LIBRA.RY
UNIVERSITY OF CALIFORNIA.
Class
m mm
^--'*,-,-.'..; r -.:.;
:'' ,
SKETCH OF
THERMODYNAMICS.
Edinburgh : Printed by Thomas and Archibald Constable
FOR
DAVID DOUGLAS.
LONDON HAMILTON, ADAMS, AND CO.
CAMBRIDGE MACMILLAN AND CO.
GLASGOW JAMES MACLEHOSE.
SKETCH
OF
THERMODYNAMICS
BY
P. G. TAIT, M.A.
FORMERLY FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE;
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH.
SECOND EDITION.
(REVISED AND EXTENDED.)
\ \ B R A
^ OF THE
UNIVERSITY
EDINBURGH:
DAVID DOUGLAS.
1877.
[The rights o_f Translation and Reproduction arc reserved. ]
PREFACE.
Two considerations led to the first publication of
this little work. Representations were made by
various men of science, especially engineers, to the
publishers of the North British Review, to the effect
that it was desirable that two articles of mine, on the
' Dynamical Theory of Heat ' and on ' Energy,' which
appeared in that Journal in 1864, should be reprinted
in a separate form. And I felt the want of a short
and elementary text-book on these subjects, to be
used in my class, until the publication of the volume
of Thomson and Taifs Natural Philosophy in which
they will be treated.
The semi-historical character of the articles has
been retained, although it is to a certain extent
unsuited for a text-book. Want of time, and the
desire to republish them in a form not very different
from the original one, have been my reasons.
I have added, for the benefit of the reader who has
some knowledge of mathematics, developments of a
somewhat more advanced character, mainly taken
from the scattered papers of Sir W. Thomson.
The science of Thermodynamics is now securely
founded upon bases almost as simply enuntiated, and
quite as impregnable, as Newton's Laws of Motion ;
vi Preface.
and the opposition which it even yet ^occasionally
meets with is therefore quite as absurd as would be
a denial of the main conclusions of the Principia.
Since the appearance of my articles in the North
British Review, I have been led to examine very
carefully the history of the subject, and have conse-
quently made several alterations. Nevertheless, in
attempting to give even a rough sketch of the history
of a grand physical theory, especially one of so
modern a date, I have been convinced by experience
that it is almost impossible to be strictly impartial,
however we may strive to be so.
Take an instance or two. In almost all French and
German works we read of the ' gaseous laws of
Mariotte and Gay-Lussac.' British authors usually
call them the laws of Boyle and Dalton. It is pro-
bable that both are wrong, the British partially, the
French and Germans wholly. Boyle 1 discovered,
and published several years before the earliest work
of Mariotte 2 which I have been able to find, the law
connecting the pressure and volume of a gas at
constant temperature, employing for the purpose the
1 Defence of the Doctrine touching the Spring and Weight of the Air,
against the objections of Francis cus Linus. Appended to New Experi-
ments, physico-mechanical, etc. Second Edition, 4to. Oxford, 1662.
See also James Bernoulli, 1683, De Gravitate sEtheris, p. 92, where
he distinctly ascribes these experiments to Boyle.
2 In the Biographie Generale, the date of the treatise De la Nature
de FAir is given as 1676. In the Histoire de VAcademie, 1666-86, and
in the Biographie Universelle, the date appears as 1679. But Prof.
Jenkin informs me that even the British Museum does not afford the
means of determining the date exactly.
Preface. vii
very apparatus still used in Physical lectures. And
it was Charles l who discovered that the co-efficient
of dilatation is nearly the same in all permanent
gases. Their equable expansion for equal increments
of temperature as measured by the mercury thermo-
meter seems to have been quietly assumed, although
it is by some stated to be the essence of the so-called
Gay-Lussac Law. Dalton 2 states that ' air expands
in geometrical progression to equal increments of
temperature' as measured by a peculiar scale, which
he says very nearly agrees with the common mer-
curial scale.
When errors like these are almost universal, where
can we expect to find truth ? Even so excellent an
authority as the late M. Verdet, whose extensive
knowledge and whose love of truth were equally con-
spicuous, made statements as to the history of
Thermodynamics which he afterwards frankly ac-
knowledged to be incorrect. The following, for
instance, amongst others, was noticed in the North
British Review ; but one has only to look to the
recently published volumes of his collected works to
1 Verdet Lefons de Chimie et de Physique, 1862. Note E. See, how-
ever, Gay-Lussac himself: Ann.'de Chimie, xliii. (An x.) p. 157. ' Avant
d'aller plus loin, je dois prevenir que quoique j'eusse reconnu un grand
nombre de fois que les gaz oxigene, azote, hydrogene, et acide carbo-
nique, et 1'air atmosphe"rique, se dilatent egalement depuis o jusqu'a
80, le cit. Charles avait remarque depuis 15 ans la meme propriete
dans ces gaz : mais, n'ayant jamais public ses resultats, c'est par le
plus grand hasard que je les ai connus. ... II me parait done qu'on
ne peut conclure de ces experiences la vraie dilatation des gaz. '
2 Chemical Philosophy, 1808.
viii Preface.
find a great many more. It is not fair to the memory
of such a man to publish his lectures without at least
indicating the corrections which death alone pre-
vented him from making :
' A une somme donnee d'actions chimiques de nature
donne'e doit correspondre un degagement constant de
chaleur, quelle que soit la constitution de la pile et du
circuit oil les deux phe'nomenes se produisent a la fois.
Cette conclusion theorique a ete verifiee par une remar-
quable experience de M. Favre.'
This was no theoretical deduction at all, but an
experimental result given by Joule in 1843 [see sect.
93 of this work] ; while the earliest of Favre's experi-
ments dates from 1853. I refer to it here as a good
instance of the way in which the contents of Joule's
magnificent, but much neglected, papers of a quarter
of a century ago are being rediscovered and attri-
buted to others.
I cannot pretend to absolute accuracy, but I have
taken every means of ensuring it, to the best of my
ability ; though it is possible that circumstances may
have led me to regard the question from a somewhat
too British point of view. I have tried to preserve a
happy medium between the old absurd British con-
tempt for all things foreign, and the still more absurd
custom (of comparatively recent origin) which gives to
foreigners all that the writer is not in a position to
claim for himself or for some of his particular friends.
But, even supposing the worst, it appears to me that
unless contemporary history be written with some
Preface. ix
II little partiality, it will be impossible for the future
' historian to compile from the works of the present
day a complete and unbiassed statement. Are not
both judge and jury greatly assisted to a correct
verdict by the avowedly partial statements of rival
pleaders ? If not, where is the use of counsel ? ^
But that I may show the reader that I am in no
way prejudiced though I have formed an opinion
I quote, from some critical remarks which Prof.
Helmholtz has been kind enough to make on my
introductory chapters, an admirable statement of a
part at least of the case for the other side. It will
be remembered that there is no dispute about dates,
there is a difference of opinion as to the validity of
processes, and the credit to be assigned to their
authors.
' . . . ich 1 muss sagen, dass mir in dieser Bezie-
hung die Entdeckungen von Kirchhoff in diesem
Felde (Radiation and Absorption) als einer der lehr-
reichsten Falle in der Geschichte der Wissenschaft
erscheinen, eben auch deshalb weil viele andere
Forscher schon vorher dicht am Rande derselben
Entdeckungen gewesen waren. Kirchhoff's Vor-
ganger verhalten sich zu ihm in diesem Felde unge-
fahr so, wie in Bezug auf die Erhaltung der Kraft
1 Freely translated, it is as follows : . . . I must say that, in this
connection, Kirchhoffs discoveries about radiation and absorption
appear to me one of the most instructive cases in the history of science ;
and the more so that many other investigators had already approached
close to the verge of these discoveries. Kirchhoffs predecessors in
this field bore to him much the same relation as, in the Conservation
x Preface.
R. Mayer, Colding, und Seguin zu Joule und W.
Thomson.
' Was nun R. Mayer betrifft, so kann ich allerdings
den Standpunct, den Sie ihm gegeniiber eingenom-
men haben, begreifen, kann aber doch diese Gelegen-
heit nicht hingehen lassen, ohne auszusprechen, dass
ich nicht ganz derselben Meinung bin. Der Fort-
schritt der Naturwissenschaften hangt davon ab, dass
aus den vorhandenen Thatsachen immer neue Induc-
tionen gebildet werden, und dass dann die Folgerun-
gen dieser Inductionen, soweit sie sich auf neue
Thatsachen beziehen, mit der Wirklichkeit durch das
Experiment verglichen werden. Ueber die Nothwen-
digkeit dieses/zweiten Geschaftsj kann kein Zweifel
sein. Es wird auch oft dieser zweite Theil einen
grossen Aufwand von Arbeit und Scharfsinn kosten,
und dem, der ihn gut durchfuhrt, zum hochsten
Verdienste gerechnet werden. Aber der Ruhm der
Erfmdung haftet doch an dem, der die neue Idee
gefunden hat ; die experimentelle Priifung ist nach-
of Energy, Mayer, Colding, and Seguin bore to Joule and W.
Thomson.
As concerns Mayer, I can of course understand the point of view
from which you regard him ; but I cannot allow this opportunity to
pass without saying that I am not quite of the same opinion. The pro-
gress of Natural Science depends upon the constant formation of new
inductions from known facts, and the comparison of these inductions,
so far as they lead to new consequences, with reality by means of ex-
periment. There can be no doubt of the necessity of this second step.
It often requires an extensive application both of labour and talent,
and it brings the greatest credit to him who executes it well. But the
glory of the discovery belongs to him who hits upon the new idea ; the
subsequent experimental verification is often a mere mechanical pro-
Preface. xi
her offenbar eine viel mechanischere Art der Leistung.
Auch kann man nicht unbedingt verlangen, dass der
Erfmder der Idee nothwendig verpflichtet sei, auch
den zweiten Theil der Arbeit auszufuhren. Damit
wiirden wir den grossten Theil der Arbeiten mathe-
matischer Physiker verwerfen. Auch W. Thomson
hat eine Reihe theoretischer Arbeiten iiber Carnot's
Gesetz und dessen Consequenzen gemacht, ehe er ein
einziges Experiment dariiber anstellte, und keinem
von uns wird einfallen, deshalb jene Arbeiten gering
schatzen zu wollen.
' R. Mayer war nicht in der Lage, Versuche anstel-
len zu konnen ; er wurde von den ihm bekannten
Physikern zuriickgewiesen (noch mehrere Jahre spater
ging es mir ebenso), er konnte kaum Raum fur die
Veroffentlichung seiner ersten zusammengedrangten
Darstellung gewinnen. Sie werden wissen, dass er
in Folge dieser Zuriickweisungen zuletzt geisteskrank
wurde. Es ist jetzt schwer sich in den Gedanken-
cedure. And we must not unconditionally require that the discoverer
of the idea should necessarily be bound to carry out the second part of
the work. For we should thus have to reject the greater part of the
work of mathematical physicists. Even W. Thomson published a
series of theoretical investigations about Carnot's law and its conse-
quences, before he had made a single experiment connected with it,
and it would not occur to any of us to value these investigations lightly
in consequence.
Mayer was not in a position to make experiments ; he was repulsed
by the physicists with whom he was acquainted (several years later I
was similarly treated), and could scarcely procure room for the publica-
tion of his first compressed exposition. You must know that in con-
sequence of these repulses his mind at last became affected. It is
difficult now to transport oneself back into the circle of thought of that
xii Preface.
kreis jener Zeit zuriick zu versetzen und sich klar zu
machen, wie absolut neu damals die Sache erschien.
Mir scheint dass auch Joule lange hat um Anerken-
nung seiner Entdeckung kampfen mussen.
' Obgleich also niemand leugnen wird, dass Joule
viel mehr gethan hat, als Mayer, und dass in des
letzteren ersten Abhandlungen viele Einzelheiten
unklar sind, so glaube ich doch, muss man Mayer
als einen Mann betrachten, der unabhangig und
selbstandig diesen Gedanken gefunden hat, der den
grossten neueren Fortschritt der Naturwissenschaft
bedingte, und sein Verdienst wird jedenfalls dadurch
nicht geringer, dass gleichzeitig ein Andrer in einem
anderen Lande und Wirkungskreise dieselbe Ent-
deckung gemacht, und sie nachher freilich besser
durchgefiihrt hat, als er.'
With a great part of this I cordially agree, and,
had I to write these chapters afresh, I should pro-
bably advert less strongly than I have done to the
defects and errors of Mayer's earliest paper. But as
my remarks have, for the most part, been already
published, and as I am still convinced of their justice,
time, and to perceive clearly how absolutely new the matter then
appeared. ' It seems to me that even Joule had to struggle long for the
recognition of his discovery.
Thus, although no one will deny that Joule has done far more than
Mayer, and that in the early writings of the latter many points are not
clear, I believe that Mayer must be considered as a man who indepen-
dently and for himself discovered this thought which has produced the
grandest recent advance of natural science ; and his deserts are by no
means diminished by the fact that, simultaneously, another, in another
country and sphere of action, made the same discovery, arid indeed has
since developed it better than he.
Preface. xiii
I think it best to retain them, giving Mayer, however,
the benefit of the able and weighty advocacy of
Helmholtz. It will be seen, however, that those who
with astonishing want of consistency, refuse Stokes
and Stewart any credit as against KirchhofF, while
giving all to Mayer as against Joule, obtain no
sympathy from such a man as Helmholtz.
There are, of course, many others besides Mayer,
whose claims are here discussed. To some of these
my statements may not be satisfactory. I can only
say to such that, as I am not consciously unfair, I
shall heartily acknowledge any misstatements regard-
ing their claims which they may prove me to have
made. But where an experimenter's claims are
founded upon mere verifications of facts deduced from
a theory already securely based upon other facts and
therefore certain, I avoid as far as possible all notice
of them. Such experimenters have claims resembling
those of Falstaff to the death of Hotspur !
I have to express my gratitude to numerous
friends, but especially to Professors Jenkin and Clerk-
Maxwell, and the late Professor Rankine, to whose
kind assistance the volume is largely indebted for
accuracy and completeness. The subject is one
of vast importance, but very few indeed are yet
acquainted with even its most elementary facts ;
and by many of these it is not yet accepted as true.
Besides, the developments which it has received
during the last twenty-five years have to be sought
for in scattered papers in the scientific journals or
xiv Preface.
the transactions of learned societies. I can but hope
that I have made the student's progress easier by
giving him a general sketch of it, with copious refer-
ences to the works in which he will find farther
information.
Reasons similar to those which led to the publica-
tion of the first edition of this work appear to call for
a second. Several books on Thermodynamics (and
especially the very valuable treatise by Clerk-Max-
well) have appeared since 1868, but none that I have
met with takes at all the same ground as this.
It has been carefully revised ; in some places con-
siderably modified, and in others extended. To have
introduced results, like those of Andrews on the
critical temperature of a gas, or of J. Thomson and
Willard Gibbs on the triple point, etc., would have
been impossible without greatly enlarging the volume.
The semi-historical form of the work has rendered
classification by divisions of the subject-matter im-
possible. I hope that the introduction of a Table of
Contents, which Mr. Scott Lang has kindly drawn up
for me, will remove any inconvenience arising from
this cause.
Here my Preface might have ended, had not some
remarks been made upon the first edition by a philo^}
sopher whose high scientific position requires that I
should explain why I have not altered the passages
objected to.
Preface. xv
Having written the work in good faith, and after
somewhat extensive reading ; and having, in conse-
quence of some objections made by Professor Clausius
to my first article in the North British Review, been
fortunate enough to induce the late Professor Mac-
quorn Rankine to re-write for me the chief paragraphs
dealing with Professor Clausius' work, I was some-
what surprised to find myself accused by the latter of
misrepresentation. 1 In the new edition of his Ab-
handlungen recently published, Professor Clausius has
modified this accusation to the charge that my book
was written with the view of claiming the Dynamical
Theory of Heat as far as possible for the British
nation. He adds that he can bring forward the most
definite grounds for this opinion. As such a purpose
never occurred to me, I have anxiously tried to find
what portions of my book can be considered open to
such a charge. The result of my inquiries is not
likely to be satisfactory to Professor Clausius, for I
have now, after careful revision of the whole docu-
mentary evidence, found it necessary to cancel certain
additions which I had made to the paragraphs written
for me by Rankine. These paragraphs, I now see,
were correct as they originally stood. But, thinking
them too severe on Professor Clausius' claims, I rashly
added some mitigating passages, which I have now
1 The correspondence will be found at full length in the Philosophical
Magazine, 1872, and it will be seen that Professor Clausius fancies him-
self to have received even worse treatment from Clerk-Maxwell than from
myself. The first only of my letters was sent to Poggendorff's Annalen.
xvi Preface.
been obliged to retract as unsupported by evidence.
This part of Professor Clausius' attack was, therefore,
really directed against Rankine's statements, although
I had (unjustifiably as I now see) softened them
down before publishing them.
Professor Clausius adds that my book is sehr geschickt
abgefasst. Read by the light of the context this can
only mean that it is skilled special pleading. It is
curious to see how complex and artful one may be
considered who keeps ingenuously to facts.
Professor Clausius also says that the expression
( 178, below) is his own, and that in claiming it for
Sir W. Thomson I referred to an article (On a
Universal Tendency in Nature to the Dissipation of
Mechanical Energy. Phil. Mag., vol. iv., 1852) ' in
which neither that expression nor any expression of
like meaning can be found.' This is very strange
indeed. Professor Clausius ought to have seen at
once that the problems proposed and solved in that
article of Thomson's must have involved in their
solution the expression in question even if formulae
had not been given. Thomson distinctly states in
the article the conditions under which energy is
dissipated. In connection with one of his statements
he solves an important problem connected with the
oeconomic working of a steam-engine, and gives the
expression
Preface. xvii
- f s dt
Rw or we J J Tfi
for the portion, of the heat w, which is ' absolutely
and irrecoverably wasted.' The whole matter is
contained in this. At the time of publication of that
article, Thomson and Joule were engaged in the
necessary work of trying by experiment whether
(another of the formulae in the article) agreed suffi-
ciently with the ordinary air-thermometer reckoning
of temperature to be a convenient assumption. With
this, the above expression for the waste becomes
or (in my notation, 178)
There are obvious misprints in the other formulae of
the article (which are corrected in the list of Errata
in the next volume of the Phil. Mag.), but these do
not affect the meaning of the dissipation, nor the
expression given for it which is merely the integral
of that last written.
In Pogg. Ann., Heft 9., 1877, which I have just
seen, Professor Clausius attacks the Demon-theory of
Clerk-Maxwell, as applied in my Recent Advances in
Physical Science to show the inconclusiveness of his
attempt to establish the second Law of Thermo-
xviii Preface.
dynamics. Professor Clausius says he has not seen
Clerk-Maxwell's own statement of his theory ; which
is strange, inasmuch as it is contained in that very
work of Clerk-Maxwell's 1 of which he so strongly
complained in 1872. In his quotations he altogether
ignores the chief point urged against him in my
Lectures, viz., that what demons could do on a large
scale really goes on without the help of demons
(though on a very small scale) in every mass of gas
that is, of course, if the kinetic theory of gaseous
pressure be true. [See the text below, 53.] For
this reason, and also because Professor Clausius an-
nounces that he has not yet completed his statements,
I do not think it necessary for the present to say more
on this subject.
But this discussion has brought to my recollection
a paper read by Sir W. Thomson to the Royal
Society of Edinburgh, on the Kinetic Theory of the
Dissipation of Energy? Had I thought of this paper
in time, I should have inserted some valuable extracts
from it in the present edition.
P. G. TAIT.
COLLEGE, EDINBURGH, November 1877.
1 Theory of Heat, 1st ed. 1871, p. 308; 4th ed. 1875, p. 328.
2 Proc. R.S.E., or Nature, April 1874.
CONTENTS.
CHAPTER I.
PAGE
HISTORICAL SKETCH OF THE DYNAMICAL THEORY
OF HEAT, i
CHAPTER II.
HISTORICAL SKETCH OF THE SCIENCE OF ENERGY, 56
CHAPTER III.
SKETCH OF THE FUNDAMENTAL PRINCIPLES OF
THERMODYNAMICS 105
NOTE A, . . . . . . -144
NOTE B 149
NOTE C 155
NOTE D . . . .156
INDEX, ... .... 159
CHAPTER I.
HISTORICAL SKETCH OF THE DYNAMICAL
THEORY OF HEAT.
1. What is Heat? The metaphysical arguments on this
question, which, in countless heaps, encumber the shelves
of mediaeval libraries, would merely stupify the reader, and
tend to prevent him from easily understanding the actual
facts, and their bearings. From the earliest times man's
apprehension of the causes and connections of natural
phenomena has been rendered uncertain and imperfect by
his wilfully ignoring the great fact that Natural Philosophy
is an experimental, and not an intuitive, science. No
d priori reasoning can conduct us demonstratively to a single
physical truth ; we must endeavour to discover what t's, not
speculate on what might have been, or presumptuously
decide what ought to have been. Hence it matters not to
us what Aristotle or Bacon may have laid down, or Locke
and Descartes imagined, with regard to the nature of heat.
2. It would be of little use to waste time in a preliminary
sketch of the early history of our subject. Such a sketch
might, perhaps, be made very attractive, but the materials
for it do not yet appear to have been collected. The rapid
march of modern discovery renders it not only useless, but
destructive, to the progress of the Natural Philosopher to
spend much time or trouble in exploring the beginnings of
his science. While he gropes about, seeking the source, his
contemporaries are borne, with ever-increasing swiftness,
along the broadening and deepening current of the river, to
the ' great ocean of truth which lies unexplored before them.'
2 Historical Sketch of the
3. In the physical world we are cognisant of but four
elementary ideas besides the inevitable Time and Space.
They are Matter, Force, Position, and Motion; and Matter is
said to possess Energy in virtue of its position or its motion.
Of these, motion is merely change of position ; and every
change of motion of matter is commonly said to be due to
force. In reality force is merely the rate at which energy
is transferred or transformed per unit of length in a given
direction, or the rate at which it would be transferred or
transformed if some obstacle were removed. The notion
of force is directly suggested to us by our so-called ' mus-
cular sense ' and it is thus difficult to get rid of the idea
that there is such a thing as force. But the direct evidence
of our senses, uninterpreted by reason, is usually wholly
misleading, when not meaningless, in physical science. Now
it is evident that to one or other of the four elementary ideas
above every distinct physical conception must be referred.
To which does heat belong? The old notions of heat
were that it was Matter; or, according to some Philoso-
phers, Force. It is only within about a century that proofs
have been gradually arrived at that when a body is hot its
particles are in Motion; and that the so-called ' Latent
Heat' of Black may possibly not be heat at all, but may
depend on Position; in other words, that heat is in all
cases a form of Energy. These may be startling state-
ments, as they are now advanced, but they will be fully
explained, and to some extent developed, in the course
of the work.
4. Thus it appears that, of the four available hypotheses
as to the nature of Heat, the two necessarily erroneous ones
have, till lately, been almost universally adopted. So much
for the trustworthiness of the metaphysical treatment of a
physical question ! Such a lesson should never be lost sight
of; so deserved and so complete a refutation of the sophistical
nonsense of the schoolmen, and so valuable a warning to the
' Philosopher ' who may be disposed to a priori argument as
Dynamical Theory of Heat. 3
more dignified and less laborious than experiment, can
scarcely occur again. Even the despised perpetual-motionist
has more reason on his side than the metaphysical pretender
to discovery of the laws of nature ; he, to his cost but to
his credit also appeals to experiment to test the validity of
his principle ; but the mighty intellect of his rival scorns
such peddling with apparatus, to it all truth is intuitive ; nay
more, what it cannot comprehend cannot be truth. But
the days of its authority have nearly expired happily for
human progress.
5. When heat was considered to be matter, under the
name of Caloric, it was of course regarded as uncreatable
and indestructible by any process at the command of man.
And there can be no doubt that many very plausible explana-
tions of curious physical phenomena were arrived at by
the labour and ingenuity of the partisans of this theory.
Thus it was natural to suppose that, when caloric entered a
body, or rather combined with it, the body should in general
expand j and, even when heating produced contraction,
there were analogies, quite sufficient to bear out the theory,
supplied by such mixtures as alcohol and water, or alloys as
copper and tin ; where the bulk of the compound is con-
siderably less than the sum of the bulks of the components :
and, as Faraday has shown in the combination of potassium
and oxygen, sometimes less than the bulk of either. Con-
duction of heat, or transference of caloric from one body to
another, or from part to part of the same body, also pre-
sented no difficulty. So it was with the experiments which
led to what was called (from the principles of this theory)
the Specific Heat of bodies ; it had merely to be assumed
that different bodies required different proportions of caloric
to be mixed with them to produce equal effects in the form
of change of temperature. Thus, the specific heat of water
being called i, that of mercury is '033, i.e. a pound of
water requires 30 times more caloric to be mixed with it to
produce a given change of temperature (measured by the
4 Historical Sketch of the
thermometer) than a pound of mercury. The fact that in
heating ice no rise of temperature is observed, however much
heat may have been applied, until the whole of the ice is
melted and similar phenomena observed in every case of
melting or liquefaction, as well as in boiling or vaporization
led Black to propound the doctrine of Latent Heat. The
fundamental ideas of this doctrine, that water differs from
ice at the same temperature simply by the admixture of a
definite equivalent of caloric ; that the steam which escapes
from boiling water, though showing the same temperature
when tested by the thermometer, contains a vastly greater
amount of calorie ; and similar ideas for all similar cases,
were thus easily and directly reduced to the caloric theory.
The additional quantity of caloric was supposed simply to
change the molecular state of the body, without altering its
temperature : hence it was said to be latent. There need
be no hesitation, so far as these, and many other, pheno-
mena of heat are concerned, in pronouncing the explanations
given by the material theory quite satisfactory, although in
several cases they are certainly cumbrous, and difficult of
application. The following extract from Black's Lectures^
gives a very fair idea of some of the best of these specula-
tions and attempts at explanation :
* Thus, to consider in the first place, the slaking of lime, the forma-
tion of lime-water, and some of the qualities of lime-water : The
calcareous earth, in its quicklime state, or deprived of its air, as it has
an attraction for water, will be found to resemble the salts in several
particulars in the mode of this attraction. The salts, if we take them in
their purest state, are disposed to combine with water in two different
ways with a certain quantity of water they unite closely, and with
considerable force to constitute the crystals of salts in which the water
is joined with the particles of salt in such a manner as to become solid
along with them. There are some of the salts which become very hot
in uniting with this portion of water : such are Glauber's salt, Epsom
salt, fixed alkali, and several others. This heat is supposed by most
authors to came out of the salts ; I am rather inclined to think it comes
from the water. After this, if more water be added, the salt unites
1 Edinburgh, 1803. Vol. ii. p. 73.
Dynamical Theory of Heat. 5
with it in a different manner, so as to become fluid along with it, or
form a solution, or liquid, in which the salt is dissolved in the water ;
and, in this part of the process, cold is produced. In the same
manner, if water be added to quicklime, a certain quantity of it is
attracted by the quicklime, and deprived of its fluidity with violence
and heat ; and it adheres to the lime with considerable force, consti-
tuting with it a dry powder, which is called slaked lime. But if this
slaked lime be mixed with a much larger quantity of water, a part of it
is dissolved and composes with the water a lime-water.
* The heat produced in slaking lime is just one of the numberless
examples of the emersion of latent heat. And if any person should
think that the heat produced in some of these instances is too great to
be explained in this way, let him consider that the 140 degrees, which
escape from water in congelation, refers only to the difference between
the heats necessary for appearing in the forms of water and ice ; but we
have no authority to say that the same abstraction of heat from the same
quantity of water will suit its subsequent appearance in a crystal of
Glauber's salt, Epsom salt, or nitre. A much greater emersion may be
necessary or a much less ; therefore, till the experiment be tried, we
cannot say how much heat must emerge before the water can unite with
quicklime in a solid form. And, let it be further remarked, that the
heat extricated in this crystallization can be 'very little diminished by
the subsequent solution, because there is very little lime dissolved in
the lime- water.'
6. But another class of common phenomena afforded no
such easy application of the theory namely, the develop-
ment of heat by friction or percussion ; and it must be
allowed that many of the warmest supporters of the caloric
hypothesis frankly admitted that their explanations of these
effects were not quite satisfactory. The general tendency of
these explanations was towards assuming a change in the
capacity for caloric to be produced by the disintegration
caused by friction or by the compression caused by impact
though it was excessively difficult to see how two such
opposite processes could each produce a diminution of the
capacity. And although the difficulty is lessened by con-
sidering a change in latent heat as well as in capacity to be
produced by attrition or condensation, it is by no means
removed.
6 Historical Sketch of the
7. The mischievous consequences of long persistence in a
false theory were perhaps never better exemplified than in
the case of this supposed materiality of heat ; for so com-
pletely were the scientific men of last century imbued with
it, that when Davy gave a conclusive proof of the actual
creation of heat in a very simple experiment, his consequent
argument against the materiality of heat (or the existence of
caloric) attracted little attention, and was treated by many of
those who condescended to notice it as a wild and extrava-
gant speculation. It is certain that even Davy himself was
led astray in his argument, by using the hypothesis of
change of capacity as the basis of his reasoning, and that
he might have been met successfully by any able Calorist
who, though maintaining the materiality of heat, might have
been willing to throw overboard one or two of the less essen-
tial tenets of his school of philosophers.
8. But Davy's experiment, rightly viewed, is completely
decisive of the question ; and, in spite of the imperfection
of his reasoning from it (due entirely to the prevailing
sophisms of the Calorists), was perfectly satisfactory to him-
self. He developed, in a singularly brief and lucid form,
the fundamental principles of the true theory, in a tract,
forming part of the Contributions to Physical and Medical
Knowledge, principally from the West of England, collected by
Thomas Beddoes, M.D., published at Bristol in 1799.
9. Davy commenced by causing two pieces of ice to rub
against each other, until both were almost entirely melted
by the friction. Here water somewhat above the freezing
point was produced. Davy's confused argument from this
is as follows : ' From this experiment it is evident that
ice by friction is converted into water, and according to the
supposition its capacity is diminished; but it is a well-known
fact that the capacity of water for heat is much greater than
that of ice ; and ice must have an absolute quantity of heat
added to it before it can be converted into water. Friction,
consequently, does not diminish the capacities of bodies for
Dynamical Theory of Heat. 7
heat.' To show that no heat was abstracted from surround-
ing bodies, he proceeded to cause two pieces of metal to
rub against each other by means of clockwork, the whole
apparatus being placed on a block of ice, which had some
unfrozen water in a canal on its surface, and enclosed in a
very perfect vacuum, produced by the now well-known ap-
plication of carbonic acid gas and caustic potash. Here
again heat was developed by the friction, but it did not
come from the ice (for the water in contact with it was not
frozen), nor from surrounding bodies (for in this case it
must have passed through, and melted, the ice, but the ice
remained unaltered). From these perfectly conclusive ex-
periments, Davy proceeds thus :
' Heat, then, or that power which prevents the actual contact of the
corpuscles of bodies, and which is the cause of our peculiar sensations
of heat and cold, may be defined a peculiar motion, probably a vibra-
tion, of the corpuscles of bodies, tending to separate them. It may
with propriety be called the repulsive motion.'
'Bodies exist in different states, and these states depend on the
differences of the action of attraction, and of the repulsive power, on
their corpuscles, or, in other words, on their different quantities of
attraction and repulsion.'
10. Let us here remark, incidentally, what an immense
simplification is at once introduced into our conception of
the laws which regulate the intermolecular forces in bodies.
Davy, by a single sentence or two, thus demolished for ever
the ingeniously unnatural speculations of Boscovich and his
school, who represented the law of the force exerted by one
molecule or particle of a body on another, by a most com-
plex alternation of attractions and repulsions, succeeding
each other as the distance between the two was gradually
diminished, a law so inconsistent with the simplicity of that
of gravitation, that it is amazing that it was ever seriously
propounded.
11. Davy, in fact, makes this very application, and illus-
trates the effect of the repulsive motion in balancing the
attraction of cohesion in bodies by the very apt comparison
8 Historical Sketch of the
to that of the orbital motion of a planet which prevents its
being drawn nearer to the sun. We shall not attempt to
follow his further development of this discovery, where he
falls into an ingenious mistake in consequence of his belief
in the corpuscular theory of light. It has nothing to do
with our subject; yet, though now known to be erroneous,
it is worthy of its author.
12, The rest of this short tract, so far as it relates to heat,
is concerned with the laws of communication of heat, which
are shown to be quite analogous to those of the communi-
cation of motion. It was not, however, so far as we know,
till 1812 that Davy distinctly laid down, in a thoroughly
comprehensive form, the law of the phenomenon. In his
Chemical Philosophy, published in that year, he enuntiates
the following perfectly definite and most important pro-
position :
* The immediate cause of the phenomenon of heat, then, is
motion, and the laws of its communication are precisely the
same as the laws of the communication of motion?
The immense consequences of this statement we shall
presently consider, after we have briefly described the
labours of a contemporary of Davy, who almost succeeded,
in 1798, in demonstrating the immateriality of heat; but
whose work is especially valuable as containing the. first
recorded approximation to the measurement of heat in
terms of ordinary mechanical units, which, singularly
enough, does not appear to have been attempted by Davy.
13. In the Philosophical Transactions for 1798, there is
a most instructive paper by Count Rumford, 1 entitled An
1 This remarkable man (Benjamin Thompson) was driven to Europe
(for his loyalty) when the British colonies in America rebelled. He
effected various important reforms in Bavaria, and chose the title by
which he was generally known, and which was conferred on him for
these services, from the village (now called Concord} in New Hamp-
shire, where he had been obliged to leave his wife and his infant
daughter. Biographic Gtnerale. See also Memoir by Ellis, 1876.
Dynamical Theory of Heat. 9
Inquiry concerning the Source of the Heat which is excited by
Friction. The author's experiments were made at Munich
while he superintended the boring of cannon in the Arsenal ;
indeed, he remarks, that 'very interesting philosophical
experiments may often be made, almost without trouble or
expense, by 'means of machinery contrived for the mere
mechanical purposes of the arts and manufactures.' He
was struck with the very great heat developed by the fric-
tion or attrition of the steel borer on the brass casting ; and
especially, in comparing it with the very small quantity of
chips or powder removed from the metal, justly observing
that it was inconceivable that a mere change of the capacity
for heat in so small a relative quantity of brass could
develop heat sufficient in some cases to boil a large quantity
of water.
* In reasoning on this subject,' he says, ' we must not forget to con-
sider that most remarkable circumstance, that the source of the heat
generated by friction in these experiments appeared evidently to be
inexhaustible. '
' It is hardly necessary to add that anything which any insulated
body, or system of bodies, can continue to furnish without limitation,
cannot possibly be a material substance, and it appears to me to be
extremely difficult, if not quite impossible, to form any distinct idea of
anything capable of being excited and communicated in the manner
that heat was excited and communicated in these experiments, except
it be MOTION.'
Had Rumford only completed his experiment, by dis-
solving separately in an acid the brass turnings, and an
equal weight of the metal in large fragments, he would have
been entitled to the sole credit of the experimental dis-
covery of the true nature of heat.
We shall have occasion again to allude to the contents
of this extremely lucid and philosophical paper ; meanwhile
we may merely observe, that Rumford pointed out other
methods to be employed in determining the amount of
heat produced by the expenditure of mechanical power,
i o Historical Sketch of the
instancing particularly the agitation of water or other liquids,
as in churning.
14. It may be well to pause for a moment at this stage,
and carefully consider to what extent the true theory of
heat had really been advanced about the commencement
of the present century. And it is easy to see from the pre-
ceding pages that the following important facts were then
completely acquired to science :
I. That Heat is Motion (or rather, in strict modern
phraseology, Energy).
II. That the laws of its communication are the same
as those of the communication of Motion (or, in
modern and more expressive phraseology, Energy).
III. Hence that the laws of the communication of Heat
are those laid down by Newton with such expres-
sive brevity in the Scholium to his Third Law of
Motion.
IV. Hence that Heat has a definite mechanical value,
and may be converted into mechanical effect, and
vice versa.
V. That the determination of the accurate value of the
mechanical equivalent of a given amount of heat,
is a question to be resolved by experiment.
VI. That Rumford had obtained an approximation (a
pretty close one as we now know) to the value of
this equivalent.
VII. That this equivalent may be determined by ex-
pending work in the boring or friction of solids,
or in agitating liquids.
15. For the benefit of such readers as may not be ac-
quainted with the elements of mechanics, it will be useful
to give a few explanations of some of the preceding state-
ments, especially with the view of showing their logical
sequence. I. and II. are simply Davy's own expression of
his experimental conclusion. As to III. Newton shows,
though not jui precisely the same words, that when work is
Dynamical Theory of Heat. 1 1
expended solely in setting a body in motion, the energy of
the motion is the measure of the work expended. This
grand statement of Newton's will be considered in the next
chapter. Work is here used in the ordinary engineering
sense of so many ' foot-pounds/ i.e. so many pounds raised
one foot. From this it follows that the heat present in a
body is really a certain definite amount of energy of motion,
which is equivalent to a certain definite amount of mecha-
nical effect or work. This is statement IV. With reference
to VI., which is the only other requiring explanation, it is
easily calculated from the data of one of Rumford's experi-
ments (viz., that the work of one horse for 2h. 3om. raised,
by 1 80 Fahr., the temperature of a mass equivalent in
capacity for heat to 26-58 Ibs. of water), that it requires
about 940 foot-pounds of work to be expended to raise
the temperature of a pound of water i Fahr. [We have
somewhat altered the result first deduced by Joule from this
experiment; for we have used 30,000 instead of 33,000 foot-
pounds per minute as the value of a horse-power the latter,
or Watt's estimate, being now allowed to be too great.]
No account was taken of the heat lost by radiation and
evaporation, which must have been considerable from the
high temperature produced, and the duration of the ex-
periment ; so that, as Rumford himself noticed, this value
must be too high. It is now known to be about 20 per
cent, too great ; still it is a most remarkable result.
16. It does not follow that, if the chief fundamental laws
and principles of a science are known, the development of
them is an easy matter. Take, for instance, the law of
gravitation. It is scarcely possible to conceive a simpler
expression than this for the mutual action of two particles ;
yet, even for the simplest possible application, the motion
of one particle about another, the numerical details are very
troublesome ; and when we have three mutually attracting
particles, the problem (so far as exact solution is concerned)
completely transcends the power of known mathematical
1 2 Historical Sketch of the
processes. It is, of course, infinitely more formidable when
we consider the mutual action of the particles of a body ;
and without the aid of hypotheses, suggested by experiment,
such a case would be incapable of even approximate treat-
ment. Thus we are prepared to find that for the practical
application of the above facts regarding heat, hypotheses
(of a kind suggested by experiment) will always be required
until we know the mechanical constitution of bodies, and
have immensely improved our mathematical methods.
17. For a considerable portion of the present century,
Davy's discoveries about heat were neglected, or only
casually mentioned ; but this was of comparatively little
consequence, as their early reception might have kept back
for a time the grand developments which must next be
mentioned immense strides in the theoretical and mathe-
matical treatment of the subject, and to a considerable
extent independent of the nature of heat. These are due
to Fourier and Sadi Carnot, and it may well be said that it
is in great part attributable to their remarkable works that
the true theory of heat, when revived about thirty-five years
ago, received so rapidly its present enormous development.
18. Fourier's Theoriedela Chaleur, composed before 1812,
is one of the most exquisite mathematical works ever
written, abounding in novel processes of the highest origi-
nality as well as practical utility. It is devoted solely (so
far as its physical applications are concerned) to the pro-
blems of the Conduction and Radiation of heat. Whatever
may eventually be found to be the true laws of conduction
and radiation, Fourier gives the means of completely solving
any problem involving these processes only, and applies his
methods to various cases of the highest interest. He works
out in detail these important cases with the particular
assumption that the flux of heat is proportional to the rate
at which the temperature changes along the line of flow ;
and to a coefficient, depending on the nature of the body,
called its Conductivity. It is only very recently indeed
Dynamical Theory of Heat. 13
that Forbes x has shown that the conductivity of a body for
heat diminishes as its temperature increases ; and thus that
the details of Fourier's solutions (in which the conductivity
is assumed to be constant) are not strictly accurate when
great differences of temperature are involved. But, besides
the fact that Fourier has shown how to adapt his methods
to any experimental data, the solutions he has given are
approximate enough for application to many most interest-
ing investigations, such as the secular cooling of the earth,
underground temperature as depending on solar radiation,
etc. By this powerful method, Fourier has reduced the
treatment of any question involving transference of heat by
conduction or radiation to a perfectly definite form ; and
he must therefore stand, in the history of the subject, as one
of its greatest promoters.
19. Very different in form and object from the systematic
treatise of Fourier, is the profound and valuable essay of
Sadi Carnot, Reflexions sur la Puissance Motrice du Feu,
published in 1824.2 The author endeavours to determine
how it is that heat produces mechanical effect, and though
some of his assumptions are not correct, he investigates the
question in an exceedingly able and instructive manner.
Starting with a correct principle, which, obvious as it is, has
been sadly neglected by many later writers, he is led into
error by assuming the materiality of heat. But with true
philosophical caution he avoids committing himself to this
hypothesis, though he makes it the foundation of his attempt
to discover how work is produced from heat. He says :
* If a body, after having experienced a certain number of transforma-
tions, be brought identically to its primitive physical state as to density,
temperature, and molecular constitution, it must contain the same
quantity of heat as that which it initially possessed ; or, in other words,
1 British Association Report, 1852. Trans. R.S.E. 1862-5.
2 An Account of Car not 1 s Theory of the Motive Power of Heat, etc.
by W. Thomson. Trans. R.S.E. 1849.
14 Historical Sketch of the
the quantities of heat lost by the body under one set of operations are
precisely compensated by those which are absorbed in the others. This
fact has never been doubted ; it has at first been admitted without
reflection, and afterwards verified in many cases, by calorimetrical
experiments. To deny it would be to overturn the whole theory of
heat, in which it is the fundamental principle. It must be admitted,
however, that the chief foundations on which the theory of heat rests
would require a most attentive examination. Several experimental
facts appear nearly inexplicable in the actual state of this theory.'
This fundamental principle of Carnot is evidently axio-
matic (so far as regards the quantity of heat in the body
when it is restored to its original state) : but there is a
serious error as regards the equality of the quantities of
heat received and given out by the body during the trans-
formations.
20. The erroneous portion of the above statement of
Carnot is contained in the clause, ' in other words, the
quantities of heat lost by the body under one set of opera-
tions are precisely compensated by those which are absorbed
in the others.' This is only true when as much work has
been expended, in bringing back the body to its primitive
state, as has been done by it in expanding. But we must
remark the peculiar merit of Carnot's reasoning, which con-
sists in the idea of bringing the body back to its initial state,
as to temperature, density, and molecular condition, after a
cycle of operations, before making any assertion, as to the
amount of heat which it contains. We shall be enabled to
appretiate the value of this idea better when we see what
others have been led to by ignoring it.
21. Thus, from Carnot's point of view, it is evident that
the motive power of heat depends upon its being transferred
from one body to another through the medium by whose
change of volume or form the external mechanical effect is
produced, as this medium is supposed to remain at the end
of the operation in precisely the same state as at the com-
mencement. He gives, as analogous, the instance of work
derived from water falling from a higher to a lower level.
Dynamical Theory of Heat. 1 5
Hence, for the production of mechanical effect, we are to
look to the successive communication of heat to, and abs-
traction of heat from, the particular medium employed ; and
to illustrate this it is natural to consider the steam-engine as
the most extensive practical application of the principle.
22. Carnot's reasoning may easily be made intelligible
without mathematical details. In the simple case given
below, all that is attempted is to show that in the ascent of
the piston in the cylinder, more work is done against external
forces than is required to be done by them to produce the
descent and restore the piston to its first position. And in
order that Carnot's axiom may be applied with strictness, and
yet with simplicity, it is better to consider a hypothetical,
than the actual, engine.
23. Suppose we have two bodies, A and B, whose tem-
peratures, S and T 7 , are maintained uniform, A being the
warmer body, and suppose we have a stand, C, which is a
non-conductor of heat. Let the piston and the sides of the
cylinder be also non-conductors, but let the bottom of the
cylinder be a perfect conductor ; and let the cylinder contain
a little water, nearly touching the piston when pushed down.
Set the cylinder on A ; then the water will at once acquire
the temperature S, and steam at the same temperature will
be formed till the space above the water is saturated, so that
pressure must be exerted to prevent the piston from rising.
Let us take this condition as our starting-point for the cycle
of operations.
First, Allow the piston to rise gradually ; work is done
by the pressure of the steam which goes on increasing in
quantity as the piston rises, so as always to be at the same
temperature and pressure. And heat is abstracted from A,
namely, the latent heat of the steam formed during the
operation.
Second, Place the cylinder on C, and allow the steam to
raise the piston farther. More work is done, more steam is
formed, but the temperature sinks on account of the latent
1 6 Historical Sketch of the
heat required for the formation of the new steam. Allow
this process to go on till the temperature falls to J 1 , the
temperature of the body B.
Third, Now place the cylinder on B ; there is of course
no transfer of heat. But if we now press down the piston,
we do work upon the contents of the cylinder, steam is
liquefied, and the latent heat developed is at once absorbed
by B. Carry on this process till the amount of heat given to
B is exactly equal to that taken from A in the first operation,
and place the cylinder on the non-conductor C. The tem-
perature of the contents is now T, and the amount of caloric
in them is precisely the same as before the first operation.
Fourth, Press down the piston farther, till it occupies the
same position as before the first operation ; additional work
is done on the contents of the cylinder, a farther amount of
steam is liquefied, and the temperature rises.
Moreover, it rises to S exactly, by the fundamental axiom,
because the volume occupied by the water and steam is the
same as before the first operation, and the quantity of
caloric they contain is also the same as much having been
abstracted in the third operation as was communicated in
the first while in the second and fourth operations the
contents of the cylinder neither gain nor lose caloric, as
they are surrounded by non-conductors.
Now, during the first two operations, work was done by
the steam on the piston, during the last two work was done
against the steam ; on the whole, the work done by the
steam exceeds that done upon it, since evidently the tem-
perature of the contents, for any position of the piston in
its ascent, was greater than for the same position in the
descent, except at the initial and final positions, where it is
the same. Hence the pressure also was greater at each
stage in the ascent than at the corresponding stage in the
descent, from which the theorem is evident.
Hence, on the whole, a certain amount of work has been
communicated by the motion of the piston to external
Dynamical Theory of Heat. 17
bodies; and, the contents of the cylinder having been
restored exactly to their primitive condition, we are entitled
to regard this work as due to the caloric employed in the
process. This we see was taken from A and wholly trans-
ferred to B. It thus appears that caloric does work by being
let down from a higher to a lower temperature. And the
reader may easily see that if we knew the laws which con-
nect the pressure of saturated steam, and the amount of
caloric it contains, with its volume and temperature, it
would be possible to apply a rigorous calculation to the
various processes of the cycle above explained, and to ex-
press by formulae the amount of work gained on the whole
in the series of operations, in terms of the temperatures (S
and T) of the boiler and condenser of a steam-engine, and
the whole amount of caloric which passes from one to the
other.
24. Though the above process is exceedingly ingenious
and important, it is to a considerable extent vitiated by the
assumption of the materiality of heat which is made through-
out. To show this, it is only necessary to consider the
definition given for the limit to which the condensation is
to be carried in the third operation; being that as much
heat is to be emitted during it as was taken in during the
first operation. But it is quite easy, as seems to have been
first remarked by J. Thomson in 1849, to put Carnot's
statement in a form which is rigorously correct, whatever be
the nature of heat. J. Thomson says
'We should not say, in the third operation, "compress
till the same amount of heat is given out as was taken in
during the first." But we should say, "compress till we
have let out so much heat that the farther compression
(during the fourth stage) to the original volume may give
back the original temperature." '
It is preferable, therefore, to go through Carnot's cycle
again with the slight change of starting with the fourth of
the preceding operations. This we leave to the student.
B
UNIVERSITY
1 8 Historical Sketch of the
The reason will be easily apprehended by him when he
reads the remarks on Watt's Indicator Diagram given in
the third chapter.
It is but bare justice, however, to acknowledge that
Carnot himself was by no means satisfied with the caloric
hypothesis, and that he insinuates, as we have already seen,
more than a mere suspicion of its correctness. The student
may easily see the difficulty, if he considers that heat may
be let down (by conduction merely) between bodies of
different temperatures, without doing any work.
25. But we owe to Carnot much more than this, as will
how be shown ; deferring to a later section an examination
of the curious particulars in which his results for the steam-
engine, or the air-engine, differ from those now received.
26. If we carefully examine the above cycle of operations
we easily see that they are reversible, i.e. that the trans-
ference of the given amount of caloric back again from B to
A, by performing the same operations in the opposite order,
requires that we expend on the piston, on the whole, as
much work as was gained during the direct operations.
This most important idea is due to Carnot, and from it he
deduces his test of a perfect engine, or one which yields
from the transference of a given quantity of caloric from one
body to another (each being at a given temperature) the
greatest possible amount of work. And the test is simply
that the cycle of operations must be reversible.
To prove it we need only consider that, if a heat-engine
M could be made to give more work by transferring a given
amount of caloric from A to B, than a reversible engine N
does, we may set M and N to work in combination, M
driven by the transfer of heat, and in turn driving IV, which
is employed to restore the heat to the source. The com-
pound system would thus in each cycle produce an amount
of work equal to the excess of that done by J/ over that ex-
pended on IV, without on the whole any transference of
heat, which is entirely contradictory of experience.
Dynamical Theory of Heat. 19
Carnot, therefore, proved, upon his assumptions, that the
ratio of the work done by a perfect (i.e. a reversible) engine
to the heat taken from the source is a function of the tem-
peratures of the source and condenser only ; because no
mention is necessarily made of any particular substance.
When these temperatures are nearly equal, this function is
expressible by the product of their difference into a function
of either, which is called Carnofs Function.
W. Thomson, so early as I848, 1 seized upon this remark-
able proposition, and made it the basis of the earliest
suggestion of an absolute Thermometric Scale ; absolute in
the sense of being based upon strict thermodynamic prin-
ciples, and entirely independent of the properties of any
particular substance. This extremely valuable suggestion
will be carefully considered in our third chapter.
27. The remarkable consequences deduced by W. Thom-
son, by a combination of the methods and results of
Fourier and Carnot, with reference to the Dissipation of
heat, and the final transformations of all forms of energy,
though properly belonging to this part of the development
of the subject, are left to a future page, so that the chrono-
logical order may be adhered to as closely as possible in
presenting the most important additions to the science.
28. A little before the publication of Carnot's work, a
second method of procuring mechanical effect from heat
was discovered by Seebeck. It consists in the production
of electro-motive force by the action of heat on hetero-
geneous conducting matter, and the employment of the
current thus produced to move a galvanometer needle ; or,
in later improvements, to drive an electro-magnetic engine.
It is not alluded to by Carnot j and it will tend greatly to
the simplicity of this explanatory narrative if the considera-
tion of the other physical agents, which the grand principle
of conservation of energy has shown to be so intimately
1 Proc. Camb. Phil. Soc. (Phil. Mag. 1848).
2O Historical Sketch of the
related to heat, be deferred to another chapter. We shall,
therefore, at present, confine ourselves as strictly as possible
to the relation between heat and mechanical effect, which
is, however, only one branch of the dynamical theory.
29. Clapeyron, in I834, 1 recalled attention to Carnot's
reasoning, and usefully applied the principle of Watt's dia-
gram of energy to the geometrical exhibition of the different
quantities involved in the cycle of operations by which work
is derived from heat by the temporary changes it produces
in the volume or molecular state of bodies. He also, first,
gave a representation of Carnot's processes in an analytical
form. But for nearly twenty years after the appearance of
Carnot's treatise little appears to have been done with
reference to the theoiy of heat.
30. Then there appeared, almost simultaneously, a group
of three or four speculators and experimenters whose relative
claims have been since pressed, in some cases, with con-
siderable warmth, though it seems not very difficult to
estimate them so far as the discovery either of the true
theory, or the mechanical equivalent, of heat is concerned.
In 1837, Mohr 2 pointed out very clearly many of the neces-
sary consequences of the establishment of the undulatory
nature of radiant heat ; and showed how the work-equivalent
of heat might be found from the two specific heats of air.
In 1839, Seguin, a relative of the celebrated Montgolfier
(from whom indeed he says he derived his ideas on the
subject of heat), in a very curious work on railways, 3 gave
data from which it is easy to deduce 650 kilogrammetres 4
as the mechanical equivalent of heat. 5 In 1842, Mayer 6
1 y&urnal de VEcole Royale Polytechnique, vol. xiv. (Scientific Me-
moirs, I.)
2 Liebig's Annalen Ansichten ilber die Natur der Warme. Trans-
lated in Phil. Mag. August 1876.
8 Sur V Influence des Chemins de Fer.
* That is, a kilogramme of water must fall 650 metres to have its
temperature raised by i C.
5 Phil. Mag. Oct. 1864. 6 Liebig's Annalen.
Dynamical Theory of Heat. 2 1
assigned 365 kilogrammetres for the value of that physical
constant. It is curious to observe that the methods em-
ployed by the two latter were almost identical : that of
Seguin being founded on the principle that the work given
out by any body dilating, and thereby losing heat, is the
equivalent of the heat lost ; while that of Mayer is, that the
heat developed by compression is the equivalent of the work
expended in compressing the body. Neither makes the
slightest limitation as to the nature of the substance to be
experimented on, both their statements are perfectly general ;
and, it may be added, not only inaccurate, but (with certain
exceptions) not even roughly approximate. 1 Mayer pro-
fesses to found his process on a species of metaphysical
reasoning as to the indestructibility of force (Kraft] we
have already seen what value is to be attached to specula-
tions of this nature. 2 Besides, Mayer gives, as an analogy
to the compression of a body and the consequent production
of heat, the fall of a stone to the earth or the impact of a
number of gravitating masses, and the consequent heating
of all. This, it need scarcely be said, is inadmissible. His
hypothesis might possibly have been a law of nature, but it
never could have had any analogy with the gravitation case
1 Seguin and Mayer have still followers. For instance ' Qu'un gaz
ou tout autre corps soit reduit, par le travail du a des forces exterieures,
a diminuer de volume, il y aura en meme temps production d'une
quantite d chaleur qui sera dans le meme rapport constant avec le
travail mecanique depenseV Combes, Thborie Mecanique de la Chaleur.
Paris, 1867, 8vo.
2 Mayer does not accept the conclusions deduced by Davy and Rum-
ford from their experiments, and now almost universally received. lie
says, * Wir mochten vielmehr das Gegentheil folgern, dass um zu
Warme werden zu konnen, die Bewegung sey sie eine einfache, oder
eine vibrirende, wie das Licht, die strahlende Warme, etc., aufJidren
musse, Bewegung zu seyri.' Thus, according to Mayer, heat is
potential energy, unless indeed there be some mysterious tertium quid.
For a farther examination of this question, and especially for the
opinions of Joule and Colding upon the so-called claims of Mayer,
see Tait, Recent Advances in Physical Science.
2 2 Historical Sketch of the
he compares it to. Besides, so far as air is concerned, the
hypothesis had been given only five years before by Mohr
in the very same journal. Mayer does not even allude to
Mohr's paper.
31. But what it most concerns us to note here is, that all
three entirely ignore Carnot's fundamental principle, viz.,
that no deduction whatever can be made as to the relation
between heat and mechanical effect, when the body operat-
ing or operated upon is in different states at the beginning
and end of the experiment. 1 Take, for instance, the second
operation in the cycle of Carnot as above explained. Yet
it is to be observed that Seguin distinctly pointed out that
steam, which has done work in an engine, ought not to heat
the water in the condenser so much as if it had been
directly led into it ; and made numerous but indecisive
experiments to prove the truth of this statement (conclusively
verified by Him in 1862).
32. The numerical data requisite for the application of
either of these erroneous methods were known at the time
for only one or two bodies, and, even for these, very
inaccurately. So that it is not at all remarkable that the
equivalents above given are far from exact. Seguin reasoned
from steam, Mohr, and after him Mayer, from air. It
happens that this paucity of data led Mayer to choose a
substance which Joule afterwards experimentally proved to
be capable of giving, even with the erroneous hypothesis,
a result not far from the truth ; but, even if Mayer had
in 1842 possessed accurate data, and had therefore been
fortunate enough to obtain an approximate result instead of
a very inexact one, his determination could never have been
correctly called more than a happy guess founded upon a
total neglect of sound reasoning. It has been stated by
some writers that Mayer is the author of the Dynamical
Theory of Heat ; and that he deduced in 1842, by a simple
1 Thomson, Trans. R.S.E. 1851, p. 291, Note.
Dynamical Theory of Heat. 23
calculation, as accurate a value of the dynamical equivalent
as Joule arrived at in 1849, after seven years of laborious
experiment. It is difficult to perceive the grounds on
which such statements are made. The dynamical theory,
as we have already seen, was established by Davy and
Rumford. Mayer, three years later than Seguin, and five
years later than Mohr, enuntiated and applied, like them, a
false principle, and got (from the experiments of others on
the specific heats of air) a widely erroneous result, which
was improved, not by its author but by Joule, two or three
years afterwards ; who, after finding the true result by a
legitimate process, proved by experiments on air 1 that Mayer
ought to have got a good approximation.
33. Another of the group is Cold ing, a Danish engineer,
whose results were published in 1 843.2 He appears to
have been led by a species of metaphysical reasoning to
the idea of the conservation of energy ; but, unlike some
other speculators, to have appealed to experiment before
publishing his views. The value (350 kilogram metres) of
the equivalent of heat which he thus obtained from friction
experiments, is not much more accurate than that deduced
from Rumford's data, and is not to be compared with
Joule's of the same year. Still Colding evidently went to
work in the right way, and deserves an amount of credit to
which Mohr, Seguin, and Mayer have no claim.
34. It must be premised that much of Joule's work
(which dates back as far as 1840 in the Proc. JR.S.) has
reference to the general theory of conservation of energy,
and that his first determinations of the dynamical equiva-
lent of heat were obtained by means of the magneto-electric
machine, so that, in accordance with the definite object of
the present chapter, only such of his investigations will be
now noticed as strictly bear on the immediate relation
between heat and mechanical effect.
1 Phil. Mag. 1844, ii. 2 Phil. Mag. Jan. 1864.
24 . Historical Sketch of the
35. His earliest published experiments of this class are
described in the appendix to a paper published in 1843 in
the Philosophical Magazine, having been read before the
British Association at its meeting in Cork. The valuable
discoveries contained in this paper do not properly belong
to the present subject, but will be carefully considered in
the second chapter. In the appendix, however, there is
described an experimental method of directly determining
the mechanical equivalent of heat, so simple, and yet so
effective, as to deserve careful consideration. It consisted
in working up and down in a closed cylinder, filled with
water, a piston formed of a number of capillary tubes
bound together, so as to constitute a mass with visible
pores. The friction of the water when forced to pass
through these tubes of course developed heat, which, as
well as the work spent in moving the piston, was carefully
measured. It is very remarkable that, from the series of
experiments, agreeing well with one another, which were
made with this simple apparatus, Joule deduced as the
dynamical equivalent of heat (that is, of the heat required
to raise the temperature of a pound of water i F.)
770 foot-pounds, 1
differing by only about a quarter per cent, from the results
of his subsequent and far more elaborate determinations.
The close agreement of the results of successive trials was
quite sufficient to justify him in publishing this, as in all
probability a very close approximation to the desired value
of the equivalent.
[It is curious that the mean of the values deduced from
1 To compare this with the estimates of Seguin and Mayer, and the
result of Colding, we must remember that a metre is 3 '28 feet, and a
centrigrade degree ths of a degree Fahrenheit, while the unit of heat
is capable of raising the temperature of a kilogramme of water by i
C. So that Joule's numerical result, expressed in kilogrammetres, is
9* 77 =422 nearly.
5 x 3-28 '
Dynamical Theory of Heat. 25
Rumford's and Colding's experiments, the two legitimate
ones whose publication preceded that of this result of
Joule's, differs from it by only about 2 J per cent]
36. Before leaving this part of our subject it may be
desirable to complete the enumeration of the results of
Joule's direct experiments for the determination of the
mechanical equivalent, as they are certainly superior in
accuracy to those of any other experimenter.
Repeating in 1845 an d 1847 his experiments on the
friction of water but now by means of a horizontal paddle,
turned by the descent of known weights he obtained
results gradually converging, as in each successive set of
experiments extraneous causes of error were more com-
pletely avoided or allowed for. The value of the equivalent
deduced in 1847 from a great number of experiments with
water was 781*5 foot-pounds, and with sperm oil, 782*1.
In the paper of 1845, we find his first speculations as to the
absolute zero of temperature, or the temperature of a body
absolutely deprived of heat. The most interesting of his
results are, that the absolute zero of temperature is 480
Fahr. below the freezing-point of water, and that a pound
of water at 60 Fahr. possesses, in virtue of its heat, me-
chanical energy to the enormous amount of at least 415,000
foot-pounds. Changes have since been shown to be neces-
sary in these numbers, but they are comparatively unim-
portant. And it must be regarded as one of the most
extraordinary results of physical science, that a pound of
water at ordinary temperatures contains heat capable (if it
could be applied) of raising it against gravity to a height
of at least 80 miles.
37. Finally, in 1849, Joule published the results of his
latest and most elaborate experiments, of which, after what
has been already said, the results only need be given :
From friction of Water, 772-692 foot-pounds.
Mercury, 7 7 4*083
Cast-iron, 774*987
26 Historical Sketch of the
The conclusions of this valuable paper, after all allowance
is made for slight but inevitable losses of energy, by sound
and other vibrations, are thus given :
ist, The quantity of heat produced by the friction of bodies,
whether solid or liquid, is always proportional to the quantity
of work expended.
2d, The quantity of heat capable of increasing the tempera-
ture of a pound of water (weighed in vacuo, and taken at
between 55 and 60) by i Fahr., requires for its evolution
the expenditure of a mechanical force represented by the fall of
772 Ibs. through the space of one foot.
It is only necessary to observe, that the determination is
for the value of gravity at Manchester, and must of course
be diminished for higher, and increased for lower latitudes,
according to the well-known law.
38. As no one has yet pretended to rival in accuracy the
experiments of Joule above mentioned, and as his cele-
brated result of 1843, so very close to the truth, preceded
all other recent sound attempts to determine the mechanical
equivalent of heat, the results of direct methods since em-
ployed by other observers may be passed over, with the
remark, that they agree more or less perfectly with those of
Joule.
39. We now come to the consideration of the method
suggested by Mohr, Seguin, and Mayer, with which Joule
seems to have occupied himself experimentally in 1844.
We shall briefly describe his experiments, though not in the
order in which they were made, this change being required
for the continuity of our exposition. Joule, repeating in a
greatly improved form an old experiment of Gay Lussac,
compressed air to twenty atmospheres or so in a strong
vessel, which was afterwards screwed to another previously
exhausted. A very perfect stop-cock prevented all passage
of air from one to the other until it was desired. The
whole was placed in a vessel of water, which was stirred to
bring it to a uniform temperature. On opening the stop-
Dynamical Theory of Heat. 27
cock, the air rushed from the first vessel to the second, so
that in a short time the pressure was the same in both. On
measuring the temperature of the surrounding water again,
no change was perceptible? at least after the proper correc-
tions, determined by separate experiments, had been made
for the amount of heat produced by the stirring, etc., during
the operation. This is a most important result, as will be
shown immediately, though it is as well to say at once that
it is not absolutely exact, as is proved by subsequent ex-
periments capable of even greater accuracy than that just
described. The condensed air has been allowed to expand
without doing work on external bodies, and though its
volume has been greatly increased, no heat has been lost,
though we might have imagined such would be the case.
From this we are entitled to conclude, that the heat
developed by compressing a gas is (to the amount of
approximation already mentioned) the equivalent of the
mechanical effect expended in the compression, and thus
that the assumption made by Mohr, Seguin, and Mayer,
unwarrantable as it is for bodies in general, is very nearly
true for air. Why, then, was Mayer's value of the me-
chanical equivalent so erroneous? Simply because the
direct determination of the specific heat of air is an exceed-
ingly difficult and delicate operation, and had been only
very roughly effected before 1842. Rankine and Thomson
first theoretically assigned the true value, founding their
calculations on Joule's experimental results from the friction
of fluids, f Joule, by a direct experiment, obtained a closely
accordant value ; and finally Regnault, also by direct experi-
ment, obtained exactly the number predicted from theory.
1 In Gay Lussac's experiment, the temperatures were measured by
thermometers suspended in the centre of each vessel. One was
observed to rise as much as the other fell ; but it was found that very
different effects were produced when the thermometers were not in the
centres of the vessels ; so that no quantitative deduction could be made
from the experiment.
28 Historical Sketch of the
40. What actually took place in Joule's experiment was,
the air in the first vessel, suddenly expanding, produced
mechanical effect in forcing a portion of its mass with great
velocity into the second vessel ; this it did at the expense of
its store of energy in the form of heat. Thus the first
vessel was cooled to a certain extent. The air rushing into
the second vessel produced, by friction against the con-
necting tube and the sides of the vessel, and amongst its
own particles, and by the condensation produced by each
successive arrival of air upon all which preceded it, a de-
velopment of heat. Thus the second vessel was heated.
Now it is obvious that we are not at liberty (without experi-
mental proof) to assume that the loss of heat in the first
vessel will be exactly, or even nearly, equal to the gain in
the second. But as experiment has shown them to be
almost equal, either the heat produced by condensing air,
or the cold produced by its expansion from a condensed
state, may legitimately be taken as one of the data for an
approximate determination of the mechanical equivalent.
The last cited paper of Joule's contains five sets of
careful experiments made for this purpose by one or
other of these methods. The extreme results are 823
and 760 foot-pounds respectively; the mean of the last
three sets, chosen as the most likely to be correct, giving
the number 798 foot-pounds only about 3^ per cent, too
great.
The student ought carefully to notice here that the
process employed in Joule's experiment ( 39) is essentially
an irreversible one. The expanded gas has lost no heat to
external bodies, and it has done no external work, yet its
power of doing work has been notably diminished, and it
could not be restored to its original condition without
expenditure of work from an independent source, and sub-
sequent removal of an equivalent amount of heat. In fact,
the whole energy contained in the gas remains unchanged
in amount, but part of its convertibility into work has been
Dynamical Theory of Heat. 29
lost. This will be more easily understood when we have
considered the Dissipation of Energy.
41. It may now be asked, does the dynamical theory of
heat necessitate any serious change in the important results
deduced by Carnot from the caloric hypothesis? This
question was answered with greater or less detail in 1849,
1850, and 1851 respectively, by Rankine, Clausius, and W.
Thomson.
42. Rankine's first investigation of the principles of the
mechanical action of heat appeared in a paper received
by the Royal Society of Edinburgh in December 1849,
and read in February I850. 1 It is based on what he calls
the 'Hypothesis of Molecular Vortices;' that is to say,
the supposition that the motions of which Davy showed
thermometric heat to consist are of the nature of vortices,
whirls, or circulating streams. That is the part of the
hypothesis which is specially connected with the pheno-
mena of the mechanical action of heat; but in order to
connect these with some other phenomena, Rankine makes
the further supposition, that the whirling matter is diffused
in the form of atmospheres round nuclei, which may be
either bodies of a special kind, or centres of condensation
and attraction in the atmospheres ; and that radiance,
whether of heat or light, consists in the transmission of a
vibratory motion of the nuclei, by means of forces which
they exert on each other. The quantity of heat in a body
is the energy of its molecular vortices ; the absolute tempera-
ture of the body is the same energy divided by a specific
co-efficient for each particular substance. A perfect gas is
a substance in which the elastic pressure is sensibly that
which varies with the centrifugal force of the vortices only ;
and the intensity of that pressure, according to the known
principles of mechanics, must be proportional directly to the
energy of the vortices and inversely to the space that they
1 Transactions, vol. xx. p. 147.
30 Historical Sketch of the
occupy. In substances not perfectly gaseous the elasticity
is modified by attractive or cohesive forces. When the
deviation from the perfectly gaseous state is small, the effects
of such forces may be approximately represented by series
in terms of the reciprocal of the absolute temperature. 1
43. Sensible heat, according to Rankine, is the energy
employed in varying the velocity of the whirling par-
ticles ; latent heat, the work done in varying the dimen-
sions of their orbits when the volumes and figures of
the spaces in which they whirl are changed. The force
which keeps any particle in its orbit is equal and opposite
to the centrifugal force of that particle ; therefore the
work done in varying the orbits of the particles is propor-
tional to their centrifugal force ; therefore to the energy of
the vortices ; therefore to the absolute temperature ; and to
compute that quantity of work, or latent heat, when a body
undergoes a given variation of dimensions, the absolute
temperature is to be multiplied by the corresponding varia-
tion of a certain function of the dimensions and elasticity of
the body ; which function is computed by taking the rate
of variation with temperature, of the external work done
during the kind of change of dimensions under considera-
tion. Such is an outline of the method by which Rankine
deduces the gene ral equation of the mechanical action of heat,
from the Hypothesis of Molecular Vortices, by means of
known dynamical principles.
44. The quantity whose variation, being multiplied by
the absolute temperature, gives the latent heat corresponding
to a given change of dimensions at that temperature, is ex-
pressed inRankine's earlier papers by symbols, but is not desig-
nated by a special name. In a paper, read in January 1853,2
1 Rankine had previously published an example of the use of such
series in a paper on the * Elasticity of Vapours,' Edin. Phil. Journal,
July 1849, and he also applied them with success to the elasticity of
carbonic acid, and some other gases. Phil. Mag. Dec. 1851.
a Trans. R.S.E. vol. xx. p. 569.
Dynamical Theory of Heat. 3 1
he proposes the name 'Heat-Potential/ and in a paper
received by the Royal Society of London in December
1853, and read in January 1 1854, he gives to the same
quantity, with a certain additional term depending on change
of temperature, the name of ' Thermodynamic Function ;' a
name which has since been adopted by various other authors.
In Rankine's paper of 1849, the chief applications of the
general equation of thermodynamics are as follows : The
values of apparent as distinguished from real specific heat,
for gases and vapours under various circumstances; the
demonstration that the apparent specific heat of a vapour
kept constantly at the pressure of saturation while its volume
varies is negative for most fluids at ordinary temperatures ;
in other words, that steam (for example) tends to become par-
tially liquefied when it works expansively, contrary to what
had been previously believed ; and the demonstration that
the total heat of evaporation of a perfect gas increases with
temperature at a rate equal to the specific heat of the gas at
constant pressure. In a paper read on the 2d December
1850,2 he deduced from Joule's -Equivalent the value 0*24
for the specific heat of air at constant pressure, and con-
cluded that the previously received value, 0-2669 must be
erroneous; this was verified by Joule's experiments, com-
municated to the Royal Society on the 23d of March 1852,
and by Regnault's experiments, communicated to the
Academy of Sciences in 1853. In a paper read on the
2ist April 185 1, 3 he 'deduced from the general equation of
thermodynamics, as given in his paper of 1849, Thomson's
law (54 below) of the efficiency of a perfect heat engine :
that the whole heat expended is to the heat which disap-
pears in doing mechanical work, as the absolute temperature
at which heat is received to the difference between the
temperatures at which heat is received and rejected.
45. In Rankine's paper of 1849, groups of circular
1 Phil. Trans. 1854.
2 Trans. R.S.E. vol. xx. p. 191. 3 Ibid. p. 205.
32 Historical Sketch of the
vortices were supposed to be arranged in spherical layers
round the atomic nuclei, in order to simplify the investiga-
tion. On the 1 8th December 1851 he read a paper, 1 in
which it is shown that precisely the same results as to the
relation between heat, elasticity, and mechanical work follow
from the supposition of molecular vortices of any figure
arranged in any way.
In a long series of papers he applied the principles of
thermodynamics to various practical questions relating to
the steam-engine and other heat engines ; and he was the
author of the first separate treatise in which the science of
thermodynamics was set forth with a view to its practical
application. 2
We have treated Rankine's theoretical investigations at
considerable length here, because they are founded on a
peculiar hypothetical basis ; and, so far as method is con-
cerned, are widely separated from those of Clausius,
Thomson, and others, which have among them some slight
resemblance as regards fundamental assumptions and mode
of investigation.
46. The first paper of Clausius 3 on the Mechanical Action
of Heat was read to the Berlin Academy of Sciences in
February 1850, and printed in PoggendorfFs Annalen for
March and April of the same year. It is divided into two
parts ; the first relating to the mechanical action of heat in
perfect gases ; the second to the mechanical action of heat
in substances in general. In the first part, Clausius makes
use of the first law of thermodynamics only, viz., that of
the equivalence of heat and mechanical work, without any
reference to a second law ; and thus he arrives at a general
equation of the mechanical action of heat in perfect gases,
1 Trans. R.S.E., p. 425.
2 A Manual of the Steam- Engine and other Prime Movers, 1859.
8 His papers are collected as Abhandlungen ilber die mechanische
Wdrme-theorie. Leipsic, 1864-7 (2d ed., 1876). See also Clatisius on
Heat, Van Voorst, 1867.
Dynamical Theory of Heat. 33
containing a certain unknown function. He endeavours to
distinguish between the external work done by an elastic
substance in expanding against external pressure, and what
he calls the internal work done by the particles of the body
in altering their relative positions against the cohesive forces
which they exert on each other ; and he states, as being
highly probable, Mayer's hypothesis, modified as follows ;
in perfect gases the internal work is inappretiably small
This supposition enables him to assign a definite form to
the function which was unknown in the previous equation,
and so to obtain a more definite equation for the mechanical
action of heat in perfect gases ; and from the latter equation
he deduces various results which are in conformity with
experiment.
47. In the second part of the paper, Clausius for the first
time makes use of ' Carnot's Law/ modified in such a way
as to bring it into harmony with the fact of the equivalence
of heat and mechanical energy ; viz., that when a substance
performs mechanical work, by going through a reversible
cycle of changes at the end of which it returns to its original
condition, the ratio borne by the quantity of heat which
disappears in performing work, to the whole quantity of heat
expended, is a function solely of the temperatures at which
the changes take place. By combining Carnot's law, as thus
modified, with the results obtained in the first part of the
paper, involving the supposition that the internal work in per-
fect gases is inappretiable, he arrives at the conclusion that
the probable value of ' Carnot's Function' 1 is the reciprocal
of the absolute temperature as measured on a perfect gas
thermometer; and thus he obtains the mathematical expres-
sion of the second law of thermodynamics, as the expression
of three principles combined, viz. :
(i.) The equivalence of heat and mechanical work ;
1 The same law had been previously suggested by Joule as probable
in a private letter to Thomson, dated 9th December 1848. (See also
54 below.)
C
34 Historical Sketch of the
(2.) The principle of Carnot, modified to harmonise with
that of the equivalence of heat and mechanical work ;
(3.) Se'guin and Mayer's hypothesis restricted to perma-
nent gases (and since proved to be very approximately true
for them by the experiments of Joule and Thomson).
48. Clausius applies his general equations to the evapora-
tion of liquids ; and arrives, amongst other results, at the
conclusion that most saturated vapours, when working
expansively at ordinary temperatures, tend to become
partially liquefied ; a conclusion arrived at simultaneously
( 44) by Rankine.
49. Passing over some papers relating to matters of detail
connected with the mechanical theory of heat, the next
paper of Clausius in which fundamental principles of that
theory are investigated, is ' On an altered form of the second
law of the mechanical theory of heat.' 1 The peculiar form
of the second law of Thermodynamics here referred to (but
which was first given by Thomson in 1851, see Chap, in.)
is called by the author 'the Law of the equivalence of
transformations.' It is expressed in words to the following
effect that ' In all cases in which a quantity of heat is trans-
formed into work, and the bodies by means of which that
transformation is effected return at the end of the operation
to their original condition, another quantity of heat must at
the same time pass from a hotter to a colder body ; and the
proportion which the latter quantity of heat bears to the
former depends solely upon the temperatures of the bodies
between which it passes, and not upon the nature of the
intervening bodies.' This is evidently Carnot's principle,
adapted to the mechanical theory of heat, and expressed
in a different way. In this paper Clausius investigates the
properties of a function which he calls Aequivalenzwerth,
and whose value is found by dividing the quantity of heat
expended in producing a given change in a given substance
by a certain function of the temperature at which that change
1 Fogg. Ann. Dec. 1854. Abhandlungen, p. 127.
Dynamical Theory of Heat. 35
takes place ; which function of temperature Clausius shows,
from the probability that internal work in perfect gases is
inappretiable, to be probably proportional simply to the
absolute temperature, as measured by a perfect gas ther-
mometer. (The Aequivalenzwerth of Clausius is nearly
identical with the Thermodynamic function of Rankine ;
but there are some points of difference which are explained
in the later papers of Clausius.) 1 The properties of the
function here called Aequivalenzwerth, and in later papers
Entropie (closely connected with Thomson's previously pub-
lished Theory of Dissipation), form the subject of a long
series of investigations. The special investigations of
Clausius as to the internal work, and his function called the
Disgregation, will be briefly considered in the third chapter.
50. 'The investigations of both these writers funda-
mentally involve various hypotheses, which may or may
not be found by experiment to be approximately true, and
which render it difficult to gather from their writings what
parts of their conclusions, especially with reference to air
and gases, depend merely on the necessary principles of the
dynamical theory.' 2
51. One of the most valuable of the results which, as
we have just seen, was obtained almost simultaneously by
Rankine and Clausius, is as follows : If saturated steam
at any high temperature is allowed to expand, pressing out
a piston, in a vessel impervious to heat, it cools so as to
keep always at the temperature of saturation and, besides,
a portion of it liquefies. This result appears at first sight
inconsistent with the paradoxical experiment long known,
that high-pressure steam escaping into the air through a
small orifice does not scald the hand, or even the face, of a
person exposed to it ; while, on the contrary, low-pressure
steam inflicts fearful burns. W. Thomson has explained
1 See Rankine ' On Thermodynamic and Metamorphic Functions,
Disgregation, and real Specific Heat.' Phil, Mag. Dec. 1865.
2 Thomson, Trans. R.S.E. 1851, p. 281.
3 6 Historica I Sketch of th e
the difficulty thus : The steam rushing through the orifice
produces mechanical effect, immediately wasted in fluid
friction, and consequently reconverted into heat, from which,
by Regnault's numerical data, it follows that the issuing steam
(in the case of the high-pressure, but not of the low-pressure,
boiler) must be over 212 Fahr. in temperature, and dry.
52. In its new form, the theory of the motive power of
heat is based upon the two following propositions : the first
of which, though really announced by Davy, was only
definitely received in science in consequence of Joule's
experiments ; the second is the proposition of Carnot
(already given, 26, with its demonstration on the caloric
theory), adapted by Thomson to the dynamical theory.
I. When equal quantities of mechanical effect are pro-
duced by any means whatever from purely thermal sources,
or lost in purely thermal effects, equal quantities of heat are
put out of existence, or are generated.
II. If an engine be such that, when it is worked back-
wards, the physical and mechanical agencies in every part
of its motions are all reversed, it produces as much
mechanical effect as can be produced by any thermo-
dynamic engine, with the same temperatures of source and
refrigerator, from a given quantity of heat.
53. In order to prove the second proposition (which
regards the Transformation of heat, as the first regards the
Conservation of energy), we must consider in what respect
Carnot's proof has become inapplicable, and we find it to be
this : we have no right now to assume, as he did, that in a
complete cycle of operations in which his fundamental
condition is satisfied (i.e. the medium brought exactly to
its primitive state) as much heat has been given out to the
refrigerator as has been absorbed from the source ; because
the first of our new propositions shows that this is only true
when the medium has had as much work done upon it as
it has exerted on external bodies. Clausius attempted to
prove the proposition in 1850, by a process strictly analo-
Dynamical Theory of Heat. 37
gous to that of Carnot already given, but based solely on
the observed fact that heat tends to pass from warmer to
colder bodies. Some years later 1 he asserts that in this
passage he had assumed the following axiom, ' // is impos-
sible/or a self-acting machine ', unaided by any external agency ',
to convey heat from one body to another at a higher tempera-
ture? Thomson 2 in 1851 gave a satisfactory proof based on
the axiom, that '// is impossible, by means of inanimate
material agency, to derive mechanical effect from any portion
of matter by cooling it below the temperature of the coldest of
the surrounding objects?
In fact, we see by. 26 that, if we make ^transfer back
to the source a quantity of heat equal to that taken from it
by M y the compound system could only do work by JV's
taking more heat from the refrigerator than M gives to it.
Thomson introduces his assumption in an exceedingly
guarded way, and this course has since been fully justified
by Clerk-Maxwell, who has shown that, if the molecules of
hot bodies are in a state of motion with unequal velocities,
finite beings, without any expenditure of work, could under
conceivable conditions transfer heat from a cold body to a
hot one \ and that the process actually goes on, though to
a very small extent, in every mass of gas. 3
54. Carnot showed that, on his principles, the amount of
work done by the transference of a given amount of heat
increases indefinitely with the increasing difference of tem-
peratures of the source and refrigerator ; and of course it
follows from this that the air-engine, in which a much
greater range of temperature may be employed with safety
than in the steam-engine, should be the more effective of
the two. The introduction of the true theory leaves this
1 Abhandlungen, p. 50; with date 1864. See Phil. Mag. 1872, i.
pp. 106, 338, 443, 516 ; and ii. 117, 240.
2 On the Dynamical Theory of Heat, etc. , by W. Thomson, Trans.
R.S.E. March [7 and April 4, 1851.
8 See Tait, Recent Advances in Physical Science.
38 Historical Sketch of the
result unaffected except in degree ; in fact it shows that the
work to be derived from a given amount of heat leaving
the source increases indeed with the excess of temperature
of the source over the reservoir ; but, far from increasing
indefinitely as Carnot's theory showed, it has as a superior
limit, which it never reaches, the mechanical equivalent of
the heat which leaves the source. In fact, temperature is
now denned by the statement that in the working of a
reversible heat-engine the ratio of the heat taken in to that
ejected is that of the absolute temperature ( 36) of the
source to the absolute temperature of the refrigerator. 1 We
may consider either the proposition that a reversible engine
is perfect, or the definition of temperature which that pro-
position has enabled us to give, as the Second Law of Ther-
modynamics. A great many attempts have of late been
made to show that this second law is merely a form of
Hamilton's Principle of Varying Action a pure principle of
dynamics. It is obvious, from the fact that if we could at
any moment exactly reverse the motion of every particle, we
should make a dynamical system (however complex) go back
through all its previous states of motion, that such a deduc-
tion from Hamilton's principle can only be made by a
method of averages which virtually assumes the degradation
of energy, a consequence of the law to be proved.
Thus, in the most favourable circumstances, the steam-
engine, and even the air-engine, are exceedingly imperfect ;
giving at most only about one-fourth of the mechanical
equivalent of the heat spent. The theory of what have been
called Caloric Engines, where ether, or chloroform, or some
such easily vaporised liquid, is used in connection with air
or steam to utilise as much as possible of the applied heat,
has been given by various investigators, including those last
mentioned, but it appears that in practice the method has
not realised the anticipations of its proposers.
55. A most remarkable result of the application of
1 W. Thomson, Trans. R.S.E. 1851.
Dynamical Theory of Heat. 39
Carnot's reasoning was given by J. Thomson in I84Q. 1
From this reasoning it is obviously demonstrable, as shown
by W. Thomson, that water at the freezing-point may, with-
out any expenditure of work on the whole, be converted into
ice by a mechanical process. For a mass of water retains the
temperature of freezing unchanged until it is all converted
into ice, and according to Carnot's, and even to the dyna-
mical, theory, no work is required to make heat pass from
one body to another at the same temperature. J. Thomson
first remarking that this result seemed to involve the possi-
bility of producing work from nothing (since water expands
with great power in the act of freezing), was led, on the only
way of escape from a conclusion which no naturalist could
admit, to the conclusion that the temperature at which
water freezes must depend, as the boiling-point had long
been known to do, upon the pressure ; and he showed that
the freezing-point of water must be lower by o 0< oi35 Fahr.
for each additional atmosphere of pressure. This very
curious theoretical deduction was verified, to its numerical
details, by means of CErsted's Piezometer, by W. Thomson, 2
and it has been successfully applied by the Thomsons and
Helmholtz to explain the extraordinary plasticity of glacier
ice discovered by the careful measurements of Forbes.
Hopkins and Bunsen have since verified experimentally
another consequence of the same theory, viz., that in cases
where bodies contract on solidifying, as is the case with
sulphur, wax, etc., the melting-point is raised by increase of
pressure.
56. The complete theory of all such cases had, however,
been previously given by W. Thomson, who was the first
(after Clapeyron) to recall attention to the work of Carnot ;
developing many of its more important consequences so far
that the form, for instance, of Carnot's function follows at
once from his paper, if the conversion of heat into work be
1 Trans. R.S.E. 1849. - Proc. R.S.E. 1850.
4O Historical Sketch of the
assumed. He seems to have been at first unwilling to
encounter the new problems suggested by the true theory, 1
but having in 1851 obtained a satisfactory basis for the
Second Law, he advanced with tremendous strides, leaving
(so far as the theory has yet been developed) little but prac-
tical applications or experimental verifications of his results
to be made by his contemporaries. Dissipation, Restora-
tion, General Laws of Transformation, Thermo-electricity,
Magnecrystallic action, etc. etc., were all comprehensively
investigated by him in little more than a year. It is
greatly to be regretted that Thomson's scattered papers,
which, while models of brevity and distinctness, evince a
marvellous clearness of perception, and, above all, are
always based directly upon ascertained facts, 2 have not
been reprinted in a collected form. Till this is done, few
will be aware, either of the immense extent to which he
has pushed the theory, or of the number of discoveries in
which he anticipated those to whom they are usually
assigned. In so elementary a work as the present, many
of the most important of these points must be passed over
with the merest allusion,
57. In his (already cited) paper of 1851, on the Dyna-
mical Theory of Heat, without encumbering himself with,
1 'If we [abandon Carnot's fundamental axiom, a view which is
strongly urged by Mr. Joule] we meet with innumerable other difficulties,
insuperable without farther experimental investigation, and an entire
reconstruction of the theory of heat, from its foundation.' (Trans,
R.S.E., 1849, p. 545, Note.)
2 ' Unter den Naturforschern, welche ihr Streben vorzugsweise darauf
gerichtet haben, die Naturwissenschaft von alien metaphysischen
Erschleichungen und von alien willkiirlichen Hypothesen zu reinigen,
sie im Gegentheil immer mehr zum reinen und treuen Ausdruck der
Gesetze der Thatsachen zu machen, nimmt Sir W. Thomson eine der
erst en Stellen ein, und er hat gerade dieses Ziel vom- Anfange seiner
wissenschaftlichen Laufbahn an in bewusster Weise verfolgt.' Helm-
holtz, Preface to 'the German edition of Thomson and Tait's Natural
Philosophy.
Dynamical Theory of Heat. 41
or limiting the generality of his results by, any hypothesis,
Thomson applies the fundamental propositions of the dyna-
mical theory (already given) to all bodies, and deduces
many very curious and important results regarding the
specific heats of all substances; with special conclusions
agreeing with those of Rankine and Clausius for * perfect '
gases, and for mixtures of portions of a body in different
states but at the same temperature, as ice and water, or
water and saturated steam. Among these we may mention
the following : When a substance contracts as its tempera-
ture rises (as is the case, for instance, with water between
its freezing-point and its point of maximum density), its
temperature will be lowered by a sudden compression. In
two most valuable experimental papers 1 by Joule, Thom-
son's formulae are completely verified (within the limits of
experimental error) for substances of the most dissimilar
qualities. One very curious result is afforded by india-
rubber, which, when suddenly extended, becomes warm;
and, in agreement with Thomson's conclusions, is found,
when stretched by a constant weight, to contract on being
heated, and to raise the weight.
58. We have several times alluded to the fact, that the
amount of heat developed by the compression of air is only
approximately equal to the equivalent of the work expended
in compressing it, although in Joule's experiment of 1844 it
appeared to be exactly equal to it. There is, as before
observed, no a priori reason for the existence of any such
equality, unless we assume the kinetic theory of gases
to be true, and that there are no internal forces (though
Clausius considers it to be a probable deduction, in the
case of a non-liquefiable gas, from the laws of Boyle and
Charles as to the relations between pressure, volume, and
temperature), for it is quite conceivable that a gas or
1 On some Thermo-dynamic Properties of Solids , and On the Thermal
Effects of compressing Fluids. Phil. Trans. 1859.
42 Historical Sketch of the
other body might exist in which the whole work expended
in compressing it, is employed in overcoming repulsive
forces among its particles, and would therefore be wholly
stored up as mechanical power in the compressed gas,
without any change of temperature whatever. That heat,
nearly equivalent to the work expended in compres-
sion, is actually developed, shows us that the mutual
molecular forces among particles of a gas are exceedingly
small, and that the pressure of a gas is due almost entirely
to the ' repulsive motion ' of Davy.
69. Daniel Bernoulli, in the tenth section of his Hydro-
dynamica, explains the pressure of air by the impact of its
particles on the sides of the vessel containing it. This idea
was developed at greater length by Le Sage and Prevost, 1
by Herapath, 2 Joule, 3 and Kronig.* Joule, in fact, by very
simple reasoning, arrived at the now recognised fact that
the average velocity of the particles of hydrogen at oC.
must be about 6055 feet per second. Clausius, 5 who
has taken up this theory more in detail, considers a gas
to be a collection of molecules always in motion, and
deflecting each other from their courses only when at a
very small distance from each other, so that the course of
each molecule consists of a series of nearly rectilinear por-
tions. The deflexions due to the encounters with other
molecules are supposed to be so sudden that the encounters
may be compared to collisions. He has determined the
relation which exists among the following quantities :
the mean length of the path of a molecule between suc-
cessive encounters, the distance of their centres at collision,
and the number of molecules in unit of volume, and has
shown that this theory is not inconsistent with any known
1 Deux Traites de Physique Mecanique. Geneve, 1818.
2 Mathematical Physics. Whittaker & Co. , 1847.
3 Some Remarks on Heat and the Constitution of Elastic Fluids. Oct.
3, 1848. (See /%7. Mag. 1857, ii.)
4 Pogg. Ann. 1856. 5 Phil. Mag. Feb. 1859.
Dynamical Theory of Heat. 43
phenomena, provided we admit that the energy of each
molecule consists partly of the motion of its centre of
inertia, and partly of rotation or internal vibration, these
two portions of the energy being on an average in a
definite ratio depending on the nature of the molecules.
This theory in the hands of Clausius explains the relations
between the volume, temperature, and pressure of a gas, its
cooling by expansion, and its two specific heats, together
with the slowness of the diffusion of gases, and of the con-
duction of heat.
60. Clerk- Maxwell, 1 taking up the theory of Clausius,
showed that it also led to the law, previously established by
Gay-Lussac on chemical considerations, that if two gases
have the same volume, pressure, and temperature, the
number of their molecules is the same. He also applied
the theory to explain the internal friction of gases, and
endeavoured to estimate the probable length of the path
described by a molecule between successive collisions. He
was thus led to the conclusion that the viscosity of a gas
is independent of its density, an unexpected result, which,
however, was shown by Stokes to be deducible from the
experiments of Graham 2 on the Transpiration of Gases.
Clausius, 3 in examining Maxwell's theory of the conduc-
tion of heat in a gas, pointed out some oversights in that
theory, and improved and extended the method of investi-
gation and O. E. Meyer 4 investigated the viscosity of air,
both experimentally, and as a mathematical consequence of
the theory of elastic molecules.
61. Maxwell, 5 however, was induced by his own experi-
ments 6 on the viscosity of air and other gases at different
temperatures, and by the results obtained by Graham on
transpiration, to substitute for the theory of elastic molecules,
that of molecules repelling one another according to the
1 Phil. Mag. i860. Jan. and July. - Phil. Trans. 1846.
3 Pgg' Ann. Jan. 1862. 4 Pogg. Ann. 1865.
6 Phil. Trans. 1867. 6 Phil. Trans. 1806.
44 Historical Sketch of tlie
inverse fifth power of the distance. He also succeeded, by
means of this assumption, in simplifying the mathematical
theory, and in adapting it to the cases of diffusion of gases,
and conduction of heat in a gas, and in explaining the
apparently anomalous results obtained by Graham relating
to the viscosity' of mixed gases. 1 The time has hardly yet
come, however, in which much is to be expected from such
hypotheses ; we are as yet almost completely ignorant of the
ultimate structure of the molecules or particles of matter,
though the thoroughly satisfactory explanation 2 which the
kinetic theory of gases furnishes of the motions of Crookes'
Radiometer has recently supplied very strong additional
arguments for the truth of that theory. The results of the
dynamical theory of heat are, however, quite independent
of particular assumptions like these as to the constitution
of bodies.
62. Maxwell has obtained three results, which, though
they are deduced from the theory of moving molecules, are
independent of the mode in which these molecules are
supposed to act on one another.
The first is the condition of equilibrium of energy between
two sets of molecules of unequal mass, namely, that the
energy of translation of a molecule of either kind must be
the same. This is the dynamical expression of Gay-Lussac's
Law of Equivalent Volumes.
The second is the condition of equilibrium of a vertical
column of mixed gases, namely, that the density of each
gas at any point is ultimately the same as if no other gas
were present. This is the law laid down by Dalton as to
the distribution of gases in equilibrium.
The third is the condition of equilibrium of temperature
1 The most recent experiments of Kundt and Warburg, however,
seem to show that the viscosity increases according to a power of the
absolute temperature which is different for different gases, but in no
case exceeds 077.
2 Dewar and Tait, Proc. R.S.E., or Nature, July 1875. Stoney,
Phil. Mag., March and April 1876.
Dynamical Theory of Heat. 45
in a vertical column of gas, namely, that the temperature
must be the same throughout, and it follows from this, by
the dynamical theory of heat, that in all substances gravity
has no effect in making the lower part permanently hotter
or colder than the upper. These results have been con-
firmed by the elaborate investigations of Boltzmann.
63. A method of experimentally discovering, with very
great accuracy, the relation between the heat produced and
the work spent in the compression of a gas, was suggested
by Thomson in 185 1, 1 and employed with some modifica-
tions in a series of experiments, which he has since carried
on in conjunction with Joule. Their results have been
from time to time published in the Philosophical Transac-
tions during the last twelve years, with the title Thermal
Effects of Fluids in Motion. The principle of this method is
excessively simple ; it consists merely in forcing the gas to
be experimented on through a porous plug, and observing
its temperature on each side of the plug. These tempera-
tures should (after the requisite corrections) be exactly
equal if the heat developed by compression is equal to the
work expended, and not otherwise. By this process it is
found that no gas perfectly satisfies the criterion ; and, as we
might expect, the liquefiable gases are those which most
diverge from it. By means of a sufficient series of such
experiments, carried on at different temperatures and pres-
sures, complete theoretical data for a gas-engine have been
obtained ; and the extensive and valuable experiments of
Regnault (with additions, as to the density of steam at high
pressures, derived by Joule and Thomson from their air
experiments) have furnished corresponding data for the
steam-engine; so that the theoretical treatment of these
important instruments is now at all events approximately
complete. But it is no part of the plan of this work to enter
into details of application.
1 Trans. R.S.E.
46 Historical Sketch of the
64. A yet more important result, viz., that, to a very close
approximation, Carnot's function ( 26) is inversely as the
temperature from the absolute zero measured on the air-
thermometer (an idea first suggested by experiment to Joule,
but which was assumed by others from mere hypothetical
reasoning), was thus definitely introduced into science.
65. As already mentioned, the direct relation between heat
and mechanical effect has alone been considered in the pre-
sent chapter. The far more extensive results which have
been arrived at with reference to indirect relations, and the
consideration of the relations which have been proved to
exist between heat and all other forms of energy will be
taken up in Chapter n. What has been given is almost
entirely confined to the thermo-elastic properties of liquids
and gases. W. Thomson 1 has published an extremely
general investigation of the laws of this subject, including
crystalline solids; but to give a satisfactory account of it
would involve details and difficulties far too great for any
but a very small class of readers.
66. There remains, however, one interesting portion of
the subject, which, though having most important bearings
upon the subject of energy and its distribution through the
universe, is in part a branch of Thermo-dynamics. This is
the consideration, already alluded to, of the Dissipation of
Energy. 2 But in accordance with the plan of the work, it
must be considered at present only as regards heat and
mechanical effect. In the first place, heat in a conducting
body tends to a state of dissipation or diffusion, never to
a concentration at one or more places. This is a direct
consequence of the law of propagation of heat in a solid.
Fourier's mathematical investigations point also to the fact
that a uniform distribution of heat or a distribution tending
1 Quarterly Math. Journal, April 1855.
2 On a Universal Tendency in Nature to the Dissipation of Mechanical
Energy. By W. Thomson. Proc. R.S.E. 1852, and Phil. Mag.
1852, ii.
Dynamical Theory of Heat. 47
to become uniform, must have arisen from some primitive
distribution of heat of a kind not capable of being produced
by known laws from any previous distribution. 1 When Car-
not's method, correctly adapted to the dynamical theory of
heat, was applied by Thomson to the transformations of
heat into work, and work into heat, it led him to the follow-
ing amongst other propositions.
When heat is created by a reversible process, there is also
transference, from a cold body to a hot one, of a quantity
of heat, bearing to that created a definite ratio depending
on the temperatures of the two bodies.
When heat is created by a non-reversible process (such
as friction) there is a dissipation of energy, and a full resto-
ration of it to its primitive condition is impossible.
From these it follows that any restoration of mechanical
effect, from the state of heat, requires the using of more
heat than the equivalent of the work obtained, this surplus
going into a colder body. No further comment on this can
be made at present, but in next chapter it will form a most
important feature.
67. We have, as yet, said nothing of Radiant heat, of
which the caloristic idea seems to have been exactly ana-
logous to the Corpuscular Theory of Light. Davy actually
speculates on the combinations of light and oxygen, in the
very paper in which he destroyed the notion of the materi-
ality of heat ! The first really extensive, and on the whole
trustworthy, experiments on radiant heat are those of Leslie,
but we need not trouble ourselves with his theoretical
speculations. The experiments of Forbes and Melloni
showed so complete a resemblance between the laws of
reflection, refraction, polarisation, absorption, etc., of light
and radiant heat, that no doubt could remain as to their
identity. And as light had, chiefly by the theoretical and
experimental investigations of Young and Fresnel, been
shown to consist in the undulations of some highly elastic
1 Thomson, Cambridge Mathematical Journal , 1843.
48 Historical Sketch of the
medium pervading all space ; it followed that radiant heat
also is energy and not matter. Dark radiant heat differs
from light merely as a grave note does from a shrill one.
Light was shown by Leslie to heat bodies which absorb it,
and on this principle he constructed his photometer. The
paper of Mohr, already referred to ( 30), contains, along
with much error, many of the more obvious consequences
of the establishment of the identity of light and radiant heat.
68. The law of exchanges, as it was called by Prevost,
who first enuntiated it, explained what was erroneously
called the radiation of cold, i.e. that a piece of ice brought
near the bulb of a thermometer cooled it, with other more
complex but perfectly analogous experimental results. He
considered that all bodies radiate heat, but the more the
higher is their temperature, so that, in the simple case
above mentioned, the thermometer gave more heat to the
ice than it received from it a perfectly satisfactory expla-
nation. This theory has since been greatly extended by
Stewart, Kirchhoff, and De la Provostaye, who have inde-
pendently arrived at the conclusion that the radiating power
of a body for any definite ray of heat is equal to its absorb-
ing power for the same. Light and radiant heat being
identical, we may (with Melloni) speak of the colours of
different kinds of radiant heat, and then the analogy, with
corresponding phenomena in the case of light, becomes at
once evident. A very curious example of the truth of this
proposition, noticed by Stewart, is furnished by heating to
whiteness a common earthenware plate, with a strongly-
marked pattern, and looking at it in the dark, when we see,
instead of a dark pattern on a white ground, a white pattern
on a dark ground ; those parts which, when the plate is cold,
appear dark, do so in consequence of their absorbing the
incident light more freely than the white parts ; and when
heated to whiteness, they appear bright because they radiate
better. Stewart has also noticed, as another excellent proof,
the fact that coloured glasses lose their colour in the fire.
Dynamical Theory of Heat. 49
De la Provostaye and Desains showed that, as most sub-
stances polarise (more or less completely) light reflected
obliquely from their surfaces, so the light they radiate
obliquely when heated to redness is partially polarised ; but
in the plane perpendicular to that of polarisation of the
reflected rays. Kirchhoff and Stewart independently ob-
served the beautiful phenomenon that the light radiated by
a heated tourmaline plate (which polarises transmitted light)
is polarised in the same plane as the light which the tour-
maline absorbs. Kirchhoff derived from his investigation,
and verified by conclusive experiments, the physical explana-
tion of the production of the dark lines in the solar spectrum,
which had, however, been previously suggested by Stokes,
and of which Brewster had, long before, pointed out one of
the causes, viz., the earth's atmosphere. The very amazing
results of spectrum analysis cannot, however, properly be
discussed here.
69. KirchhofFs earliest publication 1 on this subject is of
somewhat later date than that of Stewart, but it is more ex-
plicitly connected with the dynamical theory. The equality
of radiation and absorption for every separate wave-length
is here, for the first time, directly shown to be a consequence
of the second law of thermodynamics. Also it is shown
that as the temperature of a body becomes higher, not only
does its radiation include rays of greater refrangibility, but
the intensity of its radiation for every ray increases. Hence
Kirchhoff is enabled to explain how it happens that a flame
of lower temperature than the source is necessary for the
production of absorption bands in the spectrum. Thus, for
instance, in the spectrum of the Drummond or Oxy-hydrogen
lime light no weakening of the double line D is produced
by the interposition of a Bunsen gas-flame containing ignited
sodium vapour ; the radiation from the absorbing flame
being sometimes more than sufficient to make up for the
absorption. The light from the incandescent carbon points
1 Berlin Academy, October, December, 1859. Fogg. Ann. 1860.
D
50 Historical Sketch of the
of the electric lamp, however, exhibits these lines strongly
if passed through the Bunsen flame, even when a piece of
metallic sodium burns fiercely in it. To produce the ab-
sorption bands in the spectrum of the Drummond light the
comparatively cool flame of a spirit lamp (with salted wick)
must be employed.
70. Stewart's 1 theoretical treatment of the subject is
virtually based on the assumption (founded on experiment)
that a thermometer placed within an enclosure formed of
any materials and kept at a constant temperature, will
finally assume that temperature, no matter by what screens
the thermometer be surrounded. [This really involves,
in one of its many forms, the second law of Thermo-
dynamics.]
From this he shows, by very simple reasoning, that the
absorption of a body at a given temperature must be equal
to its radiation for every description of heat. He then ex-
perimentally proves that a plate of rock-salt, which absorbs
little heat, radiates little. That both glass and rock-salt,
cold, are more opaque to radiations from hot masses of
their own substance than to radiations from any other body
at the same temperature. And finally, that a thick plate of
rock-salt radiates more than a thin one of the same tem-
perature. From this last property he proves the existence
of internal radiation, and shows that in non-crystalline
bodies it is proportional to the square of the refractive
index ; a result also arrived at later, as a consequence of
the second law of thermodynamics, by Clausius, 2 who
employs a method analogous to that of KirchhofT. It
must be remarked, however, that this method of Kirch-
hoff s is merely a particular application of the splendid
general investigations of Sir W. Rowan Hamilton. 3 Before
the appearance of Kirchhoff's paper, Stewart had begun to
1 Trans. R.S.E. 1858-9. 2 Pogg. 1861, Abhandhmgen, p. 322.
3 Theory of Systems of Rays (1824). Tram. R.I. A. 1828, 30, 31, 37.
On a General Method in Dynamics. Phil. Trans. 1834, 35.
OF
Dynamical Theory ty ffSlETT 5 1
extend his experiments to light, and had found, though not
published, that a green glass when heated gives out reddish
light and vice versa ; and that a glass of any colour laid on
glowing coals gives its own colour while colder than the
coals, loses apparently all colour when it has acquired the
temperature of the coals, and gives the complementary
colour if hotter than the coals behind it.
71. The connection of the whole subject with the long-
known proposition in geometrical optics that it is impossible
by any series of reflections and refractions, to obtain an
image brighter than the object, must be at once obvious.
72. Leslie's result, that a body, such as coloured glass, is
heated by absorbing light, has recently received a most
interesting extension from the discovery by Stokes 1 of the
physical cause of certain curious phenomena observed by
Brewster and Herschel, in solutions of quinine and certain
kinds of fluor-spar, from the latter of which the phenomena
have been called by the general name Fluorescence. The
physical fact is simply this, that these and other bodies,
especially the green colouring matter of leaves and ' canary '
glass coloured with Oxide of Uranium, radiate as altered
light part of the light which they absorb. This is, to a
certain extent, analogous to Leslie's result, because the
light radiated is lower in the scale than that absorbed, and
is in general most freely produced from light so high in the
scale as to be invisible to the eye (just as very shrill sounds,
such as the chirp of the cricket, are inaudible to many ears).
The most important application of this discovery has been
to the rendering visible these invisible rays, and thus study-
ing through a wider range of refrangibility the radiations
from any source. The phenomena of Phosphorescence, when
not traceable to chemical combination, evidently belong to
the same class with those of fluorescence, and have been
recently studied with great care by Becquerel, who has
obtained many remarkable results. Another transforma-
1 OntheChangeof the Refrangibility of Light. Phil. Trans. 1852.
5 2 Historical Sketch of the
tion of radiant heat, seemingly opposite in character to this,
is Calcescence or Calorescence, where radiant heat from an
intensely heated body, deprived of its luminous portion by
absorption, and concentrated by a lens or mirror, produces
incandescence even in incombustible bodies. This was
predicted by Akin, 1 who described a process by which it
might probably be obtained under very favourable circum-
stances ; but the experiment was first successfully performed
by Tyndall. 2 The theory of this singular result is not yet
quite clear, as it depends essentially upon an artificial
arrangement producing extreme discontinuities of tempera-
ture. It is obviously connected with the fundamental
principle of Kirchhoffs theory, ( 69,) that a body when
incandescent gives out a greater amount of dark heat than
it does at any lower temperature.
73. Let us now, taking for granted the dynamical theory
of heat, consider very briefly the explanations which it
furnishes of many important phenomena, not alluded to in
the preceding semi-historical sketch, because their explana-
tion is very evident as soon as the true theory has been
found.
74. Thus, for instance, Heat of Combination, as it is
called, is obviously now to be explained as arising from
the mechanical effect of the force of chemical affinity
whatever may be the nature and origin of that force just as
a stone falling to the ground under the action of the earth's
attraction generates heat by the impact. From this explana-
tion also follow as obvious truths, the laws of this subject,
experimentally arrived at by Andrews, Hess, and others.
75. When a salt is deposited in crystals from a super-
saturated solution, we have, in general, evolution of heat ;
formerly this was attributed to the latent heat of solution,
but it is now easily seen to be, like ordinary latent heat,
dependent on the change of relative position of the mole-
1 Brit. Ass. Reports, 1863. Phil. Mag. 1864-5.
2 Phil. Mag. and Phil. Trans. 1864-5.
Dynamical Theory of Heat. 53
cules involved. The contrary effect is of course produced
when a salt is dissolved, and even when two crystalline
solids, as ice and salt, liquefy in the act of combining.
Hence the justice of the popular outcry against the common
process of destroying ice on the pavements by sprinkling
salt upon it; as, though the ice is melted, a great addi-
tional lowering of temperature is produced. Hence also
the effect of the combinations called ' freezing mixtures,'
which are of many kinds ; from the simplest, such as the
solution of nitrate of ammonia in water, to the most com-
plex, such as the mixture of solid carbonic acid and ether
in vacua.
76. As was cursorily noticed at the commencement of
this chapter, the so-called latent heat probably depends
upon molecular arrangement; the heat, which is lost to
the thermometer, disappears in producing, or is transformed
into, the work of tearing asunder the particles of a solid or
liquid, and placing them in the positions of less relative
constraint which they occupy in a liquid or a vapour respec-
tively. It is conceivable, however, that it also may be
motion, but of a kind not tending to diffusion. But it is
too early to speculate, with any prospect of useful results,
on such a subject. We give, in the last chapter, a few
remarks upon the speculations of Clausius.
77. The heat of the sun, and the internal heat of the
earth both of which, by the principle of dissipation, must
be now far less than they were ages ago are to be traced
almost entirely to their origin in the primaeval distribution
of matter through space, at creation, and the subsequent
transformation into heat of the energy with which the
various portions which compose the sun or a planet im-
pinged on each other in meeting.
But the consideration of such immense and important
transformations must be deferred to the second chapter,
where they will be found to flow naturally from the known
laws of transformation and transference of energy.
5 4 Historical Sketch of the
78. Reviewing, for a moment, the path we have so far
pursued, the successive steps, most important in a historical
point of view, of the foundation and development (not the
applications) of the science may be recapitulated. They
are these :
First, Newton's grand general statement of the laws of
transference of mechanical energy from one body or system
to another (1687).
Second, Davy's proof that heat is a form of energy subject
to these laws (1799).
Third, Rumford's close approximation to a measure of
the mechanical equivalent (1798).
Fourth, Fourier's great work on one form of dissipation
of energy (1812).
Fifth) Carnot's fundamental principle, his cycles of opera-
tion, and his test of a perfect engine (1824).
Sixth, Thomson's introduction of an absolute Thermo-
dynamic scale of thermometry (1848).
Seventh, Joule's exact determination of the mechanical
equivalent of heat, and the general reception of the true
theory in consequence of his experiments (1843-9).
Eighth, The adaptation, by Rankine and Clausius, and
subsequently, with greater accuracy, generality, and freedom
from hypothesis, by Thomson, of mathematical investigation
(partly based on Carnot's methods) to the true theory ; the
establishment of the second law by Thomson ; with Joule's
experimental verification of Thomson's general results
(1849-51).
Ninth, Thomson's theory of dissipation (1852).
As regards the true theory of the connection of heat with
mechanical effect, this list contains all the most important
direct steps, nearly in chronological order ; but it is to be
remembered that experimental investigation, mainly due to
Joule, has indissolubly connected by laws of equivalence all
forms of energy, including even such mysterious forms as
are observed in electro-chemistry and electro-magnetism ;
Dynamical Theory of Heat. 5 5
so that a complete account of the dynamical theory of heat
necessarily involves, what we propose to give in the next
chapter, an account of the grand law of natural philosophy
known as the CONSERVATION OF ENERGY.
79. In the brief sketch above given, a vast amount of
valuable matter has been of necessity omitted, but no direct
step of real consequence to the development of the true
theory of heat has been left unnoticed. Where the results
of early experiments were sufficiently accurate, subsequent
more perfect ones (such as the spendid series of researches
by Regnault, and the valuable large-scale investigations of
Hirn) have been merely alluded to ; and many curious, but
not very important, points have not been mentioned. The
details of such a history as this would fill volumes.
CHAPTER II.
HISTORICAL SKETCH OF THE SCIENCE OF ENERGY.
80. IN the preceding chapter the absurdity of attempting
to base extensions of Natural Philosophy upon mere meta-
physical speculations was very briefly considered ; and it
was shown that without direct experimental proof, or the
less direct but still conclusive proof furnished by rigorous
mathematical deductions from experimental results, nothing
can with any show of reason be predicated of the laws of
Nature. Experience is our only guide in these investiga-
tions, for there can evidently be no d priori reason (to our
intelligence) 1 why matter should be subject to one set of
laws rather than another, so long at least as each of these
codes is consistent with itself. The caloric or material
theory of heat was particularly instanced as not only un-
justifiable in itself, but (while it was received) antagonistic
to all real progress. The corpuscular, or material, theory
of light furnishes another excellent example. The prepos-
terous nonsense which was gravely enuntiated, and greedily
accepted, with regard to the nature and laws of light, and
the elaborately absurd properties assigned to its supposed
particles in order to fit them for their every-day work,
would be almost inconceivable to a modern reader, were it
not that equally, or even more, extravagant dicta of the
' great inexperienced ; have been, and are even now being,
1 See ' Herbert Spencer versus Thomson, and Tait ' Nature, March 26,
1874-
Historical Sketch of the Science of Energy. 5 7
propounded by self-constituted interpreters of the original
designs of Nature. Look at the dictum of the antient
philosophers, who accounted for the planetary motions by
saying ' circular motion is perfect.' Or, to come to modern
times, let us see how the gigantic intellect of a Hegel can
annihilate the conclusions even of Newton. ' The motion
of the heavenly bodies is not a being pulled this way and
that, as is imagined. They go along, as the antients said,
like blessed gods. The celestial conformity is not such a
one as has the principle of rest or motion external to itself.
It is not right to say, because a stone is inert, and the
whole earth consists of stones, and the other heavenly
bodies are of the same nature as the earth, therefore the
heavenly bodies are inert. This conclusion makes the
properties of the whole the same as those of the part.
Impulse, pressure, resistance, friction, pulling, and the like,
are valid only for other than celestial matter.' 1 We no-
where find these preposterous dicta more prevalent, or
more pernicious, than in the history of the grand question
which we are about to discuss. We have no more reason,
before experiment settles the question, to fancy Energy
indestructible than the Calorists had for believing in the
materiality of heat. The philosophers who said that
' Nature abhors a vacuum ' had at least an experimental
basis for their guidance; and, if they had limited the
generality of their statement to the class of circumstances
really involved in their experiments, we might have smiled
at the peculiarity of the language in which their conclusion
was expressed, but we must have allowed their meaning to
be correct.
81. But when we find, in modern times, conclusions,
however able, drawn without experiment from such a text
as ' Causa cequat effectumj we feel that the writer and his
supporters are, as regards method, little in advance of the
1 See Whewell On HegeFs Criticism of Newton's Principia. Camb.
Phil. Trans. 1849.
58 Historical Sketch of the
science of the dark ages. This is one of the fundamental
characteristics of all the writings of Mayer, 1 and therefore
we may for the present leave them unnoticed, although we
shall afterwards have occasion to consider them as furnish-
ing a most admirable development of the consequences of
an unwarranted assumption. For, while there can be no
doubt that the works of Mayer contain highly original and
profound deductions from his premises, deductions of a
most important character as regards the system of the
universe, it is certain that those premises were unjustified
by experiment, and therefore that his method was not
merely unphilosophic but even inconsistent with true
science.
82. Let it not be imagined that we undervalue the assist-
ance which science often receives from what appear at first
to be the wildest speculations so long as these are not
elaborately enuntiated as a priori laws, but are confined to
their only legitimate use, the suggestion of new methods of
interrogating nature by experiment. By all means let
philosophic minds indulge in any vagaries they may choose
to foster, but let these be carefully distinguished from facts
established by experiment, and let them be kept as private
magazines from which, when required, may be extracted an
idea leading to an experimental research. In perhaps one
case in a million, the expected result may follow : but, in
the many cases in which it does not occur, there are
thousands of chances (which will not be lost to the careful
experimenter) of discovering something utterly unlocked
for. Instances of this without number may be given. The
discovery of electro-magnetism by Orsted was arrived at
by his fancy that a conducting wire might when heated by
1 Bemerkungen iiber die Krdfte der unbelebten Natur. Liebig's
Annalen, 1842. Die organise he Bewegung in ihrem Zusammenhange
mit dem Stoffwechsel, Heilbronn, 1845. Beitrdge zur Dynamik des
Himmels, Heilbronn, 1848. Republished in a collected form, with
the title Die Mechanik der Wdrme, Stuttgart, 1867.
Science of Energy. 59
an electric current act on a magnet. Kepler's Laws were
deduced by means of an almost incredible amount of numer-
ical calculation based upon the supposition of the existence
of all sorts of harmonies, perfect solids, etc. etc., in the solar
system. In chemistry this mode of procedure has been
long recognised as leading to most important results, since,
in the attempt to produce directly some particular com-
pound, it often happens that the experimenter is gratified
by the appearance of some other which he had never dreamt
of as capable of existing, or at least of being obtainable by
his process. Mayer, therefore, and others who have followed
a course similar to his, cannot be considered as having any
claims to the credit of securely founding the science of
Energy ; though their works have become of great value as
developments and applications, since the science has been
based upon correct reasoning and rigorous experiments.
83. Particular cases of the Conservation of Energy were
experimentally discovered, but without any reference to the
great principle, at early stages of the progress of electricity,
electro-chemistry, heat of combination, and various other
branches of science ; and many curious cases of Trans-
formation and Dissipation of Energy had also been observed.
To these we shall advert after we have given a brief sketch
of the Laws of Energy and the history of their discovery ;
as we shall then be enabled to classify them properly, and
to show their mutual connection.
84. Before entering upon the history of this development,
we may premise a few words on Dynamical Theories in
General.
An exact Science is one the nature and connection of
whose phenomena may be expressed in exact terms.
A dynamical theory of a science is one which explains its
phenomena by the existence of bodies acting on one another
with determinate forces, and moving in a determinate
manner. A theory which ascribes to these bodies, forces,
or motions any qualities differing from those of the bodies,
60 Historical Sketch of the
forces, and motions treated of in pure dynamics is not a
dynamical theory.
A dynamical theory consists of four parts.
The first part considers the possible motions of the
system without reference to the forces producing them.
This is called Kinematics.
The second considers in what cases one system of forces is
equivalent to another. If one of these systems is reversed,
the whole will be in equilibrium. Hence this investigation
has been called Statics. It is independent of the nature of
the bodies on which the forces act.
The third part treats of the effect of forces on the motions
of material bodies. This may be called Kinetics.
The fourth part considers the conditions under which
forces act between the different parts of the system, and
thus transmit energy from one part to another. This may
be called Energetics.
85. In some departments of physical science we have
ascertained the energy required to produce certain effects,
without being able to measure on any sound principle either
the magnitude of the effect, or the force required to pro-
duce it.
In ordinary kinetics, the effect is sometimes measured as
the number of feet through which the resistance is overcome ;
the resistance expressed in pounds weight is the other factor,
if the energy is expressed in foot-pounds.
In hydrostatics, when a fluid is forced into a vessel, the
volume of the fluid, and the pressure at which it is forced
in, are the factors of the energy.
In the transfer of electricity, the quantity transferred, and
the electromotive force opposing the transfer, are the factors.
But in chemistry, though the total energy let loose during
the combination of two given substances can be ascertained
in many cases from the heat produced, the measurement of
the force which produces this effect has not been so clearly
understood. To measure this force in pounds weight, it
Science of Energy. 6 1
would be necessary to know the distance between the com-
bining molecules at every stage of the combination, and for
this we have no data whatever. W. Thomson, however, in
his Mechanical Theory of Electrolysis^ has given a method
by which in many cases the force of ' chemical affinity,' so
much studied in former times, and so neglected now, may
be expressed in a perfectly definite measure. This measure
is that of electromotive force, and the numerical value
of the electromotive force which expresses the resultant
chemical affinity involved in a given reaction is equal to the
mechanical value of the whole heat evolved during this
reaction, when one electro-chemical equivalent of each sub-
stance enters into the combination. 1
Thomson has also done a service of primary importance
to the corresponding part of thermodynamics. The con-
dition of the transfer of energy from A to B depends on the
relative value of a and /3, where a and /3 are functions of the
state of A and B. If A and B are vessels containing fluid,
and put in communication, then a and /3 are the pressures
in these vessels, and the value of a /3 determines the
direction and force of the transfer of fluid. If A and B are
electrified, a and ft will be their potentials. If A and B are
hot bodies, a and /? will be their temperatures. Temperature,
therefore, as has long been understood, is a quantity which
determines whether a body shall part with its heat to other
bodies, and temperature as measured by any particular ther-
mometer is a quantity which satisfies this condition. But
in the ordinary measure of temperature, though we may
assert that the temperature of A is greater or less than that
of B) we cannot assert that the temperature of A exceeds
that of B as much as the temperature of C exceeds that of
D, merely because the differences measured in degrees of
our thermometer are the same, for one thermometric sub-
1 See Clerk-Maxwell and Jenkin on Elementary Relations between
Electrical Measurements, Brit. Ass. Report, 1863, Art. 54.
62 Historical Sketch of the
stance differs from another in its law of expansion. By the
establishment of an absolute scale of temperature ( 26),
however, we may treat differences of temperature with the
same mathematical completeness as differences of pressure
or of potential.
86. In order that the reader may understand clearly the
terms which it is essential to employ in giving a strictly
accurate, although popular, view of the laws of Energy, it
will be useful to give preliminary examples of various forms
of energy constantly presenting themselves to his notice.
Let us consider, for instance, gunpowder. It contains, in a
dormant form, an immense store of energy, or, in common
mechanical language, it can do an immense amount of
work. Its use in blasting is simply to do at little expense,
and in a short time, an amount of work which it would take
many labourers a considerable time to perform. In virtue
of the arrangement of its chemical constituents it possesses,
in a small compass, this store of work-producing power.
Again, in order that water in a reservoir may be capable of
supplying motive power to mills or other machinery, it must
be capable of descending from a higher to a lower level, for
no work can be got out of still water, unless it have a head
as it is technically called. When the driving-weight of a
clock has run down, the clock stops ; and in order that the
weight may be rendered again efficient in maintaining the
motion of the wheels and pendulum, it must be wound up,
or . placed in such a position, relatively to the earth, that
work can be got out of it in consequence of that position.
In an air-gun we have a store of energy laid up in the form
of compressed air ; in a cross-bow, a wound-up watch, or
the lock of a cocked gun in the form of a bent spring ; in
a charged Leyden jar in the form of a distribution of
electricity; in a voltaic battery in the arrangement of
chemical elements or compounds ; in a labourer, primed
for work in the form of a proper supply of food. In all
such cases, where the energy is dormant, it is called Poten-
Science of Energy. 63
tial Energy y 1 and its amount is measured by the work
which it is capable of doing, and which it will do if
properly applied.
87. The unit for measurement of work usually employed
by British engineers is the foot-pound ; and though this
varies in amount from one locality to another, it is in such
general use, and so convenient when absolute accuracy is
not required, that it will be employed throughout this
chapter. It is the amount of work required to raise a pound
a foot high. It is evident that to raise any mass to a given
height, the amount of work required is proportional to the
number of pounds in the mass, and also to the number of
feet through which it is to be raised. Thus to raise a cwt.
a furlong high requires the same expenditure of work (73,920
foot-pounds) as to raise a stone-weight a mile high, or a
pound 14 miles. And the potential energy of the raised
mass, or the work which can be got out of it in virtue of its
position, is precisely equivalent to the work which has been
employed in raising it.
[The French or metrical unit of work is one kilogramme
raised through one metre in the latitude of Paris ; and is
called a kilogrammetre. Neglecting the dependence of
gravity upon latitude, the value of a kilogrammetre is 7*2331,
or a little less than seven and a quarter, foot-pounds.]
88. But if the mass be allowed to fall, we may remark
that it gains velocity as it descends, and that the square of
the velocity acquired at any point of the path is propor-
tional to the space through which the mass has fallen.
Also when a projectile is discharged vertically upwards it
possesses no potential energy at the commencement of its
flight, but it has, in virtue of its motion, energy, or power of
1 The term Energy is due to Young, Potential Energy to Rankine.
The idea of Potential Energy seems to have been first distinguished by
L. N. M. Carnot, who speaks of force vive latente (Principes . . . de
V Equilibre et du Mouvement, Paris, 1803), and by W. Thomson, who
called it Statical Energy.
64 Historical Sketch of the
doing work. To measure this energy, we must find how
much work it is capable of producing, and we find that it is
proportional to the square of the velocity. That is, a pro-
jectile discharged upwards will rise to four times the height
if its initial velocity be doubled, to nine times if trebled,' and
so on. If we now introduce the term Kinetic l Energy to
signify the amount of work which a mass can do in virtue
of its motion, we must measure it by half the product of the
mass into the square of its velocity ; and the ordinary
formulae for the motion of a projectile show that, neglecting
the resistance of the air, the sum of the Potential and
Kinetic Energies remains constant during the flight. There
is perpetual transformation of kinetic into potential energy,
as the projectile rises, and a retransformation as it descends.
89. An excellent illustration is furnished by the simple
case of the oscillation of a pendulum, where the energy
originally given to the bob, either in a kinetic form by pro-
jecting it from its lowest position, or in a potential form by
drawing it aside from the vertical, and then letting it fall, is
constantly transformed and retransformed every quarter
oscillation.
90. The observations above made on these very simple
cases are found to be completely borne out in more com-
plex ones, as, for instance, in the oscillations of an elastic
body, such as the balance-spring of a watch, a tuning-fork,
etc. Here the potential energy consists in a deformation
of the elastic body, as in bending a spring, etc. etc. All
this, however, is on the supposition that the bodies are
perfectly elastic, and that there is no external resistance to
the motion.
91. The complete theory of all such cases was enuntiated
in a perfect form by Newton in the Principia as a scholium
to his Third Law of Motion ; in which he not only laid down
the so-called Principle of Vis- Viva, and D'Alembert's Prin-
1 'Energy,' by Thomson and Tait Good Words, 1862.
Science of Energy. 65
ciple, for which others long afterwards obtained great credit;
but stated, so far, as the development of experimental science
in his time permitted, the great law of Conservation of
Energy. This remarkable passage -appears, until lately, to
have escaped notice; or, at least, not to have received
sufficient consideration. It is as follows :
1 Si cestimetur agentis actio ex ejus vi et velodtate conjunctim;
et similiter resistentis reactio c&stimetur conjunctim ex ejus
partium singularum velocitatibus et viribus resistendi ab earum
attritione, coh&sione, pondere, et acceleratione oriundis ; erunt
actio et reactio, in omni instrumentorum usu, sibi invicem
semper aquales.'
By the context it is easy to see that the actio here spoken
of by Newton is precisely what is now called rate of doing
work, or horse-power. Also the reactio, as far as acceleration
is concerned, is precisely what is now known as rate of
increase of kinetic energy. Newton's statement is therefore,
in modern phraseology, equivalent to this :
Work done on any system of bodies has its equivalent in the
form of work done against friction, molecular forces, or gravity,
if there be no acceleration ; but if there be acceleration, part of
the work is expended in overcoming resistance to acceleration,
and the additional kinetic energy developed is equivalent to the
work so spent.
As we have already seen, when part of the work is done
against gravity, as in raising a weight, or against molecular
forces, as in bending a spring, it is stored up as potential
energy; and the recoil of the spring, or the fall of the
weight, are capable at any future time of restoring the work
expended in producing these effects. But in Newton's
time, and long afterwards, it was supposed that work spent
in friction was absolutely lost. Now, by the experimental
researches of Davy, Rumford, and Joule, we know that it is
merely transformed into other and more inscrutable, but
equivalent, quantities of energy in the forms of heat and
electrification or electric motion.
E
66 Historical Sketch of the
92. But, before passing to these higher considerations, it
may be well to exemplify Newton's great discovery, by
applying it to such common cases of transformation of
energy as have been already mentioned, or are constantly
observed, and which are not much influenced by the pro-
duction of heat or electricity. Thus, in the case of the
simple pendulum, when it is at one end of its range, it has
potential energy, in virtue of which work can be done upon it
by gravity. This is wholly expended in producing acceleration
of motion as the bob descends ; and, when it has reached
its lowest position, the kinetic energy produced is equivalent
to the work so done, that is, to the potential energy lost.
As it rises again, work is done against gravity, which is
stored up as potential energy ; but the work so done comes
from the store of kinetic energy possessed by the bob ; and
when this is exhausted, the bob rests for an instant, to pursue
a similar course of transformations. With the change of a
word or two, the same explanation applies to the oscillations
of the balance-spring of a watch. In the case of a tuning-
fork, however, the oscillations more rapidly diminish in
energy; but here we have still the law of conservation,
because part is by imperfect elasticity changed into heat
within the substance of the fork, and what is lost to the fork
is communicated to the air, becoming transformed into the
kinetic energy of sound. Its ultimate fate will occupy us
presently.
93. The leading dates in the history of the foundation (not
the development} of the science of energy, besides those given
in the preceding chapter, are few and comparatively definite. 1
In January i843, 2 Joule showed that mechanical work
can be converted into an equivalent of heat mediately by
the induced currents of the magneto-electric machine, and
thus that current electricity is a form of energy subject to
1 See Tait, Recent Advances in Physical Science.
2 Memoirs of 'the Lit. and Phil. Soc. Manchester, vol. vii.
Science of Energy. 6 7
the law of conservation. This step enabled him to apply
his previous investigations (whose publication dates from
1840) regarding electrolysis to the establishment of the
principle of energy in chemical action. Thus, to quote
only a few sentences, he says
* However we arrange the voltaic apparatus, and whatever cells of
electrolysis we include in the circuit, the whole caloric of the circuit is
exactly accounted for by the whole of the chemical changes.'
* The mechanical and heating powers of a current are proportional
to each other.'
' I have little doubt that by interposing an electro-magnetic engine
in the circuit of a battery, a diminution of the heat evolved per equi-
valent of chemical change would be the consequence, and in propor-
tion to the mechanical power obtained.'
94. In August 1843, Joule read to the British Associa-
tion, at Cork, a paper entitled * On the Calorific Effects of
Magneto- Electricity, and the Mechanical Value of Heat?
This was inserted in the Philosophical Magazine, in October
and succeeding months of the same year. The main object
of the paper is the determination of the mechanical equiva-
lent of heat by causing a small electro-magnetic arrange-
ment to revolve between the poles of a larger electro-magnet,
and measuring the heat developed in the smaller coil after
the expenditure of a known amount of work in turning it.
He displayed great resources as an experimenter in deduc-
ing from this combination results, which, considering the
extreme difficulty of the process, agreed wonderfully well
with each other, and which led to a mean value (838 foot-
pounds) of the dynamical equivalent of heat (only) 8J per
cent, too high. He has shown that some error was to be
expected from the impossibility of measuring, and taking
account of, the fraction of the whole heat developed, which
fell to the share of the large electro-magnet. But he care-
fully proved that heat is developed in the whole circuit, and
that it is not merely transferred by induction from one part
of the circuit to another : thus supplying an additional
proof to that of Davy, of the immateriality of heat. This
68 Historical Sketch of the
experiment has since been converted by Foucault and
others into a very striking lecture-room illustration of the
transformation of work into heat.
The appendix to this paper contains the wonderful
approximation (770 foot-pounds) to the value of the dyna-
mical equivalent of heat deduced by friction of water,
which was examined in the preceding chapter.
95. Thus, in all the scientifically legitimate steps which
the early history of the principle records, Joule had the
priority. His work has been much extended by others,
especially Clausius, Helmholtz, Mayer, Rankine, and
Thomson, in the developed applications of the principle in
many directions. To their results the reader's attention
will presently be directed ; but he should clearly recognise
the fact that the experimental foundation of the principle
in its generality, and the earliest suggestions of many of its
most important applications, belong unquestionably to Joule.
Trained to accurate experiment and profound reflection in
the school of Dalton, the pupil has not only immortalised
himself, but has added to the fame of the master.
96. In an admirable tract by Helmholtz 1 (who must be
classed as one of the most successful of the early promoters
of the science of energy on legitimate principles), the whole
subject is based upon Newton's principle, with one or other
of the following postulates :
(a) Matter consists of ultimate particles which exert upon
each other forces whose directions are those of the lines
joining each pair of particles, and whose magnitudes depend
solely on the distances between the particles.
(b) ' The Perpetual Motion' is impossible.
This is, of course, a strictly logical foundation for the
science of Energy, if it be taken for granted as an experi-
mental result that the perpetual motion is impossible ; or if
1 Ueber die Erhaltung der Kraft Berlin, 1847. Translated in
Taylors Scientific Memoirs, 1853.
Science of Energy. 69
we could be sure that the ultimate parts of matter act on
each other in the manner assumed. Unfortunately, it must
be confessed that we know nothing as to the ultimate
nature of matter, and (a) is not in the present state of
experimental science more than a very improbable hypo-
thesis. Again, to assume (b) is apparently to beg the
question, to assume in fact that the Conservation of Energy
applies not only to such cases as Newton had already
treated, but to the more mysterious actions of heat, electri-
city, etc. 1 And though Joule's experiments have shown
that even for these the principle holds good : there is, it is
to be feared, still a fond hope entertained by many that the
perpetual motion may perhaps yet be obtained by electrical
processes. This has received a sort of countenance from
the fact, that the best-known complete hypothesis (that of
Weber) on which the mutual actions of electric currents
have yet been explained, requires the admission of mutual
forces between moving quantities of electricity, which are
not consistent with (a). 2 But before the facts discovered
1 That Helmholtz, even in 1847, five years after Mayer's paper
(which, is by some said to have settled the question) appeared, regarded
the inquiry as a merely speculative one, on which experiment alone
could decide, is evident from his remark : * In den Fallen, wo die
moleciilaren Aenderungen und die Electricitatsentwicklung moglichst
vermieden sind, wiirde sich diese Frage so stellen, ob fur einen gewis-
sen Verlust an mechanischer Kraft jedesmal eine bestimmte Quantitat
Warme entsteht, und inwiefern eine Warmequantitat einem Aequiva-
lent mechanischer Kraft entsprechen kann.'
2 In Poggendorffs Annalen, 1848,- vol. 73, Weber pointed out that
his very remarkable law of electric attraction does give a potential
in the sense that the electric force in any direction upon a particle
of electricity is the rate of diminution, per unit of length in that direc-
tion, of a certain function. It follows that when the system has
been brought back to its original configuration and its original veloci-
ties, no work on the whole has been done. Clerk- Maxwell has shown
that it is on this account that Weber's Theory is consistent with the
production of induced currents. But this potential involves relative
velocities as well as relative positions, and cannot therefore be properly
70 Historical Sketch of the
by Joule, all such objections must give way; just as the
corpuscular theory of light, even if we had not had the
undulatory theory to take its place, must have at once
been abandoned when it was found that light moves faster
in air than in water. Our real difficulty in such a case
as this is not with regard to the truth of the Conserva-
tion of Energy, but with regard to the nature of electricity ;
and Weber's result merely shows that electricity does not
consist of two sets of particles, vitreous and resinous, not
that there is a loop-hole for escape from the grand law
of Energy. Such a digression as this is not without its
use, if it give any reader a more complete idea of the
nature of the difficulties with which science is at present
most encumbered ; that they consist more in our ignorance
of the nature 'of matter arid energy than of the grand laws
to which their actions are ultimately subject.
97. The Theory of Energy, as at present developed, con-
templates its Conservation, Transformation, and Dissipation.
The principle of Conservation of Energy asserts that the
whole amount of energy in the universe, or in any limited
system which does not receive energy from without, or part
with it to external matter, is invariable.
The Transformation of Energy is the enuntiation of the
experimental fact, that in general any one form of energy
may by suitable processes be transformed, wholly or in part,
to an equivalent amount in any other given form. 1 It is
called potential energy. Weber's formula has been very ably discussed
by Helmholtz in Crete's Journal, but to give an idea of his reasoning
requires higher mathematics than I can venture to introduce in this
volume. I may merely mention that he shows that Weber's result is
in certain cases inconsistent with electric equilibrium.
1 Under the title Correlation of the Physical Forces, a great many
of these transformations of energy were discussed by Grove in 1842,
and he has since published a very curious work on the subject. Mrs.
Somerville in 1834, in her work ' On the Connection of the Physical
Sciences,' 1 seems to have been among the first to call attention to the
generality of such transformations.
Science of Energy. 7 1
subject, however, to laws analogous to, and including, that
of Carnot, and to limitations which are supplied by
The Dissipation of Energy. No known natural process is
exactly reversible, and whenever an attempt is made to
transform and re transform energy by an imperfect process,
part of the energy is necessarily transformed into heat and
dissipated, so that it cannot be wholly retransformed into
energy of visible motion of bodies. It therefore follows, that
as energy is constantly in a state of transformation, there is
a constant degradation of energy to the final unavailable
form of uniformly diffused heat ; and that this will go on
as long as transformations occur, until the whole energy of
the universe has taken this final form. 1
98. The remainder of the chapter will be devoted to a
semi-historical enumeration of cases occurring in nature or
experiment, and exemplification of the above laws in the
circumstances of each case.
99. The simplest cases are, of course, those of abstract
dynamics ; when we consider motion of a material system
under the action of any forces, but unresisted by friction.
The pendulum, balance-spring, projectiles, etc., have already
been noticed. As another instance, we may refer to the
motion of a planet about the sun. When in perihelion, that
is, when its potential energy is least, its velocity, and
therefore its kinetic energy, is greatest. In the case of a
comet moving in a parabolic orbit, the whole energy at any
time is equal to the potential energy at an infinite distance
from the sun ; and thus as the comet recedes from the sun,
the velocity, and with it the kinetic energy, become less and
less, tending ultimately to zero. That a cannon ball, fired
horizontally in vacua, may just rotate about the earth, its
velocity must be such as it would acquire by falling under
1 Thomson ' On a Universal Tendency in Nature to the Dissipation of
Mechanical Energy' Proc. Royal Soc. Edin., and Phil. Mag., 1852.
7 2 Historical Sketch of the
the action of ordinary terrestrial gravity (at the surface)
through a space equal to half the earth's radius ; about
five miles per second. In this case it would complete a
revolution in about 85 minutes, or the seventeenth part of
24 hours.
100. In all these cases the potential energy involved,
whether it depend upon molecular forces, as in a spring, or
upon external forces, as gravity, is of the same species as
that of a raised weight ; and the only form of kinetic energy
contemplated is that of visible motion. And here there is
constant transformation from one of these forms to the other,
and back again, for ever, without loss by dissipation, as the
process is in each case exactly reversible. They give us,
therefore, little insight into the more complex phenomena
to which we proceed. They are all summed up in the law of
conservation of Vis Viva, which we have already seen to be
merely a different form of statement of one of Newton's
discoveries. But in the ordinary text-books, the loss of vis
viva in the impact of imperfectly elastic bodies is asserted,
and its amount calculated ; not a hint being given that the
so-called loss is merely a transformation, partly, no doubt,
into the potential form of distortion of the impinging bodies,
but mainly into the kinetic form heat. The same text-
books also assert that there is no loss of vis viva in the im-
pact of perfectly elastic bodies. This is, of course, true, but
not in the sense in which it is asserted, since in the case of
impact of perfectly elastic bodies, a portion of the vis viva
of each would be changed, in general, into vibrations of the
body itself, and would, therefore, not appear as part of the
vis viva of the body considered as moving as a whole.
Take, for instance, the case of a bell and its clapper, both
supposed perfectly elastic.
101. As an example of the simpler cases of the loss by
friction, we may consider the experiment originally suggested
by Rumford, tried in a very imperfect manner by Mayer,
and completely worked out by Joule. When a mass of
Science of Energy. 73
water in an open vessel is made to rotate by stirring, its free
surface assumes a paraboloidal form ; and therefore the
energy communicated to it is partly kinetic and partly
potential, the latter being a temporary transformation of a
portion of the former. But, if it be left to itself for a short
time, it comes to rest with its surface horizontal, so that
both of these forms of energy have disappeared ; and the
water is, in all respects, except its temperature and the
effects depending thereon, precisely as it was before stirring.
Hence, if it be allowed to communicate its excess of tem-
perature to surrounding bodies, ,it will remain precisely as
before the operation, and by Carnot's axiom we are entitled
to regard the heat it has given out as the exact equivalent
of the work spent upon it. But the results of this process
were detailed in the preceding chapter. As another illus-
tration we may state, that when we see water flowing silently
and unaccelerated down the bed of a stream, the potential
energy is by fluid friction transformed into an increase of
the temperature of the water, and thus wasted, so far as
regards the production of useful work.
102. Sound has been already alluded to as the form in
which part of the energy of a tuning-fork is wasted. Sound
consists in fact of a state of air precisely analogous to the
state of the matter of the vibrating fork; comprising a
certain amount of potential energy in the form of compres-
sion or dilatation of air, analogous to the strain in the dis-
torted steel ; and a complementary amount of kinetic energy
in the vibrations of the particles of air. If air had no vis-
cosity, the transference of energy to it from the fork would
be simply a case of impact, easily reduced to a question of
abstract dynamics ; and the energy so transferred would be
propagated without loss in a form partly potential and partly
kinetic, in spherical waves through the atmosphere. The
energy of a complete wave in any such hypothetical case is,
curiously enough, always equally divided between the two
forms : and since, as the wave spreads, the amount of energy
74 Historical Sketch of the
in a given volume of air must be inversely proportional to
the whple volume of air occupied by the wave, the intensity
diminishes inversely as the square of the distance from the
centre of disturbance. There is, of course, in the portion
of the wave where the air is condensed, a rise of tempera-
ture, but in the rarefaction of the air in the other half of the
wave, an equivalent fall of temperature occurs ; so that, to
a first approximation, the mean temperature is unchanged
by the disturbance. But, in the actual case, the viscosity
of the air due to fluid friction is constantly converting a
portion of the energy of the wave into heat by an irreversible
process, and therefore the intensity of sound diminishes
more rapidly than the law of the inverse square of the
distance (which may hold, so far as experiments have yet
shown, for light and radiant heat in the interstellar space)
would require, its energy being constantly wasted in raising
the mean temperature of the air. 1 All motions of air, whether
sounds or winds, therefore, are ultimately transformed into
heat, and thus dissipated and lost, though not destroyed.
Whether there is anything analogous to this in the case of
undulatory motions in the inter-planetary ether is a grand,
but as yet almost entirely unattempted, inquiry.
103. But in actual experience the results of even the
simplest theoretical cases of abstract dynamics are never
realised. For, besides the friction between solids, and the
viscosity of fluids just considered, every motion of matter is
resisted by the all-pervading ether ; 2 and, on account of
the generation of electric currents, which in their turn
become heat, there is, in general, resistance to motion of
conducting matter in a magnetic field. The consideration
of these more recondite effects will be entered upon a little
1 Stokes on the ' Internal Friction of Fluids in Motion. ' Camb.
Phil Trans. 1845. See also Phil. Mag., 1851, i. p. 305.
2 Stewart and Tait, ' On the Heating of a Disc by Rapid Rotation in
Vacua.' Proc. R. S. 1865-6.
Science of Energy. 7 5
later ; but we will endeavour to render the transition as
gradual as possible.
104. We will now, partly following Helmholtz, consider
in order the application of the laws of 97 to the various
forms of physical energy in the more common cases which
have not as yet been particularly referred to, merely men-
tioning that he commences with a brief sketch of the
applications (already given above) of Newton's principle to
cases of abstract dynamics. Among these is one which we
have not yet noticed, viz., that Fresnel, in deducing hypo-
thetically the laws of polarisation of light by reflection and
refraction, made the conservation of Vis Viva the founda-
tion of his investigations, and arrived at results which are
at least very close approximations to truth.
105. The direct relations between mechanical energy and
heat have been sufficiently considered in the preceding
chapter, and they are therefore merely alluded to here in
order to maintain the continuity of the sketch. The
indirect relations between energy of all kinds and heat will
appear continually in the applications which follow.
106. We now pass to the consideration of the bearing of
the laws of energy upon the production of ordinary (so-
called) frictional electrification. There are two common
methods by which electrification of high tension is directly
produced, viz., by the ordinary electric machine, and by the
electrophorus.
107. When any two bodies of different kinds are brought
into contact, there is a certain amount of exhaustion of the
potential energy of chemical affinity between them (similar
to that of water which has reached a lower, from a higher,
level) and the equivalent of this is, partly at least (for it is
not yet known how bodies having chemical affinity attract
each other at a distance), developed in the new potential
form of a separation of the so-called electric fluids ; one
of the bodies receiving a positive, and the other an equal
negative, charge. The quantity of electricity, so developed,
76 Historical Sketch of the
depends upon the nature and the form of the bodies :
and is determined by the simple law (whose terms will be
presently explained), that the difference of electric potentials
in the two bodies, if they be conductors, and possibly in
the parts in contact, if they be non-conductors, depends
only on the nature of the bodies.
108. So long as the bodies remain in contact, it is
impossible to collect from them any of this electricity by
means of metallic conductors ; but since, in virtue of their
opposite charges, the bodies attract each other more than
before, more work has to be employed in separating them
than was gained in allowing them to come together. The
equivalent of the excess of work appears in the mutual
potential energy of the separated electricities. This is, in
all probability, the source of the electricity usually ascribed
to friction : so that the extra work required to turn an
electric machine when in good order, supposing the true
friction the same, would be (speaking roughly, and making
no allowance for sparks, noise, production of ozone, etc.)
directly as the square of the quantity of electricity produced.
The machine, therefore, acts by contact of dissimilar bodies,
producing a separation of electricities, and the application
of mechanical energy so as to tear these farther asunder.
And it is probable that all friction, perhaps not excepting
that caused by actual abrasion, is due to the production of
electricity. 1
1 Thomson (Bakerian Lecture, 1856, Phil. Trans., footnote to second
page) says, ' It appears highly probable that the first effect of the force
by which one solid is made to slide upon another, is electricity set into
a state of motion ; that this electric motion subsides wholly into heat
in most cases, either close to its origin .and instantaneously, as when
the solids are both of metal ; or at sensible distances from the actual
locality of friction, and during appretiable intervals of time, as when
the substance of one or both the bodies is of low conducting power for
electricity ; and that it only fails to produce the full equivalent in heat
for the work spent in overcoming the friction, when the electric currents
Science of Energy. 7 7
109. The electrophorus gives us a good instance of the
direct conversion of work into electric potential energy.
When the metallic disc is lifted from the excited plate of
resin, work requires to be expended to overcome the attrac-
tion of the electricity in the plate for the opposite electricity
developed by induction in the disc ; and the equivalent of
this work appears as the potential energy of the electricity
thus detached. Hence, when we charge a Leyden jar,
whether by the ordinary machine or by the electrophorus,
the energy of the charge is a transformation of the work
expended by the operator.
110. The potential, at any point, of a distribution of elec-
tricity, is the work required to convey unit of positive electri-
city, against the electric repulsions, from an infinite distance
to that point. From this definition it is evident that the
difference of the potentials at any two points is the work
required to carry unit of negative electricity from one to the
other ; and therefore, by the definition of work, the attraction
on unit of negative electricity at any point in any direction is
the rate of increase of the potential at that point per unit of
length in that direction. Hence the potential must have the
same value at all points of a conducting body, for otherwise
there would be (at points where its value changed) a result-
ant electric force, which observation proves never to exist
in the interior of a conductor. Thus the potential of any
conductor is the work required to remove a unit of negative
electricity from any point of its surface to an infinite
distance; or, what is -easily shown to be numerically equi-
valent to this, it is the amount of electricity which must be
given to a sphere of unit-radius connected with the con-
ductor by a long fine wire ; so that there may be no tendency
to transference of electricity along the wire.
are partially diverted from closed circuits in the two bodies, and in the
space between them, and are conducted away to produce other effects
in other localities.'
78 Historical Sketch of the
111. For any solitary conductor, as it is obvious that a
small and a large charge will be similarly distributed over
it, the potential is proportional to the quantity of electricity
in the charge. In fact the charge is the product of the
potential into a quantity called the capacity, which depends
upon the form and dimensions of the conductor, and its
position relatively to other conductors. And it is easily seen
that the potential energy of the charge is the work which
would have to be expended in bringing the charge, by
successive small instalments, from an infinite distance, to
the surface of the conductor. Helmholtz showed that this
is half the product of the charge and the potential ; hence,
as the potential is proportional to the charge, the potential
energy is, ceteris paribus, proportional to the square of the
charge. Helmholtz also showed that, if there be more than
one conductor, the whole energy is half the sum of the
products of the charge and potential of each.
112. A precisely similar process is applicable to such a
conductor as a Leyden jar ; and, in fact, to any statical
distribution of electricity. We thus see how the law, dis-
covered independently by Joule, Lenz and Jacobi, and Riess,
that the heat evolved by an electric discharge depends,
ceteris paribus, on the square of the quantity of electricity
in the charge ; or by Joule that, in voltaic electricity, it
depends on the square of the quantity of the current ;
accords with the conservation of energy.
113. The result of in, with Green's law of the capacity
of a jar, shows that in a jar of given material and form, and
with a given charge, the potential energy is inversely as the
surface of the jar, and also directly as the thickness of the
glass. The former of these statements gives an instructive
example of the dissipation of energy. Thus, if a charge be
divided between two equal jars, by simultaneously connecting
the pairs of outer and inner coatings, half of the charge
passes from the one jar to the other, and in doing so gene-
rates heat, sound, and light, each of which corresponds to a
Science of Energy. 79
loss of energy. The whole amount of electricity still
remains, but, being diffused over a greater surface, it has
less energy than before in proportion to the diminished
potential. Thus, with equal charges, and equal thickness
of glass, a small jar will give a more powerful shock than a
large one.
114. It has already been mentioned that contact of two
bodies, such as zinc and copper, developes a constant
difference of potential between them. From the explana-
tions subsequently given with reference to the potential, we
see that this is equivalent to saying that at the surface of
contact of two metals there is perpetually a force tending to
separate the two electricities in a direction perpendicular to
that surface, while at points ever so little within either of
the bodies there is no such force. The only way in which
we can conceive this to take place is by supposing that the
surfaces in contact are equally and oppositely electrified.
The effect of such an arrangement of electricity is nil on
points in either of the bodies, but at the surface of separa-
tion it accounts for the force to which is due the difference
of potentials in passing from one body to the other. If this
be the true explanation, it will follow, as Helmholtz has
pointed out, that bodies differ from each other in the
amount of the forces, sensible only at insensible distances,
which they exert upon positive and negative electricity.
By no fixed arrangement of simple conductors can a current
of electricity be produced; in fact it is obvious that if
such were the case, the current would continue for ever,
constantly producing heat by the resistance to conduction,
which is of course impossible.
115. By means, however, of a very simple arrangement,
not involving electrolysis, Thomson 1 has shown how to
collect the electricity developed in either of two metals in
contact ; but, as the principle of energy requires, mechanical
1 Proc. R. S. 1867. (N. B. Rev. 1864.)
8o Historical Sketch of the
energy has to be expended. He allows water, or copper
filings, to drop from a copper can, the drops falling (without
touching it) through a vertical zinc cylinder which is in
metallic contact with the can. Each drop carries with it
electricity induced by electrostatic induction in the air
between the zinc and copper : and, if they be collected in
an insulated dish, the latter may be charged to any extent.
The apparatus is, in fact, an electrical machine worked by
gravity ; and the energy of the charge acquired by the
insulated body on which the drops fall is accounted for by
a deficiency in the heat produced by their impacts. We
may contrast this experiment with the common one of
accelerating the flow of water from a pierced can by
electrifying it. In the last-mentioned case the loss of
potential energy by the dissipation of the charge appears in an
increase of heat produced by the impact of the falling drops.
For, in the one case, electrostatic action causes the drops
to fall less rapidly than they would if not electrified ; in the
other, more rapidly.
116. But the voltaic arrangement furnishes by far the
most powerful effects which can be obtained from the
fundamental separation of electricities by contact. By
interposing between two metals which have been electrified
by contact, a compound liquid (or electrolyte), these metals
are at once reduced to the same potential, a result which
could not have been obtained by connecting them by any
metallic conductor. By the passage of the electricity a
portion of the electrolyte is decomposed, and the potential
energy thus developed is equal to that possessed by the
electricity while separated in the metals. Bring the metals
into contact again, and the same series of operations may
be repeated. This state of things is directly obtained if we
close the circuit by connecting the metals by a wire, and
then we have constant separation of electricities at the point
of contact of different metals, and constant recombination,
attended with decomposition, through the electrolyte.
Science of Energy. 8 1
117. This is an exceedingly imperfect view of the action
of the galvanic battery, but it gives a general idea of the
fundamental processes, and must suffice for the present at
least, since the consideration of such complex phenomena
as polarisation of the electrodes, etc., would lead us into
details far too recondite for an elementary treatise. One
or two very singular results of Joule's early investigations
may be mentioned. It was shown by Faraday, that if the
current from a battery passes through any number of cells,
filled with any different electrolytes, the quantities of the
various components set at liberty in a given time in each of
the cells are proportional to the chemical equivalents of
these components and that the quantity of zinc dissolved
in each cell of the battery is determined by the same law.
Besides the electrolytic action, there is of course a develop-
ment of heat in the circuit. Hence, if the energy of
chemical affinity consumed in the battery be less than that
restored in the decomposing cell, we should have a produc-
tion from nothing of energy in the forms of heat and chemical
affinity. It appears from Thomson's calculations 1 that the
electromotive force required for the decomposition of water
is 1*318 times that furnished by a single cell of Daniell's
battery.
He says, ' Hence at least two cells of Daniell's battery are required
for the electrolysis of water ; but fourteen cells of Daniell's battery
connected in one circuit with ten electrolytic vessels of water with
platinum electrodes would be sufficient to effect gaseous decomposition
in each vessel.'
118. In Joule's paper of 1843, on tne neat of electrolysis,
he showed that heat is generated in the circuit in different
quantities by the electrical evolution of equal quantities of
hydrogen at equal surfaces of different metals, thereby
removing the difficulty arising from the fact, that in different
batteries all with the same more oxidisable metal, the
1 On the Mechanical Theory of Electrolysis. Phil. Mag. 1851.
/ " V - N OFTHE * ^V
UNIVERSITY I
82 Historical Sketch of the
electromotive force is found to vary with the other metal.
Thomson, 1 by applying the principle of energy to some
experimental results of Faraday, showed theoretically and
experimentally that a feeble continued current passing out
of an electrolytic cell by a zinc electrode, must generate
exactly as much more heat at the zinc surface than the
same amount of current would develop in passing out of an
electrolytic cell by a platinum electrode, as a zinc-platinum
pair working against great external resistance would develop
in the resistance wire by the same amount of current. Thus,
let a circuit be formed of three cells, each of water acidulated
with sulphuric acid ; with plates of zinc and platinum
immersed in No. i, zinc and tin in No. 2, and zinc and
zinc in No. 3 ; the platinum of No. i being connected with
the zinc of No. 2, the tin of No. 2 with one of the zincs of
No. 3, and the other zinc of No. 3 with the zinc of No. i.
There will be precisely the same chemical action in each of
the three cells ; yet No. 2 will give only about half the
electromotive force that No. i does ; and No. 3 will give
precisely none. That a tin-zinc element should give only
about half the electromotive force of a platinum-zinc element,
with precisely the same chemical action, and precisely the
same mode and quantity of hydrogen evolved, had been
felt as an objection to the electro-chemical theory, and
prominently put forward as such by Poggendorff. The
investigations of Joule and Thomson, just referred to, com-
pletely explain the difficulty, by proving that where the
current leaves the liquid by the zinc plate in No. 3 cell of
the circuit we have imagined, it experiences a reverse
electromotive force exactly equal to the whole electromotive
force of No. i cell ; and where it leaves the liquid of No. 2
cell by the tin plate, it there experiences a reverse electro-
motive force equal to the excess of the direct electromotive
force of No. i above that of No. 2 ; and that these reverse
1 British Association Report, 1852.
Science of Energy. 83
electromotive forces are the reactions of work done in
generating heat at the zinc and tin electrodes over and
above that (if any, whether positive or negative) at the
platinum electrode of No. i. It is to be remarked, farther,
that the whole heat of the chemical action in No. 3 is
developed in the cell itself, and that the excess of this above
that developed in No. i is exactly equal to the thermal
value of the work done externally by No. i. The three
papers just mentioned contain an immense amount of
valuable matter which cannot possibly be given in such a
work as this.
119. The conservation of energy would hold in the case
of the mutual actions of permanent magnets if their mag-
netisation were perfectly 'rigid,' because such magnetic
attractions and repulsions can be completely accounted for
by a distribution of an imaginary magnetic matter, each
unit of which attracts or repels another with a force whose
law is the same as that of gravitation ; and which therefore
satisfies the criterion (a) ( 96) required by Helmholtz's
investigation. But the perpetual-motionists have not yet
given up attempts to construct self-driving engines by means
of permanent magnets.
120. The force exerted by a voltaic current upon a
magnet at rest is precisely the same as that exerted by a
uniformly and normally magnetised open shell bounded by
the circuit, and of strength proportional to that of the
current, and is therefore also subject to the law of conser-
vation. But if the magnet be allowed to oscillate under the
influence of the current, it comes sooner to rest than it
would do under the influence of the equivalent magnetic
shell. In fact, if the experiment were made in vacuo, the
needle would ultimately come to rest in the former case, but
would maintain its oscillations undiminished for ever in the
latter. In the former it evidently loses energy, in the latter
it does not. [The hypothetical magnetic shell is supposed
to be a non-conductor, and to be unaffected by magnetic
84 Historical Sketch of the
induction.] Now, with the principle of conservation to
guide us, let us inquire what is the difference between the
two cases. Experiment shows that the motion in the former
case differs from that in the latter very much as if the
magnet were moving in a resisting medium the resistance
being (ceteris paribus) dependent on the rate of motion at
each instant. This alteration of the mutual action of cur-
rent and magnet of course implies an alteration of the
strength of the current, or what comes to the same thing,
the superposition upon it of another current which depends
solely upon the motion of the magnet, and is therefore inde-
pendent of the strength of the original current in the circuit.
More heat is, on the whole, generated in the circuit than
that due to the loss of energy by chemical combination in
the battery : and this is exactly equivalent to the corre-
sponding loss of energy by the magnet. In the Appendix (D)
below will be found an extract from Helmholtz's pregnant
pamphlet, which gives a very clear view of the relation of
electro-magnetism and magneto-electricity deducible from
the conservation of energy.
121. Now we might suppress the battery in the closed
circuit, and the conservation of energy immediately suggests
the question, Does the presence of this conducting body
alter the amount of work necessary to produce a given
motion of the magnet ? Long ago, Arago observed that, if -a
copper plate be placed under a vibrating magnetic needle,
the oscillations are very rapidly diminished, and the needle
comes to rest much sooner than when left to itself. This
Damper, as it is called, is still employed in galvanometers
of faulty construction, where the great moment of inertia of
the needles, and the small resistance opposed to their motion
by the air, render their oscillations long-continued, and their
observation tedious, and for many rapidly-changing pheno-
mena their use nil. [There are, of course, galvanometers
specially constructed to measure the ' time-integral ' of the
electro-magnetic force produced by the discharge of a con-
Science of Energy. 85
denser, etc. But these require no damper.] Subsequently,
Arago showed that if the disc be made to rotate, it carries
the needle with it. Faraday cleared up the whole subject in
1831, by his fine discovery of the induction of electric cur-
rents in the relative motion of a magnet and a conductor.
The damper acts by the reaction (upon the needle) of the
currents produced by the relative motion : which Lenz
showed to be such as in all cases to resist that motion ; and
it is their energy, and, afterwards, that of the heat into which
(by resistance to conduction) they are finally transformed,
which forms the equivalent to the loss of energy by the
vibrating needle.
122. We now see the complete explanation of the pheno-
mena of mutual action of currents and magnets which we
have already mentioned ; and whose full agreement with the
theory of energy was experimentally shown by Joule in 1843.
The magneto-electric machine, which depends entirely upon
this principle, is employed on a large scale in many im-
portant applications; for instance, it is employed in pro-
ducing chemical decomposition, as in electroplating ;
physiological effects, as in the ordinary medico-electric
machines ; and light, as in the successful trials at the South
Foreland Lighthouse, where an electric spark, much more
luminous than the ordinary oil-lamp, was maintained by the
work expended by a small steam-engine in turning before
a series of electro-magnetic coils a wheel, to whose cir-
cumference a great number of powerful steel magnets
was attached. It is also applied, on certain telegraphic
lines, to the production of electric currents for the purpose
of signalling.
123. It is only with the relative motion of the magnet
and conductor that we are concerned, and therefore, although
we have hitherto supposed the magnet to move in presence
of the conductor, precisely similar effects will be produced
if the conductor move in presence of the magnet. Thus,
when we consider that the earth acts as an immense magnet
86 Historical Sketch of the
on all bodies near its surface, it is obvious that in general
all motions of electric conductors are resisted by the earth's
action upon the currents developed in them by their motion.
Faraday suggested the application of this principle to the
construction of a magneto-electric machine in which the
earth takes the place of the usual permanent magnets. The
apparatus consists simply of a copper disc made to rotate
about an axis, and the electricity is collected by two wires,
one of which touches the rim of the disc, while the other is
connected with the axis. More work is required to turn
this disc than would be required to turn with the same speed
an equal disc of non-conducting matter, and this excess of
work is entirely transformed into electric currents. If the
axis of the disc be in the direction of the dipping-needle, the
greatest possible amount of current-electricity is generated.
If, instead of a conducting disc, a circular coil of wire be
employed, rotating about a diameter, no current will be
produced when the axis is in the line of dip. This result
has been used by Thomson to ascertain the dip, and fur-
nishes in fact a method which may probably be made very
much more sensitive and accurate than that aiforded by the
instrument in common use : being entirely unaffected by
friction, which is a most serious impediment to the working
of the dipping-needle. But it is interesting to notice, as an
immediate deduction from what has just been said, that the
heat developed in all moving machinery is partly due to
true friction, partly to the viscosity of air, and partly to the
earth's magnetism. Thus, for instance, a gyroscope will
spin longer if its axis be placed in the line of dip than in any
other position, supposing all other circumstances the same.
124. There can be little doubt of the fact that magnetism
consists in something of the nature of electric currents
surrounding each separate molecule of the magnetic, or
magnetised, body; especially since Ampere, by his con-
struction of solenoids (or helical arrangements of conducting
wires), produced, without iron or other magnetic metal, all
Science of Energy. 87
the phenomena of magnetic attractions, etc. Whether
these currents exist naturally in all bodies, and are merely
reduced by magnetising force to parallelism, or whether
they are created by the magnetising force, matters little to
the conservation of energy, so long as it is possible to show
that in magnetising any body, and therefore endowing it
with a certain amount of potential energy as regards other
magnetic or magnetisable bodies and electric currents, a
certain equivalent of energy is spent. Now this expendi-
ture is always incurred, but quantitative determinations are
wanting as to how much is spent in magnetising, how much
in heat, sound, etc., which always accompany the magnetisa-
tion of iron. A very good instance of the conservation of
energy is supplied by the fact, that even the softest iron
takes time to acquire the full amount of magnetism due to
any action of currents or other magnets ; and that when the
magnetising force is removed, it does not instantly lose its
magnetism. If, therefore, a piece of soft iron be allowed
slowly to approach a magnet, and be then rapidly withdrawn
from it, the mutual attraction during the second part of the
operation is greater at each stage than during the first, and
therefore work must (on the whole) be spent in the process.
The iron is restored to its former position, and in a little
time its magnetism is lost. The work spent during the
operation (neglecting the induced currents due to the relative
motion, which are probably the same in iron as they would
be in an equal mass of any non-magnetic substance of the
same conductivity, and which tend to the same ultimate
form) is entirely transformed into heat. If a similar experi-
ment be made with a piece of unmagnetised steel, we have
in the energy of the magnetism which it permanently receives
the equivalent of the work spent.
125. That magnetism, whether in a magnetic or a diamag-
netic body, depends upon motion, was shown by Thomson 1
1 Proceedings of the Royal Society, 1856.
8 8 Historical Sketch of the
to follow as a necessary consequence of Faraday's beautiful
discovery of the rotation of the plane of polarisation of a
polarised ray of light produced by media under the influence
of a powerful magnet. The general correctness of Ampere's
hypothesis regarding the nature of magnetism may be con-
sidered as decisively established by this dynamical theory.
Faraday had observed the effect in diamagnetic bodies only :
but it was afterwards discovered, by Verdet, that the effect
of a paramagnetic body is to produce rotation of the plane
of polarisation in the opposite direction to that in a diamag-
netic under the same conditions. It seems most probable,
notwithstanding this discovery of Verdet's, that the rotations
constituting magnetisation in a diamagnetic body, are in the
same directions, but of less amount, than in the surround-
ing medium ; although the opposite has been held by many
naturalists.
126, The commonly received opinion, that a diamagnetic
body in a field of magnetic force takes the opposite polarity
to that produced in a paramagnetic body similarly circum-
stanced, is thus attacked by Thomson by an application
of the principle of energy. Since all paramagnetic bodies
require time for the fall development of their magnetism, and
do not instantly lose it when the magnetising force is
removed, we may of course suppose the same to be true for
diamagnetic bodies ; and it is easy to see that in such a
case a homogeneous non-crystalline diamagnetic sphere
rotating in a field of magnetic force would, if it always
tended to take the opposite distribution of magnetism to
that acquired by iron under the same circumstances, be
acted upon by a couple constantly tending to turn it in the
same direction round its centre, and would therefore be a
source of the perpetual motion.
127. Among the various applications of the Science of
Energy, the proof of the mutual dependence of the different
kinds of electromagnetic phenomena is interesting, as ex-
pressing in a distinct form the ideas which were gradually
Science of Energy. 89
developed by Orsted and Ampere, and which constitute
the scientific connection of Faraday's splendid chain of
discoveries.
Helmholtz, in his tract on the Conservation of Energy,
and W. Thomson, in his Mechanical Theory of Electrolysis?
working independently of each other, arrived at a proof on
strictly mechanical principles, that if the phenomena dis-
covered by Orsted and Ampere be assumed, that is, if a
wire carrying an electric current is impelled across the lines
of magnetic force according to the observed laws, then the
phenomenon discovered by Faraday necessarily follows,
namely, a conductor moved across the lines of magnetic
force experiences an electromotive force whose intensity
can be completely determined by the application of the
equation of energy.
In this way Thomson showed that the unit of electro-
motive force already adopted by W. Weber, independently
of the principle of conservation of energy, is the only unit
consistent with that principle.
128. Thomson 2 has also remarked that the energy of an
electric current is kinetic energy, that is, it depends on the
motion of matter. The matter in motion is not however
simply in the conducting wire ; for the energy of the current
depends on the form of the wire and the media in its neigh-
bourhood, as well as on the length and thickness of the wire.
He looks for this motion not merely in the wire itself, but
also in the surrounding space, wherever the electromagnetic
action extends, and he has given reasons for supposing that
the motion is of the nature of rotation round the lines of
magnetic force as axes.
Thomson, therefore, regards the medium which surrounds
magnets and conductors as the seat of rotatory motions of
great energy, which by their centrifugal force cause the
1 Trans. British Ass., 1848; Phil. Mag. Dec. 1851.
2 Nichol's Cyclop&dia, Art. 'Magnetism.'
9O Historical Sketch of the
magnetic attractions. He has also shown that to account
for the transmission of light and heat from the sun, we
must admit that the interplanetary medium has a density
by no means inappretiable.
129. This method of looking for the origin of electrical
effects in the surrounding medium, as well as in the visible
apparatus, is that which under the name of the method of
Lines of Force is used so much by Faraday in his re-
searches. Clerk-Maxwell 1 has expressed this method in
mathematical language, and, by means of particular hypo-
theses 2 as to the molecular vortices, has shown how the
various phenomena may be connected with one another.
He seems, however, 3 to have since discarded these
hypotheses, and to rely only on the principle of energy
applied to investigate the properties of the medium which
he supposes to be the cause of electromagnetic effect.
130. He assumes that there is a medium capable of
transmitting light and heat, and therefore capable of storing
up two kinds of energy, that of motion and that of elastic
resilience, both which are exemplified in the case of
luminous waves. The medium, if capable of these motions
and stresses, may also be capable of others, and these may
produce visible phenomena. Thomson has shown that the
action of magnetism on polarised light indicates a state of
motion wherever magnetic lines exist. Now, every current
is surrounded by such lines, whose intensity depends on
that of the current. There will, therefore, be a certain
inertia to be overcome in starting the current, and a certain
persistence in the current when started, just as in any piece
1 On Faraday's Lines of Force, Camb. Phil. Trans. 1857. [For
this, and a great deal more than can be even indicated in our narrow
limits, see Clerk-Maxwell's splendid work on Electricity and Mag-
netism, which was published in 1872.]
2 On Physical Lines of Force, Phil. Mag. 1861-2.
3 Dynamical Theory of the Electromagnetic Field, Phil. Trans.
1865. Electricity and Magnetism, 1872.
Science of Energy. 9 1
of wheelwork, the inertia of every wheel adds apparent
inertia to the motions of the driving-wheel. From this
Clerk-Maxwell has deduced, by Lagrange's dynamical equa-
tion, 1 the known laws of the induction of currents, and of
the attraction of currents.
131. The force by which the motion of the medium is
transmitted from one part of the field to another is called
the electromotive force. If we suppose that when the
electromotive force acts on a dielectric, it produces a kind
of polarisation, or, as he calls it, an electric displacement,
depending on the nature of the medium, then energy of a
different kind will exist in the medium, similar to that
which exists in a strained elastic body, and measured by
the half product of the electromotive force and the electric
displacement. From these assumptions he has deduced all
the known laws of electricity and magnetism, except Ohm's
Law of Conduction, which remains a primary fact.
132. On this theory of the electromagnetic field, Clerk-
Maxwell has founded an electromagnetic theory of light.
He determines from the equations representing the known
laws of electricity the rate of propagation of any kind of
disturbance. The physical quantities involved in this cal-
culation have been already determined by W. Weber, and
the resulting velocity of propagation of electromagnetic dis-
turbance differs less from the mean of the various estimates
of the velocity of light than these do from each other. This
result goes far to strengthen the theory that both light and
electricity are phenomena of a medium, by showing that the
medium which is assumed to explain the one set of pheno-
mena is capable of explaining the other. By taking the
more general case of a medium having different properties
in different directions, the electromagnetic theory leads to
the conclusion that only two velocities of propagation are
possible, both corresponding to transverse disturbances, and
1 Thomson and Tait. Nat. Phil., 293.
9 2 Historical Sketch of the
that the disturbance normal to the wave, which forms so
great a difficulty in the ordinary form of the undulatory
theory, is incapable of being propagated. The experiments
of Holtzmann on spheres of crystallized sulphur agree
with this theory.
Another result of his theory is that the dielectric capacity
of a substance is equal to the square of its index of refrac-
tion. The experiments of Holtzmann, 1 Schiller, 2 and
Silow, 3 seem to show that this is true very exactly for gases,
and approximately for solids and liquids. But Hopkinson 4
has shown that in glass there is no approach to agreement.
133. In the preceding chapter, Seebeck's discovery of the
production of electric currents by unequal heating in any
non-homogeneous conductor was merely mentioned; we
must now consider, as a case of the conservation of energy,
the transformation of heat into work which would be effected
by applying such currents to drive an electromagnetic engine.
134. If the ends of an iron wire be attached by twisting
or soldering to the extremities of the copper wire of a
galvanometer, and one of these junctions be heated, the
galvanometer indicates the passage of a current in the
circuit in a direction from copper to iron through the heated
junction. The first application of the theory of energy to
this phenomenon is of course as follows : Since heating
the junction produces the energy of the current, part of the
heat must be expended in this process; though it is of
course entirely recovered as heat in the circuit, if the current
be not employed to do external work. The existence of
the current from copper to iron is thus associated with
absorption of heat in the junction ; agreeing with Peltier's
remarkable discovery that if an electric current be passed
through a circuit of iron and copper, originally at the same
temperature throughout, it produces cold when passing
from copper to iron, and heat when passing from iron to
1 Vienna Sitzungsb. 1870. 3 Pogg. clvi.
2 Pogg. clii. 535. 4 Froc. R.S. 1877.
Science of Energy. 9 3
copper. If the two junctions be maintained each at a con-
stant temperature, a constant current passes from the warmer
to the colder junction through the iron wire ; and by the
principle of conservation of energy, the heat developed in
the circuit (together with the equivalent of the external work
done, if the current be employed to drive an electro-magnetic
engine) must be equal to the excess of the heat absorbed at
the warmer junction over that given out at the colder, pre-
cisely as in the case of a heat-engine. So far the process
presents no difficulties. But it was discovered by Gumming 1
in 1823, that not only is the strength of the current not
generally proportional to the difference of temperatures of the
junctions, but that if the difference be sufficiently great the
current may, in many cases, pass in the opposite direction.
In the copper-iron circuit, if the temperature of one junction
be at the neutral point (about 270 C., for the exact tempera-
ture varies with the specimens of the metals), the current
passes through it from copper to iron, whether the other junc-
tion be colder or warmer. Thomson 2 applied the principle
of energy to this case, and derived from it the conclusion
that one of three things must happen, the most unexpected
of which he found by experiment to be the actual one, viz.,
the startling result that a current passing in an iron bar or
wire from a hot to a cold part produces a cooling, but in copper
a heating effect. This very remarkable discovery, which, taken
in connection with that of Peltier, gives the key to the whole
subject of Thermo-electricity, has been recently made the
subject of a valuable experimental investigation by Le Roux, 3
who has found the so-called ' specific heat of electricity ' to
be null in lead. Tait 4 has since shown that in general, for
ordinary ranges of temperature this electric convection of heat
1 Camb. Phil. Trans.
- Bakerian Lecture Phil. Trans. 1855 ' On the Electrodynamic
Properties of Metals.'' Also Proc. R. S. ., Dec. 1851,
8 Annales de Chimie, 1867.
4 Proc. R. S. E. 1868, 1871-2 ; Trans. R. S. E. 1873; Rede Lecture
Natttre, 1873.
94 Historical Sketch of the
is proportional to the absolute temperature. This seems
to be at least approximately the case for the great majority
of ordinary metals through very wide ranges of temperature,
almost in fact up to their melting points. But iron and
nickel exhibit the curious phenomenon of change of sign of
their Thomson effect. This takes place at least twice in
each of these metals as their temperature is gradually raised.
Thus we can construct an ordinary thermo-electric circuit of
two metals in which there shall be no Peltier effect at all. [In
this case the current is maintained wholly by the Thomson
effect ; which, if the second metal be properly chosen, may
be confined entirely to the iron or nickel.] This may also be
effected, of course, by making a circuit of any three metals,
and raising the junction of each two to their, neutral point.
135. The theory of such phenomena (and of others far
more complex, involving, for instance, crystalline arrange-
ment), in complete accordance with the conservation of
energy, has been given by Thomson, 1 but it would be incon-
sistent with the character of this work to enter into any
details on such a subject. A similar remark must be made
regarding his application of the principle to the subject of
Thermo-magnetism, or the relation of the magnetisability of
various substances to their temperature ; one or two of his
results may, however, be mentioned. Thus, iron, at a mode-
rate or low red heat must experience a heating effect when
allowed to approach a magnet, and a cooling effect when
slowly drawn away from it ; while in cobalt, at ordinary tem-
peratures, exactly the opposite effects must be produced.
Similar effects must in general be produced when a doubly-
refracting crystal is turned in the neighbourhood of a magnet.
136. Magnus showed that sudden contact between the ends
of a wire, at different temperatures, produces a temporary
current, which, in copper, is from the cold to the warm end
across the junction, but in the opposite direction in platinum.
137. This meagre sketch of the general application of the
1 Trans. Royal Soc. Edin. 1854.
Science of Energy. . 95
principle to the chief phenomena of experimental physics
(an application which is every day indicating how to
co-ordinate some newly discovered fact, and even occa-
sionally to predict the result of a perfectly novel experi-
mental combination) will be closed by the brief consideration
of an instance or two which must be familiar to most readers.
Thus, in the case of the galvanic battery employed to decom-
pose water, we have the potential energy of chemical affinity
in the battery to begin with. This is probably first trans-
formed into electric motion; in fact, according to Joule,
heat of combination, like that of friction, is in all likelihood
due to resistance to conduction. Part of it, then, becomes
heat, which is developed simultaneously in all parts of the
circuit, and the rest is expended in producing potential
energy in the form of the explosive mixture of oxygen and
hydrogen. Thus, if the poles be connected, first directly by
a wire, and secondly with the decomposing cell interposed
in the circuit, and the action be allowed to go on in each
case till the same given quantity of zinc has been dissolved
in the battery the heat developed in the whole circuit will
be greater in the first case than in the second, by a quantity
which can at any future time be obtained by exploding the
mixed gases. The sound produced (with the mechanical
energy of the fragments of the eudiometer, if it should burst)
ultimately becomes heat, and the flash and heat of the
explosion are already in that form. Should the battery be
made to drive an electro-magnetic engine which is employed
in raising weights, in this case also less heat will be generated
in the whole circuit than is equivalent to the consumption
of zinc in the cells ; but in the form of the raised weights
this energy is stored up, to take its final transformation into
heat at any distance from the battery, and after any interval
of time however long. This is one of the finest of Joule's
discoveries, that chemical combination (i.e. combustion)
may be made to take place without generating at once its
full equivalent of heat.
138. Ruhmkorff's induction-coil is another beautiful in-
g6 Historical Sketch of the
stance of varied transformations of energy. While it is in
action we have ight, sound, heat, electricity, and motion of
gross matter, all simultaneously produced, and representing
separate portions of the potential energy which is disappear-
ing in the battery. Ultimately, in this case also, the whole
energy which thus disappears takes the final form of heat.
139. A most important question arising naturally from the
consideration of the laws of energy is that of the economic
production of any species of work. We have seen that in
all actual processes of transformation, energy must be dissi-
pated, and therefore it becomes necessary to inquire what
modes of transformation are least imperfect. In the pre-
ceding chapter we gave Thomson's formula for the proportion
of usefully applied heat in the steam or air engine. The
fraction of the whole energy which is there wasted is formed
by dividing the lower absolute temperature employed by the
higher. The reason of the superiority of the air-engine over
the steam-engine, as depending on this, has been already
pointed out. Joule had proved in I846 1 that, when a
battery drives an electro-magnetic engine, the fraction of the
chemical energy which is wasted in the form of heat is
found by dividing the strength of the current when the
machine is at work by the strength when it is at rest (which
is of course the greater of the two). And he observes that
this follows from the fact which he had previously proved,
that the heat developed is proportional to the square of the
strength of the current, combined with Faraday's discovery,
that the strength of the current is proportional to the amount
of zinc dissolved in a given time. 2
1 Scoresby and Joule on the Mechanical Pozuers of Electro-magnetism^
Steam, and Horses. Phil. Mag.
2 In symbols, Z being the amount of potential energy lost by zinc,
/ the intensity of current, R the resistance :
Z^ = RI*, when no work is done,
Z 2 = RI* + W.
These express Joule's Law. But by Faraday's Law Z z : Z 2 : : 7, : /.,.
Hence fraction of energy usefully expended = = *~ g .
Science of Energy. 97
140. Rankine 1 has shown, from general principles of
energy, that a similar formula must hold in every case of
transformation ; so that we have the means of determining
the useful effect of any combination as soon as certain easily-
attained experimental data have been found.
141. The superiority of the air-engine to the steam-engine
depends on the fact that we can, with safety, use far greater
ranges of temperature in the former than in the latter. If a
frictionless electro-magnetic engine could be constructed in
which the driving current would be very greatly reduced by
induction, and if the fuel for the battery (zinc and sulphuric
acid) could be produced at anything like the cost of a
mechanical equivalent of coal and oxygen, there can be
no doubt that the heat-engines would soon be superseded
by the electro-magnetic. But this is, as yet, perfectly hope-
less ; for, although the faster the electro-magnetic engine
turns the smaller is the proportionate waste of energy as far
as the battery is concerned, yet the waste by ordinary
friction becomes enormously increased.
142. Very few remarks upon the physiological applica-
tions of the laws of energy need be made here, since the
subject has been most ably discussed by Helmholtz, in a
series of lectures at the Royal Institution, of which copious
abstracts have been published in an accessible form. 2
In the appendix to Joule's paper of 1843 already referred
to, we find the following most suggestive sentence :
' If an animal were engaged in turning a piece of machinery, or in
ascending a mountain, I apprehend that, in proportion to the muscular
effort put forth for the purpose, a diminution of the heat evolved in the
system by a given chemical action would be experienced. '
Mayer's pamphlet of 1845 adds considerably to the develop-
ment of the question. He speculates acutely on the merely
1 General Law of the Transformation of Energy. Phil. Mag. 1853.
The Science of Energetics. Edin. Phil. Jour. 1855.
2 Medical Times and Gazette^ April 1864.
G
98 Historical Sketch of the
directive agency of the so-called Vital Force, and gives some
excellently chosen illustrations of his views. Recent re-
searches in chemical synthesis have broken down many of
the supports on which the old theory of vital force rested,
and the mode of its action remains in consequence exceed-
ingly obscure. But there can be little doubt that, as Joule
suggested (in his paper of 1846 already quoted from), an
animal more closely resembles an electro-magnetic, than a
heat, engine. And it is wonderful that it is a far more
economic engine than any which we are yet able to con-
struct. The first idea of this seems to have been enter-
tained by Rumford, for he expressly shows, in his paper
quoted from in the preceding chapter, that the amount of
work done by a horse is much greater than could be pro-
cured by employing its food as fuel in a steam-engine.
Simple illustrations of the application of conservation of
energy to animal processes are found in hybernating animals,
which expend a great part of their substance during the
winter in maintaining the animal heat : and in the greater
supply or choicer quality of food required by convicts in
penal servitude, than by others who are merely imprisoned.
143. Between animals on the one hand, and the majority
of plants on the other, there is a fundamental difference in
the application of the laws of energy. In the animal we
have chemical combination attended with the production
of heat, muscular energy, etc., as transformations of the
potential energy of the food (in which, of course, the air
inhaled is to be included). In plants, on the other hand,
carbonic acid and water, the energy of whose constituents
has been lost in animals, are again decomposed, and their
potential energy stored up afresh, so that they are once
more adapted for food or fuel. It is obvious that this
process would be inconsistent with the conservation of
energy unless the plant during its growth were supplied
with energy from external sources sufficient to account for
the energy apparently restored. This external supply is
Science of Energy. 99
given to plants in a radiant form from the sun. Their
green leaves absorb readily, and almost completely, those
portions of the light which fall on them which are capable
of producing chemical changes. This is beautifully illus-
trated by the processes of photography. The green light
which leaves scatter or allow to pass through them, pro-
duces scarcely any effect on the most sensitive photographic
preparations ; and one of the greatest imperfections of the
beautiful art of Daguerre and Talbot, the unnatural black-
ness of the foliage in photographic landscapes, is due to this
cause. So far as is yet known, this is a defect which cannot
be wholly cured. Thus it appears that we may compare
(roughly) an animal supplied with food to a galvanic battery,
in which chemical affinity is exhausted in producing electric
motion, heat, and mechanical work ; while a plant resembles
a cell containing an electrolyte, or a photographic plate, in
either of which energy supplied from without in the form
of electricity or light is transformed into a restoration of
potential energy of chemical affinity. Of course the analo-
gies are by no means complete, but they are sufficient to
give the reader a rough idea of the essential difference
between the two forms of organic life. For, though by far
the greater portion of the energy of the food supplied to an
animal is dissipated directly or indirectly as heat, a portion
is stored up as potential energy in its flesh, which in turn is
employed as the food of man or other animals, or even of
the animal itself. And a corresponding deviation, but in
the opposite direction, takes place in plants, where radiant
kinetic energy is to a certain extent devoted to the forma-
tion of complex products, which, though necessary to
animal life, cannot be produced in the animal system.
144. The energy at present directly available to man for
the production of mechanical work is almost entirely poten-
tial, and consists mainly of
1. Fuel ;
2. Food of Animals ;
ioo Historical Sketch of the
3. Ordinary water-power ;
4. Tidal water-power ;
These will presently be considered more closely ; but we
have also energy in a kinetic form, as
5. Winds and Ocean-currents ;
6. Hot Springs and Volcanoes ; etc. etc.
145. The immediate sources of these supplies are four :
I. Primordial Potential Energy of Chemical Affinity,
which probably still exists in native metals, possibly in
native sulphur, etc., but whose amount, at all events near
the surface of the globe, is now very small.
II. Solar Radiation.
III. The energy of the earth's rotation about its axis.
IV. The internal heat of the earth.
146. Thus, as regards (i.), our supplies of fuel for heat-
engines are, as was long ago remarked by Herschel and
Stephenson, mainly due to solar radiation. Our coal is
merely the result of transformation in vegetables, of solar
energy into potential energy of chemical affinity. So, on a
small scale, are diamond, amber, and other combustible
products of primeval vegetation. Though (II.) thus accounts
for the greater part of our store, (I.) must also be admitted,
though to a very subordinate place.
As to (2.), the food of all animals is vegetable or animal,
and therefore ultimately vegetable. This energy then de-
pends almost entirely on (II.) This also was stated long
ago by Herschel.
Ordinary water-power (3.) is the result of evaporation,
the diffusion and convection of vapour, and its subsequent
condensation at a higher level. It also is mainly due
to (II.)
Tidal water-power (4.), although not yet much used, is
capable, if properly applied, of giving valuable supplies of
energy. As the water is lifted by the attraction of the sun
and moon, it may be secured by proper contrivances at its
higher level, and there becomes an available supply of
Science of Energy. 101
energy when the tide has fallen again. Any such supply is,
however, abstracted from the energy of the earth's rota-
tion (III.)
Winds and ocean currents (5.), both employed in navi-
gation, and the former in driving machinery, are, like (3.),
direct transformations of solar radiation (II.)
So far as this brief and imperfect summary (which it
would be easy to extend indefinitely) goes, there remain to
be considered only (6.) Hot Springs and Volcanoes, due
to (IV.), but of which no application to useful mechanical
purposes has yet been attempted.
147. We must next very briefly consider the origin of
these causes, with the exception of (I.), which is of course
primary ; though possibly related to gravitation. Helm-
holtz, Mayer, and Thomson come to our assistance, and
suggest as the initial form of the energy of the universe the
potential energy of gravitation of matter irregularly diffused
through infinite space. By simple calculations it is easy to
see that, if the matter in the solar system had been originally
spread through a sphere enclosing the orbit of Neptune, the
falling together of its parts into separate agglomerations,
such as the sun and planets, would far more than account
for all the energy they now possess, in the forms of heat and
orbital and axial revolutions. It is not necessary to enter
here into details as to the amount of each of these forms of
energy in the members of the solar system. The reader
will find them given with more or less detail in the writings
of the three authors just named. Thomson's numerical
results, with reference to the l Age of the Sun's Heat^ are
amongst the most recent, and are probably the most accu-
rate of any that have been given on this vast subject. It is
sufficient to observe that these calculations entirely forbid
the supposition once entertained, that the sun's heat is due
to chemical combination (or combustion). If the sun's
1 Macmillatt's Magazine, 1862.
IO2 Historical Sketch of the
whole mass were composed (in the most effective propor-
tions) of the known bodies which would give the greatest
heat of combination, the entire heat that could be developed
by their union would but supply the sun's present loss by
radiation for 5000 years. But geological facts show that
for hundreds of thousands of years the sun has been radi-
ating at its present, if not at a much higher, rate. The
potential energy of gravitation is the only known antecedent
capable of accounting for the common facts of the case.
And the sun still retains so much potential energy among
its parts, that the mere contraction by cooling must be
sufficient (on account of the diminution of potential energy)
to maintain the present rate of radiation for ages to come.
In other words, the capacity of the sun's mass for heat, on
account of the enormous pressure to which it is exposed, is
very great. Thus (on the least, and most, favourable as-
sumptions), from seven to seven thousand years must elapse,
at the present rate of expenditure, before the temperature
of the whole is lowered by one degree Centigrade.
148. As regards the transformation of energy, this pre-
sumed origin of the sun's radiation is most instructive, and
we have only to notice the as yet unexplained relations
which have been observed to exist between solar spots on
the one hand, and two such distinct phenomena as terres-
trial magnetism and planetary configurations on the other, to
show that the grand subject has as yet been barely sketched ;
and that every step towards filling in the details will be of
importance as well as novelty in science.
149. As regards dissipation of energy, all the members of
the solar and stellar systems are of course in the position of
hot bodies cooling. The smaller bodies would of course be
less heated by the agglomeration of their constituents than
the larger ; and, even if they had been equally heated, would
cool faster. The original fluidity of all the larger masses is
attested by their nearly spherical forms, rendered more or
less oblate by their axial rotations. Dissipation by radia-
Science of Energy. 1 03
tion takes place very freely until the surface cools sufficiently
to solidify to some little depth ; and is then, on account of
the low conductivity of rock masses, reduced to a very slow
rate. Though a great portion of the interior of the earth
must be still at a high temperature, the surface temperature
is not perceptibly increased by conduction through the
crust. The sun, however, has been calculated to give out
energy so profusely, that the radiation from one square foot
of its surface amounts to 7000 horse-power. This estimate
is probably too low, as no account is taken of possible
absorption by the matter which fills all space between the
earth and sun.
150. But while the heat of the sun and planets is thus
being lost by dissipation, the energy of their axial and
orbital motions is, on account of resistance, being gradually
converted into heat. This process is so slow that its effects
have as yet been observed only on one of the smaller
comets, but it is so persistent that all the planets will in
time fall in to the sun, whose store of energy will thus
be for a short time recruited. One noticeable point in
Mayer's Celestial Dynamics is the effect of tidal friction in
dissipating the energy of a planet's axial rotation, an effect
which Adams and Delaunay have recently proved to exist
in the case of the earth. 1 [J. Thomson had taught this
eight years before Mayer. It appears, however, that the
first suggestion of such an effect is due to Kant. 2 ] The
general tendency of tides on the surface of a planet is to
retard its rotation till at last it turns always the same face
to the tide-producing body : and it is probable, as seems to
have been first noticed by Helmholtz, 3 that the remarkable
fact that satellites generally turn the same face to their
1 Thomson and Tait's Nat. Phil, 830.
2 Nature, vii. p. 241.
3 Wechselwirkung der Naturkrdfte Konigsberg, 1854. Translated
in Phil. Mag. 1856, i.
IO4 Historical Sketch of the Science of Energy.
primary is to be accounted for by tides produced by the
primary in the satellite while it was yet in a molten state.
151. Numerous and beautiful though they have been,
especially in the writings of Mayer, the applications of the
laws of energy to the solar system are yet merely in their
infancy ; and, till they have been carried into further detail,
we can expect to make but little from their application to
stellar or nebulous systems, of which our knowledge is so
small in comparison.
152. In this short account of the discovery and develop-
ment of the grand laws of nature, so far as they are yet
understood, the illustrations have been confined to the
simplest cases; and the reader must not imagine that he
has been introduced to more than a small fraction of the
known facts which have been directly shown to agree with
them. It is as if, in treating of the theory of gravitation, his
information had been restricted to the proof that Kepler's
laws of the planetary motions follow from it, and that it
enables naturalists to compare the masses of the earth
and sun; without his being made aware that lunar and
planetary perturbations, precession and nutation, and far
more recondite facts, are also perfectly explained by it.
For the same reason our account contains but a small
number of names, few philosophers being mentioned save
those who have made really novel and considerable addi-
tions to our knowledge of the subject of energy; though
there are many others, both experimenters and mathema-
ticians, whose work is of great importance, whether as
regards the minuter details, or the more practical applica-
tions of the whole theory.
CHAPTER III.
SKETCH OF THE FUNDAMENTAL PRINCIPLES OF
THERMODYNAMICS.
153. THE graphic method introduced by Watt, and still
extensively employed, especially in the testing of steam
engines, supplies a valuable geometrical representation of
the changes of volume and pressure of the working substance,
and of the work done during these changes. It was em-
ployed, as we have seen ( 29), by Clapeyron ; and it has
since been ably applied by Rankine, to whose paper 1 the
reader is referred for detailed information. We introduce
it here, as it is easily intelligible even to those who cannot
follow the analytical investigations which we must give
further on.
154. Let the successive volumes of the working substance
(which may be gas, vapour, or liquid, or even liquid and
vapour together, provided it have at every instant the same
pressure throughout) be re-
presented by lengths (OA)
measured along the line Ox,
and the corresponding pres-
sures by lines (AB) parallel
to Oy. The extremities (B)
of these lines will trace the
curve called Wctfs Diagram
of Energy : and the funda- A
mental property is that any area such as A B B' A',
1 Phil. Trans. 1854.
io6 Sketch of the Fundamental Principles
bounded by the curve, two ordinates, and the axis Ox,
represents the external work done by the substance during
its|expansion from volume OA to volume OA '. Hence,
as we may draw any curve whatever from B to B', and so
apply heat to the working substance that this curve shall
represent the relation of pressure to volume during the ex-
pansion, it is evident that the work done cannot generally
be expressed in terms of the initial and final conditions of
the substance.
155. For the present it is sufficient to suppose that the
working substance is confined in a cylinder with a movable
piston. Suppose S to be the area of the piston, and ab (in
the figure) the mean value of the pressure per square inch
during the expansion, then S.ab is the mean value of the
whole pressure upon the piston. Also, if o- represent the
space through which the piston moves while the volume
increases from OA to OA', we have
S.a-=AA'.
Multiplying both by ab, we have
S.ab. J?", etc., indicate the state of the body
when two, three, etc., units of heat have been communicated
to it at temperature T\.
From the points BB'B", etc., draw curves Bb$, B'b'B',
etc., indicating the relation between volume and pressure,
when no heat is communicated to or taken from the
substance. These curves are called Adiabatic lines.
of Thermodynamics. 113
Let Pft'P" be a curve corresponding to a certain other
fixed temperature T OJ say that of ice melting at given ,
pressure, and let bb'b" correspond to some other tempera-
ture, which we may call /. It is required to determine the
value of t in terms of 7\, T Q , and the areas Bb', bf$' t and ,
JB/3'. Here Bft is the work done by a perfect engine,
working between the temperatures 7\ and T Q , for each unit
of heat supplied. And thus, by 26, we see that if the
body be compressed from the state /3' to the state j8, at
constant temperature T , the heat given out bears to one
unit of heat the ratio of the heat given out to that taken in
by any reversible engine working between the temperatures
TI and 7V This is the thermodynamic problem.
169. Thomson's earliest method ( 26) was to define
equal differences of temperature, as those of the source and
refrigerator in a reversible engine when the percentage of
work produced from a given amount of heat is the same.
But this definition had the inconvenience of giving a scale
differing greatly from the mercurial, air, and other ordinary
thermometers ; the degrees defined by it corresponding to
larger and larger intervals on the air thermometer as the
temperature is higher. Besides, on such a scale, the tem-
perature of a body totally deprived of heat is negative-
infinite.
170. The observation of Joule ( 47 footnote, 64), as to
the probable form of Carnot's function in terms of tempera-
ture by the air thermometer, was therefore afterwards made
the foundation of the following definition of absolute tem-
perature.
Car not 's function is inversely proportional to the temperature
from absolute zero.
Thomson has put this in another, and apparently different,
form though, as will be seen in 175, it is really the
same. He says : * The temperatures of two bodies are
proportional to the quantities of heat respectively taken in
and given out in localities at one temperature and at the
H
1 14 Sketch of the Fundamental Principles
other, respectively, by a material system subjected to a
complete cycle of perfectly reversible thermo-dynamic oper-
ations, and not allowed to part with or take in heat at any
other temperature : or, the absolute values of two tempera-
tures are to one another in the proportion of the heat taken
in to the heat rejected in a perfect thermo-dynamic engine,
working with a source and refrigerator at the higher and
lower of the temperatures respectively." 1
Thus defined, the absolute scale of temperature is of
immense importance. It is independent of the substance
on which we operate ; our theoretical investigations become
marvellously simplified ; and the new scale has been experi-
mentally shown by Joule and Thomson to differ but slightly
from that of the ordinary air thermometer.
171. The elements referred to in 170 being exhibited
in the diagram of 168, the definition of 170 gives
But if TI be the temperature of saturated steam under
pressure equal to that of 76o mro of mercury at the freezing
point, at the sea level in latitude 45, and T Q be that of
melting ice, we have by definition 7\ 7* = ioo.
Hence
" ' 3 h'
-, or, finally,
100
Experiments on any one substance determine T Q . Its value
is probably nearer to 274 than to any other integer.
172. Referring again to the diagram of 1 68, let H^ , If, H^
be the quantities of heat which must be supplied to the
1 Trans. R.S.E., May 1854.
of Thermodynamics. 1 1 5
body in expanding at constant temperatures T lf f, T Q
respectively from B to B' ', from b to b", and from ft to /3".
Our definition of temperature ( 170) gives at once
H i H HQ
y\ m i~T % *
Hence there is a quantity, characteristic of the substance
at all points on the same adiabatic line, such that, if < be
its value along Bbf$, and its value along B"b"P", we
may write
< = o for the adiabatic which passes through the point of
the diagram corresponding to the standard state of the
substance.
173. The lines in the diagram of 168 have thus the
equations
t= const. (Isothermal.)
<>= const. (Adiabatic.)
and it is clear that, while / and may be treated as abso-
lutely independent of one another, the volume, pressure,
energy, etc., of the working substance are determined when
/ and are given.
These remarks will enable the student to follow easily
some parts of the analysis below especially the investiga-
tions of Rankine and Clausius.
174. Simple as these geometrical processes are, the
following analytical method is really simpler. Let the
quantity (d^) of heat required to alter the volume (v) and
temperature (f) of unit mass of the working substance by
infinitesimal quantities dv and dt be represented by
&q=Mdv+Ndt (i).
Then the whole external work done by the substance,
1 1 6 Sketch of the Fundamental Principles
less the dynamical equivalent of the heat supplied, is
pdv-J(Mdv+Ndt} (2),
\ip be the pressure of the working substance. 1
Hence the whole loss of energy during any series of
changes is J\(p-JM}dv-JNdt\ (3),
where, in order that one definite cycle may be represented,
t and v must be assigned in terms of some one independent
variable, such as the time.
Again, the first law of thermodynamics shows us that in
any complete cycle of operations the external work done is
equal to the heat which has disappeared, i.e. there is no
loss of energy : so that the above integral (3) must vanish
for a complete cycle whatever be the relation between v and
/. Hence the quantity under the integral sign must be the
complete differential of a function of the two independent
variables v and /. This gives at once the condition
<* ' dv
or -4-= - !dM
which is the complete analytical statement of the first law.
By 162 we see that the second law may put in the form
dp_JM (
Tt~~T (5) '
using the definition of temperature as in 170.
1 That/dz/ is the work done during the change of volume v to v + dv
at pressure p may be proved as follows. Let n be the infinitesimal
displacement of the bounding surface of the fluid at any point, measured
in the direction of the normal, then the whole increase of volume is
dvffndS
where dS is an element of the bounding surface. But, because the
pressure is the same throughout ( 154) we have from this equation
pdv =pffndSffnpdS.
The latter form expresses the whole work done by the outward force
(pdS) on each element of the surface, acting through the space (n) by
which that element is displaced in the direction of the force.
of Thermodynamics. 1 1 7
These equations (4) and (5), contain the whole theory.
Thus it appears that the expression
dM dN
dt ' dv
does not in general vanish, and therefore that, in (i), d^ is
not a complete differential of a function of two independent
variables v and t.
But it is known that there is always what is called an
Integrating Factor in such cases. Call it 0. Then to find
it we have
Whence
Hence we have
.Jfo -JB JdN dM\ Qdp
Mj -- N-^ - = : --- -77- = -- F-77 =
dt dv \dv dt J J dt
(by (5) i.e. by second law)
^
t
As there is no necessary relation in general between J/and N
let = ,.
dv
Then = -and6 where $ nas tne
meaning assigned in 172.
This is another way of writing in symbols the Second Law.
175. We now proceed to the investigation of the quantity
of heat required, under different circumstances as to tem-
perature, for the production of a given amount of work.
Consider an engine whose range is finite to be made up
of an infinite number of engines with infinitesimal range,
and let q+dq and q be the quantities of heat taken in and
1 1 8 Sketch of the Fundamental Principles
given out by that in which the working substance is at the
temperature /.
By the first law the work done is
Jdq,
. By the second law it is ( 161)
JCqdt,
or, by 170,
Hence, by equating these values of the same quantity,
given by the two laws separately, we have
dq__dt_
4~ * '
Hence q is proportional to /, or
q=At
where A is some constant.
Thus in a reversible engine with finite range, where q
units of heat are taken in at the temperature /, and ^ given
out at the lower temperature / OJ we have (as in 171)
q - q *-A
~t~T-
or, LC^i (6).
? to
The work done is, of course,
and this, by means of (6) can be at once put in the form
so that, as in 54, the percentage, of the heat taken from
the source, which is realised as useful work is the range of
the engine divided by the higher absolute temperature.
This pure numerical ratio is sometimes called the Efficiency
of Thermodynamics. 119
of the engine; but the Duty, the term more commonly
employed in the vernacular of engineers, is the number of
foot-pounds of work obtained from each pound of coal
consumed, a considerably more complex quantity.
176. The important equation (6) was extended by its
discoverer 1 as follows. If a material system experience a
continuous action, or a complete cycle of operations, of a
perfectly reversible kind, the quantities of heat which it
takes in at different temperatures are subject to a homo-
geneous linear equation, of which the co-efficients are the
reciprocals of these temperatures. If q n be the heat taken
in at temperature t n (to be reckoned as negative when heat
is given out), this is expressed by the formula
or S=o. (7).
To prove this, conceive now, in addition to this given
system, an engine emitting a quantity q lt of heat at tempera-
ture A, and taking in the corresponding quantity q l at
t\
temperature f t then an engine emitting the quantity
j-q\ + q-i at / 2 , and taking in the corresponding quantity
//! j.*Vt temperature /, ; another emitting tj
V/i tj \t\ t z
at / and taking in the corresponding quantity
and so on. These n 2 engines constitute a material
1 Trans. K.S.E. (May) 1854. See also Proc. R.S.E., Dec. 1851,
or Phil. Mag., 1852 ; Mechanical Theory of Thermo-electric Currents t
Eq. (6).
1 20 Sketch oj the Fundamental Principles
system, which causes, by reversible operations, an emission
of heat ^! at temperature / 1; q^ at t z , and q n - z at / n _ 2 ; and
taking in
at temperature t n -i- Now this system, along with the given
one, constitutes a complex system, causing, on the whole,
neither absorption nor emission of heat at the temperatures
/!, / 2} etc., or at any other temperatures than / n _j, t n \ but
giving rise to an absorption or emission equal to
at /_!, and an emission or absorption equal to zb q n at t n .
This complete system fulfils the criterion of reversibility ;
and having only two temperatures at localities where heat
is taken in or given out, is subject to the second law as
expressed in (6) : so that we must have
or .+
which may be considered as a general expression of the
second law of thermodynamics. The first law has the
corresponding expression
where W denotes the aggregate amount of work spent in
producing the operations.
For convenience of reference we may repeat these for-
mulae in the following concise form
o (8)
of Thermodynamics. 121
4= (?)
If the engine or system be not reversible, this last equation
is no longer true. In such a case, the left-hand member is
obviously a positive quantity, if we consider heat taken in
as positive, and suppose the engine to be one in which work
is produced from heat.
177. If we suppose the temperatures of different parts of
the working substance to alter gradually during the process,
it is obvious that we must write (7) in the form
on the supposition that the cycles are reversible. This
integral is of remarkable importance in the theory of heat.
But, before entering on an examination of it, we may ad-
vantageously consider the subject from a somewhat different
point of view.
178. The real dynamical value of a quantity, d^, of heat
isya^, whatever be the temperature of the body which con-
tains it. But the practical value is only ( 54, 175)
where t is the temperature of the hot body, and / the lowest
available temperature. This value may be written in the
form
Hence, in any cyclical process whatever, if q be the whole
heat taken in, and q<> that given out, the practical value is
Now, if the cycle be reversible, the practical value is
122 Sketch of the Fundamental Principles
by the first law; so that, in this particular case (at least
unless / =o),
*.,
But in general this integral has a finite positive value,
because in non-reversible cycles the practical value of the
heat is always less than
Hence the amount of heat lost needlessly, i.e. otherwise
than to the refrigerator, or in producing work, is
This is Thomson's 1 expression for the amount of heat
dissipated during the cycle. It is, of course, an immediate
consequence of his important formula for the work of a per-
fect engine (175).
[It is very desirable to have a word to express the Avail-
ability for work of the heat in a given magazine ; a term for
that possession, the waste of which is called Dissipation^
179. If an irregularly heated body be enclosed in a non-
conducting envelop, there will be dissipation by thermal
conduction, as all parts of the body gradually arrive at a
common temperature. Thomson in 18532 gave a second
investigation of the amount of work which can be obtained
from such a body by means of perfect thermodynamic
* We have used the notation = \cdrn log
fcdm log /
Or Tf, fcdm ,
and W=j(cdm(t- T).
181. We may still farther simplify the equations by
assuming that the body is homogeneous, or that c is the
same for every element, and, as before, independent of the
temperature. Even if the body be heterogeneous, yet if the
1 24 Sketch of the Fundamental Principles
specific heat do not depend on the temperature, we may
put for the portion of any one kind of matter which has the
same temperature throughout,
cdm.
and then m lt etc., are the water equivalents of these portions.
Let, then, a mass m l of the body be given at temperature
/i, etc., and we have
2(w log /) i
When there are but two equal masses, at temperatures / and
/', the values are
T= rf
Such results as this may of course be easily obtained by a
much simpler process than the general one adopted above.
182. We may now speak of the Available Energy (A. E.)
of a single unequally heated body ; and of the mutual A. E.
of the parts of a system. And the following theorems are
easily proved :
The A. E. of a system is the sum of the A. E. of its parts,
together with their mutual A. E., when their individual A. E.
have been exhausted.
The mutual A. E. of any number of equal masses (whose
specific heats do not vary with temperature) is proportional
to the excess of the arithmetic, over the geometric, mean of
their absolute temperatures.
The A. E. of the universe tends continually to zero.
183. Clausius, 1 who published results equivalent to those
of 176 at a somewhat later date, calls such a term as
the equivalence-value of the transformation of the quantity q
1 Tail, Proc. R.S.E. 1867-8. 2 Pogg. Ann, (December) 1854.
of Thermodynamics. 125
of heat into work, or vice versa, at the temperature /. And
then equation (6) becomes the theorem of the equivalence of
transformations : which he expresses in words as follows :
If two transformations which, without necessitating any
other permanent change, can mutually replace one another,
be called equivalent, then the generation of the quantity of
heat q of the temperature t from work has the equivalence-
value
"7'
and the passage of the quantity of heat q, from the tempera-
ture /! to the temperature / 2 > has the equivalence-value
Instead of the absolute temperatures as above ( 170)
defined, Clausius introduces an unknown function, T, of the
temperature ; and, at the end of his paper, gives reasons,
( 48) for considering it probable that T is simply the abso-
lute temperature as measured on a perfect gas thermometer.
He does not seem (at least in any of his earlier papers) to
adopt or even refer to the absolute definition of tempera-
ture. This is one instance illustrating the general remark
quoted in 50 above from Thomson.
In his later papers Clausius calls by the name Entropy the
quantity
integrated from the standard state (o) to the actual state (i).
There is a difficulty about its exact definition, as will be
seen from 172 and footnote to 178.
184. From the equation
or, as it may be written,
= o,
126 Sketch of the Fundamental Principles
we may of course at once reproduce the formula (5) of
( 174) by introducing again the first law. For the equa-
tion indicates that the quantity under the integral sign is,
as already shown in 1 74, the complete differential of a
function of two independent variables, so that
<5\//
dp_^dM_dN^M
Jdt ~ dt dv" t'
From these we obtain
dN_T L ( dt L
dv~J\ dt\dt
185. From ( 174) it is evident that ^Vis the specific heat
at constant volume. To deduce a relation between the two
specific heats, let K be the specific heat at constant pressure.
Then
Kdt=Mdv+Ndt
the relation between v and / being such that the pressure is
constant, a condition which gives us the equation
.
dv dt
Eliminating the differentials dv and dt, we have
dp
df J dp
dv dv
(10).
of Thermodynamics. 1 2 7
186. In the case of the ideal perfect gas, we have, by the
law of Charles, combined with that of Boyle,
dp p
suppose: so that d~~~ '
dp p JM ,
and '
so that
For such a substance, therefore, we have
V
an absolute constant.
This property of permanent gases was arrived at by
Carnot in his original work. It gives by the equation pre-
ceding (K-N}p=RM.
Also, by '( 184),
dN
- r =0 (I2k
dv
If the gas expand or be compressed at constant tempera-
ture
, , . v dw
Mdv =-= = ~Y,
that is, the whole work spent is converted into heat, or the
whole heat supplied is converted into work.
If it have its volume changed in a non-conducting vessel
Taking v and p as independent variables this may be written
o = R Mdv -f- Nd (pv)
= Kpdv + Nvdp,
or px v K cons t. a >,
1 28 Sketch of the Fundamental Principles
as the reader may easily prove for himself. This is the
relation between the pressure and volume of air in a sound-
wave.
187. We may now take some of Thomson's more general
applications of the dynamical theory to the effects of com-
pression and distortion of bodies as regards the heat de-
veloped ( 57). For this purpose we must express the
compressibility, etc., in terms of the alterations produced,
and the forces producing them.
188. If a substance, at pressure / and volume v, be com-
pressed by an increase of pressure, S/, to volume v-\-&v, the
temperature being kept constant, the compressibility is
Hence, if K represent the reciprocal of the compressibility,
which may be called elasticity of volume, or cubical elasti-
city,
In the ideal perfect gas this is/. Hence arises the con-
fusion as to the meaning of the term ' elasticity.' For the
co-efficient of elasticity of volume is proportional to/ whether
/ or < be constant. In many books the elasticity of a gas
is merely another name for its pressure.
189. Again, if e be the co-efficient of cubical dilatation by
heat at constant pressure,
_
, i dv i dt i dp , N
Thus e= -=-= -- -T-== -- (14).
v dt v dp K dt
Thus in the ideal perfect gas e= -
of Thermodynamics. 129
190. If we substitute, in terms of these quantities, the
values of the differential co-efficients of p, the general
equations of 184, 185 take the more easily intelligible
forms
d , . N .d , .
tv(Ke}e (16).
t( K e) (17).
Of these, the first gives a singular relation between the rate
at which the specific heat at constant volume is altered by
a change of volume at constant temperature, and the rate
at which the pressure at constant volume alters with the
temperature. The third shows the amount of heat developed
by compression when the temperature is kept constant.
But it may be usefully applied, in connection with the second,
to find the change of temperature produced by compression
when a substance is enclosed in a non-conducting vessel.
For, if 8Y be the change of temperature due to the com-
pression Sv } we have
or &
which may also be written
*=
.
JKt.K.e.ve
From any of these expressions we see that when a sub-
stance contracts as its temperature rises (as is the case, for
instance, with water between its freezing-point and its point
of maximum density), its temperature is lowered by sudden
compression. For in such a substance e is negative.
Another useful formula, also given by Thomson, 1 and
immediately deducible from the above, is
(18).
1 Proc. R. S. 1857, or Phil. Mag. 1858, i. 541. See also Comptes
Rendus, Oct. 1864, fot another elementary investigation.
1 30 Sketch of the Fundamental Principles
191. Consider next the working substance to consist of a
mixture of a mass i x of some body in one molecular
state, and a mass x of the same in another state, at the
same temperature, but containing more latent heat. These
may be, for instance, water and saturated steam, or ice
and water.
Let the volumes of unit mass of the body in these two
states be respectively Fand Fi, their specific heats (that is
to say, the quantity of heat required to raise the tempera-
ture of unit mass of either form of the substance one degree,
under the condition that the two forms remain in equilibrium
during the process) c and c\, L the latent heat of unit mass
in the second state, and / the common pressure which
depends solely on the temperature and on the nature of the
body.
Then, if v be the joint volume,
F(i-#)+Fi*=0 (19).
Also, J/and N having the same meaning as in 174,
M'dv=L-j-dv,
dv
and
But, by differentiation of (19) we have
dx i
(20)
Combining these with the preceding equations, we have
and ('->+
N=c(i -xi + 'dx-L - __ (23).
.jr v
/
of Thermodynamics r'- \ 3 1
But we have ( 174)
dt ~\ dt dv
and, by (22) and (23), this becomes
a result from which x has of course disappeared.
Also, we have, by the expression for Carnot's function,
viz. :
* d P
or, by ( 170),
Substituting in (24) we obtain, finally,
dL L
(27) -
192. These equations give us among other results the
means of determining the effect of pressure on the melting,
or boiling, point of a substance.
Thus
so that an increase of pressure raises the melting, or boiling,
point if V-L is greater than V (as it is in the case of water
and steam), but lowers it where (as in ice and water) V^ is
less than V (\ 55).
By means of an equation identical with (24), Clausius
132 Sketch of the Fundamental Principles
obtained the result that the latent heat of water is diminished
by pressure.
And Regnault's experiments have shown that - for steam
J T t
is greater than ^-f
was measured in 172.
196. When a substance, by the application of heat, is
made to expand from v to v-\-dv, and to rise in temperature
from / to t+dt, the amount of heat supplied is ( 174)
Mdv+Ndt;
and the amount of work done on external bodies is
pdv,
Hence the amount by which the energy JE, present in the
body, has been increased is
dE=J(Mdv+Ndf}-pdv. (28).
We have already seen ( 174) that this expression is a com-
1 Trans. R. S. E., 1851, 'On the Quantities of Mechanical Energy
contained in a Fluid in different states. '
of Thermodynamics. 135
plete differential of two independent variables, in conse-
quence of the first law of thermodynamics. Hence we may
write
(29).
and, by eliminating E by differentiation, we obtain again
the results of 174.
197. We may put the equation (28) into many other
forms, some of which are of considerable use. Thus, by
(5) we have
and, putting w=f v v is another. (See 172.) To define it
. ( Let the body in a given state be confined in a
* < non-conducting vessel, and let its pressure and
\ volume be altered till its temperature is / -
Now let its temperature be maintained constant
at / , and let it be compressed to volume v 0t
Operation j
B.
and let it give out q units of heat in the pro-
cess ; then if < is the value of in the original
state, and > , in the state V Q , t Q ,
Hence, if < is given for a particular state of the substance,
< has a perfectly definite value for every other state.
Also, &q=td=Mdv+Ndt
and dW=pdv.
But in any complete cycle
or fpdvJftd$=o
so that, by 196,
pdv +Jtd<$> = dE.
Let E Q be the energy of the body at V Q and / , and let the
body do W units of work in the two operations A and B,
and give out q units of heat in operation B, then if E is the
original energy,
W+Jq.
Thus we have two new functions of the state of the body,
or five in all
146 Sketch of the Fundamental Principles
Among these we have the equations
p --- ( being constant),
and t=- -- , (v being constant), etc.
/ d
i , . T i dt\. I dp\
which give AT] = (-72}
\4*7+C>OMt \dJ
By the ordinary processes of the differential calculus we
may easily vary the form of these relations. For instance
dp
v const.
so that M\ =(*) etc. etc.
\ PJ4 const. \ 9/> const.
205. These hints may assist the student in the perusal of
the works of Rankine and Clausius ; and some such assist-
ance we have felt to be very desirable, as both authors, in
the more theoretical and speculative parts of their investiga-
tions, are somewhat diffuse and difficult of comprehension.
206. We conclude the chapter with a few investigations
taken from Thomson's 1 general paper on the ' Thermo-elastic
Properties of Matter.' These refer to any homogeneous
solid homogeneously strained.
207. In the first place, we see, by 176, that however
such strain be effected, and through whatever stages the
body passes in returning to its first condition, no heat will,
on the whole, have been absorbed or given out, provided
the temperature has been kept constant throughout the
whole operation, and therefore the quantity of heat absorbed
for any given change, at constant temperature, does not
depend on the way in which that change has been effected.
For in this case (7) becomes
l- ? =o.
208. Hence, if x, y, z, , 17, be any six quantities which
1 Quarterly Math. Journal, April 1855.
of Thermodynamics. 141
define the state of the body as regards strain, 1 and if
the heat absorbed, while the body is made to pass, at con-
stant temperature /, from a? , y 0y z 9 , f c , >?o, Co, to x,y, z, , 17, f,
we must have
H=t(x,y, z, , 17, f, t)-t(x ,y , So-, o, i/o, Co, ')'
Hence also the corresponding increase of the total energy
of the body ( 196) will be expressed by
EE Q = t = (x,y> Z, f, 77, f, /) <#>(^ i^o, ^o, fo, ^?o, fo, 0-
where, if w denote the work done by the applied forces, we
have
'= w +/zr. (31).
Thus, also, the work required to strain the body through
any given change of state, at a constant temperature, is
independent of the succession of strains by which that
change is effected.
209. To introduce the consideration of change of tempe-
rature, let us take the following reversible -cycle of opera-
tions, in which (as in 158) it may be considered as infinitely
small.
(I.) Raise the temperature from t to t+df, at the constant
state # , j , *o> o, ??o, Co-
The amount of heat required is
1 dE *dt
J dT a
(II.) Change the state of strain from # , y 0t etc., to x, y,
etc., at temperature t-\-dt.
The heat required is
(III.) Lower the temperature to /, at the constant state
x, y, etc.
1 Thomson and Tail's Natural Philosophy^ 669.
142 Sketch of the Fundamental Principles
The heat required is
(IV.) Restore the original state # , J , etc., keeping the
temperature at /.
The heat required is
-H.
'Hence by 176 we have (neglecting terms in (dt)'*)
H_
d '+ -"'
Tt '~~dT t+dt ~~t
4/^-i^=
Eliminating e by means of (3 1 ), we have
' t dw
as in 193, remembering that w has now the negative sign,
as it represents work done from without on the body.
Also, substituting this value of Zfin (31), we have
E-E Q =t=wt^j- (34),
which is the generalised form of the result of 198.
210. Various extremely important applications of these
equations can easily be given, as will be seen by referring
to Thomson's paper, but we confine ourselves to the very
simple one which follows.
If the strain be very small, the work required to produce
it may be expressed by
w=Pdx+Qdy+Rdz+Sd^+ Tdi}+ Ud^ (35),
where 1 #, y, z are taken as rectangular co-ordinates parallel
1 Thomson and Tait, 671.
of Thermodynamics. 143
to the edges of a cubical unit-volume of the solid. P, Q, It
are the normal pressures on the sides of this cube, S, T, U
the tangential components of the shearing stress.
Substituting in (33) we have
/ IdP, d'.dR, .dS,* .dT, .dU
Hence, as in 192, if P [or any other of the co-efficients in
dp
(35)] diminish as the temperature increases (so that -^ is
negative) cold will be produced (i.e. heat must be supplied
to keep the temperature constant) during the body's yielding
to the effect of the stress denoted by P-; and heat if it be
strained in the opposite way. The reverse will be the case
if P, etc., increase with /, as is the case, for instance, with
gaseous bodies, where P represents the pressure.
NOTE A ( 96).
(a.) The equations of motion of a particle of mass m,
under the action of any system of forces, are
where X, Y, Z are the components of the entire force found
by resolving it parallel to the several axes of x, y, z.
From them we at once deduce the equation of energy
Kdx_d*x
dt dP
dy d-y dz d i z\_dx dy <&
' dt'dP dt dt 1 ] ~di^~ .~dt^~ ~dt
If the velocity of m be called v, the first integral of this
equation may be written
(Xdx+Ydy+Zdz),
or, more definitely, v being the velocity of m when its co-
ordinates are x 0) y , Z Q ,
Now the Conservation of Energy requires that the value of
v should be always the same for the same values of x,y,z\
that is, that the change of kinetic energy in moving from
any one position to another should not depend upon the
particular path by which that transference is effected. Ana-
lytically, this is equivalent to saying that the integration of
the right-hand member of the equation must be capable of
being effected without any assumed relation or relations
between x, y, and z. Thus the expression
Xtix+ Ydy+Zdz
Note A. 145
must be the complete differential of a function of three
independent variables. We may therefore write
dV=Xdx+ Ydy+Zdz,
the negative sign being introduced for reasons of convenience.
We have, therefore,
X dV Y d Zz d ^
-A - " j , JL ~~ , Zy ^~ j ,
dx dy dz*
the differential co-efficients being partial. These show, by
the fundamental principles of the Differential Calculus, that
we have
dX dY_ dY dZ_ dZ dX__
fy~K-* y to~$- <&~&*
which are the usual analytical conditions for the existence
of a complete differential of three independent variables.
Either of the two last groups of equations gives the rela-
tions between X, Y, Z, which are required by the Conser-
vation of Energy.
(b.) If the only forces acting are such as tend to a fixed
centre, and depend on the distance from that centre only,
these conditions are fulfilled.
For let a, &, c be the co-ordinates of the centre, R the
force it exerts on m when their mutual distance is r\ we have
But
dr dr dr
r-y-=xa, r=y&, r-j- =zc, so that
dx dy dz
Y v dr v v dr 7 n dr
A = K -j-, r= JC-j-1 &=* -K-r,
dx dy dz
and
Xdx+ Ydy+Zdz=-Rdr.
This is a complete differential, because by hypothesis R
is a function of r only.
K
146 Note A.
(c.) If there be more than one fixed centre it is obvious
that we have
Xdx + Ydy+Zdz=V(Rdr)
every term of which sum is a complete differential.
(d.) When some of the centres are in motion independently
of the mass m, the conservation of energy does not neces-
sarily hold good, as it is possible by means of such moving
centres to keep m constantly revolving with ever increasing
velocity in some definite path.
(e.) But if the moving centres be themselves masses of
the system, and be acted on by m, and by the fixed centres,
only, the conservation of energy is true of the whole moving
system. In this form the problem becomes that of several
mutually acting particles m lt m z , ...... acted on by fixed
centres.
Let the co-ordinates of m l be x l9 y*, z l9 those of m 2 , x z ,
yi i
because action and reaction are equal and opposite. Hence
the above terms may be condensed into
where the differentiation is performed on the supposition
that the co-ordinates of m l and m n both change.
The resulting equation may therefore be written
dx d*x , dy d-y , dz
where R a is the force exerted by the fixed centre , , f
upon the particle distant r a from it, S the mutual force
between two particles whose mutual distance is r. Here
every term of each of the sums on the right is, separately, a
complete differential. The integral may be written
148 Note A.
where the left-hand member is the whole kinetic energy of
the moving particles ; and the right consists of two parts,
one due to the fixed centres, the other to the mutual action
of the particles.
(/) The converse, that, if the force acting on a single
particle tends to a fixed centre, and if the conservation of
energy hold, the force depends only on the distance from
that centre, is easily proved. For we must have
= - \Rdr
where it is to be shown that R is a function of r only. We
have, at once,
dv -dr dv r,dr dv ^dr
mv-j- = R-T) mv-j- R ; mv = R -
doc dx" dy dy } dz dz
Hence
or _dR dr dR dr
dx dy dy dx
with other two equations of the same form the three
forming the usual analytical conditions that R should be a
function of r only.
(g.) This theorem cannot be extended to any indefinite
number of centres : as it is usually possible, by means of
centres whose attraction is different in different directions,
to build up an arrangement producing on external matter a
conservative system of forces only.
(h.) When there is but one fixed centre of force the
conservation of energy for a single particle acted on by this
centre requires that the force should be exerted in the line
joining the particle with the centre, and should depend
upon its length only.
For it is obvious that the relative position of the particle
and the centre depends merely on their distance. That is,
v is a function of r only. Hence
Note 1). 149
%mv* = V
where V is a function of r ; and therefore
mvdv
dVfdr, , = attraction of infinite plate. Between two such parallel
plates, whose potentials are Fand V^ and whose surface
V V
densities are p and p , the attraction = 27r(/o/o ) = - --
But if /^=o, there is no force on points in second plate,
V
= o, /) =/o; .. 47r/o= If S be the common
surface, 4vQ=7-, as above.]
(n.) Suppose a number of additional concentric spherical
conducting shells, insulated from one another and uncharged,
to be interposed between the two already considered, the
condition that the force vanishes in the substance of each
requires that the inner surface of each be charged by in-
duction with a quantity Q and the outer surface with
+ Q. Hence, if there be but one, of which p is the inner
radius and r the thickness, our equation for the potential
of the innermost shell becomes
Note B. 153
. f _ f = near , y .
The same formula will therefore apply to any number of
such shells provided r be put for their joint thickness.
Hence the effect of such a substitute for the air is virtually
to reduce the distance between the coatings of the jar. A
similar effect, but to a less degree, would remain if each of
these shells were reduced to detached fragments insulated
from one another. This may give the student a hint in
understanding how, by the polarisation of its particles, one
dielectric may differ from another in specific inductive
capacity.
(.) Thus it appears that, whatever be the dielectric, we
have for a Ley den jar the formula
where / is directly as the thickness of the dielectric, and
inversely as its specific inductive capacity.
(p.} The potential energy of the charge is to be found by
allowing the separated electricities on the coatings to recom-
bine, and reckoning the work gained.
At any time when the charges on the coatings are
reduced to q and q, the potential is
qt
Hr
But if electricity, to the amount dq, now pass from the
inner to the outer coating, the work gained is
vdf,
and therefore the whole work, or potential energy of the
charge, is
(Q . t (Q ,
vdq = -^\ qdq
Jo O J
or
The principle here made use of is easily applicable to the
154 Note B.
proof of Helmholtz' general proposition ( in), but it
requires a rather higher analysis than we have employed in
this work.
(q.) These simple formulae enable us to solve many
important questions on the subject.
For instance, since
C 7/2
we have w^LL^ ,
showing how the potential energy of a fully charged jar
depends upon the extent of coated surface, the virtual
thickness of the dielectric, and the intensity of the machine
employed.
(r.) Again, suppose a charge to be divided between two
equal jars, by connecting the interior coatings, and also the
exterior coatings, of a charged and an empty jar.
At first we have, as before,
-
After division, we have in each jar, so that the potential
falls to half its value.
The work which is stored up is changed from
to 2 x7T
and is also reduced to half. But, as the conservation of
energy requires, the apparently lost half of the original poten-
tial energy is expended either mainly in a spark between the
inner coatings of the full and empty jars ; or, if a conductor
of great resistance has been introduced between them,
directly in the production of heat. This consideration
shows what a large amount of the energy contained in a
charged jar is usually wasted in the form of a spark when
conductors of small resistance are employed to discharge it.
NoteC. 155
NOTE C ( 119).
The potential at any point x, y, z, due to a magnetic
pole of m units placed at the origin, is
m
r'
where r=*J x z -\-y 2 +z*.
If the pole be moved through a small space in a line
whose direction-cosines are A, /z, v, the expression for the
potential at x, y, z becomes
m mil ^ d . d . d
Hence another pole of m units, situated at the point
whose co-ordinates are A, /*, v, produces at x,
y, z the potential
m mil ^d
A-=
r 2\ dx
From these expressions we see that a small magnet of
length /, which has m units of magnetism at each pole
whose middle point is at the origin, and whose direction-
cosines are A, p, v, produces at x, y, z the potential
7 ( . d d d\i
ml\ A- }-[*- \-v-j- P-
\ dx r dy dzjr
Hence if we now call x, y, 2 the co-ordinates of the middle
point of the first small magnet, and if at x' 9 y', z another
small magnet be placed, having a length /, poles of m' units,
and direction-cosines A', //, v', the mutual potential energy
of the two magnets is
,,J\ d , d , d\( w d , , d , ^\i
mmll( A-J-+U +v 1 A'-7-,+Ai -TT+i/- ,
\ dx^ r dy^ dz)\ dx^ r dy dzjr
where r is now to be treated as
156 Note D.
From this expression for elementary magnets we may, by
the usual processes of sextuple integration obtain at once
the expression for the mutual potential energy of any two
magnetised masses of steel in the form given by Thomson
(Phil. Trans. 1852).
It is easy to derive from it, by the principles of energy,
the position of equilibrium of either of the magnets (sup-
posed free to turn about its middle point) when the other
is fixed in a given position.
We may also easily form the equations of motion of one
magnet about its middle point when the other is made to
move in a given manner ; or when both are started with
any initial motions about their middle points and then left
to influence each other. Such questions form an interesting
and useful exercise for the student.
NOTE D ( 120-122).
The following is a version of an extract from Helmholtz
Ueber die Erhaltung der Kraft (1847).
When a magnet moves under the influence of a current,
the kinetic energy which it thereby acquires must be derived
from the consumption of the energy in the circuit. This
consists, during the time dt, of lEdt units of heat, or JIEdt
units of work, / being the intensity of the current, and E
the electromotive force in the circuit. Of this the portion
JPRdt) where R is the resistance of the circuit, is developed
as heat in the conductor (by Joule's law 139). That
acquired by the magnet is
Note D. 157
if V represent the potential energy of the magnet with
reference to unit current in the circuit. We have, there-
fore,
JIEdt=JPRdt+ J~dt
whence we obtain
i dV
We recognise in the quantity -j- a new electromotive force,
that of the induced current. It always works in a direction
opposite to that which would set the magnet in motion in
the direction in which it is moving, or would increase its
velocity. Since this electromotive force is independent of
the intensity of the current, it must remain the same if there
were originally no current in the conductor.
If its intensity vary, the complete current induced during
a given time is
where V, denotes the initial, and V tl the final, value of V.
If the magnet be brought up from a great distance
F
and is independent of the path and of the velocity of the
magnet.
INDEX.
\Thefigures refer to the Sections^
Absolute scale of temperature, 26 [168].
,, zero of temperature, 36.
Absorption bands in spectrum, 69.
Adams, tidal friction, 150.
Air, heat developed by compression of, 39,
58, 62.
specific heat of, 39, 44 [185].
Air-engine, efficiency of, 54. Superiority
to steam-engine, 139.
Akin on calorescence, 72.
Ampere, Solenoids, 124.
Andrews, heat of combination, 74.
Arago, effect of conductor in presence of
moving magnet, 121.
Available energy, 144.
Becquerel, phosphorescence, 72.
Bernoulli, D., pressure of gases, 59.
Black, latent heat, 5.
Boltzmann, kinetic equilibrium, 62. Di-
electric capacity, 132.
Brewster, atmospheric lines in spectrum,
68.
Calcescence or calorescence, 72.
Caloric theory of heat, 5, 6.
Capacity, thermal, 5-9, 13. Electric, in.
Specific inductive, 132.
Carnot, axiom, 19. Cycle of operations,
23. Efficiency of engine, 54. Function,
26. This is inversely as absolute tem-
perature, 64, 170. Reversible engine
perfect, 26.
Cell, energy of galvanic, 116-118.
Charge of electricity, in.
Chemical energy of battery, 139.
Clapeyron, 29.
Clausius, mechanical action of heat, 46-49.
Proof of Second Law, 53, 199. Pressure
of gas, 59. Conduction of heat in a
gas, 60. Disgregation and Internal
Work, [199-201],
Colding, 33.
Combination, heat of, 74.
Conduction of heat by gases, 60.
Conductivity, thermal, changes with tem-
perature, 1 8.
Conservation of energy, 97. Illustrations,
99, et seq.
Contact electricity, energy of, 107.
Cosmical energy, 149, 150.
Crookes' radiometer, 61.
Gumming, thermo-electric inversions, 134.
Currents and magnets, mutual action, 120,
123, 127.
Dalton's Law, 62.
Davy, 7-12, 78.
Decomposition, electrolytic, of water, 117.
Delaunay, tidal friction, 150.
Density of interplanetary medium, 128.
Desains, polarisation of radiated light, 68.
Dewar on Crookes' radiometer, 61.
Diamagnetic body, polarity of, 126.
Dielectric, inductive capacity of, 132.
Disgregation, 199.
Dissipation of energy, definition, 66.
Example, 97. Cosmical, 149.
Duty of an engine, [175].
Dynamical theory, definition, 84.
Efficiency of heat-engine, 54. Rankine on,
44, 139. Of electromagnetic engine, 139.
Electricity, nature of, 96.
voltaic, 116.
Electric convection, 134.
Electric potential, no.
Electrification by water dropper, 115.
]6o
Index:.
Electrophorus, energy of charge, 109.
Electrolysis, 117. Heat developed by,
118.
Electromotive force, 131.
Energetics, definition, 84.
Energy, available, 144.
,, conservation of, 97.
dissipation of, 97.
kinetic, 88.
,, laws of, 97.
forms of potential, 86.
,, sources of, 145, 146.
transformation of, 97.
,, of an electric current, 128.
,, of a galvanic battery, 139.
of a charge of electricity, i n.
of contact electricity, 107-8.
of cosmos, 147.
,, of electrophorus, 109.
of a Leyden jar, 113.
of a pendulum, 89, 92.
of permanent magnets, 119.
of a projectile, 88.
of sound waves, 102.
of sun and planet, 99.
of unequally heated body, 179.
of water rotating in vessel, 101.
physiological applications of, 142.
Entropy, 48, 178.
Exchanges, theory of, 68.
Faraday on electrolysis, 117. Induction
of electric currents, 121. Magneto-
electric machine, 122, 123. Magnet on
polarised light, 125. Conductor moved
across lines of magnetic force, 127.
Fluorescence, 72.
Forbes, thermal conductivity, 18. Identity
of light and radiant heat, 67.
Fourier (Theorie de la Chaleur), 18, 78.
Freezing mixtures, 75.
Fresnel, nature of light, 67.
Galvanic cell, theory of, 116. Energy of,
118.
Gas, specific heats of perfect, 185.
Gases, kinetic theory of pressure of, 59.
Repulsion theory, 61. Viscosity of, 60.
Gay Lussac, experiment with compressed
air, 39. Law of volumes, 60, 62.
Heat a form of energy, 3. Davy, 9.
Rumford, 13. Due to absorption of
light, 67.
Heat developed by compression of air, 39,
58. Of gases, 63.
Heat developed by electric current, 112,
i39-
Heat developed by electric discharge, 112.
,, of combination, 74, 137.
, , source of the sun's, 77, 147.
Helmholtz, conservation of energy, 96.
Action of magnets on conductors, 127.
Physiological applications of the laws of
energy, 142. Source of energy, 146-
148. Tidal friction, 150. Moon's period
of rotation, 150.
Herapath, pressure exerted by a gas, 59.
Herschel, fluorescence, 72.
Hess, heat of combination, 74.
Hirn, direct proof of disappearance of
heat when work is done, 31.
Hydrogen, velocity of particles of, 59.
Ice, effect of pressure on melting-point, 55.
India-rubber, stretched, contracts on being
heated, 57.
Internal radiation, 70.
Immateriality of heat, proof of Davy, 9.
Rumford, 13. Joule, 94.
Joule, mechanical equivalent, friction of
water, 32, 37. Expansion of air, 39. And
conservation of energy, 93-95. Absolute
zero of temperature, 36. Efficiency of
electro-magnetic engine, 139. Heat de-
veloped by compression of gases, 63.
Heat of electrical current, 112 ; of elec-
trical discharge, 112. Heat of electro-
lysis, 1 1 8. Kinetic theory of gases, 59.
Magnets and currents, action on one
another, 122, 127. Specific heat of air,
39-44. Velocity of hydrogen particles,
59. Vital Force, 142.
Kant, tidal friction, 150.
Kilogrammetre, 87.
Kinematics, 84.
Kinetic energy, 88.
Kinetics, 84.
Kinetic theory of gases, 59.
Kirchoff, theory of exchanges, 68, 69, 72.
Kronig, pressure of a gas, 59.
Latent heat, old theory, 5. New theory, 76.
Laws of Thermodynamics : I. Law, 37.
Clausius' and Thomson's proofs of Second
Law, 53. II. Law, 54.
Index.
161
Laws of Energy, 97.
Le Roux, specific heat of electricity, 134.
Le Sage, pressure of a gas, 59.
Leslie, radiant heat, heating by absorption
of light, 67.
Leyden jar, energy of, 113.
Light, nature of, 67.
,, electro-magnetic theory of, 132.
Liquefaction of ice by pressure, 55.
Liquefaction of steam, etc., worked ex-
pansively, 44, 48, 51.
Loss of vis viva in impact, zoo.
Lowering of temperature by sudden com-
pression, 57.
Luminiferous medium, density of, 128.
Clerk-Maxwell on, 130, 131.
Magnet and currents, mutual action of,
120, 123, 127,
Magnet, effect of, on polarised light, 125.
Magnetism, nature of, 125.
Magneto-electricity, 120.
Magnus, 135.
Maxwell, Clerk-, Demons, 53. Gases,
kinetic theory, 60. Repulsion theory,
61,62. Lines of force, 129. Luminiferous
medium, 130. Electromagnetic theory
of light, 132.
Mayer, speculations on mechanical equi-
valent, 30, 32. Cause of his erroneous
result, 39. Original source of energy,
142. Tidal friction, 150.
Mechanical equivalent of heat, Rumford,
15. Mohr, Seguin, Mayer, 30. Col ding
32. Joule, 35-37, 93-95-
Melloni on radiant heat, 67.
Melting-point, effect of pressure on, 55.
Meyer on viscosity of air, 60.
Mohr, mechanical equivalent, 30, 32.
Newton, 15, 78, 91.
Peltier, effect, 134.
Pendulum, energy of, 89, 92.
Perfect engine a reversible one, 26.
Phosphorescence, 72.
Photography, 143.
Physiological application of laws of energy,
142, 143.
Polarised light, action of magnet on, 125.
Potential electric, no.
Potential energy, 86.
Potential of a conductor, no.
Pressure and melting-points, sj.
Prevost, pressure of a gas, 59 ; theory of
exchanges, 63.
Radiant heat identical with light, 67.
Radiometer, Crookes', 61.
Rankine, molecular vortices, 42. Ther-
modynamic function, 44. Specific heat
of air, 44. Efficiency of heat-engine,
44. Transformation of energy, 140.
Regnault, specific heat of air, 39.
Reversible cycle, conditions for, 158.
Reversible engine, 26.
Ruhmkorff coil, 138.
Rumford, 13.
Scholium, to III. Law of Motion, 91.
Seebeck, thermo-electricity, 28, 133.
Seguin on mechanical equivalent, 30,
31, 32.
Sound waves, energy of, 102.
Sources of energy, 145.
Specific heat of air, 39, 44, 185.
Spectrum analysis, 68.
Statics, 84.
Stephenson on sources of energy, 146.
Stewart Balfour, theory of exchanges, 68.
Equality of radiating and absorbing
powers, internal radiation, 70.
Stokes, viscosity of gases, 60. Spectrum
analysis, 68. Fluorescence, 72.
Stoney, explanation of radiometer, 61.
Sun's heat, origin of, 77, 147.
Tail, motion of the radiometer, 61. Multi-
ple neutral points, and electric convec-
tion, 134.
Talbot, photography, 143.
Temperature, definition of, 54 ; as a.
quantity, 85, 163-171.
Temperature, absolute scale of, 26.
,, absolute zero of, 36.
Thermodynamic function, 44, 204.
Thermodynamics, Laws of, 52.
Thermo-electricty, 26, 134.
Thermo-magnetism, 136.
Thomson, James, correction of Carnot's
cycle of operations, 24. Effect of pres-
sure on melting-point of ice, 55. Tidal
friction, 150.
Thomson, Sir William, absolute tempera-
ture, 26, 78, 85, 169, 171. Chemical
affinity, measure of, 85. Contact elec-
tricity, 115. Current, energy of elec-
tric, 128. Diamagnetic body, polarity
l62
Index.
of, 126. Heat developed in compres-
sion of gases, 63. Heat developed by
electrolysis, 118. Laws of Thermo-
dynamics, 66. Law II., proof of, 53.
Lowering of melting-point of ice by
pressure, 55. Lowering of temperature
by sudden compression, 57. Magnets
on conductors, action of, 127. Magnetic
dip, 123. Magnetism depends on
motion, 125. Peltier effect, Thomson
effect,i34. Sources and origin of energy,
147. Thermodynamics generally, 56,
57-
Tidal friction, 150.
Transformation of work into heat, 93.
Transformation of energy, 97. Illustra-
tions, galvanic battery, 137, etc.
Unit of Work, 87.
Verdet, effect of magnet on polarised
light, 125.
Viscosity of gases, 60, 61.
Vis -viva, principle of, 91. Loss of, in im-
pact, 100.
Voltaic electricity, 116.
Watt's diagram, 29, 153.
Weber, theory of electricity, 96.
Young, nature of light, 67.
ERRATA.
P. 24, footnote, for centrigrade read centigrade.
P. 92, lines 3 and 7, for Holtzmann read Boltzmann.
SUnibcrsitg
THOMAS AND ARCHIBALD CONSTABLE, PKINTEKS TO HEK MAJESTY.
9 CASTLE STREET,
EDINBURGH, January 1878.
LIST OF BOOKS
PUBLISHED BY DAVID DOUGLAS.
BAILDON Morning Clouds: being divers Poems
by H. B. BAILDON, B.A. Cantab., Author of ''Rosamond," etc.
Ex. fcap. 8vo, 5s.
Bible Readings.
Extra fcap. 8vo, 2s.
BLACKIE- Lyrical Poems.
By JOHN STUART BLACKIE, Professor of Greek in the University
of Edinburgh. Crown 8vo, cloth, 7s. 6d.
BLACKIE The Language and Literature of the
Scottish Highlands. In 1 vol. crown 8vo, 6s.
"The way to a mother's heart is through her children; the way to &
people's heart is through its language." Jean Paul RicJiter.
"Ein Buch, das ich auch deutschen Lesern, und zwar in einem betrSchtlich
weitem Umfange, nicht angelegentlich genug empfehlen kann. . . . Mit F. A.
Wolfs lebendiger Auffassung vom Ursprunge der homerischen Epen vor der
Seele weisz er die echten lyrischen Schopfungen der Vorzeit mit ihrem
Ansatz zu epischer Fassung zu erkennen und greift aus den Resten des
wahren Ossian einige kostliche Perlen heraus, wie sie zu Anfang des 16.
Jahrhunderts der Dechant Macgregor von Lismore aufzeichnete. . . . Das
letzte Capitel enthalt eine Reihe anziehender Beispiele gaelischer Dichtung
aus den letzten hundert Jahren. . . . Auch liber die ktimmerlich gedeihende
Prosa werden lehrreiche Angaben hinzugefugt." Dr. ReinTwld Pauli.
BLACKIB Pour Phases of Morals : Socrates, Aris-
totle, Christianity, and Utilitarianism. Lectures delivered
before the Royal Institution, London Fcap. 8vo, second edi-
tion, 5s.
2 BOOKS PUBLISHED BY DAVID DOUGLAS.
BLACKIB Songs of Religion and Life.
Fcap. 8vo, 6s.
' ' The poems in this volume may be regarded as a Second Edition of the
second part of my 'Lays and Legends of Ancient Greece,' which has long
been out of print, along with other Poems not hitherto published, and a
few from a volume of ' Lyrical Poems ' previously published, all having one
common object, viz. , ' the cultivation of religious reverence without sectarian
dogmatism, and of poetical sentiment tending not so much to arouse the
imagination or to take the fancy, as to purify the passions and to regulate
the conduct of life.' " Preface.
BLACKIB On Self-Culture : Intellectual, Physical,
and Moral. A Vade-Mecum for Young Men and Students.
Tenth edition. Fcap. 8vo, 2s. 6d.
" Every parent should put it into the hands of his son." Scotsman.
" Students in all countries would do well to take as their vade-mecum a little
book on self-culture by the eminent Professor of Greek in the University of
Edinburgh." Medical Press and Circular.
" An invaluable manual to be put into the hands of students and young
men." Era.
"Written in that lucid and nervous prose of which he is a master."
Spectator.
BLACKIB On Greek Pronunciation.
Demy 8vo, 3s. 6d.
BLACKIB On Beauty.
Crown 8vo, cloth, 8s. 6d.
BLACKIB Musa Burschicosa.
A Book of Songs for Students and University Men. Fcap. 8vo,
2s. 6d.
BLACKIB War Songs of the Germans.
Feap. 8vo, price 2s. 6d. cloth ; 2s. paper.
BLACKIE Political Tracts.
No. 1. GOVERNMENT. No. 2. EDUCATION. Price Is. each.
BLACKIB Homer and the Iliad.
In three Parts. 4 vols. demy 8vo, price 42s.
BOOKS PUBLISHED BY DAVID DOUGLAS. 3
BOWBN Daily Meditations by Rev. G. Bowen of
Bombay. With Introductory Notice by Rev. W. HANNA, D.D.,
Author of "The Last Day of our Lord's Passion." Third
edition, small 4to, cloth, price 5s. ; or limp roan, red edges,
price 7s. 6d.
" Among such books we shall scarcely find another which exhibits the same
freshness and vividness of idea, the same fervour of faith, the same intensity
of devotion. ... I count it a privilege to introduce in this country a book so
fitted to attract and to benefit." Extract from Preface.
" These meditations are the production of a missionary whose mental history
is very remarkable. . . . His conversion to a religious life is undoubtedly oue
of the most remarkable on record. They are all distinguished by a tone of true
piety, and are wholly free from a sectarian or controversial bias." Morning
Post.
BROWN John Leech and other Papers.
By JOHN BROWN, M.D., F.R.S.E. Crown 8vo. [In the Press.
BROWN Locke and Sydenham, and other Papers.
Extra fcap. 8vo, 7s. 6d.
BROWN Horse Subsecivse.
Ninth edition. Extra fcap. 8vo, 7s. 6d.
BROWN Letter to the Rev. John Cairns, D.D.
Second edition. Crown 8vo, sewed, 2s.
BROWN Arthur H. Hallam ;
Extracted from "Horse Subsecivse." Fcap. sewed, 2s. ; cloth,
2s. 6d.
BROWN Rab and his Friends ;
Extracted from " Horae Subseciva)." Forty-ninth thousand.
Fcap. sewed, 6d.
BROWN Rab and his Friends.
Cheap Illustrated edition. Square 12mo, ornamental wrapper, Is.
4 BOOKS PUBLISHED BY DAVID DOUGLAS.
BROWN Rab and his Friends.
With Illustrations by Sir George Harvey, R.S.A., Sir J. Noel
Paton, R.S.A., and J. B. New edition. Demy quarto, cloth, 6s.
BROWN Marjorie Fleming: A Sketch.
Fifteenth thousand. Fcap. sewed, 6d.
BROWN Our Dogs ;
Extracted from "Horse Subsecivae. " Nineteenth thousand.
Fcap. sewed, 6d.
BROWN-" With Brains, Sir;"
Extracted from "Horse Subsecivae." Fcap. sewed, 6d.
BROWN Minchmoor.
Fcap. sewed, 6d.
BROWN Jeems the Doorkeeper : A Lay Sermon.
6d.
BROWN The Enterkin.
6d.
CAMPBELL My Indian Journal,
Containing descriptions of the principal Field Sports of India,
with Notes on the Natural History and Habits of the Wild
Animals of the Country. By Colonel WALTER CAMPBELL,
Author of "The Old Forest Ranger." 8vo, with Illustrations,
16s.
CUMMING-Wild Men and Wild Beasts. Adven-
tures in Camp and Jungle. By Lieut. -Colonel GORDON GUMMING.
With Illustrations by Lieut. -Col. BAIGRIE and others. Second
edition. Demy 4to, price 24s.
Also, a cheaper edition, with Lithographic Illustrations.
8vo, 12s.
BOOKS PUBLISHED BY DAVID DOUGLAS. 5
CHALMERS Life and Works of Rev. Thomas
Chalmers, D.D., LL.D.
MEMOIRS OF THE REV. THOMAS CHALMERS. By Rev. W.
HANNA, D.D., LL.D. Cheap edition. 2 vols. crown 8vo,
cloth, 12s.
DAILY SCRIPTURE READINGS. Cheap edition. 2 vols. crown
8vo, 10s.
ASTRONOMICAL DISCOURSES, Is.
COMMERCIAL DISCOURSES, Is.
SELECT WORKS, in 12 vols., crown 8vo, cloth, per vol., 6s.
LECTURES ON THE ROMANS. 2 vols.
SERMONS. 2 vols.
NATURAL THEOLOGY, LECTURES ON BUTLER'S ANALOGY, ETC. 1 vol.
CHRISTIAN EVIDENCES, LECTURES ON PALEY'S EVIDENCES, ETC. 1 vol.
INSTITUTES OF THEOLOGY. 2 vols.
POLITICAL ECONOMY, WITH COGNATE ESSAYS. 1 vol.
POLITY OF A NATION. 1 vol.
CHURCH AND COLLEGE ESTABLISHMENTS. 1 vol.
MORAL PHILOSOPHY, INTRODUCTORY ESSAYS, INDEX, ETC. 1 vol.
CHIBNB Lectures on Surgical Anatomy.
By JOHN CHIENE, Assistant-Surgeon, Royal Infirmary, Edin-
burgh. In 1 vol. 8vo. With numerous Illustrations drawn on
Stone by BERJEAU. [In the Press.
CONSTABLE Archibald Constable and his Literary
Correspondents : a Memorial. By his Son, THOMAS CONSTABLE.
3 vols. 8vo, 36s., with Portrait.
" The cream of a generation of interesting men and women now gone from
among us these are the subjects of this important memoir. " Saturday Review.
"These three volumes are decidedly additions to our knowledge of that
great and brilliant epoch in the history of letters to which they refer."
Standard.
"He (Mr. Constable) was a genius in the publishing world The
creator of the Scottish publishing trade. " Times.
' ' These three volumes are of a singular and lasting interest. "Nonconformist.
.
6 BOOKS PUBLISHED BY DAVID DOUGLAS.
" The third volume (Sir Walter Scott) of this elaborate and interesting history
is almost an independent work." Athenceum.
"We heartily commend this book to the notice of all readers." Guardian.
DASENT Tales from the Norse.
By Sir GEORGE WEBBE DASENT, D.C.L. Third edition, with
Introduction and Appendix. In 1 vol. demy 8vo. [In the Press.
DUN Veterinary Medicines ; their Actions and
Uses. By FINLAY DUN. Fifth edition, revised and enlarged.
8vo. [Immediately.
D UNBAR Social Life in Former Days ;
Chiefly in the Province of Moray. Illustrated by Letters and
Family Papers. By E. DUNBAR DUNBAR, late Captain 21st
Fusiliers. 2 vols. demy 8vo, 19s. 6d.
BRSKINE Letters of Thomas Erskine of Lin-
lathen, from 1800 tiU 1840. Edited by WILLIAM HANNA, D.D. ,
Author of the "Memoirs of Dr. Chalmers," etc. In 1 vol.
crown 8vo, 7s. 6d.
" Here is one who speaks out of the fulness of a large living human heart ;
whose words will awaken an echo in the hearts of many burdened with the
cares of time, perplexed with the movements of the spirit of our time, who
will speak to their deepest needs, and lead them to a haven of rest. " Daily
Review.
" It does one good to come in contact with so saintly a man, and Dr. Hanna
has certainly conferred a benefit on the Church at large by editing this
volume." Edinburgh Courant.
"'How high must that peak have been which caught the light so early/
were the words with which a writer in the Contemporary Review, in sketching
the life of Thomas Erskine, shortly after his death, characterised his position,
his spirit, and his influence. " Nonconformist.
ERSKINE Letters of Thomas Erskine of Lin-
lathen, from 1840-1870. Second Series, completing the Work.
Edited by the Rev. W. HANNA, D.D. In 1 vol. crown 8vo,
7s. 6d.
BOOKS PUBLISHED BY DAVID DOUGLAS. 7
BRSKINE The Unconditional Preeness of the
Gospel. New edition, revised. Crown 8vo, 3s. 6d.
ERSKINE An Essay on Faith.
Fourth edition. 12mo, 3s.
ERSKINE The Spiritual Order, and other Papers
selected from the MSS. of the late THOMAS ERSKINE of Lin-
lathen. Second edition. Crown 8vo, cloth, 5s.
"It will for a few have a value which others will not the least understand.
But all must recognise in it the utterance of a spirit profoundly penetrated
with the sense of brotherhood, and with the claims of common humanity."
Spectator.
" Very deserving of study. "Times.
Vide BIBLE READINGS and FRAGMENTS OF TRUTH.
FINLAY Essay to which was awarded the First
Mackenzie Prize for the best Essay on the best means of Im-
proving the Relations between Capital and Labour. By JAMES
FAIRBAIRN FINLAY, M. A. DemySvo, Is.
FLETCHER Autobiography of Mrs. Fletcher (of
Edinburgh), with Letters and other Family Memorials. Edited
by her Daughter. Second edition. Crown 8vo, 7s. 6d.
" This is a delightful book. It contains an illustrative record of a singularly
noble, true, pure, prolonged, and happy life. The story is recounted with a
candour, vivacity, and grace which are very charming. " Daily Review.
FLEURY L'Histoire d'Angleterre.
Par M. LAME FLEURY. 18mo, cloth, 2s. 6d.
FLEURY L'Histoire de France.
Par M. LAME FLEURY. New edition. ISmo, cloth, 2s. 6d.
FORBES The Deepening of the Spiritual Life.
By A. P. FORBES, D.C.L., Bishop of Brechin. Fifth edition.
ISmo, cloth, price Is. 6d. ; or paper covers, Is. ; calf, red edges,
3s. 6d.
8 BOOKS PUBLISHED BY DAVID DOUGLAS.
FORBES Kalendars of Scottish Saints, with Per-
sonal Notices of those of Alba, etc. By ALEXANDER PENROSE
FORBES, D.C.L., Bishop of Brechin. 1 vol. 4to, price 3, 3s.
A few copies for sale on large paper, 5, 15s. 6d.
"A truly valuable contribution to the archaeology of Scotland." Guardian.
"We must not forget to thank the author for the great amount of informa-
tion he has put together, and for the labour he has bestowed on a work which
can never be remunerative." Saturday Review.
"His laborious and very interesting work on the early Saints of Alba,
Laudonia, and Strathclyde. " Quarterly Review.
Fragments of Truth. Being the Exposition of several
passages of Scripture. Third edition. Extra fcap. 8vo, 5s.
GAIRDNER On Medicine and Medical Education.
By W. T. GAIRDNER, Professor of the Practice of Medicine in
the University of Glasgow. Three Lectures, with Notes and an
Appendix. 8vo, 3s. 6d.
GAIRDNER Clinical and Pathological Notes on
Pericarditis. By W. T. GAIRDNER, Professor of the Practice of
Medicine in the University of Glasgow. 8vo, sewed, Is.
GIBSON, C. P.-Cheerfulness.
By CHARLES P. GIBSON. In 1 vol. fcap., 3s. 6d.
"It depicts, in very graphic and glowing terms, much of the scenery of
this northern district of England, and is therefore sure to be prized very
highly by those Northumbrians into whose hands it may happen to fall.
Apart, however, from its local interest, it has peculiar merits ot its own,
and no one can read it without feeling that his own spirit has been en-
livened and elevated by so doing. Its pictures remind us very forcibly of
those of Thomson, Cowper, and Burns." Newcastle Daily Journal.
GORDON The Roof of the World;
Being the narrative of a journey over the high plateau of
Tibet to the Russian Frontier and the Oxus sources on Pamir.
By Lieut. -Col. T. E. GORDON, C.S.I. With numerous Illus-
trations. Royal 8vo, 31s. 6d.
BOOKS PUBLISHED BY DAVID DOUGLAS.
GORDON The Home Life of Sir David Brewster.
By his Daughter, Mrs. GORDON. Second edition. Crown 8vo, 6s.
"With his own countrymen it is sure of a welcome, and to the savants of
Europe, and of the New World, it will have a real and special interest of its
own." Pall Mall Gazette.
GORDON Workers.
Fourth thousand. Fcap. 8vo, limp cloth, Is.
GORDON Work ; or, Plenty to do and How to do it.
Thirty -fifth thousand. Fcap. 8vo, cloth, 2s. 6d.
GORDON Little Millie and her Four Places.
Cheap Edition. Fifty-fifth thousand. Limp cloth, Is.
GORDON Sunbeams in the Cottage; or, What
Women may do. A Narrative chiefly addressed to the Work-
ing Classes. Cheap edition. Forty-fourth thousand. Limp
cloth, Is.
GORDON Prevention ; or, An Appeal to Economy
and Common Sense. 8vo, 6d.
GORDON The Word and the World.
Twelfth edition. Price 2d.
GORDON Leaves of Healing for the Sick and
Sorrowful. Fcap. 4to, cloth, 3s. 6d. Cheap edition, limp
cloth, 2s.
GORDON The Motherless Boy;
With an Illustration by Sir J. NOEL PATON, R.S.A. Cheap
edition, limp cloth, Is.
"Alike in manner and matter calculated to attract youthful attention, and
to attract it by the best of all means sympathy. " Scotsman.
GRAHAM" Mystifications."
By Miss STIRLING GRAHAM. Fourth edition. Edited by
JOHN BROWN, M.D. With Portrait of " Lady Pitlyal." Fcap.
8vo, 3s. 6d.
1 BOOKS PUBLISHED BY DAVID DOUGLAS.
HANNA The Life of our Lord.
By the Rev. WILLIAM HANNA, D.D., LL.D. 6 vols., hand-
somely bound in cloth extra, gilt edges, 80s.
Separate vols., cloth extra, gilt edges, 5s. each.
1. THE EARLIER YEARS or OUR LORD. Eighth thousand.
2. THE MINISTRY IN GALILEE. Third edition.
3. THE CLOSE OF THE MINISTRY. Sixth thousand.
4. THE PASSION WEEK. Fifth thousand.
5. THE LAST DAY OF OUR LORD'S PASSION. Forty-seventh
thousand.
6. THE FORTY DAYS AFTER THE RESURRECTION. Ninth
thousand.
HANNA The Resurrection of the Dead.
By WILLIAM HANNA, D.D., LL.D. Second edition. One vol.
fcap. 8vo, 5s.
JOHNNY GIBB of Gushetneuk, in the Parish of
Pyketillim : with Glimpses of the Parish Politics about A.D. 1843.
Fourth edition, with a Glossary. Ex. fcap. 8vo, 2s.
" It is a grand addition to our pure Scottish dialect ; ... it is not merely
a capital specimen of genuine Scottish northern dialect ; but it is a capital
specimen of pawky characteristic Scottish humour. It is full of good hard
Scottish dry fun." Dean Ramsay.
Notes and Sketches Illustrative of Northern Rural
Life in the Eighteenth Century, by the Author of "Johnny
Gibb of Gushetneuk." In 1 vol. fcap. 8vo, 2s.
Life among- my Ain Folk.,
By the Author of "Johnny Gibb of Gushetneuk." 12mo,
cloth, 2s. 6d.
KENNEDY Pilate's Question, "Whence art thou?"
An Essay on the Personal Claims asserted by Jesus Christ, and
how to account for them. By JOHN KENNEDY, M.A., D.D.,
London. Crown 8vo, 3s. 6d.
KER Sermons by the Rev. John Ker, D.D., Glas-
gow. Eleventh edition. Crown 8vo, 6s.
" A very remarkable volume of sermons." Contemporary Review.
BOOKS PUBLISHED BY DAVID DOUGLAS. 1 1
LAING Lindores Abbey, and the Burgh of New-
burgh : their History and Annals. By ALEXANDER LAING,
F.S.A. Scot. 1 vol. small 4to. With Index, and thirteen Full-
page and ten Woodcut Illustrations, 21s.
" This is a charming volume in every respect." Notes and Queries.
".The prominent characteristics of the work are its exhaustiveness and the
thoroughly philosophic spirit in which it is written. "Scotsman.
LANCASTER Essays and Reviews.
By the late HENRY H. LANCASTER, Advocate ; with a Prefatory
Notice by the Rev. B. JOWETT, Master of Balliol College,
Oxford. Demy 8vo, with portrait, 14s.
LAURIE On the Philosophy of Ethics. An Analy-
tical Essay. By S. S. LAURIE, A.M., F.R.S.E., Professor of
the Theory, History, and Practice of Education in the Univer-
sity of Edinburgh. Demy 8vo, 6s.
" Mr. Laurie's volume now before us is in substance, though not in form, a
reply to Mr. Mill's Utilitarianism. Mr. Laurie has the metaphysical head and
the metaphysical training of his countrymen, and has brought both to bear
with great force on the problem proposed." Saturday Review.
LAURIE Notes on British Theories of Morals.
Demy 8vo, 6s.
"His criticisms are candid and highly instructive, e.g. those of the views of
Bentham, Mill, and Bain. He manifests great aptitude in detecting radical
defects, in exposing logical inconsistencies, and in detecting the legitimate
tendencies of philosophical systems. " British Quarterly.
MACK AY Memoir of Sir James Dalrymple, First
Viscount Stair. A Study in the History of Scotland and Scotch
Law during the Seventeenth Century. By M. J. G. MACKAY,
Advocate. 8vo, 12s.
MACPHERSON Omnipotence belongs only to the
Beloved. By Mrs. BREWSTER MACPHERSON. 1 vol., extra
fcap., 3s. 6ti.
MACPHERSON-G-ifts for Men.
1 vol. ex. fcap. Svo, 6s.
12 BOOKS PUBLISHED BY DAVID DOUGLAS.
MACLAG-AN The Hill Forts, Stone Circles, and
other Structural Remains of Ancient Scotland. By C. MAC-
LAGAN, Lady Associate of the Society of Antiquaries of Scotland.
With Plans and lUustrations. 1 vol. fol., 31s. 6d.
"We need not enlarge on the few inconsequential speculations which rigid
archaeologists may find in the present volume. We desire rather to commend
it to their careful study, fully assured that not only they, but also the general
reader, will be edified by its perusal." Scotsman.
MAXWELL Soliman the Magnificent, and the
Turks in the Sixteenth Century. By Sir WILLIAM STIRLING
MAXWELL, Bart., K.T., and M.P. Illustrated by numerous
Facsimiles of rare contemporary Drawings and Portraits.
In 1 vol. folio. [In the Press.
MAXWELL Antwerp Delivered in MDLXXVII. :
a Passage from the History of the Netherlands, illustrated with
Facsimiles of a rare Series of Designs by Martin de Vos, and
of Prints by Hogenberg, the Wiericxes, etc. By Sir WILLIAM
STIRLING MAXWELL, Bart., K.T., and M.P. In 1 vol. folio.
[In the Press.
MICHIE History of Loch Kinnord.
By the Rev. J. G. MICHIE. Demy 8vo, 2s. 6d.
MILN Researches and Excavations at Carnac
(Morbihan), the Bossenno, and Mont St. Michel. By JAMES
MILN. In 1 vol. royal 8vo, with Maps, Plans, and numerous
Illustrations in Wood- Engraving and Chromo-Lithography.
MOREHEAD Memorials of the Life and Writings
of the Pvev. ROBERT MOREHEAD, D.D., formerly Rector of
Easington, Yorkshire, and previously Dean of Edinburgh.
Edited by his Son, CHARLES MOREHEAD, M.D. Or. Svo, 7s. 6d.
NAPIER "The Lanox of Auld:" an Epistolary
Review of "The Lennox, by William Fraser." To which is
added, A Postscriptive Memorie of the House of Merchiston.
By MARK NAPIER. With Woodcuts and Plates. 1 vol. 4to.
[In preparation.
PATRICK, R. W. COCHRAN-Records of the Coin-
age of Scotland, from the earliest period to the Union. Col-
BOOKS PUBLISHED BY DAVID DOUGLAS. 13
lected by R. W. COCHRAN PATRICK of Woodside. Only Two
Hundred and Fifty Copies printed. Now ready, in 2 vols. 4to,
with 16 Full-page Illustrations, Six Guineas.
" The future Historians of Scotland will be very fortunate if many parts of
their materials are so carefully worked up for them and set before them in so
complete and taking a form." Athenceum.
"When we say that these two volumes contain more than 770 records, of
which more than 550 have never been printed before, and that they are illus-
trated by a series of Plates, by the autotype process, of the coins themselves,
the reader may judge for himself of the learning, as well as the pains, bestowed
on them both by the Author and the Publisher." Times.
"The most handsome and complete Work of the kind which has ever
been published in this country." Numismatic Chronicle, Pt. IV., 1876.
Popular Genealogists ;
Or, The Art of Pedigree-making. Crown 8vo, 4s.
" We have here an agreeable little treatise of a hundred pages, from an anony-
mous but evidently competent hand, on the ludicrous and fraudulent sides of
genealogy. The subject has a serious and important historical character,
when regarded from the point of view of the authors of The Governing
Families of England. But it is rich in the materials of comedy also. . . . We
are glad to see such a step taken in the good work as the publication of
the essay which has suggested this article, and which we commend to those
who want a bit of instructive and amusing reading." Pall Mall Gazette.
RBNTON, W. Oils and Water Colours.
By WILLIAM RENTON. 1 vol. fcap. 5s.
"The book is obviously for the Artist and the Poet, and for every one who
shares with them a true love and zeal for nature's beauties. "Scotsman.
"To have observed such a delicate bit of colouring as this, and to have
written so good a sonnet in the ' strict style,' as that we have quoted, shows
that our author has no common powers either as an observer or a writer."
Liverpool Albion.
"To those minds that really hold this joy in beauty, Mr. Kenton's book
will undoubtedly give delight." Northern Ensign.
ROBERTSON Historical Essays in connection
with the Land and the Church, etc. By E. WILLIAM ROBERT-
SON, Author of "Scotland under her Early Kings." In 1 vol.
Svo, 10s. 6d.
14 BOOKS PUBLISHED BY DAVID DOUGLAS.
ROBERTSON Scotland under her Early Kings.
A History of the Kingdom to the close of the 13th century.
By E. WILLIAM ROBERTSON. In 2 vols. 8vo, cloth, 36s.
" Mr. Robertson's labours are of that valuable kind where an intelligent
and. thorough sifting of original authorities is brought to bear upon a portion
of history handed over hitherto, in a pre-eminent degree, to a specially
mendacious set of Mediaeval Chroniclers, and (not so long ago) to a specially
polemical and uncritical class of modern Historians. He belongs to the school
of Innes and Skene and Joseph Robertson, and has established a fair right to
be classed with the Reeves and Todds of Irish historical antiquarianism, and
the Sharpes, and Kembles, and Hardys in England. " Guardian.
SHAIRP Studies in Poetry and Philosophy.
By J. C. SHAIRP, LL.D., Principal of the United College of
St. Salvator and St. Leonard, St. Andrews. Third Edition.
1 vol. fcap. 8vo, 6s.
SHAIRP On Poetic Interpretation of Nature.
By PRINCIPAL SHAIRP, LL.D. Second Edition. In 1 vol. ex.
fcap. 8vo, 6s.
SHAIRP Culture and Religion.
By PRINCIPAL SHAIRP, LL.D. Fifth Edition. Fcap. 8vo, 3s. 6d.
SHAIRP Wordsworth's Tour in Scotland in 18O3,
in company with his Sister and S. T. Coleridge ; being the
Journal of Miss WORDSWORTH, now for the first time made
public. Edited by PRINCIPAL SHAIRP, LL.D. Second Edition,
1 vol. crown 8vo, 6s.
" If there were no other record of her than those brief extracts from her
Journal during the Highland Tour, which stand at the head of several of her
brother's poems, these alone would prove her possessed of a large portion of
his genius." North British Review.
SIMPSON The Near and the Par View, and other
Sermons. By Rev. A. L. SIMPSON, D.D., Derby. 1 vol. ex.
fcap. 8vo, 5s.
SKENE The Four Ancient Books of Wales,
Containing the Cymric Poems attributed to the Bards of the
Sixth Century. By WILLIAM ., . SKENE. With Maps and
Facsimiles. 2 vols. 8vo, 36s.
"Mr. Skene's book will, as a matter of course and necessity, find its place
on the tables of all Celtic antiquarians and scholars." Archccologia Cambrensis.
BOOKS PUBLISHED BY DAVID DOUGLAS. 15
SKENE The Coronation Stone.
By WILLIAM F. SKENE. Small 4to. With Illustrations in
Photography and Zincography. 6s.
SKENE-Celtic Scotland.
A History of Ancient Alban. By WILLIAM F. SKENE. Vol. I.
Book I. History and Ethnology. Illustrated with Maps. 15s.
" It is a book of solid and good work, and which ought to be thankfully
welcomed by all who are engaged in any minute study of the early history of
Britain." Pall Mall Gazette.
"This volume is the first instalment of a work which will bring the early
history of Scotland out of the clouds and mists of artificially constructed
systems of history, exaggerated tradition, and legendary fiction, and into a
real, if still somewhat dim, historic light. "Edinburgh Courant.
"Daist es denn in der That ein Fortschritt, wenn ein Gelehrter, der sich
die schwierigen, aber unerlaszlichen Sprachkenntnisse erworben und seit
Jahren mit Sichtung der vertrauenswerthen Ueberlieferung von den Trugge-
bilden, welche alles Keltische so leicht bedecken, befaszt hat, die bedeutende
Aufgabe in die Hand nimmt nach strenger Methode die wirklichen Thatsachen
jener Anfangsjahrhunderte hinzustellen Linguistik, Ethnographie,
Topographic und Kritik der historischen Quellen greifen fiir diese wichtige
Epoche des Uebergangs wirkungsvoll in einander, wie es meines Wissens
bisher in keiuem andereii Werke geschehn 1st." Gottingische geleJirCe
Anzeigen. Dr. B. PAULI.
SKENE Celtic Scotland.
A History of Ancient Alban. Vol. II. Book II. Church and
Culture. In 8vo. With Maps, 15s.
SKENE Celtic Scotland.
A History of Ancient Alban. Vol. III. Book III. Land and
People. [In preparation.
SMALL Scottish Woodwork of the Sixteenth and
Seventeenth Centuries* Measured, Drawn, and Lithographed
by J. W. SMALL, Architect. In one folio volume with 100
Plates. [In the Press.
SMITH Shelley : a Critical Biography.
By GEORGE BARNETT SMITH. Ex. fcap. 8vo, 6s.
1 6 BOOKS PUBLISHED BY DAVID DOUGLAS.
SMYTH Life and 'Work at the Great Pyramid.
With a Discussion of the Facts ascertained. By C. PIAZZI
SMYTH, F.R.SS.L. and E., Astronomer-Royal for Scotland.
3 vols. demy 8vo, 56s.
SMYTH An Equal-Surface Projection for Maps of
the World, and its Application to certain Anthropological
Questions. By C. PTAZZI SMYTH, F.R.SS.L. and E., Astronomer-
Royal for Scotland. 8vo, 3s.
SOUTHESK Britain's Art Paradise ; or, Notes on
some Pictures in the Royal Academy, 1871. By the EARL OF
SOUTHESK. Svo, sewed, Is.
SOUTHESK Saskatchewan and the Rocky
Mountains. Diary and Narrative of Travel, Sport, and Adven-
ture, during a Journey through part of the Hudson's Bay
Company's Territories, in 1859 and 1860. By the EARL OF
SOUTHESK, K.T., F.R.G.S. 1 vol. demy Svo, with Illustrations
on Wood by WHYMPER, 18s.
SOUTHESK Herminius.
A Romance. By I. E. S. Fcap. Svo, 6s.
SPENS The Sanitary System of Scotland : its
Defects and Proposed Remedies. By WALTER COOK SPENS,
one of the Sheriff-Substitutes of Lanarkshire. Demy Svo, 6s.
STRACHAN-What is Play?
A Physiological Inquiry. Its bearing upon Education and
Training. By JOHN STRACHAN, M.D., Jun. In 1 vol. fcap., Is.
TAIT Sketch of Thermodynamics.
By P. G-. TAIT, M.A., Professor of Natural Philosophy in the
University of Edinburgh. Second edition, Revised and Ex-
tended. Cr. Svo, 5s.
WILSON- Reminiscences of Old Edinburgh.
By DANIEL WILSON, LL.D., F.R.S.E., Professor of History and
English Literature in University College, Toronto, Author of
"Prehistoric Annals of Scotland," etc. etc. 2 vols. post Svo.
B R A
or THE
UNIVERSITY
RETURN TO the circulation desk of any
University of California Library
or to the
NORTHERN REGIONAL LIBRARY FACILITY
Bldg. 400, Richmond Field Station
University of California
Richmond, CA 94804-4698
ALL BOOKS MAY BE RECALLED AFTER 7 DAYS
2-month loans may be renewed by calling
(510)642-6753
1-year loans may be recharged by bringing books
to NRLF
Renewals and recharges may be made 4 days
prior to due date
DUE AS STAMPED BELOW
JUN 1 - 1993
a 2003
SENT ON ILL
JAN 2 1995
U. C. BERKELEY
NOV a o 2002
46442
GENERAL LIBRARY - U.C. BERKELEY
BDQDflM32 c ll,
m
IP
1