where 

J  denotes  Joule's  equivalent,  or  772  foot-pounds  per  British  unit  of 
heat  (a  degree  of  Fahrenheit  in  a  pound  of  liquid  water) ; 

h,  The  heat  which  disappears  during  the  evaporation  of  1  lb.  of  the 
liquid;  that  is,  its  latent  heat  of  evaporation  in  British  units; 

r,  The  absolute  temperature  (=  temperature  on  Fahrenheit's  scale 
+  461°'2  Fahr.); 

P,  The  pressure  under  which  the  evaporation  takes  place  ; 

V,  The  volume  of  1  lb.  of  the  vapour ; 

v,  The  volume  of  1  lb.  of  the  liquid. 

As  the  latent  heat  of  evaporation  of  various  fluids  is  much  more 
accurately  known  by  direct  experiment  than  the  volume  or  density  of  their 
vapours,  the  most  useful  form  in  which  the  equation  (8)  can  be  put,  is 


THE  DENSITY   OF  STEAM.  421 

that  of  a  formula  for  computing  the  volume  of  a  vapour  from  its  latent 
heat,  viz.: — 

v-.  +  H      ....    (0.) 

r 

dr 

8.  Results  of  this  formula  were  calculated  by  Messrs.  Joule  and  Thomson,  . 
and  by  Professor  Clausius  for  steam,  and  showed,  as  had  been  expected,  a 
greater  density  and  less  volume  than  the  law  of  the  perfectly  gaseous 
condition  would  give.  Some  results  of  the  same  kind,  and  leading  to  the 
same  conclusion,  were  also  computed  by  me,  and  published  in  the  Philoso- 
phical Transactions  for  1853-54.  But  for  some  years  no  attempt  was  made 
by  any  one  to  make  a  table  for  practical  use  of  the  volumes  of  steam  from 
equation  (9);  because  the  scientific  world  were  in  daily  expectation  of 
the  publication  of  direct  experimental  researches  on  that  subject  by 
M.  Eegnault. 

9.  At  length,  in  the  spring  of  1855,  having  occasion  to  deliver  to  the 
class  of  my  predecessor,  Professor  Gordon,  a  course  of  lectures  on  the 
mechanical  action  of  heat,  and  finding  it  necessary  to  provide  the  students 
with  a  practical  table  of  densities  of  steam  founded  on  a  more  trustworthy 
basis  than  the  assumption  of  the  laws  of  the  perfectly  gaseous  condition,  I 
ventured  upon  the  step  of  preparing  a  table  of  the  densities  of  steam  for 
every  eighteenth  degree  of  Fahrenheit's  scale,  from  86°  to  410°  inclusive, 
with  the  logarithms  of  those  densities  and  their  differences,  arranged  so  as 
to  enable  the  densities  for  intermediate  temperatures  to  be  calculated  by 
interpolation.  Those  tables  were  published  in  a  lithographed  abstract  of 
the  course  of  lectures  before  mentioned,  which  is  now  out  of  print.  The 
same  tables,  however,  have  since  been  revised,  and  extended  to  every  ninth 
degree  of  Fahrenheit,  from  32°  to  428°,  and  have  been  printed  at  the  end 
of  a  work  On  Prime  Movers.  An  account  of  the  original  tables  was  read 
to  the  British  Association  in  1855.* 

10.  In  the  unpublished  paper  before  mentioned  as  having  been  read 
to  this  Society  in  1854,  the  densities  of  the  vapours  of  ether  and  bisulphuret 
of  carbon,  under  the  pressure  of  one  atmosphere,  as  computed  by  equation 
(9),  are  shown  to  agree  exactly  with  those  calculated  from  the  chemical 
composition  of  those  vapours. 

11.  The  method  of  using  equation  (9)  to  calculate  the  volume  of  one 
pound  of  steam,  is  as  follows : — 

I.  Calculate  the  total  heat  of  evaporation  of  steam  from  32°,  at  a  given 
temperature  T°  on  Fahrenheit's  scale,  by  Eegnault's  well-known  formula, 

*  The  reason  for  using  9°  Fahr.  as  the  interval  of  temperature  is,  that  it  is  equal  to 
5°  Centigrade  and  to  4°  Reaumur,  so  that  the  tables  can  be  applied  with  ease  to  any- 
one of  those  three  scales. 


422  THE  DENSITY  OF   STEAM. 

1091-7  +  0-305  (T°-  32°)    .         .         .     (10.) 

II.  From  that  total  heat  subtract  the  heat  required  to  raise  1  lb. 
of  water  from  32°  to  T°  Fahr.,  viz., 

f     cc/T; 

J  32' 

c  being  the  specific  heat  of  water,  the  remainder  will  be  the  latent  heat 
of  evaporation  of  1  lb.  of  steam  at  T°;  that  is  to  say, 

h  =  1091-7  +  0-305  (T  -  32°)  -  P    cdT.         .     (11.) 

J  32° 

In  computing  the  value  of  the  integral  in  this  formula,  use  has  been 
made  of  an  empirical  formula  founded  on  M.  Kegnault's  experiments  on 
the  specific  heat  of  water,  as  to  which,  see  the  Transactions  of  this 
Society  for  1851,  viz. : — 

P    c  d  T  =  T  -  T'  +  0-000000103  {(T  -  39°-l)3  -  (T  -  39°-l)3}     (11  A.) 

III.  The  absolute  temperature  is  given  by  the  formula, 

r  =  T  +  461°-2  Fahr.        .         .  .     (12.) 

IV    The  value  of  r  -      is  deduced  from  the  following  formula,  first 

a  t 

published  in  the  Edinburgh  Philosophical  Journal  for  July,  1849  (See  p.  1)  :— 

Com.  log.  P  =  A -»;  .         .         ■      (13.) 

r        t- 

from  which  it  follows  that 

/'P  =  2-302GP(B  +  ^);  .  .      (14.) 

drt  V         t- / 

the  values  of  the  constants  for  steam  being, — 

A,  for  pounds  of  pressure  on  the  square  foot,  .         8'2591 

log.  B  for  Fahrenheit's  scale         .         .  .  =3-43642 

log.  C  „  „  .         .  .  =5-59873. 

V.  The  volume  of  a  pound  of  liquid  water  at  the  temperature  T  may 
be  computed  with  sufficient  accuracy  for  the  present  purpose  by  the 
following  formula : — 

v  nearly  =  0-00801  Q^  +  ^).      .         .     (15.) 

VI.  These  preliminary  calculations  having  been  made,  the  formula  9 
can  now  be  applied  to  the  calculation  of  the  volume  of  one  pound  of 


THE   DENSITY   OF   STEAM. 


423 


steam  (making  J  =  772);  and  by  this  process  the  tables  already  mentioned 
were  computed. 

Comparison  of  the  Eesults  of  Theory  with  those  of  Messrs. 
Fairbairn  and  Tate's  Experiments. 

12.  The  experiments  of  Messrs.  Fairbairn  and  Tate  on  the  density  of 
steam  are  described  in  a  paper  which  was  read  to  the  Eoyal  Society  of 
London,  as  the  Bakerian  Lecture,  on  the  10th  of  May,  1860,  and  published 
in  the  Philosophical  Transactions  for  ^hat  year.  The  results  of  those 
experiments  give  what  is  called  the  "  relative  volume "  of  steam :  that  is, 
the  ratio  which  its  volume  bears  to  that  of  an  equal  weight  of  water  at 
the  temperature  of  greatest  density,  39°-l  Fahr.;  but  in  the  following 
table  of  comparison,  each  of  those  relative  volumes  is  divided  by  62"425, 
the  weight  of  a  cubic  foot  of  water  at  390,1  in  lbs.,  so  as  to  give  the 
volume  of  1  lb.  of  steam  in  cubic  feet.  The  numbers  of  the  experiments 
are  the  same  as  in  the  original  paper;  those  made  at  temperatures  below 
212°  being  numbered  from  1  to  9,  and  those  made  at  temperatures  above 
212°  from  1' to  14'. 

Comparison  of  the  Theory  with  the  Experiments  of  Messrs. 
Fairbairn  and  Tate. 


Volume  of  One  Pound  of  Steam 

in  Cubic  Feet 

Difference 

Number  of 
Experiment. 

Temperature 
Fahrenheit. 

Difference. 

Exper.  Vol. 

By  Theory. 

By  Exper. 

1 

136-77 

182-20 

132-60 

-0  40 

i 

2 

155  33 

8510 

85-44 

-0-34 

i 

3 

159-36 

77-64 

78-86 

-  1'22 

_     1 

6  5 

4 

170-92 

6016 

59  62 

+  0-54 

+  TT7 

5 

17148 

5943 

59-51 

-0-08 

1 

~TTT 

6 

174  92 

5518 

55  07 

+  011 

+  T5T 

7 

1S2-30 

47-28 

48-87 

-1-59 

_     1 

S 

188-30 

41-81 

42  03 

-0  22 

i 

T5T 

9 

198-78 

33  94 

34  43 

-0-49 

1 
?T7 

1' 

242  90 

15-61 

15  23 

+  0-12 

+  Tf^ 

2' 

244-82 

14-77 

14  55 

+  0-22 

+     CTT 

3' 

245-22 

1467 

14-30 

+  0-37 

+     & 

4' 

255-50 

12-39 

12-17 

+  0-22 

+    eV 

5' 

263-14 

10-96 

10-40 

+  0-56 

+  tV 

6' 

267  21 

10-29 

10-18 

+  0-11 

+  A 

7 

269-20 

9  977 

9-703 

+  0-274 

+  A 

8' 

274  76 

9-158 

9-361 

-0  203 

i 

9' 

273-30 

9  367 

8-702 

+  0-665 

+    A 

10' 

279-42 

8-539 

8-249 

+  0-290 

+    Tg 

11' 

2S2-58 

8-145 

7-964 

+  0-181 

+    A 

12' 

287*25 

7-603 

7  340 

+  0-263 

+    T5 

13' 

292-53 

7  041 

6-93S 

+  0-103 

+    BIT 

14' 

288-25 

7-494 

7-201 

+  0-293 

+    3fV 

424  THE  DENSITY  OF   STEAM. 


Remarks  on  the  Differences  between  the  Theoretical  and 
Experimental  Results. 

13.  The  differences  between  the  theory  and  the  experiments  as  to  the 
volumes  of  steam  at  temperatures  below  212°  are,  with  few  exceptions, 
of  very  small  relative  amount;  and  they  are  at  the  same  time  so  irregular 
as  to  show  that  they  must  have  mainly  arisen  from  causes  foreign  to  the 
data  used  in  the  theoretical  computations. 

1-1.  Above  212°  also,  the  differences  show  irregularity,  especially  in  the 
case  of  experiments  8'  and  9',  where  a  fall  of  temperature  is  accompanied 
by  a  diminution  instead  of  an  increase  in  the  volume  of  one  pound  of 
saturated  steam,  as  determined  by  experiment.  But  still  those  differences, 
presenting  as  they  do,  in  every  case  but  one,  an  excess  of  the  theoretical 
above  the  experimental  volume,  show  that  some  permanent  cause  of 
discrepancy  must  have  been  at  work ;  although  they  may  not  be  regular 
enough  to  determine  its  nature  and  amount,  nor  large  enough  to  constitute 
errors  of  importance  in  practical  calculations  relating  to  steam-engines. 

15.  So  far  as  it  is  possible  to  represent  those  differences  by  anything 
like  a  formula,  they  agree,  in  a  rough  way,  with  a  constant  excess  of  about 
0-21:  of  a  cubic  foot  in  the  theoretical  volume  of  one  pound  of  steam 
above  the  experimental  volume;  and  this  also  represents,  in  a  rough  way, 
the  difference  between  the  curves  whose  ordinates  express  respectively 
the  results  of  the  theoretical  formula  and  those  of  an  empirical  formula 
deduced  from  the  experiments,  so  far  as  those  curves,  as  shown  in  a  plate 
annexed  to  the  paper  referred  to  (See  Plate  II.),  extend  through  the  limits 
of  actual  experiment  on  steam,  above  212°. 

1G.  As  the  principles  of  the  mechanical  theory  of  heat  may  now  be 
considered  to  be  established  beyond  question,  it  is  only  in  the  data  of  the 
formula  that  we  can  look  for  causes  of  error  in  the  theoretical  calculation. 
I  shall  now  consider  those  data,  with  reference  to  the  probability  of  their 
containing  numerical  errors. 

I.  Total  Heat  of  Evaporation. — It  is  very  improbable  that  this  quantity, 
as  computed  by  M.  Regnault's  formula,  involves  any  material  error. 

II.  Sensible  Heat  of  the  Liquid  Water. — The  empirical  formula  from 
which  this  quantity  is  computed  was  determined  from  experiments  by 
M.  Regnault  which  agree  extremely  well  amongst  themselves.  (For  the 
investigation  of  the  formula,  see  Trans.  Boy.  Soc.  Edin.,  Vol.  XX.,  p.  441). 
The  subtraction  of  the  sensible  heat  from  the  total  heat  leaves  the  latent 
heat,  upon  which  the  increase  of  volume  depends;  hence,  to  account  for 
an  error  in  excess  of  the  formula  for  the  volume  by  means  of  an  error  in 


THE  DENSITY  OF  STEAM.  425 

the  computation  of  the  sensible  heat,  it  must  be  supposed  that  the  specific 
heat  of  liquid  water  above  212°  increases  much  more  rapidly  than  M. 
Eegnault's  experiments  show,  so  as  to  produce  a  correspondingly  more 
rapid  diminution  in  the  latent  heat  of  evaporation.  It  is  easily  computed, 
for  example,  that  to  account  for  an  error  in  excess  of  0*24  of  a  cubic  foot 
in  the  volume  of  one  pound  of  steam  at  266°,  by  an  error  in  defect  in 
the  sensible  heat,  we  must  suppose  that  error  to  amount  to  about  2 1 
British  thermal  units  per  pound  of  water ;  but  such  an  error  is  altogether 
improbable. 

III.  Absolute  Temperature. — The  position  of  the  absolute  zero  may  be 
considered  as  established  with  a  degree  of  accuracy  which  leaves  no  room 
for  any  error  sufficient  to  account  for  the  differences  now  in  question. 

d  P 

IV.  Function  r  . — The  same  may  unquestionably  be  said  of  this 

(It 

function  ;  which  represents  the  mechanical  equivalent  of  the  latent  heat  of 
evaporation  of  so  much  water  as  fills  one  cubic  foot  more  in  the  vaporous 
than  in  the  liquid  state. 

V.  The  volume  of  one  pound  of  the  liquid  water  is  itself  too  small  to 
affect  the  question. 

VI.  The  received  value  of  the  mechanical  equivalent  of  a  unit  of  heat 
cannot  err  by  so  much  as  -^  part  of  its  amount. 


Conclusions. 

1 7.  It  appears,  then,  that  none  of  the  data  from  which  the  theoretical 
calculations  are  made  are  liable  to  errors  of  a  magnitude  sufficient  to 
account  for  the  differences  between  the  results  of  those  calculations  and 
the  results  of  Messrs.  Fairbaim  and  Tate's  experiments,  small  as  those 
differences  are  in  a  practical  point  of  view.  Neither  does  there  appear 
to  have  been  any  cause  of  error  in  the  mode  of  making  the  experiments. 
There  remains  only  to  account  for  those  differences,  the  supposition  that 
there  was  some  small  difference  of  molecular  condition  in  the  steam  whose 
density  was  measured  in  the  experiments  of  Messrs  Fairbairn  and  Tate, 
above  212°,  and  the  steam  whose  total  heat  of  evaporation,  as  measured 
by  M.  Eegnault,  is  the  most  important  of  the  data  of  the  theoretical 
formula, — a  difference  of  such  a  nature  as  to  make  a  given  weight  of 
steam  in  Messrs.  Fairbairn  and  Tate's  experiments  occupy  somewhat  less 
space,  and  therefore  require  somewhat  less  heat  for  its  production,  than 


42G  THE   DENSITY   OF   STEAM. 

the  same  weight  of  steam  in  M.  Regnault's  experiments  at  the  same 
temperature.  That  difference  in  molecular  condition,  of  what  nature 
soever  it  may  have  been,  was  in  all  probability  connected  with  the  fact, 
that  in  the  experiments  of  Messrs.  Fairbairn  and  Tate  the  steam  was  at 
rest,  whereas  in  those  of  M.  Eegnault  it  was  in  rapid  motion  from  the 
boiler  towards  the  condenser.  It  is  obvious,  however,  that  in  order  to 
arrive  at  a  definite  conclusion  on  this  subject,  further  experimental 
researches  are  required. 


THE  SECOND   LAW   OF   THERMODYNAMICS.  427 


XXIII— ON  THE  SECOND  LAW  OF  THERMODYNAMICS.* 

1.  It  has  long  been  established  that  all  the  known  relations  between  heat 
and  mechanical  energy  are  summed  up  in  two  laws,  called  respectively 
the  first  law  and  the  second  law  of  thermodynamics :  viz. — 

First  Law. — Quantities  of  heat  and  of  mechanical  energy  are  con- 
vertible at  the  rate  very  nearly  of  772  foot-pounds  to  the  British  (or 
Fahrenheit- avoirdupois)  unit,  or  424  kilogrammetres  to  the  French  (or 
Centigrade-metrical)  unit  of  heat. 

Second  Law. — The  quantity  of  energy  which  is  converted  from  one 
of  those  forms  to  the  other  during  a  given  change  of  dimensions  and 
condition  in  a  given  body,  is  the  product  of  the  absolute  temperature  into 
a  function  of  that  change,  and  of  the  kind  and  condition  of  the  matter  of 
the  body. 

By  absolute  temperature  is  here  to  be  understood  temperature  measured 
according  to  a  scale  so  graduated  that  the  temperature  of  a  homogeneous 
body  shall  vary  in  the  simple  proportion  of  the  quantity  of  energy  it 
possesses  in  the  form  of  sensible  or  thermometric  heat. 

2.  The  laws  of  thermodynamics,  as  here  stated,  are  simply  the 
condensed  expression  of  the  facts  of  experiment.  But  they  are  also 
capable  of  being  viewed  as  the  consequence  of  the  supposition,  that  the 
condition  of  bodies  which  accompanies  the  phenomena  of  sensible  heat 
consists  in  some  kind  of  motion  amongst  their  particles. 

3.  The  first  law  would  obviously  follow  from  the  supposition  of  any 
kind  of  molecular  motion  whatsoever,  and  it  therefore  affords  of  itself  no 
reason  for  preferring  one  supposition  as  to  the  kind  of  molecular  motion 
which  constitutes  sensible  heat  to  another. 

4.  But  if  there  be  molecular  motions  in  bodies,  it  is  certain  that, 
although  all  such  motions  are  capable  of  conversion  into  that  which 
constitutes  sensible  heat,  some  of  them  are  not  accompanied  by  sensible 
heat.  For  example,  the  motion  (supposed  to  be  vibratory  and  wave-like) 
which  constitutes  radiance,  whether  visible  or  invisible,  is  not  accompanied 
by  sensible  heat,  and  only  produces  sensible  heat  by  its  absorption;  that 

*  Read  before  the  British  Association  at  Birmingham,  September,  1S65,  and 
published  in  the  Philosophical  Magazine,  October,  1805. 


428  THE   SECOND   LAW  OF  THERMODYNAMICS. 

is,  in  the  language  of  hypothesis,  by  its  conversion  into  some  other  hind 
of  motion;  while,  on  the  other  hand,  in  the  production  of  radiance 
sensible  heat  disappears. 

5.  The  object  of  the  present  paper  is  to  give  an  elementary  proof  of 
the  proposition,  that  the  second  law  of  thermodynamics  folloivs  from  the 
supposition  that  sensible  hint  consists  in  any  land  of  steady  molecular  motion 
villi  in  limited  spaces. 

G.  The  term  "  steady  motion "  is  here  used  in  the  same  sense  as  in 
hydrodynamics,  to  denote  motion,  whether  of  a  continuous  fluid  or  of  a 
system  of  detached  molecules,  in  which  the  velocity  and  direction  of 
motion  of  a  particle  depend  on  its  position  only;  so  that  each  particle  of 
the  series  of  particles  which  successively  pass  through  a  given  position 
assumes  in  its  turn  the  velocity  and  direction  proper  to  that  position. 
In  other  words,  steady  motion  may  be  denned  as  motion  in  a  set  of 
streams  of  invariable  figure. 

"When  steady  motion  takes  place  in  matter  that  is  confined  within  a 
limited  space,  the  streams  in  which  the  particles  move  must  necessarily 
return  into  themselves,  and  be  circulating  streams,  being  in  that  respect 
of  the  nature  of  whirls,  eddies,  or  vortices. 

7.  Steady  motion  keeps  unaltered  the  distribution  of  the  density  of 
the  moving  matter;  and  it  therefore  keeps  unaltered  the  forces  depending 
on  such  distribution,  whether  of  the  nature  of  pressure  or  of  attraction. 
In  that  respect  it  differs  from  unsteady  motion,  such  as  vibratory  and 
wave-like  motion. 

8.  Conceive  a  limited  space  of  any  figure  whatsoever  to  be  filled  with 
matter  in  a  state  of  steady  motion.  The  actual  energy  of  any  particle  of 
that  matter  is  the  product  of  its  mass  into  the  half-square  of  its  velocity; 
and  the  actual  energy  of  the  whole  mass  of  matter  is  the  sum  of  all  those 
products;  and  because  of  the  steadiness  of  the  motion,  the  actual  energy 
of  the  particle  which  at  any  instant  whatsoever  occupies  a  given  position 
is  some  definite  fraction  of  the  whole  actual  energy,  depending  upon  that 
position,  and  upon  the  distribution  of  matter  within  the  space;  but  the 
scale  of  absolute  temperature  is  defined  as  being  so  graduated  that  the 
whole  actual  energy  of  the  matter  within  the  space  is  the  product  of 
the  absolute  temperature,  the  mass  of  matter,  and  some  function  of  the 
sort  and  distribution  of  the  matter ;  therefore,  the  half-square  of  the  velocity 
of  the  particle  which  at  any  instant  occupies  a  given  position  in  the  space  con- 
sidered, is  equal  to  the  absolute  temperature  multiplied  by  some  function  of  that 
position,  and  of  the  sort  and  distribution  of  the  matter. 

9.  Suppose  now  that  the  dimensions  of  the  limited  space  in  which  the 
moving  matter  is  enclosed,  and  the  distribution  of  that  matter,  undergo 
an  indefinitely  small  change  by  the  application  of  suitable  forces,  and 
that  after  that  process   the  motion   becomes    steady  as   it  was  before. 


THE   SECOND  LAW  OF   THERMODYNAMICS.  429 

Then  the  dimensions  and  position  of  each  circulating  stream  will  have 
been  altered;  and  the  work  done  in  effecting  that  alteration  will  consist 
of  energy  converted  between  the  forms  of  potential  energy  of  the  applied 
forces,  and  actual  energy  of  the  molecular  motions — that  is,  between  the 
forms  of  mechanical  energy  and  of  heat.  Consider  now  a  point  in  one 
of  the  circulating  streams  before  the^  change,  and  let  fall  from  it  a 
perpendicular  upon  the  same  stream  after  the  change.  The  work  done 
in  shifting  the  path  of  the  particle  which  at  any  instant  occupies  that 
point,  is  the  product  of  the  perpendicular  displacement  of  the  stream 
into  the  force  exerted  along  that  perpendicular.  But  the  perpendicular 
displacement  of  the  stream  is  a  function  of  the  position  of  the  point 
shifted,  the  distribution  of  matter  in  the  space,  and  the  change  of 
dimensions  and  distribution;  and  the  force  is  equal  and  opposite  either 
to  the  centrifugal  force  of  the  particle  or  to  one  of  its  components,  and 
is  therefore  proportional  to  the  square  of  the  velocity  of  the  particle,  and 
to  some  function  of  its  position,  and  of  the  sort  and  distribution  of 
matter  in  the  body.  Therefore,  the  energy  transformed  in  shifting  the  path 
of  any  particle  is  proportional  to  the  square  of  its  velocity,  and  to  some  function 
of  its  position,  of  the  sort  and  distribution  of  matter  in  the  space  considered,  and 
of  the  change  in  dimensions  of  that  space  and  in  the  distribution  of  the  matter. 

10.  But  the  square  of  the  velocity  of  the  particle  which  at  any  instant 
occupies  a  given  position  has  already  been  shown  to  be  proportional  to 
the  absolute  temperature,  and  to  some  function  of  that  position  and  of 
the  sort  and  distribution  of  the  matter ;  therefore,  if  sensible  heat  consists 
in  any  kind  of  steady  molecular  motion  within  limited  spaces,  the  conversion  of 
energy  during  any  change  in  the  dimensions  of  such  spaces,  and  in  the  distri- 
bution of  matter  in  them,  is  the  product  of  the  absolute  temperature  into  some 
function  of  that  change  and  of  the  sort  and  distribution  of  the  matter. 

11.  In  a  paper  "On  the  Mechanical  Action  of  Heat,"  published  in  the 
Transactions  of  the  Royal  Society  of  Edinburgh  for  1850  (See  p.  %3Jj),  the 
author  deduced  the  second  law  of  thermodynamics,  in  the  form  above 
stated,  from  the  hypothesis  of  a  particular  sort  of  steady  molecular  motion 

viz.,  revolution  in   circular  streams  or  vortices.     In  a  paper  "  On  the 

Centrifugal  Theory  of  Elasticity,"  published  in  the  same  Transactions  for 
1851  (Seep.  4.9),  he  deduced  the  same  law  from  the  hypothesis  of  steady 
molecular  motion  in  circulating  streams  of  any  figure  whatsoever,  being  a 
proposition  substantially  identical  with  that  set  forth  in  the  present  paper; 
but  as  the  demonstration  in  the  paper  of  1851  involved  tedious  and  intri- 
cate symbolical  processes,  he  has  written  the  present  paper  in  order  to 
show  that  the  demonstration  can  be  effected  very  simply. 

12.  It  is  obvious  that  the  steadiness  of  the  supposed  molecular  motions 
is  the  essential  condition  which  makes  the  second  law  of  thermodynamics 
deducible  from  a  mechanical  hypothesis;  and  that  no  kind  of  unsteady 


430  THE   SECOND   LAW  OF  THERMODYNAMICS. 

motion,  such  as  vibratory  or  wave-like  motion,  would  lead  to  the  same 
results.  If,  then,  it  be  admitted  as  probable,  that  the  phenomena  of  heat 
are  due  to  unseen  molecular  motions,  it  must  also  be  admitted,  that 
while  the  motions  which  constitute  radiance  may  be  vibratory  and  wave- 
like, the  motions  which  constitute  sensible  or  thermometric  heat  must  be 
steady  and  like  those  of  circulating  streams. 

13.  The  function  by  which  the  absolute  temperature  is  multiplied  in 
calculating  the  conversion  of  energy  between  the  mechanical  and  the 
thermic  forms,  is  the  variation  of  what  the  author  has  called  the  mcta- 
morphic  function*  being  one  term  of  the  thermodynamic  function,}  which 
corresponds  to  what  Professor  Clausius  calls  cntrop'tv.% 


A  P  P  E  X  D  I  X. 

The  following  is  the  symbolical  expression  uf  the  demonstration  given 
in  the  paper. 

Let  m  stand  for  the  specific  properties  of  the  sort  of  matter  which  is 
in  a  state  of  steady  motion  within  a  limited  space ; 

f  for  the  figures  and  dimensions  of  that  space,  and  of  the  paths 
described  by  the  particles  contained  in  it ;  and  8/  for  any  indefinitely 
small  change  of  such  figures  and  dimensions ; 

p  for  the  position,  relatively  to  the  centre  of  the  matter  contained  in 
the  space,  of  a  point  which  is  fixed  so  long  as  S/  =  0.  Because  the 
motion  is  steady,  each  particle  of  matter  which  successively  arrives  at  the 
point  p  assumes  the  velocity,  direction,  and  curvature  of  motion  proper 
to  that  point.  Let  v  be  that  velocity,  and  r  the  radius  of  that  curvature ; 
then  for  a  particle  of  mass  unity,  in  the  act  of  traversing  p, 

•> 
Actual  energy  of  mass  1  =  —  =  kr,  .         .     (1.) 

where  r  is  a  quantity  upon  whose  uniformity  throughout  the  space  the 
steadiness  of  the  motion  depends,  and  k  a  function  of  (m,  f,  p) ;  and 

r2        2  k  T 
Centrifugal  force  of  mass  1  =       =  — — ; .         .     (2.) 

r  r  v 

2  Tt 
in  which  r,  and  consequently  — ,  are  functions  of  (m,  f,  p). 

*  See  "  On  the  Science  of  Energetics,"  page  209. 
+  Philosophical  Transactions,  1S54. 

X  Ueber  verschiedene  fur  die  Anwendung  beqneme  Formen  der  Hauptghkhungen  der 
mechanischen  Warmetheorie,  April,  1865. 


THE  SECOND   LAW  OF  THERMODYNAMICS.  431 

Now,  let  the  change  denoted  by  8/  take  place,  and  let  the  steadiness  of 
motion  be  restored :  let  8  n  be  the  length  of  a  line  drawn  through  the 
original  position  of  the  point  p,  so  as  to  be  perpendicular  to  the  path  of 

A 

the  stream  of  particles  which  formerly  traversed  p;  and  let  rn  be  the 

angle  made  by  8  n  with  r.  Then  8  n  and  r  n  are  both  functions  of 
(to,  f,  p,  bf).  Also,  the  work  done,  or  energy  converted,  for  a  unit  of 
mass  at  the  point  p,  while  the  path  of  the  particles  that  traverse  p  is 
shifted  through  8  n,  is  as  follows  : — 

A 
v2    ~             A        2  k t.  8 n.  cos  rn  »«   ,,» 

-.  c  n.  cos  rn  = =  t  X  function  of  (m,f,p,  bf).  (3.) 

The  energy  converted  during  the  change  Sf,  throughout  the  whole  space 
considered,  is  the  sum  of  the  quantities  of  energy  converted  for  each  unit 
of  mass  within  the  space.  But  r  by  definition  is  uniform ;  and  the  sum 
of  a  set  of  functions  of  p  is  a  function  of  /  and  m ;  therefore,  the  whole 
energy  converted  is 

A 
_    2  h.  Sn.  cos  r  n  .  ,,      ,  <*  fX  ,., 

r.2. =   X  functions  of  (?'/?,/,  bj);   .     (4.) 

r 

and  because  bf  is  indefinitely  small,  the  preceding  expression  is  equivalent 
to  the  following  : 

Energy  converted  =  r  .  function  (m,f)  .Bf=  t  .  8F  (m,f).       (5.) 

Let  t  be  called  absolute  temperature,  and  this  is  the  second  law  of  thermo- 
dynamics.    It  is  to  be  observed  that  /  may  be,  and  often  is,  a  function  of  r. 


432  THE   SECOND  LAW  OF  THERMODYNAMICS. 


XXIV.— OX  THE  WANT  OF  POPULAR  ILLUSTRATIONS  OF 
THE  SECOND  LAW  OF  THERMODYNAMICS  * 

1.  The  science  of  thermodynamics  is  based  on  two  laws,  the  first  of 
which  states  the  fact  of  the  mutual  convertibility  of  heat  and  mechanical 
energy,  while  the  second  shows  to  what  extent  the  mutual  conversion  of 
those  two  forms  of  energy  takes  place  under  given  circumstances.  In  the 
course  of  the  last  few  years  the  first  law  has  been  completely  "popularised;" 
it  has  been  amply  explained  in  books  and  lectures,!  composed  in  a  clear 
and  captivating  style,  and  illustrated  by  examples  at  once  familiar  and 
interesting,  so  as  to  make  it  easily  understood  by  those  who  do  not  make 
science  a  professional  pursuit. 

2.  The  second  law,  on  the  other  hand,  although  it  is  not  less  important 
than  the  first,  and  although  it  has  been  recognised  as  a  scientific  principle 
for  nearly  as  long  a  time,  has  been  much  neglected  by  the  authors  of 
I  id]  mlar  (as  distinguished  from  elementary)  works;!  and  the  consequence 
is  that  most  of  those  who  depend  altogether  on  such  works  for  their 
scientific  information  remain  in  ignorance,  not  only  of  the .  second  law, 
but  of  the  fact  that  there  is  a  second  law ;  and  knowing  the  first  law 
only,  imagine  that  they  know  the  whole  principles  of  thermodynamics. 
The  latter  is  the  worst  evil  of  the  two  :  "  a  little  learning "  is  not  "  a 
dangerous  thing  "  in  itself,  but  becomes  so  when  its  possessor  is  ignorant 
of  its  littleness. 

3.  In  the  present  paper  I  do  not  pretend  to  supply  that  want,  but 
rather  to  point  out  its  existence  to  authors  who  possess  the  faculty  of 
popularising ;  in  order  that  they,  by  means  of  lectures,  writings,  and 
lecture-room  experiments,  may  convey  a  general  knowledge  of  the  nature 
and  results  of  the  second  law  of  thermodynamics  to  those  who  feel  an 
interest  in  science  without  making  it  a  regular  study. 

4.  Before  considering  how  the  second  law  can  best  be  stated  and 
explained,  it  may  be  well  to  point  out  how  far  it  is  possible  to  proceed 

*  From  The  Engineer  of  June  28,  1867. 

f  Such,  for  example,  as  Dr.  Tyndall's  Heat  Considered  as  a  Mode  of  Motion. 
X  The  explanation  of  the  second  law  of  thermodynamics  in   Dr.    Balfour  Stewart's 
excellent  treatise  on  heat  is  elementary ;  but  it  is  not,  nor  does  it  profess  to  be,  popular. 


THE  SECOND   LAW   OF   THERMODYNAMICS.  433 

with  the  solution  of  questions  in  thermodynamics  by  means  of  the  first 
law  alone,  without  the  aid  of  the  second  law.  The  first  law  informs  us 
that  when  mechanical  work  is  done  by  means  of  heat,  a  quantity  of  heat 
disappears,  bearing  a  constant  ratio  to  the  quantity  of  work  done — viz., 
that  of  one  British  unit  of  heat  (or  one  degree  of  Fahrenheit  in  a  pound 
of  water)  for  every  722  foot-pounds  of  mechanical  work  done.  In  order, 
therefore,  to  calculate  how  much  heat  will  disappear  during  a  given 
change  in  the  dimensions  of  a  substance  under  the  action  of  given  forces, 
it  is  necessary  to  know  the  quantity  of  work  done  during  such  change ; 
and  the  cases  in  which  the  expenditure  of  heat  can  be  calculated  by 
means  of  the  first  law  alone,  are  those  and  those  only,  in  which  the  work 
done  can  be  directly  measured ;  that  is  to  say,  in  which  the  work 
is  sensibly  altogether  external,  or  done  against  forces  exerted  between  the 
body  under  consideration  and  other  bodies,  and  in  which  no  part  or  no 
sensible  part  of  the  work  is  internal,  or  done  against  forces  exerted  upon 
each  other  by  the  particles  of  the  body,  and  therefore  incapable  of  direct 
measurement. 

5.  The  only  phenomenon  which  fulfils  that  condition  is  the  expansion 
of  a  perfect  or  sensibly  perfect  gas ;  that  is,  of  a  substance  in  a  condition 
such  that  its  pressure  at  a  given  temperature  is  proportional,  or  sensibly 
proportional,  to  its  density  simply.  To  illustrate  this  by  an  example  :  let 
us  suppose  one  pound  of  atmospheric  air  (which,  though  not  absolutely 
a  perfect  gas,  may  be  treated  as  such  for  practical  purposes)  at  the 
temperature  of  melting  ice  (32°  Fahr.),  to  be  contained  in  a  cylinder 
whose  sectional  area  is  equal  to  one  square  foot,  being  confined  in  that 
cylinder  by  means  of  a  piston  loaded  Avith  a  pressure  amounting  to 
4233  pounds.  Then,  from  the  experiments  of  Regnault  on  the  pressure 
and  density  of  air,  it  is  known  that  the  length  of  the  cylindrical  space 
occupied  by  the  air  is  6*1 93  ft. 

Next,  let  the  load  on  the  piston  be  gradually  diminished  until  it  is 
reduced  to  2116|  lbs.,  being  one-half  of  its  original  amount,  and  let  the 
question  to  be  solved  be — what  quantity  of  heat  must  be  communicated  to  the 
air,  in  order  that  its  temperature  may  remain  constant  during  the  expansion 
which  accompanies  the  diminution  of  pressure  ?  The  solution  is  as  follows: — 
As  the  temperature  is  constant,  and  the  air  is  treated  as  perfectly  gaseous, 
the  product  of  the  pressure  on  the  piston  into  the  volume  of  the  air  •  that 
is  to  say,  4233  x  6-193  =  26215,  remains  constant  during  the  expansion; 
and  the  external  work  done  in  driving  the  piston  against  the  gradually 
diminishing  load  is  found  by  multiplying  that  product  by  the  hyperbolic 
logarithm  of  2,  the  rate  of  expansion,  that  is  to  say — 

Work  done  =  26215  x  '69315 
=  18171  foot-pounds. 

2  E 


434  THE   SECOND   LAW  OF   THERMODYNAMICS. 

That  is  the  external  work,  and  the  internal  work  is  practically  inappreci- 
able ;  therefore,  that  also  is  the  mechanical  equivalent  of  the  quantity  of 
heat  required  in  order  to  keep  the  temperature  of  the  air  constant;  and 
dividing  18171  by  772,  that  quantity  of  heat  in  ordinary  British  thermal 
units  is  found  to  be  23*54  j  that  is,  as  much  heat  as  would  raise  the 
temperature  of  23'54  lbs.  of  water  one  degree  of  Fahrenheit;  and  such  is 
the  value  of  the  latent  heat  of  expansion  of  1  lb.  of  air  in  doubling  its 
volume  at  the  constant  temperature  of  32°  Fain-.  Here,  then,  the  solution 
has  been  obtained  by  means  of  the  first  law  of  thermodynamics  only, 
without  the  aid  of  the  second. 

6.  It  is  otherwise  Avhen  the  external  work  is  accompanied  by  the 
performance  of  internal  work  to  a  practically  important  extent;  in  other 
words,  when  we  deal  with  substances  which  cannot  be  treated  as  perfectly 
gaseous,  such  as  fluids  in  the  act  of  evaporating.  For  example,  let 
there  be,  as  before,  a  cylindrical  vessel,  whose  sectional  area  is  one  square 
foot,  and  let  it  contain  1  lb.  of  water  in  the  liquid  state,  at  the  temperature 
of  212°  Fahr.,  which  will  occupy  a  length  of  the  cylinder  equal  to  "017 
of  a  foot.  Let  the  water  be  confined  by  means  of  a  piston,  the  load  upon 
which,  in  order  to  be  just  sufficient  to  confine  the  water,  must  be  equal  to 
the  mean  atmospheric  pressure  on  a  square  foot,  that  is  to  say,  211G3  lbs. 
If  additional  heat  be  now  communicated  to  the  water,  without  altering  the 
load  on  the  pistan,  it  is  well  known  that  its  temperature  does  not  rise,  but 
that  it  passes  by  degrees  into  the  state  of  steam,  driving  the  piston  before 
it.  By  the  time  the  water  has  entirely  assumed  the  state  of  steam,  it 
occupies  2G"3G  ft.  in  length  of  the  cylinder;  so  that  the  piston  has  been 
driven  through  26*343  ft.  against  the  constant  load  of  21 16*3  lbs.  Let 
the  question  proposed  be,  to  calculate  from  those  data  the  expenditure  of 
heat.     The  external  work  has  the  following  value  : — 

211G-3  x  2G-343  =  55750  foot-pounds, 

and  this  is  equivalent  to  7 2" 2  units  of  heat  nearly.  But  besides  the 
external  work  done  in  driving  the  piston,  there  is  internal  work  done  in 
overcoming  the  cohesion  of  the  particles  of  water ;  and  that  internal  work 
is  incapable  of  direct  measurement. 

7.  Here  it  is  that  the  second  law  of  thermodynamics  becomes  useful ; 
for  it  informs  us  how  to  deduce  the  whole  amount  of  work  done — internal 
and  external — from  the  knowledge  which  Ave  have  of  the  external  work. 
That  law  is  capable  of  being  stated  in  a  variety  of  forms,  expressed  in 
different  words,  although  virtually  equivalent  to  each  other.  The  most 
convenient  form  for  the  present  purpose  appears  to  be  the  following: — 

To  find  the  whole  work,  internal  and.  external,  multiply  the  absolute  tempera- 
ture at  which  the  change  of  dimensions  takes  place,  oy  the  rate  per  degree  at 
which  the  external  work  is  varied  ly  a  small  variation  of  the  temperature. 


THE   SECOND   LAW  OF  THERMODYNAMICS.  435 

By  absolute  temperature  is  meant  temperature  measured  from  the  absolute 
zero,  or  point  of  total  privation  of  heat,  which  is  known  by  theoretical 
deduction  from  experimental  data  to  he  about  4610,2  below  the  ordinary 
zero  of  Fahrenheit's  scale. 

8.  To  apply  the  second  law  to  the  present  problem,  suppose  the  tempera- 
ture at  which  the  given  increase  of  volume  (viz.,  26-343  cubic  feet)  takes 
place,  to  be  lowered  by  one  degree  of  Fahrenheit.  Then,  from  tables  and 
formulas  founded  on  Regnault's  experiments,  we  know  that  the  pressure 
of  the  steam  is  diminished  by  42*05  lbs.  on  the  square  foot.  Hence,  the 
external  work  is  diminished  by  reduction  of  temperature  at  the  following 
rate : — 

F  =  42"05  x  26'343  =  1 107'7  foot-pounds  per  degree  of  Fahrenheit. 

The  absolute  temperature  is 

t  =  212°  x  461°-2  =  G73°-2  Fahr.; 

and,  therefore,  the  whole  work,  internal  and   external,  done  during  the 
evaporation  of  1  lb.  of  water  at  the  temperature  of  212°  Fahr.  is, 

t  F  =  673°-2  x  1107-7  =  745,800  foot-pounds  very  nearly. 

To  reduce  this  quantity  to  British  thermal  units,  divide  by  772;  the 
result  is 

745,800  -r-  772  =  966, 

being  the  latent  heat  of  evaporation  of  steam  at  212°  Fahr. 
The  total  work  just  calculated  is  made  up  as  follows : — 

Foot-pounds. 
External  Avork,  computed  in  Article  6,  .         .        55,750 

Internal  work, 690,050 


Total  work, 745,800 

from  which  it  appears  that  the  external  work  done  in  evaporating  water 
under  the  mean  atmospheric  pressure  is  less  than  *l\  per  cent,  of  the 
whole  work;  the  remainder,  or  92^  per  cent.,  being  internal  work. 

9.  The  second  law  may  also  be  applied  to  solve  the  inverse  problem, 
of  deducing  from  the  expenditure  of  heat  in  a  given  process,  and  from 
the  relations  between  pressure  and  temperature,  the  change  of  dimensions 
with  which  that  process  is  accompanied;  and  such  has  been  the  use 
chiefly  made  of  that  law  in  the  actual  history  of  thermodynamics. 
Previous  to  the  publication,  in  September,  1859,  of  the  experiments  of 


43G  THE  SECOND   LAW   OF  THERMODYNAMICS. 

Messrs.  Fairbairn  and  Tate,  there  did  not  exist  any  accurate  determinations 
of  the  density  of  saturated  steam  at  different  temperatures  ;  and,  therefore, 
some  of  the  writers  on  thermodynamics  found  it  necessary  to  calculate 
that  density  theoretically,  by  the  help  of  the  second  law,  from  the  latent 
heat  of  evaporation,  which  was  very  accurately  known  through  the  experi- 
ments of  Regnault.     The  following  is  an  example  of  such  calculations  : — 

Latent  heat  of  evaporation  of  1  lb.  of  water  at  212°  Fahr., 

as  known  by  experiment,  in  ordinary  British  thermal  units,          9GG 

Applying   the    first   law,    that  is,    multiplying   by    Joule's 

equivalent,  .          .         .          .         .  .         .         .         .  772 


The  value  of  that  latent  heat  in  foot-pounds   of  work  is 

found  to  be  ........   7  1  -"» 752 

Dividing  that  quantity  of  work  by  the  absolute  temperature, 
G73°-2  Fahr.,  the  work  per  degree  of  absolute  temperature 
is  found  to  be,  in  foot-pounds,         .         .         .          .          .1107*7 

But  the  variation  of  pressure  per  degree,  in  pounds  on  the 

square  foot,  is       .......         .     42'0r> 

Therefore,  the  increase  of  volume  of  1  lb.  of  water  in  the  act  of  evapo- 
rating, at  212°  Fahr.,  is 

1107-7  -=-  42-05  =  26-343  cubic  feet ; 

To  which,  adding  the  volume  of  the  water  in  the 

liquid  state  .......     0-017         „ 


There  is  found  the  volume  of   1  lb.  of  atmospheric 

steam, 26-360  cubic  feet. 

10.  The  example  by  which  the  second  law  of  thermodynamics  has 
here  been  illustrated,  has  been  puposely  chosen  of  a  very  simple  kind  ■ 
but  that  law  enables  the  most  complex  questions  respecting  the  expenditure 
of  heat  required  to  produce  a  given  mechanical  effect  to  be  solved ;  and 
the  solution  is  always  effected  in  the  same  manner :  that  is  to  say,  by 
deducing  the  total  work,  internal  and  external,  from  the  manner  in  which 
a  small  variation  of  temperature  affects  the  external  work. 

11.  The  law  of  the  efficiency  of  a  perfect  heat  engine,  may  be  stated  thus  : 
If  the  substance — for  example,  air  or  water — which  does  the  work  in  a 
perfect  heat  engine  receives  all  the  heat  expended  at  one  fixed  temperature, 
and  gives  out  all  the  heat  which  remains  unconverted  into  work  at  a 


THE  SECOND   LAW   OF  THERMODYNAMICS.  4-37 

lower  fixed  temperature,  the  fraction  of  the  whole  heat  expended  which 
is  converted  into  external  work  is  expressed  by  dividing  the  difference 
between  those  temperatures  by  the  higher  of  them,  reckoned  from  the 
absolute  zero.  Now  this  is,  in  fact,  the  second  law  of  thermodynamics 
expressed  in  other  words ;  but  whether  the  demonstration  of  that  fact, 
that  is,  of  the  substantial  identity  of  the  second  law,  as  stated  in  Article  7, 
with  the  law  stated  in  the  preceding  sentence,  is  capable  of  being  put 
in  a  popular  form  is  doubtful,  seeing  that  it  involves  the  notion  of  limiting 
ratios.  The  applications,  however,  of  the  law  of  the  efficiency  of  a 
perfect  heat  engine  are  very  simple  and  easy.  For  example,  it  informs 
us  that  if  the  steam  which  drives  an  engine  receives  all  the  heat  expended 
upon  it  at  temperatures  not  exceeding  248°  Fahr.  (corresponding  to  the 
absolute  temperature  248°  +  461°-2  =  709°-2),  and  if  all  the  heat  not 
converted  into  external  work  is  given  out  by  that  steam  at  temperatures 
not  below  104°  Fahr.  (being  an  ordinary  temperature  of  condensation), 
the  efficiency  of  that  engine,  being  the  fraction  of  the  whole  heat  expended 
that  is  converted  into  external  work,  cannot  possibly  exceed  the  following 
value — 

248  -  104     144 


248  +  461-2   709-2 


•203. 


The  same  law  informs  us  that  in  order  that  the  whole  heat  expended  in  a 
heat  engine  may  be  converted  into  external  work,  it  is  necessary  that  the 
temperature  of  the  condenser  or  refrigerator  should  be  the  absolute  zero — 
a  temperature  unattainable  by  human  means.  Thus,  a  knowledge  of  the 
second  law  of  thermodynamics,  as  applied  to  the  efficiency  of  heat  engines, 
is  a  safeguard  against  the  formation  of  projects  for  increasing  the  perfor- 
mance of  such  engines  beyond  the  highest  possible  limits. 

1 2.  There  seems,  then,  to  be  no  difficulty  in  explaining  and  illustrating, 
in  a  popular  way,  the  applications  of  the  second  law  to  various  scientific 
and  practical  questions,  and  the  agreement  of  its  results  with  those  of 
experiment,  which  agreement  is  the  real  proof  of  its  being  true.  But 
it  appears  by  no  means  so  easy  to  demonstrate  popularly  the  connection 
between  the  second  law  of  thermodynamics  with  the  idea  of  "  heat  as  a 
mode  of  motion."  That  connection  consists  in  the  fact  that  the  second 
law  of  thermodynamics  necessarily  follows  from  the  established  laws  of 
dynamics  when  they  are  applied  to  the  supposition  that  the  sort  of  motion 
which  constitutes  heat  is  a  whirling  or  circulating  motion  of  the  jwticles  of 
bodies— &  supposition  otherwise  called  the  "  hypothesis  of  molecular  rortkes." 
The  original  demonstrations  of  that  fact,  which  appeared  in  February, 
1850,  and  December,  1851  {Seep.  49),  involve  algebraical  processes  that 
are  quite  beyond  the  range  of  a  popular  explanation ;  and  to  understand 
even  the  elementary  proof  without  algebra,  which  was  read  to  the  British 


438  THE   SECOND   LAW   OF   THERMODYNAMICS. 

Association  in  18G5  (See  p.  4-27),  requires  the  habit  of  scientific  reasoning. 
It  is  much  to  be  wished  that  some  means  could  be  devised  for  making  that 
proposition  as  widely  understood  as  the  first  law  of  thermodynamics 
now  is;  for  as  matters  now  stand,  the  popular  knowledge  of  thermodynamics 
is,  as  regards  the  second  law,  eighteen  years  behind  the  actual  state  of 
that  science. 


THE   SECOND   LAW   OF   THERMODYNAMICS.  43*) 


XXV.— EXAMPLES  OF  THE  APPLICATION  OF  THE  SECOND  LAW 
OF  THERMODYNAMICS  TO  A  PERFECT  STEAM- 
ENGINE  AND  A  PERFECT  AIR-ENGINE* 

1 .  The  following  examples  are  intended  to  illustrate  the  application  of  the 
second  law  of  thermodynamics  to  a  perfect  steam-engine  and  a  perfect  air- 
engine,  expending  the  same  quantity  of  heat,  and  working  between  the 
same  limits  of  temperature,  viz. : — Quantity  of  heat  expended  per  stroke, 
reduced  to  an  equivalent  quantity  of  mechanical  work,  in  foot-pounds, 

68420. 

Degrees  of  Fahrenheit. 
Temperatures.  Ordinary  scale.       Absolute  scale. 

Upper  limit,  .  .  .  .266°  7270,2 

Lower  limit,          ....      104°  565°'2 


Difference,  or  range,       .  .      1G2°  162O-0 

2.  The  phrase  "  perfect  engine  "  is  here  used  to  denote  an  engine  which 
realises  the  greatest  quantity  of  mechanical  work  possible  with  the  given 
expenditure  of  heat  and  between  the  given  limits  of  temperature,  being 
the  limit  towards  which  actual  engines  may  be  made  to  approach  through 
the  progress  of  practical  improvements.  Such  an  engine  must  fulfil  the 
following  conditions  : — There  must  be  no  waste  of  heat  through  conduction 
or  radiation,  or  of  work  through  friction;  and  the  whole  heat  expended 
must  be  communicated  to  the  working  substance  at  the  higher  limit  of 
temperature.  In  other  words,  the  elevation  of  the  temperature  of  the 
working  substance  from  the  lower  to  the  higher  limit  must  be  effected 
without  any  expenditure  of  heat ;  for  whatever  heat  is  expended  in  pro- 
ducing elevation  of  temperature,  is  either  wholly  or  partially  lost  as  regards 
the  performance  of  mechanical  work. 

3.  There  are  two  ways  of  effecting  the  elevation  of  temperature  without 
expenditure  of  heat.  One  is  to  raise  the  temperature  by  compression  of 
the  working  substance  in  a  non-conducting  cylinder ;  the  mechanical  work 
necessary  for  that  purpose  being  obtained  by  means  of  the  expansion  of 
the  working  substance  in  a  non-conducting  cylinder  while  its  temperature 
is  falling  back  from  the  higher  to  the  lower  limit.  The  other  way  is  by 
conduction,  viz. : — to  make  the  working  substance,  while  its  temperature 
is  falling,  communicate  its  heat  to  a  set  of  metal  tubes,  or  to  a  network, 

*  From  The  Engineer  of  August  2,  1S67- 


440 


THE   SECOND   LAW   OF   THERMODYNAMICS. 


called  a  "  regenerator  "  or  "  economiser,"  from  which  the  heat  is  given  out 
again  at  the  proper  time  to  the  working  substance  in  order  to  raise  its 

k,  temperature.  Supposing  those  two 
processes  to  be  carried  out  in  a 
theoretically  perfect  manner,  the  re- 
sults as  regards  the  economy  of  heat 
are  exactly  the  same.*  In  the  follow- 
ing examples  the  steam-engine  will 
be  supposed  to  act  by  compression, 
and  the  air-engine  by  the  aid  of  a 
perfect  regenerator. 

4.  According  to  the  second  law  of 
thermodynamics,  the  efficiency  of  each 
of  those  engines  is  the  same,  viz. : — 


162 

727"2 


=  0-2228; 


and  the   work   realised  per 
also  the  same,  viz.  : — 


stroke  is 


68420x0-2228  =  15244  foot-pounds. 

The  object  of  the  following  calcula- 
tions is  to  show  in  detail  what 
becomes  of  the  difference  between 
the  whole  energy  expended  in  the 
form  of  heat  and  that  obtained  in  the 
form  of  mechanical  work,  viz. ; — 

G8420— 15244  =  531 7G  foot-pounds. 

1  ieing  the  rejected  or  necessarily  lost 
energy. 

5.  The  diagrams  of  both  engines 
are  represented  in  the  figure.  Abso- 
lute pressures  are  supposed  to  be  re- 
presented in  pounds  on  the  square 
foot  by  ordinates  measured  parallel 
to  OP;  volumes  occupied  by  the 
working  substance,  in  cubic  feet,  by 
distances  measured  parallel  to  0  V. 

•  The  fact  that  those  results  are  the  same  is  illustrated  in  the  case  of  air-engines  by 
numerical  examples,  which  may  be  found  in  A  Manual  of  the  Steam-Eiufine  and  other 
Prime  Movers,  pages  347  to  369.  In  practice  the  regenerator  answers  best  for  an  air- 
engine,  because  of  the  very  large  space  required  for  the  other  process. 


THE   SECOND  LAW  OF   THERMODYNAMICS.  441 

The  diagram  of  the  steam-engine  is  marked  by  full  lines  and  capital 
letters ;  that  of  the  air-engine  by  dotted  lines  and  italic  letters.  In  the 
case  of  the  steam-engine  the  whole  work  done  by  the  steam  in  driving  the 
piston  is  represented  by  the  area  FCDE;  the  work  expended  in  com- 
pressing part  of  the  steam  and  feeding  the  boiler  by  the  area  FBAE; 
and  the  indicated  work  by  the  area  BCDA. 

In  the  case  of  the  air-engine  the  whole  work  done  by  the  air  in  driving 
the  piston  is  represented  by  the  area,  f  bed;  the  work  expended  in  raising 
the  pressure  of  the  air  and  feeding  the  engine  by  the  area  fade  ;  and  the 
indicated  work  by  the  area  b  e  d  a. 

6.  To  begin  in  detail  with  the  steam-engine.  From  the  limits  of 
temperature  given  in  Article  1,  it  is  found,  by  means  of  the  proper  table 
or  formula,  that  the  limits  of  pressure  are  as  follows : — 

Lb.  on  the  Lb.  on  the 

Sq.  In.  Sq.  Ft. 

Absolute  pressure  of  admission,  0  F,     39-25  5652 

Absolute  back  pressure,  0  E,  .         .        106  152-G 

7.  The  volume  of  water  in  the  liquid  state  which  is  used  per  stroke  is 
represented  by  F  B,  a  distance  too  small  to  be  seen  in  the  diagram ;  the 
volume  of  the  same  water  when  in  the  state  of  steam  at  the  higher  limit 
of  pressure  by  F  C ;  and,  consequently,  the  increase  of  volume  of  the  water 
in  the  act  of  evaporating  by  B  C.  That  increase  of  volume  is  produced 
by  communicating  to  the  water  the  whole  heat  expended.  Now  68420 
foot-pounds  is  the  mechanical  value  of  the  latent  heat  of  evaporation, 
under  the  higher  of  the  given  pressures,  of  so  much  Avater  as  fills  one  cubic 
foot  more  in  the  state  of  steam  than  it  does  in  the  liquid  state ;  *  so  that 
B  C  represents  one  cubic  foot.  The  steam  then  expands  in  a  non-conduct- 
ing cylinder,  without  receiving  or  giving  out  heat,  until  its  pressure  and 
temperature  fall  to  their  lower  limits.  C  D  represents  the  expansion 
curve,  and  E  D  the  volume  occupied  at  the  end  of  the  expansion,  partly 
by  steam  and  partly  by  a  small  quantity  of  water  which  spontaneously 
liquefies  during  the  expansion.  Part  of  the  steam  represented  in  volume 
by  D  A,  is  then  condensed  by  conduction  of  heat,  at  the  lower  limit  of 
temperature ;  and  a  volume  of  steam  represented  by  A  E,  less  the  volume 
of  the  water  in  the  liquid  state,  is  left  uncondensed,  in  order  that  by  its 
compression  into  the  liquid  state  heat  enough  may  be  produced  to  raise 

*  Formula  for  that  latent  heat,  in  units  of  work  :  — 
■where  p  is  the  absolute  pressure  and  t  the  absolute  temperature. 


442  THE   SECOND   LAW  OF   THERMODYNAMICS. 

the  temperature  of  the  feed-water  to  the  higher  limit.  The  curve  of  com- 
pression, showing  how  the  pressure  increases  as  the  volume  is  diminished, 
is  represented  by  A  B,  and  this  completes  the  double  stroke  or  revolution 
of  the  engine.  The  following  are  the  successive  volumes  occupied  by  the 
steam  at  different  periods,  neglecting  the  volume  of  the  liquid  water,  as 
being  so  small  compared  with  that  of  the  steam  that  it  is  unnecessary  to 
take  it  into  account  for  the  present  purpose.* 

Cubic  Feet. 
Volume  on  admission,  F  C,  taken  as  sensibly  equal  to 

BC, 100 

Volume  at  end  of  expansion,  E  D,         ....    24*68 

Volume  at  beginning  of  compression,  E  A,     .         .         .       t"08 

The  weight  of  water  expended  per  stroke  is  0'095G  lb. 

8.  The  areas  shown  in  the  diagram,  as  computed  by  the  proper  formula1, 
given  in  the  footnote  to  the  preceding  article,  are  as  follows: — 

*  The  following  are  formulae  for  these  volumes,  and  for  the  work  represented  by  the 
areas  in  the  diagram,  first  demonstrated  by  the  author  in  the  Philosophical  Trans- 
actions for  December,  1853,  and  given  also  in  A  Manual  of  the  Steam-Engine  and  other 
Prime  Mov<  rs.  Let  t  and  t'  denote  the  higher  and  lower  limits  of  absolute  tempera- 
ture ;  p  and  \>  the  corresponding  pressures ;  L  and  L'  the  corresponding  values  of 

l>       ;  J,  Joule's  equivalent  of  the  specific  heat  of  liquid  water  (772  foot-pounds  per 

degree  of  Fahrenheit  in  a  pound  of  water) ;  H,  the  value  in  foot-pounds  of  the  latent 
heat  of  evaporation  of  1  lb.  of  water  at  the  higher  limit  (—  745,S00  -  0-7  J  (T  -  A) 
nearly,  where  A  is  the  boiling  point  under  one  atmosphere) ;  then — 

E  A       J  t'  L  .         .       t 

bc=  wn' ******'* 

E  D  _  E  A      t'  L 

BC~BC  +  fL'; 


„  ,,  _  „      JL  \  .       ./,       .         .        t\)       L(7  -  t') 
F  C  D  E  =   H  -\t  -  t  ( 1  +  hyp.  log.  t,J  {  +  -^— ^ ; 

FBAE=^j«-*'(l+hyp.log.^j; 
B  C  D  A  =  L£_Z_*1>. 

All  these  formula;  were  independently  demonstrated  by  Clausius  in  1S55.  (See  his 
Abhandlungen  iiber  die  mechanische  Wdrmetheorie.)  If  the  water  at  the  lower  limit  of 
pressure  in  the  example  Mere  all  in  the  state  of  steam,  it  would  occupy  29 "9  cubic  feet. 
The  difference  between  this  and  24-6S — viz.,  5-22  cubic  feet,  shows  what  proportion  of 
the  steam  is  spontaneously  liquefied  during  the  given  expansion  in  a  non-conducting 
cylinder. 


THE   SECOND   LAW  OF   THERMODYNAMICS.  4i3 

Foot-pounds. 
Total  work  obtained  through  the  action  of  the  steam 
in  driving  the  piston,  allowing  for  back  pressure, 
FCDE  = 16690 

Work  expended  through  compressing  part  of  the  steam 

into  the  liquid  state,  F  B  A  E  =    .         .         .         .1446 


Difference,    being     the     indicated    work    per     stroke, 

BCDA  = 15244 

as  already  calculated  in  Article  4,  by  the  second  law  of  thermodynamics. 
The  lost  or  rejected  heat, 

68420  -  15244  =  53176  foot-pounds, 

is  the  heat  abstracted  during  the  condensation  of  the  volume  of  steam 
represented  by  A  D ;  and  it  is  impossible  to  get  back  any  part  of  that 
heat,  because  it  is  all  abstracted  at  the  lower  limit  of  temperature,  and 
heat  will  not  pass  from  a  colder  to  a  hotter  body. 

9.  To  proceed  now  to  the  case  of  the  air-engine.  The  limits  of  pressure 
do  not,  as  in  the  case  of  steam,  depend  on  the  limits  of  temperature,  but 
may  be  fixed  according  to  convenience.  In  general,  the  lower  limit  of 
absolute  pressure  is  the  atmospheric  pressure,  which  we  will  estimate  in 
the  present  example  at  2116-3  lbs.  on  the  square  foot,  or  14*7  lbs.  on  the 
square  inch;  and  this  is  represented  by  Oe  in  the  diagram.  The  upper  limit 
of  absolute  pressure  being  arbitrary,  we  will  assume  it  in  the  present  case 
to  be  three  atmospheres,  or  6348-9  lbs.  on  the  square  foot,  or  44*1  lbs.  on 
the  square  inch.  It  will  presently  be  seen  that  the  weight  of  air  to  be 
used  per  stroke  depends  on  the  proportion  borne  by  the  upper  limit  of 
pressure  to  the  lower.  That  weight  of  air — to  be  afterwards  determined 
— is  drawn  into  the  air-pump  at  the  atmospheric  pressure  and  at  the 
lower  limit  of  temperature ;  and  it  occupies  a  volume  represented  in  the 
diagram  by  e  d.  It  is  then  compressed  into  a  smaller  volume,  represented  by 
fa,  so  that  it  rises  to  the  higher  limit  of  pressure,  and  transferred  from 
the  pump  to  the  working  cylinder ;  and,  in  order  that  this  compression 
and  transfer  may  cause  the  smallest  possible  expenditure  of  work,  the  air 
must  be  kept  during  the  compression  at  the  lower  limit  of  temperature  by 
means  of  a  proper  refrigerating  apparatus  for  abstracting  all  the  heat  that 
the  compression  generates.  None  of  that  heat  can  be  got  back,  for  it  is 
all  abstracted  at  the  lower  limit  of  temperature.  The  temperature,  then, 
being  uniform,  the  volume  varies  inversely  as  the  absolute  pressure ;  fa, 
in  the  present  example,  is  one-third  of  e  d,  and  the  curve  d  a  is  a  common 
hyperbola.      The  work  expended  in  the  air-pump,  which  is  the  exact 


444-  THE   SECOND   LAW   OF   THERMODYNAMICS. 

equivalent  of  the  heat  generated  there  and  abstracted  by  the  refrigerating 
apparatus,  is  represented  by  the  area  fade,  and  is  computed  in  foot-pounds 
by  multiplying  together  the  following  factors: — The  constant  53-15;  the 
hyperbolic  logarithm  of  the  ratio  compression  (hyp.  log  3  =  1'0986);  the 
absolute  temperature  at  which  the  compression  takes  place  in  degrees  of 
Fahrenheit  (in  this  case  5650,2  Fahr.);  and  the  weight  of  air  used,  in 
pounds. 

10.  The  air  on  its  way  from  the  pump  to  the  working  cylinder  passes, 
without  change  of  pressure,  through  a  perfect  regenerator,  in  which  all  the 
heat  given  out  by  the  previous  supply  of  air  is  stored  up;  and  it  thus 
rises  to  the  higher  limit  of  absolute  temperature,  and  at  the  same  time 
undergoes  dilatation  from  the  volume  fa  to  the  volume///,  which  volumes 
are  to  each  other  in  the  ratio  of  the  limits  of  absolute  temperature;  viz.: — 

fa  :  fh  :  :  5G5-2  :  727*2   :  :  1   :   128G7. 

11.  The  next  process  is  that  the  air  continues  to  expand  and  drive  the 
piston  until  its  pressure  falls  to  the  lower  limit,  its  volume  at  the  same 
time  increasing  to  that  represented  by  c  c;  and  it  is  then  finally  expelled, 
giving  out  to  the  regenerator  to  be  used  over  again  the  heat  correspond- 
ing to  the  difference  between  the  upper  and  lower  limits  of  temperature. 
In  order  that  the  greatest  quantity  of  work  possible  may  be  obtained  from 
the  expansion  represented  by  the  curve  be,  and  in  order  also  that  the 
air  during  its  expulsion  may  give  out  to  the  regenerator  a  quantity  of  heat 
sufficient  to  raise  the  temperature  of  the  next  supply  of  air  to  the  higher 
limit,  the  temperature  must,  by  the  supply  of  a  sufficient  quantity  of  heat, 
be  maintained  uniform  during  the  expansion  represented  by  be;  and  that 
quantity  of  heat  constitutes  the  whole  expenditure  in  a  perfect  air-engine. 
Such  being  the  case,  the  volume  varies  inversely  as  the  pressure ;  e  c,  in  the 
present  example,  is  three  times  fb,  and  the  curve  be  is  a  common  hyper- 
bola. It  is  evident,  moreover,  that  the  volumes  represented  by  e  d  and  e  c 
are  to  each  other  in  the  ratio  of  the  limits  of  absolute  temperature,  already 
given.  The  whole  work  obtained  by  the  action  of  the  air  in  the  cylinder, 
which  is  the  exact  equivalent  of  the  whole  heat  expended,  is  represented 
by  the  area/Jce,  and  is  computed  in  foot-pounds  by  multiplying  together 
the  following  factors.  The  constant  53-15  ;  the  hyperbolic  logarithm  of  the 
ratio  of  expansion  (hyp.  log.  3  =  1-098G);  the  absolute  temperature  at 
which  the  expansion  takes  place  (in  this  case  727°-2  Fahr.) :  and  the  weight 
of  air  used,  in  pounds. 

1 2.  The  area  bed  a  represents  the  indicated  work  per  stroke,  being  the 
excess  of  the  work  obtained  in  the  cylinder  above  the  work  expended  in 
the  air-pump  ;  and  the  proportion  which  it  bears  to  j 'bee  (representing  the 
whole  expenditure  of  heat)  is  obviously  that  of  the  range  of  temperature 


THE   SECOND   LAW   OF   THERMODYNAMIC'S.  445 

to  the  higher  absolute  temperature,  as  already  stated  in  Article  3,  viz.  : 

162 

— — -  =  0'2228.     In  other  words,  the  areas  shown  in  the  diagram  repre- 
727-2  °  l 

sent  the  following  quantities  : — ■ 

Foot-pounds. 
Total  expenditure  of  heat  per  stroke,  / b  c  e  =         .         .   68420 

Heat  produced  by  compression  and  abstracted  by  the 

refrigerator,  fa  de  =   .         .         .         .         .         .53176 


Indicated  work  per  stroke,  b  c d a  =      .         .         .         .1524  + 

The  indicated  work  per  stroke  may  be  calculated  independently  by  multi- 
plying together  the  following  factors: — The  constant  53-15;  the  hyperbolic 
logarithm  of  the  ratio  of  expansion  (hyp.  3  =  1-0986);  the  range  of 
absolute  temperature  (162°  Fahr.);  and  the  weight  of  air  used,  in  pounds. 
1 3.  The  weight  of  air  used  per  stroke  is  determined  by  the  considera- 
tion, that  it  is  to  be  sufficient  to  absorb  when  expanding  to  three  times 
its  initial  volume  at  the  absolute  temperature  7 2  7° '2  Fahr.,  a  quantity  of 
heat  equivalent  to  68420  foot-pounds.     The  calculation  is  as  follows  : — 

Constant  factor  depending  on  the  nature  of  the  gas  in 

foot-pounds  per  degree  of  Fahrenheit,  .  .     53- 15 


Multiply  by  the  absolute  temperature, 


7273-2F. 


Product  in  foot-pounds,  being  equal  to  the  product  of 
the  absolute  pressure  in  pounds  on  the  square 
foot,  and  the  volume  of  1  lb.  of  air  in  cubic  feet,        38651 

Multiply  by  the  hyperbolic  logarithm  of  the  rate  of 

expansion,  hyp.  log.  3  =  .         .         ■         •       1*0986 

Product  being  the  heat  absorbed  by  each  pound  of 

air  in  foot-pounds,    .  .  .  .  •  •        42516 


Heat  expended  68420  _  1-6113  lb 

dividedby- 42510  " 

of  air  used  per  stroke.  With  a  different  ratio  of  expansion  the  weight 
of  air  required  per  stroke,  would  be  different,  varying  inversely  as  the 
logarithm  of  that  ratio.  Had  any  other  gas  been  employed  instead  of  air 
the  only  difference  in  this  quantity  would  have  been  that  arising  from  a 


410 


THE  SECOND   LAW   OF   THERMODYNAMICS. 


different  value  of  the  constant  factor.     The  following  are  the  volumes 
occupied  by  the  air  at  different  periods  : — 


ed 

J" 
Jh 


Cubic  Feet. 

^2-87 

7-62 

981 
29-43 


14.  The  following  is  a  summary  of  the  comparison  between  the  steam- 
engine  and  the  air-engine  in  the  example  given: — 


Per  Stroke. 

Steam-Engine. 
POO)  in >umls. 

Air- Engine. 
Foot-pounds. 

Difference. 

Heat  expended,     ..... 
Heat  rejected 

Heat  transformed  into  mechanical  work, 

Work  obtained  by  expansion, 
Work  expended  in  compression.     . 

Indicated  work, 

( 16420 
53176 

•  120 

53176 

0 
0 

l  :.•_•  1 1 

15244 

0 

0-2228 

0-2228 

0 

16690 

1440 

GS420 
53176 

51730 
51730 

15244 

15244 

0 

The  quantities  in  the  column  headed  "  difference,"  represent  a  diminu- 
tion of  the  work  obtained  by  expansion,  and  an  exactly  equal  saving  in 
the  work  expended  in  compression,  produced  by  the  mutual  attraction  of 
the  particles  of  water. 

15.  It  may  be  useful  to  point  out  what  effects  Avould  be  produced  by 
departing  from  that  condition  of  maximum  efficiency  which  requires  that 
the  elevation  of  temperature  shall  be  produced  without  the  expenditure  of 
heat.  In  the  case  of  the  steam-engine  there  would  be  a  gain  of  indicated 
work,  represented  by  the  area  FBAE  =  144G  foot-pounds,  so  that  the 
indicated  work  per  stroke  would  be  16690  foot-pounds.  At  the  same 
time  there  would  be  an  additional  expenditure  of  heat,  calculated  as 
follows : — 


THE   SECOND   LAW  OF   THERMODYNAMICS.  4-47 

Joule's  equivalent  of  the  specific  heat  of  liquid  water,  772 

Number  of  degrees  of  Fahr.  by  which  the  temperature 

of  the  feed-water  has  to  be  raised,      .         .         .  162° 

Weight  of  water  used  per  stroke  (lb.),       .         .         .     0-0956 


Product  of  those  three  factors,  being  the  additional 

expenditure  of  heat  per  stroke  in  foot-pounds.     .      11956 

Expenditure  of  heat  as  formerly  calculated,         .  .      68420 


Total, 80376 


Efficiency,  as  diminished  by  omitting  the  compression, 
16690  _ 
80376  ~ 


0-208 


being  about  fifteen-sixteenths  of  the  greatest  possible  efficiency  with  the 
given  limits  of  temperature. 

16.  In  the  case  of  the  air-engine,  suppose  the  regenerator  omitted,  so 
that  the  whole  elevation  of  temperature  of  the  air  has  to  be  produced  by 
heat  supplied  from  the  furnace ;  then,  in  the  present  example,  there  would 
be  an  additional  expenditure  of  heat,  calculated  as  follows  : — 

Dynamical  equivalent  of  the  specific  heat  of  air  under 
constant  pressure,  in  foot-pounds  per  degree  of 
Fahr., 183-45 

Elevation  of  temperature,  .         .         .         .  162°  F. 

Weight  of  air  used  per  stroke,  the  rate  of  expansion 

being  3, 1-6113  lb. 


Product  of  those  three  factors,  being  the  additional 

expenditure  of  heat  in  foot-pounds,    .         .         .      47886 

Expenditure  of  heat  as  formerly  calculated,         .         .      68420 


Total, 116306 

Efficiency,  as  diminished  by  altogether  omitting  the 

1  5944 
regenerator  .  '"-  =     .         .         .         .         .0-131  nearly, 
0  116300 


448  THE  SECOND   LAW  OF  THERMODYNAMICS. 

or  about  six-tenths  of  the  greatest  possible  efficiency  with  the  given  limits 
of  temperature. 

17.  It  is  evident  that  the  regenerator,  or  some  equivalent  apparatus, 
cannot  be  omitted  consistently  with  economy  of  heat  in  an  air-engine. 
An  actual  regenerator,  however,  will  never  succeed  in  storing  and  giving 
out  the  whole  supply  of  heat  required  for  the  elevation  of  temperature. 
Suppose,  for  the  sake  of  illustration,  that  in  the  present  example  the 
regenerator  stores  and  gives  out  nine-tenths  of  the  heat  required,  leaving 
one-tenth  to  be  supplied  from  the  furnace,  then  we  have  the  following 
expenditure  of  heat  per  stroke  : — 

Foot-pounds. 
Expenditure  a*  originally  calculated,  .  .  .      G8420 

Additional  expenditure  through   imperfect  action  of 

the  regenerator,         ......        4789 


Total 73209 

Efficiency,  as  diminished  by  imperfect  action  of  the 

15244 

regenerator  = 0-208  nearly, 

or  about  fifteen-sixteenths  of  the  greatest  possible  efficiency  with  the  given 
limits  of  temperature. 

18.  The  law  which  the  preceding  examples  illustrate  leads  to  the  con- 
clusion that  if  by  means  of  air-engines  greater  economy  of  fuel  than  in 
steam-engines  is  to  be  attained,  it  must  be  by  the  following  means  : — 
Working  with  a  greater  range  of  absolute  temperature  than  is  practicable 
or  safe  in  steam-engines,  and  using  the  products  of  combustion  directly 
to  drive  the  piston,  so  as  to  save  nearly  the  whole  of  the  heat  that  is 
wasted  in  a  steam  boiler,  or  in  an  air-engine  in  which  the  products  of 
combustion  are  not  so  used ;  and  it  is  probable  that  the  latter  is  the  more 
easily  practicable  of  the  two  means  of  economising  heat. 

19.  The  following  examples  illustrate  the  relation  between  the  efficiency 
of  a  heat  engine  and  the  consumption  of  fuel  per  indicated  horse-power 
per  hour.  The  fuel  is  supposed  to  be  nearly  pure  carbon,  the  mechanical 
equivalent  of  the  whole  heat  produced  by  the  combustion  of  one  pound  of 
it  being  11,000,000  foot-pound'. 


THE   SECOND   LAW   OF   THERMODYNAMICS. 


449 


Efficiency. 

Indicated  Work 

per 
Pound  of  Carbon. 

• 

Pound  Carbon  per 

Indicated  Horse-power 

per  Hour. 

1 

11,000,000 

0-18 

0-5 

5,500,000 

0-36 

0-25 

2,750,000 

0-72 

0-20 

2,200,000 

0-90 

0-15 

1,650,000 

1-20 

0-12:. 

1,375,000 

1-44 

o-ioo 

1,100,000 

1-80 

0-075 

825,000 

2-40 

0-050 

550,000 

3-60 

0-025 

275,000 

7-20 

SUPPLEMEX  T. 


20.  I  think  it  desirable  to  add  the  following  explanations  and  illustra- 
tions of  the  principle,  that  in  an  air-engine,  under  all  circumstances 
whatsoever,  the  heat  produced  hj  the  compression  of  the  air  is  wholly  and 
unavoidably  lost — a  principle  which  is  a  necessary  consequence  of  the  fact 
that  heat  never  passes  directly  from  a  colder  to  a  hotter  body. 

21.  The  heat  produced  by  the  compression  must  either  be  abstracted 
from  the  air  as  fast  as  it  is  produced,  so  as  to  keep  the  temperature  of  the 
air  constant  during  the  compression,  or  allowred  to  accumulate  and  raise 
the  temperature  of  the  air.  In  the  former  case  (which  is  that  described 
in  Article  9)  it  is  at  once  evident  that  the  heat,  being  abstracted  by 
means  of  an  external  substance,  such  as  water  or  air,  that  is  colder  than 
the  lowest  temperature  of  the  working  substance,  can  never  be  transmitted 
to  the  working  substance  again. 

22.  In  the  latter  case,  when  the  heat  produced  by  the  compression  is 
not  abstracted,  but  allowed  to  accumulate  and  raise  the  temperature  of  the 
air,  the  air  is  passed  through  the  regenerator  or  econo  miser  on  its  way 

2  F 


450  THE  SECOND  LAW   OF  THERMODYNAMICS. 

towards  the  working  cylinder  at  an  increased  temperature  higher  than  the 
lower  limit.  Therefore,  the  air  which  has  done  its  work  and  is  escaping 
through  the  regenerator  in  the  opposite  direction  is  not  reduced  below  the 
same  increased  temperature,  for  it  is  only  through  the  entering  air  being 
at  a  lower  temperature  that  any  heat  can  be  abstracted  from  the  escaping 
air.  Therefore,  the  escaping  air  carries  off  to  waste  a  quantity  of  heat 
equal  to  the  quantity  of  heat  produced  by  compression  and  allowed  to 
accumulate  in  the  compressed  air,  so  that  the  effect  of  such  accumulation 
is  neutralised,  and  the  total  expenditure  of  heat  remains  unaltered. 

23.  Moreover,  the  increase  of  temperature  of  the  air  undergoing  com- 
pression causes  an  increase  in  the  quantity  of  work  expended  in 
compressing  it  above  the  quantity  of  work  which  would  be  required  if 
the  temperature  were  kept  constant ;  and  thus  the  indicated  work  is 
diminished,  and  the  engine  ceases  to  be  one  of  maximum  efficiency 
between  the  given  limits  of  temperature. 

24.  To  illustrate  this  by  an  example,  I  will  suppose  that  the  compres- 
sion of  the  air  takes  place  in  a  non-conducting  cylinder,  and  is  carried 
to  an  extent  such  that  the  temperature  of  the  air  is  raised  from  the  lower 
limit  to  tin-  higher  limit  by  means  of  the  compression  alone.  Then  it  is 
known  that  the  pressure  of  the  air,  instead  of  varying  inversely  as  the 
volume  simply,  varies  inversely  as  that  power  of  the  volume  whose 
index  is  l-408,  while  the  absolute  temperature  varies  inversely 
as  that  power  of  the  volume  whose  index  is  0'408.  In  Fig.  2  (as 
in  the  former  figure)  let  volumes  in  cubic  feet  be  represented  by 
distances  parallel  to  OV,  and  pressures  in  pounds  on  the  square  foot 
by  ordinates  parallel  to  OP,  so  that  the  areas  of  diagrams  represent 
quantities  of   work   in    foot-pounds.       Let    O  E   represent   the   original 


Fit 


absolute  pressure,  E  D  the  original  volume .  of  the  air,  and  0  F  the 
increased  pressure  at  the  end  of  the  compression.  Then,  if  the  tempera- 
ture were  kept  uniform,  the  compression  curve  of  the  diagram  woidd  be  a 
common  hyperbola  D A,  such  that  OF  xFA=  OE  x  ED.     But  the 


THE   SECOND   LAW  OF   THERMODYNAMICS.  451 

rise  of  temperature  causes  the  compression  curve  to  assume  a  steeper 

figure,  D  B,  of  which  the  law  has  already  been  stated.     The  absolute 

FB 
temperature  at  the  end  of  the  compression  is  increased  in  the  ratio  ^rr', 

and  the  work  done  in  driving  the  compressing  pump  is  represented  by  the 

area  of  the  diagram  EFBD,  and  is  equivalent  to  the  quantity  of  heat 

required  in  order  to  produce  the  rise  of  temperature  from  the  lower  to  the 

higher  limit  under  constant  pressure.      Now,  taking  the  same  data  as 

before,  viz.  : — 

Degrees  of  Fahrenheit. 
Temperatures.  Ordinary  scale.       Absolute  scale. 

Upper  limit,         .         .         .         .266°  727°*2 

Lower  limit,         ....     104°  565°*2 


Difference,    .         .         .         .162°  162°*0 

Ratio  in  which  the  absolute  temperature  is  to  be  increased, 
FB       727°*2 


FA       565°*2 


=  1-2867; 


Dynamical  value  of  the  specific  heat  of  air  under  constant  pressure, 
183*45  foot-pounds  per  degree  of  Fahrenheit. 

Original  pressure,  OE  =  2116*3  lbs.  on  the  square  foot ; 

Original  volume  of  one  pound  of  air,  14*19  cubic  feet; 

We  obtain  the  following  results  : — 

ED 

Ratio  of  volumes,    :=-=  —  1*8546  ; 
Jb  x> 

■p.*      *  0F       ED       ED     FB       i-^±r 

Ratio  of  pressures,  ^  =  ^  =  ^  '  YJ.= 

X  1*2867  =  2*3862; 

Pressure  at  end  of  compression  OF  =  5050  pounds  on  the  square 
foot; 

Volume  of  one  pound  of  air  at  end  of  compression,  7*65  cubic  feet; 

Work  done  in  compressing  pump  per  pound  of  air, 

162°  x  183*45  =  29719  foot-pounds. 

25.  The  air  being  now  at  the  upper  limit  of  temperature,  let  it  be 
transferred  to  the  working  cylinder,  and  there  expanded  without  allowiug 


4-32  THE  SECOND   LAW   OF   THERMODYNAMICS. 

the  temperature  to  fall  below  that  limit,  until  the  pressure  falls  to  the 
atmospheric  pressure  0  E,  and  then  let  it  be  discharged.  The  expansion- 
curve  B  C  will,  as  in  Article  11,  be  a  common  hyperbola,  so  that  the  ratio 
of  expansion  will  be 

^-^-2-3862 
BF-OE-Jd8bJ' 

and  the  volume  of  one  pound  of  air  at  the  end  of  the  expansion  will  be 
18 '2  6  cubic  feet.  The  area  of  the  diagram  FBCE  will  represent  at  once 
the  work  done  in  driving  the  piston,  and  the  whole  expenditure  of  heat. 
The  method  of  computing  that  area  has  already  been  stated  in  Article  11, 
and  the  calculation  in  the  present  case  is  as  follows,  for  each  pound  of  un- 
used : — 

Constant  factor,    .         .          .         .         .         .          .       53-15 

Absolute  temperature  at  which  the  expansion  takes 

place, 727°-2  F. 

Hyperbolic  logarithm   of    the   ratio   of   expansion; 

hyp.  log.  2*3862  = 0*8697 


Product  of  those  three  factors,  being  the  work  done 
and  heat  expended  in  the  working  cylinder,  per 
pound  of  air  used,  in  foot-pounds,        .  .  .         336  K> 

26.  The  air  finally  escapes  at  the  higher  limit  of  temperature,  and 
therefore  carries  to  waste  the  whole  of  the  heat  which  was  employed  in 
raising  its  temperature,  having  been  produced  by  compression.  The  value 
of  that  heat  in  foot-pounds  per  pound  of  air  has  been  already  found,  in 
Article  24,  to  be  29719. 

27.  The  indicated  work  represented  by  the  area  B  C  D  is  consequently, 
in  foot-pounds  per  pound  of  air, 

33615  -  29719  =  389G ; 

and  the  efficiency  of  the  engine, 

3896 
33615  =  0'11Gnear1^' 

or  about  one-half  of  the  greatest  possible  efficiency  with  the  given  limits 
of  temperature. 

28.  In  order  that  the  present  example  may  be  the  more  easily  com- 
pared with  the  former  examples,  we  will  suppose  that  the  heat  to  be 
expended   per    stroke    is,   as   before,   equivalent   to    68420    foot-pounds. 


THE  SECOND   LAW   OF  THERMODYNAMICS.  453 

Then  the  expenditure  of  air  per  stroke  required  in  order  to  take  up 
that  heat  during  its  expansion  is  found  to  be 

||^=  20354  lb,; 

whence  the  following  results  are  obtained  : — 

Foot-lbs. 
Waste    heat   per   stroke    equivalent    to    the  work    of 
driving    the    compressing    pump,    162°  x   183'45 
X  2-0354  = G0490 

Indicated  work  per  stroke,  3896  X  2-0354  =     .         .       7930 

7930 
Efficiency,—- —  =  0T16,  as  already  found. 

Volumes   successively  occupied  by  the   air  in  cubic   feet  per 
stroke : 

ED  =  2889;    FB  =  15'58;   EC  =  3717. 

29.  The  case  of  Article  9 — where  the  whole  of  the  heat  generated  in 
the  compressing  pump  is  abstracted  as  fast  as  it  is  produced — and  that  of 
the  example  just  described — in  which  the  whole  of  that  heat  is  at  first 
employed  in  raising  the  temperature  of  the  air  while  an  equal  quantity  of 
heat  goes  to  waste  with  the  escaping  air  that  has  done  its  Avork — form 
two  extremes,  and  between  those  extreme  cases  there  may  lie  an  indefinite 
number  of  intermediate  cases,  in  which  part  of  the  heat  generated  by  com- 
pression is  abstracted  at  once,  and  part  employed  in  raising  the  tempera- 
ture of  the  air.  It  will  be  readily  understood  that  in  all  those  intermediate 
cases  the  result  is  the  same  as  in  the  two  extreme  cases — the  whole  of  the 
heat  generated  by  the  compression  of  the  air  goes  to  waste  either  at  once 
or  with  the  escaping  air,  and  none  of  it  is  available  for  conversion  into 
indicated  work;  nor  would  it  be  possible  that  it  should  become  so 
available,  unless  it  were  possible  for  heat  to  be  directly  transferred  from 
a  colder  body  to  a  hotter  body. 


454  THE  WORKING  OF  STEAM   IN   COMPOUND   ENGINES. 


XXVI.— OX  THE  WORKING  OF  STEAM  IN  COMPOUND 

ENGINES* 

1.  Principal  Kinds  of  Compomd  Engines. — By  a  compound  steam-engine  is 
meant  one  in  which  the  mechanical  action  of  the  steam  commences  in  a 
smaller  cylinder  and  is  completed  in  a  larger  cylinder.  Those  cylinders 
are  respectively  called,  for  convenience,  the  high-pressure  cylinder  and  the 
low-pressure  cylinder.  Two  classes  of  compound  engines  will  be  con- 
sidered— first,  those  in  which  the  steam  passes  directly  or  almost  directly 
from  the  high-pressure  to  the  low-pressure  cylinder,  the  forward  stroke  of 
the  latter  cylinder  taking  place  either  exactly  or  nearly  at  the  same  time 
with  the  return  stroke  of  the  former  cylinder ;  and,  secondly,  those  in 
which  the  steam,  on  its  way  from  the  high-pressure  to  the  low-pressure 
cylinder,  is  stored  in  a  reservoir,  so  that  any  convenient  fraction  of  a 
revolution  (such,  for  example,  as  a  quarter  revolution)  may  intervene 
between  the  ends  of  the  strokes  of  the  cylinders.  As  to  the  latter  class 
of  engines,  reference  may  be  made  to  a  paper  by  Mr.  E.  A.  Cowper  in 
the  Transactions  of  the  Institution  of  Naval  Architects  for  18G4,  page  24S. 
Sometimes,  especially  in  the  first  class  of  compound  engines  (those  without 
reservoirs),  there  are  a  pair  of  low-pressure  cylinders  whose  pistons  move 
together,  and  which  act  like  one  cylinder  divided  into  two  parts. 

2.  AilrtmtiKjcs  of  Compound  Engines. — As  regards  the  theoretical  effi- 
ciency of  the  steam,  the  compound  engine  possesses  no  advantage  over 
an  engine  with  a  single  cylinder  of  the  dimensions  of  the  low-pressure 
cylinder,  working  with  the  same  pressure  of  steam  and  the  same  rate  of 
expansion.  The  advantages  which  it  does  possess  are  the  following : — 
First,  in  point  of  strength,  the  action  of  the  steam  when  at  its  highest 
pressure  takes  place,  in  the  compound  engine,  upon  a  comparatively  small 
piston,  thus  diminishing  the  amount  of  the  greatest  straining  force  exerted 
on  the  mechanism  and  framing ;  secondly,  in  point  of  economy  of  heat 
and  steam,  in  a  single-cylindered  engine  it  is  necessary,  in  order  to  prevent 
liquefaction  and  re-evaporation  of  the  steam,  and  consequent  waste  of 
heat,  that  the  whole  metal  of  the  cylinder  should  be  kept,  by  means  of  a 
steam  jacket,  at  a  temperature  equal  to  that  of  the  steam  when  first 

*  From  The  Engineer  of  March  11,  1870. 


THE  WORKING   OF   STEAM   IN   COMPOUND   ENGINES. 


455 


admitted;  whereas,  in  a  compound  engine,  it  is  the  smaller  or  high- 
pressure  cylinder  only  which  has  to  be  kept  at  so  high  a  temperature,  it 
being  sufficient  W-'keep  the  larger  or  low-pressure  cylinder  at  the  tem- 
perature corresponding  to  the  pressure  at  which  the  steam  passes  from 
the  high-pressure  to  the  low-pressure  cylinder.  Thirdly,  in  point  of 
economy  of  work :  the  whole  of  the  force  exerted  by  the  piston  rod  upon 
the  crank  in  a  single-cylindered  engine  takes  effect  in  producing  friction 
at  the  bearings ;  whereas,  in  compound  engines,  the  mechanism  can  be 
so  arranged  that  the  forces  exerted  by  the  piston  rods  on  the  bearings 
shall,  to  a  certain  extent,  balance  each  other,  thus  diminishing  the  friction. 
When  there  are  a  pair  c_ 
of  low-pressure  cylinders 
with  a  high  -  pressure 
cylinder  between  them 
(as  in  the  engines  of 
H.M.S.  "Constance,"  by 
Messrs.  Randolph,  Elder, 
and  Co.)  the  balance  can 
be  made  almost  perfect. 
These  remarks  apply  not 
only  to  the  forces  due 
to  the  pressure  of  the 
steam,  but  to  those  pro- 
duced by  the  reaction  or  inertia  of  the  pistons  and  of  the  masses  which, 
move  along  with  them.  The  advantages  which  have  been  stated  are 
obviously  greatest  with  high  rates  of  expansion. 

3.  Combination  of  Diagrams. — When  the  diagrams  of  the  high  and  low- 
pressure  cylinders  of  a  compound  engine  are  taken  by  means  of  one 
indicator  they  have  the  same  length  of  base;  and  when  arranged  in 
the  customary  way  for  inspection  they  present  appearances  which  are 
represented  in  Fig.  1  for  engines  without  reservoirs,  and  in  Fig.  2  for 
engines  with  reservoirs.  In  each  Fig.  A  A  is  the  atmospheric  line,  0  B 
the  zero  line  of  absolute  pressure,  and  the  length  0  P  on  that  line  is  the 
common  length  of  the  diagrams  of  both  cylinders,  as  originally  drawn. 
The  diagram  of  the  high-pressure  cylinder  is  represented  in  Fig.  1  by 
CDRH,  and  in  Fig.  2  by  CDKL;  that  of  the  low-pressure  cylinder, 
as  drawn  by  the  indicator,  is  represented  in  Fig.  1  by  k  i  h,  and  in  Fig.  2 
by  I  i  g  h.  In  combining  the  diagrams  of  the  two  cylinders  into  one 
diagram,  it  is  to  be  borne  in  mind  that  when  the  area  of  a  diagram  is 
considered  as  representing  the  work  done  by  the  steam  on  the  piston  at 
one  stroke,  the  length  of  the  base  of  the  diagram  is  to  be  considered  as 
representing  the  effective  capacity  of  the  cylinder :  that  is,  the  space  swept 
through  by  the  piston   at  one  stroke.     Hence,  in  order  to  prepare  the 


Fi&  1. 


456 


THE   WORKING   OF   STEAM   IN   COMPOUND   ENGINES. 


diagram  of  the  low-pressure  cylinder  for  combination  "with  that  of  the 
high-pressure  cylinder,   the  lengths  of  its  base,  and  of  every  line  in  it 


OP  li 

Fig.  2. 

parallel  to  its  base,  are  to  be  increased  in  the  ratio  in  which  the  effective 
capacity  of  the  low-pressure  cylinder  is  greater  than  that  of  the  high- 
pressure  cylinder.*     (When  there 
are  a  pair  of  low-pressure  cylinders 
combined  with  one   high-pressure 
cylinder,    they    are    equivalent    to 
one  low-pressure  cylinder  of  double 
the  capacity.)     In  each  of  the  Figs. 
1  and  2,  then,  the  base  0  P  is,  in 
the  first  place,  to  be  produced  to 
B,  making  0  B  greater  than  0  P 
in  the  proportion  above-mentioned. 
To   complete   the    preparation    of 
the  low-pressure  diagram  draw,  in 
each  case,  a  series  of  lines  across  it 
parallel  to  the  base  0  B,  such  as 
the  dotted  lines  in  each  Fig.,  of 
which  one  is  marked  h  i  e.     Let  c  denote  the  ratio  0  B  -4-  0  P.     Then, 
in  the  case  of  an  engine  without  a  reservoir  (Fig.  1)  draw  P£  perpendi- 
cular to  0  B,  cutting  all  the  parallel  dotted  lines,  and  on  each  of  those 
lines  (such  as  s  rq)  lay  off  sq  =  c'rs.     A  curve  kqej,  drawn  through 
*  Manual  of  the  Steam-Engine  and  other  Prime  Movers,  page  334. 


THE  WORKING  OF   STEAM   IN   COMPOUND   ENGINES. 


457 


the  points,  such  as  q,  thus  found,  will  be  the  required  boundary  of  the 
enlarged  low-pressure  diagram,  kqejksJc,  which,  being  joined  on  to  the 


C 

D 

T 

R 

Sfl 

V 

A 

T' 
H 

Ssft' 

A 

A* 

R' 

I 

G 

» 

Lb 

Fig.  4. 


high-pressure  diagram  C  D  K  H,  makes  the  combined  diagram.  When 
the  engine  has  a  reservoir,  draw  0  I  (Fig.  2)  perpendicular  to  0  B,  and 
crossing  all  the  parallel  dotted  lines,  and  on  each  of  those  lines  (such  as 


srk)  lay  off  sk  =  c'sr.  A  curve  Ikefg,  drawn  through  the  points, 
such  as  h,  thus  found,  will  be  the  required  boundary  of  the  enlarged  low- 
pressure  diagram,  which,  being  joined  on  to  the  high-pressure  diagram 
C  D  K  L,    makes   the    combined   diagram.      In    a   theoretically   perfect 


458  THE   WORKING   OF   STEAM   IX   COMPOUND  ENGINES. 

engine,  in  which  the  steam  passed  from  the  high-pressure  to  the  low- 
pressure  cylinder  without  change  of  pressure  or  temperature,  the  two 
diagrams  would  join  exactly  at  the  boundaries  K  L  and  k  I  in  Fig.  2,  or 
KH  and  h  h  in  Fig.  1,  so  as  to  form  one  diagram  identical  with  that 
produced  by  the  same  quantity  of  steam  working  between  the  same  limits 
of  pressure  in  the  larger  cylinder  only.  But  in  actual  engines  there  is 
sometimes  a  gap  between  the  high  and  low-pressure  diagrams,  as  in  the 
Figs.  1  and  2 ;  and  sometimes,  when  the  steam  reservoir  is  heated,  they 
overlap  each  other. 

4.  llatcs  of  Expansion. — In  Figs.  1  and  2  let  D  be  the  point  where  the 
cut-off  takes  place  in  the  high-pressure  cylinder;  draw  D  c  parallel  to  B  0; 

OP  O  P> 

then  -=-  is  the  rate  of  expansion  in  the  high-pressure  cylinder;  p-y3  is  the 

total  increase  of  volume  in  passing  from  the  high-pressure  to  the  low- 
pressure  cylinder  ;  and  the  product  of  those  quantities, 

OP    OB  _  OB 

ri) '  or  ~  7d' 

is  the  total  rate  of  expansion. 

5.  Construction  of  Theoretical  Expansion-Diagrams  for  Proposed  Engines. — 
In  constructing  the  theoretical  diagram  of  a  proposed  steam-engine,  certain 
well-known  assumptions  are  made  in  order  to  simplify  the  figure  and  the 
calculations  founded  upon  it.  In  the  first  place,  the  pressure  of  the  steam 
during  its  admission  is  assumed  to  be  constant,  so  that  the  uppermost 
boundary  of  the  diagram,  as  in  Figs.  4  and  5,  is  a  straight  line,  C  D, 
parallel  to  the  zero  line,  0  B,  the  height  0  C  representing  the  absolute 
pressure  of  admission.  .Secondly,  the  back  pressure  is  assumed  to  be 
constant ;  so  that  the  lower  boundary  of  the  diagram  also  is  a  straight 
line  F  G  parallel  to  0  B,  the  height  0  G  representing  the  mean  absolute 
back  pressure,  as  estimated  from  the  results  of  experience.  Thirdly,  it  is 
commonly  assumed  that  at  the  beginning  of  the  forward  stroke  the  pressure 
rises  suddenly  from  the  back  pressure  to  the  pressure  of  admission,  so  that 
the  first  end-boundary  of  the  diagram  is  a  straight  line  G  C  perpendicular 
to  0  B.  Fourthly,  it  is  assumed  that  at  the  end  of  the  forward  stroke  the 
pressure  falls  suddenly  from  the  pressure  at  the  end  of  the  expansion  (or 
final  pressure)  to  the  back  pressure,  so  that  the  second  end-boundary  of  the 

diagram  is  a  straight  line,  E  F,  perpendicular  to  0  B.  Fifthly,  for  the 
expansion  curve  (D  E  in  Figs.  4  and  5),  which  completes  the  boundaries 
of  the  diagram,  there  is  assumed  a  line  of  the  hyperbolic  class.  Thus  the 
area  of  an  assumed  theoretical  diagram  of  the  work  of  the  steam  in  a  pro- 
posed engine  is  made  up  of  a  hyperbolic  area  C  D  E  H,  and  a  rectangular 
area  E  F  G  H.  The  form  of  the  expansion  curve  depends  on  a  number  of 
circumstances,  such  as  the  initial  pressure  and  temperature  of  the  steam, 


THE   WORKING  OF   STEAM  IN   COMPOUND   ENGINES.  459 

the  proportion  of  water  (if  any)  admitted  along  with  it  in  the  liquid 
state,  the  communication  of  heat  between  the  steam  and  the  metal 
of  the  cylinder,  the  communication  of  additional  heat  to  the  steam  during 
its  expansion  by  the  help  of  a  steam  jacket.  Writers  on  thermodynamics 
have  determined  the  exact  form  of  that  curve  in  various  cases,  such  as 
that  of  steam  originally  dry,  expanding  in  a  non-conducting  cylinder; 
that  of  steam  originally  containing  a  given  proportion  of  moisture  expand- 
ing in  a  non-conducting  cylinder ;  that  of  steam  originally  dry  supplied 
during  the  expansion  with  heat  just  enough  to  keep  any  part  of  it  from 
condensing;  that  of  steam  supplied  during  the  expansion  with  heat 
sufficient  to  keep  it  at  a  constant  temperature.  For  elementary  methods 
of  approximating  to  the  results  of  the  exact  methods  in  such  cases,  see 
The  Engineer  of  the  5th  January,  1866.  For  most  practical  purposes  the 
common  hyperbola  forms  a  good  approximation  to  the  true  expansion 
curve,  and  it  is  convenient  because  of  the  simplicity  of  the  processes  for 
finding  its  figure,  whether  by  calculation  or  by  construction.  To  find  by 
calculation  the  series  of  absolute  pressures  corresponding  to  a  given  series 
of  volumes  assumed  by  the  steam,  on  the  supposition  that  the  expansion 
curve  is  a  common  hyperbola,  multiply  the  initial  absolute  pressure  by  the 
initial  volume;  divide  the  product  by  any  one  of  the  given  series  of 
volumes ;  the  quotient  will  be  the  corresponding  absolute  pressure.  To 
find  a  series  of  points  in  the  common  hyperbola,  in  Fig.  3,  draw  the  two 
axes  0  X  and  0  Y  perpendicular  to  each  other;  0  X  to  form  a  scale  of 
volumes,  and  to  represent  the  zero  line  of  absolute  pressure;  0  Y  to  form 
a  scale  of  absolute  pressures.  On  0  X  lay  of  0  A  to  represent  the  initial 
volume  of  the  steam;  also  OB',  OB",  &c,  to  represent  a  given  series  of 
volumes  occupied  by  the  same  steam  during  its  expansion.  On  0  Y  lay 
off  0  C  to  represent  the  initial  absolute  pressure  of  the  steam.  Through 
C  draw  the  straight  line  C  a  V  b" ,  &c,  parallel  to  0  X,  and  through  the 
points  A,  B',  B",  &c,  draw  the  series  of  straight  lines  A  a,  B'  b',  B"  b",  &c, 
parallel  and  equal  to  0  C.  From  0  draw  the  series  of  diverging  straight 
lines  0  b' ,  0  b",  &c,  and  mark  the  series  of  points  C,  C"  &c,  where  they 
cut  A  a.  From  these  points,  and  parallel  to  0  X,  draw  the  series  of 
straight  lines  C  D',  C"  D",  &c,  and  mark  the  series  of  points  D',  D",  &c, 
where  they  cut  the  series  of  straight  lines  B'  b',  B"  b",  &c.  These  points, 
together  with  the  point  a,  will  be  points  in  the  required  hyperbola, 
a  D'  D",  &c,  which  is  taken  as  an  approximation  to  the  expansion  curve. 

6.  Calculation  of  Mean  Absolute  Pressure  and  of  Indicated  Work  in  a 
Theoretical  Diagram. — Suppose  that  in  Fig.  3  0  B""  represents  the  final 
volume  of  the  steam,  so  that  D""  is  the  end  of  the  expansion  curve,  and 
that  B""  D""  represents  the  final  absolute  pressure.  The  intermediate 
volumes  0  B',  0  B",  &c,  are  to  be  so  chosen  that  the  points  B',  B",  &c, 
shall  divide  A  B""  into  an  even  number  of  equal  intervals.     Multiply  the 


400 


THE   WORKING   OF   STEAM    IN    COMPOUND    ENGINES. 


series  of  absolute  pressures  represented  by  A  a,  B'  V,  &c.,  by  "  Simpson's 
multipliers,"  which  are,  for  the  initial  and  final  pressures,  1 ;  and  for  the 
intermediate  pressures,  4  and  2  alternately;  so  that,  for  example,  for  four 
intervals  and  five  absolute  pressures,  the  multipliers  are  1,  4,  2,  4,  1;  for 
six  intervals  and  seven  absolute  pressures,  1,  4,  2,  4,  2,  4,  1,  and  so  on. 
Add  the  products  together;  divide  the  sum  by  three  times  the  number  of 
intervals.  Multiply  the  quotient  1  >y  the  rate  of  expansion  less  one ;  to 
the  product  add  the  initial  absolute  pressure ;  divide  the  sum  by  the  rate 
of  expansion ;  the  quotient  will  be  the  required  mean  absolute  pressure 
nearly. — Example :  Kate  of  expansion,  5  ;  expansion  divided  into  8  equal 
intervals;  initial  absolute  pressure,  37*8  lbs.  on  the  square  inch. 

Absolute  Pressures.  Multipliers.  |         Products. 

Initial  37"M)  1  37-80 

r  37-S  x  \  =  25-20 
37-8  X  \  =  18-90 
37-8  X  |  =  15-12 
37-8  x  I  =  12-60 
37-8  X  f  =  10-80 
37-8  x  ]  =  9-45 
.  37-8  X  |=  8-40 
Final    37*8  X  I  =    7"56 


Inter- 
mediate 


Divide  by  8  intervals  x  3  : 

Quotient 
Multiply  by  rate  of  expansion  5  —  1 

Product 
Add  initial  absolute  pressure  . 

Divide  by  rate  of  expansion     . 

Mean  absolute  pressure  nearly 


4 

100-80 

o 

37-80 

4 

G0-48 

2 

25-20 

4 

4320 

•> 

18-90 

I 

33-60 

1 

7-56 

1\ 

)  3G5-34  sum 

15-2225 
4 

60-89 

37-80 

98-69  sum 
19-738 


The  remainder  left  after  subtracting  the  back  pressure  from  the  mean  absolute 
pressure  is  the  mean  effective  pressure,  which,  being  multiplied  by  the  area 
of  the  piston,  and  by  the  distance  moved  through  by  the  piston  in  a  given 
time,  gives  the  indicated  work  of  the  steam  in  that  time.* 

*  A  well-known  formula  for  the  ratio  of  the  mean  to  the  initial  absolute  pressure  is 

— — — yP-    °"  r;  r  being  the  rate  of  expansion.     For  a  graphic  approximate  solution  of 
v 

the  same  question,  see  The  Engineer  for  April  13,  1SG6. 


THE   WORKING   OF   STEAM   IX   COMPOUND    ENGINES.  4G1 

7.  Theoretical  Combined  Diagrams. — By  the  process  described  in  the 
preceding  section  there  may  be  constructed  the  approximate  theoretical 
diagram  of  steam  working  with  a  given  initial  pressure  and  a  given  rate 
of  expansion,  and  against  a  given  mean  back  pressure.  In  the  case  of  a 
proposed  compound  engine,  that  theoretical  diagram  is  to  be  regarded  as 
the  combined  diagram  of  the  two  cylinders,  and  it  is  to  be  divided  into 
two  parts,  representing  the  parts  of  the  indicated  work  done  in  the  two 
cylinders  respectively.  In  each  of  the  two  Figs.  4  and  5,  the  theoretical 
combined  diagram  is  represented  by  C  D  E  F  G,  0  C  being  the  initial,  and 
B  E  the  final  absolute  pressure,  0  G  =  B  F  the  mean  back  pressure,  A  A 
the  atmospheric  line,  0  B  the  zero  line  of  absolute  pressure,  H  E  =  F  G  = 
O  B  the  effective  capacity  of  the  large  cylinder,  C  D  the  initial  volume  of 

"F  TT 

steam  admitted  per  stroke,  and    r— -  the  total  rate  of  expansion.     The 

dividing  line  which  marks  the  boundary  between  the  high-pressure  and 
low-pressure  theoretical  diagrams  is  represented  in  Fig.  4  by  K  H,  and  in 
Fig.  5  either  by  k  L,  or  by  Jc  I,  or  by  some  line  near  those  lines,  as  will 
afterwards  be  more  fully  explained. 

8.  Theoretical  Diagrams  of  a  Compound  Engine  without  a  Reservoir. — 
When  the  steam  passes  directly,  without  loss  of  pressure  or  of  heat,  from  the 
high-pressure  to  the  low-pressure  cylinder,  the  dividing  line  of  the  theoreti- 
cal compound  diagram  is  found  by  the  following  process.  In  0  B  (Fig.  4) 
lay  off  0  P  to  represent  the  effective  capacity  of  the  high-pressure  cylinder. 
Through  P,  parallel  to  0  C,  draw  the  straight  hue  P  J  K,  cutting  the  back- 
pressure line  in  J,  and  the  expansion  curve  in  K ;  then  K  will  be  one  end 
of  the  dividing  line.  Through  the  lower  end  E  of  the  expansion  curve, 
and  parallel  to  B  0,  draw  E  H,  cutting  0  C  in  H ;  then  H  will  be  the 
other  end  of  the  dividing  line.  To  find  intermediate  points,  draw,  parallel 
to  0  B,  a  series  of  straight  lines,  such  as  T  B  Q,  T'  R'  Q',  across  the  part 
of  the  diagram  which  lies  below  the  point  K,  and  in  each  of  those  lines, 
for  example,  in  Q  R  T,  lay  off  R  S,  bearing  the  same  proportion  to  R  Q 
that  P  0  bears  to  P  B  ;  the  points  thus  marked,  such  as  S  and  S',  will  be 
in  the  required  dividing  line  K  H.  The  areas  of  the  two  parts  of  the 
theoretical  diagram,  CDKH,  and  K  E  F  G  H,  being  measured  by  ordinary 
methods,  wall  show  the  comparative  quantities  of  work  done  in  the  high- 
pressure  and  low-pressure  cylinders  respectively.  The  advantages  of  the 
compound  engine  in  point  of  diminution  of  stress  and  friction  are  most 
fully  realised  when  those  quantities  of  work  are  equal;  that  is,  when  the 
line  KH  divides  the  area  CDEFG  into  two  equal  parts;  for  then  the 
mean  values  of  the  forces  exerted  through  the  two  piston-rods  are  equal ; 
hence  the  proportion  borne  by  the  effective  capacity  of  the  high-pressure 
cylinder  to  that  of  the  low-pressure  cylinder  ought  to  be  chosen  so  as  to 
realise  that  condition  as  nearly  as  possible.     An  exact  rule  for  that  pur- 


4G2  THE  WORKING   OF  STEAM  IN   COMPOUND   ENGINES. 

pose  would  be  too  complex  to  be  useful  for  practical  purposes.  The 
following  empirical  rule  has  been  found  by  trial  to  give  a  good  rough 
approximation  to  the  required  result  in  ordinary  cases  of  compound 
engines  without  reservoirs :  Make  the  ratio  in  which  the  low-pressure 
cylinder  is  larger  than  the  high-pressure  cylinder,  equal  to  the  square  of 
the  cube  root  of  the  total  rate  of  expansion  ;  for  example,  if  the  total  rate 
of  expansion  is  to  be  8,  let  the  low-pressure  cylinder  be  four  times  the 
capacity  of  the  high-pressure  cylinder  (this  rule  was  tirst  given  in  Ship- 
building, Theoretical  and  Practiced,  page  275).  When  a  table  of  squares 
and  cubes  is  at  hand,  look  for  the  total  rate  of  expansion  in  the  column  of 
cubes,  the  required  ratio  will  be  found  in  the  column  of  squares. 

9.  Theoretical  Diagrams  of  a  Compound  Engine  with  a  Reservoir. — To 
realise  theoretical  perfection  in  the  working  of  an  engine  with  an  inter- 
mediate steam  reservoir,  that  reservoir  should  be  absolutely  non-conduct- 
ing, so  that  the  steam  may  pass  from  it  into  the  low-pressure  cylinder  at 
exactly  the  same  pressure  and  volume  at  which  it  is  received  from  the 
high-pressure  cylinder.  Supposing  this  condition  to  be  realised,  let  Op 
in  Fig.  5,   represent   the  volume  of  steam  admitted  into  the  low-pressure 

O  P 
cylinder  at  each   stroke,   so   that  -^ —  is  the  rate  of  expansion  in  that 

cylinder;  then  Op  will  also  represent  the  effective  capacity  of  the  high- 
Op     .  ... 

pressure  cylinder,  and  will  be  the  rate  of  expansion  in  it;  and  if^Z; 

be  drawn  parallel  to  OC,  so  as  to  cut  the  expansion  curve  in  /,•,  this  point 
will  be  one  end  of  the  required  dividing  line.  To  find  other  points  on 
that  line  under  the  same  theoretical  conditions,  combined  with  the  sup- 
position that  the  forcing  of  the  steam  into  and  its  delivery  out  of  the 
reservoir  take  place  at  certain  times,  produce  B  0,  making  ONof  a  length 
representing  the  capacity  of  the  reservoir  :  then  in  0  C  lay  off  0  L  greater 
than  pi;  in  the  proportion  in  which  TSp  is  greater  than  NO;  L  will  be 
the  other  end  of  the  dividing  line  Tc  L,  which  line  will  be  an  expansion 
curve  for  steam  of  the  initial  volume  represented  by  N  0  =  M  L,  and 
initial  absolute  pressure  represented  by  OL  =  NM,  and  may  be  con- 
structed by  the  method  of  Sec.  5.  The  high-pressure  diagram  will  be 
CDiL,  and  its  lower  boundary,  h L,  will  represent  the  increase  of  pressure 
during  the  process  of  forcing  the  steam  from  the  high-pressure  cylinder 
into  the  reservoir ;  the  low-pressure  diagram  will  be  L  h  E  F  G,  and  its 
upper  boundary  L  k  will  represent  the  diminution  of  pressure  during  the 
process  of  delivering  the  steam  from  the  reservoir  into  the  low-pressure 
cylinder.  But,  in  reality,  the  entrance  of  the  steam  into  and  its  delivery 
from  the  reservoir  take  place  partly  at  the  same  time,  and  the  metal  of 
the  reservoir  abstracts  heat  from  the  entering  steam,  and  gives  heat  back 
to  the  escaping  steam;  the  practical  result,  as  shown  by  the  diagrams 


THE   WOKKING  OF   STEAM  IN   COMPOUND   ENGINES.  4G3 

published  in  Mr.  Cowper's  paper  already  referred  to.  being  that  the 
pressure  of  the  steam  in  the  reservoir  is  nearly  constant,  so  that  the  upper 
boundary  of  the  low-pressure  diagram  nearly  coincides  with  a  straight 
line,  I 1;  parallel  to  0  B.  The  same  straight  line  also  coincides  nearly  with 
the  lower  boundary  of  the  high-pressure  diagram. 

In  the  engines  experimented  on  by  Mr.  Cowper,  the  effective  capacity 
of  the  high-pressure  cylinder  was  somewhat  smaller  than  the  volume  of 
steam  admitted  at  each  stroke  into  the  low-pressure  cylinder,  being  repre- 
sented, for  example,  by  0  P  instead  of  by  Op ;  and  the  final  pressure 
P  K  in  the  high-pressure  cylinder  was  greater  than  the  pressure  in  the 
reservoir. 

The  high-pressure  diagram  was  thus  made  to  resemble  CDK^  in 
Fig.  5,  leaving  a  sort  of  notch,  JLqJc,  between  it  and  the  low-pressure 
diagram  1 K  E  F  G ;  but  it  appears  that  this  loss  of  area  was  compensated 
by  the  effect  of  the  steam-jacket  enveloping  the  reservoir,  which,  by 
imparting  additional  heat  to  the  steam,  caused  the  low-pressure  diagram 
to  be  of  a  fuller  form  in  the  part  It  E  than  that  bounded  by  the  theoretical 
expansion  curve. 

In  this  case  a  rough  approximation  to  an  equal  division  of  work  between 
the  high  and  low-pressure  cylinders  may  be  obtained  by  making  the  rate 
of  expansion  in  the  low-pressure  cylinder  equal  to  the  square  root  of  the 
total  rate  of  expansion. 


4G4  ON  THE   THEORY   OF   EXPLOSIVE   GAS-ENGINES. 


XXVIL— ON  THE  THEORY  OF  EXPLOSIVE  GAS-ENGINES.* 

1.  Thcrmochjnamkal  Propositions. — In  calculations  respecting  the  prac- 
tical use  of  heat  engines,  it  is  convenient  to  employ  rules  in  which  the 
pressures  and  volumes  alone  of  the  working  substance  are  taken  explicitly 
into  account,  so  as  to  avoid  the  necessity  for  computing  temperatures. 
Such  rules  exist  in  the  case  of  the  steam-engine.  The  object  of  the 
present  communication  is  to  explain  a  similar  set  of  rules  applicable  to 
explosive  gas-engines.  They  are  based  mainly  on  the  following  estab- 
lished propositions  in  thermodynamics:  Let /.•  denote  the  ratio  in  which 
the  specific  heat  of  a  substance  in  the  perfectly  gaseous  state  under 
constant  pressure,  exceeds  the  specific  heat  of  the  same  substance  at 
constant  volume.     Then — 

First  proposition. — When  a  mass  of  that  substance  passes  from  the 
absolute  pressure  p  and  volume  v,  to  the  absolute  pressure  p'  and  volume  v, 
the  dynamical  equivalent  of  the  sensible  heat  absorbed  by  it  (that  is,  heat 
employed  in  producing  elevation  of  temperature,  as  distinguished  from 
heat  which  disappears  in  doing  work)  has  the  following  value  : 

p  r  —  i  >  v 
~l:  -  1     ' 

The  pressures  and  volumes  are  supposed  to  be  given  in  such  measures 
that  the  product  of  a  pressure  and  volume  may  be  expressed  in  units  of 
work.  For  example,  if  volumes  are  given  in  cubic  feet,  pressures  should 
be  given  in  pounds  on  the  square  foot,  in  order  that  the  product  of  a 
pressure  and  a  volume  may  be  expressed  in  foot-pounds. 

Second  proposition. — "When  a  mass  of  the  same  substance  performs 
work  by  expanding  without  transfer  of  heat,  the  pressure  falls  in  such  a 
manner  that  p  vk  is  a  constant  quantity. 

The  value  of  h  for  atmospheric  air  is  1'408;  it  is  very  nearly  the  same 
for  oxygen  and  nitrogen;  and  it  does  not  differ  much  from  1*4  in  the 
gaseous  mixture  resulting  from  the  explosion  of  coal  gas  and  air  in  the 
ordinary  proportions:  a  mixture  of  which  about  three-fourths  consists  of 
nitrogen.       Consequently,  throughout  this    communication,  1*4  =  f  will 

*  From  Th?  Engineer  of  July  27,  1S6G. 


ON   THE  THEORY   OF   EXPLOSIVE   GAS-ENGINES.  465 

be  taken  as  a  value  of  k,  sufficiently  near  to  the  truth  for  practical  purposes 
where  minute  accuracy  is  neither  necessary  nor  possible;  so  that 

_1       _  5         k       _  7 
k  -  1  ~~  2'  h-  1  ~~  2" 

2.  Rules  as  to  Heat  and  Expansion. — The  following  rules  are  the 
immediate  consequences  of  the  two  propositions  just  stated: 

I.  A  mass  of  a  gaseous  mixture,  occupying  the  constant  volume  v,  has 
its  pressure  increased  from  p  to  %>\  the  quantity  of  heat  in  units  of  work 
required  to  effect  that  change  is 

2  r  (/'  -  Vi- 
lli A  mass  of  a  gaseous  mixture,  under  the  constant  pressure  p,  has 
its  volume  increased  from  v  to  v.  The  quantity  of  heat  employed  in 
this  case  to  produce  rise  of  temperature  is,  as  before,  in  units  of  work, 
|.  p  (yf  _  v}  •  and  at  the  same  time  the  work  done  through  the  expansion 
is  p  (v  —  v),  and  an  equivalent  quantity  of  heat  disappears;  so  that  the 
whole  quantity  of  heat  required,  in  units  of  work,  is 

-  p  (Y  -  r). 

III.  A  mass  of  a  gaseous  substance  performs  work  by  expanding  from 
the  volume  v  to  the  greater  volume  r  v  without  transfer  of  heat,  r  being 
the  rate  of  expansion.  Then,  if  the  original  absolute  pressure  is  pv  the 
final  absolute  pressure  will  be 

lh  =1\     ' 
The  following  table  gives  some  results  of  this  rule  : — 


Eate  of  Expansion. 

Cut-off. 

Final  Pressure. 
Initial  Pressure. 

r 

1 
r 

pa  _ 
Pi 

r"* 

5 

0-2 

0-105 

4 

0-25 

0-144 

3£ 

0-3 

0-185 

2': 

0-35 

0-230 

21 

0-4 

0-277 

2| 

0-45 

0-327 

2 

0-5 

0-379 

2  G 


466 


ON   THE   THEOHY    OF   EXPLOSIVE   GAS-ENGINES. 


The  exact  calculation  of  r  ?  requires  the  aid  of  logarithms.  In  the 
absence  of  logarithms  an  approximate  value  may  be  computed  by  the 
following  empirical  formula  : — 


y'-'  =  0-54  f-  +  \)  -  0-025  nearly; 


which  is  correct  to  about  one  per  cent,  when  r  is  not  less  than  2,  nor  greatei 
than  7;  but  should  not  be  used  beyond  those  limits. 

3.  Diagram. — The  general  character  of  the  indicator-diagram  of  an 
explosive  engine  is  shown  by  the  lines  marked  ACEGHA  in  the 
figure. 

The  base  of  the  figure,  0  V,  represents  a  scale  of  volumes,  on  which 

0  B  may  be  taken  to  denote  one  cubic  foot 
of  a  suitable  explosive  mixture  introduced 
into  a  cylinder  at  the  atmospheric  pressure 
represented  by  the  ordinates  0  A  =  B  C. 
In  symbols,  let  p0  stand  for  the  atmospheric 
pressure :  then  A  C  will  represent  the  line 
drawn  by  the  indicator-pencil  during  the 
introduction  of  one  cubic  foot  of  the  explo- 
sive mixture.  Suppose  that  the  admission 
is  now  cut  off,  and  the  mixture  fired  by  a 
spark;  and  suppose  also  that  the  time 
occupied  by  the  explosion  is  very  small, 
compared  with  the  time  occupied  by  a  stroke 
of  the  piston :  then  the  sudden  increase  of 
pressure  produced  by  the  explosion  may  be 
approximately  represented  by  C  E,  the 
absolute  pressure  immediately  afterwards 
being  represented  by  BE.  The  gaseous 
mixture  of  products  of  the  explosion  then  expands,  driving  the  piston 
before  it ;  let  E  G  be  the  expansion  curve,  so  that  0  F  is  the  final  volume, 
and  F  G  the  final  absolute  pressure  of  the  gas.  G  H  represents  the  fall 
to  the  atmospheric  pressure  upon  opening  the  eduction  valve,  and  H  A 
the  expulsion  of  the  gaseous  mixture  against  the  atmospheric  pressure, 
so  that  the  work  done  by  each  cubic  foot  of  explosive  mixture  is  repre- 
sented by  the  area  C  E  G  H  C. 

4.  Total  and  Available  Heat  of  Explosion.— -The  total  heat  of  explosion 
may  be  calculated  theoretically  from  the  composition  of  the  explosive 
mixture  employed,  by  the  aid  of  data  obtained  from  such  experiments 
as  those  of  Favre  and  Silbermann.  For  example,  according  to  information 
given  by  Dr.  Letheby  {Engineer,  28th  June,   18GG,  p.  448),  the  mixture 


ON   THE   THEORY   OF   EXPLOSIVE   GAS-ENGINES.  4G7 

found  to  answer  best  in  Lenoir's  gas-engine  is  composed  of  eight  parts  by 
volume  of  air  to  one  of  common  coal  gas.  From  Dr.  Letheby's  analysis 
of  the  gas,  and  the  known  values  of  the  total  heat  of  combustion  of  its 
constituents,  it  appears  that  the  total  heat  of  explosion  of  one  cubic  foot  of 
the  mixture  is  equivalent  nearly  to  56,900  foot-pounds. 

To  find  the  available  heat  of  explosion,  it  is  necessary  to  have  recourse 
to  experiments  on  actual  gas-engines.  Let  px  be  the  absolute  pressure 
immediately  after  explosion ;  then,  according  to  Rule  I.  of  Article  2,  the 
available  heat  of  the  explosion,  in  units  of  work,  per  cubic  foot  of  explosive 
mixture,  is  expressed  by 

2  Oi  -  Po)- 

Now,  from  experiments  quoted  by  Dr.  Letheby,  it  appears  that  px  =  about 
5  atmospheres  on  an  average,  so  that 

Pi  ~  Po  ~  ^  atmospheres  =  8-164  lbs.  on  the  square  foot; 

and,  consequently,  the  available  heat  of  explosion  per  cubic  foot  is 

J-  x  8464  =  21160  foot-pounds. 

The  difference  between  this  and  the  total  heat  of  explosion  represents 
the  loss  which  occurs  through  conduction  and  imperfect  combustion. 
The  ratio  of  the  available  to  the  total  heat.  viz. : — 

^2  =  0-372, 
56900 

may  be  called  the  efficiencij  of  the  explosion. 

5 
In  the  diagram  the  available  heat  of  explosion  is  represented  by  -   X 

the  area  of  the  rectangle  AD  EC;  and  according  to  an  established  pro- 
position in  thermodynamics,  it  may  also  be  represented  as  follows  {Philo- 
sophical Transactions  for  1854)  {Seep.  339):— Through  E  and  C  draw  a 
pair  of  adiabatic  curves,  E  L  and  C  N;  that  is,  curves  of  expansion  without 
transfer  of  heat;  then  the  heat  required  to  produce  the  rise  of  pressure 
C  E  at  the  constant  volume  O  B  is  represented  by  the  limit  to  which  the 
area  NOEL  between  those  curves  approaches  as  the  curves  are  prolonged 
indefinitely  towards  N  and  L. 

5.  Final  Pressure. — The  pressure  at  the  end  of  the  expansion  represented 
by  F  G  may  be  approximately  computed  by  Rule  III.  of  Article  2.  For 
example,  if  the  rate  of  expansion  is  2,  the  table  shows  that  p2  =  0-379^; 
so  that   if  Pi  =  5   atmospheres  =  10580  lbs.  on   the    square    foot,    we 


4G8  ON  THE  THEORY   OF   EXPLOSIVE   GAS-ENGINES. 

shall  probably  have,  with  the  expansion  2,  the  final  absolute  pressure 
p^  =  4010  lbs.  on  the  square  foot  nearly,  or  1894  lbs.  on  the  square  foot 
above  the  mean  atmospheric  pressure. 

The  following  rule  serves  to  determine  the  rate  of  expansion  rx  required 
in  order  to  make  the  final  pressure  be  equal  to  the  atmospheric  pressure, 
or  nearly  so  : — 

-teV-  (iv.) 


4V 

and  this  is  the  rate  of  expansion  which  realises  the  greatest  indicated 

work.    For  example,  let  —  =  5;    then  rx  =  5'  =  3*1  G  nearly.      In  the 

2'o 

Q  A       OP 
diagram  this  ratio  is  represented  by   l  "    =   -      ;  Q  being  the  point  where 

the  line  of  atmospheric  pressure  cuts  the  expansion  curve. 

The  expansion  curve  is  here  assumed  to  coincide  sensibly  with  an 
adiabatic  curve. 

G.  Indicated  Work. — Draw  (4  K.I  parallel  to  TO.  Then  the  area 
CEGIIC,  representing  the  work  done  by  each  cubic  foot  of  explosive 
mixture,  consists  of  a  rectangular  pari  CKGH,  and  a  triangular  part  KEG. 

The   work   represented    by   the   rectangular    part   is   simply   (r  —  1) 

The  work  represented  by  the  triangular  part  is  determined  by  the  aid 
of  Rules  I.  and  II.  of  Article  2  as  follows: — Conceive  an  adiabatic  curve 
K  M  to  pass  through  the  point  K.  Then  the  area  K  E  G  is  the  difference 
between  the  limits  of  the  indefinitely-prolonged  areas  MKEL  and 
MKGL.  But,  according  to  a  principle  stated  in  Article  4,  the  limit  of 
the  area  MKEL  represents  the  quantity  of  heat  required  to  produce  the 
increase  of  pressure  K  E  at  the  constant  volume  0  B ;  and,  according  to 
Bule  I.,  that  quantity  of  heat,  in  units  of  work,  is  expressed  by 

5  5 

-  X  rectangle  JDEK  =  -  (j\  —  j>2) ; 

also,  according  to  the  same  principle,  the  limit  of  the  area  MKGL 
represents  the  quantity  of  heat  required  to  produce  the  increase  of  volume 
KG  at  the  constant  pressure  F  G;  and,  according  to  Rule  II.,  that  quantity 
of  heat,  in  units  of  work,  is  expressed  by 

-  X  rectangle  B  K  G  F  =  -(/•—  l)p2; 
whence  the  work  represented  by  the  area  KEG  is  found  to  be 


OX   THE   THEORY   OF   EXPLOSIVE   GAS-EXGIXES.  4G9 

and,  combining  with  this  the  work  represented  by  the  rectangle  CKGH, 
the  whole  work  per  cubic  foot  of  explosive  mixture  is  found  to  be 
expressed  as  follows  : — 

W  =  I  (ft  -  ft)  -  I  (r  -  l)ft  +  (r  -  1)  (ft  -  ft).    .     (V.) 

For  example,  in  the  previous  calculations  we  have  r  =  2;  ft  =  10580; 
ft  =  4010;  ft  =  2116;  and,  consequently,  W  =  16425-14035  +1894 
=  4284  foot-pounds  per  cubic  foot  of  explosive  mixture. 

The  mean  effective  pressure  (ft)  is  given  by  dividing  the  work  done  by 
the  space  swept  by  the  piston;  that  is  to  say, 

W 
Pe  =  y (VI) 

Thus,  in  the  example  already  given 

4284 

p  =  -     -=2142  lbs.  on  the  square  foot. 

7.  Efficiency. — The  term  "efficiency  of  the  expansion"  may  be  used  to 
denote  the  ratio  of  the  work  done  to  the  available  heat  of  explosion;  that 
is  to  say, 

2  W 


5  ilh  ~  lh) 
Its  value  in  the  example  is 

4284 


(VII) 


21160 


==  0*203  nearly. 


If  the  efficiency  of  the  expansion  be  multiplied  by  the  efficiency  of  the 
explosion,  already  mentioned  in  Article  4,  the  product  is  the  resultant 
efficiency  of  the  heat,  whose  value,  in  the  example,  is 

0-203  x  0-372  =  0-075  nearly; 

so  that  1\  per  cent,  of  the  whole  heat  of  explosion  is  converted  into 
mechanical  work. 

8.  Greatest  Efficiency. — As  already  stated  in  Article  5,  the  greatest 
efficiency  of  the  expansion  occurs  when  the  final  pressure  is  equal  to  the 
atmospheric  pressure.  The  diagram  of  work  is  then  represented  by 
C  E  Q,  and  the  quantity  of  work  per  cubic  foot  of  explosive  mixture  is 
found  by  making  p,  =  ft  in  formula  (V.)  Let  W,  be  that  quantity  of 
work;  then, 


470  ON  THE   THEORY  OF   EXPLOSIVE  GAS-ENGINES. 

W1  =  §(2>1-i>0)-^(r1-l)iv   .        .     (VIE) 

The  corresponding  value  of  the  efficiency  of  the  explosion  is — 

Wi        =1      7(rt-l)a, 
•5  (ft  -  2\>)  5  (Pl  -  ^J  " 

In  the  example  chosen  we  have  r  =  3T6,  and,  consequently,  the  work 
per  cubic  foot  of  the  mixture  is 

W,    =  21160  -  16000  =  5160  foot-pounds  per  cubic  foot; 

the  moan  effective  pressure, 

-  ■-.-  =  1636  lbs.  on  the  square  foot; 
d*16 

and  the  efficiency  of  the  expansion, 

2  W,  5160 

=  0-2  1  !. 


Hlh-l'o)        21160 

The  resultant  efficiency  of  the  heat  is 

0-244  x  0-372  =  009, 

so  that  nine  per  cent,  of  the  whole  heat  of  explosion  is  converted  into 
mechanical  work. 

9.  Remarks. — In  the  preceding  calculations  of  work  and  efficiency  no 
deduction  is  made  for  friction,  nor  for  any  increase  in  the  back  pressure 
which  may  arise  from  resistance  to  the  escape  of  the  waste  gases.  The 
allowances  to  be  made  for  such  losses  can  be  deduced  from  practical  trials 
alone.  Hence,  the  results  of  the  formula?  are  theoretical  limits,  which  may 
be  aimed  at  in  practice,  but  probably  cannot  be  absolutely  attained. 


ON  THE   EXPLOSIVE   ENERGY*  OF   HEATED   LIQUIDS.  471 


XXVIIL— OX  THE  EXPLOSIVE  ENERGY  OF  HEATED 
LIQUIDS* 

1.  Reference  to  Theoretical  Investigations. — In  contemplation  of  the  revival 
of  the  application  (first  invented  by  Perkins)  of  the  sudden  evapora- 
tion of  highly  heated  liquid  water,  in  order  to  propel  projectiles,  it  may 
be  useful  to  give  a  summary  of  the  rules  for  calculating  the  utmost 
theoretical  effect  of  a  given  fluid  when  so  employed,  under  given  circum- 
stances. For  the  theoretical  deduction  of  those  rules  from  the  laws 
of  thermodynamics  I  have  to  refer  to  two  independent  investigations, 
made  respectively  by  myself  and  by.  Clausius;  the  former  published 
in  the  Philosophical  Transactions  for  1854  (See  p.  339);  the  latter  in 
Poggendorff's  Annalcn  for  1856.  The  rules  themselves,  with  some 
tables  of  their  results,  having  reference  to  the  bursting  of  steam-boilers, 
have  also  been  published  in  the  Transactions  of  the  Institution  of 
Engineers  in  Scotland  for  1863-4,  Vol.  VII.,  page  8  ;  and  in  the  Philosophical 
Magazine  for  1863,  Vol.  XXVL,  pages  338  and  436.  In  a  subsequent 
communication  I  propose  to  consider  the  case  when  the  fluid  passes  into 
the  state  of  vapour  before  its  admission  into  the  gun. 

2.  General  Formulae  for  all  Fluids. — Suppose  a  closed  boiler  to  be  entirely 
filled  with  a  fluid  in  the  liquid  state,  at  a  certain  absolute  temperature  tv 
Let  the  absolute  temperature  t2,  being  lower  than  tv  be  the  boiling  point 
of  that  fluid  in  a  boiler  open  to  the  atmosphere.  Let  a  given  mass  of  the 
liquid  be  made  to  escape  from  the  boiler,  and  to  perform  work  by  expand- 
ing partly  or  wholly,  as  the  case  may  be,  into  the  state  of  vapour,  and 
driving  a  solid  body  (such  as  a  bullet)  before  it,  until  its  pressure  falls  to 
that  of  the  surrounding  atmosphere,  and  its  absolute  temperature  (con- 
sequently) to  t2.  Then  the  energy  exerted  by  that  mass  of  fluid  is  equi- 
valent to  the  raising  of  its  own  weight  to  the  height  given  by  the  following 
equation  : — 

U  =  Kt2(n-  1  -  hyp.  log.  n),         .         .     (1.) 

in  which  K  denotes  the  dynamical  value  of  the  specific  heat  of  the  fluid  in 

*  From  The  Engineer  of  November  11,  1870. 


472  OX   THE   EXPLOSIVE   ENERGY  OF   HEATED  LIQUIDS. 

the  liquid  state ;  and  n  =   \  the  ratio  in  which  the  initial  absolute  tem- 

perature  is  greater  than  the  final.  Moreover,  the  following  formula  gives 
the  excess  of  the  space  filled  by  each  unit  of  weight  of  the  fluid  at  the  end 
of  the  expansion,  above  the  space  filled  by  an  unit  of  weight  of  the 
liquid : 

dt2 

(J  ii 
in  which    ,  -  denotes  the  rate  at  which  the  pressure  of  saturation  varies 

with  the  boiling  point,  at  the  final  temperature. 

Absolute  temperatures  are  given,  as  is  well-known,  by  adding  461°-2  to 
temperatures  on  the  ordinary  Fahrenheit's  scale,  or  274°  to  temperatures 
on  the  ordinary  Centigrade  scale. 

3.  Formula-  for  Water. — For  water  the  values  of  the  co-efficients  in  the 
formulae  are  as  follows,  very  nearly  : — 

K  L  =  520,000  feet ;  * 

-  =  18-38  cubic  feet  Tier  lb. ;  t 
dt% 

or  to  a  rough  approximation,  about  1100  times  the  volume  of  the  hot 
liquid  water.  Hence  we  have  the  following  formula;  for  water ;  energy 
of  the  explosion  in  foot-pounds  per  pound  of  water  : — 

U  =  520,000  («  -  1  -  hyp.  log.  ft).  .         .     (1  A.) 

Space  swept  by  the  explosion,  or  final  volume  of  the  water  and  steam 
in  cubic  feet  to  the  pound. 

s  =  18-38  hyp.  log.rc;      .         .         ,     (2A.) 
or,  in  terms  of  the  volume  of  the  liquid  water, 

1100  hyp.  log.  ft,  nearly.     .  .  .     (2B.) 

4.  Examples. — To  illustrate  the  results  of  the  preceding  formula?,  the 
two  following  examples  are  given,  in  which  the  values  assumed  for  n  are 
respectively  2  and  2|.  The  pressures  corresponding  to  the  temperatures 
given  by  those  ratios  are  not  known  by  experiment.     The  pressures  given 

*  15S,500  metres,  nearly. 

t  1"M7  cubic  metres  per  kilogramme,  nearly. 


o 

91 

^4 

673°2 

0  K  Q°.!> 

bio    A 

374° 

374° 

1346°-4 

1514°-7 

748° 

841°-5 

ON  THE   EXPLOSIVE   ENERGY   OF  HEATED   LIQUIDS.  4"3 

in  the  following  table  of  results  are  calculated  on  the  assumption  that  the 
formulas  which  are  found  to  be  accurate  up  to  the  limits  of  experiment, 
are  applicable  also  to  temperatures  far  beyond  those  limits ;  hence,  those 
pressures  are  to  be  viewed  as  in  a  great  measure  conjectural.  This  affects 
the  safety  of  the  boiler  and  of  the  gun;  but  not  the  energy  of  the 
explosion,  nor  the  final  volume  of  the  fluid ;  for  these  two  quantities  vary 
with  the  temperature  only. 

Example  I.  Example  II. 

Eatio  of  initial  to  final  absolute  tem- 
perature,  ..... 

Final  absolute  temperature,  Fahr., 

Final  absolute  temperature,  Cent., 

Initial  absolute  temperature,  Fahr., 

Initial  absolute  temperature,  Cent., 

Initial     temperature,    ordinary     scale, 

Fahr., 885°-2  1053°-5 

Initial    temperature,    ordinary    scale, 

Cent., 474°  567°'5 

Energy  of  the  explosion,  foot-pounds 

per  pound  of  water,    .          .         .  159,562  228,189 

Final  volume — cubic  feet  per  pound  of 

Avater  and  steam,        .         .         .  12-74  14'9 

Final  volume — ratio  to  initial  volume 

of  water,  nearly  .         .         .  760  890 

Conjectural  absolute  pressure  in  boiler, 

pounds  on  the  square  inch,  .  7,180  13,345 

Ditto,  ditto,  in  atmospheres,        .         .  490  908 

The  values  of  the  energy  of  the  explosion  in  the  two  examples  agree 
very  nearly  with  the  least  and  greatest  values  found  by  experiment  for  the 
energy  of  the  explosion  of  1  lb.  of  gunpowder ;  hence  the  examples  may 
be  taken  as  showing  the  conditions  which  must  be  fulfilled  in  order  that 
1  lb.  of  heated  water  may  produce  the  same  effect  as  1  lb.  of  gunpowder. 
In  both  examples  the  initial  pressures  are  so  high  that  the  only  safe  form 
of  boiler  is  a  coil  of  tube  of  small  bore  compared  with  its  thickness.  This 
was  the  form  employed  by  Perkins. 

5.  Expenditure  of  Heat, — The  expenditure  of  heat  required  in  order  to 
produce  the  elevation  of  temperature  of  each  unit  of  mass  of  liquid  from 


474  ON   THE   EXPLOSIVE    ENERGY   OF   HEATED   LIQUIDS. 

the  temperature  of  the  feed  to  that  at  which  it  escapes  from  the  boiler  is 
expressed  in  dynamical  units  as  follows  : — 

H  =  K^~g,     ....     (3.) 

in  which  tz  denotes  the  absolute  temperature  of  the  feed.  Let  this  latter 
temperature  bear  the  ratio  n1  to  the  absolute  temperature  of  the  atmos- 
pheric boiling  point ;  then  we  may  express  the  same  expenditure  of  heat 
in  the  following  manner  : — 

11  =  Kt,(n  -  n1);       .         .  .     (3  A.) 

and  for  water,  the  value  of  this  in  foot-pounds  per  pound  is  very  nearly 

II  =  520,000  (n-n1).     .         .         .     (3  B.) 

The  value  of  nl  for  water  ranges,  in  ordinary  cases,  between  0*7  and  0'8. 
Assuming  it  to  be  0*75,  the  expenditure  of  heat  in  the  two  preceding 
examples  is  found  to  have  the  values  given  in  the  following  table  : — 


Heat  expended 

Example  I. 

Example  II. 

Foot-pounds  per  pound. 

650,000 

780,000 

Units  of  evaporation,     . 

0-873 

1-047 

The  difference  between  the  quantities  in  the  first  line  and  the  values  of 
the  energy  of  explosion,  are  the  quantities  of  heat  which  go  to  waste  with 
the  escaping  steam  and  water  after  the  explosion,  viz  : — 

Waste  heat.  Example  I.  Example  II. 

Foot-pounds  per  pound,         .         .          490,438  551,811 

6.  Efficiency  of  the  Explosion. — This  term  may  be  used  to  express  the 
ratio  borne  by  the  energy  of  the  explosion  to  the  whole  expenditure  of 
heat.     Its  value  is  as  follows  : — 

U       w  —  1  -  hyp,  log,  n 

Jl  n  -  nl  •...     [*.)    . 

And  it  is  to  be  observed  that  this  value  depends  solely  on  the  ratios  borne 
to  the  absolute  temperature  of  the  atmospheric  boiling  point,  by  two  other 
absolute  temperatures — viz.,  that  of  the  feed  water,  and  that  of  the  liquid 
just  before  it  escapes  from  the  boiler.  In  the  two  examples  the  values  of 
the  efficiency  of  the  explosion  are  respectively — 

Example  I.  Example  II. 

0-245  0-293 

7.  Bemarh. — The  preceding  formula?  all  proceed  on  the  assumption  that 
the  specific  heat  of  the  liquid  is  sensibly  constant.     This  is  not  perfectly 


ON   THE   EXPLOSIVE   ENERGY   OF  HEATED  LIQUIDS.  475 

accurate,  for  the  specific  heat  of  every  liquid  increases  slowly  with  the 
temperature.  The  effects  of  that  increase  are  shown  in  the  original 
theoretical  investigations  referred  to  at  the  commencement;  but  for 
practical  purposes  it  is  unnecessary  to  take  them  into  account. 

The  formula?  also  take  no  account  of  the  retarding  effect  of  friction  on 
the  bullet,  nor  of  the  inertia  of  the  air  which  it  drives  before  it  in  the 
barrel  of  the  gun,  nor  of  the  loss  of  energy  which  may  take  place  through 
the  abstraction  of  heat  from  the  water  by  the  metal  of  the  barrel :  those 
being  quantities  which  can  be  determined  by  direct  experiment  alone. 

The  initial  temperatures  assumed  in  the  examples  have  been  chosen  so 
as  to  make  the  explosive  energy  of  the  water  nearly  equivalent  to  that  of 
an  equal  weight  of  gunpowder.  By  choosing  a  lower  initial  temperature 
the  initial  pressure  may  be  moderated ;  but  the  explosive  energy  of  a 
given  weight  is  at  the  same  time  diminished ;  and  a  greater  mass  of  water 
must  be  used  in  order  to  obtain  a  given  amount  of  energy,  thus  increasing 
the  proportionate  quantity  of  energy  which  is  lost  in  propelling  the 
explosive  material  itself,  as  the  following  section  will  show. 

8.  Efficiency  of  Projection. — This  term  may  be  used  to  denote  the  pro- 
portion which  the  energy  of  the  bullet  at  the  instant  of  its  leaving  the 
gun  bears  to  the  whole  energy  of  the  explosion. 

Let  m  denote  the  ratio  which  the  mass  of  the  bullet  bears  to  the  mass  of 
the  explosive  material ;  M  the  ratio  which  the  whole  mass  that  recoils 
bears  to  the  mass  of  the  explosive  material ;  v  the  velocity  of  the  bullet  at 
the  instant  when  the  action  of  the  explosion  ceases,  so  that  the  energy  of 

nil  Q)" 

the  bullet  at  that  instant,  per  unit  of  mass  of  explosive  material,  is  — — ;  then, 

neglecting  friction  and  the  inertia  of  the  air,  &c,  it  can  be  shown  that  the 
energy  of  the  explosion  of  an  unit  of  mass  of  explosive  material  is  disposed 
of  in  the  following  maimer  : — 

+  3V         2M+1^\2M+1//J  K    ' 

9 

m  v 
On  the  right  hand  side  of  this  equation  the  first  term  — —  is  the  energy 

of  the  bullet.  "  9 

The  second, 

M  v*  /2  m  +  1 


2  a  \2  M  +  1 

is  the  energy  of  the  mass  which  recoils. 
And  the  third, 


476  OX   THE   EXPLOSIVE   ENERGY   OF   HEATED  LIQUIDS. 

'2m+l\«\ 
,2M  +  1/    / 

is  the  energy  of  the  projectile  motion  of  the  products  of  explosion,  at  the 
instant  when  they  cease  to  act  on  the  bullet.  Hence,  the  counter-efficiency 
of  projection,  being  the  reciprocal  of  the  efficiency,  or  in  other  words,  the 
ratio  in  which  the  whole  energy  of  the  explosion  is  greater  than  that  of 
the  bullet,  is  expressed  as  follows  : 

_  M  f2  m  +  1 

C~      +  m\2M+l 


J-jl_»fL±I+(*»+iyi    .  .     (0.) 

3  m  (  :>M  +  1        \2M+  1/   3  . 


from  which  it  appears  that  the  energy  lost  through  the  projection  of  the 

products  of  explosion  is  greater,  the  greater  the  proportion  —  borne  by 

the  mass  of  the  explosive  material  to  that  of  the  bullet,  and  that  when 
the  proportionate  weight  M  of  the  recoiling  mass  is  very  great,  that  lost 

energy  is  approximately  equal  to  the  fraction  - —  of  the  energy  of  the 
bullet  3wi 

For  example,  let  m  =  8,  and  M  =  1000  ;  then  the  three  terms  of  the 
counter-efficiency  of  projection  have  the  values  shown  in  the  following 
equation  to  three  places  of  decimals  ; 

c  =  1  +  0-009  +  0-0-11  =  1-050; 

that  is  to  say,  the  energy  lost  in  the  recoil  is  0-009,  and  the  energy  lost  in 
projecting  the  products  of  explosion  0-041  of  the  energy  of  the  bullet; 
the  latter  being  by  far  the  more  important  loss ;  and  hence  it  is  desirable 
not  to  increase  unnecessarily  the  comparative  weight  of  the  explosive 
material. 

9.  Batio  of  Final  Volume  of  Steam  to  Volume  of  Bullet. — In  section  3  of 
this  communication,  equations  (2a)  and  (2b),  expressions  have  been  given 
for  the  space  (s)  filled  by  an  unit  of  weight  of  the  mixture  of  water  and 
steam,  when  it  has  expanded  until  its  pressure  is  equal  to  that  of  the 
atmosphere.  Let  w  denote  the  heaviness  of  the  material  of  which  the 
bullet  is  made;  then,  m  being,  as  before,  the  ratio  of  the  mass  of  the  bullet 
to  the  mass  of  the  fluid  which  drives  it,  the  ratio  in  which  the  final  volume 
of  the  fluid  exceeds  the  volume  of  the  bullet  is  given  by  the  following 
expression  : — 

-  =  —  X  18-38  hyp.  log.  n ;    .  .  .     (7.) 

when  w  is  stated  in  pounds  to  the  cubic  foot. 


ON   THE   EXPLOSIVE   ENERGY   OF   HEATED   LIQUIDS.  477 

If  lead  be  the  material  of  the  bullet,  we  have  w  =  712  nearly;  and  if 
iron  or  steel,  w  —  about  480.    Hence  are  deduced  the  following  formulae: — 

„     ,     .                    ws       13087  hyp.  log.  n   ^ 
For  lead,     .         .     —  = ■ ;    I 

m  in 

>       (7  A.) 
_  .        ,    bs        8822  hyp.  log,  n 

For  iron  and  steel,    -  -  =  —  — . 

m  in  J 

When  these  formula?  are  applied  to  the  two  examples  given  in  section  4, 
the  bullet  being  supposed,  as  in  section  8,  to  have  eight  times  the  mass 
of  the  explosive  material  (so  that  m  =  8),  the  following  results  are 
obtained  : — 

Ratio  of  final  volume  of  water  and  steam  to  volume  of  bullet. 

Example  I.  Example  II. 

Lead  bullet,       .         .         .         .  1134  1327 

Iron  or  steel  bullet.    .         .         .  7G4  894 

Such  would  be  the  ratio  which  the  volume  of  the  gun-barrel  would  have 
to  bear  to  that  of  the  bullet,  in  order  to  render  available  the  Avhole  of  the 
energy  developed  by  the  expansion  of  the  steam.  It  is  obvious  that 
barrels  of  such  dimensions  are  purely  ideal,  being  many  times  longer  than 
the  greatest  length  that  it  is  possible  to  use  in  practice.  It  therefore 
becomes  necessary  to  limit  the  barrel  to  a  practicable  length,  and  to 
sacrifice  part  of  the  energy  due  to  the  expansion  of  the  steam. 

10.  Full-presmre  Steam  &im. — In  the  following  investigation  the  sup- 
position is  made  that  the  communication  between  the  boiler  and  the 
gun-barrel  remains  full  open  during  the  whole  time  of  the  motion  of  the 
bullet  along  the  barrel ;  and  it  is  further  assumed  that  at  the  instant 
when  the  bullet  quits  the  muzzle  the  barrel  is  filled  with  fluid  of  uniform 
pressure  and  density,  which,  consequently,  is  at  that  instant  moving  with 
a  velocity  equal  to  that  of  the  bullet  (r).  The  pressure  in  the  boiler  must 
be  higher  than  that  in  the  barrel  to  the  extent  required  in  order  that  the 
expansion  of  the  fluid  in  passing  from  the  higher  to  the  lower  pressure 
may  be  sufficient  to  produce  a  velocity  of  outflow  equal  to  that  of  the 
bullet ;  and  so  far  the  action  of  the  fluid  is  expansive ;  but  its  action  in 
driving  the  bullet  is  equal  simply  to  that  due  to  the  difference  between  the 
pressure  in  the  barrel  and  the  atmospheric  pressure,  acting  as  in  a  non- 
expansive  steam-engine.  This  apparatus  may  be  called  a  "  full-pressure 
steam  gun." 

The  ratio  in  which  the  volume  of  the  barrel  exceeds  that  of  the  bullet 
is  supposed  to  be  fixed  according  to  practical  convenience. 

11.  Calculation  of  Driving  Pressure. — Let  b  denote  the  volume  of  the 
bullet;   B,  that  of  the  space  through  which  the  bullet  sweeps  in  the 


478  ON   THE   EXPLOSIVE   ENERGY   OF   HEATED   LIQUIDS. 

barrel.     Let  ps  be  the  absolute  intensity  of  the  pressure  which  resists 

the  motion  of  the  bullet :  being  that  of  the  atmosphere,  with  an  addition, 

to  be  determined  by  experiment,  for  friction  and  for  the  inertia  of  the 

air  expelled  in  front  of  the  bullet.     Let  p0  be  the  absolute  intensity  of 

the  pressure   of  steam  in  the  barrel.      Then  p.,  —  pz,  is  the  effective 

intensity  of  the  driving  pressure.     The  weight  of  the  bullet  is  wb,  and 

i  •  i  .     .     lobv2   .    .  .         .  ,       .      .  . 

the  energy  impressed  on  it  is  — — ,  being  equal  to  the  work  of  raising 

o 

it  to  the  height  —   due  to  its  velocity  of  discharge     The  energy  exerted 

in  driving  the  bullet  is  that  due  to  the  pressure  whose  effective  intensity 
•s  I' •  ~  !'■:,•  acting  through  a  space  of  the  volume  B;  therefore,  by 
ci  Hutting  these  quantities  of  energy  as  follows, — 

v  I  /■'-' 
(p2-p3)B  =  —  ; 

we  obtain  the  following  formula  for  the  required  effective  intensity  of  the 
driving  pressure  : — 

h       '■- 

that  is  to  say,  the  excess  of  the  absolute  driving  pressure  (j>.,)  above  the  resisting 
presswre  (p3)  is  equivalent  to  the  weight  of  a  column  af  tin  metal  of  which  the 
bullet  is  made,  whose  height  is  less  than  the  height  due  to  the  velocity,  in  the  same 
proportion  in  which  the  volume  of  the  bullet  is  less  than  the  volume  of  the  space 
through  which  the  bullet  siveeps  in  the  barrel. 

For  example,  let  v  =  1G05  ft.  per  second;  let  us  assume  B  =  100  b; 
and  let  the  material  of  the  bullet  be  lead,  so  that  w  =712  lbs.  per  cubic 
foot ;  then  we  have  the  following  results  : — 

v2 
Height  due  to  velocity,  -  -  =  40,000  ft. 

Effective  driving  pressure,^  —p3;  lb.  on  the  square  foot,  284,800 

„                    „                    „          lb.  on  the  square  inch,  1,978 
Absolute  driving  pressure,  if  friction  and  the  inertia  of  the 

air  be  neglected;  lb.  on  the  square  inch,            .          .  1,993 

so  that  in  this  example  Ave  may  conclude  that  the  absolute  intensity  of  the 
driving  pressure  required  would  be  2,000  lbs.  on  the  square  inch,  or 
thereabouts;  or  between  135  and  136  atmospheres.  For  other  proportions 
of  the  volume  of  the  barrel  to  that  of  the  bullet,  the  effective  pressure 
required,  of  course,  varies  inversely  as  the  volume  of  the  barrel. 

1 2.  Calculation  of  Pressure  in  Boiler. — The  pressure  in  the  boiler  must 
be  such  that  a  mass  of  water  escaping  from  the  boiler,  and  expanding  from 


ON   THE  EXPLOSIVE  ENERGY   OF  HEATED  LIQUIDS.  479 

that  pressure  until  the  pressure  of  the  mixed  water  and  steam  falls  to  that 
in  the  barrel,  shall  acquire  a  velocity  equal  to  that  of  the  bullet.  Hence, 
let  U  denote  the  absolute  temperature  corresponding  to  the  absolute 
driving  pressure  p2,  as  found  by  means  of  suitable  formulae  or  tables ; 
tx  =  n  t2,  the  absolute  temperature  of  the  water  just  before  it  escapes  from 
the  boiler,  and  K  (as  before)  the  dynamical  value  of  the  specific  heat  of 
liquid  water;  then  (as  in  equation  (1)  of  section  2),  we  have 

Kt2(n  -  1  -  hyp.  log.  n)=   *-;         .         .     (9.) 

and  this  transcendental  equation  is  to  be  solved  by  approximation,  so  as 
to  find  n,  and  thence  t1  =  n  t2.  When  the  absolute  temperature  in  the 
boiler  has  thus  been  found,  the  corresponding  pressure  px  may  be  calculated 
by  the  help  of  formula?,  or  of  tables. 

In  applying  the  foregoing  rules  to  such  examples  as  that  already  given 
in  section  11,  great  uncertainty  arises  from  a  cause  formerly  referred  to — 
viz.,  that  the  pressures  and  temperatures  lie  far  beyond  the  range  of  the 
experiments  from  which  the  formula?  for  the  pressure  and  temperature  of 
steam  were  deduced. 

By  the  use  of  an  already  known  formula,"'  the  absolute  temperature 
corresponding  to  the  absolute  pressure  of  2,000  lbs.  on  the  square  inch  is 
found  to  be  1104°  Fahr.  =  613°  Cent.;  corresponding  to  6-13°  Fahr.,  or 
339°  Cent,  on  the  ordinary  scales.  The  corresponding  value  of  Kt2  is 
852,000  ft;  and  by  solving  equation  (9)  by  approximation  we  obtain  the 
following  results,  which,  however,  are  to  a  great  extent  conjectural : 
hyp.  log.  n  =  0'292  nearly;  n  =  T339  nearly;  absolute  temperature  in 
boiler,  tx  =  1478°  Fahr.  nearly  =821°  Cent,  nearly  (or,  on  the  ordinary 
scales,  1017°  Fahr.,  or  547°  Cent.)  Absolute  pressure  in  boiler,  about 
11,770  lbs.  on  the  square  inch,  or  about  800  atmospheres. 

13.  Expenditure  of   Water,  and   Heat- Efficiency. — The  weight  of  water 
expended  per  shot,   supposing  that  there  is  no  waste,  is  expressed  by 
p 
-,  in  which  s„  denotes  the  volume  filled  by  each  unit  of  weight  of  the 

mixture  of  water  and  steam  in  the  barrel ;   and  the  ratio  which  that 

weight  bears  to  the  weight  w  b  of  the  bullet  is  given  by  the  following 

formula : — 

1        JB_ B 

m       wbs0  d  /.,  ,  /1AN 

b  w  K  .    --  hvp.  log.  n.     .  .     (10.) 

dP-z 

For  the  reason  already  given  the  value  of    ,   -   is  very  uncertain ;  but, 

dp.2 
as  before,  a  conjectural  value  may  be  computed  by  means  of  the  ordinary 

*  See  Manual  of  the  Steam-Engine  and  other  Prime  Movers,  p.  237. 


4S0  OX   THE   EXPLOSIVE   ENERGY  OF  HEATED   LIQUIDS. 

formula/''      In  the  example  already  given  we  find  the  following  results : — 
5.,  =  0'12  cubic  foot  per  lb.  of  water; 

1  =  117; 

m 

m  =  0'854. 

The  dynamical  equivalent  of  the  expenditure  of  heat  for  each  unit  of 
weight  of  bullet  is  expressed  as  follows  : — 

H  =  K('1"'1>:  .        .        .    (11.) 

m  x 

in  which  /,  is  the  temperature  of  the  feed ;  and  in  the  example  given  the 
value  of  this  quantity  of  heat,  subject  to  the  causes  of  uncertainty  already 
mentioned,  is  found  to  be  about  810,000  foot-pounds  per  pound  weight  of 
bullet.  The  energy  of  each  pound  weight  of  bullet  is  4,000  foot-pounds, 
so  that  the  efficiency  and  counter-efficiency  of  the  gun  are  respectively  as 
follows  : — 

Efficiency,  "049  ;     Counter-efficiency,  20*25. 

14.  FulPpresswre  Dry  Steam  Gun, — If,  instead  of  a  coil  of  tube  entirely 
filled  with  highly -heated  liquid  water,  we  assume  it  to  be  practicable  to 
use  a  boiler  having  sufficient  steam  space  to  enable  the  gun  to  be  supplied 
with  dry  steam,  the  calculation  of  the  driving  pressure  required  in  the 
barrel  is  exactly  the  same  with  that  already  given  in  section  11,  giving 
about  2,000  lbs.  on  the  square  inch  in  the  example  chosen.  The  density 
of  the  steam  in  the  barrel,  the  weight  expended  per  shot,  the  boiler 
pressure,  the  total  expenditure  of  heat  per  shot,  and  the  efficiency, 
may  be  calculated  by  means  of  known  formulae,  subject  to  the  uncertainty 
arising  from  the  pressures  and  temperatures  being  beyond  the  range  of 
previous  experiments.  The  following  are  the  results  in  the  example : — 
s2  =  0*24  cubic  foot  per  lb.,  showing  that  with  the  boiler  quite  full  of 
liquid  water  half  the  fluid  in  the  barrel  is  liquid, 

m—  1*72;  -  =  0*582. 
m 

Boiler  pressure,^,  about  3,600  lbs.  on  the  square  inch;  expenditure  of 
heat  per  shot,  705,000  foot-pounds;  counter-efficiency,  19;  efficiency, 
0*052. 

15.  Remarks. — The  boiler-pressure,  as  well  as  the  driving  pressure 
required  in  order  to  produce  a  given  velocity  in  a  given  bullet,  varies 
inversely  as  the  capacity  of  the  barrel ;  and  hence  it  is  obvious  that  the 
safe  and  effective  working  of  steam  guns  depends  mainly  on  the  practica- 
bility of  making  and  using  very  long  gun-barrels. 

*  See  Manual  of  the  Steam-Engine  and  other  Prime  Movers,  p.  323. 


PART    III. 

PAPERS  RELATING  TO  WAVE-FORMS,  PROPULSION  OF 
VESSELS,  STABILITY  OF  STRUCTURES,  &c. 


PART    III. 

PAPERS    RELATING    TO    WAVE    FORMS,    PROPULSION    OF 
VESSELS,  STABILITY  OF  STRUCTURES,  &c. 


XXIX.— ON  THE  EXACT  FORM  OF  WAVES  NEAR  THE 
SURFACE  OF  DEEP  WATER.* 

1.  The  investigations  of  the  Astronomer-Royal,  and  of  some  other 
mathematicians,  on  straight-crested  parallel  waves  in  a  liquid,  are  based 
on  the  supposition  that  the  displacements  of  the  particles  of  the  liquid 
are  small  compared  with  the  length  of  a  wave.  Hence  it  has  been  very 
generally  inferred  that  the  results  of  those  investigations  are  approximate 
only,  when  applied  to  waves  in  which  the  displacements,  as  compared 
with  the  length  of  a  wave,  are  considerable. 

2.  In  the  present  paper,  I  propose  to  prove  that  one  of  those  results 
(viz.,  that  in  very  deep  water  the  particles  move  with  a  uniform  angular 
velocity  in  vertical  circles  whose  radii  diminish  in  geometrical  progression 
with  increased  depth,  and,  consequently,  that  surfaces  of  equal  pressure, 
including  the  upper  surface,  are  trochoidal)  is  exact  for  all  displacements, 
how  great  soever. 

3.  I  believe  the  trochoidal  form  of  waves  to  have  been  first  explicitly 
stated  by  Mr.  Scott  Russell ;  but  no  demonstration  of  its  exactly  fulfilling 
the  conditions  of  the  question  has  yet  been  published,  so  far  as  I  know. 

4.  In  A  Manual  of  Applied  Mechanics  (first  published  in  1858),  page 
579,  I  stated  that  the  theory  of  rolling  waves  might  be  deduced  from  that 
of  the  positions  assumed  by  the  surface  of  a  mass  of  water  revolving  in  a 
vertical  plane  about  a  horizontal  axis ;  as  the  theory  of  such  waves,  how- 
ever, was  foreign  to  the  subject  of  the  book,  I  did  not  then  publish  the 
investigation  on  which  that  statement  was  founded. 

5.  Having  communicated  some  of  the  leading  principles  of  that  investi- 
gation to  Mr.  William  Froude  in  April,  1862,  I  learned  from  him  that  he 
had  already  arrived  independently  at  similar  results  by  a  similar  process, 
although  he  had  not  published  them. 

*  Read  before  the  Eoyal  Society  of  London  on  November  27,  1S62,  and  published  in 
the  Philosophical  Transactions  for  1S63. 

2H 


482 


THE  EXACT  FORM  OF  WAVES. 


6.  Proposition  I. — In  a  mass  of  gravitating  liquid,  whose  particles  revolve 
uniformly  in  vertical  circles,  a  wavy  surface  of  trochoidal  profile  fulfils  the  con- 
ditions of  uniformity  of  pressure — such  trochoidal  profile  being  generated  by 
rolling,  on  tlie  underside  of  a  straight  line,  a  circle  tohose  radius  is  equal  to  the 
height  of  a  conical  pendulum  that  revolves  in  the  same  period  with  the  piarticles 
of  liquid. 

In  Fig.  1  let  B  he  a  particle  of  liquid  revolving  uniformly  in  a 
vertical  circle  of  the  radius  CB,  in  the  direction  indicated  hy  the  arrow 


Fie.  l. 


X ;  and  let  it  make  n  revolutions  in  a  second.     Then  the  centrifugal  force 
of  B  (taking  its  mass  as  unity)  will  he  4  it2  n2  .  C  B. 

Draw  CA  vertically  upwards,  and  of  such  a  length  that  centrifugal 
force  :  gravity  : :  C  B  :  AC;  that  is  to  say,  make 


AC  = 


9 


4  77-  »2 


which  is  the  well-known  expression  for  the  height  of  a  revolving  pendulum 
making  n  revolutions  in  a  second. 

Then  A  C  heing  in  the  direction  of  and  proportional  to  gravity,  and  C  B 
in  the  direction  of  and  proportional  to  centrifugal  force,  A  B  will  be  in  the 
direction  of  and  proportional  to  the  resultant  of  gravity  and  centrifugal 
force;  and  the  surface  of  equal  pressure  traversing  B  will  he  normal 
to  AB. 

The  profile  of  such  a  surface  is  obviously  a  trochoid  L  B  M,  traced  by 


THE  EXACT  FORM  OF  WAVES. 


483 


the  point  B,  which  is  carried  by  a  circle  of  the  radius  C  A  rolling  along 
the  underside  of  the  horizontal  straight  line  H  A  K.     Q.E.D. 

7.  Corollaries. — The  length  of  the  wave  whose  period  is  one-wth  of  a 
second,  is  equal  to  the  circumference  of  the  rolling  circle ;  that  is  to  say 
(denoting  that  length  by  X), 

\  =  2tt.  CA 


2tt?i2 


the  period  of  a  wave  of  a  given  length  X  is  given  in  seconds,  or  fractions 
of  a  second,  by  the  equation 

-  =      l2irX 
n        *     9     ' 

and  the  velocity  of  propagation  of  such  wave  is 

2  irn        \2  tt  j  ' 

results  agreeing  with  those  of  the  known  theory. 

8.  Proposition  II. — Let  another  surface  of  uniform  pressure  he  conceived 
to  exist  indefinitely  near  to  the  first  surface;  then,  if  the  first  surface  is  a  surface 
of  continuity,  so  also  is  the  second. 

By  a  surface  of  continuity  is  here  meant  one  which  always  passes  through 
the  same  set  of  particles  of  liquid,  so  that  a  pair  of  such  surfaces  contain 
between  them  a  layer  of  particles  which  are  always  the  same. 

The  perpendicular  distance  between  a  pair  of  surfaces  of  uniform  pres- 
sure is  in  this  case  inversely  proportional  to  the  resultant  of  gravity  and 
centrifugal  force ;  that  is  to  say,  to  the  normal  A  B.  Hence,  if  a  curve 
I  hfra  be  drawn  indefinitely  near  to  the  curve  L  B  M,  so  that  the  perpen- 
dicular distance  between  them,  B/,  shall  everywhere  be  inversely  propor- 
tional to  the  normal  A  B,  the  second  curve  will  also  be  the  profile  of  a 
surface  of  uniform  pressure. 

Conceive  now  that  the  whole  mass  of  liquid  has,  combined  with  its 
wave-motion,  a  uniform  motion  of  translation,  with  a  velocity  equal  and 


m>^— >- 


Fig.  2. 


opposite  to  that  of  the  propagation  of  the  waves.     The  dynamical  con- 
ditions of  the  mass  are  not  in  the  least  altered  by  this ;  but  the  forms  of 


484  THE   EXACT  FORM  OF   WAVES. 

the  waves  are  rendered  stationary  (as  we  sometimes  see  in  a  rapid  stream), 
and,  instead  of  a  series  of  waves  propagated  in  the  direction  shown  by  the 
arrow  P,  we  have  an  undulating  current  running  the  reverse  way,  in  the 
direction  shown  by  the  arrow  Q.  (This  is  further  illustrated  by  Fig.  2.) 
According  to  a  well-known  property  of  curves  described  by  rolling,  the 
velocity  of  the  particle  B  in  that  current  is  proportional  to  the  normal 
A  I  J,  and  is  given  by  the  expression  2  ir  n  .  AB. 

Consider  the  layer  of  the  current  contained  between  the  surfaces  LBM 
and  /  b  m.  In  order  that  the  latter  of  those  surfaces,  as  well  as  the  former, 
may  be  a  surface  of  continuity,  it  is  necessary  and  sufficient  that  the 
thickness  of  the  layer  B/  at  each  point  should  be  inversely  as  the  velocity ; 
and  that  condition  is  already  fulfilled ;  for  B/  varies  inversely  as  A  B,  and 
A  I ;  varies  as  the  velocity  of  the  current  at  B ;  therefore,  LBM  and  I  bin  are 
not  only  a  pair  of  surfaces  of  uniform  pressure,  but  a  pair  of  surfaces  of 
continuity  also.     Q.E.D. 

9.  Corollary. — The  surfaces  of  uniform  pressure  are  identical  with  sur- 
faces of  continuity  throughout  the  whole  mass  of  liquid. 

10.  Corollary. — Inasmuch  as  the  resultant  of  gravity  and  centrifugal 
force  at  B  is  represented  by 

AB 

the  excess  of  the  uniform  pressure  at  the  surface  Ibm  above  that  at  the 
surface  LBM  is  given  by  the  expression 

.  AB         -=r- 

il  i)  =  w  .  =  .  By, 
1  AC         J 

in  which  w  is  the  heaviness  of  the  liquid,  in  units  of  weight  per  unit  of 
volume.  By  omitting  the  factor  w,  the  pressure  is  expressed  in  units  of 
height  of  a  column  of  the  liquid. 

11.  Proposition  III. — The  profile  of  the  lower  surface  of  the  layer  referred 
to  in  the  preceding  proposition  is  a  trochoid  generated  by  a  rolling  circle  of  the 
same  radius  with  that  which  generates  the  first  trochoid;  and  the  tracing-arm  of 
the  second  trochoid  is  shorter  than  that  of  the  first  trochoid  by  a  quantity  bearing 
the  same  proportion  to  the  depth  of  the  centre  of  the  second  rolling  circle  below 
the  centre  of  the  first  rolling  circle,  which  the  tracing-arm  of  the  first  rolling 
circle  bears  to  the  radius  of  that  circle. 

At  an  indefinitely  small  depth  A  a  below  the  horizontal  line  H  A  K,  draw 
a  second  horizontal  line  h  a  Tc,  on  the  underside  of  which  let  a  circle  roll 
with  a  radius  c  a  =  C  A,  the  radius  of  the  first  rolling  circle  ;  so  that  the 
indefinitely  small  depths    C  c  =  A  a.      To  find    the   tracing-arm    of   the 


THE  EXACT  FORM  OF  WAVES.  485 

second  rolling  circle,  draw  eel  parallel  to  CB,  the  tracing-arm  of  the 
first  circle ;  in  cd  take  c7  =  CB,  and  cut  off  e b  =  e d ;  b  will  be 
the  tracing-point,  and  c  b  the  tracing-arm  required ;  for,  according  to  the 
principle  laid  down  in  the  enunciation,  Ave  are  to  have 

_      _  CB 

CB  —  cb  =  eb  =  Cc  .  77-7- 
C  A 

Let  the  second  circle  roll  that  b  will  trace  a  trochoid  /  b  m.  From  b  let 
fall  bf  perpendicular  to  AB  produced ;  B/  will  be  the  indefinitely  small 
thickness  at  B  of  the  layer  between  the  two  trochoidal  surfaces. 

The  proposition  enunciated  amounts  to  stating  that  B/  is  everywhere 
inversely  proportional  to  the  normal  A  B ;  so  that  I  b  m  is  the  profile  of  a 
surface  of  uniform  pressure  and  of  continuity. 

To  prove  this,  join  B~e  and  ef.  Then  B e  is  parallel  toACr,  and  equal 
to  Cc,  and  def  is  evidently  an  isosceles  triangle,  ef  being  =  ed.  Let 
A  B  (produced  if  necessary)  cut  the  circle  of  the  radius  C  B  in  G ;  then 
C  G  is  parallel  to  ef,  and  the  indefinitely  small  triangle  Bef  is  similar  to 
the  triangle  A C G ;  consequently,  AC  :  A G  :  :  B e  =  Cc  :  B/;  or 

B/-Gc-AC; 
but,  by  a  well-known  property  of  the  circle, 


— -       AC2-CB2 

AG == ; 

AB 

and,  therefore, 

AC2  -  CB2 


B/  =  Cc 


AC- AB 


that  is  to  say,  the  thickness  of  the  layer  varies  inversely  as  the  normal  A  B ; 
and  the  second  trochoid,  I  b  m,  is  therefore  the  profile  of  a  surface  of  uniform 
presswe  and  of  continuity.     Q.E.D. 

12.  Corollaries. — The  profiles  of  the  surfaces  of  uniform  pressure  and  of 
continuity  form  an  indefinite  series  of  trochoids,  described  by  equal  rolling 
circles,  rolling  with  the  same  speed  below  an  indefinite  series  of  horizontal 
straight  lines. 

The  tracing-arms  of  those  circles  (each  of  which  arms  is  the  radius  of 
the  circular  orbit  of  the  particles  contained  in  the  trochoidal  surface  which 
it  traces)  diminish  in  geometrical  progression  with  increase  of  depth, 
according  to  the  following  laws  : — 


486  THE  EXACT  FORM  OF  WAVES. 

For  convenience,  let  C  c  be  denoted  by  d  k,  C  B  by  r,  and  c  b  by  r  —  d  r; 
then, 

dr  —  dh  .  -r-pz  =  dk  . — , 

AC  2  /r X 

and  the  integration  of  this  equation  gives  the  following  result : — 

Let  k  denote  the  vertical  depth  of  the  centre  of  the  generating  circle  of 
a  given  surface  below  the  centre  of  the  generating  circle  of  the  free  upper 
surface  of  the  liquid  ; 

r0  the  tracing-arm  of  the  free  upper  surface  ( =  half  the  amplitude  of 
disturbance) ; 

j'j  the  tracing-arm  of  the  surface  whose  middle  depth  is  k;  then, 

h  _  2  *  J: 

a  formula  exactly  agreeing  with  that  found  for  indefinitely  small  disturb- 
ances by  previous  investigators. 

1 3.  Proposition  IV. — The  centres  of  the  orbits  of  the  particles  in  a  given 
surface  of  equal  pressure  stand  at  a  higher  level  than  the  same  particles  do  when 
tin  liquid  is  still,  by  a  height  which  is  a  third  proportional  to  the  diameter  of  the 
rolling  circle  and  the  traemg-arm  or  radius  of  the  orbits  of  the  particles,  and 
which  is  equal  to  the  height  due  to  the  velocity  of  revolution  of  the  particles. 

If  the  liquid  were  still,  the  given  surface  of  equal  pressure  would  become 
horizontal.  To  find  the  level  at  which  it  would  stand,  we  must  first  find 
what  relation  the  mean  vertical  depth  of  a  given  layer  of  particles  bears  to 
the  depth  C  c  =  d  k  between  the  centres  of  the  rolling  circles  that  generate 
its  boundaries. 

The  length  of  the  arc  of  the  curve  LBM  described  in  an  indefinitely 
short  interval  of  time  d  t  is 


2Trn.AB.dt, 

and  the  thickness  of  the  layer  being 


=-,       ..     AC'-C  B2 

ijf  =  dk.  — — =^ , 

J  AC.AB 

let  the  product  of  those  quantities  be  divided  by  the  distance  through 
which  the  centre  of  the  rolling  circle  moves  in  the  same  time,  viz. — 


2  it  n.  AC  .dt, 

and  the  result  will  be  the  mean  vertical  depth  of  the  layer,  which  being 
denoted  by  d  k0,  we  have 


THE  EXACT  FORM  OF  WAVES.  487 

«>="-o-g)="-(i-ry-"-  0-*-% 

The  difference  by  which  the  mean  vertical  thickness  of  the  layer  falls 
short  of  the  difference  of  level  of  the  rolling  circles  of  its  upper  and  lower 
surfaces,  is  given  by  the  following  expression, 

2  k 

0       AC2 

and  this  being  integrated  from  oo  to  Jc,  gives  the  depth  of  the  position  of 
a  given  particle,  when  the  liquid  is  still, .  below  the  level  of  the  centre  of 
the  orbit  of  the  same  particle  when  disturbed,  viz. — 


2  A 


AC  _       9"       _  *r 


2AC  2AC         A 

or,  a  third  proportional  to  the  diameter  of  the  rolling  circle  and  the  radius  of  the 
orhit  of  the  particle  ;  also 

r2       _  4  7r2w2r2 
2AC  _       2flr 

is  the  height  due  to  the  velocity  of  revolution  of  the  particles.     Q.E.D. 

1  3a.  Corollary. — The  mechanical  energy  of  a  wave  is  half  actual  and 
half  potential, — half  being  due  to  motion,  and  half  to  elevation.  In  other 
words,  the  mechanical  energy  of  a  wave  is  double  of  that  due  to  the  motion 
of  its  particles  only,  there  being  an  equal  amount  due  to  the  mean  eleva- 
tion of  the  particles  above  their  position  when  the  water  is  still. 

14.  Corollary. — The  crests  of  the  waves  rise  higher  above  the  level  of 
still  water  than  their  hollows  fall  below  it ;  and  the  difference  between  the 
elevation  of  the  crest  and  the  depression  of  the  hollow  is  double  of  the 
quantity  mentioned  in  Proposition  IV.,  that  is  to  say,  it  is 

r2        _    27TT2 

AC  ~      A 

15.  Corollary  as  to  Pressures. — An  expression  has  already  been  given  in 
Art.  10  for  the  difference  of  pressure  at  the  upper  and  under  surfaces  of  a 
given  layer.  Substituting  in  that  expression  the  value  of  the  thickness  of 
the  layer,  we  find 


AB  AC2~CB2  /        CF\ 

dp  =  w.Tf].dk.  -TKTab    =  w  *  dkV  ~  ACU  =  W  '  dk° 


488  THE  EXACT  FOEM  OF  WAVES. 

(as  the  preceding  corollary  shows),  being  precisely  the  same  as  if  the 
liquid  were  still ;  and  hence  it  follows  that  the  hydrostatic  pressure  at  each 
individual  particle  during  wave-motion  is  the  same  as  if  the  liquid  were  still. 

1G.  In  Proposition  III.  it  has  been  shown,  by  geometrical  reasoning 
from  the  mechanical  construction  of  the  trochoid,  that  a  wave  consisting  of 
trochoidal  layers  satisfies  the  condition  of  continuity.  It  may  be  satisfac- 
tory also  to  show  the  same  thing  by  the  use  of  algebraic  symbols.  For 
that  purpose  the  following  notation  will  be  used  : — 

Let  the  origin  of  co-ordinates  be  assumed  to  be  in  the  horizontal  lino 
containing  the  centre  of  the  circle  which  is  rolled  to  trace  the  profile  of 
cycloidal  waves,  having  cusps,  and  being  (as  Mr.  Scott  Russell  long  ago 
pointed  out)  the  highest  waves  that  can  exist  without  breaking.  In  such 
waves,  the  tracing-arm,  or  radius  vector,  of  the  uppermost  particles  is  equal 
to  the  radius  of  the  rolling  circle ;  and  that  arm  diminishes  for  each  suc- 
cessive layer  proceeding  downwards. 

Let  x  and  y  be  the  co-ordinates  of  any  particle,  x  being  measured  hori- 
zontally against  the  direction  of  propagation,  and  y  vertically  downwards. 

Let  h  (as  before)  be  the  vertical  co-ordinate  of  the  centre  of  the  given 
particle's  orbit ;  h  the  horizontal  co-ordinate  of  the  same  centre. 

Let  R  be  the  radius  of  the  rolling  circle,  a  the  angular  velocity  of  the 
tracing-arm  ( =  2  w  n),  so  that 

2  ir  R  =  A 

is  the  length  of  a  wave,  and 

a  R  =  n\ 

is  the  velocity  of  propagation. 

Let  0  denote  the  phase  of  the  wave  at  a  given  particle,  being  the  angle 
which  its  radius  vector,  or  tracing-arm,  makes  with  the  direction  of  +  y, 
that  is,  with  a  line  pointing  vertically  downwards. 

Let  i  denote  time,  reckoned  from  the  instant  at  which  all  the  particles 
for  which  h  =  0  are  in  the  axis  of  y;  then 

0  =  at  +  ± (1.) 

Then  the  following  equations  give  the  co-ordinates  of  a  given  particle  at 
a  given  instant : 

_k 
x  =  h  +  R  e      K  sin  0;       .         .         .     (2.) 

_h 

y  =  Jc+Re      R  cos  &        .         .         .     (3.) 


THE  EXACT  FORM  OF  WAVES. 


489 


Let  u  and  v  denote  the  vertical  and  horizontal  components  of  the  velocity 
of  the  particle  at  the  given  instant ;  then 


cos0=       a(y-h);     .     (4.) 

sin  6  =  —  a  (.'•  —  //).      .     (5.) 
The  well-known  equation  of  continuity  in  a  liquid  in  two  dimensions  is 

•     (6.) 


dx 
u  =  Tt  = 

a  R  .  e 

k 
R 

d  y 
V  =  dl  = 

—  a  R  .  e 

k 
R 

d  u      d  v 

d  x      dy 


and  from  equations  (4)  and  (5)  it  appears  that  we  have  in  the  present 
case, 

du  ,djo_      (     dk      d_h\  _     /_  d_k      R  d  Q\  . 

dx      dy~       \     dx      d  y)           \     dx         dy  .)'  ^  '' 

In  the  original  formulas,  k  and  B  are  the  independent  variables.  When 

x  and  y  are  made  the  independent  variables  instead,  we  have,  by  well- 
known  formula?, 


d  k 


j  d x       d x    dk\ 
1  tk  ~  d~0  '  dy  [ 

dOJ 


sin  0 


and 


I 


r 


dx) 


rf  y  ;  d  0      d  &  '  d  x  f 


2/fc 

1 

—  e 
k 

R 

e 

"R 

sin  6 

dkj       R\l-c 


(8) 


so  that  the  equation  of  continuity  (6)  is  exactly  verified. 

17.  Another  mode  of  testing  algebraically  the  fulfilment  of  the  con- 
dition of  continuity  is  the  following.  It  is  analogous  to  that  employed  by 
Mr.  Airy;  but  inasmuch  as  the  disturbances  in  the  present  paper  are 
regarded  as  considerable  compared  with  the  length  of  a  wave,  it  takes 
into  account  quantities  which,  in  Mr.  Airy's  investigation,  are  treated  as 
inappreciable. 

Consider  an  indefinitely  small  rhomboidal  particle,  bounded  by  surfaces 


490  THE  EXACT  FORM  OF  WAVES. 

for  which  the  values  of  h  and  h  are  respectively  h,  h  +  d  h,  l;  h  +  d  k. 
Then  the  area  of  that  rhomboid  is 


(d  x    d  y       d  x    d  y\ 
Th'Tk~  d~k'tiJdh-cUc; 


and  the  condition  of  continuity  is  that  this  area  shall  be  at  all  times  the 
3ame  ;  that  is  to  say,  that 


d_(  dx    dj_      c l  ..•    d_y\  _ 

ItKdh'dk      dh'dh)        "        '        '     l   ; 


d 

Upon  performing  the  operations  here  indicated  upon  the  values  of  the 
co-ordinates  in  equations  (2)  and  (3),  the  value  of  the  quantity  in  brackets 
is  found  to  be 

_  lh 

1  -e    '   *;      .          .         .         .     (10.) 

which  is  obviously  independent  of  the  time,  and  therefore  fulfils  the  con- 
dition of  continuity. 


A  P  P  E  X  D  I  X. 

On  the  Friction  between  a  Wave  and  a  Wave-shaped  Solid. 

Conceive  that  the  trough  between  two  consecutive  crests  of  the 
trochoidal  surface  of  a  series  of  waves  is  occupied,  for  a  breadth  which 
may  be  denoted  by  z,  by  a  solid  body  with  a  trochoidal  surface,  exactly 
fitting  the  wave-surface ;  that  the  solid  body  moves  forward  with  a  uniform 
velocity  equal  to  that  of  the  propagation  of  the  waves,  so  as  to  continue 
always  to  fit  the  wave-surface;  and  that  there  is  friction  between  the  solid 
surface  and  the  contiguous  liquid  particles,  according  to  the  law  which 
experiment  has  shown  to  be  at  least  approximately  true — viz.,  varying  as 
the  surface  of  contact,  and  as  the  square  of  the  velocity  of  sliding. 

Conceive,  further,  that  each  particle  of  the  liquid  has  that  pressure 
applied  to  it  which  is  required  in  order  to  keep  its  motion  sensibly  the 
same  as  if  there  were  no  friction;  the  solid  body  must  of  course  be  urged 
forwards  by  a  pressure  equal  and  opposite  to  the  resultant  of  all  the 
before-mentioned  pressures. 

The  action,  amongst  the  liquid  Jparticles,  of  pressures  sufficient  to  over- 
come the  friction,  will  disturb  to  a  certain  extent  the  motions  of  the 
liquid  particles,  and  the  figures  of  the  surfaces  of  uniform  pressure;  but  it 
will  be  assumed  that  those  disturbances  are  small  enough  to  be  neglected, 


THE  EXACT  FORM  OF   WAVE£.  491 

for  the  purposes  of  the  present  inquiry.  The  smallness  of  the  pressures 
producing  such  disturbances,  and  consequently  the  smallness  of  those 
disturbances  themselves,  may  be  inferred  from  the  fact,  that  the  friction 
of  a  current  of  water  over  a  surface  of  painted  iron  of  a  given  area  is  equal 
to  the  weight  of  a  layer  of  water  covering  the  same  area,  and  of  a  thick- 
ness which  is  only  about  '0036  of  the  height  due  to  the  velocity  of  the 
current. 

Those  conditions  having  been  assumed,  let  it  now  be  proposed,  to  find 
approximately  the  amount  of  resultant  pressure  required  to  overcome  the  friction 
between  the  wave  and  the  wave-shaped  solid. 

This  problem  is  to  be  solved  by  finding  the  mechanical  work  expended 
in  overcoming  friction  in  an  indefinitely  small  time  d  t,  and  dividing  that 
work  by  the  distance  through  which  the  solid  moves  in  that  time. 

Taking,  as  before,  as  an  independent  variable  the  phase  6,  being  the 
angle  which  the  tracing-arm  CB  =  r  (Fig.  1)  makes  with  a  line  pointing 
vertically  downwards,  the  length  of  the  elementary  arc  corresponding  to 
an  indefinitely  small  increment  of  phase  d  6  is 

id  6, 

where  q  is  taken,  for  brevity's  sake,  to  denote  the  normal  A  B. 
The  area  of  the  corresponding  element  of  the  solid  surface  is 

sqdd. 

The  velocity   of  sliding  of  the  liquid  particles  over  that  elementary  ' 
surface  is 

a  q, 

d  0 
in  which  a,  as  before,  denotes  r— -,  the  angular  velocity  of  the  tracing-arm. 

Hence,  let  p  denote  the  heaviness  (or  weight  of  unity  of  volume)  of  the 
liquid,  and  /  its  coefficient  of  friction  when  sliding  over  the  given  solid 
surface ;  the  intensity  of  the  friction  per  unit  of  area  is 

p  ^  r 

That  friction  has  to  be  overcome,  during  the  time  d  t,  through  the  distance 

aqdt  =  qdO. 

Multiplying  now  together  the  elementary  area,  the  intensity  of  the 
friction,  and  the  distance  through  which  it  is  overcome  in  the  time  d  t, 
we  find  the  following  value  for  the  work  performed  in  that  time  in 
overcoming  the  friction  at  the  given  elementary  surface, 


402  THE  EXACT   FORM  OF   WAVES. 

•2  (j  z  2g 

Now,  during  the  time  d  t,  the  solid  advances  through  the  distance 

a  R  d  t  =  R  3  0 

(E,  as  before,  being  the  radius  of  the  rolling  circle);  and  dividing  the 
elementary  portion  of  work  expressed  above  by  that  distance,  we  find  the 
following  value  for  an  elementary  portion  of  the  pressure  required  to 
overcome  the  friction, 

dP=f£*.£i.de.     .     .     .  (i.) 

2  (j       R 

The  total  pressure  required  to  overcome  the  friction  is  found  by 
integrating  the  preceding  expression  throughout  an  entire  revolution ; 
that  is  to  say, 

p  =  fp«2:  r  „. 


2  g  R 


f  <fde.      .      .     .  (2.) 

J  0 


To  obtain  this  integral  the  following  value  of  the  square  of  the  normal 
q  or  A  B  is  to  be  substituted, 

f  =  B?  +  r2  +  2  Rr.  cos  ft 

whence, 

riT  r2*  /         2r-        r4  (  r'2\r 

/„  q',,e  =  R'/o  0  +  w  +  i?  +  H1  +  v)k ■ cos- e 


and 


=  ^f^-(-^-  •     •  « 


*9 

The  following  modification  of  this  expression  is  sometimes  convenient: — 

Let  V  =  a  E  denote  the  velocity  of  advance  of  the  solid; 
X  =  2  7T  E,  as  before,  its  length,  being  the  length  of  a  wave; 

T 

Sin  j3  —  ^  the  sine  of  the  greatest  angle  made  by  a  tangent  to  the 
xv 

trochoidal  surface  with  the  direction  of  advance ;  then 


THE  EXACT  FORM  OF  WAVES.  493 

P  =  fjf^  .  \z  (1  +  4  sin2  /3  +  sin4  /3)  *   .          .     (4.) 
2<7 

It  is  to  be  observed  that  the  resistance  P,  as  determined  by  the 
preceding  investigation,  being  deduced  from  the  amount  of  work  per- 
formed against  friction,  includes  not  only  the  longitudinal  components  of 
the  direct  action  of  friction  on  each  element  of  the  surface  of  the  solid, 
but  the  longitudinal  components  of  the  excess  of  the  hydrostatic  pressure 
against  the  front  of  the  solid  above  that  against  its  rear,  which  is  the 
indirect  effect  of  friction.  The  only  quantities  neglected  are  those  arising 
from  the  disturbances  of  the  figures  of  the  surfaces  of  equal  pressure, 
which  quantities  are  assumed  to  be  unimportant,  for  reasons  already 
stated.  The  consideration  of  such  quantities  would  introduce  terms  into 
the  resistance  varying  as  the  fourth  and  higher  powers  of  the  velocity. 


Note.— Added  October,  18G2. 

The  investigation  of  Mr.  Stokes  (Camb.  Trans.,  Vol.  VIII.)  proceeds  to 
the  second  degree  of  approximation  in  shallow  water,  and  to  the  third 
degree  in  water  indefinitely  deep.  In  the  latter  case  he  arrives  at  the 
result,  that  the  crests  of  the  waves  rise  higher  above  the  level  of  still 
water  than  the  troughs  sink  below  that  level,  by  a  height  agreeing  with 
that  stated  in  Art.  14  of  this  paper,  and  that  the  profile  of  the  waves  is 
approximately  trochoidal. 

Mr.  Stokes  also  arrives  at  the  conclusion,  that,  when  the  disturbance 
is  considerable  compared  Avith  the  length  of  a  wave,  there  is  combined 
with  the  orbital  motion  of  each  particle  a  translation  which  diminishes 
rapidly  as  the  depth  increases.  No  such  translation  has  been  found 
amongst  the  results  of  the  investigation  in  the  present  paper;  and  hence. 


*  This  formula  (neglecting  sin4  /3  as  unimportant  in  practice)  has  been  used  to 
calculate  approximately  the  resistance  of  steam-vessels,  and  its  results  have  been  found 
to  agree  very  closely  with  those  of  experiment,  and  have  also  been  used  since  1858  by 
Mr.  James  R.  Napier  and  the  author  with  complete  success  in  practice,  to  calculate 
beforehand  the  engine-power  required  to  propel  proposed  vessels  at  given  speeds. 
The  formula  has  been  found  to  answer  approximately,  even  when  the  lines  of  the  vessel 
are  not  trochoidal,  by  putting  for  fi  the  mean  of  the  values  of  the  greatest  angle  of 
obliquity  for  a  series  of  water-lines.  The  method  of  using  the  formula  in  practice, 
and  a  table  showing  comparisons  of  its  results  with  those  of  experiment,  were 
communicated  to  the  British  Association  in  1861,  and  printed  in  the  Civil  Engineer  and 
Architect's  Journal  for  October  of  that  year,  and  in  part  also  in  the  Mechanics' 
Magazine,  The  Artisan,  and  The  Engineer.  The  ordinary  value  of  the  coefficient  of 
friction/  appears  to  be  about  '0036  for  water  gliding  over  painted  iron.  The  quantity 
Xc(l  +  4sin2/3  +  sin4 /J)  corresponds  to  what  is  called,  in  the  paper  referred  to,  the 
augmented  surface. 


494  THE  EXACT  FORM  OF  WAVES. 

it  would  appear  that  Mr.  Stokes's  results  and  mine  represent  two  different 
possible  modes  of  wave-motion.* 

The  simplicity  with  which  an  exact  result  is  obtained  in  the  present 
paper,  is  entirely  due  to  the  following  peculiarity : — Instead  of  taking  for 
independent  variables  (besides  the  time)  the  undistwrbed  co-ordinates  of  a 
particle  of  liquid,  there  are  taken  two  quantities,  h  and  /.',  which  are 
functions  of  those  co-ordinates,  of  forms  which  are  left  indeterminate  until 
the  end  of  the  investigation,  h  then  proves  to  be  identical  with  the 
undisturbed  horizontal  co-ordinate;  but/.;  proves  to  be  a  function  of  the 
undisturbed  vertical  co-ordinate,  for  which  there  is  no  symbol  in  our 
present  notation,  being  the  root  of  the  transcendental  equation 


2  I: 
R    . 


/•    _  /• L'L      e        «   —  0 

0  2  Ii  ~ 

in  which  I:0  is  the  undisturbed  vertical  co-ordinate  (see  Art.  13).  Hence 
it  is  evident  that,  had  J:Q  instead  of  k  been  taken  as  the  independent 
variable,  the  rpuestion  of  wave-motion  considered  in  this  paper  could 
not  have  been  solved  except  by  a  complex  and  tedious  process  of  approxi- 
mation. 

*  Note,  added  June,  1863.— The  difference  between  the  cases  considered  by  Mr. 
Stokes  and  by  me  is  the  following:— In  Mr.  Stokes's  investigation,  the  molecular 
rotation  is  null ;  that  is  to  say 

\d  x     <l  y) 
while  in  my  investigation  it  is  constant  in  each  layer,  being  the  following  function  of  /.-, 

_   2* 

R 


xl<lv     du\  _      a  r'r,  e  . 

*\dx~dy)~  _  .-•     l"-J 

i  >  ■      .."  .      it 


lt- 
From  this  last  equation  it  follows  that 

d  Id  v  _  d  u\  _  _  _ 

d  t  \d  X      d  if) 

and  therefore  that  the  condition  of  continuity  of  pressure  is  verified. 


ON  PLANE  WATER-LINES.  495 


XXX.— ON  PLANE  WATER-LINES  IN  TWO  DIMENSIONS. 
(See  Plates  III.  &  IV.)* 

Section  I. — Introduction,  and  Summary  of  Known  Principles. 

1.  Plane  Water-Lines  in  two  Dimensions  defined. — By  the  term  "Plane 
Water-Line  in  two  Dimensions  "  is  meant  a  curve  which  a  particle  of 
liquid  describes  in  flowing  past  a  solid  body,  when  such  flow  takes  place 
in  plane  layers  of  uniform  thickness.  Such  curves  are  suitable  in  practice 
for  the  water-lines  of  a  ship,  in  those  cases  in  which  the  vertical  displace- 
ments of  the  particles  of  water  are  small  compared  with  the  dimensions  of 
the  ship;  for  in  such  cases  the  assumption  that  the  flow  takes  place  in 
plane  layers  of  uniform  thickness,  though  not  absolutely  true,  is  sufficiently 
near  the  truth  for  practical  purposes,  so  far  as  the  determination  of  good 
forms  of  water-line  is  concerned.  As  water-line  curves  have  at  present  no 
single  word  to  designate  them  in  mathematical  language,  it  is  proposed,  as 
a  convenient  and  significant  term,  to  call  them  Neoids  (from  vijoe,  the 
Ionic  genitive  of  vavq). 

2.  General  Principles  of  the  Flow  of  a  Liquid  past  a  Solid. — The  most 
complete  exposition  yet  published,  so  far  as  I  know,  of  the  principles  of 
the  flow  of  a  liquid  past  a  solid,  is  contained  in  Professor  Stokes's  paper 
"  On  the  Steady  Motion  of  an  Incompressible  Fluid,"  published  in  the 
Transactions  of  the  Cambridge  Philosophical  Society  for  1842.  So  far  as 
those  principles  will  be  referred  to  in  the  present  paper,  they  may  be 
summed  up  as  follows  : — 

When  a  liquid  mass  of  indefinite  extent  flows  past  a  solid  body  in  such 
a  manner  that  as  the  distance  from  the  solid  body  in  any  direction 
increases  without  limit,  the  motion  of  the  liquid  particles  approaches 
continually  to  uniformity  in  velocity  and  direction,  the  condition  of  per- 
fect fluidity  requires  that  the  three  components  u,  v,  w  of  the  velocity  of  a 
liquid  particle  should  be  the  three  differential  coefficients  of  one  function 
of  the  co-ordinates  (<p) ;  viz. — 

d(h  dch  d  6  ,„  , 

d  z'         'dy        ,  .  d  z  v    ' 

*  Read  before  the  Royal  Society  of  London  on  November  26,  1S63,  and  published  in 
the  Philosophical  Transactions  for  1S64. 


dV 
dx 

d  (/> 
d  x 

+ 

dV 
dy  ' 

d  <i> 
dy 

+ 

dV 

d  Z 

49 G  ON   PLANE  WATER-LINES. 

and  the  condition  of  constant  density  requires  that  the  said  function  should 
fulfil  the  following  condition, 

d2<J>       d~<j>       cP<f>  . 

By  giving  to  the  function  <-/>  a  series  of  different  constant  values,  a  series 
of  surfaces  are  represented,  to  which  each  water-line  curve  is  an  orthogonal 
trajectory,  so  that  if  U  =  constant  be  the  equation  of  a  series  of  surfaces 
each  containing  a  continuous  series  of  water-line  curves  (and  one  of  which 
surfaces  must  be  that  of  the  solid  body),  the  function  U  must  satisfy  the 
following  condition, 

£  =  0;.        ■     (3.) 

a  z 

or  if  d  s  be  an  elementary  arc  of  a  water-line  curve,  and  /,  >/,  z  its  co-ordi- 
nates, the  following  conditions  must  be  satisfied. 

dx     d  y'     d  '.'      d  <j)    d  <J>    d  (J> 
d  s ' '  d  s' '  d  s'       dx'  dy     d  v 

and  these  are  the  most  general  expressions  of  the  geometrical  properties  of 
water-line  curves  in  three  dimensions. 

"When  the  inquiry  is  restricted  to  motion  in  two  dimensions  only,  x  and 
y,  the  terms  containing  d  :  and  d  :'  disappear  from  the  preceding  equations; 
and  it  also  becomes  possible  to  express  the  same  conditions  by  means  of 
equations  of  a  kind  which  are  more  convenient  for  the  purposes  of  the 
present  investigation,  and  which  are  as  follows:  Conceive  the  plane  layer 
of  liquid  under  consideration  of  thickness  unity,  to  be  divided  into  a 
series  of  elementary  streams  by  a  series  of  water-line  curves,  one  of  which 
must  be  the  outline  of  the  solid  body ;  let  U  =  constant  be  the  equation 
of  any  one  of  those  curves,  U  being  a  function  of  such  a  nature  that  d  U 
is  the  volume  of  liquid  which  flows  in  a  second  along  a  given  elementary 
stream;  then  the  components  of  the  velocity  of  a  particle  of  liquid  are 

dV  dV  (K  N 

u  =  -=—;  v  — r—\     .  .  .     (o.) 

dy  d  x 

the  condition  of  continuity  is  satisfied;  and  the  condition  of  perfect 
fluidity  requires  that  the  function  U  should  fulfil  the  following  equation, 

d  z2         d  //- 
(When  the  motion  of  the  liquid  is  not  subject  to  the  condition  of  being 


ON  PLANE  WATER-LINES.  497 

uniform  in  velocity  and  direction  at  an  infinite  distance  in  every  direction 
from  the  solid,  it  is  sufficient  that 

,   0   H — =— =-  =  function  ot  U ; 

or        dy- 

"but  cases  of  that  kind  do  not  occur  in  the  present  paper.)  * 

3.  Notation. — It  is  purely  a  question  of  convenience  whether  the  in- 
finitely distant  particles  of  the  fluid  are  to  be  regarded  as  fixed  and  the 
solid  as  moving  uniformly,  or  the  solid  as  fixed,  and  the  infinitely  dis- 
tant particles  of  the  fluid  as  moving  uniformly  with  an  equal  speed  in  the 
contrary  direction.  Throughout  the  present  paper  the  solid  will  be 
supposed  to  move  along  the  axis  of  a-;  so  that  v  will  represent  the  trans- 
verse component  of  the  velocity  of  a  particle  of  liquid  on  either  supposition. 
The  longitudinal  component  of  the  velocity  of  a  liquid  particle  relatively 
to  the  solid  will  be  denoted  by  u;  and  when  that  particle  is  at  an  infinite 
distance  from  the  solid,  by  c;  so  that  when  the  infinitely  distant  part  of 
the  liquid  is  regarded  as  fixed,  the  solid  is  to  be  conceived  as  moving  with 
the  velocity  —  c;  and  the  longitudinal  component  of  the  velocity  of  a 
liquid  particle  relatively  to  the  indefinitely  distant  part  of  the  liquid  will 
be  denoted  by  u  —  c. 

It  is  convenient  to  regard  the  function  U  as  equivalent  to  an  expression 
of  the  following  kind, 

U  =  b  c, (7.) 

c  being  the  uniform  velocity  of  flow  at  an  infinite  distance,  and  b  what  the 
value  of  y  would  be  for  the  water-line  under  consideration  if  the  solid 
were  removed;  in  which  case  that  line  would  become  a  straight  line 
parallel  to  the  axis  of  x.  This  enables  us  to  substitute  for  equations  (5) 
and  (6)  the  following,  in  which  proportionate  velocities  only  are  con- 
sidered : — 


u       d  b  v  d  b 

c       d  y'         c  dx' 


(8.) 


f\  +  U  =  0 M 

a  y       a  .>- 

4.  General  Characteristics  of  Water-Line  Functions. — Since  at  an  infinite 
distance  from  the  solid  body  we  have  u  =  c,  v  =  0,  it  follows  that,  if  the 

*  Professor  William  Thomson,  in  1858,  completed  an  investigation  of  the  motion  of 
a  solid  through  a  perfect  liquid,  so  as  to  obtain  expressions  for  the  motion  of  the  solid 
itself,  involving  twenty-one  constants  depending  on  the  figure  and  mass  of  the  solid 
and  the  density  of  the  liquid ;  but  as  that  investigation,  though  on  the  eve  of  publi- 
cation,  has  not  yet  been  published,  I  shall  not  here  refer  to  it  further. 

2l 


498  ON  PLANE  WATER-LINES. 

origin  of  co-ordinates  be  taken  in  or  near  the  solid  body,  b  must  be  a 
function  of  such  a  kind  that,  when  either  x  =  go.  or  y  =  oo, 

b  =y. 

Hence,  in  a  great  number  of  cases  that  function  is  of  the  form 

b  =  y  +  ¥(x,y)-}.         .  .     (10.) 

where  F  is  a  function  which  either  vanishes  or  becomes  constant  when  x 
or  y  increases  indefinitely. 

It  is  plain  that  when  the  function  b  takes  this  form,  the  term  F  is  the 
function  for  the  motions  of  the  liquid  particles  relatively  to  still  water;  that 
is  to  say, 

u  —  c  _  d  b       *_dFm       v_      db  _         dFt 
c       ~  d  y  d  y'       c  d  x  ~         d  x'  '     ^     '* 

and  also  that  the  term  F  fulfils  the  equation 

^  +  ^1=0  (12) 

df^  da?  •        •     {     } 

When  the  solid  is  symmetrical  at  either  side  of  the  axis  of  x  (as  it  is  in 
all  the  cases  that  will  be  considered  in  this  paper),  the  axis  of  x  itself,  so 
far  as  it  lies  beyond  the  outline  of  the  solid,  is  a  water-line.  Hence  it  is 
necessary  that  the  equation  of  that  axis,  viz. — 

y  =  o, 

should  be  one  of  the  solutions  of  the  equation  )*        .         .     (13.) 

b  =  y  +  F  {x,  y)  =  0,      j 

and,  consequently,  that  F  should  vanish  with  y. 

The  vanishing  of  F  when  x  =  go,  indicates  that  every  straight  line 
given  by  the  equation  y  =  b  either  forms  part  of,  or  is  an  asymptote  to,  a 
water-line  curve. 

The  vanishing  of  F  when  y  =  go,  indicates  that  the  farther  the  water- 
lines  are  from  the  generating  solid,  the  more  nearly  they  approximate  to 
parallel  straight  lines. 

Every  water-line  curve  is  itself  the  outline  of  a  solid  capable  of  moving 
smoothly  through  a  liquid. 

5.  Water-Line  Curves  generated  by  a  Circle,  or  Cyclogenous  Neoids. — 
Conceive  that  a  circular  cylinder  of  indefinite  height,  and  of  the  radius  /, 
described  about  the  axis  of  z,  moves  through  the  liquid  along  the  axis  of 
x.  Then  it  is  already  known  that  the  general  equation  of  the  water-line 
curves  is  the  following, 


ON  PLANE  WATER-LINES.  499 

»  =  »(1 -?£?).   '      '      '   (u) 

giving  a  series  of  curves  of  the  third  order.  When  b  =  0  this  equation 
resolves  itself  into  two,  viz. — 

y  =  0;  x"  +  f  =  P; 

the  first  of  which  represents  the  axis  of  x,  and  the  second  the  circular  out- 
line of  the  cylinder.  For  each  other  value  of  b,  equation  (14)  represents 
a  curve  having  two  branches:  one  of  them  is  an  oval,  contained  within 
the  circle,  and  not  relevant  to  the  problem  in  question:  the  other,  being 
the  real  water-line,  is  convex  in  the  middle  and  concave  towards  the  ends, 
and  has  for  an  asymptote  in  both  directions  the  straight  line  y  =  b. 

For  brevity's  sake,  let  x2  +  y2  =  r2.     Then  the  component  velocities  of 
a  particle  of  water  relatively  to  the  solid  are  given  by  the  equations 

u  _  d  b  Z2     '  2  P  f  _         I  {£  -  #£\ 

c       a  y  r  r*  r* 

y.   (i5.) 

v  _        d  b  _         2  Z2  x  y 

c~~       d  x  ~~  r*     ' 

and  the  square  of  their  resultant  by  the  equation 

u-  +  s>      i  -,      i  \   ,  4 1  y 


^-(»-J)-+^         •  w 


while  the  component  and  resultant  velocities  relatively  to  still  water  are 
given  by  the  following  equations: — 

u       i  _  p  (y2  _  ^     v_       2Pxy     J{(u  -  c)2  +  v2}       I        n7, 
c  r4  '    c~  r"     '  c  r2'       K     '' 

As  a  convenient  name  for  water-line  curves  of  this  sort,  it  is  proposed 
to  call  them  Cyclogenous  Neoids,  that  is,  ship-shape  curves  generated  from  a 
circle. 

The  water-line  surfaces  generated  by  a  sphere  are  known ;  but  no  use 
will  be  made  of  them  in  this  paper.* 


Section  II. — Properties  of  Water-Line  Curves  generated  from 
Ovals,  or  Oogenous  Neoids. 

6.  Derivation  of  other  Water-Line  Curves  from  Cyclogenous  Neoids. — When 
a  form  of  the  function  F  has  been  found  which  satisfies  equation  (12)  of 

*  See  Paper  by  Dr.  Hoppe,  Quart.  Journ.  Math.,  March,  1858. 


500  ON  PLANE  VrATER-LINES. 

Art.  4  (that  is  to  say,  which  fulfils  the  condition  of  liquidity),  an  endless 
variety  of  other  forms  of  that  function  possessing  the  same  property  may 
be  derived  from  the  original  form  by  differentiation  and  integration. 

The  original  form,  and  also  the  derived  forms,  must  possess  the  pro- 
perties of  vanishing  for  x  =  go  and  for  y  =  co,  and  of  becoming  =  0,  or  a 
constant  for  y  =  0.  The  first  of  those. properties  excludes  trigonometrical 
functions,  and  consequently  exponential  functions  also,  which  are  always 
accompanied  by  trigonometrical  functions,  and  leaves  available  functions 
of  the  nature  of  potentials.  The  second  property  excludes  derivation  by 
means  of  differentiation  and  integration  with  respect  to  y,  and  leaves 
available  differentiation  and  integration  with  respect  to  %. 

The  original  form  of  the  function  F  which  will  be  used  in  this  paper  is 
that  appropriate  to  cyclogenous  neoids,  or  water-line  curves  generated  from 
a  circle,  as  given  in  equation  (14)  of  Art.  5,  viz. : — 

F  =  -4  X  constant. 
r- 

"When  one  or  more  differentiations  with  respect  to  x  are  performed  on 
this  function,  and  the  results  substituted  for  F  in  equation  (10),  there  are 
obtained  curves  which  arc  real  water-lines,  but  which  are  not  suitable  for 
the  figures  of  ships,  some  of  them  being  lemniscates,  others  shaped  like  an 
hour-glass,  and  others  looped  and  foliated  in  various  ways.  It  is  otherwise 
as  regards  integration  with  respect  to  x;  for  that  operation,  being  performed 
once,  gives  the  expression  for  the  ordinate  in  a  class  of  curves  all  of  which 
resemble  possible  forms  of  ships,  and  which  are  so  various  in  their*  pro- 
portions, that  every  form  of  ships'  water-lines  which  has  been  found  to 
succeed  in  practice  may  be  closely  imitated  by  means  of  them.  As  that 
class  of  curves  consists  of  certain  ovals,  and  of  other  water-lines  generated 
from  those  ovals,  it  is  proposed  to  call  them  Oogenous  Ncdids  (from 
'  Q.  oyev  //c). 

7.  General  Equation  of  Oogenous  Neoids. — The  integration  with  respect 
to  x,  already  referred  to,  is  performed  as  follows: — The  co-ordinates  of  a 
particle  of  water  being  x  and  y,  let  x'  denote  the  position  of  a  movable 
point  in  the  axis  of  a':  then  the  function  to  be  integrated  is 


(X  -  */)«  +  f 


for  all  values  of  x  between  two  arbitrary  limits.  Let  2a  denote  the 
distance  between  those  limits:  the  most  convenient  position  for  the  origin 
of  co-ordinates  is  midway  between  them,  so  as  to  make  the  limits 

'  =  +  '-',  ■'''  =  —  a  respectively. 
Then  the  following  is  the  integral  sought: 


ON   PLANE  WATER-LINES.  501 

x'  =  +  a                  ,  . 

^-—5 »  =  tan    1 \-  tan    x .      (18.) 

(x-x)2  +  y2                     y  y 


a 

/ 


a 


This  quantity  evidently  denotes  the  angle  contained  between  two  lines 
drawn  from  the  point  (x,  y)  to  the  points  (+ a,  0)  and  (—a,  0).  For 
brevity's  sake,  in  the  sequel  that  angle  will  be  occasionally  denoted  by  9; 
the  points  (+  a,  0)  and  (—  a,  0)  will  be  called  the  foci;  and  their  distance 
a  from  the  centre  will  be  called  the  eccentricity. 

Substituting  this  integral  in  the  general  equation  (10),  we  find,  for 
the  water-line  curves  now  under  consideration,  the  following  equation, 
which  is  the  general  equation  of  oogenous  neo'ids: — 

o  =  y  -f0  =  y  -/(tan-ii-p  +  tan^^)       (19.) 

The  coefficient  /  denotes  an  arbitrary  length,  which  will  be  called  the 
parameter. 

8.  Geometrical  Meaning  of  that  Equation. — The  equation  (19)  represents 
a  curve  at  each  point  of  which  the  excess  {y  —  b)  of  the  ordinate  (y)  above 
a  certain  minimum  value  (b)  is  proportional  to  the  angle  (6)  contained 
at  that  point  between  two  straight  lines  drawn  to  the  two  foci.  Except 
when  b  =  0,  the  curve  has  an  asymptote  at  the  distance  b  from  the  axis 
of  x,  and  parallel  to  that  axis.  Since  the  value  of  b  is  not  altered  by 
reversing  the  signs  of  x,  and  is  only  changed  from  positive  to  negative 
by  reversing  the  sign  of  y,  it  follows  that  each  curve  consists  of  two 
halves,  symmetrical  about  the  axis  of  y;  and  that  there  are  pairs  of 
curves  symmetrical  about  the  axis  of  x. 

In  Plate  III.,  Fig.  1,  therefore,  which  represents  a  series  of  such  curves, 
one  quadrant  only  of  the  space  round  the  origin  or  centre  O  is  shown, 
the  other  three  quadrants  being  symmetrical.  A  is  one  of  the  foci,  at 
the  distance  O  A  =  a  from  the  centre;  the  other  focus,  not  shown  in  the 
figure,  is  at  an  equal  distance  from  the  centre  in  the  opposite  direction. 
BL  is  one  quadrant  of  the  primitive  oval;  and  the  wave-like  curves 
outside  of  it  are  a  series  of  water-lines  generated  from  it,  having  for 
their  respective  asymptotes  the  series  of  straight  lines  parallel  to  O  X,  and 
whose  distances  from  O  X  are  a  series  of  values  of  b. 

The  equation  (19)  embraces  also  a  set  of  curves  contained  within  the 
oval,  and  all  traversing  the  two  foci;  but  as  these  curves  are  not  suited 
for  the  forms  of  ships'  water-lines,  no  detailed  description  of  them  needs 
be  given. 

9.  Properties  of  Primitive  Oval  Neo'ids. — When  in  equation  (19)  b  is 
made  =  0,  so  that  the  equation  becomes 

y-fO  =  0,      .         .         .         .     (20.) 


502  ON  PLANE  WATER-LINES. 

there  are  two  solutions;  one  of  which,  viz.  y  =r  0,  represents  the  axis  of  x, 
agreeably  to  the  condition  stated  in  Art.  4,  equations  (13).  The  other 
solution  represents  the  oval  L  B. 

The  greater  semi-axis  of  that  oval,  0  L,  will  be  called  the  base  of  the 
series  of  water-lines  generated  by  the  oval,  and  denoted  by  /;  its  value  is 
found  as  follows : 

"='i+/'(ta-»J-+ta -»_£_) 

ay  ay  \  a  —  x  a  +  x/ 

-  i    |    /  f        a~x  ,  a  +  x        \. 

but  at  the  point  L  we  have 

dy 
and,  therefore, 

0   =    1    +/(      '         +         >      ); 

\a  —  I       a  -f  V 
whence 

P  =  a2  +  2  af.   .         .         .         .     (21.) 

To  find  the  parameter  /  when  the  base  I  and  eccentricity  a  are  given, 
we  have  the  formula 

/=^=^ (22.) 

The  half-breadth,  or  minor  semi-axis  of  the  oval,  0  B  =  y0,  is  the  root 
of  the  following  transcendental  equation,  found  by  making  x  =  0  in 
equation  (19), 

2/0-2/tan-1^=rO,      .         .         .     (23.) 
'Jo 

which  may  be  otherwise  written  as  follows : — 

tan|a       -  =  0.       .         .         .     (23  a.) 

When  the  minor  semi-axis  y0  and  eccentricity  a  are  given,  the  parameter 
/  is  found  by  the  equation 

/-  *         i-         '         •         •     (MO 

2  tan  "  x  - 

and  thence  the  base  Z'can  be  computed  by  equation  (21). 


ON  PLANE  WATEK-LINES.  503 

When  the  base  I  and  half-breadth  y0  are  given,  the  eccentricity  a  is 
found  by  solving  the  following  transcendental  equation: — 

a  yQ  _  (p  _  a2)  tan  *  1  —  =  0.      .         .     (24  A.) 

An  oval  neoid  differs  from  an  ellipse  in  being  fuller  towards  the  ends  and 
natter  at  the  sides;  and  that  difference  is  greater  the  more  elongated  the  oval  is. 

10.  Varieties  of  Oval  Neo'ids,  and  extreme  cases. — The  eccentricity  a 
may  have  any  value,  from  nothing  to  infinity;  and  the  base  I  may  bear 
to  the  half-breadth  y0  any  proportion,  from  equality  to  infinity.  When 
the  eccentricity  a  =  0,  the  two  foci  coalesce  with  the  centre  0;  the  base  I 
becomes  equal  to  the  half -breadth  b,  the  oval  becomes  a  circle  of  the 
radius  I;  and  the  water-lines  generated  by  it  become  cyclogenous  neo'ids, 
already  described  in  Art.  5. 

As  the  eccentricity  increases,  the  oval  becomes  more  elongated.  In 
Plate  IV.,  Fig.  3,  PL  is  an  oval  whose  length  is  to  its  breadth  as  ^3  :  1, 
its  focus  being  at  A0.  The  oval  BL  in  Plate  III.,  Fig.  1,  is  more  elongated, 
its  length  being  to  its  breadth  as  17:6  nearly.  When  the  eccentricity 
is  infinite,  the  centre  0  and  the  farther  focus  go  off  to  infinity,  leaving 
only  one  focus.  The  parameter  /  becomes  equal  to  the  focal  distance  L  A. 
The  oval  is  converted  into  a  curve  bearing  the  same  sort  of  analogy  to  a 
parabola  that  an  oval  neoid  bears  to  an  ellipse;*  but  instead  of  spreading 
to  an  infinite  breadth  like  a  parabola,  it  has  a  pair  of  asymptotes  parallel 
to  the  axis  of  x,  and  at  the  distance  ±  irf  to  either  side  of  it;  and  each 
generated  water-line  has  two  parallel  asymptotes,  at  the  respective  dis- 
tances b  and  b  +  irf  from  the  axis  of  x.  The  properties  of  these  curves 
may  be   easily  investigated  by  placing  the  origin  of  co-ordinates  at  the 

focus  A,  and  substituting,  in  equation  (19),  tan  ~  *  -  for  0 ;  but  as  their 

figure  is  not  suitable  for  ships'  water-lines,  it  is  unnecessary  here  to  discuss 
them  in  detail ;  and  the  same  may  be  said  of  a  class  of  curves  analogous 
to  hyperbolas,  whose  equation  is  formed  by  putting  —  instead  of  + 
between  the  two  terms  of  the  right-hand  member  of  equation  (18). 

11.  Graphic  Construction  of  Oval  and  Oogenous  Neo'ids. — For  the  sake 
of  distinctness,  the  processes  of  drawing  these  curves  are  represented 
in  two  figures, — Fig.  2  showing  the  preliminary,  and  Fig.  1  the  final 
processes  (see  Plate  III.) 

The  axis  0  Y  is  to  be  divided  into  equal  parts  of  any  convenient  length 
(which  will  be  denoted  by  8  y  in  what  follows),  and  through  the  divisions 
are  to  be  drawn  a  series  of  straight  lines  parallel  to  0  X.  (It  is  convenient 
to  print  those  lines  from  a  copper-plate  divided  and  ruled  by  machinery.) 
They  are  shown  in  Fig.  1  only,  and  not  in  Fig.  2,  to  avoid  confusion. 
*  This  curve  is  identical  with  the  quadratrix  of  Tschirnhausen. 


504  ON  PLANE  WATER-LINES. 

Suppose,  now,  that  the  problem  is  as  follows : — The  base  0  L  and  eccen- 
tricity 0  A  being  given,  it  is  required  to  construct  the  oval  neo'id  and  the  water-lines 
generated  by  it. 

Through  the  focus  A  (Plate  III.,  Fig.  2)  draw  A  D  perpendicular  to 
OX:  about  0,  with  the  radius  OL,  describe  the  circular  arc  L  D,  cutting 
AD  in  D;  from  D  draw  DE  perpendicular  to  OD,  cutting  OX  in  E; 
then  (as  equation  (22)  shows)  AE  will  be  =2/,  the  double  parameter. 

About  A,  with  the  radius  AE=  2/  thus  found,  describe  a  circle  cutting 

A  D  in  F.     Then  commencing  at  F,  lay  off  on  that  circle  a  series  of  arcs, 

each  equal  to  2  By  (the  double  of  the  length  of  the  equal  divisions  of  the 

axis  O  Y).     Through  the  points  of  division  of  the  circle  draw  a  series 

of  radii,  A  Gv  A  G2,  &c,  cutting  the  axis  0  Y  in  a  series  of  points  (some 

of  Avhich,  from  G3  to  G10,  are  marked  in  Fig.   2).*     (These  radii  make, 

•xi.  xi     v       a  -r.  •       r        i      8y    2S?/    3?)/   .    , 

with  the  line  AD,  a  scries  ot  angles,  -jr,  — —f  — j*,  &jc.) 

Then  about  each  of  the  points  in  the  axis  0  Y  thus  found,  with  the 
outer  leg  of  the  compasses  starting  from  the  focus  A,  describe  a  series 
of  circles  (shown  in  Plate  III.,  Fig.  1),  AC1?  AC.,,  AC3,  Arc. 

Each  of  those  circles  traverses  the  two  foci ;  and  the  equation  of  any 
one  of  them  is 

-Y=fO  =  ncy,         .  ■      .         .     (25.) 

where  0  denotes  the  angle  made  at  any  point  of  the  circle  by  straight 
lines  drawn  to  the  two  foci,  and  n  has  the  series  of  values  1,  2,  3,  &c. 
Since  F,  as  explained  in  Art.  4,  is  the  characteristic  function  for  the 
motion  of  the  liquid  particles  relatively  to  still  water,  it  is  plain  that 
each  of  the  circles  for  which  F  :=  constant  is  a  tangent  to  the  directions 
of  motion  of  all  the  particles  that  it  traverses. 

The  paper  is  now  covered,  as  in  Fig.  1,  with  a  network  made  by  a 
series  of  straight  lines,  whose  equations  are  of  the  form  y  =  n'  By,  crossed 
by  a  series  of  circles,  whose  equations  are  of  the  form/0  =  nSy. 

Consequently,  any  curve  drawn  like  those  in  Plate  III.,  Fig.  1,  diagonally 
tlrrough  the  corners  of  the  quadrangles  of  that  network,  will  have  for  its 
equation 

y  -fB  —  (n'  -n)Sy  =  b, 

and  will  accordingly  be  an  oogenous  neo'id,  having  for  its  asymptote  the 
line  y  =  b. 

*  When  the  parameter  is  small,  it  is  sometimes  advisable  to  use  a  circle  (such  as  a 
protractor)  with'a  radius  which  is  a  larger  multiple  of  the  parameter  than  double,  the 
length  of  the  divisions  being  increased  in  the  same  proportion ;  or  the  points  on  the 
axis  0  Y  may  be  laid  down  by  means  of  their  distances  from  O,  calculated  by  the 
formula  OG  =  a.  cotan  6. 


ON  PLANE   WATER-LINES.  505 

The  primitive  oval  is  drawn  by  starting  from  the  point  L,  and  traversing 
the  network  diagonally.  As  many  curves  as  are  required  can  be  drawn 
by  the  eye  with  great  precision,  and  the  whole  process  is  very  rapid  and 
easy  (see  Appendix). 

When  the  problem  is,  with  a  given  base  and  eccentricity  to  draw  an  oogenous 
iieoid  through  a  given  point  in  the  axis  OY,  such  as  P,  the  process  is  modified 
as  follows : — The  axis  0  Y  must  be  so  divided  that  P  shall  be  at  a  point 
of  division.  Then,  up  to  the  describing  of  the  circle  about  A  with  the 
radius  A  E,  the  process  is  the  same  as  before.  Then,  join  A  P  (Plate  III., 
Fig.  2),  and  draw  Kg,  making  the  angle  P Kg  =  APO,  and  cutting  the 
axis  0  Y  in  a  point  (such  as  G10),  which  will  be  the  centre  of  the  circle 
traversing  A  and  P.  Then,  on  the  circumference  of  the  circle  about  A, 
from  g  towards  F,  lay  off  a  series  of  arcs  each  =  2  By,  through  the  points 
of  division  draw  radii  cutting  the  axis  0  Y  in  the  points  G9,  G8,  &c,  and 
complete  the  process  as  before. 

12.  Graphic  Construction  of  Cyclogenous  and  Parabologenous  Neo'ids, — When 
the  eccentricity  vanishes  and  the  oval  becomes  a  circle,  all  the  circles 
composing  the  network  become  tangents  to  0  X  at  the  point  0.  They 
pass  through  the  points  where  the  primitive  circular  water-line  is  cut  by 
the  equidistant  parallel  lines.  Their  radii  are  in  harmonic  progression; 
the  equation  of  any  one  of  them  is  of  the  form 

-¥  =  -J^-,  =  nBy,     .        .         .     (26.) 
xr  +  y 

n  having  the  series  of  values  1,  2,  3,  &c. ;  and  its  radius  is  given  by  the 
formula 

4- (26a-) 

n  by 

When  there  is  but  one  focus,  as  in  the  infinitely  long  curve  described 
in  Art.  10,  the  network  of  circles  is  changed  into  a  set  of  straight 
lines  radiating  from  the  focus,  and  making  with  A  X  the  series  of  angles 
given  by  the  formula 

fO  =  nBy (27.) 

13.  Component  and  Resultant  Velocities  of  Gliding.— The  component 
and  resultant  velocities  with  which  the  liquid  particles  glide  along  the 
water-lines  are  given  by  the  following  equations,  in  terms  of  the  eccen 
tricity  a,  the  parameter  /,  and  the  co-ordinates  : — 


c  ~  dy  ~      ^  (a  -  xf  +  if  "*"  (a  +  xf  +  f- 

v___  cVb fy  fy 

c~    "  dx  (a  -  x)2  +  f       (a  +  x)2  +  y1 


y  (28.) 


ON   PLANE   WATER-LINES. 


d^  "^  dx*  ~      ^  (a  -  xf  +  if  "*"  («  +  x)2  +  ^   | 


if  a2 

+  {(a-z)2  +  f}.{(a  +  xf  +  If^ 


(28.) 


J 


At  the  point  of  greatest  breadth    (that  is,  at  the  axis  of  y)  these 
expressions  take  the  following  values : — 


c  a?  +  y0  a2  +  y*       a-  +  %     c 


0.      (28a.) 


These  equations  are  applicable  to  a  whole  series  of  water-lines  (such  as 
those  shown  in  Fig.  1),  including  the  generator  oval,  and  are  the  best 
suited  for  solving  questions  relating  to  such  a  series. 

But  when  one  particular  ivater-line  is  in  question,  it  is  sometimes  more 
convenient  to  use  another  set  of  equations,  formed  from  the  equations  (28) 
by  the  aid  of  the  following  substitutions,  in  which  0,  as  before,  denotes 
y  —  h 


J 


{(a-x)*  +  f}.{(a  +  x)*  +  f\  =^; 

{(a  -  xf  +  y2}  +  {(a  +  xf  +  if)  =  2  a2  +  2  x- 

+  2  if  =  4  a2  +  -4  ay  cotan  0 ; 
-  +  if  =  or  -f  2  a  y  cotan  0 ; 
•''  =  */{<&  —  IT  +  2  ay  cotan  6). 


y  (29.) 


These  substitutions  being  made  in  the  equations  (28),  give  the  following 
results : — 


-  =  1  +  f-  sin2  0  -  *-  ■  cos  0  sin  0  =  1  +  J- 

c  a  y  'la 


/cos  2  0       /sin  2  0; 


2a 


2y 


=  -  *—  sin2  6  =  -  —J  {a2  -  y2  +  2  ay  cotan  0}  sin2  0  :    v. 
a«  air   K  ' 


u2  +  v2  2  f  2  /"  Z-2 

-^.iL  =  1  +  tl  sin2  0  _  tL  CoS  0  sin  0+4,  sin2  0 

=  1  +~  +  ~^>  -  (^+|C)cos2  0-/sm2  0. 

a       2  ?/-        \  a       2yV  y  ^ 


(30.) 


ON  PLANE  WATER-LINES.  507 

14.  Trajectories  of  Normal  Displacement,  and  of  Swiftest  and  Slowest 
Gliding. — By  the  "trajectory  of  normal  displacement"  is  meant  a  curve 
traversing  all  the  points  in  a  series  of  water-lines  at  which  the  directions 
of  motion  of  the  liquid  particles  relatively  to  still  water  are  perpendicular 
to  the  water-lines;  or,  speaking  geometrically,  a  curve  traversing  all  the 
points  at  which  the  circles  AC1?  A  C2,  &c,  of  Fig.  1,  Plate  III,  cut  the 
water-lines  at  right  angles.  To  find  the  form  of  that  trajectory  it  is 
sufficient  to  make 

?  +  *.!?  =  0;'.  .  .  .     (31.) 

employing  the  values  of  those  ratios  given  by  the  equations  (28).  This 
having  been  done,  it  appears,  after  some  simple  reductions,  that  the 
equation  of  the  trajectory  of  normal  displacement  is  the  following, 

x\-y*  =  P,.         .         .         .         (32.) 

being  that  of  a  rectangular  hyperbola  LM,  Fig.  1,  having  its  vertex  at  L, 
and  its  centre  at  0.  Hence,  that  curve  is  similar  for  all  oogenous  and 
endogenous  neoids  whatsoever,  being  independent  of  the  eccentricity,  and  is 
identical  for  all  oogenous  and  cyclogenous  neoids  having  the  same  base  /. 

By  the  "  trajectory  of  swiftest  and  slowest  gliding "  is  meant  a  curve 
traversing  every  point  in  a  series  of  water-lines  at  which  the  velocity  of 
gliding,  *Ju2  -f  v2,  is  a  maximum  or  a  minimum  for  the  water-line  on 
which  that  point  is  situated.  To  find  the  equation  of  that  curve,  it  is 
necessary  to  solve  the  following  equation, 

v  +  a  _(n    i  + .  _  d\  t*  +  a  =  0_    (33 0 


c  d  t  V      c2     /        \c  '  d  x       c  '  d  yJ    \      c 

9       i  9 

the  expression  employed  for   r2 —  being  that  given  by  the  third  of  the 

equations  (28).  After  a  tedious  but  not  difficult  process  of  differentiation 
and  reduction,  which  it  is  unnecessary  to  give  in  detail,  an  equation  is 
found  which  resolves  itself  into  three  factors,  viz. — 

x=0,  .         .         .         •     (34.) 

being  the  equation  of  the  axis  0  Y,  and 

J  IT+y1  +  y±+/P  +  yi  =  0,      .        .    (35.) 

being  the  equations  of  the  two  branches  L  N  and  L  P  of  a  curve  of  the 
fourth  order.  This  curve,  too,  is  independent  of  the  eccentricity,  and 
therefore  similar  for  all  oogenous  and  cyclogenous  neoids  whatsoever,  and 
identical  for  those  having  the  same  base  I.     It  has  also  the  following 


508  ON   PLANE   WATER-LINES. 

properties : — The  straight  line  joining  L  with  P  makes  an  angle  of  30° 
with  the  axis  OX;  there  are  a  pair  of  straight  asymptotes  through  0, 
making  angles  of  30°  to  either  side  of  0  X;  and  the  two  branches  of  the 
curve  cut  0  X  in  the  point  L,  at  angles  of  45°. 

15.  Graphic  Construction  <>f  those  Trajectories. — The  curves  described  in 
the  preceding  article  are  easily  and  quickly  constructed,  with  the  aid 
<  if  the  series  of  equidistant  lines  parallel  to  0  X,  as  follows  : — In  Fig.  2, 
Plate  III.,  let  S  T  be  any  one  of  those  lines.  With  the  distance  S  L  in 
the  compasses,  lay  off  SH  on  that  line;  H  will  be  a  point. in  the  hyper- 
1  k >la  L  M.  Also  from  S  lay  off,  on  the  axis  of  y,  SI  and  S  J,  each  equal 
to  the  same  distance  S  L.  About  the  centre  0,  with  the  radius  01,  draw 
a  circular  arc  cutting  ST  in  K ;  this  will  be  a  point  in  the  branch  L  1ST. 
About  the  centre  0,  with  the  radius  0  J,  draw  a  circular  arc  cutting  ST 
in  /• ;  this  will  be  a  point  in  the  branch  L  P. 

1G.  Properties  of  tJie  Trajectory  of  Swiftest  and  Sloicest  Gliding. — The 
branch  LN  traverses  a  series  of  points  of  slowest  gliding,  where  the 
water-lines  are  farthest  apart ;  the  branch  L  P  traverses  a  set  of  points  of 
swiftest  gliding,  where  the  water-lines  are  closest  together ;  from  0  to  P 
the  axis  of  y  traverses  points  of  slowest  gliding,  and  beyond  P,  points  of 
swiftest  gliding. 

Hence  every  complete  oogenous  neoid  which  cuts  the  axis  of  y  between 
0  and  P,  contains  two  points  of  SAviftest  and  three  of  slowest  gliding; 
and  every  complete  oogenous  or  cyclogenous  neoid  which  cuts  the  axis  of 
y  at  or  beyond  P  contains  only  one  point  of  swiftest  and  two  of  slowest 
gliding. 

17.  Water-Lines  of  Smoothest  Gliding,  or  IAssoncoids. — At  the  point  P 
itself,  situated  at  the  distance 

OP=-*=        ....     (36.) 

from  the  centre,  two  maxima  and  a  minimum  of  the  velocity  of  gliding 
coalesce;  and  therefore  not  only  the  first,  but  the  second  and  third 
differential  coefficients  of  the  velocity  of  gliding  vanish ;  from  which  it 
follows  that  the  velocity  of  gliding  changes  more  gradually  on  those 
water-lines  which  pass  through  the  point  P,  than  on  any  other  class  of 
oogenous  or  cyclogenous  neoids. 

It  is  proposed,  therefore,  to  call  this  class  of  water-lines  Lisroneoids  (from 
Xiggoq). 

The  oval  neoid,  whose  length  is  to  its  breadth  as  ^3  :  1,  is  itself  a 
lissoneoid;  and  every  series  of  water-lines  generated  by  an  oval  more 
clongeded  than  this  contains  one  lissoneoid ;  for  example,  in  the  series  of 
water-lines  shown  in  Fig.  1,  the  lissoneoid  is  marked  P  Q. 

The  eccentricity  of  the  oval  lissoneoid  is  computed  by  solving  equation 


The  radius  of  that  circle  is 


L      A3     A,    A,     A„ 


ON   PLANE  WATER-LINES. 


509 


(24  A)  of  Art.  9,  when  y0 


J~3' 


and  it  is  found  to  be 


a  =  -732  I,  or  nearly  (  *J  3  -  1 )  I. 


(36  a.) 


By  giving  the  eccentricity  values  ranging  from  *732Z  to  I,  there  are 
produced  a  series  of  lissoneoids  ranging  from  the  oval  P  L,  in  Fig.  3, 
Plate  IV.,  whose  focus  is  at  A0,  to  the  straight  line  PN,  whose  focus 
coalesces  with  L.  P  Qv  P  Q2,  and  P  Q3  are  specimens  of  the  intermediate 
forms,  having  their  foci  respectively  at  Av  A2,  and  A3.  For  a  reason 
which  Avill  be  explained  in  Section  III.,  those  curves  are  not  shown  beyond 
the  trajectory  of  slowest  gliding. 

P 
The  greatest  speed  of  gliding,  for  a  lissoneoid,  is  found  by  making  y\=  — 

in  equation  (28a)  of  Art.  13  •  that  is  to  say, 


At1 


3  a2  +  l- 


(37.) 


18.  Orbits  of  the  Particles  of  Water. — The  general  expressions  for  the 
components  of  the  velocity  of  a  liquid  particle  relatively  to  still  water 
have  been  given  in  equation  (11)  of  Art.  4;  and  to  apply  those  to  the 
case  of  oogenous  neo'ids,  it  is  only  necessary  to  modify  the  equations  (28) 


of  Art.  13,  by  introducing  the  expression  for 
as  follows  : — 


-  instead  of  that  for  -, 
c 


u  —  c 


(t2  -  a2) .  (a2  -x2  +  y2) 
-  {(a -*)«  +  *»}.{ (a +  s)«  +  3f»}; 

v  —  2  (I2— a2)  x  y 

~c=  {(a-  xf  +  /}  .  {(a  +  xf  +  fY 

{u  —  c)2  +  v- (I2  -  a2)2 


y  -  (38.) 


{(a-xy  +  tf}  .{{a  +  of  +  y2} 


From  the  last  of  these  equations  it  appears  that  the  velocity  of  a  particle 
relatively  to  still  water  is  inversely  as  the  product  of  its  distances  from  the  two 
foci. 

The  only  other  investigation  which  will  here  be  made  respecting  the 
orbit  of  a  particle  of  water,  is  that  of  the  relation  between  its  direction 
and  curvature  at  a  given  point,  and  its  ordinate  y. 

It  has  already  been  explained,  in  Art.  11,  that  the  direction  of 
motion  of  a  particle  is  a  tangent  to  a  circle  traversing  it  and  the  two  foci. 
The  radius  of  that  circle  is 


510  ON   PLANE  WATER-LINES. 


.    y  —  b       sin  6 ' 
sin  - — -z — 

and  if  <\>  be  taken  to  denote  the  angle  which  the  direction  of  the  particle's 
motion  relatively  to  still  water  makes  with  the  axis  of  x,  it  is  easily  seen 
that 

cos  0  =  cos  6  —  -  sin  0.      .         .         .     (39.) 
While  that  angle  undergoes  the  increment  d  <p,  the  particle  moves  through 

(I  V 

an  arc  of  its  orbit  whose  length  is     .        ;  consequently  the  curvature  of 

sin  <p 

that  orbit  at  the  arc  in  question  is 

1       sin  <j>  dd>  d .  cos  ch       ( 1        1  \    .  V         a 

-  =  — ^—^-  = =-— ^  =(-+-)  sm  6  +     -  cos  6 

p  dy  dy  \f      (/ /  fa 

=  »-?—«.  j-t— -sine+ycosd}     .         .     (40.) 
/-  —  a-     L     2  a  J 

For  cyclogenous  neoids,  we  obtain  the  value  of  this  expression  by 

making 

?/  —  h 

sin  6  =  : — 7—,  cos  0  =  1, 

substituting  P  -  o2  for  2 fa,  and  then  making  a  =  0;  the  result  being 
as  follows, 

S-ft-i)'     •    •    •  «*» 

that  is  to  say,  the  curvature  of  the  orbit  varies  as  the  distance  of  the  particle 
from  a  line  parallel  to  the  axis  of  x,  and  midivay  between  that  axis  and  the 
undisturbed  position  of  the  particle.  This  is  the  property  of  the  looped  or 
coiled  elastic  curve;  therefore,  when  the  water-lines  are  cyclogenous,  the  orbit 
of  each  particle  of  water  forms  one  loop  of  an  elastic  curve. 

The  general  appearance  of  such  an  orbit  is  shown  in  Fig.  6,  Plate  III. 
The  arrow  D  shows  the  direction  of  motion  of  the  solid  body.  The  dotted 
line  A  C  is  supposed  to  be  at  the  distance  b  from  the  axis  of  x.  The 
particle  starts  from  A,  is  at  first  pushed  forwards,  then  deviates  outwards 
and  turns  backwards,  moving  directly  against  the  motion  of  the  solid  body 
as  it  passes  the  point  of  greatest  breadth,  as  shown  at  B.  The  particle 
then  turns  inwards,  and  ends  by  following  the  body,  and  coming  to  rest 
at  C,  in  advance  of  its  original  position. 

When  the  water-lines  are  oogenous,  the  equations  (39)  and  (40)  show 


ON   PLANE   WATER-LINES.  511 

that  the  orbit  is  of  the  same  general  character  with  the  looped  elastic 
curve  in  Fig.  6,  but  differs  from  it  in  detail  to  an  extent  which  is  greater 
the  greater  the  eccentricity  a;  and  the  difference  consists  mainly  in  a 
flattening  of  the  loop,  so  as  to  make  it  less  sharply  curved  at  B. 

When  the  eccentricity  increases  without  limit,  the  orbit  approximates 
indefinitely  to  a  "  curve  of  pursuit,"  for  which 

n  1        sin  9 

<t>  =  e,-  =  -j~.     .     .     .  (40b.) 

19.  Trajectory  of  Transverse,  Displacement. — Of  Speed  of  Gliding  equal  to 
Speed  of  Ship. — Orthogonal  Trajectories. — The  trajectories  described  in  this 
article  differ  from  those  described  in  Arts.  14,  15,  and  16  by  being 
dependent  upon  the  eccentricity,  and  therefore  not  similar  for  all  sets  of 
oogenous  neo'ids. 

By  the  "trajectory  of  transverse  displacement"  is  meant  the  curve 
traversing  all  the  points  at  which  the  liquid  particles  are  moving  at  right 
angles  to  the  axis  0  X,  relatively  to  still  water.  It  is  determined  from 
the  first  of  the  equations  (28),  by  making 

--1  =  0; 

c 

from  which  is  easily  deduced  the  following  equation, 

&-tf  =  a\  .         .         .         (41.) 

being  that  of  a  rectangular  hyperbola,  with  its  centre  at  0  and  its  vertex 
at  the  focus  A. 

The  trajectory  of  the  points  where  the  speed  of  gliding  is  equal  to  the 
speed  of  the  solid  body,  is  found  from  the  third  of  the  equations  (28)  by 
making 

Ul±^  -1  =  0. 
cr 

Its  equation  is 

x*-f  =  !-t^,  .         .         .      (42.) 

being  that  of  a  rectangular  hyperbola,  with  its  centre  at  0  and  its  vertex 
between  A  and  L,  at  a  distance  from  0  equal  to  half  the  hypothenuse  of 
a  right-angled  triangle  whose  other  sides  are  equal  to  the  base  and  the 
eccentricity  respectively. 

Let  q  =  constant  be  the  equation  of  one  out  of  an  indefinite  number  of 
orthogonal  trajectories  to  a  set  of  oogenous  neoids.  The  function  q,  as  is 
well  known,  must  satisfy  the  equation 


512  ON   PLANE  WATER-LINES. 

dq    db       dq    db _ 
dx'  dx       dy'dy 

Eeferriiig  to  equation  (19)  of  Art.  7  for  the  value  of  b,  it  is  easily  seen 
that  this  condition  is  fulfilled  by  the  following  function, 

which  has  also  the  following  properties, 

dq  _db       u     dq  db       v     d2q   ,d?q  _  (aa\ 

dx~  dy       c'  dy  dx       c'  dx*      dy2 

Every  orthogonal  trajectory  has  a  straight  asymptote  parallel  to  the 
axis  of  y,  and  expressed  by  the  equation  x  =  q. 

The  perpendicular  distance  between  two  consecutive  orthogonal  trajec- 
tories, like  that  between  two  consecutive  water-lines,  is  inversely  propor- 
tional to  the  velocity  of  gliding;  hence,  if  a  complete  set  of  orthogonal 
trajectories  were  drawn  on  Fig.  1,  they  would  divide  it  into  a  network  of 
small  rectangles,  the  dimensions  and  area  of  any  one  of  which  would  be 
expressed  as  follows: — 

C  d  I)        v        c  d  '/  _C2  d  b  (1  <j  (\r  \ 


s/ir  +  r         x/V  +  v2      '    U2  +  V2' 

For  a  series  of  cyclogcnous  neoids,  the  equation  of  the  orthogonal  trajec- 
tories takes  the  following  form, 


'i  =  •  (i  +  m>  ■     •     •  (i^ 


x-  +  f 

20.  Disturbances  of  Pressure  and  Level. — Let  h  denote  the  head  at  a 
given  particle  of  liquid,  being  the  sum  of  its  elevation  above  a  fixed  level, 
and  of  its  pressure  expressed  in  units  of  height  of  the  liquid  itself.  In  a 
mass  of  liquid  which  is  at  rest,  the  head  has  a  uniform  value  for  every 
particle  of  the  mass ;  let  that  value  be  denoted  by  h0.  Then,  when  the 
mass  of  liquid  is  in  the  state  of  motion  produced  by  the  passage  of  a  solid 
through  it,  the  head  at  each  particle,  according  to  well-known  principles, 
undergoes  the  change  expressed  by  the  following  equation, 

h  -  h0  =  *-*-*    •        •        .     («.) 

being  the  height  due  to  the  difference  between  the  squares  of  the  speed 
of  the  solid  body  and  of  the  speed  of  gliding ;  and  in  an  open  mass  of 


ON   PLANE  WATER-LINES.  513 

water  with  a  vessel  floating  in  it,  that  change  will  take  place  by  alterations 
in  the  level  of  surfaces  of  equal  pressure.  The  trajectory  of  slowest 
gliding,  LN  (Plate  III,  Fig.  1),  will  mark  the  summit  of  a  swell  thus 
produced,  and  so  also  will  the  axis  of  y  between  0  and  P ;  while  the 
trajectory  of  swiftest  gliding  0  P,  and  the  axis  of  y  beyond  P,  will  mark 
the  bottom  of  a  hollow.  These  are  the  principal  vertical  disturbances 
which,  throughout  this  investigation,  have  been  assumed  to  be  so  small, 
compared  with  the  dimensions  of  the  body,  as  not  to  produce  any  appre- 
ciable error  in  the  consequences  of  the  supposition  of  motion  in  plane  layers. 

21.  Integral  on  which  the  Friction  depends. — Suppose  a  portion  of  an 
oogenous  neo'id  to  be  taken  for  the  water-line  of  part  of  the  side  of  a 
vessel,  which  part  is  of  the  depth  8  z,  and  that  the  resistance  arising  from 
friction  between  the  water  and  the  vessel  is  to  be  expressed — the  law  of 
that  friction  being,  that  it  varies  as  the  square  of  the  velocity  of  gliding, 
and  as  the  extent  of  rubbing  surface. 

That  resistance  is  to  be  found  (as  already  explained  in  a  paper  on  Waves, 
published  in  the  Philosophical  Transactions  for  1863)  (Seep.  4^1)  by  deter- 
mining the  work  performed  in  a  second  in  overcoming  friction,  and  dividing 
by  the  speed  of  the  vessel;  for  thus  is  taken  into  account  not  only  the  direct 
resistance  caused  by  the  longitudinal  component  of  the  friction,  but  the 
resistance  caused  indirectly  through  the  increase  of  pressure  at  the  bow, 
and  diminution  of  pressure  at  the  stern,  assuming  the  vertical  disturbance 
to  be  unimportant. 

Then,  for  a  part  of  the  water-line  which  measures  longitudinally  d  x, 
the  extent  of  surface  is 


?  ..    v  u2,  + 


d  x ; 


the  friction  on  the  unit  of  surface  is 

K  W  (%'  +  v2) 

where  W  is  the  weight  of  a  unit  of  volume  of  water,  and  K  a  coefficient 
of  friction ;  and  that  friction  has  to  be  overcome  through  the  distance 
tju*  +  1?,  while  the  vessel  advances  through  the  distance  c,  giving  as  a 
factor 

s/  u2  +  v"- 


Those  three  factors  being  multiplied  together,  and  the  result  put  under 
the  sign  of  integration,  give  the  following  expression  for  the  resistance, 

2  K 


514  ON   PLANE  WATER-LINES. 

B_««*a,     !(t+^f.idx..  .      (46  A.) 

2  g  J  \     r      J      u 

Another  form  of  expression  for  the  same  integral  is  obtained  by  putting 

dv  or   f-dQ  instead   of   -dx:  and  a  third  form  by  putting  for  the 
v  .v  u 

elementary  area  of  the  rubbing  surface  the  following  value, 

da-    i  .,'  ■     .,'lT,       ■ 
V«"  +  v 

where  dq  is  the  distance  between  the  asymptotes  of  a  pair  of  orthogonal 
trajectories,,  as  explained   in   Art.    19.       This  gives   for   the  resistance 


l9mju_+*dqm  ^     m 


K  AY  c2  ^        f  u2  +  v 

In  preparing  these  formula?  for  integration,  it  is  necessary  to  express 
the  function  to  be  integrated  in  terms  of  constants  and  of  the  independent 
variable  only,  x,y}  0,  or  q,  as  the  case  may  be;  for  example,  if  y  or  0  is 
the  independent  variable,  the  expression  of  the  function  to  be  integrated 
is  to  be  taken  from  the  equations  (30)  of  Art.  13. 

Owing  to  the  great  complexity  of  that  function,  its  exact  integration 
presents  difficulties  which  have  not  yet  been  overcome,  although  a  probable 
approximate  formula  for  the  resistance  has  been  arrived  at  by  methods 
partly  theoretical  and  partly  empirical,  as  to  which  some  further  remarks 
will  be  made  in  the  third  section  of  this  paper.* 

There  is  one  particular  case  only  in  which  the  exact  integration  of 
equation  (46  a)  is  easy,  that  of  a  complete  circular  water-line  of  the 
radius  /;  and  the  result  is  as  follows  : — 

K  W  r2 

R=   \U      S,~  X  21'.  I  .         .     (48.) 

l'2.  Statement  of  the  General  Problem  of  the  Water-Line  of  Least  Friction. — 
It  is  evident  that,  by  introducing  under  the  sign  of  integration  in  ecmation 
(18)  of  Art.  7  an  arbitrary  function  of  x',  the  integral  may  be  made 
capable  of  representing  an  arbitrary  function  of  x  and  y,  and  will  still 
satisfy  the  condition  of  perfect  liquidity;  and  thus  the  equation 


»='+/::^*»tf=*  •  (48a-> 


(x  —  x'f  +  f 

may  be  made  to  represent  an  arbitrary  form  of  primitive  water-line. 

*  See  The  Civil  Engineer  and  Architect's  Journal  for  October,  1861,  The  Philosophical 
Transactions  for  1863,  TJie  Transactions  of  the  Institution  of  Naval  Architects  for 
1S64,  and  a  Treatise  on  Shipbuilding,  published  in  1S64. 


ON   PLANE  WATER-LINES.  515 

To  find  therefore,  by  the  calculus  of  variations,  the  water-line  enclosing 
a  given  area  which  shall  have  the  least  friction,  will  require  the  solution 
of  the  following  problem : — To  determine  the  function  ^  (x)  so  that,  with 
a  fixed  value  of  the  integral  fx  d  y,  the  integral  in  equation  (46  a)  shall  be 
a  minimum. 

22  a.  Another  Class  of  Plane  Water-Line  Equations. — A  mode  of 
expressing  the  conditions  of  the  flow  of  water  in  plane  layers  past  a  solid 
differing  in  form  from  that  made  use  of  in  the  preceding  parts  of  this 
paper,  consists  in  taking  for  independent  variables,  not  the  co-ordinates  of 
the  water-lines  themselves,  x  and  y,  but  the  co-ordinates  of  their  asymptotes 
(b),  and  of  the  asymptotes  of  their  orthogonal  trajectories  (q).  These  new 
variables  are  connected  with  x  and  y,  and  with  the  velocity  of  gliding,  by 
the  following  equations: — 


u-  +  v2 d  q    db       d  q    d  b 


d  x '  dy       d  y  '  d  x       d  x    d  y       d  y    d  x 
d  q'  d  b       d  q    d  b 


(49.) 


It  can  be  shown  that,  in  order  to  satisfy  the  condition  of  liquidity,  we 
must  have 

Jj~  db'     J~  dq'         '  '  "      V    J 

where  xp  denotes  a  function  of  b  and  q,  such  that 


and,  consequently,  that 


d  b-        d  q- 


_/**V+?*4Y     .      .   (52.) 


m2  +  vl       \d  b  d  q)         \d  ft2/ 

The  curves  to  which  this  method  of  investigation  leads  are  inferior  to 
oogenous  neoids  as  water-lines  for  ships,  because  they  have  comparatively 
sharp  curvature  amidships,  which  causes  them  to  have  small  capacity  for 
then  length  and  breadth,  and  would  give  rise  to  comparatively  sudden 
changes  in  the  speed  of  gliding.  They  will  therefore  not  be  further 
discussed  in  the  present  paper,  except  to  state  that  the  simplest  of  them 
is  the  well-known  cissoid. 


Section  III. — Eemarks  on  the  Practical  Use  of  Oogenous 
Water-Lines. 

23.  Previous   Systems    of  Water-lines. — Owing  principally  to  the  great 
antiquity  of  the  art  of  shipbuilding,  and  the  immense  number  of  practical 


510  ON   PLANE   WATER-LINKS. 

experiments  of  which  it  lias  been  the  subject,  that  part  of  it  which  relates 
to  the  forms  of  water-lines  has  in  many  cases  attained  a  high  degree  of 
excellence  through  purely  empirical  means.  Excellence  attained  in  that 
manner  is  of  an  uncertain  and  unstable  kind;  for  as  it  does  not  spring 
from  a  knowledge  of  general  principles,  it  can  be  perpetuated  by  mere 
imitation  only. 

The  existing  forms  of  water-lines,  whose  merits  are  known  through 
their  practical  success,  constitute  one  of  the  best  tests  of  a  mathematical 
theory  of  the  subject;  for  if  that  theory  is  a  sound  one,  it  will  reproduce 
known  good  forms  of  water-line ;  and  if  it  is  a  comprehensive  one,  it  will 
reproduce  their  numerous  varieties,  which  differ  very  much  from  each 
other. 

The  geometrical  system  of  Chapman  for  constructing  water-lines  is 
wholly  empirical;  it  consists  in  the  use  of  parabolas  of  various  orders, 
chosen  so  as  to  approximate  to  figures  that  have  been  found  to  answer  in 
practice,  and  it  has  no  connection  with  any  mechanical  theory  of  the 
motion  of  the  particles  of  water. 

The  first  theory  of  Blips'  water-lines  which  was  at  once  practically 
useful,  and  based  on  mechanical  principles,  was  that  of  Mr.  Scott  Russell, 
explained  in  the  first  and  second  volumes  of  the  Transactions  of  the 
Institution  »(  Naval  Architects.  It  consists  of  two  parts;  the  first  has 
reference  to  the  dimensions  of  water-lines  intended  for  a  given  maximum 
speed,  and  prescribes  a  certain  relation  between  the  length  of  those  lines 
and  the  length  of  a  natural  wave  which  travels  with  that  speed;  the 
second  part  relates  to  the  form  of  those  lines,  and  prescribes  for  imitation 
the  figures  of  certain  natural  waves,  as  being  lines  along  which  water  is 
more  easily  displaced  than  along  other  lines.  The  figures  thus  obtained 
are  known  to  be  successful  in  practice;  but  it  is  also  well  known  that 
there  are  other  figures  which  answer  well  in  practice,  differing  considerably 
from  those  wave-lines ;  and  it  is  desirable  that  the  mathematical  theory 
of  the  subject  should  embrace  those  figures  also.  It  may  further  be 
observed,  that  the  figure  of  the  solitary  wave,  as  investigated  experi- 
mentally by  Mr.  Scott  Russell  {Reports  of  the  British  Association,  1845), 
and  mathematically  by  Mr.  Earnshaw  (Comb.  Trans.,  1845),  is  that  of  a 
wave  propagated  in  a  canal  of  small  breadth  and  depth  as  compared  with 
the  dimensions  of  the  wave,  and  in  which  particles  of  water  originally  in 
a  plane  at  right  angles  to  the  direction  of  motion,  continue  to  be  very 
nearly  in  a  plane  at  right  angles  to  the  direction  of  motion,  so  as  to  have 
sensibly  the  same  longitudinal  velocity.  This  state  of  things  is  so  different 
from  the  circumstances  of  the  motions  of  the  particles  in  the  open  sea, 
that  it  appears  desirable  to  investigate  the  subject  with  special  reference 
to  a  mass  of  water  of  unlimited  breadth  and  depth,  as  has  been  done  in 
the  previous  sections  of  this  paper. 


ON   PLANE  WATER-LINES.  517 

2-f .  Variety  of  Forms  of  Oogenous  Neoids,  and  their  Likeness  to  good  known 
Forms  of  Water-line. — The  water-lines  generated  from  ovals  which  have 
been  described  in  the  second  section  of  this  paper,  are  remarkable  for  the 
great  varieties  of  form  and  proportions  which  they  present,  and  for  the 
resemblance  of  their  figures  to  those  of  the  water-lines  of  the  different 
varieties  of  existing  vessels.  There  is  an  endless  series  of  ovals,  having 
all  proportions  of  length  to  breadth,  from  equality  to  infinity;  and  each 
of  those  ovals  generates  an  endless  series  of  water-lines,  with  all  degrees 
of  fulness  or  fineness,  from  the  absolute  bluffness  of  the  oval  itself  to  the 
sharpness  of  the  knife-edge.  Further  variations  may  be  made  by  taking 
a  greater  or  a  less  length  of  the  curve  chosen. 

The  ovals  are  figures  suitable  for  vessels  of  low  speed,  it  being  only 
necessary,  in  order  to  make  them  good  water-lines,  that  the  vertical 
disturbance  (as  explained  in  Art.  20)  should  be  small  compared  with  the 
vessel's  draught  of  water.  At  higher  speeds  the  sharper  water-lines, 
more  distant  from  the  oval,  become  necessary.  The  water-lines  generated 
by  a  circle,  or  "  cyclogenous  neoids."  are  the  "  leanest "  for  a  given 
proportion  of  length  to  breadth;  and  as  the  eccentricity  increases,  the 
lines  become  "fuller."  The  lines  generated  from  a  very  much  elongated 
oval  approximate  to  a  straight  middle  body  with  more  or  less  sharp  ends. 
In  short,  there  is  no  form  of  water-line  that  has  been  found  to  answer  in 
practice  which  cannot  be  imitated  by  means  of  oogenous  neoids. 

25.  Discontinuity  at  the  Bow  and  Stern. — Best  limits  of  Water-Lines. — 
Amongst  the  endless  variety  of  forms  presented  by  oogenous  wa^er-lines, 
it  may  be  well  to  consider  whether  there  are  any  which  there  are  reasons 
for  preferring  to  the  others.  One  of  the  questions  which  thus  arise  is 
the  following: — Inasmuch  as  all  the  water-line  curves  of  a  series,  except 
the  primitive  oval,  are  infinitely  long  and  have  asymptotes,  there  must 
necessarily  be  an  abrupt  change  of  motion  at  either  end  of  the  limited 
portion  of  a  curve  which  is  used  as  a  water-line  in  practice,  and  the  question 
of  the  effect  of  such  abrupt  change  or  discontinuity  of  motion  is  one  which 
at  present  can  be  decided  by  observation  and  experiment  only.  Now 
it  appears  from  observation  and  experiment,  that  the  effect  of  the  discon- 
tinuity of  motion  at  the  bow  and  stern  of  a  vessel,  which  has  an  entrance 
and  run  of  ordinary  sharpness  and  not  convex,  extends  to  a  very  thin  layer 
of  water  only ;  and  that  beyond  a  short  distance  from  the  vessel's  side 
the  discontinuity  ceases,  through  some  slight  modification  of  the  water- 
lines,  of  which  the  mathematical  theory  is  not  yet  adequate  to  give  an 
exact  account.  * 

*  In  confirmation  of  this,  experiments  made  on  the  steamers  "Admiral"  and  "Lance- 
field,"  by  Mr.  J.  R.  Napier  and  the  author,  may  be  specially  referred  to.  The  water- 
lines  of  the  "Admiral"  are  complete  trochoids,  and  tangents  to  the  longitudinal  axis 
at  the  bow  and  stern.     The  engine-power  required  to  drive  her  at  her  intended  speed 


518  ON   PLANE   WATER-LINES. 

Still,  although  the  effect  of  the  discontinuity  in  increasing  resistance 
may  not  yet  have  been  reduced  to  a  mathematical  expression,  and  although 
it  may  be  so  small  that  our  present  methods  of  experimenting  have  not  yet 
dotected  it,  it  must  have  some  value;  and  it  is  desirable  so  to  select  the 
limits  of  the  water-line  as  to  make  that  value  as  small  as  possible.  In 
order  that  the  abrupt  change  of  motion  may  take  place  in  as  small  a  mass 
of  water  as  possible,  it  would  seem  that  the  limits  of  the  water-line 
employed  in  practice  should  be  at  or  near  the  point  of  slowest  gliding; 
that  is,  where  the  water-line  curve  is  cut  by  the  trajectory  of  slowest 
gliding  I  A',  in  Plate  III.,  Fig.  1,  and  Plate  IV.,  Fig.  3,  as  explained  in 
Arts.  14,  15,  and  1G;  and  that  conclusion  is  borne  out  by  the  figures  of 
many  vessels  remarkable  for  economy  of  power. 

2G.  Preferable  Figwes  of  Water-Lmest — In  forming  a  probable  opinion 
as  to  which,  out  of  all  the  water-lines  generated  by  a  given  oval,  is  to  be 
preferred  to  the  others,  regard  is  to  be  had  to  the  fact,  that  every  point 
of  maximum  disturbance  of  the  level  of  the  water,  whether  upwards  or 
downwards,  that  is  to  say,  every  point  of  maximum  or  minimum  speed 
of  gliding  (see  Art.  20),  forms  the  origin  of  a  wave,  which  spreads  out 
obliquely  from  the  vessel  (as  may  easily  be  observed  in  smooth  water), 
ami  so  transfers  mechanical  energy  t'>  distant  particles  of  water,  which 
energy  is  lost.  Hence  such  points  should  be  as  few  as  possible;  and  the 
changes  of  motion  at  them  should  be  as  gradual  as  possible;  and  these 
conditions  arc  fulfilled  by  the  curves  described  in  Art.  17,  by  the  name  of 
"  lissoneoids,"  being  those  which  traverse  the  point  P  in  the  figures,  and 
which  may  have  any  proportion  of  length  to  breadth,  from  v  3  to  infinity. 

27.  Approximate  Rules  for  Construction  ami  Calculation. — The  description 
of  those  curves,  already  given  in  Art.  1 7,  has  been  confined  to  those 
properties  which  are  exactly  true.  The  following  rules  are  convenient 
approximations  for  practical  purposes,  when  the  proportion  of  length  to  breadth 
is  not  less  than  4  :  1  (see  Plate  IV.,  Figs.  3  and  4). 

I.  A  tangent  to  the  curve  at  Q,  the  point  of  slowest  gliding,  passes 
very  nearly  through  the  point  P  of  greatest  breadth. 

II.  The  area  P  (t)  K  enclosed  within  the  water-line  is  very  nearly  equal 
to  the  rectangle  of  the  breadth  P  E  and  eccentricity  a.  (When  the  length 
is  not  less  than  six  times  the  breadth,  this  rule  is  almost  perfectly  exact.) 


was  computed  from  the  frictional  resistance,  according  to  principles  explained  in 
publications  already  referred  to  in  the  note  to  Art.  2 1 ;  and  the  result  of  the  calculation 
was  closely  verified  by  experiment.  The  waterdines  of  the  "Lancerield"  are  only 
partly  trochoidal,  being  straight  from  the  point  of  contrary  flexure  to  the  bow,  so  that, 
instead  of  being  tangents  there  to  the  longitudinal  axis,  they  form  with  it  angles  of 
about  13i°.  Yet  the  same  formula  which  gave  the  I'esistance  of  the  "Admiral"  has 
been  found  to  give  also  the  resistance  of  the  "Lancefield"  without  any  addition  on 
account  of  the  discontinuity  of  motion  at  the  bow. 


ON   PLANE   WATER-LINES.  519 

III.  For  the  trajectory  of  slowest  glidiDg,  L  N,  there  may  be  substituted* 
without  practical  error,  a  straight  line  cutting  the  axis  0  X  in  L  at  an 
angle  of  45° ;  and  when  this  has  been  done,  the  eccentricity  0  A  or  a  is 
almost  exactly  equal  to  the  length 

and  this  of  course  is  also  the  ratio  of  the  area  to  the  circumscribed  rect- 
angle. The  base  OL  or  /  also  is  very  nearly  equal  to  (the  sum  of  the 
length  and  breadth)   X  'G34. 

IV.  Hence  the  following  approximate  construction  :  Given,  the  common 
length  Q II  of  a  set  of  water-lines  of  smoothest  gliding,  which  are  to  have 
a  common  termination  at  Q,  and  their  breadths  R  P1?  R  P2,  R  P;;,  &c  : 
required,  to  find  their  areas,  bases,  and  foci. 

Through  Q,  and  R  draw  the  straight  lines  Q  U  and  R  U,  making  the 
angles  RQU  =  45°,  QRU  =  30°.  Through  their  intersection  U  draw 
U  V  perpendicular  to  R  Q.  All  the  required  foci  will  be  in  U  V ;  and 
R  V  will  be  the  length  of  the  rectangles  equivalent  to  each  of  the  water- 
line  areas ;  so  that 

aBaaBj  QR1  =  RV  x  RPX, 

area  P.2  Q  R2  =  R  V  x  R  P,, 

&c.  &c. 

Through  P1?  P.2,  P3,  &c,  draw  lines  parallel  to  R  U,  cutting  Q  U  in  Lv  L2, 
L  „  &c.  :  these  points  will  be  the  ends  of  the  bases  required,  through  which 
draw  the  bases  Lx  0^  L2  02,  L3  03,  &c,  parallel  to  Q  R,  and  cutting  V  U 
in  Av  Ac,,  A3,  &c. :  these  will  be  the  required  foci. 

The  bases  and  foci  and  the  points  Px,  P2,  P3,  &c,  being  given,  the  water- 
lines  are  to  be  constructed  by  the  rules  given  in  Art,  11. 

28.  Lissonedids  compared  with  Trochoids. — In  Fig.  5,  Plate  IV.,  the  full 
line  P  Q  is  a  lissoneo'id,  and  the  dotted  line  P<?  a  trochoid  of  the  same 
breadth  and  area.  The  curves  lie  very  near  together  throughout  their 
whole  course — the  only  difference  being,  that  the  trochoid  is  slightly  less 
full  and  more  hollow  than  the  lissoneo'id,  but  at  the  same  time  the  trochoid 
is  the  longer,  and  has  a  greater  frictional  surface.  Had  the  entrance  of 
the  trochoid  consisted  of  a  straight  tangent  from  its  point  of  contrary 
flexure  (as  in  the  bow  of  the  "  Lancefield,"  mentioned  in  the  note  to  Art. 
25),  the  two  curves  would  have  lain  still  closer  together.  The  same  like- 
ness to  a  trochoid  is  found  in  all  lissoneoids  whose  length  is  more  than 
about  3tt  times  the  breadth. 

29.  Combinations  of  Boiv  and  Stern. — Although  there  is  reason  to  believe 
that  water-lines  of  equal  length  and  similar  form  at  the  bow  and  stern, 


520  ON    PLANE   WATER  LINES. 

such  as  are  produced  by  using  one  neoid  curve  throughout,  are  the  best 
on  the  whole,  still  the  naval  architect,  should  he  think  tit,  can  combine 
two  different  oogenous  neoids  for  the  bow  and  stern ;  or,  according  to  a 
frequent  practice,  he  may  adapt  the  figure  of  the  stern  to  motion  of  the 
particles  in  vertical  layers  instead  of  horizontal  layers ;  provided  he  takes 
care  in  every  case  that  the  midship  velocity  of  gliding  (u0,  as  given  by 
equation  (28a)  of  Art.  13)  is  the  same  for  each  bow  water-line  and  stern 
water-line  at  their  point  of  junction. 

30.  Provisional  Formula  for  Resistance. — Until  the  difficulty  of  integra- 
tion, mentioned  in  Art.  30,  shall  have  been  overcome,  or  until  more 
exact  experimental  data  than  we  have  at  present  shall  have  been  obtained, 
the  following  provisional  formula,  analogous  to  that  which  has  been  found 
to  agree  with  the  results  of  experiment  on  trochoidal  and  nearly  trochoidal 
lines,  as  well  as  some  others,  may  be  considered  as  a  probable  approxima- 
tion for  lissoneoids, 

R  =  ^.(l+4<«^)LG;         .     (53.) 

where  G  is  the  mean  girth  of  the  vessel  under  water ;  L  her  total  length ; 
//0  the  midship  velocity  of  gliding,  found,  for  a  lissoneoid,  by  equation 
(37)  of  Art.  17;  c  the  speed  of  the  ship;  W  the  heaviness  of  water; 
and  K  a  coefficient  of  friction  (=  about  '0036  for  a  clean  surface  of  paint). 


APPENDIX. 

Note  to  Artidv  11. — The  general  process  of  constructing  a  series  of 
curves  whose  equation  is  (f>  (x,  y)  +  xp  (x,  y)  =  constant,  by  drawing  lines 
diagonally  through  a  network  consisting  of  two  sets  of  curves  whose  equa- 
tions are  respectively  <p  (x,  y)  =  constant  and  ip  (x,  y)  =  constant,  is  due 
to  Professor  Clerk  Maxwell. 

Summary  of  the  Contents. 

Section  I. — Introduction,  and  Summary  of  known- 
Principles. 

Art. 

1.  Plane  "Water-Lines  in  two  Dimensions  defined. 

2.  General  Principles  of  the  Flow  of  a  Liquid  past  a  Solid. 

3.  Xotation. 

4.  General  Characteristics  of  Water-Line  Functions. 

5.  "Water-Line  Curves  generated  by  a  Circle,  or  Cyclogenous  Xeoids. 


ON   PLANE   WATER-LINES.  521 


Section  II. — Properties  of  Water-Line  Curves  generated 
from  Ovals,  or  Oogenous  Neoids. 

Art. 
G.  Derivation  of  other  Water-Line  Curves  from  Cyclogenous  Neoids. 

7.  General  Equation  of  Oogenous  Neoids. 

8.  Geometrical  Meaning  of  that  Equation. 

9.  Properties  of  Primitive  Oval  Neoids. 

10.  Varieties  of  Oval  Neoids,  and  extreme  cases. 

11.  Graphic  Construction  of  Oval  and  Oogenous  Neoids. 

12.  Graphic  Construction  of  Cyclogenous  and  Parabologenous  Neoids. 

1 3.  Component  and  Kesultant  Velocities  of  Gliding. 

14.  Trajectories  of  Normal  Displacement,  and  of  Swiftest  and  Slowest 

Gliding. 

1 5.  Graphic  Construction  of  those  Trajectories. 

16.  Properties  of  the  Trajectory  of  Swiftest  and  Slowest  Gliding. 

1 7.  Water-Lines  of  Smoothest  Gliding,  or  Lissoneoids. 

18.  Orbits  of  the  Particles  of  Water. 

1 9.  Trajectory  of  Transverse  Displacement. — Of  Speed  of  Gliding  equal 

to  Speed  of  Ship. — Orthogonal  Trajectories. 

20.  Disturbances  of  Pressure  and  Level. 

21.  Integral  on  which  the  Friction  depends. 

22.  Statement  of  the  General  Problem  of  the  Water-Line  of  Least  Friction. 
22a.  Another  Class  of  Plane  Water-Line  Equations. 

Section  III— Remarks  on  the  Practical  Use  of  Oogenous 
Water-Lines. 

23.  Previous  Systems  of  Water-Lines. 

24.  Variety  of  Forms  of  Oogenous  Neoids,  and  their  likeness  to  good 

known  Forms  of  Water-Line. 

25.  Discontinuity  at  the  Bow  and  Stern. — Best  limits  of  Water-Lines. 

26.  Preferable  Figures  of  Water-Lines. 

27.  Approximate  Rules  for  Construction  and  Calculation. 

28.  Lissoneoids  compared  with  Trochoids. 

29.  Combinations  of  Bow  and  Stern. 

30.  Provisional  Formula  for  Resistance. 


522  PRINCIPLES   RELATING   TO   STREA.M-LINES. 


XXXL— ELEMENTARY  DEMDN<STRA!TT0N8  OF  PRINCIPLES 
RELATING  TO  STREAM-LINES.* 

1.  Object  of  this  Cffrwnvmcaiiow. — The  object  of  this  communication  is  fcb 
explain  some  very  elementary  demonstrations  of  certain  propositions  in 
hydrodynamics  which  bear  upon  important  practical  questiona  The  pro- 
positions  themselves  are  not  new;  but  all  the  previously  published,  demon- 
strations of  them  with  which  I  am  acquainted  involve  the  use  of  mathe- 
matical methods  of  some  difficulty,  and  especially,  of  the  solution  of 
differential  equations  of  the  second  order.  The  demonstrations  now  given 
(which  have  hitherto  been  made  public  in  the  form  of  lectures  only),  are 
intended  to  enable  persons  who  have  not  mastered  the  higher  mathematics 
to  understand  the  propositions  in  question,  and  to  satisfy  themselves  of 
their  truth. 

2.  Stream-Lines  explained. — A  stream-line  is  the  line,  whether  straight 
or  curved,  that  is  traced  by  a  particle  in  a  current  of  fluid.  In  Avhat  is 
termed  a  "steady  current,"  each  individual  stream  -line  preserves  its  figure 
and  position  unchanged,  and  marks  the  path  or  track  of  a  filament,  or 
continuous  series  of  particles  that  follow  each  other.  The  direction  of 
the  motions  in  different  parts  of  a  steady  current  may  be  represented  to 
the  eye  by  drawing  the  group  of  stream-lines  traced  by  different  particles 
in  that  current,  and  indicating  by  one  or  more  arrows  in  which  of  two 
contrary  directions  the  motion  takes  place.  The  wavy  lines  in  Fig.  3 
{See  p.  527)  represent  an  example  of  this. 

3.  Relation  between  Velocity  and  Transverse  Area. — It  is  obvious  that  if 
the  area  of  a  transverse  section  of  a  current  be  multiplied  by  the  mean 
velocity  of  the  particles  of  fluid  in  the  act  of  traversing  that  transverse 
section,  the  product  will  be  the  flow;  that  is,  the  volume  of  fluid  which 
passes  through  that  transverse  section  in  an  unit  of  time ;  and  conversely, 
that  if  the  flow  be  divided  by  the  transverse  area,  the  quotient  will  be  the 
mean  velocity  of  the  particles  that  traverse  that  section.  By  a  trans- 
verse section  is  to  be  understood  a  surface  that  cuts  all  the  stream -lines  at 
right  angles.  Moreover,  in  a  liquid  of  invariable  density  (to  which  class  of 
fluids   alone   this  communication    is    restricted),   the  flow  through    each 

*  From  The  Engineer  of  Oct.  16,  18G8. 


PKINCIPLES  RELATING  TO   STREAM-LINES.  523 

transverse  section  of  a  steady  current  is  of  equal  volume ;  therefore,  in  a 
steady  current,  the  mean  velocity  of  the  particles  at  a  given  transverse  section  is 
inversely  proportional  to  the  area  of  that  section. 

4.  Elementary  Streams. — A  current  may  be  conceived  mentally  to  be 
divided,  by  insensibly  thin  partitions,  following  the  course  of  the  stream- 
lines, into  a  number  of  elementary  streams;  and  the  positions  of  those 
partitions  may  be  conceived  to  be  so  adjusted  that  the  volumes  of  flow  in 
all  the  elementary  streams  shall  be  equal.  The  use  of  this  conception  is 
to  represent  to  the  mind  the  velocity,  as  well  as  the  direction  of  motion, 
of  the  particles  in  different  parts  of  the  current ;  for  it  is  obvious  that  in 
a  set  of  elementary  streams  of  equal  flow,  the  velocity  of  a  particle  at  any 
point  is  inversely  proportional  to  the  area  of  the  transverse  section,  through  that 
point,  ■  of  the  elementary  stream  to  which  the  particle  belongs.  This  is  the 
principle  which,  when  expressed  in  the  symbols  of  the  differential  calculus, 
is  called  "  the  equation  of  continuity  "  of  a  liquid." 

5.  Component  Velocities. — The  component  velocity  of  a  particle  in  a  direc- 
tion oblique  to  its  actual  direction  of  motion,  may  be  found  by  the  help  of 
elementary  streams ;  for  it  is  only  necessary  to  divide  the  elementary 
volume  of  flow  by  the  area  of  an  oblique  section  of  the  elementary  stream, 
made  by  a  plane  perpendicular  to  the  direction  of  the  required  component. 
For  example,  in  Fig  1,  let  A  A,  A' A',  be  the  boundaries  of  an  elementary 
stream,  and  let  C  C  be  a  plane  cutting  it  obliquely ;  then,  if  the  volume 
of  flow  be  divided  by  the  area  of  the  oblique  section  made  by  the  plane 
C  C,  the  quotient  will  be  the  component  velocity  of  a  particle  in  a 
direction  perpendicular  to  that  plane,  being  less  than  the  total  velocity 
along  the  stream  in  the  same  ratio  in  which  the  area  of  a  transverse 
section  of  the  stream  is  less  than  the  area  of  the  oblique  section. 

6.  Representation  of  the  Elementary  Streams  in  a  Layer  of  a  Current. — In 
considering  the  motion  of  the  particles  of  one  layer  of  a  current,  we  may 
conceive  the  paper  of  a  diagram,  such  as  Fig.  3,  to  represent  either  one  of 
the  actual  surfaces  of  that  layer,  if  it  is  plane,  or  if  it  is  not  plane,  the 
same  surface  developed,  that  is,  spread  out  flat.  The  layer  may  be  con- 
ceived to  be  divided  into  elementary  streams  of  equal  flow  by  partitions 
perpendicular  to  the  surfaces  of  the  layer;  and  in  the  diagram  those 
partitions  will  be  represented  by  stream-lines,  such  as  the  wavy  lines  in 
Fig.  3.  Such  a  diagram  exhibits  to  the  eye  the  velocity,  as  well  as  the 
direction,  of  the  motions  of  the  particles  in  every  part  of  the  layer ;  for,  if 
the  layer  is  uniformly  thick,  the  velocity  of  any  particle  is  inversely  pro- 
portional simply  to  the  perpendicular  distance  between  the  two  adjacent 

*  Let  S  Q  denote  the  volume  of  flow  in  each  of  the  elemental  streams  of  which  a 
steady  current  of  liquid  consists;  and  at  a  given  point  let  SS  be  the  transverse  area  of 

an  elementary  stream,  and  v  the  velocity ;  then  v  =  ^. 


524  PRINCIPLES   RELATING   TO   STREAM-LINES. 

stream-lines ;  and  if  the  thickness  of  the  layer  varies  at  different  points, 
that  velocity  is  inversely  proportional  to  the  same  perpendicular  distance 
multiplied  by  the  thickness  of  the  layer. 

7.  ( 'omposition  of  Elenu  ntary  Stream*. — If  a  layer  of  liquid  is  acted  upon 
at  the  same  time  by  two  sets  of  forces,  which,  if  acting  separately,  would 
produce  currents  consisting  of  two  different  sets  of  elementary  streams, 
the  combined  action  of  those  two  sets  of  forces  will  produce  currents  con- 
sisting of  a  third  set  of  elementary  streams,  which  may  he  regarded  as  the 
resultant  of  the  two  former  sets.  The  stream-lines  marking  the  boundaries 
of  the  first  two  sets  of  elementary  streams  may  be  called  the  component 
stream-lilies,  and  those  marking  the  boundaries  of  the  third  set,  the 
resultant  stream-lines.  Then  the  principle  which  connects  the  resultant 
mi  lent  with  the  component  currents  is  as  follows  : — 

The  resultant  stream-lines  pass  diagonally  through  all  the  unities  of  the 
network  formed  by  the  component  stream-lines. 

For  example,  in  Fig.  1,  A  A,  A' A',  are  a  pair  of  lines  belonging  to  one 
set  of  component  stream-lines,  and  B  B,  B'B',  a  pair  belonging  to  another 


-----,  ,T>,--- 


»•   ,---'"C 


1'. 


Fig.  1. 


set.  The  line  C  C,  drawn  through  two  of  the  intersections,  is  one  of  the 
set  of  resultant  stream-lines ;  and  the  lines  parallel  to  C  C,  drawn  through 
D  and  D',  are  two  more.  Also,  in  Fig.  2,  the  straight  dotted  lines 
diverging  from  A,  and  the  straight  dotted  lines  converging  towards  B, 
are  two  sets  of  component  stream-lines;  and  the  curved  lines  which 
traverse  the  intersections  of  the  straight  lines  are  the  resultant  stream- 
lines. For  a  third  example,  in  Fig.  3,  the  straight  lines  parallel  to  X  0, 
and  the  arcs  diverging  from  A,  are  two  sets  of  component  stream-lines ; 
and  the  wavy  lines  drawn  through  the  intersections  of  the  first  two  sets 
are  the  resultant  stream-lines. 

To  demonstrate  this  principle  it  is  to  be  considered — First,  as  regards  the 
direction  of  the  resultant  stream-lines:  that  in  Fig.  1,  CC  represents  a  plane, 
which  is  an  oblique  section  at  once  of  the  elementary  stream  A  A,  A' A', 
and  of  the  elementary  stream  B  B,  B'  B'.  The  forces  which  produce  the 
elementary  stream  A  A,  A'  A',  tend  to  send  a  certain  volume  of  liquid  per 
second  through  that  oblique  section,  from  the  side  next  D  to  the  side  next  D'. 
The  forces  which  produce  the  elementary  stream  B  B,  B'  B',  tend  to  send 


PRINCIPLES   RELATING    TO   STREAM-LINES.  525 

an  equal  volume  of  liquid  through  the  same  oblique  section  in  the 
contrary  direction.  Therefore,  the  effect  of  the  combination  of  the  forces 
is  that  there  is  no  flew:  through  the  oblique  section  C  C ;  therefore  C  C  is 
part  of  one  of  the  resultant  stream-lines.  Secondly,  as  to  the  number  and 
closeness  of  those  stream-lines,  it  is  to  be  considered  that  D  D'  also 
represents  a  plane,  which  is  an  oblique  section  at  once  of  both  the  elemen- 
tary streams.  The  forces  which  produce  the  elementary  stream  A  A,  A'  A', 
tend  to  send  a  certain  volume  of  liquid  per  second  through  that  section  in 
a  certain  direction ;  and  the  forces  which  produce  the  elementary  stream 
B  B,  B'  B',  tend  to  send  an  equal  volume  per  second  through  in  the  same 
direction.  Therefore,  the  effect  of  the  combination  of  the  forces  is  that  a 
double  volume  per  second  passes  through  the  section  D  D' ;  therefore,  the 
space  between  D  and  D'  contains  two  resultant  elementary  streams;  there- 
fore, each  of  the  points  D  and  D'  is  traversed  by  one  of  the  resultant 
stream-lines.  Thus  it  is  proved  that  all  the  intersections  of  the  com- 
ponent stream-lines  are  traversed  by  resultant  stream-lines. 

8.  Condition  of  Perfect  Fluidity.  —  The  characteristic  property  of  a 
perfect  fluid — in  other  words,  a  fluid  absolutely  free  from  viscosity — is 
that  the  particles  have  no  tendency  to  preserve  any  definite  figure,  and  are 
incapable  of  exerting  any  force  against  a  surface  which  they  touch  except 
normal  pressure  ;  that  is  to  say,  pressure  in  a  direction  at  right  angles  to 
that  surface.  One  consequence  of  this  is  that  no  particle  of  a  perfect 
fluid  can  have  rotation  impressed  upon  it ;  for  normal  pressure  can 
impress  rotation  only  on  a  body  which  tends  to  preserve  a  definite  figure. 
No  existing  fluid  is  absolutely  free  from  viscosity;  and  therefore  the 
mechanical  consequences  of  the  supposition  of  perfect  fluidity  are  not 
realised  exactly,  but  only  approximately.  Nevertheless,  there  are  cases  in 
which  the  errors  caused  by  neglecting  viscosity  are  unimportant;  and 
hence  the  use  of  investigating  the  properties  of  stream-lines  in  a  perfect 
fluid. 

9.  Rectilinear  Motions  in  a  Perfect  Fluid. — Another  way  of  stating  the 
absence  of  rotation  in  the  motion  of  a  perfect  fluid  is  to  say  that  any  two 
particles  which  move  side  by  side  in  straight  lines  must  move  with  equal  velocities; 
for  if  their  velocities  are  different  the  larger  particle  formed  by  uniting 
them  is  in  a  state  of  rotation,  one  side  moving  faster  than  the  other. 

There  are  three  modes  of  rectilinear  motion  in  a  perfectly  fluid  liquid 
which  fulfil  this  condition,  and  by  combining  which  an  immense  number 
of  modes  of  curvilinear  motion  may  be  generated ;  and  all  those  curvi- 
linear resultant  motions  fulfil  the  condition  of  perfect  fluidity,  because 
their  components  do  so.  Those  three  modes  of  rectilinear  motion  are  the 
following : — 

I.  Motion  in  parallel  straight  lines,  with  an  uniform  velocity.  Here 
the  elementary  streams  are  everywhere  of  equal  transverse  area. 


526 


PRINCIPLES   RELATING   TO   STREAM-LINES. 


II.  Motion  in  straight  lines  converging  towards  or  diverging  from  an 
axis,  to  which  they  are  all  perpendicular.  Here  we  may  consider  the 
motion  of  the  particles  in  a  layer  of  uniform  thickness  perpendicular  to 
the  axis.  The  elementary  streams  in  such  a  layer  are  of  the  form  of 
Avedges,  separated  from  each  other  hy  planes  radiating  from  the  axis, 
and  making  equal  angles  with  each  other.  The  area  of  a  transverse 
section  of  an  elementary  stream  varies  directly  as  the  distance  from  the 
axis,  and  the  velocity  of  a  particle  varies  inversely  as  that  distance. 

III.  Motion  in  straight  lines  converging  towards  or  diverging  from  a 
central  point.  Here  the  elementary  streams  are  of  the  forms  of  cones  or 
•  if  pyramids,  having  their  summits  at  the  central  point,  and  of  such 
shapes  and  sizes  as  to  divide  the  surface  of  a  sphere  described  about  that 
point  into  equal  areas.  The  area  of  a  transverse  section  of  an  elementary 
stream  varies  directly  as  the  square  of  the  distance  from  the  central  point, 
and  the  velocity  of  a  particle  consequently  varies  inversely  as  the  square 
of  that  distance. 

It  is  easy  to  see  that  in  the  last  two  modes  of  motion  the  elementary 
streams  cannot  actually  extend  in  a  pointed  form  to  the  axis  or  to  the 
central  point,  but  must  be  deflected  in  its  neighbourhood,  so  as  to  afford 
an  inlet  or  an  mulct  for  the  liquid,  as  the  case  may  be. 

10.  Rectilinear  Stream-Lines  in,  an,  Uniformly  Thick  Layer. — The  stream- 


Fig.  2 


lines  which  represent  an  uniform  straight  current  in  an  uniformly  thick 
layer  of  liquid  are  simply  parallel  equidistant  straight  hues,  such  as  those 
shown  in  Fig  3.  The  stream-lines  which,  in  a  similar  layer,  represent  a 
current  diverging  from  or  converging  towards  an  axis,  are  straight  lines 


PRINCIPLES   RELATING  TO   STREAM-LINES. 


527 


radiating  from  a  point,  and  making  equal  angles  with  each  other,  like 
either  of  the  two  sets  of  dotted  straight  lines  in  Fig  2. 

11.  Circular  Stream-Lines  in  an  Uniformly  Thick  Layer. — The  simplest 
example  of  a  set  of  resultant  stream-lines  is  that  obtained,  as  in  Fig.  2,  by 
combining  together  a  pair  of  equal  and  similar  sets  of  radiating  stream- 
lines, one  set  diverging  from  a  point  at  A,  and  the  other  converging  towards 
a  point  at  B.  Those  two  points  may  be  called  foci.  According  to  well- 
known  geometrical  principles,  the  resultant  stream-lines,  which  traverse 
the  intersections  of  the  network  formed  by  the  two  sets  of  radiating 
stream-lines,  are  a  series  of  circles,  each  of  which  traverses  the  foci  A  and 
B,  the  only  exception  being  the  straight  line  through  A  and  B.  The 
radii  of  those  circles  are  proportional  to  the  secants  of  a  series  of  angles, 
increasing  by  equal  intervals  from  0°  to  90°.  These  resultant  stream- 
lines represent  the  motion  of  a  layer  of  liquid  of  uniform  thickness,  under 
the  action  of  forces  which  urge  the  particles  to  move  from  an  axis  at  A,, 
and  towards  another  axis  at  B. 

In  a  paper  published  in  the  Philosophical  Transactions  for  1863  (See  p. 
495),  the  properties  of  those  circular  stream-lines  traversing  two  foci  Avere 
arrived  at  by  the  integration  of  a  differential  equation  of  the  second  order. 
They  have  now  been  demonstrated  by  a  very  elementary  method ;  and  to 
do  so  was  one  of  the  chief  objects  of  the  communication. 


Fig.  3. 


12.  Various  Resultant  Stream-Lines  in  a  Layer  of  Uniform  Thickness. — By 
compounding  the  circular  stream-lines  of  Art.  11  with  the  equidistant 
straight  stream-lines  of  Art.  10,  and  drawing  curves  through  the  angles  of 
the  network,  an  endless  variety  of  stream-lines  is  obtained  of  figures 
closely  resembling  the  lines  of  ships  of  various  degrees  of  fineness,  and  of 
various  proportions  of  length  to  breadth.  These,  under  the  name  of 
Nedials  (or  ship-like  curves),  have  been  fully  explained  and  illustrated  in 
previous  papers  and  publications;  and  especially  in  the  paper  already 
referred  to  as  having  been  published  in  the  Philosophical  Transactions  for 
1863,  from  which  Fig.  3  is  copied;  and  it  is  unnecessary  to  enter  into 


528 


PRINCIPLES   RELATING   TO    STREAM-LINES. 


details  respecting  them  here.  The  figure  shows  only  one  quadrant  of  the 
complete  set  of  stream-lines,  the  other  three  quadrants  being  symmetrical 
to  the  first.  The  curves  diverging  from  the  focus  A  are  circular  stream- 
lines, and  they  converge  to  another  focus  at  an  equal  distance  from  0,  in 
the  opposite  direction. 


Fis.    4. 


13.  Parallel  Straight  Stream-Lines  in  a  Wedge-shaped  Layer. — When  a 
current  flows  past  a  solid  of  revolution,  the  figure  and  arrangement  of  the 
stream-lines  are  to  be  determined  by  considering  the  motion  in  a  wedge- 
shaped  layer  of  indefinite  length  and  breadth,  having  its  edge  at  the  axis 
of  the  solid.  The  thickness  of  such  a  layer  varies  as  the  distance  from 
the  edge.  In  Fig.  4,  let  0  X  represent  the  axis  of  the  solid  and  edge  of 
the  wedge-shaped  layer,  and  let  the  paper  represent  one  of  the  plane 
surfaces  of  the  layer.  The  stream-lines  representing  an  uniform  straight 
current  must  be  so  arranged  as  to  divide  the  layer  into  elementary 
streams  of  uniform  transverse  area;  and,  in  order  that  they  may  do  so,  the 
squares  of  their  distances  from  the  axis  0  X  must  increase  by  uniform 
differences.  Let  0  A  be  the  total  breadth  which  it  is  desired  to  sub- 
divide into  elementary  streams.  Make  0  B  =  0  A,  and  divide  it  into 
as  many  equal  parts  as  there  are  to  be  elementary  streams.     On  0  B  as  a 


PRINCIPLES   RELATING  TO   STREAM-LINES.  529 

diameter  draw  a  semicircle,  and  from  the  points  of  division  of  0  B  draw 
ordinates  perpendicular  to  it,  and  cutting  the  semicircle.  Then  lay  off 
from  0,  along  0  A,  a  series  of  distances  equal  respectively  to  the  chords 
measured  from  0  to  the  points  of  division  of  the  semicircle ;  the  required 
stream-lines  will  be  straight  lines  drawn  parallel  to  0  X  through  the 
points  of  division  of  0  A. 

14.  Radiating  Straight  Stream-Lines  in  a  Wedge-shaped  Layer. — To  divide 
such  a  wedge-shaped  layer,  as  has  been  described  in  the  preceding  article, 
into  equal  elementary  streams  radiating  from  a  point  B  in  the  axis  X  0  B, 
lay  off  along  the  axis  as  many  equal  divisions  as  there  are  to  be  elemen- 
tary streams  in  one  quadrant  of  the  space  round  B.  Let  B  0  be  the 
distance  containing  all  those  divisions.  About  B,  with  the  radius  B  C, 
draw  the  quarter  circle  0  C ;  and  from  the  points  of  division  B  0,  and 
perpendicular  to  it,  draw  ordinates  cutting  the  quarter  circle.  Then  draw 
lines  radiating  from  B  to  the  points  of  division  of  the  quarter  circle.  These 
will  be  the  required  stream-lines  for  one  quadrant  of  the  space  round  B. 
Those  of  the  other  quadrants  are  symmetrical  to  them.  The  reason  for 
this  construction  is  the  well-known  geometrical  proposition,  that  if  0  C  is 
the  trace  of  a  spherical  surface,  and  if  the  dotted  ordinates  are  the  traces 
of  a  set  of  parallel  planes  perpendicular  to  the  radius  0  B,  and  dividing  it 
into  equal  parts,  those  planes  divide  the  spherical  surface  into  zones  of 
equal  area. 

15.  Compound  Stream-Lines  in  a  Wedge-shaped  Layer. — By  compounding 
two  sets  of  straight  stream-lines,  like  those  shown  in  the  lower  part  of 
Fig.  4,  radiating  from  a  pair  of  foci  in  the  same  axis  (that  is,  in  the  edge 
of  the  wedge-shaped  layer),  and  drawing  curves  diagonally  through  the 
network,  there  are  obtained  a  set  of  oval  stream-lines,  representing  the 
motion  of  a  current  which  diverges  in  all  directions  from  one  of  the  foci, 
and  converges  towards  the  other.  These  ovals  all  pass  through  the  foci, 
and  are  arranged  like  the  circular  stream-lines  of  Fig.  2.  It  may  be 
mentioned  that  they  are  of  the  same  figure  with  the  lines  of  force  of  a 
two-poled  magnet. 

Then,  by  combining  these  oval  stream-lines  with  the  parallel  straight 
stream-lines  of  the  upper  part  of  Fig.  4,  there  are  obtained  a  great  variety 
of  curved  lines,  representing  the  stream-lines  of  a  current  flowing  past  a 
solid  of  revolution.  Their  figures  resemble  in  a  general  way  those  of  the 
stream-lines  of  an  uniformly  thick  layer,  exemplified  in  Fig.  3. 


2l 


>30  THE   THERMODYNAMIC  THEORY  OF  WAVES. 


XXXII.— OX  THE  THERMODYNAMIC  THEORY  OF  WAVES  OF 
FINITE  LONGITUDINAL  DISTURBANCE.* 

1.  The  object  of  the  present  investigation  is  to  determine  the  relations 
which  must  exist  between  the  laws  of  the  elasticity  of  any  substance, 
whether  gaseous,  liquid,  or  solid,  and  those  of  the  wave-like  propagation 
of  a  finite  longitudinal  disturbance  in  that  substance ;  in  other  words,  of 
a  disturbance  consisting  in  displacements  of  particles  along  the  direction 
of  propagation,  the  velocity  of  displacement  of  the  particles  being  so  great 
that  it  is  not  to  be  neglected  in  comparison  with  the  velocity  of  propaga- 
tion. In  particular,  the  investigation  aims  at  ascertaining  what  conditions 
as  to  the  transfer  of  heat  from  particle  to  particle  must  be  fulfilled  in 
order  that  a  finite  longitudinal  disturbance  may  be  propagated  along  a 
prismatic  or  cylindrical  mass  without  loss  of  energy  or  change  of  type  : 
the  word  type  being  used  to  denote  the  relation  between  the  extent  of 
disturbance  at  a  given  instant  of  a  set  of  particles,  and  their  respective 
undisturbed  positions.  The  disturbed  matter  in  these  inquiries  may  be 
conceived  to  be  contained  in  a  straight  tube  of  uniform  cross-section  and 
indefinite  length. 

2.  Mass-Velocity. — A  convenient  quantity  in  the  present  investigation 
is  what  may  be  termed  the  mass-velocity  or  somatic  velocity — that  is  to  say, 
the  mass  of  matter  through  which  a  disturbance  is  propagated  in  a  unit 
of  time  while  advancing  along  a  prism  of  the  sectional  area  unity.  That 
mass-velocity  will  be  denoted  by  m. 

Let  S  denote  the  bulkiness,  or  the  space  filled  by  unity  of  mass,  of  the 
substance  in  the  undisturbed  state,  and  a  the  linear  velocity  of  advance  of 
the  wave;  then  we  have  evidently 

a  =  m  S.  ....     (1.) 

3.  Cinematical  Condition  of  Permanency  of  Type. — If  it  be  possible  for  a 
wave  of  disturbance  to  be  propagated  in  an  uniform  tube  without  change 
of  type,  that  possibility  is  expressed  by  the  uniformity  of  the  mass- velocity 
m  for  all  parts  of  the  wave. 

Conceive  a  space  in  the  supposed  tube,  of  an  invariable  length  Ax,  to 

*  Read  before  the  Royal  Society  of  London  on  Dec.  16,  1869,  and  published  in  the 
Philosophical  Transactions  for  1870. 


THE  THERMODYNAMIC  THEORY  OF  WAVES.         531 

be  contained  between  a  pair  of  transverse  planes,  and  let  those  planes 
advance  with  the  linear  velocity  a  in  the  direction  of  propagation.  Let 
the  values  of  the  bulkiness  of  the  matter  at  the  foremost  and  aftermost 
planes  respectively  be  denoted  by  s1  and  s2,  and  those  of  the  velocity  of 
longitudinal  disturbance  by  u^  and  m2.  Then  the  linear  velocities  with 
which  the  particles  traverse  the  two  planes  respectively  are  as  follows  : 
for  the  foremost  plane  ux  -  a,  for  the  aftermost  plane  u2  -  a.  The  uni- 
formity of  type  of  the  disturbance  involves,  as  a  condition,  that  equal 
masses  of  matter  traverse  the  two  planes  respectively  in  a  given  time, 
being  each,  in  unity  of  time,  expressed  by  the  mass-velocity;  hence  we 
have,  as  the  cinematkal  condition  of   uniformity  of  type,   the  following 

equation : 

a  —  u,       a  —  u.?       a  ,n  N 

m.     .  .  .     (2.) 


Sj  s2  S 

Another  way  of  expressing  the  same  condition  is  as  follows : 

A«=-j»As.    .        .        .        •     (3.) 

4.  Dynamical  Condition  of  Permanency  of  Type. — Let^q  and_p2  be  the  inten- 
sities of  the  longitudinal  pressure  at  the  foremost  and  aftermost  advanc- 
ing planes  respectively.  Then  in  each  unit  of  time  the  difference  of 
pressure,  p9  -  pv  impresses  on  the  mass  m  the  acceleration  u2  -  uv  and 
consequently,  by  the  second  law  of  motion,  we  have  the  following  value 
for  the  difference  of  pressure  : 

Ih  ~  lh  =  m  (u2  -  l'i)-         ■         ■         •     (40 

Then,  substituting  for  the  acceleration  u2  —  u1  its  value  in  terms  of  the 
change  of  bulkiness  as  given  by  equation  (3),  we  obtain,  for  the  dynamical 
condition  of  permanency  of  type,  the  following  equation, 

p.2  -1\  =  m2  (s1  -  Sg),          .         .         •     (5.) 

which  may  also  be  put  in  the  form  of  an  expression  giving  the  value  of 
the  square  of  the  mass-velocity,  viz. — 

»« = -  !*  =  - d/.    .    •    •  (so 

As  d  s 

The  square  of  the  linear  velocity  of  advance  is  given  by  the  following 
equation : 

a2  =  m2S2=-S2^.        .         •         •     (7.) 
d  s 

The  integral  form  of  the  preceding  equations  may  be  expressed  as  follows. 
Let  S,  as  before,  be  the  bulkiness  in  the  undisturbed  state,  and  P  the 


532  THE  THERMODYNAMIC   THEORY  OF  WAVES. 

longitudinal  pressure ;  then  in  a  wave  of  disturbance  of  permanent  type 
Ave  must  have  the  following  condition  fulfilled  : 

p  +  m-  s-P+  m2  S.  (8.) 

5.  Waves  of  Sudden  Disturbance. — The  condition  expressed  by  the  equa- 
tions  of  the  preceding  section  holds  for  any  type  of  disturbance,  continu- 
ous or  discontinuous,  gradual  or  abrupt.  To  represent,  in  particular,  the 
case  of  a  single  abrupt  disturbance,  we  must  conceive  the  foremost  and 
aftermost  advancing  planes  already  mentioned  to  coalesce  into  one.  Then 
P  is  the  longitudinal  pressure,  and  S  the  bulkiness,  in  front  of  the  advanc- 
ing plane  ;  p  is  the  longitudinal  pressure,  and  s  the  bulkiness,  behind  the 
advancing  plane;  and  the  advancing  plane  is  a  wave-front  of  sudden  com- 
pression or  of  sudden  m  refaction*  according  as  p  is  greater  or  less  than  P. 
The  squares  of  the  mass-velocity  and  of  the  linear  velocity  of  advance  arc 
respectively  as  follows  : 

2       P  ~  P 


S- 


(9.) 


p-P 


=  m*S*=Z ^.S2.      .         .         .     (10.) 

S  —  s 

The  velocity  of  the  disturbed  particles  is  as  follows : 

u  =  m(S  -  s)  =  J'  ~  -    =  V(p-P).(S-s);    •     (ii.) 

and  it  is  forward  or  backward  according  as  the  wave  is  one  of  compression 
or  of  rarefaction. 

The  energy  expended  in  unity  of  time,  in  producing  any  such  wave,  is 
expressed  by  p  u ;  for  the  wave  may  be  conceived  to  be  produced  in  a 
tube  closed  at  one  end  by  a  movable  piston  of  inappreciable  mass,  to 
which  there  is  applied  a  pressure  p  different  from  the  undisturbed  pressure 
P,  and  Avhich  consequently  moves  with  the  velocity  u.  The  way  in  which 
that  energy  is  disposed  of  is  as  follows:  actual  energy  of  the  disturbance, 

— —  j   work  done  in  altering  bulkiness,    — ~— - —  ;    and  the 

equation  of  the  conservation  of  energy  is 

*  Note,  added  1st  August,  1870. — Sir  William  Thomson  has  pointed  out  to  the 
author,  that  a  wave  of  sudden  rarefaction,  though  mathematically  possible,  is  an 
unstable  condition  of  motion ;  any  deviation  from  absolute  suddenness  tending  to 
make  the  disturbance  become  more  and  more  gradual.  Hence  the  only  wave  of  sudden 
disturbance  whose  permanency  of  type  is  physically  possible,  is  one  of  sudden  com- 
pression ;  and  this  is  to  be  taken  into  account  in  connection  with  all  that  is  stated  in 
the  paper  respecting  such  waves. 


THE   THERMODYNAMIC  THEORY   OF  WAVES.  533 

*«  =  *{tf  +  (p  +  P)(B-«)}.       .         .     (11  A.) 

6.  Thermodynamic  Conditions. — While  the  equations  of  the  two  preced- 
ing sections  impose  the  constancy  of  the  rate  of  variation  of  pressure  with 

bulkiness  during  the  disturbance  f  ~  =  —  m"  J  as  an  indispensable  con- 
dition of  permanency  of  type  of  the  wave,  they  leave  the  limits  of  pressure 
and  of  bulkiness,  being  four  quantities,  connected  by  one  equation  only 

\X^ ±3*  r=  *—  — —  r=  m2).     Two  only  of  those  quantities  can  be  arbi- 

\  5-i      *""    On  Oi   S 

trary ;  therefore,  one  more  equation  is  required,  and  that  is  to  be  deter- 
mined by  the  aid  of  the  laws  of  thermodynamics. 

It  is  to  be  observed,  in  the  first  place,  that  no  substance  yet  known 

dp  „ 

fulfils  the  condition  expressed  by  the  equation  — -  =  —  m-  =  constant, 

CL  s 

between  finite  limits  of  disturbance,  at  a  constant  temperature,  nor  in  a 
state  of  non-conduction  of  heat  (called  the  adiabatic  state).  In  order,  then, 
that  permanency  of  type  may  be  possible  in  a  wave  of  longitudinal  dis- 
turbance, there  must  be  both  change  of  temperature  and  conduction  of 
heat  during  the  disturbance. 

The  cylindrical  or  prismatic  tube  in  which  the  disturbance  is  supposed 
to  take  place  being  ideal,  is  to  be  considered  as  non-conducting.  Also, 
the  foremost  and  aftermost  transverse  advancing  planes,  or  front  and 
back  of  the  wave,  which  contain  between  them  the  particles  whose  pres- 
sure and  bulkiness  are  in  the  act  of  varying,  are  to  be  considered  as  non- 
conducting, because  of  there  being  an  indefinite  length  of  matter  before 
the  foremost  and  behind  the  aftermost  plane,  to  resist  conduction. 

The  transfer  of  heat,  therefore,  takes  place  wholly  amongst  the  particles 
undergoing  variation  of  pressure  and  bulkiness ;  and  therefore  for  any 
given  particle,  during  its  passage  from  the  front  to  the  back  of  the  wave, 
the  integral  amount  of  heat  received  must  be  nothing;  and  this  is  the  thermo- 
dynamic condition  which  gives  the  required  equation.  That  equation  is 
expressed  as  follows : 

f2r^  =  0;     .  .    •      .  •      (12.) 

J  <px 

in  which  r  denotes  absolute  temperature,  and  0  the  "  thermodynamic 
function."  The  value  of  that  function,  as  explained  in  various  papers 
and  treatises  on  thermodynamics,  is  given  by  the  following  formula :      \> 

(p  =  JclmxloS.T  +  X(r)+(-jV,     ■         •     02  a.) 


534         THE  THERMODYNAMIC  THEORY  OF  WAVES. 

in  which  J  is  the  dynamical  value  of  a  unit  of  heat ;  c,  the  real  specific 
heat  of  the  substance ;  \  (r),  a  function  of  the  temperature  alone,  which 
is  =  0  for  all  temperatures  at  which  the  substance  is  capable  of  approxi- 
mating indefinitely  to  the  perfectly  gaseous  state,  and  is  introduced  into 
the  formula  solely  to  provide  for  the  possible  existence  of  substances 
which  at  some  temperatures  are  incapable  of  approximating  to  the  per- 
fectly gaseous  state ;  and  U,  the  work  which  the  elastic  forces  in  unity  of 
mass  are  capable  of  doing  at  the  constant  temperature  r.  The  substitution 
for  the  integral  in  equation  (12)  of  its  value  in  terms  of  p  and  s  for  any 
particular  substance,  gives  a  relation  between  the  limits  of  pressure  p1 
and  ]>.„  and  the  limits  of  bulkiness  sx  and  s2,  which  being  combined  with 
equation  (5),  or  with  any  one  of  the  equivalent  equations  (6),  (8),  or  (9), 
completes  the  expression  of  the  laws  of  the  propagation  of  waves  of  finite 
longitudinal  disturbance  and  permanent  type  in  that  particular  substance. 

7.  Assumption  as  to  Transfer  of  Heat. — In  applying  the  principles  of  the 
preceding  section  to  the  propagation  of  waves  of  longitudinal  disturbance, 
it  is  obviously  assumed  that  the  transfer  of  heat  takes  place  between  the 
various  particles  which  are  undergoing  disturbance  at  a  given  time,  in 
such  a  manner  as  to  ensure  the  fulfilment  of  the  dynamical  condition  of 
permanency  of  type.  It  appears  highly  probable,  that  how  great  soever 
the  resistance  of  the  substance  to  the  conduction  of  heat  may  be,  that 
assumption  as  to  the  transfer  is  realised  when  the  disturbance  is  sudden, 
as  described  in  sec.  5 ;  for  then  particles  in  all  the  successive  stages  of  the 
change  of  pressure  and  bulkiness  within  the  limits  of  the  disturbance  are 
at  inappreciable  distances  from  each  other ;  so  that  the  resistance  to  the 
transfer  of  heat  between  them  is  inappreciable. 

But  when  the  disturbance  is  not  sudden,  it  is  probable  that  the  assump- 
tion as  to  the  transfer  of  heat  is  fulfilled  in  an  approximate  manner  only; 
and  if  such  is  the  case,  it  follows  that  the  only  longitudinal  disturbance  which 
can  be  propagated  with  absolute  permanence  of  type  is  a  sudden  disturbance. 

8.  Combination  of  the  Dynamic  and  Thermodynamic  Equations. — In  every 
fluid,  and  probably  in  many  solids,  the  quantity  of  heat  received  during  an 
indefinitely  small  change  of  pressure  dp  and  of  bulkiness  d  s  is  capable  of 
being  expressed  in  either  of  the  following  forms  : 

r  d  <j>  d  r  ,       ,        d  t  , 

— =J-  =  csT~dp  +  cpT-  d  s; 
J  s  d  p  v  d  s 

in  which  c  and  cp  denote  the  specific  heat  at  constant  bulkiness  and  at 

constant  pressure  respectively ;  and  the  differential  coefficients  j—  and  -=— 

of  the  absolute  temperature  are  taken,  the  former  on  the  supposition  that 
the  bulkiness  is  constant,  and  the  latter  on  the  supposition   that   the 


THE   THERMODYNAMIC  THEORY  OF  WAVES.  535 

pressure  is  constant.     Let  it  now  be  supposed  that  the  bulkiness  varies 

with  the  pressure  according  to  some  definite  law ;  and  let  the  actual  rate 

d  s 
of  variation  of  the  bulkiness  with  the  pressure  be  denoted  by  -=- .     Then 

equation  (12)  may  be  expressed  in  the  following  form : 
(P2  (     (It,        dr    ds\ 

Now,  according  to  the  dynamic  condition  of  permanence  of  type,  we 
have  by  equation  (6), 

d  s  _         1 
dp  m2' 

which,  being  substituted  in  the  preceding  integrals,  gives  the  following 
equations  from  which  to  deduce  the  square  of  the  mass-velocity : 


C">*35-*r3=*-     •  ™ 


It  is  sometimes  convenient  to  substitute  for  cp  -=—  the  following  value, 

which  is  a  known  consequence  of  the  laws  of  thermodynamics  : 

dr  dr    ,   r  dp  ,-  „     v 

v  d  s        s  d  s       J  d  t 

the  differential  coefficient  ~~  being  taken   on  the  supposition  that  s  is 

d  t 

constant.     The  equations  (13)  and  (13  a)  are  applicable  to  all  fluids,  and 
probably  to  many  solids  also,  especially  those  which  are  isotropic. 

The  determination  of  the  squared  mass-velocity,  m2,  enables  the  bulki- 
ness s  for  any  given  pressure  p,  and  the  corresponding  velocity  of 
disturbance  u,  to  be  found  by  means  of  the  following  formula?,  which  are 
substantially  identical  with  equations  (8)  and  (3)  respectively : 

S  =  S+^/;  .         •         •     (U.) 

m 


u 


=  »<S -«)=*--*     .         .         .     (15.) 


Equation  (15)  also  serves  to  calculate  the  pressure  p  corresponding  to  a 
given  velocity  of  disturbance  u.  It  may  here  be  repeated  that  the  linear 
velocity  of  advance  is  a  =  m  S  (equation  1). 


53G 


THE  THERMODYNAMIC  THEORY  OF  WAVES. 


9.  Application  to  a  Perfect  Gas. — In  a  perfect  gas,  the  specific  heat  at 
constant  volume,  cs,  and  the  specific  heat  at  constant  pressure,  cp,  are  both 

c 
constant;  and,  consequently,  bear  to  each  other  a  constant  ratio,  J\  whose 

cs 

value  for  air,  oxygen,  nitrogen,  and  hydrogen  is  nearly  1*41,  and  for 
steam-gas  nearly  1*3.  Let  this  ratio  be  denoted  by  y  Also,  the  differ- 
ential coefficients  which  appear  in  equations  (13)  and  (13  a)  have  the 
following  values : — 

(I   T  T  8  S 


dp       p       J  (cp-  r.)       J  (y-  l)c8} 
V  P 


dr 

d  s 


J  (<*-',)     J(y-iK'     r 


(16.) 


';  v  _p  _  J  (')>  -  O  =  J  Or  -  !) 

d  T        T  s  s 


AVhcn  these  substitutions  are  made  in  equation  (13),  and  constant  common 
factors  cancelled,  it  is  reduced  to  the  following: 


I    "  dp  .  [m?  s  —  yv}  =  Q- 
J  Pi 


(17.) 


But  according  to  the  dynamical  condition  of  permanence  of  type,  as  ex- 
pressed in  equation  (8),  we  have  in2  s  =  m2  S  +  P  —  p ;  whence  it  follows 
that  the  value  of  the  integral  in  equation  (17)  is 

jP'2dp  .  {m2S  +  P  -  (y  +  l)p)  =  (m2S  +  P)  [p2-pj 
J  Pi 


-7|J(^-i'l)  =  0; 


which,  being  divided  by  p2  —  pv  gives  for  the  square  of  the  mass-velocity 
of  advance  the  following  value  : 


///- 


|{(v+D-HiLl-p}-  •    •  <ia> 


The  square  of  the  linear  velocity  of  advance  is 

flP«rfSP*s{(y+l).8L±&-.p}..  .     (19.) 

The  velocity  of  disturbance  u  corresponding  to  a  given  pressure  p,  or, 


THE  THERMODYNAMIC   THEORY   OF  WAVES.  537 

conversely,  the  pressure  f  corresponding  to  a  given  velocity  of  disturbance, 
may  be  found  by  means  of  equation  (15). 

Such  are  the  general  equations  of  the  propagation  of  waves  of  longi- 
tudinal disturbance  of  permanent  type  along  a  cylindrical  mass  of  a  perfect 
gas  whose  undisturbed  pressure  and  bulkiness  are  respectively  P  and  S. 
In  the  next  two  sections  particular  cases  Avill  be  treated  of. 

10.  Wave  of  Oscillation  in  a  Perfect  Gas. — Let  the  mean  between  the 
two  extreme  pressures  be  equal  to  the  undisturbed  pressure ;  that  is,  let 

lh+lh  =  p.     ....     (20.) 


.     (21) 


then  equations  (18)  and  (19)  become  simply 

s  ' 

and 

a2  =  yPS;       ....     (22.) 

the  last  of  which  is  Laplace's  well-known  law  of  the  propagation  of  sound. 
The  three  equations  of  this  section  are  applicable  to  an  indefinitely  long 
series  of  waves,  in  which  equal  disturbances  of  pressure  take  place  alter- 
nately in  opposite  directions. 

11.  Wave  of  Permanent  Compression  or  Dilatation  in  a  Tube  of  Perfect 
Qas. — To  adapt  equation  (18)  to  the  case  of  a  wave  of  permanent  com- 
pression or  dilatation  in  a  tube  of  perfect  gas,  the  pressure  at  the  front 
of  the  wave  is  to  be  made  equal  to  the  undisturbed  pressure;  and  the 
pressure  at  the  back  of  the  wave  to  the  final  or  permanently  altered 
pressure.  Let  the  final  pressure  be  denoted  simply  by  ]) ;  then  pt  =  Pr 
and  jp2  =  ]) ;  giving  for  the  square  of  the  mass-velocity 

«2  =  g{(y+i)f  +  (y-i)|}>  •       •    (23-) 

for  the  square  of  the  linear  velocity  of  advance 

ft2  =  m2S2  =  s{(y+l)f +(y-l)|},         •     (24.) 
and  for  the  final  velocity  of  disturbance 

"  ~1?~T~  =  (■?  ~  PW  -  v P  V     (25  ) 

Equations  (23)   and  (24)  show  that  a  wave  of  condensation  is  pro- 
pagated faster,  and  a  wave  of  rarefaction  slower,  than  a  series  of  waves  of 


53S         THE  THERMODYNAMIC  THEORY  OF  WAVES. 

oscillation.  They  further  show  that  there  is  no  upper  limit  to  the  velocity 
of  propagation  of  a  wave  of  condensation;  and,  also,  that  to  the  velocity  of 
propagation  of  a  wave  of  rarefaction  there  is  a  lower  limit,  found  by 
making  p  =  0  in  equatious  (23)  and  (24).  The  values  of  that  lower 
limit,  for  the  squares  of  the  mass-velocity  and  linear  velocity  respectively, 
are  as  follows  : — 

«"(i>  =  o)  =  (Y7g)F;   .      .     ..   (20.) 

aMj,  =  0)  =  (y-31)P*;.        .        .    (27.) 

and  the  corresponding  value  of  the  velocity  of  disturbance,  being  its 
negative  limit,  is 


•<f-o>=  -Vlf-4}'  ■  <2&) 


7 

It  is  to  be  borne  in  mind  that  the  last  three  equations  represent  a  state 
of  matters  which  may  be  approximated  to,  but  not  absolutely  realised. 

Equation  (25)  gives  the  velocity  with  which  a  piston  in  a  tube  is  to  be 
moved  inwards  or  outwards,  as  the  case  may  be,  in  order  to  produce  a 
change  of  pressure  from  P  to^,  travelling  along  the  tube  from  the  piston 
towards  the  farther  end.  Equation  (25)  may  be  converted  into  a  quad- 
ratic equation,  for  finding  p  in  terms  of  u ;  in  other  words,  for  finding 
what  pressure  must  be  applied  to  a  piston  in  order  to  make  it  move  at  a 
given  speed  along  a  tube  filled  with  a  perfect  gas,  whose  undisturbed 
pressure  and  bulkiness  are  P  and  S.  The  quadratic  equation  is  as 
follows  : 

,_(iP  +  3-M,,),_:^.Prf  +  p_0. 

and  its  alternative  roots  are  given  by  the  following  formula : 

The  sign  +  or  —  is  to  be  used,  according  as  the  piston  moves  inwards, 
so  as  to  produce  condensation,  or  outwards  so  as  to  produce  rarefaction. 
Suppose,  now,  that  in  a  tube  of  unit  area,  filled  with  a  perfect  gas  whose 
undisturbed  pressure  and  volume  are  P  and  S,  there  is  a  piston  dividing 
the  space  within  that  tube  into  two  parts,  and  moving  at  the  uniform 
velocity  u :  condensation  will  be  propagated  from  one  side  of  the  piston, 
and  rarefaction  from  the  other ;  the  pressures  on  the  two  sides  of  the 
piston  will  be  expressed  by  the  two  values  of  p  in  equation  (29) ;  and  the 


THE  THERMODYNAMIC   THEORY  OF  WAVES.  539 

force  required  in  order  to  keep  the  piston  in  motion  will  be  the  difference 
of  these  values  ;  that  is  to  say, 

.*'-■  -vivp+Hr}--  •  <*) 

Two  limiting  cases  of  the  last  equation  may  be  noted  :  first,  if  the  velocity 
of  the  piston  is  very  small  compared  with  the  velocity  of  sound,  that  is,  if 

— p  is  very  small,  we  have 


A p  nearly  =  2u.  J(~t-);       •         •     (30a.) 


secondly,  if  the  velocity  of  the  piston  is  very  great  compared  with  the 

yP 

velocity  of  sound,  that  is,  if  ^— r2  is  very  small,  we  have 

Ap  nearly  =  *7  "t^"'.  •         ■     (30  b.) 

12.  Absolute  Temperature. — The  absolute  temperature  of  a  given  particle 
of  a  given  substance,  being  a  function  of  the  pressure  p  and  bulkiness  s, 
can  be  calculated  for  a  point  in  a  wave  of  disturbance  for  which  p  and  s 
are  given.  In  particular,  the  absolute  temperature  in  a  perfect  gas  is  given 
by  the  following  well-known  thermodynamic  formula : 

.  .  .      (31.) 


and  if,  in  that  formula,  there  be  substituted  the  value  of  s  in  terms  of  p, 
given  by  equations  (8)  and  (18)  combined,  we  find,  for  the  absolute  tem- 
perature of  a  particle  at  which  the  pressure  is  p,  in  a  wave  of  permanent 
type,  the  following  value  : 

__PS_    (y +1X^+^-2/.         /32N 
r-Jfe-^)'(y  +  1)(i>i+K>i>-2p2'     "    {'^ 

P  s 

in  which  the  first  factor  = r  is  obviously  the  undisturbed  value  of 

J  (Cp  -  <y 

the  absolute  temperature.     For  brevity's  sake  let  this  be  denoted  by  T. 

The  following  particular  cases  may  be  noted.     In  a  wave  of  oscillation, 
as  defined  in  sec.  10,  we  have  pt  +  p.2  =  2  P;  and,  consequently, 

,  =  T.fr+1)  £."-**.  •  •     (32A.) 


540         THE  THERMODYNAMIC  THEORY  OF  WAVES. 

In  a  wave  of  permanent  condensation  or  rarefaction,  as  described  in  sec. 
11,  let  2>!  =  P,  p.-,  =  P;  then  the  final  temperature  is 

_T    (y  +  l)Pj>  +  (y -I)/'"-' 
•(y+l)Pp  +  (y-l)P2' 

13.  Types  of  Disturbance  capable  of  Permanence. — In  order  that  a  par- 
ticular type  of  disturbance  may  be  capable  of  permanence  during  its 
propagation,  a  relation  must  exist  between  the  temperatures  of  the 
particles  and  their  relative  positions,  such  that  the  conduction  of  heat 
between  the  particles  may  effect  the  transfers  of  heat  required  by  the 
thermodynamic  conditions  of  permanence  of  type  stated  in  sec.  6. 

I  luring  the  time  occupied  by  a  given  phase  of  the  disturbance  in 
traversing  a  unit  of  mass  of  the  cylindrical  body  of  area  unity  in  which 
the  wave  is  travelling,  the  quantity  of  heat  received  by  that  mass,  as 
determined  by  the  thermodynamic  conditions,  is  expressed  in  dynamical 
units  by 

t  d  <•/,. 

The  time  daring  which  that  transfer  of  heat  takes  place  is  the  reciprocal 

of  the  mass-velocity  of  the  wave.     Let    ,     be  the  rate  at  which  tem- 
m  a  X 

perature  varies  with  longitudinal  distance,  and  /.■  the  conductivity  of  the 
substance,  in  dynamical  units;  then  the  same  quantity  of  heat,  as  deter- 
mined by  the  laws  of  conduction,  is  expressed  by 


1       7/.'; 
— ■  .  a 
m 


(<:;:)• 


The  equality  of  these  two  expressions  gives  the  following  general  differ- 
ential equation  for  the  determination  of  the  types  of  disturbance  that  are 
capable  of  permanence : 

d 


mrd.  <p  =  d.(l:CjQ.       .         .         .     (33.) 


The   following   are   the   results    of  two   successive  integrations   of   that 
differential  equation  : 

.     (33  a.) 


(It       A  +  tii/t  d  (j> 

'  =  b  +  /a+"/W'    •      •    <33E-> 


THE  THERMODYNAMIC  THEORY  OF  WAVES.         541 

in  which  A  and  B  are  arbitrary  constants.  The  value  of  A  depends  on 
the  magnitude  of  the  disturbance,  and  that  of  B  upon  the  position  of  the 
point  from  which  x  is  reckoned.  In  applying  these  general  equations  to 
particular  substances,  the  values  of  t  and  <p  are  to  be  expressed  in  terms 
of  the  pressure  p,  by  the  aid  of  the  formula?  of  the  preceding  section, 
when  equation  (33  b)  will  give  the  value  of  x  in  terms  of  p,  and  thus  will 
show  the  type  of  disturbance  required. 

Our  knowledge  of  the  laws  of  the  conduction  of  heat  is  not  yet  sufficient 
to  enable  us  to  solve  such  problems  as  these  for  actual  substances  with 
certainty.  As  a  hypothetical  example,  however,  of  a  simple  kind,  Ave  may 
suppose  the  substance  to  be  perfectly  gaseous  and  of  constant  conductivity. 
The  assumption  of  the  perfectly  gaseous  condition  gives,  according  to  the 
formulas  of  the  preceding  sections, 

PS  (y  +  1)  (ft  +.p2)P  ~  if 


and 


'(y-l)Jc/(y+l)(ft+dp2)P-2P2 

7  ,  y  +  1      fft  +lh        1  ? 


It  is  unnecessary  to  occupy  space  by  giving  the  whole  details  of  the 
calculation ;  and  it  may  be  sufficient  to  state  that  the  following  are  the 
results.     Let 

m       ft  +  ft  _  „ 


1              2        ~~  " 

ft  "ft  -  a  ■ 

9       —  'n  > 

then 

dx       clx                 k              (y  -  1)  (ft  +  ft)  -  4  q      /gl  x 
dp  ~  dq  ~  (y  +  1)  mJcs  '                  q\  -  q2 

h           f(y-l)(ft+ft)    h    'lft s<h±l 

x~  (y  +  i)fnJc-(.         a  a                  °qi-q 

+  2hy1xlog.(l-|)}.           •         .     (34  A.) 

In  equation  (34  a)  it  is  obvious  that  x  is  reckoned  from  the  point  where 
q  =  0 :  that  is,  where  the  pressure  p  =  2^j^ ;  a  mean  between  the 
greatest  and  least  pressures.     The  direction  in  which  x  is  positive  may  be 


542         THE  THERMODYNAMIC  THEORY  OF  WAVES. 

either  the  same  with  or  contrary  to  that  of  the  advance  of  the  wave ; 
the  former  case  represents  the  type  of  a  wave  of  rarefaction,  the  latter 
that  of  a  wave  of  compression.      For  the  two  limiting  pressures  when 

a  —  ±  ?„  —  becomes  infinite,  and  x  becomes  positively  or  negatively 
*  l   aq 

infinite ;  so  that  the  wave  is  infinitely  long.     The  only  exception  to  this 

is  the  limiting  case,  when  the  conductivity  k  is  indefinitely  small ;  and 

i  dx  .   .  „  ., 

then  we  have  the  following  results  :  when  p  =  pv  or  p  =  p2,  j-  is  infinite, 

dx 
and  x  is  indefinite;  and  for  all  values  of  p  between  px  and^;2,  -3-  anc*  x 

are  each  indefinitely  small.  These  conditions  evidently  represent  the  case 
of  a  wave  of  abrupt  rarefaction  or  compression,  already  referred  to  in  sees, 
(i  and  7. 

Si  itlement  (Dec.,  18G9). 

Note  as  to  previous  investigations. — Four  previous  investigations  on  the 
subject  of  the  transmission  of  waves  of  finite  longitudinal  disturbance  may 
be  referred  to,  in  order  to  show  in  what  respects  the  present  investigation 
was  anticipated  by  them,  and  in  what  respects  its  results  are  new. 

The  first  is  that  of  Poisson,  in  the  Journal  de  VEcole  Poll/technique,  Vol. 
VII.,  Cahier  14,  p.  319.  The  author  arrives  at  the  following  general 
equations  for  a  gas  fulfilling  Mariotte's  law  : — 

>1<P        A  .       deb) 

,,,=■' v  - at  -  j,1  r 

d(j>  d  0       1     dj?  _  n 

dJ  +  a7l.r  +  2'   d.r  ~   J' 

in  which  <p  is  the  velocity-function ;  -^  the  velocity  of  disturbance,  at 
the  time  t,  of  a  particle  whose  distance  from  the  origin  is  x ;  a  is  the  limit 
to  which  the  velocity  of  propagation  of  the  wave  approximates  when  ^ 

becomes  indefinitely  small,  viz.  J'll°,  pQ  being  the  undisturbed  pressure 

VdPo 

and  p0  the  undisturbed  density ;  and  /  denotes  an  arbitrary  function. 

This  equation  obviously  indicates  the  quicker  propagation  of  the  parts  of 

the  wave  where  the  disturbance  is  forward  (that  is,  the  compressed  parts), 

and  the  slower  propagation  of  the  parts  where  the  disturbance  is  backward 

(that  is,  the  dilated  parts'). 


THE  THERMODYNAMIC  THEORY  OF  WAVES.         543 

The  second  is  that  of  Mr.  Stokes,  in  the  Philosophical  Magazine  for 
November,  1848,  3rd  series,  Vol.  XXXIII,  p  349,  in  which  that  author 
shows  how  the  type  of  a  series  of  waves  of  finite  longitudinal  disturbance 
in  a  perfect  gas  alters  as  it  advances,  and  tends  ultimately  to  become  a 
series  of  sudden  compressions  followed  by  gradual  dilatations. 

The  third  is  that  of  Mr.  Airy,  Astronomer-Eoyal,  in  the  Philosophical 
Magazine  for  June,  1849,  3rd  series,  Vol.  XXXIV.,  p.  401,  in  which  is 
pointed  out  the  analogy  between  the  above-mentioned  change  of  type  in 
waves  of  sound,  and  that  which  takes  place  in  sea-waves  when  they  roll 
into  shallow  water. 

The  fourth  and  most  complete,  is  that  of  the  Rev.  Samuel  Earnshaw, 
received  by  the  Eoyal  Society  in  November,  1858,  read  in  January,  1859, 
and  published  in  the  Philosophical  Transactions  for  1860,  page  133.  That 
author  obtains  exact  equations  for  the  propagation  of  waves  of  finite  longi- 
tudinal disturbance  in  a  medium  in  which  the  pressure  is  any  function  of  the 
density ;  he  shows  what  changes  of  type,  of  the  kind  already  mentioned, 
must  go  on  in  such  waves ;  and  he  points  out,  finally,  that  in  order  that 

cl  v              cl  V 
the  type  may  be  permanent  p2  — -  ( = 7     m  *ne  notation  of  the  present 

It  f)  Oj  s 

paper)  must  be  a  constant  quantity ;  being  the  proposition  which  is 
demonstrated  in  ail  elementary  way  near  the  beginning  of  the  present 
paper.  Mr.  Earnshaw  regards  that  condition  as  one  which  cannot  be 
realised. 

The  new  results,  then,  obtained  in  the  present  paper  may  be  considered 
to  be  the  following : — The  conditions  as  to  transformation  and  transfer  of 
heat  which  must  be  fulfilled,  in  order  that  permanence  of  type  may  be 
realised,  exactly  or  approximately ;  the  types  of  wave  which  enable  such 
conditions  to  be  fulfilled,  with  a  given  law  of  the  conduction  of  heat ; 
and  the  velocity  of  advance  of  such  waves. 

The  method  of  investigation  in  the  present  paper,  by  the  aid  of  mass- 
velocity  to  express  the  speed  of  advance  of  a  wave  is  new,  so  far  as  I  know ; 
and  it  seems  to  me  to  have  great  advantages  in  point  of  simplicity,  enabling 
results  to  be  demonstrated  in  a  very  elementary  manner,  which  otherwise 
would  have  required  comparatively  long  and  elaborate  processes  of 
investigation. 


541  THE   EFFICIENCY   OF   FROPELLERS. 


XXXIIL— ON  THE  THEORETICAL  LIMIT  OF  THE  EFFICIENCY 

OF  PROPELLERS.  * 

1.  The  following  statement  of  a  certain  theoretical  limit  towards  which 
the  efficiency  of  propellers  may  be  made  to  approximate  by  mechanical 
improvements,  and  of  certain  causes  which  make  the  actual  efficiency  fall 
short  of  that  limit,  although  it  involves  no  new  principle,  may  be  useful 
in  the  present  state  of  the  question  of  propulsion. 

To  avoid  complexity,  let  the  water  be  still  when  the  action  of  the  pro- 
pelling apparatus  begins;  so  that  its  velocity  relatively  to  the  vessel 
(which  may  be  called  the  velocity  of  feed  of  the  propelling  apparatus),  is 
simply  equal  and  opposite  to  the  speed  of  the  vessel.  Let  that  velocity  be 
denoted  by  v. 

•1.  Let  .s  be  the  true  slip,  or  acceleration,  or  additional  velocity,  impressed 
on  the  water  by  the  propelling  apparatus ;  so  that  v  +  s  is  what  may  be 
called  the  velocity  of  discharge  from  the  propelling  apparatus,  relatively  to 
the  vessel. 

3.  If  W  is  the  weight  of  the  mass  of  water  acted  upon  in  each  second, 
and  g  the  acceleration  produced  by  gravity  in  one  second,  the  reaction  of 
the  water,  equal  and  opposite  to  the  resistance  of  the  ship,  is  well  known 
to  be  given  by  the  following  formula, 

E  =  ^i;      ....         (1.) 

and  the  effective  power,  or  useful  work  per  second,  done  in  driving  the  ship, 
by  the  formula, 

IU  =  ^-S (2.) 

9 

4.  When  the  apparatus  first  takes  up  a  supply  of  water,  then  carries  it 
for  a  time  along  with  the  vessel,  and  then  discharges  it,  the  reaction  R 

W 

may  be  the  resultant  of  a  forward  reaction  —  (v  -f-  s)  exerted  by  the 

*  From  The  Engineer  of  Jau.  11,  1S67. 


THE   EFFICIENCY   OF   PROPELLERS.  545 

W  r 
water  when  discharged,  and  a  backward  reaction—  — ,  exerted  by  the 

9 

water  when  received;  but  in  this,  as  in  other  cases,  the  resultant  reaction 
.    Ws 

IS   . 

9 

5.  In  order  that  the  loss  of  work  may  be  the  least  possible,  the  pro- 
pelling instrument  should  be  so  contrived  as  to  act  on  each  particle  of 
water  with  a  velocity  at  first  simply  equal  to  the  velocity  of  feed  r,  and 
gradually  increasing  at  an  uniform  rate  up  to  the  velocity  of  discharge 
v  +  s.  If  this  condition  were  fulfilled,  the  mean  velocity  with  which  the 
propelling  apparatus  would  have  to  work  against  the  reaction  E  would  be 

v  +  - ;  and  the  total  work  per  second  Avould  be 

/         *\       Wvs   ,   ^Ys2  ,0, 

in  which  equation  the  first  term  is  the  useful  work  per  second,  as  already 
given  in  equation  (2),  and  the  second  term  is  the  lost  work,  reduced  to  a 
minimum;  for  it  is  easy  to  see  that  this  lost  work  is  simply  the  actual 
energy  of  the  discharged  water,  moving  astern  with  the  velocity  s  relatively 
to  still  water ;  and  that  quantity  of  energy  must  necessarily  be  lost  under 
all  circumstances. 

6.  The  corresponding  value  of  the  efficiency,  or  ratio  of  the  useful  to 
the  total  work,  is 


s 

!)  +  - 

'    2 


(4.) 


and  this  is  the  theoretical  limit  to  the  efficiency  of  a  propeller. 

7.  It  is  certain  that  no  actual  propelling  instrument  has  ever  attained 
the  limit  of  efficiency  stated  above.  It  is  probable  that  the  nearest 
approach  to  the  theoretical  limit  of  efficiency  is  made  by  the  oar ;  for  the 
skilful  rower  pulls  with  a  nearly  uniform  force,  and  thus  produces  a  gradual 
acceleration  of  the  water  laid  hold  of  by  the  blade. 

In  the  following  articles  are  described  some  causes  of  additional  loss  of 
work,  irrespective  of  friction.  Those  causes  may  be  briefly  enumerated 
thus  : — Suddenness  of  change  from  the  velocity  of  feed  to  the  velocity  of 
discharge;  transverse  motions  impressed  on  the  water;  and  waste  of  the 
energy  of  the  feed  water ;  and  the  effect  of  each  of  them  is  to  waste  work- 
in  the  production  of  eddies. 

8.  Suddenness  of  the  change  from  the  velocity  of  feed  to  the  velocity 
of  discharge  operates  to  the  full  extent  in  every  case  in  which  the  pro- 
peller, instead  of  beginning  its  action  with  the  velocity  of  feed  r,  and 

2  M 


54G  THE   EFFICIENCY"   OF   PllOPELLEIlS. 

gradually  increasing  its  speed  to  the  velocity  of  discharge,  v  +  s,  acts 
throughout  with  the  velocity  of  discharge  v  +  s.  Thus  the  total  work 
per  second  becomes 

R(»  +  s)  = 1 :  .  .  (•).) 

9  0 

so  that  the  lost  work,  instead  of  being  simply  equal  to  the  actual  energy 
of  the  water  discharged  per  second,  is  increased  to  double  that  quantity 
of  energy;  and  thus  besides  the   unavoidable  loss   of  work,  there  is  a 

or  unnecessary  loss  of  work  per  second,  expressed  by    „         The 

corresponding  value  of  the  efficiency  is 

6. 
+  s 

The  object  of  such  inventions  as  Woodcroft's  gaining  pitch  screw,  and 
Mangin's  screw,  is  to  diminish  waste  of  the  kind  that  has  now  been 
ibed  ;  and  in  Kuthven's  form  of  centrifugal  pump  the  same  principle 
appeals  to  be  Kept  in  view.  The  same  is  also  the  object  of  making 
paddles  feather,  so  as  to  enter  the  water  edgewise.  It  is  probable  that 
the  object  is  partly  attained  by  all  those  inventions,  but  by  none  of  them 
wholly;  and  such  being  the  case,  the  loss  of  work  may  be  expressed  by 

— —  ;  c  being  a  multiplier  not  exceeding  unity,  depending  on  the  mode 

of  action  of  the  particular  propeller  employed.  It  is  probable  that  in  a 
well-designed  centrifugal  pump  c  may  be  very  small ;  while  for  ordinary 
paddles  and  screws  it  is  =  1. 

9.  Transverse  motions  are  impressed  on  the  discharged  water  by  all 
forms  of  the  screw  and  paddle. 

Let  u  denote  the  transverse  component  (whether  vertical,  horizontal,  or 
inclined)  of  the  velocity  of  the  discharged  water.     Then,  if  that  motion  is 

W  "- 

impressed  gradually,  the  Avork  wasted  per  second  in  producing  it  is  — —  , 

W  u2 
and  if  more  or  less  suddenly  (1  +  cx)   ;  cx  being  a  multiplier  not 

exceeding  unity ;  and  the  latter  is  the  more  common  case.  The  jet  pro- 
peller is  free  from  this  cause  of  waste  of  work, 

10.  Waste  of  the  energy  of  the  feed  water  may  occur  in  those  cases  in 
which  the  water  acted  upon  by  the  propelling  apparatus  is  received  into 
the  vessel,  and  carried  along  with  her  before  being  discharged ;  that  is  to 
say,  in  certain  forms  of  jet  propeller.  The  feed  -water  has,  relatively  to 
the  ship,  the  velocity  v;  and  in  order  that  the  energy  due  to  that  relative 
velocity  may  not  be  wasted,  it  is  necessary  either  that   each  particle   of 


THE   EFFICIENCY   OF   PROPELLERS. 


547 


water  should  begin  to  be  acted  upon  by  the  propelling  apparatus  without 
losing  any  part  of  that  relative  velocity  (as  in  the  case  of  the  screw  and 
the  paddle),  or  that  any  loss  of  velocity  should  be  compensated  by  a  cor- 
responding increase  of  pressure,  to  co-operate  with  the  propelling  apparatus 
in  producing  the  velocity  of  discharge  v  +  s.  For  example,  if  the  feed 
water  is  taken  into  a  space  in  which  it  is  sensibly  at  rest  relatively  to  the 
ship,  it   should  produce  by  its  impulse  on  the  water  previously  in  that 

space  the  whole  head  of  pressure  due  to  its  relative  velocity  — ,  otherwise 

energy  will  be  wasted  in  producing  eddies  in  the  confined  water,  to  an 

f  "W  »i2 

amount  per  second  which  may  be  expressed  by  ' — - —  ;  /  being  a  multi- 
plier whose  value  may  range  from  an  insensibly  small  fraction  to  unity, 
according  to  the  degree  of  suddenness  with  which  the  velocity  of  feed 
is  checked. 

11.  The  multiplier  /  may  even  take  values  greater  than  unity,  if  the 
feed  water  is  "throttled:"  that  is,  if  it  is  drawn  through  openings  so 
narrow  that  the  velocity  becomes  for  a  time  greater  than  v,  and  then  falls 
suddenly  by  the  water  entering  a  large  receiver. 

12.  The  following  is  a  summary  of  the  results  arrived  at  in  the  pre- 
ceding articles  : — 

Ratio  to  the 
Useful  Work. 


A.  Useful  work  per  second, 

B.  Work   unavoidably   lost,    being  the 

energy  of  the  discharged  water, 

C.  Additional    work    wasted,    through 

suddenness   of  action   of  the  pro- 
peller on  the  water, 

D.  Work  wasted  through  transverse  motion  of 

the  water  if  produced  gradually,     . 

E.  Additional  work  wasted  if  transverse  mo- 

tion is  produced  suddenly, 

F.  Work  Avasted,  through  loss  of  energy  of 

feed  water,  .... 


nv 

W  v  s 
(J 

E.9 

W  s2 

2 

2</ 

oR 

s     fWr 

2~ 

2  9 

I 

W«2 

2<7 

)- 

c^Vu2 

£ 

2  9 

)I 

fWv2 

%9 

■1  r  S 


c,  u 


2vs 

ii 

2s 


1 3.  The  following  particular  case  may  be  specially  mentioned.     Sup- 
pose that  the  velocity  of  discharge  is  impressed  gradually  (so  that  c  =  0), 


.rj4<S  THE   EFFICIENCY   OF   PROPELLERS. 

that  there  is  no  transverse  motion  of  the  discharged  water  (so  that  u  =  0), 
and  that  all  the  energy  due  to  the  velocity  of  feed  is  lost  (so  that  /  =  1). 
Then  the  total  work  per  second  is — 

being  the  actual  energy  corresponding  to  the  velocity  of  discharge:  and 
the  corresponding  efficiency  is — 

¥+7y     ....       (8.) 

In  this  case  the  losl  work  becomes  a  minimum,  and  the  efficiency  a 
maximum*  when  s  =  v;  and  such  is  very  nearly  the  case  in  the 
"  Nautilus"  and  the  "  Vvaterwitch." 

14.   In  the  following  example  the  data  assumed  are — 

W  =  5  tons,  or  11,200  lbs.  per  second  ; 

'•     =  15  ft.  per  second  :  .-•  =  15  ft.  per  second  ; 

s<.  that  /•  +  s  —  30  ft.  per  second,  and  R  =  5217  lbs.:  the  velocity  is 
supposed  to  be  impressed  gradually;  and  n  =  <>.     Then — 

Ratio  t<>       Foot-pounds 
il  Work,     per  second.         H.P. 

A.  Useful  work 1  78,255  I  li' 

B.  Necessary  loss  of  work,  ....',  39,127^  71 

F.   Additional  loss  if  energy  of  feed  water  is 

wholly  wasted,       .....',  39, 1 27£  71 

Total  work,  including  the  above  losses,  but 
exclusive  of  friction  (the  efficiency 
being  0*5), 2  15G,510  284 

( '.  Additional  loss  if  the  velocity  s  is  im- 
pressed suddenly,    ....  i  39,127^  71 

Total  work  with  that  addition,  but  still  ex- 
clusive of  friction  (the  efficiency 
being  CM), 2£         195,637* 


*  This  case  of  maximum  efficiency  has  been  pointed  out  by  Mr.  It.  D.  Napier.     See 
Engineer,  November  30,  18GG,  page  424. 


THE    EFFICIENCY   OF    PROPELLERS.  549 

1 5.  As  another  example,  let — 

W  =  10  tons,  or  22,400  lbs.  per  second; 

v     ='15  ft.  per  second  ;  s  =  71  ft.  per  second  ; 

(so  that  v  +  *  —  22i  ft.  per  second,  and  R  =  5217  lbs.,  as  before);  let 
n  —  0  ;  and  let  the  velocity  s  be  impressed  suddenly.     Then — 

Ratio  to      Foot-pounds 
Use 

A.  Useful  work.         ..... 

['>.   Necessary  loss  of  work, 

( '.    Additional    loss   through   suddenness  of 
action,   ...... 

Total  work,  exclusive  of  friction    (the   effi- 
ciency being  0'07),  .  .       U  117,882^        213§ 


Work 

:.     per  second. 

H.P. 

1 

78,255 

142 

i 

4 

19,5  6  3| 

35| 

I 

4 

19,503-2 

35| 

550  DESIGN   AND    CONSTRUCTION   OF   MASONRY  DAMS. 


XXXIV.— REPORT  ON  THE  DESIGN  AND  CONSTRUCTION 
OF  MASONRY  DAMS. 

1.  Subjects  of  Report. — I  have  carefully  considered  the  letter  of  Captain 
Tulloch,  U.K.,  Executive  Engineer  of  the  Municipality  of  Bombay,  dated 
the  10th  December,  1870,  on  the  subject  of  masonry  dams  or  reservoir 
walls  of  great  height,  and  also  the  papers  on  the  same  subject  by  M. 
GraefFand  by  M.  Delocre,  which  appeared  in  the  Aimales  des  Pants  et 
Chaussets.  These  las!  I  have  studied  both  in  the  original  and  in  the  very 
faithful  translation  by  Mr.  J.  G.  Fife.  I  have  also  made  mathematical 
investigations  as  to  the  proper  figure  and  dimensions  of  such  dams,  which 
are  given  in  an  appendix  to  this  report. 

J.  Mnl<  rial. — As  regards  the  material  best  suited  for  a  reservoir  wall  or 
embankment,  I  consider  that  it  musl  be  determined  by  the  nature  of  the 
foundation.  That  foundation  should  be  sound  rock,  if  practicable ;  and 
should  a  rock  foundation  be  unattainable,  firm  impervious  earth.  It  may 
be  doubted  whether  any  earthen  foundation  is  thoroughly  to  be  relied  on 
where  the  depth  of  water  exceeds  100  or  120  feet.  It  is  not  advisable 
to  build  a  high  masonry  dam  on  an  earthen  foundation;  for  the  base  of 
the  dam  must  lie  spread  to  a  width  sufficient  to  distribute  the  pressure,  so 
that  it  shall  not  be  more  intense  than  the  earthen  foundation  can  bear ; 
and  this  involves  the  use  of  a  quantity  of  material  which  would  lead  t<» 
immoderate  expense,  if  the  material  used  were  masonry. 

3.  Mode  of  Building. — In  the  case  of  a  rock  foundation,  the  proper- 
material  is  unquestionably  rubble  masonry,  laid  in  hydraulic  mortar ;  and 
the  opinion  of  M.  Graeff  that  continuous  courses  in  building  that  masonry 
are  to  be  avoided,  is  fully  corroborated  by  experience;  for  the  bed-joints 
of  such  courses  tend  to  become  channels  for  the  leakage  of  the  water. 

4.  Precaution. — The  very  fact,  however,  of  the  irregular  structure  of  that 
masonry  renders  necessary  unusual  care  and  vigilance  in  superintending  its 
erection,  in  order  to  insure  that  every  stone  shall  be  thoroughly  and  firmly 
bedded,  and  that   there  shall  be  no  empty  hollows  in  the  interior  of  the 

*  From  The  Emjlnecr  for  Jan.  5,  1872. 


DESIGN    AND   CONSTRUCTION   OF  MASONRY   DAMS.  551 

wall,  nor  spaces  filled  with  mortar  alone  where  stones  ought  to  be  placed. 
The  practice  of  "grouting,"  or  filling  hollows  by  pouring  in  liquid  mortar, 
should  be  strictly  prohibited.  Should  it  be  resolved  to  insert  in  the  face 
of  the  wall  headers,  or  long  bond-stones,  with  or  without  projecting  ends 
to  form  corbels,  as  in  the  dam  of  the  river  Furens,  those  stones  ought  to 
be  laid  with  their  lengths  not  horizontal,  but  normal  to  the  face  of  the  watt. 

5.  Principles  determining  Profile. — With  respect  to  the  profile  of  the  wall, 
its  figure  is  in  the  main  to  be  determined  by  principles  nearly  the  same 
with  those  laid  down  by  the  French  engineers  already  referred  to,  and  put 
in  practice  in  the  dams  of  the  rivers  Furens  and  Ban ;  that  is  to  say,  the 
intensity  of  the  vertical  pressure  at  the  inner  face  of  the  wall  should  at  no 
point  exceed  a  certain  limit  when  the  reservoir  is  empty,  and  the  intensity 
of  the  vertical  pressure  at  the  outer  face  of  the  wall  should  at  no  point 
exceed  a  certain  limit  when  the  reservoir  is  full. 

(».  Limits  of  Vertical  Pressure. — In  the  theoretical  investigations  of  M. 
Delocre,  and  the  practical  examples  given  by  M.  Graeff,  the  same  limit  is 
assigned  to  the  intensit}T  of  the  vertical  pressure  at  both  faces  of  the  wall. 
But  it  ajmears  to  me  that  there  are  the  following  reasons  for  adopting  a 
lower  limit  at  the  outer  than  at  the  inner  face.  The  direction  in  which 
the  pressure  is  exerted  amongst  the  particles  close  to  either  face  of  the 
masonry,  is  necessarily  that  of  a  tangent  to  that  lace  :  and,  unless  the  face 
is  vertical,  the  vertical  pressure  found  by  means  of  the  ordinary  formula 
is  not  the  whole  pressure,  but  only  its  vertical  component ;  and  the  whole 
pressure  exceeds  the  vertical  pressure  in  a  ratio  which  becomes  the  greater 
the  greater  the  "  batter,"  or  deviation  of  the  face  from  the  vertical.  The 
outer  face  of  the  Avail  has  a  much  greater  batter  than  the  inner  face ; 
therefore,  in  order  that  the  masonry  of  the  outer  face  may  not  be  more 
severely  strained  when  the  reservoir  is  full,  than  that  of  the  inner  face 
when  the  reservoir  is  empty,  a  lower  limit  must  lie  taken  for  the  intensity 
of  the  vertical  pressure  at  the  outer  face  than  at  the  inner  face. 

7.  Weight  of  Wall  to  be  Thrown  Inwards. — The  proposal  of  the  executive 
engineer  to  throw  the  weight  of  the  wall  farther  inwards  than  in  the 
French  designs,  tends  to  realise  the  principles  just  stated,  and  so  far  I  fully 
approve  of  it,  and  have  carried  it  out  in  the  profile  which  accompanies  this 
report. 

8.  Wall  not  to  Overhang  Inwards. — I  do  not,  however,  concur  with  the 
executive  engineer  in  the  proposal  to  throw  the  weight  of  the  wall  so  far 
inwards  as  to  make  it  overhang,  for  the  following  reason — the  additional 
stability  against  the  horizontal  thrust  of  the  water  gained  by  giving  the 
wall  an  overhanging  batter  inwards,  is  not  that  due  to  the  whole  weight  of 
the  overhanging  masonry,  but  only  to  the  excess  of  that  weight  above  the 
weight  of  water  which  it  displaces ;  in  other  words,  about  half  the  effect 
of  the  weisjht  of  the  overhanging  mass  of  masonry  in  giving  stability  is 


552  DESIGN  AND  CONSTRUCTION   OF   MASONRY    DAMS. 

lost  through  its  buoyancy,  and  hence  the  additional  stability  gained  by 
making  the  wall  overhang  inwards  is  not  proportionate  to  the  additional 
load  thrown  upon  the  lower  parts  of  the  inner  face;  and  more  stability 
would  be  gained  by  placing  a  given  mass  of  masonry,  so  as  to  form  an 
uniform  addition  to  the  thickness  of  the  wall,  than  by  making  it  overhang 
inwards. 

9.  Limits  of  Vertical  Pressure,  how  Fixed. — In  choosing  limits  for  the 
intensity  of  the  vertical  pressure  at  the  inner  and  outer  faces  of  the  wall 
represented  by  the  accompanying  profile,  I  have  not  attempted  to  deduce. 
the  ratio  which  those  quantities  ought  to  bear  to  each  other  from  the 
theory  of  the  distribution  of  stress  in  a  solid  body;  for  the  data  on  which 
any  such  theoretical  determination  would  have  to  be  based  are  too  uncer- 
tain. The  limits  which  1  have  chosen  arc  as  follows,  and  they  are  given, 
in  the  first  place,  in  feet  of  a  vertical  column  of  masonry  whose  weight 
would  be  equivalent  to  the  pressure,  and  are  then  reduced  to  various  other 
measures : — 

Limits  of  vertical  pressure  at 

Feel  of  masonry,  .... 

Feet  of  water,        .... 

Pounds  on  the  square  foot  (nearly), 

Metres  of  masonry  (nearly), 

Metres  of  water  (nearly), 

Kilog.  on  the  square  centimetre  (nearly 

In  choosing  these  two  limits  I  have  been  guided  by  the  consideration  of 
the  following  facts.  As  regards  the  inner  face,  where  the  deviation  of  the 
direction  of  the  stress  from  the  vertical  is  unimportant,  it  is  certain,  from 
practical  experience,  that  rubble  masonry  laid  in  strong  hydraulic  mortar, 
and  on  good  rock  foundations,  will  safely  bear  a  vertical  pressure  equiva- 
lent to  the  weight  of  a  column  of  masonry  1G0  feet  high,  if  not  higher. 
As  regards  the  outer  face,  the  practical  data  given  by  M.  GraefFshow  that 
masonry  of  the  same  quality  in  the  sloping  outer  face  of  a  dam  will  safely 
bear  a  pressure  whose  vertical  component,  as  found  by  the  ordinary  rules, 
is  equivalent  to  the  weight  of  a  column  125  feet  high. 

10.  Diminution  of  Vertical  Pressure  toivards  Foot  of  Slope. — The  same 
reasons  which  show  that  the  intensity  of  the  vertical  component  of  the 
pressure  ought  to  be  less  for  a  battering  than  for  a  vertical  face,  show  also 
that  this  intensity  ought  gradually  to  diminish  at  the  lower  part  of  the 
outer  face,  where  the  batter  gradually  increases.  In  the  present  state  of 
our  knowledge  we  should  not  be  warranted  in  forming  any  definite 
theory  as  to  the  law  which  this  diminution  ought  to  follow ;  and,  there- 


Inner  face. 

•  inter  face. 

160 

1  25 

320 

250 

20,000 

1  5,625 

49 

38 

98 

76 

9-8 

7-6 

DESIGN   AND   CONSTRUCTION   OF   MASONRY   DAMS.  553 

fore,  in  preparing  the  accompanying  design,  I  have  thought  it  best  to 
be  guided  in  this,  as  in  the  previous  case,  by  practical  examples,  and  to 
consider  it  sufficient  to  make  the  law  of  diminution  such,  that  at  the 
depth  of  150  feet  below  the  surface,  the  intensity  of  the  vertical  com- 
ponent of  the  pressure  at  the  outer  face  becomes  nearly  equal  to  what  it  is 


ORDTE.TO  INNFTRgL- 


ORDINATES  TO  OUTER  FACE 
\11.40'  FEET 


to       no       oo 


at  the  same  depth  in  the  outer  face  of  the  dam  across  the  Furens— viz., 
107  feet  of  masonry,  or  about  6|  kilogrammes  on  the  square  centimetre. 

11.  Tension  to  be  avoided— I  have  kept  in  view  another  principle,  not 
referred  to  by  the  French  authors— viz.,  that  there  ought  to  be  no  practi- 
cally appreciable  tension  at  any  point  of  the  masonry,  whether  at  the  outer 
face  when  the  reservoir  is  empty,  or  at  the  inner  face  when  the  reservoir 
is  full.  Experience  has  shown  that  in  structures  of  brickwork  and 
masonry  that  are  exposed  to  the  overturning  action  of  forces  which 
fluctuate  in  amount  and  direction  (as  when  a  factory  chimney  is  exposed 
to  the  pressure  of  the  wind),  the  tendency  to  give  way  first  shows  itself 


554  DESIGN   AND   CONSTRUCTION    OF   MASONRY   DAMS. 

at  that  point  at  which  the  tension  is  greatest.  In  order  that  this  principle 
may  be  fulfilled,  the  line  of  resistance  should  not  deviate  from  the  middle 
of  the  thickness  of  tin-  wall  to  an  extent  materially  exceeding  one-sixth  of 
the  thickness.  In  other  words,  the  lines  of  resistance  when  the  reservoir 
is  empty  and  full  respectively,  should  both  lie  within,  or  but  a  small 
distance  beyond,  the  middle  third  of  the  thickness  of  the  wall. 

12.  Horizontal  Curvature  of  Wall. — As  regards  the  effect  of  giving  the 
wall  a  curvature  in  plan  convex  towards  the  reservoir,  I  look  upon  this  as 
a  desirable,  and  in  many  cases  an  essential  precaution,  in  order  to  prevent 
the  wall  from  being  bent  by  the  pressure  of  the  water  into  a  curved  shape 
concave  towards  the  water,  and  thus  having  its  outer  face  brought  into  a 
state  of  tension  horizontally,  which  would  probably  cause  the  formation  of 
vertical  fissures,  and  perhaps  lead  to  the  destruction  of  the  dam.  I  con- 
sider, however,  that  calculations  of  stability  which  treat  the  dam  as  a 
horizontal  arch  are  so  uncertain  as  to  he  of  very  doubtful  utility;  audi 
would  not  rely  upon  them  in  designing  the  profile.  In  fixing  the  radius 
of  horizontal  curvature,  1  consider  that  the  engineer  should  be  guided  by 
the  form  of  the  gorge  in  which  the  dam  is  to  be  built,  making  that  radius 
as  short  as  may  be  consistent  with  convenience  in  execution,  and  with 
making  the  ends  of  the  dam  abut  normally  against  the  sound  rock  at  the 
sides  of  the  gorge. 

L3.  Summary  of  Conditions  to  be  fulfilled  by  Profile;  Logarithmic  Curves 
chosen. — The  conditions  which  have  been  observed  in  designing  the  accom- 
panying  profile  may  be  summed  up  as  follows: — A.  The  vertical  pressure 
at  the  inner  face  not  to  exceed  1G0  feet  of  masonry.  B.  The  vertical 
pressure  at  the  outer  face  not  to  exceed  12.*)  feet  of  masonry  at  the  point 
where  it  is  most  intense,  and  to  diminish  in  going  down  from  that  point. 
C.  The  lines  of  resistance  when  the  reservoir  is  full  and  empty  respectively, 
to  lie  within  or  near  to  the  middle  third  of  the  thickness  of  the  wall. 
These  are  limiting  conditions,  and  do  not  prescribe  exactly  any  definite 
form.  In  choosing  a  form  in  order  to  fulfil  them  without  any  practically 
important  excess  in  the  expenditure  of  material  beyond  what  is  necessary, 
I  have  been  guided  by  the  consideration  that  a  form  whose  dimensions, 
sectional  area,  and  centre  of  gravity  under  different  circumstances,  are 
found  by  short  and  simple  calculations,  is  to  be  preferred  to  one  of 
a  more  complex  kind,  when  their  merits  in  other  respects  are  ecpial ; 
and  I  have  chosen  logarithmic  curves  for  both  the  inner  and  the  outer 
faces. 

14.  Rule  us  h<  Thicknesses. — The  constant  subtangent  common  to  both 
curves  (marked  A  D  in  the  figure)  is  80  feet ;  this  bears  relations  to  the 
vertical  pressures  which  are  stated  in  the  appendix.  The  thickness  C  B 
at  120  feet  below  the  top  is  84  feet;  and  of  this  one-fourteenth, 
A  C  =  G  feet,  lies  inside  the  vertical  axis  0  X,  and  thirteen-fourteenths, 


DESIGN    AND   CONSTRUCTION    OF   MASONRY   DAMS.  555 

AB  =  78  feet,  outside  that  axis.  The  formula  for  the  thickness  t  at  any 
depth  x  below  the  top  is  as  follows  : 

■'*  ~~  -''i 

'  =  '1''    a    ;     •       •       •       •    (l.) 

or,  in  common  logarithms, 

log.  /  ---  log.  /,  +  0-4343 -      '''''         .         .     (1  a.) 

a 

in  which  a  denotes  the  subtangent  (80  feet);  and  /x  the  given  thickness 
(84  feet)  at  the  given  depth  (.<■_,  =  120  feet)  below  the  top.  The  thickness 
at  the  top  is  18*74  feet. 

15.  Horizontal  Ordinate*. — In  the  profile,  horizontal  ordinates  are  drawn 
at  every  10  feet  of  depth,  from  the  top  down  to  180  feet,  and  their  lengths, 
from  the  vertical  axis  O  X  to  the  inner  and  outer  faces  respectively,  are 
marked  in  feet  and  decimals.  In  each  case  those  ordinates  are  respectively  ' 
one-fourteenth  and  thirteen-fourteenths  of  the  thickness.  Intermediate 
ordinates,  at  intervals  of  5  feet,  can  easily  be  calculated,  if  required,  by 
taking  mean  proportionals  between  the  adjacent  pairs  of  ordinates  at  the 
intervals  of  10  feet. 

16.  Sectional  Areas. — The  sectional  area  of  the  wall,  from  the  top  down 
to  any  given  depth,  is  found  by  multiplying  the  constant  subtangent 
(ct  =  80  feet)  by  the  difference  (/  —  /0)  between  the  thicknesses  at  the 
top  and  at  the  given  depth  ;  that  is  to  say, 

"tdx  =  a(t-Q.  .         .         .     (2.) 

17.  Line  of  Resistance  wlien  Reservoir  is  Empty. — The  vertical  line  through 
the  centre  of  gravity  of  the  part  of  the  wall  above  a  given  horizontal 
plane,  stands  midway  between  the  middle  of  the  thickness  at  the  given 
horizontal  plane  and  the  middle  of  the  thickness  at  the  top  of  the  wall ; 
and  thus  have  been  found  points  in  the  curve  marked  "Line  of  resistance. 
reservoir  empty." 

18.  Moment  of  Pressure  of  Water. — Supposing  the  reservoir  filled  to  the 
level  of  the  top  of  the  wall,  the  moment  of  the  pressure  exerted  horizon- 
tally by  the  water  against  each  unit  of  length  of  Avail,  from  the  top  down 
to  a  given  depth  (x),  is  found  by  multiplying  the  weight  of  a  cubic  unit  of 
water  by  one-sixth  of  the  cube  of  the  depth ;  and  if  we  take,  for  con- 
venience, the  weight  of  a  cubic  unit  of  masonry  as  the  unit  of  weight,  and 
suppose  the  masonry  to  have  twice  the  heaviness  of  water,  this  gives  us, 
for  the  moment  of  horizontal  pressure 

M  =  ~ (3-) 


55G  DESIGN    AND   CONSTRUCTION    OF   MASONRY    DAMS. 

19.  l.in'  of  Resistance  when  "Reservoir  is  Full. — The  moment  of  horizontal 
pressure,  expressed  as  above  stated,  being  divided  by  the  area  of  cross- 
section  above  the  given  depth,  gives  the  horizontal  distance  at  the  given 
depth  between  the  lines  of  resistance  with  the  reservoir  empty  and  full 
respectively;  thai  is  to  say, 


3V1 

itdx       Via  (I  -  Q 


(±0 


and  thus  have  been  found  points  in  the  curve  marked  "  lane  of  resistance, 
•  eservoir  full." 

:M>.  Vertical  Component  of  Water-Pressure  neglected.  In  the  preceding 
formulae  the  pressure  of  the  water  against  the  inner  face  of  the  wall  is 
treated  as  if  it  were  wholly  horizontal  as  in  the  investigations  <>t 
*  M.  Graeff  and  M.  Delocre).  In  fact,  however,  that  pressure,  being  normal 
to  tin'  inner  face  of  the  wall,  has  a  small  inclination  downwards,  and. 
therefore,  contains  a  small  vertical  component,  which  adds  to  the  stability 
of  the  wall.  The  neglect  of  thai  vertical  component  is  an  error  on  the 
-.it'e  side. 

21.  Intensity  of  Vertical  Pressurt  in  Masonry. — To  find  the  mean  inten- 
sity of  the  vertical  pressure  on  a  given  horizontal  plane  in  the  masonry, 
expressed  in  feet  of  masonry,  divide  the  sectional  area  by  the  thickness  at 
the  given  plane  :   that  is  to  say, 


'***     -„M    _^ 

/ 


(i-;-  •     •  (so 


To  find  the  greatest  intensity  of  that  vertical  pressure,  according  to  the 
ordinary  assumption  that  it  is  an  uniformly  varying  sfress — in  other  words, 
that  it  increases  at  an  uniform  rate  from  the  face  farthest  from  the  line  of 
resistance  to  the  face  nearest  to  that  line,  the  mean  intensity  is  to  be 
increased  by  a  fraction  of  itself  expressed  by  the  ratio  which  the  deviation 
of  the  line  of  resistance  from  the  middle  of  the  thickness  bears  to  one-sixth 
of  the  thickness;  that  is  to  say,  let  p  denote  that  greatest  intensity, 
expressed  in  feet  of  masonry,  and  r  the  deviation  of  the  line  of  resistance 
from  the  middle  of  the  thickness  :  then, 

When  that  deviation  is  appreciably  greater  than  one-sixth  of  the  thickness, 
the  preceding  rule  is  no  longer  applicable  ;  but  this  case,  as  already  ex- 
plained, ought  not  to  occur  in  a  reservoir  wall.  The  assumption  on  which 
this  rule  is  based,  of  an  uniform  rate  of  variation  of  that  component  of  the 


DESIGN   AND   CONSTRUCTION   OF   MASONRY   DAMS.  557 

pressure  which  is  normal  to  the  pressed  surface,  is  known  to  be  sensibly 
correct  in  the  case  of  beams,  and  is  probably  Aery  near  the  truth  in  walls 
of  uniform  or  nearly  uniform  thickness.  Whether,  or  to  what  extent,  it 
deviates  from  exactness  in  walls  of  varying  thickness  is  uncertain  in  the 
present  state  of  our  experimental  knowledge. 

22.  Profiles  fur  Different  Depths. — The  range  of  different  depths  to  which 
the  same  profile  is  applicable  without  any  waste  of  material  extends  from 
the  greatest  depth  shown  on  the  figure  180  feet,  up  to  110  feet 
or  thereabouts.  Fur  depths  between  110  feet  and  80  or  90  feet,  or 
thereabouts,  the  waste  of  material  is  unimportant.  For  depths  to  any 
considerable  extent  less  than  90  feet,  the  use  of  a  part  of  the  same  profile 
gives  a  surplus  of  stability.  Fur  example,  if  the  depth  be  50  feet,  the 
quantity  of  material  is  greater  than  that  which  is  necessary  in  the  ratio  of 
I -4  to  1  nearly.  For  the  shallow  parts,  however,  at  the  ends  of  a  dam 
that  is  deep  in  the  centre,  I  think  it  preferable  to  use  the  same  profile  as 
in  the  deep  parts,  notwithstanding  this  expenditure  of  material,  in  order 
that  the  full  advantage  of  the  abutment  against  the  sides  of  the  ravine 
may  be  obtained.  In  the  case  of  a  dam  that  is  less  deep  in  the  centre 
than  120  feet,  the  following  rule  may  be  employed:  construct  a  profile 
similar  to  that  suited  to  a  depth  of  120  feet,  with  all  the  thicknesses  and 
ordinates  diminished  in  the  same  proportion  with  the  depth.  The  intensity 
of  the  vertical  pressure  at  each  point  will  be  diminished  in  the  same  pro- 
portion also,  but  this  does  not  imply  waste  of  material,  the  whole  strength 
of  the  material  being  required  in  order  that  there  will  lie  no  appreciable 
tension  in  any  part  of  the  wall. 


A  P  P  E  X  I)  I  X. 

Mathematical  Principles  of  the  Profile  Curves. 

I.  Principles  Relating  to  all  Form*  of  Profile.— L%t  t,  as  before,  be  the 
thickness  of  the  wall  in  a  horizontal  plane  at  the  depth  x  below  the  top ; 
then,  taking  the  weight  of  a  cubic  unit  of  masonry  as  the  unit  of  weight, 
the  weight  of  each  unit  of  length,  of  the  wall  above  that  plane  i^ 
expressed  by 

/    tdx. 

In  order  that  there  may  be  no  appreciable  tension  at  the  outer  edge  of  the 
given  plane  when  the  reservoir  is  empty,  nor  at  the  inner  edge  when  it  is 


558  DESIGN   AND   CONSTRUCTION   OF   MASONRY   DAMS. 

full,  the  centre  of  resistance  of  that  plane  ought  not  to  deviate  from  the 
middle  of  the  thickness  by  more  than  about  one-sixth  of  the  thickness 
inwards  when  the  reservoir  is  empty,  outwards  when  it  is  full. 

Let   //  denote  the-  deviation  of  the  centre  line  of  the  thickness  of  the 

wall  outwards  from  a  vertical  axis  0  X  ;  so  that  y  —  -  and  y  +  -  are  the 

ordinates  of  the  inner  and  outer  faces  of  the  Avail  respectively  ;  and  when 
x  =  0,  let  y  =  yQ.  The  line  of  resistance  when  the  reservoir  is  empty 
cut-  the  horizontal  plane  at  the  depth  x,  in  a  point  vertically  "below  the 
centre  of  gravity  of  the  part  of  the  Avail  above  that  plane;  and  in  order 
that  the  weight  of  the  wall  may  lie  thrown  as  far  inwards  as  is  consistent 
with  there  being  no  appreciable  tendon  at  the  outer  face  when  the 
reservoir  is  empty,  the  deviation  of  that  line  of  resistance  from  the 
middle  of  the  thickness  of  the  wall  ought  not  materially  to  exceed  one- 
sixth  of  the  thickness;  hence,  if  i\  he  taken  to  denote  the  inward 
deviation  in  question, 


r,  =  y  -  -jj-   -  =  or  <^-  nearly.      .  .      (A.) 

Jo 

Lei  "'  be  the  ratio  in  which  the  masonry  is  heavier  than  water.  Then 
the  moment  of  the  horizontal  pressure  of  the  water  above  the  same  plane 
on  each  unit  of  the  length  of  wall  is, 

M         r     ■ 

h  W 

The  vertical  component  of  that  pressure  is  neglected,  as  explained  in  the 
body  of  the  report.  The  extent  to  which  the  centre  of  resistance  at  the 
given  horizontal  plane  is  shifted  outwards  by  the  pressure  of  the  water  is 


>\  +  r  =  -^-  =  - 
tdx 

J  o 


in  which  /■  denotes  the  outward  deviation  of  the  line  of  resistance  from 
the  middle  of  the  thickness  when  the  reservoir  is  full ;  and  the  condition 
that  the  centre  of  resistance,  when  the  reservoir  is  full,  is  not  to  deviate 
from  the  middle  of  the  thickness  by  more  than  about  one-sixth  of  the 
thickness,  is  expressed  by  the  following  formula  : — 


DESIGN  AND   CONSTRUCTION   OF   MASONRY   DAMS.  559 

7 \-   \    ijtd  x 

6»       !/  y  t 

y  =  or  <^-  nearly.  .     (C.) 


f 

J  o 


t  a  x 


The  formulas  (A)  and  (C)  express  the  condition  that  there  shall  be  no 
practically  important  tension  in  the  masonry  at  any  horizontal  plane. 
Let  p1  and  j>  be  the  vertical  pressures  at  the  inner  and  outer  faces 
respectively  at  the  depth  x :  and  Px  and  P  the  limits  which  those 
pressures  are  not  to  exceed.  Then  we  have,  as  another  pair  of  equa- 
tions to  be  satisfied, 

ft  =  (l  +  5i)£^!  =  or  <Pr       .         .     (D.) 
1  +  6r\/>* 


P=  (}  +  —)■/<>      _=or<P.         .         .     (E.) 
z 

II.  Principles  Relating  to  the  Logarithmic-Curve  Profile. — As  a  means  of 
satisfying  the  equations  of  condition  to  a  degree  of  approximation 
sufficient  for  practical  purposes,  let  the  inner  and  outer  boundaries  and 
the  centre  line  of  the  profile  be  all  three  logarithmic  curves,  with  the 
vertical  axis  0  X  for  their  common  asymptote,  and  having  one  common 
constant  subtangent  a.  It  may  be  remarked  that  one  reason  for  adopting 
the  logarithmic  curve  is  its  giving  a  thickness  at  the  top  of  the  wall 
sufficient  for  the  formation  of  a  roadway ;  and  that  another  reason  is,  its 
giving  values  to  the  intensity  of  the  pressure  at  the  outer  face  below  the 
point  of  maximum  pressure,  which  diminish  as  the  batter  increases.  Let 
the  ratio  borne  by  the  deviation  y  of  the  centre  line  of  the  thickness  from 

the  vertical  axis  to  the  thickness  t  be  expressed  by  c  =  —. 

Then  we  have  the  following  equations  : — 

X 

t  =  t0e" (F.) 

X 

ij  =  ct  ■=  ctQea (G.) 

X 

rtdx  =  at0(e«  -l)  =  a(t-t0).     .         .     (II.) 


rn  =-£-[€     -  1  )  =  c-— —_——-.  .         .     (Iv.j 


560  DESIGN   AND   CONSTRUCTION    OF   MASONRY   DAMS. 

r  (t  -  g 


r 


Qwa(t-t0) 

x 


6  wa  /,i  [ea  —  1 

p,  =o(l  -f  "  ")  i  1  +  3c(  1  -  «  "«)  J        •     (M.) 

p  =  A  \1  -i      "  -  3  c  VI  -''     VI    ■         •     (N.) 

When  the  values  given  above  are  substituted  in  the  expressions  of  con- 
ditions. A.  C,  D.  and  E,  the  formulae  obtained  are  of  a  kirn!  incapable  of 
.solution  by  any  direct  process.  They  can,  however,  be  solved  approxi- 
mately without  much  difficulty  by  the  process  of  trial  and  error  ;  and  such 
is  the  method  by  which  the  dimensions  of  the  profile  sent  with  the  report 
have  been  obtained;  the  constants  employed  being 

w  =  2;  1",        160  feet;  P  =  125  feet. 

The  general  nature  of  the  process  of  approximation  followed  may  be 

dp  ..... 

summed  ui»  as  follows: — Bv  making    ,     =  0.   an   equation   is  obtained 
1  (I  x 

involving  the  value  of    ,  which  makes p  a' maximum.     That  equation  shows 

that  as  a  first  approximation  to  that  value  we  may  take  -.     This  first 

approximation  is  inserted  in  equation  (K)  ;  and  by  making  i\  =  -,  there  is 
deduced  from  that  equation  an  approximate  value  of  c.  Then,  in  equation 
(M),  by  inserting  the  approximate  values  of  and  of  c,  and  making  px  =  1\ 
(the  limit  of  px),  there  is  obtained  an  approximate  value  of  a  ;  and  by 
making  r  =  -  in  equation  (L).  an  approximate  value  of  tff     The  several 

first  approximate  values  being  then  inserted  in  ~  =  0,  there  is  obtained 

a  corrected  value  of  L,  which  is  found  to  be  about      ,  ;  and   thence   by 
a  ° 

means  of  equation  (N),  the  actual  maximum  value  of  p  is  computed,  and 


DESIGN  AND  CONSTRUCTION   OF  MASONRY   DAMS.  5G1 

found  to  fall  slightly  within  the  prescribed  limit.  Finally,  as  a  test  of  the 
approximations,  equations  (K),  (L),  (M),  and  (N)  are  applied  to  a  series  of 
values  of  x,  extending  from  the  top  to  the  bottom  of  the  wall.  As  to  the 
degree  of  approximation  obtained,  the  greatest  values  px  and  p  are  respec- 
tively 154  feet  and  124  feet,  instead  of  160  feet  and  125  feet;  and  there 
are,  as  the  drawing  shows,  some  small  deviations  of  the  lines  of  resistance 
beyond  the  middle  third  of  the  thickness,  but  not  sufficient  to  be  of 
practical  importance. 


2n 


5G2  OX  BARYCENTRIC  PEESPECTIVE. 


XXXV.— ON  THE  APPLICATION  OF  BARYCENTRIC 

PERSPECTIVE  TO  THE  TRANSFORATION 

OF  STRUCTURES.* 

I.  This  paper  contains  the  substance  of  some  remarks  which  I  made  at 
the  recent  meeting  of  the  British  Association,  on  the  elegant  investigation 
by  Professor  Sylvester  of  the  principles  of  Barycentric  Perspective  and 
Homalographic  Projection. 

2.  In  the  Proceedings  of  the  Royal  Society  for  the  Gth  March,  1856, 
I  published  a  theorem  called  that  of  "  The  Transformation  of  Structures," 
which  may  be  briefly  expressed  as  follows: — 

If  a  structure  of  a  given  figure  be  balanced  and  stable  under  forces  repre- 
tented  by  given  lines,  then  will  any  structure  whose  figure  is  a  parallel  projection 
>f  the  original  figure  be  balanced  and  stable  under  forces  represented  by  the 
corresponding  projections  of  the  lines  representing  the  original  forces. 

3.  By  a  parallel  projection  of  a  figure  is  meant  a  figure  derived  from 
the  original  figure  by  altering  the  co-ordinates  in  uniform  proportions,  or 
by  substituting  oblique  for  rectangular  co-ordinates;  and  it  is  called 
parallel  because  to  every  pair  of  equal  and  parallel  lines  in  the  original 
figure  there  correspond  a  pair  of  equal  and  parallel  lines  in  the  trans- 
formed figure.  For  example,  every  orthographic  projection  of  a  plane 
figure  is  a  parallel  projection  ;  all  ellipsoids  are  parallel  projections  of  each 
other  and  of  a  sphere,  &c. 

4.  That  theorem  was  applied  in  A  Manual  of  Applied  Mechanics  to  the 
deduction  of  the  figures  of  a  skew  arch  and  of  a  ramping  arch  from  that 
of  a  common  arch,  of  an  equilibrated  rib  from  a  common  catenary,  of 
arches  for  supporting  earth  from  arches  for  supporting  the  pressure  of  a 
liquid,  &c. 

5.  Its  applications,  however,  were  limited  by  the  condition  of  parallel 
projection;  and  there  were,  consequently,  many  conceivable  transformations 
of  structures  to  which  it  could  not  be  applied. 

6.  The  theorems  discovered  by  Mr.  Sylvester  now  afford  the  means  of 

*  From  the  Philosophical  Magazine  for  Xov.,  1S63. 


OX   BARYCENTR1C   PERSPECTIVE.  563 

greatly  extending  the  art  of  designing  structures  by  transformation  from 
structures  of  more  simple  figures ;  for  they  obviously  give  at  once  the 
solution  of  the  question — given  the  figure  of  a  structure  which  is  balanced 
and  stable  under  a  load  distributed  in  a  given  way;  given  also  any  perspective 
or  Iwmalographic  projection  of  that  figure;  to  find  how  the  load  mud  he  dis- 
tributed on  the  transformed  structure,  in  order  that  if  also  may  be  balanced 
and  stable. 

7.  This  is  not  the  first  instance  in  which  theorems  of  pure  science 
have  proved  to  be  capable  of  practical  applications  unexpected,  perhaps, 
by  their  discoverers. 


5G4  ON   THE   EQUILIBRIUM   <  >F   POLYHEDRAL   FRAMES. 


XXXVL— PRINCIPLE  OF  THE  EQUILIBRIUM  OF 
POLYHEDRAL  FRAM  ES.* 

The  following  theorem  is  the  extension  to  polyhedral  frames  of  a  principle 
which  is  proved  for  polygonal  frames  in  -/  Manual  of  Applied  Mechanics, 
Art,  150. 

Theorem. — If  plains  diverging  from  a  point  or  line  be  drawn  normal 
to  the  lines  of  resistance  of  the  bars  of  a  polyhedral  frame,  then  the  I 
of  a  polyhedron  whose  edges  lie  in  those  diverging  planes  (in  such  a 
manner  that  those  faces,  together  with  the  diverging  planes  which  contain 
their  edges,  form  a  set  of  contiguous  diverging  pyramids  or  wedges)  will 
represent,  and  be  normal  to,  a  system  of  forces  which,  being  applied  to 
the  summits  of  the  polyhedral  frame,  will  balance  each  other — each  such 
force  being  applied  to  the  summit  of  meeting  of  the  bars  whose  lines  of 
resistance  arc  normal  to  the  set  of  diverging  planes  that  enclose  that  face 
of  the  polyhedron  of  forces  which  represents  and  is  normal  to  the  force  in 
question.  Also,  the  areas  of  the  diverging  planes  will  represent  the 
stresses  along  the  bars  to  whose  lines  of  resistance  they  are  respectively 
normal. 

It  is  obvious  that  the  polyhedron  of  forces  and  the  polyhedral  frame 
are  reciprocally  related  as  follows  : — their  numbers  of  edges  are  equal, 
and  their  corresponding  pairs  of  edges  perpendicular  to  each  other ;  and 
the  number  of  faces  in  each  polyhedron  is  ecpial  to  the  number  of  summits 
in  the  other. 

*  From  the  Philosophical  Magazine  for  Feb.,  1SC4. 


ON    A   PROPERTY   OF   CURVES.  5G5 


XXXVII.— OX  A  PROPERTY  OF  CURVES  FULFILLING 

THE  CONDITION  %&  +  %4  =  0.* 

(l,i-       dy2- 

1 .  In  a  paper  "  On  Streara-Lines,"  published  in  the  Philosophical  Magazine, 
for  October,  18G4,  I  stated,  and,  in  a  Supplement  to  the  same  paper, 
published  in  the  Philosophical  Magazine  for  January,  1865,  I  proved  the 
proposition  that  "  all  waves  in  which  molecular  rotation  is  nidi,  begin  to 
break  when  the  two  slopes  of  the  crest  meet  at  right  angles." 

2.  I  have  now  to  state  the  purely  geometrical  proposition  of  which 
that  mechanical  proposition  is  a  consequence.  If  a  plane  curve  which  fulfils 
the  condition 

<P<f>    ,d1<f>_ 
dor       dy 

cuts  itself  in  a  double  point,  it  does  so  at  rigid  angles. 

3.  The  following  is  the  demonstration.  It  is  well-known  that  the 
inclination  of  any  plane  curve  to  the  axes  at  an  ordinary  point  is  given  by 
the  equation 

rr  dx  +  -v2-  dy  =  0; 

d  x  dy     J 

also,  that  at  a  double  point  -  ^  and    -.—  both  vanish,  so  that  the  inclina- 
1  d  x  d  y 

tions  of  the  two  branches  to  the  axes  are  given  by  the  two  roots  of  the 

quadratic  equation 

d v?  d x  dy  d y 

whence  it  follows  that  the  product  of  the  two  values  of  -—^  which  are  the 
two  values  of  the  tangent  of  the  inclination  to  the  axis  of  x,  is 
*  From  the  Proceedings  of  the  Royal  Society  for  1SG7. 


560  ON  A   PROPERTY   OF   CURVES. 

</"'  <<> 
il  ./•'■ 

=  35 

ll  lf- 

In  a  curve  which  fulfils  the  before-mentioned  condition,  the  value  of  that 
product  is  —  1 ;  and  when  such  is  the  case  with  the  product  of  the  tangents 
of  two  angles,  the  difference  of  those  angles  is  a  right  angle;  therefore,  the 
two  branches  cut  each  other  at  right  angles.      <L>.H.D. 

4.  The  proposition  just  demonstrated  is  so  simple  and  so  obvious,  that 
1  was  at  first  disposed  to  think  it  must  have  been  known  and  published 
previously;  and  had  1  not  been  assured  by  several  eminent  mathematicians 
that  it  had  not  been  previously  published  to  their  knowledge.  I  should  not 
have  ventured  to  put  it  forth  as  new. 


Supplement  to  the  preceding  Paper 

Professor  Stokes.  D.C.L.,  has  pointed  out  to  me  an  extension  of  the 
preceding  theorem — viz..  ///<</  "/  every  multiple  point  in  a  plane  curve  which 
fulfils  the  condition 

'/J  fl>  +  &  <!>  =  0 

,i  y1       1 1  if1 

the  branches  make  equal  angles  with  each  other;  so  that,  for  example,  it'  w 
branches  cut  each  other  at  a  multiple  point,  they  make  with  each  other 

2  n  equal  angles  of 

x  °  a 

The  following  appears  to  me  to  be  the  simplest  demonstration  of  the 
extended  theorem:  At  a  point  where  n  branches  cut  each  other  tin- 
following  equation  is  fulfilled  by  all  curves  : 

(dx  .     +<///,       <j>  =  ". 

\      i.i  .'•  a  i//    ' 

Let  0  be  the  angle  made  by  any  branch  with  the  axis  of  .'• ;  then 

(  cos  6  -j-  +  sin  0  -,-  )  d)  =  0. 
\  ax  a  11/    r 


OX   A   PROPERTY   OF   CURVES.  567 

But  in  a  curve  which  fulfils  the  equation 


d?$       <F<p  _ 

d  /-*  ^  d  if         ' 


we  have 


dy      ^      l  '  dx' 


whence  it  follows  that  in  such  a  curve  the  equation  of  a  multiple  point  of 
11  branches  is 

|  (cos  6  +  J  -  1  .  sin  0)  ^  |B0  =  0. 

Choose  for  the  axis  of  x  a  tangent  to  one  of  the  branches  at  the 
multiple  point.  Then  it  is  evident  that  the  preceding  equation  is 
satisfied  by  the  2  n  values  of  0  corresponding  to  the  2  ?ith  roots  of 
unity  j  that  is  to  say,  by 

77      2  77  ( 2  11   —    1 )  7T 

6  =  0,,        ,  &c * —    — '—', 

therefore,  the  n  branches  make  with  each  other  2n  equal  angles  of  -. 
Q.E.D. 


BELL  AND  BAIS,   PBISTERS,    GLASGOW. 


.A.     SELECTION 

FROM 

CHARLES    GRIFFIN    AND    COMPANY'S 

CATALOGUE. 


General  Catalogue  of  Standard  Publications,  in  all  Departments 
of  Literature,  forwarded  on  Application. 


10  STATIONERS'  HALL  COURT,  LONDON. 


A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 


Published  with  the  Approval  of  the  Director-General  of  Telegraphs 

in  India. 

Crown  Svo,  cloth,  15s.    Second  Edition. 

A      M A  NUAL      OF 

TELEGRAPH    CONSTRUCTION: 

THE   MECHANICAL   ELEMENTS  OF    ELECTEIC  TELEGRAPH 
ENGINEERING. 


JOHN    CHRISTIE    DOUGLAS, 

MEM.    -    ■       11  '  i.i.i:Al'H    LNi.IM.1k-:     RAST    IM'IV    GOVT.    TELBGBAPL    DEPARTMENT. 

This  Work  lias  been  prepared  to  supply  the  existing  want  of  a  Text-Book 
on  the  Mechanical  Principles  and  Practice  of  Telegraph  Engineering,  and 
is  the  only  treatise  in  the  language  specially  devoted  to  the  subject.  It 
gives  the  lvsult  of  many  years'  practical  experience  on  the  part  of  the 
Author,  and  is  intended  for  use  by  the  Student  as  well  as  by  the  Engineer 
actually  engaged  in  Design  and  Construction. 

GENERAL     CONTENTS. 

Part  [.—GENERAL  PRINCIPLES  OF  STRENGTH  AND  STABILITY,  compris- 
ing the  Strength  of  Materials ;  the  1  distribution  of  Load  and  Stress  in  Telegraph 
Structures,  Buch  as  Poles  simple,  strutted,  tied,  stayed,  coupled,  and  trussed; 
the  ( atenary,  with  application  of  its  Formula;  to  the  cases  of  Wires  and 
( 'uliles:  Theory  of  the  Submersion  of  Cables,  &c 

PART  [I.-  PROPERTIES  AND  APPLICATIONS  OF  MATERIALS,  OPERA- 
TIONS, AND  MANIPULATION,  including  the  Principles  and  Practice 
of,  and  Numerical  Data  for,  designing  Simple  Structures,  such  as  Poles  of 
Iron  and  Wood;  Iron  and  Wooden  Masts  simple  and  compound;  Specifica- 
tions for  Wire,  &c ;  Soldering;  Surveying;  the  liaising  of  heavy  Masts; 
Insulating  Materials  and  their  Applications,  &c. 

Part  IIL— TELEGRAPH  CONSTRUCTION,MAINTENANCE,  AND  ORGANISA- 
TtON,  treating  of  the  Application  of  the  Information  conveyed  in  Parts  I. 
and  II.  to  the  case  of  Combined  Structures,  including  the  Construction  of 
Overground,  Subterranean,  and  Subaqueous  Lines;  Office  Fittings;  Estimat- 
ing; Organisation,  <.V'e. 


"  Mr.  Douglas  deserves  the  thanks  of  Telegraph  Engineers  for  the  excellent  '  Manual '  now  before  us 
...  he  has  ably  supplied  an  existing  want  .  .  .  the  subject  is  treated  with  great  clearness  and 
judgment    .    .    .    good  practical  information  given  in  a  clear,  terse  style." — Engineering. 

"Mr.  Douglas's  work  is,  we  believe,  the  first  nf  its  kind.  .  .  .  The  author  is  evidently  a  practical 
Telegraphic  Engineer.  .  .  .  The  amount  of  information  given  is  such  as  to  render  this  volume  a  most 
useful  guide  to  any  one  who  may  be  engaged  in  any  branch  of  Electric  Telegraph  Engineering  "— 
Athenxum. 

"The  book  is  calculated  to  be  of  great  service  to  Telegraphic  Engineers  .  .  .  the  arrangement  is 
so  judicious  that  with  the  aid  of  the  full  Table  of  Contents,  reference  to  any  special  point  snould  be 
easy." — Iron.  • 


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A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 


Royal  8uo,  764  pp.,  cloth.     With  over  200  Illustrations  drawn  to  Scale,  and 
reduced  in  many  instances  from   Working  Drawings.     Price  34s. 

ELEMENTS  OF  METALLURGY: 

A  Practical  Treatise  on  the  Art  of  Extracting  Metals  from  their  Ores. 

BY 

J.  ARTHUR  PHILLIPS,  C.E.,  F.C.S.,  F.G.S., 

Ancien  Eleve  c'.e  1'EcoIe  des  Mines,  Paris. 


The  Fourth  Edition  of  "  The  Manual  if  Metallurgy"  Enlarged,  Remodelled, 
and  Hi  writU  n. 


GENERAL     CONTENTS. 

I.  A  Treatise  on  Fuels  and  Refractory  Materials. 

II.  A  Description  of  the  principal  Metalliferous  Minerals  with  their  Distri- 
bution. 

III.  Statistics  of  the  amount  of  each  Metal  annually  produced   throughout  the 

world,  obtained  from  official  sources,  or,  where  this  has  not  been  practicable, 
from  authentic  private  information. 

IV.  The  Methods  of  Assaying  the  different  Ores,  together  with  the  Processes  of 

Metallurgical  Treatment,  comprisimg-- 

Eefractoby  Materials.  Iron.  Aluminium. 

Fire  Clays.  Cobalt.  Copper. 

Fuels,  &c.  Nickel.  Tin. 

Antimony.  Mercury.  Gold. 

Arsenic.  Bismuth.  Silver. 

Zinc.  Lead.  Platinum.  &c. 


"'Elements  of  Metallurgy'  possesses  intrinsic  merits  < >r  tin-  highest  degree.  Such  a  work  is  pre- 
cisely wanted  by  the  great  majority  of  students  aud  practical  workers,  and  its  very  compactness  is  m 
itself  a  first-rate  recommendation.  "The  author  has  treated  with  great  skill  the  metallurgical  operations 
relating  to  all  the  principal  metals.  The  methods  are  described  with  surprising  clearness  and  exactness 
placing  an  easily  intelligible  picture  of  each  process  even  before  men  of  less  practical  experience,  ana 
illustrating  the  most  important  contrivances  in  an  excellent  aud  perspicuous  manner.  .  .  .  in  our 
opinion  the  best  work  ever  written  on  the  subject  with  a  view  to  its  practical  treatment.  —  Westminstei 

'"'"•'in  this  most  useful  and  handsome  volume,  Mr.  Phillips  has  condensed  a  large  amount  of  valuable 
practical  knowledge.  We  have  not  only  the  results  of  scientific  inquiry  most  cautiously  set  lorm, Dm 
the  experiences  of  a  thoroughly  practical  man  very  clearly  given.  .  .  .  A  careful  study  of  the i  first 
division  of  the  book,  on  Fuels,  will  be  found  to  be  of  great  value  to  every  one  m  training  foi  the  piacticai 
applications  of  our  scientific  knowledge  to  any  of  our  metallurgical  operations.  —Atliena'""l-nn.  n-  ,  .  . 

"For  twenty  vears  the  learned  author,  who  might  well  have  retired  with  honour  on  account  otni> 
acknowledged  success  and  high  character  as  an  authority  in  Metallurgy,  has  been  making  note 3,  b ot bas 
a  Mining  Engineer  and  a  practical  Metallurgist,  and  devoting  the  most  valua b  e  portion  ot  his  ti  i e i  to m 
accumulation  of  materials  for  this,  his  Masterpiece.  There  can  be  no  possible  doubt  that  Eh  menMOT 
Metallurgy '  will  be  eagerly  sought  for  by  Students  in  Science  and  Art,  as  well  as  by  Practical  workers 
in  Metals  .  .  .  Tw,  hundred  and  fifty  pages  are  devoted  exclusively  to  the  Metallurgy  ot  lion,  in 
which  every  process  of  manufacture  is  treated,  and  the  latest  improvements  accurately  <  eta  le  .  .  . 
The  arrangement  of  subjects  is  practically  clear,  aud  calculated  to  facilitate  the  ready  discoveiy  bj 
Students  of  any  special  knowledge  they  may  seek  to  acquire.  —Colliery  Guardian.  n_n__f  nf  tim„ 

"The  value  of  this  work  is  almost  inestimable.    There  can  be  no  question  that  the  amount  o time 
and  labour  bestowed  on  it  is  enormous.    .    .    .    There  is  certainly  no  Metallurgical  Treatise  m^ 
language  calculated  to  prove  of  such  general  utility  to  the  Student  really  seeking  sou  ml  ,.r.  c  he!  1       orm. 
Hon  upon  the  subject,  and  none  which  gives  greater  evidence  ot  the  extensive  metallmgical  knowledge 
of  its  author."— Minim/  Journal.  ,,  ,  .    „„  „      'Kaj  fl,,.'(nla, 

"Such  a  work  was  much  needed,  and  the  need  could  hardly  have  been  better  supplied  than  it  has 
been  bv  Mr  Phillies  " — Ouarlerhi  Journal  of  Science.  ,     „,_„•,. 

"Mr  Philips  deserves  well  of  the  Metallurgical  interests  of  this  country,  for  having produced a wo  k 
which  is  equally  valuable  to  the  Student  as  a  Text-hook,  and  to  the  practical  Smelter  as  a  Standard 
Work  of  Reference.  .  .  .  The  Illustrations  are  admirable  examples  of  \iV  ood  Engraving.  —Chemical 
Xews. 


London  :  CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


4  A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 

Published  with  Concurrence  of  the  Surueyors-General  of  New  South   Wales 

and  Victoria. 
Half-Bound,  Folio,  Price   30s. 

TRAVERSE    TABLES: 

Computed  to  4  Places  Decimals  for  every  Minute  of  Angle  up  to  100  of  Distance, 

FOR   THE   USE  OF  SURVEYORS  AXD  ENGINEERS. 
BY 

R.      LLOYD     GURDEN, 

authorised  sueveyob  foe  the  governments  of  new  south  wales  and  victoria. 


DISTINCTIVE      FEATURES: 

1.  The  Tal>lcs  arc  calculated  to  Single  Minutes,  and  to  100  of  Distance. 

2.  The  Traverses  are  given  to  four  places  of  decimals,  so  that  the  Sines  and  Cosines 

•  for  a  distance  of  twelve  miles  can  be  ascertained  correctly  to  within  half  an  inch. 

3.  Simplicity  and  economy  of  labour  in  calculation.     Oxe  OPENING  OF  THE  TABLES 

gives  the  information  which,  when  Bought  by  the  usual  method,  involves  four 
references  to  the  book  of  logarithms,  two  additions,  and  the  writing  out  of  forty  - 
eight  nunc  figures  than  arc  required  in  the  use  of  Traverse  Tables. 


"The  rperience  inexact  Scbvet-wobe  will  best  know  how  to  appreciate  the  enormous 

amount  of  labour  represented  by  this  ■  '■■    The  computations  enable  the  user  to  ascertain  the 

sines  ami  cosines  for  a  distance  of  twelve  miles  l"  within  half  an  inch,  and  this  i.v  REFERENCE  to  btjt  •  >NE 
Table,  in  place  of  the  nana]  fifteen  minute  computations  required.  This  alone  is  evidence  of  the  assist- 
ance which  the  Tables  en  i  user,  and  as  every  Surveyor  in  active  practice  has  felt  the  want  of 
Bnch  assistance,  few  knowing  of  their  publication  will  remain  without  them." — Engineer. 

"  We  cannot  sufficiently  admire  the  heroic  patience  of  the  author,  who,  in  order  to  prevent  error, 
•h  result  by  two  different  modes,  and,  before  the  work  was  finally  placed  in  the  Printer's 
hauds,  repeated  the  operation  for  a  third  time  on  revising  the  Proofs." — Engine  i 

"Mr.  Gdbdes  is  to  be  thanked  for  the  extraordinary  labour  which  he  has  bestowed  on  facilitating 
the  work  of  the  Surveyor.  .  .  .  An  almost  unexampled  instance  of  professional  and  literary  industry 
.  .  .  When  the  anxious  and  laborious  work  of  one  man  affords  the  means  of  such  a  saving  of  toil  for 
all  those  who  avail  themselves  of  his  work,  the  patient  and  careful  tabulator  deserves  the  name  of  a 
benefactor  of  his  profession,  and  of  a  good  servant  of  his  fellows." — Allien  x  it  in. 

"These  Tables  are  characterised  by  ABSOLUTE  simplicitt,  and  the  saving  of  time  effected  by  their 
use  is  most  material.  .  .  .  The  Author  has  done  much  to  reduce  the  cost  and  burden  of  the  Surveyor's 
work.  Every  one  connected  with  Engineering  or  Survey  should  be  made  aware  of  the  existence  of  this 
elaborate  and  useful  set  of  Tables." — Builder. 

"  From  the  enormous  amount  of  time  and  labour  which  the  Tables  will  save,  they  may  safely  be 
recommended  to  every  Surveyor  and  Engineer." — Mining  Journal. 

"  Up  to  the  present  time,  no  Tables  for  the  use  of  Surveyors  have  been  prepared  which  in  minute- 
ness of  detail  can  be  compared  with  those  compiled  by  Mr.  R.  L.  Ourhen.  .  .  .  "With  the  aid  of 
this  book  the  toil  of  calculation  is  reduces  to  a  minimum;  and  not  only  is  time  saved,  but  the  risk  of 
error  is  avoided.  .  .  .  The  profession  is  under  an  obligation  to  Mr.  Gukdeh  for  ensuring  that  in  the 
calculation  of  triangles  and  traverses  inaccuracies  are  for  the  future  impossible.    .    .    .    Mr.  Guuden's 

BOOK  HAS  BUT  TO    BE   KNOWN,    AND    NO   ENGINEERS    OR  SURVEYOR'S  OFFICE  WILL    BE   WITHOUT   A  CorV."— 

Architect. 


London  :  CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 


THE   CIRCLE   OF   THE   SCIENCES: 

A  SERIES  OF  POPULAR  TREATISES 

ON   THE 

NATURAL  AND  PHYSICAL  SCIENCES,  AND  THEIR  APPLICATIONS. 

BY 

Professors  Owen,  Axsted,  Young,  and  Tenxaxt;  Drs.  Latham,  Edward  Smith, 

Scofferx,  Bushnan,  and  Bronxer;  Messrs.  Mitchell,  Twisden, 

Dallas,  Gore,  Imeay,  Martin,  Sparling,  and  others. 

Complete  in  Nine  Volumes,  illustrated  with  many  thousand  Engravings  on  Wood. 
( Irown  Svo.     Cloth  lettered.     5s.  each  volume. 


Vol.  1.— ORGANIC  NATURE.— Part  I.  Animal  and  Vegetable  Physiology ;  the 
Skeleton  and  the  Teeth;  Varieties  of  the  Human  Race.  By  Professor  Owen,  Dr.  Latham,  and  Dr. 
BuSHXAN. 

Vol.  2.— ORGANIC  NATURE.— Part  II.  Structural  and  Systematic  Botany,  and 
Natural  History  of  the  Animal  Kingdom — Invertebrated  Animals.  By  Dr.  Edward  Smith  and 
William  S.  Dallas,  F.L.S. 

Vol.  3.— ORGANIC  NATURE— Part  III.  Natural  History  of  the  Animal  Kingdom 
— Vertebrated  Animals.    By  William  S.  Dallas,  F.L.S. 

Vol.  4.— INORGANIC  NATURE.— Geology  and  Physical  Geography;  Crystallo- 
graphy: Mineralogy.  Meteorology,  and  Atmospheric  Phenomena.  By  Professor  Axsted,  Rev.  W. 
Mm  HELL,  M.A.,  Professor  Tenhaut,  and  Dr.  Scofferx. 

Vol.   5.  —  PRACTICAL     ASTRONOMY,    NAVIGATION,    AND     NAUTICAL 

Astronomy.    By  High  Breem,  Greenwich  Observatory,  Professor  Youxg,  and  E.  J.  Lowe,  F.B.A.S. 

Vol.  6.— ELEMENTARY  CHEMISTRY.— The  Imponderable  Agents  and  Inorganic 
Bodies.    By  Jonx  Scoffers,  M.D. 

Vol.   7.— PRACTICAL   CHEMISTRY— Monographs  on  Electro-Metallurgy;    the 

Photographic  Art:  Chemistry  of  Food  and  its  Adulterations;  and  Artificial  Light.  By  George 
(tore,  Birmingham,  Johx  Scofferx,  M.D.,  Dr.  Edward  Broxxer,  Bradford,  Marcus  Spaelixg,  and 
Jonx  Martix. 

Vol.  8.— MATHEMATICAL  SCIENCE.— Philosophy  of  Arithmetic;  Algebra  and 

its  Solutions;  Plane  Geometry;  Logarithms;  Plane  and  Spherical  Trigonometry;  Mensuration 
and  Practical  Geometry,  with  use  of  Instruments.  By  Professor  Youxg,  Bev.  J.  F.  Twisdex,  M.A., 
Sandhurst  College,  and  Alexander  Jardixe,  C.E. 

Vol.  9.— MECHANICAL  PHILOSOPHY— The  Properties  of  Matter,  Elementary 
Statics;  Dynamics;  Hydrostatics:  Hydrodynamics;  Pneumatics;  Practical  Mechanics  ;  and  the 
Steam  Engine.     By  the  Bev.  Walter  Mitchell.  M.A.,  J.  E.  Young,  and  Johx  Imrat. 


In   Separate   Treatises.     Cloth. 


1.  Axsted's   Geology   and    Physical    Geo- 

graphy  2 

2.  Breem's  Practical  Astronomy,   .        .        .2 

3.  Bronner  and  Scoi  fern's  Chemistry  of 

Food  and  Diet, 1 

4.  Bushnan's    Physiology  of   Animal    and 

Vegetable  Life, 1 

o.  Gore's  Theory  and  Practice  of  Electro- 
Deposition,        1 

6.  Imray's  Practical  Mechanics,     .        .        .    1 

7.  Jardine's  Practical  Geometry,  .  .    1 

5.  Latham's     Varieties     of     the     Human 

Species, 1 

9.  Mitchell  and  Texxaxt's  Crystallography 
and  Mineralogy 3 

10.  Mitchell's  Properties  of  Matter  and  Ele- 

mentary Statics 1 

11.  Owen's  Principal  Forms  of  the  Skeleton 

and  the  Teeth,  ...  .       .    1 


12.  Scofferx's  Chemistry  of  Light,  Heat,  and 

Electricity. 3 

13.  Scoffern's  Chemistry  of  the  Inorganic 

Bodies 3 

14.  Scofferx's  Chemistry  of  Artificial  Light,    1 

15.  Scofferx  and  Lowe's  Practical  Meteor- 

ology  

10.  Smith's  Introductionto  Botany:  Structural 
and  Systematic, 

17.  Twisden's  Plane  and  Spherical  Trigono- 

metry,         

18.  Twisdex  on  Logarithms 1 


0 
6 

1    6 


19.  Young's  Elements  of  Algebra. 

20.  Young's  Solutions  of  Questions  in  Algebra, 

21.  Youxg's  Navigation  and  Nautical  Astro- 

nomy,        ...  ... 

22.  Young's  Plane  Geometry 

23.  Young's  Simple  Arithmetic, 

24.  Young's  Elementary  Dynamics, 


London  :  CHARLES  GEIFFIN  &,  CO.,  10  Stationers'  Hall  Court. 


6  A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 

TECHNICAL  AND  SCIENTIFIC  PUBLICATIONS. 


ELECTRO-METALLURGY. 

ELECTRO-METALLURGY  (A  Manual  of).     By  James  Napier, 

F.R.S.E.,  P.C.S.     Comprising — 

Electrotype  Processes.  ,        Bronzing. 

Electro-plating.  Coating  with  Copper. 

Electro-gilding.  Deposition  op  other  Metals. 

A    HlSTORY  OF   I  HE  A.RT,  &C,  &C. 

Wicli  Illustrations,  crown  Svo,  cloth,  7s.  6d.     Fifth  Edition, 

"The  Fifth  Edition  hap  all  the  advantogen  of  a  \nv  work,  and  of  a  proved  and  tried  friend.    .    . 
A  work  calculated  to  inspire  invention.'1— Vr<a  'Itrand  Watch 

DYEING 

DYEING  AND  DYEING  RECEIPTS  (A  Manual  of).     By  James 

Napier,  F.R.S.E.,  F.C.S.     Comprisu 

1.  Chemistry  op  Dyeing.  4.  AnimalDyes. 

2.  Mordants  and  Alterants.  •">.  Aniline  Colours. 

:;.  Vegetable  Dyes.  !    G.  Practical  Manipulation. 

With  Diagrams  and  ."7  Specimens  of  Dyed  Cotton,  Silk,  and  Woollen  Fabrics. 

Demy  8vo,  cloth  hevelled,  21s.     Third  Edition. 

"  Exceedingly  valdabu  to  the  Practical  Dyer VMannalol  •  reference  to  all  who 

wish  tu  keep  pace  with  the  scientific  disco  vt  riea  of  the  time."    .-""i  u.ii  oj  Applied  Science. 


CHEMISTRY. 
CHEMICAL    RECREATIONS:    A   Popular  Manual  of  Experi- 

mental  Chemistry.     By  John  Joseph  Grippin,  F.C.S.     With  540  Engravings 
of  Apparatus.     Crown  4to,  cloth.     Tenth  Edition. 

Part  I.  Elementary  Chemistry.    Price  2s. 

Part  II.  The  Chemistry  ok  the  Non-Metallic  Elements,  including  a  Comprehen- 
sive Course  of  Class  Experiments.     Price  10s.  (id. 

Or,  complete  in  one  volume,  cloth,  /jilt  lop,  12*.  61. 


PRACTICAL  CHEMISTRY  (A  Manual  of):  The  Analysis  of 

Foods   and  the   Detection   of   Poisons.      By  A.   Wynter  Blyth,    F.C.S., 
M.B.C.S,  Public  Analyst  for  the  County  of  Devon. 

Part  I.  FOODS  :  Sugar,  Starches  ;  Flour,  Bread  ;  Milk,  Butter,  Cheese;  Tea, 
Coffee,  Cocoa  ;  Alcohol  and  Alcoholic  Liquids  ;  Condiments. 

Part  II.  POISONS  :  Organic  and  Inorganic,  their  Detection  and  Estimation. 
Crown  Svo,  cloth,  with  Numerous  Tables  and  Diagrams,  price  12s.  Gd. 

"  Will  be  used  by  every  Analyst." — Lancet. 

"  Stands  unrivalled  for  completeness  of  information." — Sanitary  Record. 


London  :  CHARLES  GRTFFIX  t  CO.,  10  Stationers'  Hall  Court. 


A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 

GENERAL  SCIENTIFIC  PUBLICATIONS. 


GEOLOGY. 

A   MANUAL    OF    GEOLOGY: 

PRACTICAL    AND    THEORETICAL. 

BY 

JOHN  PHILLIPS,  M.A.,  P.P.S.,  F.G.S., 

LATE  PROFESSOR  OF  GEOLOGY  IX  THE  UNIVERSITY  Of  OXFOKH. 

REVISED  AND  EDITED  BY 

ROBEET      ETHERIDGE,     F.  R.  S.,    F.  G.  S., 

<>F  THE  MF   I'.IM  OY  PRACTICAL  GEOLOGY, 

AND 

H.     G  OVIER     SEELEY,     F.  R.  S, 

OF   KING'S   COLLEGE,    LONDON. 

WITH    NUMEROUS    ILLUSTRATIONS. 
{In  Preparation.) 


NATURAL    HISTORY. 

THE  STUDENT'S  NATURAL   HISTORY;  a  Dictionary  of  the 

Natural  Sciences :  Botany,  Conchology,  Entomology,  Geology,  Mineralogy, 
Palaeontology,  and  Zoology.  By  W.  Baird,  M.D.,  F.L.S.,  late  of  the  British 
Museum.  With  a  Zoological  Chart,  showing  the  Distribution  and  Range  of 
Animal  Life,  and  over  Two  hundred  and  lift}7  Illustrations.  Demy  Svo.  (Jloth 
gilt,  10s.  6d. 

'•The  work  is  a  very  useful  one.  and  will  contribute,  by  its   cheapness  and  comprehensiveness,  to 
foster  the  extending  taste  for  Natural  Science. " — Westminster  Review. 


THE  NATURAL  HISTORY  OF  THE  INANIMATE  CREATION, 

recoi'ded  in  the  Structure  of  the  Earth,  the  Plants  of  the  Field,  ami  the  Atmos- 
pheric Phenomena.  By  Professor  Axsted,  M.A.,  F.E.S.  "With  numerous 
Illustrations.     Large  post  Svo.     Cloth,  Ss.  6d. 

A  POPULAR  HISTORY  OF  THE  ANIMAL  CREATION:  being 

a  Systematic  and  Popular  Description  of  the  Habits,  Structure,  and  Classifica- 
tion of  Animals.  By  W.  S.  Dallas,  F.L.S.  With  coloured  Frontispiece  and 
many  hundred  Illustrations.     Crown  Svo.     Cloth,  Ss.  Gel.     2srcw  Edition. 


London  :  CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 


GENERAL  SCIENTIFIC  PUBLICATIONS. 


HYGIENE    AND    PUBLIC    HEALTH. 


A  DICTIONARY  OF  HYGIENE  AND  PUBLIC  HEALTH 

BY 

ALEXANDER  WYNTER  BLYTII,   .M.R.C.S.,  F.C.S., 

AHALl  ST  FOR  THH  i  OUH  i\    OP  N 
□PBISING  — 

I.  samiakv  Chemistry.  •':.  Sanitary  Legislation. 

•_'.  Sanitary  Engineering.  4.  Epidemk  and  Epizootic  Diseases. 

5.  Eygiene     Military,   Naval.    Private,    Public,   School. 
Royal  Sco,  cloth  bevelled,  with  Illustrations,  pru 

"  A  work  of  EXTBE1IB  vai  I  'ally  interested  in  Sanitation."— Medical  Timet  ami  Gazette. 


/'<■'  i  'ii. 

DOMESTIC  MEDICINE  AND  HOUSEHOLD  SURGERY 

(A  DICTIONARY  OF).  By  Spencek  Thomson,  M.D.,  Edin.,  L.R.C.S. 
With  Appendix  on  the  Management  o]  the  Sick-room,  and  many  Hints  for 
the  Debt  and  Comfort  of  Invalids. 

Large  Svo,  clotJi,  with  nurm  '       tralions,  price  8s.  6d. 

"Dr.  Thomson  lias  iu\  public  a  vast  amount  of  useful  professional 

knowledge.'' — Dublin  Journal  of  Mi 


MILITARY    SURGERY. 
[INTERNATIONAL   PRIZE    ESSAY. 

THE   SURGEON'S   POCKET-BOOK:    an  Essay  on  the 

Best  Treatment  of  the  Wounded  in  War  ;  for  which  a  prize  was  awarded  by 
Her  Majesty  the  Empress  of  Germany.  Specially  adapted  to  the  PUBLIC 
M  LDICAL  SERVICES.  By  Surgeon-Major  •).  II.  Porter,  Hon.  Assoc,  of  the 
Order  of  St.  John  of  Jerusalem,  late  Assistant-Professor  of  Military  Surgery  in 
the  Army  Medical  School. 
l6mo,  roan,  with  152  Illustrations,  p\       7  .  M.     Second  Edition,  revised  and  enlarged. 

'Every  medical  officer  is  recommended  to  ha-.  jeon's  Pocket-Book,'  by  Surgeon-Major 

Porter,  accessible,  to  refresh  his  memory  and  fortify  his  judgment"— Precis  of  Field-Service  Medical 
Arrangements  for  Afghan  War. 

"  This  capital  little  book    ...    of  the  greatest  practical  value.    ...   A  surgeon  with  this  Manual 
in  his  pocket  becomes  a  man  of  resource  at  once." — Westminster  Review. 

Published  under  the  Sanction  of  the  National  Society  for  Aid  to  the 
Ski:  and  Wounded  in   War. 

A   MANUAL   OE   INSTRUCTION   EOR   ATTENDANTS 

OX  THE   SICK  AND  WOUNDED   IN  WAR.      By  Staff-Assistant-Surgeon 
A.  Moffitt,  of  the  Royal  Victoria  Hospital,  Netley. 

With  numerous  Illustrations,  post  Svo,  cloth,  5s. 


London  :  CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


A  Selection  from  Charles  Griffin  and  Company's  Catalogue.  9 


GENERAL    SCIENTIFIC    PUBLICATIONS. 


MEDICINE. 


WORKS  BY  PROFESSOR  AITKEN,  M.D.,  F.R.S., 

Professor  of  Pathology  in  the  Army  Medical  School; 
Examiner  in  Medicine  for  the  Military  Medical  Services  of  the  Queen. 


THE  SCIENCE  AND  PRACTICE  OF  MEDICINE.     In  Two  Volumes,  Royal 
Svo,  cloth,  with  Map,  and  Numerous  Illustrations.     Seventh  Edition,  thoroughly 
revised,  in  part  re-written,  and  greatly  enlarged.     Price,  42s. 
"  The  Standard  Text-Book  in  the  English  language.    .    .    .    There  is  no  work  more  indispensable 
for  the  Practitioner  and  Student"— Edin.  Medical  Journal. 

OUTLINES  OF  THE  SCIENCE  AND  PRACTICE  OF  MEDICINE.  A 
Text-Book  for  Students.     In  Crown  Svo.     Second  Edition  in  preparation. 

"  Well  digested,  clear,  and  well  written,  the  work  of  a  man  conversant  with  every  detail  of  his 
subject,  and  a  thorough  master  of  the  art  of  teaching."— British  Medical  Journal. 

THE  GROWTH  OF  THE  RECRUIT,  and  the  Young  Soldier,  with  a  view 
to  the  Selection  of  "Growing  Lads"  and  their  Training.      2s.  6d. 

"This  little  work  should  be  in  the  hands  of  all  instructors  of  youth,  and  all  employers  of  youthful 
labour." — Lancet. 

OUTLINE  FIGURES  OF  THE  TRUNK  OF  THE  HUMAN  BODY,  on  which 
to  indicate  the  areas  of  physical  signs  in  the  Clinical  Diagnosis  of  Disease.  For 
the  use  of  Students  and  Practitioners  of  Medicine.     Is.  6d. 


IMPERFECT  DIGESTION :  Its  Causes  and  Treatment.  By  Arthur  Leared, 
M.D.,  F.B.C.P.,  late  Senior  Physician  to  the  Great  Northern  Hospital.  Post 
Svo,  Cloth,  4s.  6d.     Sixth  Edition. 

"  It  now  constitutes  about  the  best  work  on  the  subject."— Lancet. 

"Dr.  Leared  has  treated  a  most  important  subject  in  a  practical  spirit  and  popular  manner."- 
Medical  Times  and  Gazette. 

"A  useful  manual  of  the  subject  upon  which  it  treats,  and  we  welcome  it  as  an  addition  to  our 
Medical  Literature."— Dublin  Quarterly  Journal  of  Medical  Science. 


London  :  CHAELES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


10  A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 

UNIVERSITY  TEXT-BOOKS. 


MR.    CRUTTWELL'S     CLASSICAL     WORKS. 

Second  Edition,  crown  Svo,  cloth,  8s.  6d. 

1.  A  HISTORY  OF  ROMAN   LITERATURE,  from  the  Earliest  Period 

to  the  Times  of  the  An1  I      I  ,  T.  Cruttweix,  M.A.,  Fellow  of  Morton  College,  Oxford, 

Bead  Master  of  Malvern  Colli 
"Nothing  at  .'ill  equal  to  Li  has  hitherto  been  published  in  England." — British  Quarterly  Review. 

"A  most  serviceable— imlt •<•  I  indispensable — guide  for  the  Btudent.    .    .    .    The  'general  reader' 
will  be  both  charmed  and  instructed. " — Satut 

COMPANION    VOLUME. 

2.  SPECIMENS  OF  ROMAN  LITERATURE,  from  the  Earliest  Period 

Antonines. 

Paki  L— SOMAN  THOUGHT— Beligion,  Fhbvosofhi  and  Science,  Ari   ind  Lettkes. 

LI.— SOMAN  STYLE— Descriptive,  Rhetorical,  and  Humorous  Passages. 

With  Bimi  implete. 

Edited  by  0.  T.  Cbuttwbix,  MA.,  Morton  College  Oxford;    md  Pi  ixb  Banton,  M.A.,  some  time 

(  vuivn  svo,  cloth,  10s.  6d. 

•  \  work  with  a  Btanding-ground  of  its  own.    .    .    .    Not  only  useful,  but  necessary,  for  many 
■  iiinl  judgrj  I  in  arranging  the  plan  and  in  the  selection  of 

-ages  calls  for  hearty  commendation." — Saturday  / 


PROFESSOR    RAMSAY'S    CLASSICAL   WORKS. 
In  crown  Svo,  cloth. 

1.  A  MANUAL  OF  ROMAN  ANTIQUITIES:  Law,  Constitution,  Army 

ivv,  Public  and  Social  Life,  Agriculture,  Religion,  &c.    Fur  the  Uscof  Advanced  Students. 

By   \  ',  late   Professor  of    Humanity  in  the 

r'niv  numerous  Engravings,  and  very  copious  Index.    Eleventh 

■  lid. 

2.  AN  ELEMENTARY  MANUAL  OF  ROMAN  ANTIQUITIES.    Adapted 

for  Junior  i  Illustrations.    Seventh  Edition,    4s. 

3.  A  MANUAL  OF  LATIN  PROSODY.     Illustrated  by  Copious  Examples 

anii  Critical  I  •■  Edition.    5s. 


DR.    BRYCE'S    VIRGIL. 
VIRGILII  OPERA.   Edited  by  A.  Hamilton  Bryce,  D.C.L.,  LL.D.,  Senior 

Dublin.     Text  from  Heine  and  Wagner.    English 
riginal  and  Selected,  from  the  leading  German  and  English  Commentators.    Illustra- 
tions from  the  Antique.    Thirteenth  Edition.    In  1  Vol.,  fcap.  8vo,  cloth,  lis. ;  or  in  Three  Parts: — 
Part     I.— BUCOLICS  ahd  GEOBGICS,         2s.  64 
Part    IL— THE  .T.XEin,  Books  I.— VI.,        2s.  (id. 
Part  III.— THE  iENEID,  Books  VIL— XII.,  2s.  Gd. 
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UNIFORM    WITH    THE  ABOVE. 

HORATII  OPERA.     Edited  by  Joseph  Cukrie,  formerly  Head  Classical 

\v  Academy.    Text  from  Oiu  u.n  3.    English  Notes,  Original  and  Selected,  from 
mmentators.    Illustrations  from  the  Antique.    In  1  Vol.,  fcap.  8vo,  cloth,  5s. ;  or  in 
Paris:— 

Pai-.t    I.— CABMlNA,  3s. 

Part  II.— SATIRES  and  EPISTLES,  3s. 

"  The  notes  are  excellent  and  exhaustive."'— Quarterly  Journal  of  Education. 


THE  VOCABULARY  OF  PHILOSOPHY :  Mental,  Moral,  and  Metaphysi- 
cal   With  Quotations  and  References  for  the  Use  of  Students.    By  William  Fleming,  D.D.,  late 
Professor  of  Moral  Philosophy  in  the  University  of  Glasgow.    Revised  and  Edited  by  Henrt 
od,  LL.D.,  Professor  of  Moral  Philosophy  in  the  University  of  Edinburgh.    Crown  Svo, 
cloth,  bevelled,  10s.  6d.    Third  Edition. 


London  :  CHARLES  GEIFFIX  &  CO.,  10  Stationers'  Hall  Court. 


A  Selection  from  Charles  Chiffin  and  Company's  Catalogue.         11 

STANDARD  PRESENTATION  WORKS. 


THE    STANDARD     DICTIONARY     OF     QUOTATIONS. 

First  Series,  Thirtieth  Edition.     Second  Series,  Sixth  Edition. 

MANY  THOUGHTS  OF  MANY  MINDS: 

A  TREASURY  OF  REFERENCE. 

Consisting  of  Selections  from  the  Writings  of  the  most  Celebrated  Authors. 

FIRST  AND  SECOND  SERIES,  Compiled  and  Analytically  Arranged 

By  HENRY  SOUTHGATE. 


In  square  Svo,  elegantly  printed  on  toned  paper. 

Presentation  Edition,  Cloth  and  Gold,         .         .         12s.  6d.  each  Volume. 
Library  Edition,  Half-bound  Eoxburghe,      .         .         14s.  , ,  , , 

Do.  Do.       Morocco  Antique,       .         .        .         21s.  ,,  ,, 

Each  Series  is  complete  in  itself,  and  sold  separately. 


'  The  produce  of  years  of  research." — Examiner. 

1 A  treasure  to  every  reader  fortunate  enough  to  possess  it." — Journal  of  Education. 

•  A  Magnificent  Gift-Book,  appropriate  to  all  times  and  seasons." — Freemason's  Magazine. 

'  There  is  positively  nothing  in  the  language  that  will  bear  a  moment's  comparison  with  '  Many 

Thoughts.' " — Manchester  Advertiser. 
'The  Second  Series  fully  sustains  the  deserved  reputation  of  the  First.  "— John  Bull. 


BY    THE    SAME    AUTHOR. 

In  Square  8vo,  elegantly  printed  on  Toned  Paper,  10s.  6c?., 

SUGGESTIVE   THOUGHTS  ON  RELIGIOUS   SUBJECTS. 

A  DICTIONARY  OF  QUOTATIONS  AND  SELECTED  PASSAGES  FEOM  THE  BEST 
WRITERS,  ANCIENT  AXD  3IODEEN.  . 

FOR  THE   USE  OF   THE  CLERGY  AND  OTHERS. 

COMPILED   AND    ANALYTICALLY   ARRANGED 

By    HENRY    SOUTHGATE. 


PROFESSOR  CRAIK'S   ENGLISH    LITERATURE. 

A  HISTORY  OF  ENGLISH  LITERATURE  AND  OF  THE  ENGLISH  LANGUAGE 
FROM  THE  NORMAN  CONQUEST, 

With  numerous  Excerpts  and  Specimens. 

By    GEORGE    LILLIE    CRAIK,    LL.D., 

Late  Professor  of  History  and  English  Literature,  Queen's  College,  Belfast. 

"  Professor  Craik's  book  goin<?,  as  it  does,  through  the  whole  history  of  the  language,  probably  takes 
a  place  quite  by  itself.  The  great  value  of  the  work  is  its  thorough  comprehensiveness.  It  is  always 
clear  and  straightforward,  and  deals  not  in  theories  but  m  facts.  —Saturday  heview. 

LIBRARY  EDITION,  in  Two  Vols.,  royal  Svo,  handsome  cloth,  25s.     New  Edition. 

A    MANUAL    OE    ENGLISH   LITERATURE,    for  the  Use  of  Colleges, .Civil 
Service,  and  other  Competitive  Examinations.     Selected  from  the  larger  \Y  ork, 
by  Dr.  Craik.     Crown  Svo,  cloth,  7s.  6d.     Eighth  Edition. 
"A  Manual  of  English  Literature  from  so  experienced  and  well-read  a  scholar  as  Professor  Craik 
needs  no  other  recommendation  than  the  mention  of  us  existence.  —Spectator. 

London  :  CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


12  A  Selection  from  Charles  Griffin  and  Company's  Catalogue. 

WORKS     BY 

W.  J.  MACQUORN  RANKINE,  C.E.,  LL.D.,  F.R.8. 

I,  A  MANUAL  OF  APPLIED  MECHANICS: 

Comprising  the  Principles  of  Statics  and  Cinematics,  and  Theory  of  Structures, 
Mechanism,  and  Machines.  With  numerous  Diagrams.  Crown  8vo,  cloth, 
12s.  6d.     Ninth  Edition. 

"Cannot  fail  to  be  adopted  as  a  text-book.  .  .  .  The  whole  of  the  information  is  bo  admirably 
arranged,  that  there  is  every  facility  for  reference."— Mining  Journal. 

II.  A  MANUAL  OF  CIVIL  ENGINEERING: 

Comprising  Engineering  Surveys,  Earthwork,  Foundations,  Masonry,  Carpentry, 
Metal  Work,  Roads,  Railways,  Canals,  Rivers,  Waterworks,  Harbours,  &c. 
With  numerous  Tables  and  Illustrations.  Crown  Svo,  cloth,  16s.  Thirteenth 
Edition. 

"  Far  surpasses  in  merit  everv  existing  work  of  the  kind.  As  a  'Manual'  for  the  hands  of  the  pro- 
fessional Civil  Engineer  it  is  sufficient  and  unrivalled;  and  even  when  we  say  this,  we  fall  short  of  that 
high  appreciation  of  Dr.  Rankine's  labours  which  wo  should  like  to  express."— The  Engineer. 

III.  A  MANUAL  OF  MACHINERY  AND  MILLWORK: 

Comprising  the  Geometry,  Motions,  Work,  Strength,  Construction,  and  Objects  of 
Machines,  &c.  Illustrated  with  nearly  300  Woodcuts.  Crown  Svo,  cloth, 
12s.  6d.     Fourth  Edition. 

"Fully  maintains  the  high  reputation  which  Professor  Rankine  enjoys  as  a  scientific  writer ;  higher 

S raise  it  is  difficult  to  award  to  any  book.    ...    It  cannot  fail  to  be  a  lantern  to  the  feet  of  every 
nginecr." — The  Engineer. 

IV.   A  MANUAL   OF   THE   STEAM-ENGINE   AND 
OTHER  PRIME  MOVERS: 

With  numerous  Tables  and  Illustrations,  and  a  Diagram  of  the  Mechanical  Properties 
of  Steam.     Crown  Svo,  cloth,  12s.  6d.     Ninth  Edition. 

V.  USEFUL  RULES  AND  TABLES: 

For  Architects,  Builders,  Carpenters,  Coachbuilders,  Engineers,  Engravers,  Founders, 
Mechanics,  Shipbuilders,  Surveyors,  Wheelwrights,  &c.  Crown  Svo,  cloth,  9s. 
Fifth  Edition. 

"A  necessity  of  the  Engineer."— Athenxum.  ,  „     ™~  •      T  .   „„, 

"  Undoubtedly  the  most  useful  collection  of  engineering  data  hitherto  produced.  —Mining  Journal. 

VI.  A  MECHANICAL  TEXT-BOOK: 

A  Practical  and   Simple  Introduction  to  the   Study  of  Mechanics.     By   Professor 
Rankine  and  E.  F.  Bamber,  C.E.     With  numerous  Illustrations.     Crown  Svo, 
cloth,  9s.     Second  Edition. 
»**  The  "Mechanical  Text-Book"  was  designed  by  Pbofessor  Rankine  as  an  Introduction  to  the 

above  Series  of  Manuals. 


London:   CHARLES  GRIFFIN  &  CO.,  10  Stationers'  Hall  Court. 


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