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«

THE

MECHANICAL THEORY OF HEAT,

WITH ITS

APPLICATIONS TO THE STEAM-ENGINE

AND TO THE

PHYSICAL PROPERTIES OF BODIES.

[E UNIVKRSITY OF ZURICH.

1\ ARCHER 5IRST, F.R.S.,

PR0FK8S0R OP MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.

WITH AN INTRODUCTION BY

PROFESSOR TYNDALL.

LONDON:
JOHN VAN VOORST, 1 PATERNOSTER ROW

MDCCCLXVII.

c — \

HARVARD I

UNlVERSIiY I

LIBRARY I

PBINTED BY TAYLOE AND FBANC18,
RID LION COUBT, FLEET STREET.

INTRODUCTION.

Nearly seventeen years ago I translated for the Philosophical
Magazine the first of this series of Memoirs, by Professor
Clausius, on the Mechanical Theory of Heat. A short time
afterwards the Essay of Professor Helmholtz, Ueber die Erhal-
tung der Krafts was placed in my hands : I translated it, and
had it published in the continuation of ^Taylor's Scientific
Memoirs/ It was thus my fortune to introduce to the sci-
entific public of England the earliest writings of two of the
most celebrated contributors to the great theory in question.
For many years subsequent to the period here referred to, I
was careful to translate, or to have translated, every paper
published by these two writers; and the fact that the fol-
lowing series of these Memoirs is thought worthy of being
presented in a collected form to the English public, proves
that I did not overestimate their importance. I have been
asked by its publisher to write a line or two of introduction
to the present volume. This I could not refiise to do, though
I feel how superfluous it must be ; for the name and fame of
Pyofessor Clausius stand as high in this country as in his own.
My Introduction therefore shall be confined to this brief
statement of my relationship to his writings. They fell into
my hands at a time when I knew but little of the Mechanical
Theory of Heat. In those days their author was my teacher ;
and in many respects I am proud to acknowledge him as

my teacher still.

John Tyndall.
London, May 1867.

a2

i

AUTHOR'S PEEFACE.

It has been repeatedly represented to me (and from very dif-
ferent quarters) that the memoirs on the Mechanical Theory of
Heat which, since the year 1850, 1 have published fipom time to
time, principally in Poggendorflf^s Annalen, Bxe not easily acces-
sible to all who wish to read them, the interest taken in the
•Mechanical Theory of Heat having in recent times greatly
augmented in circles where physical Journals are not usually
found. Accordingly I have thought it advisable to collect and
republish those Memoirs. In so doing I have also sought to
remedy certain defects which have hitherto diminished their
utility.

My memoirs ^' On the Mechanical Theory of Heat '^ are of
different kinds. Some are devoted to the development of the
generai theory and to the application thereof to those properties
of bodies which are usually treated of in the doctrine of heat.
Others have reference to the application of the mechanical
theory of heat to electricity. The latter contain many exposi-
tions peculiar to the doctrine of electricity, and they form a
separate group, the study of which is not requisite for under-
standing the former. Other memoirs, again, have reference to
the conceptions I have formed of the molecular motions which
we call heat. These conceptions, however, have no necessary
connexion with the general theory, the latter being based solely
on certain principles which may be accepted without adopting
any particular view as to the Dature of molecular motions. I
have therefore kept the consideration of molecular motions
quite distinct from the exposition of the general theory.

f

yi author's pbbvacs.

The memoirs constitating these three different groups did
not^ however, appear exactly in their present order; partly
in consequence of the direction of my own studies, and partly
for other reasons, I found it desirable to pass during their
publication from one group to another. Hence has arisen the
disadvantage that a reader desirous of becoming acquainted
only with the theory, freed as much as possible from hypotheses,
cannot know in advance which memoirs are requisite, and which
are unnecessary for his purpose. This disadvantage is remedied
in the present reprint by simply separating the memoirs into
groups, as above explained.

The present collection contains the memoirs which belong to
the first group ; in them the mechanical theory of heat is deve-
loped from certain simple axiomatic principles, and is applied to
a series of phenomena depending upon heat. I have also in-
cluded the application of the theory to steam-engines, because
this application may be conveniently associated with the ex-
positions occurring in these memoirs, and especially with those
which have reference to vapours*.

The memoirs which treat of the application to electricity, and
those which relate to my conceptions of molecular motions, I
intend subsequently to collect in like manner. The memoirs
contained in this collection, however, are quite independent
of the others, and form in themselves a complete and con-
nected whole.

Another disadvantage which, as I frequently found, dimi-
nished the usefulness of my memoirs, arose from the fact that
many passages therein were with difficulty understood. The
mechanical theory of heat has introduced new ideas into science,
differing from the earlier accepted views, and accordingly re-
quiring special mathematical treatment. An instance of this,
especially worthy of mention, is a certain kind of differential
equations which I have used in my researches, and which differ
from the ordinary ones in one essential point : misconception

* [The ninth memoir of the present edition having been published in G^-
many subsequent to the appearance there of the first Part of the Collected
Memoirs, was not included therein. It is now published for the first time
in English ; and, at the Author's suggestion, its appropriate place in the entire
series of Memoirs is here assigned to it. — T. A. H.]

vu

might easily arise if this difference were not sufficiently observed.
The signification of, and the mode of treating these differential
equations have, indeed, long been known to mathematicians
through the researches of Monge ; but, from the fact that an
energetic attack on my theory originated in a misconception of
the true nature of these equations, it would appear that they have
not been sufficiently well studied. In order to avoid similar
misunderstandings in fiiture, I gave at the time a more detailed
explanation of the subject ; as this, however, was not published
in Poggendorff's Annalen, in which my other memoirs appeared,
but in Dingler's Polytechnic Journal (which contained that
attack), it may possibly have been seen by few of my readers.

In order, once for all, to remove any difficulty of this nature,
the present collection is preceded by a mathematical intro-
duction, in which the treatment of the differential equations in
question is discussed in a manner similar to that adopted
in Dingler's Journal. I have also in many places added notes
and appendices, in order to elucidate passages in the text.

The memoirs are reprinted verbatim in their original form.
The mechanical theory of heat, to the establishment and develop-
ment of which these memoirs have, as I believe, essentially con-
tributed, is of so great importance that it has already frequently
given rise to discussions on priority. Under these circum-
stances it appeared to me advisable to allow myself no altera-
tions ; fior even unimportant ones, having reference solely to
modes of expression, might possibly give rise to the thought that
I intended thereby either to take credit, ultimately, for some-
thing which did not appear in the original memoirs, or to sup-
press something which was there inserted*.

The notes and appendices now given for the first time are
plainly recognizable as such. In order to distinguish these
notes from those which were previously published, the former

* [It is scarcely necessaiy to state that in the present English edition this
role has not been adhered to. The translations of the original memoirs,
which are h^re reprinted from the Philosophical Magazine, were made by
different persons; and in order to secure the necessary uniformity m termi-
nology, verbal alterations were frequently requisite. All such alterations,
however, have been made with Prof. Clausius's sanction, to whom the proofs
have all been submitted for revision. — ^T. A. H.]

VIU

are enclosed in square brackets; and to every note containing
more than a mere reference is added the date. To the appen-
dices also dates have been affixed.

Should apparently superfluous repetitions be here and there
detected^ it must be remembered that the memoirs were published
at different times during the course of fourteen years, and that
often, between two memoirs which directly follow each other in
this edition, I had published several others bearing upon different
subjects. It was necessary in such cases to recapitulate such
portions of the antecedent memoirs as were deemed essential to
the comprehension of the new one, or requisite for bringing the
reader into the proper train of thought.

R, Clausius.

Zurich, August 1864.

CONTENTS.

MATHEMATICAL INTRODUCTION*.
On thb treatment of dipfebential equations which abjb not

DIBBCTLY JNTEGJELABLEj pp. 1-13.

Page

Notation 1

Coiidition of immediate integrability, and treatment of a differential

equation when this condition is not fulfilled 4

Example from analytical mechanics 7

Difference between the results obtained in the two cases 9

Greneralizations relative to the form of the equation and to the mode of

treating it 10

Differential equations involving more than three variables 11

FIRST MEMOIR.
On tbb moving fobcb ov heat and thb laws of heat which mat

BB DEDUCED THBREFBOM^ pp. 14r-69.

Historical remarksf 14

First Fundamental Theorem in the Mechanical Theory of Heat.

Enunciation of the theorem^ and general considerations thereon 18

Deduction of the analytical expression of the theorem ; special form of

this expression for perfect gases 21

Investigation of the form of the expression in the case of vapours .... 30
Conclusion deduced from tiie form of the expression for vapours; pre-
cipitation of vapours by expansion 34

Incidental assumption relative to perfect gases 37

Specific heats of gases 38

Deportment of gases during changes of volume 41

* [In Arts. 4 and 6 of this Introduction it is tacitly amumed that the functions
F (a, y) and F {x, y, z) given in the equations (11) and (18) have each but one
value for given values of the variables. It is easy to see what modifications will
ensue on abandoning this assumption. — 1867.]

t [Here and elsewhere ]l£ayer*s name has been incorrectly written Meyer. — T. A.H. ]

X CONTENTS.

Second Fundamental Theorem in the Mechanical Theory of Heat,

PreTious fonn of the theorem 43

Modification of the theorem^ and new mode of establishing it 43

Analytical expression of the theorem, especiallj for gases and yapours. . 45
Combination of the equation deduced from this theorem in the case of
gases, with certain consequences of the first fundamental theorem and

of the incidental assumption. Determination of Gamot's function . 48

Verification of this conclusion relative to Camot's function 49

Conclusions deduced from the principal equation in the case of yapours 62

Examination of Roche's formula for the tension of yapours 62

Deportment of saturated aqueous yapour with reference to the law of

Mariotte and Gaj-Lussac 63

Empirical formula for yapour-yolumes 60

Comparison of the deportment of yapour with that of carbonic acid . . 60
Calculation of the densities of saturated yapour at different tempera-
tures 61

Equation for the determination of the specific heat of saturated yapour 66

Determination of the mechanical equiyalent of heat 66

APPENDICES TO FIRST MEMOIR.
Appendix A. Coi^lbted deduction of the expression fob the

EXPENDED HEAT GIVEN IN EQUATION (3) (p. 27), pp. 60-75.

Qeneral differential equation for the heat taken up by a body 60

Application to two different kinds of changes of yolume 72

Expression for the expended heat 76

Appendix B. Integbation of the diffebential equation (II) (p. 28),

pp. 76-78.

Difference between this equation (11.) and ordinary differential equations
of the second order. Mode of integration 76

Conyenient form of the integral obtained by the introduction of the
function U 77

Appendix C. On the density of satubated aqueous vapoub,

pp. 78-80.

Modified form of the empirical formula for yapour-yolumes 70

Comparison with the results of Fairbaim and Tate \ 70

NOTE
On the influence of pbessube upon the fbeezing of liquids^
pp. 80-83.
Theoretical yiews and experiments of Professors James and William

Thomson 80

Application of the second fundamental theorem of the mechanical theory

of heat to the phenomenon of freezing 80

Application of the first fundamental theorem 82

CONTENTS. XI

Pag«

Another mode of change of the freezing-point 83

APPENDIX TO PRECEDING NOTE.

On thb diffebence between the lowebino of the fbeezing-point
ithich is caused by change of pbessvbe and that which mat
occub without any such change^ pp. 83-89.

Deduction of the equation relative to freezing at different temperatures
when the pressure remains unchanged 84

Deduction of the equation relative to freezing at different temperatures
when the pressure varies in a corresponding manner 87

Kemark on a change of the heat of fusion which takes place near (f , , 89

SECOND MEMOIR.
On the depobtment of vapoub dubing its expansion undeb dif-

FEBENT CntCUMSTANCES^ pp. 90-100.

Professor W. Thomson's remark on steam issuing from a boiler 90

Distinction between various cases of expansion 91

Treatment of the first case 91

Treatment of the second case 92

Treatment of the third case 97

APPENDIX TO SECOND MEMOIR.

On the yabiations of pbessube in a spbbading stbeam of gas,

pp. 100-103.

Observation of the difference of pressure^ and the cause of the latter . . 100
Professor W. Thomson's statements. . ; 102

THIRD MEMOIR.
On the theobetic connexion of two empibical laws bblating to

THE TENSION AND THE LATENT HEAT OF DIFFEBENT YAPOUBS,
pp. 104-110.

Relation between the several tension series ; the statements of Dalton

and Faraday thereon 104

Groshans's equation 106

Empirical law relative to the latent heat of vapour 106

Connexion between the two laws 107

Deduction of the latent heat of the vapour of a liquid from its tension

series, and vice versd 109

FOURTH MEMOIR.

On a modified fobm of the second fundamental theobem in the
mechanical thboby of heat, pp. 111-136.

Object of the memoir Ill

Xll CONTENTS.

Page

Concise re-statement of the first fundamental theorem^ and deduction

of the fundamental equation dependent thereon 112

On the form giyen to the second fundamental theorem in the First

Memoir^ and the reason why this form is still incomplete 116

Principle upon which the demonstration depends 117

Description of a completed cyclical process 118

Connexion between the two transformations which occur in the cyclical
process. Introduction of the equivalence-value of tranrforma-
tions. Definition of the positive and negative characters of the

Utter. 121

Mathematical expression of equivalence-values *. 123

Theorem of the equivalence of transformations 125

Expression for the equivalence- value of all the transformations in any -

given cyclical process 127

For every reversible cyclical process the algebraical sum of aU transfor-
mations must be zero. Equation which expresses this theorem . . 127
Special form of this equation ; comparison of the function of the tem-
perature introduced therein^ with Camot's function 131

For non-reversible cyclical processes the sum of all the transformations

is necessarily positive]. 138

Determination of the introduced function of temperature 134

FIFTH MEMOIR.
On the application op thb mechanioal theoby op heat to the

STEAM-ENGINE, pp. 136-207.

Iteasons which render a new theory of the steam-engine necessary. . . . 136

Difierent kinds of periodically working machines 139

Keduction of internal actions to a cyclical process 140

Fundamental equations for cyclical processes 141

Non-reversible changes of condition 142

General application of the equations^ which hold for cyclical processes,

to thermo-dynamic machines 145

Development of the principal equations applicable to vapour at a

maximum density 147

Changes of volume of a mass, consisting of vapour and liquid, enclosed
in a vessel impermeable to heat. Determination of the magni-
tude of the vaporous portion, of the volume, and of the work

done as functions of the temperature 151

General examination of the working of a steam-engine ; statement

of certain simplifying conditions 157

Determination of the work done during a cyclical process 159

Special forms of the expression for the work in the case of machines
without expansion, and in that of machines with complete expansion 161
Opposite procedure for the determination of the work 162

CONTENTS. xiii

Page

Comparifion of the steam-esgine with a perfect thermo-dynamic

machine 164

Statement of further imperfections requiring especial consideration . . 166
Brief explanation of Pambour's method of calculating the work of a

steam-engine , 167

Determination of the change suffered by a mass^ consisting of liquid
and vapour, and issuing from the boiler into the cylinder; when
vicious space exis^ and the pressure in the cylinder is not the

same as that in the boiler 172

Magnitude of the imcompensensated tranformation involved therein . . 177
Determination of the work done during a cyclical process, the above-
named imperfections being taken into account . , 178

Determination of the work by the opposite method 180

On the pressure in the cylinder at various stages of the working ; sim-
plifications of the equations which have reference thereto 181

Transformation of the equations to suit the case where the data consist
of the volumes; instead of the corresponding temperatures of the

working mass 183

Reduction of the work to a unit of weight of vapour 185

Treatment of the equations in numerical calculations 185

Values of the specific heat of water, and of the latent heat of aqueous

vapour employed in these calculations 192

Numerical calcidation of the work of a steam-engine without ex-
pansion 194

Numerical calculation of the work of a steam-engine with expansion . . 198

Reduction of the values to a unit of consumed heat .... 202

Friction taken into consideration 202

Table containing; for aqueous vapour, the values of the pressure p,

of its differential coefficient % and of the product T . ^ 204

at at

APPENDIX TO FIFTH MEMOIR.

On some appboximatb fobmul^ employed to facilitate calcula-
tions, pp. 208-214.

Quantity of heat which must be imparted to vapour, when expanding

in fuU work; in order to prevent partial condensation 208

Condensation of vapour; and work done during expansion in a vessel

impermeable to heat 211

Different modes of expansion of vapour 212

SIXTH MEMOIR.
On the application op the theorem op the equivalence op tbans- y

POBMATIONS TO INTEBIOB WORK, pp. 215-250.

Object of the memoir ^215

Expression of the second fundamental theorem in its previous form . . 216

XIV CONTENTS.

Page
Law of the dependence of the active force of heat upon the tempera-
ture. Introduction of the conception '' DiBgregation " 219

Distinction drawn between reTersible and irreyersible changes 223

Mathematical expression of the enunciated law 224

On a differential equation^ of similar form, proceeding from the equa-
tions hitherto known, and on the mode in which Pro£ Kankine

has transformed it 228

Theorem concerning the heat actually present in a hpdy 235

Previous opinions on this subject 237

Application of the theorem to chemical combinations 240

Theorem of the equivalence of transformations in its extended form . . 242
Consideration of the non-reversible changes, and of the uncompensated

transformations which then present themselves .244

Transformation-value of the heat of a body. Changes of temperature
caused by changes of disgregation. Impossibility of attaining the
absolute zero of temperature 247

APPENDICES TO SIXTH MEMOIR.
Appendix A. On tebminolooy, pp. 250-256.

On the various names which have been proposed for the function U.

Energy of a body 250

Thermal content of a body 252

Proposed introduction of the term ''Ergon " to denote work measured

by a thermal unit 253

Interior and exterior ergon. Ergonal content of a body 254

The expression, ergonized heat proposed instead of latent heat 255

Appendix B. On the specific heat of gases at constant volume,

pp. 256-266.

How £Eur the specific heat of gas at constant volume can serve as an ap-
proximate measure of the true capacity for heat 256

Convenient units for the expressicm of the specific heat of gas 258

The principal equations having reference to perfect gases collected. . . . 269
Calculation of the specific heat at constant volume from that at con-
stant pressure 261

Table 266

SEVENTH MEMOIR.
On an axiom in the mechanical theory op heat, pp. 267-289.

Historical account of the circumstances under which the axiom was
enunciated, that heat cannot of itself pass from a colder to a
warmer body .... 267

Zeimer's interpretation of this axiom 271

CONTENTS. XV

Page

Treatment of the second fundamental theorem in the mechanical theory

of heat given in Professor Hankine's memoirs 272

Progressive establishment and development^ in the present memoirs^
of the several theorems connected with the second fundamental
theorem 277

Him's objection to the axiom 280

Demonstration of the agreement between the axiom and the operation

conceived by Him 284

The same subject considered by the application of the transforma-
tion-value of the heat contained in the body 285

EIGHTH MEMOm.
On the concbntbation op bays of heat and light, and on the

LIMITS OF ITS ACTION, pp. 290-326.

Origin of the memoir. Bankine's views on the concentration of rays

of heat in contradiction with the axiom in the Seventh Memoir. . 290

Insufficiency of the previous determination of the mutual radiation be-
tween two surfaces for the case now under consideration 292

Determination of corresponding points and of corresponding surface-
elements in three planes intersected by rays 297

Determination of the mutual radiation in the case where no concen-
tration of rays takes place 305

Determination of the mutual radiation between two elements which

are optical images of each other 311

Relation between the enlargement and the ratio of the apertures of an

elementary pencil of rays 316

General determination of the mutual radiation between two surfaces in

which any concentrations whatever occur 319

Summary of principal results 326

NINTH MEMOIR.

On several convenient forms of the fundamental EQUATIONS OF
THE MECHANICAL THEORY OF HEAT, pp. 327-365. #

Equations expressing the two fundamental theorems of the mechanical
theory of heat, and the various properties of the magnitude^ oc-
curring therein 827

Differential equations deducible from the foregoing equations when the

condition of the body is determined by two variable quantities . . 331
Special forms of the equations in the case where the sole external force

is a pressure actiag uniformly on the sur£a,ce of the body 336

Treatment in the case of a homogeneous body 338

Treatment in the case of a perfect gas 346

Treatment in the case of a body which consists of two parts in different

states of aggregation 347

XVI CONTENTS.

ConBideration of the energy of a body and of an allied quantity desig-
nated by the tenn entropy 358

Equations for the determination of the energy and entropy of a body. . 358

Changes which occur in a non-reversible manner 362

Application of the two fundamental theorems of the mechanical theory

of heat to the entire condition of the universe 364

APPENDIX TO NINTH MEMOIR.

On the dbtbbmination op the enebgy and entbopy op a body,

pp. 366-376.

Deduction of the partial differential equations necessary for the determi-
nation of energy and entropy, and properties of the magnitude oc-
curring therein which has been termed the eigonal difierence .... 366

Establishment of the complete differential equations 372

Special forms of the equations for the case where the sole external force

is a pressure uniformly distributed over the whole surface 374-

ON THE

MECHANICAL THEORY OF HEAT.

MATHEMATICAL INTRODUCTION.
ON Ae treatment op differential equations which are

NOT DIRECTLY INTEGRABLE*.

1. A differential equation of the form

dz=^<f>{x,y)dw+ylr{a;,l/)dy (1)

being given, we may, for brevity, introduce the letters M and
N as representatives of the arbitrary functions ^ (ap, y) and
'^(^>y) o^ tlie variables a and y, and thus write that equation in
the somewhat more convenient form

dz^Mda-\-T^dy (1^)

This equation indicates by how much the magnitude z is in-
creased, when X and y receive the infinitesimal increments re-
presented by dx and dy ; a decrement being here, of course, con-
sidered as a negative increment. The above two functions, by
which' the differentials dx and dy are multiplied, represent the
partial differential coefficients of z according to x and to y. De-
noting, therefore, these partial differential coefficients by the

fractional forms ^r aiid -j-, we may write

•£=(t>{x,y)=U,
^=:'f{ar,y)=N.

(2)

* The principal part of this introduction is contained in a note^ published
by me, in Dingler's Polytechnisches Jmimal, vol. cl. p. 29 (1858).

-—■ ^

Z MATHEMATICAL INTRODUCTION.

This representation of partial differential coefficients by the

simple fractions -r-y ^ is, in a cexAain sense, objectionable. For

if in the equation (1) or (1*) we substitute these fractions for the
functions in question, the equation

dz=^dx+^dy (3)

is obtained, in which the same symbol dz appears three times
with three different meanings. On the right of the equation dz
denotes, y?r«/, the increment of z when, y remaining constant, x
alone is increased by dx ; and secondly, the increment received by
z when, without changing a?, y is increased by dy ; whilst on the
left of the equation dz represents the total increment of z due to
the simultaneous reception by x and y of the increments dx and
dy, respectively. This diversity in the interpretation of one and
the same symbol, arising from the different combinations into
which it enters, vitiates the expressiveness of the equation.

In consequence of this, various changes in the notation of
partial differential coefficients have been proposed. In order to
distinguish the partial differential coefficients from others, Euler
enclosed the above simple fractions in brackets, and his method
is still frequently adopted. In this notation the equation (3) as-
sumes the form

^=(£)^Ht)'» (»■)

Other mathematicians give, as a suffix to the symbol d in the
numerators of the above fractions, the variable to whose varia-
tion the differential coefficient is due ; in this notation the equa-
tion would be written thus :

rfr=^£fc+§5dy (3b)

dx dy

Others again, following the example of Jacobi, use the symbol b
in place o{ din the numerator as well as in the denominator of
the fraction which represents a partial differential coefficient. In
this maimer our equation acquires the form

''=^d'^-^^'y- (^")

Of these three notations that of (3^), wherein suffixes are em-

UNINTEGRABLE DIFFERENTIAL EQUATIONS. 3

ployed, is perhaps the most rational ; for it is precisely the nume-
rators of the representative fractions which admit of diflPerent in-
terpretations, and the latter are clearly and unequivocally ex-
pressed by means of these suffixes. Nevertheless the incessant
addition of a suffix constitutes an inconvenience, which, though
trivial in individual cases, becomes much graver when partial dif-
ferential coefficients are frequently employed. It must also be
observed that, in the cases which most frequently occur, the ori-
ginal and most convenient notation gives rise to no ambiguity.
For whena? and y denote two mutually independent variables upon
whose values that of z depends, it is manifest that the dz in the

dz
numerator of the fraction — - cannot be understood to denote

dx

other than that increment of z which is due to the increment dx
of the variable x which appears in the denominator. Any altera-
tion which may simultaneously take place in the value of the
other variable y must, together with the consequent variation
of z, be perfectly independent of the differential dxy so that the

dz
fraction j- would have no definite meaning whatever were the

above variation of z included in that of which dz is here the

symbol. It is consequently of little importance whether, in the

representation of partial differential coefficients, we give prefer-

dz dz
ence to the ordinary fractional forms ^-> -^^ or, for the sake of

greater clearness, to one of the above described modified forms
of notation.

In one case only is it necessary to have recourse to a distinc-
tive symbol in order to avoid misconception. It sometimes
happens, for instance, that the magnitudes x and y, upon whose
values that of z depends, are not independent of each other, but
that the value of one is affected by that of the other ; in other
words, that the former may be regarded as a function of the latter.
If y, for example, be considered as a function of x, then, in the
event of x increasing by dx, the simultaneous increment dy
of y cannot be regarded as arbitrary, but must be treated as a
magnitude whose value is also determined by the differential dx,

du
and capable of representation by the expression -. - dx. By sub-

b2

4 MATHEMATICAL INTRODUCTION.

stitution^ the differential equation (3) would now take the form

ax ay aw

Dividing throughout by dx we obtain an equation which, if
we also denote the quotient on the left by a simple fraction,
would read thus :

^^dz dz dy ,

dx'~ dx dy dx^

dz
here, however, the fraction t- on the left has a very different

meaning from that of the like fraction on the right.

In such cases the two fractions must in some way or other be
distinguished. To do so, we may either employ, for the partial
differential coefficients on the right, one of the three notations
above described, or we may employ a different symbol for the
fraction on the left. For the last purpose, mathematical authors

have proposed to write, in place of -i-,

.,, 1 , d[z) dz

either^rf^, or-^, or^.

Since cases of this kind however occur, comparatively speaking,
but seldom, it is of little importance which of these methods of
notation is adopted. In fact, whenever necessary, it will be easy
to add an explanatory remark as to the meaning to be attached
to any chosen symbol.

I have thought it necessary to enter into these details con-
cerning the different systems of notation now in use, because it
is precisely in investigations where familiar ideas are departed
from, that a diversity of notation is most liable to give rise to
misconceptions.

%, Returning to the differential equation

given in (1) and (1*), let us now inquire if, and how the magni-
tude z can be determined therefrom.

Differential equations of this form cannot all be regarded as
of like kind ; according to the constitution of the functions M
and N they are, on the contrary, divisible into two classes which

UNINTEGBABLE BIFFERENTIArL EQUATIONS. 5

differ firom each other essentially, not only with respect to the
treatment which they require, but also with reference to the
results to which they lead. To the first class belong the cases
where the functions in question satisfy the condition

dy'^ dx' ^^^

and the second class embraces all cases where this equation of
condition is not satisfied by the two functions.

When the equation (4) is fulfilled, the expression on the right
of the given differential equation (1) or (1*) is integrable; that
is to say, it is the complete differential of a ftinction of w and y,
in which these two variables may be regarded as independent of
each other ; and by integration an equation can be obtained of
the form

5r=F(a?, y)H- const (5)

When the condition expressed by the equation (4) is not ful-
filled, the expression on the right of the given differential equa-
tion is not integrable, whence we conclude that z cannot be ex-
pressed as a function of x and y so long as these variables are
regarded as independent, one of the other. In fact, if we were to
assume

we should have

dx dx
^^dz^ d¥{x,y) ^
dy dy '

whence would result

dU^ d^Yjx, y)
dy dxdy '

dii_ ^F{x, y)
dx"^ dydx '

But since, when the two variables of a function are indepen-
dent of each other, the result of differentiating, successively, ac-
cording to both is not affected by the order in which these differ-
entiations are effected, we have necessarily

d^F{x.y) _( PF{x,y) ^
dxdy "" dydx

1

6 MATHEMATICAL INTRODUCTION.

SO that from the two preceding equations the equation (4) follows
as a consequence, and thus contradicts the hypothesis from which
we started.

In such a case, therefore, the integration is impossible on the
assumption that the variables x and y preserve their property of
mutual independence. K, on the other hand, we assume any
determinate relation whatever to exist between the two magni-
tudes, in consequence of which one may be represented as a
function of the other, the integration of the given differential
equation will be thereby rendered practicable. For if we put

/(^,y)=o, (6)

where / represents any function whatever, we can by means of
this equation express either variable in terms of the other, and
then eliminate the variable thus expressed, together with its dif-
ferential, from the differential equation (1) . The general form
given in the equation (6) embraces, of course, the special case,
where one of the magnitudes a?, y ceases to be variable ; for then
its differential, by becoming equal to zero, at once vanishes
from the differential equation, and the magnitude itself becomes
replaced by its constant value.

Let us now suppose one of the variables, say y, together with
its differential, eliminated fix)m the differential equation (1) by
means of the equation (6), and the former thereby reduced to
the form

dz^<b{x)da: \

the equation thus modified will obviously give, by integration,
another of the form

^=F(^)+ const (7)

Accordingly, the two equations (6) and (7) must together be re-
garded as constituting a solution of the given differential equa-
tion. Since the function /(^,y) which appears in (6) is an arbi-
trary one, and to every different form of this fimction must
correspond, in general, a different fimction F {x) in (7), it is
manifest that there will be an infinite number of solutions of
the kind under consideration.

The form of the equation (7) , it is to be observed, is suscepti-
ble of several modifications. If, by means of the equation (6), x
had been expressed in terms of y ; and then, together with. its

UNINTEGRABLE DIFFERENTIAL EQUATIONS. 7

diflfereutial, eliminated from the given differential equation, the
form in question would have been

€fe=a>i(y)rf2^;
&om which, by integration, an equation of the form

^=F,(y)+ const {7»)

would have been obtained. Precisely the same equation would
be arrived at by substituting for aCy in the result (7) obtained
by the first method, its value in ^ as given by the equation (6).
Again, os might be only partially eliminated from (7) . For in-
stance, the function F(a?) will in general contain x in two or
more different combinations (or rather, it may be always made
to do so, by substituting for x equivalent expressions such as

(1— a)^ + aa?, -—-, &c. . . .), and when this is the case the value

of X expressed in y may always be substituted in some combina-
tions, whilst others are allowed to remain tmaltered. The equa-
tion would thereby assume the form

^=F2(a?, y)+const.,; (7^)

which may be regarded as the more general one, embracing both
the other forms as special cases.

It is obvious, however, that the three equations (7), (7*), and
(7^) , each of which only holds in combination with (6) , do not
constitute different solutions, but merely different expressions of
one and the same solution.

3. In order clearly to appreciate the essential difference
between the cases when the given differential equation belongs
to the first, and when it belongsuto the second class^ — ^that is to
say, when the condition (4) is, and when it is not fulfilled, — ^we
will consider an example which, partly by its relation to an
already well-known subject, and partly also by its susceptibility
of geometrical representation, is well suited to famish a clear
conception of the matter.

Conceive a moveable point j9 in a fixed plane, and let its posi-
tion at any stated moment be determined by rectangular coor-
dinates X and y. Acting on the point, and tending to move it
in the plane, is a force whose intensity and direction are different
at different parts of the plane. Required the work done by this
force when the point moves imder its influence.

8 MATHEMATICAL INTRODUCTION.

Let d^ be an element of the path described by the point, S the
component of the force acting thereon which fSedls in the direction
of this path, and dW the element of work performed by the force
during this small motion. This last element will be determined
by the equation

dW^Sds, (8)

to which, however, another form may be given more convenient
for our present purpose. Let P be the whole force acting in the
immediate neighbourhood of the arc-element ds^ and if> the angle
between this element and the direction of that force. Then, ob-
viously,

S=cos^.P,
so that

rfW=cos^.Prfi (9)

Now if X and Y denote the two components of the force P in the
directions of the coordinate axes, the cosines of the angles between
these directions and that of P wiU be expressed by

p and p.

Moreover, if by rfa? and dy we understand the increments which
the a? and y of the point p receive when the latter describes the
arc-element ds, the cosines of the angles between the direction
of this element and those of the coordinate axes wiU be expressed

by

do? J dy
-J- and -f-
ds ds

Hence for the cosine of the angle <^ between the force P and the

arc-element ds we have the expression

^ X da^ Y dy

which, when substituted in (9), gives the equation

dW=Xda; + Ydy • • (10)

This is a differential equation of the same form as those given
in (1) and (1*^), the notation alone being a little changed.
Instead of z the letter W is used, as more appropriate for the
representation of work ; and M and N, which before were abbre-
viated symbols for the functions ^(o?, y) and '^(.a?, y), are now
replaced by X and Y, the customary representatives of the com-

UNINTEORABLE DIFFERENTIAL EQUATIONS. 9

ponents of the force P, and are again abbreyiations for arbitrary
fiinctions of the coordinates x and y ; for^ as abeady remarked^
the force P varies in intensity and direction^ according to some
arbitrary law, with the position which the moveable pointy, upon
which it acts, occupies in the plane.

Before proceeding, by the integration of this equation, to de-
duce the work corresponding to any finite motion, the question
arises : does it satisfy the condition

dy dx

analogous to the equation (4) ? Should it do so, we may de-
duce at once an equation of the form

W=F(^, y)4-con8t.; (11)

but if it should not satisfy this condition, then in order to be able
to integrate, we must first assume a relation to exist between the
variables x and y ; so that finally we shall obtain a system of
equations of the form

/(^,y)=o, 1 ,^.

■W=F(^,y)-f const./ ^ ^

4. The geometrical signification of these two different results
is easily recognized.

Suppose the point p to move from a given initial position
«2?o> yo *^ * given final one ^j, y^. Then in the first case the
work done by the acting force during this motion may be at once
ascertained without the necessity of inquiring into the nature of
the path thereby described. This work, in fact, is expressed by
the difference

Although the point, therefore, may pursue very different paths
when moving from one position to another, the quantity of work
thereby performed by the acting force, being independent of the
path, is perfectly determined so soon as the starting-point and
the terminus are given.

In the second case it is otherwise. Of the two equations (12),
which have reference thereto, the first is arbitrary, and the
second can only be determined when the first is given, since the
form of the function F(a?, y) varies obviously with that which is

10 MATHEMATICAL INTBODUCTION.

given to the fanctifmf{x, y). The first equation is that of scHne
curve^ so that the above relation may be expressed^ geometri-
cally^ by sayings in the present case^ the work done by the acting
force during the motion of the point p can only be determined
when the whole course of the curve on which it moves is known.
The initial and final points of the motion being previously known,
the first of the above equations must be chosen so that the curve
thereby represented may pass through these two points; this
curve, however, may have an infinity of different forms to which,
notwithstanding the coincidence of the extremities, will corre-
spond an infinity of different quantities of work.

K, for instance, the point p be compelled to describe a closed
curve, and thus to return to its initial position, the coordinates
^u Vi being respectively equal to Xq, y^, the total work done, in
the first case, will be zero ; in the second, however, it need not
be so, but may have any positive or negative value whatever.

The example here borrowed firom analytical mechanics shows
also very clearly how a magnitude which is incapable of expression
as a function of x and y (so long as the latter are regarded as
variables independent of each other) may still have, for partial
differential coefficients according to x and y, determinate func-
tions of these variables. For it is manifest that, strictly speak-
ing, the components X and Y must be termed the partial differ^
ential coefficients, according to x and y, of the work W ; since,
when X increases by dx, y remaining constant, the work increases
by ILdx ; and when y increases by rfy, x remaining unaltered, the
work augments by Ydy. Now whether W be a magnitude ge-
nerally expressible as a function of x and y, or whether it can
only be determined on knowing the path described by the point,
we may always employ the ordinary notation for the partial dif-
ferential coefficients of W, and write.

rfW_Y.

(13)
=Y

dy

5. When the functions <^ {x, y) and '^ (x, y) , or M and N, which
ixicwr in the differential equation (1) or (1*), fail to satisfy the
condition (1), it has been shown that the integration may be

J

UNINTEGRABLE DIFFERENTIAL EQUATIONS. 11

eflfected by assuming a relation to exist between the variables x
and y of the form

/(«,y)=o.

The same object is achieved in a more general manner^ however,
by assuming the existence of an equation, not merely between x
and y, but involving all the variables a?, y, r, or rather any one or
more of them, and therefore expressible in the form

/(a7,y,2r)=0 (14)

If by means of this equation one of the variables be eliminated
from the given differential equation, another dijBPerential equation
will be obtained which may always be integrated.

Indeed in order to exhaust all possible cases of complete dif-
ferential equations of the first order in three variables, still fur-
ther extensions of the above considerations would be necessary.

The differential equation (1) is itself a limited form of the kind
in question, inasmuch as functions of all three variables, in-
stead of the two variables >, y, might therein present themselves.
When the differential equation has this more general form, which
may be thus written :

«^(a?,y,j2r)(ir + 'i^(a?,y,2r)rfy-fx(^>y^^)^'2r=0, . (15)

the condition to be satisfied in order that integration may be
possible without the assumption of any further relation between
the variables, assimies a more complicated form than that given
in (4) . It should, moreover, be observed that in the case of the
non-ftdfilment of this condition, and the consequent impossibility
of actual integration, the relation which must be assumed, or
be involved in some imposed condition, in order to be able to inte-
grate, need not have the form of a primitive, but may itself be
a differential equation. In the treatment of the equations, too,
as well as in the manner of expiressing the result, many modifi-
cations may present themselves.

It is not necessary, however, to enter here into all these ex-
tensions, since the preceding exposition will suffice to render in-
telligible the equations hereafter to be developed, as well as the
operations to which such equations will be subjected.

6. I may mention, lastly, that the preceding considerations,
relative to complete differential equations involving three varia-
bles, may be extended in a similar manner to complete differen-

12 MATHEMATICAL INTRODUCTION.

tial equations in four or more variables^ and that amongst the
latter we are thereby led to the detection of corresponding dif-
ferences. In illustration of this I will give but one simple spe-
cial case^ well known in mechanics, and closely related to the
example already considered.

Let p be a moveable point in space whose rectanglar coordi-
nates at any particular moment are x^ y, z. Conceive a force P
to act on this point with an intensity and in a direction which
may be different at different places in space. I propose to de-
termine the quantity of work done by the force during any as-
signed motion.

Let d!9 be an element of the path described by the point, and
^ the angle at which this path is inclined to the direction of the
force. The element of work rfW wiU be again given by the
equation

rfW=cos^.P&.

In order to give another form to this expression, we may denote
by X, Y, Z, the three components of P in the directions of the,
coordinate axes j in which case the fractions

X Y Z

p> p^ p>

will represent the cosines of the angles which the direction of the
force makes with the directions of the three coordinate axes. If,
further, dx, dy, dz be the increments of the coordinates x, y, z, due
to the description of the path-element ds, the cosines of the angle
between the element ds, and the three coordinate axes will be
expressed, respectively, by

dx dy dz
da ds ds
Hence is deduced, for the cosine of the angle ^ between the
directions of the path and the force, the value
J a dx Y dy Z dz
^ P ds^-p &^P ds
Substituting this value in the above expression for rfW, we have
the differential equation

dW=Xdar+Ydy + Zdz (16)

for the determination of the work. The magnitudes X, Y, Z

J

UNINTEGRABLE'DIFPERENTIAL EQUATIONS. 13

which here present themselves are perfectly arbitrary functions
of x,y,2; for whatever values .these three components may have
at different points in space^ a force P always results therefrom.

In treating this equation the following three conditions at once
enter into consideration :

^2^=^ ^=:^ ^=f^. (17)

dy dx' dz dy^ dx dz '

and the question arises whether or not the functions X, Y, Z
fulfil them.

When the three equations of condition are satisfied, the ex-
pression on the right of (16) is the complete differential of a func-
tion of X, y, Zy wherein these three variables may be regarded as
mutually independent. The integration therefore may be at once
effected, and an equation thereby obtained of the form

W=F(^,y,;2r)+ const. ...... (18)

If we now conceive the point jo to move from a given initial
point x^y y^, Zq to a given terminal point x^, y^, z^y the work done
' by the force during the passage will be represented by the dif-
ference

This work, therefore, is completely determined by the positions
of the extreme points between which motion has occurred, and
hence it follows that the work done by the force is always the
same whatever path may have been followed in passing from the
one position to the other.

When the three conditions (17) are not fulfilled, the integra-
tion cannot be performed with the same generality. The inte-
gration will be rendered possible, however, so soon as the path
pursued by the moving point p is known. If between the ex-
treme points we conceive several curves to be drawn, and the
point/? compelled to move thereon, we shall obtain a definite
amount of work corresponding to each curve, but these quanti-
ties of worK need not, as in the previous case, be equal to one
another ; in fact they wUl, in general, have different values.

11* FIRST MEMOIR.

FIRST MEMOIR.

ON THE MOVING FORCE OF HEAT AND THE LAWS OF HEAT WHICH
MAY BE DEDUCED THEREFROM^.

The steam-engine having furnished us with a means of con-
verting heat into a motive power^ and our thoughts being
thereby led to regard a certain quantity of work as an equivalent
for the amount of heat expended in its production^ the idea of
establishing theoretically some fixed relation between a quantity
of heat and the quantity of work which it can possibly produce,
from which relation conclusions regarding the nature of heat
itself might be deduced, naturally presents itself. Already, in-
deed, have many successful efforts been made with this view ;
I believe, however, that they have not exhausted the subject,
but that, on the contrary, it merits the continued attention
of physicists ; partly because weighty objections lie in the way
of the conclusions already drawn, and partly because other con-
clusions, which might render efficient aid towards establishing
and completing the theory of heat, remain either entirely unno-
ticed> or have not as yet found sufficiently distinct expression.

The most important investigation in connexion with this sub-
ject is that of S. Camot f-

Later still, the ideas of this author have been represented
analytically in a very able manner by Clapeyron J.

Camot proves that whenever work is produced by heat and a

♦ Communicated in the Academy of Berlin, Feb. 1850; published in Pog-
gendorfTs Annalen, March-April 1860, vol. Ixxix. pp. 368, 500, and trans-
lated in the Philosophical Magazine, July 1851, vol. ii. pp. 1, 102.

t Reflexions mr la puissance motrice dufeu, et sur les machines propres H de-
velopper cette puissance, par S. Camot. Paris, 1824. I have not been able to
procure a copy of this work ; I know it solely through the writings of Clapey-
ron and Thomson, from which latter are taken the passages hereafter cited.
[At a later date I had an opportunity of studying the work itself, and of thus
confirming the views, regarding its contents, which I had previously formed
from a perusal of the writings referred to. — 1864.]

X Journal de I'Ecole Polytechnique, vol. xiv. 1834 ; Pogg. Ann. vol. lix. ;
and Taylor's Scientific Memoirs, Part III. p. 347.

MOVING FORCE OF HEAT. • 15

permanent alteration of the body in action does not at the same
time take place, a certain quantity of heat passes from a warm
body to a cold one ; for example, the vapour which is generated
in the boiler of a steam-engine, and passes thence to the con-
denser where it is precipitated, carries heat from the fireplace to
the condenser. This transmission Camot regards as the change
of heat corresponding to the work produced. He says expressly,
that no heat is lost in the process, that the quantity remains un-
changed ; and he adds, ^^ This is a fact which has never been dis-
puted ; it is first assumed without investigation, and then con-
firmed by various calorimetric experiments. To deny it, would
be to reject the entire theory of heat, of which it forms the
principal foundation.^^

I am not, however, sure that the assertion, that in the pro-
duction of work a loss of heat never occurs, is sufficiently esta-
blished by experiment. Perhaps the contrary might be asserted
with greater justice ; that although no such loss may have been
directly proved, still other facts render it exceedingly probable
that a loss occurs. If we assume that heat, like matter, cannot
be lessened in quantity, we must also assume that it cannot be
increased ; but it is almost impossible to explain the ascension
of temperature brought about by friction otherwise than by
assuming an actual increase of heat. The careful experiments
of Joule, who developed heat in various ways by the application
of mechanical force, establish almost to a certainty, not only the ^
possibility of increasing the quantity of heat, but also the fact
that the newly-produced heat is proportional to the work ex-
pended in its production. It may be remarked further, that
many facts have lately transpired which tend to overthrow the
hypothesis that heat is itself a body, and to prove that it con-
sists in a motion of the ultimate particles of bodies. If this be
so, the general principles of mechanics may be applied to heat ;
this motion may be converted into work, the loss of vis viva in
each particular case being proportional to the quantity of work
produced.

These circumstances, of which Carnot was also well aware, and
the importance of which he expressly admitted, pressingly de-
mand a comparison between heat and work, to be undertaken
with reference to the divergent assumption that the production

16 FIRST MEMOIR.

of work is not only due to an alteration in the distribution of
heat^ but to an actual consumption thereof; and inversely^ that
by the expenditure of work heat may be produced.

In a recent memoir by Holtzmann^^ it seemed at first as if
the author intended to regard the subject from this latter point
of view. He says (p. 7), " the effect of the heat which has been
communicated to the gas is either an increase of temperature
combined with an increase of elasticity, or a mechanical work,
or a combination of both ; a mechanical work being the equiva-
lent for an increase of temperature. Heat can only be measured
by its effects ; and of the two effects mentioned, mechanical
work is peculiarly applicable here, and shall therefore be chosen
as a standard in the following investigation. I name a unit of
heat, the quantity which, on being communicated to any gas, is
able to produce the quantity of work a ; or to speak more defi-
nitely, which is able to raise a kilogrammes to a height of one
metre.'' Afterwards, at page 12, he determines the numerical
value of the constant a, according to the method of Meyer t, and
obtains a number which exactly corresponds to that obtained in
a totally different manner by Joule. In carrying out the theory,
however, that is, in developing the equations by means of which
his conclusions are arrived at, he proceeds in a manner similar
to Clapeyron, so that the assumption that the quantity of heat is
constant is still tacitly retained.

The difference between the two ways of regarding the subject
has been seized with much greater clearness by W. Thomson,
who has applied the recent investigations of Regnault, on the
tension and latent heat of steam, to the completing of the memoir
of Camot J. Thomson mentions distinctly the obstacles which lie
in the way of an unconditional acceptance of Carnot's theory,
referring particularly to the investigations of Joule, and dwelling
on one principal objection to which the theory is liable. If it be
even granted that the production of work, where the body in
action remains in the same state after the production as before,

• Ueber die Wdrme und Elastidtat der Oase tmd Ddmpfe, von . C Iloltz-
mann. Manheim, 1845. Also Poggendorif's Annalen, vol. Ixxii. a ; and Tay-
lor's Scientific Memoirs, Part XIV. p. 189

t Awn, der Chem. und Pharm.j vol. xlii. p. 239.

X Transactions of the Royal Society of Edinburgh, vol. xvi.

MOVING FORCE OF HEAT. 17

is in all cases accoinpanied by a transmission of heat from a warm
body to a cold one, it does not follow that by every such trans-
mission work is produced, for the heat may be carried over by
simple conduction ; and in all such cases, if the transmission
alone were the true equivalent of the work performed, an abso-
lute loss of mechanical force must take place in nature, which is
hardly conceivable. Notwithstanding this, however, he arrives
at the conclusion, that in the present state of science the prin-
ciple assumed by Camot is the most probable foundation for an
investigation on the moving force of heat. He says, " If we
forsake this principle, we stumble immediately on innumerable
pther difficulties, which, without further experimental investiga-
tions, and an entirely new erection of the theory of heat, are
altogether insurmountable.^^

I believe, nevertheless, that we ought not to suffer ourselves
to be daunted by these difficulties -, but that, on the contrary, we
must look steadfastly into this theory which calls heat a motion,
as in this way alone can we arrive at the means of establishing
it or refuting it. Besides this, I da not imagine that the diffi-
culties are so great as Thomson considers them to be; foip
although a certain alteration in our way of regarding the subject
is necessary, stiU I find that this is in no case contradicted by
proved facts. It is not even requisite to cast the theory of
Camot overboard ; a thing difficult to be resolved upon, inas-
much as experience to a certain extent has shown a surprising
coincidence therewith.* On a nearer view of the case, we find
that the new theory is opposed, not to the real fundamental
principle of Camot, but to the addition " no heat is lost ;^' for it
is quite possible that in the production of work both may take
place at the same time ; a certain portion of heat may be con-
sumed, and a further portion transmitted from a warm body to
a cold one ; and both portions may stand in a certain definite
relation to the quantity of work produced. This will be made
plainer as we proceed ; and it will be moreover shown, that the
inferences to be drawn from both assumptions may not only exist
together, but that they mutually support each other.

18 FIRST MEMOIR.

I. Deductions from the principle of the equivalence of heat and

work.

We shall fco'bear entering at present on the nature of the mo-
tion whieh may be supposed to exist within a body^ and shall
assume generally that a motion of the particles does exists and
that heat is the measure of their vis viva. Or yet more generally,
we shall merely lay down one maxim which is founded on the
above assumption : —

In all cases where work is produced by heat, a quantity of heat
proportional to the work done is consumed ; and inversely, by the
expenditure of a like quantity of work, the same amount of heat
may be produced.

Before passing on to the mathematical treatment of this maxim^
a few of its more immediate consequences may be noticed^ which
haye an influence on our entire notions as to heat^ and which are
capable of being understood^ without entering upon the more
definite proofs by calculation which are introduced further on.

We often hear of the total heat of bodies^ and of gases and
vapours in particular^ this term being meant to express the sum
of the sensible and latent heat. It is assumed that this depends
solely upon the present condition of the body under considera-
tion; so that when all other physical properties thereof^ its
temperature^ density, &c. are known, the total quantity of heat
which the body contains may also be accurately determined**
According to the above maxim, however, this assumption cannot
be admitted. If a body in a certain state, for instance a quan-
tity of gas at the tempeirature Iq and volume Vq, be subjected to
various alterations as regards temperature and volume, and
brought at the conclusion into its original state, the sum of its
sensible and latent heats must, aeQording to the above assump^
tion, be the same fis before j hence, if during any portion of the
procesfi heat be communicated firom mthout, the quantity thus
received nmst be given gff again during some other portion of

* [The above may perhaps be more clearly expressed thus : — By total heat
was formerly meant the total quantity of heat which must be imparted to a
body in order, from any given initial condition, to bring it to any other, and
it was thereby implied that, the initial condition being known, the quan-
tity of heat in question is completely determined by the present condition of
the body, no matter in what manner the body may have been brought to this
condition. — 1864.]

^

MOVING FORCE OF HEAT. 19

the process. With every alteration of volume, however, a cer-
tain quantity of work is either produced or expended by the gas ;
for by its expansion an outward pressure is forced back, and on
the other hand, compression can only be effected by the advance
of an outward pressure. If, therefore, alteration of volume be
among the changes which the gas has undergone, work must be
produced and expended. It is not, however, necessary that at
the conclusion, when the original condition of the gas is again
established, the entire amount of work produced should be exactly
equal to the amount expended, the one thus balancing the other ;
an excess of one or the other will be present if the compression
has taken place at a lower or a higher temperature than the ex-
pansion, as shall be proved more strictly further on. This excess
of produced or expended work must, according to the maxim,
correspond to a proportionate excess of expended or produced
heat, and hence the amount of heat refunded by the gas cannot
be the same as that which it has received.

There is still another way of exhibiting this divergence of our
maxim from the common assumption as to the toteU heat of bo-
dies. When a gas at t^ and Vq is to be brought to the higher
temperature /| and the greater volume V|, the quantity of heat
necessary to effect this would, according to the usual hypothesis,
be quite independent of the manner in which it is communicated.
By the above maxim, however, this quantity would be different
according as the gas is first heated at the constant volume Vq and
then permitted to expand at the constant temperature /], or first
expanded at the templerature t^ and afterwards heated to /|, or
expansion and heating alternated in any other manner, or even
effected simultaneously ; for in all these cases the work done by
the gas is different.

In like manner, when a quantity of water at the temperature
/q is to be converted into vapour of the temperature /j and the
volume rj, it will make a difference in the amount of heat ne-
cessary if the water be heated first to ^, and then suffered to eva-
porate, or if it be suffered to evaporate by t^ and the vapour
heated afterwards to ti and brought to the volume Vi ; or finally,
if the evaporation take place at any intermediate temperature.

From this and from the immediate consideration of the
maxim, we can form a notion as to the light in which latent heat

c2

20 FIRST MEMOIR.

must be regarded. Referring again to the last example^ we dis-
tinguish in the quantity of heat imparted to the water during
the change the sensible heat and the latent heat. Only the former
of these, however, must we regard as present in the produced
steam ; the latter is, not only as its name imports, hidden from
our perceptions, but has actually no existence ; during the alte-
ration it has been converted into work.

We must introduce another distinction still as regards the
heat expended. The work produced is of a twofold nature. In
the first place, a certain quantity of work is necessary to over-
come the mutual attraction of the particles, and to separate them
to the distance which they occupy in a state of vapour. Secondly,
the vapour during its development must, in order to procure
room for itself, force back an outer pressure. We shall name
the former of these interior work, and the latter exterior work,
and shall distribute the latent heat also under the same two
heads.

With regard to the interior work, it can make no difference
whether the evaporation, takes place at Iq or at ^i, or at any other
intermediate temperature, inasmuch as the attraction of the par-
ticles must be regarded as invariable*. The exterior work, on
the contrary, is regulated by the pressure, and therefore by the
temperature also. These remarks are not restricted to the ex-
ample we have given, but are of general application ; and when
it was stated above, that the. quantity of heat necessary to bring
a body from one condition into another depended, not upon the
state of the body at the beginning and the end alone, but upon
the manner in which the alterations had been carried on through-
out, this statement had reference to that portion only of the la-
tent heat which corresponds to the exterior work. The remainder

* It must not be objected here that the cohesion of the water at ti is less
than at t^, and hence requires a less amount of work to overcome it. The
lessening of the cohesion implies a certain work performed by the warming
of the water as water, and this must be added to that produced by evapora-
tion. Hence it follows, at once, that only part of the quantity of heat which
water receives fiom without when heated, is to be regarded as heat in the free
state, the rest being consumed in diminishing cohesion. This view is in ac-
cordance with the circumstance that water has so much higher a specific heat
than ice, and probably also than steam. [The views briefly referred to in this
note will receive full consideration in a subsequent memoir. — 1864.]

MOVING FORCE OF HEAT. 21

of the latent heat and the entire amount of sensible heat are in-
dependent of the manner in which the alteration is eflfected.

When the vapour of water at ^j and Vi is reconverted into
water at t^, the reverse occurs. Work is here expended^ inas-
much as the particles again yield to their attraction^ and the
outer pressure once more advances. In this case^ therefore^ heat
must be produced ; and the sensible heat which here exhibits it-
self does not come from any retreat in which it was previously
concealed^ but is newly produced. It is not necessary that the
heat developed by this reverse process should be equal to that
consumed by the other ; that portion which corresponds to the
exterior work may be greater or less according to circumstances.

We shall now turn to the mathematical treatment of the sub-
ject, confining ourselves, however, to the consideration of per-
manent gases, and of vapours at their maximum density ; as be-
sides possessing the greatest interest, our superior knowledge of
these recommends them as best suited to the calculus. It will,
however, be easy to see how the maxim may be applied to other
cases also.

Let a certain quantity oi permanent gas, say a unit of weight,
be given. To determine its present condition, three quantities
are necessary \ the pressure under which it exists, its volume,
and its temperature. These quantities stand to each other in a
relation of mutual dependence, which, by a union of the laws of
Mariotte and Gay-Lussac"'*', is expressed in the following equa-
tion :

/w=R(a+/), (I)

where j9, v and t express the pressure, volume and temperature
of the gas in its present state, a a constant equal for all gases, and

R also a constant, which is fully expressed thus, ^\ where /?o>

O + fQ

Vq, and /q express contemporaneous values of the above three
quantities for any other condition of the gas. This last constant
is therefore different for different gases, being inversely propor-
tional to the specific weight of each.

It must be remarked, that Regnault has recently proved, by a
series of very careful experiments, that this law is not in all

* This shall be expressed in future briefly thus — ^the law of M» and G. j^
and the law of Mariotte alone thus — the law of M.

23 FIRST MEMOIR.

strictness correct. The deyiations^ however^ for the permanent
gases are yery small^ and exhibit themselyes principally in those
cases where the gas admits of condensation. From this it would
seem to follow^ that the more distant^ as rq;ardB pressure and
temperature^ a gas is firom its point of condensation^ the more
correct will be the law. Whilst its accuracy, therefore, for per*
manent gases in their common state is so great, that in most in*
vestigations it may be regarded as perfect, for every gas a limit
may be imagined, up to which the law is also perfectly true ; and
in the following pages, where the permanent gases are treated as
such, we shall assume the existence of this ideal conditkm*.

The value ~ for atmospheric air is found by the experiments

both of Magnus and Regnault to be =s0*003665, the tempera-
ture being expressed by the centesimal scale reckoned firom the
fireezing-point upwards. The gases, however, as abready men-
tioned, not following strictly the law of M. and G., we do not

always obtain the same value for - when the experiment is re-
peated under different circumstances. The number given above
is true for the case when the air is taken at a temperature of (f
under the pressure of one atmosphere, heated to a temperature
of 100°, and the increase of expansive force observed. K, how-
ever, the pressure be allowed to remain constant, and the in-
ecrease of volume observed, we obtain the somewhat higher value
0*003670. Further, the values increase when the experiments
are made under a pressure exceeding that of the atmosphere, and
decrease when the pressure is less. It is clear from this that
the exact value for the ideal condition, where the differences
pointed out would of course disappear, cannot be ascertained.
It is certain, however, that the number 0*003e65 is not far fi'om
the truth, especially as. it very nearly agrees with the value found
for hydrogen, which, perhaps of all gases, approaches nearest the

ideal condition. Retaining, therefore, the above value for ~, we

a
have a =273.

One of the quantities in equation (I), for instance j9, may be re-

* [In my later memoira the gases relative to which the existence of this
ideal condition is assumed ore termed perfeci gases. — 1864.}

J

MOVING FORCE OF HEAT.

23

garded as a fanction of the two others ; the latter will then be the
independent variables which determine the condition of the gas"^.

We will now endeavour to ascertain in what manner the quan-
tities which relate to the amount of heat depend upon v and /.

When any body whatever changes its volume^ the change is
always accompanied by a mechanical work produced or expended.
In most cases^ howev6r^ it is impossible to determine this with
accuracy^ because an unknown interior work usually goes on at
the same time with the exterior. To avoid this difficulty^ Camot
adopted the ingenious contrivance before alluded to : he allowed
the body to undergo various changes^ and finally brought it into
its primitive state ; hence if by any of the changes interior work
was produced^ this]|was sure to be exactly nullified by some other
change ; and it was certam that the quantity of exterior work
which remained over and above was the total quantity of work
produced. Clapeyron has made this very evident by means of a
diagram : we propose following his method with permanent gases
in the first instance^ introducing^ however^ some slight modifi-
cations rendered necessary by our maxim.

In the annexed figure let oe Fig. 1.

represent the volume^ and ea the
pressure of the unit-weight of
gas when the temperature is t ;
let us suppose the gas to be con-
tained in an expansible bag^ with
which, however, no exchange of
heat is possible. K the gas be
permitted to expand, no new heat

* /

* [Clapeyron ia his Tesearches generally selected v and p for his two inde-
pendent variables — a choice which accords best with the graphic represen-
tation, about to be described, wherein v and J9 constitute the coordinates* I
have preferred, however, to consider v and t as the independent variables, and
to regard ^ as a function thereof ; since in the theory of heat the temperature
t is especially important, and at the same time very suitable for determination
by direct measurements, accordingly it is ordinarily regarded as a previously
known magnitude upon which depend the several other magnitudes which
there enter into cmisideration. For the sake of uniformiiy I have everywhere
abided by this choice of independent variables ; it need scarcely be remarked,
however, that occasionally the equations thus established would assume a
somewhat simpler form, if instead of v and t, v and ^ or ^ and p were intro-
duced theroia as independent variables. — 1864.]

24 FIRST MEMOIR.

being added^ the temperature will fall. To avoid this^ let the
gas during the expansion be brought into contact with a body A
of the temperature t, from which it shall receive heat sufficient
to preserve it constant at the same temperature. While this ex-
pansion by constant temperature proceeds^ the pressure decreases
according to the law of M.^ and may be represented by the or-
dinate of a curve a b, which is a portion of an equilateral hyper-
bola. When the gas has increased in volume from oeix^of, let
the body A be taken away^ and the expansion allowed to proceed
still further without the addition of heat ; the temperature will
now sink^ and the pressure consequently grow less as before.
Let the law according to which this proceeds be represented by
the curve b e. When the volume of the gas has increased from
of to Off, and its temperature is lowered from / to t, let a pressure
be commenced to bring it back to its original condition. Were
the gas left to itself^ its temperature would now rise ; this^ how-
ever^ must be avoided by bringing it into contact with the body B
at the temperature t, to which any excess of heat wiU be imme-
diately imparted^ the gas being thus preserved constantly at r.
Let the compression continue till the volume has receded to h,
it being so arranged that the decrease of volume indicated by the
remaining portion h e shall be just sufficient to raise the gas from.
rtotyiS during this decrease it gives out no heat. By the first
compression the pressure increases according to the law of M.^
and may be represented by a portion cd o{ another equilateral
hyperbola. At the end the increase is quicker^ and may be re-
presented by the curve da. This curve must terminate exactly
•in a ; for as the volume and temperature at the end of the ope-
ration have again attained their original values^ this must also
be the case with the pressure, which is a function of both. The
gas wiU therefore be found in precisely the same condition as at
the commencement.

In seeking to determine the amount of work performed by
these alterations, it will be necessary, for the reasons before as-
signed, to direct our attention to the exterior work alone. During
the expansion, the gas prodttces a work expressed by the integral
of the product of the differential of the volume into the corre-
sponding pressure, which integral is represented geometrically by
the quadrilaterals e a i/, and/ 6 eg. During the compression.

MOVING FOKCE OF HEAT.

25

^

however, work will be expended , which is represented by the qua-
drilaterals g cdh and hdae. The excess of the former work
above the latter is to be regarded as the entire work produced by
the alterations, and this is represented by the quadrilateral ab c d.

If the foregoing process be reversed, we obtain at the conclu-
sion the same quantity aicrf as the excess of the work expended
over ^Bi prodtAced.

In applying the foregoing Fig. 2.

considerations analytically, we
will assume that the various
alterations which the gas has
undergone have been infinitely
small. We can then consider
the curves before mentioned to
be straight lines, as shown in

the accompanying figure. In o e h f g

determining its superficial content, the quadrilateral aba dms^y
be regarded as a parallelogram, for the error in this case can
only amount to a differential of the third order, while the area
itself is a differential of the second order. The latter may there-
fore be expressed by the product ef. b *, where k marks the point
at which the ordinate bf cuts the lower side of the parallelogram.
The quantity i A: is the increase of pressure due to raising the tem-
perature of the gas, at the constant volume o/, firom t to ^, that is
to say, due to the differential t—r^dt. This quantity can be ex-
pressed in terms of v and / by means of equation (I.) , as follows :

, ^dt
dp^ •

If the increase of volume cf be denoted by rfv, we obtain the
area of the quadrilateral, and with it

The wtyrk produced^ (1)

We must now determine the quantity of heat consumed during
those alterations. Let the amount of heat which must be im-
parted during the transition of the gas in a definite manner from
any given state to another, in which its volume is v and its tem-
perature ty be called Q ; . and let the changes of volume occurring
in the process above described, which are now to be regarded se-

/

26 FIB8T MEMOIR.

parately^ be denoted as follows: tfhy dv, hg by dfv, eh by hv, and
fg by £'v. During an expansion from the volnme oe^v to
of:=iV'\-dVf at the constant temperature t, the gas must receire
the quantity of heat expressed by

(s>»"

and in accordance with this^ during an expansion from oh^v-\-hj
to og^V'\-ho'\-dfv at the temperature t—dt, the quantity

In our case^ however^ instead of an expansion^ a compression
has taken phtce ; hence this last expression must be introduced
with the negative sign. During the expansion from o/to o^,
and the compression from o A to o e^ heat has been neither re-
ceived nor given away ; the amount of heat which the gas has

* [In ibis memoir I have for the sake of greater deamess employed Eufer's
notation for partial differential coefficients in which the fractions which re-
present the latter are placed between brackets. This precaution was perhaps
lumecessary, since in most cases^ as was observed in the Introduction, no mis-
conception can arise even when the brackets are ofbitted ; nevertheless in the
present reprint of the memoir the original notation has been retained. In ac-
cordance with the equation (3a) of the Introduction the complete di£&iential
equation of Q would here be

For a g^ven quantity of gaff, and indeed for every other body whose condition
is defined by its temperature and volume, the two partial differential coeffi-
cients (^j, (^\ must be regarded as perfectly determinate functions of t

and V, for the quantities of heat are perfectly defined which a body must re-
ceive when, hom a given condition, its temperature is raised under constant
volume, or its volume is increased without any alteration of temperature, a
counter-pressure corresponding to its elastic force being thereby overcome.
Whether Q itself, however, is also a magnitude which can be represented as
a function of t and v, in which these variables are independent of each other,
or whether a fiirther relation between these variables must be given in order to
determine Q, depends upon the circumstance mentioned in the Introduction ;
viz^ whether or not the condition (4), which in the present notation is thus
written,

dvXdi) dtXdor
is satisfied ; the object of the following development is to decide this ques-
tion.— 1864.]

MOVING FORCE OF HEAT. 27

received over and above that which it has communicated^ or, in
other words^ the quantity of heat consumed, will therefore be

The quantities St; and d!v must now be eliminated; a conside-
ration of the figure furnishes us with the following equation :

During its compression from ohiooe, consequently during its
expansion under the same circumstances from oe \o oh, and
during the expansion from of\oog, both of which cause a de-
crease of temperature dt, the gas neither receives nor communi-
cates heat : from this we derive the equations

From these three equations and equation (2) the quantities
dh, Bvj and Sfv may be eliminated ; neglecting during the pro-
cess all differentials of a higher order than the second^ we obtain

n<,*«<.,p«*^ = g(f)-Kf)]**. . (8)

Turning now to our maxim^ which asserts that the production
of a certain quantity of work necessitates the expenditure of a
proportionate amount of heat^ we may express this in the form
of an equation^ thus :

The heat expended _> . .

The work produced"" ' ^ ^

where A denotes a constant which caresses the equivalent of heat
for the unit of workf. The expressions (1) and (3) being in-
troduced into this equation^ we obtain

* [With lefeienoe to the deduction of the equation (8), see also the Ap^
pendix A. at the end of the present memoir.]

t [This magnitude, which will often present itself in the sequel, may, in ac-
cordance with a modem custom, be briefly termed the cahr^ equivalent of
worA.— 1864]

28 FIRST MEMOIR.

or

TiUni =^'

V

dt\dvJ dv\dt) V ^^^^

dAdv) dv\dt) \dtr

in which form it frequently presents itself in subsequent memoirs. — 1864.]

t [By an oversight the order of this equation was not stated in the ori-
ginal edition. — 1864.]

X [With respect to the manner in which, by integration, the equation (II a)
may be deduced from the equation (11), see Appendix B to this Memoir.

That the differential equation (IE a) of the first order actually corresponds
to the differential equation (II) of the second order may moreover be easily
shown, conversely, by differentiating (II a), and thence deducing (II). In
fact, if for dU we write the complete expression

(?)*-(Sh

This equation may be regarded as the analytical expression of
the above maxim applicable to the case of permanent gases ^. It
shows that Q cannot be a function of v and / as long as the two
latter are independent of each other. For otherwise, according
to the known principle of the differential calculus, that when a \

function of two variables is differentiated according to both, the i

order in which this takes place is a matter of indifference^ the
right side of the equation must be equal 0.

The equation can be transformed to a complete differential
equation of the first order f and of the following form :

dQ=dV + A.n^^dv, .... (Ila) !

where U denotes an arbitrary function of v and ^ J. This differ-

* [The equation (11) may obviously be generalized so as to apply not only
to a gas, but to any other body whatever whose condition is determined by
its temperature and volume, and upon which the sole external forces which
act consist of pressures normal to the surfiEU^, of equal intensity at all points
of the latter, and differing so slightly j&om the body's force of expansion as to
admit, in calculation, of being considered equal thereto. This generalization is

effected by merely substituting for — the differential coefficient l^\ ; whicli
latter, in the special case of gases, is equal to — . The equation then becomes

V

MOVING FORCE OP HEAT. 29

ential equation is of course unintegrable until we find a second
condition between the variables^ by means of which t may be
expressed as a fdnction of v. This is due, however, to the last
member alone, and this it is which corresponds to the exterior
work efiected by the alteration ; for the differential of this work
ispdv, which, when/? is eliminated by means of (I), becomes

V

It follows, therefore, in the first place, from (II a), that the
entire quantity of heat, Q, absorbed by the gas during a change
of volume and temperature may be decomposed into two portions.
One of these, U, which comprises the sensible heat and the heat
necessary for interior work, if such be present, fulfils the usual
assumption, it is a function of v and t, and is therefore deter-
mined by the state of the gas at the beginning and at the end of
the alteration; while the other portion, which comprises the
heat expended on exterior work, depends, not only upon the state

and similarly for dfQ the complete expression
the equation (ILa.) becomes transformed to

(f)*+{S)*-(f)*+[©*-'«"4-'>.

whence, by comparison, the following equalities may be deduced :

(§)=©.

\dv) \dv)

a+<

+A.R

V

On differentiating the first of these expressions according to v, and the second
according to t, it is to be noticed that the magnitude U being, by a previous
statement, a Unction of t and v, the condition

dv\dt) dt\dvl

is satisfied. Each of the quantities involved in the last equation, therefore,
may be denoted by -^^ ; so that

jrf/dQ\ (PIT
dvXdt) dtdv

dAdvl dtdv V '
But the first of these two equations being subtracted from the second, leads at
once to the equation (11). — 1864.]

80

FiaST MEMOIR.

of the gas at these two limits^ but also upon the manner in which
the alterations have been effected thronghoat. It is shown above
that the same conclusion flows directly from the maxim itself.

Before attempting to render this equation suitable for the deduc-
tion of further inferences^ we will develope the analytical expres-
sion of the maxim applicable to vapoun at their maximum density.

In this case we are not at liberty to assume the correctness of
the law of M. and O.^ and must therefore confine ourselves to the
maxim alone. To obtain an equation from this^ we will again
pursue the course indicated by Camotj and reduced to a diagram
by Clapeyron. Let a vessel impervious to heat be partially filled
with water, leaving a space above for steam of the maximum
density corresponding to the temperature t. Let the volume of
both together be represented in the annexed figure by the
abscissa o e, and the pressure of the Fig. 3.

steam by the ordinate e a. Let the
vessel be now supposed to expand,
while both the liquid and steam
are kept in contact with a body A
of the constant temperature t. As
the space increases, more liquid is
evaporated, the necessary amount
of latent heat being supplied by
the body A; so that the temperature, and consequently the
pressure of the steam, may remain unchanged. When the en-
tire volume is increased in this manner from o c to o/, an exterior
work is produced which is represented by the rectangle ea hf.
Let the body A be now taken away, and let the vessel continue to
expand without heat being either given or received. Partly by
the expansion of the steam already present, and partly by the
formation of new steam, the temperature will be lowered and the
pressure become less. Let the expansion be suffered to continue
until the temperature passes from t to t, and let o g represent the
volimie at this temperature. If the decrease of pressure during
this expansion be represented by the curve b c, the exterior work
produced by it will be represented by/i eg.

Let the vessel be now pressed together so as to bring the liquid
and vapour to their origiual volume o e, and during a portion of
the process let the vessel be in contact with a body, B, of the

MOVING FORCE OF HEAT. 31

temperature t, to which any excess of heat shall be immediately
imparted^ and the temperature of the liquid and vapour kept con*
stant at r. During the other portion of the process^ let the body
B be withdrawn so that the temperature may rise ; let the first
comp-ession continue till the volume has been reduced to o A^ it
being so arranged that the remaining space h e shall be just suf-
ficient to raise the temperature from t to /. During the first
decrease of volume the pressure remains constant at ff c, and the
quantity of exterior work expended is equal to the rectangle g c
d h. During the last decrease of volume the pressure increases,
and may be represented by the curve d a, which must terminate
exactly in the point a, as the original temperature t must again
correspond to the original pressure ea. The exterior work ex-
pended in this case is ^hdae.

At the end of the operation both fluid and vapour are in the
same state as at the commencement, so that the excess of the
exterior work produced over the amount expended expresses the
total amount of work accomplished. This excess is represented
by the quadrilateral abed, the content of which must therefore
be compared with the heat eapended at the same time.

For this purpose let it be as- Fig, 4.

«umed, as before, that the de**
scribed alterations are infinitely
small, and under this view let the
process be represented by the an-
nexed figure, in which the curves
a d and b c shown in fig. 3 have
passed into straight lines. With

^

regard to the area of the qua- o e h f g

drilateral abed, it may be again regarded as a parallelogram, the
area of which is expressed by the product ef.b k. Now if, when
the temperature is /, the pressure of the vapour at its maximum
tension be equal to p, and the diflference of temperature /— t be
expressed by di, we have

* [In the equations coneaponding to sutuxated vapoiur th^ differential co*

efficient ® is written without brackets, since the pressure is now no longer a

at
flinction of the temperature and volume, but of the temperature solely. — 1864.]

32 FIRST MEMOIR.

ef is the increase of yolume caused by the passing of a certain
quantity of liquid represented by dm into a state of vapour. Let
the yolume of the unit of steam at its maximum density for the
temperature t be called s^ and the volume of the same quantity
of liquid at the temperature / be called <r ; then is

and hence the area of the rectangle^ or

The work produced ^{8-'a)-^dmdt.. . . (5.)

To express the amount of heat^ we will introduce the following
notation : — Let the quantity of heat rendered latent by the pas-
sage of a unit weight of liquid at the temperature t, and under a
corresponding pressure into a state of vapour^ be called r, and
the specific heat of the liquid c ; both of these quantities^ as also

the foregoing s, <r, and -^ being fimctions of t. Finally, let

the quantity of heat which must be communicated to a unit
weight of vapour of water to raise it from the temperature t to
/4-rf/ (the vapour being preserved by pressure at the maximum
density due to the latter temperature without precipitation) be
called hdt, where h likewise represents a function of t. We shall
reserve the question as to whether its value is positive or nega-
tive for future consideration*.

If /x be the mass of liquid originally present in the vessel^
and m the mass of the vapour ; further, dm he the mass eva-
porated during the expansion from oe to of, and d!m the mass
precipitated by the compression from op to oh, we obtain in the
first case the quantity

rdm

of latent heat which has been extracted from the body A ; and
in the second case, the quantity

(s-m'^y-

* [Tlie magnitude h here introduced is predsely the tpecyic heat of the va-
pour at Us maximun density, or, in other words, the epeci/ic heat of the saturated
vapour, which may be regarded as a peculiar kind of specific heat just as well
as is the specific heat at a constant volume or the specific heat under constant
pressure. — 1864. ]

MOVINO FORCE OF HEAT. 83

of sensible heat which has been imparted to the body B, By the
other expansion and compression heat is neither gained nor lost ;
hence at the end of the process we have

The heat escpended = rdm — ( ^-";57 dtjd'm. . . (6)

In this equation the differential cPm must be expressed through
dm and dt ; the conditions under which the second expansion
and the second compression have been carried out enable us to
do this. Let the mass of vapour precipitated by the compression
from oh to oe, and which therefore would develope itself by ex-
pansion from oe to ohy be represented by Sm, and the mass de-
veloped by the expansion from 0/ to off by Sfm ; then, as at the
conclusion of the experiment the original mass of fluid and of
vapour must be present, we obtain in the first place the equation

dm + S'm^^d'm+Sm.

Further, for the expansion from oe to oA, as the temperature
of the liquid mass fi and the mass of vapour m must thereby be
lessened, the quantity dt without heat escaping, we obtain the
equation

rBm^^fi.cdt—m.hdt^O ;

and in like manner for the expansion from of to off, as here we
have only to set (i—dm and m + dm in the place of /i and m, and
Sfm in the place of Sm, we o))tain

rSfm'-{fi—dm)cdt'—{m + dm)hdt=0.*

If from these three equations and equation (6.) the quantities
d'm, Bm, and B^m be eliminated, and all differentials of a higher

* [With respect to these two equations, whose use is to determine the relation
which exists between dm or ^m and dt, a remark may be made of a similar
kind to those contained in the Apijendix A, which relate to the deduction of
the equation (3). To be strictly accurate up to differentials of the second
order, the expressions for 5m or d'm ought to contain another term with the
factor d^, just as do the expressions for bv and b*v in the equations (m) and
(n) oi Appendix A. Since this term would be the same in both equations,
however, it would again disappear from the equation

rf'm —dm-\- b'm — 5m,
which determines <f m, and thus be wholly without influence on our result.
Consequently it is unnecessary here to take this term into further considera-
tion.— 1864.]

D

84 Wn»T MEMOIB.

order tha« the second be n^lected, we haye

The heat expended=i{^'^c—h\dmdt. . . . (7>

The formulse (7) and (5) must now be united, as in tbe case
of permanent gases, thus :

and bence we obtain, as the analytical expression of the maxim,
applicable to vapours at their maximum density, the equation

J+c-A=A(*-<r)^ (HI)

If, instead of the above maxim, the assumption that the quan-
tity of heat is constant be retained, th^, according to (7), in-
stead of equation (III) we must set

|+«-*=0* (8)

And this equation, although not exactly in the same form, has
been virtually used heretofore to determine the value of the quan-
tity h. As long as the law of Watt is regarded as true, that the
sum of the latent and sensible heat of a quantity of steam at its
maximum density is the same for all temperatures, and conse-
quently that

* [As before remarked, it would follow fiK>m this assumption that when a
body suffers a series of changes such that it thereby returns finally to its initial
state, the quantity of heat which it receives horn without during one portion
of these changes must be equal to the quantity which it gives off during the
remaining changes. Now the difference between the received and imparted
quantities of heat in the previously described cycle of infinitely small changes
is, according to equation (7), represented by

and this expresaion, equated to lero, leads at once to the equation

which is another form of the equation (8). — 1864.]

t [The law of Watt mentioned in the text, and formerly acc^ted as true.

MOVING FORCE OF HEAT. 86

it must be inferred that for this liquid h also is equal ; this, in-
deed, has often been asserted, by saying that when a quantity
of vapour at its maximum density is compressed in a vessel im-
pervious to heat, or suflTered to expand in the same, it will remain
at its maximum density. As, however, Regnault* has corrected
the law of Watt so that we can set with tolerable accuracy

^+c=0-305,t

the equation (8) gives for h also the value 0'305. It follows
from this, that a portion of the steam in the impermeable vessel
must be precipitated by compression, and that it cannot retain ,
its maximum density after it has been suffered to expand, as its
temperature does not decrease in a ratio corresponding to the
decrease of density.

Quite otherwise is it if, instead of equation (8), we make use
of equation (III). The expression on the right-hand side is
from its nature always positive, and from this follows in the first
place that h is less than 0-305. It wiU be afterwards shown
that the value of the said expression is so great that h becomes
even negativeX. Hence we must conclude that the above quan-

asserts that the sum of the two quantities of heat required to raise the unit of
weight of water from 0° to the temperature t, and then to convert it into va-
pour at this temperature, is independent of this temperature t. Accordingly
we should have

r+y c (ft sa const.,
an equation which, by differentiation, leads to the equation

^ven in the text — 1864.]

• M&m, de VAcad. voL xxi., 9th and 10th Memoirs.

t [Eegnault has found that the sum of free and latent heat is not constant^
as by the law of Watt it should be, but that with increasing temperature it
increases in a manner approximately expressed by the equation

r4- /*% <ft=606-6+0-306 tf
from which the equation

$4-c=0-306
at

follows by differentiation. — 1864.]

X [In order to decide whether the equation

*=J-,c-A(.-.)|,

d2

86 nitST MSMOIR.

tity of vapour will be partially precipitated^ not by the compres-
sion, but by the expansion ; when compreflsed, its temperature
rises in a quicker ratio than that corresponding to the increase
of density^ so that it does not continue at its maximum density.
The result is indeed directly opposed to the notions generally
entertained on this subject ; I believe, however, that no experi-
ment can be found which contradicts it. On the contrary, it
harmonizes with the observations of Pambour better than the
common notion. Pambour found^ that the steam issuing from
a locomotive after a journey always possesses the temperature
for which the tension observed at the same time is a maximum.
From this it follows that h is either 0, as was then supposed,
because this agreed with the law of Watt, which was considered
correct at the time, or that h is negative. If h were positive, then
the temperature of the issuing steam must have been too high
in comparison with its tension, and this could not have escaped
Pambour. If, on the contrary, in agreement with the above, h
be negative, too low a temperature cannot occur, but a portion
of the vapour will be converted into water so as to preserve the
remainder at its proper temperature. This portion is not neces-
sarily large, as a small quantity of vapour imparts a compara-
tively large quantity of heat by its precipitation ; the water thus
formed is probably carried forward mechanically by the steam,
and might remain unregarded ; the more so, as, even if observed,
it might have been imagined to proceed from the boilerf.

deduced from (HI), gives a positive or a negative value of h, the numerical
value of A must be known ; and since nothhig has been said in the previous
part of the memoir with respect to the numerical determination of this con-
stant, I have not here entered into the determination of the magnitude h, but
have referred the question to the sequel. In the second part of the memoir^

an expression for the product A(«— <r) -^ will be given which involves kno^vim

magnitudes solely, and whose substitution in the foregoing equation leads to
another, from wMcfa, not only the sign, but also the magnitude of A, as a f\md-
tion of the temperature, can be at once determined. — 1864]

* Traits dm kaomo^oes, Sod edit, and Thiorie des machines A tapeur^
2nd edit

t [The proeeas to which the observation of Pambour refers is too compli-
cated to furnish a convenient and accurate comparison with the theoretic^al
results obtained above. Accor^^gly the observation in question is cited, not
with a view of supplying a reliable verification of those theoretical results, but

MOVING FOBCE OF HEAT. 37

So far the consequences have been deduced from the above
maxim alone^ without any new assumption whatever being made.
Nevertheless^ by availing ourselves of a very natural incidental
assumption^ the equation for permanent gases (II a) may be ren-
dered considerably more productive. Gases exhibit in their de-
portment^ particularly as regards the relations of volume^ tem-
perature^ and pressure expressed by the laws of M. and G.^ so
much regularity as to lead us to the notion that the mutual
attraction of the particles which takes place in solid and liquid
bodies is in their case annulled ; s o that while with s olids and
liquids the heat necessary to eflfect an expmsion has to contend
with both an interior and an exterio r resiistance^ the latter only
is eifective in the case of gases. If this be tke case, then, by the
expansion of a gas, only so much heat can be rendered latent as
is necessary to exterior work. Further, there is no reason to
suppose that a gas, after it has expanded at a constant tempera-
ture, contains more sensible heat than before. If this also be
admitted, we obtain the proposition, when a permanent gas ex^
panda.at a constant temperature, it absorbs only as much heat as
is necessary to the exterior work produced by the expansion — a
proposition which is probably true for each gas in the same de-
gree as the law of M. and G is true for that gas''^.

From this immediately follows

(f).A.K«-f', ...... („

for, as already mentioned, R dv represents the quantity of

merely to show that it accords better with the latter than with the views pre-
viously entertained. — 1864.]

* [Several authors before me regarded the heat which disappears during
the expansion of a gas as simply equivalent to the work done in overcoming
pressure. As far as I know, however, I was the first to enunciate the theorem
in its complete form ; according to which it is asserted that in general exterior
and interior work are both simultaneously done when a body expands, but
that in the special case of a permanent gas the law of M. and G. sanctions the
assumption of an infinitesimal amount of interior work ; further, that the de-
gree of accuracy to which this assumption can lay claim, when applied to a cer-
tain gas, is the same as that which would attend the application thereto of
the law of M. and G. ; and, finally, that the theorem involves the additional
assumption that the heat actually present in the gas is independent of its den-
8ity.~1864.]

88 nB8T HXMOIR.

exterior work produced by the expaiiBioii dv. According to tliis^
the function U^ which appears in equation (II a), cannot contain
V, and hence the equation changes to

dQ==cdt+AR^dv, (II i)

wherein e can only be a function of t* ; and it is even probable
that the quantity c, which denotes the specific heat of the gas at
a constant yolume^ is itself a constant.

To apply this equation to particular cases^ the peculiar con-
ditions of each case must be brought into connexion therewith^
so as to render it infiegrablet* We shall here introduce only a
few simple examples^ which possess either an intrinsic interest^
or obtain an interest by comparison with other results connected
with this subject.

In the first place^ if in equation (II b) we put^ successively^
v= const, and j9=s const.^ we shall obtain the specific heat of the
gas at a constant volume^ and its specific heat under a constant
pressure. In the former case dv=0^ and (II b) becomes

f «■ OO)

• [In fiwst, from the equation (II a), written in the form
it follows immediately that
Now if^ on the other hand| the equation (9), viz.

O"

be tiue^ a necessaiy consequence of the coexistence of it and the previous
equation is that

accordingly the function IJ, for perfect gases, must be.independent of v^ Sub-
stituting the symbol e for the differential coefficient (^ ), which of course

shares with U the property of being independent of v, the equation QIh)ia
at once obtained. — 18^.]

t [The equation (Jib) belongs, in fact, to the class of equations, described
in the Introduction, which only admit of being integrated on ftiMiiTw^Tig a se-
cond equation to exist between the variables^ whereby the sequence of the
changes becomes determined.^-1864.]

MOVING FORCE OP HEAT. 39

In the latter case, from the condition /)= Const.; we ol)tairi with
help of equation (I),

, Rdt

P

or

dv dt

»

V a+t

which by substitution in (11 i), the specific heat under a constant
pressure being denoted by c'j gives us

^=c'=c+AR* (10 a)

Prom this it may be inferred that the difference of both specific
heats for each gas is a constant quantity AR f. But this quantity
also expresses a simple relation for different gases. The com-
plete expression for R is ^\^ , ^here p^^ % and t^ denote anV

* [It will be easily understood why the fraction -^, in the equations (10)
and (10 a), is written without the brackets which ordinarily enclose the

fractions -^ and -^. For when, from the commencement, a condition is in-
dt dv '

troduced which implies the constancy of v or of ^^ the sequence of changes

through which the gas can pass is thereby so far fixed, that the increment of

Q is completely determined by the increment of one of the variables L In

such cases, therefore, the fraction -^ does not represent the partial difier-
ential coefficient of a magnitude whose value depiends upon those of two inde-
pendent variables, but corresponds to the fraction -=-, treated at page 4 of the

Introduction, which stands on the left of the diiFerential equation wherein y
was considered as a function of x. In fact, it is obvious that, in the equations

(10) and (10 a), the symbol -^ has two different meanings, arising from the

distinct conditions to which the equations have reference. — 1864.]

t [The difference between the two specific heats c and d being constant,
the conclusion above arrived at, with reference to the specific heat at a con-
stant volume, also holds for the specific heat under constant pressure, so that
the latter is likewise independent of the density, and probably also of the
temperature of the gas. At the time my memoir appeared, this" conclusion
was objected to on the ground of its being at variance with some of the ob-
servations of Suermann, and of De la Roche and B^rard, which at that time
were pretty generally accepted as trustworthy ; since then, however, it has
been verified by the experiments which, in 1853, were published by Kegnault.
—1864.]

40 FIRST MEMOIR.

three contemporaneous values otp, v, and t for a unit of weight
of the gas in question ; and from this it follows^ as already men-
tioned in establishing equation (I), that B is inversely propor-
tional to the specific gravity of the gas ; the same must be true
of the difference c'^cssAR^ as A is the same for all gases.

If it be desired to calculate the specific heat of the gas^ not re-
lative to the imit of weight, but (in accordance with the method
more in use) to the unit of volume , say at the temperature t^ and
the pressure p^y it is only necessary to divide c and (f by v^. Let
these quotients be expressed by 7 and y^ and we obtain

7'-7=— =A-^, (11)

In this last expression nothing appears which is dependent on
the pecuKar nature of the gas ; the difference of the specific heats
relative to the unit of volume is therefore the same for all gases.
This proposition has been deduced by Clapeyron from the theory
of Camot; but the result, that the difference cf—ch constant,
is there not arrived at; the expression found for it having still
the form of a function of the temperature.

Dividing both sides of equation (11) by 7, we obtain

'-'=?•=$;;• ■ • <'^>

wherein, for brevity, * is put in the place of -. This is equal

to the quotient — ; and through the theoretic labours of La-
c

place on the transmission of sound through air, has attained a
peculiar interest in science. For different gases, tlierefore, the
excess qf this quotient above unity is inversely proportional to
the specific Tieat, at constant volume, the latter being calculated
relative to the unit of volume. ThiB proposition has been proved
experimentally by Dulong* to be so nearly correct, that its
theoretic probability induced him to assume its entire truth, and
to use it in an inverse maimer in calculating the specific heats
of various gases, the value of k being first deduced from obser-
vation. It must, however, be remarked, that the proposition
is theoretically safe only so far as the law of M. and G. holds

* Ann. de Chim, et de Phys,^ xli. ; and Pogg, Ann., xvL

MOVING FORCE OP HBAT. 41

good ; which, as regards the various gases examined by Didong,
was not always the case to a sufficient degree of accuracy.

Let us suppose that the specific heat c at constant volume is
constant for every gas; a supposition which we have already
stated to be very probable ; this will also be the case when the
pressure is constant, and hence the quotient of both specific heats

— = A: must be also constant. This proposition, which Poisson, in

agreement with the experiments of Gay-Lussac and Welter, has
assumed to be correct, and made the basis of his investigations
on the tension Jtnd heat of gases*, harmonizes very well with our
present theory, while it is not possible to reconcile it with the
theory of Camot as heretofore treated.

In equation {lib) let Cl= const., we then obtain the following
equation between v and / :

crf^+A.R— ^rft7=0; (13)

from which, when c is regarded as constant, we derive

AB

V « . (a+/)=const. ;

AR c'
or, since according to equation (10 o), — = — 1=A:--1,

c c

t;*-i(a+^)= const.

Let three corresponding values of t?, t, and j9 be denoted by Vq,
^Q, and j9q ; we obtain from this

^=(^Y"' (14)

By means of equation (I) let the pressure p, first for v and then
for t, be introduced here, we thus obtaiu

(mj=©* • ••••••• US)

^r(?)* '"»

These are the relations which subsist between volume, tempe-
rature, and pressure when a quantity of gas is compressed, or is

* TratU de Micanique, 2nd edit. voL ii. p. 646.

43 FIRST MEMOIR.

saffered to expand in a holder impervioas to heat. Thesd equa-
tions agree completely with those developed by Poisson for the
same case"^^ the reason being that he also regarded k as constant.
Finally^ in equation (II b) let /ssconst.^ the first member at
the right-hand side disappears^ and we have remaining

dQ^AR^tldv; (17)

V

from which follows

Q=3AB(a + logv+const.)
or when the values of v, p, t, and QL, at the oommraiceHiea^f
the experiment^ are denoted by Vq, Pq, t^ and Q^

Q-Qo=AR(a+/o)log^ . . . . (18)

Prom this, in the first place, we derive the proposition deve-
loped also by Camot ; when a gas, untkont aUeratian of tempera^'
ture, changes its volume, the quantities of heat developed or ab^
sorbed are in arithmetical progression, while the volumes are in
geometrical progression.

Further, let the complete expression for R= ^^^ be set in

equation (18), and we obtain

Q-Qo=Apot^ologJ (19)

If we apply this equation to different gases, not directing our
attention to equal weights of the same, but to such quantities as
at the beginning embrace a common volume v^y the equation
will in all its parts be independent of the peculiar nature of thie
gas, and agrees with the known proposition to which Dulong,
led by the above simple relation of the quantity *— 1, has given
expression : that when equal volumes of different gases at the same
pressure and temperature are compressed or expanded an equal
fractional part of the volume^ the sam^ absolute amxmnt of heat is
in all cases developed or absorbed. The equation (19) is, however,
much more general. It says besides this, that the quantity of
heat is independent of the temperature at which the alteration of
volume takes place, if only the quantity of gas applied be always
so determined that the ori^al volumes Vq at the different tem-
peratures shall be equal ; further, that when the original pressure
is in the different cases different, the quantities of heat are thereto
proportional,

* TraiU de Micaniqtte, vol. ii. 647.

MOVING FOBCE OF HEAT. 43

II. Consequences of the principle of Camot in combination vntH

the preceding.

Camot^ as already mentioned^ has regarded the production of
work as the equivalent of a mere transmission of heat from a warm
body to a cold one, the quantity of heat being thereby undimi^
nished.

The latter portion of this assumption^ that the quantity of
heat is undiminished^ contradicts oxir maxim^ and must there*
fore, if the latter be retained, be rejected. The former portion^
however, may remain substantially as it is. For although we
have -no need of a pecuUar equivalent for the produced work,
after we have assumed as such an actual consumption of heat, it
is nevertheless possible that the said transmission may take place
contemporaneously with the consumption, and may likewise stand
in a certain definite relation to the produced work. It remains
therefore to be investigated whether this assumption, besides
being possible, has a sufficient degree of probability to recom-
mend it.

A transmission of heat from a warm body to a cold one cer-
tainly takes place in those cases where work is produced by heat,
and the condition fulfilled that the body in action is in the same
state at the end of the operation as at the commencement. In
the processes described above, and represented geometrically in
figs. 1 and 3, we have seen that the gas and the evaporating water,
while the volume was increasing, received heat from the body A,
and during the diminution of the volume yielded up heat to the
body B, a certain quantity of heat being thus transmitted firom
A to B ; and this quantity was so great in comparison with that
which we assumed to be expended, that, in the infinitely small
alterations represented in figs. 2 and 4, the latter was a differ-
ential of the second order, while the formed was a differential of
the first order. In order, however, to bring the transmitted
heat into proper relation with the work, one limitation is still
necessary. As a transmission of heat may take place by con-
duction without producing any mechanical effect when a warm
body is in contact with a cold one, if we wish to obtain the
greatest possible amount of work firom the passage of hefit be-
tween two bodies, say of the temperatures / and t, the matter

44 FIRST MEMOIR.

most be so arranged that two substances of different tempera-
tures shall never come in contact with each other.

It is this maximum of work that must be compared with the
transmission of the heat ; and we hereby find that it may reason-
ably be assumed^ with Camot^ that the work depends solely upon
the quantity of heat transmitted^ and upon the temperatures /
and T of both bodies A and B^ but not upon the nature of the
substance which transmits it. This maximum has the property,
that, by its consumption, a quantity of heat may be carried firom
the cold body B to the warm one A equal to that which passed
from A to B during its production. We can easily convince our-
selves of this by conceiving the processes above described to be
conducted in a reverse manner; for example, that in the first
case the gas shall be permitted to expand by itself until its tem-
perature is lowered from t to r, the expansion being then con-
tinued in connexion with B ; afterwards compressed by itself
until its temperature is again t, and the final compression effected
in connexion with A. The amount of work expended during the
compression will be thus greater than that produced by the ex-
pansion, so that on the whole a loss of work will take place ex-
actly equal to the gain which accrued from the former process.
Further, the same quantity of heat will be here taken away from
the body B as in the former case was imparted to it, and to the
body A the same amount will be imparted as by the former pro-
ceeding was taken away from it ; from which we may infer, both
that the quantity of heat formerly consumed is here produced,
and also that the quaatity which formerly passed from A to B
now passes from B to A.

Let us suppose that there are two substances, one of which is
able to produce more work by the transmission of a certain
amount of heat, or what is the same, that in the performance of
a certain work requirts a less amount of heat to be carried from
A to B than the other ; both these substances might be applied
alternately; by the first work might be produced according to
the process abpve described, and then the second might be applied
to consume this work by a reversal of the process. At the end
both bodies would be again in their original state ; ftoiiher, the
work expended and the work produced would exactly annul each
other, and thus, in agreement with our maxim also, the quantity

MOVING FORCE OF HEAT. 45

of heat would neither be increased nor diminished. Only with
regard to the distribution of the heat would a difference occur^ as
more heat would be brought from. B to A than from A to B^ and
thus on the whole a transmission from B to A would take place.
Hence by repeating both these alternating processes^ without
expenditure of force or other alteration whatever, any quantity
of heat might be transmitted from a cold body to a warm one ;
and this contradicts the general deportment of heat, which every-
where exhibits the tendency to annul differences of temperature,
and therefore to pass from a warmer body to a colder one*.

Prom this it would appear that we are /Aeor^/ica% justified in
retaining the first and really essential portion of the assumption
of Camot, and to apply it as a second maxun in connexion with
the former. It will be immediately seen that this procedure
receives manifold corroboration from its consequences.

This assumption being made, we may regard the maximum
work which can foe effected by the transmission of a unit of heat
from the body A at the temperature t to the body B at the tem-
perature T, as a function of t and t. The value of this ftinction
must of course be so much smaller the smaller the difference
/ — T is; and must, when the latter becomes infinitely small {=dt),
pass into the product of dt with a frmction of t alone. This
latter being our case at present, we may represent the work
under the form

wherein C denotes a function of t only f.

* [The principle here assumed, that heat cannot of itself pass from a colder
to a warmer body, and by means of which I have theoretically established the
relation between the work gained and the heat transmitted, is to be regarded
as a principle of the same importance as the one, in virtue .of which it is as-^
sumed that neither work nor heat can be produced from nothing. In conse-
quence of the different opinions of other authors I afterwards thought it ne-
cessary to make this principle the subject of a special memoir, which will be
found in the sequel. — 1864.]

t [It will perhaps be well to illustrate somewhat further what is here
stated in the text.

When any substance whatever undergoes a complete cycle of changes, heat
being thereby withdrawn from a body A of the temperature ty and when of
this heat a portion is consumed by the production of work and the remaining
portion transmitted to a body B of the temperature r, then, according to the

46 FIRST MEMOIR.

To apply this result to the case of permanent gases^ let ns
once more turn to the process represented by fig. 2. During
the first expansion in that case the amount of heat^
fdQ\

(g)*-

aboye prindples, the latter portion, that is to say the quanlity of heat trans-
mitted from A to B, must bear to the amount of work produced (provided
the latter be the above-mentioned maximum) a certain definite ratio which
will depend upon the temperatures of the two bodies A and B, but not upon
the nature of the interposed substance or upon that of its changes. Conse-
quently an equation of the following form must exist :

Work produced jtr* \ /^\

-^^—^^^,l>(t,r), ........ (a) I

wherein <f>(t, r) denotes a generally true function of the two temperatures t and i

t; it is, in fact, the function which, as stated in the text, represents the max-
imum of the produced work corresponding to the unit of transmitted heat.

Let the temperature t of the body A be now regarded as given^ the tempe-
rature r of the body B being at the same time susceptible of any values what-
ever. It is readily seen that when the difference ^— r is smaller, the work
which corresponds to the transmission of the unit of heat will also be smaller,
and that when the difference of temperature is infinitesimal, in which case it
may be represented by dt, the work will also be an infinitesimal quantity of
the same order. Ima^e then t—dt to he substituted for r in the function
^(<, r) which represents the work, and this function to be subsequently ex-
panded in a series arranged according to increasing powers of dt. No term of
this series wiU contain a. power of dt lower than the first, so that, neglecting
terms which contain higher powers of dt, we may write - |

4>(t,t-dt)^ir(t)dt, \

where the function ^(^) is likewise a generally true one. On proceeding to I

further calculations it is found that the equations assume a somewhat more
convenient form when, in place of writing the function ^(t) itself, a new

symbol is introduced for its reciprocal -j-^ ; the letter C having already been

employed by Clapeyron for this purpose I have provisionally retained it. Ac- j

cordingly, . 1

the equation (a), in the case where the bodies A and B have the temperatures
t and t—dt, becomes thus transformed :

Work produced _1 j/ /v\

Heat transmitted" C ^ ^ i

The function of the temperature denoted by C is frequently called Camot's '

function. An opportunity will present itself in the course of this memoir of
determining the form of this function. Its expression will then be found
sufficiently simple to admit of direct introduction into the equations, and 1

that done the symbol C will of course become superfluous. — 1864.]

i

MOVINO IfORCE OF HEAT. . 47

passed ttom. A to the gas ; and during the first compression the
following portion thereof was yielded to the body B :

or

(S)*-[s(f)-i(f)]**-

The latter quantity is therefore the amount of heat transmitted.
As, however, we can neglect the differential of the second order
in comparison with that of the first, we retain simply

The quantity of work produced at the same time was

ndv.dt
,

V

and from this we can construct the equation
-n dv.dt

.=7^ • dtj

(Wy-C

^*r

or

/rfQ\_ R.C ^ ^jYj

\dvJ V

Let us now make a corresponding application to the process
of evaporation represented by fig. 4. The quantity of heat in
that case transmitted from A to B was

(r-l «).'»,
or

rdm-(^+c-h\dmdt;

* [This equation may be generalized in the same way as was the equation
(II) in a previous note. In fact, replacing the fraction - by the differential

coefficient f ^j, which for gaseous bodies has the same value, the more ge-
neral equation

is obtained.— 1804.]

48 FIRST MBMOIB.

for which^ neglecting the differentials of the second order^ we
may simply pnt

The quantity of work thereby produced was

at
and hence is obtained the equation

{s-a) ^.dm.dt ,
at _ 1 ^/

or

r=C.(,-«r)|* (V.)

These, idthough not in the same form, are the two analytical
expressions of the principle of Camot as given by Clapeyron. In
the case of vapours, the latter adheres to equation (V.), and con-
tents himself with some immediate applications thereof. For
gases, on the contrary, he makes equation (IV.) the basis of a
further development ; and in this development alone does the
partial divergence of his result from ours make its appearance f.

We will now bring both these equations into connexion with
the results furnished by the first fundamental principle, com-
mencing with those which have reference to permanent gases.

* [This equation also is merely a special fonn of the equation

(§)-°®>

for in the present case we may put

\dv/ s-tr'
since the heat which must be imparted to the body under consideration, con-
sisting of liquid and vapour, during its increase of volume is precisely the heat
rendered latent by the production of vapour.

The differential coefficient -^ is written in the equation (V) without

brackets, for the manifest reason before alluded to. — 1864.]

+ [Clapeyron, in fact, when treating the equation (IV) and the more ge-
neral one given in a previous note, sturts from the hypothesis that the mag-
nitude Q is completely determined by the state of the body at the moment
under consideration, and consequently that it can be at once represented by a
function of the two variables (p and v in his case) upon which the condition
of the body depends. In this sense he effected the integration. — 1864.]

MOVING FORCE OF HEAT. 49

Confining ourselves to that deduction which has the maxim
alone for basis^ that is to equation (II a), the quantity U which
stands therein as an ^bitrary function of v and t may be more
folly determined by (IV) ; the equation thus becomes

rfQ=[B+K/^-A)logt;]rf/+^.rfi;, (lie)

in which B remains as an arbitrary function of / alone*.

If, on the contrary, we regard the incidental assumption alsO
as correct, the equation (IV) will thereby be rendered unneces-
sary for the nearer determination of (II a) , inasmuch as the same
object is arrived at in a much more complete manner by equa-
tion (9), which flowed immediately from the combination of the
said assumption with the original maxim. The equation (IV),
however, frimishes us with a means of submitting both princi-
ples to a reciprocal test. The equation (9) was thus expressed,

/dQ\
Kdv)'

* [This equation is obtained in the following manner. From the equation
(XI a)f that is from

V

may be deduced

Hence, replacing (^J by its value given in (TV), we have

V \av/ V

(f) = [C-A(«+0]?.

This, integrated according to v, gives

U=[C-A(a+0] Rlog«+(^(0,
where ^(<) denotes an arbitrary function of t Differentiating the last
equation completely, and putting B in place of the difierential coefficient

\^ f which, like <l>(t) itself, is also to be regarded as an arbitrary function of

t, we have

<nj==Q^-A )iiiog v+b] (?<+rc- A (a+o]- ^t?.

But if this expression for dU be substituted in the equation (II a) the term
AR ^i- will disappear, and the equation (II c) of the text will remain. —
1864.] ^

50

FIRST MEMOIR.

and when we compare this with equation (IV) ^ we find that
both of them express the same thing ; with this diflTerence only,
that one of them expresses it more definitely than the other. In
(IV.) the function of the temperature is expressed in a general
manner merely, whereas in (9) we have instead of C the more
definite expression A{a+t),

To this surprising coincidence the equation (V) adds its testi-
mony, and confirms the result that A{a-ht) is the true expres-
sion for the function C. This equation is used by Clapeyron and
Thomson in determining the values of C for particular tempe-
ratures. The temperatures chosen by Clapeyron were the boiling-
points of aether, of alcohol, of water, and of oil of turpentine. He

employed the values of ~ , s and r, determined by experiment for

these liquids at their boiling-points ; and setting these values in
equation (V), he obtained for C the numbers contained in the se-
cond column of the following Table. Thomson, on the contrary,
limited himself to the vapour of water; but considered it at
various temperatures, and in this way calculated the value of
C for every single degree from 0° to 230° Cent. The observa-
tions of Regnault had furnished him with a secure basis as re-
gards the quantities -^ and r ; but for other temperatures than

the boiling-point, the value of « is known with much less certainty.
In this case, therefore, he felt compelled to make an assumption
which he himself regarded as only approximately correct, using
it merely as a preliminary help until the discovery of more exact
data. The assumption was, that the vapour of water at its
maximimi density follows the law of M. and G. The numbers
thus foimd for the temperatures used by Clapeyron, as reduced
to the French standard, are exhibited in the third column of the
following table : —

Table I.

1.

t in Cent, degrees.

2.
C according to Clapeyron.

s.
C according to Thomson.

lOO

156-8

0733
0-828
0-897
0-930

0-718
0*814
0-855
0-952

MOVING FORCE OF HEAT. .51

We see that the values of C found in both cases increase, like
those of A(a + /), slowly with the temperature. They bear the
same ratio to each other as the numbers of the following series :

1; 113; 1-22; 1-27;
1; 112; 117; 1-31;

and when the ratio of the values of A{a+t) (obtained by setting
a =273) corresponding to the same temperatures are calculated,
we obtain

1; 114; 1-21; 1-39.

This series of relative values deviates jfrom the former only so far
as might be expected from the insecurity of the data from which
those are derived : the same will also exhibit itself further on in
the determination of the absolute value of the constant A.

Such a coincidence of results derived from two entirely differ-
ent bases cannot be accidental. Bather does it frimish an im-
portant corroboration of both, and also of the additional inci-
dental assumption.

Let us now turn again to the application of equations (IV)
and (V) ; the former, as regards permanent gases, has merely
served to substantiate conclusions already known. For vapours,
however, and for other substances to which we might wish to
apply the principle of Camot, the said equation furnishes the
important advantage, that by it we are justified in substituting
everywhere for the function C the definite expression A(a-f/)*.

The equation (V) changes by this into

r^A{a + t).{s-.a)^; . . . . (Va)

we thus obtain for the vapour a simple relation between the
teniperature at which it is formed, the pressure, the volume, and

* [In this manner we arrive at the definite and simple expression for the
function C of the temperature to which allusion was made in a previous, note
(p. 46), and which when first introduced had no determined form. Since
this function, in virtue of its signication, must have a general validity, it is
obvious that the expression for it which has been found on considering spe-
cially the expansion of a perfect gas, may also be applied to all other sub-
stances, and to all kinds of changes whereby these substances are able to pro-
duce work through the expenditure of heat. Whenever, therefore, by the
interposition of any variable substance, heat is transferred from a body A of

E 2

52 FIRST MEMOIR.

the latent heat> and can make use of it in drawing still further
conclusions. <

Were the law of M. and G. true for vapours at their maxi-
mum density*, we should have

p8szB.{a + t) (20)

By means of this equation let s be eliminated from (V a) ; neg- j

lecting the quantity a, which, when the temperature is not very
high, disappears in comparison with s, we obtain

Idp r

If the second assumption, that r is constant, be here made, we \

obtain by integration j

1 P_ r(^-lOO)

^^;?,"AR(a + 100) (a + ty '

where pi denotes the tension of the vapour at 100°. Let

.-100=r, « + 100=«,and^^^-r^^=/9;
we have then j

log£=^ (21) \

^Pi a + T ^ '

This equation cannot of course be strictly correct, because the |

two assumptions made during its development are not so. As

the temperature < to a body B of the temperature t^-dt, the relation between

the transmitted heat and the maximum of the work possibly produced thereby ^

may be expressed by the equations

Work produced __ dt
Ifeat tramniiUed A(o+<)*
In a similar manner the general equation given in the note to equation j

(IV) (p. 47), can now be written thus : — |

whereby the quantity of heat is completely determined which a body must
absorb when, at a constant temperature, it changes its volume under the in-
fluence of an external pressure equal to its own force of expansion. — 1864.]

* [The sole object of the inaccurate assumptions made, merely enpasst^nt,
in this paragraph is to elucidate further the formula for the tension of va-
pours which was established by Roche, and considered, from theoretical
points of view, by Holtzmann and other authors ; and to show, on the one
hand, why the formula is approximately correct, and on the other, why it is
not strictly so. — 1864.]

MOVING FORCE OP HEAT. pS

^ however the latter approximate at least in some measure to the

truth, the formula ^-^ expresses in a rough manner, so to speak,

the route of the quantity log ^ ; and from this it may be per-
ceived how it is, when the constants a and 13 are regarded as
i arbitrary, instead of representing the definite values which their

meaning assigns to them, that the above may be used as an em-
pirical formula for the calculation of the tension of vapours,
i without however considering it, as some have done, to be com-

pletely true theoretically.

Our next application of equations- (Vfl) shall be to ascertain
^ how far the vapour of water, concerning which we possess the

most numerous data, diverges in its state of maximum density from
• the law of M, and G, This divergence cannot be small, as car-

bonic acid and sulphurous acid gas, long before they reach their
points of condensation, exhibit considerable deviations.
The equation (V a) can be brought to the following form :

pdt
Were the law of M. and G. strictly true, the expression at the
left-hand side must be very nearly constant, as the said law
would, according to (20), immediately give

^ a+t

where instead of s we can, with a near approach to accuracy, set
the quantity *— o*. By a comparison with its true values calcu-
lated from the formula at the right-hand side of (22), this ex-
pression becomes peculiarly suited to exhibit every divergence
from the law of M. and G. I have carried out this calculation
for a series of temperatures, using for r and p the numbers given
by Regnault*.

In the first place, with regard to the latent heatj the quantity
of heat \ necessary, according to Regnault f^ to raise a imit of
weight of water from 0° to r, and then to evaporate it at this.

* Mem. deVAcad. de VInst, de France^ vol. xxi. (1847).
t Ibid, M^, IX. 5 also Po^g. Ann., vol. Ixxviii.

54 FIRST MEMOIR.

temperature^ may be represented with tolerable accuracy by the
following formula :

X=:606-5+0-305/ (23)

In accordance^ however, with the meaning of X, we have

^=^•+1'

cdt (23a)

For the quantity c, which is here introduced to denote the spe-
cific heat of the water, Regnault* has given, in another investi-
gation, the following formula :

c=l4-000004./+00000009.f«. . . . (23A)

By means of these two equations we obtain firom (23) the fol-
lowing expression for the latent heat :

r=606-5-0-695./-0-00002./«-00000003./8t. • (24)

Further, with regard to the pressure, Regnault j: has had re-
course to a diagram to obtain the most probable values from his
numerous experiments. He has constructed curves in which the
abscissse represent the temperature, and the ordinates the pres-
sure/?, taken at diflferent intervals from —33° to 230^. From
100° to 230° he has drawn another curve, the ordinates of which

* Mim, de VAcad, de Find, de France, Mem. X. |

t In the greater munber of his experiments Regnanlt has observed, not so
much the heat which becomes Itxient during evaporation, as that which be-
comes sensible by the precipitation of the vapour. Since, therefore, it has
been shown, that if the maxim regarding the equivalence of heat and work ^.

be correct, the heat developed by the precipitation of a quantity of vapour is
not necessarily equal to that which it had absorbed during evaporation, the
question may occur whether such differences may not have occurred in Keg-
nault's experiments also, the given formula for r being thus rendered useless.
I believe, however, that a negative may be returned to this question ; the
matter being so arranged by Kegnault, that the precipitation of the vapour
took place at the same pressure as its development, that is, nearly under the
pressure corresponding to the maximum density of the vapour at the observed
temperature ; and in this case the same quantity of heat must be produced
during condensation as was absorbed by evaporation.

[In a subsequent memoir I have proposed to employ, instead of the equa-
tion (24), the following equation for the latent heat:

r= 607-0-708.^.
It is more convenient for calculation, and gives very nearly the same value
for r as the equation (24) itself.— 1864.]

X Ibid. MSm. VHI.

MOVING FORCE OT HEAT.

55

r

represent, not p itself, but the logarithms of p. From this dia-
gram the following values are obtained; these ought to be re-
garded as the most immediate results of his observations, while
the other and more complete tables which the memoir contains
are calculated from formulae, the choice and determination of
which depend in the first place upon these values.

Table II.

^ in Cent, degrees
of the air-ther-
mometer.

p in millimetres.

/inCent. deerees

of the air-ther-
mometer.

according to the
cunre of the

according to the
eurve of the

numbers.

logarithms*.

o

—20

0*91

IIO

1073-7

1073-3

— 10

2-o8

120

1489-0

14907

4-6o

130

20290

2030-5

10

9-16

140

2713-0

2711-5

20

1739

150

3572-0

3578-5

30

3^-55

160

4647-0

4651-6

40

54*91

170

5960-0

59567

1°

9198

180

7545-0

75370

60

14879

190

9428-0

94a5-4

70

233-09

200

11660*0

11679-0

80

354-64

210

14308-0

143250

90

5»5*45

220

173900

17390-0

100

760-00

230

20915-0

20927-0 1

To carry out the intended calculations from these data, I have

first obtained from the Table the values of - . ■— for the tempe-

p dt ^

ratures —15°, —5°, 5°, 15°, &c. in the following manner. As
the quantity - . -^ decreases but slowly with the increase of tem-
perature, I have regarded the said decrease for intervals of 10°,
that is, from —20° to —10°, from —10° to 0°, &c. as uniform,

* This column contains, instead of the logarithms derived immediately from
the curve and given by Regnault, the corresponding numbers, so that they
may be more readily compared with the values in the column preceding.

t [It would have been more convenient to employ the values of vapour-
tensions, calculated by Regnault, from degree to degree, by help of an empi-
rical formula, and collected in his well-known larger table. On attempting
to do so, however, I found that it would be more appropriate for my present
object to return to the values here tabulated, and which were obtained with-
out the aid of an empirical formula, from immediate measurements of the
curves drawn according to observations ; for these values represent with the
greatest purity the results of observations, and are consequently particularly
well adapted for comparison with theoretical results. — 1864.]

56 FIBST MEMOIR.

BO that the value due to 25'' might be considered as a mean be-
tweenthatof2(f andthatof30°. As^ . ^=^^^^^,1 washy
this means enabled to use the following formula :

\p' dtjuo 10

or (1 dp\ Iiogpsup—^gPMP . (25)

V*j««"" lOM ' ' '

wherein, by Log, is meant Briggs^s logarithms, and by M, the
modulus of his system. With the assistance of these values of

- . -£, and those of r given by equation (24), as also the value

273 of a, the values assumed by the formula at the right-hand
side of (22) are calculated, and will be found in the second co-
lumn of the following Table. For temperatures above 100°, the

Table III.

1.

^ in Cent, degrees
of the air-ther-
mometer.

Ai<-«')STi*

4.
Differences.

a.

According to the values
OMenred.

3.
equation (27).

o

- 5
5

15
»5
35
45

11

95
105

"5
125

135
H5

165

185

195

205
215
225

30*61

29*21

30*93

30-60

30-40

30*23

30*10

2998

29*88

29-76

2965

2949
»9*47 2950
29*16 29*02
28*89 *^'93
28*88 29-01
28*65 28-40
28*16 28*25
28*02 28*19
27*84 27*90
27*76 27*67
27*45 »7*»o
26*89 2694
26*56 2679
26-64 26*50

30*61

30-54
30*46
30*38
30*30
30*20
30-10
30*00
29*88
29*76
29*63
29*48

29*33
29*17

llVo
2860
28*38
28*14
27-89
27-62

17-33
27*02
26*68
26*32

O'OO

+ 1-33
-0-47
—0*22
—0*10

— 003
0-00

-I-002
0*00
O'OO

— 0-02

— 0*01

— 0*14 —0*17

-)-o*oi +0-15
-i-o*io -i-o*o6
— o-o8 — 0-2I

— 0*05 -f0*20
+0-22 -i-0*I3

-^0*12 —0-05

-j-0-05 —O'OI

—0*14 —0*05
—0*12 +0-13
+0*13 +0*08

-fO*I2 —0*11

—0*32 — o-i8

MOVING FORCE OF HEAT. 57

two series of numbers given above for p are made use of singly^
and the results thus obtaraed are placed side by side. The sig-
nification of the third and fourth columns will be more particu-
larly Explained hereafter.

^ We see directly from this Table that Ap (*— <r) —j^ is not con-

\ stant^ as it must be if the law of M. and Q. were valid^ but that

I it decidedly decreases with the temperature. Between 35° and

I 95° this decrease is very uniform. Before 35°, particularly in

' the neighbourhood of 0°, considerable irregularities take place ;

which, however, are simply explained by the fact, that here the

[ pressure p and its differential quotient | are very smaU, and

hence the trifling inaccuracies which might attach themselves to
the observations can become comparatively important. It may
be added, farther, that the curve by means of which, as men-
tioned above, the single values o{p have been obtained, was not
drawn continuously from —33° to 100°, but to save room was
r broken off at 0°, so that the route of the curve at this point

cannot be so accurately determined as within the separate por-
tions above and below 0°. From the manner in which the di-
vergences show themselves in the above Table, it would appear
that the value assumed for^ at 0° is a little too great, as this

would cause the values of Ap{S'—cr) — - to be too small for the

temperatures immediately under 0°, and too large for those above
it*. From 100° upwards the values of this expression do not
decrease with the same regularity as between 35° and 95°. They

* [It must be remembered that the values of
are calculated by the formula

AK.-T)^

(a+0'i.|

given in equation (22). K now the value ofp, and consequently also the value
of log/), corresponding to 09 be too great, we must assume that the values of

the differential coefficient of log p, that is to say of - . ^, will be too great im-
mediately under 0°, and too small immediately above 0°, in consequence of

58 FIBST MSMOIR.

show, however^ a ^aia*a/ correspondence ; and particularly when
a diagram is made, it is found tluit the curve which, within
those limits, connects almost exactly the points, as determined
from the numbers contained in the foregoing Table, may be car-
ried forward to 230°, so that the points are uniformly distributed
on both sides of it.

Taking the entire Table into account, tilie route of this curve
may be expressed with tolerable accuracy by the equation

which the values of the above formula whkh contains - ^ in the denomi-

p at

natoT will necessarily be incorrect in an opposite sense. With reference to tem-
peratures under 0^ another circumstance must also be mentioned. For tem-
peratures under 0^ I have, in my calculations, applied the values of the va-
pour-tension j?, given by Regnault's observations, also to the case when the
vapour is in contact with liquid water, as of course it may be, since under cer-
tahi conditions water may remain liquid at a temperature far below 0^. Ac*
oordingly I have considered the magnitude r in the numerator of the formula
to be, at all temperatures, the quantity of heat consumed in the evaporation
of liquid water. If, on the contrary, we assume that those values of p given
by observation have reference, for temperatures under 0°, to the case where
vapour Ib in contact with ice ; then, for these temperatures, r must be under-
stood to denote the quantity of heat which is consumed in the evaporation of
ice. For the temperature (P itself the latter quantity of heat is obtained by
amply adding to the heat consumed in the evaporation of liquid water the
latent heat of fusion, that is to say 606-5+ 79 s 685*5. For temperatures under
QP this method, it is true, is not quite accurate ; nevertheless it must be very
nearly so when, in applying it, the differences in the latent heat effusion are
considered which correspond to diiFerent temperatures. The value of

Ap(«-(r) -±-.

corresponding to the temperature —5°, when calculated in this manner ac-
cording to the above formula, gives the number 32*93, instead of 29*21, as found
by the previous calculation. Comparing this number 32*93 with the series
of numbers which correspond to the positive temperatures 6°, 16°, &c., we
find that its deviation from the course of the latter is of an opposite kind to,
and indeed somewhat greater than, the deviation of the previously calculated
number 29*21. Regnault's values, therefore, regarded in either of the ways,
lead to irregularities in the course of the numbers. The occurrence of such ir-
regularities at low temperatures, is explained, as has already been observed, by
the fact that the vapour-tensions are then so small, that errors of observation,
though absolutely small, may become relatively great ; less weight, therefore,
must be attached to the numbers in the above Table which refer to low tem-
peratures, than to those which correspond to the mean and to the higher tem-
peratures. — 1864.]

MOVING FORCB OF HEAT. .59

Ao(«-<7)-?L.=w-«c*'; .... (26)

in which e denotes the base of the Napierian logarithms, and m,
n, and k are constants. When the latter are determined &om
the values given by the curve for 45°, 125° and 205°, we obtain

w=81-549; n=10486; *=0-007138; . . (26a)

and when, for the sake of convenience, we introduce the loga-
rithms of Briggs, we have

Logr31-549-4p(*-o-) -^1=:0-0206+0-003100/. (27)

Prom this equation the numbers contained in the third column
are calculated, and the fourth column contaias the differences
between these numbers and those contained in the second.

Prom the data before us we can readily deduce a formula
which will enable us more definitely to recognize the manner in
which the deportment of the vapour diverges from the law of M.
and G. Assuming the correctness of the law, ifps^ denote the
value o{ps for 0°, we must put, in agreement with (20),

ps __g-f^
ps^" a

and we thereby obtain for the differential quotients -_ ( £— j
a constant quantity, that is to say, the known coefficient of ex-
pansion -=0-003665. Instead of this we derive from (26), when

in the place of *— or we set s itself simply, the equation

ps _ m'-n.€^ o-M. .^g.

PSq m—n ' a ^ ' '

and from this follows

The differential quotient is therefore not a constant, but a func-
tion which decreases with the increase of temperature; and
which, when the numbers given by (26 a) for m, n and k, are
introduced, assumes among others the following values : —

60

FIRST MEMOIR.

Table IV.

t.

^©-

i.

m-

t

o

o

0*00342

70

0-00307

140

0-00244

lO

0-00338

80

0-00300

150

0*00231

20

0-00334

90

0*00293

160

0-00217

30

0*00329

100

0-00285
0*00276

170

0-00203

40

0*00325

IIO

180

0-00187

50

0*00319

120

0-00266

190

0*00168

60

0-00314

130

0-00256

200

0-00X49

We see from this that the deviations from the law of M. and
Gr. are small at low temperatures ; at high temperatures^ how-
ever, for example at 100° and upwards, they are no longer to be
neglected.

It may, perhaps, at first sight appear strange that the values

found for --(£1.] are less than 0-003665, as it is known that

for those gases which deviate most from the law-of M. and G.,
as carbonic acid and sulphurous acid, the coefficient of expan-
sion is not smaller hut greater. The diflferential quotients before
calculated must not however be regarded as expressing literally
the same thing as the coefficient of expansion, which latter is
obtained either by suflfering the volume to expand under a co»-
stant pressure, or by heating a constant volume, and then obser-
ving the increase of expansive force ; but we are here dealing
with a third particular case of the general differential quotient

— f =^ j, where the pressure increases with the temperature in

the ratio due to the vapour of water which retains its maximum
density. To establish a comparison with carbonic acid, the same
case must be taken into consideration.

At 108° steam possesses a tension of 1 metre, and at 129^° a
tension of 2 metres. We wiU therefore inquire how carbonic acid
acts when its temperature is raised 21 i°, and at the same time the
pressure increased from 1 to 2 metres. According to Regnault*,
the coefficient of expansion for carbonic acid at a constant press-
ure of 760 millims. is 0*003710, and at a pressure of 2520 millims.
it is 0*003846. For a pressure of 1500 millims. (the mean be-
tween 1 metre and 2 metres) we obtain, when we regard the in-
* M^m de VAcad», vol. xxi. Mem. I,

MOVING FORCE OF HEAT. 61

crease of the coefficient of expansion as proportional to the in-
crease of pressure, the value 0*003767. If therefore carbonic acid
were heated under this mean pressure from to 21 J% the quantity

^ would be thus increased from 1 to 1 +0003767 x 21*5

= 1*08099. Further, it is known from other experiments of
Regnault"^, that when carbonic acid at a temperature of nearly
(f, and a pressure of 1 metre, is loaded with a pressure of
1*98292 metre, the quantity /w decreases at the same time in
the ratio of 1 : 0'9914f6 ; according to which, for an increase of
pressure from 1 to 2 metres, the ratio of the decrease would be
1 : 0*99131 . If now both take place at the same time, the increase
of temperature from to 21 J, and the increase of pressure from

1 metre to 2 metres, the quantity -^ must thereby increase

very nearly from 1 to 1*08099x0*99131 = 1 071 596; and from
this we obtain, -as the mean value of the differential quotient

dt \pVq/

0071596^0.00333^

21*5

We see, therefore, that for the case under contemplation a value
is obtained for carbonic acid also which is less than 0*003665;
and it is less to be wondered at if the same result should occur
with the vapour at its maximum density.

If, on the contrary, the real coefficient of expansion for the
vapour were sought, that is to say, the number which expresses
the expansion of a certain quantity of vapour taken at a definite
temperature in the state of maximum density, and then heated
under a constant pressure, we should certainly obtain a value
greater y and perhaps considerably greater, than 0*003665.

Prom the equation (26) the relative volumes of a unit weight
of steam at its maximum density for the different temperatures,
as referred to the volume at a fixed temperature, is readily esti-
mated. To calculate from these the absolute volumes with suffi-
cient exactitude, the value of the constant A must be established
with greater certainty than is at present the casef.

* M^, de rAcad., vol. xxi. Mem. VI.

t [At the time I wrote this Joule had not stated which value of the me-

62 FIRST MEMOIR.

The question now occurs^ whether a single volume maj not
be accurately estimated in some other manner^ so as to enable
us to infer the absolute values of the remaining volumes from their
relative values. Already^ indeed^ have various attempts been made
to determine the specific gravity of water vapour ; but I believe
for the case in hand^ where the vapour is at its maximum den-
sity, the results are not yet decisive. The numbers usually given,
particularly that foimd by Gay-Lussac, 0*6235, agree pretty well
with the theoretic value obtained from the assumption, that two
measures of hydrogen and one of oxygen give by their combina-
tion two measures of vapour, that is to say, with the value

2 X 006926-f l'10568 ^Q,gog

These numbers, however, refer to observations made, not at those
temperatures where the pressure used was equal to the maximum
expansive force, but at higher ones. In this state the vapour
might nearly agree with the law of M. and G., and hence may
be explained the coincidence of experiment with the theoretic
values. To make this, however, the basis from which, by appli-
cation of the above law, the condition of the vapour at its max-
imum density might be inferred, would contradict the results
before obtained ; as in Table IV. it is shown that the divergence
at the temperatures to which these determinations refer are too
considerable. It is also a fact, that those experiments where the
vapour at its maximum density was observed have in most cases
given larger numbers ; and Begnault* has convinced himself^,
that even at a temperature a little above 30°, when the vapour
was developed in vacuo, a satisfactory coincidence was first ob-
served when the tension of the vapour was 0*8 of that which
corresponded to the maximum density due to the temperature
existing at the time ; with proportionately greater tension, the

chanical equivalent of heat he considered to be most in accordance with the
results of all his experiments. Taking experimental difficulties into conside-
ration, the values yielded by his various methods of observation agreed suffi-
ciently well with each other to leave no doubt in the mind as to the accuracy
of the theorem relative to the equivalence of heat and work, but not well
enough to enable me to deduce therefrom a value capable of being employed
with safety in the calculation of vapour-volumes. — 1864.]
• Ami, de Chim, et de Phys,, ser. 3, vol. xv. p. 148.

MOVING FORCE OF HEAT. 63

numbers were too large. The case, however, is not finally set at
rest by these experiments ; for, as remarked by Regnault, it is
doubtful whether the divergence is due to the too great specific gra-
vity of the developed vapour, or to a quantity of water condensed
upon the sides of the glass balloon. Other experiments, wherein
the vapour was not developed in vacuo but saturated a current of
air, gave results which were tolerably free* from these irregulari-
ties ; but from these experiments, however important they may
be in other respects, no safe conclusion can be drawn as to the
deportment of the vapour in vacuo.

The following considerations wiU perhaps serve to fill up to
some extent the gap caused by this uncertainty. The Table (IV.)
shows that the lower the temperature of the vapour at its maxi^
mum density, the more nearly it agrees with the law of M. and
G. ; and hence we must conclude that the specific gravity for
low temperatures approaches more nearly the theoretic value
than for high ones. If therefore, for example, the value of 0*622
for OP be assumed to be correct, and the correspondrag values d
for higher temperatures be calculated from the following equa-
tion, deduced from (26),

rf=:0-622 -^:::^,t, (so)

* Ann, de Chim, et de Fhys,, ser. 3, vol. xv. p. 158.

t [The magnitude d denotes the density of the vapour compared with that
of atmospheric air of the same temperature, and under the same pressure.
Now if 8 represent, as before, the volume of a imit of weight of the vapour,
and V the volume of an equal weight of atmospheric air of the same tempera-
ture, and under the same pressure, we may put

8

But, according to the law of M. and G.,

p a '
where Pq and Vq have reference to the temperature 0°; and again, according
to the equation (26), <r being neglected therein,

Ap a ^ ^

Now these values of v and « being substituted in the above fraction, we have

64

FIRST MEMOIR.

we shall obtain far more probable values than if we had made
use of 0-622 for all temperatures. The following Table gives
some of these.

Table V.

i.

9P,

Mf*.

10«<*.

i5a<».

Mt».

d.

o*6ia

0*631

0645

0-666

0698

Strictly speakings however^ we must proceed still further. In

Table III. it is seen that the values of Ap{$^a)

a + f

as the

temperature decreases^ approach a limit which is not attained
even by the lowest temperatures in the Table ; and not until this
limit be reached can we really admit the validity of the law of
M. and G.^ or assume the specific gravity to be 0*622. The
question now occurs^ what is this limit ? Could we regard the
formula (26) to be true for temperatures under —15^ also^ it
would only be necessary to take that value to which it approaches
as an asymptote, m=31'549, and we could then replace (30) by
the equation

rf=0-622.

m

m—n^'

(31)

From this we should derive for 0° the specific gravity 0*643
instead of 0*622, and the other numbers of the above Table would
have to be increased proportionately. But we are not jus-
tified in making so wide an application of the formula (26), as
it has been merely derived empirically from the values contained
in Table III. ; and among these, the values belonging to the
lowest temperatures are insecure. We must therefore for the ♦

present regard the limit of A(«— o-) ^ as unknown, and con-

and hence, for the temperature (P,

•♦0 •

tn—n

EUmmatingi^oVo from these equations, we arrive at the equation

^=rfn-

from which, on replacing d^ by its value 0*622, the equation (30) is at once
obtained.— 1864.]

MOYINO FORCE OW HEAT.

:&

tent ourselves with an approximation sunilaj^ to that furnished
by the numbers in the foregoing Table ; so much howeYCr '^fjB
may conclude^ that these numbers are rather too small than too
large*. -. . :

JBy combining (Va) with the equation (III), which was de-
rived from the first fundamental principle^ we can eliminate

A(«— «r) ^, and thus obtain the equation

r

dt'

at a-^t

m

bj means of which^ the quantity b, described above as negative,
can be more nearly determined. For c and r let the expres-
sions in(23'i) and (24) be substituted, and for a the number 273;
we then obtain

A=0-305-

606-5 -0-695f^000002f«-00000003/»

t;

(83)

273+^
and from this we derive among others the following values for h :

TXBLE VI. J:;

*.

•^

60«>. .

100*. .

160«..

20l»». . .

k.

-i*9i6

-1-465

-1x33

-0-879

-.p'676

* [For a comparison of the theoretical determination of the density of vapour,
as here expounded^ with more recent results of ohservations^ see Jippendix C]
t [When we employ the simplified formula
r=607-0-708 . «,

gi^en in the note to equation (24), and retain the yalue 0*306 given by Reg-

j„-

nault for the sum ;^+<^; the equation for h assumes the simpler form

607-^0-708. <

273-f<

800-3

A=0-305-
which may also he written thus :

A=1013--— - ,
2734-^

This formula for A is a still more convenient one. — 1864.]

X [The conclusion, that A is a negative magnitude, was also drawn by
Kankine, in a memoir published almost at the same time as my own, in the
Transactions of the Royal Society of Edinburgh (vol. xx.), wherein the mag-
nitude itself is represented by K,. The above equation (32), however, which
serves for the exact numerical calculation of A, was not established by Ran-
kine, since he was not then in possession of the necessary second Amda-*
mental theorem of the mechadical theory of heat. The equation employed by

0B muitMEVoiE^

Ia % tnftA&er similtt to that already portaad in tbe ease of
aqUdWi l^apotit^ the equation (Y a) might be applied to the vapouA
ot oHm fluids^ and the results tiiUs obtained compared with each
other, as is done in Table I. with the numbers calculated by CJlfr-
peyrM. We will ttbt, howerer, toter farther upon this appli-
ealion^.

We must now endeavour to determine, at least approximately,
the numerical value of the bonstant A, or, what is more usual,

the value of the fraction j-; in other words, to determine the

equivalent ofwchrkfar the unH qfheatfk

Parsuing the same course as ihat of Meyer and Holtemann^ we
can in the first place make use of equation (IO41), developed for
permanent gases. This equation was

r'as^-fAR)

-J
and when for c the equivalent expression j is introduced, we have

i-(j^^ f'*'

For atmbsp^ferie air, thb riumbfer 0*267, as given by De Laroche
and Beriurd, is genwaHy assuiitied for cf ; and for *, as given by

Dulong, 1-421, For the determination of R= ^o^o we know

that the pressure of one atmosphere (760 millims.) on a squa^
metre amounts to 10333 kils. ; and the volume of 1 kil. of atmo-
spheric air under the said pressure and at the temperature of the

him for the detetmiiiatioli bf this inajgnituAe, numbetecT (30) in his toembii',
would agree with my equation (lU), deduced from the first fundamental
theorem, had not Eanldne, contrary to myself, assimied the law of M. and G-.
to he true for saturated vapomrs. — 1864.]

* [The experimental data when this was. written heing too incomplete and
unsafe, further pursuit of the subject appeared inappropriate. Kegnault, how-
ever, having now publi^ed the second series of his extremely valuable inves-
tigations (lUlaUom de» Expiriences, t. ii.), in winch the vapour-tension,
the latent heat of evaporation, and the specific heat for a considerable
number of liquids are determined in the same manner as was done for aqueous
vapour in the first series, it would be easy to extend to vapours of other
liquids the calculatioBs which above have reference to aqueous vapour.—
1864.]

t [Now called, more briefly, the mechanical eg^uivalent ofheaL — 1864.]

MOVIN0 roses f>j seat* 97

fyssmg-^mt is ?F 0*7733 of a cubie melxe. fVom Uiio fidl(S>^]^

andlieiioe

that 19 to B»j, hj tke fxpenditiire of o&e «Biit of hea^ (the qoaiipi-
tity wMch rakes 1 Idl. of vater from 6^ to 1^) a Trei^ <^f 879
kils. can l)e raised to a height c^ 1 metre. This value, however,
on acconnt of the uncertainty of the numbers 0*267 and 1*421,
is deserving of little confidence. Holtzmann gives as the limits
between which he is in doubt the numbers 343 and 429*.

The equation (Va) developed for vapours can be made use of
for the same purpose. If we apply it to the vapour of water,
the foregoing determinations, whose result is expressed in equa-
tion (26), may be used. If, for example, the temperature 100^
be chos^i, and for j? the corresponding pressure of one atmo-
sphere = 10333 kils. be substituted in the above equation, we
obtain

^=257.(j?-cr).t (35)

-* [The remark in the text on the uncertainty of the experimental data em-
ployed in thils oalcuLation; has been recently verified by Regnault's finding
^t tjhe figpecific heat of atmosp^^c i^r js represented by 0*2876, instead of
by the number 0*267, which was previously considered to be the most trustr
worthy. By introducing the former into the above calculation the value 416,
instead of 370, is obtained for the mechanical equivalent of heat. If more-
over we «>eplac^ the number 1*4(21 by 1*410, which probably more nearly ezr
presses the true proportion between the two specific heats, we obtain ^4 as
the result of the calculation. I may also here remark that the number 29*26
of the text requires changing to ^*27 : this, however, has no influence
upon the given result, since the latter is calculated only to three figures. —

ie64.]

t [This equation may obviously be directly deduced fix)m the equation
(V a) } for the latter gives at once

1 c+ol

X= r (•-»■)•

^e diffoential coefficient ~£, hwe involTed, has, According toRegnault, at

}.0(1P, the valae 27-200, expressed in millimeties of mercury, and when this
number is i«Bduc^;tp l^o measiue of pressure above employed, ti, e. to kilo-

f2

68 FIBST IffSMOIR.

If it be now aMtuned witli Ghty-Lussac that the specific gravity
of aqueouB vapour is 0*6235, we obtain «= 1*696, and hence

~=487.
A

Similar results are obtained from the values of C contained in

Table I., which Clapeyron and Thomson have calculated from

equation (Y). If these be regarded as the values of A{a-\-i)

corresponding to the adjacent temperatures, a series of nimibers

are obtained for -j-, all of which lie between 416 and 462.
A

It has been mentioned above, that the specific gravity of

aqueous vapour at its maximum density given by Gay-Lussac is

probably a little too small, and the same may be said of the

specific gravities of vapours generally. Hence the value of —

derived from these must be considered a little too large. If the
number 0*645 given in Table V. for the vapour of water, and
from which we find «= 1*638, be assumed, we obtain

which value is perhaps still too great, though probably not
much. As this result is preferable to that obtained from the
atmospheric air, we may conclude that the equivalent of work for
the unit of heat is the raising of something over 400 His. to a
height of 1 metre.

With this theoretic result, we can compare those obtained by
Joule from direct observation. From the heat produced by
magnetoa-electricity he found

i=460*
A

From the quantity of heat absorbed by atmospheric air dunng
its expansion,

logrammes on a squaie metre^ becomes 360*8. On substituting furtiier, for
a+t and r, the values 373 and 536'6, corresponding to the temperature 100%
the equation (35) is obtained. — 1864.]

* Phil. Mag. YoL xziii. p. 441. The English measure has been reduced
td the French standard. . -j- ibid. vol. xxvi. p. 381. '

I

MOVING FOBCE OP HEAT. : 69

and as mean of a ^eat number of experiments in which the heat
developed by the friction of water, of mercury, and of cast iron
was observed,

1=425*

A

The coincidence of these three numbers with each other, not-
withstanding the difficulty of the experiments, dispels all doubt
as to the correctness of the principle which asserts the equiva-
lence of heat and work ; and the agreement of the same with the
number 421 corroborates in like manner the truth of Camot's
principle in the form which it assumes when combined with our
first fundamental principle.

APPENDICES TO FIRST MEMOIR [1864].
APPENDIX A. (Page 27.)

COMPLETED DEDUCTION OF THE EXPRESSION FOR THE EXPENDED
HEAT GIVEN IN EQUATION (3).

In developing the expression for the expended heat given in the
equation (3) of the text, certain magnitudes have been left un^
considered which have no influence on the result, and which in
order to simplify the calculus are usually disregarded in aU si-
milar cases. One disadvantage of this procedure, however, is
that to the reader doubts may thereby arise as to the accuracy of
the result. On this account I deem it desirable to supply here
a somewhat more complete deduction of the equation (3) .
• In doing so it must he remembered that the following deve-
lopment, as well as that given in the text, holds not only for a
gas, but also for every other body whose condition is determined
by its temperature aud its volume, and whose variations of vo-
lume occur in such a manner that force and resistance differ so
little from one another las to justify, in calculation, the assump-
tion of their equality. We shall assume, moreover, that the sole

• Phil. Mag. vol. xxxv. p. 534,

79 VIB8T HXMOni^ MFXNDIX A.

bxterior force whieh influences tbe ohangeft of volume aets every-
where normftlljr and equally upon the surface^ so that m general
it may be termed a pressure^ inasmuch as any pull which may
possibly take place may be regarded as a negative pressure.

Let us consider the quantity of heat dQ, which a body must
receive duribg ail increase of temjierature equal to dt, and an
aUgibentation of volume equal to dv. Foi^ a differential which
depends^ as dQ does> on the differentials of two independent va^
riables^ it is eustomary to employ the equation

^=(§)*^(w)*' w

which by the introduction of simple symbols for the partial dif-
ferential coefficients, that is to say by putting

(f)=".

(b)

may be thus written :

dQ^Mdt+^dv (c)

Strictly speaking, however, this equation is incomplete. The
complete expression for dQ contains an infinity of terms, of the
successive orders one, two, three, &c., in reference to the differ-
entials dt and dv. By actually introducing the terms of the sex-
cond order, and merely indicating the remaining ones, the equa-
tion for dQ becomes

.a=M*+N*4((f)*.^[(f).(f)].,*

4-^^W} + &c (d)

Now it is clear that when an expression contains terms oi the
first order in the differentials, all accompanying terms of the se-
cond or of higher order may be neglected. Accordingly the two
first terms on the right of the above equation are th6 only oniss
which are usually written. When in any calculation, however,
the terms of the first order cancel each other, so that among the
terms of the final result those of the second are the lowest in
order, then from the comm^eement all terms of the second

EXPRESSION FOB T^B BXPBKDEB HEAT. 71

order must be taken into consideration^ and it is only those of
t^e third and higher orders which can be neglected. This occurs
in the case under consideration^ since the expression for the ex-
pended heat^ containing the product dv dt eia a factor^ is neces-
sarily of the second order. The calculation^ given in the text,
was condact«d^ it is true, in such a manner that only those
t^rms of the second order were neglected which were without
influence on the final result, nevertheless for the sake of com-
pleteness and rigour, these terms in the following calculation
shall also be written.

When any relation whatever is given between the variables t
and V, in virtue of which the one may be regarded as a function
of the other, the equation (d) may be written so that the terms
on the right proceed simply according to ascending powers of §
single variable. If t, for instance, be regarded as a function of
r, and the following symbols be introduced for the differential
coefficients of t according to v,

dt ^ dH y .

then we shall have

rf/=frft; + r^%fcc , , . , (e)

whereby th« equation (d) will become

If, on the other hand, v be regarded as a fiinetion of t and we in-
troduce the symbols

we sliall have

rft;«iyrf/+i/ +8d§ , ••••(f)'

and accordingly

<ra=,M.K^**{(f)^[P^(f)],

These equations a,re to be appUed to the four chaoEiges to which

72 FIRST MEMOm^ APPENDIX A.

'the gas or the body xmder consideration is to be subjected^ and
which are to proceed according to two diflPerent laws. ^

We consider first the changes of volume which occur at a
constant temperature. In this case the differential coefficients
of / according to v, that is to say^ the magnitudes ^, P, &c.^ must
be put equal to zero. Consequently in order to determine the
quantity of heat which the body must receive during an expan-
sion dv from its initial state without change of temperature^ we
may employ the equation (f) in a simplified form ; the terms
which contain the factors f , f, &c. . . being omitted. Stopping
at terms of the second order^ we thus arrive at the equation

'«=N*.(f)f.

in order to express^ in a similar manner^ the quantity of heat
which the body must receive when at the temperature t—dt it
expands from the volume r + Sv to the volume v-\-Sv-td!v, we
must replace dv in the foregoing equation by cPv, and in place

of N and (-7- I introduce the values which these magnitudes

possess at the slightly changed temperature i—dt, and the
somewhat altered volume v + Sv. Assuming these values to be
expressed in series proceeding according to powers of dt and dp^
we need only retain terms of the first order in the case of N,
since the, latter quantity is multiplied by a differential in the
above equation^ and all subsequent terms in N would merely
lead, in ^Q/ to terms of a higher order than the second. Ac-
eordingly in place of N. we have to put

In the valueof f -7- V which in the above equation is multij^lied

by the square of a differential/ we may for tlie same reason omit
terms of the first order, and simply retain the original value

(-i-f- Accordingly if we represent the quantity of heat re-
ceived during this expansion by d'Q, we have the equation

BXFRESSIOK FOR THE EXPENDED HEAT. 73

Subtracting this quantity of heat from the former, we obtain the
heat expended during the whole process, that is

rfQ-rf'Q=Nrf»-rN+(^S»-/'^)rf/ld'»

\dv) 2 '

(i)

This expression differs only in the last term from the one num-
bered (2) in the text, and this term is easily recognized to be
only apparently of the second order, for the differentials dv and
dfv can only differ from one another- by a magnitude infinitely
small relative to their own proper values, so that the difference
dv^^d^v^ is an infinitesimal of an order higher than the
second.

We proceed now to- changes of volume of another description,
—to changes produced without either communicating heat to the
body or abstracting it therefrom. In this case the temperature
must change with the volume, and one of these magnitudes being
chosen as the independent variable, we have to determine the
differential coefficients of the other. We shall consider v as a
function of t^ and determine the differential coefficients rj, vj &c.
of the former. To this end we must employ the equation (h),
and put therein rfCl=0, whereby we shall have

0=(M.N„*.{(f).[(f).(f)],l

Since this equation must hold for any value of dt^ the factor of
each power of dt must vanish. Equating to zero the factor of
the first power of dty we have

whence we deduce

The magnitude y\ is thereby determined as a function of t £|jid v%
The next differential coefficient tj might be similarly determined
by equating to zero the factor of the second power of dt ; it is
not necessary, however, actually to 'perform this calculation.

(k)

74 FIBSV mVOIBy APPSHDIX A.

flince 1/ may also be found by differentiating the expression for
ff, already found completely^ according to / ; that is to say we
may differentiate according to / and v, and regarding t; as a func-
tion of t^ put -ji^^V' 'The succeeding differential coefficients of

t; according tot/if required^ would have to be calculated in a
similar manner.

Now to determine the magnitude Sv by which the volume of
the body must increase from its initial value^ in order that the
temperature may sink from t to t^dt, we must employ the
equation (g)^ and write therein Bv in place of dv, and ^dt in
place of dt. By so doing and contenting ourselves with terms
whose order does not exceed the second^ we obtain the equation

Sv^^ffdt-^f/^ (m)

Similarly^ to find the value of B'v, that is to say how much the
body^ starting from the volume v + dv, must expand in order
that the temperature may fall from t to t^dt, we must replace

17 in the foregoing expression by its changed value i7+(t^ Kfo.

The corresponding change of 7/ need not be considered, since
the only terms which could arise therefrom wotdd be of an order
higher than the second. We have therefore

Besides these equations for Sv and ifv, another must exist in-
volving the four changes of volume which the body suffers suc-
cessively, during the process. This is the equation which ex-
presses the condition that the body ultimately returns to its
initial volume, and which is thus written :

dv-i-Sfv^iSv + d'v . . (p)

From this it follows that

^v^dv-^-Sfv^^,
an ^nation which, on substituting for Sv and B'v their respective
values as akeady founds becomes

d!v^dv-(^^<hdt (p)

fiXPBiSSKW fdl TtfS Sn«yi>Sl» HEAT. 7^5

We now return to equation (i)^ which represents the hetii ex^
pended during the whole process^ aikl substitute therein the va*
lues of St; and dfv as given by the equations (m) and (p) • Ne-
glecting all terms of an order higher than the second^ we thus
find

.Q^d'Q=[(f)+(^^^)>rf/; . . . . (q)

and if in this we replace 17 by its value given in (1), We have

which^ by introducing in place of M and N the original symbols
for partial differential coefficients^ becomes

^-'"='=[r^)-i(§)]**- • • • «

Thi)9 is th6 equation (3) givesi in the teict> to re-establish
which^ in a somewhat more rigorous manner^ was the object of
the present Appendix.

APPElfDiXB. (Page 28.)

iKTSORAtlON 07 TfiB DIFFBRENTIAL SQt^ATtOK (11).

It will perhaps be useAil to elucidate somewhat more fiiUy the
manner of obtaining the equation (II a) from the equation (II).
The equation (11)^ which in the text is thus written^

dt\dv}--dvKdt) JT' • • • • i^Aj

may be called a partial differential equation of the second order,
althoug^,it differs somewhat from the ordinary equations of this
k^d^ since in the latter it is usual to assume^ tacitly^ the fulfil"
ment <^ the condition

'^KdtJ dkdvJ'

In ord^ to pass^ by integration^ from the equation (C) to a
differential equation of the first order^ we may proceed as
follows. In the first place we take any fdnction whatevcar of t
and V as a.representative 6f one -of the two partial differential

76 ^ PiaST MEMOIB, APPENDIX B.

coefficients (-^) and f ^\ For instance, M being any such
function, we put

Q=M «

and introduce tins value into the equation (II). The substi-
tuted term being then removed from the left to the right of the
equation, we have

^ /dQ\/dM\AR
. . dt \dv/ \dv) V '

Integrating this equation according to t, and under the hypo-
thesis that V remains constant, we find

(S)=J(f>'+*''^*W' • • • ("

where ^(v) denotes an arbitrary function of v. Having thus
obtained an expression for the partial differential coefficient

J -T— J, we next form the complete differential eqtMtion of the
first order,

and substitute therein the assumed function M for (-^ j,and

the expression just obtained for ( -r- ). We thus arrive at the
equation

dQ=Mdt-\-[((^yt + Ani-^(l>{v)'\dv. . . . (c)
The expression • .

which forms a constituent part of the right-hand side of thid
equation, is at once seen to be the complete differential of a
function of t and v ; for the factor of dt, when differentiated ac-
cording to V, gives the same result^ (—-), as does the differ-
entiation, accordiog'to t, of the factor of rfr. For this expres-
sion, therefore,- we may introduce the symbol dS ; and since M
represents an arbitrary function oi t and v, and ^(v) an arbitrary
function, of 17^ S itself must be regarded as a perfectly arbitrary

INTEf^RATION Of EQUATION (II) . 77

ftinction of/ and v. The introduction of the symbol into the
equation (c) gtven

rfa=rfS+AR-rft; (d)

For the further treatment of this equation it will be conve-
nient to introduce, in place of the simple magnitude t in the
last term, the sum a + t, where a is the constant defined in the
text. To do this the last equation may be written in the form

V V

or rather thus :

rfQ=rf(S^ARalogt;)+AR^rft;, ..(e)

which latter may be simplified by putting

S-AIUlogt;=U, ....... (f)

where U is again an arbitrary function of / and t;, since an alge-
braical sum which consists of an arbitrary, and of a known func-
tion of the same variables must itself be regarded as an arbi-
trary function of these variables. By introducing this new
symbol U into the equation (e), we obtain the equation (II a) of
the text, that is to say,

dCL^dV-hAIi^^dv. .... (Ila)

The object of the introduction of the sum a + ^ in place of the
quantity t, is to render the last term susceptible of a simple
mechanical meaning. In fact, from the equation

jw=R(a + /),
which applies to permanent gases, it follows that

AB.^dv=Apdv; (g)

and Bmce pdv denotes the exterior work done during the expan-
sion dv, the last term of the equation (Ila) obviously represents
the heat-equivalent of the exterior work.

The more general diflferential equation of the second order,

dt{di^j''di{'wrH'i)' • • • • ("')

given in the first note on p. 28, may be treated in the same man-*
ner. as. we have just treated the equation (II), and thereby the fol-

78 n»T wsnmn, jkrrnfDiz C.

lowing complete fUffenittifll eqmitioii of ihe irpk or^ oMfWi^*

rfQ=dU+A/?(fo (IFi)

The function U.^ here introduced^ ie of great importance in the
theory of heat ; it will firequejitlj come under di8ci:yBsion in the
following mejnoirs. As stated in the text^ it inyplyes two of the
three quantities of heat which enter into cooaideration whoi a
hodj changes its condition ; these are l3ie augmentation of the
so-called senMIe or actuaHy prefeni heat^ and the heat expended
in iniericr work.

APPENDIX C. (Page 66.) j

ON TVS DENSITY Oy flAT^TJUTSP ▲QTTBOPP VAPOUR. |

The conclusions drawn in the text^ relative to the deyiation
from the law of M. and G. presented by saturated vapours^ and
which at that time stood isolated^ inasmuch as it was the uni-
Tersal custom to apply the law in question also to vapours, have
since been experimentally verified by Fairbairn and Tate*.
The following summary of a note communicated by me to the
Academy of Sciences at Paris t, will sho^ how far these results 1

of observation agree with my formula.

Under (30), in the text, is given the equation

dttcO'^22 -r.9

wherem d (denotes the density of the tMtturated aqueous vapour, '

HI comparifiom with atmoi^faeric air at the same tem^ieiBture |

and under the «ame inressure, and m, «i, k axe three constants
having the values

m=31J549, n^VO^S^, *;=0007133.
By means of this equation the values of d were calculated which
are contained in Table V. of l3ie text (p. 64). It 8 be the
volume of a kilogramme of saturated vapour^ and v that of a ki-
logranmie of atmospheric air at the same temperature and pres-
sure, the fraction - ms^ .be put instead of d. The jrecdproail

fraction, therefore, will, according to the foregoing equation, have
the value 9^ m^n^ ^

t;°'0-622.(m-w)'

* Proceedings of the Hoyal Sodiety, 1860 ; and Phil. Mag. Fomrtii 'Series,
vd. sad. p. '290. t Cwn^ Me/ndm^ys^. liL p. 706 (April 1661).

DENSzrr or satvbiAtxd Ajomzovn tapoub.

79

Tbis eqnfttkoi maj be mnttcan in the more comremenit form

i=M-N«^ (a)

where the €oi»st&nti M^ N^ « have the following TaluAi^ depen-
dent on the Yidnes of m, n, k previously given^

M= 1-6680, N=005527, «=1007164.
Strictly speaking the diflferjence *— o-, where a is the Tohnne of a
kilogramme of water, should enter into the foregoing equations
instead of the quantity t, since this difference occurs in the
equation (26), from which (30) is deduced. The volume of
water being very small, however^, when compared with that of
vapour, the quantities 9 and ^•-irmay,in an approximate calcu«
lation, be regarded as equal to one another.
In the following Table the values of *, calculated from the above

formula for -, are placed side by side with those deduced by

Fairbaim and Tate from their observations, and with the Talues
formerly assumed as corresponding to the equation

8_ 1

V 0-622'

Tempenttnre in
degrtMOebtigrade.

fhe former ■•-
kumptkm.

the equation (a).

obterration.

68-52

8-38

823

8-27

5*41

5*9

•5*33

7076

4*94

4-83

4-91

77-18

3-84

3-74

3-72

77-49

3-79

369

3T1

7940

3-5*

3-43

3*43

83-50
86-83

3-02

294

3-05

2-68

260

a-62

92-66

2-18

211

2-J5

117-17

0-991

0-947

0*941

118-23

0-961

0-917

0-906

118-46

0-954

0-911

0*891

ia4'a7

0-809

0-769

=0-758

128*41

0718

0-68I

0-648

130-67

0-674

0-639

0-634

13178

0-654

0*619

0*604

134-87

0-602

0-569

0-583

137+6

0-562

.0-530

0-514

139-21

0-537

0-505

0-496

141-81

0-502

0472

0457

142-36

0-495

0-465

o-44«

1447+

0-466

0*437

o-43»

80 ON THE INFLUENCE OF PBE88UBE .

From this Table it will be seen that the observed Talnes agree
much better with those calculated from my equation than with
the formerly assumed values * and further^ that the differences
which still exist between the observed values and those of my
formula are generally of such a character that the observed
values differ from the formerly assumed ones still more than do
the values of my formula.

ON THE INFLUENCE OF PRESSURE UPON THE
FREEZING OF LIQUIDS*.

Mr. William Thomson has described an experimental inves-
tigation^ conducted by himself t> and originating in a theoretic
view entertained by his brother, James Thomson. The latter
had concluded, from the known principle of Carnot, that by an
increase of pressure the freezing-point of water must be lowered,
which view was completely verified by experiment.

Some time ago I published a theoretic memoir X^ i^ which
the principal part of Camot^s law is retained, but altered in
one minor particular. This alteration rendered certain of the
conclusions heretofore deduced from the principle impossible,
while others remained valid ; the latter being those whose cor-
rectness or high probability had been demonstrated by expe-
riment. Now as the above conclusion regarding the freezing-
point of fluids has also been substantiated experimentally, and
thus in a scientific point of view has obtained a greater signifi-
cance than one woidd be inclined at first sight to attribute to so
small a difference, I feel myself called upon, in behalf of my
theory, to show that my alteration of Camot's principle is in no
way opposed to this result §. Moreover, by a simultaneous ap-

* Note published in Poggendorff's Annalen, September 1850, vol. Izzzi.
p. 168 ; and translated in the Philosophical Magazine, S. 4. vol. ii. p. 548.

t Proceedings of the Royal Society of Edinburgh, February 1850 j and
Phil. Mag. S. 3. vol. xxxvii. p. 123.

} [First Memoir of this collection.]

§ I need hardly mention that I have here no thought of disputing with
Mr. J. Thomson the priority of his ingenious application of the principle of
Camot.

UPON THE FREEZING OF LIQUIDS. 81

plication of the first Aindamental principle which I have assumed^
a new conclusion is arrived at which^ although practically unim-
portant on account of the smallness of the numbers which it in-
volves, nevertheless deserves expression on account of its theoretic
interest.

A lengthened analysis of the subject is not here necessary.
The considerations dwelt upon in my former paper regarding
the evaporation"^, may be applied almost verbatim to the freezing
of a liquid. We have only to conceive the vessel impervious to
heat to be filled with the body partly in the solid and partly in
the licjuid state, instead of, as in the former case, partly in the
liquid and partly in the vaporiform state ; and then, instead of
permitting a fresh portion of the liquid to evaporate, to allow a
portion of it to freeze, &c.

One of the two principal equations deduced therefrom was

r=A(a + 0{»-«r)^j (Va)

and this holds good for the freezing also, p and t again denoting
the pressure and temperature, and a- the volume of a unit of
weight of the liquid, whereas s denotes the volume of a unit of
weight of a solid body (instead of vapour, as in the former case) ,
and r the latent heat of the freezing (instead of the evaporation).
The latter, however, must be here taken as negative, because by
freezing, heat will be liberated, and not rendered latent. We
have therefore

dt__ A{a'^t) {s--a) .,v

dp r ^ ^

Let the value of — , given by Joule in his last investigation f as

A.

the most probable result of all his experiments, that is 423-55
(772 English) , be here substituted, as also for a the number
273; further, with regard to the water, if=0, r=79, cr=0-001,
and * =0*001087 ; and, finally, letp be expressed in atmospheres,
instead of in kilogrammes, pressing upon a square metre, we
then obtain

^=-000733,
dp

* [First Memoir, pp. 80 and 47.]

t Phil. Trans, of the Royal Society of London for the year 1850, part 1.
p. 61.

a

82 ON THE INFLUENCE OF PRBiSURE

which may be regarded as equal to the value oalculated by
James Thomson, and corroborated by William Thomson, namely
-00076.

The other principal equation deduced from the principle of the
equivalence of heat and work was

J+<r-A=A(*-<r)* (IH)

To apply this to the case of freezing, we must regard c
and h as two quantities which differ from the specific heats
of the liquid and solid body only so far as they express, not
the heat which must be imparted to a body when it is simply
warmed, but that which is necessary when the pressure varies
with the temperature in the manner indicated by equation
(1). This difference, however, cannot be considerable, since
Begnault* has found that water, by an additional pressure
of 10 atmospheres, does not increase i^th of a degree Cent,
in temperature; besides this, as the differences for c and h
take place both in the same sense, and hence in the difference
c— A are subtracted, we can set with a near approach to accu-
racy for c— A the difference of both specific heats simply f. If

the value of -^ estimated from (1) be substituted in (III), and

if the sign of -^ be changed like that of r in the former case,
we have ^^c^h^r (2)

From this we must conclude, that when the freezing-point
changes, the latent heat must also change ; iar water c=l, and,
according to Person {, A =0-48. Hence we have

J=0-52+0-29=0-81;
at

that is to say, when the freeeing-ppint of water is lowered by

pressure, the latent heat decreases 0*61 for every degree.

We must not confound this result with that i^eady expressed

by Person §. From the circumstance that the specific heat at

« Mim, de VAead. de VInst. de France, vol. zxi. M^m. VII.
t [In one of the following Memoirs a more accurate determination will be
given.— 1866.]

X CoMptes Rendus, vol. xxx. p. 626.

§ Ibid. vol. xxiii. p. 336, and Poggendorffs Annalen, vol. Ixx. p. 302.

UPON THE FREEZING OF LIQUIDS. 83

ice is less than that of water, the latter concluded with great
probability, that when the freezing-point, without increasing the
pressure, is simply lowered by preserving the fluid perfectly mo-
tionless, the latent heat must then be less than at 0**. This
decrease may be expressed by the equation

the above equation (2) therefore shows that, when the freezing-
point is lowered by pressure, the latent heat, besides the change
due to the last-mentioned cause, suffers a still further diminution

expressed by the quantity ; this ia the case of water is == 0*29,

U "1" »

and it is this quantity which corresponds, as equivalent, to the
exterior work accomplished.

The recent observation of Person*, that ice does not melt
completely at a definite temperature, but becomes softer imme-
diately before it reaches the melting-point, I have left unnoticed,
as its introduction would merely render the development more
difficult, without serving any important end ; for the decrease of
latent heat, which corresponds, as equivalent, to the produced
work, must be independent of the little irregularities which may
take place during the melting.

APPENDIX TO PRECEDING NOTE (1864).

ON THE DIFFERENCE BETWEEN THE LOWERING OF THE FREEZING-
POINT WHICH IS CAUSED BY CHANGE OF PRESSURE AND THAT
WHICH MAY OCCUR WITHOUT ANY SUCH CHANGE.

It win not, perhaps, be without advantage to examine some-
what more closely what has been said at the end of the preceding
Note. AUusion was there made to the well-known phenomenon
of the lowering of the freezing-point of water brought about, not
by increasing the pressure, but by protecting the water from all
agitation ; and it was asserted that in this case the l^atent heat,
or rather the heat rendered sensible on solidificatioii, must change
according to a law different from that which obtains when the^

* Comptes ItendiiSy vol. xxx. p. 526.

g3

84 APPENDIX TO PRECEDING NOTE.

freezing-point is lowered by pressure. The correctness of this
assertion^ and of the equation relative thereto, will be rendered
manifest by the following considerations.

It was frdly demonstrated in the First Memoir that the heat
which must be imparted to (or abstracted from) a body in order
to bring it from a given initial condition to another determinate
one, may be divided into three parts ; these are the quantity of
heat which serves to increase that which is actually present in
the body (the so-called sensible heat), the quantity expended on
interior work, and the quantity expended on exterior work. It
was stated that the two first parts are completely determined by
the initial and final conditions of the body, and that for this de-
termination it is not necessary to know in what manner the
changes of the body have occurred, in other words, what path has
been pursued by the body in passing from one condition to the
other. If, therefore, we include both these quantities of heat in
one symbol U, as was done in the First Memoir, we shall thereby
obtain a magnitude which, on the supposition that the initial con-
dition of the body is known, depends only upon its present condi-
tion, and not at all upon the manner in which it has been brought
into this condition. The third quantity of heat, however, that
expended on exterior work, depends not only upon the initial and
final states of the body, but also upon the whole series of changes
which it has undergone. The exterior work being represented by
W, the heat expended in its production will be AW, and on adding
to the latter the other two quantities of heat, we obtain the sum

U-fAW
as the representative of the total heat which must be imparted
to the body during its several changes.

Now let us conceive a unit of weight of water to be given at
the temperature 0°, and let it be required to convert it into ice at
a certain temperature t^ below zero, the pressure remaining con-
stantly equal to that of the atmosphere, and to express the quan-
tity of heat which must be withdrawn fix)m the mass in order to
do so.

The simplest way of producing this change would be to allow
the water to freeze at 0°, and then to cool the ice so formed to
the temperature /j. The process, however, may be also con-
ducted in another way. We will allow the water, in its liquid

DIFFERENT VARIATIONS OF THE FREEZING-POINT, 85

state, to be cooled to a temperature t between 0^ and ^„ and then
at this temperature to be solidified. When water which has been
cooled to a temperature below zero freezes, a considerable quan-
tity solidifies suddenly, and the heat thereby produced or ren-
dered sensible raises the whole mass to 0° again, after which the
further solidification proceeds gradually at the latter tempera-
ture. Nevertheless, although not actually feasible, we will con-
ceive the sensible heat to be withdrawn from the mass during
its solidification just as quickly as it is generated, so that the
whole mass may freeze at one and the same temperature t. The
ice thus produced shall then be subjected to a ftirther cooling
down to the temperature ty

In finding an expression for the quantity of heat which must
be withdrawn from the mass during this process, we shall em-
ploy the following symbols : —

/ the heat rendered sensible during gelation,

d the specific heat of the water,

A' the specific heat of the ice,

tr' the volume of a unit of weight of water,

y the volume of a unit of weight of ice.
The letters, it will be observed, are the same as in the foregoing
Note, they are here accented because they have now slightly dif-
ferent values. In the preceding Note, in fact, they had reference
to the case where the pressure increased according to a certain law
with the diminution of temperature, whereas now the pressure is
supposed to remain constantly equal to one atmosphere."

Accordingly the heat which serves to bring the water from the
temperature 0° to the temperature t will be represented by the
integral

Now the temperature t being, by hypothesis, lower than 0°, t is
a negative quantity, and with it the value of the integral also •
this expresses the fact that the heat in question is not imparted
to, but withdrawn from the body. In a similar manner the
quantity of heat which serves to depress the temperature of the
ice thus formed from / to t^ is expressed by the integral

'' h'dt.

f

86 APPENDIX TO PRECEDING NOTE.

Lastly, r' represents the heat rendered sensible during solidifica-
tion, and to it a negative sign must be affixed in order to indi-
cate that this quantity of heat must also be withdrawn firom the
body.

The algebraical sum of these three quantities constitutes the
required expression for the total heat under consideration, and
since this latter quantity is also expressible by the sum pre-
viously determined, we have the equation

-r'-|-fV*+rA'*=U+AW.

(a)

The exterior work "W still remains to be determined. The
initial volume of the mass coincides* with that of a unit of
weight of water at the temperature 0^, and its final volume is
that of a unit of weight of ice at the temperature /j. These two
volumes, as special values of ^ and «', being represented by a^
and *,', the increment of volume will be expressed by */— o-q'.
Since this increment of volume takes place under the constant
pressure /7o of one atmosphere, the corresponding work will be
expressed by the product /?o(*/""0 simply, and the temperature
at which freezing may have taken place is here a matter of in-
difference. By substituting this expression for W in the pre-
ceding equation, the latter takes the form

We will next differentiate this equation according to the in-
termediate temperature t at which freezing took place. Since
now the magnitude U, in every case, depends solely upon the
initial and final conditions, and since, in the special case now
under consideration, the heat expended on exterior work is like-
wise independent of the intermediate temperature f, we may, in
differentiating, consider the whole of the right-hand side of the
equation as constant. The result, therefore, will be

or

w='^-*' ■ • «=)

This is, in reality, the last equation of the preceding Note ; the

DIFFERENT VARIATIONS OF THE FREEZING-POINT. 87

notation alone is slightly difierent^ inasmuch as the accents
which^ for the sake of better definition^ have been here intro-
duced were not there employed ; it being assumed that the dif-
ferent significations of the several quantities^ due to the peculiar
circumstances^ were self-evident, even in the absence of distin-
guishing marks.

In order to render perfectly manifest the essential points of
difference between the case just considered, where the lowering
of the freezing-point is occasioned solely by protecting the water
from agitation, and the case where the freezing-point is lowered
by increasing the pressure, I wiQ here also re-establish the equa-
tion (2) of the preceding Note, and in so doing retain the same
method of reasoning which has just led to the equation (c) .

Given, once more, a unit of weight of water at 0° to be con-
verted into ice of the temperature /), but in such a manner that
during the diminution of the temperature the pressure shall in-
crease according to the law expressed by the equation (1) of the
preceding Note. Since, under these circumstances, the diminu-
tion of the temperature t likewise represents the depression of
the freezing-point of the water, every temperature t between 0°
and /} may be assumed as that at which freezing takes place.
Conceive the water, therefore, to be cooled in the liquid condition
from QPtot, then to be frozen at this temperature /, and finally
to be cooled, in its solid state, from t to t^. The quantity of heat
which must be withdrawn from the mass during this process will,
on again calculating abstracted heat as a negative quantity im-
part^ to the mass, be represented by the algebraical sum

Equating this sum to the expression U + AW, which applies ge-
nerally to all changes, we have, corresponding to (a), the follow-
ing equation :

-r + Pcdf^+r^Arf^ssU + AW. . . . (d)

The exterior work W must here be determined anew, and its
determination under the present assumed circumstances will not
be so simple as in the previously considered case, since the
pressure, instead of being constant, is now dependent upon the
temperature. During the cooling of the water from (f to t the

-'*^i

cdt+X^hdt.

I

88 APPENDIX TO PRECEDING NOTE.

volume changes from o-q to <r under variable pressure ; during

the process of gelation the volume changes from o- to * under 1

constant pressure ; and as the ice finally cools from / to /, the |

volume changes from « to «| again under variable pressure. The

total work therefore is I

or, otherwise expressed, !

where p is the function of the temperature which defines the !

pressure. j

On substituting this expression for W in equation (d), we have

-r+f 'crfZ+r^ Arf/=U-hA[/>(«-(r) + f ';>^rf/+ C'p^ *]. (e)

We will now diflferentiate this equation according to ^, as we
formerly did the equation (b), and remember, in doing so, that
the quantity U is independent of the intermediate temperature L
We thus obtain the equation

= A(.-.) |,
whence we deduce

|=c-A-A(,-<r)| (f)

do T

Replacing therein the expression A(«— o-) ^ by ——37^ ^ ac-
cordance with the equation (1) of the preceding Note, we ob-
tain the equation there marked (2), namely,

%-'-'>*iri- *)

On comparing the formation of the equation (f) or (g) with
that of the equation (c), it will be seen that the principal diflFer-
ence between the two cases corresponding thereto arises from
the circumstance that in the latter the exterior work is inde-
pendent of the intermediate temperature at which freezing oc-
curs, whereas in the former it is dependent thereon. In the

DIFFERENT VARIATIONS OF THE FREEZING-POINT. 89

equations (f) and (g), therefore, there occurs a term expressive
of the variability of the heat expended on exterior work, whilst
in the equation (c) this term is absent. Moreover a small dif-
ference also arises from the fact that the quantities d and A' have
not exactly the same values as c and h. In another place I shall
have occasion to return to the consideration of this difference,
and an opportunity wiU then present itself of determining its
numerical value.

In the last paragraph of the preceding Note allusion was made
to Person^s remark, that ice near the temperature 0° is softer
than at lower temperatures, and that this circumstance must exert
an appreciable influence on the latent heat. If the cohesion of
the ice change, of course the interior work inseparable from the
act of fusion or solidification, and with it the heat corresponding
to this work, will likewise change. At the same time, however, it
must be remembered that a certain amount of interior work is
necessary in order to diminish the cohesion of the ice, and that the
heat expended in this work must necessarily be contained in the
specific heat of the ice. We must conclude, therefore, that when

the differential coefficient — f or — 1 considerably decreases in

the vicinity of 0°, that the quantity h (or A') which occurs, with
a negative sign, on the right side of the foregoing equations in-
creases just as considerably. The truth of the equations them-
selves cannot at aU be impaired by this internal deportment, for
these equations were established on perfectly general principles,
without predicating anything whatever relative to the iatemal
deportment of ice and water during changes of temperature.

It is scarcely necessary to mention, in conclusion, that the
preceding developments, which have been applied to water
merely for the sake of an example, are equally applicable to
every other liquid. With respect to the circumstances where
the difference $—<t comes into consideration, a behaviour ana-
logous to that of water, or opposite thereto, will present itself,
according as the substance under examination occupies a greater
or a less volume in the solid than it does in the liquid state.

90 SECOND MEMOIR.

SECOND MEMOIR.

ON THE DEPORTMENT OF VAPOUR DURING ITS EXPANSION UNDER
DIFFERENT CIRCUMSTANCES'^.

Not long ago^ Mr. Bankinef aiid myself]: gave utterance
almost contemporaneoosly to the proposition^ — ^that when. the
saturated vapour of water^ contained in a vessel impervious to
heat^ is subjected to compression^ it does not remain saturated^
but can part with a certain quantity of heat without being pre-
cipitated; and conversely^ when^ under the same circumstances^
the vapour is suffered to expand^ to preserve it from precipita-
tion a certain amount of heat must be imparted from without.

In connexion with this proposition, Mr. W. Thomson, in a
letter to Mr. Joule, refers to the fact " that the hand may be
held with impunity in a current of steam issuing from the safety-
valve of a high-pressure boiler'' §. From this he concludes that
the stream of vapour carries no water along with it, and holds
that this conclusion must contradict the above proposition, if
the existence of a source of heat from which the vapour shall
receive a quantity sufficient to preserve it from precipitation
cannot be established. This source he finds in the friction which
takes place during the issue of the steam from the orifice.

Although Mr. Thomson himself observes, in the course of his
letter, that, according to the mechanical theory of heat, different
states of the vapour are induced by different methods of expan-
sion, still in making the remark cited above he does not appear
to have taken this circumstance into account. He, in fact, ap-
plies the proposition to a case, to which, according to its deve-

* Published in Poggendorft**8 Annalen, Feb. 1851, voL Ixxxii. p. 263, and
translated in the Philosophical Magazine, May 1851^ S. 4. voL i. p. 398.

t Transactions of the Royal Society of Edinburgh, vol. zx. part 1. p. 147 ;
and Pogg. Ann, vol. Ixxxi. p. 172 (abstract).

t Pogg. Ann, vol. Ixxix. pp. 368 and 500 ; Monatsberichte der K, Pteuss,
Acad, der Wise, Feb. 1850 (abstract) ; and Phil. Mag. S. 4. vol. ii. pp. 1 and
102. [FntST MBMom of this collection.]

§ Phil. Mag. vol. xxxvii. p. 387 ; and Pogg. Ann, vol. Ixxxi. p. 477.

DEPORTMENT OF VAPOUR DURING EXPANSION. 91

lopment, it is altogether inapplicable. For vapour escaping
from a boiler into the air the theory would give a totally diflPerent
result, which latter may be likewise easily deduced.

From the innumerable modifications to which the expansion
of the steam may be subjected, I will choose three which may
be considered the most important, and in which the essential
differences exhibit themselves with peculiar clearness.

We will consider the matter as subjeqted successively to the
two foUowiQg conditions : — ^first, that the vapour during its ex-
pansion has to overcome a resistance which corresponds to its
entire expansive power ; and secondly, that it escapes into the
open air, in which case the pressure of the atmosphere alone is
opposed to it. We will further consider the two cases embraced
by the last condition ; namely, that in which the vapour is sepa-
rated from water and left to itself to expand, and that in which
the vessel which contains the vapour contains water also, which
by its evaporation always replaces the quantity of vapour which
escapes.

First, then, suppose a unit of weight of vapour at its maxi-
mum density to be contained in a vessel separated from water*,
and let the vapour expand itself by pushing back a piston, for
instance. Let us suppose that the vapour in each stage of its
expansion exerts against the piston the entire expansive force due
to that stage. To effect this, it is only necessary that the piston
should recede so slowly, that the vapour which foUows it can
always adjust its expansive force to that of the vapour in the
remaining portion of the vessel. During the expansion so much
heat is to be communicated to the vapour, or abstracted from it,
as is necessary to its preservation in the saturated gaseous state.
The question is, what quantity of heat is here necessary ?

To this case the proposition expressed by Mr. Bankine and
myself applies. The work performed by the vapour in this in-
stance, and the quantity of heat consumed in its production, are
so considerable, that, were this heat supplied from the vapour
itself, the latter would be cooled to an extent that would render
the retention of the gaseous condition impossible. It will there-
fore be necessary to communicate heat to it from without.

* For the sake of brevity I will always speak of water, although the same
reasoning holds, substantially, for all other liq.uids.

92 SECOND MEMOIR.

The quantity of heat to be communicated^ which corresponds
to an alteration of temperature dt, I have expressed in my former
memoir by hdt, if here A is a negative quantity ; so that the pro-
duct, Mt for increasing temperatures is negative, and for de-
creasing temperatures is positive. The value of A in the case of
water I have expressed as a function of the temperature t in
equation (33)*, thus :

r r. nr.^ 6065 - 0*695/ -000002f«-00000003/»^
A=0-305 ^ ^^^^ 1. .

If, therefore, the quantity of heat necessary to be communicated
to the unit-weight of vapour, when its temperature changes from
/, to t^ be called Q„ we have

lj = pArf/,

(1)

and from this we can readily calculate the value of Q, for each
particular case. For example, let the tension of the vapour at
the beginning be five or ten atmospheres, and let the expansion be
carried on until the tension sinks to one atmosphere. According
to Regnault^s determination, we must put /|= 152*^*2 or = 180°-3,
and t^=^ lOQP ; on doing so we obtain the values

Qi=521 or =74-9 units of heat. ... (I)
Secondly, let us again assume that a imit-weight of saturated
vapour at the temperature /„ above 100°, is enclosed in a vessel
separated from water, and that an orifice is made in the vessel
through which the vapour can issue into the atmosphere. Let
us follow it at the other side of the orifice until a distance is
attained where its expansive force is exactly equal to the atmo-
spheric pressure, the vapour being supposed to remain immixed
with air, and inquire how much heat must be imparted to the
entire mass of vapour during its passage, so that it may remain
throughout gaseous and saturated.

* [See p. 66.]

t [In a note appended to this equation I remarked that by means of a sim-
plified empirical formula proposed by me for the latent heat r, and which very
well represents the results of Kegnault's observations, the equation for h as-
sumes the form

and thereby becomes more convenient for calculation. — 1864.]

I DEPORTMENT OF VAPOUR DURING EXPANSION. V3

The interior work which the vapour has to execute during this
expansion is exactly the same as in the first case ; for here the
state of the vapour at the commencement and at the end is the
same as there. The exterior work, on the contrary, is much less ;
for while, in the first case, the resistance at the commencement
was equal to the tension which corresponds to the temperature
/j, and decreased slowly to one atmosphere, in the present in-

j stance the resistance is only one atmosphere from beginning to

end. The amount of heat converted into work is therefore in
the present case less, and hence a much smaller quantity is

^ required from without to preserve the vapour gaseous.

That this difierence in regard to the quantity of heat con-
sumed actually occurs, is already established with complete di-

I stinctness by the experiments of Joule with atmospheric air*.

He found that by pumping air into a rigid vessel, the mode of
compression here being analogous to the first of the above two

' cases, much more heat was developed than disappeared when the

L compressed air was permitted to stream into a space where the

pressure of one atmosphere was exerted, the process here being
analogous to our second case. These two quantities were nearly
in the ratio of the quantities of work calculated according to the
foregoing principles.

In order to carry out the calculation in our case, we must, in
reality, besides the resistance of the atmosphere, take two other
quantities into account ; namely, the resistance due to the fric-
tion of the vapour as it issues, and the work which must be ex-
pended to communicate to the vapour the motion which it still
possesses at the point where its tension is equal to the pressure
of the atmosphere. To overcome the friction, a certain quantity
of heat must be consumed ; by the friction, however, heat will
be again developed ; and although a portion of this is conducted
away by the surrounding mass, still the remaining portion com-
municates itself to the vapour. It is here, however, evident that
the eflfect of friction does not, as Mr. Thomson supposes, exhibit
itself in a gain of heat, but, on the contrary, in a loss of heat ;
the latter, however, not corresponding to the entire quantity of
work expended in overcoming the friction, but only to a portion

* "On the Chaliges of Temperature produced by the Rarefaction and
Condensation of Air, by J. P. Joule," Phil. Mag. S. 3. vol. xxvi. p. 369.

94 SKCOND HEMOIB.

I

thereof. We will neglect tliis, and also the loss arising from the
second circumstance alluded to^ which is undoubtedly inconsi-
derable*, — ^in this way the calculation is rendered very simple.
It is here necessary to subtract from the amount of heat

I ' hdt foimd in the former case, the heat which corresponds to

the difference of the quantities of exterior work produced in both
cases. Let p be the tension of the vapour for the temperature t^ ^

and s the volume of the unit of weight belonging to this tempe- i

rature. Further, letp^ and j^^ ^ the values otp, and 8^ and s^
the values of s at the commencement and at the end of the ope^ I

ration, ^2 ^^^g^ according to our assumption, the pressure of |

one atmosphere ; the exterior work then is —

in the first case = 1 'pch,

in the second case = I ^p^.

The corresponding amounts of heat are obtained by multi-
plying these quantities by the heat-equivalent of the unit of
work, which equivalent I have formerly denoted by A. K Q,

* [The velocity of the current of vapour, and the vis viva corresponding
thereto are different at different distances from the orifice. In the orifice
itself the velocity is considerable ; it is due of course to the difference between
the pressure in the vessel, and that in the orifice. Beyond the orifice, in the
space where the stream of vapour spreads out, the velocity diminishes again
quickly. The cause of this diminution of velocity will be discussed in the
appendix to this memoir. Without any such special examination^ however,
we may safely conclude that with the decrease of vis viva in the current is
associated an increase in the vis viva of the molecules of the vapour ; in other
words, that the destruction of vis viva in the current is accompanied by the
generation of heat. Now when, as in the present case, our object is, not to
follow individually the various phenomena which present themselves during
the several phases of the vapour's efflux, but merely to determine the total
quantity of heat which must be imparted to the vapour in order that it may,
without partial condensation, remain precisely at its maximum density, we
may from the commencement leave out of consideration both the heat ex-
pended in the production of motion, and that which is generated by the de-
crease of the motion ; for the two having opposite signs will in our calcula-
tions cancel each other. If, moreover, we assume that at the place where
we finally examined the vapour the velocity of the current is so small as to
justify our neglecting the vis viva corresponding thereto, then we need not in
our calculations pay any attention whatever to the velocity of the current. —
1864.]

DEPORTMENT OF VAPOUR DURING EXPANSION. 95

express the quantity of heat sought^ or that which is required
by the unit of vapour as it issues, we must put

Q,=PM^-Af%(&+Ar%2^ (2)

J'l J*i Jh

It is, however, evident that
and

=i?«(*2-<r)-i?i(*i-cr)-i " {8'-<r)-£dt,

where a- is an arbitrary constant*, for which we will substitute
the volume of a unit of weight of water, since the alteration of the
latter with the change of temperature may be so much the more
neglected, inasmuch as the entire volume of the water is scarcely
deserving of notice. This expression introduced into (2) gives

Q,=£[A+A(*-<r)|]rf/+A^,(,,-.r)(l-|i) . . (3)

The sum A+ A(*— o-) -^ is, according to equation (III) of my

df*
former memoir f, =^+^^ ^^'^ ^^^ ®^^^°^ again, according to

the determinations of Regnault, is nearly a constant quantity,
viz. 0*305. Equation (3) thus passes into

Q^^^O'SO&{t,^t^+Ap,{s,^a)(l^^). . . (4)

v Pi/

The only unknown quantity here is A^i(*i — «*), and this can be
expressed as a function of the initial temperature by means of

* [In fact, if in place of ds we put the equivalent difierential rf(«— o-), a-
being regarded as any constant quantity whatever, we shall have, as is well
known, the equation

j]pi?(«-. <r) =sX«- 0-) -]<«•- (r)(^,

and if we here conceive the integration to be effected between the determinate
limits which correspond to the assumed extreme temperatures t^ and t^f we
shall obtain the equation given in the text. — 1864.]
t [See p. 34.]

96 SECOND MEMOIR.

equation (26) of my former memoir* ; so that for every initial
pressure and the corresponding initial temperature the value of |

a^ may be calculated. Supposing^ for example^ the pressure at |

the commencement^ as in the former case^ to be five or ten at-
mospheres^ we obtain j
Q,= 19-5 or =170 units of heat. . . . (II) |
As Q, is a positive quantity^ it follows that in this case also heat ,
must not be withdrawn, but on the contrary communicated, to i
preserve the vapour from partial precipitation ; which, however, I
would take place not only at the orifice, but also within the j
vessel. The quantity of vapour thus precipitated would be
smaller than in the former case, inasmuch as Q^ is less than Qj.

It may appear singular that the equation (II) gives for an
initial pressure of five atmospheres a greater quantity of heat
than for ten atmospheres. This is explained by the &ct, that
under a pressure of five atmospheres the volume of the vapour
is already so small, and imder ten atmospheres is reduced to so
small an amoimt, that the increase of work thus rendered ne-
cessary during the issue of the vapour is more than compensated
by the excess of the sensible heat in the one state over that in the
other, the vapour being heated in one case to 180°'3, and in the
other case to 152°-2.

The second case which we have just considered, can be applied
with some degree of approximation to the case of vapour issuing,
without eoepansion from the cylinder of a high-pressure engine
after the completion of work; provided we assume that the vapour,
as long as it remains in connexion with the boiler, is completely
gaseous and at the same time completely saturated. In engines
where the expansive principle is applied, the first case becomes
applicable from the moment when the steam is shut oflf and the
piston is driven by expansion alone. Strictly speaking, the case
applies to those engines only in which the expansion continues

• [See p. 69. In place of the equation (26), which constitutes an empi-
rical formula adapted for numerical calculation, the principal equation (Va)
p. 51, may, of course, be employed. The latter gives at once the equation

A(»-<r)= L-5-,

(«+0|

in which the quantities on the right are all knov/nfrom observation. — 1864.]

DEPORTMENT OF TAFOITR DURING EXPANSION.

97

until the pre8sm*e within is equal to that of the atmosphere ; and
even here the correspondence would not be perfectly exacts inas-
much as the heat developed by the friction of the piston must
certainly be considerable*.

We will finally apply ourselves to the
consideration of the third case^ that is to
say, to the case to which the remark of
Mr. Thomson refers. Let the vessel
ABCD (see the accompanying figure)
be supposed to be filled with water to
EP,.and from here upwards to be fiUed
with vapour. Let PQ be the orifice, con-
nected with which is a neck PQKM, f]|^
which widens slowly, and renders the
expansion of the vapour more regular.
This is not essential, but is merely as- ^^
sumed to render the conception of the matter easier. By the
application of a proper source of heat, let the water be preserved
at the constant temperature /, so that the vapour which escapes
shall be continually replaced by newly developed vapour, the
state of things as regards the issue of the vapour being thus pre-
served stationary.

Let G H J represent a surface in which the vapour which passes
has, everywhere, the expansive force j»„ the temperature t^y and
the volume S„ which exist within the vessel, and with which the
new vapour is developedf. Let KLM, on the other hand, re-
present a surface in which the vapour which passes has, every-
where, the expansive force jOj, equal to one atmosphere, the vapour

* In connexion with the proposition which applies to the first case, I cited
in my former memoir the experiment made hy Pambour with the steam pro-
ceeding from a high-pressure engine after the completion of work. I deemed
it sufficient to notice the fact, that Pambour did not find a higher tempera-
ture than that which corresponded to the pressure observed at the same time,
although according to the common theory he must have done so. To require
from such observations that they shall exhibit the exact quantity of water
mixed with the vapour which the theory gives, would, for the reasons given
above, and on account of many other simultaneous causes of disturbance, be
unjustifiable.

t [In the figure given in the first editions of this memoir the surface GHJ
is drawn too near the orifice. Notice was given of this in the corrigenda of
the volume of PoggendorfTs Annaien, wherein the memoir appeared. — 1864]

98 SECOND KEMOIR.

being supposed to be unmixed urith air. During the passage of
the vapour from GH J to KLM let heat be continually with-
drawn or communicated, so that the vapour may remain com-
pletely gaseous and quite saturated, and hence at the surface
ELM have the temperature t^=100° exactly, and the corre-
sponding volume s^. The question is, what quantity of heat
Q9 must be imparted to, or withdrawn from, the issuing vapour
so that this condition shall be fulfilled.

The interior work performed by the vapour during its issue in
the present instance is exactly the same as in the other cases.
With regard to the exterior work, however, an entirely new cir-
cumstance enters into the consideration, which renders this
case essentially different from the former ones.

We must here, in fact, consider the quantity of work pro-
duced at both the surfaces G H J and ELM. Through the surface
GH J the vapour is driven with the volume *i and the pressure
j»i, it therefore produces the work

Pi • ^1-
This work proceeds from the vapour within the vessel, and more«
over only from that portion of it which, during the time of issue,
is developed anew. To obtain room for itself, this presses the
neighbouring stratum forwards, this the next, and so on. The
intervening layers thus serve merely to transmit the force from
the surface of the water to the orifice. The quantity of heat
consumed in the production of this work is contained in the
latent heat of the developed steam, and need not in the present
consideration be further taken into account.

If now in the surface ELM exactly the same work be pro-
duced as in GH J, then in the interval between both surfaces no
proper work is produced, inasmuch as in this case there would
be merely a transmission of work from one surface to the other.
If, on the contrary, the work accomplished at the surface ELM
be different from that produced at GH J, the difference must be
referred to the said interval. But through ELM the unit of
weight of steam with the volume 8^ and the pressure /?2 is driven,
and hence produces the work

Pi'Sq.

The work performed in the intervening space is then

Pi ' *2 Pi ' ^v

DEPORTMJBNT OF VAPOUR DURING EXPANSION. 99

.which is a negative quantity. This shows, that, during the
passage from surface to surface, a portion of the exterior work
already completed is actually lost again*.

The quantity p^'S^—p^Si must be treated as the quantity

I ^p^ds in the second case ; in this way we obtain the following

equation, which corresponds to equation (2) :

08= j 'M^— Aj ''pd8'hA.{p^.s^—pi.8{).

(5)

Subjecting this equation to the same process as that applied in
the deduction of equation (4) from equation (2), and neglecting
the terms which contain the factor <r, we obtaia

0,= -.0-305 (/i-^2)t (6)

* [The deportment of the vapour in the space between the two surfaces
GH J and KL M is by no means simple, inasmuch as the velocity of the stream
from the first surface to the orifice PQ is greatly accelerated, whilst that from
the orifice to the second surface is, approximately, quite as strongly retarded.
As abeady remarked in a previous note, however, it is not necessary, in de-
termining the total quantity of heat which must be communicated to the va-
pour, to take into consideration the peculiarities of this deportment ; for it
may be predicated with certainty that heat will be expended during the in-
crease of the vis viva of the stream, and generated during its decrease. It will
suffice, therefore, to know what takes place at the two limiting surfaces
chosen for consideration, and the vis viva in these surfeu^es being small enough
to be neglected, we need only take cognizance of the mechanical work which
is performed on the passage of the vapour by the pressure which there exists,
the latter being estimated in the direction of the stream. — 1864.]

t [As already remarked when transfonning the equation (2), the following
equation holds :

f*' ^=j,,(,, - t) -i,.(*. - a) -f !'(.- <r) I *,

and in virtue of it the equation (6) takes the form

On substituting herein the expression -£-{-0 for its equal A+A(«-(r) ^
and neglecting the term which contains the factor <r, we have

Q,

-ji^e^")-

This equation, when specially applied to water by putting, with Eegnault,
J+c=0-305,

leads to the equation (6), — 1864.]

H 2

100 SECOND MEMOIR^ APPENDIX.

Calculating from this the numerical value of Q3 for an initial
pressure of five or ten atmospheres^ we obtain

Qg= — 15*9 or = — 24*5 units of heat^ respectively . (Ill)

The value of Q3 being negative, it follows that in this case
heat is not to be communicated, but, on the contrary, must be
unthdratvn, the quantity being the same as that found by apply-
ing the common theory of heat. If this withdrawal up to the
place under consideration be not sufficiently effected, then the
vapour at this place will have a temperature which exceeds 100° ;
and hence, if water be not mechanically carried along with the
vapour, the latter must be completely dry.

It is thus shown that the friction is not necessary to the ex-
planation of the fact adduced by Mr. Thomson ; the effect of this
friction, as already mentioned, being exactly opposite to what he
supposes it to be. The loss of heat arising from this cause is
not reckoned above. In such cases as the issuing of steam from
the safety-valve of a high-pressure engine, this loss is by no
means capable of effecting the consumption of the quantity of
heat found by equation (6).

APPENDIX TO SECOND MEMOIR [1864].

ON THE VARIATIONS OF PRESSURE IN A SPREADING STREAM OF GAS.

In the preceding memoir it was stated that the velocity of the
stream diminishes considerably, from the orifice PQ to the sur-
face KLM, in the gradually widening neck of the vessel drawn
on p. 97. The force which causes this diminution must be
sought for in the difference between the pressure which prevails
near the orifice, and that which exists at the surface KLM.

Such a difference of pressure
invariably occurswhena stream
of gas spreads out; its existence
may be detected readily by
means of a well-known little
instrument. Let AB in the ad-
joining figure be a harrow tube
fitted, by means of a cork, into
a wider tube CDEF ; so that

VARIATIONS OF PIUfiSSURE IN A STREAM OF GAS. 101

a stream of air, driven through the narrow tube from A to B,
can spread itself out in the wider tube, before it reaches the free
atmosphere. Just below the mouth of the small tube a siphon-
shaped tube GHK is fitted to the wide tube, and partially filled
with a liquid. On blowing through the narrow tube AB, the
liquid in the branch HG of the siphon-shaped tube is seen to
rise; thereby showing that during the blast the pressure in the
wide tube near the mouth of the smaU one is less than that of
the surroimding atmosphere. Hence the pressure at B in the
stream of air which flows in the wide tube from B to EP is
smaller than at the mouth of the latter, where the atmospheric
pressure exists ; it is this diflference of pressure which retards
the current and causes so great a diminution in its velocity that
the same quantity of air which in a given time passes through
the small section at B, is able throughout that time to fill the
wider section EF.

The origin of this diflference of pressure may be thus ex-
plaiued : — ^The stream of air in the neighbourhood of the orifice
B, where it has not yet spread itself so far as to occupy, in a uni-
form manner, the whole section of the wide tube, seeks to carry
with it the still air at its side, and in consequence of this effort
a portion of the circumjacent air is removed at the commence-
ment of the current, and a rarefaction thereby ensues which
continues as long as the current lasts.

A deportment precisely similar to the one observed in the small
apparatus just described, must also present itself in the gradually
widening neck PQKM of the vessel previously alluded to. Here
also, in the neighbourhood of the narrow orifice P Q, the pressure
must be less than at the broader part of the neck, and the stream
of vapour must be retarded in its passage from the narrow, to
the broader parts. When the gradually widening neck is with-
drawn, which separates from the exterior air the stream of va-
pour during its expansion and consequent retardation, that is to
say, when the vapour passes directly, with its full velocity of
efflux, into the atmosphere, a slight diflference arises from the cir-
cumstance that the stream of vapour continually sets in motion
a certain quantity of the circumjacent air, the air carried with it
being continually replaced by new air streaming in from the
surrounding space. On the whole, however^ the phenomena

102 SECOND MEMOIR^ APPENDIX.

under these circumstances must be similar to those previously
considered.

In the letter written by Mr. W. Thomson to Mr. Joule^ and *\

cited in the preceding memoir^ Mr. Thomson commences with
comparing the case where steam issues from the safety-yalye of !

a boiler^ with that where vapour^ contained in a vessel without
liquid^ expands by overcoming a resistance corresponding to its
entire expansive force. To prevent partial condensation in the \

latter case^ a certain quantity of heat must be imparted to the \

vapour, and Mr. Thomson holds the opinion that it is '^ by the
friction of the steam as it rushes through the orifice '^ that this 1

quantity of heat, necessary to prevent partial condensation during 1

the efflux of the steam, is produced. I understood by the ex-
pression above quoted, the friction of the vapour in the orifice <
itself, that is to say against the fixed walls thereof as the vapour
rushes past them, and it is to Mr. Thomson^s views thus inter-
preted that reference is made in the preceding memoir. On
comparison with the words used, my interpretation of the ex-
pression will be found to be a very natural one, and Mr. Thom-
son himself, in his reply*, nowhere states that I have misin-
terpreted that expression. In this reply, however, he has chosen
another form of expression, when alluding to his former expla-
nation, and that without either stating his reasons for so doing,
or even drawing attention to the difference. He there says,
in fact, that in his former explanation he stated that the quan-
tity of heat in question was generated ^^ by the fluid friction in
the neighbourhood of the aperture." If from the beginning Mr.
Thomson had used the latter form of expression, which I cannot
consider as identical with the previous one, the difference of
opinion between us would have been to some extent, if not
wholly, avoided. It has, in fact, been already stated that the
difference of pressure, which retards the stream of vapour or
other expanding gas (with which retardation a development of
heat is associated), arises from the circumstance that the stream
of gas seeks to carry with it the surrounding gaseous particles,
and hence it is clear that the phenomenon must be referred, ul-
timately, to the action of the particles of gas upon one another ;
to this action, although it is not of a very simple kind, the term
-* PhU. Mag. S. 4. vol. i. p. 474.

VARIATIONS OF PRESSURE IN A STREAM OF GAS. 103

Jriction may be applied. It was not my intention to dispute, in
the least degree, the influence of this friction, occurring in the
stream of gas beyond the orifice ; I merely objected to the too
great part which, as it appeared to me, was ascribed to the fric-
tion against the walls of the orifice.

It is now, indeed, of little importance whether the diflFerence
of meaning which gave rise to the preceding memoir was an es-
sential one, or whether, and to what extent, it arose merely from
the use of an inappropriate form of expression. The memoir itself
will scarcely be aflfected thereby, and quite apart from the way in
which it originated, I trust that the developments it contains,
and particularly the precise distinction which* is therein drawn
between the three cases treated, and the consideration, in the
last case, of the work done in the surface GH J (not alluded to
by Mr. Thomson), will not be without scientific interest.

104 THIRD MEMOIR.

THIRD MEMOIR.

ON THE THEORETIC CONNEXION OF TWO EMPIRICAL LAWS RELA-
TING TO THE TENSION AND THE LATENT HEAT OF DIFFERENT

A SUPERFICIAL contemplation of the tension series, experimen-
tally developed for the vapours of diflferent liquids^ suffices to
show that a certain uniformity exists therein; and hence the
various effi)rts which have been made to ascertain a definite law
by means of which the series which holds good for one liquid,
water for instance, might be apjdied to other liquids.

A very simple law of this nature was expressed by Dalton.
Calling those temperatures which belong to equal tensions car- .
responding temperatures, the law ran thus :— In the case of any
two liquids tJie differences between the corresponding temperatures
are all equal.

This law agrees pretty well with experience in the case of
those liquids whose boiling-points are not far apart ; for those,
however, which possess very diflferent degrees of volatility, it is
inexact. This is shown by a comparison of the vapour of mer-
cury with that of water, according to the observations of Avo-
gradof. Still more decidedly does the divergence exhibit itself
in the investigations of Faraday J on the condensation of gases.

lu the " Additional Remarks '^ to his memoir, Mr. Faraday,
after having disproved the applicability of the law of Dalton to
gases, expresses himself as follows : — " As far as observations
upon the following substances, namely, water, sulphurous acid,

* PubUshed in PoggendorflTs Annalen, February 1851, vol. bczzii. p. 274 ;
and translated in the Pliilosopliical Magazine, December 1851, S. 4. vol. ii.
p. 483.

t Abstracts in Aum, de Chim, et de Phys, xlix. p. 869 j andPogg. Ann, vol.
xxvii. p. 60. Complete in M&m, de VAcad, de Turin, voL xxxvi.

t Pbil. Trans, of the Eoy. Soc. of London for 1845, p. 155 5 and Pogg. Ann,
vol. Ixxii a. p. 193.

CONNEXION OF TWO EMPIRICAL LAWS.

105

cyanogen^ ammonia, arseniuretted hydrogen, sulphuretted hy*
drogen, muriatic acid, carbonic acid, olefiant gas, &c., justify any
conclusion respecting a general law, it would appear that the
more volatile a body is, the more rapidly does the force of its va-
pour increase by further addition of heat, commencing at a given
point of pressure for all ;" and further on, ^' there seems every
reason therefore to expect that the increasing elasticity is di^
rectly as the volatility of the substance, and that by further
and more correct observation of the forces a general law may
be deduced, by the aid of which and only a single observation
of the force of any vapour in contact with its liquid, its elasticity
at any other temperature may be obtained ''*.
What Faraday here expresses with evident reserve and caution.

♦ [By the more recent appearance of the second volume of Regnault's
Experimental Besearches, in which extensive tension series are given for a
considerable number of substances, an opportunity is afforded of testing more
accurately than it was possible to do according to previous data, the mutual
relations of the several tension series. In order to elucidate what has been
said in the text, I will here tabulate some of the numbers, given by Reg-
naulfs observations, which are most suitable for comparison. The first hori-
zontal line contains the boiling-points of the several substances, that is to say,
those corresponding temperatures to which belongs a vapour-tension of one
atmosphere. The second line likewise contains corresponding temperatures
solely, namely, those to which a vapour-tension of five atmospheres belongs.
The lowest line, finally, contams the differences between the numbers in ^e
two first lines.

Sulphur-
ous Add.

Ether.

Sulphide

AloohoL

Water-

Mercury.

Sulphur.

Boiling-points ..

o
— lO

o
35

46

78

100

357

448

Temperatures for
a tension of five
atmospheres . .

33

89

106

125

152

458

568

Differences

43

54

60

47

5a

Id

IXO

According to Dalton's law, the differences in the last horizontal line should
be equal to one another ; this, however, is manifestly not the case, the dif-
ferences being, in general, greater the higher the boiling-point of the liquid.
The latter progression, it is true, is not so regular that each difierence is
greater than the preceding one ; but on the whole, and particularly so far as
liquids are concerned whose boiling-points lie fat asunder, it is umnistakeable.
—1864.]

106 THIRD MEMOIR.

we find again in the form of an equation in a later memoir by '

M. Oroshans*. The equation (3) of the said memoir contains^ I

implicitly^ the following law : — If all temperatures be counted from I

—273° C. (that is^ from that temperature which is expressed \

by the inverse value of the coefficient of expansion for atmo-
spheric air)> then for any two liquids the corresponding tempera-
tures are proportional.

Although this carries with it a great degree of probability, at f

least as an approximate law, and is undoubtedly proved by the \

experimental researches of Avogrado and Faraday to be preferable \

to the law of Dalton, still the maimer in which M. Groshans
deduces his equations leaves much to be desired. He bases the i

deduction upon two equations which can only be regarded as '

approximately correct, inasmuch as they contain the expression i

of the law of Mariotte and Gay-Lussac for vapours at their
maximum density. For the further development, however, he
makes use of the following proposition : — If in the case of any
two vapours the temperatures are so chosen that the tensions
of both are equal, then, if the density of each vapour at the
temperature in question be measured by its density at the boiling-
point, these densities are equal. This proposition is introduced
by the author in the memoir alluded to without any proof what-
ever. In a later memoirf, however, he says that he was led to
the above conclusion by observing that in the case of seven dif-
ferent bodies composed of/^C + g'H+rO the density of the va-
pour at the boiling-point compared with the density of steam at
100° could be expressed by the formula

^~ 3 '
and immediately afterwards J he states, that ^^ there are several
bodies to which the formula

j.Ji±q±r

3

is inapplicable.^' From this it appears that the foundation on

which, the proposition rests cannot be regarded as estabUshed.

It seems to me, that although the law mentioned above has ob-

* Pogg. Ann, vol. Ixxviii. p. 112.

t Pogg. Ann, vol. Ixxix. p. 290. % Ibid, p. 292.

CONNBXION OP TWO EMPIRICAL LAWS. 107

tained from M. Groshans a more definite form than in Faraday's

expression^ its probable. vaKdity is in no way augmented thereby.

^ In this state of uncertainty every new. point of view from

^ which a more extended insight as to the deportment of liquids

during evaporation may be obtained is deserving of attention ;

and hence it will not perhaps be without interest, to establish

such a connexion between the above law as regards the tension

) and another law regarding the latent heat, — the latter being also

■ empirically established in a manner totally independent of the

^ former — ^that the one shall appear to be a necessary consequence

of the other.

I refer to the law, that the latent heat of a unit of volume of
vapour developed at the boiling-point is the same for all liquids.
f Although this has not been completely corroborated by the ex-

periments hitherto made, and even if it were perfectly true, could
not be so corroborated, our knowedge of the volumes of vapours
at their maximum density beiQg too scanty, still, an approxima-
tion is observed which it is impossible to regard as accidental.
We will therefore for the present assume the law to be correct,
and thus make use of it for further deductions.

In the first place, it is clear that if the law be true for the
boiling-points of aU liquids, it must also be true for every other
system of corresponding temperatures ; for the boUing-poiats de-
pend merely upon the accidental pressure of the atmosphere, and
hence the law can be immediately extended thus : the latent heat
calculated for the unit of volume of vapour is, for all liquids^ the
same function of the tension. Let r be the latent heat of a unit of
weight of vapour at the temperature t, the volume of the unit of
weight for the same temperature being «, the latent heat of a unit

r
of volume will then be expressed by the fraction - ; let ^ be the

s

corresponding tension ; the law will then be expressed by the

equation

-=f{p), (I)

in which / is the symbol of a function which is the same for all

liquids.

r
Let this function be substituted for - in the equation (Va) of

8

108 THIRQ MEMOIR.

my memoir '^ On the Moying Force of Heaf *, by neglecting I

therein the volume o* of a unit of weight of liquid as compared I

with that of vapour, .we thus obtain I

/(;,)=A(a+0f'

where A and a are two constants^ the latter denoting the number
278, so that a + ^ is the temperature of the vapour reckoned from
—273°. If, for the sake of brevity, we call this quantity T,
we have ^

rfT_A^ i

and from this we obtain by integration i

in which P is the symbol of another function, which is likewise
the same for all liquids, and c is an arbitrary constant which must '

be determined for each liquidt. Let us suppose this equation
solved for J?, it will assume the form

1^=<A(^T), (II)

♦ [See p. 61. The equation (Va) here cited is I

r=A(a+0(*-^)J, j

whence |

in which, with great approximation, we may put - for — ^. — 1864.]

s 8 — (r

t [By integration, in fact, we have in the first place

where A; is an arbitrary constant, and j^j represents, for all liquids, one and the j

same arbitrary initial tension; for instance, a tension of one atmosphere. i

From this equation it follows that

/* pA . dp

or

/*p A . dp

Introducing here the abbreviations

p p A dp

-k
esse , I

the equation in the text is at once obtained. — 1864.]

CONNEXION OF TWO EMPIRICAL LAWS. 109

where af) is the symbol of a third ftinction, which is the same for
all liquids.

This equation is evidently the mathematical expression of the
law of tension mentioned above ; for to apply the fiinction which
in the case of any one liquid determines the tension from the tem-
perature, to any other liquid, it is only necessary to multiply the
temperature by a different constant, which constant is easily
found when the tension for a single temperature is known.

It is thus shown, that, in so far as the validity of equation
(Va) is granted, the two laws expressed by the equations (I) and
(II) are so connected with each oth^ that when one of them is
true, the other must necessarily be true also.

But in case both laws are only approximations to the truth,
as to me appears most probable, the equation (Vfl), which by
introducing T instead of t becomes

enables us at least to conclude, from the manner and degree of
divergence between two vapours with regard to their latent heat,
what divergence there is between their tension series, and vice
versd*. Thus, for instance, in comparing water with other li-
quids, it is observed that, relatively to its boiling-point, the ten-
sion of the vapour of the former increases more quickly with the
temperature than the tension of other vapours. There is a com-
plete coincidence between this fact and that observed by An-
drews t, that the vapour of water possesses a greater latent heat
than an equal volume of the vapour of any other liquid which

* [For if the fraction -^ relative to the boiling-point, and almost iden-
tical with ^ has a greater value for a certain liquid than it has for others, we
must conclude that, for the former, the product T ^ is also greater, and hence

that the vapour-tension near the value of one atmosphere increases more
quickly with increasing temperature than one would anticipate from the
height of the liquid's boiling-point. In a similar manner, from the circum-
stance that an exceptionally quick increase of the yapour-tension of a liquid
takes place, we should conclude that the latent heat, calculated according to
the volume of the vapour, has an unusually large value. — 1864.] ^
t Quarterly Journal of the Chem. Soc. of London, No. 1. p. 27.

110 THIED MBMOIK.

Andrews examined^ alcohol excepted. From this we perceire
that it is by no means advantageous for the application of the
above two laws to choose^ as is generally done, water as the liquid j

of comparison; but that, on the contrary, the comparison of \

water with liquids of lower boiling-points is peculiarly calculated j

to support the law of Dalton''^.

* [Since, in the case of water, the fraction - is greater than for most

other liquids, the tension of aqueous vapour must, according to the foregoing a

note, increase more quickly in the neighhouriiood of the boiling-point, and, .

as a consequence of this, the difference between two corresponding tempera- '

tures must be smaller than the height of the boiling-point would lead us to |

expect ; so that in this respect water approaches the liquids having lower |

boUing-points. The same remark must apply still more forcibly to alcohol, j

for which liquid the fraction - is still greater. This is verified, in &ct, by |

the small Table given in a note on p. 105 ; for on comparing the differences
between the systems of corresponding temperatures tiiere selected for con-
sideration, it will be found that these differences are smaller for water and
alcohol than for sulphide of carbon and ether, although the latter substances
have lower boiling-points than the former. — 1864.]

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. Ill

FOURTH MEMOIR.

ON A MODIFIED FORM OF THE SECOND FUNDAMENTAL THEOREM IN
THE MECHANICAL THEORY OF HEAT*.

In my memoir "On the Moving Force of Heat, fec.^'t^ I have
shown that the theorem of the equivalence of heat and work,
and Camot^s theorem, are not mutually exclusive, but that, by
a small modification of the latter, which does not affect its prin-
cipal part, they can be brought into accordance. With the
exception of this indispensable change, I allowed the theorem of
Carnot to retain its original form, my chief object then being,
by the application of the two theorems to special cases, to arrive
at conclusions which, according as they involved known or un-
known properties of bodies, might suitably serve as proofs of the
truth of the theorems, or as examples of their fecundity.

This form, however, although it may suffice for the deduction
of the equations which depend upon the theorem, is incomplete,
because we cannot recognize therein, with sufficient clearness,
the real nature of the theorem, and its connexion with the first
fundamental theorem. The modified form in the following pages
will, I think, better ftdfil this demand, and in its applications
will be found very convenient.

Before proceeding to the examination of the second theorem,
I may be allowed a few remarks on the first theorem, so far as this
is necessary for the supervision of the whole. It is true that I
might assume this as known from my former memoirs or from
those of other authors, but to refer back would be inconvenient ;
and besides this, the exposition I shall here give is preferable
to my former one, because it is at once more general and more
concise.

* Published in Poggendoff 's Annalenf December 1854, vol. xciii. p. 481 ;
translated in the Journal de MathSmatiqueSy vol. xx. Paiis, 1865, and in the
Philosophical Magazine, August 1856, S. 4. vol. xii. p. 81.

t [First Memoir of this Collection.]

.Hf

112 FOUETH MBMOim.

Theorem of the eguivalenee of Heat and Work.

Whenerer a moving force generated by heat acts against an-
other force, and motion in the one direction or the other ensues,
positive work is performed by the one force at the same time that
negative work is done by the other. As this work has only to
be considered as a simple quantity in calculation, it is perfectly
arbitrary, in determining its sign, which of the two forces is
chosen as the indicator. Accordingly in researches which have
a special reference to the moving force of heat, it is customary
to determine the sign by counting as positive the work done
by heat in overcoming any other force, and as negative the work
done by such' other force. In this manner the theorem of the
equivalence of heat and work, which forms only a particular case
of the general relation between vis viva and mechanical work,
can be briefly enunciated thus : —

Mechanical work may be transformed into heat, and conversely
heat into work, the magnitude of the ofie being always proportional
to that of the other.

The forces which here enter into consideration maybe divided
into two classes : those which the atoms of a body exert upon each
other, and which depend, of course, upon the nature of the body,
and those which arise from the foreign influences to which the
body may be exposed. According to these two classes of forces
which have to be overcome (of which the latter are subjected
to essentially diflFerent laws), I have divided the work done by
heat into interior and exterior work.

With respect to the interior work, it is easy to see that when
a body, departing from its initial condition, suffers a series of
modifications and ultimately returns to its original state, the
quantities of interior work thereby produced must exactly can-
cel one another. For if any positive or negative quantity of
interior work had remained, it must have produced an opposite
exterior quantity of work or a change in the existing quantity
of heat ; and as the same process could be repeated any number
of times, it would be possible, according to the sign, either to
produce work or heat continually from nothing, or else to lose
work or heat continually, without obtaining any equivalent ; both
of which cases are universally allowed to be impossible. But if
at every return of the body to its initial condition the quantity

i

\

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 113

of interior work is zero, it follows, further, that the interior work
corresponding to any given change in the condition of the body
is completely determined by the initial and final conditions of
the latter, and is independent of the path pursued in passing
from one condition to the other. Conceive a body to pass succes-
sively in different ways from the first to the second condition, but
always to return in the same maimer to its initial state. It is
evident that the quantities of interior work produced along the
different paths must all cancel the common quantity produced

f during the return, and consequently must be equal to each other.

It is otherwise with the exterior work. With the same initial
and final conditions, this can vary just as much as the exterior
influences to which the body may be exposed can differ.

I Let us now consider at once the interior and exterior work

produced during any given change of condition. If opposite in
sign they may partially cancel each other, and what remains
must then be proportional to the simultaneous change which
has occurred in the quantity of existing heat. In calculation,
however, it amounts to the same thing if we assume an alteration
in the quantity of heat equivalent to each of the two kinds of
work. Let Q, therefore, be the quantity of heat which must be
imparted to a body during its passage, in a given manner, &om

' one condition to another, any heat withdrawn from the body
being counted as an imparted negative quantity of heat. Then
Q may be divided into three parts, of which the first is employed
in increasing the heat actually existing in the body, the second
in producing the interior, and the third in producing the ex-
terior work. What was before stated of the second part also
applies to the first — ^it is independent of the path pursued in the
passage of the body from one state to another : hence both parts
together may be represented by one function U, which we know
to be completely determined by the initial and final states of the
body. The third part, however, the equivalent of exterior work,
can, like this work itself, only be determined when the precise
manner in which the changes of condition took place is known.
If W be the quantity of exterior work, and A the equivalent'
of heat for the unit of work, the value of the third part wiU
be A . W, and the first fundamental theorem will be expressed
by the equation Q=U + A.W (I)

114 FOUBTH MEMOIR.

When the sereral changes are of such a natnre that through
them the body returns to its original condition, or when^ as
we shall in future express it^ these changes form a cyclical .

process^ we have ]

U=0,

and the foregoing equation becomes

Q=A.W (1) J

In order to give special forms to equation (I), in which it ^

shall express definite properties of bodies^ we must make special ^

assumptions with respect to the foreign influences to which the
body is exposed. For instance, we will assume that the only
active exterior force, or at least the only one requiring consi- '

deration in the determination of work, is an exterior pressure |

which (as is always the case with liquid and gaseous bodies, |

when other foreign forces are absent, and might at least be the
case with solid bodies) is everywhere normal to the surface, and
equally intense at every point thereof. It will be seen that under
this condition it is not necessary, in determining the exterior
work, to consider the variations in form experienced by the '

body, and its expansion or contraction in different directions, j

but only the total change in its volume. We will further assume ]

that the pressure always changes very gradually, so that at any '

moment it shall differ so little &om the opposite expansive force
of the body, that both may be counted as equal. Thus the
pressure constitutes a property of the body itself, which can be
determined from its other contemporaneous properties.

In general, under the above circumstances, we may consider the
pressure as well as the whole condition of the body, so far as it is ;

essential to us, as determined so soon as its temperature t and i

volume V are given. We shall make these two magnitudes, there- |

fore, our independent variables, and shall consider the pressure
p as well as the quantity U in the equation (I) as fimctions of
these. If, now, t and v receive the increments dt and rfv, the cor-
responding quantity of exterior work done can be easily ascer- ^
tained. If any increase of temperature is not accompanied by
a change of volume, no exterior work is produced ; on the other
hand, if, with respect to the differentials, we neglect terms higher
than the first in order, then the work done during an incre-

J>

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 115

ment of volume dv will be pdv. Hence the work done during
a simultaneous increase of t and v is

dW==pdv,

and when we apply this to the equation (I), we obtain

dQ=dV + A.pdv (2)

On account of the term A .pdv, this equation can only be inte-
^ grated when we have a relation given, by means of which t may

be expresse4 as a function of v, and therefore p S/aa, function of v
alone*. It is this relation which, as above required, defines the
manner in which the changes of condition take place.

The unknown function U may be eliminated from this equa-
tion. ' When written in the form

^dt+-dv=-dtf(-+A.p)dvf,

we easily see that it is divisible into the two equations

dQ_dV
dt dt*
and

dv dv
Let the first of these be differentiated according to v and the
second according to t. In doing so we may apply to U the well-
known theorem, that when a function of two independent varia-

* [In fact since dU is itself a complete differential, the magnitude A . pdv
must also be one, in order that the whole of the expression on the right
may be so; but this can only be the case when j9 is expressible as a function
ofv alone.— 1864.]

t [In this and the following memoirs the notation for partial differential
coefficients is somewhat different from that employed in the first memoir;
the brackets, which were there used for the sake of clearness, are here omitted,
since, as stated in the Introduction, no misunderstanding can be thereby
produced. The same simplified notation is also retained in the present
reprint, in order to preserve unchanged the form of the memoirs. It would
certainly have been more convenient to the reader had I, on collecting into
one volume the memoirs vrritten at various epochs, adopted one and the
same notation throughout; nevertheless every mathematician is so accus-
tomed to see first one and then the other notation employed, that the transi-
tion will probably be scarcely noticed ; at all events, it will not render the
memoirs themselves less intelligible, or seriously impair the facility with
which one may be compared with another. — 1864.]

i2

/V

116 FOURTH MEMOIR.

bles is successively differentiated according to both^ the order in
which this is done does not affect the result. This theorem,
however, does not apply to the magnitude Q, so that for it we
must use symbols which will show the order of differentiation.
This is done in the following equations : —

dfdQ\ <PU ^
dvKdtJ dtdv'

dtKdvJdtdv^ dt'
By subtraction, we have

^©-i©=^-i' • • • • <"

an equation which no longer contains U.

The equations (2) and (3) can be still further specialized by
applying them to particular classes of bodies. In my former
memoir I have shown these special applications in two of the
most important cases, viz. permanent gases and vapours at a
maximum density. On this account I will not here pursue the
subject further, but pass on to the consideration of the second
fundamental theorem in the mechanical theory of heat.

Theorem of the equivalence of transformations.

Camot's theorem, when brought into agreement with the first
fundamental theorem, expresses a relation between two kinds of
transformations, the transformation of heat into work, and the
passage of heat from a warmer to a colder body, which may be
j / regarded as the transformation of heat at a higher, into heat at a
lower temperature. The theorem, as hitherto used, may be
enunciated in some such manner as the following : — In all cases
where a quantity of heat is converted into work, and where the
body effecting this transformation ultimately returns toils original
condition f another quantity of heat must necessarily be transferred
from a warmer to a colder body ; and the magnitude of the last
quantity of heat, in relation to the first, depends only upon the tem~
peratures of the bodies between which heat passes, and not upon
the nature of the body effecting the transformation.

In deducing this theorem, however, a process is contem-

■-^

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 117

plated whicli is of too simple a eliaracter ; for only two bodies
losing or receiving heat are employed^ and it is tacitly assumed
that one of the two bodies between which the transmission of heat
takes place is the source of the heat which is converted into work.
Now by previously assuming^ in this manner^ a particular tem-
perature for the heat converted into work, the influence which
a change of this temperature has upon the relation between the
two quantities of heat remains concealed, and therefore the
theorem in the above form is incomplete.

It is true this influence may be determined without great
difficulty by combining the theorem in the above limited form
with the first fundamental theorem, and thus completing the
former by the introduction of the results thus arrived at. But
by this indirect method the whole subject would lose much of its
clearness and facility of supervision, and on this account it
appears to me preferable to deduce the general form of the
theorem immediately from the same principle which I have
already employed in my former memoir, in order to demonstrate
the modified theorem of Carnot.

This principle, upon which the whole of the following deve-
lopment rests, is as follows : — Heat can never pass from a colder
to a warmer body without some other change y connected therewith,
occurring at the same time*. Everything we know concerning

* [The principle may be more briefly expressed thus: Meat cannot by
itgdf pass from a colder to a warmer body ; the words '^ by itself/' (von selbst)
however, here require explanation. Their meaning will, it is true, be rendered
sufficiently clear by the expositions contained in the present memoir, never-
theless it appears desirable to add a few words liere in order to leave no
doubt as to the signification and comprehensiveness of the principle.

In the first place, the principle implies that in the inmiediate interchange
of heat between two bodies by conduction and radiation, the warmer body
never receives more heat from the colder one than it imparts to it. The
principle holds, however, not only for processes of this kind, but for aU
others by which a transmission of heat can be brought about between two
bodies of different temperatures, amongst which processes must be particu-
larly noticed those wherein the interchange of heat is produced by means of
one or more bodies which, on changing their condition, either receive heat
from a body, or impart heat to other bodies.

On considering the results of such processes more closely, we find that in
one and the same process heat may be carried from a colder to a warmer
body and another quantity of heat transferred from a warmer to a colder body
without any other permanent change occurring. In this case we have not a

\

118 FOURTH MEMOIR.

the intercliaiige of heat between two bodies of different tempera-
X \1 tures confirms this ; for heat everywhere manifests a tendency to
^ I equalize existing differences of temperature, and therefore to
pass in a contrary direction, i. e. from warmer to colder bodies.
Without fiirther explanation, therefore, the truth of the prin-
ciple will be granted.

. For the present we will again use the well-known process first
conceived by Camot and graphically represented by Clapeyron,
with this difference, however, that, besides the two bodies be- \

tween which the transmission of heat takes place, we shall assume
a' third, at any temperature, which shall fttmish the heat con-
verted into work. An example being the only thing now re-
quired, we shall choose as the changiug body one whose changes
are governed by the simplest possible laws, e. g, a permanent
gas^. Let, therefore, a quantity of permanent gas having the
temperature / and volume v be given. In the adjoining figure
we shall suppose the volume represented by the abscissa o A, and
the pressure exerted by the gas at this volume, and at the tem-

smiple transmission of heat from a colder to a warmer body, or an ttscending
transmission of heat^ as it may be called^ but two connected transmissions of
opposite characters, one ascending and the other descending, wbich compen-
sate each other. It may, moreover, happen that instead of a descending
transmission of heat accompanying, in the one and the same process, the
ascending transmission, another permanent change may occur which has
the peculiarity of not being reversible without either becoming replaced by
a new permanent change of a similar kind, or producing a descending trans-
mission of heat. In this case the ascending transmission of heat may be said
to be accompanied, not immediately, but mediately, by a descending one, and
the permanent change which replaces the latter may be regarded as a com-
pensation for the ascending transmission.

Now it is to these compensations that our principle refers ; and with the
aid of this conception the principle may be eJso expressed thus : an tmcom-
pensated transmission of heat from a colder to a warmer body can never occur.
The term ^'uncompensated " here expresses the same idea as that which was
intended to be conveyed by the words *'by itself" in the previous enuncia-
tion of the principle, and by the .expression '^ without some other change,
connected therewith, occurring at the same time " in the original text. — 1864.]

* [It will readily be understood that everything here said, by way of ex-
ample, concerning a gas applies, essentially, to every other body whose con-
dition is determined by its temperature and volume. Of course the shapes
of the curves, representing the decrease of pressure corresponding to an aug-
mentation of volume, differ for different bodies ; in other words, the aspect
of the figure will depend upon the choice of the body. — 1864.]

MODIFIED FORM pF SECOND FUNDAMENTAL THEOREM.

119

perature t, by the ordinate A a. This gas we subject, succes-
sively, to the following operations : —

1. The temperature t of the gas is changed to t^, which, for

Fig. 7.

the sake of an example, may be less than t. To do this, i;he
gas may be enclosed within a surface impenetrable to heat, and
allowed to expand without either receiving or losing heat. The
diminution of pressure, consequent upon the simultaneous in-
crease of volume and decrease of temperature, is represented by
the curve ab; so that, when the temperature of the gas has
reached /j, its volume and pressure have become oi and ib
respectively.

2. The gas is next placed in communication with a body Kj,
of the temperature t^, and allowed to expand still more, in such
a manner, however, that all the heat lost by expansion is again
supplied by the body. With respect to this body, we shall
assume that, owing to its magnitude or to some other cause, its
temperature does not become appreciably lower by this expen-
diture of heat, and therefore that it may be considered constant.
Consequently, during expansion the gas will also preserve a
constant temperature, and the diminution of the pressure will
be represented by a portion of an equilateral hyperbola b c. The
quantity of heat fiirnished by K, shall be Qj.

3. The gas is now separated from the body K^ and allowed to
expand still farther, but without receiving or losing heat, until
its temperature has diminished from t^ to t^. The consequent
diminution of pressure is represented by the curve c rf, which is
of the same nature as a b.

4. The gas is now put in communication with a body K^

120 FOURTH MEMOIR.

having the constant temperature t^, and compressed ; all the I

heat thus produced in it being imparted to Kj. This com- j

pression is continued until K^ has received the same quantity of |

heat Qj as was before furnished by K^. The pressure will in- J

crease according to the equilateral hyperbola de. \

5. The gas is then separated from the body K^ and com-
pressed^ without being permitted to receive or lose heat^ until i
its temperature rises from t^ to its original value /, the pressure j
increasing according to the curve ef. The volume o « to which 1
the gas is thus reduced is smaller than its origiaal volume o h, *
for the pressure which had to be overcome in the compression '
d e, and therefore the work to be spent, were less than the cor- |
responding magnitudes during the expansion be; so that, in

order to restore the same quantity of heat Q^, the compression I

must be continued further than would have been necessary
merely to annul the expansions.

6. The gas is at length placed in communication with a body
K, of the constant temperature t, and allowed to expand to its
origiual volume o A, the body K replacing the heat thus lost, the
amount of which may be Q. When the gas reaches the volume
h with the temperature t, it must exert its original pressure;
and the equilateral hyperbola, which represents the last diminu-
tion of pressure, wiU precisely meet the point a.

These six changes together constitute a cyclical process, the
gas ultimately returning to its original condition. Of the three
bodies K, K^ and K^, which throughout the whole process are
considered merely as sources or reservoirs of heat, the two first
have lost the quantities of heat Q and ft^ and the third has
received the quantity Q^ or, as we may express it, Q^ has been
transferred from K^ to K:^, and Q has disappeared. The last
quantity of heat must, according to the first theorem, have been
converted into exterior work. The pressure of the gas during
expansion being greater than during compression, and therefore
the positive amount of work greater than the negative, there has
been a gain of exterior work, which is evidently represented by
the area of the closed figure abcdef. If we call this amount
of work W, then, according to equation (1),

Q=A.W*.

* [The cyclical process here described differs from the one described at

MODIFIED FORM OF SECOND FITNDAMENTAL THEOREM. 121

The whole of the above-described cyclical process may be re-
versed or executed in an opposite manner by connecting the gas
with the same bodies and^ under the same circumstances as be-
fore, executing the reverse oi)erations, i. e. commencing with the
compression af, after which would foUow the expansions /c and
e dy and lastly the compressions* dc, cb, and ba. The bodies K and
Kj will now evidently receive the quantities of heat Q and Qj,
and Kg will lose the quantity Q.^. At the same time the nega-
tive work is now greater than the positive, so that the area of
the closed figure now represents a loss of work. ' The result of
the reverse process, therefore, is that the quantity of heat Q^ has
been transferred from K^ to K^, and the quantity of heat Q,
generated from work, given to the body K.

In order to learn the mutual dependence of the two simulta-
neous transformations above described, we shall first assume that
the temperatures of the three reservoirs of heat remain the same,
but that the cyclical processes through which the transforma-
tions are effected are different. This will be the case when,
instead of a gas, some other body is submitted to similar trans-
formations, or when the cyclical processes are of any other kind,
subject only to the conditions that the three bodies K, K^ and
K, are the only ones which receive or impart heat, and of the
two latter the one receives as much as the other loses. These
several processes can be either reversible, as in the foregoing
case, or not, and the law which governs the transformations will
vary accordingly. Nevertheless the modification which the law
for non-reversible processes suffers may be easily applied after-
wards, so that at present we will confine ourselves to the con-
sideration of reversible cyclical processes.

page 23 of the first Memoir, and there graphically represented in ^q. 1, only
by the circumstance that three, instead of two bodies, serving as reservoirs of
heat, now present themselves. K we assume the temperature t of the body
K to be equal to the temperature t^ of the body Kj, we may dispense with
the body K altogether, and instead thereof employ the body K^ ; the result
of this would be that the body Ki would give up, on the whole, the quantity
Q+Qi of heat, and the body K^ would receive the quantity Q^. It would
then be said that of the total quantity of heat given up by the body E„ the
portion Q is transformed into work, and the other part Q^ is transferred to
the body Kg ; but this occurred in the previously described process, so that
the latter must be regarded as a special case of the one here described. — 1864.]

122 FOURTH MEMOIR.

With respect to all these it may be proved firom the foregoing
principle, that the quantity of heat Qp transferred from K^ to
Kq, has always the same relation to Q, the quantity of heat
transformed into work. For if there were two such processes
wherein, Q being the same, Qj was different, then the two pro-
cesses could be executed successively, the one in which Q^ was
smaller in a direct, the other in an opposite manner. Then the
quantity of heat Q, which by the first process was converted into
work, would be again transformed into heat by the second pro-
cess and restored to the body K, and in other respects every-
thing would ultimately return to its original condition ; with this
sole exception, however, that more heat would have passed from
E^ to Kj than in the opposite direction. On the whole, there-
fore, a transmission of heat from a colder body K<2 to a warmer
Kj has occurred, which in contradiction to the principle before
mentioned, has not been compensated in any manner.

Of the two transformations in such a reversible process either
can replace the other, if the latter is taken in an opposite direc-
tion ; so that if a transformation of the one kind has occurred,
this can be again reversed, and a transformation of the other
kind may be substituted without any other permanent change
being requisite thereto. For example, let the quantity of heat
Q, produced in any manner whatever from work, be received by
the body K; then by the foregoing cyclical process it can be again
withdrawn from K and transformed back into work, but at the
same time the quantity of heat Q^ will pass from K| to K^ ; or
if the quantity of heat Qj had previously been transferred from
Ki to Kg, this can be again restored to Kj by the reversed cyclical
process whereby the transformation of work into the quantity of
heat Q of the temperature of the body K will take place.

We see, therefore, that these two transformations may be
regarded as phenomena of the same nature, and we may call two
tranformations which can thus mutually replace one another
equivalent. We have now to find the law according to which
the transformations must be expressed as mathematical magni-
tudes, in order that the equivalence of two transformations may
be evident from the equality of their values. The mathematical
value of a transformation thus determined may be called its
equivalence-value (Aequivalenzwerth) .

MODIFIED FORM OP SECOND FUNDAMENTAL THEOREM. 123

With respect to the direction in which each transformation is
to be considered positive, it may be chosen arbitrarily in the one,
' but it will then be fixed in the other, for it is clear that the trans-

it formation which is equivalent to a positive transformation must

I itself be positive. In future we shall consider the conversion of

work into heat and, therefore^ the passage of heat from a higher
to a lower temperature as positive transformations'*^.
i With respect to the magnitude of the equivalence-value, it is

^ first of all clear that the value of a transformation from work

^ into heat must be proportional to the quantity of heat produced ;

and besides this it can only depend upon the temperature.
Hence the equivalence-value of the transformation of work into
the quantity of heat Q, of the temperature /, may be represented
. generally by

wherein /(/) is a fimction of the temperature, which is the same
for all cases. When Ql is negative in this formula, it will indi-
cate that the quantity of heat Q is transformed, not from work
into heat, but from heat into work. In a similar manner the
value of the passage of the quantity of heat Q, from the tem-
perature ti to the temperature t^ must be proportional to the
quantity Q, and besides this, can only depend upon the two
temperatures. In general, therefore, it may be expressed by

QL.^{t,,t,),

wherein F(/j, t^ is a function of both temperatures, which is
the same for all cases, and of which we at present only know
that, without changing its numerical value, it must change its
sign when the two temperatures are interchanged ; so that

■p{t„t,) = -nt„Q (4)

In order to institute a relation between these two expressions,
we have the condition, that in every reversible cyclical process
of the above kind, the two transformations which are involved
must be equal in magnitude, but opposite in sign ; so that their
algebraical sum must be zero. For instance, in the process for

• * [The reason why this choice of the positive and negative senses is pre-
ferable to the opposite one, will become apparent after the theorems relative
to the transformations have been enunciated. — 1864.1

124 FOTJBTH MEMOIR.

a gas^ BO fiilly described aboye^ the quantity of heat Q^ at the
temperature t, was converted into work; this gives — Q./(/) 1

as its equivalence-value^ and that of the quantity of heat Q^
transferred firom the temperature t^ to t^, will be Qj . ¥{1^, t^), so j

that we have the equation |

-Q./(f)+Q..F(fi,g=0 (5)

Let us now conceive a similar process executed in an opposite i

manner, so that the bodies K^ and K^, and the quantity of heat 1

Q^ passing between them^ remain the same as before ; but that ;

instead of the body K of the temperature t, another body K' of
the temperature f be employed ; and let us call the quantity of
heat produced by work in this case &, — ^then, analogous to the I

last, we shall have the equation j

Q'./(0+Q..r(<^g=o. ..... (6) ]

Adding these two equations, and applying (4), we have

-Q./(0 + Q'./(O=O (7)

If now we regard these two cyclical processes together as one
cyclical process, which is of course allowable, then in the latter ,

the transmissions of heat between K^ and K^ will no longer |

enter into consideration, for they precisely cancel one another, i

and there remain oidy the quantity of heat Q, taken from K and |

transformed into work, and the quantity Q! generated by work ,

and giveii to K'. These two transformatibns of the same kind, j

however, may be so divided and combined as again to appear as
transformations of different kinds. If we hold simply to the
fact that a body K has lost the quantity of heat Q, and another
body K' has received the quantity Q', we may without hesitation ^ j

consider the part common to both quantities as transferred from
K to K'ySmi regard only the other part, the excess of one quan-
tity over the other, as a transformation from work into heat, or
vice ver$d. For example, let the temperature f be greater than
/, so that the above transmission, being a transmission from the
colder to the warmer body, will be negative. Then the other I

transformation must be positive, that is, a transformation from j

work into heat, whence it follows that the quantity of heat Q!
imparted to K' must be greater than the quantity Q lost by K. |

If we divide Q' into the two parts

QandQ'-Q, '

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 125

the first will be the quantity of heat transferred from K to K',
and the second the quantity generated from work.

According to this view the double process appears as a pro-
cess of the same kind as the two simple ones of which it
consists ; for the circumstance that the generated heat is not
imparted to a third body, but to one of the two between
which the transmission of heat takes place, makes no essential
difference, because the temperature of the generated heat is
arbitrary, and may therefore have the same value as the tem-
perature of one of the two bodies ; in which case a third body
would be superfluous. Consequently, for the two quantities of
heat Q and Q'— Q, an equation of the same form as (6) must
hold, ». e.

(Q'-Q)./(O+a.F(/,O=0,

Eliminating the magnitude Q! by means of (7), and dividing by
Q, this equation becomes

F(^,0=/(0-/W, ..... (8)

so that the temperatures t and f being arbitrary, the function
of two temperatures which applies to the second kind of trans-
formation is .reduced, in a general manner, to the function of
one temperature which applies to the first kind.

For brevity, we will introduce a simpler symbol for the last
function, or rather for its reciprocal, inasmuch as the latter will
afterwards be shown to be the more convenient of the two.
Let us therefore make

M=^ (9)-

so that T is now the unknown function of the temperature
involved in the equivalence-values. Further, Tj, T^, &c. shall
represent particular values of this function, corresponding to the
temperatures t^, t^, &c.

According to this, the second ftmdamental theorem in the
mechanical theory of heat, which in this form might appro-
priately be called the theorem of the equivalence of tranaforma'-
tions, may be thus enunciated :

If two transformations which, without necessitating/ any other
permanent change, can mutually replace one another^ be called

126 FOURTH MEMOIR.

equivalent, then the generation of the quantity of heat Q of the
temperature tfrom work, has the equivalence-value

Q

r

and the passage of the quantity of heat Q from the temperature
t^ to the temperature t^ has the equivalence-value

'<k-T}

wherein T is a Junction of the temperature, independetU of the
nature of the process by which the transformation is effected.
If to the last expression we give the form

it is eyident that the passage of the quantity of heat Qi, from the
temperature t^ to the temperature t^, has the same equivalence-
value as a double transformation of the first kind^ that is to say^
the transformation of the quantity Q from heat at the tempera-
ture t^ into work, and from work into heat at the temperature t^.
A discussion of the question how far this external agreement is
based upon the nature of the process itself would be out of
place here*; but at all events, in the mathematical determina-
tion of the equivalence-value, every transmission of heat, no
matter how eflfected, can be considered as such a combination of
two opposite transformations of the first kind.

By means of this rule, it will be easy to find a mathematical
expression for the total value of all the transformations of both
kinds, which are included in any cyclical process, however com-
plicated. For instead of examining what part of a given quan-
tity of heat received by a reservoir of heat, during the cyclical
process, has arisen from work, and whence the other part has
come, every such quantity received may be brought into calcu-
lation as if it had been generated by work, and every quan-
tity lost by a reservoir of heat, as if it had been converted into
work. Let us assume that the several bodies Kj, K^, Kg, &c.,
serving as reservoirs of heat at the temperatures t^, t^, t^, &c.,
have received during the process the quantities of heat Q^ Q^,
Qg, &c., whereby the loss of a quantity of heat will be counted

[* This subject is discussed in one of the subsequent memoirs. — 1864.]

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 127

as the gain of a negative quantity of heat ; then the total value
N of all the transformations will be

^ N=^+^ + ^+&c..=2§ (10)

It is here assumed that the temperatures of the bodies K^ Kg, Kg,
&c. are constant, or at least so nearly constant, that their varia-
tions may be neglected. When one of the bodies, however,
.^ either by the reception of the quantity of heat Q itself, or

f through some other cause, changes its temperature during the

process so considerably that the variation demands considera-
tion, then for each element of heat rfd we must employ that
temperature which the body possessed at the time it received it,
^ whereby an integration will be necessary. For the sake of

[ generality, let us assume that this is the case with all the

^ bodies ; then the foregoing equation will assume the form

N=j-*

(11)

wherein the integral extends over all the quantities of heat
received by the several bodies.

If the process is reversible, then, however complicated it may
be, we can prove, as in the simple process before considered,
that the transformations which occur must exactly cancel each
other y so that their algebraical sum is zero.

For were this not the case, then we might conceive all the
transformations divided into two parts, of which the first gives
the algebraical sum zero, and the second consists entirely of
transformations having the same sign. By means of a finite or
infinite number of simple cyclical processes, the transforma-
tions of the first part must admit of being cancelled"^, so that
the transformations of the second part would alone remain

* [By a simple cyclical process is here to be understood one in which, as
above described, a quantity of heat is transformed into, or arises from work,
whilst a second quantity is transferred from oiie body to another. Now it
may be readily shown that every two transformations whose algebraical sum
is zero may be cancelled by means of one or two simple cyclical processes.

In the first place, let the two given transformations be of different kinds.
For instance, let the quantity of heat Q at the temperature t be transformed
nto work, and the quantity Qj be transferred from a body Kj of the tempera-

128 rOlTRTH MEMOIR.

without any other change. Were these transformations ne^
gative, i. e. transformations from heat into work^ and passages
of heat from lower to higher temperatures^ then of the two
kinds the first could be replaced hj transformations of the

ture ^] to a body K^ of the temperature i^, whereby we will aBsume, since
our intended exposition will be thereby fiELcilitated, that Q and Q| denote the
abaohste values of the quantities of heat, so that the positive or negative cha-
racter of each transformation must be denoted explicitly by a prefixed + or
— sign. Suppose, moreover, that the magnitudes of the two qiumtities of
heat are related to one another in the manner expressed by the equation

-?.Q.(i-^).a

Conceive the cyclical process above described to be performed in a contrary
manner, so that the quantity of heat Q at the temperature t arises from work,
and another quantity of heat is transferred from the body K, to the body K|.
This latter quantity must then be precisely the quantity Q, which enters into
the above equation, and thus the given transformations are cancelled.

In the next place, let a transformation from work to heat, and another from
heat to work be given ; for instance, let the quantity of heat Q, at the tem-
perature,^, be generated by work, and the quantity Q', at the temperature t\
be converted into work, and suppose the two quantities to be so related to
one another that

T T- "•

Conceive the above-described cyclical process to be first performed, whereby
the quantity of heat Q at the temperature t is converted into work, and
another quantity Q| transferred from a bodyE^ to another body E^. After-
wards conceive a second cyclical process of the opposite kind to be performed,
in which the last-named quantity of heat Qi is transported back from K, to
K^, and, besides this, a quantity of heat of the temperature f is generated
from work. This conversion of work into heat must then, apart from its
sign, be equivalent to the preceding conversion of heat into work, since both
are equivalent to one and the same transmission of heat. The heat at the
temperature f, which has arisen from work, must consequently be just as
great as the quantity Q' involved in the last equation, and the given trans-
formations are thus cancelled.

In the last place, let two transmissions of heat be given ; for instance,
let the quantity of heat Qi be transferred from a body E^, of the temperature
<i, to a body E, of the temperature t^, and let another quantity Q'^ be con-
veyed from a body E'^, of the temperature t'^, to a body E\, of the temperature
t\j and suppose these two quimtities to stand to each other in the relation

Hti-)^Hk-T^y'-

Conceive now two cyclical processes to be performed, in one of which the
quantity of heat Q^ is carried from Ej to Ei, and thereby the quantity Q at
the temperature t generated by work, whilst in the second the same quan-

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 129

latter kind*, and ultimately transmissioiiB of heat from a lower
to a higher temperature would alone remain, which would be
compensated by nothing, 8ind therefore contrary to the above

S principle. Further, were those transformations j»o«i^it?c, it would

only be necessary to reverse the operations in order to render
them negative, and thus we should again obtain the foregoing
impossible case. Hence we conclude that the second part of

[ the transformations can have no existence.

Consequently the equation

\

Jf =0 (ID

is the analytical expression, for all reversible cyclical processes,
of the second fondamental theorem in the mechanical theory of
heat.

The application of this equation can be considerably extended
by giving to the magnitude / involved in it a somewhat diflferent
signification. For this purpose, let us consider a cyclical pro-
cess consisting of a series of changes of condition made by a

tity of heat Q is reconverted into work, and thereby another quantity trans-
ferred from K\ to K'g. This other quantity must then be precisely that
which is denoted by Q/, and the two given transformations are thus cancelled.

If now, instead of two^ any number of transformations were given^ having
an algebraical sum equal to zero^ we could always separate and combine them
so as to obtain, solely^ groups consisting each of two transformations whose
algebraical sum is equal to zero ; and the two transformations of each such
group could then, as has just been shown, be cancelled by means of one or
two simple cyclical processes. K continuous changes of temperature should
present themselves in the given original process, so that the quantities of
heat given up and received would have to be divided into infinitesimal
elements, the number of the groups which would have to be formed, and
consequently also the niunber of simple cyclical processes, would be infinite ;
as far as the principle is concerned, however, this makes no difference. —
1864.]

* [For if the given transformation consist in the conversion into work of
the quantity of heat Q at the temperature t, we have, as already explained
in the text in reference to the opposite case, merely to conceive the above-
described cyclical process performed in a contrary manner, whereby the
quantity of heat Q at the i;emperature t will be generated by work ; and at
the same time another quantity Q^ will be transferred from a body Ky of the
temperature t^f to a body Kj of the higher temperature ty The given trans-
formation from heat to work will thus be cancelled, and replaced by the
transmission of heat from K, to K^. — 1864.]

180

FOURTH MIMOIR.

body which ultimately returns to its original state^ and for sim-
pliciiy^ let us assume that all parts of the body have the same
temperature ; then in order that the process may be reversible,
the changing body when imparting or receiving heat can only
be placed in communication with such bodies as have the same
temperature as itself, for only in this case can the heat pass in
an opposite direction. Strictly speaking, this condition can
never be fulfilled if a motion of heat at all occurs ; but we may
assume it to be so nearly fulfilled, that the small differences of
temperature still existing may be neglected in the calculation.
In this case it is of course of no importance whether /, in the
equation (II), represents the temperature of the reservoir of
heat just employed, or the momentary temperature of the
changing body, inasmuch as both are equal. The latter signi-
fication being once adopted, however, it is easy to see that any
other temperatures may be attributed to the reservoirs of heat

without producing thereby any change in the expressign j-=j-

which shall be prejudicial to the validity of the foregoing equation.
As with this signification of t the several reservoirs of heat need
no longer enter into consideration, it is customary to refer the
quantities of heat, not to them, but to the changing body itself,
by stating what quantities of heat this body successively receives
or imparts during its modifications. If hereby a quantity of
heat received be again counted as positive, and a quantity im-
parted as negative, all quantities of heat will of course be afiected
with a sign opposite to that which was given to them with
reference to the reservoirs of heat, for every quantity of heat
received by the changing body is imparted to it by some reser-
voir of heat ; nevertheless, this circumstance can have no influ-
ence upon the equation which expresses that the value of the
whole integral is zero. From what has just been said, it fol-
lows, therefore, that when for every quantity of heat dCi which
the body receives or, if negative, imparts during its changes,
the temperature ci the body at the moment be taken into cal-
culation, the equation (II) may be applied without fiirther con-
sidering whence the heat comes or whither it goes, provided the
process be in other respects reversible.

To the equation (II) thus interpreted we can now give a

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM^ 131

more special form^ as was formerly done to equation (I), in
which form it shall express a particiQar property of the body.
We shall thus obtain an equation essentially the same as the
well-known one deduced by Clapeyron from the theorem of
Camot*. With respect to the nature of the changes, we shall
assume the same conditions as before led to the deduction of
the equations (2) and (3) from (I), and which also suffice for
the fulfilment of equation (II) f. Hence, the condition of the
body being defined by its temperature t and volume v, we have

at dv

Inasmuch as by (II) |-7=- must always equal zero, ^whenever

/ and V assume their initial values, the expression under the
integral sign, which by the foregoing equation becomes

1 dQ.^1 rfQ,

must be a complete differential, if / and v are independent vari-
ables ; and the two terms of the expression must consequently
satisfy the following condition.

rf/1 dQ\d/l dQ\
dA^T' dvhdv\f' dtJ'

* Joum, de VEcole Pokftechnique, tome xiv.

t [These conditions were that the sole exterior force in operation is a
pressure acting everjrwrhere with the same intensity upon, and perpendi-
cularly to the surface, and that this pressure always differs so little from the
expansive force of the body that Ihe two may be regarded as equal to one
another in all calculations. Hence it follows that the changeable body may
be again compressed imder the same pressure as that under which it expands,
and consequently that its changes of volume have occurred in a reversible
manner. A certain temperature t was likewise ascribed to the entire change-
able body — an assumption which implies that all parts of the latter have one
and the same temperature, or at all events, that the differences of tempera-
ture which present themselves are small enough to be neglected. Hence
it follows that within the body no transmissions of heat occur from warmer
to colder places, which are of sufficient importance to be taken into calcula-
tion. All the changes, therefore, of which the body is susceptible may be
regarded as reversible, and for the truth of the equation (II) nothing more
than this is reqtdsite. — 1864.]

k2

or

132 Toxmn memoib.

From this we obtain

rfT
1 d/dQt\ do, dt__l d^fdQ\
f • JAWJI^ ' 1«~T • dv\ dt)

S-f-'-[^(S)-i(f)]- • • • ('»'

Substituting^ from equation (3), the value of the expression
within the [ ]j we obtain the desired equation,

^ ^-A T* (131

■^.^-A.T^, (13)

which, on account of the relation
may be written thus

dt^dT dt'

^-A T^* (lSd\

If we compare this result with the before-mentioned equation
established by Clapeyron, we shall at once see the relation which
exists between the function T, here introduced, and that used by
Clapeyron, denoted by C, and known as Camof s function, which
I have also used in former memoirs. This relation may be ex-
pressed thus :

* [I may here lemiirk that the equation (12) may be transfonned in the
some way as the equation (13) has abeady been truisfonned. For putting
thevein

dQ_dQ dT
dt'^dT'W
didQ\_d/dQ\ dT

and dividing throughout by ^, we have

dldQ\_d/dQ\ dT
dt\dv)''^\'d^) • ST'

by ^, we have

d(i^Trd/dq\ d(dQ\n,

or^ otherwise written,

d(dQ\d(dQl\l dQ .-

Hereby the meaning of the equation (11) is expressed^ even more simply
than in (12)^ in the form of a partial differential equation of the second order.

—1864.]

MODIFIED FOKH OF SBCONO FUNDAUBNTAL THXOBBH.

133

rfT

dt A.

(14)

We proceed now to the consideration of non-reversible cycKcal
processes.

In the proof of the previous theorem, that in any reversible
cyclical process, however complicated, the algebraical sum of all
the transformations must be zero, it was first shown that the sum
could not be negative, and afterwards that it could not be^o^
tive, for if so it would only be necessary to reverse the process in
order to obtain a negative sum. The first part of this proof re-
mains unchanged even when the process is not reversible ; the
second part, however, cannot be applied in such a case. Hence
we obtain the following theorem, which applies generally to all
cyclical processes, those that are reversible forming the Umit : —

The algebraical mm of all transformations occurring in a cyclical
process can only be positive.

A transformation which thus remains at the conclusion of
a cyclical process without another opposite one, and which
according to this theorem can only be positive, we shall, for
brevity, call an uncompensated transformation.

The different kinds of operations giving rise to uncompensated
transformations are, as far as external appearances are concerned,
rather numerous, even though they may not differ very essen-
tially. One of the most frequently occurring examples is that
of the transmission of heat by mere conduction, when two bodies
of different temperatures are brought into immediate contact ;
other cases are the production of heat by friction, and by an
electric current when overcoming the resistance due to imper-
fect conductibility, together with all cases where a force, in doing
mechanical work, has not to overcome an equal resistance, and
therefore produces a perceptible external motion, with more or

* [The equation established by Clapeyroii; when written in the form given
to it in the notes to the equations (IV) and (V) of the first Memoir (pp. 47,
48), is ^^

dv '

and on comparing this expression for ^with the one which results from the
equation (13), we obtain the equation (14). — 1864.]

i-ff^

184 VOURTH MEMOIR.

less velocity^ the vis viva of which afterwards passes into heat.
An instance of the last kind may be seen when a vessel filled
with air is suddenly connected with an empty one ; a portion of
air is then propelled with great velocity into the empty vessel
and again comes to rest there. It is well known that in this
case just as much heat is present in the whole mass of air after
expansion as before^ even if differences have arisen in the several
parts^ and therefore there is no heat permanently converted into
work. On the other hand^ however^ the air cannot agaia be
compressed into its former volume without a simultaneous con-
version of work into heat.

The principle according to which the equivalence-values of
the uncompensated transformations thus produced, are to be
determined^ is evident from what has gone before, and I will
not here enter further into the treatment of particular cases.

In conclusion, we must direct our attention to the fonction T,
which hitherto has been left quite undetermined ; we shall not
be able to determine it entirely without hypothesis, but by
means of a very probable hypothesis it will be possible so to do.
I refer to an accessory assumption already made in my former
memoir, to the effect that a permanent gas, when it expands at a
constant temperaturey absorbs only so much heat as is consumed by
the exterior work thereby performed. This assumption has been
verified by the later experiments of Begnault, and in all proba-
bility is accurate for all gases to the same degree as Mariotte
and Gay-Lussac's law, so that for an ideal gas, for which the
latter law is perfectly accurate, the above assumption will also
be perfectly accurate.

The exterior work done by a gas during an expansion dv,
provided it has to overcome a pressure equivalent to its total ex-
pansive force p, is equal to pdv, and the quantity of heat absorbed

dCt
thereby is expressed by -^dv. Hence we have the equation

da .

and by substitutiag this value of -j- in the equation (13), the
latter becomes

MODIFIED FORM OF SECOND FUNDAMENTAL THEOREM. 185

(15)

dt_dt

T-J

But^ according to Mariotte and Gay-Lnssac^s law^

o= . const..

where a is the inverse value of the coefficient of expansion of
the permanent gases^ and nearly equal to 278^ if the temperature
be given in Centigrade degrees above the freezing-point. Elimi-
nating^ from (15) by means of this equation, we have

whence, by integration,

T=:(a+/) . const (17)

It is of no importance what value we give to this constant, be-
cause by changing it we change all equivalence-values propor-
tionally, so that the equivalences before existing wiU not be
disturbed thereby. Let us take the simplest value, therefore,
which is unity, and we obtain

T:=a+t (18)

According to this, T is nothing more than the temperature
counted from c^, or about 273° C. below the freezing-point;
and, considering the point thus determined as the absolute zero
of temperature, T is simply the absolute temperature. For this
reason I introduced, at the commencement, the symbol T for the
reciprocal value of the function / {f) . By this means all changes
which would otherwise have had to be introduced in the form of
equations, after the determination of the frmction, are rendered
unnecessary ; and now, according as we feel disposed to grant
the sufficient probability of the foregoing assumption or not, we
may consider T as the absolute temperature, or as a yet unde-
termined fimction of the temperature. I am inclined to believe,
however, that the first may be done without hesitation.

136 FIFTH MEMOIR

FIFTH MEMOIR. <

ON THE APPLICATION OF THE MECHANICAL THEORY OF HEAT TO
THE STEAM-ENOINE^.

1. As our present modified views respecting the nature and
deportment of heat^ which constitute the mechanical theory of
heat; had their origin in the well-known fact that heat may be
employed for producing mechanical work, we may naturally an-
ticipate that the theory so originated will in its turn help to
place this application of heat in a clearer light. At all events
the more general views thus obtained must enable us to pro-
nounce safely upon the efficiency of the several machines for
thus applying heat, as to whether they already perfectly fulfil
their purpose, or whether and to what extent they are capable
of being perfected.

Besides these reasons, which apply to all thermo-dynamic
machines, there are others, applicable more particularly to the
most important of them, the steam-engine, which appear to
render a new investigation of the latter, conducted according to
the principles of the mechanical theory of heat, desirable. It is
precisely with respect to vapour at a maximum density that this
new theory has led us to laws which differ essentially from those
formerly accepted as true, or at least introduced into former
calculations.

2. I may here be allowed to refer to a fact proved by Bankine
and myself, that when a quantity of vapour, at its maximum den-
sity and enclosed by a surface impenetrable to heat, expands and
thereby displaces a moveable part of the enclosing surface, e, g.
a piston, with its full force of expansion, a part of the vapour
must undergo condensation; whereas in most works on the
steam-engine, amongst others in the excellent work of De

• Published in Poggendorff's Annalen, March and April 1866, vol. xcvii.
pp. 441 and 513 ', translated in the Philosophical Magazine, S. 4. vol. xii.
pp. 241, 338 and 426 ; and in Silliman's Journal, S. 2. vol. xxii. pp. 180 and
364, vol. xxiii. p. 25.

\

THEORY OP THE STEAM-ENGINE. 137

Pamboiir*, Watf s theorem^ that under these circumstances the
vapour remains precisely at its maximum density^ is assumed as
a fundamental one.

Further, in the absence of more accurate knowledge, it was
formerly assumed, in determining the volumes of the unit of
weight of saturated vapour at different temperatures, that vapour
even at its maximum density still obeys Mariotte's and Gay-
Lussac^s laws. In opposition to this, I have already shown in
my first memoirf on this subject, that the volumes in question
can be calculated from the principles of the mechanical theory
of heat under the assumption, that a permanent gas when it ea?-
pands at a constant temperature only absorbs so much heat as is
consumed in the external work thereby performed, and that these
calculations lead to values which, at least at high temperatures,
differ considerably from Mariotte^s and Gay-Lussac^s laws.

Even the physicists who had occupied themselves more espe-
cially with the mechanical theory of heat, did not at that time
coincide with this view of the deportment of vapour. William
Thomson in particular opposed it. In a memoir J presented to
the Eoyal Society of Edinburgh a year later, in March 1851,
he only regarded this result as a proof of the improbability of
the above assumption which I had employed.

Since then, however, he and J. P. Joule have together under-
taken to test experimentally the accuracy of this assumption §.
By a series of well-contrived experiments, executed on a large
scale, they have in fact shown that, with respect to the perma^
nent gases, atmospheric air and hydrogen, the assumption is so
nearly true, that in most calculations the deviations from exacti-
tude may be disregarded. With carbonic acid, the non-perma-
nent gas they investigated, the deviations were greater. This is
in perfect accordance with the remark I made on first making the
assumption, which was that the latter would probably be found
to be accurate for each gas in the same measure as Mariotte's and
Gay-Lussac's laws were applicable thereto. In consequence of

* ThSorie des Machines H Vapeur, pax le Oomte F. M. G. de Pambour.
Paris, 1844.

t [First Memoir of this collection.]

X Transactions of the Koyal Socie^ of Edinburgh, vol. xx. part 2, p. 261.

§ Phil. Trans, vol. cxliii. part 3, p. 367 j and vol. cxliv. part 2, p 321.

1S8 FIFTH MEMOIR.

these experiments^ Thomson now calculates the volumes of satu-
rated vapours in the same manner as myself. There is reason
to believe, therefore, that the accuracy of this method of calcu-
lation will be gradually more and more recognized by other
physicists.

3. These two examples will suffice to show that the principles
upon which our former theory of steam-engines was founded have
suffered such essential modifications through the mechanical
theory of heat, as to render a new investigation of the subject
necessary.

In the present memoir I have attempted to develope the prin-
ciples of the calculation of the work of the steam-engine in
accordance with the mechanical theory of heat. I have, how-
ever, limited myself to the steam-engines now in use, without at
present entering into a consideration of the more recent and cer-
tainly very interesting attempts to employ vapour in a super-
heated state.

In recording the results of my investigation, I shall only
assume, on the part of the reader, an acquaintance with my last
memoir, ''On a modified Form of the Second Fundamental
Theorem in the Mechanical Theory of Heat'^*. This wiU of
course necessitate the deduction, in a somewhat different man-
ner, of results which are no longer new, but have already been
found by myself or others ; I believe, however, that this re-
petition, by leading to greater unity and facility of comprehen-
sion, wiU not be found superfluous. At the proper places I shall,
to the best of my ability, cite the papers wherein these results
first appeared.

4. The expression '' a machine is driven by fieat" is not of
course strictly accurate. By it we must understand, that, in
consequence of the changes produced by heat upon some kind
of matter in the machine, the parts of the latter are set in motion*
We shall refer to this matter as that which manifests the action
of heat.

If a continuously-acting machine is in uniform action, all

accompanying changes occur periodically, so that the condition

which at a given time prevails in the machine and all its parts

returns at equal intervals. Hence the matter which manifests

* [Fourth Memoir of this collection.]

THEORY OF THE STEAM-ENGINE. 139

the action of the applied heat must at such regularly-recurring
periods be present in the machine in equal quantity, and in the
same state. This condition can be fulfilled in two diflferent ways.

First. One and the same quantity of matter may always re-
main in the machine, when the changes of condition which this
matter suffers during the action of the machine will be such, that
at the end of each period it will regain its original condition and
recommence the same cycle of changes.

Secondly. The machine may always expel the matter which
served to produce the effect during a period, and in its place
receive from an external source just as much matter of the same
kind.

5. The last method is the one usually employed m most
machines. This is the case, for instance, in machines with
heated air as at present constructed ; for after every stroke, the
air which moved the piston in the driving cylinder is expelled
into the atmosphere, and in its place an equal quantity of air
from the same source is received into the feeding cylinder. Si-
milarly in steam-engines without condensers, steam is driven
from the cylinder into the atmosphere, and in its place fresh
water is pumped from a reservoir into the boiler.

Further, a similar method is at least partially adopted even in
steam-engines provided with a condenser as usually constructed.
In them the water condensed from the steam is only partially
pumped back into the boiler, for being mixed with the cooling
water, a part of the latter also reaches the boiler. The remaining
part of the condensed water, together with the remaining part
of the cooling water, has to be got rid of.

The first method has lately been employed in steam-engines
propelled by two vapours, e, g. those of water and sether. In
these machines the steam is condensed solely by contact with
metallic tubes filled with liquid sether, and the water thus pro-
duced is then completely pumped back into the boiler. In the
same manner the vapour of the sether is condensed in metal tubes^
which are merely surrounded by cold water, and subsequently
it is pumped back into the first space intended for the vapori-
zation of the sether. In order to maintain a uniform action,
therefore, only so much fresh water and aether is necessary as
will replace the leakage consequent upon imperfect construction.

140 FIFTH MEMOIR.

6. In a machine of this kind^ where the same matter is con-
tinually re-employed^ the several changes which this matter suf-
fers daring a period must^ as above stated^ form a closed cyde^
or^ according to the nomenclature in my former memoir^ a cy-
clical process.

On the contrary^ machines in which a periodical reception and
expulsion of matter occurs are not necessarily subject to this
condition, though they may alsofiilfil it by expelling the matter
in the same condition in which it was received. This is the case
in steam-engines with condensers, where the water is ultimately
expelled from the condenser in the liquid state, and at the same
temperature as it had when introduced from the condenser into
the boiler*.

In other machines, the condition, when expelled, is different
from what it was when received. For example, heated-air ma-
chines, even when provided with regenerators, expel the air at
a higher temperature than it formerly had ; and steam-engines
without condensers receive water in the liquid, and expel it in
the gaseous form. Strictly, therefore, the complete cyclical pro-
cess is not fulfilled in these cases ; nevertheless we may always
conceive a second machine appended to the given one which
shall receive the matter from the first, reduce it in some manner
to its original condition, and then expel it. Both machines may
then be regarded as constituting one and the same machine,
which will fulfil the above condition. In many cases this addi-
tion may be made without introducing greater complexity into
the investigation. For example, a steam-engine with a con-
denser at a temperature of 100^ C. maybe substituted for a ma-
chine without a condenser, provided we assume the latter to be
fed with water at 100° C.

Hence, if we assume that machines which do not fulfil the
above condition are theoretically completed in the above manner,
we may apply the theorems concerning cyclical processes to
all thermo-dynamic machines, and thereby arrive at conclusions

♦ The cooling water,' which enters the condenser at a low, and leaves it at
a high temperature, is not here taken into consideration, inasmuch as it does
not form a part of the matter manifesting the effect of the applied heat, hut
merely constitutes a negative source of heat

THEORY OP THE STEAM-ENGINE. 141

wluch are quite independent of the nature of the processes
executed by the several machines.

7. In my former memoir I have represented the two funda-
mental theorems which hold good in every cyclical process by
the following equations : —

Q=A.W, (I)

Jf =-N, (II)

wherein the letters have the same signification as before, viz. —

A is the thermal equivalent of the unit of work.

W represents the external work performed during the cyclical
process. *"

Q signifies the heat imparted to the changeable body during
a cyclical process, and dCi an element of the same, whereby any
heat withdrawn from the body is to be considered as an imparted
negative quantity of heat. The integral in the second equation
is extended over the whole quantity Q.

T is a function of the temperature which the changing matter
has at the moment when it receives the element of heat rfQ ; or
should the temperature of different parts of the body be different,
a function of the temperature of the part which receives dCL.
With respect to the form of the function T, I have shown ia my
former memoir that it is probably the temperature itself reckoned
from a point which may be determined from the reciprocal value
of the coefficient of expansicm of an ideal gas, and which must
be in the neighbourhood of —273° C; so that if / represents
the temperature above the freezing-point,

T=27S + t (1)

In the present memoir T will always have this signification, and
for brevity will be called the absolute temperature. It may be
here remarked, however, that the conclusions do not essentially
depend upon this signification, but remain true even when T is
considered as an undetermined function of the temperature.

Lastly, N denotes the equivalence-value of aU the uncompen-
sated transformations'^ involved in a cyclical process.

* One species of uncompensated transformations requires further remark.
The sources from which the changing matter derives heat must have higher
temperatures than itself; and, on the other hand, those from which it derives

142 FIFTH MEMOIR.

8. If the process is such that it can be reversed in the same
manner^ then N=0. If, however, one or more changes of con-

?-

negative quantities of heat, or which deprive it of heat, must have lower
temperatures than itself Therefore whenever heat is interchanged between
the changing body and any source whatever^ heat passes immediately from the
body at a higher to the one at a lower temperature^ and thus an uncompen-
sated transformation occurs which is greater the greater the difierence be-
tween the temperatures. In determining such uncompensated transforma-
tions^ not only must the changes in the condition of the variable matter be
taken into consideration^ but also the temperatures of the sources of heat which
are employed ; and these uncompensated transformations will be included in
N or not, according to the signification which is attached to the temperature
occurring in equation (11). If thereby the temperature of th^^offroe cfheat
belonging to cQ is understood, the above changes will be included in N. If,
however, agreeably to the above definition, and to our intention throughout
this memoir, the temperature of the changing matter is imderstood, then the
above transformations are excluded from N. One more remark must be added
concerning the minus sign prefixed to N, which did not appear in the same
equation in my former memoir. This difference arises from the different ap-
plication of the terms negative and positive with respect to quantities of heat.
Before, a quantity of heat received by the changeable body was considered as
negative because it was lost by the source of heat ; now, however, it is con-
sidered as positive. Hereby every element of heat embraced by the integral^
and consequently the integral itself, changes its sign \ and hence, to preserve
the correctness of the equation, the sign on the other side must be changed.
[The reason why, in different investigations, I have changed the significa-
tions of positive and negative quantities of heat, is that the points of view
from which the processes in question are regarded, differ according to the na-
ture of the investigations. In purely theoretical investigations on the trans-
formations between heat and work, and* on the other transformations con-
nected therewith, it is convenient to consider heat generated by work as po-
sitive, and heat converted into work as negative. Now the heat generated by
work during any cyclical process must be imparted to some body serving as a
reservoir or as a source of heat, and the heat converted into work must be
withdrawn from one of these bodies. Quantities of heat will receive appro-
priate signs in theoretical investigation, therefore, when the heat gained by a
reservoir is calculated as positive, and that which it loses as negative. There
are investigations, however, in which it is not necessary to take into special
consideration the reservoirs or sources which receive the heat that is generated,
or furnish the heat that is consumed by work, the condition of the variable
body being the chief object of research. In such cases it is customary to re-
gard the heat received by the changing body as positive, and the heat which
it loses as negative ; to deviate from this custom, for the sake of consistency,
would be attended with many inconveniences. Researches on the interior
processes in a steam-engine are of the latter kind, and accordingly I have
deemed it advisable to adopt the customary choice of signs. — 1864.]

THEORY OF THE STEAM-ENGINE. 143

dition occur in a cyclical process which are not reversible, then
uncompensated transformations necessarily arise, and the mag-
nitude N has consequently a determinable and necessarily posi-
tive value.

Amongst the operations to which the last remark is applicable,
is one which in the following will be often mentioned. When
a quantity of gas or vapour expands, and thereby overcomes a
pressure equivalent to its total expansive force, it may be again
compressed info its former volume by employing the same power,
when all the phenomena which accompanied the expansion will
take place in an inverse manner. This is not the case, however,
when the gas or vapour does not, during its expansion, encounter
all the resistance it is capable of overcoming ; when, for instance,
it issues from a vessel in which the pressure is greater than in
the one into which it enters. In this case a compression, under
circumstances similar to those accompanying expansion, is im-
possible.

By equation (II) we can determine the sum of all the uncom-
pensated transformations in a cyclical process. As, however, a
cyclical process may consist of several changes of condition in the
given matter, of which some have occurred in a reversible, and
others in an irreversible manner, it is often interesting to know
how much of the whole sum of uncompensated transformations
has resulted from changes of each kind. For this purpose let
us conceive the matter, after the changes of condition which has
to be examined in this manner, reduced to its original condition by
any reversible operation. We shall thereby obtain a small cyclical
process, to which the equation (II) will be just as applicable as
to the whole. Consequently, if we know the quantities of heat
which the matter has received during the process, and the tem-

peratures which correspond thereto, the negative integral — |-rfr

will give the uncompensated transformation involved therein.
But as the uncompensated transformation involved in the given
change of condition could not have been increased by the above
reduction, which was executed in a reversible manner, it will be
fully represented by the above expression"*^.

* [Let us suppose the changeable body to be a quantity of gas, and that
one of the changes which this gas has suffered consists of an expansion, with-

144 FIFTH MEMOIR.

Having thus investigated all the parts of the- whole cyclical
process which are not reversible, and found the values N„ N^,
&c., which must all be positive, their sum will give the magni-

out change of temperature, from the volume v^ to the volume v^. As already
stated in the text, this expansion may occur in several ways. The gas may
so expand that at every moment the pressure which it has to overcome cor-
responds to its expansive force at that moment ; or it may be allowed to ex-
pand without overcoming any resistance whatever, by suddenly placing an
empty vessel in connexion with the one in which it occupied the volume v^ ;
or lastly, it may, during its expansion, have to overcome a resistance less than
that which corresponds to its own expansive force. If we wish to know the
magnitude of the uncompensated transformation involved in this change of
volume, we have merely to conceive the gas to be again compressed, at a con-
stant temperature, from the volume v^ to the volume v^, and to determine the
quantities of heat received and withdrawn during the cyclical process thus
completed.

If during its expansion the gas has the ftdl resistance to overcome, it must
receive just as much heat as it afterwards gives off during compression, so
that we obtain for the cyclical process the equation

If the gas has no resistance to overcome during expansion, and if we more-
over assume it to be a perfect gas, it need not receive any heat during expan-
sion. During compression, however, it must give off a quantity of heat equal
to that which is generated by the exterior work necessary for compression.
For each element of the change of volume this will be represented by A.pdv,
where I? denotes the pressure, and the positive or negative sense is already
expressed in the formula itself, since a quantity of heat to be received is po-
sitive, and one to be given off is negative. We must put then

l A.pffo
T •

-J^-J./

Now, according to the law of M. and G.,

RT

where B is an already known constant, so that

-Jt-^j.;'?-'^"*?-

This, therefore, is the value of the uncompensated transformation when a per-
fect gas has expanded from the volume v^ to the volume v^ without having
had to overcome any exterior resistance.

If, lastly, the gas, on expanding, has to overcome some, but not the full re-
sistance, it will on so doing receive some heat, but not so much as it will give
off on compression. We should now obtain, for the uncompensated trans-
formation corresponding to the expansion, a value between zero and the one
•last calculated.— 1864.]

THEORY OF THB STEAM-ENGINE. 145

tude N corresponding to the whole cyclical process^ without its
being necessary to take into consideration those parts which are
known to be reversible.

9. If we now apply the equations (I) and (II) to the cyclical
process which occurs during a period in a thermo-dynamic
machine^ it will be at once evident that, the whole quantity of
heat communicated during this period to the matter in the
machine being given, the corresponding amount of work can be
immediately determined from the first equation without its being
necessary to know the nature of the operations constituting the
cyclical process.

In an equally general manner the work may be determined
from other data by a combination of both equations.

We will assume that the quantities of heat successively im-
parted to the changing material, as well as the temperatures at
the times of reception, are given, and that only one temperature.
To, remains at which a certain as yet unknown quantity of heat
was imparted or, if negative, abstracted. The sum of all the
known quantities of heat shall be represented by Qi, and the
unknown quantity of heat by Q^.

We will divide the integral in equation (II) into two parts, of
which the one shall extend over the known quantity of heat Q„
and the other over the unknown quantity Q,,. In the latter part,
T having a constant value T^, the integration may be immediately
effected, and as result we have

9s.

To
The equation (II) thus becomes

f

.Jo

whence results

T To" '

Q„=-T..J7'f-T,.N.
Further, seeing that in our case

we have from equation (I),

W=l(Q, + Qo).

146 FIFTH MEMOIR.

Substituting the above-found value of Qo in this equation^ it
becomes

W=1(q,-To(J''^-To.n). ... (2)

If, as a special case, the whole cyclical process is reversible,

then

N=0,

and the above equation becomes

This expression di£Fers from the preceding one only in the absence

T
of the term — ^N. Now as N can only be positive, this term

A,

must necessarily be negative ; and thus we see that, under the
above conditions with respect to the communication of heat, the
greatest possible amount of work is obtained when the whole
process is reversible ; and that every circumstance which renders
one of the operations in the cyclical process not reversible, dimi-
nishes the amount of work, — a conclusion which results easily
from a direct consideration of the subject.

The equation (2) leads to the value of the amount of work in a
manner opposite to that usually followed. The amounts of work
done in the several operations are not separately determined
and then added together, but, instead of this, the maximum of
work is first found, and the losses occasioned by the several
imperfections of the process are subsequently deducted from it.

If, with respect to the communication of heat, we introduce a
still more limited condition, and assume that the whole quantity
of heat Q^ is also imparted to the body at a constant tempera-
ture Tj, then the integration which embraces this quantity of
heat may also be executed, and gives

T,

whereby the equation (3) for the maximum of work assumes the
form

A • Ti ^^^

In this special form the equation has already been deduced by
William Thomson and Bankine from a combination of Camot's

THEORY OF THE STEAM-ENGINE. 147

theorem^ as modified by me, and the theorem of the equivalence
of heat and work*.

10. Before we proceed, from these considerations which apply
to aU thermo-dynamic engines, to treat of the steam-engine,
we must first premise something concerning the deportment of
Tapours at a maximum density.

In a memoir of mine, published as early as 1850, " On the
Moving Force of Heat,'' kc.f, I have already established the
equations which show the application of the two fundamental
theorems of the mechanical theory of heat to vapours at a maxi-
mum density, and I have there employed these equations in de-
ducing several consequences. But as in my last memoir, ^^ On
a modified Form of the Second Fundamental Theorem of the
Mechanical Theory of HeatJ,'' I proposed a somewhat different
mode of treating the whole subject, it appears preferable to me
to assume the last memoir only as known. I shall therefore
deduce those equations once more, in a different manner, by
means of the results established in my last memoir.

It was there assumed, in order to apply the general equations
which were first established to a somewhat more special case,
that the only foreign force, acting upon the changing material,
which required consideration in determining the external work,
was an external pressure equally intense at all points of the sur-
face, and directed everywhere at right angles to the same ; and
fiirther, that this pressure always changed so slowly, and con-
sequently at each moment differed so little from the opposite
expansive force of the body, that in calculation the two might be
considered equal. Let then^ be the pressure, v the volume,
and T the absolute temperature of the body. We introduce the
last instead of t, the temperature counted from the freezing-
point, because thereby the formulae assume a simpler form. The
equations already established in this case are

d(dGi\ d(dQ\ . dp .^.^.

f =A.Tg5. ....... (IV)

♦ Phil. Mag. July 1851.
t [First Memoir of this Collection.]
X [Fourth Memoir of this Collection.]

J [Instead of this equation (IV), which is identical with (13 «) of the Fourth

l2

148 FIFTH MEMOIR.

These equations shall next be applied to the still more special
case of vapours at their maximum density.

11. Let M be the mass of the matter whose vapour is to be
considered^ and which is placed in a perfectly closed expansible
vessel. Let the part m be in a vaporous^ and the rest,M— w, in
a liquid state. This mixed mass shall be the changing body to
which the foregoing equations are to be referred.

The condition of the mass^ as far as it here enters into con-
sideration^ is perfectly determined as soon as its temperature
T and its volume v, L e. the volume of the vessel, are given..
For, according to hypothesis, the vapour is always in contact
with the liquid, and therefore remains at its maximum density ;
so that its condition, as well as that of the liquid, depends only
upon the temperature T. It only remains to be seen, therefore,
whether the magnitude of each of the parts in different conditions
is perfectly determined, from the condition that both parts
together exactly fill the space enclosed by the vessel. Let s
represent the volume of the imit of weight of vapour at its maxi-
mum density where the temperature is T, and a that of the unit
of weight of liquid, then

t;=m«+ (M— m)<r
=m(*— o-)+Mo'.
The magnitude 8 never occurs hereafter except in the combina-
tion «— <r, so that we will introduce another letter for this dif-
ference, and make

tf=«— <r, (5)

Memoir (p. 132), the equation (12) of that memoir may be written, which
latter at once assumes the fomi (12 a), there given in a note, on regarding T
as the absolute temperature, and introducing it into the differential coefficients
instead of the temperature estimated from the freezing-point. For the sake
of reference I will here write all three equations, and that in the following
order:

d(dQ\ d(dQ\^l dQ ^

orKdv)

I dp

rft; • rfT ^"^^

Of these equations (a) expresses the first fundamental theorem, and (b) the
second frmdamental theorem employed in its modified form. The equation (c)
is obtained by combining, both these theorems. — 1864.]

THEORY OF THE STEAM-ENGINE. 149

in consequence of which the foregoing equation becomes

rssmw+M<r, (6)

and we have

v— Mo- ,^.

m±= (7)

By this equation m is expressed as a function of T and v, be-
cause u and a- are functions of T.

12. In order to be able to apply equations (III) and (IV) to

our case, we must next determine the magnitudes --r- and -t^*

If the volume of the vessel increases by dv, then the quantity
of heat which must be imparted to the mass in order to maintain
a constant temperature will be generally expressed by

dCL.

-J- dv.
dv

But this quantity of heat is expended solely in the vaporization

which takes place during the expansion ; so that if r represents

the heat required to vaporize the unit of mass, the above quan«

tity of heat may also be represented by

dm ,

dv *

and we have

dQ dm
dv dv

But according to (7),

dm 1
dv u'

hence

dOL r

^=- v . . . (8)

dv u ^ '

Let us next assume that, whilst the volume of the vessel
remains constant, the temperature of the mass increases by rfT ;
then the general expression for the requisite quantity of heat
will be

This quantity of heat consists of three parts : —

(1) The liquid part M— m of the whole mass suffers an incre-
ment of temperature rfT, for which, c being the specific heat of

150 FIFTH MEMOIR.

the liquid^ the quantity of heat

(M-m)crfT
is necessary.

(2) The vaporoufi part m will also undergo an increment of
temperature dT, but it will be thereby compressed so as still to
remain at its maximum density for the increased temperature
T + rfT. For an increment of temperature rfT, we will represent
hy h*dT the quantity of heat which must be imparted to the
unit of mass of vapour during its contraction^ in order that at
every density it may have precisely that temperature for which
this density is a maximum. The value and even the sign of the
magnitude A is at present unknown. The quantity of heat ne-
cessary in our case will therefore be

mhdT.

(3) During the elevation of temperature^ a small quantity of
liquid^ represented generally by

— rfT

becomes vaporized^ for which the quantity of heat

^dT
is necessary. Herein, according to equation (7),

dm r— Mq- 5^_M da

rfT"" ~ifi rfT "tt'Sr

m ^fcf^^M da^

"" tt rfT tt^'rfT'
so that by substitution the last expression becomes

\u rfT u rfT/

Equating the sum of these three quantities of heat and the
former expression ^^^ rfT, we obtain the equation

13. As indicated by equation (III), the above expression for

-— must be differentiated according to T, and the expression for
dv

THEORY OF THE STEAM-ENGINE. 151

— according to v. The magnitude M is constant, the magni-

tudes Uy 0-, r, c and h are all functions of T alone, and only the
magnitude w is a function of T and v, so that

dTKdvJ u'dT u^'dr' . • • • (10)

rf/rfQ\ A ^r^ du\dm
diAdTjV ^ u'dTJdv'

or, substituting for -^ its value -,

d /rfQ\ A~c _ r du /,,x

dvKdTJ u w« dT

By substituting the expressions given in (10), (11), and (8)
in (III) and (IV), we obtain the required equations, which re-
present the two principal theorems of the mechanical theory of
heat as applied to vapours at their maximum density. These are

^+c-A=A.«^ ....... (V)

r=A.Tu±; (VI)

and from a combination of both we have

5+'-*=}* ™

14. By means of these equations we will now treat a case,
which in the following will so frequently occur, as to render it
desirable at once to establish the results which have reference
thereto.

* [These equations, written in the following order,

^+c-.=A.«^. . . ... (a)

S+-*=T (^>

r=A-'I«%, (c)

correspond to the three equations (a), (b), (c) of the last note (p. 148). The first,
containing the quantity h, is therefore a consequence of the first fundamental
theorem, and the second follows from the modified form of the second funda-
mental theorem ; the third equation, which does not contain A, arises from a
combination of both these fundamental theorems. — 1864.]

152 FIFTH MEMOIR.

Let us suppose that the vessel before considered, containing
the liquid and vaporous parts of the mass, changes its volume
without heat being imparted to, or tvithdraum Jrom, the mass.
Then, simultaneously with the volume, the temperature and
magnitude of the vaporous port of the mass will change ; and
besides this — seeing that during the change of volume the
pressure of the enclosed vapour is active, which pressure during
expansion overcomes, and during contraction is overcome by an
external force — a positive or negative amount of external work
will be done by the heat which produces the pressure.

Under tJiese circumstances, the magnitude of the vaporous part
m, the volume v, and the work W shall be determined as Junctions
of the temperature T.

16. It has already been shown that, in order that the volume
and temperature may suffer any infinitely small increments dv
and dTy a quantity of heat expressed by the sum

r^dv+[(l>lL-fn)e+mh+r^]dT

must be imparted to the mass. In consequence of the present
condition, according to which heat is neither imparted to, nor
abstracted from the mass, this sum must be set equal to zero.
Accordingly, writing dm in place of

dm , , dm j^

we obtain the equation

rrfw+w(A-c)rfT + McrfT=0 (13)

But by (12),

- dr r

dr
so that by again writing dr in the place of ^ rfT, r being a fiinc-

tion of T alone, we have

rrfm -h w<fr -^ rfT + McrfT = 0,
or

rf(mr)-^rfT+McrfT=0^ (14)

* [It is manifest that the expressions in tlie left; of the equations (13) and
(14), which are respectively equal to zero when heat is neither imparted to

THEORY QF THE STEAM-ENGINE. 153

This equation^ divided by T, becomes

or

'^(f)+Mc^=0 (15)

Inasmuch as the specific heat of a liquid changes only very
slowly with its temperature^ we will in future always consider
the magnitude c as constant. In this case the above equation
can be immediately integrated^ and gives

^+M<:logT=const.;

or if T„ r^, and m^ be the initial values of T, r, and w,

^=^-Mclogl* (VII)

If r may be considered as a known function of the tempera-
ture^ as through Begnault^s experiments it may be in the case of
steam^ then by means of this equation m is also expressed as a
function of the temperature.

In order to give some idea of the deportment of this fimction^
I have, for one particular case, collected together a few calculated
values in the following Table. For instance, it is assumed that

the mass nor abstracted from it, most in general be equated to c{Q. For every
change of volume and temperature, therefore, whereby the quantity of the va-
porous part likewise changes in a corresponding manner, we have the equa-
tions

(?Q=:rrffii+»n(A-c)cfr-l-Modrr

=rf(mr)-^drr4-McdT,

the frequent applicability of which is obyious.— 1864.]

« [If the constancy of e be not assumed, the integral of the equation (15)
willbe ^

wherein the integration indicated in the last term may be effected sa soon as
c is given as a function of the temperature. All those equations in the sequel,
which contain an integral in whose development c was regarded as a constant,
are susceptible of a similar modification. I have not thought it necessary ac-
tually to write the equations in this form, since the modification in question
is a self-evident one. — 1864.]

154

FIFTH MEMOIR.

at the commencement the vessel contains no water in a liquid
state^ but is exactly filled with vapour at a maximum density^ so
that m,=M; and that an expansion of the vessel now takes
place. If the vessel has to be compressed^ then the assumption
that at the commencement it contained no liquid could not be
made^ because in such a case the vapour would not remain at a
maximum density^ but would become over-heated by the heat
generated through compression. During expansion^ however^
not only does the vapour remain at a maximum density^ but a
part of it is actually condensed ; and it is the diminution of m
consequent thereon which is exhibited in the Table. The initial

temperature is supposed to be 150^ C.^ and the values of tf aire

given which correspond to the periods when, by expansion, the

temperature is reduced to 125°, 100°, &c As before, in

order to distinguish it fix)m the absolute temperature T, the tem-
perature counted from the freezing-point is represented by t.

t.

i5o»

1*5-

I0O«

75°

50*

»5-

m
M

1

0-956

0*911

0-866

0-821

0776

16. In order to express the relation which exists between the
volume V and the temperature, we must employ the equation
(6), according to which

t;=mM+M<r.

The magnitude <r herein involved, which represents the volume
of a unit of weight of liquid, changes very little with the tempe-
rature; and these small changes may be the more safely neglected,
because the whole value of <r is very small in comparison to u ;
we shall consequently consider o-, as well as the product M<r, as
constant. The product mu therefore alone remains to be deter-
mined. For this purpose we have only to substitute the value
of r, as given in equation (VI), in equation (VII), and we obtain

dp

i»tt-^=miWi(^^j --^logjr. . . . (Vni)

dp

The differential coefficient ^ which here appears, is to be con-

THEORY OF THE STEAM-ENGINE.

155

sidered as known, ^ itself being given as a function of the tem-
perature. Hence the product mu is determined by this equation,
and by the addition of M<r the required value of v wiU also result
from it.

The same suppositions being made as before, the following

Table shows a series of values of the fraction — calculated from

this equation. For the sake of comparison the values of —

are also appended which would be obtained if the two assumptions
formerly made in the theory of the steam-engine were correct ;
that is to say, (1) that the vapour during expansion remaias
without partial condensation at a maximum density ; (2) that it
follows Mariotte's and Oay-Lussac's laws. According to these
hypotheses, we should have

», p T,

/.

iSo*^

i»5^

IOO*>

75"

50°

25°

P Tj

I
I

1-93

390
416

9-»3

257
»97

887
107-1

17. We have still to determine the work done during the
change of volume. In order to do so, we have the general
equation

W=Pj9rft; (16)

But, considering a constant, we have from equation (6),

therefore

pdv^pd{mu)j

for which we may abo write

pdv-d{fmp)-mu!^dT (17)

In the place of mu -^ we might here substitute the expression
given in (VIII), and then integrate; but the result is at once

156 FIFTH MEMOIR.

obtained in a rather more convenient form by the following sub-
stitution. According to (VI),

and through the application of equation (14), this becomes

mtt^rfT=~ ld{mr) + UcdT].

By means of this (17) becomes

pdv=d{rmp) --^ \d{mr) -f McrfT]*;

and integrating this equation, we have

W=mtt/i-»ii«,;i,-f ^ [m,r,-mr+Mc(T,-T)], (IX)

whence, the magnitudes mr and mu being abready known from

former equations, W may be calculated.

* I haVe also made this calculation for the above special case,

and given the values of rjrj, ». e. of the work done during expan-
M

sion by the unit of mass, in the following Table. A kilogramme is

chosen as unit of mass, and a kilogramme-metre as unit of work.

For — , the value 423*55, as found by Joule, is employedf.

A.

For the sake of comparison with the numbers in the Table, it
may be well to state that when 1 kilogramme of water is eva-
porated at the temperature of 150% and under the corresponding
pressure, the quantity of work done by the vapour during its
formation in overcoming the external counter-pressure has the
value 18700.

* [If instead of assuming <r to be constant, it be thought desirable to obtain
an accurate expression iorpdv, it will be necessary merely to supply the ex-
pression in the text with the additional term "Mpda-^ — 1864.]

t -^ is the equivalent of work for the unit of heat ; and the above number

denotes, therefore^ that the quantity of heat which can raise a kilogramme
of water from (F to 1° C, when converted into mechanical work, gives an
amount equal to 423*55 kilogramme-metres.

j

THEORY OF THE BTEAU-ENQINE.

157

/.

,500

i»S°

ioo«

75°

SO"

«S°

M

11300

23200

35900

49300

63700*

18. We proceed now to the consideration of the steam-engine
itself.

In the adjoining fig. 8, Fig. 8.

which is intended merely
to facilitate our oversight
of the whole series of
operations involved in the
working of a common
steam-engine^ A repre-
sents the boiler whose
contents are maintained
by the source of heat at a
constant temperature Tj.
A part of the steam passes
from the boiler to the cy-
linder B and raises the
piston a certain height. The cylinder and boiler are next dis-
connected, and the vapour contained in the former raises the
piston still higher by its own expansion. After this the cylinder
is put in communication with the space C, which shall represent
the condenser. We shall suppose the latter to be kept cold by
external cooling, and not by injected water, which, as before re-
marked, causes no essential diflference in the results, and yet sim-
plifies our problem. The constant temperature of the condenser
shall be Tq. During the connexion of the cylinder with the con-
denser the piston retraces the whole of its former path, and thus
all the vapour which did not immediately pass by itself into the
condenser is driven into it, and there becomes condensed. In
order to complete the cycle of operations, it is now necessary to
convey the liquid produced by condensation back again into the
boiler. This is done by means of the small pump D, whose ac-
tion is so regulated, that at every ascent of the piston just as much

* [With respect to certain formiilae of approximation^ which have been em-
ployed by Zeuner in order more easily to calculate the results given above
and in the Second Memoir, see the Appendix to the present memoir.]

158 PIFTH MEMOIR.

liquid is withdrawn from the condenser as entered it by the above
condensation ; and during the descent of the piston this same
quantity of liquid is forced back into the boiler. As soon as this
liquid is again raised in the boiler to the temperature Tj, every-
thing is once more in its initial condition^ and the same series of
operations can commence again. Here^ therefore^ we have a
complete cyclical process.

In ordinary steam-engines the steam enters the cylinder not
only at one end^ but alternately at both. But the only differ-
ence produced thereby is^ that during an ascent and descent of
the piston^ two circular processes take place instead of one ; and
in this case even the determination of the work for one of the
processes is sufficient^ because from it the total amount of work
done during any time can be deduced'^.

19. In making this determination^ we shaU^ as is indeed usual
in such cases^ consider the cylinder as impenetrable to heat, so
that we may neglect the interchange of heat which takes place
during a stroke between the walls of the cylinder and the vapour.

The mass in the cylinder can only consist of vapour at a
maximum density, together with some admixed liquid. For it is
evident from the foregoing that, during its expansion in the
cylinder, after the latter is cut off from the boilel*, the vapour
cannot pass into the over-heated condition, but must, on the
contrary, be partially condensed, provided no heat reaches it
from an external source. In other operations hereafter to be
mentioned, where this over-heated state might certainly occur,
it wiU be prevented by the smaU amount of liquid which the
vapour always carries with it into the cylinder, and with which
it remains in contact.

The quantity of liquid thus mixed with the vapour is incon-
siderable ; and as it is for the most part distributed tliroughout
the vapour in small drops, so that it can readily participate in
any changes of temperature which the vapour may suffer during
expansion, we shall incur no great inaccuracy if, in calculation,
we consider the temperature at any moment as the same through-
out the whole of the mass in the cylinder.

Further, in order to avoid complicating our formulas too much,

* The space on one side of the piston is a little diminished by the piston-
rod, but an allowance can easily be made for this small difference.

THEORY OF THE STBAM-ENOINE. . 159

we will for the present determine the total amount of work done
hy the vapour pressure, without taking into consideration how
much of this work is useful, and how much is again consumed
by the machine itself in overcoming friction, and in working
any pumps, which, besides the one in the figure, may be neces-
sary to the efficiency of the machine. This part of the work may
be afterwards determined and deducted, as will subsequently be
shown.

With respect to the friction of the piston in the cylinder, how-
ever, we may remark, the work consumed in overcoming it can-
not be considered as totally lost. For heat is generated by this
friction, and consequently the interior of the cylinder kept warmer
than it would otherwise be, and thus the force of the vapour
increased.

Lastly, inasmuch as it is advisable first to study the actions
of the most perfect machines before examining the influence of
the several imperfections which practically are always unavoid-
able, we will add to these preliminary considerations two more
suppositions, which shall afterwards be again relinquished. First,
the canal from the boiler to the cylinder, and that from the
cylinder to the condenser, or to the atmosphere, shall be so wide,
or the speed of the machine shall be so slow, that the pressure
in the part of the cylinder in connexion with the boiler shall be
equal to that in the boiler itself, and similarly the pressure on the
other side of the piston shall be equal to the pressure in the
condenser or to the atmospheric pressure; and secondly, no
vicious space shall be present.

20. Under these circumstances, the quantities of work done
during a cyclical process can be written down, without further
calculation, by help of the results above attained ; and for their
sum they give a simple expression.

Let M be the whole mass which passes from the boiler into

the cylinder during the ascent of the piston, and of it let m^ be

the vaporous, and M— mj the liquid part. The space occupied

by this mass is

niiUi + Mcr ;

where u^ is the value of u corresponding to Tj. The piston is
raised therefore until this space is left free under it ; and as this
takes place under the action of the pressure pi, corresponding to

160 FIFTH MEMOIR.

Ti^ the work performed daring this first operation is

Wi=mi«i/?i + Moy, (18)

The expansion which now follows is continued until the tem-
perature of the mass enclosed in the cylinder sinks from Tj to a
second given value T^. The work thus done, which shall be W^,
is given immediately by equation (IX), if T, be taken therein as
the final temperature, and for the other magnitudes involved in
the equation the corresponding values be substituted, thus

W,=»i«ttgp^-mittijpi+l [mirj-mgra -l-Mc(Ti-T2)]. . (19)

By the descent of the piston, which now commences, the mass,
which at the close of the expansion occupied the volume

WgMg + Ma-,
is driven from the cylinder into the condenser, and has to over-
come the constant pressure /?o« The negative work hereby done
by this pressure is

W3=— «ijt«aPo— Mo^o (20)

Whilst the piston of the small pump now ascends, so as to
leave the free space Mcr under it, the pressure Pq in the con-
denser acts favourably and does the work,

W4=M<iyo (21)

Lastly, during the descent of this piston, the pressure pi in
the boiler must be overcome, and therefore it does the negative
work

W^rr-Mcipi (22)

By adding these five magnitudes together we obtain the fol-
lowing expression for the work done by the vapour pressure, or,
as we may say, by heat, during a cyclical process :

W'=l [wi,r,-m,r,+Mc(T,-T,)] +^^(ft-/^o). (X)

With respect to the magnitude m^ which must be eliminated
from this equation, it will be observed that, if for u^ we substi-
tute the value

^'

as given in (VI), it only occurs in the combination m^^, and for

THEORY OP THE STEAM-ENOINE. 161

this product we have from equation (VII) the expression

T T

mjrjrr Wir, ^-McT, log ^.

By employing this expression^ therefore, we obtain an equation
the right-hand side of which contains only known quantities ;
for the masses w, and M, and the temperatures Tj, T^, and Tq
are assumed to be immediately given, and the magnitudes r, p,

and -^ are supposed to be known functions of the temperature.

21. If in the equation (X) we set T2=Tp we find the amount
of work, for the case that the machine works without expansion,
to be

W=m,u,{p,^Po) (23)

If, on the contrary, we suppose the expansion to be continued

until the vapour sinks from the temperature of the boiler to that of

the condenser, — ^which case cannot of course be strictly realized,

'but rather forms a limit which it is desirable to approach as much

as possible, — ^we have only to set l!^=T!f^, when we obtain

W'=^Kr^-moro+Mc{T,-To)]. . . . (24)

Eliminating mgr^ by means of the equation before given, in
which we must also set Tj=Tq, we have

W'=i-[»»,r,TiZ^+Mc(T,-To+Tolog^)]*. (XI)

* The above equations, representing the amount of work under the two
simplifying conditions introduced at the close of Art. 19, were developed by me ^
some time ago, and publicly commimicated in my lectures at the Berlin Uni-
versity as early as the summer of 1854. Afterwards, on the appearance, in
1855, of the Philosophical Transactions for the year 1864, 1 found therein a
memoir of Rankine's, " On the Geometrical Representation of the Expansive
Action of Heat, and the Theory of Thermo-dynamic Engines," and was sur-
prised to learn that at about the same date Rcmkine, quite independently, and
in a different manner, ^arrived at equations which almost entirely agreed with
mine, not only in their essential contents, but even in their forms; RanHne,
however, did not take the circumstance into consideration, that, when enter-
ing the cylinder, a quantity of liquid is mixed with the vapour. By the earlier
publication of this memoir I lost, of course, all claim to priority with respect to
this part of my investigations ; nevertheless the agreement was so far satisfac-
tory as to furnish me with a guarantee for the accuracy of the method I had
employed.

162 FIFTH MEMOIR.

22. If to the foregoing equation we give the form

then the two products Mc(Tj— Tq) andm^r^ which appear therein
together represent the quantity of heat furnished by the source
of heat during a cyclical process. For the first is the quantity
of heat which is necessary to raise the temperature of the liquid
mass M, coming from the condenser, from Tq to T^ ; and the
latter is the quantity consumed in vaporizing the part m^ at the
temperature T^. As m^ is but little smaller than M, the last
quantity of heat is far greater than the first.

In order more conveniently to compare the two factors with
which these two quantities of heat are multiplied in equation (25) ,
we will alter the form of the one which multiplies Mc(Tj— Tq).
If, for brevity, we make

•»^=%r^^ (26)

then

To ^l-z

and

Tr

-To z '

so thai

; we have

T.

= 1-^;

•

^Tj-T„

•^1

= l + i^log(l-

-z)

'^ z (l+2

4^

"1.3^2.3 ' 3.4

4-.&C

Hence the equation (25) or (XI) becomes
W'=«,.r..J+Mc(T,-To).£(ji3+3^ + ^+&c....). (27)

It is easy to see that the value of the infinite series, which
distinguishes the factor of the quantity of heat Mc(T^— T^) from
that of the quantity of heat m^r^, varies from i to 1, as z increases
from to 1.

23. In the case last considered, where the vapour by expan-
sion cools down to the temperature of the condenser, we can

1

THEORY OF THE STEAM-ENOINE. 1^3

easily obtain the expression for the work done in another man-
ner, without considering the several operations which constitute
the cyclical process.

For in this case every part of the cyclical process is reversible.
We can imagine that the vaporization takes place in the con-
denser at the temperature Tq, and that the mass M, of which m^
is vaporous and M— wiq liquid, enters the cylinder and raises the
piston ; farther, that by the descent of the piston the vapour is
first compressed imtil its temperature is raised to T^, and then
that it is forced into the boiler; and lastly, that by means of the
small pump the mass M is again conveyed in the liquid form
from the boiler to the condenser, and allowed to cool there to
the original temperature Tq. The matter here passes through
the same conditions as before, but in an opposite order. All
communications and abstractions of heat take place in opposite
order, but in the same quantity and at the same temperature of
the mass ; all quantities of work have opposite signs, but the
same numerical value.

Hence it follows that in this case no uncompensated trans-
formation is involved in the cyclical process, and we must con-
sequently set N=0 in equation (2), by which we obtain the fol-
lowing equation, — already given in (3), with the exception that
W is here put in the place of W, —

-4^-.rf>

In our present case, Q^ denotes the quantity of heat imparted to
the mass M in the boiler, that is.

In determining the integral I ^-=-, the two quantities of heat

Mc(Tj— Tq) and m^r^ contained in Q^ must be separately con-
sider^. In order to execute the integration extending over the
first quantity, we have but to give to the element of heat dQ, the
form McrfT, and this part of the integral is at once expressed by

During the communication of the latter quantity of heat, the

M 2

164 FIFTH MEMOIR.

temperature is constant and equal to T^ and consequently the
part of the integral referring to this quantity is simply

■tT'

By substituting these values, the foregoing expression for W
becomes

W'=^[m,r, + Mc(T,-T<^-To(^+Mclog^)]

=Wm,r, TL-To^.Mc(T,-To+Tolog^)] ;

and this is the same expression as that contained in equation (XI),
which was before obtained by the successive determination of the
several quantities of work done during the cyclical process.

24. From this it foUows that, if the temperatures at which the
matter manifesting the action of heat receives heat from the source
of heaty or imparts heat to some external object , are considered as
previously given, then the steam-engine, under the conditions
made in deducing the equation (XI), is k perfect machine ; that
is to say, for a certain amount of imparted heat it furnishes as
much work as, according to the mechanical theory of heat, is
possible at those temperatures.

It is otherwise, however, when those temperatures, instead of
being given, are also considered as a variable element, to be taken
into consideration in judging the machine.

One uncompensated transformation not included in N, which,
with respect to the economy of heat, causes a great loss,
arises from the fact that the liquid, during the processes of heating
and evaporation, has far lower temperatures than the fire, and
consequently the heat which is imparted to it must pass from a
higher to a lower temperature. The amount of work which can
be produced by the steam-engine from the quantity of heat
mjri-hMc(Ti— T^) = Clp is, as may be seen from equation (27),
somewhat smaller than

A • 1\
If, on the contrary, tre could impart the same quantity of heat
Qj to a changeable body at the temperature of the fire, which
may be T', whilst the temperature during the abstraction of heat

THEORY OP THE STEAM-ENOINE.

165

remained T^, as before, then by equation (4) tbe greatest possible
amount of work to be gained in such a case would be

Hi

A

T'-T

T'

0.

In order to compare the values of these expressions in a few
examples, let the temperature t^ of the condenser be fixed at
50° C, and for the boiler let us assume the temperatures H0°,
150°, and 180° C, of which the two first correspond approxi-
mately to the low- and the ordinary high-pressure machines
respectively, and the last may be considered as the limit of the
temperatures hitherto employed in steam-engines. In these
cases the fraction dependent upon the temperatures has the fol-
lowing values :— T-

'l-

IIO°

1500

180°

0-157

0-236

0-287

whereas the corresponding value for the temperature of the fire
f, assuming the latter to be only 1000*" C, is 0-746.

25. We may here easily discern, what has already been ex-
pressed by S. Camot and several other authors, that in order to
render machines driven by heat more efficient, attention must be
particularly directed towards the enlargement of the interval of
temperature between T^ and Tq.

For instance, machines driven by heated air wiU only attain a
decided advantage over steam-engines when a method is found
of allowing them to work at a far higher temperature than steam-
engines, in consequence of the danger of explosions, can bear.
The same advantage, however, could be attained with over-heated
vapour ; for as soon as the vapour is separated from the liquid,
it is just as safe to heat it farther as to heat a permanent gas.
Machines employing vapour in this condition may possess many
of the advantages of the steam-engine besides those of air-ma-
chines, so that a practical improvement may sooner be expected
fix)m these than from air-machines.

In the machines above mentioned, where, besides water, a
second more vaporizable substance was employed, the interval
Tj— Tq is increased by lowering Tq. It has already been sug-

166 Fiirrii memoir.

gested that this interval might be increased in a similar manner
on the upper side, by the addition of a third liquid less vapor-
izable than water. In such a case the fire would be immediately
applied to the evaporation of the least vaporizable of the three
substances, the condensation of this to the evaporation of the
second, and the condensation of the second to the evaporation of
the third. Theoretically, there is no doubt that such a combi-
nation would be advantageous; the practical difficulties, how-
ever, which would have to be overcome in realizing such a scheme
cannot of course be predicted.

26. Besides the above-mentioned defect, arising out of the
very nature of our ordinary steam-engines, these machines suffer
from many other imperfections, which may be ascribed more
immediately to defective construction.

One of these has already been considered in the foregoing
development, and allowed for in equation (X), that is to say, the
expansion cannot be continued nearly far enough to allow the
vapour in the cylinder to reach the temperature of the condenser.
If, for example, we assume the temperature of the boiler to be
150°, and that of the condenser to be 50°, then the Table in
Art. 16 shows that, for the above purpose, the expansion must be
prolonged to twenty-six times the original volume ; whereas in
practice, owing to many inconveniences. attending great expan-
sions, three or four, and at most ten times the original volume
is attained.

Two other imperfections, however, are expressly excluded in
the foregoing : these are, first, that the pressure of the vapour in
one part of the cylinder is smaller than in the boiler y and in the
other part greater than in the condenser i and secondly, the pre-
sence of vicious space.

We must consequently extend our former considerations so as
to include these imperfections.

27. The influence exercised by the difference 'of pressure in
the boiler and cylinder upon the work performed, has hitherto
been most completely treated of by Pambour in his work on the
Theorie des Machines a Vapeur. Before entering upon the sub-
ject myself, therefore, I may be allowed to state the most essen-
tial parts of his treatment, altering only the notation, and neg-
lecting the magnitudes which have reference to friction. By

THEORY OF THE STEAM-ENGINE. 167

this means it will be easier, on the one hand^ to judge how far
this treatment is no longer in accordance with our more recent
knowledge of heat, and, on the other, to add to it the new me-
thod of treatment which, in my opinion, must be substituted for
the former one.

28. The two laws which, as was before mentioned, were for-
merly very generally applied to steam, form the basis of Pam-
bour's theory. The first of these is WatVs law, according to
which the sum of the latent and sensible heat is constant.
From this law it was concluded that when a quantity of steam
at its maximum density is enclosed within a surface impenetrable
to heat, and the volume of the enclosing space is either increased
or diminished, the steam will neither become over-heated nor
partially condensed, but will remain precisely at its maximum
density ; and it was further assumed that this would take place
quite independently of the manner in which the change of vo-
lume occurred, whether thereby the steam had, or had not, to
overcome a pressure corresponding to its own expansive force.
Pambour supposed that the steam in the cylinder of a steam-
engine deported itself thus ; and at the same time he did not
assume that the particles of water, which in this case are mixed
with the steam, could exert any appreciable influence.

Further, in order to establish a more accurate relation between
the volume and the temperature, or the volume and the pressure
of steam at a maximum density, Pambour applied, secondly,
Mariotte's and Gay-Lussac's laws. If, with Gay-Lussac, we
assume the volume of a kilogramme of steam at 100° C, and
at its maximum density, to be 1*696 cubic metres, under a pres
sure of one atmosphere, which latter amounts to 10,333 kilo-
grammes on every square metre, then from the above law we
obtain the equation

. = 1.696.12???. _^Z^±L, (28)

p 273 + 100' ^ ^

where, with reference to the same units, v and p represent the
volume and the pressure corresponding to any other tempera-
ture t. Herein it is only necessary to substitute in place of p
the values given in the tension series in order to have, according
to the above assumption, the proper volume for each temperature.

168 FIFTH MEMOIR.

29. In order, however, to be able conveniently to calculate
the value of the integral

J^

pdvy

which plays an important part in the formula for the work done
by a steam-engine, it was necessary to find the simplest possible
formula between v and/? alone.

If, by means of the ordinary empirical formulae for p, the
temperature t were eliminated firom the above equation, the
results would prove to be too complicated; hence Pambour
preferred forming a special empirical formula for this purpose,
to which, according to the proposal of Navier, he gave the fol-
lowing general form : —

-if/ '^>

wherein B and b are constants. He then sought to determine
these constants, so that the volmnes calculated from this for-
mula might agree as nearly as possible with those calculated
from the foregoing one. As this could not be done with suffi-
cient accuracy, however, for all the pressures which occur in
steam-engines, he established two different formulae for machines
tvith and without condenser.
The first of these was

"t^-- • ■ ■ • • f^""

which agrees best with the above formula (28) between § and 3i
atmospheres, but is also applicable for a somewhat wider in-
terval, from about ^ to 5 atmospheres.
The second, for machines without condensers, is

v^ 3^^ , (296)

3020+/

which is most correct between 2 and 5 atmospheres, though the
range of its applicability extends from about 1^ to 10 atmo-
spheres.

30. The magnitudes which depend upon the dimensions of
the steam-engine, and enter into the determination of the work,
shall be here, somewhat differently from Pambour's method.

THEORY OF THE STEAM-ENGINE. 169

represented in the following manner. Let v* be the whole space
left free to the vapour during a stroke in the piston, the vicious
space being also included. Let the vicious space form a frac-
tional part e of the whole space, so that this space itself will
be represented by ev', and that described by the surface of the
piston by (1— e) v'. Further, let the part of the whole space
left free to the vapour up to the moment of disconnecting the
cylinder and boiler (also inclusive of vicious space) be repre-
sented by ev. Consequently the space described by the surface
of the piston during the entrance of the vapour will be expressed
by (e— €)t;', and that described by it during expansion will be
(l-e) »'.

In order to determine, in the next place, the amount of work
done during the entrance of the vapour, the pressure acting in
the cylinder during this time must be known. This is at any
rate smaller than the pressure in the boiler, otherwise no influx
of vapour could occur; but the magnitude of the difference
cannot in general be stated j for it depends not only upon the
construction of the engine, but also upon the engine-driver,
how far he has opened the valve in the tube leading from the
boiler, and with what velocity he drives the machine. These
things being changed, the above diflFerence may vary between
wide limits. Further, the pressure in the cylinder need not be
constant during the whole time of influx, because the velocity
of the piston may vary, as well as the magnitude of the influx
orifice left free by the valve or the slide.

With respect to the last circumstance, Pambour assumes that
the mean pressure to be brought into calculation in determining
the work may, with sufficient accuracy, be set equal to that
which exists in the cylinder at the end of the influx, and at the
moment of disconnexion from the boiler. Although I do not
think it advisable to introduce such an assumption — which is
only adopted for numerical calculation in the absence of more
certain data — at once into the general formulae, yet here, whilst
explaining his theory, I must adopt his method.

Pambour determines the pressure existing in the cylinder at
the moment of disconnexion by means of the relation, esta-
blished by him, between volume and pressure ; assuming at the
same time that the quantity of steam which passes from the

170 FIFTH MEMOIE.

I

boiler into the cylinder in a unit of time, and therefore the
quantity which passes during a stroke of the piston, is known ;

from special observations. As before, we will represent by M
the whole mass which enters the cylinder during a stroke, and
by m the vaporous part of the same. As this mass, of which
Pambour only considers the vaporous part, fills the space ei/ at
the moment of disconnexion, we have, according to (29), the
equation

where p^ represents the pressure at the same moment. Prom '

this equation we deduce (

Multiplying this magnitude by (e— €)t/, the space described
by the surface of the piston up to the same moment, we obtain
the following expression for the first part of the work : —

W^=w».B/-Ilf-t/(e-€)i. . . . (31)

The law according to which the pressure changes during the
expansion which now follows, is also given by equation (29).
If at any moment v represents the variable volume, and p the
corresponding pressure, then

V

This expression must be substituted in the integral
\pdv,

>

and the integration efiected between the limits i?=et/ and t;= «/ ;
whence, as the second part of the work, we obtain

W2=mB.logi-i/(l-c)*. . . . (32)

In order to determine the negative work done by the reacting
pressure during the descent of the piston, this reaction must
itself be known* Without at present inquiring into the relati(m
which exists between the reaction and the pressure in the con-

THEORY OF THE STEAM-ENGINE. 171

denser, we will represent the mean reaction by p^, so that the
work done by it will be expressed by

Ws=-vf(l-e)p^ (33)

There yet remains the work necessary to convey the quantity
M of liquid back again into the boiler. Pambour has not sepa-
rately considered this work, but has included it in the Mction
of the machine. As I have included it in my formulae, however,
in order to have the cycle of operations complete, I will also
here add it for the sake of easier comparison. As shown by
equations (21) and (22), established in a former example, this
work will on the^ whole be expressed by

W4=-.Mcr(i?,-;?o)> (34)

where j»j and j^q respectively represent the pressures in the boiler
and condenser. This expression, it is true, is not quite correct
for our present case, because by Pq we do not understand the
pressure in the condenser itself, but in the parts of the cylinder
in communication with the condenser. Nevertheless we wiU
retain the expression in its present form, for owing to the small-
ness at <r, the whole expression has a value scarcely worth con-
sideration ; and the inaccuracy, being again small in comparison
to the value of the expression itself, may with still greater im-
punity be disregarded.

By adding these four separate amounts of work together, we
find the whole work done during the cyclical process to be

W'=mB(^^+logi)-t/(l-6)(A+;?o)-M(r(;.,-;.o)- • (35)

31. If, lastly, we wish to refer the work to the unit of weight
of vapour instead of to a single stroke, during which the quan-
tity m of vapour acts, we have only to divide the foregoing value

M

by m. We will put I in place of the fraction — , which expresses

the relation which the whole mass entering the cylinder bears to
the vaporous part of the same, and whose value is consequently

a little greater than unity ; V in place of the fraction — , or the

whole space offered to the unit of weight of vapour in the cylin-

W
der ; and W in place of the fraction — , or the work correspond-

^

1^2 FIFTH MEMOIR.

ing to the unit of weight of vapour. We thus obtain

W-B(^%log^^)-V(l-6)(4+j,o)-/<r(/,,-i,o). . (XII)

Only one term of this equation depends upon V, and it con-
tains V as factor. As this term is negative, it follows that the
work which we can obtain from the unit of weight of vapour is,
all other circumstances being the same, greatest when the vo-
lume offered to the vapour in the cylinder is smallest. The
least value of this volume, which we may approach more and
more although we may never quite reach, is that which is found
by assuming that the machine goes so slowly, or that the influx
canal is so wide, that the same pressure j^, exists in the cylinder
as in the boiler. This case therefore gives the ma2dmum of
work. If with equal influx of vapour the velocity of motion is
greater, or with equal velocity of motion the influx of vapour is
smaller, we obtain from the same quantity of vapour a less
quantity of work.

32. Before we now proceed to consider connectedly the same
series of processes according to the mechanical theory of heat,
it will be best to submit one of the same, which requires especial
investigation, to a separate treatment in order at once to esta-
blish the results which have reference thereto. I refer to the
entrance of vapour into the vicious space and into the cylinder,
when it has there to overcome a smaller pressure than that tvith
which it was forced out of the boiler. In this investigation I
can proceed according to the same principles as those which
I have employed in a former memoir* when treating similar'
cases.

• " Ueber das Verhalten dee Dampfes bei der Aufldehnung unter verschie-
den€£Q Umstanden " [Second Memoir of this Collection]. With reference to
this memoir, and to a notice connected therewith, which appeared in the
Philosophical Magazine^ Helmholtz, in his report published in the Fort-
sckritte der Phydk, by the Physical Society of Berlin (years 1860 and 1861,
p. 682), is of opinion that the principle is in many points incorrect. I have
not, however, been able to understand the reasons he adduces in support of
this opinion. Views are ascribed to me which I never held, and, on the
other hand, theorems enunciated which I never disputed, and which, indeed,
partially constitute the basis upon which my own researches in the mecha-
nical theory of heat are founded j at the same time so great a generality is
maintained throughout, that I found it impossible to recognize how far those

THEORY OF THE STEAM-ENGINE. 173

The vapour from the boiler first enters the vicious space^ here
compresses the vapour of small density which still remains from
the former stroke, fills up the space thus becoming free, and
then presses against the piston, which, in consequence of its
assumed comparatively small charge, recedes so quickly that the
vapour cannot follow it quickly enough to reach the same den-
sity in the cylinder as it had in the boiler.

If saturated vapour alone issued from the boiler, it must under
such circumstances become over-heated in the cylinder, for the
vis viva of the entering -mass is here converted into heat ; as the
vapour, however, carries with it some finely divided drops of
water, a part of the latter will be evaporated by the surplus heat,
and thus the remaining vapour will be maintained in its satu-
rated condition.

We must now consider the following problem : — Given, first,
the initial condition of the whole mass under consideration, viz.
that which was previously in the vicious space, as well as that
more recently arrived from the boiler ; secondly, the magnitude
of the work done by the pressure acting upon the piston during
the entrance of the vapour ; and thirdly, the pressure in the
cylinder at the moment of cutting off the same from the boiler :
to determine how much of the mass in the cylinder at this moment
is vaporotts.

33. Let fjb be the whole mass in the vicious space before the
entrance of the fresh vapour, and, for the sake of generality, let
us suppose that the part ^q of it is vaporous and the rest liquid.
For the present letp^ and Tq represent respectively the pressure
of this vapour and its corresponding absolute temperature, with-
out implying, however, that these are exactly the same values as
those which refer to the condenser. As before, p^ and T^ shall
be the pressure and temperature in the boiler, M the mass
issuing from the boiler into the cylinder, and m^ the vaporous

views ought to follow from my words, and these theorems contradict my con-
clusions. I do not therefore feel myself called upon to defend my former
researches against this censure. As the following development, however,
rests precisely upon the same views which before served me, Helmholtz will
probably again find the same inaccuracy of principle. In such a case I shall
look forward' to his objections, and request him merely to enter somewhat
more specially into the subject.

174 FIFTH MEMOIR.

part of the latter. As we have already remarked^ the pressure
upon the piston during the entrance of the vapour need not be
constant. The mean pressure //, may be defined as that by
which the space described by the surface of the piston^ during
the entrance of the vapour^ must be multiplied in order to obtain
the same work as is actually done with the variable pressure.
Further, let j^^ and T^ be the pressure and corresponding tem-
perature in the cylinder at the moment of cutting it off from
the boiler ; and lastly, m^ the magnitude to be determined, that
is to say, the vaporous part of the whole mass M -f /* now in the
cylinder.

To determine this magnitude, let us conceive the mass M+/L6
reduced in any manner to its original condition. For instance,
thus : let the vaporous part m^ be condensed in the cylinder by
depressing the piston, whereby we shall suppose that the latter
can also enter the vicious space. At the same time let heat be
constantly withdrawn from the mass in such a manner that the
temperature T^ may remain constant. Then of the whole liquid
mass, let the part M be forced back into the boiler, where it may
assume its original temperature T^ By this means the condi-
tion of the mass within the boiler is the same as it originally
was, for of course it is of no importance whether precisely the
same mass m^ which was before vaporous, is again so now, or
whether another equally great mass has taken its place*. With

* pb fact at the end of the operation there is in the boiler just as much
liquid water and just as much steam, both at the temperature T^, as there
was at the beginning ; so that the original condition, so far as is necessary
for our consideration, is reestablished; for we are concerned solely with
the magnitudes of the vaporous and of the liquid portions of the whole mass,
and have not to inquire which of the several molecules there present
belong to the vaporous, and which to the liquid portion. If it were re-
quired that exactly the same molecules should constitute the vaporous por-
tion at the end, as at Ihe beginning of the operation, it would merely be
necessary to assume, first, that the water forced back into the boiler is not
only equal in quantity to that which originally quitted it, but that this water
consists of the same molecules ; and secondly, that of this water, after it has
attained the temperature T^, the formerly vaporous portion m^ again va-
porizesy an exactly equal quantity of that already present being precipitated.
For this purpose, of course, no heat need be imparted to, or withdrawn from
the total mass in the boiler; since the heat consumed in evaporation , and
that generated by precipitation, would compensate each other. — 1864.]

THEORY OF THE STEAM-ENGINE. 175

respect to the remaining part ^, let it be at first cooled in the
liquid state from T^ to Tq^ and at this temperature let the part
fiQ become vaporous, to do which the piston must recede so that
this vapour can again ocQupy its original volume.

34. In this manner the mass M +/i has gone through a com-
plete cyclical process, to which we may apply the theorem, that
the sum of all the quantities of heat received by the mass during
a cyclical process must be equivalent to the whole amoimt of
external work thereby performed.

The following quantities of heat have been successively con-
sumed : —

(1) To raise the temperature of the mass M in the boiler from
T, to T|, and at the latter temperature to evaporate the part m^

mir^ + McCTi-Tj).

(2) To condense the part m, at the temperature T^,

(3) To cool the part /i from Tj, to To,

-^c(T,-To).

(4) To evaporate the part jXq at the temperature T^,

Hence the total quantity of heat is

Q=m,r,-.m3r3 4-Mc(T,-T,) + /ioro-/^(T2-To). . (36)

The quantities of work may be found as follows : —
(1) In order to find the space described by the surface of the
piston during the entrance of the vapour, we know that at the
end of that time the whole mass M + /x occupies the space

mj,ti2+(M-f/A)o-.
Prom this we must deduct the vicious space. As at the com-
mencement, this was filled by the mass /lc, of which /Aq was vapo-
rous, at the temperature T^, its volume is

/AqWoH-zxct.

Deducting this from the foregoing magnitude, and multiplying
the difference by the mean pressure y^, we have for the first
amount of work.

176 FIFTH MEMOIR.

(2) The work expended in condensing the mass m^ is

(3) The work expended in forcing back the mass M into the
boiler is

(4) The work expended in evaporating the part fi^ is

By adding these four magnitudes^ we obtain for the whole work
W the following expression : —

If in the equation (I), which was
Q=A . W,
we substitute the values of Q and W thus found, and then bring
the terms involving m^ to one side of the equation, we have

m,[r,-^Au,{p\-^p,)'\ =m,r,4-Mc(T,-T,) •{•fi^^-fj^iT.^To)

-^AfjLoUo{p\-Po)+AMa{p,-p\) (XIII)

By means of this equation the magnitude m^ is expressed in
terms of other magnitudes, all of which are supposed to be
given.

35. If the mean pressure /?'i were considerably greater than
the final pressure p^, it might happen that the value of m^ would
be less than m^+fiQ, which would denote that a part of the vapour
originally present had become condensed. This would be the
case, for instance, if we were to suppose that, during the time
the vapour was entering the cylinder, the pressure there was
nearly equal to that in the boiler, and that by the expansion of
this vapour already in the cylinder, the pressure ultimately sunk
to the smaller value p^. On the contrary, ifp\ were but a little
greater, or indeed smaller than p^, then for m^ we should find a
value greater than m^+zi-Q. The latter ought to be considered
as the rule in steam-engines, and amongst others it holds for the
special case o{p\ =p^ assumed by Pambour.

We have thus arrived at results which differ essentially from
Pambour^s views. Whilst he assumes that the two different
kinds of expansion which successively take place in the steam-
engine are governed by one and the same law, according to which

J

THEORY 07 THE STEAM-ENGINE. 177

the original quantityof vapour is neither increased nordiminished^
but always remains exactly at its maximum density, we have found
two diflferent equations which point to different deportments. By
the equation (XIII), fresh vapour must be produced by the first
expansion during the entrance of the steam; and according to
the equation (VII), a part of the then existing vapour must
become condensed when the further expansion takes place, after
disconnecting the cylinder and boiler, during which time the
work done by the vapour corresponds to its full expansive
force.

As these two opposite actions, consisting of an increase and a
diminution of vapour, which must also exercise opposite influ-
ences on the work performed by the machine, partially cancel
one another, the ultimate result may, under certain circumstances,
be approximately the same as that to which Pambour^s simpler
assumption leads. We must not, however, on this account neg-
lect to consider this difference when once established, especially
if we wish to ascertain in what manner a change in the construc-
tion or driving of the steam-engine wiU affect the magnitude of
its work.

36. According to what was said in Art. 8, we can easily deter-
mine the uncompensated transformation which occurs in the ex-
pansion by referring the integral contained in the equation

N=-Jf

to the several quantities of heat expressed in Art. 34.

The quantities of heat m^^y^m^r^y and fi^^ are imparted at
the constant temperatures T^, Tg, and Tq, so that these parts of
the integral are, respectively,

^£}. _^V^ and ^Q^o

The parts of the integral arising from the quantities of heat

Mc(Ti— Tg) and— /ic(T2— To),are found, by the method adopted

in Art. 23, to be

T T

Mc log f^ and — /ac log ^.

By putting the sum of these magnitudes in place of the above

178 FIFTH MEMOIR.

integral, we obtain for the nncompensated transformation the
value

N=-^+^»-Mclog^-^o+Mclog|. . (38)

37. We can now return to the complete cyclical process which
occurs in an acting steam-engine, and consider the several parts
thereof in the same manner as before.

The mass M, of which the part m^ is vaporous and the rest
liquid, issues from the boiler, where the pressure is supposed to
be p^y into the cylinder. As before, the mean pressure acting
in the cylinder during this time shall be j^'p and the final pres-
sure jOg.

The vapour now expands until its pressure sinks from p^ to a
given value p^ and consequently its temperature from Tg to Tg.

After this the cylinder is put in communication with the con-
denser, where the pressure is p^y and the piston returns through
the whole of the space it has just described. When the motion
is somewhat quick, the reaction which it now experiences will
be somewhat greater than j^^; to distinguish it from the latter
value, we wiU represent the mean reaction by pf^.

Similarly, the pressure of the vapour which remains in the
vicious space after the piston's motion is completed wiU not ne-
cessarily be equal either- ibp^ or to jo'q, and must consequently
be represented by another symbol jo"q. It may be greater or less
than p'q, according as the communication with the condenser is
cut off somewhat before, or somewhat after the conclusion of the
piston's motion j for in the first case the vapour would be a little
further compressed, whereas in the latter case it would liave
time to expand a little more by paxtiaUy passing into the con-
denser.

Lastly, the mass M is conveyed back from the condenser into
the boiler, when, as before, the pressure p^ acts favourably, and
the pressure p^ has to be overcome.

38. The expressions for the amounts of work done in these
processes will be quite similar to those in the simpler case before
considered, except that a few simple changes in the indices of
the letters will have to be made, and the magnitudes which refer
to the vicious space will kave to be added. In this manner we
obtain the following equations.

THEORY OP THE STEAM-ENGINE. 179

During the time vapour is passing into the cylinder, we have,
according to Art. 34,

W,= Kw3-fM<r-/^Oy,, .... (39)

where w"q is simply substituted for Wq.

By putting M H-^it in place of M, we have, from equation (IX) ,
during the expansion from the pressure j^^ to tte pressure je?3,

W3=m3M3;?3-w^2tt^2 + ^[w^27•2-w^3r3+ {M + fi) ciT^-^Ts)! - (40)

During the return of the piston, when its surface is traversing
the whole space occupied by the mass M-h/-t at the pressure j!?3,
diminished by the vicious space fiQU^^Q-^-jj^a-, we have

W3=-(m3M3+Mo--/Xott"o)jo'o. . . . (41)
Lastly, during the conveyance of the mass M back into the
boiler, we have

W,= -M<T{p,-po) (42)

Consequently the whole work done is

W=~[m,r,-m3r3+ (M+Ai)c(T,-T3)] +m^u^{p\ -^p^) \ ^^^

+ %««3(P8-yo)-M<r(p,-/,4-yo-i^o)--Mo<(p'i--i>o)J
* The masses m^ and m^ which are here involved, are given by
the equations (XIII) and (VII), provided in the former we put
p"q in the place of j»q, and change the magnitudes Tq, Tq, and Uq in a
similar manner, and in the latter we substitute M.+fi for M.
Nevertheless, although it is possible to eliminate m^ and m^ by
means of these equations, I will here merely replace m^ by its
value ; it being more convenient in calculation to consider the
equation which thus results in connexion with the equations
(XIII) and (VII) before obtained. The following, therefore,
is the most general form of the system of equations which serve
to determine the work done by the steam-engine : —

W^=^[m,r,-m3r3+Mc(T,-T3)+/.or"o-M^(T3-n)]

+^31^3(^8-^0) +/^o«*"o(i>'o-i>' 0) -M<r(yo-/>o). V/VTTn
m^[r^ + Au^{p\--p^] =m,r, + Mc(T,-T2) +/./'o ('^^^^^

-MT2-T"o) + A/.o<(/>',-A) + AM<r(p,-yp,

^3^^2^(M + ^)elog|.

n2

180 FIFTH MEMOIR.

39. Before endeavouring to render these equations more con-
venient for application^ it may not be without interest to show
how^ for an imperfect steam-engine, the same expressions may
be arrived at by a method before alluded to^ and opposite to the
one just applied. In order to avoid prolixity in this digression^
however, we will consider two only of the imperfections provided
for in the above equations, viz. the presence of vicious space, and
the existence of a smaller pressure in the cylinder than in the
boiler during the time that the vapour is passing into the former.
On the other hand, we shall assume the expansion to be com-
plete, therefore T3=To, and the magnitudes Tq, Tq, and T'q to
be equal.

In this determination we shall have to employ the equation
(2), to which we will give the following form : —

^=K'^-nrf)4"N-

The first term on the right-hand side of this equation denotes
the work which could be obtained from the employed quantity
of heat Qj, which in our case is represented by m^r^ -f- Mc(Tj — Tq) ,
did not the two imperfections exist. This term has been already
calculated in Art. 23, and found to be \

i[m,r,+Mc{T,-To)-To(^ + Mclog^J].

The second term denotes the loss of work caused by those two
imperfections. The magnitude N contained therein has been
calculated in Art. 36, and is represented by the expression in
equation (38).

Substituting these two expressions in the foregoing equation^
we have

W'=i[m,rj-^».^,+Mc(Tj-To)-(M+M)cTolog^+y«o»-o] • (44)

That this equation actually agrees with the equations (XIV),
may be easily seen by using the third equation in (XIV) in
order to eliminajte W3 from the first, and then setting T3=Tq
=T'o=TV
In the same manner we might make allowance for the loss of

THEORY OF THE STEAM-ENGINE. 181

work occasioned by incomplete expansion. To do so it would
only be necessary to calculate the uncompensated transformation
which occurs during the passage of the vapour from the cylinder
to the condenser, and to include it in N. By this calculation,
which need not here be executed, we obtain precisely the expres-
sion for the work which is given in (XIV) .

40. In order next to be able to use the equations (XIV) in
a numerical calculation, it will be necessary first to determine the
magnitudes y^ j^q, and j»"q more precisely.

With respect to the manner in which the pressure in the
cylinder varies during the entrance of the steam, no general law
can be instituted, because the entrance canal is opened and closed
in such a variety of ways in different machines. Hence no defi-
nite general value can be found for the relation between the
mean pressure y^, and the final pressure /?2, as long as the latter
is strictly interpreted. Nevertheless this will be possible if the
signification of j^g be slightly changed.

The cylinder and boiler cannot of course be instantaneously
disconnected; more or less time is always required to move the
necessary valves or slides, and during this interval the vapour in
the cylinder expands a little, because the orifice being diminished,
less steam enters than that which corresponds to the velocity
of the piston. In general, therefore, we may assume that at the
end of this time the pressure is already somewhat smaller than
the mean pressure p\.

But if, in. calculation, instead of restricting ourselves to the
end of the time necessary for closing the entrance canal, we allow
ourselves a little freedom in fixing the time of disconnexion, we
shall be able to obtain other values for p^. We can imagine the
point of time so chosen, that if, previously thereto, the whole
mass M had entered, the pressure at that moment would have
been precisely equal to the mean pressure calculated up to the
same time. By substituting this instantaneous disconnexion
in place of the actual gradual one, we incur but an insignificant
error, as far as the amount of work is concerned. We may
therefore, with this modification, adopt Pambour's assumption,
that p\=P29 reserving, however, for special consideration in
each particular case the proper determination, according to the
existing circumstances, of the moment of disconnexion.

182 FIFTH MEMOIR.

41. With respect to the reaction //oat the return of the piston,
it is evident that, other circumstances being the same, the dif-
ference p'o'-Po '^'^ ^® smaller the smaller j^^ is. In machines
with a condenser, therefore, it will be smaller than in maclunes
without a condenser, where p^ is equal to one atmosphere. In
locomotives, the most important machines without condensers,
there is usually a particular circumstance tending to magnify
this difference. The steam, instead of being allowed to pass off
into the atmosphere through a tube as short and wide as pos-
sible, is conducted into the chimney and there made to issue
through a somewhat contracted blowpipe in order to create an
artificial draft.

In this case an exact determination of the difference is essen-
tial to the accuracy of the result. In doing so, regard must be
had to the fact, that in one and the same machine the difference
is not constant, but dependent upon the velocity with which it
works; and the law which governs this dependence must be
ascertained. Into these considerations, and into the investiga-
tions which have already been made upon the subject, I will not
here enter, however, because they do not concern the present ap-
plication of the mechanical theory of heat.

In machines where the vapour from the cylinder is not thus
employed, and particularly in machines with a condenser, je/q dif-
fers so little from Pq, and therefore can change so little with the
working velocity, that it is suflBlcient for most investigations to
assume a mean value for p'^.

Seeing, further, that the magnitude /?q occurs only in one term
of the equations (XIV) , which term involves the factor o*, it can
have but a very small influence on the amount of work ; so that
without hesitation we may put, in place otp^, the most probable
value otp'o*

As already mentioned, the pressure p"q in the vicious space
may vary very much, according as the cylinder is cut off from
the cojidenser before or after the end of the piston^s motion. But
here, again, this pressure, and the magnitudes dependent thereon,
occur only in terms of the equations (XIV), which involve
the small factors /jl and fiQ ; so that we may dispense with an ac-
curate determination of this pressure, and rest satisfied with an
approximate evaluation. In cases where no particular circum-

THEORY OF THE STEAM-ENGINE. 183

stances are present to cause p^\ to differ essentially from
p\y their diflference, like the diflFerence between p\ and p^,
may be neglected, and the most probable value of the mean re-
action in the cylinder may be assumed as the common value of
all the magnitudes. This value may be represented simply
byi^o-

By introducing these simplifications, the equations (XIV)
become

W'=l[m,ri-m3r3+Mc(T,-T3)+/.oro-/tc(T3-To)]

-fWl3W3(/?3— ^o),

m,r,=wi,r, + Mc(T,-T,)+^o^o-/^^(T3-To) ^P^^^u^{p^^p^) \ (XV)
+ AM<r(;?i-^2),

^=^+(M+/.)clog|

42. In these equations it is assumed that the four pressures,
P\i P2y Ps> ^^^ Poy ^^ what amounts to the same, the four tem-
peratures Tj, Tj, T3, and Tq are given, as well as the masses
M, wij, fi and fi^, of which the first two must be known from
direct observation, and the last two may be approximately deter-
mined from the magnitude of the vicious space. In practice,
however, this condition is only partially fulfilled, so that in
calculation we must have recourse to other data.

Of the four pressures, only two, jt?^ bjxA Pq, can be assumed as
known. The first is given immediately by the manometer on
the boiler, and the second may at least be approximately deduced
from the indications of the manometer attached to the condenser.
The two others, p^ and p^, are not given ; but in their place we
know the dimensions of the cylinder, and at what position of the
piston the cylinder is cut off from the boiler. From these we
may deduce the volumes occupied by the steam at the moment
of disconnexion and at the end of the expansion, and these two
volumes will then serve as data in place of the pressures p^
and p^.

We must now bring the equations into such a form that the
calculation may be made by means of these data.

43. Let v', as in the explanation of Pambour's theory, again
be the whole space, including vicious space, set free during one

184 FIFTH MEMOIR.

stroke in the cylinder; ex/ the space set free up to the time of
disconnexion from the boiler; and er' the vicious space. Then^
according to what was before said^ we have the following equa-
tions : —

m,V3+ (M+/A)<r=«/,

1113113+ (M+/Lfr)cr= ^,

The magnitudes /a and o- are both so small that we may at once
neglect their product^ so that the above become
m j«a = «;' — M<r, >

Mo—- J

Further, according to equation (VI),
rrsATujf,
where, on account of its subsequent frequent occurrence, a single

letter g is introduced in place of the differential coefficient -^.

Accordingly, in the above system of equations we may express
r, and ^3 in terms of u^ and 113 ; and then, as the masses m^ and
m^ will only occur in the products mjn^ and m3ti3, we may sub-
stitute the values of the latter as given in the first two of
equations (45).

Similarly, by means of the last of these equations, we may
eliminate the mass /l6q ; and as to the other mass fi, although it
may be a little greater than /l^q, yet the terms which contain it
as a factor are altogether so unimportant, that we may without
hesitation give it the same value as we have found for /i^; in
other words, for the numerical calculation we may give up the
assumption, made for the sake of generality, that the mass
in the vicious space is partially liquid and partially vapo-
rous, and suppose that the mass in question consists entirely of
vapour.

The substitutions here mentioned may be made in the general
equations (XIV), as well as in the simplified equations (XV).
As they present no difficulties, however, we will here limit our-
selves to the last, in order to obtain the equations in a form
convenient for numerical calculation.

THEORY OF THE STEAM-ENGINE.

185

After this change the equations become

K_M,)v..a!l±Mc(Tii:3aH.^('i=£^ +,°_^)

+M<r (/>,-/>,),
{V-M<r)ffs= («/-M<T)ir, + (M+^)-|log ^.

44. In order to refer these equations^ which now express the
work done in a stroke or by the quantity m^ of vapour, to the
unit of weight of vapour, we have to proceed in the same manner
as when the equations (35) were changed into (XII) ; that is to
say, we divide each of the three equations by m^, and set

i».

— =V, and — =W.

Hereby the equations become

w=!i±i£2.=ia_(v-j,)(T^.-,.+^j+.vrir^=^, ■

(V-to)j»3=(«V-fa),.+((+ ^)jlog^-

45. These equations may be applied in the following manner
to the calculation of the work. From the intensity of evapora-
tion, supposed to be known, and from the velocity with which
the machine is at the same time driven, we determine the volume
• V which corresponds to the unit of weight of vapour. By means
of this value we calculate the temperature T^ from the second
equation, afterwards the temperature T3 from the third, and
lastly, we employ the temperature T3 to determine the work
from the first equation.

In doing so, however, we encounter a peculiar difficulty. In
order to calculate T^ and T3 from the two last equations, they
ought in reality to be solved according to these temperatures.
But they contain these temperatures not only explicitly, but

186 FIFTH MEMOIR.

implicitly, p and ff being functions of the same. If, in order to
eliminate these magnitudes, we were to replace p by one of the
ordinary empirical formulae which express the pressure of a vapour
as a fiinction of its temperature, and ff by the diflTerential co-
efficient ofp, the equations would become too complicated for
further treatment. We might, it is true, like Pambour did,
help ourselves by instituting new empirical formula more con-
venient for our purpose, which, if not true for all temperatures,
would be correct enough between certain limits. Instead of here
making any such attempt, however, I will draw attention to
another method, by which, although the calculation is some-
what tedious, the several parts thereof are capable of easy execu-
tion.

46. When the tension series for the vapour of any liquid is
known with sufficient accuracy, the values of the magnitudes
ff and T^ for the several temperatures can be calculated from it,
and arranged in tables in the same maimer as is usually done
with the values o{p.

In the case of steam, hitherto almost solely used in ma-
chines, and for the iaterval of temperature extending from 40°
to 200^ C, between which the application takes place, I have,
with the help of Regnault^s tension series, made such a calcu-
lation.

Strictly, I ought to have diflferentiated according to t the for-
mulae which Regnault used in calculating the several values of
p below and above 100° C, and then to have calculated ff by
means of the new formulae thus obtained. But as it appeared
to me that those formula did not fulfil their purpose perfectly
enough to justify so large an amount of labour, and as the cal-
culation and institution of another suitable formula would have
been still more tedious, I contented myself with using the num-
bers ab^ady calculated for the pressure in order approximately
to determine the diflRerential coefficient of the pressure. For
example, j9i4g andj^j^g being the pressures for the temperatures
146° and 148°, I have assumed that the magnitude

2
represents with sufficient accuracy the value of the diflTerential
coefficient for the mean temperature 147°.

THEORY OP THE STEAM-ENGINE. 187

In doing this, I have, for temperatures above 100°, used the
numbers given by Regnault himself*. With respeqt to the
values below 100°, Moritzf has lately drawn attention to the
fact that the formula employed by Regnault between 0° and 100°
was, especially in the vicinity of 100°, somewhat incorrect in
consequence of his having used logarithms of seven places in
calculating the constants. In consequence of this, Moritz has
calculated those constants with logarithms often places, basing
liis calculations on the same observed values ; and he has pub-
lished the values of p (as far as they diflFer from Regnault's,
which only occurs above 40°) thus deduced from the .corrected
formulae. I have used these values.

As soon as g is calculated for the several temperatures, the
calculation of T . ^ also is attended with no further difficulty,
because T is determined from the simple equation

I have given the values of ^ and T . ff thus found in a Table
at the end of this memoir. For the sake of completeness, I have
also added the corresponding values o{ p; those above 100°
being calculated by Regnault, and those below by Moritz. To
each of these three series of numbers are attached the differ-
ences between every two successive numbers; so that from
the Table the values of the three magnitudes can be found for
every temperature ; and conversely, for any given value of one
of the three magnitudes the corresponding temperature can be
seen.

After what was before said of the calculation of g, it need
scarcely be mentioned that the numbers of this Table are not to
be considered as quite exact ; they are only communicated in the
absence of better ones. As, however, the calculations with refer-
ence to steam-engines are always based upon rather uncertain
data, the numbers can without hesitation be used for this pur-
pose, there being no fear that the uncertainty of the result will
be much increased thereby J.

* MSm» de VAcad. des Sciences, voL xxi. p. 626.

t BuUetin de la Classe PhyskO'mathematiqtie de VAcad, de St. P^tet^a-'
bourg, vol. xxi. p. 41.

X [Since the differential coefficient ^ frequently presents itself in calcu-

at

188 nVTH HEMOIB.

As to the method of application^ however, another remark is
still necessary. In the equations (XVII) ^ it is assumed that the
pressure/? and its differential coefficient g are expressed in kilo-
grammes to a square metre ; whereas in the Table the same unit
of pressure, a millimetre of mercury, is retained as that referred
to in Begnault's tension series. In order, notwithstanding this,
to be able to apply the Table, it is only necessary to divide every

lations connected with yapour, it is of interest to know how far the con-
yenient method of determining it, employed by me, is trustworthy. I will
therefore here collect a few numbers for the sake of comparison.

In calculating the values of the yapour-tensions for temperatures above
100^, contained in his Tables, Regnault employed the formula

Logj^sa— &«x— 0/3*,
wherein Log refers to common or Briggs*s logarithms, x denotes the tempera-
ture calculated from — 2(P, so that ;r=<+20, and the five constants are
given by the equations

0=6-2640348,
Log 6=01307743,
Log 0=06924351,
Log 0=1-994049292,
Log/3=l-998343862.

On deducing an equation for -^ from this formula for j>, we have

wherein « and /3 have the same values as before, and the new constants A
and B are given by the equations

Log A= 2-6197602,

LogB=2-6028403.
On calculating from the above equation the value of the difierential co-
efficient ^ mentioned, by way of example, in the text and having refer-
ence to the temperature 147°, we find

By the above approximate method of determination we have, according to
Begnault's Tables,^ the tensions

i?i^=3392-98,

-Pue=3212-74,
and thence

5 2~

This approximate value, as is at once seen, agrees so well with the more accu-

THEORY OF THE STEAM-ENGINE. 189

term in those equations^ which does not contain either^ or ^ as
factor, by the number 13*596. This number, which is nothing
more than the specific gravity of mercury at 0° C, compared
with water at its maximum density, wiU for the sake of brevity
be represented by A:*.

This change of the formulae, however, scarcely increases the
calculation, inasmuch as it is equivalent to substituting every-
where, in place of the constant factor -j-, — ^which, according to

rate one calculated from the above equation, that it may without hesitation be
employed in calculations connected with the steam-engine.

With respect to temperatures between QP and 100°, Regnault employed the
following formula for calculating the vapour-tensions : —
Log jp SB a+ ba^—c^.

The constants, according to the improved calculations of Moritz, have the
following values : —

a«4-73d3707,
Log 6=21319907112,
Log c=0-6117407675,
Log«=0-006864937162,
Log /3 = 1-996725636856.

From this formula an equation for ^ may be again deduced, of the form

wherein the constants «, p have the values above indicated, and A and B are
given by the formulso

LogA=4-6930586,

Log B=2-8513123.
On calculating from this equation the value of ^ corresponding, for in-
stance, to a temperature of 7(P, we find

and by the approximate method of determination we have

?n~£s?=10113;

a numher which again agrees satisfactorily with the one calculated from
the more accurate equation. — 1864.]

* [To express a pressure of p millimetres of mercury, in kilogrammes per
square metre, the number p must be multiplied by the weight of a column of
mercury, having a height of one millimetre and a base equal to one square

190 FIFTH MEMOIR.

Jotde, has the value 423*65 already mentioned, — ^the other con-
stant,

jk=W£='''''''' • • • • (^)

when, instead of the work W, the magnitude -r- will be found

in the first instance, and will subsequently merely have to be
multiplied by *.

47. Let us now return to the equations (XVII), and consider
first the second of them.

This equation may be written in the following form : —

'r^2=C + ait,-t,)-bip,-p^, . . . (47)
wherein the magnitudes C, a, and b are independent of t^, and
have the following values : —

cV-Zo-

Of the three terms on the Tight-hand side of (47), the first
far exceeds the others ; hence it will be possible, by successive
approximation, to determine the product T^^a^ and thence alsa
the temperature t^.

In order to obtain the first approximate value of the product,
which we will call Tff', let us on the right side of (47) set /^ in
the place of t^, and corresponding thereto p^ in place o{p^y then

Ty=C (48)

metre. The volume of such a column is the -nsVu^th part of a cubic metre,
in other words a cubic decimetre. Now a cubic decimetre of water at the
maximum density weighs 1 kilogramme, and consequently a cubic decimetre
of mercury at 0^ weighs 13*596 kilogranames. This is the factor, therefore,
with which the number j9 must be multiplied in the case under consideration.
In our equations, however, it will of course amount to the same thing if,
instead of multiplying the terms which contain the factor p, or the differen-
tial coefficient of p by 13*696, we divide the remaining terms by the same
number.— 1864.]

THEORY OF THE STEAM-ENGINE. 191

The temperature f, corresponding to this value of the product,
can be sought in the Table. In order to find a second ap-
proximate value of the product, the value of f just found, and
the corresponding value of the pressure p\ are introduced into
(47) in the places of t^ and p^y whereby, having regard to the
former equation, we have

TV'=TV + a(^,-0 -«(/>! -i>')- • . . (48 a)

As before, the temperature f, corresponding to this value of
the product, is given by the Table. If this does not with suffi-
cient exactitude represent the required temperature t^ the same
method must be repeated. The newly-found values f and y
must be substituted in (47) in place of t^ and p^, whereby with
the assistance of the two last equations, we have

T'iyf=TY-^a{t*^t")-b{p'-p"),. . . (48*)

and in the table we can find the new temperature f.

We might proceed in this manner for any length of time,
though we shaU find that the third approximation is alrieady
within Y^dth, and the fourth within j^^^^th of a degree of the
true value of the temperature t^.

48. The treatment of the third of the equations (XVII) is
precisely similar. If we divide by V— Zc, and for facility of cal-
culation introduce Briggs's logarithms (Log) in place of natural
logarithms (log) by dividing by M the modulus of this system,
the equation will take the form

^3=C+aLogJ^ (49)

wherein C and a are independent of T3, and have the following
values : —

a=

/ €V\

M.Ak(V-l<T)'

(49 a)

Again, in equation (49) the first term on the right is greatest,
80 that vre can apply the method of successive approximation.

192 PIFTH MEMOIR.

In the first place^ T, is put in the place of Tq, and we obtain the
first approximate value oSy^, viz.

/=C, (50)

from which we can find the corresponding temperature f in the
Tables, and thence the absolute temperature V. This is now
substituted for T, in (49), and gives

/=y + aLog^, (50 a)

whence T" is found. Similarly we obtain

/'=y' + aLog^, (50*)

and so forth.

49. Before proceeding to the numerical application of the
equations (XVII), the magnitudes c and r alone remain to be
determined.

The magnitude c, which is the specific heat of the liquid, has
hitherto been treated as constant in our development. Of course
this is not quite correct, for the specific heat increases a little
with increasing temperature. If, however, we select as a common
value the one which is correct for about the middle of the inter-
val over which the temperatures involved in the investigation
extend, the deviations cannot be important ones. In machines
driven by steam, this mean temperature may be taken at 100° C. ;
this being, in ordinary high-pressure engines, about equally
distant from the temperature of the boiler and that of the con-
denser. In the case of water, therefore, we will employ the
number which, according to Begnault, expresses its specific
heat at 100°, and thus set

c=10180 (51)

In the determination of r we shall start from the equation
X=606-5 + 0-805.^,

given by Begnault as expressing the whole quantity of heat
necessary to raise the unit of weight of water from OP to the
temperature /, and afterwards to evaporate it at that tempera-
ture. According to this definition, however.

-j;

cdt+r,

so that

THEORY OF THE STEAM-ENOINE.

r=606-5-f0-305./-l cdt,

193

-i'

In order to obtain precisely Regnanlt's value of r, we ought
to substitute for c in the above integral, the function of the tem-
perature which Regnault determined. For our present purpose,
however, I think it will suffice to give to c the constant value
above selected, by means of which

r

Jo

crf/=1013.^.

and the two terms in the above equation involving t combine to
form the single one —0-708 . t.

At the same time we must alter the constant term of the equa-
tion a little, and determine it so that the formula will correctly
express that observed value of r which in all probability is most
accurate. As a mean of thirty-eight observations, Regnault
found the value of X at 100° to be 636-67. Deducting the
quantity of heat necessary to raise the unit of weight of water
from 0° to 100°, which, according to Regnault, amounts to 100*5
units of heat, and contenting ourselves with one decimal, there
remains

rioo=586-2*

Employing this value, we obtain the following formula :—

r= 607 -0-708./ (52)

The following comparison of a few values calculated here-
from, with the corresponding ones given by Regnault in his
tables t> will show that this simplified formula agrees suffi-
ciently well with the more accurate method of calculation above
alluded to : —

t.

0°.

SO-. ■

ioo«.

150^

200<».

r according to equation (52)
r according to Regnault ...

607-0
6065

571-6
571-6

536-2
5365

500-8
5007

465-4
464-3

• In his tables Regnault gives, instead of this, the number 538-5 ; the
reason is, however, that instead of the above value 636*67 for X at 100^, he
used the round number 637 in his calculations.

t M&fn. de VAcad, des Sciences, vol. xxi. p. 748.

194 FIFTH MEMOIE.

50. In order to be able to distinguish between the effects of
the two different kinds of expansions to which the two last of
the equations (XVII) refer, it will perhaps be best to consider
in the first place a steam-engine in which only one of them takes
place. We will commence, therefore, with one of the machines
which are said to work untkout expansion.

In this case, c, which expresses the relation of the volumes
before and after expansion, equals 1, and at the same time
Tg=T2; so that the equations (XVII) assume a simpler form.

The last of these equations becomes an identity, and therefore
vanishes. Further, many terms of the first will admit of elimi-
nation, because they now become like the corresponding terms
of the second, ifrom which they before differed only by contain-
ing Tg instead of T,. . Introducing the above-mentioned quantity
k at the same time, we now obtain

W "1

y=V(l-6)(;?j-/?o)-^(i?x-i^o)>

^ (XVIII).

The first of these two equations is exactly the same as the one
which we also obtain by Pambour^s theory, if in (XII) we make
€=1, and introduce V instead of B. The second equation,
however, differs from and replaces the simple relation between
volume and pressure assumed by Pambour.

51. To the quantity €, which occurs in these equations and
represents the vicious space as a fractional part of the whole
space set free to the vapour, we will give the value 0*05 . The
quantity of liquid which the vapour carries with it on entering

* [If in the two first "equations in (XVII) we naake essl, l!i=l^^ and intro-
duce the quantity kj the second eqiiatipn at once reduces itseK to the second
equation in (XVIII). The first equation, however, assumes at first the form

W_ r,+fe(Tx-T,) ^ 1. ii.->ia ^°-^^^"^°>

But if in place of (N—la) Tg^a we here substitute the value given by the
second equation, a lerms disappear which contain Ak as divisor, and the
remaining terms have merely to be arranged according to the factors V and Ur^
in order to obtain the first equation in (XVIII). — 1864.]

THEORY OP THE STEAM-ENGINE. 195

the cylinder varies in different machines. Pambour states that
it amounts on the average to 0*25 in locomotives, but in sta-
tionary engines to much less, probably only to 0*05 of the whole
mass entering the cylinder. In our example we will make use
of the latter number, according to which the ratio of the whole
mass entering the cylinder is to the vaporous part of the same
as 1 : 0*95. Further, let the pressure in the boiler be five atmo-
spheres, to which the temperature 152°*22 belongs, and let us
suppose that the machine has no condenser, or,, in other words,
let it have a condenser with the pressure of one atmosphere.
The mean reaction in the cylinder is accordingly greater than
one atmosphere. As before mentioned, the difference in loco-
motives may be considerable, but in stationary engines it is
smaller. With respect to stationary engines, Pambour has
altogether neglected this difference ; and as our only object at
present is to compare the new formulae with those of Pambour,
we will also disregard the difference, and letp^ equal one atmo-
sphere.

In this example, therefore, the following values will have to
be made use of in equations (XVIII) : —

(53)

c=0-05,

'=0^5= 1«^^'
/?i=3800,
Po^760.
To these must be added the values

/t=13-596,

cr=0001,

which are the same for all cases ; and then in the first of the
equations (XVIII), besides the required value of W, the magni-
tudes V BxiAp^ alone will remain undetermined.

52- We must now examine, in the first place, the least possible
value of V. .

This value corresponds to the case where the pressure in the
cylinder is the same as that in the boiler, so that we have merely
to put p^ in the place of p^ in the last of equations (XVIII) in
order to obtain

o2

196 FIFTH MEMOIB.

v-^ (^)

In order at once to give an example of the influence of the
vicious space^ I have calculated two values of this expression,
corresponding respectively to the cases where no vicious space
exists (e=0), and where, according to supposition, €=:0*05.
These values, expressed as fractions of a cubic metre to one kilo-
gramme of vapour from the boiler, are

0-3637 and 03690.

The latter value is greater than the former, because, first, the
vapour entering the vicious space with great velocity, the vis
viva of its motion is converted into heat, which in its turn
causes the evaporation of a part of the accompanying liquid;
and secondly, because the vapour before present in the vicious
space, contributes to the increase of the ultimate quantity of
vapour.

Substituting both the above values of V in the first of equa-
tions (XVIII), and in the one case again making €=0, whilst
in- the other €=0'05, we have as the corresponding quantities of
work expressed in kilogramme-metres, the numbers

14990 and 14450.

According to Pambour's theory, it makes no difference whether
a part of the volume is vicious space or not ; in both cases this
volume is determined from the equation (29 b) by giving to p the
particular value |?i. By so doing we obtain

0-3883.

This value is greater than the one (0*3637) before found for
the same quantity of vapour, because hitherto the volume of
vapour at its maximum density was esteemed greater than, ac-
cording to the mechanical theory of heat, it can be, and this
former estimate also finds expression in equation (29 b) .

If, by means of this volume, we determine the work under
the two suppositions 6=0 and €=0*05, we have

16000 and 15200.

THEORY OP THE STEAM-ENGINE. 197

As might have been concluded immediately from the greater
volume, these quantities of work are both greater than those
before found, but not in the same ratio ; for, according to our
equations, the loss of work occasioned by vicious space is less
than it would be according to Pambour^s theory.

53. In a machine of the kind here considered, which Pambour
actually examined, the velocity which the machine actually pos-
sessed, compared with the minimum velocity calculated, accord-
ing to his theory, for the same intensity of evaporation and the
same pressure in the boiler, gave the ratio 1*275 : 1 in one ex-
periment, and in another, where the charge was less, 1*70 :1.
These velocities would in our case correspond to the volumes
0*495 and 0*660. As an example of the determination of work,
we will now choose a velocity between these two, and set simply,

V=0-6.

In order next to find the temperature t^ corresponding to this
value of V, we employ the equation (47) under the following
special form : —

Ta^2=26577 + 56-42 . {t,^Q -0*0483 . (Pi-p^). (55)

EflFecting, by means of this equation, the successive determina-
tions of t^ described in Art. 47, we obtain the following series of
approximate values : —

if =133*01,

/" =134*43,

^'" = 134-32,

/""= 134-33.
Further approximate values would only differ from each other in
higher decimal places ; so that, contenting ourselves with two
decimal places, the last number may be considered as the true
value of t^. The corresponding pressure is
j9a=2308*30.
Applying these values of V and p^, as well as those given in
Art. 51, to the first of the equations (XVIII), we obtain

W= 11960.
Pambour^s equation (XII) gives for the same volume 0*6, the
work

W= 12520.

198

FIFTH MEMOIR.

54. In order to show more clearly the dependence of the work
upon the volume^ and at the same time the difference which
exists between Pambour's and my own theory in this respect, I
have made a calculation^ similar to the last, for a series of other
volumes increasing uniformly. The results are comprised in the
following Table. The first horizontal row of numbers, separated
from the rest by a line, contains the values found for a machine
without vicious space. In other respects the arrangement of the
Table will be easily understood.

V.

t,.

W.

Aoooiding to Pamboor.

V.

w.

0-3637

152*22

14990

0-3883

16000

0-3690

0-4

tl

o*7
08
09

I

152*22
14912
14083

134-33
12903

"455
120*72
117*36

14450

14100

13020

1 1960

10910

98^0

8860

7840

03883
0*4

Zl

07
08
09

I

15200

15050
13780
12520
II250

9980

8710
7440

We see that the quantities of work calculated according to
Pambour's theory diminish more quickly with increasing volume
than those calculated from our equations ; for at first the former
are considerably greater than the latter, afterwards they approach
thereto, and finally they are actually less than the latter. 'The
reason is, that according to Pambour^s theory, the same mass,
as at first, always remains vaporous during expansion; whilst,
according to our theory, a part of the liquid accompanying the
vaporous mass afterwards evaporates, and the more so the
greater the expansion.

55. In a similar manner we will now consider a machine
which works with expansion, and we will further select one with
a condenser.

With reference to the magnitude of the expansion, we will
suppose that the cylinder is cut off from the boiler when the
piston has completed one-third of its journey. Then for the
determination of e we have the equation

I

J

THEORY OF THE STEAM-ENGINE. 199

e-e=|(l-6);
whence, retaining the former value, 0*05, of e,
e=^i=0-3666

As before, let the pressure in the boiler be five atmospheres.
By good arrangement the pressure in the condenser may be
kept below one-tenth of an atmosphere. As it is not always so
small, however, and as the reacting pressure in the cylinder
always exceeds it a little, we will assume the mean reaction to
be one-fifth of an atmosphere (or 152 millims.), to which the
temperature ^q=60°'46 corresponds. Retaining the former
assumed value of /, therefore, the quantities requiring applica-
tion in this example are

e =0-36667,1
6=005,

/=1053, "t (^6)

/?j=3800,
i?o=152.

In order to calculate the work, we now only require the value
of V to be given. To guide our choice, we must first know the
least possible value of V, which we can find, as before, from the
second of the equations (XVII.) by putting p^ ^ ^^ place of
/?2, and changing the other quantities dependent on p accord-
ingly. In this manner we find for the present case the value

1010.

Starting from this, we will assume, as a first example, that the
actual velocity of the machine^s motion has to this miaimum a
ratio of 3 : 2 nearly ; so that setting

V=l-5,

we will determine the work for this velocity.

56. The temperatures t^ and /g must now be determined by
setting this value of V in the two last of equations (XVII). For
the machine without a condenser, the determination of t^ has
been sufBiciently explained ; and as the present case differs from

200 FIFTH MEMOIR.

that one only by a different value for e, which was there equal
to 1, it will be sufficient to state here that the final result is

The equation (49), which serves to determine t^ now takes
the form

^8=26-604+51-515Logj3, .... (57)

and from it we obtain the following approximate values : —

/' = 99-24,
t" =101-98,
/'" =101-74,
^'" = 101-76.

We may consider the last of these values, from which the fol-
lowing ones would only differ in higher places of decimals, as
the proper value of t^ ; and we may use it, together with the
known values of /j and t^, in the first of the equations (XVII) .
By so doing we find

W = 81080.

When, assuming the same value of Y, we calculate the work
according to Pambour's equation (XII), — ^whereby, however, the
values of B and b are not taken from equation (29 d), as in the
machine without condenser, but from equation (29 a) intended
for machines with condensers, — ^we find

W = 32640.

57. In a manner similar to the foregoing I have also calcu-
lated the work for the volumes 1*2, 1*8, and 2*1. Besides this,
in order to illustrate by an example the influence which the
several imperfections have upon the work, I have added the fol-
lowing cases : —

(1) The case of a machine having no vicious space, and where
at the same time the pressure in the cylinder during the en-
trance of the vapour is equal to that in the boiler, and the ex-
pansion is carried so far that the pressure diminishes from its
original value j», to p^. If we further suppose that p^ is exactly
the pressure in the condenser, this case wUl be the one to which
equation (XI) refers, and which for a given quantity of heat —

THBOKY OF THE STBAM-ENGINE.

201

the temperatures at which the heat is received and imparted
being also considered as given — ^famishes the greatest possible
quantity of work.

(2) The case of a machine^ again^ having no vicious space^
and when the pressure in the cylinder is again equal to that in
the boiler, but where the expansion is not, as before, complete,
but only continued until the ratio e : 1 is obtained. This is the
case to which equation (X) refers ; only in order to determine
the amount of expansion, the change caused by the same in the
temperature of the vapour was before supposed to be known,
whilst here the expansion is determined according to the volume,
and the change of temperature must be afterwards calculated
therefrom.

(3) The case of a machine with vicious space and incomplete
expansion, and where, of the former favourable conditions, the
OBly one which remains is, that during the entrance of the vapour
the pressure in the cylinder is the same as in the boiler, so that
the volume has its smallest possible value.

To these cases may be added the one already mentioned, where
the last favourable condition is relinquished, and the volume has
a greater than its minimum value.

For the sake of comparison, all these cases, with the exception
of the first, are also calculated according to Pambour's theory.
The reason of the exception is, that the equations (29 a) and
(296) do not here suffice; for even the one which is intended
for small pressures cannot be applied below one-half, or at most
one-third of an atmosphere, whereas here the pressure ought to
decrease to one-fifth of an atmosphere.

The following are the numbers given by our equations in the
first of the above cases : —

Volume before
expansion.

Volume after
expansion.

w.

0-3637

6345

50460

For all the rest of the above cases the results are given in the
following Table, where the numbers referring to a machine
without vicious space, are again separated from the rest by a
horizontal line. The volumes after expansion are alone given.

202

FIFTH MEMOIR.

because the corresponding ones before expansion^ being in all
cases smaller in the proportion oteil, may be easily found : —

V.

<..

^,.

W.

Aooording to Fambour.

V.

W.

0992

152*22

»i37i

34300

1*032

36650

I*OZO
1*2

M

152*22
14563
X37-43

I31'02

125*79

1 1368
108*38
101*76

9655
92*30

32430
31870
31080
30280
29490

1*032

1-2

'•5

34090
33570
32640
31710
30780

58. The quantities of work in this Table^ as well as those in
the former Table for machines without condensers^ refer to a
kilogramme of vapour coming from the boiler. It is easj^
however^ to refer the work to a unit of heat furnished by the
source of heat; for every kilogramme of vapour requires as
much heat as is necessary^ first to raise the mass / (somewhat
more than one kilogramme) from the temperature it had when
entering the boiler up to the general temperature of the same,
and then at that temperature to convert a kilogramme of it into
vapour. This quantity of heat can be calculated from former
data.

59. In conclusion^ I will add a few remarks on friction, re-
stricting myself^ however, to a justification of my having hitherto
disregarded friction in the developed equations, by showing that
instead of introducing the same at once into the first general
expressions for the work, as Pambour has done, it may also^ ac-
cording to the same principles, and according to the manner of
other authors, be afterwards brought into calculation.

The forces which the machine has to overcome during its
action may be thus distinguished : — (1) The resistance exter-
nally opposed to it, and in overcoming which it performs the
required useful work. Fambour calls this resistance the charge
of the machine* (2) The resistances which have their source in
the machine itself, so that the work expended in overcoming
them is not externally of use. All these resistances are in-
cluded in the term friction; although, besides friction in its

THEORY OF THE STEAM-ENGINE. 203

more limited sense^ they comprise other forces^ particularly the
resistances caused by pumps belonging to the machine^ exclu-
sive of the one which feeds the boiler, and which has already
been considered.

Fambour brings both these kinds of resistances into calcula-
tion as forces opposing the motion of the piston ; and in order
conveniently to combine them with the pressures of the vapour
on both sides of the piston, he also adopts a notation similar to
the one ordinarily used for vapour pressures ; that is to say, the
symbol denotes, not the whole force, but that part of it which
corresponds to a unit of surface of the piston. In this sense let
the letter R represent the charge.

A farther distinction must still be made in the case of friction,
for it has not a constant value in each machine, but increases
with the charge. Accordingly Pambour divides it into two
parts: that which is already present when the machine moves
without charge, and that which the charge itself occasions.
With respect to the last, he assumes that it is proportional to
the charge. Accordingly, the friction referred to the unit of
surface is expressed by

/+8.R,
where / and S are magnitudes which, although dependent upon
the construction and dimensions of the machine, are, according
to Pambour, to be considered as constant in any given machine.
We can now refer the work of the machine to these resisting
forces instead of, as before, to the driving force of steam ; for the
negative work done by the former must be equal to the positive
work done by the latter, otherwise an acceleration or retardation
of motion would ensue, which would be contradictory to the
hypothesis of uniform motion hitherto made. During the time
that a unit of weight of vapour enters the cylinder, the surface
of the piston describes the space (1 — e) V, hence for the work W
we obtain the expression

W=(1-€)V[(1 + S).RH-/].
On the other hand, the tiseful part of this work, which for
distinction from the whole work shall be symbolized by (W),
is expressed thus,

(W)=(1-€)V.R.

204

FIFTH MEMOIR.

Eliminating R from this equation by means of the former, we
have the equation

(W)= ^-y-'^-/ . (58)

by means of which, V being known, the useful work (W) can be
deduced from the whole work W as soon as the quantities /
and 8 are given.

I will not here enter into Pambour's method of finding the
latter quantities, as this determination still rests upon a too in-
secure basis, and as friction is altogether foreign to the subject
of this memoir.

Table containing the values, fob steam, of p, its differ-
ential COEFFICIENT -$7=^9} AND THE PRODUCT T .^ EXPRESSED

dt

IN MILLIMETRES OF MERCURY.

i

inCen-
tigrads
degreet

p.

A.

9-

A.

^•9-

A.

9
40

41
42

43

44

:i

49

5'

5*
S3
54

\l

57
58

60
61
6»

54

S4"90«

57-909
61-054

6+-34S
67789
71-390
75156
79091
83103

£7-497
gtfo

96-659
101-541
106-633
111-94:^

i^r475
123-241

1*9147
135*501
143,^021
141786

155-834
163*164
1 7073 J
178707
186 93S

3-003

3H5
3191

3-444
3*601

3766

r935
4^111

4-194
4-4«3
4-679
4-3»i
5092
5*309
5^533
5766
6*006
6-154
6*510
6775
7-048
7-330
7"6ii

7-91*
8-131

^"935
3-074
3'ii8
3*367
3*521
3^683
3-850

+"013
4103
4^338
4*5»r
4780
4*987
5*100
S'4ii

SM9
5-886

6-130
6-182
6*642
6-911
7-189
7'47S
7771
8-076
8-390

0*139

0-144

0-149 1

0-155

o-[6i

167

o'r73

0*180

0*185

0*193

0*199

0*107

0-113

o-iii

o'liS

0-137

0-144

0*152

0-160

0"l6g

0-278

oiSfi
02 9 6
0305
0-314

919

965
1014
1064
1116
1171
iii3
1187
1349
>4n
14S0

^549
1611
169s
^773
1853
1936
2013
1112
2105
2301
1401
2504
2611
1721
2836

46
49
50
51
55
57

6z
64
67
69
71
74
78
80

83
87
89

93
96
100
103
107
III
114

THEORY OF THE STEAM-ENGINE.

205

Table [continued).

in Cen-
tigrade
degrees.

T.^r.

65
66

67
6S
65
70
71
71
73
74

76

78
79
£g
Si
Si
H
84
SS
86

«7
SS

89
90

91

92

n
94

95

98
99

loo
lot

IDI
103
104

I05

106
107
loS
109
no
III

113
114

US

186-958
,95-488
204-368
113-586

13^3154
i33'o8i
a43'38o
154-060
1651 31
176-608

xas'soo

goo- 8 10

313579
316-789

340464
354-616

36915*
384-404
400" 068
41 6'x6i
433-001
450^01
468-175
4*6638

S0570S
525*391

545-715
566^690

588-333
610-661
633-691
657-443
6&! 931

707-174
733191

760-00

7S7S9

8i6'Oi
845 -a S

S7S'4i

906-41

938-31

971-14

1 004^9 1

1039-65

1075-37

1 1 11*09

1149-83

jr8r6i

1118*47

116^-41

8-550

8-83o

9-118

9-568

9918

10198

io-63o

11071

11-476

11*891

11-310

11-759
13-210
13^675
14-151
14-641
15-146
15-664
16-194
16740
17-199
17-874
18-463
19067
19-637
10-313
io'975
11-643
11-318
13-031
13*751
14-483

15-^43
lO'o 1 7
16809

27-59
iS-41
19-27
30*13
31*00
31-90
3x83
3377
3474
35'7i
36*71
3774
38-7S
39"S6
40-94

8-390

S7'S
9049

9"393
9-748
10-113
10^489
io-g76
11-174
11*684
11*106

ii'S39
11-984
13-441

^3-9'3
14397

14894
*5*4<^S

iS'9^9
16*467
17019
17^586
18-168
18*765

19377
10-005
10*649

1! 309
11-985
11-679
13-391
14-119
I4-S65
15-630
26-413

a 7 "loo
x8'oo5
18-S45
19700
30565
31-450
31-365
33-300

34155
35*130
36-110

37-13*^
38160

39-310
40-400
41-500

0-3^5
0134
o'344
o"355
0-365
0-376
0-387
0-398
0-410
0-411

o'433
0-445
0*458
0-471
0-4S4
0497
0-51 1

053S
0-551

0*577
0-581
o'597
0-611
0'6i8
0-644
0-660
0676
0-694
0*711
0-71 8

0747
0*765
0783
0*787
0805
0-840
0-855
0*865
o'885
0-915
0-935
0-955

0-975
0-990
I -010
1*030
i'o6o
1080

I'lOO

1836
1954
3077
3103

3334
3469
3608

3751
3901

4054
4ii3
4376
4544
4718
4897
S081
5171
S469
5671

5879
6093
6313
6540

6774
7014
7161
7516

777«
8047

8313

8608

8900

9100

9509

9816

10146

10474

10817

1 1 1 67

11513

ir883

11366

11654

13051

13458

13871

14x96

14730

15173

'5^35
16101

J18

113
116
131
"35

'39
144
149

m
159

let

174
179
185
190

^97
101
108
114
no

3tl7

134

140
14S

154
161
169
176
185
191
300
309
317
320
328

343
350
356
367

37*
388

397
4^7
4" 4
4.14

434

443

457
467

206

FIFTH MEMOIR.

Table {continved).

in Cen-
tigrade
degree^.

T.g,

"IS
116
117
ijS
119
110
t%i

f%l

1x9

130
131

134

136

137
13B
139
I40
14.1
142

H3
144

145
146

HI
148

150
tsi
152

<S4

158
159

160
i€i
162
163
164
165

I i69'4T

1311-47
135466
1399*01

i444'S5
r49t'i3

IS19'^5
15»8H7
163896
169076
J 743-8 S
1798-35
1854 10
1911-47
1970- IS
1030-18
1091-90
1155-03
1219-69
21^5 92

135373
1413^16
249413
1567'Qo
1641-44
1717-63

*795VS7
1875*30
2956-86
3040*16

3i=5"SS
31 r 1-74
33orS7
3391*98
3486-09
3581-13
3^78*43
377774
387918
3981-77
40H8-56
4196-59
4306-&8
44' 9^5
4S34-36
4651-61
4771-18
4893^36
5017-91
S«44-97
5^7454

41-06

43*9

44-36
4S'S3
4673

4797
49a*
5^H9
51 So
53 11
5447
SS'85
S7'17
58-68
60*13
6 1 '61
63-13
64-66
66*13
67S1

^9*43
71-07
71-77

74'44
76-19

77'94
7973
8156
83-40
85-29

87'X9
89-13
gl'li
93-11
95-14
97-40

99'3i
101-44
103-59

I0S79
10803
110-29
iii'57
114-91
117-26
119-66
111-08
124*55
117*06
119*57

41.500
41-625
43775
44"94S
46-130

47350
48-595

49"S55

5»'46o

53795
55160
56-560

57975
59-405
60-S75
61-375
63-895

65445
67-010
6S'6io
70*150
7 1 -910
73-605

75*315
77-065

78-835
80*645
81*480

84"34S
86^240
S3'i6o
90- J 10
91*110

94-'»S
961 70

98-155
100-375
ioz-515
104-690
106*910
109-160
1 11-430
113740
116-085
118-460
110-B70

li3-3'5
125-805
128*315
130-860

I 125

1150
1-170
1-185

I'llO

1-245
I "160
1-390
1-315
''335
'-365
I '400
1-415
"'430
1-470
1-500
1-510
i"55'>
*^575
J -600
1*630
1-670
1-685
1 7 10
1-750
1-770
|-8jo

1*35

1-865
1-895
1-920
1*960
1-990
1*015
2*045
2*085

I'llO

1*140

»''75
i-lio
1*150
1-270
1-310
1-345
1*375
1-410
1-445

1-490
1-510

i"S45

16101
16581
17072

'7574
18083
18609
19146
19693
20253
10827
21410
21009
12624
23248
23881
14533
25199

25877
26571
27277
27997
28732
19487
30252
31030
31828
31638
33468
343 >*
35^7*
36048

3^939
37S50
38778
39711
406S0
41 660
41659
43671
44703
45757
46830

479' 5
49021

50149
51293
5145S
53642
54851
56073
S73«7

479
491
50Z

509
516

537

560

574
583

599
615
614
633
652
666
678

694
706
720
735

765
778

798
810
830

844
860

376
891
911
928

943
959
980

999
loii
1031
1054
1073
1085
1 107
1127
"44
1165
11S4
1209
1222
1244^

THEORY OF THE STEAM-ENGINE.

Table {continued).

207

in Cen-
tigrade
degrees.

T.ff.

165
166
167
j6S
165
170
171
172
173
174

176

177
I7S
179
180
iSi
18*
i3j
184

ise
187

i3S
1S9
190
J91
192
195
194

196
197
198
199

£O0

5^74" 54
5406-69

5541-43
567882
5813-90
5961-66
6io7"i9
6^55-48
6406-60

65^0" SS
6717-43
6877'ai
7039-97
7 205 '72,
737454

7546"39
7721-37

7S99"S*
3 o So- 34
8265^40
8453-23
8644-35
8S33-82
9036^63

9*37'95
9^270
965093
936271
ioo7S'o4
I0297'0i
10519^63
10745-95
io976'oo
1 1 3,09' 8a
11447-46
1163S-96

132 15

13474
i3r39
140-08
142*76

14553
148-29
151-12

"S3'95
15688
J 5979
162-75
165-75
i68-3o
171^37
174-98
173- IS
181-32
134-56

1S7-33
191-12

>94"47
197S6

3&OI'27

204-75

208-15
2II-7S
115-33

213-97
222-62

226'32

230-05
235^82

237-64
241-50

130-860

133^445

136-065

133-735

141-420

144-145

T46-9IO

149705

15^^535
»55>iS
^8-335
161-270
164-250
167-275

i7o'335
173-425
176-565

179-735
182-940
iS6 195
1 89-425
191-795
196-165
199*565
203-010
206-490
210-005

ii3'555
217150
220-795
224-470
22S-185
231 -93s
a35"730
239-570
H3"45S

2-585
2*620
2-670
2685
2-725
2-765
2*795
2-830
2-330
2920
2-935
2-980
3025
3-060
3-090
3*140
3-170
3-205

3155

3-280
3-320

3-400

3*445
3-480

3"S'5
3'55^
3^595
3'*45
3 ^^7 5
3-715
3750

379 S
3-340
3-885

57317
5^5^^
59868
61182
62508
63856
65228
66618
6S030
69470

70934
72410
73912

75441
76991

78561
80160

81779
334^1
35091
86779

S3493
90236
91999

93791
95605
97441

99303
101I9X

103111
105052
107018
109009
III 0^9
113077
ii5«S4

1265
12S6

1334
1326
1348

1374
1390
1412
1440
1464
1476
1502
1529
1550
1570
1599
1619
1642
1670
i63S
1714

1741
1763
1791
1814

1837
iS&i
1889
1919

1941
1966
1991
2020
2048

2Q77

206 FIFTH MEMOIR^ APPENDIX.

APPENDIX TO FIFTH MEMOIR [1864].

ON SOME APPROXIMATE FORMULA EMPLOYED TO FACILITATE
CALCULATIONS.

With a view of elucidating the mechanical theory of heat
and its applications to the phenomena which present themselves
in the steam-engine^ Zeuner^ in his work on the subject'^^ has
treated a series of problems havings for the most part^ reference
to the same cases which have been treated by me in the imme-
diately preceding (p. 151 to 157), and in the Second Memoir.
In doing so, Zeuner enters into somewhat greater detail than I
have done, and he seeks to facilitate the calculations by intro-
ducing approximate formulae ; of some of the latter, however,
I cannot approve.

Zeuner thus enunciates the Problem I : — '' A cylinder con-
tains w, kilogrammes of vapour, and (M— w^) kilogrammes of
water. Suppose the mass to expand slowly, so that the pres-
sure on the piston which has to be overcome by the vapour
is at each moment equal to the tension of the latter ; the tem-
perature of the mass falls from ti to t^ and the tension from p^ to
p^. What wc^rk does the vapour perform, and how much heat must
be received or given oSin order that the quarUity of vapour mj may
remain constant y and consequently that neither condensation of
vapour nor evaporation of water may take place during ex-
pansion ?^*

The quantity of heat which must be imparted to the mass
within the cylinder during its expansion and consequent change
of temperature in order to prevent both condensation and eva-
poration, may be at once expressed by means of the specific heat
of water, say c, and the specific heat of saturated vapour, for
which latter I have already used the symbol h. For an infini-
tesimal change of temperature dt we have

dOi^i^K—m^cdt + mJidt, .... (a)
and for a finite change, say from t^ to t^,

Q=(M-w,)rcrf/+mij M/ (b)

* Grrundziige der meehanischen Warmetheorief von G. Zeuner. Freiberg,
1860.

APPROXIMATE FORMULAE. 209

The first term on the right, which has reference to the liquid
water, is self-intelligible, since the specific heat of water is a
magnitude given immediately by observation ; we may conse-
quently leave this term out of consideration. The quantity h
which appears in the second term has been determined by the
equation

which is numbered (32) in my First Memoir (p. 65) . In it we
have merely, as is there done, to put, for c and r, the values given
by Regnault^s experiments, in order to have h expressed as a
ftmction of the temperature. It will be readily seen that we
have here to do with the case which I have already treated as the
first case in the Second Memoir (p. 91 et seq.), and where, by
way of example, I actually calculated the integral for two expan-
sions ; that is to say, for expansions whereby the pressure dimi-
nishes firom five and from ten atmospheres, respectively, down to
one atmosphere, and consequently the temperature falls from
152°-2 and from 180°-3, respectively, down to 100°.

Zeuner, on the other hand, by introducing various simplifica-
tions in order to facilitate his calculations, arrives at the follow-
ing equation, numbered (122) in his work : —

Q=(20433w,-10224M)(/,-g (d)

He remarks that this equation only holds, immediately, for
those temperatures w;hich are more fully defined in § 30 of his
work. Now in this paragraph he states that a certain approx-
imate formula there given holds perfectly between the tempera-
tures 100° and 150°, and that beyond these limits even, from
about 60° to 180°, the deviations are small. After referring to
this paragraph he adds, "I believe that our present experi-
mental knowledge of the deportment of vapour is so uncertain
and fluctuating, particularly beyond the limits of temperature
stated in § 30, that, provisionally, the above expression, de-
duced by methods of approximation, may without hesitation be
generally applied.^'

I propose, by a numerical example, to compare Zeuner^s ap-
proximate formula with my expression. To do so we will take
the case where precisely the whole mass in the cylinder is

210 FIFTH MEMOIR^ APPENDIX.

vaporous. In this case we have mi=M, and Zenner's equa-
tion (d) becomes

Q=10209M(^,-g, (e)

whilst the preceding equation (b) becomes

°-"£

hdt (f)

Suppose now the initial temperature to be 180°, and that the
vapour expands so that its temperature falls successively to 170°,
160", 150°, and so on to 60°. During this expansion heat must
be imparted to the vapour in order to prevent its partial precipi-
tation ; let us calculate how much must be imparted during the
successive falls often degrees on the assumption that M=l.

For each of these intervals Zeuner's equation gives the value

10-209;

my equation, on the other hand, gives the following series of
values : —

773; 814; 857; 901; 949; 998;
10-50; 11-04; 11-62; 1222; 12-87; 1355.

From this it will be seen that in the interval from 180° to 60°,
within which according to Zeuner the deviations should be small,
the true values, instead of being constant and equal to 10*209,
vary between 7*73 and 13*55. Even in the small interval be-
tween 150° and 100°, within which Zeuner considers his for-
mula to be perfectly true, the true values vary between 9*01
and 11-04.

Although I quite admit that to render the mechanical theory
of heat useful to practical mechanicians it is necessary to sim-
plify its use by the calculation of tables, and the establish-
ment of the simplest possible approximate formulae, and con-
sequently that Zeuner's efforts to that end must be very wel-
come to them, still I cannot think that they will be satisfied
with formulae which correspond so little with the actual state
of things as does that here adduced.

The questions discussed in Zeuner^s Problem II. are those to
which the equations (VII) and (IX) of the preceding memoir
refer. The same expansion as in Problem I. is assumed to

i

APPROXIMATE FORMULA. 211

take place^ but with this difference, that no heat is thereby im-
parted to, or abstracted from the mass in the cylinder, and under
these circumstances it is required to determine in what manner
the quantity of vapour present is changed through expansion,
and what work is performed.

After giving my equations (VII) and (IX) (the latter in a
somewhat modified form), Zeuner again proceeds to simpli-
fications, and arrives at two approximate formulae, one for the
quantity of vapour fi which is newly developed or, if nega-
tive, precipitated during the expansion, and the other for the
heat L consumed by exterior work. If in the first of these for-
mulae we substitute for a quantity p which therein occurs, Zeu-
ner's own expression for that quantity, in exactly the same
manner as he himself has done in the second formula, the two
become

^=(M-2m,)^^, '.(g)

L=[Mc-(M-»»J^](;^-g, . . . . (h)
wherein », fi, c are three constants having the respective values

/3= 0-7882,

c= 10224.

(i)

In order to compare the results of these formulae with those
which follow from my own equations, we will suppose that at
the commencement of the expansion precisely the whole mass
in the cylinder is vaporous, so that m^=zM; and we will fiir-
ther introduce into the equation (h), in place of the heat L con-
sumed by work, the work W itself in accordance with the rela-
tion W=x« The two equations then are

'*=-^^^ 0^)

w=^(^-g (1)

Assuming 150° to be the initial, and 125°, 100°, 75°, succes-
sively, the final temperatures, we may, for the purpose of com-
parison, employ the numerical values already calculated from

p2

212

PIFTH MEMOIR^ APPENDIX.

my equations^ and collected iu the small tables on pp. 154 & 157.
If m denote the quantity of vapour present at the end of the
expansion^ we have, in fact, to put /liasw— M, and hence

—: = ^— 1 ; and in this equation we may substitute the values

m

of r^ given in the first Table.

The Tables in question extend

beyond the interval now under consideration, in fact reach down
* to 25°, but in this comparison we will only employ the numbers
down to a temperature of 75°, in order not to overstep the
Umits for which Zeuner has determined his formulae. By so
doing the following corresponding numbers are obtained : —

Final temperature.

1250.

100».

75«.

A* r aocording to equation (Vll)
M \ according to equation (k)

W r according to equation (IX)
M \ aocording to equation (1)

-0*044
—0*041

1 1300
10800

—0*089
-0*079

23*00
21700

-0*134
-0-115

35900
32500

Here, therefore, within the small interval from 150° to 75®,
not inconsiderable deviations already manifest themselves.
Zeuner expresses the opinion that it is on the equations, repre-
senting work, which he has established, that a new theory of the
steam-engine may be based ; and that their simplicity and con-
venient form, as well as the circumstance that the results which
they famish agree quite satisfactorily with those which follow
from my own equations, certainly recommend them as suitable
for this object. I must confess, however, that in the preceding
numbers I find no agreement sufficiently satisfactory to con-
vince me that the equations in question would form a suitable
foundation for the new theory of the steam-engine.

In his Problem III. Zeuner considers the case where a quan-
tity of vapour and water contained in a vessel is suddenly ex-
posed to a pressure diflferent from that which exists in that vessel ;
this occurs, for instance, when a vessel in which the pressure
exceeds one atmosphere is suddenly put in communication with
the atmosphere so that the vapour can issue therefrom. The
question now is how much heat must be imparted to the mass

•APPROXIMATE FORMULA. 213

during its change of volume^ and the associated change of tem-
perature, in order that neither precipitation of vapour nor eva-
poration of water may take place, but that exactly the original
quantity of vapour may preserve its state of vapour at the max-
imum density.

This case allies itself to those which I have treated in the
Second Memoir as second and third cases. I have there distin-
guished the two cases where the vessel contains vapour solely,
and where (like the boiler of a steam-engine) it contains not
only vapour, but likewise water, which during the efflux of the
former further evaporates, and thus replaces the vapour lost.
Zeuner thinks he is able to represent these two cases in a very
simple manner, by one equation.

In fact, the equation (144) of his book contains an expression
for the heat to be imparted to the mass which essentially agrees
with the one developed by me for the second case, and given in
equation (3) of my memoir (p. 95), although its form is some-
what diflFerent in consequence of the modified notation therein
adopted. To this quantity of heat, however, is added that which
must be imparted to the water in order to change its temperature
from ti to /g, and in this manner is formed the expression for
the total quantity of heat, which must be imparted to the mass
consisting of vapour and water. By the addition of this term,
having reference to the water, Zeuner imagines he has rendered
the equation, developed by me for the second case, also applicable
to the third case. I cannot, however, agree with him.

The equation formed by Zeuner implies that the water con-
tained in the vessel changes its temperature in the same manner
as the vapour. If we consider a boiler, however, from which
steam issues through an aperture, for instance, through the
safety valve, the issuing vapour will suflFer on expanding a very
quick and considerable diminution of temperature, in which
diminution the mass in the boiler takes no part. The issuing
vapour, therefore, must be considered separately on deciding
whether, when no heat is communicated to it or withdrawn
from it, it is overheated or partially precipitated, and on deter-
mining how much heat must be imparted to, or withdrawn from
it in order to maintain it exactly at its maximum density.
For this purpose I hold that the procedure adopted by me is

214 FIFTH MEMOIE^ APPENDIX.

qidte appropriate; namely that of considermg the issuing .

vapour between two surfaces, in the first of which exists the |

pressure prevailing in the interior of the vessel, whilst in the
second the pressure is equal to that of the atmosphere, and the
velocity of efflux has already so fax diminished that the vis
viva corresponding to the ipotion of the stream of vapour may
be neglected.

1

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 215

SIXTH MEMOIR.

ON THE APPLICATION OF THE THEOREM OF THE EQUIVALENCE OF
TRANSFORMATIONS TO INTERIOR WORK*.

In a memoir published in the year 1854t, wherein I sought to
simplify to some extent the form of the developments I had pre-
viously published, I deduced, from my fundamental proposition
that heat cannot, by itself , pass from a colder into a warmer body,
a theorem which is closely allied to, but does not entirely coin-
cide with, the one first deduced by S. Camot from considera-
tions of a different kind, based upon the older views of the na-
ture of heat. It has reference to the circumstances under which
work can be transformed into heat, and conversely, heat con-
verted into work ; and I have called it the Theorem of the Equi-
valence of Transformations* I did not, however, there commu-
nicate the entire theorem in the general' form in which I had
deduced it, but confined myself on that occasion to the publica-
tion of a part which can be treated separately from the rest, and
is capable of more strict proof.

In general, when a body changes its state, work is performed
externally and internally at the same time, — the exterior work
having reference to the forces which extraneous bodies exert
upon the body under consideration, and the interior work to the
forces exerted by the constituent molecules of the body in ques-
tion upon each other. The interior work is for the most part
so little known, and connected with another equally imknown
quantity J in such a way, that in treating of it we are obliged in

* Communicated to the Naturforschende GeseUschaft of Zurich, Jan. 27th,
1862 ; published in the Vierteljahrschrift of this Society, vol. vii. p. 48 j in
Poggendopft''s Annalen, May 1862, vol. cxvi. p. 73 -, in the Philosophical
Magazine, S. 4. vol. xxiv. pp. 81, 201 ; and in the Journal des MathSmatiques
of Paris, S. 2. vol. vii. p. 209.

t " On a modified form of the second Fimdamental Theorem in the Me-
chanical Theory of heat." [Fourth Memoir of this collection, p. 116.]

t [In fact with the increase of the heat actually present in the body. — 1864.]

216 SIXTH MEMOIE.

some measure to trust to probabilities; whereas the exterior '

work is immediately accessible to observation and measurement, \

and thus admits of more strict treatment. Accordingly, since, i

in my former paper, I wished to avoid everything that was hy- \

pothetical, I entirely excluded the interior work, which I was |

able to do by confining myself to the consideration of cyclical i

processes — that is to say, operations in which the modifications
which the body undergoes are so arranged that the body finaUy |

returns to its original condition. In such operations the inte- i

rior work which is performed during the several modifications, -

partly in a positive sense and partly in a negative sense, neu- |

tralizes itself, so that nothing but exterior work remains, for I

which the theorem in question can then be demonstrated with !

mathematical strictness, starting from the above-mentioned fun- i

damental proposition. j

I have delayed till now the publication of the remainder of
my theorem, because it leads to a consequence which is con-
siderably at variance with the ideas hitherto generally entertained
of the heat contained in bodies, and I therefore thought it desi- .

rable to make stiU further trial of it. But as I have become I

more and more convinced in the course of years that we must |

not attach too great weight to such ideas, which in part are |

founded more upon usage than upon a scientific basis, I feel that ]

I ought to hesitate no longer, but to submit to the scientific |

public the theorem of the equivalence of transformations in its
complete form, with the theorems which attach themselves to it.
I venture to hope that the importance which these theorems,
supposing them to be true, possess in connexion with the theory t

of heat will be thought to justify their publication in their '

present hypothetical form. '

I will, however, at once distinctly observe that, whatever
hesitation may be felt in admitting the truth of the following
theorems, the conclusions arrived at in my former paper, in re-
ference to cyclical processes, are not at all impaired.

1. I wiQ begin by briefly stating the theorem of the equivalence
of transformations, as I have already developed it, in order to
be able to connect with it the following considerations.

When a body goes through a cyclical process, a certain amount i

of exterior work may be produced, in which case a certain quantity ^

i

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 217

of heat must be simultaneously expended; or, conversely, work
may be expended and a corresponding quantity of heat may be
gained. This may be expressed by saying : — Heat can be trans-
formed into work, or work into heat, by a cyclical process.

There may also be another effect of a cyclical process : heat
may be transferried from one body to another, by the body which
is undergoing modification absorbing heat from the one body
and giving it out again to the other. In this case the bodies
between which the transfer of heat takes place are to be viewed
merely as heat reservoirs, of which we are not concerned to
know anything except the temperatures. If the temperatures
of the two bodies differ, heat passes, either from a warmer to a
colder body, or from a colder to a warmer body, according to
the direction in which the transference of heat takes place.
Such a transfer of heat may also be designated, for the sake of
uniformity, a transformation, inasmuch as it may be said that
heat of one temperature is transformed into heat of another tem-
perature.

The two kinds of transformations that have been mentioned
are related in such a way that one presupposes the other, and
that they can mutually replace each other. If we call transfor-
mations which can replace each other equivalent, and seek the
mathematical expressions which determine the amount of the
transformations in such a manner that equivalent transforma-
tions become equal in magnitude, we arrive at the following ex-
pression : — If the quantity of heat Q of the temperature t is pro-
duced from work, the equivalence-value of this transformation is

Q

and if the quantity of heat Q passes from a body whose tempera-
ture is t, into another whose temperature is t^, the equivalence-value
of this transformation is

<,-%)■

where T is a ftinction of the temperature which is independent
of the kind of process by means of which the transformation is
effected, and Tj and Tg denote the values of this ftinction which
correspond to the temperatures t^ and t^- I have shown by sepa-

218 SIXTH MEMOIR.

rate considerations that T is in all probability nothing more than
the absolute temperature.

These two expressions further enable us to recognize the posi-
tive or negative sense of the transformations. In the firsts Q is
taken as positive when work is transformed into heat^ and as
negative when heat is transformed into work. In the second,
we may always take Q as positive, since the opposite senses of
the transformations are indicated by the possibility of the differ-
ence =; — ps- being either positive or negative. It will thus be

Ig Ij

seen that the passage of heat from a higher to a lower tempera-
ture is to be looked upon as a positive transformation, and its
passage from a lower to a higher temperature as a negative
transformation.

K we represent the transformations which occur in a cyclical
process by these expressions, the relation existing between them
can be stated in a simple and definite manner. If the cyclical
process is reversible^ the transformations which occur therein
must be partly positive and partly negative, and the equivalence-
values of the positive transformations must be together equal to
those of the negative transformations, so that the algebraic sum
of all the equivalence-values becomes =0. If the cyclical process
is not reversible, the equivalence- values of the positive and nega-
tive transformations are not necessarily equal, but they can only
differ in such a way that the positive transformations predomi-
nate. The theorem respecting the equivalence-values of the
transformations may accordingly be stated thus : — JTie algebraic
sum of all the transformations occurring in a cyclical process can
only be positive, or, as an extreme case, equal to nothing.

The mathematical expression for this theorem is as follows.
Let rfQ be an eleirient of the heat given up by the body to any
reservoir of heat during its own changes (heat which it may
absorb from a reservoir being here reckoned as negative), and T
the absolute temperature of the body at the moment of giving
up this heat, then the equation

Jf =0 (1)

must be true for every reversible cyclical process, and the relation

J

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 219

f-

'¥>0 (la)

must hold good for every cyclical process which is in any way
possible.

2. Although the necessity of this theorem admits of strict ma-
thematical proof if we start &om the fundamental proposition
above quoted, it thereby nevertheless retains an abstract form,
in which it is with diflSculty embraced by the mind, and we feel
compelled to seek for the precise physical cause, of which this
theorem is a consequence. Moreover, since there is no essen-
tial difference between interior and exterior work, we may
assume almost with certainty that a theorem which is so ge-
nerally applicable to exterior work cannot be restricted to this
alone, but that, where exterior work is combined with interior
work, it must be capable of application to the latter also.

Considerations of this nature led me, in my first investigations
on the mechanical theory of heat, to assume a general law re-
specting the dependence of the active force of heat on tempera-
ture, among the immediate consequences of which is the theorem
of the equivalence of transformations in its more complete
form, and which at the same time leads to other important con-
clusions. This law I will at once quote, and will endeavour to
make its meaning clear by the addition of a few comments. As
for the reasons for supposing it to be true, such as do not at
once appear from its internal probability will ,gradually become
apparent in the course of this paper. It is as follows : —

In all cases in which the heat contained in a body does mecha-
nical work by overcoming resistances , the magnitude of the resist-
ances which it is capable of overcoming is proportional to the ab-
solute temperature.

In order to understand the significance of this law, we require
to consider more closely the processes by which heat can perform
mechanical work. These processes always admit of being re-
duced to the alteration in some way or another of the arrange-
ment of the constituent parts of a body. For instance, bodies
are expanded by heat, their molecules being thus separated
from each other : in this case the mutual attractions of the
molecules on the one )iand, and external opposing forces on the
other, in so far as any such are in operation, have to be over-

220 SIXTH MEMOIR.

come. Again^ the state of aggregation of bodies is altered by
heat^ solid bodies being rendered liquid^ and both solid and
liquid bodies being rendered aeriform: here likewise internal
forces^ and in general external forces also^ have to be overcome.
Another case which I will also mention^ because it differs so
widely from the foregoing^ and therefore shows how various are
the modes of action which have here to be considered^ is the
transfer of electricity from one body to the other, constituting
the thermo-electric current, which takes place by the action of
heat on two heterogeneous bodies in contact.

In the cases first mentioned, the arrangement of the molecules
is altered. Since, even while a body remains in the same state
of aggregation, its molecules do not retain fixed unvarying po-
sitions, but are constantly in a state of more or less extended
motion, we may, when speaking of the arrangement of the mole^
ctdes at any particular time, understand either the arrangement
which would result from the molecules being fixed in the actual
positions they occupy at the instant in question, or we may sup-
pose such an arrangement that each molecule occupies its mean
position. Now the effect of heat always tends to loosen the
connexion between the molecules, and so to increase their mean
distances from one another. In order to be able to represent this
mathematically, we will express the degree in which the molecules
of a body are separated from each other, by introducing a new
magnitude, which we will call the disgregation of the body, and by
help of which we can define the effect of hes^t as simply tending to
increase the disgregation. The way in which a definite measure of
this magnitude can be arrived at will appear from the sequel.

In the case last mentioned, an alteration in the arrangement
of the electricity takes place, an alteration which can be repre-
sented and taken into calculation in a way corresponding to the
alteration of the position of the molecules, and which, when it
occurs, we will consider as always included in the general expres-
sion change of arrangement, or change of disgregation.

It is evident that each of the changes that have been named
may also take place in the reverse sense, if the effect of the
opposing forces is greater than that of the heat. We will
assume as likewise self-evident that, for the production of work,
a corresponding quantity of heat must always be expended, and

i

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 221

conversely, that, by the expenditure of work, an equivalent quan-
tity of heat must be produced,

3. If we now consider more closely the various cases which
occur in relation to the forces which are operative in each of
them, the case of the expansion of a permanent gas presents
itself as particularly simple. We may conclude from certain pro-
perties of the gases that the mutual attraction of their molecules
at their mean distances is very small, and therefore that only a
very slight resistance is oflfered to the expansion of a gas, so that
the resistance of the sides of the containing vessel must main-
tain equilibrium with almost the whole eflfect of the heat. Ac-
cordingly the externally sensible pressure of a gas forms an
approximate measure of the separative force of the heat con-
tained in the gas ; and hence, according to the foregoing law,
this pressure must be nearly proportional to the absolute
temperature. The iatemal probability of the truth of this re-
sult is indeed so great, that many physicists since Gay-Lussac
and Dalton have without hesitation presupposed this propor-
tionality, and have employed it for calculating the absolute tem-
perature.

In the above-mentioned case of thermo-electric action, the
force which exerts an action contrary to that of the heat is like-
wise simple and easily determined. For at the point of contact
of two heterogeneous substances, such a quantity of electricity
is driven from the one to the other by the action of the heat,
that the opposing force resulting from the electric tension suffices
to hold the force exerted by the heat in equilibrium. Now in
a former memoir " On the application of the Mechanical Theory
of Heat to the Phenomena of Thermal Electricity*'"^, I have
shown that, in so far as changes in the arrangement of the mole-
cules are not produced at the same time by the changes of
temperature, the difference of tension produced by heat must be
proportional to the absolute temperature, as is required by the
foregoing law.

In the other cases that are quoted, as well as in most others,

the relations are less simple, because in them an essential part is

played by the forces exerted by the molecules upon one another,

forces which, as yet, are quite unknown. It results, however,

* Poggendorft*s Annalen, vol. xc. p. 513.

222 SIXTH MEMOIR

from the mere consideration of the external resistances which
heat is capable of overcoming, that in general its force increases
with the temperature. If we wish, for instance, to prevent the
expansion of a body by means of external pressure, we are obliged
to employ a greater pressure the more the body is heated ; hence
we may conclude, without having a knowledge of the interior
forces, that the total amount of the resistances which can be over-
come in expansion, increases with the temperature. We cannot,
however, directly ascertain whether it increases exactly in the
proportion required by the foregoing law, without knowing the
interior forces. On the other hand, if this law be regarded as
proved on other grounds, we may reverse the process, and
employ it for the determination of the interior forces exerted
by the molecules.

The forces exerted upon one another by the molecules are not
of so simple a kind that each molecule can be replaced by a
mere point ; for many cases occur in which it can be easily seen
that we have not merely to consider the distances of the mole-
cules, but also their relative positions. If we take, for example,
the melting of ice, there is no doubt that interior forces,
exerted by the molecules upon each other, are overcome, and ac-
cordingly increase of disgregation takes place ; nevertheless the
centres of gravity of the molecules are on the average not so
far removed from each other in the liquid water as they were in
the ice, for the water is the more dense of the two. Again, the
peculiar behaviour of water in contracting when heated above
0° C, and only beginning to expand when its temperature
exceeds 4°, shows that likewise in liquid water, in the neigh-
bourhood of its melting-point, increase of disgregation is not
accompanied by increase of the mean distances of its molecules.

In the case of the interior forces, it would accordingly be
difficult — even if we did not want to measure them, but only to
represent them mathematically — ^to find a fitting expression for
them which would admit of a simple determination of magni-
tude. This difficulty, however, disappears if we take into calcu-
lation, not the forces themselves, but the mechanical work
which, in any change of arrangement, is required to overcome
them. The expressions for the quantities of work are simpler
than those for the corresponding forces ; for the quantities of

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 223

work can be all expressed^ without further secondary statements^
by njombers which, having reference to the same unit, can be
added together, or subtracted from one another, however various
the forces may be to which they refer.

It is therefore convenient to alter the form of the above law
by introducing, instead of the forces themselves, the work done
in overcoming them. In this form it reads as follows : —

The mechanical work which can be done by heat during any
change of the arrangement of a body is proportional to the abso^
lute temperature at which this change occurs.

4. The law does not speak of the work which the heat does,
but of the work which it can do ; and similarly, in the first form
of the law, it is not of the resistances which the heat overcomes,
but of those which it can overcome that mention is made. This
distinction is necessary for the following reasons : —

Since the exterior forces which act upon a body while it is
undergoing a change of arrangement may vary very greatly,
it may happen that the heat, while causing a change of ar-
rangement, has not to overcome the whole resistance which
it would be possible for it to overcome. A well-known and
often-quoted example of this is afforded by a gas which expands
under such conditions that it has not to overcome an oppo-
sing pressure equal to its own expansive force, as, for in-
stance, when the space filled by the gas is made to communi-
cate with another which is empty, or contains a gas of lower
pressure. In order in such cases to determine the force of the
heat, we must evidently not consider the resistance which actu-
ally is overcome, but that which can be overcome.

Also in changes of arrangement of the opposite kind, that
is, where the action of heat is overcome by the opposing forces, a
similar distinction may require to be made, but in this case only
as far as this — that the total amoimt of the forces by which the
action of the heat is overcome may be greater than the active
force of the heat, but not smaller.

Cases in which these differences occur may be thus charac-
terized. When a change of arrangement takes place so that
the force and coimterforce are equal, the change can likewise
take place in the reverse direction under the influence of the same
forces. But if it occurs so that the overcoming force is greater

224 SIXTH MEMOIR.

than that which is overcome^ the change cannot take place in the
opposite direction under the influence of the same forces. We
may say that the change has occurred in the first case in a rever-
sible manner, and in the second case in an irreversible maimer.

Strictly speaking, the oyercoming force must always be more
powerful than the force which it overcomes ; but as the excess
of force does not require to have any assignable value, we may
think of it as becoming continually smaller and smaller, so that
its value may approach to nought as nearly as we please. Hence
it may be seen that the case in which the changes take
•^ I place reversibly is a limit which in reality is never quite reached,
^ but to which we can approach as nearly as we please. We may

therefore, in theoretical discussions, still speak of this case as one
which really exists ; indeed, as a limiting case it possesses special
theoretical importance.

I will take this opportunity of mentioning another process in
which this distinction is likewise to be observed. In order for
one body to impart heat to another by conduction or radiation
(in the case of radiation, wherein mutual communication of heat
takes place, it is to be understood that we speak here of a body
which gives out more heat than it receives), the body which
parts with heat must be warmer than the body which takes up
heat ; and hence the passage of heat between two bodies of dif-
ferent temperature can take place in one direction only, and not
in the contrary direction. The only case in which the passage
of heat can occur equally in both directions is when it takes
place between bodies of equal temperature. Strictly speaking,
however, the communication of heat from one body to another of
the same temperature is not possible ; but since the difference of
temperature may be as small as we please, the case in which it
is equal to nothing, and the passage of heat accordingly rever-
sible, is a limiting case which may be regarded as theoretically
possible.

5. We will now deduce the mathematical expression for the
above law, treatiag in the first place the case iu which the
change of condition undergone by the body under consideration
takes place reversibly. The result at which we shall arrive for
this case will easily admit of subsequent generalization, so as to
include also the cases in which a change occurs irreversibly.

EXTENSION OP SECOND FUNDAMENTAL THEOREM. 225

Let the body be supposed to undergo an infinitely small
change of condition, whereby the quantity of heat contained in
it, and also the arrangement of its constituent particles, may be
altered. Let the quantity of heat contained in it be expressed
by H, and the change of this quantity by rfH. Further, let
the work, both interior and exterior together, performed by the
heat in the change of arrangement be denoted by rfL, a magni-
tude which may be either positive or negative according as the
active force of the heat overcomes the forces acting in the con-
trary direction, or is overcome by them. We obtain the heat
expended to produce this quantity of work by multiplying the
work by the thenifel-equivalent of a unit of work which we
may call A; hence it is ArfL.

The sum rfH+ArfL is the quantity of heat which the body
must receive from without, and must accordingly withdraw from
another body during the change of condition. We have, how-
ever, already represented by dCi the infinitely small quantity of
heat imparted to another body by the one which is undergoing
modification, hence we must represent in a corresponding man-
ner, by — rfCl, the heat which it withdraws from another body.
We thus obtain the equation

~rfQ=rfH-fArfL,
or

rfQ-frfH + AdL=0* (1)

In order now to be able to introduce the disgregation also into
the formulae, we must first settle how we are to determine it as a
mathematical quantity.

By disgregation is represented, as stated in Art. 2, the degree

• In my previous memoirs I have separated from one another the interior
and the exterior work performed by the heat during the change of condition
of the body. If the former be denoted by di, and the latter by cHV, the
above equation becomes

JQ4-<?H+A(fl[+A<fW=0 (a)

Since, however, the increase in the quantity of heat actually contained in a
body, and the heat consumed by interior work during a change of condi-
tion, are magnitudes of which .we conunonly do not know the individual
values, but only the sum of those values, and which resemble each other in
being fully determined as soon as we know the initial and final conditions
of the body, without our requiring to know how it has passed from the one
to the other, I have thought it advisable to introduce a function which shall

Q

226 SIXTH MEMOIR.

of dispersion of the body. Thus^ for example, the disgregation
of a body is greater in the liquid state than in the solid^ and
greater in the aeriform than in the liquid state. Further, if part
of a given quantity of matter is solid and the rest liquid, the
disgregation is greater the greater the proportion of the whole
mass that is liquid; and similarly, if one part is liquid and the
remainder aeriform, the disgregation is greater the larger the
aeriform portion. The disgregation of a body is fiilly deter-
mined when the arrangement of its constituent particles is
given; but, on the other hand, we cannot say conversely that
the arrangement of the constituent particles is determined when
the magnitude of the disgregation is kn<f^. It might, for
example, happen that the disgregation of a given quantity of
matter should be the same when one part was solid and one part
aeriform, as when the whole mass was liquid.

We wiU now suppose that, with the aid of heat, the body
changes its condition, and we will provisionally confine ourselves
to such changes of condition as can occur in a continuous and

represent the sum of these two magnitudes, and which I have denoted by U.
Accordingly

dU=rfH+Acn, (5)

and hence the foregoing equation becomes

<?Q+rfU+AJW=Oj (c)

and if we suppose the last equation integrated for any finite alteration of
condition, we have

Q4,U+AW=0. (d)

These are the equations which I have used in my memoirs published in
18^ and in 185^, partly in the particular form which they assume for tlie
permanent gases, and partly in the general form in which they are here
given, with no other difierence than that I there took the positive and ne-
gative quantities of heat in the opposite sense to what I have done here, in
order to attain greater correspondence with the equation (I) given in Art. 1.
The function U which I introduced is capable of manifold application in
the theory of heat, and, since its introduction, has been the subject of very
interesting mathematical developments by W. Thomson and by Kirchhoff
(see Philosophical Magazine, S. 4. vol. ix. p. 523, and Poggendorff's An-
fuUen, vol. ciii. p. 177). Thomson has called it "the mechanical energy of
a body in a given state," and Kirchhoff " Wirkungsfimction." Although I
consider my original definition of it as representing the sum of the heat culded
to the qwmtity already present and of that expended in interior ufork, starting
from iany given initial state (pp. 29 and 113), as perfectly exact, I can still have
no objection to make against an abbreviated mode of expression. [See the
Appendix A. On Terminology at the end of this memoir]

{

J

EXTENSION OF SECOND Ft7NDAMENTAL THEOREM. 227

reversible manner, and we will also assume that the body has a
uniform temperature throughout. Since the increase of disgre-
gation is the action by means of which heat performs work, it
follows that the quantity of work must bear a definite ratio to
the quantity by which the disgregtttion is increased; we will
therefore fix the still arbitrary determination of the magnitude
of disgregation so that, at any given temperature, the increase
of disgregation shall be proportional to the work which the heat
can thereby perform. The influence of the temperature is de-
termined by the foregoing law. For if the same change of dis-
gregation takes place at difierent temperatures, the correspond-
ing work must be proportional to the absolute temperature.
Accordingly, let Z be the disgregation of the body, and dZ an
infinitely small change of it, and let rfL be the corresponding
infinitely small quantity of work, we can then put

dL=KTrfZ,
or

where K is a constant dependent on the unit, hitherto left
undetermined, according to which Z is to be measured. We

wUl choose this unit of measure so that K=-^, and the equation

becomes rfZ=^ (2)

If we suppose this expression integrated, from any initial con-
dition in which Z has the value Zg, we get

Z=Zo+A

^Jf (8)

The magnitude Z is thus determined, with the exception of a
constant dependent upon the initial condition that is chosen.

If the temperature of the body is not everywhere the same,
we can regard it as divided into any number we choose of
separate parts, refer the elements dZ and dh in equation (2)
to any one of them, and at once substitute for T the value
of the absolute temperature of that part. If we then unite by
summation the infinitely small changes of disgregation of the
separate parts, or by integration, if there is an infinite number
of them, we obtain the similarly infijiitely small change of

q2

228 SIXTH MEMOIR.

disgregation of the entire body, and from this we can obtain,

likewise by integration, any desired finite change of disgregation.

We will now return to equation (1), and by help of equation

(2) we will eliminate from it the element of work dlj. Thus

we get

rfQ4-rfH+TrfZ=:0; (4)

or, dividing by T,

'^^pE^az^o (5)

If we suppose this equation integrated for a finite change of
condition, we have

p+««+j,z=0 ,11)

Supposing the body not to be of uniform temperature through-
out, we may imagine it broken up again into separate parts,
make the elements dQ, rfH, and dZ in equation (5) refer, in
the first instance, to one part only, and for T put the absolute
temperature of this part. The symbols of integration in (II)
are then to be understood as embracing the changes of all the
parts. We must here remark that cases in which one conti-
nuous body is of different temperatures at different parts, so that
a passage of heat immediately takes place by conduction from the
warmer to the colder parts, must be for the present disregarded,
because such a passage of heat is not reversible, and we have
provisionally confined ourselves to the consideration of reversible
changes.

Equation (II) is the required mathematical expression of the
above law, for all reversible changes of condition of a body ; and
it is clearly evident that it also remains applicable, if a series of
successive changes of condition be considered instead of a single
one.

6. The differential equation (4), whence equation (II) is
derived, is connected with a differential equation which results
from the already known principles of the mechanical theory of
heat, and which transforms itself directly into (4) for the parti-
cular case in which the body under consideration is a perfect gas.

We will suppose that there is given any body of variable
volume, upon which the only active external force is the pressure
exerted on the surface. Let the volume which it assumes under

EXTENSION OV SECOND FUNDAMENTAL THEOREM. 229

I this pressure p, at the temperature T (reckoned from the abso-

i lute zero) be v, and let us suppose that the condition of the

body is folly determined by the magnitudes T and v. If we now

dQ
denote by -^dv the quantity of heat which the body must take

up in order to expand to the extent of dv, without change of
temperature (for the sake of conformity with the mode in
which the signs are used in the other equations occurring in this
section, the positive sense of the quantity of heat is here taken
' diflFerently from what it is in equation (4), in which heat given

up by the body, and not heat communicated to it, is reckoned
positive), the following weU-known equation, from the mecha-
nical theory of heat, will hold good : —

y dv" dT'

Let us now suppose that the temperature of the body is changed
by rfT, and its volume by dVy and let us call the quantity of heat
which it then takes up dCi; we may then write

For the magnitude here denoted by -77^, which represents the
specific heat under constant volume*, we can put the letter c,
and for -j- the expression already given. Then we have

rfQ=crfT+AT^rfv (6).

oT

The only external force which the body has to overcome on

► expanding, being p, the work which it performs in so doing is

pdv, and the magnitude^^rfv indicates the increase of this work

with the temperature.

If we now apply this equation to a perfect gas, the specific
heat under constant volume is in this case to be regarded as the
real specific heat [capacity for heat] f, which gives the increase

* [Provided the weight of the body under consideration be regarded as a
unit of weight.— 1864.]

t [I must here say a few words relative to the expressions real specific heat
and real capacity for heat.

According to the mechanical theory of heat, it is not necessary that the

230 SIXTH MEMOIR.

in the quantity of heat actually present in the gas ; for here no

heat i& consumed in work^ since exterior work is only performed xl

lieat imparted to a body should afterwaids be actually present in the form
of heat in that body ; a portion of the heat, in fiict, may be consumed on interior
or exterior work. Accordingly it is necessary to consider, not only the dif-
ferent specific heats of bodies, which indicate how much heat must be im-
parted to a body for the purpose of heating it under different circumstances
(e, g, the specific heat of a solid or liquid body at the ordinaiy atmospheric
pressure, the specific heat of a gas at constant volume or under constant
pressure, and the specific heat of a vapour at its maximum density), but also
another magnitude indicating by how much the heat actually present in a
unit-weight of the substance in question — ^in other words, the rur viva of its '

molecular motions — ^is increased by an elevation of one degree in temperature. |

Hitherto I have, in accordance with Rankine's custom, called this magni- {

tude the real tpecifie heat ; I must confess, however, that to me the expres- ,

sion does not appear to be quite appropriate. It implies, in fact, that the
other magnitudes above named — ^for instance the specific heat of a gas under j

constant pressure— have improperly received the name specific heat. But in ^

the words specific heat there is nothing whatever on which to found the ob-
jection that the quantity of heat which must be imparted to a unit-weight of
gas, under constant pressure, in order to raise its temperature one degree
ought not to be termed the specific heat imder constant pressure. It is
otherwise, however, with the expression capacity for heat. These words
clearly indicate that the heat in question is that which the body can contain.
Accordingly the expression is a very appropriate one for denoting by how much
the heat actually present in the body is increased by the elevation of tem-
perature ; whilst, strictly speaMng, it is not applicable to the total quantity of
heat which must be imparted to a body during its elevation of temperature,
of which quantity a portion is transformed into work^ and the remaining por-
tion alone remains as heat in the body.

I should deem it advantageous, therefore^ to distinguish the two expressions
specific heat and capacity for heat, which have hitherto been employed synony-
mously in physical treatises ; to give the name ^ecific heat to the total quan-
tity of heat which must be imparted to the body to elevate its temperature,
so that there will be different Mnds of specific heat according to the different
conditions imder which this elevation occurs, and to reserve the name capa-
city for heat for the heat actually present in a body. Since this distinction,
however, has not yet been made, the term real may provisionally be added,
and the expression real capa^^y for heat used to denote the magnitude which
indicates by how much the heat actually present in a unit-weight of the
substance is increased by an elevation of one degree in temperature.

In order to introduce this terminology, I have in the text employed the
words real capacity for heat wherever the expression real specific heat appears
in the original. This is of course a merely verbal alteration and has no effect
on the meaning, since, hitherto, physicists have drawn no distinction what-
ever between specific heat and capacity for heat ; to indicate the alteration,
however, the latter of these expressions, whenever it is substituted for the
former, is placed between square brackets. — 1864.]

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 231

when increase of volume occurs, and interior work has no
existence in the case of perfect gases. We may therefore re-
gard cdT as identical with rfH. We have further, for the per-
fect gases, the equation

where B is a constant, and hence we get

-^flto = — rft? = Rd . log r.
dT V ^

Equation (6) is thus transformed into

rfQ=dH+ARTrf.logt?*. ' . . . . (i)

This equation agrees, disregarding the difference in the sign of
rfQ (which is caused only by the different way in which we have
chosen to employ the signs + and — in this case), with equa-
tion (4), and the function there represented by the general sym-
bol Z has, in this particular case, the form AR log v.

Rankine, who has written several interesting memoirs on the
transformation of heat into workf, has proposed a transforma-
tion of equation (6), for all other bodies, similar to that above
given for perfect gases only J. To this end he writes, only with
slightly different letters,

rfQ=*rfT + ATrfP, (8)

where * denotes the real [capacity for heat] of the body, and P is
a magnitude to the determination of which Rankine appears to
have been led chiefly by the circumstance mentioned above, that

the quantity -^dv which occurs in equation (6) represents the

increase of exterior work done during the infinitely small
change of volume which accompanies an increase of temperature.
Rankine defines the magnitude P as ^^ the rate of variation of

* [If we retain for cTEL the product cdT, and for d log v the expression
~, the equation (7) will have the form

dQ^cdT+AR^^dv,

and become the same as the equation (II b) which presented itself in the
Rrst Memoir, at p. 38.— 1864.]

t Philosophical Magazine, S. 4, vol. v. p. 106 ; Edinburgh New Philo-
sophical Journal, New Series, vol. ii. p. 120 ; Manual of the Steam-engine.

X Manual of the Steam-engine, p. 310.

232 SIXTH MBlfOIR.

performed eflPective work with temperature;'*' and denoting the
exterior work which the body can do in passings at a given tem-
perature^ from a given former condition into its present one^
bjr U, he puts

F=g (9)

In the discussion which immediately follows^ of the case in
which the exterior work consists only in overcoming an external
pressure^ he gives the equation

V^(pdv,

whence follows

^-iP- (i«)

The integals which here occur are to be taken from a given
initial volume to the actually existing volimie, the temperature
being supposed constant. Introducing this value of F into
equation (8), he writes it in the following form : —

rfQ=(* + ATr^rft;)rfT + AT^rft?. . . (11)

His reason for taking an infinitely large volume as the initial
volume is not stated^ although the choice of the initial volume
is evidently not a matter of indifference.

It is easy to see that this manner of modifying equation (6)
is very different from my development; the results are abo dis-
cordant; for the quantity F is not identical with the corresponding

quantity -^Z in my equations, but only coincides with it in that

part which could be deduced from data already known ; that is
to say, the last term of equation (6) gives the differential coeffi-
cient, according to v, of the magnitude which has to be intro-
duced, since, to secure the coincidence of this term and the last
in the modified equation, we must in any case put

dv Adv dT ^ '

Rankine has, however, as may be seen from equation (10),
* [Conceive a function of T and t? to be introduced ; it may be denoted at

EXTENSION OF SECOND PUNDAMENTAL THEOREM. 233

formed the magnitude F by simply integrating, according to t?,
this expression for the differential coefficient of F according to

V. In order to see in what way the magnitude j- Z differs ^from

this, we will modify somewhat the expression for Z given in the
preceding Art.

According to equation (2),

?rfZ=rfL.
A

rfL denotes here the interior and exterior work, taken together,
which is performed when the body undergoes an infinitely slight
change of condition. We will denote the interior work by dl ;
and since when the condition of the body is determined by its
temperature T and its volume r, I must be a function of these
two quantities, we may write

The exterior work, assuming it to consist merely in overcoming
an external pressure, is represented by pdv. Hence, if we fur-
ther decompose the differential dZ into its two parts, we may
write the above equation thus : —

TrfZ„. TdZ^_dI_,^,/dI

5-„ , TrfZ, rfl ,_ /dl . \,

pleasure by either F or by -^Z, but la either case let it have the property of
satisfying the equation

or

«?Q=AjrfT+TJZ.

These equations may then be written in the form

rfQ=(A+ATg)dT+ATfrf.,
or

and on supposing them to exist simultaneously with (6), and remembering
that the factors of dv must be the same in all three equations, we at once

conclude that the magnitudes F and — Z must satisfy the condition expressed

A

in (12).— 1864.]

284 SIXTH HKMOIB.

whence tre have

T«rZ_rfI
ArfT~<fr'
• TrfZ_rfI

A dv "rfr"*"^'

(13)

Differentiating the first of these equations according to v^ and
the second according to T^ we get

T (PZ _ (PI
AdTdv^dTdv'

IdZ H d^Z _ dn dp
A rfv "*"A dTdv" dTdv"^ dT'

The first of these equations subtracted &om the second^ gives

l^dZ^^
A dv "rfT"

The differential coefficient of Z according to v consequently
folfils the condition given in (12) ; the first of the equations
(13) gives at the same time the differential coefficient accord-
ing to T ; and putting these two together ^ we obtain the complete
differential equation

ldZ=]^^dT+pv.. .... (14)

To obtain the quantity -^Z, we must integrate this equation.

It is easy to see that this integral will in general differ by a
function of T from that which would be obtained by integrating

only the last term*. It is only when ^=0 that the two

integrals may at once be regarded as equals and then^ in order
that the foregoing equation may be integrable^ it follows that

-^=0; this case occurs in perfect gases.

I believe that what I can claim as new in my equation (II)

* [In order to integrate the equation (14), we will assume an initial con-
dition in which the temperature and volume have the values Tq and v^, and
denote by Z,, the corresponding value of Z. Let us suppose then that the
temperature first changes from T^ to any given value T, and that, subse-
quently, the volume changes, without further alteration of temperature, from

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 235

is just this^ that the magnitude Z which th6re occurs has
acquired, through my developments, a definite physical mean-
ing, whence it follows that it is ftdly determined by the arrange-
ment of the constituent particles of the body which exist at any
given instant. Thus only does it become possible to deduce from
this equation the important conclusion which follows.

7. We will now investigate the manner in which, from equa-
tion (II), it is possible to arrive at the equation (I) previously
given in Art. 1, which equation must hold, according to the
ftindamental theorem that I have already enunciated, for every
reversible cyclical process.

When the successive changes of condition constitute a cyclical
process, the disgregation of the body is the .same at the end of
the operation as it was at the beginning, and hence the follow-
ing equation must hold good : —

JrfZ=0 (15)

Equation (II) is hereby transformed into

j"^?a+^=o. (16,

In order that this equation may accord with equation (I),
namely,

the following equation must hold for every reversible cyclical
process : —

Jf =0 (Ill)

It is this equation which leads to the consequence referred to

Vf^ to V, By pursuing this course of changes we obtain by integration the
following equation

It is manifest that the integral

which here presents itself is the function of T, mentioned in the text, as that
by which the magnitude -jZ differs from the magnitude F, as defined by the
equation (10).— 1866.]

236 SIXTH MEMOIR.

in the introductory Art.* as at variance with commonly received
views. It can, in fact, be proved that, in order that this equa-
tion may be true, it is at once necessary and sufficient to assume
the following theorem : —

The quantity of heat actually present in a body depends only
on its temperature, and not on the arrangement of its constituent
particles.

It is at once evident that the assumption of this theorem suf-
fices for equation (III) ; for if H is a function of the temperature

only, the diflferential expression -=- takes the form/(T)rfT, in

which /(T) is obviously a real Amction which can have but one
value for each value of T, and the integral of this expression
must obviously vanish if the initial and final values of T are the
same.

The necessity of this theorem may be demonstrated thus.

In order to be able to refer the changes of condition to
changes of certain magnitudes, we will assume that the manner
in which the body changes its condition is not altogether arbi-
trary, but is such that the condition of the body is deter-
mined by its temperature, and by any second magnitude
which is independent of the temperature. This second mag-
nitude must plainly be connected with the arrangement of
the constituent particles : we may, for example, consider the
disgregation of the body as such a magnitude ; it may, how-
ever, be any other magnitude dependent on the arrangement of
the constituent particles. A case which often occurs, and one
which has been frequently discussed, is that in which the volume
of the body is the second magnitude, which can be altered in-
dependently of the temperature, and which, together with the
temperature, determines the condition of the body. We will
take X as a general expression for the second magnitude, so
that the two magnitudes T and X together determine the con-
dition of the body.

Since, however, the quantity of heat H, present in the body,
is a magnitude which in any case is completely determined by
the condition of the body at any instant, it must here, where
the condition of the body is determined by the magnitudes T

• [See p. 216.]

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 237

and X, be a function of these two magnitudes. Accordingly,
we may write the differential dK in the following form,

rfH=MrfT+Ne;X, (17)

where M and N are functions of T and X, which must satisfy
the well-known equation of condition to which the differential
coefficients of a function of two independent variables are sub-
ject ; that is, the equation

rfM rfN .,^.

Again, if the integral i-m~ is to become equal to nothing each
time that the magnitudes T and X return to the same values as
they had at the beginning, -7=- must also be the complete dif-
ferential of a fimction of T and X. And since we may write,
as a consequence of (17),

f=^+^^, (19)

we obtain, for the differential coefficients which here occur, the
equation of condition

^xw^StCt} (^^

which exactly corresponds to equation (18).

By effecting the differentiations, this equation becomes

lrfM_lrfN_N .

TdX^TdT T«^ ^ ^

and, by applying equation (18) to this, we get

N=0 (22)

According to (17), N is the differential coefficient of H accord-
ing to X ; and if this differential coefficient is to be generally
equal to nothing, H itself must be independent of X ; and suice
we may understand by X any magnitude whatever which is in-
dependent of T, and together with T determines the condition
of the body, it follows that H can only be a function of T.

8. This last conclusion appears, according to commonly re-
ceived opinions, to be opposed to well-known facts.

288 SIXTH MEMOIR.

I will choose as an illustratire example^ in the first placCj a
ease which is very familiar^ and in which the discrepancy is
particularly great, namely^ water in its various states. We may
have water in the liquid state^ and in the solid state in the form
of ice, at the same temperature ; and the above theorem asserts
that the quantity of heat contained in it is in both cases the
same. This appears to be contradicted by experience. The
specific heat of ice is only about half as great as that of liquid
water, and this appears to furnish grounds for the following
conclusion. If at any given temperature a unit-weight of ice
and a unit-weight of water in reality contained the same quantity
of heat, we must, in order to heat or cool them both, impart to
or withdraw from the water more heat than we impart to or
withdraw from the ice, so that the equality in the quantity of
heat could not be maintained at any other temperatures. A
similar difference to that existing between water and ice also
exists between water and steam, inasmuch as the specific heat of
steam is much smaller than that of water.

To explain this difference, I must recall the fact that only
part of the heat which a body takes up when heated goes to
increase the quantity of heat actually present in it, the remainder
being consumed as work. Now, I believe that the differences in
the specific heat of water in its three states of aggregation are
caused by great differences in the proportion which is consumed
as work, and that this proportion is considerably greater in the
liquid state than in the other two states*. We must, accord-
ingly, here distinguish between the observed specific heat and

* I have already enunciated this view in my first memoir on the Mecha^
nical Theory of Heat, having, in fact, inserted the following in a note [p. 20],
which has reference to the diminution of the cohesion of water with increase
of temperature : — " Hence it follows, at once, that only part of the quantity
of heat which water receives from without when heated, is to be r^^turded as
heat in the free state, the rest being consumed in diminishing cohesion. This
view is in accordance with the circumstance that water has so much higher
a specific heat than ice, and probably also than steam." At that time the
experiments of Regnault on the specific heat of gases were not yet published,
and we still found in the text-books the number 0*847, obtained by De la Boche
and B^rard, for the specific heat of steam. I had, however, already con-
cluded, on the theoretical groimds which are the subject of the present dis-
cussion, that this number must be much too high ; and it is to this conclusion
that the concluding words ^' and probably also than steam " refer.

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 239

the real [capacity for heat] with which the change of tempera-
ture rfT must be multiplied, in order that we may obtain the
corresponding increase of the quantity of heat actually present ;
and, in accordance with the above theorem, I beUeve we must
admit that the real [capacity for heat] of water is the same in
all three states of aggregation; and the same considerations
which apply to water must naturally also apply in like manner
to other substances. In order to determine experimentally
the real [capacity for heat] of a substance, it must be taken in
the form of strongly overheated vapour, in such a state of expan-
sion, in fact, that the vapour may, without sensible error, be
regarded as a perfect gas; and its specific heat must then be
determined under constant volume*.

Rankine is not of my opinion in relation to the real [capacity
for heat] of bodies in difierent states of aggregation. At page 307
of his ^ Manual of the Steam-Engine,^ he says, '' The real specific
heat of each substance is constant at all densities, so long as the
substance retains the same condition, solid, liquid, or gaseous ;
but a change of real specific heat, sometimes considerable, often
accompanies the change between any two of these conditions.'*

In the case of water in particular, he says, on the same page,
that the real specific heat of liquid water is ^^ sensibly equal " to
the apparent specific heat; whereas, according to the view above
put forth by myself, it must amount to less than half the
apparent specific heat.

If Rankine admits that the real [capacity for heat] may be
different in different states of aggregation, I do not see what
reason there is for supposing it to remain constant within the
same state of aggregation. Within one and the same state
of aggregation, e. g. within the solid state, alterations in the
arrangement of the molecules occur, which, though without
doubt less considerable, are still essentially of the same kind as
the alterations which accompany the passage from one state of
aggregation to another; and it therefore seems to me that there
is something arbitrary in denying for the smaller changes what

* [In the Appendix B. to this memoir will be found a Table containing the
specific beats at constant volume, calculated according to the principles of
tbe mechanical theory of heat, for those erases and vapours whose specific
heats under constant pressure have been obs^ .Ted by Regnault. — 1864.]

240 SIXTH MBMOia.

is admitted in respect to the greater. On this point I cannot
agree with the way in which the talented English mathematician
treats the subject; relying simply on the law established by
myself in relation to the working force of heat, it appears to
me that but one of the following cases can be possible. Either
the above law is correct, in which case the real [capacity for
heat] remains the same, not only for the same state of aggre-
gation, but for the diflTerent states of aggregation, or the law is
not correct, and in this case we have no definite knowledge
whatever concerning the real [capacity for heat], and it may
equally well be variable within the same state of aggregation as
in different states of aggregation.

9. I beUeve, indeed, that we must extend the application of
this law, supposing it to be correct, still farther, and especially
to chemical combinations and decompositions.

The separation of chemically combined substances is likewise
an increase of the disgregation, and the chemical combination of
previously isolated substances is a diminution of their disgrega-
tion ; and consequently these processes may be brought under
considerations of the same class as the formation or precipitation
of vapour. That in this case also the effect of heat is to increase
the disgregation, results &om many well-known phenomena,
many compounds being decomposable by heat into their consti-
tuents — as, for example, mercuric oxide, and, at very high
temperatures, even water. To this it might perhaps be objected
that, in other cases, the effect of increased temperature is to
favour the union of two substances — ^that, for instance, hydrogen
and oxygen do not combine at low temperatures, but do so
easily at higher temperatures. I believe, however, that the heat
exerts here only a secondary influence, contributing to bring
the atoms into such relative positions that their inherent forces,
by virtue of which they strive to unite, are able to come into
operation. Heat itself can never, in my opinion, tend to pro-
duce combination, but only, and in every case, decomposition.

Another circumstance which renders the consideration of this
case more difficult is this, that the conclusions we have been
accustomed to draw always imply that the alterations in question
can take place in a continuous and reversible manner; this,
however, is not usually the case under the circumstances which

]

EXTENSION OF SECOND FUNDAMENTAL THEOREM. 241

accompany our chemical operations. Nevertheless cases do
occur in which this condition is fulfilled, especially in the
chemical changes brought about by the action of electric force.
The galvanic current affords us a simple means of causing com*
bination or decomposition ; and in this case the cell in which the
chemical change takes place itself forms a galvanic element, the
electromotive force of which either contributes to intensify the
current, or has to be overcome by other electromotive forces ; so
that in the one case there is a production, and in the other a
consumption of work.

Similarly, I believe that we could in all cases, by producing
or expending work, cause the combination or separation of sub-
stances at pleasure, provided we possessed the means of acting
at will on the individual atoms, and of bringing them into what-
ever position we pleased. At the same time I am of opinion
that heat, leaving out of view its secondary effects, tends' in a
definite manner, in all cases of chemical change, to render the
combination of atoms more difficult, and to facilitate their sepa-
ration, and that the energy of its action is likewise regulated by
the general law above given.

Supposing this to be the case, the theorem which we have
deduced from this law must also be applicable here, and a che-
mical compound must contain exactly the same quantity of heat
as its constituents would contain at the same temperature iii the
uncombined state. Hence it follows that the real [capacity for
heat] of every compound must admit of being simply calculated
from the real [capacities] of the simple bodies^. If, further,
the well-known relation between the specific heats of the simple
bodies and their atomic weights be taken into consideration (a
relation which I believe not only to be approximately, but, in the
case of real [capacities for heat], absolutely exact), it will be
apparent what enormous simplifications the law which we have
established is capable, supposing it to be true, of introducing
into the doctrine of heat.

* [ThQAj)pendix B, already referred to, also contains the principal portions
of a Note, published by me in the Amuden der Cherme und Pharmade,
which offers an opportunity of testing how far the specific heats, at constant
volume, of a series of gases (calculated according to Kegnault's observations)
correspond to the theorem, deduced in the text, relative to the true capacities
for heat.— 1864.]

R

24SI SIXTH MEMOIR.

10. After these expositions^ I can now state the mor^ extended
form of the theorem of the equivalence of transformations.

In Art. 1 I hare mentioned two kinds of transformations :
; first, the transformation of work into heat, and vice versd', and
'' j secondly, the passage of heat between bodies of different tem-
peratures. In addition to these, we will now take, as a third
/^^ kind of transfonnation, the change in the disgregation of a

\ body, assuming the increase of disgregation to be a positive,
and the diminution of it to be a negative transformation.

We will now, in the first place, bring the first and last trans-»
formations into relation with each other; and here the same
circumstances have to be taken into consideration as have
already been, discussed in Art. 5« If a body changes its disgre-*
gation in a reversible manner, the change is accompanied by a
transformation of heat into work, or of work into heat, and we
can determine the equivalent values of the two kinds of trans*
formations by comparing together the transformations which
take place simultaneously.

Let us first assume that the same change of arrangemeni
takes place at different temperatures ; the quantity of heat which
is thereby converted into work, or is produced from work, will
then vary; in fact, according to the above law, it wiU be pro-
portional to the absolute temperature. If, now, we regard as
equivalent the transformations which correspond to one and the
same change of arrangement, it results that, for the determi<-
nation of the equivalence-values of these transformations, we
must divide the several quantities of heat by the absolute tem-
peratures respectively corresponding to them. The production
of the quantity of heat Q from work must, therefore, if it takes
place at the temperature T, have the equivalence-value

^ const. ;

and if we here take the constant, which can be assumed at will,
as equal to unity, we obtain the expression given in Artv 1.

We will assume, in the second place, that different chanpes
of arrangement take place at one and the same temperature, these
changes being accompanied by increase of disgregation j and if
we adopt as a principle that increments of disgregation whereiia
the same quantity of heat is converted into work shall be re-

I

EXTENSION OP SEOOKD FUNDAMENTAL THEOREM. 243

garded as equivalent to each other, and that their equivalence-
value shall be equal, when taken absolutely, to that of the
simultaneously occurring transformation from heat into work,
but that they shall have the opposite sign, we thus acquire a
starting-point for the determination of the equivalence-values of
changes 6f disgregation.

By combining these two rules, we can determine also the
equivalence-value of a change of disgregation occurring at
various temperatures, and we thus obtain the expression given
in Art. 5. Let, for instance, dh be an element of the work
performed during a change of disgregation, in eflFecting which
the quantity of heat AdJj is consumed, and let the equivalence-
value of the change of disgregation be denoted by Z— Zq,
we then have

Z-Zo

Finally, as to the process cited above as the second kind of
transformation — ^namely, the passage of, heat between bodies of
diflferent temperatures, — ^in the case of reversible changes of
condition it can be brought about only by heat being converted
into work at the one temperature, and work back again into
heat at the other ; it is therefore already comprised among the
transformations of the first kind. And, as I have mentioned in
my previous memoir, we may in all cases regard a transforma-
tion of the second kind as a combination of two transformations
of the first kind.

We will now return to equation (II), namely,

J-^9+«?+J^=„.

dH is here the increment of the quantity of heat present in the
body corresponding to an infinitely small change of condition,
and dO. is the quantity of heat simultaneously given up to external
bodies. The sum dCL+dK is therefore the new quantity of heat
which, supposing it to be positive, must be produced £rom work,
or if it is negative, must be converted into work. Accordingly,
the first integral in the above equation is the equivalence-value
of all the transformations which have occurred of the first kind ;

b2

244 SIXTH MEMOIR.

the second integral represents the transformations of the third
kind; and the sum of all these transformations must be, as is
expressed by the equation, equal to nothing.

Hence, in so far as reversible alterations of condition are con-
cerned, the theorem may be expressed in the following form : —

If the equivalence-value ^ be assumed for the production of the

quantity of heat Qfrom work at the temperature T, a magnitude
admits of being introduced, as a second transformation corre-
sponding thereto, which has relation to changes in the arrange-
ment of the particles of the body, is completely determined by the
initial and final conditions of the body, and fulfils the condition
that in every reversible change of condition the algebraic sum of
the transformations is equal to nothing.

11. We must now examine the inanner in which the fore-
going theorem is modified when we give up the condition that
all changes of condition are to take place reversibly.

From what has been said in Art. 4 concerning non-reversible
changes of condition, it is easy to perceive that the following
must be a general property of all three kinds of transforma-
tions. A negative transformation can never occur without a
simultaneous positive transformation whose equivalence-value is
at least as great ; on the other hand, positive transformations are
not necessarily accompanied by negative transformations of equal
value, but may take place in conjunction with smaller negative
transformations, or even without any at all.

If heat is to be transformed into work, which is a negative
transformation, a positive change of disgregation must take
place at the same time, which cannot be smaller in amount than
that determinate magnitude which we regard as equivalent. In
the positive transformation of work into heat, on the other hand,
the state of things is different. If the force of heat is overcome
by opposing forces, so that a negative change of disgregation is
brought about, we know that in this case the overcoming forces
may be greater than is required to produce the particular result.
The excess of force may then give rise to motions of considerable
velocity in the parts of the body under consideration, and
these motions may subsequently be changed into the molecular
motions which we call heat, so that in the end more work comes

EXTENSION OF SECOND FUNDAMENTAL THEOKEM. 245

to be transformed into heat than corresponds to the negative
change of disgregation brought about. In many operations,
especially in friction, the transformation of work into heat may
take place even quite independently of any simultaneous negisitive
transformation.

The relation in which the third kind of transformation,
namely change of disgregation, stands to considerations of this
nature, is implied in what has been already said. The positive
change of disgregation may indeed be greater, but cannot be
smaller, than the accompanying transformation of heat into
work j and the negative change of disgregation may be smaller,
but cannot be greater, than the transformation of work into
heat.

Finally, in so far as regards the second kind of transformation,
or the passage of heat between bodies of different temperatures,
I have thought myself justified in assuming as a fundamental
proposition what, according to all that we know of heat, must be
regarded as well-established, . namely, that the passage from a
lower to a higher temperature, which counts as a negative trans-
formation, cannot take place of itself — that is, without a simulta-
neous positive transformation. On the other hand, the passage of
heat in the contrary direction, from a higher to a lower tempera-
ture, may very well take place without a simultaneous negative
transformation.

Taking these circumstances into consideration, we will now
return once more to the consideration of the development by
means of which we arrived at equation (II) in Art. 5. Equation (2),
which occurs in the same Article, expresses the relation in which
an infinitely small change of disgregation must stand to the
work simultaneously performed by the heat, under the condition
that the change takes place in a reversible manner. In case
this last condition need not be fulfilled, the change of disgre-
gation may be greater, provided it is positive, than the value
calculated from the work; and if negative, it may be, when
taken absolutely, smaller than that value, but in this case also it
would algebraically have to be stated as greater. Instead of
equation (2), we must therefore write

rfZ^^. . (2a)

246 SIXTH MKMOIR.

Applying tluB to equation (1)^ we obtain^ instead of equation (5),
^-^^^dZ^O (5«)

The further question now arises^ what influence would it have
on the formulae, if a direct passage of heat took place between
parts of different temperature within the body in question.

In case the body is not of ujHform temperature throughout,
the differential expression occurring in equation (5fl) must not
be referred to the entire body, but only to a portion whose tem-
perature may be considered as the same throughout ; so that if
the temperature of the body varies continuously, the number of
parts must be assumed as infinite. In integrating, the ex-
pressions which apply to the separate parts may be united again
to a single expression for the whole body, by extending the in-
tegral, not only to the changes of one part, but to the changes
of all the parts. In forming this integral, we must now have
regard to the passage of heat taking place between the dif-
ferent parts.

It must here be remarked that dQ is an element of the heat
which the body under consideration gives up to, or absorbs from,
an external body which serves only as a reservoir of heat, and
that this element does not come into question now that we are
discussing the passage of heat between the different parts of the
body itself. This transfer of heat is mathematically expressed
by a decrease in the quantity of heat H in one part, and an
equivalent increase in another part ; and accordingly we require

to direct our attention only to the term -^ in the differential

expression (5 a) . If we now suppose that the infinitely small
quantity of heat ^H leaves one part of the body whose tempera-
ture is Tp and passes into another part whose temperature is T^,
there result the two following infinitely small terms,

--^and+^,

which must be contained in the integral ; and since T^ must be
greater than Tg, it follows that the positive term must in any
case be greater than the negative term, and that consequently
the algebraic sum of both is positive. Tb^ s^pae thing applies

EXTENSION OF SE.COND FUNDAMENTAL THEOEEM. M?

equally to every other element of heat transferred from one pari
to another ; and the change which the integral of the whole
differential expression occurring in (5 a) undergoes^ on account
pf this transfer of heat, can therefore only consist in the addition
pf a positive quantity to the value which would else have been
obtained. But since, as results from equation (5 a), the last
value which would be obtained, without taking this direct
transfer of heat into consideration, cannot be less than nothing,
this can still less be the case when it has been increased by
another positive quantity.

We may therefore write as a general expression, including all
the circumstances which occur in non-reversible changes, the
following, instead of equation (II) :—

J^a+^+J^>o (II«)

The theorem which in Art. 1 was enunciated in reference to cycli-
cal processes only, and was represented by the expression (I a),
has thus assumed a more general form, and may be enun-
ciated ihus :-^ '

The algebraic sum of all the transformations occurring during
any change of condition whatever can only be positive, or, as an
extreme case, equal to nothing.

In my previous paper I have spoken of two transformations
with opposite signs, which neutralize each other in the algebraic
sum, as compensating transformations. The foregoing theorem
may therefore be enunciated still more briefly as follows : —

Uncompensated transformations can only be positive*.

12. In conclusion, we will submit the integral

Cdn

J T ^
which haa been frequently used above, to a somewhat closer con-
sideration. We will call this integral, when it is taken from any

* [I will here say a word as to the manner in which I have defined
poffltive and negative transformations, since, without anticipating matters, I
T50uld not on first making choice of these signs, in the Fourth Memoir,
p. 123, state the reasons which determined my choice.

If transfers of heat between bodies of difierent temperatures were alone the
subject of contemplation, it might perhaps be thought more appropriate to
call the transfer of heat from a colder to a warmer body a positive, and the

248 SIXTH MEMOIB.

given initial condition to tlie condition actually existing^ the trans^
formation-value of the heat actually present in the body when
calculated from the given initial condition. That is^ when in any
way whatever work is transformed into heat, or heat into work,
and the quantity of heat present in the body is thereby altered,
the increment or decrement of this integral gives the equivalence-
value of the transformations which have taken place. Further,
if transfers of heat take place between parts of different tempe-
rature within the body itself, or within a system of bodies,
the equivalence-value of these transfers of heat is likewise ex-
pressed by the increment or decrement of this integral, if it is
extended to the whole system of bodies under consideration.

In order to be able actually to perform the integration which
has been indicated, we must know the relation between the
quantity of heat H and the temperature T. If we call the mass
of the body m, and its real [capacity for heat] c, we have, for
a change of temperature throughout amounting to rfT, the
equation

rfH=mcrfT • . . . (23)

According to what has been said above, the real [capacity for
heat] of a body is independent of the arrangement of its
particles ; and since an arrangement is known, namely, that in
perfect gases, for which we must regard it as established, partly
by existing experimental data, and partly as the result of
theoretical considerations, that the real [capacity for heat] is
independent of temperature, we may assume the same thing for
the other states of aggregation, and may regard the real [capacity

transfer from a wanner to a colder body a negative transformation. Since
we have to consider, however, not only transference of heat but likewise two
other kinds of transformation connected therewith, our point of view on
proceeding to a choice of signs must not be limited to making provision for
the ordimary notions regarding transfers of heat; we must, on the contrary,
seek for an appropriate distinctive feature common to all three kinds of trans-
formations. The theorem enunciated in the text furnishes this feature. Each
of the three kinds of transformation can take place, in one direction, of itself
or without compensation, but in the opposite direction only with compensa-
tioiL In nature, therefore, there is a general tendency to transformations of
a definite direction. I have taken this tendency as the normal one, and
called the transformations which occur in accordance with this tendency,
positive; and those which occur in opposition to this tendency, negative. —
1864.]

I

EXTENSION- OF SECOND FUNDAMENTAL THEOREM. 249

for heat] as always constant. Thence it follows that the amount
of heat present in a body is simply proportional to its absolute
temperature^, inasmuch as we can write

H=mcT (24)

Even when the body is not homogeneous, but consists of
different substances, all, however, at the temperature T, the
foregoing equation will still remain applicable, if for c we substi-
tute the corresponding mean value. On the other hand, if diffe-
rent parts of the body have different temperatures, we must in
the first instance apply the equation to the separate parts, and
then unite the various equations by summation. If, for the
sake of generality, we assume that the temperature varies con-
tinuously, so that the body must be conceived as divided into an
infinite number of parts, the equation takes the following form :

H = jcTrfm (25)

Applying these expressions to the integral given above for the
transformation-value of the heat in the body, and denoting the
initial temperature by Tq, we obtain> for the more simple case
in which the temperature is uniform throughout,

J -^ =mc 1^ -^=mc log ^, . . . . (26)

and, as a general expression embracing all cases,

J^=Jclog J^- rfm (27)

If the disgregation of a body is changed, without heat being
supplied to or withdrawn from it, by an external object, the
amount of heat contained in the body must be changed in con-
sequence of the production or consumption of heat attendant on
the change of disgregation, and a rise or fall of temperature
must be the result ; consequently the question may be raised.
How great must the change of disgregation be in order to
bring about a given change of temperature, it being assumed
that all changes of condition take place reversibly? In this
case we must apply equation (II), putting c^QssO, whereby it is
transformed into

j^-j---

(28)

260 BIXTH MXMOIB.

If we assume^ for tlie sake of simplicity^ that the temperature of
the entire body varies uniformly, so that T has the same value
for all parts, we may apply equation (26) to the determinatioii
of the first of the two integrals; and we thus obtain, for the
required c!hange of diBgregation, the equation

Z-Zo=mclog^* (29)

If we desired to cool a body down to the absolute zero of tepi-
perature, the correspondiag change of disgregation, as shown
by the foregoing formula, in which we should then have T=0,
would be infinitely great. Hereon is based the argument by
which it may be proved to be impossible practically to arrive at
the absolute zero of temperature by any alteration of the con-
dition of a body.

APPENDICES TO SIXTH MEMOIR [1864].
APPENDIX A. (Page 226.)
ON TERMINOLOGY.

The new conceptions which the mechanical theory of heat has
introduced into science present themselves so frequently in all
investigations ou heat^ that it has become desirable to possess
simple and characteristic names for them.

I have divided into the following three parts the heat which
must be imparted to a body in order to change its condition in
any manner whatever : first, the increased amount of heat ac-
tually present in the body ; second, the heat consumed by inte-
rior work ; and third, the heat consumed by exterior work. Of
these three quantities of heat the last can only be determined
when aU the changes are known which the body has suffered ;

* [K the above simplifying hypothesis — ^that the temperature is the same
in all parts of the body and changes in the same manner — be not made, we
shall have the equation

Z-Z,= fclog^<?m,
instead of the equation (29).— 1864.]

ON TERMINOLOGY, 251

for the determination of the two first quantities, however, a
knowledge of the entire series of changes is not necessary, an ac-
quaintance with the initial and final conditions of the body suf-
fices. Given, therefore, the initial condition, proceeding from
which the body arrives successively at any other conditions what-
ever, the first and second of the above quantities of heat may be
regarded as two magnitudes which are perfectly defined by the
condition of the body at the moment under consideration. The
same remark applies, of course, to the sum of these two quanti-
ties which I have represented by U, and which is of great im-»
portance, inasmuch as it presents itself in the first fimdamental
equation of the mechanical theory of heat^.

The definition I have given of this magnitude — the mm of
the increment of actually present heat, and of the heat consumed
by interior wor*-r-being for general purposes too long to serve
as the name of the quantity, several more convenient ones have
been proposed. As already remarked in the note on p. 226,
Thomson has to this end employed the expression " the mecha-
nical energy of a body in a given state,'' and Kirchhoff the term
" tVirkungsfunction.^' Zeuner, again, in his " GrundzUge der
mechanischen Wdrmetheorie,'' has called U " die innere Warme
des Korpers^^ (interior heat of the body).

The latter name does not appear to me to correspond quite to
the signification of the magnitude U, since only a portion of the
latter represents heat actually present in the body, in other
words, vis viva of its molecular motions, the other portion having
reference to heat which has been consumed by interior work, and
which, therefore, no longer exists as heat. I do not for a mo-
ment imagine that Zeuner had any intention to imply, by that
name, that all the heat represented by U was actually present
as heat in the body; nevertheless the name might easily be
interpreted in this sense.

Of the two other expressions mentionod above, the term
energy employed by Thomson appears to me to be very afpro-
priate ; it has in its favour, too, the circumstance that it corre-
sponds to the proposition of Bankine to include under the
common name energy, both heat and everything that heat can

* See equation (11 a) of the First Memoir, p. 28, and equation (I) of the
Fourth Memoir, p. 113.

<^

252 SIXTH MEMOIR/ APPENDIX A.

replace. I have no hesitation, therefore, in adopting, for the
quantity U, the expression energy of the body.

It must be here observed, however, that the total energy of a
body cannot be measured, it is only the increment of energy,
due to the passage of the body from any initial state to its pre-
sent condition, that is susceptible of measurement. The initial
condition being assumed as given, the increment of energy is a
perfectly defined magnitude for every other condition of the
body. The question is, are we to understand by the energy of a
body merely the increment of energy estimated from a given
initial condition, or is the energy which the body possessed at
the beginning to be included in the term ? In the latter case,
where the total energy of the body is implied, we must conceive
the increment of energy to be supplemented by the addition of
an unknown constant having reference to the initial condition.
It will not always be necessary, of course, to mention thia con-
stant expressly ; we may tacitly assume that it is included.

Since the magnitude U consists of two parts which have fre-
quently to be considered individually, it will not suffice to have
an appropriate name for U merely, we must also be able to refer
conveniently to these its constituent parts.

The first part presents no difficulty whatever ; the heat actually
present in the body may be simply called tJie heat of the body, or
the thermal content of the body {fVdrmeinhalt des Korpers),

In giving a name to the second part of U, however, we
are at once inconvenienced by a circumstance which embar-
rasses the whole mechanical theory of heat, — the fact that heat
and work are measured by different units. The unit of heat is
the quantity of heat which is necessary to raise the temperature
of a unit-weight of water from 0° to 1°, and the unit of work is
the quantity which is represented by the product of the unit of
weight into the unit of length, — ^in French measure, therefore, a
kilogramme-metre.

Nbw in the mechanical theory of heat, after admitting that
heat can be transformed into work and work into heat, in
other words, that either of these may replace the other, it be-
comes frequently necessary to form a magnitude of which heat
and work are constituent parts. But heat and work being mea-
sured by different units, we cannot in such a case say, simply.

J

ON TERMINOLOGY. 253

the magnitude is the mm of heat and work ; we are compelled
to say either the sum of the heat and the heat-equivalent of the
worky or the sum of the work and the work-equivalent of the heat,

Rankine has avoided this inconvenient mode of expression in
his memoirs by assuming as his unit of heat the quantity which
is equivalent to a unit of work. Nevertheless, although per-
fectly appropriate on theoretic grounds, it must be admitted
that great difficulties oppose themselves to the general introduc-
tion of this measure of heat. On the one hand it is always diffi-
cult to change a unit when once adopted, and on the other there
is here the additional circumstance that the heat-unit hitherto
used is a magnitude intimately connected with ordinary calori--
metric methods, and the latter being mostly based on the
heating of water, necessitate only slight reductions, and these
founded on very trustworthy measurements ; the heat-unit
adopted by Rankine, however, besides requiring the same re-
ductions, assumes the mechanical equivalent of heat to be
known, — an assumption which is only approximately correct.
Accordingly, since we cannot expect the mechanical measure
for heat to be universally adopted, we must always, when quan-
tities of heat enter into an equation, first state whether these
quantities are measured in the ordinary manner or by the me-
chanical unit, and consequently the above-mentioned inconveni-
ence would not be removed by Rankine^s procedure.

For this purpose, therefore, I will venture another proposition.
Let heat and work continue to be measured each according to its
most convenient unit, that is to say, heat according to the
thermal unit, and work according to the mechanical one. But
besides the work measured according to the mechanical unit,
let another magnitude be introduced denoting th£ work measured
according to the thermal unit, that is to say, the numerical value
of the work when the unit of work is that which is equivalent to
the thermal unit. For the work thus expressed a particular
name is. requisite. I propose to adopt for it the Greek word
{epyov) ergon^.

• The author has used the German word Werhy which is almost synony-
mous with Arbeit, but he proposes the term Ergon as more suitable for intro-
duction into other languages. The Greek word epyov is so closely allied to
the English word work, that both are quite well suited to designate two

264 SIXTH MEMOIR^ APPENHIX A.

The processes which are considered in the mechanical theory
of heat may be very conveniently described by means of this
new term. Heat and ergon are, in fact, two magnitude^ which
admit of mutual transformation and substitution, without any
alteration in the numerical values of the respective quantities
being thereby involved. Accordingly, heat and ergon may, with-
out preparation, be added to, or subtracted from, one another.

When we consider the work produced during any change in
the condition of a body, we must call it the ergon produced, if
it be measured by the unit of heat, and here again we distin-
guish interior ergon and exterior ergon. The latter, as already
stated in the memoirs, is dependent upon the entire series of
successive changes, whilst the former is completely determined
when the initial and final conditions, solely, are known. As-
suming the initial condition to be given, therefore, the interior
ergon may be regarded as a magnitude which dep^ids solely
upon the condition of the body at the moment under considera-
tion.

Analogous to the expression thermal content of the body,
we may introduce the expression ergonal content of the body.
With reference to the last conception, however, the same re-
mark applies which was previously made with reference to
energy. We may imderstand by ergonal content, either the
increment of ergon reckoned from a given initial condition, or
the total ergonal content. In the latter case we have merely
to conceive an unknown constant, having reference to the
initial state, added to the increment of ergon ; this is so obvious,
however, that in such cases we may usually assume tacitly that
the constant has been included.

The same remark also applies to the thermal content of a
body. By this term we may likewise understand either the in-
crement of heat calculated from an arbitrarily assumed initial
condition, or the total thermal content. In the latter case a con-
stant associated with that initial condition is to be added to the
heat-increment. The only difference is that in 'the case of the
ergonal content, the added constant is quite unknown, whilst

magnitudes which are essentially the same, but measured according to dif*-
ferent units.— T. A. H.

ON TERMINOLOGY. 255

in tlie case of the thermal content, the constant may be approx-
imatdy determined, seeing that the absolute zero of temperature
is to ft certain extent known.

Now the quantity U is the sum of the thermal content and
ergonal content, so that in place of the word energy, we may use
if we please the somewhat longer expression, thermal and ergo^
nal content.

In connexion with these remarks on Terminology I will
venture another suggestion. Hitherto the heat which dis-
appears when a body is fused or evaporated has been termed
latent heat. This name originated when it was thought that / , ^
the heat which can no longer be detected by our senses, when a [
body fuses or evaporates, still exists in the body in a peculiar
concealed condition. According to the mechanical theory of
heat> this notion is no longer tenable. All heat actually present
in a body is sensible heat; the heat which disappears
during fusion or evaporation is converted into work, and con-
sequently exists no longer as heat; I propose, therefore, in
place of latent heat, to substitute the term ergonized heat.

In order to distinguish, in a similar manner, the two parts of
the latent heat which I have stated to be expended, respectively,
on interior and on exterior work, the expressions interior and ex^
terior ergonized heat might be used.

It must farther be observed that of the heat which must be
imparted to a body in order to raise its temperature without
changing its state of aggregation (all of which was formerly re-
garded asyrec), a great portion falls in the same category as that
which has hitherto been called latent heat, and for which I now
propose the term ergonized heat. For, in general, the heating of
a body involves a change in the arrangement of its molecules.
This change usually occasions a sensible alteration in volume,
but it may occur even when the volume of the body remains the
same. For every change in molecular arrangement, a certain
amount of ergon is requisite, which may be partly interior and
partly exterior, and in producing this ergon, heat is consumed.
Only a part of the heat communicated to a body, therefore,
serves to increase the heat actually present therein ; the remain-
ing part constitutes the ergonized heat.

In certain cases, such as those of evaporation and fusion.

256 SIXTH MEMOIR^ APPENDIX B. ,

where the proposed term ergonized heat firequently presents
itself^ a more abbreviated form of expression may, of course, be
adopted, should it be found convenient to do so. For instance,
instead of using the expressions ergonized heat of evaporation,
and ergonized heat of fusion, we may simply say, as I have done
in my memoirs, heat of evaporation and heat offtmon,

APPENDIX B. (Page 239.)
ON THE SPECIFIC HEAT OP GASES AT CONSTANT VOLUME.

In the foregoing memoir it was stated that, in order to ob-
tain the time heat-capacity of a substance, it must be used as a
strongly over-heated vapour, and in fact in such a condition of
expansion, that the vapour without appreciable error may be '

regarded as a perfect gas, and then its specific heat at constant I

volume must be determined. Now in reality this is not, strictly
speaking, quite practicable, since permanent gases themselves,
which are furthest removed from their point of condensation,
do not exactly follow the laws of a perfect gas; and hence we ,

must certainly assume that at the temperatures at which they
can be observed, condensible gases, and still more substances,
which at the atmospheric pressure and at ordinary temperatures
are either liquid or solid, and only become gaseous at higher
temperatures, deviate still more considerably from those laws.
To this must be added the circumstance that, with chemically
constituted substances, and particularly with those of a compli-
cated and not very permanent constitution, partial chemical
changes accompany the processes of heating and cooling ; such
changes, even if they took place to so small an extent as to be
with difficulty detected, might cause the quantities of heat
taken up or given oflF by the gas during its heating or cooling
at constant volume to deviate considerably from the true heat-
capacity. Notwithstanding these imperfections, which are more
or less unavoidable, the specific heat at constant volume cor-
responding to the gaseous condition of a body is always, of all
the several specific heats of the substance, that which is most
suited to serve as the approximate measure of the true heat-ca- I

pacity, and consequently it is, in a theoretical point of view, a
magnitude of some interest.

ON THE SPECIFIC HEAT OF GASES AT CONSTANT VOLUME. 257

Now Regnault having recently determined experimentally
the specific heats at constant pressures of a considerable number
of gases and vapours, it was easy to calculate from these numbers,
according to the principles of the mechanical theory of heat, the
specific heats at constant volumes. Accordingly, immediately
after the first publication of Regnault^s results, I made these
calculations and registered the results in a table for my own
use. With reference thereto, it must be again remarked that
the method of calculation employed is only strictly correct for
2L perfect gas; nevertheless the tables give at least approximate
results for other gases. It must also be observed that the ob-
servation of the specific heat of a gas is the more diflScult, and
consequently the corresponding observation-number the less trust-
worthy the less permanent the gas is, and consequently the more
its deportment deviates from the laws of a perfect gas. Since,,
then, no greater exactitude can be demanded from the calcuXatibu
than that which the observation-numbers themselves possess,
the method of calculation employed may be regarded as per-
fectly suited to its object. In forming my Table, I have thought
it advisable to introduce a small change in one of the two series
of numbers which Regnault has given for the specific heats at
constant pressures. I have referred the numbers there to a
unit somewhat different from that employed by Regnault, ani
I have likewise chosen a corresponding different unit in one
of the two series containing specific heats at constant volumes.
Regnault, in fact, gives us the specific heats of gases in two
different ways. He first compares the weights of the gases,
and states the quantity of heat which a unit-weight of each,
gas requires in order to have its temperature raised one degree ;
this he expresses in ordinary heat-units, that is to say, in terms
of the quantity of heat which a unit- weight of water absorbs
on being heated from 0° to 1"^, In the second place he com*
pares the volxmies of the gases, and here he again uses the
ordinary unit of heat; the volume to which the numbers refer
being that which a unit-weight of atmospheric air occupies
when it is of the same temperature and under the same pres-.
sure as the gas itself under consideration.

By this choice of units the second series of numbers clearly
possesses a rather complicated signification, and its application

s

258 SIXTH MEMOIB^ APPENDIX B.

is thereby embarrassed. When we compare the yolomes of the
gases with those of atmospheric air^ it is best to choose a unit
of heat in a corresponding manner, that is to say, so as to com-
pare the quantity of heat, which the heating of a certain volume
of gas by 1° requires, with that quantity which an equal volume
of atmospheric air at the same temperature and under the same
pressure requires when equally heated. This method of express-
ing the specific heats of gases was formerly generally employed^
and consequently I have deemed it advisable to supplement the
numbers given by Begnault under the rubric ^^ en volume " by a
series of numbers, which have reference to the last-named unit
of heat ; this was easily done, inasmuch as it was merely ne-
cessary to divide those numbers by the specific heat of a unit-
weight of atmospheric air expressed in ordinary heat-units. In
a fdmilar manner I have also expressed in both ways the spe-
cific heats at constant volumes calculated by myself; so that^
on the one hand, gases are compared with equal weights of
water (the ordinary heat-unit having reference to the unit-
weight of water) , and on the other hand, they are compared with
equal volumes of atmospheric air.

The Tables which I calculated for myself in this manner were
published in the year 1861, in a paper which appeared in the
Annalen der Chemie tmd PJiarmacte^ voL cxviii. p. 106, and
which originated in a previous note by Buff"; in this paper the
method of calculation was more fiilly described. The great
importance of the able physical investigations of Begnault
justify me, I think, in here entering on this subject, and com-
municating the Tables in question. To do so, however, I must
once more recalculate the Tables; for Begnault published the
results of his investigations on the specific heats of gases provi-
sionally in the year 1853, in the Comptes Rendus, vol. xxxvi.
p. ^7Q, and in the year 1862 his investigations appeared in
a complete form in the second volume of his Relation des Ex^
pAienceSf which likewise forms the twenty-sixth volume of the
Memoirs of the Academy of Paris. The numbers in this
volume, however, are not quite the same as those which first
appeared in the Comptes Rendus ; they were somewhat changed
by subsequent corrections.

My previously published Tables having been calculated firom

ON THE SPECIFIC HEAT OT GASES AT CONSTANT VOLUME. 259

the numbers first published by Regnault, I have now to apply
my calculations to the corrected numbers.

In order to explain the calculation of the Tables, it will perhaps
be best first to]| collect briefly the principal equations having
reference to perfect gases.

The first characteristic equation of perfect gases, is that which
expresses the law of Mariotte and Gay-Lussac, and is given
in equation (I) of the First Memoir, page 21. If we intro-
duce therein the absolute temperature T instead of the tem-
perature t, calculated from the freezing-point, and represent,
as before, hjp and v the pressure and the volume referred to
the unit of weight, the equation will take the form

pv=n% (A)

wherein R is a constant to be specially determined for each
gas.

The other equation which here enters into consideration is
that which expresses the first fimdamental theorem of the me-
chanical theory of heat when applied to a perfect gas. It is
given in the equation (II) of the First Memoir, p. 38, and on
again introducing the absolute temperature instead of the tem-
perature counted from the freezing-point, takes the form

dCL=cdT'\-KR^dv, (B)

whereiA A denotes 'the calorific equivalent of work, and c, as is
manifest from the equation, the specific heat of the gas at con^
itant volume, .which may now without hesitation be regarded as
a constant, since the conclusions drawn in that memoir have
been confirmed by the experiments of Regnault.

In the First Memoir it was stated that a relation being
established by the equation (A) between the pressure, volume,
and temperature, any two of these magnitudes may be chosen
arbitrarily as independent variables, and the third then regarded
as a ftinction thereof. The two magnitudes thus chosen, then,
determine the condition of the gas, and by their variations the
changes which the gas suffers, and consequently also the
quantities of heat which the gas must thereby absorb may be
expressed ; provided, of course, that we assume, as we always did
in the equations of the First Memoir, that the changes of the

s2

260 SIXTH MEMOIR^ APPENDIX B.

gas take place in a reversible manner .In tlie differential equation
(B) the temperature and volume are chosen as independent
variables ; by the help of equation (A) we may at once deduce
from this differential equation the two other corresponding ones^
which contain on their right-hand sides^ instead of the tempera-
ture and volume, either the temperature and pressure or the
pressure and volume.

After introducing the variables T and p, the equation becomes

dQ=(c+AR)dT-AR-rfi?, . • . . (Bj)

and on introducing/? and v as variables it takes the form

dQ^l^vdp-i-^-^pdv. ..... (B2)

From equation (BJ we see that the sum

c+AR

denotes the specific heat of the gas at constant pressure. Repre-
senting this by d and replacing in a corresponding maimer AR
by the difference c—c, the foregoing three equations become

rfQ=cdT+(c'-c)-rfr, (BO

rfQ=c'rfT-.(c'-c)?rf/?, (Bg

dQ^^vdp+^pdv (By

I take this opportunity of remarking that in my memoirs I
have employed for the two specific heats, a notation different
from the Usual one. Formerly, in fact, it was customary to
denote the specific heat of a gas at constant pressure by c, and
the specific heat at constant volimie by cf. But since from the
point of view reached by the mechanical theory of heat, the
specific heat at constant volume presents itself as a simpler
magnitude (inasmuch as for a perfect gas the specific heat at
ponstant volume is the true capacity for heat, whilst the spe-
cific heat at constant pressure is the sum of the true heat-capa-
city, and of the heat consumed by the work of expansion, that

ON THE SPECIFIC HEAT OF GASES AT CONSTANT VOLUME. 261

is to say, a magnitude composed of two essentially different
parts), it appeared to me to be desirable to select the simpler
symbol for the simpler magnitude, and I have therefore denoted
the specific heat at constant volume by c, and the specific heat
at constant pressure by d.

Each of the three difierential equations, (B), (B^), (B2), may
serve for the calculations corresponding to the changes of con-
dition of a gas, and we may always select from the three difie-
rent forms the* one which is most convenient for the required
calculation. All three differential equations are unintegrable so
long as the two variables on the right-hand side are regarded as
independent of each other, and they will only become integrable
when some further relation is given, between the variables
which they involve, by means of which it will be possible to re-
duce the differential equations involving three variables to diffe-
rential equations between two variables. The nature of the
changes which the gas suffers is so determined by this relation,
that the whole series of changes become known.

The particular relation which exists between the two specific
beats is now manifest from what has been already stated. For
by the equation (10 a) of the First Memoir (p. 39), we have

c'=c-fAR, (a)

whence, on substituting for R the fraction ^, in accordance

with equation (A), and at the same time removing the mag-
nitude c to the left-hand side, and c' to the right-hand side, we
obtain

. c=c'-^v (b)

Let us suppose this equation to be specially formed for at-
mospheric air, and let us denote by c^, c/ and v^ the values of c,
c' and V for atmospheric air ; the equation then gives

whence follows

A»

Ap_ c\^c,

263 SIXTH MEMOIBj AF7ENDIX B.

This substituted in equation (b) gives

c=c'-(c'j-0^ (d)

The fraction — is the reciprocal of the density of the gas com-
pared with air at the same temperature and under the same
pressure; hence^ on denoting this density by d^ we have

c=c'-^ • («>

The diflference c\^c^ which here makes its appearance may

c'
also be determined, and to do so the ratio — ^ must first be calcu*

lated from the observed velocity of sound. In my First Memoir
I employed for this ratio the value 1*421 assumed by Dulong;
the value 1*41, however, appears to be a more correct one, the
third decimal being for the present omitted, since if introduced
it would be uncertain. Employing the latter value, therefore,
we will put

whence it will follow that

7' = 1-41, (f)

Substituting now the value c\=0 2375 as found experimentally
by Regnault*, we have

'^.=^=01684, . . . . (g)

and from this we deduce

c'^-Cj=0-2376-01684=00691. . . . (h)
On substituting this numerical value in the equation (e), we have

, 00691
c=c' -i — , or

c'^-00691

^= — d— w

* Li the second volume of his Helation des Exp&rienceSy p. 108, Eegnault
gives the following numbers corresponding to different limits of temperature:

between-80° and + 10° 0-23771.

,y (P „ +100P 0-23741.

„ 0^ „ +200° 0-23751,

ON THE SPECIFIC HEAT OF GASES AT CONSTANT VOLUME. 263

With respect to the specific heats at constant volumes cor-
responding to the unit of volume ^ I denoted them by 7 in my
First Memoir. They are obtained by dividing the specific heats
c corresponding to the unit of weight by the volume which a
unit of weight of the gas, at tlie particular temperature and
pressure, assumes. We may put therefore

v=£, ........ (k)

and for atmospheric air,

7.=?.. . • (1)

^1

The magnitude with which we are at present interested, and by
which the quantity of heat consumed in heating the gas at con-
stant volume is compared with the quantity which an equal
volume of atmospheric air requires when equally heated under
the same circumstances, is obtained on finding, from the above

two equations, the value of the fraction— . We have, in fact,

— =— . -^; (m)

, V

or substituting for c^ its value, and for the fraction -^ the letter

d as before,

yrwmi-'^' (^)

liastly, on substituting for c the expression given in (i), we have

y _ gW- 00691 . .

7," 0-1684 ^^^

The product c'd, which occurs in the equations (i) and (o),
is the above-named magnitude, given by Regnault, for the
several gases, in the column headed ^^ en volume/^ In order
therefore to obtain the specific heat of a gas at constant volume,
expressed in either of the two ways above alluded to, viz., as
compared with that of an equal weight of water or as com-
pared with that of an equal volume of air, it is merely neces-
sary to deduct the number 0*0691 from the number standing
in the column in question, and then to divide the remainder
either by the density of the gas or by the number 0*1684.

264 SIXTH MEMOIR^ APPENDIX B*

I have collected the numbers so calculated in a Table (p. 266)^
wherein the several columns have the following significa«
tions : —

Column I. The name of the gas.

Column II. The chemical constitution expressed in such a
manner that the diminution of volume accompanying the
combination is at once visible. In fact the volumes of the
simple gases which are stated, are those which must combine in
order to produce two volumes of the compound gas. The hypo-
thetical volume of carbon gas which appears therein, is that
which must be assumed in order to say that a volume of carbon
gas combines with a volume of oxygen to produce carbonic
oxide gas, and with two volumes of oxygen to produce carbonic
acid. Accordingly, when in the Table alcohol, for instance, is
denoted by C^ Hg O, it indicates that 2 volumes of the hypo-
thetical carbon gas, 6 volumes of hydrogen and 1 volume of
oxygen, give 2 volumes of alcohol vapour. In determining the
volume of sulphur gas, I have employed the specific gravity
2'23 found, at very high temperatures, by Sainte-Claire Deville
and Troost. The ordinary chemical symbols are used, irre-
spective of their volumes in a gaseous condition, for the elements
silicon, phosphorus, arsenic, titanium and tin, which enter into
the five last combinations of the Table. Since the gas-volumes
of these elements are either stiU unknown or affected with
certain, not yet sufficiently explained irregularities.

Column III. The density of the gas as given by Regnault.

Column IV. The specific heat under constant pressure, com--
pared with that of an equal weight of water, or what amounts
to the same thing, referred to the unit-weight of gas and ex-
pressed in ordiuary thermal units. These are the numbers
which Regnault has given in the column headed ^^ enpoids/^

Column V. The specific heat under constant pressure, com^
pared tvith that of an equal volume of air, and calculated by
dividing the numbers given by Regnault under the heading
^^ en volume,^' by the number 0*2375.

Column VI. The specific heat at constant volume, compared
with that of an equal weight of water, and calculated according
to equation (i).

Column VU. The specific heat at constant volume^ compared

ON THE SPECIFIC HEAT OF GASES AT CONSTANT VOLUME. 265

with that of an eqtuil volume of air, and calculated according
to equation (o) .

In the Tables published in the Ann. der Chem, und Pharm.y 1
added an eighth column^ giving the trtAC thermal capacity of the
compounds^ as compared i/^ith the true thermal capacity of an
equal volume of a simple gas.

The numbers contained in this column depend on the suppo-
sition more accurately laid down in the preceding memoir, ac-
cording to which a chemically compounded substance contains
just as much heat as the constituents would do if separated and
at the same temperature. According to this supposition, we
may very easily calculate the true thermal capacity, correspond-
ing to the unit of volume, of a compound gas, from the changes
of volume which occur during the combination, provided we
know the true thermal capacities, corresponding to the same
unit of volume, of the simple gases. For instance, let us con-
sider the three simple gases, oxygen, hydrogen, and nitrogen,
whose specific heats at constant volume may be considered as
very nearly equal to one another ; so that we may also assume
that their true thermal capacities are also equal. (The small
deviations arising from the circumstance that the gases are not
quite perfect ones are here neglected.) Now if we consider
the gaseous compounds of the gases and compare the true
thermal capacities of a unit of volume of each compound with
the true thermal capacities of the simple gases, the numerical
ratios are immediately given by the changes of volume which
occur during combination. We thus obtain for binoxide of
nitrogen, where no change of volume occurs, the true thermal
capacity 1 ; for protoxide of nitrogen and for aqueous vapour,
where the diminution of volume is in ratio 1 : f , the thermal
capacity f ; and for ammonia, where the diminution of volume
is in the ratio of 1 : i, the thermal capacity 2. In a similar
manner the true thermal capacity of any other compound gas
may be determined by the theorem under consideration.

My reasons for omitting this column completely from the
following Table, were twofold. In the first place the true
thermal capacity corresponding to the unit of volume, of each
simple component gas being known, the method of determining
that of the compound gas is so simple that the required numbers

366

SIXTH MEMOIR^ APPENDIX B.

may be immediately read from the chemical formulae^ and con*
sequently it is scarcely necessary to print them, specially ; and
in the second place, since imcertainties still exist as to the true
thermal capacities, corresponding to the unit of volume, of some
of the simple gases, a discussion thereof ought to precede any
numerical statement. I propose to return to thjis subject on
some iiiture occasion.

Nuneii of the gates <

CbasIciiJ

CDBltitU-

Deniitjr.

cotnpaKd

witb that

of on cquiLl

wei|:ht of

water*

Sj>e<iif5e hcttt under

compiLred
with that
of an equnl
Tolunie tif

ftir.

VU*

Specific beat at
eoDiLiint volume

compsTcd compared
with tkiC [ with thkt
af ail equal, of AQ equ»l
Wi;tphtof Tolume of
water. air.

Atmoapliepic dir --...,
0:cjg^TL . i P * . * . . ......

Nitrogen . * * * . .

Hjdrogen ...,..*

QhlorjTie . .,..,.

Bromme.. . ,.^ ,♦*...,.-
Biiiox.]de of nitrogen . .

C^i-boniir oxide ...

HjdrtJt^lJoric! acid , . . .
Ciirbonic acid ,*»-,. .., ,
Pnjtomdo of nitrogSTi . . . .
Amieou.^ vapour **,.♦.*.
Siilphurotis ftcid .,.-....
SitlphiLretkd hydrogen . .
Bii*ulphide of carbon . * » .

Marsh-gas ......... , .

Chloroform , . . . .

Olefiimt gas .......... . .

Ammonia ..............

Benzole . * . ^ *»...,

Oil of tnrpQntin© •»..*..,
Wood-spirit ............

Alcohol

Ether -

Svdphide of otlijle

Chloride of eth jle .......

Bromido of ethjie

Dutch liquid

Acetone ^

Acetic-etiier » ^ *

Terchloride of silicon . . .
Terohlorido of phosphorus
Twehlorido of araenic . . *
Chloride of titanium *..
Chloride of tin ..........

S;

NO
CO

UCl

ca

tf,0

so.
cm

CHO^
Call,
NH,,

c if
UH.O
C,1I,0
C,Hi,0

C,11,C1

C„ II, Br

0.;il. CI

C5, If, O"

C.H,0,

SiCL

PCI,

AaCL

Ti CI,

SnCl,

t

0*0692

S-+77^
I-013+
0-9673
1-2596
1*5101
15741
0*6119
2-2113

1*174^7
1-6258

0-9671
0-5394-
2-6941
4-6978
rjos5
rs^90
^"5573
3*1 101
22^69
3-7058

3 +'74
1*0036
3*0400
58^33
4-74^
6' 1667
6 6402
S'9654

0-1375

Q2> 1751

O'a43¥o

3-40900

0-1^099

0-05552

0-1317

0*2450

0*1851

0-2169

0-2161

04^05

0-1544

Q-2431

0-J569

0*5929

0-1567

0-4040

0-50S4

0-3754
0-5061
a*45 80

o'4534
0^4797
0-4008
0-2738
0-1896
0*2193
0-4125
o'4oo8
Q-1311

°*i347
0*1121
0-1290
0*0939

I
1*013

°*997

0-993

1*248

laSo

1*013

0-998

o'gSi

1*39

IHS

1-44
1*20
174

175
116
4-26

lO'Ol

a"i3
3*03
516
5-15

*"57
i'96

3-4^
S-13

2 69
2*96
3'6i
3^54

o'i634
0-1551
0*1727

2'41I

00918

0*0429

01652

0*1736

01 304

0-172

0181

0*370

0^113

0*184

0-131

0-468

0-140

0-359

0*391

o'35o

0-491

o"39S
0-410
0-453
0*379
0143
0*171

0*1 og
0-378
0-378
0*110
Q'llO
O'lOI
0-119
0-086

2

1*018
0*996
0*990
''350

'■395
i-oi8
0-997
0*975

1*55.
j'64
1-36
1-62
1*29
1*04
"■54
3 '43
i'o6

'■37
5*60

>3"7i
2-6o

3-87

6-37

6-99
3-11
3-76
4-14

4-50
6-82

4*21

V19

377
4'^7
4'S9

ON AN AXIOM IN THE MECHANICAL THEORY OV HEAT. 267

SEVENTH MEMOIK.

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT"**".

1. When I wrote my First Memoir on the Mechanioal Theory
of Heatf, two diflferent views were entertained relative to the
deportment of heat in the production of mechanical work. One
was based on the old and widely spread notion, that heat is a
peculiar substance, which may be present in greater or less
quantity in a body, and thus determine the variations of tem-
perature. Conformably with this notion was the opinion that,
although heat could change its mode of distribution by passing
from one body into another, and could farther exist in different
conditions, to which the terms latent and free were applied,
yet the quantity of heat in the whole mass could neither
increase nor diminish, since matter can neither be created nor
destroyed.

Upon this view is based the paper published by S. Camot J,
in the year 1824, wherein maclunes driven by heat are subjected
to a general theoretical treatment. Carnot, in investigating
more closely the circumstances under which moving force can
be produced by heat, found that in all cases there is a passage
of heat from a body of higher into one of a lower temperature ;
as in the case of a steam-engine where, by means of steam^
heat passes from the fire or from a body of very high tem-
perature, to the condenser, a space containing bodies of lower
temperature. He compared this manner of producing work
with that which occurs when a mass of water falls from a high^
to a lower level, and consequently, in correspondence with the
expression "une chute d^eau/' he described the fall of heat

* Eead at a Meeting of the Swiss Association, held at Samaden, August
26th, 1863, and published in Poggendorft^s Annalm, November 1863, vol. cxx.
p. 426.

t On the Moving force of Heat, &c. (First Memoir of this collection).

t E^flexionssui la puissance motrioe.du feu.

268 SEVENTH MSMOIB.

from a higher to a lower temperature as '^ une chute du calo^
rique"^.

Regarding the subject from this point of view, he lays down
the theorem that the magnitude of the work produced always
bears a certain general relation to the simultaneous transfer of
heat, t. e, to the quantity of heat which passes over, and to
the temperatures of the bodies between which the transfer takes
place, and that this relation is independent of the nature of the
substances through which the production of work and the trans-
fer of heat are effected. His proof of the necessity of such a
relation is based on the axiom that it is impossible to creute
a movififf force out of nothing^ or in other words, that perpettial
motion is impossible.

The other view above referred to is that heat is not invariable
in quantity; but that when mechanical work is produced by
heat, heat must be consumed, and that, on the contrary, by
^ // the expenditure of work a corresponding quantity of heat can
^ -j be produced. This view stands in immediate connexion with
the new theory respecting the nature of heat, according to
which heat is not a substance but a motion. SiQce the end of
the last century various writers, amongst whom Rumford, Davy,
and Seguin may be mentioned, have accepted this theory ; but
it is only since 1842 that Mayer of Heilbronn, Colding of
Copenhagen, and Joule of Manchester examined the theory
more closely, founded it, and established with certainty the law
of the equivalence of heat and work.

According to this theory, the causal relation involved in the
process of the production of work by heat is quite different from
that which Camot assumed. Mechanical work ensues from the
conversion of existing heat into work, just in the same manner
as, by the ordinary laws of mechanics, force is overcome, and
work thereby produced, by motion which already exists ; in the
latter case the motion suffers a loss, in vis viva, equivalent to
the work done, so that we may say that the vis viva of motion
has been converted into work. Carnot^s comparison, therefore,
in accordance with which the' production of wotk by heat cor-
responds to the production of work by the falling of a mass of

* See page 2d of Camot's paper.

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 269

water, — and, in fact, the fall of a certain quantity of heat from a
higher to a lower temperature may be regarded as a cause of
the work produced, was no longer admissible according to
modem views. On this account it was thought that one of
two alternatives must necessarily be accepted; either Camot's ;
theory must be retained and the modem view rejected, accord- | i .
ing to which heat is consumed in the production of work, or, i7 ^>^
on the contrary, Camot^s theory must be rejected and the j
modem view adopted. ,1

2. When at the same period I entered on the investigation of
this subject, I did not hesitate to accept the view that heat
must be consumed in order to produce work. Nevertheless I
did not think that Camot^s theory, which had found in Clapeyron
a very expert analytical expositor, required total rejection; on
the contrary, it appeared to me that the theorem established
by Camot, after separating one part and properly formulising
the rest, might be brought into accordance with the more
modem law of the equivalence of heat and work, and thus
be employed together with it for the deduction of important
conclusions. The theorem of Camot thus modified was treated
by me in the second part of the above-cited memoir, in the
first part of which I had considered the law of the equiva-»
lence of heat and work.

In my later memoirs I succeeded in establishing simpler and
at the same time more comprehensive theorems by pursuing
further the same considerations which had led me to the first
modification of Camot^s theorem. I wiU not now enter, how-»
ever, upon these extensions of the theory, but wUl limit myself
for the present to the question how, in accordance with the
law of the equivalence of heat and work, the necessity can be
demonstrated of the other theorem in its modified form.

The axiom employed by Camot in the proof of his theorem,
and which consists in the impossibility of creating moving force,
or, more properly expressed, mechanical work out of nothing,
could no longer be employed in establishing the modified theo-r
rem. In fact, since in the latter it is already assumed that to
produce mechanical work an equivalent amount of heat must
be consumed, it follows that the supposition of the creation of
work is altogether out of the question, no matter whether a

270 BEYSNTH MXHOIR*

transfer of heat from a warm to a colder body does or does not
accompany the consumption of heat.

On the other hand^ I found that another and^ in my opinion^ a
more certain basis can be secured for the proof by reversing the
sequence of reasoning pursued by Camot, and by accepting as an
axiom a theorem^ in a somewhat modified form^ which niay be
regarded as a consequence of his assumptions.

In fact^ after establishing from the axiom that work cannot
be produced from nothing, the theorem that in order to pro-
duce work a corresponding quantity of heat must be transferred
from a warmer to a colder body, Camot to be consistent could
not but conclude that, in order to transfer heat from a colder
to a warmer body, work must be expended. Although we must
now abandon the argument which led to this result, and not«
withstanding the fact that the result itself in its original form
is not quite admissible, it is nevertheless manifest that an
essential difTerence exists between the transfer of heat from a
warmer to a colder body and the transfer from a colder to a
warmer, since the first may take place spontaneously under
circumstances which render the latter impossible.
> On investigating the subject more closely, and taking into
consideration the known properties and actions of heat, I came
to the conviction that the difference in question had its origin
in the nature of heat itself, inasmuch as by its veiy nature it
must tend to equalize existing differences of temperature. Heat
accordingly incessantly strives to pass from warmer to colder
bodies, and a passage in a contrary direction can only take
place under circumstances where simultaneously another quantity
of heat passes from a warmer to a colder body, or when some
change occurs which has the peculiarity of not being reversible
without causing on its part such a transfer from a warmer to a
colder body. This change which simultaneously takes place is
consequently to be regarded as the equivalent of that transfer of
heat from a colder to a warmer body^ so that it cannot be said
that the transfer has taken place of itself (von selbst).

I thought it permissible, therefore^ to lay down the axiom, that

Heat cannot of itself pass from a colder to a warmer body,
and to employ it in demonstrating the second fundamental
theorem of the mechanical theory of heat*

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 271

3. This axiom has met with very diflFerent receptions on
the part of the scientific public. To some it appeared to be
so self-evident as to render its express statement nnneces-*
sary, to others, on the contrary, its correctness appeared to be
doubtful.

The first opinion is, I find, expressed in the very meritorious
paper published by Zeuner in 1860, under the title " Grundziige
der Mechanischen Warmetheorie,^^ in which he seeks to expound
this theory, so far as at that time developed, in as connected and
simple a manner as possible, in order to extend an acquaintance
with the principal results of this theory to those to whom the
original memoirs were either not accessible, or were difficult of
perusal in consequence of the mathematical developments therein
contained.

In this work Zeimer gives my proof of the second fundamental
theorem essentially in the form in which it was reproduced by
Beech*. In one poiat, however, his exposition differs from the
latter ; for Reech gives the theorem that heat cannot of itself
pass from a colder to a warmer body, expressly as an axiom laid
down by me, and he bases his demonstration thereon. Zeuner,
however, does not mention this theorem at all, but merely shows
that, if for any two bodies the second fundamental theorem of
the mechanical theory of heat were not true, it would be
possible by means of two cyclical processes performed in
opposite ways with these two bodies, to transfer heat firom a
colder to a warmer body without any other change; and he then
remarks t, ^' since we can repeat both processes any number of
times by employing the two bodies alternately in the manner
described, it would foUow that, without expending either work
or heat, we could continually transfer heat from a body of lower
to a body of higher temperature; which is absurd.*' Few
readers, I believe, will admit that the impossibility here alluded
to of the transfer, without any other change, of heat from a
colder to a warmer body is as self-evident as the words ^^ which
is absurd '* would imply. In the conduction of heat as well

♦ R^apitulation tr^s^succinte des recherches alg^briques feites sur la
throne des effete m^caniques de la chaleur par difidrents auteurs. — Joum, de
Liouville, S. 2. vol. i. p. 58.

t Page 24 of his work.

272 SEVENTH MEMOIR.

as in its radiation under ordinary circumstances^ we may cer-
tainly say that this impossibility is established by daily ex-
perience. But even in the radiation of heat, the question may
arise whether it would not be possible, by artificially concen*
trating rays of heat by means of mirrors or lenses, to generate
a temperature higher than that possessed by the bodies which
emit the rays, and thus to cause heat to pass itito a warmer
body*. The subject is still more complicated in cases where
heat is transformed into work and work into heat, either by
processes such as friction and resistance of the air, or by the
circumstance that one or more bodies suffer changes of condition
in which are involved both positive and negative, interior and
exterior work, — ^where, in fact, according to the customary mode
of expression, heat is rendered latent or free ; which heat the
changing bodies may withdraw from or communicate to other
bodies of different temperatures.

I cannot but think that when in such cases, and however
complicated the processes may be, it is asserted that heat never
passes from a colder to a warmer body without some permanent
change occurring which may be regarded as an equivalent thereof,
this theorem ought not to be treated as quite self-evident; it
ought rather to be introduced as a new axiom upon whose
acceptance or rejection the admissibility of the proof depends.

4. The opposite view, however, is more frequently met with,
that this theorem is not sufl&ciently trustworthy to serve as a ,

basis of demonstration, or even that it is incorrect.

With reference hereto, I must first explain Bankine^s manner
of treating the subject. In a memoir which appeared almost at
the same time as minef, Rankine developed the theory of a i

* I have treated this subject, which in other respects is interesting, in a |

separate paper read before the Scientific Society of Zurich last June^ and
which will appear in a forthcoming Number of these Annals. [It forma the
Eighth Memoir of this collection.]

t It was communicated to the Royal Society of Edinburgh in the same
month (February 1850) in which my paper, was read to the Academy of |

Berlin. Rankine states in a communication to Poggendorflf that his paper i

was sent in in October 1849. Its publication, however, took place somewhat j

later than mine did. It appeared in the ' Transactions of the Royal Society
of Edinburgh/ vol. xx. p. 147, and was republished, with some changes, in.
the Phil. Mag. S. 4. vol. vii. 18o4.

J

ON AN AXIOM IN THB MECHANICAL THEORY OF HEAT. 273

peculiar molecular motion assumed by him and termed molecular
vortices, and from it he deduced conclusions concerning the de-
portment of bodies, particularly of gases and vapours, which
agree in some measure with those at which I arrived in the
first part of my memoir by means of the law of the equivalence
of heat and work. The subject of the second part of my memoir,
Camot*s theorem, as modified by me, and its consequences, is,
however, not contained in Bankine's memoir.

In a subsequent paper communicated to the Boyal Society of
Edinburgh in April 1851, and appended as Section 5 to his
former memoir*, Bankine occupies himself with that second
fundamental theorem. He there remarks with reference to my
modified theorem, that he ^^ had at first dovbts as to the reason-
ing ^^ by which I maintained itf, but that he was induced by
W. Thomson, to whom he had communicated his doubts, to
examine the subject more closely. He arrived thereby at the
conclusion that this theorem ought not to be treated as an in-
dependent principle in the theory of heat ; but that it might be
deduced as a consequence of the equations established by him in
the first section of his memoir.

I must, however, confess that I cannot regard as satisfactory
the demonstration of the theorem thereupon given by Bankine.

5. In the heat which must be communicated to a body in
order to raise its temperature, Bankine distinguishes, as I also
have done, two different portions ; one of which serves to increase
the heat actually present in a body, and the other is consumed
by work. The latter portion includes the heat consumed by
interior as well as by exterior work.

For the heat consumed by work Bankine employs an ex-
pression deduced by him in the first section of his paper from
the hypothesis of molecular vortices. I need not here enter
more closely into this mode of deduction, since the circumstance
that it depends on a peculiar hypothesis concerning the con-
stitution of molecules, and on their mode of motion, is sufficient
to produce the conviction that complicated considerations must
necessarily arise of a nature to raise doubts as to the degree of
its trustworthiness. In my memoirs I have taken especial care

♦ Phil. Mag. S. 4. vol. vii. p. 249. t Jbid. p. 261,

T

274 SEVENTH MEMOIR.

to base the development of the equations which enter into the
mechanical theory of heat upon certain general axioms^ and not
upon particular views regarding the molecular constitution of
bodies^ and accordingly I should be inclined to regard my treat-
ment of the subject as the more appropriate one, even were the
above-mentioned circumstance the only one which could be
adduced against Bankine's demonstration. But the determina-
tion of the first portion of the heat communicated to a body,
f . e, of the portion which serves to increase the heat actually
present therein, is still more uncertain.

Bankine represents, by the product Idt simply, the increment
of heat present in the body when its temperature / is increased
by dty no matter whether the volume of the body does or does
not change at the same time ; and in his demonstration he treats
the magnitude f, which he terms '^ the real specific heat"*y as a
magnitude independent of the volume. We seek in vain, however,
in his memoir for a sufficient reason for this procedure, in fact,
statements appear therein which are in direct contradiction
thereto.

In the introduction to his memoir he gives in equation (XIII)
an expression for the real thermal capacity, which contains a
factor ky of which he saysf, ''The coefficient h (which enters
into the value of specific heat) being the ratio of the rw moa of
the entire motion impressed on the atomic atmospheres by the
action of their nuclei, to the m» viva of a peculiar kind of
motion, may be conjectured to have a specific value for each
substance, depending in a manner yet unknown' on some circum-
stance 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.'' According to
this, Bankine is of opinion that the real thennal capacity of
one and the same substance may be different in different
states of aggregation; and for the assumption that it is

* [In a note to the Sixth Memoir^ p. 229, 1 have proposed to employ the
term real capacity for heat, instead of real specific heat, since the former,
or the equivalent expression real thermal capacity, literally signifies that the
heat under consideration is really contained in the body. — 1864.]

t Phil. Ma^. S. 4. vol. vii. p. 10.

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 275

invariable in the same state of aggregation, the only reason he
gives is,, that no known experiment contradicts it.

At page 307 of a more recent work by Rankine, entitled ^' A
Manual of the Steam-engine and other Prime Movers/^ London
and Glasgow, 1859, occurs a still more definite statement on
this subject, already cited by me on a former occasion, to the
following effect, '^ a change of real specific heat, sometimes con-
siderable, often accompanies the change between any two of those
conditions/^ i. e. the three states of aggregation. The magnitude
of the differences between the real thermal capacities of one and
the same substance in different states of aggregation, held by
Bankine to be possible, is manifest from a statement on the
same page to the effect that in liquid-water the specific heat
determined by observation, termed by him the apparent specific
heat, is nearly equal to the real specific heat. . Now as Bankine
knows very well that the observed specific heat of water is twice
as great as that of ice, and more than twice as great as that of
vapour, and since the real specific heat (thermal capacity) of ice
and of vapour can certainly not be greater than the observed
one, it follows that Rankine must assimie that the real thermal
capacity of water exceeds twice that of ice and of vapour.

If we now inquire how, in accordance with this assumption, for
a body whose temperature t is increased by dt, and whose
volume V is increased by dv, the corresponding increment of heat
actually present in the body is to be expressed, we must proceed
as follows.

In the case where the body, during its change of volume,
suffers no change of condition, the increment of actually present
heat would no doubt be expressible, as Rankine states, by a
simple product of the form Idt ; but different values would have
to be ascribed to the factor f for different states of aggregation.
Where the body, however, changes its state of aggregation as
well as its volume ; for instance, in the case often considered,
where a quantity of matter is given partly in a liquid and
partly in a vaporous condition, and where during a change of
volume the magnitude of both these parts is altered, either by
the partial evaporation of the liquid, or by the partial condensa-
tion of the vapour, we shoiQd not be able to represent by a
simple product dt the increase of heat which accompanies a

t2

276 SEVENTH MEMOIR.

change both of temperature and volume; we should^ on the
contrary, have to employ for the latter an expression of the
form

in fact, if the real thermal capacity of a substance were different
in different states of aggregation, we should necessarily conclude
that the quantity of heat present therein depended upon its state
of aggregation, so that equal quantities of the substance in the
solid, liquid, and gaseous states would contain different quantities
of heat; accordingly, whenever the state of aggregation of a
part of the substance changes without any change of temperature,
the total quantity of heat present therein must also change.

From this it follows that Bankine, according to his own
admission, can only regard the manner in which he expresses
the increment of heat present in the body, and in which he
treats this expression in his demonstration, as trustworthy in
cases where no changes of the state of aggregation present
themselves ; and consequently it is only in these cases that he
can claim accuracy for his proof. In all cases, therefore,
where changes of the state of aggregation present themselves,
the theorem remains unproved; and notwithstanding this, these
cases are particularly important, inasmuch as it is precisely to
them that the theorem has hitherto been most frequently applied.

We must indeed go further, and assert that the demonstration
loses hereby all trustworthiness, even in those cases where no
changes of the state of aggregation present themselves. For if
Bankine assumes that the real thermal capacity can be different
in different states of aggregation, we do not at all see on what
grounds it is to be regarded as unchangeable, when the state of
aggregation is the same. We know that in solid and liquid
bodies changes in the condition of cohesion may occur without
any change in the state of aggregation, and that in gaseous
bodies, besides the great differences in volume, the circumstance
also presents itself that they follow the laws of Mariotte and
Gay-Lussac with more or less accuracy, according as they are
more or less distant from their point of condensation. Why,
therefore, if changes in the state of aggregation have an in-
fluence on the real thermal capacity, may not to these changes

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 277

of cohesion an influence, the same in kind though less in degree,
be with equal justice ascribed ? The assumption that the real
thermal capacity is invariable in the same state of aggregation,
is consequently not merely left unestablished by Rankine, but
would be rendered in the highest degree improbable if the other
assumptions made by him were accurate.

6. The method which I have proposed for the treatment of
the second fundamental theorem in the mechanical theory of
heat, and which, notwithstanding Rankine's objection, I still
hold to be the most convenient, is essentially the following.

By means of the axiom that heat cannot of itself pass from a
colder to a warmer body, I have first proved the theorem in
question for cyclical processes, that is to say, for processes
wherein all the interior work that may possibly be performed is
subsequently cancelled, so that exterior work alone remains.
After having brought the theorem to such a form that from its
simplicity we may almost conclude with certainty that it cannot
be limited to a special class of phenomena, but must be
generally true, I have then applied it also to the interior work.
I have been thereby led to the establishment of a general law
for the dependence of the active force of heat upon the tem-
perature, according to which the effective force of heat is propor^
tional to the absolute temperature. By combining the equation
which expresses this law with the equation which I had proved
from the above axiom in the case of a cyclical process, I was
led to the conclusion, that the quantity of heat actually present
in a body must depend on its temperature solely, and not upon the
arrangement of its molecules. If this conclusion be correct, the
real thermal capacity of a body must not only be independent of
its volume when its state of aggregation remains unaltered, but,
contrary to Bankine^s expressed opinion, it must also be inde-
pendent of the state of aggregation.

The last result being obtained, the course of reasoning might
perhaps be reversed. For if we assume, from the commence-
ment, the truth both of the theorem, that the quantity of heat
actually present in a body is independent of the arrangement of
its molecules, and of the before mentioned law relative to the
dependence of the effective force of heat on the temperature, we
can prove therefrom the correctness of the equation which ex-

278 SKVENTH MBMOIB.

presses for cyclical processes the second fundamental theorem
of the mechanical theory of heat*. I cannot but think^ however,
that little encouragement would have been given to the physicist
who should have proposed to start firom the improved theorem
relative to the heat actually present^ which involves the con-

* [It will, I think, be useful here to collect together the equations which
express the aboye mentioned theorems in order to render perfectly clear the
relation which exists between them.

Let dL be the total work, or, in other words, the sum of the interior and
exterior work done by the heat during the time that the body undergoes, in
a reversible manner, an infinitesimal change of condition, and let T be the
absolute temperature of the body at the moment this change of condition
occurs, then the law that the effective force of heat is proportional to the absohde
tenyteratttre is expressed by saying that the equation

must hold for every reversible cyclical process.

The heat which must be imparted to the body during the above-mentioned
infinitesimal change of condition, and whose quantity has always been denoted
by dQ, consists of two parts, (1) the quantity <7H, which serves to increase
the heat H actually present in the body, and (2) the quantity AdL consumed
in the production of the work dh. We have, therefore, the equation

dQ^dEL+AdL,
whence we deduce

A

By the substitution of this value of dL, the above equation expressing the
law relative to the effective force of heat assumes the form

'^^^=0. (a)

9

f

"T"

The theorem that the heat actually present in a body d^ends solely upon the
temperature of the latter and not upon the arrangement of its molecules^ and
therefore that the magnitude H, which denotes this heat, is a function of T
solely, may be expressed by saying that the equation

[^=0 0.)

J^

holds for every cyclical process.

Lastly, the second fundamental theorem of the mechanical theory of Tieat is, for
every cyclical process, expressed by the equation

ff =0. (c)

f

A glance at the three equations (a), (b), (c) is sufficient to show that
each is a necessary consequence of the other two. — 1866.]

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 279

sequence, that the real thermal capacity of a body may differ so
far from the specific heat determined by observation, as to
amount to less than half of the latter. A theorem which differs
so much as this does from hitherto received notions, could only
have a prospect of recognition on being supported by reasons
which other considerations have rendered extremely probable. It
appears to me, therefore, that this reversed mode of reasoning,
although useful as elucidating the subject from another side, is
not suitable for the purpose of demonstration.

Another circumstance which gives a preference to my mode <^
of treating the subject is, that the ftmdamental theorem under
consideration is proved thereby, so far as it refers to cyclical
processes, without the assistance of any assumption whatever
concerning the interior condition of a body, and that it is only
on applying the theorem to interior work that the interior con-
dition of the body, and in particular the quantity of heat present
in the body, enters into consideration. Thence arises the ad-
vantage that the fundamental theorem, so far as it refers to
cyclical processes, can bie maintained unchanged, even by one
who entertains doubts concerning the accuracy of the conclusion
concerning the quantity of heat contained in the body*.

* [Now that the theorem that the heat actually present in a body is in-
dependent of the arrangement of its molecules, and hence that its real thermal
capacity is the same in all conditions, has gained a certain scientific founda-
tion from the developments contained in the Sixth Memoir, we may possibly
soon come to regard this theorem £rom the commencement no longer as im-
probable, but rather as theoretically probable. The theorem, too, may receive
additional verification if the attention of many physicists be directed thereto.
In this case it may hereafter appear more justifiable to deduce the equation

J-?=o,

which expresses for a reversible cyclical process the second fundamental
theorem in the mechanical theory of heat, from the theorem mentioned in
the text relative to the dependence of the efiective force of heat upon the
temperature, together with the theorem just alluded to concerning the heat
actually present in a body ; and this mode of deduction wiU possibly be not
unfrequentiy employed, since it is both convenient and easily intelligible.

In doing so, however, it must not be forgotten that the two theorems
upon which this mode of deduction is based, first acquired their credibility
firom the circumstance that they lead to an equation whose truth has been
proved in another manner. The reduction of the equation, therefore, to such

280 SEVENTH MEMOIR.

7. A more definite objection to my axiom has recently been
raised by Him^ and it is from this that the present memoir prin-
cipally originated^ since the same objection appears to other
authors to be yalid.

In a former work, entitled Recherches sur Vequivalent me-
canique de la Chaleur, Him opposed the theorem of the equi-
valence of heat and work. In his new work. Exposition ana-
lytique et expMmentale de la theorie mScanique de la CJidkur, he
withdraws his former assertions relative to the equivalence, but
opposes the axiom that heat cannot of itself pass from a colder
to a warmer body. After the appearance of this work, he ex-
tended his views on the same subject, in two articles published
in Cosmos^, 1 replied in the same joumalf ; and he thereuponf
explained that in bringing forward his objection, he only in-
tended to direct attention to an apparent contradiction, for that
essentially he agreed with me. I cannot but think, however,
that his objection arises from an incorrect interpretation of my
axiom, a misconception which certainly might easily be formed,
and which on that account renders a thorough correction more
necessary.

Hirn describes a peculiar operation invented by himself, the
result of which he believes to be in contradiction to my axiom.

Let A and B, in fig. 9, represent two cylinders of equal
cross section, connected at the bottom by a comparatively
narrow tube, and in which move air-tight pistons; let the
piston-rods be provided with teeth which fit on both sides into
the teeth of a wheel situated between them, so that when one
of the pistons descends the other must ascend to an equal ex-
tent. The total space consisting of the spaces underneath the
pistons, and of the space enclosed by the tube connecting the

theorems cannot exactly be regarded as a, proof of the equation^ but rather
as a means of rendering its physical meaning clear. It will certainly be
granted that at a time when those theorems were in nowise accepted, and when
one of them indeed was at variance with views which were very widely
entertained, this mode of deduction could not even have served to raise the
probability of the truth of the equation in the minds of the scientific public. —
1864.]

♦ Tome xxii. (Premier semestre, 1863) pp. 283 & 413.

t Ibid. p. 660. J Ibid. p. 734.

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 281

Fig. 9.

two cylinders, must consequently remain invariable during the
motion of the pistons ; since every diminution of space in the
one cylinder is accompanied by just as
great an augmentation in the other.

Let us suppose that at the commence-
ment one piston is quite at the bottom of
the cylinder B, and consequently the
other at the top of the cylinder A ; and
let us assume that the latter is filled with
a perfect gas of a given density, whose
temperature is /q- Now let the piston in
A gradually descend and consequently
that in B ascend, so that the gas shall be
gradually driven from the cylinder A to
the cylinder B. Let us conceive, more-
over, that the connecting tube through
which the gas must pass is maintained
constantly at a temperature t^ higher than
^0, so that each quantity of gas which
traverses the tube shall be thereby raised
to the temperature t,, and shall pass at

this temperature into the cylinder B. The walls of both cylinders,
on the contrary, shall be supposed to be impermeable to heat, so
that the gas within the cylinder neither receives nor loses heat,
but merely receives heat from without when traversing the
connecting tube. In order to have a definite example relative
to temperatures, we will suppose that the initial temperature of
the gas in the cylinder A is that of the freezing-point 0°; the
temperature of the connecting tube being 100°, as it would be,
if, for instance, it were surrounded by the vapour of boiling
water.

The result of this operation may now be understood without
diflSiculty.

The first small quantity of gas which passes through the
connecting tube has its temperature raised from 0° to 100°, and
consequently expands proportionally, that is to say, by about the
^§th part of its original volume. The gas which is still in the
cylinder A will thereby be somewhat compressed, and conse-
quently the pressure in both cylinders will be increased. The

282 SEVENTH MEMOIR.

next small quantity of gas which passes through the tube will
likewise expand^ and thereby compress the gas in both cylinders.
In a similar manner every succeeding quantity of gas whicb
passes through the tube contributes by its expansion not only
to the still further compression of the gas remaining in A, but
also to that of the gas already in B, which had previously ex-
panded, so that the density of the latter gradually approaches to
what it was before. The compression causes a heating of the
gas in both cylinders ; and since the quantities of gas which
successively pass into the cylinder B have all the temperature
100° at their entrance, they must subsequently acquire tempera-
tures above 100°; and, moreover, this excess of temperature
must be greater the more the quantity in question is afterwards
again compressed.

If we consider, therefore, the condition at the close of the
operation, when all the gas has been forced from A into B, it
is evident that the highest stratum of gas immediately under the
piston, which first passed over and consequently suffered the
greatest subsequent compression, must be the hottest. The
underlying strata, taken in order down to the lowest, which
possesses exactly the temperature 100^ which it assumed on
passing over, will be less and less heated.

For our present purpose it is not necessary to know the tem-
peratures of the several strata individually, it will suffice to
determine the mean temperature, which is also that temperature
which would ensue if the temperatures existing in the several
strata became equalized by conduction or by a mixture of the
quantities of gas. This mean temperature amounts to about
120°.

In an article which subsequently appeared in Cosmos, Him
completed this operation by supposing the gas in B, after being
heated, to be brought into contact with quicksilver at 0°, and
thereby cooled again to (f ; and then he supposed the gas to be
driven back from B to A under the same circumstances under
which it had arrived from A to B, and thus that it became
heated in the same manner ; then he again supposes it to be
cooled by quicksilver, subsequently again driven from A to B,
and so on, so that a periodical process is obtained by which the
gas continually netums to its initial state, and all the heat given

ON AN AXIOM IN THE MECHANICAIi THEORY OF HEAT. 283

up by the source of heat ultimately passes to the quicksilyer
used in cooling. We will not, however, here enter into the ex-
tension of this procedure, but will limit ourselves to the con-
sideration of the previously described simple operation through
which the gas is heated from 0° to a mean temperature of 120°;
for this operation already contains the essential parts whereon
the objection of Him is based.

8. In this operation external work is neither gained nor lost ;
for since the pressure in both cylinders is always the same, both
pistons are pressed upwards at each instant with equal force,
and these forces destroy one another on the toothed wheel upon
which the teeth of the piston rods work ; so that, disregarding
friction, the smallest force would be sufficient to cause a rotation
of the toothed wheel in one sense, or in the other, and thereby
a descent of the one piston and ascent of the other. The excess
of heat in the gas, therefore, cannot be generated by exterior
work, and there can of course be no question of interior work,
since the latter is altogether excluded by the hypothesis of a
perfect gas.

The process is manifestly the following. When a quantity of
gas, very small in proportion to the whole quantity under con-
sideration, becomes heated in the connecting tube and thereby
expanded, it must receive from the source as much heat as is
requisite to elevate its temperature under constant pressure. Of
this quantity one portion serves to increase the heat actually
present in the gas, and another portion is consumed by the work
of expansion. But since the expansion of the gas in the tube
necessitates a compression of that in the cylinder, just as much
heat must here be generated as was there consumed. This
second portion of the heat given up by the source, and which has
become transformed to work in the tube, appears again in the
cylinders, therefore, as heat, and serves to heat the gas still in A
above its initial temperature 0°, and to heat the gas already in
B, which at its entrance had the temperature 100°, above this
temperature, and thus to bring about the above-mentioned
excess of temperature.

Consequently, without taking the intermediate processes into
consideration, we may say that the entire quantity of heat which
the gas contains at the end of the operation above that, which it

284 SEVENTH MEMOIR.

had at firsts proceeds from the source of heat surrounding the
connecting tube. We thereby arrive at the peculiar result, that
the enclosed gas is heated above 100°, viz. to a mean temperature
of 120^, by means of a body at 100°, that is to say, by the aqueous
vapour surrounding the connecting tube. Him finds in this
a contradiction of the axiom, that heat cannot of itself pass
from a colder to a warmer body, since, according to his repre-
sentation, the heat given by the vapour to the gas has passed
from a body at 100° to another at 120°.

9. In so doing, however, he has overlooked one circumstance.
If the gas had had at the commencement a temperature of 100°,
or more, and had then been raised to a still higher tempera-
ture by vapour which had only a temperature of 100°, then
a contradiction to my axiom would certainly have presented
itself. The state of the case, however, is different. In order
that at the end of the operation the gas may be warmer than
100^, it must necessarily be colder than 100° at the commence-
ment, and in our example, where at the conclusion it has a tem-
perature of 120°, it had at the commencement a temperature 0°.
The heat imparted by the vapour to the gas, therefore, has
served partly to raise the gas from 0° to 100°, and partly to bring
it from 100° to 120°.

Now since the temperatures aUuded to in my axiom are those
which the bodies between which the transfer of heat takes
place, possess at the moment when they receive or lose heat,
and are not those which the bodies subsequently possess, we
must form the following conception of the transfer of heat which
takes place in this operation. One part of the heat given up by
the vapour has passed into the gas during the time that its
temperature was still below 100°, has passed therefore from the
vapour into a colder body ; it is only the other portion of the
heat, viz., that which serves to raise the gas above 100°, which
" has passed from the vapour to a warmer body.

If we compare this with the principle according to which,
whenever heat is to pass from a colder to a warmer body with-
out either a transformation from work into heat, or a change in
the molecular condition of a body, other heat must necessarily
pass in the same operation from a warmer to a colder body, it
will be at once manifest that a complete accordance exists. The

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT, 285

peculiarity presented by the operation which Him has con-
ceived, consists solely in the fact, that there are not two dif-
ferent bodies of which one is colder aiid the other warmer than
the source of heat, but that one and the same body, the gas,
plays in one portion of the operation the part of a colder,
and in the other portion of the operation the part of a warmer
body. This involves, however, no departure from my axiom,
it is merely a special case of the many possible cases. Him's
misconception has arisen from his directing his attention to
the final temperature solely, instead of taking into account
the diflerent temperatures which the gas possesses successively
during the course of the operation.

10. The subject may be further represented in a somewhat
different manner, and by so doing a conception will come under
discussion, which I have introduced in my last memoir''*", and
which in my opinion is of great importance in the theory of
heat ; I refer to the transformation^value of the heat contained
in a body. I wish in conclusion to consider this subject, since
an explanation of this conception will perhaps powerfully con-
tribute to prevent misconceptions of the before mentioned
kind.

In my memoirs I have given the term transformation to the
passage of heat from a body of one temperature to a body at j
another temperature, since it may be said that heat of a certain (i f .;
temperature is transformed to heat of another temperature. / 1
This process is thereby brought into parallelism with two other
processes, which may also be termed transformations, that is to
say, with the transformations of heat iuto work, and vice versd,
and with the transformation which I have designated as a
change of ^^ disgregation.^^ In order to be able to distinguish
in a suitable mathematical manner the direction of the pas-
sage of heat, I have called the passage from a warmer to a
colder body a positive transformation, and the passage from a
colder to a warmer body a negative one ; accordingly the above
axiom may be expressed by saying that a negative transformation
cannot occur of itself, that is to say, without being accompanied
in the same operation by a positive transformation, whilst a

* [Sixth Memoir of this collection.]

. I

286 SEVENTH MEMOIR.

positive transformation^ on the contrary, can very well take
place without being accompanied by a negative one.

On applying this to the above described operation, in which a
quantity of gas is heated to 120° by means of heat proceeding
from vapour at 100°, the question arises how the heat contained
in a body must be considered when we wish to determine its
temperature. Can we, in fact, consider all the heat contained
in a body of the temperature /, simply as heat of the tempera-
ture /, or must we ascribe other temperatures to this heat ?

If the first were the case, that is to say, if the heat contained
in a body of the temperature t were to be considered through-
out as heat of the temperature /, we should, through the above
described operation conceived by Him, arrive at a result in
contradiction to my axiom, for we should have to reason thus.
The heat contained in the aqueous vapour is heat of 100°. If
by means of a portion of this heat the gas is raised to 120°, this
portion is present in the gas as heat of 120°, and consequently
whatever the intermediate temperatures may have been, a
certain quantity of heat becomes ultimately transformed from
heat of 100° to heat of 120°.

This is not the conception, however, from which I started
when formulizing the theorem of the equivalence of transfor-
mations.

When a body is heated, a portion of the heat which must be
imparted to it for this purpose, is in general consumed in ex-
terior and interior work (provided the body by being heated
changes its volume and the arrangement of its molecules), and
^ » the other portion serves to increase the heat actually present
in the body. We may here neglect the first portion, and limit
ourselves solely to the consideration of the second. If we now
conceive the body to be raised from any initial temperature t^ to
any other temperature t, it will not receive all the heat which is
necessary thereto at the temperature t, but it will receive the
different elements of this heat at different and gradually in-
creasing temperatures, so that to each element of heat a definite
temperature will correspond. If, on the other hand, the body be
cooled, it will not give off all the heat which is necessary thereto
at one and the same temperature, but it will part with different
elements at different and gradually sinking temperatures.

X

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 287

Now, when we speak of the temperature of the heat contained
in a body, we must not, according to my view, ascribe one and
the same temperature to the whole quantity of heat ; we must
rather conceive this whole quantity to be divided into an
infinite number of elements, and to each element we must
consider that temperature to correspond which the body on
cooling would have at the moment when it parted with this
element, or that which on being heated it would have at the
moment when it received this element of heat.

11. In the memoir above alluded to I gave a simple mathe-
matical magnitude in which the temperatures of the several
elements of heat were taken into account in the manner re-
quired by the theorem of the equivalence of transformations,
and I termed this magnitude the transformation-value of the
body's heat.

In fact, if we conceive that the heat which serves to raise the
temperature of a given body (and of which, as before said, we
only consider the portion which is finally present as heat in the
body, and not the portion which may have been consumed in
work consequent upon the change of condition caused by heat-
ing) has proceeded, in some manner or other^ from the transfor-
mation of work into heat, we may determine the equivalence^
value of the transformation of each element of heat thus ge-
nerated. Let T denote the temperature of the body counted
from the absoltUe zero, and let us suppose this temperature to
be increased by dT, the increase which thereby takes place
in the heat actually present in the body will then be repre-
sented by the product mcdT; wherein m denotes the mass of
the body, and c its real thermal capacity. The equivalence-
value of the transformation from work to heat, from which
this element of heat has proceeded, will be expressed by the

fraction

mcdT
T •

If we apply this to the determination of the quantity of heat
which must be added to that present in the body, in order to
raise it from the given inital temperature Tq to another tempe-
rature T, the equivalence-value of the transformation from
which this quantity of heat, with its temperatures rising from

288 SEVENTH MEMOIR.

element to element has proceeded^ mU be represented by the
integral 1

Jto T '
which I have called the transformation-valve of the body's heat
estimated from a given initial temperature.

If it were required to determine the transformation-value of
the entire quantity of heat present in the body, we should have
to conceive the body to be heated from the absolute zero up to the
absolute temperature T under consideration^ and consequently
to put as the lower limit of the above integral. The value of
the integral would thereby become infinitely great, since the
product mCy which appears in the numerator, cannot vanish. In
order, therefore, to obtain a finite value for the integral, it is
necessary to start from 2i finite absolute temperature as a lower
limit, and consequently not to determine the transformation-
value of the entire heat present in the body, but only that of
the additional quantity of heat which the body at its present
temperature possesses above what it had at the temperature
chosen as the starting-point for heating.

The actual integration is rendered very simple by the help of
a deduction previously drawn by me. In fact, for reasons into
which I will not here enter, I concluded that the real thermal
capacity of a body is not only independent of its molecular
arrangement, but also of its temperature. We may consequently
place the real thermal capacity c, together with the mass m, as
factors before the sign of integration, whereby the integral
becomes

fT rfT , T

.J^^^ = mclog^^.

Should this last simplification, however, not be deemed a
sufficiently well grounded one, should it, in fact, be deemed
desirable to regard c as a still unknown function of the tempe-
rature, it would merely be necessary to change, somewhat, the
form of the expression for the tranformation- value of the body's
heat, the conception of a transformation-value would thereby
sufler no essential alteration.

By the introduction of this new conception, we can very
easily and with perfect accuracy characterize the changes which

ON AN AXIOM IN THE MECHANICAL THEORY OF HEAT. 289

may spontaneously occur in the distribution of heat. If we
conceive any process whatever such that ultimately no other
transformations remain except passages of heat between bodies
of diflferent temperatures, all other transformations which may
possibly occur in the course of the process being again cancelled
by opposite transformations which likewise occur therein, we
may enunciate the following general theorem relative to passages
of heat, and the changes in the distribution of heat consequent
thereon. The change which occurs by such a process in the
distribution of heat can only be such that the sum of the trans-
formatimir-values of the heat in the several bodies is thereby in-
creased or in the limit remains unchanged; it can never be such
that the sum of the transformation-values diminishes thereby.

If we test in this maimer the result of the operation above
considered, by which a quantity of gas has its temperature
raised from 0° to 120° by means of the heat arising from vapour
at 100°, we shall find that here also the sum of the transformation-
values of the heat contained in the vapour and in the gas in-
creases, and that accordingly the theorem of the equivalence of
transformations, and the axiom from which it is deduced, is
simply verified by this operation*.

♦ [I regret very much that I have here been compelled to dissent from
Him's expositions of the second fundamental theorem in a manner similar to
that in which^ on a former occasion, I found it necessary to declare myself
opposed to his view of the first fundamental theorem of the mechanical
theory of heat. I feel convinced, however, that no one will think of reproach-
ing Mm for having raised the objection vrhich has been discussed in the
present memoir.

The second fundamental theorem of the mechanical theory of heat, and
all that depends thereupon, is much more difficult to understand than the
first fundamental theorem, and the way in which Him has interpreted the
former is, in fact, as has already been mentioned, not an unnatural one, so
that in all probability others may also have encountered the same difficulty.
Under these circimistances the objection raised by Him was fiilly justified
from a scientific point of view ; and when, as ia the present case, an objection
of this kind is made with such clearness and precision, and accompanied by
so ingenious an illustration as the operation conceived by Him, science can
only derive profit therefrom ; such a procedure, in fact, deserves to be regarded
as a meritorious one. The exposition of the subject is very much facilitated
by the definite and clear elucidation of the apparent contradiction, and in this
manner the advantage is gained of at once and for ever settling a difficulty
which, otherwise perhaps, might have given rise to many misconceptions and
rendered necessary frequent and long discussions. — 1864.]

V

f//

290 EIGHTH MEMOIR.

EIGHTH MEMOIR.

ON THE CONCENTRATION OF RATS OF HEAT AND LIOHT^ AND ON
THE LIMITS OF ITS ACTION*.

The* starting-point of my treatment of the second fundamental
theorem in the mechanical theory of heat, was the difference which
exists between the transfer of heat from a warmer to a colder body,
and that from a colder to a warmer one ; the former may, but the
latter cannot, take place of itself. This difference between the
two kinds of transmission being assumed from the commence-
ment, it can be proved that an exactly corresponding difference
must exist between the conversion of work into heat, and the
transformation of heat into work; that heat, in fact, cannot
simply transform itself into work (another simultaneous change,
serving as a compensation, being always necessary thereto),
whereas the opposite transformation of work into heat may occur
without compensation.

A general and prevailing tendency in nature to changes of a
certain character is indicated by these principles, and the latter
may be extended in a similar maimer to a third action, affecting
the changes of the condition of bodies ; into this, however, I do
not propose to inquire here. On applying the above considera-
tions to the universe as a whole, we arrive at the remarkable
conclusion to which W. Thomson first drew attentionf, he
having then admitted the truth of the modification applied by
me to Camot's theorem, and adopted my conception of the
second fundamental theorem in the mechanical theory of heat.
For if in the universe cases continually occur, through friction
or other similar impe;diments to motion, of the conversion into
heat, that is to say, into molecular motions, of the motions with
which large masses are animated, and which are due, either

* Communicated June 22, 1863, to the Natural Science Association of
Zurich, and published in PoggendorfTs Annalen for January 1864, vol. cxxi.

p.1.

t Pha. Mag". S. 4 vol. iv. p. 304.

CONCENTRATION OP RAYS OF LIGHT AND HEAT. 291

actuallj or conceivably, to work done by natural forces; and
if, further, heat always strives to alter its distribution, so that
existing differences of temperature may be cancelled, then the
universe must gradually be approaching more and more to the
condition in which forces can produce no further motion, and
differences of temperature can no longer exist.

This conclusion suggested to Bankine his paper ^^ On the Re-
concentration of the Mechanical Energy of the Universe''"*,
wherein the question is examined whether, to counterbalance the
above processes whereby mechanical energy becomes more and
more dissipated, another of an opposite effect is not conceivable,
whereby mechanical energy may be again concentrated and
stored up in individual masses.

After having* spoken of the manifold ways in which heat may
be produced by the work of natural forces, and of the incessant
tendency of heat to distribute itself amongst bodies so as to
annul existing differences of temperature, and after adding that
the heat present in bodies has also a tendency to become con-
verted into radiant heat, so that all the bodies in the universe
continually give off more and more heat to the aether which per-
vades space, Rankine continues f : —

^^Let it now be supposed that, in aU 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 reconstituted 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
vaporized 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 diffusion, in the form of radiant heat, of all physical
energy, the extinction of the stars, and the cessation of all phe-
nomena, yet the world, as now created, may possibly be pro-

* Phil. Mag. S. 4. vol. iv. p. 358. f Ibid. p. 360.

u2

292

EIGHTH MEMOIR.

vided within itself with the means of reconcentrating its physical
energies, and renewing its activity and life/'

According to this, Rankine appears to think it possible so to
concentrate rays of heat by reflexion, that in the foci thereby
produced, a body may be heated to a temperature higher than
that of the bodies which emitted the rays. If this view were
correct, the principle assumed by me as an axiom, that heat
cannot of itself pass from a colder to a warmer body, would be
false, and as a consequence the proof, founded on this axiom,
of the second fundamental theorem in the mechanical theory
of heat would have to be rejected.

The wish to put the truth of the axiom beyond all doubt of
this kind, and the fact that, apart from this special question, the
concentration of rays of heat, with which that of luminous rays
is closely connected, is a subject offering many points of interest,
have induced me to submit to closer mathematical investigation
the laws which govern the concentration of rays, and the influ-
ence which the latter may* have on the interchange of rays be-
tween bodies. The results of this iavestigation I propose here
to commimicate.

I. Insufficiency of the previous determination of tfie mutual radi-
ation between two surfaces for the case now under con-
sideration.

1. When two bodies are placed in a medium penetrable by
rays of ieat, they transmit heat to each other by radiation. In
general, -one portion of the rays which fall on a body is absorbed,
whilst another portion is partly reflected and partly transmitted ;
and it is well known that the powers of absorption and emission
have a simple relation to each other. As it forms no part of
our present object to examine the variations and regularities
involved in this relation, we will take the simple case where the
bodies under consideration have the prop^^y of at once com-
pletely absorbing all incident rays, either at their surfaces, or
within a stratum so thin that its thickness may be neglected.
Such bodies have been denominated perfectly black by Kirch -
hoff in his well-known and remarkable memoir on the relation
between emission and absorption*.

* PoggendorflTs Annalen, vol. cix. p. 275, and Phil. Mag. S. 4. vol. xx. p. 1.

I

J

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 293

Bodies of this kind have also the greatest possible emissive
power, and it has long been assumed as certain that the intensity
of their emission depends solely upon their temperature; so
that at the same temperature all perfectly black bodies radiate
the same amount of heat from equally large portions of their
surfaces ; now, since the rays which a body emits are not homo-
geneous, but of diflferent colours, the emission must be con-
sidered specially with respect to the diflferent colours : KirchhoflF
accordingly has completed the above theorem by showing that
perfectly black bodies at the same temperature not only radiate
the same total amount of heat, but also equal quantities of every
particular kind of heat ; moreover, since, in our investigation,
these particularities are likewise to be excluded from considera-
tion, we shall always assume in future that we are solely con-
cerned with rays of a particular kind, or more strictly, with rays
whose wave-lengths vary only within infinitely small limits. For
whatever holds for one kind of rays being, in a corresponding
manner, true for every other kind, the results found for homo-
geneous heat may without difl&culty be extended to heat con-
sisting of diflferent kinds of rays.

To avoid unnecessary complications, we will likewise disregard

, all phenomena of polarization, and assume that we have to deal

with unpolarized rays solely. Helmholtz and Kirchhoflf have

explained how polarization will have to be taken into account in

considerations of this kind.

2. Let Sj^ and s^ be given surfaces of any two perfectly black
bodies of the same temperature, and upon them let any two
elements, ds^ and ds<^ be selected with a view of determining
and comparing the quantities of heat which they mutually
transmit to each other by radiation. When the medium which
surrounds the bodies and fills the intervening space is homo-
geneous, so that the rays proceed in right lines from one surface
to the other, it is easy to see that the quantity of heat which
the element ds^ sends to ds^ must be just as great as that which
ds^ sends to ds^. If, however, the medium which surrounds the
bodies is not homogeneous, but of such a character as to cause
refractions and reflexions, the procedure is less simple, and a
more careful consideration is necessary in order to convince our-
selves that here also the above perfect reciprocity still exists.
Kirchhoflf has examined this question in a very elegant manner^

294 EIGHTH MEMOIR.

and I will here briefly give his results so far as they, have refer-
ence to the case, where the rays suffer no diminution of in-
tensity on their way from one element to the other, that is to
say, where the refractions and reflexions occur without loss, and
the propagation is not accompanied by absorption. In doing
so I shall merely change, to some extent, his notation and his
system of coordinates in order to secure a closer agreement with
what will follow.

Of the infinite number of rays which one of two given points
emits, only one can in general attain the other; or should
more do so in consequence of refractions and reflexions, their
number will at all events be limited, and each can be con-
sidered individually*. The path pursued by a ray which, starting
from one of the two points, arrives at the other, is determined on
the principle that the time required to traverse it is less than
that which any other adjacent path would have demanded. This
minimtiin time, which, with Kirchhoff, we will denote by T, is
determined by the positions of the two points, a single ray being,
of course, selected for consideration in the case where many se-
parate rays present themselves.

Betuming now to the two elements rf«j and ds^ we will con-
ceive tangent planes to each surface drawn at a point of each
element, and consider dsu ds^ as elements of these planes. In
each of the latter we will introduce an arbitrary system of
rectangular coordinates, a?i, y^ and x^, y^t* If ^ each plane we
now take a point, the time T, which the ray requires to travel

* That a, point can emit an infinite number of rays may perhaps be regarded,
from a strictly mathematical point of view, as an incorrect mode of expression ;
since a surface only, not a mathematical point, can radiate heat or light. It
would accordingly be more correct to refer the radiation of light and heat to
a surface-element in the vicinity of the point, instead of to the point itself.
Nevertheless, since the notion of a ray is itself a pure mathematical abstrac-
tion, the conception of an infinity of rays proceeding from each point of a sur-
face may be retained without fear of any misunderstanding arising therefrom.
When the qtuintity of heat or light which a surface radiates has to be deter-
mined, the magnitude of the surface will of course enter into consideration,
and when the surface is divided into elements, these elements will not be
points but infinitesimal surfaces, whose magnitudes will enter as factors into
the formulflB which represent the quantities of heat or light radiated from
these surface-elements.

t Kirchhofi* placed his coordinate systems in two planes perpendicular to
the directions of the ray in the vicinity of the two surface-elements, and upon
these planes he likewise projected the elements.

«

CONCENTRATION OP RAYS OP LIGHT AND HEAT. 295

I from one to the other, is, as above remarked, determined by the

positions of the two points, and accordingly is to be regarded
as a function of the four coordinates of the two points.

This granted, the following is, according to Kirchhoff*, the
expression for the quantity of heat which the element dsi sends
to the element ds^ during the unit of time,

e^/ (PT <PT dTT d^T \^^ ^^

IT \dxi dx^ dy^ dy^ dx^ dy^ dy^ dxj ^ ^'

wherein tt denotes the ratio of the circumference to the diameter
of a circle, and e, is the intensity of the emission of the surface
«i at the element ds^, so that e, ds^ represents the total quan-
tity of heat which the element dsi radiates during the unit of
time.

In order to obtain the expression for the quantity of heat
which ds^ sends to ds^, we have only to replace e^ in the pre-
ceding expression by the intensity of emission e^ of the surface ^^ *
at the element ds^. Everything else remains unchanged; for
'the expression is symmetrical with respect to the two elements ;
since the time T which a ray requires in order to traverse the
path between two points of the two elements is the same in
whichever direction the ray moves. If, now, we assume that
the two surfaces, when at the same temperature, radiate equal
quantities of heat (that, in short, ^1=^2), it will follow that the
element ds^ must send just as much heat to ds^ as ds^ sends
to ds^.

3. It was stated above that between two given points only one
ray, or a limited number of rays, was in general possible. In
particular cases, however, it may happen that an infinite number
of the rays, which proceed from one point, may converge to the
other, and these may either fiU a part of the surface of a cone,
or of the solid angle formed by a cone. The same applies ob-
viously to rays of light as weU as to rays of heat, and in optics
it is customary to call the point to which the rays proceeding
from a given point and filling a conical space converge, the
image of the latter point, or since the first point may also be
the image of the second, the two are called conjugate foci.
When what has here been said of two isolated points applies

* Pogg. Ann. vol. cix, p. 286 ; Phil. Mag. S. 4. vol. xx. p. 9,

296 EIGHTH MEMOIE«

to all the points of two surfaces, so that every point of the one
surface has its conjugate focus at a point on the other, one
surface is said to be the optical image of the other.

The question now arises, how does the ray-interchange take
place between the elements of two such surfaces ? Is the above
reciprocity still maintained ; in other words, the temperatures
being equal, does each element of the one surface send to an
element of the other surface exactly as much heat as it receives
therefrom ? If so, one body could not raise another to a higher
temperature than its own ; if otherwise, then by the concentra-
tion of rays it would be possible for one body to raise another to
a higher temperature than it possesses itself.

Kirchhoflfs expression is not directly applicable to the case
under consideration. For if the surface s^ were the optical
image of s^, then the rays proceeding from a point p^ of the
surface s^, and filling a certain cone, would converge to a deter-
minate point p^ of the surface s^y and none of the points of ^2
adjacent top^ would receive rays fcomp^, consequently the co-
ordinates iTp y^ of the point p^^ being given, the coordinates
^2> ^2 of tlie point j»3 will no longer be arbitrary, but perfectly
determined ; and similarly, when ^3, y^ are given, the coordi-
nates ^p y^ wiU be determined. Accordingly no real magnitude
of a finite value can be represented by a difierential coefficient

of the form -z — -^, wherein, when differentiating according to

a?j, the coordinate 4?^ is to be regarded as variable, whilst the
second coordinate y^ of the same point, as well as the coordinates
a?2, y^ of the other point, are to be considered as having constant
values, and similarly, when differentiating according to a^^ the
coordinate a?^ is to be considered variable, whilst y^, w^, y^ re-
main constant.

In this case, therefore, an expression of a form somewhat
different from that of Kirchhoff^s must be found, and to this
end the following considerations, similar in kind to tnose which
led Kirchhoff to his own expression, will be of service.

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 297

II. Determination of corresponding points and -of corresponding
surface-elements in three planes intersected by rays.

4. Let three planes a, b, c be given, of which b lies between
d and c (fig 10). In each.plane let a system of coordinates be
established, and let the latter be denoted
respectively by ^^,y^; x^,y^; and a?^,y^.
If a point p^ be given in the plane a, and
a point j»j in the plane i, and we consider
a ray passing jfrom one to the other, we
shall have for the determination of its path
the condition, that the time required for its
passage thereon will be a minimum when
compared with the times of passage along
all neighbouring paths. Let T^ denote
this minimum time, which will be a function
of the coordinates o?^ y^, and a?^ y^ of the points p^ and py In
a similar manner let T^ be the time of passage of the ray be-
tween the two points p^ and p^ in the planes a and c, and T^
the time required for a ray to pass from one to the other of the
points j9j and p^ in the planes b and c. We must consider T^
as a function of the four quantities ^^, y^, x^, y^/and Tj^ as a
function of a?^, y^ o?^, y^.

Now since a ray, which passes through two planes, also cuts
in general the third, we have for each ray three intersections
which are so related to one another, that in general any one is
determined by the other two. Suitable equations for this de-
termination may be easily established by help of the above con-
dition.

We will first assume the points p^ and jo^ in the planes a and c
(fig. 10) to be immediately given; the point where the ray cuts
the intermediate plane b being still unknown, shall be repre-
sented by j^'j in order to distinguish it from other points in that
plane. We will now select any point p^ whatever in the plane
by and consider two auodliary rays, of which one passes from
Pa ^Pb^ ^^^ *^® other fromj9j top^. In fig. 10 the auxiliary
rays are denoted by broken lines, and the principal ray, proceed-

298 EIGHTH MEMOIR.

ing directly from ^^ to jo^, with which we are really concerned, is
shown by a full line*. If in accordance with the preceding we
call T^, Tj^ the times corresponding to the two auxiliary rays,
and form the sum T^+Tj^ the value of the latter will depend
upon the position of the selected point p^ and must accord*
ingly be considered as a fiinction of the coordinates o?^, y^ of
Pj^ ; since the points ^^ and p^ are supposed to be given. Now
of all the values which this sum can acquire by giving to the
point pj^ different positions in the vicinity of the point y^, the
one obtained on allowing p^^ to coincide with pf^, and thus
causing the two auxiliary rays to coincide with portions of the
direct one. proceeding from p^ to p^ must be a minimum.
Hence to determine the coordinates of the point p^j^ we have
the following two equations of condition,

~^. — ' ~~w. — • • ^^

Since the quantities T^^ and T^^, besides the coordinates
a?j, y^ of the previously unknown point, likewise contain the
coordiuates x^, y^ and x^ y^ of the points previously supposed
to be given, the two preceding equations, once established,
may be regarded simply as two equations between the six
coordinates of the three points in which the three planes are
intersected by one and the same ray.

We will now regard as previously
given, the two points p^ and je>j, in
which the ray intersects the planes
a and b (fig. 11), and as unknown
the point pf^, in. which it cuts the
plane c. We wiQ then select any
point p^ whatever in the plane c, and
consider the two auxiliary rays pass-
ing, respectively, from p^ to p^ and
from p^ to p^. In the figure, the latter
are again indicated by broken lines, and
the principal ray by a full line. Calling the times of passage along

* The paths of the rays are drawn somewhat curved in the figure, merely

CONCENTRATION OF EATS OF LIGHT AND HEAT. 299

the auxiliary rays T^^ and T^^, the diflFerence, T^— T^, will be de-
pendent upon the position in the plane c of the selected point ^^.
Of aU the values acquired by this difference when the point j9^ is
placed near the point jj/^ none will be so great as that which
results from making/?^ coincide with^^. For in this case the
ray proceeding from p^ to p^ cuts the plane b in the given
point jE?^ and consequently it consists of the rays passing from
Pa ^Pby ^^^ from p^ to p^. We may therefore put

and hence deduce the equation

for the difference under consideration. But should the point p^
not coincide with p^^, the ray proceeding from p^ to p^ would
not coincide with the two rays from^^ to/?j, and from j!?^ ^Pe^
and the direct ray between p^ and/?^ being the one which
corresponds to the least time of passage^ we should have the
inequality

whence we should deduce for the difference under consideration
the inequality

that is to say, the difference T^— T^ could in general be smaller
than in the foregoing special case, where the point p^ lay in the
production of the ray proceeding from p^ to p^ ; that particular
value of the difference, therefore, is a maaimnm^. Hence we

to indicate that the path pursued by a ray between two given points is not
necessarily the right line which connects these points, but that, in consequence
of refractions and reflexions, another route may be imposed upon it which may
consist either of a broken line composed of several straight ones, or of a
curved line, according as the mediimi traversed by the ray changes suddenly
or gradually.

* In Kirchhoff s memoir, p. 286 (p. 8 of translation), the quantity is stated
to be a minimum, which corresponds essentially to the difference last con-
sidered, — the only modification being that Kirchhoff's quantity has reference

800 SIOHTB MEMOIE.

again deduce two equations of condition ; namely,

__ 0, _^- 0.. . . (2)

If, lastly, we suppose the points p^ and p^ in the planes b and c
to be given, and regard the point where the ray cuts the plane a
as unknown, we obtain the following two equations by a process
of reasoning which, being precisely similar to the foregoing,
may be here omitted : —

We have thus arrived at three pairs of equations, of which
each pair may serve to express the mutual relation which exists
between the three points in which a ray intersects the three
planes a, b, c, and this in such a manner that, whenever two of
the points are given, the third can be found; or still more
generally, when of the six coordinates of the three points any
four are given, the remaining two may be determined.

5. We will now consider the follow-
ing problem. Let a point p^ be given
in one of the three planes, say a,, and
in a second plane 6, a surface-element
dsy If we conceive the rays which
proceed from the point p^ to the several
points of rf*j to be continued until they
reach the third plane c, they will in
general determine on the plane c an
infinitesimal surface-element ds (see
fig. 12) ; required the ratio of the surface-elements dsj^ and ds^.

In this case two, x^ and y^, of the six coordinates of the three
points in which each ray intersects the three planes are given.
Hence any values of the coordinates ^j, y^ being assumed, the

to four planes instead of three. This error is probably typographical -, what-
ever its origin, however, a substitution of the term minimum for maximum
at that place involves no further error, since the theorem, that the differential
coefficients must vanish, which is employed in the calculations which follow,
holds for a maximum as well as for a minimum.

CONCENTRATION OP RAYS OP LIGHT AND HEAT. 301

coordinates x^, y^ are thereby in general determined. In this
case, therefore, each of the coordinates x^ and y^ may be re-
garded as a function of the two coordinates x^^, y^.

If we suppose the arbitrary form of the surface-element dSj^ to
be that of a rectangle dx^ . rfy^, and seek the points in c which
correspond to the several points of its contour, we shall obtain
in the latter plane an infinitesimal parallelogram for the form
of the corresponding element ds^. Irrespective of its sign, the
magnitude of this parallelogram will be represented, ficcording
to simple geometrical principles, by

\dx, dy, dy, dwj » "'•

The circumstance that in the determination of this surface-
element the absolute magnitude alone is considered, shall be
indicated by prefixing the letters v,n. {valor numericus) to the
differential expression which may itself have a positive or
negative value. We may then write

*.-(^-|-l:-l>

(4)

To determine the relation which exists between the coordi-
nates x^, y^ and the coordinates a?^, y^, we must employ one of
the three pairs of equations in Art. 4. To this end we will first
employ the two equations (1). On differentiating the latter
according to Xj^ and to y^, and remembering that each of the
quantities denoted by T contains those two of the three
pairs of coordinates x^ y^ ; ^j, y^ ; x^, y^ which are indicated
by its suffixes, and that x^ and y^ have to be treated as functions
of ^j and y^, whilst x^ and y^ are to be regarded as constants, we
obtain the following four equations : —

802

EIGHTH MEMOIR.

<&»(&.

<i^» dx^d!f, dx^

rf«T,

drjrfyj ^ dx^dxj dy^^ dx^dy^ dy^

*• $^=0,

da>idy^

d^T^ dx d*1^ dy^_^
<fy»<^. dxt dtf^dy/dx^ '

(5)

'^(T^-'rTJ ^ITj.^^ illȣ.^. =0
W dy^dx/dtf^ dy^dy/dy^

The required relation between the elements ds^ and dg^ is now

obtained by substituting the yalues of — f, — ', -Es Jl.«, as de-

<toj rfy» *r»' dy^

termined by these equations in the equation (4) . In order to

exhibit the result of this substitution in an abbreviated form, we

will introduce the symbols

\dx^dx^ dy^dy^ dx^dy^ dy^dxj ^ '

(8)

The required relation may then be written thus :

*• = !

A, A

Similarly^ on assuming a definite point p^ to be given in the
plane c (fig. 13), we may determine the
surface-element ds^ in the plane a, which
corresponds to the given element &j in
the plane b ; in fact, the result may be
deduced from the foregoing by simply
interchanging everywhere the suffixes a
and c. If, for brevity, we likewise intro-
duce the symbol

)z=v.nA ?i

\dx„dx,

rf*T„,_ rf*T

'^^a^Vl ^Va^h

CONCENTRATION OF RAYS OF LIGHT AND H^AT.

8oa

we shall have

E

(10)

Let lis, lastly^ suppose a definite point pj^ to be given in the
plane b (fig. 14), and in the plane a let
us select any surface-element ds^. Con-
ceive rays to proceed from the several
points of the latter element, and passing
through the point p^^ to be continued to
the plane c. If we seek the magnitude
of the surface-element ds^, which all these
rays determine on the plane c, we shall
find on employing the symbols already,
introduced,

dT a: ' '

(11)

From this we see that the two surface-elements which here
correspond are related to one another in precisely the same
maimer as are the two surface-elements which we obtain when,
a definite element ds^ being given in the plane b, we first assume
a point on the plane a, and afterwards a point in the plane c, as
the starting-point, and determine each time the surface-element
in the third plane, c or a, which corresponds to dsy

6. In the calculations of the foregoing article we have em-
ployed only the first of the three pairs of available equations,
given in Art. 4. The calculations, however, may be performed
in the same manner on employing either of the other two pairs
(2) or (3) . By means of each pair we are led to three quantities,
analogous to A, C, and E, which serve to express the ratio of the
surface-elements. Of the nine quantities which, on the whole,
present themselves in this manner, it occurs three times that
two are equal to one another, so that the nine quantities are
reduced to six. Although the expressions for three of these six
quantities have already been given, I will here for the sake of
completeness give the whole series.

304

BIOBTH HXMOIK.

rf«T^ d*T^

«^T^ rf«T.

L=».n,/ *•. !l— ??. ° -^t* \

\dx^dx^ dyidy, dx^dy, dy^dyj

{sBp.n.( ?£. 21 — ??. "o Y

\<ip.«£r, rfy^rfy, dxjy, dyjyj

\dx,dx^ dv^dv^ dx dy^ dy^dxj'

,<&j dyjy^ dx^
L {dxf (dy,)« L dx^dy, J J

By means of these six quantities each ratio of two surface-
elements may be represented by three different fractions, in the
following manner : —

(I)

*,
*»

E

A

C
B'

d»a

C

B

'a~

ds.

A

F
= B =

B

(11)

As is easily seen, the three equations have reference, respectively,
to the three cases where the definite point through which the
rays must pass is taken in the plane a, in c, or in b. Of the
three vertical rows of fractions representing the ratios of the
surface-elements, the first is deduced from the equations (1), the
second from the equations (2), and the third from the equations
(8) of Art. 4.

Since the three fractions which represent a definite ratio of
two surface-elements must be equal to one another, we obtain
the following equations between the six quantities from which
the fractions are formed : —

CONCENTRATION OP RAYS OP LIGHT AND HEAT. 305

D= -T-; E=-g-; F= -^ (12)

A«=EF; B«=FD; C«=DE (13)

Now our subsequent calculations are to be made with these
six quantities ; and since each ratio of two surface-elements is
represented by three diflTerent fractions, we may select from the
latter, in each special case, the fraction which proves to be most
convenient.

III. Determination of the mutual radiation in the case where no
concentration of rays takes place.

7. We will in the first place consider the case to which Kirch-
hoflF's expression refers, by seeking to determine how much heat
two surface-elements tra;nsmit to each other, on the hypothesis
that each point of either element receives from each point of the
other but one ray, or at most but a Umited number of single
rays, each of which admits of separate consideration.

Two elements, d s^ and ds^, in the pj^ j^

planes a and c (fig. 15) being given, we
will first determine the heat which the ele-
ment & sends to the element ds.

a e

To this end let us conceive an inter-
mediate plane b to be drawn parallel to
a, and at so small a distance p from the
latter that the portion of each ray pro-
ceeding from ds^ to ds^ which lies be-
tween the two planes a and b may be re-
garded as rectilinear, and the medium between the two plan
as homogeneous. Let any point be now taken in the element ds^^
and let us consider the pencil of rays which proceed therefrom
to the element ds^ ; this pencil intersects the plane 6 in an ele-
ment dsj^ whose magnitude may be expressed by one of the three
fractions standing in the uppermost of the horizontal rows of (II) ,
Selecting the last of these fractions, we have the equation

&*=S* (14)

306 EIGHTH MEMOIR.

This quantity C may in the present case be reduced to a parti-
cularly simple form in consequence of the peculiar position of
the plane b.

We will, with KirchhoflF, choose the coordinate system in b
so that it shall correspond perfectly with the coordinate system
in the parallel plane a. That is to say, the origins of the two
coordinate systems sliall lie in a line perpendicular to both
planes, and the coordinates of one system shall be parallel to the
corresponding coordinates of the other system. The distance r
from a point x^ y^ in the one plane to a point x^ y^ in the other
wiU then be determined by the equation

r=i/p«4.(^,-^J«+(y,-y.)«. • . . (15)

Now the propagation of rays between the two planes being by
hypothesis rectilinear, the length of the path described by a
ray proceeding from one to the other of these points will be
simply the distance r between the points themselves ; and if we
represent by v^ the velocity of propagation in the neighbourhood
of the plane a, the time required for the description of the dis-
tance r will be determined by the equation

T =-^

since the velocity v^ according to our hypothesis, does not sen-
sibly change in the interval between the planes a and b. Ac-
cordingly the expression for C may be thus written :-^

Substituting here the value of r as given by (15), we bave

The equation (14), therefore, becomes

&,=t;2^Bife^. ^ {17)

If, fiirther, we denote by ^ the angle between the infinitely
small pencil of rays proceeding from a point of the element

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 307

da^y and the normal to that element, we may put

cosd= -9
r

and the foregoing equation wiU thereby assume the form

ds,^:^Bds,. ...... (18)

8. The magnitude of the surface-element ds^, being thus found,
the quantity of heat which the element ds^ transmits to the ele-
ment ds^ may be also easily expressed. For an infinitely
small pencil of rays proceeds to ds^ from each point of the
element ds^y and the conical apertures of the pencils pro-
ceeding from the several points may be regarded as equal
to one ajiother. The magnitude of the conical aperture of
each such p^icil of rays is determined by the magnitude and
position of the sur&ce^element ds^^ which the cone determines on
the plane 6. To express this conical aperture geometrically, con-
ceive a spherical surface of radius p described around the vertex
of the cone as centre. All rays being propagated in right lines
within this sphere, the aperture of the cone will be represented

by the fraction -r, if .rfo- be the surfaces-element in which the

spherical surface is intersected by the cone of rays. Now the
element ds^ being at the distance r from the vertex of the cone
^ and the normal to ds^, like its parallel the normal to ds^ ma-
Ismg with the infinitely small cone the angle ^, we have the
equation

&r_ co8^.&,

P^ ?r-^ ^^^>

from which, by substituting the value of cfo^ as given in (18),
we deduce

^^<Bds, (20)

p cosd

We have now to determine how much of the heat emitted

by the element ds^ corresponds to this infinitely small aperture ;

' in other words, how much heat the element ds^ transmits through

) that particular element d<r of the spherical surface. In the first

x2

308 EIGHTH MEMOIR.

place this quantity of heat must be proportional to the magni-
tude of the radiating element ds^; it must further be propor-
tional to the magnitude of the aperture of the cone, that is to

da-
say, to the fraction -^; and lastly, according to the known law

of radiation, it must be also proportional to the cosine of the angle
d which the infinitely small cone of rays forms with the normal.
It may therefore be expressed by the product

ecosd-g^flw^,

where € is a factor dependent on the temperature of the surface-
element. To determine this factor, we have the condition that
the total quantity of heat radiated by the element ch^, in other
words, the quantity transmitted by it to the whole hemispherical
surface above the plane a, must be equal to the product e^ ds^^
where e^ denotes the intensity of the emission of the plane a at
the position of the element ds^. We have, consequently, the
equation

where the integration is to be extended over the whole hemi-
sphere. From this it foUows that

On introducing the value of e, thus determined, into the above
expression, we obtain the following formula for the quantity of
heat transmitted by the element ds^ through da- :

TT p^ *

In order to obtain the required expression for the quantity
of heat transmitted by the element ds^ to the element ds^, we have

merely to substitute in this formula the value of the fraction —o

P
already found and given in the equation (20) . The result is

a a ^ a e

If, in exactly the same manner, we seek the quantity of heat

V

i

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 809

which, on the contrary , the element ds^ sends to ds^ and in doing
SO denote by e^ the intensity of the emission from the plane c in
proximity to the element ds^, and by v^ the velocity with which
the rays are propagated in the neighbourhood of this element,
we shall find the expression

IT

9. The expressions obtained in the preceding article are es-
sentially the same as KirchhoflF^s expression given in Art. 2,
the only difference being that the former still contain the square
of the velocity of propagation as a factor, which factor does not
appear in Kirchhoff^s expression on account of his having, for
the object in view, considered solely the velocity of propagation
in empty space and considered that as unity. Since the bodies,
however, whose mutual radiation is under consideration may
possibly be situate in different media, in which latter the veloci-
ties of propagation differ, this factor is not unessential in such
cases, and its appearance leads at once to a peculiar and theore-
tically interesting conclusion.

As mentioned in Art. 1, it has been hitherto assumed that,
for perfectly black bodies, the intensity of emission depends upon
the temperature solely ; so that, at the same temperature, equal
portions of the surfaces of two such bodies radiate equal quanti-
ties of heat. So far as I know, it has nowhere been stated that
the nature of the surrounding medium can have any influence
on the intensity of the radiation. Since in both the above expres-
sions for the mutual radiation of two elements, however, a factor
is involved, which depends on the nature of the medium, the
necessity of considering the medium is forced upon us, and at
the same time the possibility presents itself of determining its
influence.

On forming the ratio of those two expressions, and cancelling
■p
the factor —dsads, which is common to both terms, we find that

the quantity of heat which the element dSa transmits to the ele-
ment dsc, bears to that which the element ds^ transmits to the
element dSa the ratio

810 KIOHTH MEMOIR.

Now if we were to assume that the radiation is necessarily the
same at the same temperature^ even when the media adjacent to
the two elements differ^ we should haye to put ea^e^ for equal
temperatures, and the quantities of heat which the two elements
transmit to each other would then have the ratio of vl : t^, in-
stead of being equal to each other. Hence it would follow that
two bodies placed in different media, e, g. the one in water and
the other in air, do not seek to equalize their temperatures by
mutual radiation, but that the one could by radiation raise the
other to a temperature higher than its own.

If, on the contrary, the theorem stated by me as an axiom
be admitted in all its generality, namely that heat cannot of
itself pass from a colder to a warmer body, then the mutual
radiations of two perfectly black surfiEUse-elements of the same
temperature must be considered as equal to one another, and
we must put

eA^e,i?, (21)

Hence follows the proportion

^«:Ce=l^:t^2> (22)

or, since the ratio of the velocities of propagation is the inverse
of that of the coefficients of refraction of the two media, say
!»« and Uqj the proportion

^a:ec=«a:»c (23)

According to this, therefore, the radiation» of perfectly black
bodies of the same temperature are different in different media;
they are inversely proportional to the squares of the velocities of
propagation in those media, and therefore directly proportional to
the squares of their coefficients of refraction. The radiation in
water, for instance, must have to that in air the ratio of

(1^:1 = 16:9 nearly.

If we take into consideration the circumstance that in the
heat radiated from a perfectly black body there are rays of very
different colours, and if we admit that the equality of the mutual
radiation must hold not only for the total heat, but likewise for

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 811

that of each particular colour, we shall obtain proportions
like (22) and (23) for each colour; in these proportions, how-
ever, the ratios on the right, to which the ratios of radiation are
equated, will have somewhat diflferent values.

If we wish to consider, instead of perfectly black bodies, bodies
which absorb incident rays partially instead of completely, we
must introduce into the formulae, instead of emission, a fraction
having emission for its numerator and the coefficient of absorp-
tion for its denominator, and thus obtain for this fraction relations
corresponding to those which previously had reference to emis-
sion alone. Into this generalization of our result, which would
involve a discussion of the influence of ray-direction on emission
and absorption, I need not here enter, since it follows at once
fit)m an appropriate consideration of the subject.

IV. Determination of the muttuil radiation between two elements
which are optical images of each other.

10. We will now proceed to the case where it is no longer
true, as before assumed, that the planes a and e, so far as they
enter into consideration, interchange rays in such a manner, that
from each point of the one plane proceeds but one ray, or at most
a limited number of distinct rays, to each point of the other
plane. The rays which diverge from each point of one plane
may, in consequence of refractions or reflexions, become con-
vergent and meet again in the other plane ; so that correspond-
ing to a point j9^, selected for consideration in the plane o, there
may be in the plane c one or more points or lines in which an
infinite number of the rays which issued from p^ intersect, whilst
other parts of the plane c may receive no rays whatever from
that point. In such a case, of course, similar properties are
also possessed by the rays which, issuing from the plane c,
arrive at the plane a, since the same paths are pursued by the
rays which pass to and fro between the two planes.

From the infinity of different cases of this kind, we will for
the sake of greater clearness, first treat the extreme one where
all the rays which issue from the point ^^ in the plane a, and
fall within a certain finite conical space, meet again at a definite

812 EIGHTH MEMOIR.

point p^ of the plane c, as shown in ^- 16.

fig. 16. This case would occur, for
instance, if the directions of the rays
were changed by a spherical mirror
or by a lens, or by any system of
centred mirrors or lenses, and we
were to disregard the accompany-
ing spherical and chromatic aberra-
tions ; it may be remarked, indeed,
that chromatic aberration is in every
case to be excluded, since, from the commencement, we have
limited our considerations to homogeneous rays. As already
stated, the name conjugate foci will be given to two points so
related, that the rays which issue from the other converge in
each.

In such a case, with the coordinates x^ y^ of the point p^
from which each of the rays in question has issued, are simul-
taneously determined the coordinates x^, y^ of the point p^^
where the ray strikes the plane c. The other points of the
plane c, which are in the neighbourhood of p^ receive no
rays from the point p^ since there is no path leading to
them which possesses the property of being describable by a
ray in a time which is a mathematical minimum when compared
with the times of description of every other adjacent path.
Accordingly the quantity T^, which represents this minimum of
time, can have no real value for points adjacent to p^, but solely
for the point p^ itself. As a consequence of this it follows, that
the differential coefficients of T^ cannot be real and finite mag-
nitudes if formed on the assumption that the coordinates x^ y^
are constant, whilst one of the coordinates x^y y^ varies ; or, on
the contrary, that x^ y^ are constant during the variation of
^a ^^ Va' Hence we conclude, that of the six quantities

A, B, C, D, E, F determined by the six formulae (I), the three,

B, D, F, which contain differential coefficients of T^^ are not
applicable to our present inquiry.

The three other quantities. A, C, E, contain, however, dif-
ferential coefficients of T^ and T^ solely. Consequently, if we

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 313

assume the plane b to be so chosen^ that between it and the
planes a and c, so far at least as the latter enter into considera-
tion, the ray-interchange takes place in the manner previously
described, that is to say, from each point of b proceeds one, but
only one ray to each point of the planes a and c, or at most a
limited number of distinct rays, the quantities T^ and T^^ will
have real and not infinitely great values for all points which
have to be considered. The quantities A, C, and E, therefore,
are just as applicable in the present as in the former case.

In the present case, one of these magnitudes, E, takes a special
value, which may be at once deduced. The two equations (1), or

must hold with respect to the three points in which a ray inter-
sects the three planes a. A, c. , Now since in our present case
the positions of the points p^ and p^ in the planes a and c do
not suffice to determine the position of the intersection of a ray
with the plane i, inasmuch as the latter plane may be intersected
in any point within a certain finite area, the two foregoing
equations must hold for all such points, whence it follows that,
by the differentiation of these equations according to a?^ and y^,
new and equally true equations must be obtained. We have
therefore

^(T^+Tj .n. rf'CT^+T^ .n. ^^(T^+TJ _^

— ^ ^' -^i;^ — ' ~M ^^

On applying these equations to the one by which, in the
system (I), E is defined, we have

E=0 (25)

The two other quantities, A and C, have, in general, finite
values dependent upon the circumstances of each individual case ;
these values must be employed in the following determinations.

11. Let it be granted that the element ds^ of the plane a has
ds for its optical image in the plane c, so that each point of the
element ds^ forms, with a point of the element ds^, a pair of con-
jugate foci, and vice versa. We will inquire whether the quan-

314 KIOHTH MEMOIR.

titles of heat are equal which these two elements^ considered as
belonging to the surfaces of two perfectly black bodies of the
same temperature, transmit to each other.

In order, first, to determine the position and magnitude of
the image ds^ of the given element ch^, let us select any point/>j
in the intermediate plane 6, and conceive rays passing through
it from every point of the element eh^. Each of these rays
strikes the plane c in the conjugate focus of the point whence
the ray issued, hence the surface-element in which this pencil of
rays cuts the plane c is precisely the optical image of the element
ds^ ; in other words it is ds^. To express the magnitude of the
image ds^ relative to ds^ therefore, we may employ one of the
three fractions, in the lowest horizontal row of (II.) , which repre-
sent^he ratio of the two surface-elements which an infinitesimal
pencil of rays, with vertex p^^ in the intermediate plane 6, deter-
mines upon the planes a and c. Of the three fractions in ques-
tion, however, one only is suitable, the two others being indeter-
minate, so that we have the equation

S:4 «

This equation is of interest in optics, since it is the most
general equation for determining the relative magnitudes of an
object and its optical image ; it may be remarked with respect to
it, that the intermediate plane ft, to which the quantities A and
C have reference, being arbitrary, may in each particular case be
chosen so as most to facilitate the calculation.

12. The surface-element ds^ image of ds^ having been deter-
mined, let us take a surface-element (fe^ instead of a point in
the plane b, and consider the rays which the two elements ds
and ds^ transmit through it. All rays which, issuing from a
point of the element ds^ pass through the element rf*j unite
again in a point of the element ds^ ; accordingly, all rays trans-
mitted by the element ds^ through <foj precisely reach ds^, and
vice versa, the rays which ds^ sends through efe^ all strike the
element ds^. The two quantities of heat which the elements
ds^ and ds^ transmit to the element cfe^ are, therefore, also the

CONCENTRATION OF KAYS OP LIGHT AND HEAT. 815

quantities of heat whicli the elements ds^ and ds^ transmit to
each other through the intermediate element ds^^. Now these
quantities of heat may be at once found, according to previously
established principles.

In fact, for the quantity of heat which the element ds^ sends
to the element dsj^, the same expression holds which in Art. 8
was developed for the quantity of heat which the element ds^
sends to the element ds^ ; provided we therein replace ds^ and B
by ds^ and C respectively. The required expression is^ there-
fore,

Similarly the expression already found for the quantity of heat
which the element ds^ transmits to the element ds^ famishes
the quantity of heat which the element ds^ sends to the element
d»f^, on changing in the former ds^ to ds^, and replacing the
quantity B by the quantity A. Hence the expression required is

On remembering now that, according to equation (26),
Cds^^Ads^

we conclude that the two quantities here expressed are to each
other in the ratio of e^vl : ejtr^.

We arrive at precisely the same result when, in the inter-
mediate plane b, we take any other surface-element dSj^ and con-
sider the quantities of heat which the elements ds^ and ds^
transmit to each other through it. The two quantities of heat
always bear to each other the ratio of eji^ : e^rj. Now, since
the total quantities of heat which the elements ds^ and ds^
transmit to each other consist of those which they transmit
through the several elements of the intermediate plane, we con-
clude, as the final result, that the total quantities of heat which
the surface-elements ds^ and ds^ transmit to each other have
the ratio ey^ : e^^.

This is the same ratio as that which was found in Arts. 8 and 9

X

316 EIGHTH MEMOIR.

for the case where no concentration took place. It follows,
therefore, that however much the concentration of rays may
change the absolute magnitudes of the quantities of heat which
two surface-elements interchange by radiation, the ratio of these
quantities is not altered thereby.

In Art. 9 it was shown that if for ordinary unconcentrated
inter-radiation the theorem is to hold, in virtue of which heat
cannot be thereby transferred from a colder to a warmer body,
the radiation must necessarily vary in diflFerent media, and that
in such a manner that for perfectly black bodies of the same
temperature we have always

a (g

If this equation be fiilfilled, then in the present case also, where
of the two surface-elements ds^ and ds^ one is the image of the
-1 other, the quantities of heat must be equal which they transmit

to each other, and hence, notwithstanding the concentration,
neither element can raise the other to a temperature higher than
its own.

V. Relation between the enlargement and the ratio of the aper-
tures of an elementary pencil of rays.

13. As a secondary result of the preceding investigation, I
may be allowed here to develope a proportion which appears to
me to possess general interest, inasmuch as it renders manifest
a peculiar difierence in the constitution of the pencil of rays at
the object and at its image, which difference must always pre-
sent itself in a definite manner whenever object and image are
of unequal magnitude.

If we consider the infinitesimal pencil of rays which, issuing
from a point of the element dsa, pass through the element dsb of
the intermediate plane and then combine in a point of the ele-
ment ds^y we may compare the divergence which these rays pre-
sent at the point of issue with their convergence at the point of
combination. This divergence and convergence, in other
words, the apertures of the infinitesimal cones, which the rays
form at the points of issue and combination, may be imme-
diately found by the same method as that which we employed
in Art. 8.

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 317

Around each of the points in question we conceive a spherical
surface to be described with a radius so small^ that the rays
within the sphere may be regarded as rectilinear, and we con-
sider the element which the pencil intercepts on the spherical
surface. Let this surface-element be denoted by rfo-, and let p
be the radius of the sphere ; then the aperture of the infinite-
simal cone formed by the rays, considered as rectilinear, will be

represented by the fraction —5-.

In the similar case treated in Art. 8 we determined this frac-
tion by the equation (20) ; and to obtain the expression appli-
cable to our present case, it will suffice to change slightly the
letters involved in the expression there found. To express the
aperture of the cone, whose vertex is at the point of the plane a
whence the rays issue, we have to put in place of the element
dscf and the quantity B of our former expression, the element
dsb and the quantity C. In order, moreover, to indicate more
distinctly that we are considering the cone whose vertex is in
the plane o, we wiU use the symbol ^a instead of ^ to denote the
angle between the elementary pencil and the normal to the
surface-element dSa ; and for a similar reason we will provide

the fraction — g-, which represents the required aperture with the

suffix a. We have then

i^r^.^"" «

In order to obtain the corresponding formula for the aperture
of the cone, whose vertex is at the point in the plane c where the
rays unite, we have merely to replace every suffix a in the pre-
ceding formula by c, and likewise the quantity C by A. The
result is

""' Ads, (28)

(P*A cosde

From these two equations we obtain the proportion
cosdg fd(r\ ^ cos ^c (^Z\ — r A

818 EIGHTH MSMOIll.

and if to this we apply the equation (26), we find

On introducing coefficients of refraction in place of yelocities of
propagation, this proportion becomes

nl cos ^ay^J' nl cos ^^ (^j =dSc : *«• . (80)

On the right of this proportion stands the ratio of the mag-
nitudes of a surface-element of the image, and the corresponding
surface-element of the object; in short, the superficial enlarge-
ment ; it furnishes, therefore, a simple relation between the en-
largement and the ratio of the apertures of an elementary conical
pencil of rays. It is moreover obvious that for the truth of the
proportions it is not absolutely necessary that the rays should be
ultimately converffent and actually intersect in a point ; on the
contrary, they may be divergent, so that their rectilinear back-
ward productions meet in a point and form a so-called virtual
image.

If, as a special case, we assume the medium to be the same
at the point whence the rays issue and where they combine ; if,
for example, the rays proceed from an object in air and, after
suffering any refractions and reflexions whatever, give an image
in air, really or virtually, then Va^Vc and na=ne, and we have

cos

*•(?); ^ *"(?)=*•=*-

If we further introduce the condition that the elementary pencil
of rays m to make equal angles with both surface-elements, for
instance, to be normal to each, the cosines will disappear, and
we shall have

(fl^Cf).^*"^*''

In this case, therefore, the apertures of the elementary conical
pencils of rays at the object and at the image are simply in-
versely proportional to the magnitudes of the corresponding
surface-elements of object and image.

In the clear and elaborate exposition of the laws of refraction

i

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 319

in systems of spheric surfaces^ which HeLnholtz has given in
his ^ Physiological Optics ^* preparatory to examining the refrac-
tions which take placean the eye, I find on p. 50, and expanded
on p. 54, an equation which expresses the relation between the
magnitude of the image and the convergence of the rays for the
case where the directions of rays are changed by refraction, or
even by reflexion in centred spheric surfaces, and where the rays
strike the planes containing object and image normally or ap-
proximately so. So far as I know, the relation has never beftH'e
been given with the generality which appertains to the propor-
tions (29) and (30).

VI. General determination of the mutual radiation between two
surfaces in which any concentrations whatever occur,

14. The investigation must now be generalized so as to em-
brace every imaginable case of ray-concentration, and not
merely the extreme case where all the rays, issuing &oma point
of the plane a within a certain finite conical angle, are again
collected in a point of the plane c, so that a <3onjugate focus
there ensues. To define more accurately the conception of con-
centration, we will introduce the following definition. If rays,
issuing fi-om any point j^^ fall on the plane c, and have in the
neighbourhood of this plane directions such that at any point
thereon the density of the incidait rays is infinite compared
with the mean density, we shall say that at that point occurs a
concentration of the rays issuing jfrom pa*

According to this definition we may easily make the concen-
tration of rays mathematically intelligiUe. Between the point
Pa and the plane c we take any intermediate plane d, so situated
that in it no concentration of the rays issuing from j^a occurs,
and also so related to the plane c tiliat, as far as our considera-
tions extend, the pencils of rays which proceed from points in
the one suffer no concentration in the other. We then con-
ceive an infinitesimal pencil of rays proceeding from pai and
compare the magnitudes of the surface-elements dsi and ds^
which it intercepts on the planes b and c. If on doing so we
find dsc to be infinitely small in comparison with e&j, so that we

* Allgemeine Encyklopadie der Physik, edited by G. Karsten.

320 EIGHTH MEMOIR.

may put

ds

*;=«' : («^)

the fact will indicate that a concentration of rays, in the above
sense^ takes place in the plane c.

Returning now to the equations (II) of Art. 6, of which equa-
tions those in the first horizontal row have reference to the pre-
sent case, we find that of the three fractions in that row, each of
which represents the ratio of the surface-elements, the first is
applicable to the case under consideration; since, according
to the hypothesis now made relative to the position of the
intermediate plane, the magnitudes A and E are determinable
in the ordinary way. We have, accordingly, the equation

(cfo,_E

In order that this fraction representing the ratio of the two ele-
ments may vanish, the numerator E must do so, since the deno-
minator A, in virtue of our hypothesis relative to the position
of the plane b, cannot be infinitely great. Hence the equation
of condition,

E=0, (32)

is the mathematical criterion for deciding whether the rays
issuing from the point pa do or do not sufier a concentration at
the particular point of the plane c ; in the case of a concentra-
tion this equation must be fulfilled.

If, on the contrary, we now suppose a point /?c to be given in
the plane c, and we inquire whether or not the rays issuing from
this point sufier a concentration at any place on the plane a, we
find in an exactly similar manner the condition

^=0-
and since, according to (II), we may put

dsj, C
we obtain the same condition,

E=0.

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 321

It is, in fact, manifest that when rays proceeding from a point
of the plane a suffer concentration in a point of the plane c, the
rays which issue from the latter point must likewise undergo
concentration in the first plane.

Having in the equations (12) and (13) expressed the relations
which exist between the six magnitudes A, B, C, D, E, P, we
may employ these equations to find what B, D, and P become
when E=0, and A and C have values different from zero. Ac-
cording to those equations, we have

^=^' ^=¥' ^=¥' • • • (33)

so that all three magnitudes become infinitely great in the case
under consideration.

15. We will now seek to determine the ratio of the quantities
of heat which two surfaces interchange by radiation, in such a
manner that the result shall always hold, no matter whether a
concentration of rays takes place or not.

For the sake of greater generality, let any two surfaces, s^ and
s^y be given in place of the two planes a and c. Between these two
surfaces we take a third, *j, subject only to the condition that the
rays which proceed from s^ to s^ or in the contrary direction,
suffer no concentration in *j. In s^ let any element ds^ be now
chosen, and let ds^ be an element in ^^ so situated that the rays
from ds^ which pass through ds^^ reach, when prolonged, the
surface s^. This done, we will next determine hi/w much heat
the element ds^ sends to and receives from the surface s^ through
the element ds^^ of the intermediate surface.

To find the first of these two quantities of heat, we have
merely to. determine how much heat the element ds^ sends to
the element ds^^ j for according to the hypothesis relative to the
position of the element ds^, the whole of this heat, after passing
through the element ds^, reaches the surface s^. The required
quantity of heat may be at once expressed by means of the
formulae previously developed. To do so, we conceive the
tangent plane to the surface s^ drawn at a point of the element
ds^y and likewise the tangent plane to s^ at a point of the

Y

822 EIGHTH MEMOIR.

element ds^^, and consider the given surface-elements as coin-
cident with elements of these planes. On introducing coordi-
nate systems x^, y^ and a?^ y^ into these tangent planes and
forming the magnitude C, as defined by the third of the equa-
tions (I), the required quantity of heat transmitted by the
element ds^ to the element ds^, and thence to the surface s^y will
be represented by the expression

IT

Passing next to the quantity of heat which the element ds^
receives from the surface 8^ through the element cfo^, it must be
observed, with reference to the points of the surface 8^ from
which the several rays proceed, that in general the simple
relation no longer holds here, as in the special case, where the
element d8^ had an optical image ds^ in the surface *^, and was
consequently itself the optical image of ds^. If we select any
definite point /?j of the intermediate element ds^ and conceive
rays passing through this point from all points of the element
ds^, we obtain an infinitely small pencil of rays which intersect
«^ in a certain surface-element. It is from this surface-element
that rays proceed to the element d8^ through the selected point
/?j. If, however, we were to choose any other point in cb^ as the
vertex of a pencil of rays, we should obtain a somewhat dif-
ferently situated element in the surface s^. Consequently the
rays which the element ds^ receives from the surface s^ through
the several points of the intermediate element, do not all
proceed from one and the same element of the surface 8^,

Nevertheless, the magnitude of the intermediate element ds^
being arbitrary, there is nothing to prevent us from making it
as small as we please, — ^in fact an infinitesimal of a higher order
than the given element d8^. This granted, the element of the
surface 8^ which corresponds to the element d8^ will change its
position so little when the vertex of the pencil of rays moves
within the element rfi^, that the differences will be infinitesimal
when compared with the dimensions of the element itself, and

CONCENTRATION OF RAYS OF LIGHT AND HEAT. 323

may accordingly be neglected. On this hypothesis, therefore,
we may consider the portion of the surface s^, which inter-
changes rays with the element ds^ through the element ds^^, to
be identical with the element ds^, which we obtain on selecting
any point j9j of the element ds^ as the vertex of a pencil of rays
proceeding from ds^.

The magnitude of this element ds^ can be easily expressed by
foregoing principles. We conceive, as before, the tangent plane
to the surface *j at the point jo^ to be drawn, as well as the
tangent plane to the surface s^, at a point of the element ds^, and
the tangent plane to s^ at a point of ds^; and we regard the two
last surface-elements as elements of the respective tangent
planes. On introducing systems of coordinates into the three
tangent planes and forming the magnitudes A and C, as defined
by the first and third of the equations (I), we have, by (II),

a ^ a

The quantity of heat which this element ds^ transmits to the
element ds^, and which, as above explained, may be regarded as
equal to the quantity received by the element ds^ from the sur-
face 8^ through the element dSj^, is represented by

^- V? — ds^ ds.y

c 0JP e 0'

or, on substituting for ds^ the value just given, by

e^ vl — ds^ ds..

IT

On comparing this expression with the one previously found
for the quantity of heat, which the element ds^ transmits
through d$^ to the surface s^, it will be seen that the two
quantities have to each other the ratio

a a e e

Now if we suppose s^ and s^ to be surfaces of two perfectly
black bodies of the same temperature, and assume that for
such surfaces the two products e^vl and e^v^ are equal — an

y2

3

824 EIGHTH MEMOIR.

assumption which was shown to be necessary in the case of
radiation without concentration — ^the quantities of heat repre-
sented by the two. expressions under consideration will likewise
be equal to one another.

• 16. If we take a diflferent element in the intermediate sur-
face «j, again considered as an infinitesimal of higher order, the
element of the surface s^, which interchanges rays, through it,
with the element ds^, will have a different position ; but the two
interchanged quantities of heat will again be equal to one
another, and the same will be true for all other elements of the
intermediate surface.

To obtain the total quantity of heat which the element ds^
transmits to the surface 8^, and likewise the total quantity which
it receives in return, the two expressions above found must be
integrated relative to the intermediate surface s^, and the
integral extended to that portion of this surface which is en-
countered by the rays which proceed from the element ds^ to
the surface s^, and vice versd. It is manifest that the two in-
tegrals wiU be equal to one another, since for each element ds^
the two differential expressions are equal.

Lastly, the quantity of heat which the entire surface s^ inter-
changes with the surface s^ wiU be obtained by the further in-
tegration of the two expressions relative to the surface s^ and
here again the equality which exists for the several elements
ds^ wiU not be disturbed.

The theorem above established for more special cases, in virtue
of. which two perfectly black bodies of the same temperature
interchange equal quantities of heat, provided the equation
^t?a^^?e ^® applicable to them, also presents itself as the result
of considerations altogether independent of the circumstance
whether or not the rays proceeding from s^ to 5^, or from s^ to s^
suffer concentration. The only condition imposed was, that the
rays proceeding from s^ and s^ suffer no concentration in the in-
termediate surface s^^ and this condition may always be fulfilled,
inasmuch as the intermediate surface may be arbitrarily chosen.

From this result it further follows, of course, that when re-
ciprocal action takes place between a given black body and any

J

CONCENTRATION OF RAYS OP LIGHT AND HEAT. 325

number of other black bodies of the same temperature, the for-
mer receives from all the latter, put together, exactly as much
heat as it transmits to them.

17. The preceding developments were made on the supposi-
tion that the reflections and refractions therein involved occur
vrithout loss, and that no absorption takes place. It may easily
be shown, however, that the final result remains unchanged, even
when this condition is relinquished. To this end let us consider
for a moment the different ways in which a ray may become
weakened during its passage from one body to another. On
reaching the dividing surface of two media, one portion may
suffer reflection, and the other, after refraction, pursue its
course in the adjacent medium. Whichever of these two por-
tions, however, we consider as the prolongation of the original
ray, we shall have a weaker ray. Again, a ray on passing
through a medium may suffer absorption. Now in each of these
cases the law holds that the weakening effect is proportionally
the same for two rays propagated in opposite directions along
the same path. The effect of such processes will be, therefore,
to diminish to the same degree both the quantities of heat which
two bodies mutually transmit to each other, so that if these quan-
tities were equal before the weakening occurred, they will re-
main equal afterwards.

Another circumstance to be mentioned in connexion with the
above-named processes is, that a body may receive like-directed
rays proceeding from different bodies. For instance, from a
point situated in the dividing surface of two media, a body. A,
may receive two rays coincident in direction, but proceeding
from two different bodies, B and C ; and of these rays one may
have come from the adjacent medium, after having suffered re-
fraction at the point, whilst the other may have remained in the
same medium, and have suffered reflection merely. In this case,
however, the two rays must have been weakened by refraction
and reflection in such a manner that, if originally equal in in-
tensity, the sum of their intensities must afterwards be equal to
the original intensity of each. If we now conceive an equally
intense ray to proceed in an opposite direction from the body A,
it will, at the same point, be divided into two parts, one of which
will enter the adjacent medium and proceed to the body*B,
whilst the other will be reflected to the body C. The two parts

326 EIOBTH MBMOIR.

which in this manner reach B and C from A will be just as
great as the parts of rays which A receives from B and C.
Under the hypothesis of equal temperatures, therefore, the rela-
tion between the body A and the bodies B and C is such, that
it exchanges equal quantities of heat with each. In consequence
of the equality of the actions on two rays which pursue, in
opposite directions, any path whatever, the above relations must
hold in all other cases, however complicated.

Further, if, instead of perfectly black bodies, we also consider
bodies which only partially absorb incident rays, or if, instead
of homogenous heat, we consider heat due to waves of different
lengths, or lastly, if, instead of regarding all rays as unpolarized,
we have regard to the phenomena of polarization, we shall in all
cases be concerned solely with circumstances which affect, to
the same extent, the heat emitted by a body, and the heat
which it receives from other bodies.

It is not necessary to consider here all these circumstances,
for they occur also in ordinary radiation without concentration,
and the sole object of the present memoir was to investigate the
effects which may possibly arise from the concentration of rays. j

18. THie following is a brief summary of the principal results '

of the foregoing considerations : — j

1. To harmonize the effects of ordinary radiation, without '
concentration, with the principle that heat cannot of itself pass
from a colder to a warmer body, it is necessary to assume that j
the intensity of emission from any body depends not only upon

its own constitution and its temperature, but also upon the
nature of the surrounding medium ; in fact the intensities of
emission of one and the same body in different media must be
inversely proportional to the squares of the velocities with which
rays are therein propagated, or, in other words, directly propor- .

tional to the squares of the coefficients of refraction for these |

media. '

2. If this assumption as to the influence of the surrounding
medium be correct, the above principle must obtain, not only
for ordinary radiation, but also when the rays become concen-
trated in anymannerwhatever through reflections and refractions; '
for although concentration may change the absolute magnitudes j
of the quantities of heat which two bodies radiate to each other, ^^
it cannot alter the ratio of these magnitudes. i

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 327

NINTH MEMOIR.

ON SEVERAL CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS
OF THE MECHANICAL THEORY OF HEAT*.

In my former Memoirs on the Mechanical Theory of Heat,
my chief object was to secure a firm basis for the theory,
and I especially endeavoured to bring the second fundamental
theorem, which is much more difficult to understand than the
first, to its simplest and at the same time most general form,
and to prove the necessary truth thereof. I have pursued special
applications so far only as they appeared to me to be either appro-
priate as examples elucidating the exposition, or to be of some
particular interest in practice.

The more the mechanical theory of heat is acknowledged to
be correct in its principles, the more frequently endeavours are
made in physical and mechanical circles to apply it to diflferent
kinds of phenomena, and as the corresponding differential equa-
tions must be somewhat differently treated from the ordinarily
occurring differential equations of similar forms, difficulties of
calculation are frequently encountered which retard progress
and occasion errors. Under these circumstances I believe I
shaU render a service to physicists and mechanicians by bring-
ing the fundamental equations of the mechanical theory of heat
from their most general forms to others which, corresponding to
special suppositions and being susceptible of direct application
to different particular cases, are accordingly more convenient
for use.

1. The whole mechanical theory of heat rests on two ftinda-
mental theorems, — ^that of the equivalence of heat and work,
and that of the equivalence of transformations.

In order to express the first theorem analytically, let us

* Eead at the Philosophical Society of Zurich on the 24th of April, 1865,
published in the Vierteljahrsschrift of this Society, Bd. x. S. 1. j Pogg. Ann.
July, 1865, Bd. cxxv. S. 353 ; Journ. de Liouville, 2^ s^r. t. x. p. 361.

■ I

z'

NINTH MEMOIR.

contemplate any body which changes its condition^ and consider
the quantity of heat which must be imparted to it during the
change. If we denote this quantity of heat by Q^ a quantity of
heat given oflF by the body being reckoned as a negative
quantity of heat absorbed^ then the following equation holds
for the element dQ of heat absorbed during an infinitesimal
change of condition,

rfQ=rfU + ArfW (I)

Here U denotes the magnitude which I first introduced into the
theory of heat in my memoir of 1850, and defined as the sum of
the free heat present in the body, and of that consumed by interior
work*. Since then, however, W. Thomson has proposed the
term enerffy of the body for this magnitude t, which mode of
designation I have adopted as one very appropriately chosen ;
nevertheless, in all cases where the two elements comprised in
U require to be separately indicated, we may also retain the
phrase thermal and ergonal content, which, as already explained
on p. 255, expresses my original definition of U in a rather
simpler manner. W denotes the exterior work done during the
change of condition of the body, and A the quantity of heat
equivalent to the unit of work, or more briefly, the thermal
equivalent of work. According to this AW is the exterior work
expressed in. thermal units, or according to a more convenient
terminology recently proposed by me, the exterior erg(m. (See
Appendix A. to Sixth Memoir.)

If, for the sake of brevity, we denote the exterior ergon by a
simple letter,

w=AW,

we can write the foregoing equation as follows,

dQL-d\]-\-dw (la)

In order to express analytically the second fundamental
theorem in the simplest maimer, let us assume that the changes
which the body suflfers constitute a cyclical process, whereby
the body returns finally to its initial condition. By dO, we will
again understand an element of heat absorbed, and T shall

* Pogg. Arm, Bd. Ixxix. S. 385, and p, 29 of this collection.
t PhU. Mag. S. 4. vol. ix. p. 623.

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 329

denote the temperature, counted firom the absolute zero, which
the body has at the moment of absorption, or, if different
parts of the body have different temperatures, the temperature
of the part which absorbs the heat element dQ. If we divide
the thermal element by the corresponding absolute temperature
and integrate the resulting differential expression over the
whole cyclical process, then for the integral so formed the
relation

'^±0 (II)

J

holds, in which the sign of equality is to be used in cases where
aU changes of which the cyclical process consists are reversible,
whilst the sign <: applies to cases where the changes occur in a
non-reversible manner^.

2. We will first consider more closely the magnitudes occur-
ring in equation (I a) in reference to different kinds of changes
of the body.

The exterior ergon w, which is produced whilst the body
passes from a given initial condition to another definite one,
depends not merely on the initial and final conditions, but
also on the nature of the transition.

In the first place, we have to consider the exterior forces which
act on the body, and which are either overcome by, or overcome
the forces of the body itself; — the exterior ergon being positive
in the former, and negative in the latter case. The question

* In my memoir " On a Modified Fonn of the Second Fundamental Theorem
of the Mechanical Theory of Heat" (Fourth Memoir of this collection), in
which I first gave the most general expression of the Second Fundamental
Theorem for a Cyclical Process, the signs of the differentials dQ, were
differently chosen ; there a thermal element given up by a changing body
to a reservoir of heat is reckoned positive, an element withdrawn from a
reservoir of heat is reckoned negative. "With this choice of signs, which in
certain general theoretical considerations is convenient, we have to write
instead of (II),

In the present memoir, however, the choice mentioned in the text is every-
where retained, according to which a quantity of heat absorbed by a changing
body is positive, oad a quantity given off by it is negative.

330 NINTH MEMOIB.

then arises^ are these exterior forces^ at each moment^ the same
as, or different from the forces of the body ? Now although we
may assert that for one force to overcome another, the former
must necessarily be the greater ; yet since the difference between
them may be as small as we please, we may consider the case
where absolute equality exists as the limiting case, which,
although never reached in reality, must be theoretically con-
sidered as possible. When force and counter-force are different,
the mode in which the change occurs is not a reversible one.

In the second place, the change taking place in a reversible
manner, the exterior ergon likewise depends upon the inter-
mediate conditions through which the body passes when chang-
ing from the initial to the final condition, or, as it may be figu-
ratively expressed, upon the path which the body pursues when
passing from its initial to its final condition.

With the energy U of the body whose element, as well as
that of the exterior ergon, enters into the equation (I a), it
is quite different. If the initial and final conditions of the
body are given, the variation in energy is completely deter-
mined, without any knowledge of the way in which the trans-
ition from the one condition to the other took place — ^in fact
neither the nature of the passage nor the circumstance of its
being made in a reversible or non-reversible manner, has any
influence on the contemporaneous change of energy. If, there-
fore, the initial condition and the corresponding value of the
energy be supposed to be given, we may say that the energy is
fully defined by the actually existing condition of the body.

Finally, since the heat Q which is absorbed by the body
during the change of condition is the sum of the change of
energy and of the exterior ergon produced, it must like the
latter depend upon the way in which the transition of the body
from one condition to another takes place.

Now in order to limit the field of our immediate investigation,
we shall always assume, unless the contrary is expressly stated,
that we have to do with reversible changes solely.

The equation (I a) which expresses the first fundamental
theorem, holds for reversible as well as for non-reversible
changes; hence, in order to apply it specially to reversible

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 881

changes^ we have not to modify it externally in any manner^
but merely to understand that by w and Q are meant the ex-
terior ergon and quantity of heat which correspond to reversible
changes.

On applying to reversible changes the relation (II) which
expresses the second fiindamental theorem, we have not only
to understand by Q the quantity of heat which relates to
reversible changes, but also, instead of the double sign <, we
have simply to employ the sign of equality. We obtain for all
reversible cyclical processes, therefore, the equation

J¥=o.

(II a)

3. In order to be able to calculate with the equations (I a)
and (II a), we will assume that the condition of the body under
consideration is defined by any appropriate magnitudes what-
ever ; the cases which most frequently occur are those where the
condition of the body is either defined by its temperature and
volume, or by its temperature and the pressure to which it is
exposed, or finally by its volume and pressure. We will not at^
present, however, confine ourselves to any particular magnitudes,
but assume that the condition of the body is defined by any two
magnitudes, say x and y ; these magnitudes will be considered
in the calculations as independent variables. In special applica-
tions, we are of course always at liberty to identify one or both of
these variables with any one or two of the above-named magni-
tudes, temperature, volume, and pressure.

If the magnitudes a: and y define the condition of the body,
then its energy U, which depends only on the instantaneous
condition of the body, must admit of being represented as a
function of these two variables. It is otherwise with the mag-
nitudes w and Q. The differential coefficients of these magni-
tudes, which we will denote in the following maimer,

dw dw ,_.

§=»■. ?=N. (»)

are definite functions of x and y. For if we assume that

332 NINTH MEMOIR.

the variable x becomes x-^-dw^ whilst y remains unchanged,
and that this change of condition of the body takes place
in a reversible manner, a perfectly definite process is assumed,
and hence the exterior work thereby done is defined, whence

it follows, further, that the function -j- must likewise have a

ax

definite value. The same holds good if we assume that y

changes to y + dy, whilst x remains constant. But if the

differential coefficients of the exterior ergon to are definite

functions of x and y, it follows from equation (la) that

the diflPerential coefficients of the heat Q which is absorbed by

the body, must likewise be definite functions of x and y.

If we now form for dw and dQ their expressions in dx and dy

and, neglecting terms of a higher order than the first in dx and

dy, write

dw = mdx-\'ndyy (3)

dQ=Mdx+l^dy (4)

we obtain two complete diflferential equations, which cannot be
integrated so long as the variables x and y are independent of
one another, since the magnitudes m, », and M, N do not satisfy
the conditions of integrabUity : —

dm^dn , dM^cTN

dy ^ dx dy ^ dx'

The magnitudes w and Q thus belong to those which were
discussed in the mathematical introduction to this collection of
Memoirs, whose peculiarity consists in the fact that, although
their diflferential coefficients are definite functions of both inde-
pendent variables, they themselves cannot be represented by such
ftinctions, but only become defined when a further relation is
given between the variables, and by that means the sequence of,
or path pursued during the changes is prescribed.

4. If we return to equation (I a), and put therein for dw and
dCL the expressions (3) and (4), and also separate d\J into the
two parts which have reference to dx and dy, we obtain the
equation

Udx-\-^dy=(^ +^jrf^ + ^_ ^njdy.
Since this equation must hold for all values of dx and dy, it

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS.

may be resolved into the following two : —

If we differentiate the first of these equations with respect to y,
and the second with respect to x, we obtain

rfM^ cPV dm
dy "^ dxdy dy'

dN^ d^V dn
da: dy da? dx

Now to U may be applied the well-known theorem, according
to which when any function whatever is differentiated succes-
sively according to each of two variables, the order in which
the differentiations are made has no influence on the result;
accordingly

dx dy"^ dy dx

If, having regard to this last equation, we subtract the second of
the two former equations from the first, we get

flfM__rfN__rfw dn ^gv

dy dx "" dy ^ dx

In order to treat equation (II a) in a similar manner, let us
substitute in it for rfQ its value given in (4), we thus obtain

J(^^+N^y)=0.

If the integral standing on the left-hand side vanish as often as
X and y resume their original values, the expression under the
integral sign must be a perfect differential of a function of x
and y, and hence the above-named condition of integrability
must be fulfilled ; in the present case this condition is

On performing the differentiations, and remembering that the

884 NINTH UKHOIB.

temperature T of the body is also to be considered as a function

of X and y, we obtain

1 rfM_M dT_l rfN_N dT
T'rfy T«'dy~T'<te T«*rfr'

or differently arranged^

We will give to the equations (5) and (6) thus obtained^ a
somewhat different form. In order not to have too many different
letters in the formulae^ we will in future replace M and N,
which were introduced as symbols for the differential coefficient

-J- and --T-i by the differential coefficients themselves. We

farther observe that the difference on the right-hand side of (5),

which on replacing m and n by the differential coefficients -r-

and -7- becomes
dy

— f^\ ^ (dw\

dy\dx) dx\dyJ'

is a function of x and y, which may usually be regarded as
known, inasmuch as the forces operating on the body externally
being susceptible of direct observation, the exterior ergon can
be determined. We shall call this difference, which frequently
occurs in the following pages, the ergonal difference corresponding
to xy, and introduce for it a particular symbol by putting

^'9'=^dif\l^Jdx\dhf) ^^^

In consequence of this change in the notation, the equations (5)
and (6) become

dy\dx) dxKdyJ^^'v' ^^^

dyxdx) dx\dyJ'~T\dy dx dx dy;

(9)

These two equations form, for reversible changes, the analytical
expressions of the two fundamental theorems in the case where

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 835

the condition of the body is defined by any two variables. From
them a third, and simpler equation at once results, inasmuch as
it contains only the differential coeflicients of Q of the first
order, namely,

^.^_f,da

dy da; da? ay ^ ^ '

5. The three foregoing equations are particularly simple,
when we select the temperature of the body as one of the in-
dependent variables. To this end we will put y=T, so that
now the undetermined magnitude x and the temperature T are
the two independent variables. When y=T, it follows that

rfT_
dy^

dT
Moreover, with respect to the differential coefficient -r-, we

supposed on forming it that, whilst a? changes to a^+dx, the
other variable, hitherto called y, remains constant. Since at
present T itself is the other variable, which is supposed to be
constant in the differential coefficient, we have to put

dT ^
dx

The ergonal -difference corresponding to xT is now

jj, d /dw\ d (dw\ f .

^«T"5TW""^l5Ty' • • • • (11)

and on hitroducing this value the equations (8), (9) and (10)
become

lihuo)~di{l^)^'^'^" • • • • (12)

Ji(d^\_±(dQi\_l dQ ,

dT\dxJ dp\dT)~T' ds' • ' • (^**)

S=TE^ (1^)

If we introduce the product TE^j, given in (14), in place of the
differential coefficient -^ in equation (12), and differentiate

836 NINTH MEMOIR.

it, as there prescribed, according to T, we obtain the following
simple equation,

d_/dQ\ dE^

dx\dT)'^^dT ^ ^

6. We have hitherto made no special assumptions concerning
the exterior forces to which the body is subjected, and to which
the exterior ergon produced during a change of condition refers.
We will now consider more closely a case which is of very
frequent occurrence, namely, that where the only external
force, or at least the only one which is important enough to
merit notice in the calculations, is a pressure acting on the
surface of the body with equal intensity at all points, and which
is everywhere directed perpendicularly to that surface.

In this case exterior ergon is produced solely in consequence
of changes of volume in the body. If we call p the pressure on
a unit of surface, the exterior work which is done, when the
volume V increases by rfv, is

dW=pdv,

and accordingly the exterior erffon, that is to say, the exterior
work measured by thermal units, is

dwssApdv (16)

If we conceive the condition of the body to be defined by any
two variables x and y, the pressure p and the volume v must be
considered as functions of x and y. We can consequently write
the foregoing equation in the following form.

from which it follows that

dx" ^dx'

dw^ . dv
dy'^ ^dy'

(17)

Introducing these values of -r- and --r- into the expression for
E^ given in (7), performing the two differentiations therein

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 837

indicated^ and remembering at the same time that , \ =

da^dy

dy cue'

^^"^^fe'^"^^'^) ^^^^

We have to employ this value of E^ in the equations (8) and
(10).
If x and T be the two independent variables^ we obtain

=«=^(*-£-i-s> ■ • ■ w

which quite corresponds to the preceding equation. This value
has to be employed in the equations (12), (14), and (15).

The expression (18) acquires very simple forms when we choose
either the volume or the pressure as one of the independent
variables, and when we make the volume and the pressure the
two independent variables. In these cases, as may be easily seen,
the equation (18) becomes transformed, respectively, into

E.=a|> (20)

E.= -a|^ (21)

E^=A (22)

If, finally, in those cases where either the volume or the
pressure is taken as one of the independent variables, we wish
to make the temperature the other independent variable, we
need only replace y by T in equations (20) and (21).

7. Under the circumstances before alluded to, when the only
existing exterior force is a uniform and normal surface-pressure,
the independent variables most frequently used for defining the
condition of the body, are the magnitudes just mentioned, that
is to say, volume and temperature, or pressure and temperature,
or finally, volume and pressure. Although the systems of dif-
ferential equations answering to these three cases could easily be
deduced from the above general systems, yet on account of their
frequent application I lyill here place them side by side.

The first system is the one which I have always employed in
my memoirs when special cases were under consideration.

z

SfS

NINVH MEMOIK.

When V and T arb chosen as independent variables,

dTKdvJ dv\dT:J~^(fl'

_rf/rfaN d/dQ> 1 dO,
dT\do) dv\dT)~^"^'

dv --^^dT'

When p and T are chosen as independent variables,

di\dpJ dpKcm)^ dT'

^f^\ -^(^^ -1^
di:\dpJ dp\dTJ~Tdp'

(23)

When V and /? are taken as the independent variables,
dp\dv ) dv\dp) *

dp\dv) dv\dp J "^ T\dp dv dv dpJ^

dT dQ^dT dQ
dp dv dv dp

(24)

rAT.

(25)

8. The simplest of the eases to which the equations of the
preceding Article are applicable, is that where a homogeneous
body is given of absolutely uniform temperature, which is ex-
posed to a uniform and normal surface-pressure, and can, by a
change of temperature and of pressure, change its volume with-
out at the same time changing its state of aggregation.

In this case the differential coefficient -^ has a simple

physical meaning. For if we suppose the weight of the body to
be equal to unity, this differential coefficient denotes the specific

CONVElflENT FOBMS OF THE FUNDAMENTAL EQUATIONS. 339

heat at constant yolume^ or the specific heat at constant pressure^
accb^^ding as, when forming it, the volame or the pressure was
supposed to be constant.

In cases where the nature of the subject requires that
the independent variables should often be changed, and hence
where diflferential coefficients occur which only differ from one
another in that the magnitude which was supposed to be con-
stant during differentiation is not the same in all, it is con-
venient to denote this difference by an outward mark, in order
to avoid the necessity of repeated verbal explanations. I shall
do this by enclosing the differential coefficients in brackets, and
adding, in the form of a suffix with a bar over it, the magnitude
which is supposed to be constant during differentiation. Accord-
ingly, we will write the two differential coefficients which denote
the specific heat at constant volume, and the specific heat at con-
stant pressure, in the following manner : — •

/dQ\ , /dQ\

{dTjf ^^ W-

Further, of the three magnitudes, temperature, volume, and
pressure, which in our present case come under consideration
when determining the condition of the body, each is to be re-
garded as a fiinction of the other two, hence the following six
differential coefficients may be formed : —

/_^\ /^\ fdv\ fdv\ /rfT\ /rfT\
VdTjf UvJf' W^ \dpJf' \dv)p' \dpk

The suffixes, which here indicate the magnitude which in each
differentiation is supposed constant, might be omitted provided
we agreed, once for all, that of the three magnitudes T, v, and p,
the one which does not occur in the differential coefficients is
to be considered as constant. Nevertheless, for the sake of
clearness, and because differential coefficients of the same magni-
tudes occur in which the quantity supposed to be constant is
not the same as at present, we will, at least in the following
equations, retain the suffixes.

It will facilitate the calculations to be made with these six
differential coefficients, if we first determine the relations ex-
isting between them.

In the first place, it is clear that amongst the six differential

z2

340 NINTH MEMOIR.

coefficients there are three pairs of reciprocals. For example^
if we assume the magnitude v to be constant^ the? dependence
between the two other magnitudes T and p is such that each of
them may be regarded simply as a function of the other. . It is
the same with T and v ifp be assiuned to be constant^ and with
V and/> when T is supposed to be invariable. Consequently we
must put

\dpJi \dv Jp KdvJf

In order to obtain the relation between the three pairs of
differential coefficients^ we will by way of example consider p
as a function of T and v. Then we have the complete differ-
ential equation

If, now, we apply this equation to the case where/? is constant,
we have to put

rfp=0 and ^«^=(^)-^>

whereby it is transformed into

If we remove dT and then divide by (^ ).j we obtain

(l),(S)-©r-' w

With the help of this equation, in conjunction with th^ equa-
tions (26), we can represent each of the six differential coeffi-
cients as a pl*oduct or quotient of two other differential coeffi-
cients.

9. To return to the consideration of the absorption and pro-
duction of heat by the given body, let us denote the specific
heat at constant volume by c, and the specific heat at constant
pressure by C ; then, if we assume the weight of the body to be
unity, we must put

/rfQ\ /rfQ\ ^

J~ '

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 341

Further, according to equations (28) and (24),

SO that we can form the following complete differential equa-
tions,

rfQ=C(fr+AT(^)>, (28)

rfQ=CdT-AT(^).dp (29)

By comparing these two expressions of rfQ, we may at once
deduce the relation which exists between the two specific heats
c and C. In fact, from the last equation, which has reference
to T and j9 as independent variables, we can deduce an equation
wherein the independent variables are T and v. To do this,
we need only consider j9 as a function of T and t?, and ac-
cordingly vrrite

*=(!!

■"HIX*-

By the introduction of this value of ^ in equation (29) it is
transformed into

If in place of the product of two differential coefficients in the
last term of the above, we put by help of equation (27) a simple
differential coefficient, we get

On comparing this expression for rfCl with that given in (28), and
considering that the coefficients of rfT in both expressions must be
equal, we obtain the following equation expressing the relation
between the two specific heats,

''^-^\%0-. m

The differential coefficient (^), which here occurs repre-
sents the expansion of the body caused by an increase in its

342 NINTH MBMOIE.

temperature^ and is to be ocniBidered as known. The other
differential coefficient^ v^)-' ^ ^^*^ ^* ^ *^^' nsually given

directly by observation in the case of solid and liquid bodies^
but according to (27) we may put

KdTj-,

i

flfe\
rfTA

r-V

and the differential coefficient in the numerator of this fraction
is again the one before referred to, whilst the differential co-
efficient in the denominator represents, when taken with a
negative sign, the diminution of volume through increase of
pressure or the compressibility ; for a number of liquids this has
been measured, and for solid bodies it can be approximately esti-
mated fipom the coefficients of elasticity. By the introduction
of this fraction equation (30) becomes

*=c+^W (81)

Kctp/T

On using this equation for numerical calculations, it must be
remembered that the unit of volume in the differential co-
efficients is the cube of that unit of length which was employed
in the determination of the magnitude A; and the unit of
pressure is the pressure which a unit of weight exerts when
spread over a unit of surface. Hence we have to reduce to
these units the coefficients of expansion and the coefficients of
compressibility whenever the latter have been referred to other
imits, as is usually the case.

Since the differential coefficient (^ )_ is always negative, it

follows that the specific heat at constant volume must always
be less than that at constant pressiu-e. The other differential

coefficient f ^)_ is generally a positive magnitude. For water,

however, at the temperature of maximum of density it vanishes,
and accordingly the two specific heats at this temperature are

CONVBNIBNT FOBMS OF THE fUNDAMENTlL EQUATIONS. 843

equal. At all other temperatures^ under or oirer the tempera-
ture of maximum density^ the specific heat at constant volume is
less than that at constant pressure ; for even if the differential

coefficient ( jm ). have a negative value below this temperature^

the signs in the formula are not affected thereby, since the square
of this differential coefficient is alone involved therein"*^.

From the equations (28) and (29) we can easily deduce a
complete differential equation for Q, having reference to p and
V as independent variables. To do this^ we need only consider

* To give an example of the application of equation (31), we will consider
water at a few definite temperatures, and calculate the difference between
the two specific heats.

According to the observations of Kopp, whose results are tabulated in the
Lehrbuch der Fhys. und Theor. Chemie, S. 204, we have for water, if its volume
at 4^ be taken as unit, the following coefficients of expansion

at b ^0-000061,

at 25 ... . +0-00025,
at 50 ... . +0-00045.

According to the observations of Grassi (Ann, de Chim. et de Phys, 3 s^r.
t. xxxi. p. 437, and Kronig's Joum.fUr Physik dee AuslandeSj Bd. ii. S. 129),
we have for the compressibility of water the following numbers, which give
the diminution of volume, caused by an increase of pressure equal to an
atmosphere, expressed as a fraction of the volume corresponding to the
original pressure : — ©

at . . . . 0-000050,

at 25 ... . 0-000046,

at 50 ... . 0-000044.

We will now by way of example go through the calculation for the tern*
perature of 26°.

Ati unit of length we will choose the metre, and as unit of weight the
kilogramme. We then have to assume a cubic metre as the unit of volume,
and since the volume of a kilogramme of water at 4P is 0*001 cubic metre,

we must, in order to obtain (-^)-, multiply the above-given coefficient of

expansion with 0-001, hence

(^)_=0-00000025=25 . lO"®.

In accordance with the foregoing, the unit of volume, in the case of com-
pressibility, is that which water occupies at the temperature under con-
sideration and at the original pressure, which we may assume to be the
ordinary pressure of one atmosphere. This volume is, at 25°, equal to 0-001003
cubic metre. Moreover, one atmosphere is taken as the unit of pressure,
whilst we must take for our unit the pressure of a kilogranmie upon a square

344 NINTH MEMOIR.

T as a function oip and v, and accordingly put

If we substitute this value in equation (29) for rfT, we have

The difference, standing between square brackets in the last
expression, is by (30) equal to c, hence we can write the equa-
tion thus :

'^=<S)>+t'(S>/''- • ■ «

10. The three complete differential equations (28), (29), and

metre, whereby one atmosphere of pressure will be represented by 10338.
Accordingly we have

/^\ 0000046 . 0-001003 _ _^ ,q-i»

\rfp/f"~ 10333

Besides this we have, at 25°, to put

T=: 273+26=208,
and, with Joule, we will assume

• A— ^

These numerical values introduced into equation (31) give

424 45 .10-1*
In the same manner the following numbers result from the above values
of the coefficients of expansion and of compressibility at 0° and 50^;

at 5 C-r=0O006,

at 50 .... C-c=0-0358.

If for the specific heat C at constant pressure we use the values found
experimentally by Regnault, we obtain for the two specific heats the follow-
ing pairs of numbers :-

!

0-9995;

at 0^1^=1
I c=0

^,250 1 0=1-0016,
1 c=0-9918;

at 50° 1^ = 1-^2,
I c =0-9684.

J

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 345

(32) do not fulfil the condition of immediate integrabUity ; this
is at once manifest, as far as the two first are concerned, from
the equations already given. In fact, if we introduce into the
equations, which in the systems (23) and (24) stand lowest, the
letters c and C, they become

(f ).=-(sv I

whilst the equations which must be fulfilled if (28) and (29)
are integrable, are

In a similar, though somewhat more tedious manner, it may be
proved that the equation (32) is not integrable ; this, however,
is manifest from the circumstance that (32) is deduced from
(28) and (29) . The three equations belong, therefore, to those
complete diflferential equations which are described in the In-
troduction to the present collection of Memoirs, and which can
only be integrated when another relation between the variables
is given, and by that means the sequence of the changes is pre-
scribed.

Among the manifold applications which can be made of the
equations (28), (29), and (32), I will here only adduce one by
way of example. Let us assume that the body changes its
volume in a reversible manner through a change of pressure,
without thereby losing or acquiring heat. We will determine
the change of volume which, under these circumstances, is caused
by a given change of pressure, and in what manner the tempe-
rature simultaneously changes, or more generally, what rela-
tions, under these circumstances, exist between temperature,
volume, and pressure.

We obtain these relations immediately by putting JCI=0
in the three above-named equations. Equation (28) gives

cdT+AT(|f).ci«=0.

346 NIITTH MKMOIB.

If we divide the terms of this equation by dv, the fraction -j-

thereby resulting is^ for this particular case^ the differential co-
efficient of T according to v, which we will distinguish from
other differential coefficients of T according to t; by giving to
it the suffix Q. Hence we obtain

o,-^{%\ <«)

In a similar manner we obtain from equation (29)

(f),=f (^v (»)

From the equation (32) we obtain^ in the first place^

\dv)-,
for which, according to (27), we may write

(a=c(ix •'^

On introducii^ into this equation the value of c as given in (31)^
it becomes

(^Js^^C^Jx + 'cUJ^-- • • • (37)

11. When the equations of the two foregoing Articles are
applied to a perfect gas, they assume still more definite, and at
the same time very simple forms.

In this case the laws of Mariotte and Gay-Lussac give the
following relation between T, v, and p,

/?v=RT, (38)

wherein B is a constant. From this it follows that

(39)

W;~»' \dTj-p~p'

On combining the two last equations with the equations (33),

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 847

we obtain

©,=<" Or" («)

From this it follows that for a perfect gas the two specific heats
c and C can only be functions of the temperature. On other
grounds, which depend on special considerations into which I
will not now enter, it may be shown that the two specific heats
are independent of the temperature, and consequently are con-
stant; results which, with respect to specific heat at constant
pressure, have been corroborated by the experimental researches
of Begnault on permanent gases.

If we apply the two first of equations (39) to equation (30),
which gives the relation between the two specific heats, we
obtain the expression

— A — •— ^
p V

which by (38) becomes

c=C-AR . (41)

On applying the first two of equations (39), the equations (28),
(29), and (32) assume the following forms :

rfQ=crfT+AE-rft?,
rfQ=CdT-AR-^,
dQ^^vdp+^peh,

(42)

wherein, moreover, the product AB» can, in virtue of (41), be
replaced by the difierence C— c. As in the First Memoir of
this collection, and in the Appendix B to the Sixth Memoir,
several examples have already been given of the applications of
these equations, I will not here enter into further details.

12. Another case, which on account of its frequent applica-
tions is of particular interest, is that where with the changes of
condition of the body a partial change of the state of aggregation
is associated.

We wiU assume that a body is given, of which one part is in
one, and the remaining part in another state of aggregation.
For instance, one portion of the body may be conceived to be in

348 NINTH MEMOIK.

a liquid^ and the rest in a yaporons state (the density of the
latter being that which the vapoor naturally assnmes when in
contact with the liquid), or one part of the body may be in a
solid and the other in a liquid state, or one part in a solid and
the other in a vaporous state. Accordingly we will, for the sake
of generality, abstain from explicitly defining the two states of
aggregation with which we are to be concerned, and simply refer
to them as the first and second states of aggregation.

Conceive then a certain quantity of matter to be enclosed
in a vessel of a given volume, and one portion of it to be in
the first and the other in the second state of aggr^ation. If
the specific volumes (volumes referred to the unit of weight)
which the substance occupies at a given temperature in the two
states of aggregation be unequal, the two portions existing in
different states of aggregation in a given space must necessarily
have quite definite magnitudes. For when the portion which
is in the state of aggregation of greater specific volume increases
in magnitude, the pressure which the enclosed substance exerts
on the containing walls, and with it the reaction of the latter
on the substance, must likewise increase, so that a point wiU
ultimately be reached when the pressure is so great as to prevent
all fiirther passage into this state of aggregation. When this
point is reached, the magnitudes of the portions present in the
two states of aggregation cannot change further, so long as the
temperature of the mass and its volume (». e. the volume of the
vessel) remain constant. But if, whilst the temperature remains
constant, the volume of the vessel increase, the portion which is
in the state of aggregation corresponding to the greater specific
volume may increase still further at the expense of the other,
imtil the same pressure as before is again reached, and all
further change again prevented.

Hence follows a peculiarity which distinguishes this case from
all others. For if we take the temperature and the volume of
the mass as the two independent variables by which its condition
is defined, then the pressure is not a function of both these vari-
ables, but of the temperature solely. The same thing also occurs
when, instead of the volume, another magnitude is taken for the
second independent variable, which can likewise change indepen-
dently of the temperature and, together with the temperature.

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 349

define the whole condition of the body; — ^the pressure cannot
depend on such a variable. The two magnitudes, temperature
and pressure, cannot in this case be chosen as the two variables
which serve to define the condition of the body.

In conjunction with the temperature T, we will now, in order
to define the condition of the body, take any undetermined
magnitude x as our second independent variable. Accordingly,
in the expression

■^«t~^Ut £fa? da? our

which by (19) gives the ergonal diflFerence corresponding to ^,
and which wc
will become

and which we will next consider, we must put ^=0, whereby it

^-^^•1 w

By means of this the three equations (12), (13), and (14) take
the following forms : —

\ d fdQ\ d (dQC\_.dp dv

^\d^J^di\MJ''^Wr'di' • • • (^)

dT\dx) £teUTy""T'£&' • • • • (45)
dCi ^ „dj) dv

13. In order to give these equations more definite forms, we

I will call the whole weight of the substance M, and the portion of

it which is in the second state of aggregation m, so that M— m
is the weight of the portion which is in the first state of aggrega-
tion. We will, fiirther, consider the magnitude m as the indepen-

[ dent variable which, together with T, defines the condition of the

I body.

I Let the specific volume of the substance in the first state of

aggregation be denoted by cr, and the specific volume in the
second state of aggregation by s. Both magnitudes have re-
ference to the temperature T, and to the pressure correspond-

f ing to this temperature, and, like the pressure, are to be

regarded as functions of the temperature solely. If we denote,

i

850 NINTH MEMOIR.

fiuiher, the volimie which the whole mass possesses by v, we

have to put

r=(M— m)o-+fiw

or^ introducmg the symbol u for the difference s-^a,

vsnitt+Mo*^ (47)

whence follows

£=- • • (*»

Let r be the quantity of heat which must be imparted to the
mass in order that a unit of weight of it at the temperature T,
and under the corresponding pressure^ may pass from the first
state of aggregation into the second^ so that

S" («)

We will next introduce the specific heats of the substance in the
two states of aggregation into the equations. The specific heat
treated of here is neither the specific heat at constant volume nor
that at constant pressure^ but has reference to that quantity of
heat which is necessary to heat the substance when simultaneously
with the temperature the pressure changes in the manner deter-
mined by the circumstances of the case under consideration.
This kind of specific heat is denoted in the following formulae
by c for the first state of aggregation and by h for the second^^
and we have

-j^ = (M-m)c+mA,

or^ arranged differently^

^=m(A-c)+Mc. ...... (50)

From (49) and (50) it follows immediately that

d /dQ\ dr d /dQ\ , ,^,,

5tW=5t^ A^ldrJ^*"^- • • (^^^

On introducing the values given in the equations (48) to (51)
into the equations (44)^ (45)^ and (46)^ after replacing x by m

* The letter c, therefore, has in the following formulaa a meaning different
from what it had above, when it denoted the spedfio heat at constant volume.

CONVENIENT FORMS OP THE FUNDAMENTAL EQUATIONS. 851

in the latter, we have

-5^-fc-A=A«:^

^^.c-A=A«^, (52)

^+c-A=^, (53)

^=AT«^. (54)

These are the fundameiital equations, having reference to the
formation of vapour, which have already been established in the
First Memoir of this collection (pp. 34, 65, and 52).

In the previous numerical calculations, which had special re-
ference to the vaporization of water, I did not distinguish, in the
case of the liquid state of aggregation, the specific heat implied
in these equations from the specific heat of water at constant
pressure. This procedure was in fact perfectly justifiable, since
the difference between these two kinds of specific heat is less
than the error of observation occurring in the experimental de-
termination of the specific heat*.

* We can easily deduce from the above equations the relation whicli
exists between the specific heat at constant pressure and that specific heat
which corresponds to the assumption that the pressure increases with the
temperature in such a manner that it is always equal to the maximum
tension of the vapour given off by the liquid.

According to equation (29), the quantity of heat which must be imparted
to a unit-weight of the liquid while the temperature increases by (fl, and
the pressure by 4p, is determined by the equation

wherein C denotes the specific heat at constant pressure. Let ub now
assume that the pressure increases with the temperature in the same
manner as the maximum tension of the vapour, and let us denote the in-
crement of pressure which corresponds to the increment of temperature cTT

by ^ <n? ; the quantity of heat which under theee circumstances must be
ai.

imparted to a imit-weight of liquid in order to raise its temperature by cfT,

will then be represented by

On dividing this equation by cFT, the resulting fraction ^ will denote the
specific heat under consideration ; and since the latter is denoted by c in the

862 NINTH MEMOIR.

If we form the complete differential equation

and put therein the values given in (49) and (50), we deduce

rfQ=rrfm+ [w(A— c) + Mc]rfT.
On substituting for A— c its value resulting from (53), we have

rfQ=rrfm+[m(^-^) + Mc]dT,

texty we shall have I

Let ufl apply this specially to water, and at the same time, by way of example, |

adopt the temperature 100°. According to the experiments of Kopp, the ^

coefficient of expansion of water at 100^, the volume of the water at 4° being I

taken as unit, is 0*00080. We must multiply this magnitude by 0*001, in '

order to obtain (^)-> for the case where a cubic metre has been taken as I

the unit of volume and a kilogramme as the unit of weight. Hence ^

(^).=:0-00000080. !

Again, according to the tension series of Regnault, the pressure being ex- I

pressed in kilogrammes to the square metre, we have for the temperature j
lOOP,

The absolute temperature T at 100^ is nearly equal to 373, and for A we

wiU put with Joule j^ ; we thus obtain I

AT (^) _ . ^= ^ . 0*00000080 . 370=*00026. j

Hence follows

c=C-0*00026j

and if we now assume for the specific heat of water, at 100°, under constant I

pressure, the value resulting from Regnaulfs empirical formulae, we obtain |

the following values for the two specific heats under comparison : I

C=l-013,

c= 1-01274.

Hence we see that the two magnitudes are so nearly equal that it would have

been useless to take the difference between them into account in my former j

numerical calculations.

The consideration of the influence of pressure on the freezing of liquids ^

leads to a somewhat different result ; for a considerable change in the pressure ^

CONVENIENT FORMS OV THE FUNDAMENTAL EQUATIONS. 853

which equation may also be written
or still shorter.

dQ=rf(wr)-^dT + McdT, . . . (55)

rfQ

=Trf(^) + McdT (56)

I will not enter here into the applications of these equations,
since they have frequently been discussed in the First and Fifth
Memoirs of this collection.

14. All the foregoing considerations had reference to changes

changes tlie freezing-point veiy little, and hence the differential coefficient
^ has, in this case, a very great value. The procedure which I adopted in

my Note on this subject (p. 82 of this collection) is therefore not quite ac-
curate, since I there also employed, in numerical calculations, the known
values of c and h which correspond to the specific heats of water and ice
at constant pressure. In the Appendix to the Note (p. 89), I have, in fact,
already alluded to this circumstance. If we assume, in accordance with
the calculations made in that Note, that for an increment of pressure
amounting to one atmosphere the freezing-point sinks about 0^*00733, we
must put

dp_ 10333

dH" 000733'
On bringing this value into combination with the coefficients of expansion of
water and of ice at 0^ in the same manner as before, we obtain, instead of the
nimibers 1 and 0*48 which represent the specific heats of water and ice at
constant pressure, the following values : —

0=1-0-05=096,
A=0-48+014=0-62.
Applying these values to the equation

we have, instead of the result,

^=0-62+0-29=0-81,
given at p. 82, the following somewhat different one : —

^=0-33+0-29=0-62.

It may, however, be remarked, with reference to the small correction which
we have here taken occasion to introduce, that it relates only to an isolated
calculation, — ^in fact to the numerical calculation of an equation, which, as I
have stated in the Note, is practically unimportant, and only merits mention
for theoretical considerations; the equation itself and the theoretical con-
siderations referring thereto are not affected by this correction.

2a

864 NINTH MBMOIR.

which occurred in a reversible manner. We will now also take
non-reversible changes into consideration in order briefly to indi-
cate at least the most important features of their treatment.

In mathematical investigations on non-reversible changes two
circumstances^ especially, give rise to peculiar determinations of
magnitudes. In the first place, the quantities of heat which must
be imparted to, or withdrawn firom a changeable body are not
the same, when these changes occur in a non-reversible manner,
as they are when the same changes occur reviersibly. In the
second place, with each non-reversible change is associated an
imcompensated transformation, a knowledge of which is, for
certain considerations, of importance.

In order to be able to exhibit the analytical expressions cor-
responding to these two circunmtances, I must in the first place
recall a few magnitudes contained in the equations which I have
previously established.

One of these is connected with the first fundamental theorem,
and is the magnitude U, contained in equation (I a) and dis-
cussed at the beginning of this Memoir ; it represents the ther-
mal and ergonal content, or the energy of the body. To deter-
mine this magnitude, we must apply the equation (I a), which
may be thus written,

dV = dQ-dw; (57)

or, if we conceive it to be integrated, thus :

U=Uo+Q-m; (58)

Herein Uq represents the value of the energy for an arbitrary
initial condition of the body, Q denotes the quantity of heat
which must be imparted to the body, and w the exterior ergon
which is produced whilst the body passes in any manner from
its initial to its present condition. As was before stated,
the body can be conducted in an infinite number of ways from
one condition to another, even when the changes are to be
reversible, and of all these ways we may select that one which
is most convenient for the calculation.

The other magnitude to be here noticed is connected with the
second fundamental theorem, and is contained in equation (II a).

In fact if, as equation (II a) asserts, the integral I 7=- vanishes

I

CONYENIBNT FORMS OF THE FUNDAMENTAL EQUATIONS. 355

whenever the body, starting from any initial condition, returns
thereto after its passage through any other conditions, then

the expression -™- under the sign of integration must be the

complete differential of a magnitude which depends only on the
present existing condition of the body, and not upon the way by
which it reached the latter. Denoting this magnitude by S, we
can write

rfS^^; (59)

or, if we conceive this equation to be integrated for any re-
versible process whereby the body can pass from the selected
initial condition to its present one, and denote at the same time by
Sq the value which the magnitude S has in that initial condition,

'''^ (60)

>=S.+Jf

This equation is to be used in the same way for determining S
as equation (58) was for defining U.

The physical meaning of the magnitude S lias been abeady
discussed in the Sixth Memoir. If in the fundamental equa-
tion (II) of the present Memoir, which holds for all changes
of condition of the body that occur in a reversible manner, we
make a small alteration in the notation, so that the heat taken
up by the changing body, instead of the heat given off by it, is
reckoned as positive, that equation will assume the form

CdCL frfH , C

dZ. ...... (61)

The two integrals on the right are the values for the case under
consideration, of two magnitudes first introduced in the Sixth
Memoir.

In the first integral, H denotes the heat actually present in
the body, which, as I have proved, depends solely on the tem-
perature of the body and not on the arrangement of its parts.

Hence it follows that the expression -j=^ is a complete differen-
tial, and consequently that if for the passage of the body from

CdH
its initial condition to its present one we form the integral j -™-,

2a2

856 NINTH MEMOIR.

we shall thereby obtain a magnitude which is perfectly defined

by the present condition of the body, without the necessity of

knowing in what manner the transition -from one condition to

the other took place. For reasons which are stated in the Sixth

Memoir, I have called this magnitude the transformation-value

of the heat present in the body.

It is natural when integrating, to take, for initial condition,

that for which H=0, in other words, to start from the absolute I

zero of temperature ; for this temperature, however, the integral

CdH

j -jp- is infinite, so that to obtain a finite value, we must take ^

an initial condition for which the temperature has a finite
value. The integral does not then represent the transformation-
value of the entire quantity of heat contained in the body, but (
only the transformation-value of the excess of heat which the
body contains in its present condition over that which it
possessed in the initial condition. I have expressed this by call-
ing the integral thus formed the transformation-value of the
body^s heat, estimated from a given initial condition (p. 248).
For brevity we will denote this magnitude by Y. {

The magnitude Z occurring in the second integral I have j

called the disgregation o{ the body. It depends on the arrange- :

ment of the particles of the body, and the measure of an in- ^

crement of disgregation is the equivalence-value of that trans-
formation from ergon to heat which must take place in order
to cancel the increment of disgregation, and thus serve as a |

substitute for that increment. Accordingly we may say that I

the disgregation is the transformation-value of the existing J

arrangement of the particles of the body. Since in determining I

the disgregation we must proceed from some initial condition of
the body, we will assume that the initial condition selected for
this purpose is the same as that which was selected for the I

determination of the transformation-value of the heat actually I

present in the body.

The sum of the two magnitudes Y and Z, just discussed, is
the before-mentioned magnitude S. To show this, let us return i

to equation (61), and assuming, for the sake of generality, that I

the initial condition, to which the integrals in this equation ^

refer, is hot necessarily the same as the initial condition which |

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 357

was selected when determining Y and Z, but that the integrals
refer to a change which originated in any manner whatever
suited to any special investigation, we may then write the inte-
grals on the right of (61) thus :

ffi=Y-Yo and ^dZ=Z-2^

wherein Yq and Zq are the values of Y and Z which correspond
to the initial condition. By these means equation (61) becomes

J^=Y+Z-(Yo+Zo) (62)

Putting herein

Y+Z=S, (63)

and in a corresponding manner

we obtain the equation

J^=S-So, . (64)

which is merely a different form of the equation (60) , by which
S is determined.

We might call S the transformational content of the body, just
as we termed the magnitude U its thermal and ergonal content.
But as I hold it to be better to borrow terms for important
magnitudes from the ancient languages, so that they may be
adopted unchanged in all modem languages, I propose to call
the magnitude S the entropy of the body, from the Greek word
rpoinjy transformation. I have intentionally formed the word
entropy so as to be as similar as possible to the word energy ;
for the two magnitudes to be denoted by these words are so
nearly allied in their physical meanings, that a certain similarity
in designation appears to be desirable.

Before proceeding farther, let us collect together, for the sake
of reference, the magnitudes which have been discussed in the
course of this Memoir, and which have either b^en introduced
into science by the mechanical theory of heat, or have obtained
thereby a different meaning. They are six in number, and
possess in common the property of being defined by the present
condition of the body, without the necessity of our knowing
the mode in which the body came into this condition : (1) the

358 NINTH MEMOIR.

thermal content, (2) the ergonal content, (8) the sum of the two
foregoing^ that is to say the thermal and ergonal content^ or
the energy, (4) the transformation-value of the thermal content,
(5) the disgregation, which is to be considered as the transfor-
mation-value of the existing arrangement of particles^ (6) the
sum of the last two^ that is to say^ the transformational content,
or the entropy.

15. In order to determine the energy and the entropy for
particular cases^ the several expressions above given for dQ have
to be used in conjunction with the equations (57) and (59), or
(58) and (60). I will here, by way of example^ treat a few
simple cases merely.

Let the body under consideration be homogeneous and of
the same temperature throughout, upon which the only active
foreign force is a uniform and normal surface-pressure, and let
us assume that it can change its volume, with a change of tem-
perature and pressure, without at the same time suffering a
partial change of its state of aggregation ; then if the weight
of the body be taken as unit, we can employ for dCi the equa-
tions (28), (29), and (82) given in Art. 9. In these equations,
the specific heat at constant volume, there denoted by c, and the
specific heat at constant pressure, denoted by C, occur; now,
since the latter specific heat is that which is usually directly
determined by observation, we will use the equation (29) in
which it occurs, namely,

dOi^CdT-AT^dp^.

With respect to the exterior ergon we have, for an infinite-
simal change of condition diuing which the volume increases by
dv, the equation

dw=Apdv;

and if T and j9 are chosen as independent variables, we can give
this equation the form

* I write here edinply ^ instead of the symbol (4S,)- in (29), because in

a case where only T andjp occur as mdependent variables^ it is manifest that
in the differentiation with respect to T, the other variable p is supposed con-
stant.

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 859

Applying these expressions for dQ and dw to equations (57)
and (59), we obtain

^=(C-^*)«-A(T*+,*)«„

dS = ^dT-A%dp.

T

From the equation

"dT

(65)

the last in (38), it is easy to see that these two complete dif-
ferential equations are integrable, without necessarily assuming
a further relation between the variables. By performing the in-
tegration, we obtain expressions for U and S, of which each con-
tains but one undetermined constant ; this is the value which the
magnitude U or S has in the initial condition of the body selected
as one of the limits of the integration.

If the body is a perfect gas, the equations assume a simpler
form. They may be obtained either by combining equations
(65) with the equation ^=RT, which expresses the law of
Mariotte and Gay-Lussac, or by going back to the equations (57)
and (59), putting therein^ in place of rfQ, one of the expressions
before deduced for a perfect gas, and contained in the equations
(42), and introducing at the same time one of the three expres-

sions AR — dv, AR IdT dpL and Apdv for dw. If we choose

the first of equations (42), as being the most convenient; for the
present case, we have

rfU=cdT,

£?S=Cy+AR~.

(66)

Since c and AE. are constant, these equations may be immediately
integrated ; on doing so and denoting the initial values of U
and S, for which T=To and »=«<>, by Uq and Sq, we have

U=Uo+c(T-To), 1

T V \ ' ' ' (67)

S=So+clogi+ARlog^. J

We will treat, as a last special case, that to which Arts. 12 and

860 NINTH MEMOIB.

13 refer^ where the body under consideration is a mass of the
weight M, of which the portion M— m is in one, and the portion
in in a different state of aggregation, and where the pressure,
to which the whole mass is exposed, depends only on the tem-
perature.

We will assume that at first the whole mass is in the first
state of aggregation, that it has the temperature T^, and is also
exposed to the pressure corresponding to this temperature. The
values of the energy and entropy in this initial condition may be
denoted by Uq and Sq. We will next conceive the body to be
brought from this initial condition to its final one, in the follow-
ing manner : — ^Whilst the entire mass continues in the first state
of aggregation, the body is first raised from the temperature Tq
to the temperature T, and at the same time the pressure changes
so as to have at every moment the magnitude which corresponds
to the then existing temperature. Thereupon the portion of the
mass whose weight is m passes, at the temperature T, from the
first into the second state of aggregation. We will consider
these two changes separately, and in so doing employ the nota-
tion of Art. 13.

For the first-mentioned change of temperature we have to
use the equation

dQ=McdT.

The magnitude c occurring here is the specific heat of the
body in the first state of aggregation, on the assumption that
the pressure, during the change of temperature, alters in the
manner above stated. The foot-note to Art. 13 gives an account
of this magnitude, and according to what is there proved
we may without hesitation, in the case where the first state
of aggregation is liquid or solid and the second gaseous, put
for c, in all numerical calculations, the specific heat of the
liquid or solid body at constant pressure. It is only when we
are concerned with high temperatures, for which the vapour
tension increases very rapidly with the temperature, that the
difference between the specific heat c, and the specific heat at
constant pressure, becomes sufSciently great to be regarded.
On remembering that with an increase in temperature dT, an

increase i» volume M -ffh dTj and consequently the exterior

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 861

ergon MA^^dT is associated, it follows from the foregoing
^ equation that

dS=M|rfT.

For the change of the state of aggregation taking place at the
temperature T, we have

rfQ=:rrfm.

Since the increment dm of the portion in the second state of
aggregation involves an increment of volume equal to tidm, ani
^ consequently an amount of exterior ergon denoted by Apudm^

it follows from the above that

rfU = (r — Apu) dm.
If in order to replace the magnitude u by magnitudes better
known experimentally, we apply the equation (54), which may
• be thus written,

r

^

Au-

it follows that

^dT

rfU=r/l-

t^)^-

At the same time from the expression for dQ, we have directly

dS=ffydm.

The two differential equations referring to the first process
must be integrated according to T from To to T, and the two
referring to the second process according to m from to m ;
hence we obtain

S>^'f

S = So+M ^rfT + 9-*

(68)

* A few more complete mathematical developments concerning energy and
entropy will be communicated in an Appendix to this Memoir.

862 NINTH MEMOIR.

16. If we now assume that in one of the ways above in-
dicated the magnitudes U and S have been determined for a
body in its different conditions^ the equations which hold good
for non-reversible changes may be at once written down.

The first fundamental equation (Ia)> and the equation (58),
resulting from it by integration, which we will arrange thus,

Q=U-Uo+ir, • . (69)

hold just as well for non-reversible as for reversible changes ;
the only difference being, that of the magnitudes standing on
the right side, the exterior ergon w has a different value, in the
case where a change occurs in a non-reversible manner, from
that which it has in the case where the same change occurs in a
reversible maimer. With respect to the difference U— Uq this
disparity does not exist. It only depends on the initial and
final condition, and not on the nature of the transition. Con-
sequently we need only consider the nature of the transition so
far as is necessary in order to determine the exterior ergon
thereby performed; and on adding this exterior ergon to the
difference U— Uq, we obtain the required quantity of heat Q
which the body takes up during the transition.

The uncompensated transformation involved in any non-rever-
sible change may be thus obtained : —

The expression for the uncompensated transformation which
is involved in a cyclical process, is given in equation (11) of the
Fourth Memoir (p. 127). If we give to the differential dOi in
that equation the opposite sign, a quantity of heat given off by
the body to a reservoir of heat being there reckoned positive,
whilst here we consider the heat taken up by the body to be
positive, it becomes

N=-J^ (70)

If the body has suffered one change or a series of changes,
which do not form a cyclical process, but by which it has
reached a final condition which is different from the initial
condition, we may afterwards supplement this series of changes
so as to form a cyclical process, by appending other changes of
such a kind as to reconduct the body from its final to its initial
condition. We will assume that these newly appended changes.

\

CONVENIENT FORMS OF THB FUNDAMENTAL EQUATIONS. 863

by whicli the body is brought back to the initial condition, take
place in a reversible manner.

On-applying equation (70) to the cyclical process thus formed,
we may divide the iotegral occurring therein into two parts, of
which the first relates to the originally given passage of the
body from the initial to the final condition, and the second to
the supplemented return from the final to the initial condition.
We will write these parts as two separate integrals, and dis-
tinguish the second, which relates to the return, by giving to
its sign of integration a suflSx r. Hence equation (70) becomes

— j"¥-j,f-

Since by hypothesis the return takes place in a reversible
> manner, we can apply equation (64) to the second integral,

taking care, however, to introduce the difference Sq— S instead
of S — Sq (where Sq denotes the entropy in the initial condition,
and S the entropy in the final condition), since the integral here
in question is to be taken backwards from the final to the initial
condition. We have therefore to write

J:

ip =So— S.

By this substitution the former equation is transformed into

N=S-S,-J^.. ..... (71)

> The magnitude N thus determined denotes the uncompensated

transformation occurring in the whole cyclical process. But from

the theorem, that the sum of the transformations which occur

in a reversible change is null, and hence that no uncompensated

transformation can arise therein, it follows that the supposed

reversible return has contributed nothing to the augmentation

^ of the uncompensated transformation, and the magnitude N

. represents accordingly the uncompensated transformation which

' has occurred in the given passage of the body from the initial

to the final condition. In the deduced expression, the difference

S— Sq is again perfectly determined when the initial and final

conditions are given, and it is only when forming the integral

7p- that the maimer in which the passage from one to the

other took place must be taken into consideration.

r-

A'

864 NINTH MEMOIR.

17. In conclusion I wish to allude to a subject whose com-
plete treatment could certainly not take place here^ the exposi-
tions necessary for that purpose being of too wide a range^ but
relative to which even a brief statement may not be without in-
terest^ inasmuch as it will help to show the general importance
of the magnitudes which I have introduced when formulizing the
second fundamental theorem of the mechanical theory of heat.

The second fundamental theorem^ in the form which I have
given to it^ asserts that all transformations occurring in nature
may take place in a certain direction^ which I have assumed as
positive, by themselves, that is, without compensation; but that
in the opposite, and consequently negative direction, they can
only take place in such' a manner as to be compensated by
simultaneously .occurring positive transformations. The appli-
cation of this theorem to the Universe leads to a conclusion to
which W. Thomson first drew attention*, and of which I have
spoken in the Eighth Memoir. In fact, if in all the changes of
condition occurring in the universe the transformations in one
definite direction exceed in magnitude those in the opposite
direction, the entire condition of the universe must always
continue to change in that first direction, and the universe must
consequently approach incessantly a limiting condition.
J. VI The question is, how simply and at the same time definitely

^ to characterize this limiting condition. This can be done by

considering, as I have done, transformations as mathematical
quantities whose equivalence- values may be calculated, and by
algebraical addition united in one sum.

In my former Memoirs I have performed such calculations
relative to the heat present in bodies, and to the arrangement
of the particles of the body. For every body two magnitudes
have thereby presented themselves — ^the transformation-value
of its thermal content, and its disgregation ; the sum of which
constitutes its entropy. But with this the matter is not ex-
hausted ; radiant heat must also be considered, in other words,
the heat distributed in space in the form of advancing oscilla-
tions of the aether must be studied, and farther, our researches
must be extended to motions which cannot be included in the
term Heat.

♦ Phil. Mag. Ser. 4. vol. iv. p. 804.

^

'i4

CONVENIENT FORMS OF THE FUNDAMENTAL EQUATIONS. 865

The treatment of the last might soon be completed, at least
so far as relates to the motions of ponderable masses^ since
allied considerations lead us to the following conclusion. When
a mass which is so great that an atom in comparison with it
may be considered as infinitely small^ moves as a whole, the
transformation- value of its motion must also be regarded as
infinitesimal when compared with its vis viva ; whence it follows
that if such a motion by any passive resistance becomes con-
verted into heat, the equivalence-value of the uncompensated
transformation thereby occurring will be represented simply by
the transformation-value of the heat generated. Radiant heat,
on the contrary, cannot be so briefly treated, since it requires
certain special considerations in order to be able to state how
its transformation-value is to be determined. Although I have
already, in the Eighth Memoir above referred to, spoken of
radiant heat in connexion with the mechanical theory of heat, I
have not alluded to the present question, my sole intention
being to prove that no contradiction exists between the laws of
radiant heat and^ an axiom assumed by me in the mechanical
theory of heat. I reserve for future consideration the more special
application of the mechanical theory of heat, and particularly
of the theorem of the equivalence of transformations to radiant
heat.

For the present I will confine myself to the statement of one
result. If for the entire universe we conceive the same magni-
tude to be determined, consistently and with due regard to all
circumstances, which for a single body I have called entropy ^
and if at the same time we introduce the other and simpler con-
ception of energy, we may express in the following manner the
fundamental laws of the universe which correspond to the two
fundamental theorems of the mechanical theory of heat.

1. The energy of the universe is constant,

2. The entropy of the universe tends to a maanmum.

866 NINTU MEMOIB^ APPENDIX.

APPENDIX TO NINTH MEMOIR [1866].

ON THE DETERMINATION OF THE ENERGY AND ENTROPY OF A

BODY*.

It may perhaps not be inappropriate if I communicate, as an
Appendix to tlie foregoing Memoir, a few further developments,
in order to show how the equations which serve for the deter-
mination of energy and entropy may be derived directly from
the fundamental equations of the mechanical theory of heat.

1. The first fundamental equation wiU be applied in the form
which is given in (I a) of the foregoing Memoir, and the second
in the form equivalent to that given in (59) . We have, therefore,

dQ,=dU+dw, (A)

dQi=TdS (B)

The first of these two equations applies to reversible, as well as
to non-reversible changes ; the second, on the contrary, holds
good for reversible changes solely. But in order to be able to
bring the two equations into conjunction, we will suppose that
they relate to one and the same reversible change of a body.
In this case the thermal element dCL is the same in both equa-
tions, hence we can eliminate it from the equations, whereby we
obtain

TrfS=rfU + rfw {a)

We will now assume that the condition of the body is defined
by any two variables, which we will provisionally denote quite
generally by x and y. We can afterwards put in the place of
these undefijied variables definite magnitudes, such as tem-
perature, volume, pressure, or any others appropriate to the
particular investigation in view. If the condition of the body
is defined by the two variables a? and y, all magnitudes which
are defined by the actually existing condition of the body, in-
dependently of the way in which the body came into this con-

♦ The substance of this Appendix is taken from a note which was recently
published by me, and which may be found in Schlomilch's Zeitschriftfur Ma-
thematik und Fhysik, Bd. xi. S. 31, and in an English Translation in the Phil
Mag. Series 4. vol. xzxii. p. 1.

DETERMINATION OF ENERGY AND ENTROPY. 367

dition, must be capable of being expressed by fimctions of these
variables, in which the variables themselves may be regarded
as independent of each other. Accordiagly the entropy S and
the energy U are to be regarded as functions of the independent
variables x and y. In this respect, however, the external ergon
w deports itself very differently, as was repeatedly stated in the
foregoing Memoir. Although the differential coefficients of w
may be regarded as definite functions of x and y, in so far as
reversible changes only are concerned, w itself cannot be ex-
pressed by such a function, but can only be defined when,
besides the initial and final conditions of the body, the way in
which the change from the one to the other takes place is like-
wise given.

If now in equation (a) we put

it is transformed into

Since this equation must be true for arbitrary values of the dif-
ferentials dx and dy, and therefore for that case, amongst others,
in which one or other of the differentials vanishes, it resolves
itself at once into the two following equations :

rr^dS__d^ djv

dx^ dx dx'

^d^_dJ] dw

f dy" dy dy

W

Prom these equations one of the magnitudes S and U may be
eliminated by means of a second differentiation.

2. We wiU first eliminate the magnitude U, since the result-
ing equation is the simpler of the two.

For this purpose we wiU differentiate the first of the equations
(i) with respect to y, and the second with respect to x. In so

368 NINTH MEMOIR^ APPENDIX.

doing we will write the second differential coefficients of S and
U in the usual way. The differential coefficients of -r- and -t-,

on the other hand^ shall be written^ as in the foregoing Me-
moir^ thus :

d /dw\ J d fdw\

in order to express explicitly that they are not differential co-
efficients of the second order of any function of x and y. Lastly,
we shall have to remember that the magnitude T which occurs
in these equations, namely the absolute temperature of the body,
must abo be regarded as a function of x and y. On thus dif-
ferentiating we obtain.

^.^+T— = — + — f— \
dy dz dxdy dxdy dy\dxj

dx dy dydx dydx dx\dy)

Subtracting the second of these equations from the first, and re-
membering that

d^^d^ and — =^,
dxdy dydx dxdy dydx'

we have

dy dx dx dy'^dy\dxJ dx\dyJ

In the foregoing Memoir I have called the difference which
stands here on the right-hand side, the ergonal difference corre-
sponding to xy^ and denoted it by E^; so that we may put

^ ay\dxJ dx\dyj

(c)

The foregoing equation is thus transformed into

rfT rfS_rfT dS

dy ' dx dx' dy '» ^^

This is the differential equation, resulting from equation (a),
which serves to define S.

In order now to eliminate the magnitude S from the two equa-

DETERMINATION OF ENERGY AND ENTROPY. 369

tions (b), we will write them as follows : —
dS_l rfU 1 ^
da^^T' dx '^T' dx'
dS_l dV I dw
dy^T' dy ^H' dy'

Of these equations^ again^ we will differentiate the first with
respect to y, and the second with resi)ect to x, whereby we have

- dxdy'^T'dxdy T^ ' dy ' dx '^ dy\T' dxJ'

T^' dx'dy '^dxVr' dyJ

dydx T dydx T* dx dy
Subtracting the second of these equations from the first, bring-
ing all the terms of the resulting equation in which U occurs to
the left-hand side, and multiplying the whole equation by T*,
we have

dy* dx dx' dy^ IdyVS ' dx) dxVH * dyJj '
For the magnitude which here stands on the right-hand side
we will likewise employ a special symbol, putting

The last equation then becomes

dy dx dx dy '^ ^"^ '

This is the differential equation, resulting from equation (a),
which serves to define U.

8. Before pursuing fiirther the treatment of the two differential
equations (d) and (/), it will be advisable to direct attention for
a moment to the magnitudes E and E' which therein occur.

Between these two magnitudes the following relation exists^
which can be easily deduced from the expressions given in {c)
and (e),

E-=TE,-^.? + ^.^. ...(g)
** 'v dy dx dx dy ^^'

Both the magnitudes E^ and E'^ are functions of x and y.

If in order to define the body we select, instead of x and y, any

other two variables, which we may call f and ij, and form with

2b

S70

NINTH MEMOIR; APPENDIX.

(A)

them the corresponding magnitudes Ej, and E'^^, namely,

these magnitudes are of course functions of f and ij, just as the
foregoing magnitudes are functions of x and y. But if now one
of the last two expressions, c. g. the one for E^^, is compared
with the expression for the corresponding magnitude E^, we find
that they represent, not merely expressions for the same magni-
tude with reference to dififerent variables, but actually diflTerent
magnitudes. For this reason I have not called E^ simply the
ergonal difference, but the ergonal difference corresponding to xy,
whereby it is at once distinguished from E^^ or the ergonal dif-
ference corresponding to fij. The same holds good of E'^ and
EV , which are also to be regarded as two different magnitudes.
The relation existing between the magnitudes E^ and E^^ may
be found as follows. The differential coefficients occurring in
the expression given for E^^ in (A) may be arrived at by first
forming the differential coefficients in relation to the variables
X and y, and then treating each of these two variables as a func-
tion of and 1]. In this way we obtain

dw _ dw

li'^ dx

dw ^dw dx dw dy

dvi "^ dx dfi dy dii

Let the first of these two expressions be differentiated with
respect to ij and the second with respect to f , and we then ob-
tain, by the application of the same process,

dx du^ dy

d fdw\

dn<di)'

d rdw\ _

^fdw\ dx dx^ d fdw\ dx
d^\Jx)'dS'dii'^diJ\M)'d^

d fdw\ dx dy , d /dw\
dx\dyJ dr^ d^ ^ dy\dyJ

d fdw\ dx dx d fdw\
dx\dxJ di dm dy\dxJ

d fdw'\ dx dy d fdw\
'^diKd^J'dl'^'^'^U^J

dx
dy

dy dw

d^i dji;

dy dw

dri dy

dy dw
d^'^'d^

dy dw

dri dy

d^x
didn
dhf
'd^dn
d^x
d^dvi
d^y
didm'

DETERMINATION OF ENERGY AND ENTROPY. 371

If the second of these equations be subtracted from the first,
most of the terms on the right-hand side will disappear, and there
will remain only four terms, which may be thus expressed as a
product of two binomial expressions —

d fdw\ ^ d rdw\ __ fdx dy ^ dx dy\ V d rdw\ ^ d /^w?\"]

The expression standing on the left-hand side of this equation
is E^^, and that contained within square brackets on the right-
hand side is E^. Hence we finally obtain

^ _fdx dy dx dy\^ ,..

Similarly we may also find

^, _(dx dy dx dy\^, .,

If we replace' only one of the variables by a new one — ^if, for
instance, we retain the variable x while putting the variable ij in
place of y, we have in the two foregoing equations a7=0, and

consequently -Tf = l, and ^=0, whereby they become

E^=T^E^ and E' =fe . . . . {k)

If, further, the original variables are retained but their order
of succession altered, the magnitudes in question take the oppo-
site sign, as may be seen at once from the expressions (c) and
(e) ; that is to say, they become

4. We now return again to the diflferential equations (d) and
(/) that have been deduced for S and U.

These assume particularly simple forms when the temperature

T is taken as one of the independent variables. If, for instance,

dT rfT

we put T=y, it follows thence that -7- = l and ^-=0; and
^ ^ dy dx '

we have also, in place of E^ andE'^, to write E^ and E'^.
Equations {d) and (/) thus become

2b2

372 NINTH MEMOIR^ APPENDIX.

These equations can at once be integrated with respect to x, and
we thus deduce

where ^(T) and -^(T) are two arbitrary functions of T.

The last two equations can of course be easily changed back
again by putting any other variable y in place of the variable T.
For this purpose we only require to substitute for T the func-
tion of X and y which represents this magnitude. The equations
hence resulting are the same as those obtained when we start
from the more general differential equations (d) and [f)y and
apply to them the common process of integration, keeping in
mind, at the same time, that according to (k) we have to put

We have thus in what precedes arrived, by help of the par-
tial differential equations deduced from equation (a), at expres-
sions for S and U, each of which still contains an arbitrary func-
tion of T. If we wish to determine the functions, which are
there left arbitrary, we must go back to equations (A) and (B),
whence equation {a) was obtained by the elimination of dQ,,

5. Let us assume that the condition of the body is determined
by its temperature and any other variable x ; we can then give to
the two equations (A) and (B) the following form :

Since these equations must be true for any values of the dif-
ferentials rfT and dx, each resolves itself, as has been already
pointed out in a similar case above, into two equations. Of the
four equations thus arising we will here employ only those two

which can serve for the determination of ^ and of --p^ namely

(»)

rfT"

1

dQ
dlf

rfU

dO,
' dH

dw
~dT

i

DETERMINATION OF ENERGY AND ENTROPY. 373

In order to determine the two other differential coeflSicients

jQi //TT

-^ and -T-,, we Tnll apply the equations (m) deduced above.

With the aid of these expressions of the four differential co-
efficients we can form the following complete differential equa-
tions of S and U :

'^HS-^y^^^'-^-

ip)

Since the magnitudes S and U must be capable of being re-
presented by functions of T and a?, in which the two variables T
and X may be looked upon as independent of each other, the
known equation of condition of integrability must apply to both
the foregoing equations. For the first equation this is

dAT' dTJ"

rfE,.

dT
or, differently written,

dxKdTJ ^^T ^^^

which is equation (1 5) of the foregoing Memoir. For the second
equation the equation of condition is

dxKdTj dx\d^y'"^r' (^^

These two equations of condition are connected with each
other in such a manner that from either of them the necessity
of the other can be immediately deduced. Between the two
magnitudes E^ aad E'^, which occur in them, the following
relation subsists, which results from {ff) if we therein place
T=y:

E',T=TE^-;^- . . (s)

Differentiating this equation with respect to T, we have

Now bearing in mind that

y _ d_ fdw\ d_ /dw\

^^"dT^dx) dxKlTr

/

874 NINTH MEMOIB, APPENDIX.

the last equation becomes

It — "St dxKdrJ " ^^^

By aid of this equation we can immediately refer either of the
equations {q) and (r) to the other.

By the integration of the complete differential equations {p)
each of the two magnitudes S and U can be determined^ except
as to a constant that still remains unknown. ^

Of course any other variable y might be substituted for the I

variable T in these complete differential equations, if it appeared '

appropriate for special purposes to make the substitution ; this
could be done without any diflSculty, if T were supposed to be a
known function of x and y, and therefore does not require to be
further dwelt upon. z

6. All the foregoing equations are developed in such a way
that no limiting conditions are set up in relation to the external ' d

forces which act upon the body, and to which the external ergon |

has reference. We wiU now consider the particular case rather
more closely, where the only external force which acts either to
hinder or promote the change of condition in the body, and so
occasions the production of positive or negative ergon, is a pres-
sure uniformly distributed over the whole surface of the body, and t
everywhere directed perpendicularly to the surface of the body. '

In this case, in accordance with equations (17) of the foregoing
Memoir, ifp denote the pressure and v the volume, we must put

f^— A —

dy'^ ^ dy
By introducing these values into the expressions given for E^
and E' in {c) and (e), we have

*y LrfyvT dxj dxyL dyjj

In the last of these equations we will put, for shortness,

^=1' W

i

1*^

;

I

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