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Full text of "Reflections on the motive power of heat [microform] : and on machines fitted to develop that power"

SADI CARN OT 

AT THE AGE OP 17. 

(From a Portrait by Bailly. 1813.) 



REFLECTIONS 



ON THE 



MOTIVE POWER OF HEAT; 

FROM THE ORIGINAL FRENCH OF 

N.-L.-S. CARNOT, 

Graduate of the Polytechnic School. 



ACCOMPANIED BY 



AN ACCOUNT OF CARNOT'S THEORY. 

BY SIR WILLIAM THOMSON (LORD KELVIN). 



EDITED BY 



R. H. THURSTON, M.A., LL.D., DR.ENG'G ; 

Director of Sibley College, Cornell University ; 

" Officier de VInstruction Publique de France," 11 

etc., etc., etc. 




SECOND. REVISED, EDITIO1 
PIKST THOUSAND. 



JOHN WILEY & SONS, 



LONDON : CHAPMAN & HALL, LIMITED. 




Copyright, 1890, 
ROBERT H. THURSTON. 



ROBERT DRUMMOND, ELECTROTVPER AND PRINTER, NEW YORK. 



DEDICATED 

TO 

SaDf Garnet, 

PRESIDENT OF THE FRENCH REPUBLIC, 

THAT DISTINGUISHED MEMBER OF THE PROFESSION OF ENGINEERING 

WHOSE WHOLE LlFE HAS BEEN AN HONOR TO HIS 

PROFESSION AND TO HIS COUNTRY ; 

AND WHO, ELEVATED TO THE HIGHEST OFFICE WITHIN THE GIFT OF THE 

FRENCH NATION, 

HAS PROVEN BY THE QUIET DIGNITY AND THE EFFICIENCY WITH WHICH 
HE HAS PERFORMED HIS AUGUST DUTIES THAT HE IS 

A WORTHY MEMBER OF A NOBLE FAMILY, 
ALREADY RENDERED FAMOUS BY AN EARLIER SADI CARNOT, 

NOW IMMORTAL IN THE ANNALS OF SCIENCE, 

AND IS HIMSELF DESERVING OF ENROLMENT IN A LIST OF GREAT MEN 

WHICH INCLUDES THAT OTHER DISTINGUISHED ENGINEER, 

OUR OWN FIRST PRESIDENT, 

GEORGE WASHINGTON. 



CONTENTS. 



i. 

PAGE 

THE WORK OF N.-L.-SADi CARNOT. By the Editor, I 

II. 

THE LIFE OF N.-L.-SADi CARNOT. By Mons. H. 
Carnot, . . . . . . . ; . . .20 

III. 

REFLECTIONS ON THE MOTIVE POWER OF HEAT 
AND ON MACHINES FITTED TO DEVELOP THAT 
POWER. By Mons. N.-L.-Sadi Carnot, . . 37 

IV. 

ACCOUNT OF CARNOT'S THEORY. By Sir William 
Thomson (Lord Kelvin), . . . . .127 

APPENDIX. 

A. EXTRACTS FROM UNPUBLISHED WRITINGS OF 

CARNOT, ' .205 

B. CARNOT'S FOOT-NOTES, 237 

C. NOTE BY THE EDITOR, 261 

v 




PUBLISHERS' NOTE. 

THE raison d'etre of the following translation of 
the famous work of Carnot is not the usual one, 
either with the Publishers or the Editor expec- 
tation of gain in either purse or fame. Neither 
could reasonably be anticipated from the reproduc- 
tion of the work of an author of more than a half- 
century ago, in a field then unrecognized, and 
to-day familiar to but few ; and especially when, 
as is in this case the fact, the work itself has been 
long out of date as a scientific authority, even had 
it ever held such a position. It could not be pre- 
sumed that a very large proportion of even the 
men of science of the English-speaking world 
would be sufficiently familiar with the subject, or 
interested in its origin, to purchase such a relic of 
a primitive period as is this little book. Nor 
could the translation of the work, or the gather- 
ing together by the Editor of related matter, be 
supposed likely to be productive of any form of 
compensation. The book is published as matter 
of limited but most intense scientific interest, 
and on that score only. 

vii 



viii PUBLISHERS' NOTE. 

It has seemed to the Editor and to the Pub- 
lishers that the product of the wonderful genius of 
Carnot, the great foundation-stone of one of the 
most marvellous and important of modern sciences, 
the first statement of the grand though simple laws 
of Thermodynamics, as illustrated in this one lit- 
tle treatise, should be made accessible to all who 
desire to study the work in English, and preserved, 
so far as its publication in this form could ac- 
complish it, as a permanent memorial, in a foreign 
tongue, of such grand truths, and of such a great 
genius as was their discoverer. It is with this 
purpose that Publishers and Editor have co- 
operated in this project. 

The book consists, as will be seen on inspection, 
of the translation of Carnot's Reflexions sur la 
Puissance Motrice du Feu, preceded by a notice 
written by the Editor calling attention to its 
remarkable features, and its extraordinary char- 
acter as the product of a most remarkable genius ; 
and by a biographical sketch of the great author, 
written by his brother, Mons. Hyppolyte Carnot, 
which sketch we find in the French copy of the 
work as published by Gauthier-Villars, the latest 
reproduction of the book in the original toDgue. 
To the main portion of the book, Carnot's Re- 
flexions, is appended the celebrated paper of Sir 



PUBLISHERS' NOTE. ix 

William Thomson, his "Account of Carnot's 
Theory," in which that great physicist first points 
out to the world the treasure so long concealed, 
unnoticed, among the scientific literature, already 
mainly antiquated, of the first quarter of the nine- 
teenth century. The distinguished writer of this 
paper has kindly interested himself in the scheme 
of the Editor, and has consented to its insertion 
as a natural and desirable commentary upon the 
older work, and especially as exhibiting the rela- 
tions of the fundamental principles discovered and 
enunciated by Carnot to the modern view of the 
nature of thermodynamic phenomena relations 
evidently understood by that writer, but not by 
the leaders of scientific thought of his time, and 
therefore ignored by him in the construction of 
his new science. 

The Appendix contains a number of Carnot's 
own notes, too long to be inserted in the body of 
the paper in its present form, and which have 
therefore been removed to their present location 
simply as a matter of convenience in book- 
making. 

The dedication of the work to the grand- 
nephew of the author, who by a singular coinci- 
dence happens to-day to occupy the highest posi- 
tion that any citizen can aspire to reach in that 



X PUBLISHERS' NOTE. 

now prosperous Kepublic, will be recognized as in 
all respects appropriate by every reader of the work 
of the earlier Sadi Carnot who is familiar with 
the character, the history, the attainments, the 
achievements, of the later Sadi Carnot in so 
many and widely diverse fields. The Carnot 
talent and the Carnot character are equally ob- 
servable in both men, widely as they are separated 
in time and in the nature of their professional 
labors. Both are great representatives of a noble 
family, whose honor and fame they have both 
splendidly upheld. 

The Publishers offer this little book to its 
readers as a small, yet in one sense not unim- 
portant, contribution to the great cause of modern 
science, as a relic, a memorial/a corner-stone. 



NOTE BY THE EDITOR. 

" Je me suis propose de grands desseins dans ce 
petit ouvrage" as Bernardin de Saint-Pierre says in 
the preface to his pathetic story of Paulet Virginie. 
I have sought to present to the great English- 
speaking world the work of a genius hitherto only 
known to a few men of science, and not well known, 
even among the people of France, for whose credit 
he has done so much. In placing before the read- 
ers of this translation his book small of size but 
great in matter as it is I feel that I have accom- 
plished an easy task, but one of real importance. 
I have been asked, as Corresponding Member for 
the United States of the Societe des Ingenieurs 
Civils de France, to communicate to my colleagues 
scientific and professional memoirs and whatever 
may be of interest to them "en un mot,que nous 
resserrions les liens qui font des ingenieurs en ge- 
neral une seule famille." That were a pleasant 
task; but a grander and a more agreeable one still 
is that of bringing " nearer in heart and thought " 
the members of that still larger community, the 
men of science of the world, and of weaving still 

xi 



xii NOTE BY THE EDITOR. 

more firmly and closely those bonds of kindly 
thought and feeling which are growing continually 
more numerous and stronger as the nations are 
brought to see that humanity is larger and more 
important than political divisions, and that the 
labors of educated men and of the guiding minds 
in the great industries are constantly doing more 
to promote a true brotherhood of mankind inan 
ever have, or ever can, the greatest statesmen. 

When the wonderful intellectual accomplish-- 
menis of men like the elder Sadi Carnot become 
known and appreciated by the world, much more 
will have been accomplished in tliis direction. It 
is perhaps from this point of view that the impor- 
tance of such work will be most fully recognized. 
When the little treatise which is here for the first 
time published in English becomes familiar to 
those for whom it is intended, it will be, to many 
at least, a matter of surprise no less than pleasure 
to discover that France has produced a writer on 
this now familiar subject whose inspiration antici- 
pated many of the principles that those founders 
of the modern science, Eankine and Clausius, 
worked out through the tedious and difficult 
methods of the higher mathematics, and which 
were hailed by their contemporaries as marvellous 
discoveries. 



NOTE TO SECOND EDITION. 



THE present edition of this little work is im- 
proved by the removal of a few errata observed in 
the first issue, and by the addition of a recent and 
excellent portrait of Lord Kelvin, as a frontispiece 
to his era-making paper, at page 127. This pic- 
ture, taken within the last year, is thought by the 
friends of its distinguished subject to be one of the 
best yet produced. That it is satisfactory to him 
and his friends is indicated by the fact that the 
original of this reproduction was presented to the 
writer by Lady Kelvin, in 1895, immediately after 
it was taken, and the autograph supplied by her 
distinguished husband. The Editor takes this 
occasion to acknowledge cordially the letters of 
appreciation and commendation received from 
those who have agreed with M. Haton de la Gou- 
pilliere that the translation of Carnot and its 
publication in this manner, with the famous paper 
of Lord Kelvin, will be considered as worthy of 
approval by English-speaking readers as well as 
"appreciated by the whole French nation." 

xiii 




THE WORK OF SADI CARNOT, 
BY THE EDITOR. 

NICOLAS-LEONAKD-SADI CAK:NX)T was, perhaps, 
the greatest genius, in the department of physical 
science at least, that this century has produced. 
By this I mean that he possessed in highest degree 
that combination of the imaginative faculty with 
intellectual acute ness, great logical power and ca- 
pacity for learning, classifying and organizing in 
their proper relations, all the facts, phenomena, 
and laws of natural science which distinguishes 
the real genius from other men and even from the 
simply talented man. Only now and then, in the 
centuries, does such a genius come into view. 
Euclid was such in mathematics ; Newton was 
such in mechanics ; Bacon and Compte were such 
in logic and philosophy ; Lavoisier and Davy were 
such in chemistry; and Fourier, Thomson, Max- 

1 



2 THE WORK OF SAD1 CAENOT. 

well, and Clausius were such in mathematical 
physics. Among engineers, we have the exam- 
ples of Watt as inventor and philosopher, Eankine 
as his mathematical complement, developing the 
theory of that art of which Watt illustrated the 
practical side ; we have Hirn as engineer- experi- 
mentalist, and philosopher, as well ; Corliss as in- 
ventor and constructor ; and a dozen creators of 
the machinery of the textile manufactures, in 
which, in the adjustment of cam-work, the high- 
est genius of the mechanic appears. 

But Carnot exhibited that most marked charac- 
teristic of real genius, the power of applying such 
qualities as I have just enumerated to great pur- 
poses and with great result while still a youth. 
Genius is not dependent, as is talent, upon the 
ripening and the growth of years for its pres- 
cience ; it is ready at the earliest maturity, and 
sometimes earlier, to exhibit its marvellous works ; 
as, for example, note Hamilton the mathema- 
tician and Mill the logician ; the one becoming 
master of a dozen languages when hardly more 
than as many years of age, reading Newton's Prin- 
cipia at sixteen and conceiving that wonderful 
system, quaternions, at eighteen ; the other com- 
petent to begin the study of Greek at three, learn- 
ing Latin at seven and reading Plato before he 



UNIVERSITY 



THE WORK OF 8ADI CARNOT. 

was eight. Carnot had done his grandest work of 
the century in his province of thought, and had 
passed into the Unseen, at thirty-six ; his one little 
volume, which has made him immortal, was writ- 
ten when he was but twenty-three or twenty-four. 
It is unnecessary, here, to enter into the particu- 
lars of his life ; that has been given us in ample 
detail in the admirable sketch by his brother 
which is here republished. It will be quite suf- 
ficient to indicate, in a few words, what were the 
conditions amid which he lived and the relation 
of his work to that great science of which it was 
the first exposition. 

At the time of Carnot, the opinion of the 
scientific world was divided, as it had been for 
centuries, on the question of the true nature of 
heat and light, and as it still is, to a certain ex- 
tent, regarding electricity. On the one hand it 
was held by the best-known physicists that heat 
is a substance which pervades all bodies in greater 
or less amount, and that heating and cooling are 
simply the absorption and the rejection of this 
" imponderable substance " by the body affected ; 
while, on the other hand, it was asserted by a 
small but increasing number that heat is a 
"mode of motion," a form of energy, not only 
imponderable, but actually immaterial ; a quality 



4 THE WORK OF SADI CARNOT. 

of bodies, not a substance, and that it is identical, 
in its nature, with other forms of recognizable 
energy, as, for example, mechanical energy. A 
quarter of a century before Carnot wrote, the ex- 
periments of Rumford and of Davy had been cru- 
cial in the settlement of the question and in the 
proof of the correctness of the second of the two 
opposing parties ; but their work had not become 
so generally known or so fully accepted as to be 
acknowledged as representative of the right views 
of the subject. The prevalent opinion, following 
Newton, was favorable to the first hypothesis ; 
and it was in deference to this opinion that Carnot 
based his work on an inaccurate hypothesis ; 
though, fortunately, the fact did not seriously 
militate against its value or his credit and fame. 

"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." * 

The results of Carnot's reasoning are, fortu- 
nately, mainly independent of any hypothesis as 
to the nature of heat or the method or mechanism 
of development and transfer or transformation of 
its energy. Carnot was in error in assuming no 

* Tait : Thermodynamics, p. 13. 



THE WORK OF 8ADI CARNOT. 5 

loss of heat in a completed cycle and in thus ignor- 
ing the permanent transformation of a definite 
proportion into mechanical energy ; but his propo- 
sition that efficiency increases with increase of 
temperature-range is still correct ; as is his asser- 
tion of its independence of the nature of the 
working substance. 

Oarnot's "Reflexions sur la Puissance Motrice 
du Feu" published in 1824, escaped notice at the 
time, was only now and then slightly referred to 
later, until Clapeyron seized upon its salient ideas 
and illustrated them by the use of the Watt dia- 
gram of energy, and might, perhaps, have still re- 
mained unknown to the world except for the fact 
that Sir William Thomson, that greatest of modern 
mathematical physicists, fortunately, when still a 
youth and at the commencement of his own great 
work, discovered it, revealed its extraordinary 
merit, and, readjusting Carnot's principles in ac- 
cordance with the modern views of heat-energy, 
gave it the place that it is so well entitled to in 
the list of the era-making books of the age. But 
it still remained inaccessible to all who could not 
find the original paper until, only a few years 
since, it was reprinted by Gauthier-Villars, the 
great publishing house of Paris, accompanied by a 
biographical sketch by the younger brother, which 



6 THE WORK OF SADI CARNOT. 

it has been thought wise to reproduce with the 
translation of Carnot's book. In making the 
translation, also, this later text has been -followed ; 
and now, for the first time, so far as is known to 
the writer, the work of Carnot is made accessible 
to the reader in English. 

The original manuscript of Carnot has been de- 
posited by his brother in the archives of the 
French Academy of Sciences, and thus insured 
perpetual care. The work of Carnot includes not 
only the treatise which it is the principal object of 
this translation to give to our readers, but also a 
considerable amount of hitherto unpublished mat- 
ter which has been printed by his brother, with 
the new edition of the book, as illustrative of the 
breadth and acuteness of the mind of the Founder 
of the Science of Thermodynamics. 

These previously unpublished materials consist 
of memoranda relating to the specific heats of 
substances, their variations, and various other 
facts and data, and principles as well ; some of 
which are now recognized as essential elements of 
the new science, even of its fundamental part. 
The book is particularly rich in what have been 
generally supposed to be the discoveries of later 
writers, and in enunciations of principles now 
tecognized as those forming the base and the sup- 



THE WORK OF SADI CARNOT. 7 

porting framework of that latest of the sciences. 
As stated by Tait, in his history of Thermody- 
namics, the " two grand things" which Carnot ori- 
ginated and introduced were his idea of a "cycle" 
and the notion of its " reversibility," when perfect. 
" Without this work of Carnot, the modern theory 
of energy, and especially that branch of it which 
is at present by far the most important in prac- 
tice, the dynamical theory of heat, could not have 
attained its now enormous development." These 
conceptions, original with our author, have been, 
in the hands of his successors, Clausius and other 
Continental writers, particularly, most fruitful of 
interesting and important results ; and Clapeyron's 
happy thought of so employing the Watt diagram 
of energy as to render them easy of comprehen- 
sion has proved a valuable aid in this direction. 

The exact experimental data needed for numer- 
ical computations in application of Carnot's prin- 
ciples were inaccessible at the date of his writing; 
they were supplied, later, by Mayer, by Cold ing, 
by Joule, and by later investigators. Even the 
idea of equivalence, according to Hypolyte Car- 
not, was not originally familiar to the author of 
this remarkable work; but was gradually developed 
and defined as he progressed with his philosophy. 
It is sufficiently distinctly enunciated in his later 



8 THE WORK OF 8ADI CARNOT. 

writings. He then showed a familiarity with 
those notions which have been ascribed generally 
to Mayer and which made the latter famous, and 
with those ideas which are now usually attributed 
to Joule with similar result. He seems actually to 
have planned the very kind of research which Joule 
finally carried out. All these advanced views 
must, of course, have been developed by Carnot 
before 1832, the date of his illness and death, and 
ten or fifteen years earlier than they were made 
public by those who have since been commonly 
considered their discoverers. These until lately 
unpublished notes of Carnot contain equally well- 
constructed arguments in favor of the now accepted 
theory of heat as energy. While submitting to 
the authority of the greatest physicists of his time, 
and so far as to make their view the basis of his 
work, to a certain extent, he nevertheless adhered 
privately to the true idea. His idea of the equiva- 
lence of heat and other forms of energy was as dis- 
tinct and exact as was his notion of the nature of 
that phenomenon. He states it with perfect ac- 
curacy. 

In making his measures of heat-energy, he as- 
sumes as a unit a measure not now common, but 
one which may be easily and conveniently reduced 
to the now general system of measurement. He 



THE WORK OF SADI CARNOT. 9 

takes the amount of power required to exert an 
energy equal to that needed to raise one cubic 
meter of water through a height of one meter, 
as his unit; this is 1000 kilogrammeters, taken 
as his unit of motive power; while he says that 
this is the equivalent of 2.7 of his units of 
heat; which latter quantity would be destroyed 
in its production of this amount of power, or 
rather work. His unit of heat is thus seen 
to be 1 000 -f- 2. 7, or 370 kilogram meters. This 
is almost identical with the figure obtained by 
Mayer, more than ten years later, and from 
presumably the same approximate physical data, 
the best then available, in the absence of a Reg- 
nault to determine the exact values. Mayer ob- 
tained 365, a number which the later work of 
Regnault enabled us to prove to be 15 per cent, 
too low, a conclusion verified experimentally by 
the labors of Joule and his successors. Carnot was 
thus a discoverer of the equivalence of the units of 
heat and work, as well as the revealer of the prin- 
ciples which have come to be known by his name. 
Had he lived a little longer, there can be little 
doubt that he would have established the facts, as 
well as the principles, by convincing proof. His 
early death frustrated his designs, and deprived the 



10 THE WORK OF SADI CARROT. 

world of one of its noblest intellects, just when it 
was beginning its marvellous career. 

The following sentence from Carnot illustrates 
in brief his wonderful prescience; one can hardly 
believe it possible that it should have been written 
in the first quarter of the nineteenth century: 
" On pent done poser en these generate que la puis- 
sance motrice est en quantite invariable dans la 
Nature; qu' elle n'est jamais, a proprement parler, 
ni produite, ni detruite. A la verite, elle change 
de forme) c'est a direqu' elle produittantotun genre 
de mouvement, tantot un autre; mais elle n'est 
jamais aneantie." It is this man who has prob- 
ably inaugurated the development of the modern 
science of thermodynamics and the whole range of 
sciences dependent upon it, and who has thus made 
it possible to construct a science of the energetics 
of the universe, and to read the mysteries of every 
physical phenomenon of nature; it is this man who 
has done more than any contemporary in his field, 
and who thus displayed a more brilliant genius 
than any man of science of the nineteenth century: 
yet not even his name appears in the biographical 
dictionaries; and in the Encyclopaedia Britannica 
it is only to be found incidentally in the article on 
Thermodynamics. 

Throughout his little book, we find numerous 



THE WORK OF SADI CARNOT. 11 

proofs of his clearness of view and of the wonder- 
ful powers of mind possessed by him. He opens 
his treatise by asserting that " C'est a la chaleur 
que doivent etre attribute les grands mouvements 
qui frappent nos regards sur la terre; c'est a elle 
que sont dues les agitations de V atmosphere, Vas- 
cension des nuages, la chute des pluies et ties autres 
meteores, les courants d'eau qui sillonnent la surface 
du globe et dont Vhomme est parvenue a employer 
pour son usage une faible partie; en fin les tremble- 
menfs de terre, les eruptions volcaniques reconnais- 
sent aussi pour cause la chaleur" 

Carnot was the first to declare that the maximum 
of work done by heat, in any given case of appli- 
cation of the heat-energy, is determined solely by 
the range of temperature through which it fell in 
the operation, and is entirely independent of the 
nature of the working substance chosen as the 
medium of transfer of energy and the vehicle of 
the heat. His assumption of the materiality of 
heat led, logically, to the conclusion that the 
same quantity of heat was finally stored in the 
refrigerator as had, initially, left the furnace, and 
that the effect produced was a consequence of a fall 
of temperature analogous to a fall of water; but, 
aside from this error which he himself was evi- 



12 THE WORK OF 8ADI CARNOT. 

dently inclined to regard as such, his process and 
argument are perfectly correct.* 

Throughout his whole work are distributed con- 
densed assertions of principles now well recognized 
and fully established, which indicate that he not 
only had anticipated later writers in their estab- 
lishment, but that he fully understood their real 
importance in a theory of heat-energy and of heat- 
engines. In fact, he often italicizes them, placing 
them as independent paragraphs to more thor- 
oughly impress the reader with their fundamental 
importance. Thus he says : " Partout ou il existe 
une difference de temperature, il pent y avoir pro- 
duction de puissance motrice;" and again, this 
extraordinary anticipation of modern science : ' ' le 
maximum de puissance resultant de I'emploi de la 
vapeur est aussi le maximum de puissance motrice 
realisable par quelque moyen que ce soit." 

(( La puissance motrice de la chaleur est inde- 
pendante des agents mis en ceuvre pour la realiser ; 
sa quant ite est fixee uniquement par les temper a- 



* Account of Carnot's Theory of the Motive Power of 
Heat; Sir Wm. Thomson; Trans. Roy. Soc. of Edin- 
burgh, xvi. 1849; and Math, and Phys. Papers, xli. vol. 1 
(Cambridge, 1882), p. 113. In this paper the corrections due 
to the introduction of the dynamic theory are first applied. 



THE WORK OF SADI CARNOT. 13 

tures des corps entre lesquels sefait, en dernier re- 
sultat, le transport du calorique." 

" Lorsqu'un gaz passe, sans changer de tempera- 
ture, d'un volume et d'une pression determines a une 
autre pression egalement determinee, la quantite 
de calwique absorbee ou abandon?iee est toujours la 
meme, quelle que soit la nature du gaz choisi comme 
sujet a" experience." 

Perhaps as remarkable a discovery as any one of 
the preceding (and one which, like those, has been 
rediscovered and confirmed by later physicists ; 
one which was the subject of dispute between 
Clausius, who proved its truth by the later methods 
which are now the source of his fame, and the 
physicists of his earlier days, who had obtained 
inaccurate measures of the specific heats of the 
gases; values which were finally corrected by Reg- 
nault, thus proving Carnot and Clausius to be 
right is thus stated by Carnot, and is italicized 
in his manuscript and book : 

" La difference entre la clialeur specijique sous 
pression constante et la clialeur specijique sous vo- 
lume constant est la meme pour tous les gaz." 

He bases his conclusion upon the simplest of 
thermodynamic considerations. He says that the 
increase of volumes with the same differences of 
temperature are the same, according to Gay-Lussao 



14 THE WORK OF SADI CAENOT. 

and Dal ton ; and that, therefore, according to the 
laws of thermodynamics as lie has demonstrated 
them, the heat absorbed with equal augmenta- 
tions of volume being the same, the two specific 
heats are constant, and their difference as well. As 
will be seen on referring to the text, he bases upon 
this principle a determination of the specific heats 
of constant volume, taking as his values of the de- 
termined specific heats of constant pressure those 
of Delaroche and Berard, making the constant 
difference 0.300, that of air at constant pressure 
being taken as the standard and as unity. The 
establishment of this point, in the face of the op- 
position, and apparently of the facts, of the best 
physicists of his time, was one of those circum- 
stances which did so much to win for Clausius his 
great fame. How much greater credit, then, 
should be given Carnot, who not only anticipated 
the later physicists in this matter, but who must 
have enunciated his principle under far more seri- 
ous discouragements and uncertainty ! 

It must be remembered, when reading Oar- 
not, that ,all the "constants of nature" were, in 
his time, very inaccurately ascertained. It is only 
since the time of Regnault's grand work that it has 
been the rule that such determinations have been 
published only when very exactly determined. No 



THE WORK OF SADI CARNOT. 15 

change has been attempted in Carnot's figures, in 
any respect ; as it would be far less satisfactory to 
read a paraphrased work, and the exact figures are 
now easily accessible to every one, and his compu- 
tations may all be made, if desired, on the basis of 
modern data. Sir William Thomson has already 
performed this task in the paper appended. 

Throughout the whole of this treatise, small as 
it is, we find distributed a singular number of 
these anticipations of modern thermodynamic 
principles. Studying the relation of heat-energy 
to work done, he concludes : 

"La chute du calorique produit plus de puis- 
sance motrice dans les degres inferieurs que dans 
Us degres superieurs." 

We to-day admit that, since the one degree at a 
low temperature, and the corresponding quantity 
ct heat, are larger fractions of the total tempera- 
ture, and the total heat stored in the substance, 
than the one degree at a higher point on the scale 
of absolute temperature, this principle of Carnot 
has become obvious. 

In the enunciation of the essential principles of 
efficiency of the heat-engine, we find the proofs of 
this same wonderful prescience. He asserts that, 
for best effect : " (1) The temperature of the 
working fluid must be raised to the highest degree 



16 THE WORK OF SADI CARNOT. 

possible, in order to secure a commensurate range 
of temperature ; (2) The cooling must be carried 
to the lowest point on the scale that may be found 
practicable ; (3) The passage of the fluid from the 
upper to the lower limit of temperature must be 
produced by expansion;" i.e., "it is necessary 
that the cooling of the gas shall occur sponta- 
neously by its rarefaction ;" which is simply his 
method of stating the now universally understood 
principle that, for highest efficiency, the expansion 
must be adiabatic, from a maximum to a mini- 
mum temperature. He goes on to explain these 
principles, and then says that the advantage of 
high-pressure engines lies " essentiellement dans la 
faculte de rendre utile vne plus grande chute de ca- 
loriqne." This principle, as a practical system of 
operation, had already, as he tells us, been enunci- 
ated by M. Clement, and had been practised, as 
we well know, since the days of its originator, 
Watt ; but Carnot saw clearly the thermodynamic 
principle which underlies it, and as clearly states 
it, for the first time. 

He sees clearly, too, the reasons for the attempts 
of Hornblower and of Woolf, premature as they 
proved and as he also sees, in the introduction of 
the compound engine, and even suggests that this 
idea might be still further developed by the use 



THE WORK OF SADI CARNOT, 17 

of a triple-expansion engine, a type which is to- 
day just coming into use, more than a half -cen- 
tury after Carnot/s date. He recognizes the ad- 
vantages of the compound engine in better distri- 
bution of pressures and in distribution of the work 
of expansion, but does not, of course, perceive the 
then undiscovered limitation of the efficiency of 
the simple engine, due to ' ' cylinder condensation," 
which has finally led, perhaps more than any other 
circumstance, to its displacement so largely by 
the multi-cylinder machine. No one has more ex- 
actly and plainly stated the respective advantages 
to be claimed for air and the gases, used as work- 
ing fluids in heat-engines, than does Carnot ; nor 
does any one to-day better recognize the difficul- 
ties which lie in the path to success in that direc- 
tion, in the necessity of finding a means of hand- 
ling them at high temperatures and of securing 
high mean pressures. 

His closing paragraph shows his extraordinary 
foresight, and the precision with which that won- 
derful intellect detected the practical elements 
of the problem which the engineer, from the days 
of Savery, of Newcomen, and of Watt, has been 
called upon to study, and the importance of the 
work, which he began, in the development of a 
theory of the action, or of the operation, of the 



18 THE WORK OF SADI CARNOT. 

heat-engines, which should give effective assistance 
in the development of their improved forms : 

" On ne doit pas se flatter de mettre jamais a 
profit, dans la pratique, toute la puissance des com- 
bustibles. Les tentatives que Von ferait pour ap- 
procher ce resultat seraient meme plus nuisiW.es 
qu'utiles, si ellex faisaient negliger d'autres conside- 
rations importantes. L'economie du combustible 
n'est qu'une des conditions a remplir par les ma- 
chines a feu; dans beaucoup de cir Constances, elle 
n'est que secondaire: elle doit souvent ceder le pas 
a la silrete, a la solidite, a la duree de la machine, 
aupeu de place qu'ilfaut lui faire occuper, au peu 
defrais de son etablissement, etc. Savoir appre- 
cier, dans chaque cas, a leur juste valeur, les con- 
siderations de convenance et d 'economic quipeuvent 
se presenter ; savoir discerner les plus importantes 
de celles qui sont seulement accessoires, les balancer 
toutes convenablement entre elles, afin de parvenir, 
par les moyens les plus faciles, au meilleur resul- 
tat: tel doit etre le principal talent de I'homme 
appele a diriger, a co-ordonner entre eux les travaux 
de ses semblables, a les faire concourir vers un but 
utile de quelque genre qu'il soit." 

Such was the work and such the character of 
this wonderful man. Those whose desire to fol- 
low more closely and to witness the process of de- 



THE WORK OF 1SADI CARNOT. 19 

velopment of the work of which this initial paper 
of Carnot was the introductory, should study the 
contribution of Sir William Thomson to this devel- 
opment, as published in 1849, a paper which 
constitutes that physicist the virtual discoverer of 
Carnot and the godfather of the man and his 
thoughts. This paper constitutes the final chapter 
of this little book. 

From that time the additional progress so rap- 
idly made in the new science was as inevitable 
as the development of a gold-field, once the pre- 
cious metal has been found in paying quantities 
in the hitherto unvisited canons and gorges of 
a distant and unexplored mountain-range. But 
great as is the work since done, and great as have 
been the discoveries and the discoverers of later 
years, none claims our gratitude and compels our 
respect in greater degree than does the original 
discoverer 

SADI CARNOT. 



II. 

LIFE OF SADI CARNOT. 
BY M. H. CARNOT. 

As the life of Sadi Carnot was not marked by 
any notable event, his biography would have occu- 
pied only a few lines ; but a scientific work by him, 
after remaining long in obscurity, brought again 
to light many years after his death, has caused his 
name to be placed among those of great inventors. 
In regard to his person, his mind, his character, 
nothing whatever has been known. Since there re- 
mains a witness of his private life the sole witness, 
has he not a duty to fulfil ? Ought he not to 
satisfy the natural and legitimate interest which 
attaches to any man whose work has deserved a 
portion of glory ? 

Nicolas-Leonard-Sadi Carnot was born Junel, 
1796, in the smaller Luxembourg. This was that 
part of the palace where our father then dwelt as 
a member of the Directory. Our father had a 
predilection for the name of Sadi, which recalled 
to his mind ideas of wisdom and poetry. His first- 
born had borne this name, and despite the fate 

20 



LIFE OF SADI CARNOT. 21 

of this poor child, who lived but a few months, 
he called the second also Sad I, in memory of the 
celebrated Persian poet and moralist. 

Scarcely a year had passed when the proscrip- 
tion, which included the Director, obliged him to 
give up his life, or at least his liberty, to the con- 
spirators of fructidor. Our mother carried her 
son far from the palace in which violation of law 
had just triumphed. She fled to St. Omer, with 
her family, while her husband was exiled to Switz- 
erland, then to Germany. 

Our mother often said to me, " Thy brother was 
born in the midst of the cares and agitations of 
grandeur, thou in the calm of an obscure retreat. 
Your constitutions show this difference of origin/' 

My brother in fact was of delicate constitution. 
He increased his strength later, by means of va- 
ried and judicious bodily exercises. He was of 
medium size, endowed with extreme sensibility 
and at the same time with extreme energy, more 
than reserved, almost rude, but singularly cou- 
rageous on occasion. When he felt himself to be 
contending against injustice, nothing could re- 
strain him. The following is an anecdote in illus- 
tration. 

The Directory had given place to the Consulate. 
Carnot, after two years of exile, returned to his 



22 LIFE OF SADI CARNOT. 

country and was appointed Minister of War. 
Bonaparte at the same time was still in favor with 
the republicans. He remembered that Carnot had 
assisted him in the beginning of his military ca- 
reer, and he resumed the intimate relation which 
had existed between them during the Directory. 
When the minister went to Malmaison to work 
with the First Consul, he often took with him his 
son, then about four years old, to stay with 
Madame Bonaparte, who was greatly attached to 
him. 

She was one day with some other ladies in a 
small boat on a pond, the ladies rowing the boat 
themselves, when Bonaparte, unexpectedly ap- 
pearing, amused himself by picking up stones and 
throwing them near the boat, spattering water on 
the fresh toilets of the rowers. The ladies dared 
not manifest their displeasure, but the little Sadi, 
after having looked on at the affair for some time, 
suddenly placed himself boldly before the con- 
queror of Marengo, and threatening him with his 
fist, he cried "Beast of a First Consul, will you 
stop tormenting those ladies I" 

Bonaparte, at this unexpected attack, stopped 
and looked in astonishment at the child. Then 
he was seized with a fit of laughter in which all 
the spectators of the scene joined. 



LIFE OF SAD1 CARNOT. 23 

At another time, when the minister, wishing to 
return to Paris, sought his son, who had been left 
with Madame Bonaparte, it was discovered that he 
had run away. They found him a long way off, in 
a mill, the mechanism of which he was trying to 
understand. This desire had been in the child's 
mind for days, and the honest miller, not knowing 
who he was, was kindly answering all his ques- 
tions. Curiosity, especially in regard to mechanics 
and physics, was one of the essential traits of 
Sadies mind. 

On account of this disposition so early mani- 
fested, Carnot did not hesitate to give a scientific 
direction to the studies of his son. He was able 
to undertake this task himself when the monarchi- 
cal tendencies of the new government had deter- 
mined him to retire. For a few months only Sadi 
followed the course of M. Bourdon at the Charle- 
magne Lycee to prepare himself for the Poly- 
technic School. 

The pupil made rapid progress. He was just 
sixteen years old when he was admitted to the 
school, the twenty-fourth on the list. This was 
in 1812. The following year he left it, first in 
artillery. But he was considered too young for the 
school of Metz, and he continued his studies at 
Paris for a year. To this circumstance is due the 



24 LIFE OF SADI CARNOT. 

fact that he took part in March, 1814, in the 
military exploits of Vincennes, and not of the 
butte Chaumont, as almost all the historians of 
the siege of Paris declared. M. Chasles, one of 
Sadi's school-fellows, took pains to rectify this 
error at a seance of the Institute in 1869. 

If the pupils of the Polytechnic School did not 
earlier enter into the campaign, it was not because 
they had not asked to do so. I find in my broth- 
er's papers the copy of an address to the Emperor, 
signed by them December 29, 1813 : 

" SIRE : The country needs all its defenders. 
The pupils of the Polytechnic School, faithful to 
their motto, ask to be permitted to hasten to the 
frontiers to share the glory of the brave men who 
are consecrating themselves to the safety of France. 
The battalion, proud of having contributed to the 
defeat of the enemy, will return to the school to 
cultivate the sciences and prepare for new services/' 

General Carnot was at Anvers, which he had just 
been defending against the confederate English, 
Prussians, and Swedes, where the French flag yet 
floated, when he wrote to his son, .April 12, 1814 : 

" MY DEAR SADI : I have learned with extreme 
pleasure that the battalion .of the Polytechnic 
School has distinguished itself, and that you have 
performed your first military exploits with honor. 



LIFE OF SADI CARNOT. 25 

When I am recalled, I shall be very glad if the 
Minister of War will give you permission to come 
to me. You will become acquainted with a fine 
country and a beautiful city, where I have had the 
satisfaction of remaining in peace while disaster 
has overwhelmed so many other places." 

Peace being restored, Sadi rejoined his father at 
Anvers and returned with him into France. 

In the month of October he left the Polytech- 
nic School, ranking sixth on the list of young 
men destined to service in the engineer corps, 
and went to Metz as a cadet sub-lieutenant at the 
school. Many scientific papers that he wrote there 
were a decided success. One is particularly re- 
ferred to as very clever, a memoir on the instru- 
ment called the theodolite which is used in astron- 
omy and geodesy. 

I obtain these details from M. Ollivier, who was 
of the same rank as Sadi and who, later, was one 
of the founders of the EcoleCentrale. Among his 
other comrades besides M. Chasles, the learned 
geometrician just now referred to, was Gen. Du- 
vivier, lamented victim of the insurrection of 
June 1848. I ought also to mention M Robelin, 
Sadi's most intimate friend, who came to help me 
burse him during his last illness, and who pub- 



UNIVERSITY 



26 LIFE OF SADI CABNOT. 

lished a notice concerning him in the Revue ency- 
clopedique, t. Iv. 

The events of 1815 brought General Carnot back 
into politics during the " Cent Jours " which ended 
in a fresh catastrophe. 

This gave Sadi a glimpse of human nature of 
which he could not speak without disgust. His 
little sub-lieutenant's room was visited by certain 
superior officers who did not disdain to mount to 
the third floor to pay their respects to the son of 
the new minister. 

Waterloo put an end to their attentions. The 
Bourbons re-established on the throne, Carnot was 
proscribed and Sadi sent successively into many 
trying places to pursue his vocation of engineer, 
to count bricks, to repair walls, and to draw plans 
destined to be hidden in portfolios. He performed 
these duties conscientiously and without hope of 
recompense, for his name, which not long before 
had brought him so many flatteries, was hence- 
forth the cause of his advancement being long 
delayed. 

In 1818 there came an unlooked-for royal ordi- 
nance, authorizing the officers of all branches of 
the service to present themselves at the examina- 
tions for the new corps of the staff. Sadi was 
well aware that favor had much more to do with 



LIFE OF SADI CARNOT. 27 

this matter than ability, but he was weary of 
garrison life. The stay in small fortresses to 
which the nature of his work confined him did 
not offer sufficient resources to his love of study. 
Then he hoped, and his hope was realized, that a 
request for a furlough would be obtained without 
difficulty, and would insure him the leisure that 
he sought. In spite of the friendly opposition of 
some chiefs of the engineer corps, testifying to a 
sincere regret at the removal from their register 
of a name which had gained honor among them, 
Sadi came to Paris to take the examination, and 
was appointed lieutenant on the staff, January 20, 
1819. 

He hastened to obtain his furlough, and availed 
himself of it to lead, in Paris and in the country 
round about Paris, a studious life interrupted but 
once, in 1821, by a journey to Germany to visit our 
father in his exile at Magdeburg. We had then 
the pleasure of passing some weeks all three to- 
gether. 

When, two years later, death took from us this 
revered father and I returned alone to France, I 
found Sadi devoting himself to his scientific studies, 
which he alternated with the culture of the arts. 
In this way also, his tastes had marked out for 
him an original direction, for no one was more 



28 LIFE OF SADI CARNOT. 

opposed than he to the traditional and the con- 
ventional. On his music-desk were seen only the 
compositions of Lully that he had studied, and 
the concert! of Viotti which he executed. On his 
table were seen only Pascal, Moliere, or La Fon- 
taine, and he knew his favorite books almost by 
heart. I call this direction original, because it 
was anterior to the artistic and literary movement 
which preceded the revolution of 1830. As to the 
sympathy of Sadi for the author of the Provin- 
ciates, it was due not only to the respect of the 
young mathematician for one of the masters of 
science, but his devoutly religious mind regarded 
with horror hypocrisy and hypocrites. 

Appreciating the useful and the beautiful, Sadi 
frequented the museum of the Louvre and the 
Italian Theatre, as well as the Jardin des Plantes 
and the Conservatoire des Arts et Metiers. Music 
was almost a passion with him. He probably in- 
herited this from our mother, who was an excel- 
lent pianist, to whom Dalayrac and especially 
Monsigny, her compatriot, had given instruction. 
Not content with being able to play well on the 
violin, Sadi carried to great length his theoretical 
studies. 

His insatiable intellect, moreover, would not 
allow him to remain a stranger to any branch of 



LIFE OF 8ADI CARNOT. 29 

knowledge. He diligently followed the course of 
the College of France and of the Sorbonne, of 
the Ecole des Mines, of the Museum, and of the 
Bibliotheque. He visited the workshops with 
eager interest, and made himself familiar with the 
processes of manufacture; mathematical sciences, 
natural history, industrial art, political economy, 
all these he cultivated with equal ardor. I have 
seen him not only practise as an amusement, but 
search theoretically into, gymnastics, fencing, 
swimming, dancing, and even skating. In even 
these things Sadi acquired a superiority which 
astonished specialists when by chance he forgot 
himself enough to speak of them, for the satisfac- 
tion of his own mind was the only aim that he 
sought. 

He had such a repugnance to bringing himself 
forward that, in his intimate conversations with a 
few friends, he kept them ignorant of the treasures 
of science which he had accumulated. They never 
knew of more than a small part of them. How 
was it that he determined to formulate his ideas 
about the motive power of heat, and especially to 
publish them ? I still ask myself this question, I, 
who lived with him in the little apartment where 
our father was confined in the Rue du Parc-Royal 
while the police of the first Restoration were 



30 LIFE OF SADI GARNOT. 

threatening him. Anxious to be perfectly clear, 
Sadi made me read some passages of his manu- 
script in order to convince himself that it would 
be understood by persons occupied with other 
studies. 

Perhaps a solitary life in small garrisons, in the 
work-room and in the chemical laboratory, had 
increased his natural reserve. In small compa- 
nies, however, he was not at all taciturn. He took 
part voluntarily in the gayest plays, abandoning 
himself to lively chat. "The time passed in 
laughing is well spent," he once wrote. His lan- 
guage was at such times full of wit, keen without 
malice, original without eccentricity, sometimes 
paradoxical, but without other pretension than 
that of an innocent activity of intelligence. He 
had a very warm heart under a cold manner. He 
was obliging and devoted, sincere and true in his 
dealings. 

Towards the end of 1826, a new royal ordinance 
having obliged the staff lieutenants to return to 
the ranks, Sadi asked and obtained a return to the 
engineer corps, in which he received the following 
year, as his rank of seniority, the grade of captain. 

Military service, however, weighed upon him. 
Jealous of his liberty, in 1828, he laid aside his 
uniform that he might be free to come and go at 



LIFE OF 8ADI CARNOT. 31 

will. He took advantage of his leisure to make 
journeys and to visit our principal centres of 
industry. 

He frequently visited M. Clement Desormes, 
professor at the Conservatoire des Arts et Metiers, 
who had made great advances in applied chemistry. 
M. Desormes willingly took counsel with him. 
He was a native of Bourgogne, our family coun- 
try, which circumstance, I believe, brought them 
together. 

It was before this period (in 1824) that Sadi had 
published his Reflexions sur la puissance motrice, 
du feu. He had seen how little progress had been 
made in the theory of machines in which this 
power was employed. He had ascertained that 
the improvements made in their arrangement were 
effected tentatively, and almost by chance. He 
comprehended that in order to raise this important 
art above empiricism, and to give it the rank of a 
science, it was necessary to study the phenomena 
of the production of motion by heat, from the 
most general point of view, independently of any 
mechanism, of any special agent ; and such had 
been the thought of his life. 

Did he foresee that this small brochure would 
become the foundation of a new science? He 
tnust have attached much importance to it to 



32 LIFE OF SADI CARNOT. 

publish it, and bring himself out of his voluntary 
obscurity. 

In fact (as his working notes prove), he per- 
ceived the existing relation between heat and 
mechanical work ; and after having established the 
principle to which savants have given his name, 
he devoted himself to the researches which should 
enable him to establish with certainty the second 
principle, that of equivalence, which he already 
clearly divined. Thermodynamics was established 
from that time. 

But these researches were rudely interrupted by 
a great event the Revolution of July, 1830. 

Sadi welcomed it enthusiastically not, however, 
it is evident, as a personal advantage. 

Several old members of the Convention were 
still living, even of those who had become cele- 
brated ; no favor of the new government was 
accorded them. To the son of Philippe-Egalite 
was ascribed a saying which, if it was untrue, at 
least agreed well with the sentiment of his posi- 
tion: "I can do nothing for the members of the 
Convention themselves," he said, "but for their 
families whatever they will." 

However it may be, some of those about him 
vaguely questioned my brother as to his desires in 
case one of us should be called to the Chamber of 



LIFE OF SADI CARNOT. 33 

Peers, of which Carnot had been a member in 
1815. We had on this occasion a brief conference. 
Unknown to us both, this distinction could be 
offered only to a title in some sort hereditary. 
We could not accept it without forsaking the prin- 
ciples of Carnot, who had combated the heredity 
of the peerage. The paternal opinion therefore 
came to second our distaste for the proposition, 
and dictated our reply. 

Sadi frequented the popular reunions at this 
period without forsaking his role of a simple ob- 
server. 

Nevertheless he was, when occasion demanded 
it, a man of prompt and energetic action. One 
incident will suffice to prove this, and to show the 
sang-froid which characterized him. 

On the day of the funeral of Gen. Lamarque, 
Sadi was walking thoughtfully in the vicinity of 
the insurrection. A horseman preceding a com- 
pany, and who was evidently intoxicated, passed 
along the street on the gallop, brandishing his 
sabre and striking down the passers-by. Sadi 
darted forward, cleverly avoided the weapon of 
the soldier, seized him by the leg, threw him to 
the earth and laid him in the gutter, then contin- 
ued on his way to escape from the cheers of the 
crowd, amazed at this daring deed. 



34 LIFE OF SADI CARNOT. 

Before 1830, Sadi had formed part of a Reunion 
poly technique industrielle, made up of old pupils 
of the school, with a plan of study in common. 
After 1830, he was a member of the Association 
polyteclmique, consisting also of graduates, the 
object being the popular propagation of useful 
knowledge. The president of this association was 
M. de Choiseul-Praslin; the vice-presidents, MM. 
de Tracy, Auguste Comte, etc. 

The hopes of the democracy meanwhile seeming 
to be in abeyance, Sadi devoted himself anew to 
study, and pursued his scientific labors with all the 
greater energy, as he brought to bear upon them 
the political ardor now so completely repressed. 
He undertook profound researches on the physical 
properties of gases and vapors, and especially on 
their elastic tensions. Unfortunately, the tables 
which he prepared from his comparative experi- 
ments were not completed; but happily the excel- 
lent works of Victor Regnault, so remarkable for 
their accuracy, have supplied to science, in this 
respect, the blanks of which Sadi Carnot was con- 
scious. 

His excessive application affected his health 
towards the end of June, 1832. Feeling temporar- 
ily better, he wrote gayly to one of his friends who 
had written several letters to him : "My delay this 



LIFE OF SADI CARNOT. 35 

time is not without excuse. I have been sick for 
a long time, and in a very wearisome way. I have 
had an inflammation of the lungs, followed by scar- 
let-fever. (Perhaps you know what this horrible 
disease is.) I had to remain twelve days in bed, 
without sleep or food, without any occupation, 
amusing myself with leeches, with drinks, with 
baths, and other toys out of the same shop. This 
little diversion is not yet ended, for I am still very 
feeble." 

This letter was written at the end of July. 

There was a relapse, then brain fever; then final- 
ly, hardly recovered from so many violent illnesses 
which had Aveakened him morally and physically, 
Sadi was carried off in a few hours, August 24, 
1832, by an attack of cholera. Towards the last, 
and as if from a dark presentiment, he had given 
much attention to the prevailing epidemic, follow- 
ing its course with the attention and penetration 
that he gave to everything. 

Sadi Carnot died in the vigor of life, in the 
brightness of a career that he bade fair to run with 
glory, leaving memory of profound esteem and 
affection in the hearts of many friends. 

His copy-books, filled with memoranda, attest 
the activity of his mind, the variety of his knowl- 
edge, his love of humanity, his' clear sentiments of 



36 LIFE OF SADI CARNOT. 

justice and of liberty. We can follow therein the 
traces of all his various studies. But the only 
work that he actually completed is this which is 
here published. It will suffice to preserve his 
name from oblivion. 

His moral character has other claims on our 
recognition. Our only ambition here is to present 
a sketch of it. But, much better than through 
the perusal of these few pages, Sadi Carnot can be 
appreciated by reading the thoughts scattered 
through his memoranda, which are to be carefully 
collected. There are many practical rules of con- 
duct which he records for himself ; many observa- 
tions that he desires to fix in his memory ; some- 
times an impression that has just come to him, 
grave or gay ; sometimes too, though rarely, a 
trace of ill-humor directed against men or society. 
He never thought that these notes, the outpouring 
of his mind, would be read by other eyes than his 
own, or that they would some day be used to judge 
him. I find in them, for my part, touching analo- 
gies with the thoughts of my father, although the 
father and son had, unfortunately, lived almost 
always apart, by force of circumstances.* 

* See the Appendix for these memoranda, and for other 
previously unpublished matter. 



III. 

REFLECTIONS ON THE MOTIVE-POWER OF 
HEAT, AND ON MACHINES FITTED TO 
DEVELOP THAT POWER.* 

BY S. CARNOT. 

EVERY one knows that heat can produce motion. 
That it x possesses vast motive-power no one can 
doubt, in these days when the steam-engine is 
everywhere so well known. 

To heat also are due the vast movements which 
take place on the earth. It causes the agitations 
of the atmosphere, the ascension of clouds, the fall 
of rain and of meteors, the currents of water which 
channel the surface of the globe, and of which 

* Sadi Carnot's Reflexions sur la puissance motrice du 
feu (Paris, Bachelier 1824) was long ago completely ex- 
hausted. As but a small number of copies were printed, 
this remarkable work remained long unknown to the 
earlier writers on Thermodynamics. It was therefore for 
the benefit of savants unable to study a work out of print, 
as well as to render honor to the memory of Sadi Carnot, 
that the new publishers of the Annales Scientifique de 
VEcole Normale superieure (ii. series, 1. 1, 1872) published a 
new edition, from which this translation is reproduced. 

37 



38 MOTIVE POWER OF HEAT. 

man has thus far employed but a small portion. 
Even earthquakes and volcanic eruptions are the 
result of heat. 

From this immense reservoir we may draw the 
moving force necessary for our purposes. Nature, 
in providing us with combustibles on all sides, 
has given us the power to produce, at all times and 
in all places, heat and the impelling power which 
is the result of it. To develop this power, to 
appropriate it to our uses, is the object of heat- 
engines. 

The study of these engines is of the greatest 
interest, their importance is enormous, their use 
is continually increasing, and they seem destined 
to produce a great revolution in the civilized world. 

Already the steam-engine works our mines, im- 
pels our ships, excavates our ports and our rivers, 
forges iron, fashions wood, grinds grains, spins 
and weaves our cloths, transports the heaviest 
burdens, etc. It appears that it must some day 
serve as a universal motor, and be substituted for 
animal power, waterfalls, and air currents. 

Over the first of these motors it has the advan- 
tage of economy, over the two others the inestima- 
ble advantage that it can be used at all times and 
places without interruption. 

If, some day, the steam-engine shall be so per- 



MOTIVE POWER OF HEAT. 39 

fected that it can be set up and supplied with fuel 
at small cost, it will combine all desirable qualities, 
and will afford to the industrial arts a range the 
extent of which can scarcely be predicted. It is 
not merely that a powerful and convenient motor 
that can be procured and carried anywhere is 
substituted for the motors already in use, but that 
it causes rapid extension in the arts in which it is 
applied, and can even create entirely new arts. 

The most signal service that the steam-engine 
has rendered to England is undoubtedly the 
revival of the working of the coal-mines, which had 
declined, and threatened to cease entirely, in con- 
sequence of the continually increasing difficulty of 
drainage, and of raising the coal.* We should 
rank second the benefit to iron manufacture, both 
by the abundant supply of coal substituted for 
wood just when the latter had begun to grow scarce, 

*It may be said that coal-mining has increased tenfold 
in England since the invention of the steam-engine. It is 
almost equally true in regard to the mining of copper, tin, 
and iron. The results produced in a half-century by the 
steam-engine in the mines of England are to-day parallel- 
ed in the gold and silver mines of the New World mines 
of which the working declined from day to day, prin- 
cipally on account of the insufficiency of the motors em- 
ployed in the draining and the extraction of the minerals. 



40 MOTIVE POWER OF HEAT. 

and by the powerful machines of all kinds, the use 
of which the introduction of the steam-engine has 
permitted or facilitated. 

Iron and heat are, as we know, the supporters, 
the bases, of the mechanic arts. It is doubtful if 
there be in England a single industrial establish- 
ment of which the existence does not depend on 
the use of these agents, and which does not freely 
employ them. To take away to-day from England 
her steam-engines would be to take away at the 
same time her coal and iron. It would be to dry 
up all her sources of wealth, to ruin all on which 
her prosperity depends, in short, 'to annihilate that 
colossal power. The destruction of her navy, 
which she considers her strongest defence, would 
perhaps be less fatal. 

The safe and rapid navigation by steamships 
may be regarded as an entirely new art due to the 
steam-engine. Already this art has permitted the 
establishment of prompt and regular communica- 
tions across the arms of the sea, and on the great 
rivers of the old and new continents. It has made 
it possible to traverse savage regions where before 
we could scarcely penetrate. It has enabled us to 
carry the fruits of civilization over portions of the 
globe where they would else have been wanting for 
years. Steam navigation brings nearer together 



MOTIVE POWER OF HEAT. 41 

the most distant nations. It tends to unite the 
nations of the earth as inhahitants of one country. 
In fact, to lessen the time, the fatigues, the uncer- 
tainties, and the dangers of travel is not this the 
same as greatly to shorten distances?* 

The discovery of the steam-engine owed its birth, 
like most human inventions, to rude attempts 
which have been attributed to different persons, 
while the real author is not certainly known. It 
is, however, less in the first attempts that the prin- 
cipal discovery consists, than in the successive im- 
provements which have brought steam-engines to 
the condition in which we find them to-day. There 
is almost as great a distance between the first appa- 
ratus in which the expansive force of steam was 
displayed and the existing machine, as between the 
first raft that man ever made and the modern vessel. 

If the honor of a discovery belongs to the nation 
in which it has acquired its growth and all its 
developments, this honor cannot be here refused 

* We say, to lessen the dangers of journeys. In fact, 
although the use of the steam-engine on ships is attended 
by some danger which has been greatly exaggerated, this 
is more than compensated by the power of following al- 
ways an appointed and well-known route, of resisting the 
force of the winds which would drive the ship towards 
the shore, the shoals, or the rocks. 



42 MOTIVE POWER OF HEAT. 

to England. Savery, Newcomen, Smeaton, the 
famous Watt, Woolf, Trevithick, and some other 
English engineers, are the veritable creators of the 
steam-engine. It has acquired at their hands all 
its successive degrees of improvement. Finally, it 
is natural that an invention should have its birth 
and especially be developed, be perfected, in that 
place where its want is most strongly felt. 

Notwithstanding the work of all kinds done by 
steam-engines, notwithstanding the satisfactory 
condition to which they have been brought to-day, 
their theory is very little understood, and the at- 
tempts to improve them are still directed almost 
by chance. 

The question has often been raised whether the 
motive power of heat* is unbounded, whether the 
possible improvements in steam-engines have an 
assignable limit, a limit which the nature of 
things will not allow to be passed by any means 
whatever ; or whether, on the contrary, these im- 
provements may be carried on indefinitely. We 

* We use here the expression motive power to express 
the useful effect that a motor is capable of producing. 
This effect can always be likened to the elevation of a 
weight to a certain height. It has, as we know, as a 
measure, the product of the weight multiplied by the 
height to which it is raised. 



MOTIVE POWER OF HEAT. 43 

have long sought, and are seeking to-day, to ascer- 
tain whether there are in existence agents preferable 
to the vapor of water for developing the motive 
power of heat; whether atmospheric air, for ex- 
ample, would not 'present in this respect great ad- 
vantages. We propose now to submit these ques- 
tions to a deliberate examination. 

The phenomenon of the production of motion 
by heat has not been considered from a sufficiently 
general point of view. We have considered it only 
in machines the nature and mode of action of 
which have not allowed us to take in the whole 
extent of application of which it is susceptible. 
In such machines the phenomenon is, in a way, 
incomplete. It becomes difficult to recognize its 
principles and study its laws. 

In order to consider in the most general way 
the principle of the production of motion by heat, 
it must be considered independently of any mecha- 
nism or any particular agent. It is necessary to 
establish principles applicable not only to steam- 
engines* but to all imaginable heat-engines, what- 



* We distinguish here the steam-engine from the heat- 
engine in general. The latter may make use of any agent 
whatever, of the vapor of water or of any other, to develop 
the motive power of heat, 



44 MOTIVE POWER OF HEAT. 

ever the working substance and whatever the 
method by which it is operated. 

Machines which do not receive their motion from 
heat, those which have for a motor the force of 
men or of animals, a waterfall, an air-current, etc., 
can be studied even to their smallest details by 
the mechanical theory. All cases are foreseen, all 
imaginable movements are referred to these general 
principles, firmly established, and applicable under 
all circumstances. This is the character of a com- 
plete theory. A similar theory is evidently needed 
for heat-engines. We shall have it only when the 
laws of Physics shall be extended enough, general- 
ized enough, to make known beforehand all the 
effects of heat acting in a determined manner on 
any body. 

We will suppose in what follows at least a 
superficial knowledge of the different parts which 
compose an ordinary steam-engine; and we con- 
sider it unnecessary to explain what are the 
furnace, boiler, steam-cylinder, piston, condenser, 
etc. 

The production of motion in steam-engines is 
always accompanied by a circumstance on which 
we should fix our attention. This circumstance 
is the re-establishing of equilibrium in the caloric; 
that is, its passage from a body in which the 



MOTIVE POWER OF HEAT. 45 

temperature is more or less elevated, to another in 
which it is lower. What happens in fact in a 
steam-engine actually in motion? The caloric 
developed in the furnace by the effect of the com- 
bustion traverses the walls of the boiler, produces 
steam, and in some way incorporates itself with it. 
The latter carrying it away, takes it first into the 
cylinder, where it performs some function, and 
from thence into the condenser, where it is lique- 
fied by contact with the cold water which it en- 
counters there. Then, as a final result, the cold 
water of the condenser takes possession of the 
caloric developed by the combustion. It is heated 
by the intervention of the steam as if it had been 
placed directly over the furnace. The steam is 
here only a means of transporting the caloric. 
It fills the same office as in the heating of baths 
by steam, except that in this case its motion is 
rendered useful. 

We easily recognize in the operations that we 
have just described the re-establishment of equi- 
librium in the caloric, its passage from a more or 
less heated body to a cooler one. The first of 
these bodies, in this case, is the heated air of the 
furnace; the second is the condensing water. The 
re-establishment of equilibrium of the caloric 
takes place between them, if not completely, at 



46 MOTIVE POWER OF HEAT. 

least partially, for on the one hand the heated air, 
after having performed its function, having passed 
round the boiler, goes out through the chimney 
with a temperature much below that which it had 
acquired as the effect of combustion; and on the 
other hand, the water of the condenser, after hav- 
ing liquefied the steam, leaves the machine with 
a temperature higher than that with which it 
entered. 

The production of motive power is then due in 
steam-engines not to an actual consumption of 
caloric, but to its transportation from a warm 
body to a cold body, that is, to its re-establishment 
of equilibrium an equilibrium considered as de- 
stroyed by any cause whatever, by chemical action 
such as combustion, or by any other. We shall 
see shortly that this principle is applicable to 
any machine set in motion by heat. 

According to this principle, the production of 
heat alone is not sufficient to give birth to the 
impelling power: it is necessary that there should 
also be cold; without it, the heat would be use- 
less. And in fact, if we should find about us 
only bodies as hot as our furnaces, how can we 
condense steam ? What should we do with it if 
once produced ? We should not presume that we 
might discharge it into the atmosphere, as is done 



MOTIVE POWER OF HEAT. 47 

in some engines;* the atmosphere would not re- 
ceive it. It does receive it under the actual con- 
dition of things, only because it fulfils the office 
of a vast condenser, because it is at a lower tem- 
perature; otherwise it would soon become fully 
charged, or rather would be already saturated, f 

* Certain engines at high pressure throw the steam out 
iuto the atmosphere instead of the condenser. They are 
used specially in places where it would be difficult to 
procure a stream of cold water sufficient to produce 
condensation. 

f The existence of water in the liquid state here 
necessarily assumed, since without it the steam-engine 
could not be fed, supposes the existence of a pressure 
capable of preventing this water from vaporizing, con- 
sequently of a pressure equal or superior to the tension 
of vapor at that temperature. If such a pressure were 
not exerted by the atmospheric air, there would be in- 
stantly produced a quantity of steam sufficient to give 
rise to that tension, and it would be necessary always 
to overcome this pressure iu order to throw out the 
steam from the engines into the new atmosphere. Now 
this is evidently equivalent to overcoming the tension 
which the steam retains after its condensation, as effected 
by ordinary means. 

If a very high temperature existed at the surface of 
our globe, as it seems certain that it exists in its interior, 
all the waters of the ocean would be in a state of vapor 
in the atmosphere, and no portion of it would be found 
in a liquid state. 



48 MOTIVE POWER OF HEAT. 

Wherever there exists a difference of tempera- 
ture, wherever it has been possible for the equilib- 
rium of the caloric to be re-established,, it is possible 
to have also the production of impelling power. 
Steam is a means of realizing this power, but it is 
not the only one. All substances in nature can 
be employed for this purpose,, all are susceptible of 
changes of volume, of successive contractions and 
dilatations, through the alternation of heat and cold. 
All are capable of overcoming in their changes of 
volume certain resistances, and of thus developing 
the impelling power. A solid body a metallic 
bar for example alternately heated and cooled in- 
creases and diminishes in length, and can move 
bodies fastened to its ends. A liquid alternately 
heated and cooled increases and diminishes in vol- 
ume, and can overcome obstacles of greater or less 
size, opposed to its dilatation. An aeriform fluid is 
susceptible of considerable change of volume by 
variations of temperature. If it is enclosed in an 
expansible space, such as a cylinder provided with 
a piston, it will produce movements of great ex- 
tent. Vapors of all substances capable of passing 
into a gaseous condition, as of alcohol, of mercury, 
of sulphur, etc., may fulfil the same office as vapor 
of water. The latter, alternately heated and 
cooled, would produce motive power in the shape 



UNIVERSITY 



MOTIVE POWER OF HEAT. 49 

of permanent gases, that is, without ever return- 
ing to a liquid state. Most of these substances 
have been proposed, many even have been tried, 
although up to this time perhaps without remark- 
able success. 

We have shown that in steam-engines the motive- 
power is due to a re- establishment of equilibrium 
in the caloric ; this takes place not only for steam- 
engines, but also for every heat-engine that is, 
for every machine of which caloric is the motor. 
Heat can evidently be a cause of motion only by 
virtue of the changes of volume or of form which 
it produces in bodies. 

These changes are not caused by uniform tem- 
perature, but rather by alternations of heat and 
cold. Now to heat any substance whatever requires 
a body warmer than the one to be heated; to cool 
it requires a cooler body. We supply caloric to 
the first of these bodies that we may transmit 
it to the second by means of the intermediary 
substance. This is to re-establish, or at least to 
endeavor to re-establish, the equilibrium of the 
caloric. 

It is natural to ask here this curious and impor- 
tant question : Is the motive power of heat invari- 
able in quantity, or does it vary with the agent 
employed to realize it as the intermediary sub- 



50 MOTIVE POWER OF HEAT. 

stance, selected as the subject of action of the 
heat? 

It is clear that this question can be asked only 
in regard to a given quantity of caloric,* the differ- 
ence of the temperatures also being given. We 
take, for example, one body A kept at a tempera- 
ture of 100 and another body B kept at a tempera- 
ture of 0, and ask what quantity of motive power 
can be produced by the passage of a given portion 
of caloric (for example, as much as is necessary to 
melt a kilogram of ice) from the first of these 
bodies to the second. We inquire whether this 
quantity of motive power is necessarily limited, 
whether it varies with the substance employed to 
realize it, whether the vapor of water offers in this 
respect more or less advantage than the vapor of 
alcohol, of mercury, a permanent gas, or any other 
substance. We will try to answer these questions, 
availing ourselves of ideas already established. 

* It is considered unnecessary to explain here what is 
quantity of caloric or quantity of heat (for we employ 
these two expressions indifferently), or to describe how we 
measure these quantities by the calorimeter. Nor will we 
explain what is meant by latent heat, degree of temperature, 
specific heat, etc. The reader should be familiarized with 
these terms through the study of the elementary treatises 
of physics or of chemistry. 



MOTIVE POWER OF HEAT. 51 

We have already remarked upon this self-evident 
factj or fact which at least appears evident as soon 
as we reflect on the changes of volume occasioned 
by heat : wherever there exists a difference of tem- 
perature, motive-power can be produced. Recipro- 
cally, wherever we can consume this power, it is 
possible to produce a difference of temperature, 
it is possible to occasion destruction of equilibrium 
in the caloric. Are not percussion and the fric- 
tion of bodies actually means of raising their tem- 
perature, of making it reach spontaneously a 
higher degree than that of the surrounding bodies, 
and consequently of producing a destruction of 
equilibrium in the caloric, where equilibrium pre- 
viously existed ? It is a fact proved by experience, 
that the temperature of gaseous fluids is raised by 
compression and lowered by rarefaction. This is 
a sure method of changing the temperature of 
bodies, and destroying the equilibrium of the 
caloric as many times as may be desired with the 
same substance. The vapor of water employed in 
an inverse manner to that in which it is used in 
steam-engines can also be regarded as a means of 
destroying the equilibrium of the caloric. To be 
convinced of this we need but to observe closely 
the manner in which motive power is developed by 
the action of heat on vapor of water. Imagine 



52 MOTIVE POWER OF HEAT. 

two bodies A and B, kept each at a constant tem- 
perature, that of A being higher than that of B. 
These two bodies, to which we can give or from 
which we can remove the heat without causing 
their temperatures to vary, exercise the functions 
of two unlimited reservoirs of caloric. We will 
call the first the furnace and the second the re- 
frigerator. 

If we wish to produce motive power by carrying 
a certain quantity of heat from the body A to the 
body B we shall proceed as follows : 

(1) To borrow caloric from the body A to make 
steam with it that is, to make this body fulfil 
the function of a furnace, or rather of the metal 
composing the boiler in ordinary engines we here 
assume that the steam is produced at the same 
temperature as the body A. 

(2) The steam having been received in a space 
capable of expansion, such as a cylinder furnished 
with a piston, to increase the volume of this space, 
and consequently also that of the steam. Thus rare- 
fied, the temperature will fall spontaneously, as 
occurs with all elastic fluids ; admit that the rare- 
faction may be continued to the point where the 
temperature becomes precisely that of the body B. 

(3) To condense the steam by putting it in con- 
tact with the body B, and at the same time exert- 



MOTIVE POWER OF HEAT. 53 

ing on it a constant pressure until it is entirely 
liquefied. The body B fills here the place of the 
injection-water in ordinary engines, with this dif- 
ference, that it condenses the vapor without 
mingling with it, and without changing its own 
temperature.* 

* We may perhaps wonder here that the body B being 
at the same temperature as the steam is able to condense 
it. Doubtless this is not strictly possible, but the slightest 
difference of temperature will determine the condensation, 
which suffices to establish the justice of our reasoning. It 
is thus that, in the differential calculus, it is sufficient that 
we can conceive the neglected quantities indefinitely re- 
ducible in proportion to the quantities retained in the 
equations, to make certain of the exact result. 

The body B condenses the steam without changing its 
own temperature this results from our supposition. We 
have admitted that this body may be maintained at a con- 
stant temperature. We take away the caloric as the steam 
furnishes it. This is the condition in which the metal of 
the condenser is found when the liquefaction of the steam 
is accomplished by applying cold water externally, as was 
formerly done in several engines. Similarly, the water of 
a reservoir can be maintained at a constant level if the 
liquid flows out at one side as it flows in at the other. 

One could even conceive the bodies J.and B maintaining 
the same temperature, although they might lose or gain 
certain quantities of heat. If, for example, the body A 
were a mass of steam ready to become liquid, and the body 



54 MOTIVE POWER OF HEAT. 

The operations which we have just described 
might have been performed in an inverse direction 
and order. There is nothing to prevent forming 
vapor with the caloric of the body B, and at the 
temperature of that body, compressing it in such 
a way as to make it acquire the temperature of the 
body A, finally condensing it by contact with this 
latter body, and continuing the compression to 
complete liquefaction. 

By our first operations there would have been 
at the same time production of motive power 
and transfer of caloric from the body A to the 
body B. By the inverse operations there is at the 
same time expenditure of motive power and return 
of caloric from the body B to the body A. But 
if we have acted in each case on the same quantity 
of vapor, if there is produced no loss either of 
motive power or caloric, the quantity of motive 
power produced in the first place will be equal to 
that which would have been expended in the second, 
and the quantity of caloric passed in the first case 
from the body A to the body B would be equal to 
the quantity which passes back again in the second 
from the body B to the body A ; so that an indefi- 

B a mass of ice ready to melt, these bodies might, as we 
know, furnish or receive caloric without thermometrig 
change. 



MOTIVE POWER OF HEAT. 55 

nite number of alternative operations of this sort 
could be carried on without in the end having 
either produced motive power or transferred caloric 
from one body to the other. 

Now if there existed any means of using heat 
preferable to those which we have employed, that 
is, if it were possible by any method whatever to 
make the caloric produce a quantity of motive 
power greater than we have made it produce by our 
first series of operations, it would suffice to divert 
a portion of this power in order by the method just 
indicated to make the caloric of the body B return 
to the body A from the refrigerator to the furnace, 
to restore the initial conditions, and thus to be 
ready to commence again an operation precisely 
similar to the former, and so on : this would be 
not only perpetual motion, but an unlimited crea- 
tion of motive power without consumption either 
of caloric or of any other agent whatever. Such 
a creation is entirely contrary to ideas now accepted, 
to the laws of mechanics and of sound physics. 
It is inadmissible.* We should then conclude that 
the maximum of motive power resulting from the 
employment of steam is also the maximum of motive 
power realizable by any means whatever. We will 

* Note A, Appendix B. 



56 MOTIVE POWER OF HEAT. 

soon give a second more rigorous demonstration of 
this theory. This should be considered only as 
an approximation. (See page 59.) 

We have a right to ask, in regard to the propo- 
sition just enunciated, the following questions: 
What is the sense of the word maximum here ? 
By what sign can it be known that this maximum 
is attained ? By what sign can it be known whether 
the steam is employed to greatest possible advan- 
tage in the production of motive power ? 

Since every re-establishment of equilibrium in 
the caloric may be the cause of the production of 
motive power, every re-establishment of equilibrium 
which shall be accomplished without production of 
this power should be considered as an actual loss. 
Now, very little reflection would show that all 
change of temperature which is not due to a change 
of volume of the bodies can be only a useless re- 
establishment of equilibrium in the caloric.* The 
necessary condition of the maximum is, then, that 

* We assume here no chemical action between the bodies 
employed to realize the motive power of heat. The chem- 
ical action which takes place in the furnace is, in some 
sort, a preliminary action, an operation destined not to 
produce immediately motive power, but to destroy the 
equilibrium of the caloric, to produce a difference of tem- 
perature which may finally give rise to motion. 



MOTIVE POWER OF HEAT. 57 

in the bodies employed to realize the motive power 
of heat there should not occur any change of tem- 
perature which may not be due to a change of 
volume. Reciprocally, every time that this condi- 
tion is fulfilled the maximum will be attained. 
This principle should never be lost sight of in the 
construction of heat-engines ; it is its fundamental 
basis. If it cannot be strictly observed, it should 
at least be departed from as little as possible. 

Every change of temperature which is not due 
to a change of volume or to chemical action (an 
action that we provisionally suppose not to occur 
here) is necessarily due to the direct passage of the 
caloric from a more or less heated body to a colder 
body. This passage occurs mainly by the contact 
of bodies of different temperatures; hence such 
contact should be avoided as much as possible. It 
cannot probably be avoided entirely, but it should 
at least be so managed that the bodies brought in 
contact with each other differ as little as possible 
in temperature. When we just now supposed, in 
our demonstration, the caloric of the body A em- 
ployed to form steam, this steam was considered as 
generated at the temperature of the body A ; thus 
the contact took place only between bodies of equal 
temperatures ; the change of temperature occurring 
afterwards in the steam was due to dilatation, con- 



58 MOTIVE POWER OF HEAT. 

sequently to a change of volume. Finally, conden- 
sation took place also without contact of bodies of 
different temperatures. It occurred whiJe exert- 
ing a constant pressure on the steam brought in 
contact with the body B of the same temperature 
as itself. The conditions for a maximum are thus 
found to be fulfilled. In reality the operation 
cannot proceed exactly as we have assumed. To 
determine the passage of caloric from one body to 
another, it is necessary that there should be an 
excess of temperature in the first, but this excess 
may be supposed as slight as we please. We can 
regard it as insensible in theory, without thereby 
destroying the exactness of the arguments. 

A more substantial objection may be made to 
our demonstration, thus : When we borrow caloric 
from the body A to produce steam, and when this 
steam is afterwards condensed by its contact with 
the body B, the water used to form it, and which 
we considered at first as being of the temperature 
of the body A, is found at the close of the opera- 
tion at the temperature of the body B. It has 
become cool. If we wish to begin again an opera- 
tion similar to the first, if we wish to develop a 
new quantity of motive power with the same in- 
strument, with the same steam, it is necessary first 
to re-establish the original condition to restore 



MOTIVE POWER OF HEAT. 59 

the water to the original temperature. This can 
undoubtedly be done by at once putting it again 
in contact with the body A ; but there is then 
contact between bodies of different temperatures, 
and loss of motive power.* It would be impossi- 
ble to execute the inverse operation, that is, to 
return to the body A the caloric employed to raise 
the temperature of the liquid. 

This difficulty may be removed by supposing the 
difference of temperature between the body A and 
the body B indefinitely small. The quantity of 
heat necessary to raise the liquid to its former 

* This kind of loss is found in all steam-engines. In 
fact, the water destined to feed the boiler is always cooler 
than the water which it already contains. There occurs 
between them a useless re-establishment of equilibrium of 
caloric. We are easily convinced, a posteriori, that this re- 
establishment of equilibrium causes a loss of motive power 
if we reflect that it would have been possible to previously 
heat the feed-water by using it as condensing-water in a 
small accessory engine, when the steam drawn from the 
large boiler might have been used, and where the conden- 
sation might be produced at. a temperature intermediate 
between that of the boiler and that of the principal con- 
denser. The power produced by the small engine would 
have cost no loss of heat, since all that which had been 
used would have returned into the boiler with the water of 
condensation. 



60 MOTIVE POWER OF HEAT. 

temperature will be also indefinitely small and un- 
important relatively to that which is necessary to 
produce steam a quantity always limited. 

The proposition found elsewhere demonstrated 
for the case in which the difference between the 
temperatures of the two bodies is indefinitely small, 
may be easily extended to the general case. In 
fact, if it operated to produce motive power by the 
passage of caloric from the body A to the body Z, 
the temperature of this latter body being very dif- 
ferent from that of the former, we should imagine 
a series of bodies B, C, D . . . of temperatures 
intermediate between those of the bodies A, Z, 
and selected so that the differences from A to B, 
from B to C, etc., may all be indefinitely small. 
The caloric coming from A would not arrive at Z 
till after it had passed through the bodies B, C, D, 
etc., and after having developed in each of these 
stages maximum motive power. The inverse 
operations would here be entirely possible, and the 
reasoning of page 52 would be strictly applicable. 

According to established principles at the present 
time, we can compare with sufficient accuracy the 
motive power of heat to that of a waterfall. Each 
has a maximum that we cannot exceed, whatever 
may be, on the one hand, the machine which is 
acted upon by the water, and whatever, on the 



MOTIVE POWER OF HEAT. 61 

other hand, the substance acted upon by the heat. 
The motive power of a waterfall depends on its 
height and on the quantity of the liquid; the 
motive power of heat depends also on the quantity 
of caloric used, and on what may be termed, on 
what in fact we will call, the height of its fall,* 
that is to say, the difference of temperature of the 
bodies between which the exchange of caloric is 
made. In the waterfall the motive power is ex- 
actly proportional to the difference of level between 
the higher and lower reservoirs. In the fall of 
caloric the motive power undoubtedly increases 
with the difference of temperature between the 
warm and the cold bodies ; but we do not know 
whether it is proportional to this difference. We 
do not know, for example, whether the fall of ca- 
loric from 100 to 50 degrees furnishes more or less 
motive power than the fall of this same caloric from 
50 to zero. It is a question which we propose to 
examine hereafter. 

We shall give here a second demonstration of 
the fundamental proposition enunciated on page 
56, and present this proposition under a more gen- 
eral form than the one already given. 

* The matter here dealt with being entirely new, we are 
obliged to employ expressions not in use as yet, and which 
perhaps are less clear than is desirable. 



62 MOTIVE POWER OF HEAT. 

When a gaseous fluid is rapidly compressed its 
temperature rises. It falls, on the contrary, when 
it is rapidly dilated. This is one of the facts best 
demonstrated by experiment. We will take it for 
the basis of our demonstration.* 

If, when the temperature of a gas has been 
raised by compression, we wish to reduce it to its 
former temperature without subjecting its volume 
to new changes, some of its caloric must be re- 
moved. This caloric might have been removed in 
proportion as pressure was applied, so that the 
temperature of the gas would remain constant. 
Similarly, if the gas is rarefied we can avoid lower- 
ing the temperature by supplying it with a cer- 
tain quantity of caloric. Let us call the caloric 
employed at such times, when no change of tem- 
perature occurs, caloric due to change of volume. 
This denomination does not indicate that the 
caloric appertains to the volume : it does not ap- 
pertain to it any more than to pressure, and 
might as well be called caloric due to the change 
of pressure. We do not know what laws it 
follows relative to the variations of volume : it is 
possible that its quantity changes either with the 
nature of the gas, its density, or its temperature. Ex- 

* Note 13, Appendix B. 



MOTIVE POWER OF HEAT. 



63 



periment has taught us nothing on this subject. It 
has only shown us that this caloric is developed in 
greater or less quantity by the compression of the 
elastic fluids. 

This preliminary idea being established, let us 
imagine an elastic fluid, atmospheric air for exam- 
ple, shut up in a cylindrical vessel, abed. (Fig. 1), 
provided with a movable dia- 
phragm or piston, cd. Let 
there be also two bodies, A and 
B, kept each at a constant 
temperature, that of A being 
higher than that of B. Let 
us picture to ourselves now 
the series of operations which 
are to be described : 

(1) Contact of the body 
A with the air enclosed in the 
space abed or with the wall 
of this space a wall that we 
will suppose to transmit the 
caloric readily. The air be- 
comes by such contact of the 

same temperature as the body A\ cd is the actual 
position of the piston. 

(2) The piston gradually rises and takes the 
position ef. The body A is all the time in con- 




FlG. 1 



64 MOTIVE POWER OF HEAT. 

tact with the air, which is thus kept at a constant 
temperature during the rarefaction. The body A 
furnishes the caloric necessary to keep the tem- 
perature constant. 

(3) The body A is removed, and the air is then 
no longer in contact with any body capable of fur- 
nishing it with caloric. The piston meanwhile 
continues to move, and passes from the position ef 
to the position gh. The air is rarefied without 
receiving caloric, and its temperature falls. Let 
us imagine that it falls thus till it becomes equal 
to that of the body B\ at this instant the piston 
stops, remaining at the position gh. 

(4) The air is placed in contact with the body 
B\ it is compressed by the return of the piston as 
it is moved from the position gh to the position 
cd. This air remains, however, at a constant 
temperature because of its contact with the , body 
B 9 to which it yields its caloric. 

(5) The body B is removed, and the compres- 
sion of the air is continued, which being then 
isolated, its temperature rises. The compression 
is continued till the air acquires the temperature 
of the body A. The piston passes during this 
time from the position cd to the position ik. 

(6) The air is again placed in contact with the 
body A. The piston returns from the position iJc 



MOTIVE POWER OF HEAT. 65 

to the position ef ; the temperature remains un- 
changed. 

(7) The step described under number 3 is re- 
newed, then successively the steps 4, 5, 6, 3, 4, 5, 
6, 3, 4, 5 ; and so on. 

In these various operations the piston is subject 
to an effort of greater or less magnitude, exerted 
by the air enclosed in the cylinder; the elastic 
force of this air varies as much by reason of the 
changes in volume as of changes of temperature. 
But it should -be remarked that with equal 
volumes, that is, for the similar positions of the 
piston, the temperature is higher during the move- 
ments of dilatation than during the movements of 
compression. During the former the elastic force 
of the air is found to be greater, and consequently 
the quantity of motive power produced by the 
movements of dilatation is more considerable than 
that consumed to produce the movements of com- 
pression.. Thus we should obtain an excess of 
motive power an excess which we could employ 
for any purpose whatever. The air, then, has 
served as a heat-engine ; we have, in fact, employed 
it in the most advantageous manner possible, for 
no useless re-establishment of equilibrium has 
been effected in the caloric. 

All the above-described operations may be 



66 MOTIVE POWER OF HEAT. 

executed in an inverse sense and order. Let us 
imagine that, after the sixth period, that is to say 
the piston having arrived at the position ef, we 
cause it to return to the position ik, and that at 
the same time we keep the air in contact with the 
body A. The caloric furnished by this body 
during the sixth period would return to its source, 
that is, to the body A, and the conditions would 
then become precisely the same as they were at the 
end of the fifth period. If now we take away the 
body A, and if we cause the piston to move from 
ef to cd, the temperature of the air will diminish 
as many degrees as it increased during the fifth 
period, and will become that of the body B. We 
may evidently continue a series of operations the 
inverse of those already described. It is only 
necessary under the same circumstances to exe- 
cute for each period a movement of dilatation 
instead of a movement of compression, and re- 
ciprocally. 

The result of these first operations has been the 
production of a certain quantity of motive power 
and the removal of caloric from the body A to the 
body B. The result of the inverse operations is 
the consumption of the motive power produced and 
the return of the caloric from the body B to the 
body A ; so that these two series of operations annul 



MOTIVE POWER OF HEAT. 67 

each other, after a fashion, one neutralizing the 
other. 

The impossibility of making the caloric produce 
a greater quantity of motive power than that which 
we obtained from it by our first series of opera- 
tions, is now easily proved. It is demonstrated by 
reasoning very similar to that employed at page 5G; 
the reasoning will here be even more exact. The 
air which we have used to develop the motive 
power is restored at the end of each cycle of opera- 
tions exactly to the state in which it was at first 
found, while, as we have already remarked, this 
would not be precisely the case with the vapor of 
water.* 

* "We tacitly assume in our demonstration, that when a 
body has experienced any changes, and when after a cer- 
tain number of transformations it returns to precisely its 
original state, that is, to that state considered in respect to 
density, to temperature, to mode of aggregation let us 
suppose, I say, that this body is found to contain the same 
quantity of heat that it contained at first, or else that the 
quantities of heat absorbed or set free in these different 
transformations are exactly compensated. This fact has 
never been culled in question. It was first admitted with- 
out reflection, and verified afterwards in many cases by 
experiments with the calorimeter. To deny it would be 
to overthrow the whole theory of heat to which it serves 
as a basis. For the rest, we may say in passing, the main 



68 MOTIVE POWER OF HEAT. 

We have chosen atmospheric air as the instru- 
ment which should develop the motive power of 
heat, but it is evident that the reasoning would 
have heen the same for all other gaseous substances, 
and even for all other bodies susceptible of change 
of temperature through successive contractions and 
dilatations, which comprehends all natural sub- 
stances, or at least all those which are adapted to 
realize the motive power of heat. Thus we are led 
to establish this general proposition : 

The motive power of heat is independent of the 
agents employed to realize it ; its quantity is fixed 
solely by the temperatures of the bodies between 
which is effected, finally, the transfer of the caloric. 

We must understand here that each of the 
methods of developing motive power attains the 
perfection of which it is susceptible. This condi- 
tion is found to be fulfilled if, as we remarked 
above, there is produced in the body no other 
change of temperature than that due to change of 
volume, or, what is the same thing in other words, 
if there is no contact between bodies of sensibly 
different temperatures. 

Different methods of realizing motive power may 

principles on which the theory of heat rests require the 
most careful examination. Many experimental facts ap- 
pear almost inexplicable in the present state of this theory. 



MOTIVE POWER OF HEAT. 69 

be taken, as in the employment of different sub- 
stances, or in the use of the same substance in two 
different states for example, of a gas at two dif- 
ferent densities. 

This leads us naturally to those interesting re- 
searches on the aeriform fluids researches which 
lead us also to new results in regard to the motive 
power of heat, and give us the means of verifying, 
in some particular cases, the fundamental proposi- 
tion above stated.* 

We readily see that our demonstration would 
have been simplified by supposing the temperatures 
of the bodies A and B to differ very little. Then 
the movements of the piston being slight during 
the periods 3 and 5, these periods might have been 
suppressed without influencing sensibly the pro- 
duction of motive power. A very little change of 
volume should suffice in fact to produce a very 
slight change of temperature, and this slight change 
of volume may be neglected in presence of that of 
the periods 4 and 6, of which the extent is unlim- 
ited. 

If we suppress periods 3 and 5, in the series of 

* We will suppose, in what follows, the reader to be au 
courant with the later progress of modern Physics in re- 
gard to gaseous substances and heat. 



70 



MOTIVE POWER OF HEAT. 



operations above described, it is reduced to the fol- 
lowing : 

(1) Contact of the gas confined in abed (Fig. 2) 
with the body A, passage of the piston from cd to ef. 








f_ e[ 


_ 









d c' 
















I 

FlQ 


1 

. 2. 


> < 


(/ 

FlQ 


6 

. 3. 



(2) Eemoval of the body A, contact of the gas 
confined in abef with the body B, return of the 
piston from efto cd. 

(3) Removal of the body B, contact of the gas 
with the body A, passage of the piston from cd to 
ef, that is, repetition of the first period, and so on. 

The motive power resulting from the ensemble 
of operations 1 and 2 will evidently be the differ- 
ence between that which is produced by the expan- 
sion of the gas while it is at the temperature of the 
body A, and that which is consumed to compress 
this gas while it is at the temperature of the 
body B. 



MOTIVE POWER OF HEAT. 71 

Let us suppose that operations 1 and 2 be per- 
formed on two gases of different chemical natures 
but under the same pressure under atmospheric 
pressure, for example. These two gases will be- 
have exactly alike under the same circumstances, 
that is, their expansive forces, originally equal, 
will remain always equal, whatever may be the 
variations of volume and of temperature, provided 
these variations are the same in both. This results 
obviously from the laws of Mariotte and MM. Gay- 
Lussac and Dalton laws common to all elastic 
fluids, and in virtue of which the same relations 
exist for all these fluids between the volume, the 
expansive force, and the temperature. 

Since two different gases at the same tempera- 
ture and under the same pressure should behave 
alike under the same circumstances, if we subjected 
them both to the operations above described, they 
should give rise to equal quantities of motive power. 

Now this implies, according to the fundamental 
proposition that we have established, the employ- 
ment of two equal quantities of caloric; that is, it 
implies that the quantity of caloric transferred from 
the body A to the body B is the same, whichever 
gas is used. 

The quantity of caloric transferred from the 
body A to the body B is evidently that which is 



72 MOTIVE POWER OF HEAT. 

absorbed by the gas in its expansion of volume, or 
that which this gas relinquishes during compres- 
sion. We are led, then, to establish the following 
proposition : 

When a gas passes without change of tempera- 
ture from one definite volume and pressure to an- 
other volume and another pressure equally definite, 
the quantity of caloric absorbed or relinquished is 
always the same, ivhatever may be the nature of 
the gas chosen as the subject of the experiment. 

Take, for example, 1 litre of air at the tempera- 
ture of 100 and under the pressure of one atmos- 
phere. If we double the volume of this air and 
wish to maintain it at the temperature of 100, a 
certain quantity of heat must be supplied to it. 
Now this quantity will be precisely the same if, 
instead of operating on the air, we operate upon 
carbonic-acid gas, upon nitrogen, upon hydrogen, 
upon vapor of water or of alcohol, that is, if we 
double the volume of 1 litre of these gases taken at 
the temperature of 100 and under atmospheric 
pressure. 

It will be the same thing in the inverse sense if, 
Instead of doubling the volume of gas, we reduce 
it one half by compression. The quantity of heat 
that the elastic fluids set free or absorb in their 
changes of volume has never been measured by 



MOTIVE POWER OF HEAT, 73 

any direct experiment, and doubtless such an ex- 
periment would be very difficult, but there exists a 
datum which is very nearly its equivalent. This 
has been furnished by the theory of sound. It de- 
serves much confidence because of the exactness of 
the conditions which have led to its establishment. 
It consists in this : 

Atmospheric air should rise one degree Centi- 
grade when by sudden compression it experiences 
a reduction of volume of T fg-.* 

Experiments on the velocity of sound having 
been made in air under the pressure of 760 milli- 
metres of mercury and at the temperature of 6, 
it is only to these two circumstances that our 
datum has reference. We will, however, for greater 
facility, refer it to the temperature 0, which is 
nearly the same. 

Air compressed T fg-, and thus heated one degree, 
differs from air heated directly one degree only in 
its density. The primitive volume being supposed 



* M. Poisson, to whom this figure is due, has shown 
that it accords very well with the result of an experiment 
of MM. Clement and Desormes on the return of air into a 
vacuum, or rather, into air slightly rarefied. It also ac- 
cords very nearly with results found by MM. Gay-Lussaq 
and Welter. (See note, p. 87.) 



74 MOTIVE POWER OF HEAT. 

to be V, the compression of TTT reduces it to 
V-j^V. 

Direct heating under constant pressure should, 
according to the rule of M. Gay-Lussac, increase 
the volume of air T above what it would be at : 
so the air is, on the one hand, reduced to the vol- 
ume V TT7 F; on the other, it is increased to 



_ 

The difference between the quantities of heat 
which the air possesses in both cases is evidently 
the quantity employed to raise it directly one de- 
gree; so then the quantity of heat that the air 
would absorb in passing from the volume V -- T } T V 
to the volume F -\- ^V is equal to that which 
is required to raise it one degree. 

Let us suppose now that, instead of heating one 
degree the air subjected to a constant pressure and 
able to dilate freely, we inclose it within an invari- 
able space, and that in this condition we cause it 
to rise one degree in temperature. The air thus 
heated one degree will differ from the air com- 
pressed TT ^- only by its 1T -g- greater volume. So 
then the quantity of heat that the air would set 
free by a reduction of volume of yir is equal to 
that which would be required to raise it one degree 
Centigrade under constant volume. As the differ- 
ences between the volumes F T |-g F, F, and 



MOTIVE POWER OF HEAT. 75 

V -f- J T V are small relatively to the volumes 
themselves, we may regard the quantities of heat 
absorbed by the air in passing from the first of 
these volumes to the second, and from the first to 
the third, as sensibly proportional to the changes 
of volume. We are then led to the establishment 
of the following relation : 

The quantity of heat necessary to raise one de- 
gree air under constant pressure is to the quantity 
of heat necessary to raise one degree the same air 
under constant volume, in the ratio of the numbers 

rhr + irh- to TIT; 

or, multiplying both by 116 X 267, in the ratio of 
the numbers 267 + 116 to 267. 

This, then, is the ratio which exists between the 
capacity of air for heat under constant pressure 
and its capacity under constant volume. If the 
first of these two capacities is expressed by unity, 
the other will be expressed by the number 267 + 7 116 , 
or very nearly 0.700; their difference, 1 0.700 or 
0.300, will evidently express the quantity of heat 
which will produce the increase of volume in the 
air when it is heated one degree under constant 
pressure. 

According to the law of MM. Gay-Lussac and 
JDalton, this increase of volume would be the same 



76 



MOTIVE POWER OF HEAT. 



for all other gases; according to the theory demon- 
strated on page 87, the heat absorbed by these equal 
increases of volume is the same for all the elastic 
fluids, which leads to the establishment of the fol- 
lowing proposition : 

The difference between specific heat under con- 
stant pressure and specific heat under constant 
volume is the same for all gases. 

It should be remarked here that all the gases 
are considered as taken under the same pressure, 
atmospheric pressure for example, and that the 
specific heats are also measured with reference to 
the volumes. 

It is a very easy matter now for us to prepare a 
table of the specific heat of gases under constant 
volume, from the knowledge of their specific heats 
under constant pressure. Here is the table : 
TABLE OF THE SPECIFIC HEAT OF GASES. 



NAMES OF GASES. 


Specific Heat 
under 
Const. Press. 


Specific Heat 
at 
Const. Vol. 


Atmospheric Air, .... 


1.000 


0.700 


Hydrogen Gas, 
Carbonic Acid, 


0.903 
1.258 


0.603 
0.958 




0.976 


0.676 


Nitrosren . 


1 000 


700 


Protoxide of Nitrogen, . . 
Olefiant Gas . . 


1.350 
1.553 


1.050 
1.253 


Oxide of Carbon, .... 


1.034 


0.734 









MOTIVE POWER OF HEAT. 77 

The first column is the result of the direct 
experiments of MM. Delaroche and Berard on the 
specific heat of the gas under atmospheric pressure, 
and the second column is composed of the numbers 
of the first diminished by 0.300. 

The numbers of the first column and those of 
the second are here referred to the same unit, to 
the specific heat of atmospheric air under constant 
pressure. 

The difference between each number of the first 
column and the corresponding number of the sec- 
ond being constant, the relation between these 
numbers should be variable. Thus the relation 
between the specific heat of gases under constant 
pressure and the specific heat at constant volume, 
varies in different gases. 

We have seen that air when it is subjected to a 
sudden compression of T fg- of its volume rises one 
degree in temperature. The other gases through 
a similar compression should also rise in tempera- 
ture. They should rise, but not equally, in inverse 
ratio with their specific heat at constant volume. 
In fact, the reduction of volume being by hypothe- 
sis always the same, the quantity of heat due to 
this reduction should likewise be always the same, 
and consequently should produce an elevation of 
temperature dependent only on the specific heat 



78 MOTIVE POWER OF HEAT. 

acquired by the gas after its compression, and 
evidently in inverse ratio with this specific heat. 
Thus we can easily form the table of the elevations 
of temperature of the different gases for a compres- 
sion of yfg-. 

TABLE OF THE ELEVATION OP TEMPERATURE 

OF 

Oases through the Effect of Compression. 



NAMES OP GASES. 


Elevation of Temperature 
for a Reduction of 
Volume of y^. 




1.000 




1.160 




0.730 




1.035 


Nitrogen, 


1.000 


Protoxide of Nitrogen, . . . 
Olefiant Gas ...... 


0.667 
0.558 




0.955 







A second compression of T |^- (of the altered vol- 
ume), as we shall presently see, would also raise the 
temperature of these gases nearly as much as the 
first; but it would not be the same with a third, a 
fourth, a hundredth such compression. The capac- 
ity of gases for heat changes with their volume. 
It is not unlikely that it changes also with the 
temperature. 

We shall now deduce from the general proposi- 



MOTIVE POWER OF HEAT. 79 

tion stated on page 68 a second theory, which will 
serve as a corollary to that just demonstrated. 

Let us suppose that the gas enclosed in the 
cylindrical space abed (Fig. 2) be transported into 
the space a'b'c'd' (Fig. 3) of equal height, but of 
different base and wider. This gas would increase 
in volume, would diminish in density and in elastic 
force, in the inverse ratio of the two volumes abed, 
a'b'c'd'. As to the total pressure exerted in each 
piston cd, c'd', it would be the same from all quar- 
ters, for the surface of these pistons is in direct 
ratio to the volumes. 

Let us suppose that we perform on the gas in- 
closed in a'b'c'd' the operations described on page 
70, and which were taken as having been performed 
upon the gas inclosed in abed', that is, let us sup- 
pose that we have given to the piston c'd' motions 
equal to those of the piston cd, that we have made 
it occupy successively the positions c'd' correspond- 
ing to cd, and e'f corresponding to ef, and that at 
the same time we have subjected the gas by means 
of the two bodies A and B to the same variations 
of temperature as when it was inclosed in abed 
The total effort exercised on the piston would be 
found to be, in the two cases, always the same at 
the corresponding instants. This results solely from 



Of r> 



80 MOTIVE POWER OF SEAT. 

the l&w <vf Mariotte.* In fact, the densities of the 
two gases maintaining always the same ratio for 
similar positions of the pistons, and the tempera- 
tures being always equal in both, the total pressures 
exercised on the pistons will always maintain the 
same ratio to each other. If this ratio is, at any 
instant whatever, unity, the pressures will always 
be equal. 

As, furthermore, the movements of the two pis- 
tons have equal extent, the motive power produced 
by each will evidently be the same; whence we 
should conclude, according to the proposition on 

* The law of Mariotte, which is here made the founda- 
tion upon which to establish our demonstration, is one of 
the best authenticated physical laws. It has served as a 
basis to many theories verified by experience, and which 
in turn verify all the laws on which they are founded. 
We can cite also, as a valuable verification of Mariotte's 
law and also of that of MM. Gay-Lussac and Dalton, for a 
great difference of temperature, the experiments of MM. 
Dulong and Petit. (See Annales de CMmie el de Physique, 
Feb. 1818, t. vii. p. 122.) 

The more recent experiments of Davy and Faraday can 
also be cited. 

The theories that we deduce here would not perhaps be 
exact if applied outside of certain limits either of density 
or temperature. They should be regarded as true only 
within the limits in which the laws of Mariotte and of 
MM. Gay-Lussac and Dalton are themselves proven. 



MOTIVE POWER OF HEAT. 81 

page 68, that the quantities of heat consumed by 
each are the same, that is, that there passes from 
the body A to the body B the same quantity of 
heat in both cases. 

The heat abstracted from the body A and com- 
municated to the body B, is simply the heat ab- 
sorbed during the rarefaction of the gas, and after- 
wards liberated by its compression. We are therefore 
led to establish the following theorem : 

When an elastic fluid passes without change of 
temperature from the volume U to the volume V, 
and when a similar ponderable quantity of the 
same gas passes at the same temperature from the 
volume V to the volume V, if the ratio of U' to 
V is found to be the same as the ratio of U to V, 
the quantities of heat absorbed or disengaged in 
the two cases will be equal. 

This theorem might also be expressed as follows : 

When a gas varies in volume without change of 
temperature, the quantities of heat absorbed or 
liberated by this gas are in arithmetical progres- 
sion, if the increments or the decrements of volume 
are found to be in geometrical progression. 

When a litre of air maintained at a temperature 
of ten degrees is compressed, and when it is re- 
duced to one half a litre, a certain quantity of 
heat is set free. This quantity will be found always 



32 MOTIVE POWER OF HEAT. 

the same if the volume is further reduced from a 
half litre to a quarter litre, from a quarter litre to 
an eighth, and so on. 

If, instead of compressing the air, we carry it 
successively to two litres, four litres, eight litres, 
etc., it will be necessary to supply to it always equal 
quantities of heat in order to maintain a constant 
temperature. 

This readily accounts for the high temperature 
attained by air when rapidly compressed. We 
know that this temperature inflames tinder and 
even makes air luminous. If, for a moment, we 
suppose the specific heat of air to be constant, in 
spite of the changes of volume and temperature, 
the temperature will increase in arithmetical pro- 
gression for reduction of volume in geometrical 
progression. 

Starting from this datum, and admitting that 
one degree of elevation in the temperature cor- 
responds to a compression of T -\- , we shall readily 
come to the conclusion that air reduced to -fa of 
its primitive volume should rise in temperature 
about 300 degrees, which is sufficient to inflame 
tinder.* 

* When the volume is reduced TT ^, that is, when it 
becomes yyf of what it was at first, the temperature rises 
one degree. Another reduction of TT ^ carries the volume 



MOTIVE POWER OF HEAT. 83 

The elevation of temperature ought, evidently, 
to be still more considerable if the capacity of the 
air for heat becomes less as its volume diminishes. 
Now this is probable, and it also seems to follow 
from the experiments of MM. Delaroche and 
Berard on the specific heat of air taken at different 
densities. (See the Memoire in the Annales de 
Chimie, t. Ixxxv. pp. 72, 224.) 

The two theorems explained on pp. 72 and 81 
suffice for the comparison of the quantities of heat 
absorbed or set free in the changes of volume of 
elastic fluids, whatever may be the density and the 
chemical nature of these fluids, provided always 

to (Hf) a and the temperature should rise another degree. 
After x similar reductions the volume becomes (HI) 37 ' an d 
the temperature should be raised x degrees. If we suppose 
({{l) x T^, and if we take the logarithms of both, we find 

x - about 300. 
If we suppose (Hf) x = i> we find 

ar=80; 
which shows that air compressed one half rises 80. 

All this is subject to the hypothesis that the specific heat 
of air does not change, although the volume diminishes. 
But if, for the reasons hereafter given (pp. 86, 89), we re- 
gard the specific heat of air compressed one half as 
reduced in the relation of 700 to 616, the number 80 must 
be multiplied by |ff, which raises it to 90. 



84 MOTIVE POWER OF HEAT. 

that they be taken and maintained at a certain in- 
variable temperature. But these theories furnish 
no means of comparing the quantities of heat liber- 
ated or absorbed by elastic fluids which change in 
volume at different temperatures. Thus we are 
ignorant what relation exists between the heat re- 
linquished by a litre of air reduced one half, the 
temperature being kept at zero, and the heat relin- 
quished by the same litre of air reduced one half, 
the temperature being kept at 100. The knowl- 
edge of this relation is closely connected with that 
of the specific heat of gases at various temperatures, 
and to some other data that Physics as yet does not 
supply. 

The second of our theorems offers us a means of 
determining according to what law the specific 
heat of gases varies with their density. 

Let us suppose that the operations described on 
p. 70, instead of being performed with two bodies, 
A, B, of temperatures differing indefinitely small, 
were carried on with two bodies whose tempera- 
tures differ by a finite quantity one degree, for 
example. In a complete circle of operations the 
body A furnishes to the elastic fluid a certain quan- 
tity of heat, which may be divided into two por- 
tions : (1) That which is necessary to maintain the 
temperature of the fluid constant during dilata- 



MOTIVE POWER OF HEAT. 85 

tion; (2) that which is necessary to restore the tem- 
perature of the fluid from that of the body B to 
that of the body A, when, after having brought 
back this fluid to its primitive volume, we place it 
again in contact with the body A. Let us call the 
first of these quantities a and the second ~b. The 
total caloric furnished by the body A will be ex- 
pressed by a -\- b. 

The caloric transmitted by the fluid to the body 
B may also be divided into two parts : one, Z>', due 
to the cooling of the gas by the body B ; the other, 
a', which the gas abandons as a result of its re- 
duction of volume. The sum of these two quanti- 
ties is a' -j- V ') it should be equal to a -j- #, for, 
after a complete cycle of operations, the gas is 
brought back exactly to its primitive state. It has 
been obliged to give up all the caloric which has 
first been furnished to it. We have then 

a+ b = a' + b'; 
or rather, 

a - a' = V - I. 

Now, according to the theorem given on page 81, 
the quantities a and a' are independent of the den- 
sity of the gas, provided always that the ponderable 
quantity remains the same and that the variations 
of volume be proportional to the original volume. 



86 MOTIVE POWER OF HEAT. 

The difference a a' should fulfil the same condi- 
tions, and consequently also the difference V b, 
which is equal to it. But b' is the caloric neces- 
sary to raise the gas enclosed in abed (Fig. 2) one de- 
gree ; b' is the caloric surrendered by the gas when, 
enclosed in abcf, it is cooled one degree. These 
quantities may serve as a measure for specific heats. 
We are then led to the establishment of the follow- 
ing proposition: 

The client ge in the specific heat of a gas caused 
by change of volume depends entirely on the ratio 
between the original volume and the altered volume. 
That is, the difference of the specific heats does not 
depend on the absolute magnitude of the volumes, 
but only on their ratio. 

This proposition might also be differently ex- 
pressed, thus: 

When a gas increases in volume in geometrical 
progression, its specific heat increases in arith- 
metical progression. 

Thus, a being the specific heat of air taken at a 
given density, and a -\- h the specific heat for a 
density one half less, it will be, for a density equal 
to one quarter, a -f- 2h; for a density equal to one 
eighth, a -f- 37^ ; and so on. 

The specific heats are here taken with reference 
to weight. They are supposed to be taken at an 



MOTIVE POWER OF HEAT. 87 

invariable volume, but, ; s we shall see, they would 
follow the same law if they were taken under con- 
stant pressure. 

To what cause is the difference between specific 
heats at constant volume and at constant pressure 
really due ? To the caloric required to produce in 
the second case increase of volume. Now, accord- 
ing to the law of Mariotte, increase of volume of a 
gas should be, for a given change of temperature, 
a determined fraction of the original volume, a . 
fraction independent of pressure. According to 
the theorem expressed on page 76, if the ratio be- 
tween the primitive volume and the altered volume 
is given, that determines the heat necessary to pro- 
duce increase of volume. It depends solely on this 
ratio and on the weight of the gas. We must then 
conclude that : 

The difference between specific heat at constant 
pressure and specific heat at constant volume is 
alivays the same, whatever may be the density of the 
gas, provided the weight remains the same. 

These specific heats both increase accordingly as 
the density of the gas diminishes, but their differ- 
ence does not vary.* 

*MM. Gay-Lussac and Welter have found by direct 
experiments, cited in the Mecanique Celeste and in the 
Annales de Chimie et de Physique, July, 1822, p. 267, that 



88 MOTIVE POWEH OF HEAT. 

Since the difference between the two capacities 
for heat is constant, if one increases in arithmetical 
progression the other should follow a similar pro- 
gression: thus one law is applicable to specific 
heats at constant pressure. 

We have tacitly assumed the increase of specific 
heat with that of volume. This increase is indi- 
cated by the experiments of MM. Delaroche and 
Berard: in fact these physicists have found 0.967 
for the specific heat of air under the pressure of 

the ratio between the specific heat at constant pressure and 
the specific heat at constant volume varies very little with 
the density of the gas. According to what we have just 
seen, the difference should remain constant, and not the 
ratio. As, further, the specific heat of gases for a given 
weight varies very little with the density, it is evident that 
the ratio itself experiences but slight changes. 

The ratio between the specific heat of atmospheric air at 
constant pressure and at constant volume is, according 
to MM. Gay-Lussac and Welter, 1.3748, a number almost 
constant for all pressures, and even for all temperatures. 
We have come, through other considerations, to the number 
^_11 6 = 1.44, which differs from the former ^, and we 
have used this number to prepare a table of the specific 
heats of gases at constant volume. So we need not regard 
this table as very exact, any more than the table given on 
p. 89. These tables are mainly intended to demonstrate 
the laws governing specific heats of aeriform fluids. 



MOTIVE POWEH OF HEAT. 



1 metre of mercury (see Memoire already cited), 
taking for the unit the specific heat of the same 
weight of air under the pressure of O m .7GO. 

According to the law that specific heats follow 
with relation to pressures, it is only necessary to 
have observed them in two particular cases to 
deduce them in all possible cases : it is thus that, 
making use of the experimental result of MM. 
Delaroche and Berard which has just been given, 
we have prepared the following table of the specific 
heat of air under different pressures: 

SPECIFIC HEAT OF Am. 



Pressure in 


Specific Heat, 
that of Air under 


Pressure in 


Specific Heat, 
that of Air under 


Atmospheres. 


Atmospheric Pres- 
sure being 1. 


Atmospheres. 


Atmospheric Pres- 
sure being 1. 


TTFUT 


1.840 


1 


1.000 


5 \~% 


1.756 


2 


0.916 


'ET>'?f 


1.672 


4 


0.832 




1.588 


8 


0.748 


w 


1.504 


16 


0.664 


S 


1.420 


32 


0.580 


TV 


1.336 


64 


0.496 




1.252 


128 


0.412 


1 


1.165 


256 


0.328 


1 


1.084 


512 


0.244 


1 


1.000 


1024 


0.160 



The first column is, as we see, a geometrical 
progression, and the second an arithmetical pro- 
gression. 



90 MOTIVE POWER OF HEAT. 

We have carried out the table to the extremes 
of compression and rarefaction. It may be be- 
lieved that air would be liquefied before acquiring 
a density 1024 times its normal density, that is, 
before becoming more dense than water. The 
specific heat would become zero and even negative 
on extending the table beyond the last term. We 
think, furthermore, that the figures of the second 
column here decrease too rapidly. The experi- 
ments which serve as a basis for our calculation 
have been made within too contracted limits for us 
to expect great exactness in the figures which we 
have obtained, especially in the outside numbers. 

Since we know, on the one hand, the law ac- 
cording to which heat is disengaged in the com- 
pression of gases, and on the other, the law accord- 
ing to which specific heat varies with volume, it 
will be easy for us to calculate the increase of tem- 
perature of a gas that has been compressed with- 
out being allowed to lose heat. In fact, the com- 
pression may be considered as composed of two 
successive operations : (1) compression at a con- 
stant temperature ; (2) restoration of the caloric 
emitted. The temperature will rise through the 
second operation in inverse ratio with the specific 
heat acquired by the gas after the reduction of 
volume, specific heat that we are able to calculate 



MOTIVE POWER OF HEAT. 91 

by means of the law demonstrated above. The 
heat set free by compression, according to the 
theorem of page 81, ought to be represented by an 
expression of the form 

s = A + B log v, 

s being this heat, v the volume of the gas after 
compression, A and B arbitrary constants depen- 
dent on the primitive volume of the gas, on its 
pressure, and on the units chosen. 

The specific heat varying with the volume ac- 
cording to the law just demonstrated, should be 
represented by an expression of the form 

z = A' + B' log v, 

A' and B' being the different arbitrary constants 
of A and B. 

The increase of temperature acquired by the 
gas, as the effect of compression, is proportional to 

the ratio - or to the relation .,,,, It 
z A' + B' log v 

can be represented by this ratio itself; thus, calling 
it t, we shall have 

A +B logv 
~ A' + H'Iogv 

If the original volume of the gas is 1, and the 
original temperature zero, we shall have at the 



92 MOTIVE POWEH OF HEAT. 

same time t = 0, log v = 0, whence A . = ; ^ will 
then express not only the increase of temperature, 
but the temperature itself above the thermometric 
zero. 

We need not consider the formula that we have 
just given as applicable to very great changes in 
the volume of gases. We have regarded the ele- 
vation of temperature as being in inverse ratio to 
the specific heat; which tacitly supposes the specific 
heat to be constant at all temperatures. Great 
changes of volume lead to great changes of tem- 
perature in the gas, and nothing proves the con- 
stancy of specific heat at different temperatures, 
especially at temperatures widely separated. This 
constancy is only an hypothesis admitted for gases 
by analogy, to a certain extent verified for solid 
bodies and liquids throughout a part of the ther- 
mometric scale, but of which the experiments of 
MM. Dulong and Petit have shown the inaccuracy 
when it is desirable to extend it to temperatures 
far above 100.* 

According to a law of MM. Clement and De- 
sormes, a law established by direct experiment, the 
vapor of water, under whatever pressure it may 
be formed, contains always, at equal weights, the 

* Note C, Appendix B. 



MOTIVE POWER OF HEAT. 93 

same quantity of heat; which leads to the assertion 
that steam, compressed or expanded mechanically 
without loss of heat, will always be found in a 
saturated state if it was so produced in the first 
place. The vapor of water so made may then be 
regarded as a permanent gas, and should observe 
all the laws of one. Consequently the formula 

A + B log v 
~ A' + B' log v 

should be applicable to it, and be found to accord 
with the table of tensions derived from the direct 
experiments of M. Dalton. 

We may be assured, in fact, that our formula, 
with a convenient determination of arbitrary con- 
stants, represents very closely the results of experi- 
ment. The slight irregularities which we find 
therein do not exceed what we might reasonably 
attribute to errors of observation.* 

We will return, however, to our principal sub- 
ject, from which we have wandered too far the 
motive power of heat. 

We have shown that the quantity of motive 
power developed by the transfer of caloric from 
one body to another depends essentially upon the 
temperature of the two bodies, but we have not 

* Note D, Appendix B. 



94 MOTIVE POWER OF HEAT. 

shown the relation between these temperatures and 
the quantities of motive power produced. It would 
at first seem natural enough to suppose that for 
equal differences of temperature the quantities of 
motive power produced are equal ; that is, for ex- 
ample, the passage of a given quantity of caloric 
from a body, A, maintained at 100, to a body, B, 
maintained at 50, should give rise to a quantity of 
motive power equal to that which would be devel- 
oped by the transfer of the same caloric from a 
body, B, at 50, to a body, C, at zero. Such a law 
would doubtless be very remarkable, but we do not 
see sufficient reason for admitting it a priori. We 
will investigate its reality by exact reasoning. 

Let us imagine that the operations described on 
p. 70 be conducted successively on two quantities 
of atmospheric air equal in weight and volume, 
but taken at different temperatures. Let us sup- 
pose, further, the differences of temperature be- 
tween the bodies A and B equal, so these bodies 
would have for example, in one of these cases, the 
temperatures 100 and 100 h (h being indefi- 
nitely small), and in the other 1 and 1 h. The 
quantity of motive power produced is, in each case, 
the difference between that which the gas supplies 
by its dilatation and that which must be expended 
to restore its primitive volume. Now this differ- 



MOTIVE POWER OF HEAT. 95 

ence is the same in both cases, as any one can 
prove by simple reasoning, which it seems un- 
necessary to give here in detail ; hence the motive 
power produced is the same. 

Let us now compare the quantities of heat em- 
ployed in the two cases. In the first, the quantity 
of heat employed is that which the body A fur- 
nishes to the air to maintain it at the temperature 
of 100 during its expansion. In the second, it is 
the quantity of heat which this same body should 
furnish to it, to keep its temperature at one degree 
during an exactly similar change of volume. If 
these two quantities of heat were equal, there 
would evidently result the law that we have already 
assumed. But nothing proves that it is so, and we 
shall find that these quantities are not equal. 

The air that we shall first consider as occupying 
the space abed (Fig. 2), and having 1 degree of 
temperature, can be made to occupy the space abef, 
and to acquire the temperature of 100 degrees by 
two different means: 

(1) We may heat it without changing its vol- 
ume, then expand it, keeping its temperature 
constant. 

(2) We may begin by expanding it, maintaining 
the temperature constant, then heat it, when it 
has acquired its greater volume. 



96 MOTIVE POWER OF HEAT. 

Let a and 1} be the quantities of heat employed 
successively in the first of the two operations, and 
let V and a' be the quantities of heat employed 
successively in the second. As the final result of 
these two operations is the same, the quantities of 
heat employed in both should be equal. We have 

then 

a + b = a' + V, 

whence 

a' - a = b -b'. 

a' is the quantity of heat required to cause the 
gas to rise from 1 to 100 when it occupies the 
space abef. 

a is the quantity of heat required to cause the 
gas to rise from 1 to 100 when it occupies the 
space abed. 

The density of the air is less in the first than in 
the second case, and according to the experiments 
of MM. Delaroche and Berard, already cited on 
page 87, its capacity for heat should be a little 
greater. 

The quantity a' being found to be greater than 
the quantity a, b should be greater than b'. Con- 
sequently, generalizing the proposition, we should 
say: 

The quantity of heat due to the change of volume 
of a gas is greater as the temperature is higher. 



MOTIVE POWER OF HEAT. 97 

Thus, for example, more caloric is necessary to 
maintain at 100 the temperature of a certain 
quantity of air the volume of which is doubled, 
than to maintain at 1 the temperature of this 
same air during a dilatation exactly equal. 

These unequal quantities of heat would produce, 
however, as we have seen, equal quantities of 
motive power for equal fall of caloric taken at dif- 
ferent heights on the thermometric scale; whence 
we draw the following conclusion : 

The fall of caloric produces more motive power at 
inferior than at superior temperatures. 

Thus a given quantity of heat will develop more 
motive power in passing from a body kept at 1 
degree to another maintained at zero, than if these 
two bodies were at the temperature of 101 and 
100. 

The difference, however, should be very slight. 
It would be nothing if the capacity of the air for 
heat remained constant, in spite of changes of 
density. According to the experiments of MM. 
Delaroche and Berard, this capacity varies little 
so little even, that the differences noticed might 
strictly have been attributed to errors of observa- 
tion or to some circumstances of which we have 
failed to take account. 

We are not prepared to determine precisely, 



98 MOTIVE POWER OF HEAT. 

with no more experimental data than we now pos- 
sess, the law according to which the motive power 
of heat varies at different points on the ther mo- 
metric scale. This law is intimately connected 
with that of the variations of the specific heat of 
gases at different temperatures a law which ex- 
periment has not yet made known to us with suffi- 
cient exactness.* 

We will endeavor now to estimate exactly the 
motive power of heat, and in order to verify our 
fundamental proposition, in order to determine 
whether the agent used to realize the motive power 
is really unimportant relatively to the quantity of 
this power, we will select several of them succes- 
sively: atmospheric air, vapor of water, vapor of 
alcohol. 

Let us suppose that we take first atmospheric 
air. The operation will proceed according to the 
method indicated on page 70. We will make the 
following hypotheses : The air is taken under 
atmospheric pressure. The temperature of the 
body A is y^r ^ a degree above zero, that of the 
body B is zero. The difference is, as we see, very 
slight a necessary condition here. 

The increase of volume given to the air in our 

* Note E, Appendix B. 



MOTIVE POWER OF HEAT. 99 

operation will be TI7 + ^T of the primitive vol- 
ume ; this is a very slight increase, absolutely 
speaking, but great relatively to the difference of 
temperature between the bodies A and B. 

The motive power developed by the whole of 
the two operations described (page 70) will be very 
nearly proportional to the increase of volume and 
to the difference between the two pressures exer- 
cised by the air, when it is found at the tempera- 
tures 0.001 and zero. 

This difference is, according to the law of M. 
Gay-Lussac, ^Wo^o ^ * ne elastic force of the gas, 
or very nearly ^g^VinF f * ne atmospheric pressure. 

The atmospheric pressure balances at 10.40 
metres head of water ; wfop$ f this pressure 
equals -g-^VoFo X 10 m .40 of head of water. 

As to the increase of volume, it is, by supposi- 
tion, yj-g- + ^- T of the original volume, that is, of 
the volume occupied by one kilogram of air at 
zero, a volume equal to O mc .77, allowing for the 
specific weight of the air. So then the product, 



will express the motive power developed. This 
] ower is estimated here in cubic metres of water 
raised one metre, 



100 MOTIVE POWER OF HEAT. 

If we carry out the indicated multiplications, we 
find the value of the product to be 0.000000372. 

Let us endeavor now to estimate the quantity of 
heat employed to give this result ; that is, the 
quantity of heat passed from the body A to the 
body B. 

The body A furnishes : 

(1) The heat required to carry the temperature 
of one kilogram of air from zero to 0.001; 

(2) The quantity necessary to maintain at this 
temperature the temperature of the air when it 
experiences a dilatation of 



TTTT ~T 



The first of these quantities of heat being very 
small in comparison with the second, we may dis- 
regard it. The second is, according to the rea- 
soning on page 74, equal to that which would be 
necessary to increase one degree the temperature 
of one kilogram of air subjected to atmospheric 
pressure. 

According to the experiments of MM. Delaroche 
and Berard on the specific heat of gases, that of 
air is, for equal weights, 0.267 that of water. If, 
then, we take for the unit of heat the quantity 
necessary to raise 1 kilogram of water 1 degree, 



MOTIVE POWER OF SEAT. 101 

that which will be required to raise 1 kilogram of 
air 1 degree would have for its value 0.267. Thus 
the quantity of heat furnished by the body A is 

0.267 units. 

This is the heat capable of producing 0.000000372 
units of motive power by its fall from 0.001 to 
zero. 

For a fall a thousand times greater, for a fall of 
one degree, the motive power will be very nearly a 
thousand times the former, or 

0.000372. 

If, now, instead of 0.267 units of heat we employ 
1000 units, the motive power produced will be 
expressed by the proportion 

0.267 1000 , 372 

-, whence x = ^-- = 1.395. 



0.000372 x ' 267 

Thus 1000 units of heat passing from a body 
maintained at the temperature of 1 degree to 
another body maintained at zero would produce, in 
acting upon the air, 

1.395 units of motive power. 

We will now compare this result with that fur- 
nished by the action of heat on the vapor of water, 



102 MOTIVE POWER OF HEAT. 

Let us suppose one kilogram of liquid water en- 
closed in the cylindrical vessel abed (Fig. 4), be- 
tween the bottom ab and the piston 
cd. Let us suppose, also, the two 
bodies A, B maintained each at a 
constant temperature, that of A being 
a very little above that of B. Let us 
imagine now the following operations : 
(1) Contact of the water with the 
body A, movement of the piston from 
the position cd to the position ef, for- 
mation of steam at the temperature 
of the body A to fill the vacuum pro- 
duced by the extension of volume. We will sup- 
pose the space abef large enough to contain all the 
water in a state of vapor. 

(2) Removal of the body A, contact of the vapor 
with the body B, precipitation of a part of this 
vapor, diminution of its elastic force, return of 
the piston from ef to ab, liquefaction of the rest of 
the vapor through the effect of the pressure com- 
bined with the contact of the body B. 

(3) Removal of the body B, fresh contact of 
the water with the body A, return of the water 
to the temperature of this body, renewal of the 
former period, and so on. 

The quantity of motive power developed in a 



MOTIVE POWER OF HEAT. 103 

complete cycle of operations is measured by the 
product of the volume of the vapor multiplied by 
the difference between the tensions that it pos- 
sesses at the temperature of the body A and at 
that of the body B. As to the heat employed, 
that is to say, transported from the body A to the 
body B, it is evidently that which was necessary 
to turn the water into vapor, disregarding always 
the small quantity required to restore the tempera- 
ture of the liquid water from that of B to that 
of A. 

Suppose the temperature of the body A 100 de- 
grees, and that of the body .Z? 99 degrees: the 
difference of the tensions will be, according to the 
table of M. Dalton, 26 millimetres of mercury or 
O m .36 head of water. 

The volume of the vapor is 1700 times that of 
the water. If we operate on one kilogram, that 
will be 1700 litres, or l rac .700. 

Thus the value of the motive power developed 
is the product 

1.700 X 0.36 =0.611 units, 

of the kind of which we have previously made use. 

The quantity of heat employed is the quantity 

required to turn into vapor water already heated to 

100. This quantity is found by experiment. "We 



104 MOTIVE POWER OF HEAT. 

have found it equal to 550, or, to speak more 
exactly, to 550 of our units of heat. 

Thus 0.611 units of motive power result from 
the employment of 550 units of heat. The quan- 
tity of motive power resulting from 1000 units of 
heat will be given by the proportion 

550 1000 611 

whence x = -- = 1.112. 



0.611 x 550 

Thus 1000 units of heat transported from one 
body kept at 100 degrees to another kept at 99 
degrees will produce, acting upon vapor of water, 
1.112 units of motive power. 

The number 1.112 differs by about J from the 
number 1.395 previously found for the value of the 
motive power developed by 1000 units of heat acting 
upon the air ; but it should be observed that in this 
case the temperatures of the bodies A and B were 
1 degree and zero, while here they are 100 degrees 
and 99 degrees. The difference is much the same ; 
but it is not found at the same height in the ther- 
mometric scale. To make an exact comparison, it 
would have been necessary to estimate the motive 
power developed by the steam formed at 1 degree 
and condensed at zero. It would also have been 
necessary to know the quantity of heat contained 
in the steam formed at one degree. 



MOTIVE POWER OF HEAT. 105 

The law of MM. Clement and Desormes re- 
ferred to on page 92 gives ns this datum. The 
constituent heat of vapor of water being always the 
same at any temperature at which vaporization 
takes place, if 550 degrees of heat are required to 
vaporize water already brought up to 100 degrees, 
550 -f- 100 or 650 will be required to vaporize the 
same weight of water taken at zero. 

Making use of this datum and reasoning exactly 
as we did for water at 100 degrees, we find, as is 
easily seen, 

1.290 

for the motive power developed by 1000 units of 
heat acting upon the vapor of water between one 
degree and zero. This number approximates more 
closely than the first to 

1.395. 

It differs from it only T *j, an error which does not 
exceed probable limits, considering the great num- 
ber of data of different sorts of which we have 
been obliged to make use in order to arrive at this 
approximation. Thus is our fundamental law veri- 
fied in a special case.* 

* We find (Annales de Chimie et de Physique, July, 1818, 
p. 294) in a memoir of M. Petit an estimate of the motive 
power of heat applied to air and to vapor of water. This 



106 MOTIVE POWER OF HEAT. 

We will examine another case in which vapor of 
alcohol is acted upon by heat. The reasoning is 
precisely the same as for the vapor of water. The 
data alone are changed. Pure alcohol boils under 
ordinary pressure at 78.7 Centigrade. One kilo- 
gram absorbs, according to MM. Delaroche and 
Berard, 207 units of heat in undergoing transfor- 
mation into vapor at this same temperature, 78.7. 

The tension of the vapor of alcohol at one de- 
gree below the boiling-point is found to be dimin- 
ished -gig-. It is 2^ less than the atmospheric 
pressure ; at least, this is the result of the experi- 
ment of M. Betancour reported in the second part 
of V Architecture hydraulique of M. Prony, pp. 
180, 195.* 

If we use these data, we find that, in acting upon 
one kilogram of alcohol at the temperatures of 
78. 7 and 77. 7, the motive power developed will 
be 0.251 units. 

This results from the employment of 207 units 
of heat. For 1000 units the proportion must be 

207 1000 



0.254 



whence x = 1.230. 



estimate leads us to attribute a great advantage to atmos- 
pheric air, but it is derived by a method of considering the 
action of heat which is quite imperfect. 
* Note F, Appendix B. 



MOTIVE POWER OF HEAT. 107 

This number is a little more than the 1.112 re- 
sulting from the use of the vapor of water at tb.e 
temperatures 100 and 99; but if we suppose the 
vapor of water used at the temperatures 78 and 
77, we find, according to the law of MM. Clement 
and Desorme, 1.212 for the motive power due to 
1000 units of heat. This latter number ap- 
proaches, as we see, very nearly to 1.230. There 
is a difference of only ^. 

We should have liked to be able to make other 
approximations of this sort to be able to calculate, 
for example, the motive power developed by the 
action of heat on solids and liquids, by the conge- 
lation of water, and so on; but Physics as yet re- 
fuses us the necessary data * 

The fundamental law that we propose to confirm 
seems to us to require, however, in order to be 
placed beyond doubt, new verifications. It is based 
upon the theory of heat as it is understood to-day, 
and it should be said that this foundation does not 
appear to be of unquestionable solidity. New ex- 
periments alone can decide the question. Mean- 
while we can apply the theoretical ideas expressed 

* Those that we need are the expansive force acquired 
by solids and liquids by a given increase of temperature, 
and the quantity of heat absorbed or relinquished in the 
changes of volume of these bodies. 



108 MOTIVE POWER OF HEAT. 

above, regarding them as exact, to the examination 
of the different methods proposed up tc date, for 
the realization of the motive power of heat. 

It has sometimes been proposed to develop mo- 
tive power by the action of heat on solid bodies. 
The mode of procedure which naturally first occurs 
to the mind is to fasten immovably a solid body 
a metallic bar, for example by one of its extremi- 
ties ; to attach the other extremity to a movable 
part of the machine; then, by successive heating 
and cooling, to cause the length of the bar to vary, 
and so to produce motion. Let us try to decide 
whether this method of developing motive power 
can be advantageous. We have shown that the 
condition of the most effective employment of heat 
in the production of motion is, that all changes 
of temperature occurring in the bodies should be 
due to changes of volume. The nearer we come 
to fulfilling this condition the more fully will the 
heat be utilized. Now, working in the manner 
just described, we are very far from fulfilling this 
condition : change of tempeiYiture is not due here 
to change of volume ; all the changes are due to 
contact of bodies differently heated to the con- 
tact of the metallic bar, either with the body 
charged with furnishing heat to it, or with the 
body charged with carrying it off. 




MOTIVE POWER OF HEAT. 



The only means of fulfilling the prescribed con- 
dition would be to act upon the solid body exactly 
as we did on the air in the operations described on 
page 92. But for this we must be able to pro- 
duce, by a single change of volume of the solid 
body, considerable changes of temperature, that is, 
if we should want to utilize considerable falls of 
caloric. Now this appears impracticable. In 
short, many considerations lead to the conclusion 
that the changes produced in the temperature of 
solid or liquid bodies through the effect of com- 
pression and rarefaction would be but slight. 

(1) We often observe in machines (particularly 
in steam-engines) solid pieces which endure con- 
siderable strain in one way or another, and 
although these efforts may be sometimes as great 
as the nature of the substances employed permits, 
the variations of temperature are scarcely per- 
ceptible. 

(2) In the action of striking medals, in that of the 
rolling-mill, of the draw-plate, the metals undergo 
the greatest compression to which we can submit 
them, employing the hardest and strongest tools. 
Nevertheless the elevation of temperature is not 
great. If it were, the pieces of steel used in these 
operations would soon lose their temper. 

(3) We know that it would be necessary to exert 



110 MOTIVE POWER OF HEAT. 

on solids and liquids a very great strain in order to 
produce in them a reduction of volume comparable 
to that which they experience in cooling (cooling 
from 100 to zero, for example). Now the cooling 
requires a greater abstraction of caloric than would 
simple reduction of volume. If this reduction 
were produced by mechanical means, the heat set 
free would not then be able to make the tempera- 
ture of the body vary as many degrees as the cool- 
ing makes it vary. It would, however, necessitate 
the employment of a force undoubtedly very con- 
siderable. 

Since solid bodies are susceptible of little change 
of temperature through changes of volume, and 
since the condition of the most .effective employ- 
ment of heat for the development of motive power 
is precisely that all change of temperature should be 
due to a change of volume, solid bodies appear but 

111 fitted to realize this power. 

The same remarks apply to liquids. The same 
reasons may be given for rejecting them.* 

We are not speaking now of practical difficulties. 

* The recent experiments of M. Oerstedt on the com- 
pressibility of water have shown that, for a pressure of 
five atmospheres, the temperature of this liquid exhibits 
no appreciable change. (See Annales de Ohimie et de 
Physique, Feb. 1823, p. 192.) 



MOTIVE POWER OF HEAT. Ill 

They will be numberless. The motion produced 
by the dilatation and compression of solid or liquid 
bodies would only be very slight. In order to give 
them sufficient amplitude we should be forced to 
make use of complicated mechanisms. It would 
be necessary to employ materials of the greatest 
strength to transmit enormous pressure ; finally, 
the successive operations would be executed very 
slowly compared to those of the ordinary steam- 
engine, so that apparatus of large dimensions and 
heavy cost would produce but very ordinary re- 
sults. 

The elastic fluids, gases or vapors, are the means 
really adapted to the development of the motive 
power of heat. They combine all the conditions 
necessary to fulfil this office. They are easy to 
compress ; they can be almost infinitely expanded ; 
variations of volume occasion in them great 
changes of temperature; and, lastly, they are very 
mobile, easy to heat and to cool, easy to transport 
from one place to another, which enables them to 
produce rapidly the desired effects. We can easily 
conceive a multitude of machines fitted to develop 
the motive power of heat through the use of 
elastic fluids ; but in whatever way we look at it, 
we should not lose sight of the following prin- 
ciples: 



112 MOTIVE POWER OF HEAT. 

(1) The temperature of the fluid should be made 
as high as possible, in order to obtain a great fall 
of caloric,, and consequently a large production of 
motive power. 

(2) For the same reason the cooling should be 
carried as far as possible. 

(3) It should be so arranged that the passage 
of the elastic fluid from the highest to the lowest 
temperature should be due to increase of volume; 
that is, it should be so arranged that the cooling of 
the gas should occur spontaneously as the effect of 
rarefaction. The limits of the temperature to 
which it is possible to bring the fluid primarily,, 
are simply the limits of the temperature obtainable 
by combustion ; they are very high. 

The limits of cooling are found in the tempera- 
ture of the coldest body of which we can easily and 
freely make use ; this body is usually the water of 
the locality. 

As to the third condition, it involves difficulties 
in the realization of the motive power of heat 
when the attempt is made to take advantage of 
great differences of temperature, to utilize great 
falls of heat. In short, it is necessary then that 
the gas, by reason of its rarefaction, should pass 
from a very high temperature to a very low one, 
which requires a great change of volume and of 



MOTIVE POWER OF HEAT. 113 

density, which requires also that the gas be first 
taken under a very heavy pressure, or that it 
acquire by its dilatation an enormous volume 
conditions both difficult to fulfil. The first neces- 
sitates the employment of very strong vessels to 
contain the gas at a very high temperature and 
under very heavy pressure. The second necessi- 
tates the use of vessels of large dimensions. These 
are, in a word, the principal obstacles which pre- 
vent the utilization in steam-engines of a great 
part of the motive power of the heat. We are 
obliged to limit ourselves to the use of a slight fall 
of caloric, while the combustion of the coal fur- 
nishes the means of procuring a very great one. 

It is seldom that in steam-engines the elastic 
fluid is produced under a higher pressure than six 
atmospheres a pressure corresponding to about 
160 Centigrade, and it is seldom that condensa- 
tion takes place at a temperature much under 40. 
The fall of caloric from 160 to 40 is 120, while 
by combustion we can procure a fall of 1000 to 
2000. 

In order to comprehend this more clearly, let us 
recall what we have termed the fall of caloric. 
This is the passage of the heat from one body, A, 
having an elevated temperature, to another, B, 
where it is lower. We say that the fall of the 



114 MOTIVE POWER OF HEAT. 

caloric is 100 or 1000 when the difference of 
temperature between the bodies A and B is 100 
or 1000. 

In a steam-engine which works under a pressure 
of six atmospheres the temperature of the boiler is 
160. This is the body A. It is kept, by contact 
with the furnace, at the constant temperature of 
160, and continually furnishes the heat necessary 
for the formation of steam. The condenser is the 
body B. By means of a current of cold water it 
is kept at a nearly constant temperature of 40. It 
absorbs continually the caloric brought from the 
body A by the steam. The difference of tempera- 
ture between these two bodies is 160 - 40, or 120. 
Hence we say that the fall of caloric is here 120. 

Coal being capable of producing, by its combus- 
tion, a temperature higher than 1000, and the 
cold water, which is generally used in our climate, 
being at about 10, we can easily procure a fall of 
caloric of 1000, and of this only 120 are utilized 
by steam-engines. Even these 120 are not wholly 
utilized. There is always considerable loss due 
to useless re-establishments of equilibrium in the 
caloric. 

It is easy to see the advantages possessed by 
high-pressure machines over those of lower pres- 
sure. This superiority lies essentially in the power 



MOTIVE POWER OF HEAT. 



115 



of utilizing a greater fall of caloric. The steam 
produced under a higher pressure is found also 
at a higher temperature, and as, further, the 
temperature of condensation remains always about 
the same, it is evident that the fall of caloric is 
more considerable. But to obtain from high-pres- 
sure engines really advantageous results, it is 
necessary that the fall of caloric should be most 
profitably utilized. It is not enough that the steam 
be produced at a high temperature : it is also 
necessary that by the expansion of its volume 
its temperature should become sufficiently low. A 
good steam-engine, therefore, should not only em- 
ploy steam under heavy pressure, but under succes- 
sive and very variable pressures, differ- 
ing greatly from one another, and pro- 
gressively decreasing.* 

In order to understand in some sort 
a posteriori the advantages of high- 
pressure engines, let us suppose steam 
to be formed under atmospheric 
pressure and introduced into the cylin- 
drical vessel abed (Fig. 5), under the 
piston cd, which at first touches the 
bottom ab. The steam, after having FIG. 5. 
moved the piston from ab to cd, will continue 

*Note G, Appendix B. 



116 MOTIVE POWER OF HEAT. 

finally to produce its results in a manner with 
which we will not concern ourselves. 

Let us suppose that the piston having moved to cd 
is forced downward to ef, without the steam being 
allowed to escape, or any portion of its caloric to be 
lost. It will be driven back into the space abef, and 
will increase at the same time in density, elastic 
force, and temperature. If the steam, instead of 
being produced under atmospheric pressure, hud 
been produced just when it was being forced back 
into cibef, and so that after its introduction into the 
cylinder it had made the piston move from ab to 
ef, and had moved it simply by its extension of 
volume, from ef to cd, the motive power produced 
would have been more considerable than in the first 
case. In fact, the movement of the piston, while 
equal in extent, would have taken place under the 
action of a greater pressure, though variable, 
and though progressively decreasing. 

The steam, however, would have required for its 
formation exactly the same quantity of caloric, only 
the caloric would have been employed at a higher 
temperature. 

It is considerations of this nature which have led 
to the making of double-cylinder engines engines 
invented by Mr. Hornblower, improved by Mr. 
Woolf, and which, as regards economy of the com- 



MOTIVE POWER OF HEAT. 117 

bustible, are considered the best. They consist of 
a small cylinder, which at each pulsation is filled 
more or less .(often entirely) with steam, and of a 
second cylinder having usually a capacity quadruple 
that of the first, and which receives no steam ex- 
cept that which has already operated in the first 
cylinder. Thus the steam when it ceases to act 
has at least quadrupled in volume. From the 
second cylinder it is carried directly into the con- 
denser, but it is conceivable that it might be carried 
into a third cylinder quadruple the second, and in 
which its volume would have become sixteen times 
the original volume. The principal obstacle to the 
use of a third cylinder of this sort is the capacity 
which it would be necessary to give it, and the large 
dimensions which the openings for the passage of 
the steam must have. We will say no more on this 
subject, as we do not propose here to enter into the 
details of construction of steam-engines. These 
details call for a work devoted specially to them, 
and which does not yet exist, at least in France.* 

* We find in the work called De la Eichesse Minerals, by 
M. Heron de Villefosse, vol. iii. p. 50 and following, a 
good description of the steam-engines actually in use in 
mining. In England the steam-engine has been very fully 
discussed in the Encyclopedia Britannica. Some of the 
data here employed are drawn from the latter work. 



118 MOTIVE POWER OF HEAT. 

If the expansion of the steam is mainly limited 
by the dimensions of the vessels in which the dila- 
tation must take place, the degree of condensation 
at which it is possible to use it at first is limited 
only by the resistance of the vessels in which it is 
produced, that is, of the boilers. 

In this respect we have by no means attained 
the best possible results. The arrangement of the 
boilers generally in use is entirely faulty, although 
the tension of the steam rarely exceeds from four 
to six atmospheres. They often burst and cause 
severe accidents. It will undoubtedly be possible 
to avoid such accidents, and meantime to raise the 
steam to much greater pressures than is usually 
done. 

Besides the high-pressure double-cylinder en- 
gines of which we have spoken, there are also high- 
pressure engines of one cylinder. The greater part 
of these latter have been constructed by two in- 
genious English engineers, Messrs. Trevithick and 
Vivian. They employ the steam under a very high 
pressure, sometimes eight to ten atmospheres, but 
they have no condenser. The steam, after it has 
been introduced into the cylinder, undergoes 
therein a certain increase of volume, but preserves 
always a pressure higher than atmospheric. When 
it has fulfilled its office it is thrown out into the 



MOTIVE POWER OF HEAT. 119 

atmosphere. It is evident that this mode of work- 
ing is fully equivalent, in respect to the motive 
power produced, to condensing the steam at 100, 
and that a portion of the useful effect is lost. But 
the engines working thus dispense with condenser 
and air-pump. They are less costly than the 
others, less complicated, occupy less space, and can 
be used in places where there is not sufficient water 
for condensation. In such places they are of in- 
estimable advantage, since no others could take 
their place. These engines are principally em- 
ployed in England to move coal-wagons on rail- 
roads laid either in the interior of mines or outside 
of them. 

We have, further, only a few remarks to make 
upon the use of permanent gases and other vapors 
than that of water in the development of the mo- 
tive power of heat. 

Various attempts have been made to produce 
motive power by the action of heat on atmospheric 
air. This gas presents, as compared with vapor of 
water, both advantages and disadvantages, which 
we will proceed to examine. 

(1) It presents, as compared with vapor of water, 
a notable advantage in that, having for equal vol- 
ume a much less capacity for heat, it would cool 
more rapidly by an equal increase of vohime. 



120 MOTIVE POWER OF HEAT. 

(This fact is proved by what has already been 
stated.) Now we have seen how important it is to 
produce by change of volume the greatest possible 
changes of temperature. 

(2) Vapors of water can be formed only through 
the intervention of a boiler, while atmospheric air 
could be heated directly by combustion carried on 
within its own mass. Considerable loss could thus 
be prevented, not only in the quantity of heat, but 
also in its temperature. This advantage belongs 
exclusively to atmospheric air. Other gases do 
not possess it. They would be even more difficult 
to heat than vapor of water. 

(3) In order to give to air great increase of 
volume, and by that expansion to produce a great 
change of temperature, it must first be taken under 
a sufficiently high pressure; then it must be com- 
pressed with a pump or by some other means be- 
fore heating it. This operation would require a 
special apparatus, an apparatus not found in steam- 
engines. In the latter, water is in a liquid state 
when injected into the boiler, and to introduce it 
requires but a small pump. 

(4) The condensing of the vapor by contact with 
the refrigerant body is much more prompt and 
much easier than is the cooling of air. There 
might, of course, be the expedient of throwing the 



MOTIVE POWER OF HEAT, 121 

latter out into the atmosphere, and there would be 
also the advantage of avoiding the use of a refrig- 
erant, which is not always available, but it would be 
requisite that the increase of the volume of the air 
should not reduce its pressure below that_ of the 
atmosphere. 

(5) One of the gravest inconveniences of steam 
is that it cannot be used at high temperatures with- 
out necessitating the use of vessels of extraordinary 
strength. It is not so with air for which there ex- 
ists no necessary relation between the elastic force 
and the temperature. Air, then, would seem more 
suitable than steam to realize the motive power of 
falls of caloric from high temperatures. Perhaps 
in low temperatures steam may be more conven- 
ient. "We might conceive even the possibility of 
making the same heat act successively upon air and 
vapor of water. It would be only necessary that 
the air should have, after its use, an elevated tem- 
perature, and instead of throwing it out immedi- 
ately into the atmosphere, to make it envelop a 
steam-boiler, as if it issued directly from a 
furnace. 

The use of atmospheric air for the development 
of the motive power of heat presents in practice 
very great, but perhaps not insurmountable, diffi- 
culties. If we should succeed in overcoming them, 



122 MOTIVE POWER OF HEAT. 

it would doubtless offer a notable advantage over 
vapor of water.* 

As to the other permanent gases, they should be 
absolutely rejected. They have all the inconven- 
iences of atmospheric air, with none of its advan- 
tages. The same can be said of other vapors than 
that of water, as compared with the latter. 

If we could find an abundant liquid body which 
would vaporize at a higher temperature than water, 
of which the vapor would have, for the same vol- 
ume, a less specific heat, which would not attack 
the metals employed in the construction of ma- 
chines, it would undoubtedly merit the preference. 
But nature provides no such body. 

The use of the vapor of alcohol has been pro- 
posed. Machines have even been constructed for the 
purpose of using it, by avoiding the mixture of ita 
vapor with the water of condensation, that is, by 
applying the cold body externally instead of intro- 
ducing it into the machine. It has been thought 
that a remarkable advantage might be secured by 
using the vapor of alcohol in that it possesses a 
stronger tension than the vapor of water at the 
same temperature. We can see in this only a fresh 
obstacle to be overcome. The principal defect of 

* Note I, Appendix B, 



MOTIVE POWER OF HEAT. 123 

the vapor of water is its excessive tension at an 
elevated temperature ; now this defect exists still 
more strongly in the vapor of alcohol. As to the 
relative advantage in a greater production of mo- 
tive power, an advantage attributed to it, we 
know by the principles above demonstrated that it 
is imaginary. 

It is thus upon the use of atmospheric air and 
vapor of water that subsequent attempts to perfect 
heat-engines should be based. It is to utilize by 
means of these agents the greatest possible falls of 
caloric that all efforts should be directed. 

Finally, we will show how far we are from having 
realized, by any means at present known, all the 
motive power of combustibles. 

One kilogram of carbon burnt in the calorimeter 
furnishes a quantity of heat capable of raising one 
degree Centigrade about 7000 kilograms of water, 
that is, it furnishes 7000 units of heat according to 
the definition of these units given on page 100. 

The greatest fall of caloric attainable is measured 
by the difference between the temperature pro- 
duced by combustion and that of the refrigerant 
bodies. It is difficult to perceive any other limits 
to the temperature of combustion than those in 
which the combination between oxygen and the 
combustible may take place. Let us assume, how- 



124 MOTIVE POWER OF HEAT. 

ever, that 1000 may be this limit, and we shall 
certainly be below the truth. As to the tempera- 
ture of the refrigerant, let us suppose it 0. We 
estimated approximately (page 104) the quantity of 
motive power that 1000 units of heat develop be- 
tween 100 and 99. We found it to be 1. 112 units 
of power, each equal to 1 metre of water raised to 
a height of 1 metre. 

If the motive power were proportional to the 
fall of caloric, if it were the same for each ther- 
mometric degree, nothing would be easier than to 
estimate it from 1000 to 0. Its value would be 

1.112 X 1000 = 1112. 

But as this law is only approximate, and as pos- 
sibly it deviates much from the truth at high tem- 
peratures, we can only make a very rough estimate. 
We will suppose the number 1112 reduced one-half, 
that is, to 560. 

Since a kilogram of carbon produces 7000 units 
of heat, and since the number 560 is relatively 
1000 units, it must be multiplied by 7, which gives 

7 X 560 = 3920. 

This is the motive power of 1 kilogram of carbon. 
In order to compare this theoretical result with 



MOTIVE POWER OF HEAT. 



that of experiment, let us ascertain how much mo- 
tive power a kilogram of carbon actually develops 
in the best-known steam-engines. 

The engines which, up to this time, have shown 
the best results are the large double-cylinder en- 
gines used in the drainage of the tin and copper 
mines of Cornwall. The best results that have 
been obtained with them are as follows : 

65 millions of Ibs. of water have been raised one 
English foot by the bushel of coal burned (the 
bushel weighing 88 Ibs.). This is equivalent to 
raising, by a kilogram of coal, 195 cubic metres of 
water to a height of 1 metre, producing thereby 
195 units of motive power per kilogram of coal 
burned. 

195 units- are only the twentieth of 3920, the 
theoretical maximum ; consequently ^ only of the 
motive power of the combustible has been util- 
ized. 

We have, nevertheless, selected our example from 
among the best steam-engines known. 

Most engines are greatly inferior to these. The 
old engine of Chaillot, for example, raised twenty 
cubic metres of water thirty-three metres, for 
thirty kilograms of coal consumed, which amounts 
to twenty-two units of motive power per kilogram, 
result nine times less than that given above, 



126 MOTIVE POWER OF HEAT. 

and one hundred and eighty times less than the 
theoretical maximum. 

We should not expect ever to utilize in practice 
all the motive power of combustibles. The at- 
tempts made to attain this result would be far more 
hurtful than useful if they caused other important 
considerations to be neglected. The economy of 
the combustible is only one of the conditions to be 
fulfilled in heat-engines. In many cases it is only 
secondary. It should often give precedence to 
safety, to strength, to the durability of the engine, 
to the small space which it must occupy, to small 
cost of installation, etc. To know how to appreciate 
in each case, at their true value, the considerations 
of convenience and economy which may present 
themselves ; to know how to discern the more im- 
portant of those which are only accessories ; to bal- 
ance them properly against each other, in order to 
attain the best results by the simplest means : such 
should be the leading characteristics of the man 
called to direct, to co-ordinate among themselves the 
labors of his comrades, to make them co-operate 
towards one useful end, of whatsoever sort it may 
be. 








(To face p. 127.) 



IV.* 

CARNOT'S THEORY OF THE MOTIVE POWER 
OF HEAT, f 

WITH NUMERICAL RESULTS DEDUCED FROM REGNAULT'S 
EXPERIMENTS ON STEAM. J 

BY SIR WILLIAM THOMSON [LORD KELVIN], 

1. THE presence of heat may be recognized in 
every natural object ; and there is scarcely an 
operation in nature which is not more or less 

* From Transactions of the Edinburgh Royal Society, xiv. 
1849 ; Annales de Chimie, xxxv. 1852. 

f Published in 1824, in a work entitled "Reflexions BUT 
la Puissance Motrice du Feu, et sur les Machines Propres d 
Developer cette Puissance. Par S. Car not." [Note of Nov. 
5, 1881. The original work has now been republished, 
with a biographical notice, Paris, 1878.] 

\ An account of the first part of a series of researches 
undertaken by Mons. Regnault, by order of the late 
French Government, for ascertaining the various physical 
data of importance in the theory of the steam-engine, has 




128 THOMSON ON CARNOT'S 

affected by its all-pervading influence. An evolu- 
tion and subsequent absorption of heat generally 
give rise to a variety of effects ; among which may 
be enumerated, chemical combinations or decom- 
positions ; the fusion of solid substances ; the 
vaporization of solids or liquids ; alterations in the 
dimensions of bodies, or in the statical pressure 
by which their dimensions may be modified ; me- 
chanical resistance overcome ; electrical currents 
generated. In many of the actual phenomena of 
nature several or all of these effects are produced 
together ; and their complication will, if we 
attempt to trace the agency of heat in producing 
any individual effect, give rise to much perplex- 
ity. It will, therefore, be desirable, in laying the 
foundation of a physical theory of a-ny of the 
effects of heat, to discover or to imagine phe- 
nomena free from all such complication, and de- 
pending on a definite thermal agency ; in which 
the relation between the cause and effect, traced 

been recently published (under the title " Relation des 
Experiences," etc.) in the Memoires de I'Institut, of which 
it constitutes the twenty-first volume (1847). The second 
part of these researches has not yet been published. [Note 
of Nov. 5, 1881. The continuation of these researches has 
now been published ; thus we have for the whole series, 
vol. i. in 1847 ; vol. ii. in 1862 ; and vol. iii. in 1870.] 



MOTIVE POWER OF HEAT. 129 

through the medium of certain simple operations, 
may be clearly appreciated. Thus it is that 
Carnot, in accordance with the strictest principles 
of philosophy, enters upon the investigation of the 
theory of the motive power of heat. 

2. The sole effect to be contemplated in inves- 
tigating the motive power of heat is resistance 
overcome, or, as it is frequently called, " work per- 
formed" or " mechanical effect" The questions to 
be resolved by a complete theory of the subject are 
the following: 

(1) What is the precise nature of the thermal 
agency by means of which mechanical effect is to 
be produced, without effects of any other kind? 

(2) How may the amount of this thermal 
agency necessary for performing a given quantity 
of work be estimated? 

3. In the following paper I shall commence by 
giving a short abstract of the reasoning by which 
Carnot is led to an answer to the first of these 
questions ; I shall then explain the investigation 
by which, in accordance with his theory, the ex- 
perimental elements necessary for answering the 
second question are indicated ; and, in conclusion, 
I shall state the data supplied by Regnault's recent 
observations on steam, and apply them to obtain, 
as approximately as the present state of experi- 



130 THOMSON ON CARNOT'8 

mental science enables us to do, a complete solu- 
tion of the question. 

I. On the nature of Thermal agency, considered 
as a motive power. 

4. There are [at present known] two, and only 
two, distinct ways in which mechanical effect can 
be obtained from heat. One of these is by means 
of the alterations of volume, which bodies may ex- 
perience through the action of heat ; the other is 
through the medium of electric agency. Seebeck's 
discovery of thermo-electric currents enables us at 
present to conceive of an electro-magnetic engine 
supplied from a thermal origin, being used as a 
motive power ; but this discovery was not made 
until 1821, and the subject of thermo-electricity 
can only have been generally known in a few iso- 
lated facts, with reference to the electrical effects 
of heat upon certain crystals, at the time when 
Carnot wrote. He makes no allusion to it, but 
confines himself to the method for rendering 
thermal agency available as a source of mechanical 
effect, by means of the expansions and contrac- 
tions of bodies. 

5. A body expanding or contracting under the 
action of force may, in general, either produce 
mechanical effect by overcoming resistance, or re- 
ceive mechanical effect by yielding to the action 



MOTIVE POWER OF HEAT. 131 

of force. The amount of mechanical effect thus 
developed will depend not only on the calorific 
agency concerned, but also on the alteration in the 
physical condition of the body. Hence, after al- 
lowing the volume and temperature of the body to 
change, we must restore it to its original tempera- 
ture and volume; and then we may estimate the 
aggregate amount of mechanical effect developed 
as due solely to the thermal origin. 

6. Now the ordinarily-received, and almost uni- 
versally-acknowledged, principles with reference 
to "quantities of caloric" and "latent heat" lead 
us to conceive that, at the end of a cycle of opera- 
tions, when a body is left in precisely its primitive 
physical condition, if it has absorbed any heat dur- 
ing one part of the operations, it must have given 
out again exactly the same amount during the re- 
mainder of the cycle. The truth of this principle 
is considered as axiomatic by Carnot, who admits 
it as the foundation of his theory ; and expresses 
himself in the following terms regarding it, in a 
note on one of the passages of his treatise :* 

" In our demonstrations we tacitly assume that 
after a body has experienced a certain- number of 
transformations, if it be brought identically to its 

* Carnot, p. 67. 



THOMSON ON CARNOT'8 



primitive physical state as to density, temperature, 
and molecular constitution, it must contain the 
same quantity of heat as that which it initially pos- 
sessed; or, in other words, we suppose that 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." 

7. Since the time when Carnot thus expressed 
himself, the necessity of a most careful examina- 
tion of the entire experimental basis of the theory 
of heat has become more and more urgent. Es- 
pecially all those assumptions depending on the 
idea that heat is a substance, invariable in quan- 
tity; not convertible into any other element, and 
incapable of being generated by any physical 
agency; in fact the acknowledged principles of 
latent heat, would require to be tested by a most 



MOTIVE POWER OF HEAT. 133 

searching investigation before they ought to be 
admitted, as they usually have been, by almost 
every one who has been engaged on the subject, 
whether in combining the results of experimental 
research, or in general theoretical investigations. 

8. The extremely important discoveries recently 
made by Mr. Joule of Manchester, that heat is 
evolved in every part of a closed electric conductor, 
moving in the neighborhood of a magnet,* and 

* The evolution of heat in a fixed conductor, through 
which a galvanic current is sent from any source whatever, 
has long been known to the scientific world ; but it was 
pointed out by Mr. Joule that we cannot infer from any 
previously-published experimental researches, the actual 
generation of heat when the current originates in electro- 
magnetic induction; since the question occurs, is the lieat 
which is evolved in one part of the closed conductor merely 
transferred from tJiose parts which are subject to the inducing 
influence ? Mr. Joule, after a most careful experimental 
investigation with reference to this question, finds that it 
must be answered in the negative. (See a paper "On the 
Calorific Effects of Magneto-Electricity, and on the Me- 
chanical Value of Heat; by J. P. Joule, Esq." Read be- 
fore the British Association at Cork in 1843, and subse- 
quently communicated by the Author to the Philosophical 
Magazine, vol. xxiii., pp. 263, 347, 435.) 

Before we can finally conclude that heat is absolutely 
generated in such operations, it would be necessary to 
prove that the inducing magnet does not become lower in 



134 THOMSON ON CARNOT'S 

that heat is generated by the friction of fluids in 
motion, seem to overturn the opinion commonly 
held that heat cannot be generated, but only pro- 
duced from a source, where it has previously ex- 
isted either in a sensible or in a latent condition. 

In the present state of science, however, no opera- 
tion is known by which heat can be absorbed into 
a body without either elevating its temperature or 
becoming latent, and producing some alteration in 
its physical condition; and the fundamental axiom 
adopted by Carnot may be considered as still the 
most probable basis for an investigation of the mo- 
tive power of heat; although this, and with it 
every other branch of the theory of heat, may 
ultimately require to be reconstructed upon another 
foundation, when our experimental data are more 
complete. On this understanding, and to avoid a 



temperature, and thus compensate for the heat evolved in 
the conductor. I am not aware that any examination with 
reference to the truth of this conjecture has been instituted ; 
but, in the case where the inducing body is a pure electro- 
magnet (without any iron), the experiments actually per- 
formed by Mr. Joule render the conclusion probable that 
the heat evolved in the wire of the electro-magnet is not 
affected by the inductive action, otherwise than through 
the reflected influence which increases the strength of its 
own current. 



MOTIVE POWER OF HEAT. 135 

repetition of doubts, I shall refer to Carnot's funda- 
mental principle, in all that follows, as if its truth 
were thoroughly established. 

9. We are now led to the conclusion that the 
origin of motive power, developed by the alternate 
expansions and contractions of a body, must be 
found in the agency of heat entering the body and 
leaving it ; since there cannot, at the end of a com- 
plete cycle, when the body is restored to its primi- 
tive physical condition, have been any absolute ab- 
sorption of heat, and consequently no conversion 
of heat, or caloric, into mechanical effect; and it 
remains for us to trace the precise nature of the 
circumstances under which heat must enter the 
body, and afterwards leave it, so that mechanical 
effect may be produced. As an example, we may 
consider that machine for obtaining motive power 
from heat with which we are most familiar the 
steam-engine. 

10. Here, we observe, that heat enters the ma- 
chine from the furnace, through the sides of the 
boiler, and that heat is continually abstracted by 
the water employed for keeping the condenser cool. 
According to Carnot's fundamental principle, the 
quantity of heat thus discharged, during a complete 
revolution (or double stroke) of the engine, must be 
precisely equal to that which enters the water of 



136 THOMSON ON CARNOT'S 

the boiler;* provided the total mass of water and 
steam be invariable, and be restored to its primitive 
physical condition (which will be the case rigorously, 
if the condenser be kept cool by the external appli- 
cation of cold water instead of by injection, as is 
more usual in practice), and if the condensed 
water be restored to the boiler at the end of each 
complete revolution. Thus we perceive that a cer- 
tain quantity of heat is let down from a hot body, 
the metal of the boiler, to another body at a lower 
temperature, the metal of the condenser; and that 
there results from this transference of heat a certain 
development of mechanical effect. 

11. If we examine any other case in which 
mechanical effect is obtained from a thermal origin, 
by means of the alternate expansions and contrac- 
tions of any substance whatever, instead of the 
water of a steam-engine, we find that a similar 
transference of heat is effected, and we may there- 
fore answer the first question proposed, in the fol- 
lowing manner : 

The thermal agency ~by which mechanical effect 
may be obtained is the transference of heat from 
one body to another at a lower temperature. 

* So generally is Carnot's principle tacitly admitted as an 
axiom, that its application in this case has never, so far as 
I am aware, been questioned by practical engineers. (1849). 



MOTIVE POWER OF HEAT. 137 

11. On the measurement of Thermal Agency, 
considered with reference to its equivalent of 
mechanical effect. 

12. A perfect thermodynamic engine of any 
kind is a machine by means of which the greatest 
possible amount of mechanical effect can be obtained 
from a given thermal agency; and, therefore, if in 
any manner we can construct or imagine a perfect 
engine which may be applied for the transference 
of a given quantity of heat from a body at any 
given temperature to another body at a lower given 
temperature, and if we can evaluate the mechanical 
effect thus obtained, we shall be able to answer 
the question at present under consideration, and 
so to complete the theory of the motive power 
of heat. But whatever kind of engine we may 
consider with this view, it will be necessary for us 
to prove that it is a perfect engine; since the 
transference of the heat from one body to the other 
may be wholly, or partially, effected by conduction 
through a solid,* without the development of 

*When " thermal agency" is thus spent in conducting 
heat through a solid, what becomes of the mechanical 
effect which it might produce? Nothing can be lost in 
the operations of nature no energy can be destroyed. 
What effect, then, is produced in place of the mechanical 
effect which is lost ? A perfect theory of heat impera- 



138 THOMSON ON CARNOT'S 

mechanical effect; and, consequently, engines may 
be constructed in which the whole or any portion 

tively[demands an answer to this question ; yet no answer 
can be given in the present state of science. A few years 
ago, a similar confession must have been made with refer- 
ence to the mechanical effect lost in a fluid set in motion in 
the interior of a rigid closed vessel, and allowed to come to 
rest by its own internal friction; but in this case the 
foundation of a solution of the difficulty has been ac- 
tually found in Mr. Joule's discovery of the generation 
of heat, by the internal friction of a fluid in motion. En- 
couraged by this example, we may hope that the very per- 
plexing question in the theory of heat, by which we are 
at present arrested, will before long be cleared up. 
[Note of Sept., 1881. The Theory of the Dissipation of 
Energy completely answers this question and removes the 
difficulty.] 

It might appear that the difficulty would be entirely 
avoided by abandoning Carnot's fundamental axiom ; a 
view which is strongly urged by Mr. Joule (at the conclu- 
sion of his paper " On the Changes of Temperature pro- 
duced by the Rarefaction and Condensation of Air." Phil. 
Mag., May 1845, vol. xxvi.) If we do so, however, we 
meet with innumerable other difficulties insuperable 
without farther experimental investigation, and an entire 
reconstruction of the theory of heat from its foundation. 
It is in reality to experiment that we must look either 
for a verification of Carnot's axiom, and an explanation of 
the difficulty we have been considering; or for an entirely 
new basis of the Theory of Heat. 



MOTIVE POWER OF HEAT. 139 

of the thermal agency is wasted. Hence it is of 
primary importance to discover the criterion of a 
perfect engine. This has been done by Carnot, who 
proves the following proposition : 

13. A perfect thermodynamic engine is such 
that, whatever amount of mechanical effect it can 
derive from a certain thermal agency, if an equal 
amount be spent in working it baclcivards, an equal 
reverse thermal effect will be produced* 

14. This proposition will be made clearer by the 
applications of it which are given later ( 29), in 
the cases of the air-engine and the steam-engine, 
than it could be by any general explanation ; and it 
will also appear, from the nature of the opera- 
tions described in those cases, and the principles of 
Carnot's reasoning, that a perfect engine may be 
constructed with any substance of an indestructible 
texture as the alternately expanding and contract- 
ing medium. Thus we might conceive thermo- 
dynamic engines founded upon the expansions and 
contractions of a perfectly elastic solid, or of a 
liquid; or upon the alterations of volume experi 
enced by substances in passing from the liquid to 
the solid state, f each of which being perfect, would 

* For a demonstration, see 29. 

f A case minutely examined in another paper, to be laid 
before the Society at the present meeting. ' ' Theoretical 



140 THOMSON ON CARNOT'S 

produce the same amount of mechanical effect from 
a given thermal agency ; but there are two cases 
which Carnot has selected as most worthy of minute 
attention, because of their peculiar appropriateness 
for illustrating the general principles of his theory, 
no less than on account of their very great practi- 
cal importance: the steam-engine, in which the 
substance employed as the transferring medium is 
water, alternately in the liquid state and in the 
state of vapor ; and the air-engine, in which the 
transference is effected by means of the alternate 
expansions and contractions of a medium always 
in the gaseous state. The details of an actually 
practicable engine of either kind are not con- 
templated by Carnot in his general theoretical rea- 
sonings, but he confines himself to the ideal con- 
struction, in the simplest possible way in each case, 
of an engine in which the economy is perfect. He 
thus determines the degree of perfectibility which 
cannot be surpassed ; and by describing a conceiv- 
able method of attaining to this perfection by an 
air-engine or a steam-engine, he points out the 
proper objects to be kept in view in the practical 
construction and working of such machines. I now 
proceed to give an outline of these investigations. 

Considerations on the Effect of Pressure in Lowering the 
Freezing-point of Water," by Prof. James Thomson. 



MOTIVE POWER OF HEAT. 141 

CARROT'S THEORY OF THE STEAM- ENGIKE. 

15. Let CDF^E^ be a cylinder, of which the 
curved surface is perfectly impermeable to heat, 
with a piston also impermeable to heat, fitted in it ; 
while the fixed bottom CD, itself with no capacity 
for heat, is possessed of perfect conducting power. 
Let K be an impermeable stand, such that when 
the cylinder is placed upon it the contents below 
the piston can neither gain nor lose heat. Let A 
and B be two bodies permanently retained at con- 
stant temperatures, S and T, respectively, of which 
the former is higher than the latter. Let the cyl- 
inder, placed on the impermeable stand, K, be par- 
tially filled with water, at the temperature S, of the 
body A y and (there being no air below it) let the 
piston be placed in a position EF, near the surface of 
the water. The pressure of the vapor above the 
water will tend to push up the piston, and must 
be resisted by a force applied to the piston,* till 

* In all that follows, the pressure of the atmosphere on 
the upper side of the piston will be included in the applied 
forces, which, in the successive operations described, are 
sometimes overcome by the upward motion, and some- 
times yielded to in the motion downwards. It will be un- 
necessary, in reckoning at the end of a cycle of operations, 
to take into account the work thus spent upon the atmos- 
phere, and the restitution which has been made, since 
these precisely compensate for one another. 



142 



THOMSON ON CARNOT'S 



the commencement of the operations, which are 
conducted in the following manner: 

(1) The cylinder being placed on the body A, 



E 



F- 






so that the water and vapor may be retained at the 
temperature S 9 let the piston rise any convenient 



MOTIVE POWER OF HEAT. 143 

height EE^ , to a position E^F l , performing work 
by the pressure of the vapor below it during its 
ascent. 

[During this operation a certain quantity, H, of heat, 
the amount of latent heat in the fresh vapor which is 
formed, is abstracted from the body A] 

(2) The cylinder being removed, and placed on 
the impermeable stand K, let the piston rise grad- 
ually, till, when it reaches a position E^, the 
temperature of the water and vapor is T, the same 
as that of the body B. 

[During this operation the fresh vapor continually 
formed requires heat to become latent ; and, therefore, as 
the contents of the cylinder are protected from any acces- 
sion of heat, their temperature sinks.] 

(3) The cylinder being removed from K, and 
placed on B, let the piston be pushed down, till, 
when it reaches the position E Z F Z , the quantity of 
heat evolved and abstracted by B amounts to that 
which, during the first operation, was taken from A. 

[Note of Nov. 5, 1881. The specification of this 
operation, with a view to the return to the primi- 
tive condition, intended as the conclusion to the 
four operations, is the only item in which Carnot's 
temporary and provisional assumption of the mate- 
riality of heat has effect. To exclude this hypothe- 
sis, Prof. James Thomson has suggested the fol* 



144 THOMSON ON CARNOT'S 

lowing corrected specification for the third opera- 
tion : Let the piston be pushed down, till it readies 
a position E^F^ , determined so as to fulfil the con- 
dition, that at the end of the fourth operation the 
primitive temperature S shall be reached :*] 

[During this operation the temperature of the contents 
of the cylinder is retained constantly at T, and all the 
latent heat of the vapor which is condensed into water at 
the same temperature is given out to JB.] 

(4) The cylinder being removed from B, and 
placed on the impermeable stand, let the piston be 
pushed down from E^F^ to its original position EF. 

[During this operation, the impermeable stand prevent- 
ing any loss of heat, the temperature of the water and air 
must rise continually, till (since the quantity of heat 
evolved during the third operation was precisely equal to 

* [Note of Nov. 5, 1881. Maxwell has simplified the 
correction by beginning the cycle with Carnot's second 
operation, and completing it through his third, fourth, 
and first operations, with his third operation nearly as fol- 
lows : 

let the piston be pushed down to any position E 3 F 3 ; 
then Carnot's fourth operation altered to the following': 

let the piston be pushed down from E 3 F 3 until the tem- 
perature reaches its primitive value 8 ; 
and lastly, Carnot's first operation altered to the follow- 
ing : 

let the piston rise to its primitive position.] 



MOTIVE POWER OF HEAT. 145 

that which was previously absorbed) at the conclusion it 
reaches its primitive value, S, in virtue of Carnot's funda- 
mental axiom.] 

[Note of Nov. 5, 1881. With Prof. James Thomson's 
correction of operation (3), the words in virtue of " Car- 
not's Fundamental Axiom" must be replaced by "the 
condition fulfilled by operation (3)," in the description of 
the results of operation (4).] 

16. At the conclusion of this cycle of operations * 
the total thermal agency has been the letting down 
of H units of heat from the body A, at the tem- 
perature S, to B, at the lower temperature T\ and 
the aggregate of the mechanical effect has been a 
certain amount of work produced, since during the 
ascent of the piston in the first and second opera- 
tions, the temperature of the water and vapor, and 
therefore the pressure of the vapor on the piston, 
was on the whole higher than during the descent, 
in the third and fourth operations. It remains for 
us actually to evaluate this aggregate amount of 
work performed ; and for this purpose the f ollow- 

* In Carnot's work some perplexity is introduced with 
reference to the temperature of the water, w;hich, in the 
operations he describes, is not brought back exactly to 
what it was at the commencement ; but the difficulty 
which arises is explained by the author. No such difficulty 
occurs with reference to the cycle of operation described 
in the text, for which I am indebted to Mons. Clapeyron. 



146 THOMSON ON CARNOT'S 

ing graphical method of representing the mechan- 
ical effect developed in the several operations, taken 
from Mons. Clapeyron's paper, is extromely con- 
venient. 

17. Let OX and OF be two lines ,t right angles 
to one another. Along X measure off distances 
ON^ , -ZVi JV a , N^Ns , N a 0, respectively proportional 
to the spaces described by the piston during the 
four successive operations described above; and, 
with reference to these four operations respective- 
ly, let the following constructions be made: 

(1) Along Y measure a length OA, to repre- 
sent the pressure of the saturated vapor at the 
temperature S m , and draw A A l parallel to OX, and 
let it meet an ordinate through N^ , in A l . 

(2) Draw a curve A^PA such that, if ON repre- 
sent, at any instant during the second operation, 
the distance of the piston from its primitive posi- 
tion, NP shall represent the pressure of the vapor 
at the same instant. 

(3) Through A^ draw A Z A 3 parallel to OX, and 
let it meet an ordinate through N z in A 9 . 

(4) Draw the curve A 3 A such that the abscissa 
and ordinate of any point in it may represent re- 
spectively the distances of the piston from its 
primitive position, and the pressure of the vapor, 
at each instant during the fourth operation. The 



MOTIVE POWER OF HEAT. 



147 



last point of this curve must, according to CarnoVs 
fundamental principle, coincide with A, since the 
piston is, at the end of the cycle of operations, 




again in its primitive position, and the pressure of 
the vapor is the same as it was at the beginning. 

18. Let us now suppose that the lengths, ON^ , 
JVjJV,, N^NI, and N t O, represent numerically the 
volumes of the spaces moved through by the piston 
during the successive operations. It follows that 
the mechanical effect obtained during the first 
operation will be numerically represented by the 
area AA^N^O; that is, the number of superficial 
units in this area will be equal to the number of 
" foot-pounds" of work performed by the ascend- 
ing piston during the first operation. The work 
performed by the piston during the second opera- 
tion will be similarly represented by the area 



148 THOMSON ON CARNOT'S 

A^A^N^N^. Again, during the third operation a 
certain amount of work is spent on the piston^ 
which will be represented by the area A^A^N^N^ ; 
and lastly, during the fourth operation, work is 
spent in pushing the piston to an amount repre- 
sented by the area A 3 A ON 3 . 

19. Hence the mechanical effect (represented 
by the area OA A A^N^) which was obtained dur- 
ing the first and second operations, exceeds the 
work (represented by N^A^A^AO) spent during 
the third and fourth, by an amount represented 
by the area of the quadrilateral figure AA 1 A. 2 A 3 ; 
and, consequently, it only remains for us to 
evaluate this area, that we may determine the 
total mechanical effect gained in a complete 
cycle of operations. Now, from experimental data, 
at present nearly complete, as will be explained 
below, we may determine the length of the line 
AA l for the given temperature S, and a given ab- 
sorption H, of heat, during the first operation; 
and the length of A^A Z for the given lower tem- 
perature T, and the evolution of the same quantity 
of heat during the fourth operation: and the 
curves A^PA^, A 3 P'A may be drawn as graphical 
representations of actual observations. The figure 
being thus constructed, its area may be measured, 
and we are, therefore, in possession of a graphical 



MOTIVE POWER OF HEAT. 149 

method of determining the amount of mechanical 
effect to be obtained from any given thermal 
agency. As, however, it is merely the area of the 
figure which it is required to determine, it will not 
be necessary to be able to describe each of the 
curves A^PA^, A^P'A, but it will be sufficient to 
know the difference of the abscissas corresponding 
to any equal ordinates in the two; and the follow- 
ing analytical method of completing the problem 
is the most convenient for leading to the actual 
numerical results. 

20. Draw any line PP' parallel to OX, meeting 
the curvilinear sides of the quadrilateral in P and 
P'. Let denote the length of this line, and p 
its distance from OX. The area of the figure, 
according to the integral calculus, will be denoted 
by the expression 

/>*, 

e/P 3 

where p l and p 9 (the limits of integration indicated 
according to Fourier's notation) denote the lines 
OA and N t A 3 , which represent respectively the 
pressures during the first and third operations. 
Now, by referring to the construction described 
above, we see that is the difference of the volumes 
below the piston at corresponding instants of the 
second and fourth operations, or instants at which 



150 THOMSON ON CARNOT'S 

the caturated steam and the water in the cylinder 
have the same pressure p, and consequently the 
same temperature, which we may denote by t. 
Again, throughout the second operation the entire 
contents of the cylinder possess a greater amount 
of heat by H units than during the fourth ; and, 
therefore, at any instant of the second operation 
there is as much more steam as contains H units 
of latent heat than at the corresponding instant 
of the fourth operation. Hence if k denote the 
latent heat in a unit of saturated steam at the 
temperature t, the volume of the steam at the two 

TT 

corresponding instants must differ by -T-. Now, if 
(T denote the ratio of the density of the steam to 

TT 

that of the water, the volume -j- of steam will be 

K 

TT 

formed from the volume a -y- of water ; and con- 

rC 

sequently we have, for the difference of volumes of 
the entire contents at the corresponding instants, 



Hence the expression for the area of the quadri- 
lateral figure becomes 



MOTIVE POWER OF HEAT. 151 

Now, <r, k, and p, being quantities which depend 
upon the temperature, may be considered as func- 
tions of t; and it will be convenient to modify the 
integral so as to make t the independent variable. 
The limits will be from t T to t 8, and, if we 
denote by M the value of the integral, we have the 
expression 

dp 

dt. . . . (1) 

for the total amount of mechanical effect gained 
by the operations described above. 

21. If the interval of temperatures be extremely 

dp 

small, so small that (1 cr) -r- will not sensibly vary 

for values of t between I 7 and 8, the preceding 
expression becomes simply 

dp 

T). . . (2) 



This might, of course, have been obtained at once 
by supposing the breadth of the quadrilateral 
figure AA^A^A to be extremely small compared 
with its length, and then taking for its area, as an 
approximate value, the product of the breadth into 



152 THOMSON ON CARNOT'S 

the line A A 1 , or the line A^A^ or any line of in- 
termediate magnitude. 

The expression (2) is rigorously correct for any 

dp 

interval S T, if the mean value of (1 <r)-r- for 

that interval be employed as the coefficient of 
H(S-T). 

CARNOT'S THEORY OF THE AIR-ENGINE. 

22. In the ideal air-engine imagined by Carnot 
four operations performed upon a mass of air or 
gas enclosed in a closed vessel of variable volume 
constitute a complete cycle, at the end of which 
the medium is left in its primitive physical condi- 
tion; the construction being the same as that which 
was described above for the steam-engine, a body 
A, permanently retained at the temperature 8, and 
B at the temperature T\ an impermeable stand K\ 
and a cylinder and piston, which in this case con- 
tains a mass of air at the temperature S, instead 
of water in the liquid state, at the beginning and 
end of a cycle of operations. The four successive 
operations are conducted in the following manner : 

(1) The cylinder is laid on the body A, so that 
the air in it is kept at the temperature S; and the 
piston is allowed to rise, performing work. 



MOTIVE POWER OF HEAT. 153 

(2) The cylinder is placed on the impermeable 
stand K, so that its contents can neither gain nor 
lose heat, and the piston is allowed to rise farther, 
still performing work, till the temperature of the 
air sinks to T. 

(3) The cylinder is placed on B, so that the air 
is retained at the temperature T, and the piston is 
pushed down till the air gives out to the body B 
as much heat as it had taken in from A, during the 
first operation. 

[Note of Nov. 5, 1881. To eliminate the assumption of 
the materiality of heat, make Professor James Thomson's 
correction here also ; as above in 15; or take Maxwell's 
rearrangement of the cycle described in the foot-note to 
15, p. 144.] 

(4) The cylinder is placed on K 9 so that no more 
heat can be taken in or given out, and the piston 
is pushed down to its primitive position. 

23. At the end of the fourth operation the tem- 
perature must have reached its primitive value S, 
in virtue of CARNOT'S axiom. 

24. Here, again, as in the former case, we observe 
that work is performed by the piston during the 
first two operations ; and during the third and 
fourth work is spent upon it, but to a less amount, 
since the pressure is on the whole less during the 
third and fourth operations than during the first 



154 THOMSON ON CARNOT'8 

and second, on account of the temperature being 
lower. Thus, at the end of a complete cycle of 
operations, mechanical effect has been obtained ; 
and the thermal agency from which it is drawn is 
the taking of a certain quantity of heat from A, 
and letting it down, through the medium of the 
engine, to the body B at a lower temperature. 

25. To estimate the actual amount of effect thus 
obtained, it will be convenient to consider the altera- 
tions of volume of the mass of air in the several 
operations as extremely small. We may afterwards 
pass by the integral calculus, or, practically, by 
summation to determine the mechanical effect 
whatever be the amplitudes of the different motions 
of the piston. 

26. Let dq be the quantity of heat absorbed 
during the first operation, which is evolved again 
during the third; and let dv be the corresponding 
augmentation of volume which takes place while 
the temperature remains constant, as it does during 
the first operation.* The diminution of volume 

* Thus, -^ will be the partial differential coefficient, 

with respect to , of that function of wand t which expresses 
the quantity of heat that must be added to a mass of air 
when in a " standard " state (such as at the temperature zero, 
and under the atmospheric pressure), to bring it to the 
temperature t and the volume v. That there is such a 



MOTIVE POWER OF HEAT. 155 

in the third operation must be also equal to dv, or 
only differ from it by an infinitely small quantity of 
the second order. During the second operation we 
may suppose the volume to be increased by an in- 
finitely small quantity 0; which will occasion a 
diminution of pressure and a diminution of tem- 
perature, denoted respectively by GJ and r. During 
the fourth operation there will be a diminution of 
volume and an increase of pressure and temperature, 
which can only differ, by infinitely small quantities 
of the second order, from the changes in the other 
direction, which took place in the second operation, 
and they also may, therefore, be denoted by 0, GO, 
and r, respectively. The alteration of pressure 

function, of two independent variables v and t, is merely 
an analytical expression of Carnot's fundamental axiom, as 
applied to a mass of air. The general principle may be 
analytically stated in the following terms : If Mdv denote 
the accession of heat received by a mass of any kind, not 
possessing a destructible texture, when the volume is in- 
creased by dv, the temperature being kept constant, and if 
Ndt denote the amount of heat which must be supplied to 
raise the temperature by dt, without any alteration of vol- 
ume ; then Mdv -\-Ndt must be the differential of a func- 
tion of v and t. [Note of Nov. 5, 1881. In the corrected 
theory it is (M Jp) dv -\- Ndt t th&t is a complete differential, 
not Mdv + Ndt. See Dynamical Theory of Heat (Art. XLVIII. , 
below), 20. J 



156 THOMSON ON CARNOT'S 

during the first and third operations may at once 
be determined by means of Mariotte's law, since 
in them the temperature remains constant. Thus, 
if, at the commencement of the cycle, the volume 
and pressure be v and p, they will have become 
v -f- dv and pv/(v -f- civ) at the end of the first 
operation. Hence the diminution of pressure 
during the first operation is p pv/(v -f- dv) or 
pdv/(v + dv) and therefore, if we neglect infinitely 
small quantities of the second order, we hswepdv/v 
for the diminution of pressure during the first 
operation ; which to the same degree of approxima- 
tion, will be equal to the increase of pressure during 
the third. If t + T and t be taken to denote the 
superior and inferior limits of temperature, we 
shall thus have for the volume, the temperature, 
and the pressure at the commencements of the 
four successive operations, and at the end of the 
cycle, the following values respectively: 



(1) v, 

(2) v + dv, 

(3) v + dv+ 

(4) v + 0, t, p - 

(5) v, t + r, p. 



MOTIVE POWER OF HEAT. 157 

Taking the mean of the pressures at the beginning 
and end of each operation, we find 



(4) j> - ^ G*, 

which, as we are neglecting infinitely small quan- 
tities of the second order, will be the expressions 
for the mean pressures during the four successive 
operations. Now, the mechanical effect gained or 
spent, during any of the operations, will be found 
by multiplying the mean pressure by the increase 
or diminution of volume which takes place; and 
we thus find 



(4) (p - \ 



158 THOMSON ON CARNOTS 

for the amounts gained during the first and second, 
and spent during the third and fourth operations ; 
and hence, by addition and subtraction, we find 

, ,dv , ,.dv 

codv p4> , or (VGJ pep) , 

for the aggregate amount of mechanical effect 
gained during the cycle of operations. It only re- 
mains for us to express this result in terms of dq 
and r, on which the given thermal agency depends. 
For this purpose we remark that and GO are al- 
terations of volume and pressure which take place 
along with a change of temperature r, and hence, 
by the laws of compressibility and expansion, we 
may establish a relation* between them in the fol- 
lowing manner : 

Let p 9 be the pressure of the mass of air when 
reduced to the temperature zero, and confined 
in a volume v ; then, whatever be v , the product 
p Q v Q will, by the law of compressibility, remain con- 
stant ; and, if the temperature be elevated from 
to t + *> and the gas be allowed to expand freely 
without any change of pressure, its volume will be 

* We might also investigate another relation, to express 
the fact that there is no accession or removal of heat during 
either the second or the fourth operation; but it will be 
seen that this will not affect the result in the text, although 
it would enable us to determine both and GO in terms of T. 



OF THK 

UNIVERSITY 



MOTIVE POWER OF HEAT. 159 

increased in the ratio of 1 to 1 -|- E(t -\- T), where 
E is very nearly equal to .00366 (the Centigrade 
scale of the air-thermometer being referred to), 
whatever be the gas employed, according to the 
researches of Regnault and of Magnus on the ex- 
pansion of gases by heat. If, now, the volume be 
altered arbitrarily with the temperature continually 
at t -f- 7? the product of the pressure and volume 
will remain constant ; and therefore we have 

pv = p v {l + fi(t + r)}. 
Similarly, 

Hence, by subtraction, we have 
or, neglecting the product 00$, 

Hence the preceding expression for mechanical 
effect, gained in the cycle of operations, becomes 

p v . Er . dv/v. 
Or, as we may otherwise express it, 

vdq/dv ' * 

Hence, if we denote by M the mechanical effect due 
to H units of heat descending through the same 
interval T, which might be obtained by repeating 



160 THOMSON ON CARNOT'S 

TT 

the cycle of operations described above, -=- times, 

we have M = -^- . Hr ..... (3) 

vdq/dv 

27. If the amplitudes of the operations had been 
finite, so as to give rise to an absorption of H units 
of heat during the first operation, and a lowering 
of temperature from 8 to T during the second, the 
amount of work obtained would have been found 
to be expressed by means of a double definite in- 
tegral thus :* 

*= far #..&%. 

i/o ijT vdq/dv 

or 



this second form being sometimes more convenient. 

* This result might have been obtained by applying the 
usual ^notation of the integral calculus to express the 
area of the curvilinear quadrilateral, which, according to 
Clapeyron's graphical construction, would be found to 
represent the entire mechanical effect gained in the cycle 
of operations of the air-engine. It is not necessary, how- 
ever, to enter into the details of this investigation, as the 
formula (3), and the consequences derived from it, include 
the whole theory of the air-engine, in the best practical 
form; and the investigation of it which I have given in the 
text will probably give as clear a view of the reasoning on 
which it is founded as could be obtained by the graphical 
method, which in this case is not so valuable as it is from 
its simplicity in the case of the steam-engine. 



MOTIVE POWER OF HEAT. 161 

28. The preceding investigations, being founded 
on the approximate laws of compressibility and ex- 
pansion (known as the law of Mariotte and Boyle, 
and the law of Dalton and Gay-Lussac), would re- 
quire some slight modifications to adapt them to 
cases in which the gaseous medium employed is such 
as to present sensible deviations from those laws. 
Regnault's very accurate experiments show that 
the deviations are insensible, or very nearly so, for 
the ordinary gases at ordinary pressures ; although 
they may be considerable for a medium, such as 
sulphurous acid, or carbonic acid under high pres- 
sure, which approaches the physical condition of a 
vapor at saturation ; and therefore, in general, and 
especially in practical applications to real air-engines, 
it will be unnecessary to make any modification in 
the expressions. In cases where it may be necessary, 
there is no difficulty in making the modifications, 
when the requisite data are supplied by experiment. 

29.* Either the steam-engine or the air-engine, 
according to the arrangements described above, 
gives all the mechanical effect that can possibly be 
obtained from the thermal agency employed. For 

* This paragraph is the demonstration, referred to above, 
of the proposition stated in 13, as it is readily seen that 
it is applicable to any conceivable kind of therinodynamic 
engine. 



162 THOMSON ON CARNOT'S 

it is clear that in either case the operations may 
be performed in the reverse order, with every 
thermal and mechanical effect reversed. Thus, in 
the steam-engine, we may commence by placing 
the cylinder on the impermeable stand, allow the 
piston to rise, performing work, to the position 
E 3 F S ; we may then place it on the body B, and 
allow it to rise, performing work, till it reaches 
E^F^'y after that the cylinder may be placed again 
on the impermeable stand, and the piston may be 
pushed down to E^F^ ; and, lastly, the cylinder 
being removed to the body A, the piston may be 
pushed down to its primitive position. In this 
inverse cycle of operations a certain amount of 
work has been spent, precisely equal, as we readily 
see, to the amount of mechanical effect gained in 
the direct cycle described above ; and heat has been 
abstracted from B, and deposited in the body A, 
at a higher temperature, to an amount precisely 
equal to that which in the direct style was let 
down from A to B. Hence it is impossible to 
have an engine which will derive more mechanical 
effect from the same thermal agency than is ob- 
tained by the arrangement described above; since, 
if there could be such an engine, it might be em- 
ployed to perform, as a part of its whole work, the 
inverse cycle of operations, upon an engine of the 



MOTIVE POWER OF SEAT 163 

kind we have considered, and thus to continually 
restore the heat from B to A, which has descended 
from A to B for working itself; so that we should 
have a complex engine, giving a residual amount 
of mechanical effect without any thermal agency, 
or alteration of materials, which is an impossibility 
in nature. The same reasoning is applicable to 
the air-engine ; and we conclude, generally, that 
any two engines, constructed on the principles laid 
down above, whether steam-engines with different 
liquids, an air-engine and a steam-engine, or two 
air-engines with different gases, must derive the 
same amount of mechanical effect from the same 
thermal agency. 

30. Hence, by comparing the amounts of me- 
chanical effect obtained by the steam-engine and 
the air-engine from the letting down of the H 
units of heat from A at the temperature (t -j- *) to 
B at t, according to the expressions (2) and (3), 
we have 

M =(l-a)%L.ffT = ^j-.HT.. (5) 

' kdt vdq/dv 

If we denote the coefficient of Hr in these equal 
expressions by //, which maybe called '"Carnot's 
coefficient," we have 



164 THOMSON ON CARNOT'S 

and we deduce the following very remarkable con- 
clusions : 

(1) For the saturated vapors of all different 
liquids, at the same temperature, the value of 

(1 a) -~- must be the same. 

(2) For any different gaseous masses, at the 

same temperature, the value of , . f- must be 

vdq/dv 

the same. 

(3) The values of these expressions for saturated 
vapors and for gases, at the same temperature, 
must be the same. 

31. No conclusion can be drawn a priori re- 
garding the values of this coefficient JJL for different 
temperatures, which can only be determined, or 
compared, by experiment. The results of a great 
variety of experiments, in different branches of 
physical science (Pneumatics and Acoustics), cited 
by Carnot and by Clapeyron, indicate that the 
values of JJL for low temperatures exceed the values 
for higher temperatures ; a result amply verified 
by the continuous series of experiments performed 
by Regnault on the saturated vapor of water for all 
temperatures from to 230, which, as we shall 
see later, give values for /* gradually diminishing 
from the inferior limit to the superior limit of 



MOTIVE POWER OF HEAT. 165 

temperature. When, by observation, j* has been 
determined as a function of the temperature, the 
amount of mechanical effect, M, deducible from 
H units of heat descending from a body at the 
temperature S to a body at the temperature T, 
may be calculated from the expression 

rrt 

M=H C pdt,. . . . (7) 

t/S 

which is, in fact, what either of the equations (1) 
for the steam-engine, or (4) for the air-engine, be- 
comes, when the notation //, for Carnot's multi- 
plier, is introduced. 

The values of this integral may be practically 
obtained, in the most convenient manner, by first 
determining, from observation, the mean values of 
/* for the successive degrees of the thermometric 
scale, and then adding the values for all the de 
grees within the limits of the extreme temperatures 
tfand T.* 

32. The complete theoretical investigation of 
the motive power of heat is thus reduced to the 
experimental determination of the coefficient /t ; 
and may be considered as perfect, when, by any 
series of experimental researches whatever, we can 

\ 
* The results of these investigations are exhibited in 

Tables I and II. 



166 THOMSON ON CARNOT'S 

find a value of /* for every temperature within 
practical limits. The special character of the ex- 
perimental researches, whether with reference to 
gases or with reference to vapors, necessary and 
sufficient for this object, is defined and restricted 
in the most precise manner, by the expressions (6) 
for //, given above.* 

33. The object of Regnault's great work, referred 
to in the title of this paper, is the experimental de- 
termination of the various physical elements of the 
steam-engine ; and when it is complete, it will 
furnish all the data necessary for the calculation 
of /*. The valuable researches already published 
in a first part of that work make known the 
latent heat of a given weight, and the pressure, of 
saturated steam for all temperatures between 
and 230 Cent, of the air-thermometer. Besides 
these data, however, the density of saturated va- 
por must be known, in order that k, the latent 
heat of a unit of volume, may be calculated from 
Regnault's determination of the latent heat of a 
given weight. * Between the limits of and 100, 

* It is, comparatively speaking, of little consequence to 
know accurately the value of or, for the factor (1 cr) of 
the expression for >w, since it is so small (being less than 
T ^ for all temperatures between and 100) that, unless 
all the data are known with more accuracy than we can 



MOTIVE POWER OF HEAT. 167 

it is probable, from various experiments which 
have been made, that the density of vapor follows 
very closely the simple laws which are so accurately 
verified by the ordinary gases;* and thus it may 
be calculated from Regnault's table giving the 
pressure at any temperature within those limits. 
Nothing as yet is known with accuracy as to the 
density of saturated steam between 100 and 230, 
and we must be contented at present to estimate it 
by calculation from Regnault's table of pressures; 
although, when accurate experimental researches 
on the subject shall have been made, considerable 
deviations from the laws of Boyle and Dalton, on 
which this calculation is founded, may be dis- 
covered. 

34. Such are the experimental data on which 
the mean values of // for the successive degrees of 
the air- thermometer, from to 230, at present 
laid before the Royal Society, is founded. The 
unit of length adopted is the English foot; the 
unit of weight, the pound ; the unit of work, a 

count upon at present, we might neglect it altogether, and 
take dp/Mi simply, as the expression for /*, without com- 
mitting any error of important magnitude. 

* This is well established, within the ordinary atmos- 
pheric limits, in Regnault's Etudes Meteorologiques, in the 
Annales de Chimie, vol. xv. , 1846, 



168 THOMSON ON CARNOT'S 

" foot-pound ;"and the unit of heat that quantity 
which, when added to a pound of water at 0, will 
produce an elevation of 1 in temperature. The 
mean value of /* for any degree is found to a suffi- 
cient degree of approximation by taking, in place 
of (r, dp/dt and k ; in the expression 



the mean values of those elements; or, what is 
equivalent to the corresponding accuracy of ap- 
proximation, by taking, in place of cr and k respec- 
tively, the mean of the values of those elements for 
the limits of temperature, and in place of dp/dt, 
the difference of the values of p, at the same limits. 

35. In Regnault's work (at the end of the eighth 
memoir), a table of the pressures of saturated steam 
for the successive temperatures 0, 1, 2, ... 230, 
expressed in millimetres of mercury, is given. On 
account of the units adopted in this paper, these 
pressures must be estimated in pounds on the 
square foot, which we may do by multiplying each 
number of millirnetre3 by 2.7896, the weight in 
pounds of a sheet of mercury, one millimetre thick, 
and a square foot in area. 

36. The value of &, the latent heat of a cubic 
foot, for any temperature t, is found from A, the 



MOTIVE POWER OF HEAT. 169 

latent heat of a pound of saturated steam, by the 
equation 

p 1 + .00366 X 100 . 
~760 1 + .00366 X t 

where p denotes the pressure in millimetres, and 1 
the latent heat of a pound of saturated steam; the 
values of A, being calculated by the empirical for- 
mula f 

A = (606.5 -f 0.3050 -(t + .00002** + 0.0000003*'), 
given by Regnault as representing, between the 

* It appears that the vol. of 1 kilog. must be 1.69076 ac- 
cording to the data here assumed. 

The density of saturated steam at 100 is taken as ~^ 
of that of water at its maximum. Rankine takes it as T ^. 

f The part of this expression in the first vinculum (see 
Regnault, end of ninth memoir) is what is known as " the 
total heat " of a pound of steam, or the amount of heat 
necessary to convert a pound of water at into a pound 
of saturated steam at t ; which, according to " Watt's 
law," thus approximately verified, would be constant. 
The second part, which would consist of the single term 
t, if the specific heat of water were constant for all tem- 
peratures, is the number of thermic units necessary to raise 
the temperature of a pound of water from to t, and 
expresses empirically the results of Regnault's experi- 
ments on the specific heat of water (see end of the tenth 
memoir), described in the work already referred to. 



170 THOMSON ON CARNOT'S 

extreme limits of his observations, the latent heat 
of a unit weight of saturated steam. 

EXPLANATION OF TABLE I. 

37. The mean values of jn for the first, for the 
eleventh, for the twenty-first, and so on, up to the 
231st* degree of the air-thermometer, have been 
calculated in the manner explained in the preced- 
ing paragraphs. These, and interpolated results, 
which must agree with what would have been ob- 
tained, by direct calculation from Regnault's data, 
to three significant places of figures (and even for 
the temperatures between and 100, the experi- 
mental data do not justify us in relying on any of 
the results to a greater degree of accuracy), are 
exhibited in Table I. 

To find the amount of mechanical effect due to a 
unit of heat, descending from a body at a temper- 
ature 8 to a body at T, if these numbers be in- 
tegers, we have merely to add the values of ja in 
Table I. corresponding to the successive numbers. 

T+\, T+2, ....#- 2, S-l. 

* In strictness, the 230th is the last degree for which the 
experimental data are complete ; but the data for the 231st 
may readily be assumed in a sufficiently satisfactory 
manner. 



MOTIVE POWER OF HEAT. 171 



EXPLANATION OF TABLE II. 

38. The calculation of the mechanical effect, in 
any case, which might always be effected in the 
manner described in 37 (with the proper modifi- 
cation for fractions of degrees, when necessary), is 
much simplified by the use of Table II., where the 
first number of Table I., the sum of the first and 
second, the sum of the first three, the sum of the 
first four, and so on, are successively exhibited. 
The sums thus tabulated are the values of the in- 
tegrals 

/I />2 />3 /*2 

pdt, I pdt, I dt,.... I 
e/0 I/O t/0 

and, if we denote / pdt by the letter M, Table II. 

may be regarded as a table of the value of M. 

To find the amount of mechanical effect due to a 
unit of heat descending from a body at a tempera- 
ture 8 to a body at T, if these numbers be integers, 
we have merely to subtract the value of M, for the 
number T, from the value for the number S, given 
in Table II 



172 



THOMSON ON CARNOT'S 



TABLE I.* 

MEAN VALUES OF n FOR THE, SUCCESSIVE. DEGREES OF 
THE AIR-THERMOMETER FROM TO 230. 



o 


M 


o 


M 


o 


M 


o 


M 


1 


4.960 


32 


4 559 


63 


4.194 


94 


3.889 


2 


4.946 


33 


4.547 


64 


4.183 


95 


3.880 


3 


4.932 


34 


4.535 


65 


4.172 


96 


3.871 


4 


4.918 


35 


4.522 


66 


4.161 


97 


3.863 


5 


4.905 


36 


4.510 


67 


4.150 


98 


3.854 


6 


4.892 


37 


4.498 


68 


4.140 


99 


3.845 


7 


4.878 


38 


4.486 


69 


4.129 


100 


3.837 


8 


4.865 


39 


4.474 


70 


4.119 


101 


3.829 


9 


4.852 


40 


4.462 


71 


4.109 


102 


3.820 


10 


4.839 


41 


4.450 


72 


4.098 


103 


3.812 


11 


4.826 


42 


4.438 


73 


4.088 


104 


3.804 


12 


4.812 


43 


4.426 


74 


4.078 


105 


3.796 


13 


4.799 


44 


4.414 


75 


4.067 


106 


3.788 


14 


4.786 


45 


4.402 


76 


4.057 


107 


3.780 


15 


4.773 


46 


4.390 


77 


4.047 


108 


3.772 


16 


4.760 


47 


4 378 


73 


4.037 


109 


3.764 


17 


4.747 


48 


4.366 


79 


4.028 


110 


3.757 


18 


4 785 


49 


4.355 


80 


4.018 


111 


3.749 


19 


4.722 


50 


4.343 


81 


4.009 


112 


3.741 


20 


4.709 


51 


4.331 


82 


3.999 


113 


3.734 


21 


4.697 


52 


4.319 


83 


3.990 


114 


3.726 


22 


4.684 


53 


4.308 


84 


3.980 


115 


3.719 


23 


4.672 


54 


4.296 


85 


3.971 


116 


3.712 


24 


4.659 


55 


4.285 


86 


3.961 


117 


3.704 


25 


4.646 


56 


4.273 


87 


3.952 


118 


3.697 


26 


4.634 


57 


4.262 


88 


3.943 


119 


3.689 


27 


4.621 


58 


4.250 


89 


3.934 


120 


3.682 


28 


4.609 


59 


4.239 


90 ~ 


3.925 


121 


3.675 


29 


4.596 


60 


4.227 


91 


3.916 


122 


3.668 


30 


4.584 


61 


4.216 


92 


3.907 


123 


3.661 


31 


4.572 


62 


4.205 


93 


3.898 


124 


3.654 



* The numbers here tabulated may also be regarded as 
the actual values ofjufort = i t t = H, t = 2, t = 3|, etc. 



MOTIVE POWER OF HEAT. 



173 



TABLE I. (Continued.} 






p 











M 





P 


125 


3.647 


152 


3.479 


179 


3.342 


206 


3.225 


126 


3.640 


153 


3.473 


180 


3.337 


207 


3.221 


127 


3.633 


154 


3.468 


181 


3.332 


208 


3.217 


128 


3.627 


155 


3.462 


182 


3.328 


209 


3.213 


129 


3.620 


156 


3.457 


183 


3.323 


210 


3.210 


130 


3.614 


157 


3.451 


184 


3.318 


211 


3.206 


131 


3.607 


158 


3.446 


185 


3.314 


212 


3.202 


132 


3.601 


159 


3.440 


186 


3.309 


213 


3.198 


133 


3.594 


160 


3.435 


187 


3.304 


214 


3.195 


134 


3.586 


161 


3.430 


188 


3.300 


215 


3.191 


135 


3.579 


162 


3.424 


189 


3.295 


216 


3.188 


136 


3.573 


163 


3.419 


190 


3.291 


217 


3.184 


137 


3.567 


164 


3.414 


191 


3.287 


218 


3.180 


138 


3.561 


165 


3.409 


192 


3.282 


219 


3.177 


139 


3.555 


166 


3.404 


193 


3.278 


220 


3.173 


140 


3.549 


167 


3.399 


194 


3.274 


221 


3.169 


141 


3.543 


168 


3.394 


195 


3.269 


222 


3.165 


142 


3.537 


169 


3.389 


196 


3.265 


223 


3.162 


143 


3.531 


170 


3.384 


197 


3.261 


224 


3.158 


144 


3.525 


171 


3.380 


198 


3.257 


225 


3.155 


145 


3.519 


172 


3.375 


199 


3.253 


226 


3.151 


146 


3.513 


173 


3.370 


200 


3.249 


227 


3.148 


147 


3.507 


174 


3.365 


201 


3.245 


228 


3.144 


148 


3.501 


175 


3.361 


202 


3.241 


229 


3.141 


149 


3.495 


176 


3.356 


203 


3.237 


230 


3.137 


150 


3.490 


177 


3.351 


204 


3.233 


231 


3.134 


151 


3.484 


178 


3.346 


205 


3.229 







174 



THOMSON ON CARNOT'S 



TABLE II. 

MECHANICAL EFFECT IN FOOT-POUNDS DUE TO A THER- 
MIC UNIT CENTIGRADE, PASSING FROM A BODY, AT ANY 
TEMPERATURE LESS THAN 230 TO A BODY AT 0. 



Superior 
Limit of 
Temper- 
ature. 


Mechanical 
Effect. 


Superior 
Limit of 
Temper- 
ature. 


Mechanical 
Effect. 


Superior 
Limit of 
Temper- 
ature. 


Mechanical 
Effect. 





Ft.-Pouuds. 


<- 


Ft. -Pounds. 


o 


Ft.-Pounds. 


1 


4.960 


38 


179.287 


75 


337.084 


2 


9.906 


39 


183.761 


76 


341.141 


3 


14.838 


40 


188.223 


77 


345.188 


4 


19.756 


41 


192.673 


78 


-> 349. 225 


5 


24.661 


42 


197.111 


79 


353.253 


6 


29.553 


43 


201.537 


80 


357.271 


7 


34.431 


44 


205.951 


81 


361.280 


8 


39.296 


45 


210.353 


82 


365.279 


9 


44.148 


46 


214.743 


83 


369.269 


10 


48.987 


47 


219.121 


84 


373.249 


11 


53.813 


48 


223.487 


85 


377.220 


12 


58.625 


49 


227.842 


86 


381.181 


13 


63.424 


50 


232.185 


87 


385.133 


14 


68.210 


51 


236.516 


88 


389.076 


15 


72.983 


52 


240.835 


89 


393.010 


16 


77.743 


53 


245.143 


90 


396.935 


17 


82.490 


54 


249.439 


91 


400.851 


18 


87.225 


55 


253.724 


92 


404.758 


19 


91.947 


56 


257.997 


93 


408.656 


20 


96.656 


57 


262.259 


94 


412.545 


21 


101.353 


58 


266.509 


95 


416.425 


22 


106.037 


59 


270.748 


96 ; 


420.296 


2& 


110.709 


60 


274.975 


97 


424 159 


24 


115.368 


61 


279.191 


98 


428.013 


25 


120.014 


62 


283.396 


99 


431.858 


26 


124.648 


63 


287.590 


100 


435.695 


27 


129.269 


64 


291.773 


101 


439.524 


28 


133.878 


65 


295.945 


102 


443.344 


29 


138.474 


66 


300.106 


103 


447.156 


30 


143.058 


67 


304.256 


104 


450.960 


31 


147.630 


68 


308.396 


105 


454.756 


32 


152.189 


69 


312.525 


106 


458.544 


33 


156.736 


70 


316.644 


107 


462.324 


34 


161.271 


71 


320.752 


108 


466.096 


35 


165.793 


72 


324.851 


109 


469.860 


36 


170.303 


73 


328.939 


110 


473.617 


37 


174.801 


74 


333.017 


111 


477.366 



MOTIVE POWER OF HEAT, 



175 



TABLE II. (Continued.) 



Superior 
Limit of 
Temper- 
ature, 


Mechanical 
Effect. 


Superior 
jimit of 
temper- 
ature. 


Mechanical 
Effect. 


Superior 
Limit of 
Temper- 
ature. 


Mechanical 
Effect. 





Ft.-Ponnds. 





Ft.-Pounds. 


o 


Ft.-Pounds. 


112 


481.107 


152 


625.105 


192 


760.069 


113 


484.841 


153 


628.578 


193 


763.347 


114 


488.567 


154 


632.046 


194 


766.621 


115 


492.286 


155 


635.508 


195 


769.890 


116 


495.998 


156 


638.965 


196 


773.155 


117 


499.702 


157 


642.416 


197 


776.416 


118 


50*3.399 ! 


158 


645.862 


198 


779.673 


119 


507.088 


159 


649.302 


199 


782.926 


120 


510.770 


160 


652.737 


200 


786.175 


121 


514.445 


161 


656.167 


201 


789.420 


122 


518.113 


162 


659.591 


202 


792.661 


123 


521.174 


163 


663.010 


203 


795.898 


124 


525.428 


164 


666.424 


204 


799.131 


125 


529,075 


165 


669.833 


205 


802.360 


126 


532.715 


166 


673.237 


206 


805.585 


127 


536.348 


167 


676.636 


207 


808.806 


128 


539.975 


168 


680.030 


208 


812.023 


129 


543.595 


169 


683.419 


209 


815.236 


130 


547.209 


170 


686.803 


210 


818.446 


131 


550.816 


171 


690.183 


211 


821.652 


132 


554.417 


172 


693.558 


212 


824.854 


133 


558.051 


173 


696.928 


213 


828.052 


134 


561.597 


174 


700.293 


214 


831.247 


135 


565.176 


175 


703.654 


215 


834.438 


136 


568.749 


176 


707.010 


216 


837.626 


137 


572.316 


177 


710.361 


217 


840.810 


138 


575.877 


178 


713.707 


218 


843.990 


139 


579.432 


179 


717.049 


219 


847.167 


140 


582.981 


180 


720.386 


220 


850.840 


141 


586.524 


181 


723.718 


221 


853.509 


142 


590.061 


182 


727.046 


222 


856.674 


143 


593.592 


183 


730.369 


223 


859.836 


144 


597.117 


184 


733.687 


224 


862.994 


145 


600.636 


185 


737.001 


225 


866.149 


146 


604.099 


186 


740.310 


226 


869.300 


147 


607.656 


187 


743.614 


227 


872.448 


148 


611.157 


188 


746.914 


228 


875.592 


149 


614.652 


189 


750.209 


229 


878.733 


150 


618.142 


190 


753.500 


230 


881.870 


151 


621.626 


191 


756.787 


231 


885.004 



176 THOMSON ON CARNOT'S 

Note on the curves described in Clapeyron's 
graphical method of exhibiting Carnot's TJieory of 
the Steam-Engine. 

39. At any instant when the temperature of the 
water and vapor is t, during the fourth operation 
(see above, 16, and suppose, for the sake of sim- 
plicity, that at the beginning of the first and at 
the end of the fourth operation the piston is ab- 
solutely in contact with the surface of the water), 
the latent heat of the vapor must be precisely equal 
to the amount of heat that would be necessary to 
raise the temperature of the whole mass, if in the 
liquid state, from t to S.* Hence, if v' denote the 
volume of the vapor, c the mean capacity for heat 
of a pound of water between the temperatures S 



* For at the end of the fourth operation the whole mass 
is liquid, and at the temperature S. Now, this state might 
be arrived at by first compressing the vapor into water at 
the temperature t, and then raising the temperature of the 
liquid to 8 ; and however this state may be arrived at, there . 
cannot, on the whole, be any heat added to or subtracted 
from the contents of the cylinder, since, during the fourth 
operation, there is neither gain nor loss of heat. This 
reasoning is, of course, founded on Carnot's fundamental 
principle, which is tacitly assumed in the commonly-re- 
ceived ideas connected with "Watt's law," the "latent 
heat of steam," and "the total heat of steam." 



MOTIVE POWER OF HEAT. 177 

and t, and W the weight of the entire mass,, in 
pounds, we have 

kv'=c(S-t)W. 

Again, the circumstances during the second oper- 
ation are such that the mass of liquid and vapor 
possesses H units .of heat more than during the 
fourth; and consequently, at the instant of the 
second operation, when the temperature is t, the 
volume v of the vapor will exceed v' by an amount 
of which the latent heat is H, so that we have 

* '='+? ' 

40. Now, at any instant, the volume between 
the piston and its primitive position is less than 
the actual volume of vapor by the volume of the 
water evaporated. Hence, if x and x' denote the 
abscissae of the curve at the instants of the second 
and fourth operations respectively, when the tem- 
perature is t, we have 

x = v (TV, x'= v' vv' , 
and, therefore, by the preceding equations, 

. . (a) 
.. (b) 



These equations, along with y = y' = p, . . (c) 



178 THOMSON ON CARNOT'S 

enable us to calculate, from the data supplied by 
Regnault, the abscissa and ordinate for each of the 
curves described above (17) corresponding to any 
assumed temperature t. After the explanations of 
33, 34, 35, 36, it is only necessary to add that c 
is a quantity of which the value is very nearly 
unity, and would be exactly so were the capacity 
of water for heat the same at every temperature 
as it is between and 1; and that the value of 
c(S t), for any assigned values of S and t, is 
found, by subtracting the number corresponding 
to t from the number corresponding to s, in the 
column headed "Nombre des unites de chaleur 
abandonnees par un kilogramme d'eau en descen- 
dant de T a 0," of the last table (at the end of 
the tenth memoir) of Kegnault's work. By 
giving S the value 230, and by substituting suc- 
cessively 220, 210, 200, etc., for t, values for x, y, 
x', y', have been found, which are exhibited in the 
table opposite. 



MOTIVE POWER OF HEAT. 



179 



Tempera- 
tures. 


Volumes to be de- 
scribed by the pis- 
ton, to complete 
the fourth opera- 
tion. 


Volumes from the 
primitive position of 
the piston to those 
occupied at instants 
of the second opera- 
tion. 


Pressures of sat- 
urated steam, in 
pounds on the 
square foot. 


t 


x f 


X 


y = y'=P 





1269. W 


#'4-5.409.5' 


12.832 


10 


639. 6. W 


z'+2. 847.fi" 


25.567 


20 


337.3. W 


x'- 


-l.571.fi 


48.514 


30 


185. 5. W 


x'- 


- .9062.fi 


88.007 


40 


105. 9. W 


x'- 


- .5442.fi 


153.167 


50 


62.62.TF 


x'- 


- .3392.fi 


256.595 


60 


38.19.TF 


x'- 


- .2188.fi 


415.070 


70 


21.94. W 


#'+ .1456.fi 


650.240 


80 


15.38.TF 


x'+ .09962.fi 


989.318 


90 


10.09.T7 


x'+ .06994.fi 


1465.80 


100 


6.744. W 


x'+ .05026.fi 


2120.11 


110 


4. 578. IT 


x'+ .03688.fi 


2999.87 


120 


3.141. TP 


x'-\- .02758.fi 


4160.10 


130 


2.176. W 


'+ .02098.fi 


5663.70 


140 


1.519. W 


x'+ .01625.fi 


7581.15 


150 


1.058. W 


x'-\- .01271.fi 


9990.26 


160 


0.7369. W 


x'- 


h .01010.fi 


12976.2 


170 


0.5085. W 


x'- 


- .008116.fi 


16630.7 


180 


0.3454.TF 


x'- 


- .006592.fi 


21051.5 


190 


0.2267. TF 


x'- 


- .005406.fi 


26341.5 


200 


0.1409.TF 


X'- 


- .004472.fi 


32607.7 


210 


0.0784.TF 


x'4- .003729.fi 


39960.7 


220 


0.3310.TP 


a?'+ .003130.fi 


48512.4 


230 





*'+ .002643.fi 


58376.6 



Appendix. 
(Read April 30, 1849.) 

41. In p. 30 some conclusions drawn by Carnot 
from his general reasoning were noticed ; accord- 
ing to which it appears, that if the value of // for 



180 THOMSON ON CARNOT'S 

any temperature is known, certain information 
may be derived with reference to the saturated 
vapor of any liquid whatever, and, with reference 
to any gaseous mass, without the necessity of ex- 
perimenting upon the specific medium considered. 
Nothing in the whole range of Natural Philosophy 
is more remarkable than the establishment of gen- 
eral laws by such a process of reasoning. We have 
seen, however, that doubt may exist with reference 
to the truth of the axiom on which the entire the- 
ory is founded, and it therefore becomes more than 
a matter of mere curiosity to put the inferences 
deduced from it to the test of experience. The 
importance of- doing so was clearly appreciated by 
Carnot ; and, with such data as he had from the 
researches of various experimenters, he tried his 
conclusions. Some very remarkable propositions 
which he derives from his theory coincide with 
Dulongand Petit's subsequently discovered experi- 
mental laws with reference to the heat developed 
by the compression of a gas ; and the experimen- 
tal verification is therefore in this case (so far as 
its accuracy could be depended upon) decisive. 
In other respects, the data from experiment were 
insufficient, although, so far as they were available 
as tests, they were confirmatory of the theory. 
42. The recent researches of Regnault add im- 



MOTIVE POWER OF HEAT. 181 

mensely to the experimental data available for this 
object, by giving us the means of determining with 
considerable accuracy the values of // within a very 
wide range of temperature, and so affording a trust- 
worthy standard for the comparison of isolated 
results at different temperatures, derived from ob- 
servations in various branches of physical science. 
In the first section of this Appendix the theory 
is tested, and shown to be confirmed by the com- 
parison of the values of /* found above, with those 
obtained by Carnot and Clapeyron from the obser- 
vations of various experimenters on air, and the 
vapors of different liquids. In the second and 
third sections some striking confirmations of the 
theory arising from observations by Dulong, on 
the specific heat of gases, and from Mr. Joule's 
experiments on the heat developed by the com- 
pression of air, are pointed out ; and in conclu- 
sion, the actual methods of obtaining mechanical 
effect from heat are briefly examined with refer- 
ence to their economy. 

I. On the values of ju derived by Carnot and 
Clapeyron from observations on Air, and on 
the Vapors of various liquids. 

43. In Carnot's work, pp. 80-82, the mean 
value of * between and 1 is derived from the 



182 THOMSON ON CARNOT'S 

experiments of Delaroche and Berard on the spe- 
cific heat of gases, by a process approximately 
equivalent to the calculation of the value of 

/ / V for the temperature 1. There are also, in 
vdq/dv 

the same work, determinations of the values of /* 
from observations on the vapors of alcohol and 
water ; but a table given in M. Clapeyron's paper, 
of the values of ^ derived from the data supplied 
by various experiments with reference to the va- 
pors of ether, alcohol, water, and oil of turpen- 
tine, at the respective boiling-points of these 
liquids, affords us the means of comparison through 
a more extensive range of temperature. In the 
cases of alcohol and water, these results ought of 
course to agree with those of Carnot. There are, 
however, slight discrepancies which must be owing 
to the uncertainty of the experimental data.* In 
the opposite table, Carnot's results with reference 
to air, and Clapeyron's results with reference to 
the four different liquids, are exhibited, and com- 
pared with the values of /* which have been given 



* Thus, from Carnot's calculations, we find, in the case 
of alcohol 4.035, and in the case of water 3.648, instead 
of 3.963 and 3.658, which are Clapeyron's results in the 
same cases. 



MOTIVE POWER OF HEAT. 



183 









Values of M. 




Names of the 
Media. 


Temperatures. 


Values of /u.. 


deduced 
from Reg- 
nault's Ob- 


Differ- 
ences. 








servations. 









(Carnot) 






Air . ... 


0.5 


4.377 


4.960 


.383 


Sulphuric 
Ether 


(Boil, pt.) 35.5 


(Clapeyron) 

4.478 


4.510 


.032 


Alcohol 


78.8 


3.963 


4.030 


.071 


Water 


100 


3.658 


3.837 


.179 


Essence of 










Turpentine. 


156.8 


3.530 


3.449 


-.081 



above (Table I.) for the same temperatures, as de- 
rived from Regnault's observations on the vapor 
of water. 

44. It may be observed that the discrepancies 
between the results founded on the experimental 
data supplied by the different observers with ref- 
erence to water at the boiling-point, are greater 
than those which are presented between the results 
deduced frotn any of the other liquids, and water 
at the other temperatures ; and we may therefore 
feel perfectly confident that the verification is 
complete to the extent of accuracy of the obser- 
vations.* The considerable discrepancy presented 

* A still closer agreement must be expected when more 
accurate experimental data are afforded with reference to 
the other media. Mons. Regnault informs me that he is 



184 THOMSON ON CARNOT'S 

by Carnot/s result deduced from experiments on 
air, is not to be wondered at when we consider the 
very uncertain nature of his data. 

45. The fact of the gradual decrease of /* 
through a very extensive range of temperature, 
being indicated both by Kegnault's continuous 
series of experiments and by the very varied ex- 
periment on different media, and in different 
branches of Physical Science, must be considered 
as a striking verification of the theory. 

II. On the Heat developed ~by the Compression of 
Air. 

46. Let a mass of air, occupying initially a 
given volume V, under a pressure P, at a tem- 
perature t, be compressed to a less volume F', 
and allowed to part with heat until it sinks to its 
primitive temperature t. The quantity of heat 
which is evolved maybe determined, according to 
Carnot's theory, when the particular value of //, 

engaged in completing some researches, from which we 
may expect, possibly before the end of the present year, 
to be furnished with all the data for five or six different 
liquids which we possess at present for water. It is there- 
fore to be hoped that, before long, a most important test of 
the validity of Carnot's theory will be afforded. 



MOTIVE POWER OF HEAT. 185 

corresponding to the temperature t y is known. 
For, by 30, equation (6), we have 



where dq is the quantity of heat absorbed, when 
the volume is allowed to increase from v to v + dv\ 
or the quantity evolved by the reverse operation. 
Hence we deduce 



JH 



(8) 



Now, is constant, since the temperature 

remains unchanged ; and therefore we may at 
once integrate the second number. By taking it 
between the limits V and V, we thus find 



. ,. . 

where Q denotes the required amount of heat 
evolved by the compression from Vto P'. This 
expression may be modified by employing the equa- 
tions P V = P' V = 2).v Q (1 + Et) ; and we thus 
obtain 

EPV V EP'V . V 

Q = g 7 = log -- 



y 

* The Napierian logarithm of -~ is here understood. 



186 THOMSON ON CARNOT'ti 

From this result we draw the following conclu- 
sion : 

47. Equal volumes of all elastic fluids, taken at 
the same temperature and pressure, when com- 
pressed to smaller equal volumes, disengage equal 
quantities of heat. 

This extremely remarkable theorem of Carnot's 
was independently laid down as a probable experi- 
mental law by Dulong, in his " Recherches sur la 
Chaleur Specifique des Flu ides Elastiques," and it 
therefore affords a most powerful confirmation of 
the theory.* 

* Carnot varies the statement of his theorem, and illus- 
trates it in a passage, pp. 81, 82, of which the following is 
translation : 

" When a gas varies in volume without any change of tem- 
perature, the. quantities of heat absorbed or evolved by tJiis gas 
are in arithmetical progression, if the augmentation or dimi- 
nutions of volume are in geometrical progression. 

" When we compress a litre of air maintained at the tem- 
perature 10, and reduce it to half a litre, it disengages a 
certain quantity of heat. If, again, the volume be reduced 
from half a litre to a quarter of a litre, from a quarter to 
an eighth, and so on the quantities of heat successively 
evolved will be the same. 

"If, in place of compressing the air, we allow it to ex- 
pand to two litres, four litres, eight litres, etc., it will be 
necessary to supply equal quantities of heat to maintain the 
temperature always at the same degree. " 



MOTIVE POWER OF HEAT. 187 

48. In some very remarkable researches made 
by Mr. Joule upon the heat developed by the 
compression of air, the quantity of heat produced 
in different experiments has been ascertained with 
reference to the amount of work spent in the 
operation. To compare the results which he has 
obtained with the indications of theory, let us de- 
termine the amount of work necessary actually to 
produce the compression considered above. 

49. In the first place, to compress the gas from 
the volume v + dv to v, the work required is pdv, 
or, since 

PV =JV>.(1 + Et), 



Hence, if we denote by W the total amount of 
work necessary to produce the compression from 
Fto V 9 we obtain, by integration, 

W = p Q V Q (I + fit) log y,. 

Comparing this with the expression above, we find 
W_(1+M) . . 

-Q = JE~ 
50. Hence we infer that 
(1) The amount of work necessary to produce 
a unit of heat by the compression of a gas is the 
same for all gases at the same temperature; 



188 



THOMSON ON CARNOT'S 



(2) And that the quantity of heat evolved in 
all circumstances, when the temperature of the 
gas is given, is proportional to the amount of work 
spent in the compression. 

51. The expression for the amount of work nec- 
essary to produce a unit of heat is 

Ml + Et) 

E 

and therefore Kegnault's experiments on steam 
are available to enable us to calculate its value for 
any temperature. By finding the values of yu at 
0, 10, 20, etc., from Table I., and by substi- 
tuting successively the values 0, 10, 20, etc., for f, 
the following results have been obtained : 
TABLE OF THE VALUES OF 



Work requisite to 
produce a unit 
of Heat by the 
compression of 


Temperature 
of the Gas. 


Work requisite to 
produce a unit 
of Heat by the 
compression of 


Temperature 
of the Gas. 


a Gas. 




a Gas. 




Ft. -pounds. 





Ft.-pounds. 


o 


1357.1 





1446.4 


120 


1368.7 


10 


1455.8 


130 


1379.0 


20 


1465.3 


140 


1388.0 


30 


1475.8 


150 


1395.7 


40 


1489.2 


160 


1401.8 


50 


1499.0 


170 


1406.7 


60 


1511.3 


180 


1412.0 


70 


1523.5 


190 


1417.6 


80 


1536.5 


200 


1424.0 


90 


1550.2 


210 


1430.6 


100 


1564.0 


220 


1438.2 


110 


1577.8 


230 



MOTIVE POWER OF HEAT. 189 

Mr. Joule's experiments were all conducted at 
temperatures from 50 to about GO Fahr., or from 
10 to 16 Cent.; and consequently, although some 
irregular differences in the results, attributable to 
errors of observation inseparable from experiments 
of such a very difficult nature, are presented, 
no regular dependence on the temperature is ob- 
servable. From three separate series of experi- 
ments, Mr. Joule deduces the following numbers 
for the work, in foot-pounds, necessary to produce 
a thermic unit Fahrenheit by the compression of 
a gas. 

820, 814, 760. 

Multiplying these by 1.8, to get the corresponding 
number for a thermic unit Centigrade, we find 

1476, 1465, and 1368. 

The largest of these numbers is most nearly 
conformable with Mr. Joule's views of the relation 
between such experimental '''equivalents," and 
others which he obtained in his electro-magnetic 
researches ; but the smallest agrees almost perfect- 
ly with the indications of Carnot's theory ; from 
which, as exhibited in the preceding table, we 
should expect, from the temperature in Mrc Joule's 
experiments, to find a number between 1369 and 
1379 as the result.* 

* The best figure (1896) is J - 778 ft.-lbs. = 1 B.T.U., or 
J = 426.8 kgin. = 1 calorie, aud probably with great ac- 
curacy. 



190 ' THOMSON ON CARNOT'S 

III. On the Specific Heats of Gases. 

52. The following proposition is proved by 
Carnot as a deduction from his general theorem 
regarding the specific heats of gases. 

The excess of specific heat* under a constant 
pressure above the specific heat at a constant volume, 
is the same for all gases at the same temperature 
and pressure. 

53. To prove this proposition, and to determine 
an expression for the "excess" mentioned in its 
enunciation, let us suppose a unit of volume of a 
gas to be elevated in temperature by a small 
amount, T. The quantity of heat required to do 
this will be AT, if A denote the specific heat at a 
constant volume. Let us next allow the gas to 

O 

expand without going down in temperature, until 
its pressure becomes reduced to its primitive value. 

ET 

The expansion which will take place will be -- , 

1 -j- Et 

if the temperature be denoted by t ; and hence, 
by (8), the quantity of heat that must be supplied, 
to prevent any lowering of temperature, will be 



Et 



* Or the capacity of a unit of volume for heat. 



MOTIVE POWER OF HEAT. 191 

Hence the total quantity added is equal to 



But, since B denotes the specific heat under con- 
stant pressure, the quantity of heat requisite to 
bring the gas into this state, from its primitive 
condition, is equal to Br\ and hence we have 



IV. Comparison of the Relative Advantages of the 
Air-engine and Steam-engine. 

54. In the use of water-wheels for motive power, 
the economy of the engine depends not only upon 
the excellence of its adaptation for actually trans- 
mitting any given quantity of water through it, 
and producing the equivalent of work, but upon 
turning to account the entire available fall; so, as 
we are taught by Carnot, the object of a thermo- 
dynamic engine is to economize in the best possible 
way the transference of all the heat evolved, from 
bodies at the temperature of the source, to bodies 
at the lowest temperature at which the herat can be 
discharged. With reference, then, to any engine of 
the kind, there will be two points to be considered: 

(1) The extent of ihefall utilized. 



192 THOMSON ON CARNOT'8 

(2) The economy of the engine, with the fall 
which it actually uses. 

55. In the first respect, the air-engine, as Carnot 
himself points out, has a vast advantage over the 
steam-engine; since the temperature of the hot 
part of the machine may be made very much 
higher in the air-engine than would be possible in 
the steam-engine, on account of the very high 
pressure produced in the boiler, by elevating the 
temperature of the water which it contains to any 
considerable extent above the atmospheric boiling- 
point. On this account a "perfect air-engine " 
would be a much more valuable instrument than a 
" perfect steam-engine." * 

* Carnot suggests a combination of the two principles, 
with air as the medium for receiving the heat at a very 
high temperature from the furnace; ^nd a second medium, 
alternately in the state of saturated vapor and liquid water, 
to receive the heat, discharged at aii intermediate temper- 
ature from the air, and transmit it to the coldest part of 
the apparatus. It is possible that a complex arrangement 
of this kind might be invented which would enable us to 
take the heat at a higher temperature, and discharge it at a 
lower temperature than would be practicable in any simple 
air-engine or simple steam-engine. If so, it would no 
doubt be equally possible, and perhaps more convenient, 
to employ steam alone, but to use it at a very high tem- 
perature not in contact with water in the hottest part of 



MOTIVE POWER OF HEAT. 193 

Neither steam-engines nor air-engines, however, 
are nearly perfect; and we do not know in which 
of the two kinds of machine the nearest approach 
to perfection may be actually attained. The beau- 
tiful engine invented by Mr. Stirling of Galston 
may be considered as an excellent beginning for 
the air-engine;* and it is only necessary to com- 
pare this with Newcomen's steam-engine, and con- 
sider what Watt has effected, to give rise to the 
most sanguine anticipations of improvement. 

V. On the Economy of Actual Steam-engines. 

56. The steam-engine being universally em- 
ployed at present as the means for deriving motive 
power from heat, it is extremely interesting to 
examine, according to Carnot's theory, the econ- 
omy actually attained in its use. In the first 



the apparatus, instead of, as in the steam-engine, always 
in a saturated state. 

* It is probably this invention to which Carnot alludes 
in the following passage: "II a ete fait, dit-on, tout re- 
cemment en Angleterre des essais heureujt sur le de- 
veloppement de la puissance motrice par 1'action de la 
chaleur sur 1'air atmospherique. Nous ignorons entiere- 
ment ne quoi ces essais ont consiste, si toutefois ils sont 
reels." 



194 THOMSON ON CARNOT'S 

place we remark, that out of the entire "fall" 
from the temperature of the coals to that of the 
atmosphere it is only part that from the tem- 
perature of the boiler to the temperature of the 
condenser that is made available; while the very 
great fall from the temperature of the burning 
coals to that of the boiler, and the comparatively 
small fall from the temperature of the condenser 
to that of the atmosphere, are entirely lost as 
far as regards the mechanical effect which it is 
desired to obtain. We infer from this, that the 
temperature of the boiler ought to be kept as 
high as, according to the strength, is consistent 
with safety, while that of the condenser ought 
to be kept as nearly down at the atmospheric 
temperature as possible. To take the entire ben- 
efit of the actual fall, Carnot showed that the 
" principle of expansion" must be pushed to the 
utmost.* 

* From this point of view, we see very clearly how im- 
perfect is the steam-engine, even after all Watt's improve- 
ments. For to " push the principle of expansion to the 
utmost, " we must allow the steam, before leaving the cyl- 
inder, to expand until its pressure is the same as that of 
the vapor in the condenser. According to "Watt's law/ 5 
its temperature would then be the same as (actually a little 
above, as Regnault has shown) that of the condenser, and 



MOTIVE POWER OF HEAT. 195 

57. To obtain some notion of the economy which 
has actually been obtained, we may take the al- 
leged performances of the best Cornish engines, 
aud some other interesting practical cases, as ex- 
amples.* 

(1) The engine of the Fowey Consols mine was 
reported, in 1845, to have given 125,089,000 foot- 
pounds of effect, for the consumption of one 
bushel or 94 Ibs. of coals. Now the average amount 
evaporated from Cornish boilers, by one pound of 
coal, is 8 Ibs. of steam ; and hence for each 
pound of steam evaporated 156,556 foot-pounds of 
work are produced. 

The pressure of the saturated steam in the boiler 
may be taken as 3J atmospheres;! and, conse- 



hence the steam-engine worked in this most advantageous 
way has in reality the very fault that Watt found in New- 
comen's engine. This defect is partially remedied by 
Hornblower's system of using a separate expansion cylin- 
der, an arrangement the advantages of which did not 
escape Caruot's notice, although they have not been recog- 
nized extensively among practical engineers, until within 
the last few years. 

* I am indebted to the kindness of Professor Gordon of 
Glasgow for the information regarding the various cases 
given in the text. 

f In different Cornish engines, the pressure in the boiler 



196 THOMSON ON CARNOT'S 

quently, the temperature of the water will be 140. 
Now (Regnault, end of Memoire X.) the latent 
heat of a pound of saturated steam at 140 is 508, 
and since, to compensate for each pound of steam 
removed from the boiler in the working of the 
engine, a pound of water, at the temperature of 
the condenser, which may be estimated at 30, is 
introduced from the hot- well; it follows that 618 
units of heat are introduced to the boiler for each 
pound of water evaporated. But the work pro- 
duced, for each pound of water evaporated, was 
found above to be 156,556 foot-pounds. Hence 
JJ5 AV~> or 253 foot-pounds, is the amount of work 
produced for each unit of heat transmitted through 
the Fowey Consols engine. Now in Table II. we 
find 583.0 as the theoretical effect due to a unit de- 
scending from 140 to 0, and 143 as the effect due 
to a unit descending from 30 to 0. The difference 
of these numbers, or 440,* is the number of foot- 
is from 2| to 5 atmospheres; and, therefore, as we find 
from Regnault's table of the pressure of saturated steam, 
the temperature of the water in the boiler must, in all of 
them, lie between 128 and 152. For the better class of 
engines, the average temperature of the water in the boiler 
may be estimated at 140, the corresponding pressure of 
steam being 3| atmospheres. 
* This number agrees very closely with the number 



MOTIVE PO WEE OF HEAT. 197 

pounds of work that & per feet engine with its boiler 
at 140 and its condenser at 30 would produce for 
each unit of heat transmitted. Hence the Fowey 
Consols engine, during the experiments reported 
on, performed f JJ of its theoretical duty, or 57J- 
per cent. 

(2) The best duty on record, as performed by an 
engine at work (not for merely experimental pur- 
poses), is that of Taylor's engine, at the United 
Mines, which in 1840 worked regularly for sev- 
eral months at the rate of 98,000,000 foot-pounds 
for each bushel of coals burned. This is ^/j, or 
.784 of the experimental duty reported in the case 
of the Fowey Consols engine. Hence the best 
useful work on record is at the rate of 198.3 foot- 
pounds for each unit of heat transmitted, and is 
-Y^ 3 , or 45 per cent of the theoretical duty, on 
the supposition that the boiler is at 140 and the 
condenser at 30. 

(3) French engineers contract (in Lille, in 1847, 
for example) to make engines for mill-power which 
will produce 30,000 metre-pounds or 98,427 foot- 
pounds of work for each pound of steam used. If 

corresponding to the fall from 100 to 0, given in Table 
II. Hence, tlie fall from 140 to 30 of the scale of the 
air-thermometer is equivalent, with reference to motive 
power, to the fall from 100 to 0. 



198 THOMSON ON CAENOT'S 

we divide this by 618, we find 159 foot-pounds for 
the work produced by each unit of heat. This is 
36.1 per cent of 440, the theoretical duty.* 

(4) English engineers have contracted to make 
engines and boilers which will require only 3J Ibs. 
of the best coal per horse-power per hour. Hence 
in such engines each pound of coal ought to pro- 
duce 565,700 foot-pounds of work, and if 7 Ibs. of 
water be evaporated by each pound of coal, there 
would result 83,814 foot-pounds of work for each 
pound of water evaporated. If the pressure in the 

* It being assumed that the temperatures of the boiler 
and condenser are the same as those of the Cornish en- 
gines. If, however, the pressure be lower, two atmos- 
pheres, for instance, the numbers would stand thus: The 
temperature in the boiler would be only 121. Conse- 
quently, for each pound of steam evaporated, only 614 
units of heat would be required ; and therefore the work 
performed for each unit of heat transmitted would be 
160.3 foot-pounds, which is more than according to the 
estimate in the text. On the other hand, the range of tem- 
peratures, or the fall utilized, is only from 131 to 30, in- 
stead of from 140 to 30, and, consequently (Table II.), the 
theoretical duty for each unit of heat is only 371 foot- 
pounds. Hence, if the engine, to work according to the 
specification, requires a pressure of only 15 Ibs. on the 
square inch (i.e., a total steam-pressure of two atmos- 
pheres), its performance is - l $%', or 43.2 per cent of its 
theoretical duty. 



MOTIVE POWER OF HEAT. 199 

boiler be 3^ atmospheres (temperature 140) the 
amount of work for each unit of heat will be 
found, by dividing this by 618, to be 130.7 foot- 
pounds, which is -VA 5 or 29.7 per cent of the theo- 
retical duty.* 

(5) The 'actual average of work performed by 
good Cornish engines and boilers is 55,000,000 
foot-pounds for each bushel of coal, or less than 
half the experimental performance of the Fowey 
Consols engine, more than half the actual duty 
performed by the United Mines engine in 1840; 
in fact, about 25 per cent of the theoretical duty. 

(6) The average performances of a number of 
Lancashire engines and boilers have been recently 
found to be such as to require 12 Ibs. of Lanca- 
shire coal per horse-power per hour (i.e., for per- 
forming 60 X 33,000 foot-pounds), and of a num- 
ber of Glasgow engines such as to require 15 Ibs. 
(of a somewhat inferior coal) for the same effect. 
There are, however, more than twenty large en- 
gines in Glasgow at presentf which work with a 

* If, in this case again , the pressure required in the boiler 
to make the engine work according to the contract were 
only 15 Ibs. on the square inch, we should have a different 
estimate of the economy, for which see Table B, at the 
end of this paper. 

f These engines are provided with separate expansion 



200 THOMSON ON CARNOT'S 

consumption of only 6J Ibs. of dross, equivalent 
to 5 Ibs. of the best Scotch or 4 Ibs. of the best 
Welsh coal, per horse-power per hour. The 
economy may be estimated from these data, as in 
the other cases, on the assumption which, with 
reference to these, is the most probable we can 
make, that the evaporation produced by a pound 
of best coal is 7 Ibs. of steam. 

58. The following tables afford a synoptic view 
of the performances and theoretical duties in the 
various cases discussed above. 

In Table A the numbers in the second column 
are found by dividing the numbers in the first by 
8J in cases (1), (2), and (5), and by 7 in cases (4), 
(6), and (7), the estimated numbers of pounds of 
steam actually produced in the different boilers by 
the burning of 1 Ib. of coal. 

The numbers in the third column are found 
from those in the second, by dividing by 618 in 
Table A, and 614 in Table B, which are respec- 
tively the quantities of heat required to convert a 
pound of water taken from the hot-well at 30, 
into saturated steam, in the boiler, at 140 or at 
121. 



cylinders, which have been recently added to them by 
Mr. M 'Naught of Glasgow. 



MOTIVE POWER OF HEAT. 201 

With reference to the cases (3), (4), (6), (7), the 
hypothesis of Table B is probably in general nearer 
the truth than that of Table A. In (4), (6), and 
(7), especially upon hypothesis B, there is much 
uncertainty as to the amount of evaporation that 
will be actually produced by 1 Ib. of fuel. The 
assumption on which the numbers in the second 
column in Table B are calculated, is, that each 
pound of coal will send the same number of units 
of heat into the boiler, whether hypothesis A or 
hypothesis B be followed. Hence, except in the 
case of the French contract, in which the evapora- 
tion, not the fuel, is specified, the numbers in the 
third column are the same as those in the third 
column of Table A. 



202 



THOMSON ON CARNOT'S 



TABLE A. 

VARIOUS ENGINES IN WHICH THE TEMPERATURE OP THE 
BOILER is 140 C. AND THAT OF THE CONDENSER 30 C. 

Tfworetical Duty for each Unit of Heat transmitted, 440* 
foot-pounds. 



CASES. 


Work pro- 
duced for 
each Ib. of 


Work pro- 
duced for 
each Ib. of 


Work pro- 
duced for 
each unit 


Percent- 
age of 
theo- 




coal con- 


watereva- 


of heat 


retical 




sumed. 


porated. 


transmit- 
ted. 


duty. 




Ft.-lbs. 


Ft.-lbs. 


Ft.-lbs. 




(1) Fowey Consols experi- ) 
ment, reported in 1845 j 


1,330,734 


156,556 


253 


57.5 


(2) Taylor's engine at the i 
United Mines, work- v 
ing in 1840 f 


1,042,553 


122,653 


198.4 


45.1 


(3) French engines, accord- ) 
ing to contract f 




98,427 


159 


36.1 


(4) English engines, ac- ( 
cording to contract. . f 


565,700 


80,814 


130.8 


29.7 


(5) Average actual per- i 










formance of Cornish V 


585,106 


68,836 


111.3 


25.3 


engines ) 










(6) Common engines, con-~| 
suming 12 IDS. of best ', 
coal per horse-power [ 


165,000 


23,571 


38.1 


8.6 


per hour J 










(7) Improved engines withl 
expansion cylinders, 










consuming an equiva- 1 
lent to 4 Ibs. of best f 


495,000 


70,710 


114.4 


26 


coal per horse-power 










per hour 





















* [Note added March 15, 1881. Total work for thermal unit, 1390 
(Joule), 377.1 corrected by the dynamical theory, March 15, 1851. 
377.1= .2713X1390, 
853 = .1820 X 1390 = X 1390.] 



MOTIVE POWER OF HEAT. 



203 



TABLE B. 

VARIOUS ENGINES IN WHICH THE TEMPERATURE OF THE 
BOILER is 121 C.* AND THAT OP THE CONDENSER 30 C. 

Theoretical Duty for each Unit of Heat transmitted, 371 
foot-pounds. 



CASES. 


Work pro- 
duced for 
each Ib. of 
coal con- 
sumed. 


Work pro- 
duced for 
each Ib. of 
water eva- 
porated. 


Work pro- 
duced for 
each unit 
of heat 
transmit- 
ted. 


Per- 
cent- 
age of 
theo- 
retical 
duty. 




Ft. -Ibs. 


Ft.-lbs. 


Ft.-lbs. 




(3) French engines, accord- 




98,427 


160.3 


43.2 


ing to contract 










(4) English engines, ac- 
cording to contract.. 


565,700 


fit x 80,814 


130.8 


35 


(6) Common engines, con- 










suming 12 Ibs. of coal 
per horse-power per 
hour 


165,000 


fit x 23, 571 


38.1 


10.3 


(7) Improved engines with 










expansion cylinders, 
consuming an equiva- 
lent to 4 Ibs. best coal 


495,000 


iif x 70,710 


114.4 


30.7 


per horse-power per 
hour 





















* Pressure 15 Ibs. on the square inch. 




APPENDIX A. 



EXTRACTS FROM UNPUBLISHED WRITINGS 
OF CARNOT. 

I. NOTES. 

LET us first open at the memoranda relating to 
his daily occupations : 

"Plan in the morning the work of the day, and 
reflect in the evening on what has been done." 

" Carry when walking a book, and a note-book 
to preserve the ideas, and a piece of bread in order 
to prolong the walk if need be." 

"Vary the mental and bodily exercises with 
dancing, horsemanship, swimming, fencing with 
sword and with sabre, shooting with gun and pistol, 
skating, the sling, stilts, tennis, bowls; hop on one 
foot, cross the arms, jump high and far, turn on 
one foot propped against the wall, exercise in shirt 
in the evening to get up a perspiration before going 
to bed ; turning, joinery, gardening, reading while 
walking, declamation, singing, violin, versification, 
musical composition ; eight hours of sleep ; a walk 
on awakening, before and after eating ; great so- 

205 



206 APPENDIX A. 

briety ; eat slowly, little, and often ; avoid idle- 
ness and useless meditation/' 

Then come more general precepts : 

" Adopt good habits when I change my method 
of life." 

"Never turn to the past unless to enlighten the 
future. Regrets are useless/' 

"Form resolutions in advance in order not 
to reflect during action. Then obey thyself 
blindly." 

"The promptitude of resolutions most fre- 
quently accords with their justice/' 

" Yield frequently to the first inspiration. Too 
much meditation on the same subject ends by sug- 
gesting the worst part, or at least causes loss of 
precious time." 

"Suffer slight disagreeables without seeming to 
perceive them, but repulse decisively any one who 
evidently intends to injure or humiliate you." 

" One should never feign a character that he 
has not, or affect a character that he cannot sus- 
tain." 

" Self-possession without self-sufficiency. Cour- 
age without effrontery." 

" Make intimate acquaintances only with much 
circumspection ; perfect confidence in those who 



APPENDIX A. 207 

have been thoroughly tested. Nothing to do with 
others/' 

" Question thyself to learn what will please 
others/' 

" No useless discourse. All conversation which 
does not serve to enlighten ourselves or others, 
to interest the heart or amuse the mind, is hurt- 
ful." 

" Speak little of what you know, and not at all 
of what you do not know." 

"Why not say more frequently, 'I do not 
know'?" 

" Speak to every one of that which he knows 
best. This will put him at his ease, and be profit- 
able to you/' 

"Abstain from all pleasantry which could 
wound." 

" Employ only expressions of the most perfect 
propriety. " 

" Listen attentively to your interlocutor, and so 
prepare him to listen in the same way to your reply, 
and predispose him in favor of your arguments." 

" Show neither passion nor weariness in discus- 
sion." 

' ' Never direct an argument against any one. If 
you know some particulars against your adversary, 
you have a right to make him aware of it to keep 



208 APPENDIX A. 

him under control, but proceed with discretion, 
and do not wound him before others." 

" When discussion degenerates into dispute, be 
silent; this is not to declare yourself beaten." 

" How much modesty adds to merit ! A man of 
talent who conceals his knowledge is like a branch 
bending under a weight of fruit." 

"Why try to be witty? I would rather be 
thought stupid and modest than witty and pre- 
tentious." 

" Men desire nothing so much as to make them- 
selves envied." 

' ' Egotism is the most common and most hated 
of all vices. Properly speaking, it is the only one 
which should be hated." 

" The pleasures of self-love are the only ones 
that can really be turned into ridicule. " 

"I do not know why these two expressions, 
good sense and common sense, are confounded. 
There is nothing less common than good sense." 

' ' The strain of suffering causes the mind to 
decay." 

We will quote one of those misanthropic sallies 
the rarity of which we are glad to remark : 

" It must be that all honest people are in the 
galleys; only knaves are to be met with elsewhere." 



APPENDIX A. 209 

But serenity of mind returns immediately after 
the above : 

" I rejoice for all the misfortunes which might 
have happened to me, and which I have escaped." 

' i Life is a short enough passage. I am half the 
journey. I will complete the remainder as I can." 

"Hope being the greatest of all blessings, it is 
necessary, in order to be happy, to sacrifice the 
present to the future." 

" Let us not be exacting; perfection is so rare/' 

" Indulgence ! Indulgence !" 

"The more nearly an object approaches perfec- 
tion, the more we notice its slightest defects." 

"To neglect the opportunity of an innocent 
pleasure is a loss to ourselves. It is to act like a 
spendthrift." 

"Recherche pleasures cause simple pleasures to 
lose all their attractions." 

"It may sometimes be necessary to yield the 
right, but how is one to recover it when wanted ?" 

"Love is almost the only passion that the good 
man may avow. It is the only one which accords 
with delicacy." 

"Do nothing that all the world may not know." 

" The truly wise man is he who loves virtue for 
its own sake." 



210 APPENDIX A. 

" We say that man is an egotist, and neverthe- 
less his sweetest pleasures come to him through 
others. He only tastes them on condition of shar- 
ing them." 

" If one could continually satisfy his desires, he 
would never have time to desire. Happiness then 
is necessarily composed of alternatives. It could 
not exist at*a constant level." 

On the subject of nations and conquerors : 

" To each conqueror can be said, when he has 
ceased tormenting our poor globe, ' Would you 
not have been able to tilt equally well against a 
little globe of pasteboard ? ' ' 

" The laws of war, do they say ? As if war 
were not the destruction of all laws." 

" War has been represented as necessary to pre- 
vent the too rapid increase of the population, but 
war mows down the flower of the young men, 
while it spares the men disgraced by nature. 
Hence it tends to the degeneration of the species." 

Then the writer turns his shafts against medi- 
cine : 

" In some respects medicine is directly opposed 

to the will of nature, which tends to perpetuate the 

strongest and best of the species, and to abandon 



APPENDIX A. 211 

the delicate to a thousand forms of destruction. 
This is what occurs among animals and savage 
men. Only the most robust attain the adult age, 
and these only reproduce the species. Medicine 
and the aids of the social state prolong the lives of 
feeble creatures whose posterity is usually equally 
feeble. Among the Spartans, barbarous regula- 
tions put an end to the existence of mal-formed 
infants, that the strength and beauty of the race 
might be preserved. Such regulations are anti- 
pathetic to our customs; nevertheless it might be 
desirable that we should devote ourselves to the 
preservation of the human race from the causes of 
weakness and degeneracy/' 

" The decadence of the Greeks and Eomans 
without change of race proves the influence of in- 
stitutions upon customs." 

We will give here a fragment on political econ- 
omy, to show the variety contained in the pages on 
which we draw : 

" According to the system of modern economists, 
it would be desirable that the government should 
interfere as little as possible in the commerce and 
industry of the country. Nevertheless we cannot 
deny that in certain circumstances this interven- 
tion is very useful." 



212 APPENDIX A. 

" Taxes are regarded by economists as an evil, 
but as a necessary evil, since they provide for pub- 
lic expenses. Consequently, economists think that 
if the government possessed sufficient revenues, in 
domains for example, the suppression of all taxes 
would be a desirable measure." 

" Taxes are a means of influencing production 
and commerce to give to them a direction which 
they would not naturally have taken. Such an 
influence may undoubtedly have disagreeable con- 
sequences if the taxes are imposed without dis- 
crimination or exclusively for a fiscal purpose, but 
it is entirely otherwise if wisdom and tact preside 
at their institution." 

" A tax on the rent of a farm would be much 
better than a tax on the land itself. Proprietors 
then could only avoid taxes by themselves improv- 
ing their property. As it is, they merely collect 
the rents, and usually employ their surplus in un- 
productive expenditure, while the proprietary 
farmers voluntarily devote theirs to the improve- 
ment of the land." 

" A tax on the farms would then result in the 
proprietors themselves working the lands, and tnis 
would mean better cultivation, and improvements 
which would yield returns indeed, but at too re- 
mote a period for the tenant. It would tend to a 



APPENDIX A 213 

division of landed property, men of small fortune 
uniting in the purchase with capitalists who seek 
only the rent or payment for the land." 

" Great capitalists could not themselves culti- 
vate vast extents of land, and not wanting to di- 
minish their revenues by renting them, would be 
induced to sell portions suitable for cultivation by 
their new owners, and would then carry their 
money into new industrial and commercial enter- 
prises. " 

" The competition of the sellers would cause a 
momentary fall in the price of the lands, and would 
enable small farmers to become land-owners. The 
number of vast estates often badly managed would 
then be diminished, and considerable fortunes, 
changing hands more easily, would naturally pass 
into those which would be most likely to increase 
their value." 

"Proprietors, becoming cultivators to escape the 
taxes, would settle in the country, where their pres- 
ence would disseminate intelligence and comfort; 
their revenues, before spent unprofitably, would 
then pay expenses and improvements on their 
propert} r ." 

<( The establishment of such a tax would cer- 
tainly find many opponents among proprietors, 
landed non-cultivators who form in fact the influ- 



214 APPENDIX A. 

ential personnel in the state, for it is they who 
usually make the laws." 

" Perhaps it would be necessary to weaken their 
opposition by not subjecting the actual proprietors 
to the new tax, which might take effect only with 
the next change either by sale or by inheritance. 
A restriction of the right of transfer would also 
facilitate the passage from one situation to the 
other. All changes in taxes should, as a general 
thing, be made gradually, in order to avoid sudden 
changes of fortune." 

" We may consider the renting of a property 
for several years as a sale of the usufruct during 
the time of the lease. Now nine years' possession, 
for example, is equal to more than a third of the 
value of the property, supposing the annual prod- 
uct to be one twentieth of the capital. It would 
then be reasonable to apply to this sort of sale the 
laws which govern that of landed property, and 
consequently the mutation tax. The person who 
cannot or will not cultivate his soil, instead of 
alienating the property itself, binds himself to 
alienate the usufruct for a time, and the price is 
paid at stated intervals instead of all at once. 
There is farm rent." 

"Now it is by a fiction that the purchaser pays 
the mutation tax. In fact, it is always the seller 



APPENDIX A. 215 

who pays it. The buyer compares the money that 
he spends with the advantage that he gains, and 
this comparison determines it. If he did not make 
money out of it he would not buy it. When the 
registration tax did not exist, the purchaser had to 
pay the same sum for the same purpose, and this 
sum went into the pocket of the seller/' 

" Proprietors of lands, then, after all, have to 
bear the mutation taxes. All increase of these 
taxes is a loss for them, and these taxes are heav- 
ier on the small proprietors than on the large, be- 
cause their changes are more frequent. The tax 
on the farms, on the contrary, would bear more 
heavily on large estates. " 

" The tax on farms not affecting the owners of 
timber, would be made up by a tax on the felling, 
a very justifiable tax, for standing timber is landed 
property. Standing timber is often worth much 
more than the land on which it stands." 

Finally, we will give some thoughts which reveal 
the religious sentiments of Sadi Carnot: 

''Men attribute to chance those events of the 
causes of which they are ignorant. If they suc- 
ceed in divining these causes, chance disappears. 
To say that a thing has happened by chance, 



216 APPENDIX A. 

is to say that we have not been able to foresee it. 
I do not myself believe that any other acceptation 
can be given to this word. What to an ignorant 
man is chance, cannot be chance to one better in- 
structed." 

"If human reason is incapable of discovering 
the mysteries of Divinity, why has not Divinity 
made human reason more clear-sighted ?" 

"God cannot punish man for not believing 
when he could so easily have enlightened and con 
vinced him." 

"If God is absolutely good, why should 
He punish the sinner for all eternity, since 
He does not lead him to good, or give him an 
example ?" 

"According to the doctrine of the church, God 
resembles a sphinx proposing enigmas, and devour- 
ing those who cannot guess them." 

" The church attributes to God all human pas- 
sions anger, desire for vengeance, curiosity, tyr- 
anny, partiality, idleness." 

" If Christianity were pruned of all which is 
not Christ, this religion would be the simplest in 
the world." 

""What motives have influenced the writers who 
have rejected all religious systems ? Is it the con- 
viction that the ideas which they oppose are all 



APPENDIX A. 217 

injurious to society? Have they not rather in- 
cluded in the same proscription religion and the 
abuse of it ?" 

" The belief in an all-powerful Being, who loves 
us and watches over us, gives to the mind great 
strength to endure misfortune." 

" A religion suited to the soul and preached by 
men worthy of respect would exercise the most 
salutary influence upon society and customs." 



II. NOTES OF SADI CARNOT OK MATHEMATICS, 
PHYSICS, AND OTHEE SUBJECTS. 

Up to the present time the changes caused in 
the temperature of bodies by motion have been 
very little studied. This class of phenomena mer- 
its, however, the attention of observers. When 
bodies are in motion, especially when that motion 
disappears, or when it produces motive power, re- 
markable changes take place in the distribution of 
heat, and perhaps in its quantity. 

We will collect a few facts which exhibit this 
phenomenon most clearly. 

1. The Collision of Bodies. We know that in 
the collision of bodies there is always expenditure 
of motive power. Perfectly elastic bodies only form 
an exception, and none such are found in nature. 



218 APPENDIX A. 

We also know that always in the collision of 
bodies there occurs a change of temperature, an 
elevation of temperature. We cannot, as did M. 
Berthollet, attribute the heat set free in this case 
to the reduction of the volume of the body; for 
when this reduction has reached its limit the liber- 
ation of heat would cease. Now this does not oc- 
cur. 

It is sufficient that the body change form by per- 
cussion, without change of volume, to produce dis- 
engagement of heat. 

If, for example, we take a cube of lead and strike 
it successively on each of its faces, there will always 
be heat liberated, without sensible diminution in 
this disengagement, so long as the blows are con- 
tinued with equal force. This does not occur when 
medals are struck. In this case the metal cannot 
change form after the first blows of the die, and 
the effect of the collision is not conveyed to the 
medal, but to the threads of the screw which are 
strained, and to its supports. 

It would seem, then, that heat set free should 
be attributed to the friction of the molecules of 
the metal, which change place relatively to each 
other, that is, the heat is set free just where the 
moving force is expended. 

A similar remark will apply in regard to the col- 



APPENDIX A. 219 

lision of two bodies of differing hardness lead and 
iron for instance. The first of these metals be- 
comes very hot, while the second does not vary sen- 
sibly in temperature. But the motive power is 
almost wholly exhausted in changing the form of 
the first of these metals. We may also cite, as a 
fact of the same nature, the heat produced by the 
extension of a metallic rod just before it breaks. 
Experiment has proved that, other things being 
equal, the greater the elongation before rupture, 
the more considerable is the elevation of tempera- 
ture. 

(2) [The remainder is blank.] 

When a hypothesis no longer suffices to explain 
phenomena, it should be abandoned. 

This is the case with the hypothesis which re- 
gards caloric as matter, as a subtile fluid. 

The experimental facts tending to destroy this 
theory are as follows : 

(1) The development of heat by percussion or 
the friction of bodies (experiments of Rumford, 
friction of wheels on their spindles, on the axles, 
experiments to be made). Here the elevation of 
temperature takes place at the same time in the 
body rubbing and the body rubbed. Moreover, 
they do not change perceptibly in form or nature 



220 APPENDIX A. 

(to be proved). Thus heat is produced by motion. 
If it is matter, it must be admitted that the matter 
is created by motion. 

(2) When an air-pump is worked, and at the 
same time air is admitted into the receiver, the 
temperature remains constant in the receiver. It 
remains constant on the outside. Consequently, 
the air compressed by the pumps must rise in 
temperature above the air outside, and it is ex- 
pelled at a higher temperature. The air enters 
then at a temperature of 10, for instance, and 
leaves at another, 10 -f 90 or 100, for example. 
Thus heat has been created by motion. 

(3) If the air in a reservoir is compressed, and at 
the same time allowed to escape through a little 
opening, there is by the compression elevation of 
temperature, by the escape lowering of tempera- 
ture (according to Gay-Lussac and Welter). The 
air then enters at one side at one temperature and 
escapes at the other side at a higher temperature, 
from which follows the same conclusion as in the 
preceding case. 

(Experiment to be made : To fit to a high-pres- 
sure boiler a cock and a tube leading to it and empty- 
ing into the atmosphere; to open the cock a little 
way, and present a thermometer to the outlet of 
the steam; to see if it remains at 100 or more; 



APPENDIX A. 



to see if steam is liquefied in the pipe; to see 
whether it comes out cloudy or transparent.) 

(4) The elevation of temperature which takes 
place at the time of the entrance of the air into the 
vacuum, an elevation that cannot be attributed to 
the compression of the air remaining (air which 
may be replaced by steam), can therefore be at- 
tributed only to the friction of the air against the 
walls of the opening, or against the interior of the 
receiver, or against itself. 

(5) M. Gay-Lussac showed (it is said) that if 
two receivers were put in communication with 
each other, the one a vacuum, the other full of air, 
the temperature would rise in one as much as it 
would fall in the other. If, then, both be com- 
pressed one half, the first would return to its pre- 
vious temperature and the second to a much higher 
one. Mixing them, the whole mass would be 
heated. 

When the air enters a vacuum, its passage 
through one small opening and the motion im- 
parted to it in the interior appear to produce ele- 
vation of temperature. 

We may be allowed to express here an hypothe- 
sis in regard to the nature of heat. 

At present, light is generally regarded as the 



222 APPENDIX A. 

result of a vibratory movement of the ethereal 
fluid. Light produces heat, or at least accompa- 
nies the radiating heat, and moves with the same 
velocity as heat. Eadiating heat is then a vibratory 
movement. It would be ridiculous to suppose that 
it is an emission of matter while the light which 
accompanies it could be only a movement. 

Could a motion (that of radiating heat) pro- 
duce matter (caloric) ? 

No, undoubtedly; it can only produce a motion. 
Heat is then the result of a motion. 

Then it is plain that it could be produced by the 
consumption of motive power, and that it could 
produce this power. 

All the other phenomena composition and de- 
composition of bodies, passage to the gaseous state, 
specific heat, equilibrium of heat, its more or less 
easy transmission, its constancy in experiments 
with the calorimeter could be explained by this 
hypothesis. But it would be difficult to explain 
why, in the development of motive power by heat, 
a cold body is necessary ; why, in consuming the 
heat of a warm body, motion cannot be produced. 

It appears very difficult to penetrate into the 
real essence of bodies. To avoid erroneous reason- 
ing, it would be necessary to investigate carefully 



APPENDIX A. 223 

the source of our knowledge in regard to the na- 
ture of bodies, their form, their forces; to see what 
the primitive notions are, to see from what im- 
pressions they are derived ; to see how one is raised 
successively to the different degrees of abstraction. 

Is heat the result of a vibratory motion of mole- 
cules ? If this is so, quantity of heat is simply 
quantity of motive power. As long as motive 
power is employed to produce vibratory movements, 
the quantity of heat must be unchangeable; which 
seems to follow from experiments with the calo- 
rimeter; but when it passes into movements of sen- 
sible extent, the quantity of heat can no longer 
remain constant. 

Can examples be found of the production of 
motive power with actual consumption of heat ? 
It seems that we may find production of heat with 
consumption of motive power (re-entrance of the 
air into a vacuum, for example). 

What is the cause of the production of heat in 
combinations of substances? What is radiant 
caloric ? 

Liquefaction of bodies, solidification of liquids, 



'224. APPENDIX A. 

crystallization are they not forms of combinations 
of integrant molecules ? 

Supposing heat due to a vibratory movement, 
how can the passage from the solid or the liquid to 
the gaseous state be explained ? 

When motive power is produced by the passage 
of heat from the body A to the body B, is the quan- 
tity of this heat which arrives at B (if it is not the 
same as that which has been taken from A, if a 
portion has really been consumed to produce mo- 
tive power) the same whatever may be the sub- 
stance employed to realize the motive power? 

Is there any way of using more heat in the pro- 
duction of motive power, and of causing less to 
reach the body B ? Could we even utilize it en- 
tirely, allowing none to go to the body B ? If 
this were possible, motive power could be created 
without consumption of combustible, and by mere 
destruction of the heat of bodies. 

Is it absolutely certain that steam after having 
operated an engine and produced motive power 
can raise the temperature of the water of conden- 
sation as if it had been conducted directly into it? 

Reasoning shows us that there cannot be loss of 



APPENDIX A. 225 

living force, or, which is the same thing, of motive 
power, if the bodies act upon each other without 
directly touching each other, without actual col- 
lision. Now everything leads us to believe that 
the molecules of bodies are always separated from 
each other by some space, that they are never ac- 
tually in contact. If they touched each other, 
they would remain united, and consequently 
change form. 

If the molecules of bodies are never in close con- 
tact with each other whatever may be the forces 
which separate or attract them, there can never 
be either production or loss of motive power in 
nature. This power must be as unchangeable in 
quantity as matter. Then the direct re-establish- 
ment of equilibrium of the caloric, and its re-estab- 
lishment with production of motive power, would 
be essentially different from each other. 

Heat is simply motive power, or rather motion 
which has changed form. It is a movement among 
the particles of bodies. Wherever there is destruc- 
tion of motive power there is, at the same time, 
production of heat in quantity exactly proportional 
to the quantity of motive power destroyed. Ke- 
ciprocally, wherever there is destruction of heat, 
there is production of motive power. 



226 APPENDIX A. 

We can then establish the general proposition 
that motive power is, in quantity, invariable in 
nature; that it is, correctly speaking, never either 
produced or destroyed. It is true that it changes 
form, that is, it produces sometimes one sort of 
motion, sometimes another, but it is never annihi- 
lated. 

According to some ideas that I have formed 
on the theory of heat, the production of a unit of 
motive power necessitates the destruction of 2.70 
units of heats. 

A machine which would produce 20 units of 
motive power per kilogram of coal ought to destroy 
20 X 2.70 



7000 



of the heat developed by the combustion. 



20 X 2.70 8 , ,. , . . 1 

about > that 1S > less than 



7000 1000 

(Each unit of motive power, or dyname, repre- 
senting the weight of one cubic metre of water 
raised to the height of one metre.) 

Experiments to be made on Heat and Motive Power. 

To repeat Rumford's experiments in the drilling 
of a metal in water, but to measure the motive 
power consumed at the same time as the heat pro- 



APPENDIX A. 227 

duced; same experiments on several metals and 
on wood. 

To strike a piece of lead in various ways, to 
measure the motive power consumed and the heat 
produced. Same experiments on other metals. 

To strongly agitate water in a small cask or in 
a double-acting pump having a piston pierced with 
a small opening. 

Experiment of the same sort on the agitation of 
mercury, alcohol, air and other gases. To measure 
the motive power consumed and heat produced. 

To admit air into a vacuum or into air more or 
less rarefied; id. for other gases or vapors. To 
examine the elevation of temperature by means of 
the manometer and the thermometer of Breguet. 
Estimation of the error of the thermometer in the 
time required for the air to vary a certain number 
of degrees. These experiments would serve to 
measure the changes which take place in the tem- 
perature of the gas during its changes of volume. 
They would also furnish means of comparing these 
changes with the quantities of motive power pro- 
duced or consumed. 

Expel the air from a large reservoir in which it is 
compressed, and check its velocity in a large pipe in 



228 APPENDIX A. 

which solid bodies have been placed; measure the 
temperature when it has become uniform. See if 
it is the same as in the reservoir. Same experi- 
ments with other gases and with vapor formed 
under different pressures. 

To repeat Dalton's experiments and carry them 
on to pressures of thirty or forty atmospheres. To 
measure the constituent heat of the vapor within 
these limits. 

Id. on the vapor of alcohol, of ether, of essence 
of turpentine, of mercury, to prove whether the 
agent employed makes any difference in the pro- 
duction of motive power. 

Id. on water charged with a deliquescent salt, 
the calcium chloride, for instance. 

Is the law of tensions always the same? To 
measure the specific heat of vapor. 

Experiments to le made on the Tension of Vapors. 

A graduated capillary tube filled with water, 
mercury, or with oil and air. Plunge this tube 
into a bath of oil, of mercury, or of melted lead. 
To measure the temperature by an air thermometer. 

Same experiments with alcohol, ether, sulphide 
of carbon, muriatic ether, essence of turpentine, 
sulphur, phosphorus. 



APPENDIX A. 229 

Experiments on the tension of steam with a 
boiler, and a thermometric tube full of air. A 
thermometer will be placed in a tube immersed in 
the boiler, open outwards and filled with oil or 
mercury. 

Experiments by means of a simple capillary 
tube filled with three successive parts first of air, 
second of mercury, third of water or other liquid 
of which the tension can be measured (of alcohol, 
of ether, of essence of turpentine, of lavender, of 
sulphide of carbon, of muriatic ether, etc.). 
One end of the tube may be immersed in a bath 
of mercury or oil, the temperature of which is to 
be measured. The column of mercury can be made 
long enough to allow of the air being previously 
compressed or rarefied. 




FIG. 6. 

The tube will be bent into a spiral at one end, 
the straight part being graduated (thus permitting 
the tension of mercurial vapor to be measured). 

Experiments on the tension of vapors at low 




230 APPENDIX A. 

temperature, with a thermometric tube bent 
round, and filled partly with mercury, 
partly with water or alcohol. The mer- 
cury will operate by its weight. The 
upper part of the tube will be empty and 
sealed, or fully open to the atmosphere. 

The bulb will be immersed in water the 
temperature of which is to be measured. 
. 7. If the tube is sealed, the upper part 
must be cooled. 

The bulb might contain water, ether, or essence 
of turpentine. 

If the tube is sealed, the tension of mercurial 
vapor could be measured. 

Experiments on the constituent heat of vapors 
by means of a barometric tube having two en- 
larged bulbs. One of the bulbs may be im- 
mersed in cold water, and the elevation of temper- 
ature of this water will indicate the constituent 
heat of the vapor. 



FIG. 8. 




APPENDIX A. 231 

The other bulb may be warmed either by boiling 
liquid or by fire. 

Water, alcohol, steam, ether, mercury, acetic 
acid, sulphide of carbon. 

The operation may be repeated and add the results. 

Experiments to le made on Oases and Vapors. 

To measure the temperature acquired by the air 
introduced into a vacuum or space containing pre- 
viously rarefied air. 

If the vacuum is made under the glass receiver 
of an air-pump, and the cock admitting the outer 
air be t suddenly opened, the introduction of this 
air will cause a Breguet thermometer to rise to 50 
or 60. To examine the movement of this 
thermometer when the reintroduction 
takes place only by degrees, to compare 
it with the movement of the manometer. 

Construction of a manometer which 15 ] 
may give the pressure almost instanta- 
neously. 

Imagine a capillary tube bent into a 
spiral at one end, and having one ex- 
tremity closed, the other open. This 
tube will be perfectly dry and a small 
index of mercury may be introduced 
into it. The diameter of the tube will be small 




232 APPENDIX A. 

enough for the air enclosed in it to take almost 
instantly the temperature of the glass. We shall 
try to ascertain the time necessary for the estab- 
lishment of this equilibrium of temperature by 
placing the tube under the receiver of the air- 
pump, making a partial vacuum, and admitting 
the air. We shall see whether, some seconds after 
the introduction, the index perceptibly moves. 
The index must be of very light weight to avoid 
oscillation as much as possible. 

For the same reason, the capillary tube should 
be also as narrow as possible. If the straight part 
of the tube is equal to the bent part and the index 
be placed at the beginning of the bent part, for a 
pressure equal to atmospheric pressure, it would 
not be necessary to subject the instrument to a 
less pressure than -J atmosphere. It is between 
these two limits that it would serve as a measure. 

It might end in an open enlargement to prevent 
the projection of the mercury outside the tube. 
Disposed in this way, it could be used as a general 
measure for pressures between p and _/?; p being 
anything whatever. The apparatus will be fast- 
ened to a board bearing a graduated scale placed 
against the straight tube. The scale will be, for 
instance, numbered by fives or tens. A correspond- 
ing table denoting pressures would be required. 



APPENDIX A. 233 

Placing the instrument under the receiver and 
forming a partial vacuum, the index will rise into 
the enlargement. Then, admitting the air by de- 
grees and very slowly, we may note the correspond- 
ence between the heights of the ordinary mercury 
manometer and the point which will be reached 
by the lower face of the index of the instrument. 
This will answer to form a comparative table of 
the pressures and the numbers of the scale. The 
pressures would be represented by their relations 
to the observed pressure at the moment of the 
passage of the index over zero, for any other fixed 
number of the scale. 

Thus, for example, suppose that we observed on 
the manometer 400 or n millimetres of mercury 
when the index is on o, then n' when the index is 
on 1, n" when on 2, and so on. This will give the 

n' n" 

ratios ,,... which must be inscribed in the 
n n 

table. Then n could be varied at pleasure, and 
the table could still be used. 

In fact, according to the law of Mariotte, vol- 
umes preserving the same ratios, pressures should 
also preserve the same ratios to each other. 

Let p be the pressure when the index is on o, v 
the volume of air at the same moment, p' and v f 
the same pressures and volume at the moment 



234 APPENDIX A. 

when the index is on 1. Whether the air be ex- 
pelled or admitted the pressures would be instead 
of p and^/, q and q r . But there would follow 
p : p' : : v' : v and q : q' : : v' : v ; 
then p :p' : : q : q'. 

We should moreover work at a uniform tempera- 
ture and note the variations. 

If the straight part of the tube were perfectly 
calibrated, the volumes, and consequently the pres- 
sures, would form a geometrical progression, when 
the figures of the scale would be found to be in 
arithmetical progression, and a table of logarithms 
would enable one to be found from the other. 

In order to increase as required the mass of air 
enclosed in the tube the instrument must be 
placed on its side or flat, in the air-pump receivers. 
The mercury index would be placed in the lateral 
part of the enlargement of the tube, and the at- 
mospheric air would enter. The instrument 
might also be heated in this position. 

Care must be taken to admit only very dry air, 
which could be obtained by placing under the re- 
ceiver calcium chloride or any other substance 
which absorbs moisture greedily. 

Instead of bending the tube into a spiral, it 
might be bent in the middle in the form of a U, 
or it might be better to form three, four or mors 



APPENDIX A. 235 

parallel branches. Making the tube very long, the 
index would have a larger range for the same 
changes of pressure, and the results produced 
could then be measured by a slight variation in 
density in the air of the receiver. 

Comparison of the Rapidity with which the Air 
cools in the Receiver and in the Tube. 

Let us suppose, what I believe to be very near 
the truth, that the heat absorbed is proportional 
to the surface of the bodies in contact. From 
this we can infer without difficulty, that the rapid- 
ity of the cooling of the air in two cylindrical 
tubes would be inversely as their diameters. 

If the receiver is considered as a tube of two 
decimetres in diameter, and the manometer as a 
tube of one millimetre diameter, the rapidity of 
the cooling of the air would be in the ratio, very 
nearly, of 1 to 200. 

Extent of the Movement of the Index. 

Suppose the tube turned up on itself five times 
and having a total length of 1 metre; a variation 
of density equal to T V in the air will give a move- 
ment of 1 decimetre; a variation of heat of 1 de- 
gree supposed to be equivalent to a variation of 
density of ^ will give ^ of a metre, or about 



236 APPENDIX A. 

3 mm .70, quite an appreciable quantity. As to the 
time required to move the mercury index, regard 
being had to its mass, if we suppose it 1 centi- 
metre long, and the variation of pressure T J^ of an 
atmosphere, it would require about of a second 
to make it pass over one decimetre. 

Use of the Instrument in Measuring the Varia- 
tions of the Tensions of the Air under a Pneu- 
matic Receiver. 

At each stroke of the piston which expands the 
air under the pneumatic receiver when a vacuum 
is to be created, a lowering of pressure is produced, 
and undoubtedly a change of temperature. It can 
be determined approximately, at least, by observing 
the position of the manometer at the instant after 
the dilatation has taken place, and again after a 
time long enough for the temperature to have re- 
turned to its original point, that of the surrounding 
bodies. Comparison of the elastic force in the two 
cases will lead to comparison of the temperatures. 

The temperature having returned to its original 
point, we will give a second stroke of the piston 
which will rarefy the air more than the former, 
and thus we will make two observations of the 
manometer, before and after the return to the 
former temperature. And so on. 



OF THE 

UNIVERSITY 



APPENDIX B. 

CARNOT'S FOOT-NOTES. 

NOTE A. The objection may perhaps be raised 
here, that perpetual motion, demonstrated to be 
impossible by mechanical action alone, may pos- 
sibly not be so if the power either of heat or elec- 
tricity be exerted; but is it possible to conceive 
the phenomena of heat and electricity as due to 
anything else than some kind of motion of the 
body, and as such should they not be subjected to 
the general laws of mechanics ? Do we not know 
besides, a posteriori, that all the attempts made to 
produce perpetual motion by any means whatever 
have been fruitless ? that we have never succeeded 
in producing a motion veritably perpetual, that 
is, a motion which will continue forever without 
alteration in the bodies set to work to accomplish 
it ? The electromotor apparatus (the pile of Volta) 
has sometimes been regarded as capable of pro- 
ducing perpetual motion ; attempts 'have been 
made to realize this idea by constructing dry piles 
said to be unchangeable ; but however it has been 
done, the apparatus has always exhibited sensible 

337 



238 APPENDIX B. 

deteriorations when its action has been sustained 
for a time with any energy. 

The general and philosophic acceptation of the 
words perpetual motion should include not only a 
motion susceptible of indefinitely continuing itself 
after a first impulse received, but the action of an 
apparatus, of any construction whatever, capable 
of creating motive power in unlimited quantity, 
capable of starting from rest all the bodies of na- 
ture if they should be found in that condition, of 
overcoming their inertia; capable, finally, of find- 
ing in itself the forces necessary to move the whole 
universe, to prolong, to accelerate incessantly, its 
motion. Such would be a veritable creation of 
motive power. If this were a possibility, it would 
be useless to seek in currents of air and water or 
in combustibles this motive power. We should 
have at our disposal an inexhaustible source upon 
-which we could draw at will. 

NOTE B. The experimental facts which best 
prove the change of temperature of gases by com- 
pression or dilatation are the following: 

(1) The fall of the thermometer placed under 
the receiver of a pneumatic machine in which a 
vacuum has been produced. This fall is very sen- 
sible on the Breguet thermometer: it may exceed 
40 or 50. The mist which forms in this case 



APPENDIX B. 239 

seems to be due to the condensation of the watery 
vapor caused by the cooling of the air. 

(2) The inflammation of German tinder in the 
so-called pneumatic tinder-boxes ; which are, as 
we know, little pump- chambers in which the air is 
rapidly compressed. 

(3) The fall of a thermometer placed in a space 
where the air has been first compressed and then 
allowed to escape by the opening of a cock. 

(4) The results of experiments on the velocity 
of sound. M. de Laplace has shown that, in 
order to secure results accurately by theory and 
computation, it is necessary to assume the heating 
of the air by sudden compression. 

The only fact which may be adduced in opposi- 
tion to the above is an experiment of MM. Gay- 
Lussac and Welter, described in the Annales de 
Chimie et de Physique. A small opening having 
been made in a large reservoir of compressed air, 
and the ball of a thermometer having been intro- 
duced into the current of air which passes out 
through this opening, no sensible fall of the tem- 
perature denoted by the thermometer has been 
observed. 

Two explanations of this fact may be given: 
(1) The striking of the air against the walls of the 
opening by which it escapes may develop heat in. 



240 APPENDIX B. 

observable quantity. (2) The air which has jusl 
touched the bowl of the thermometer possibly 
takes again by its collision with this bowl, or 
rather by the effect of the detour which it is 
forced to make by its rencounter, a density equal 
to that which it had in the receiver, much as the 
water of a current rises against a fixed obstacle, 
above its level. 

The change of temperature occasioned in the 
gas by the change of volume may be regarded as 
one of the most important facts of Physics, be- 
cause of the numerous consequences which it 
entails, and at the same time as one of the most 
difficult to illustrate, and to measure by decisive 
experiments. It seems to present in some respects 
singular anomalies. 

Is it not to the cooling of the air by dilatation 
that the cold of the higher regions of the atmos- 
phere must be attributed? The reasons given 
heretofore as an explanation of this cold are en- 
tirely insufficient; it has been said that the air of 
the elevated regions receiving little reflected heat 
from the earth, and radiating towards celestial 
space, would lose caloric, and that this is the cause 
of its cooling; but this explanation is refuted by 
the fact that, at an equal height, cold reigns with 
equal and even more intensity on the elevated 



APPENDIX B. 241 

plains than on the summit of the mountains, or in 
those portions of the atmosphere distant from the 
sun. 

NOTE C. We see no reason for admitting, a 
prioriy the constancy of the specific heat of bodies 
at different temperatures, that is, to admit that 
equal quantities of heat will produce equal incre- 
ments of temperature, when this body changes 
neither its state nor its density; when, for example, 
it is an elastic fluid enclosed in a fixed space. 
Direct experiments on solid and liquid bodies have 
proved that between zero and 100, equal incre- 
ments in the quantities of heat would produce 
nearly equal increments of temperature. But the 
more recent experiments of MM. Dulong and 
Petit (see Annales de Chimie et de Physique ,^ob- 
ruary, March, and April, 1818) have shown that this 
correspondence no longer continues at tempera- 
tures much above 100, whether these temperatures 
be measured on the mercury thermometer or on 
the air thermometer. 

Not only do the specific heats not remain the 
same at different temperatures, but, also, they do 
not preserve the same ratios among themselves, so 
that no thermometric scale could establish the con- 
stancy of all the specific heats. It would have been 
interesting to prove whether the same irregulari- 



242 APPENDIX B. 

ties exist for gaseous substances, but such experi- 
ments presented almost insurmountable difficul- 
ties. 

The irregularities of specific heats of solid bodies 
might have been attributed, it would seem, to the 
latent heat employed to produce a beginning of 
fusion a softening which occurs in most bodies 
long before complete fusion. We might support 
this opinion by the following statement: According 
to the experiments of MM. Dulong and Petit, the 
increase of specific heat with the temperature is 
more rapid in solids than in liquids, although the 
latter possess considerably more dilatability. The 
cause of irregularity just referred to, if it is real, 
would disappear entirely in gases. 

NOTE D. In order to determine the arbitrary 
constants A, B, A', B' , in accordance with the 
results in M. Dalton's table, we must begin by com- 
puting the volume of the vapor as determined by 
its pressure and temperature, a result which is 
easily accomplished by reference to the laws of 
Mariotte and Gay-Lussac, the weight of the vapor 
being fixed. 

The volume will be given by the equation 

267 + tf 

v = c , 

P 
in which v is this volume, t the temperature, p the 



APPENDIX S. 



243 



pressure, and c a constant quantity depending on 
the weight of the vapor -and on the units chosen. 
We give here the table of the volumes occupied by 
a gramme of vapor formed at different tempera- 
tures,, and consequently under different pressures. 



t 


P 


V 


or degrees Centi- 
grade. 


or tension of the vapor 
expressed in millime- 
tres of mercury. 


or volume of a gramme 
of vapor expressed 
in litres. 





mm. 


lit. 





5.060 


185.0 


20 


17.32 


58.2 


40 


53.00 


20.4 


60 


144.6 


7.96 


80 


352.1 


3.47 


100 


760.0 


1.70 



The first two columns of this table are taken 
from the Traite de Physique of M. Biot (vol. i., p. 
272 and 531). The third is calculated by means 
of the above formula, and in accordance with the 
result of experiment, indicating that water vapor- 
ized under atmospheric pressure occupies a space 
1700 times as great as in the liquid state. 

By using three numbers of the first column and 
three corresponding numbers of the third column, 
we can easily determine the constants of our equa- 
tion 

A + B log v 

~ A' + B' log v 



244 APPENDIX B. 

We will not enter into the details of the calcula- 
tion necessary to determine these quantities. It 
is sufficient to say that the following values, 



A' = 19.64, 
B= -1000, B' = 3.30, 

satisfy fairly well the prescribed conditions, so that 
the equation 

_ 2268 - 1000 log v 
' 19. 64 + 3.30 log v 

expresses very nearly the relation which exists be- 
tween the volume of the vapor and its tempera- 
ture. We may remark here that the quantity B' 
is positive and very small, which tends to confirm 
this proposition that the specific heat of an elastic 
fluid increases with the volume, but follows a slow 
progression. 

NOTE E. Were we to admit the constancy of 
the specific heat of a gas when its volume does not 
change, but when its temperature varies, analysis 
would show a relation between the motive power 
and the thermometric degree. We will show how 
this is, and this will also give us occasion to show 
how some of the propositions established above 
should be expressed in algebraic language. 

Let r be the quantity of motive power produced 
by the expansion of a given quantity of air passing 



APPENDIX 3. 245 

from the volume of one litre to the volume of v 
litres under constant temperature. If v increases 
by the infinitely small quantity dv, r will increase 
by the quantity dr, which, according to the nature 
of motive power, will be equal to the increase dv 
of volume multiplied by the expansive force which 
the elastic fluid then possesses; let p be this ex- 
pansive force. We should have the equation 

dr = pdv (1) 

Let us suppose the constant temperature under 
which the dilatation takes place equal to t degrees 
Centigrade. If we call q the elastic force of the 
air occupying the volume 1 litre at the same tem- 
perature t t we shall have, according to the law of 
Mariotte, 

- = " whence p = -. 
Ip v 

If now P is the elastic force of this same air at the 
constant volume 1, but at the temperature zero, 
we shall have, according to the rule of M. Gay- 
Lussac, 



whence 

P 267 



246 APPENDIX S. 



p 

If, to abridge, we call N the quantity 1 ^ E > the 

~ 



equation would become 

^ t + 267 

P = N- ^> 

whence we deduce, according to equation (I), 



ar, 

dr = N -- dv. 

v 

Regarding t as constant, and taking the integral of 
the two numbers, we shall have 

r = N(t + 267) log v + C. 

If we suppose r = when v = 1, we shall have 
(7=0; whence 

r = N(t + 267) log v. . . . (2) 

This is the motive power produced by the expan- 
sion oi the air which, under the temperature t, has 
passed from the volume 1 to the volume v. If in- 
stead of working at the temperature t we work in 
precisely vtto name manner at the temperature 
t -j- dt, the power developed will be 

r + dr = N(t + dt + 267) log v. 
Subtracting equation (2), we have 

dr = Nlogvdt. .... (3) 

Let e be the quantity of heat employed to maintain 
the temperature of the gas constant during its 



APPENDIX B. 247 

dilatation. According to the reasoning of page 69, 
Sr will be the power developed by the fall of the 
quantity e of heat from the degree t -f- td to the 
degree t. If we call u the motive power developed 
by the fall of unity of heat from the degree t to the 
degree zero, as, according to the general principle 
established page 68, this quantity u ought to de- 
pend solely on i, it could be represented by the 
function Ft, whence u = Ft. 

When t is increased it becomes t + td, u be- 
comes u + du ; whence 



Subtracting the preceding equation, we have 
du = F(t + df) - Ft = F'tdt. 

This is evidently the quantity of motive power 
produced by the fall of unity of heat from the 
temperature t + dt to the temperature t. 

If the quantity of heat instead of being a unit 
had been e, its motive power produced would have 
had for its value 

edu = eF'tdt ..... (4) 
But edu is the same thing as dr\ both are the 
power developed by the fall of the quantity e of 
heat from the temperature t -j- dt to the tempera- 
ture t; consequently, 

edu = dr, 



248 APPENDIX B. 

and from equations (3), (4), 

eF'tdt = N \ogvdt; 
or, dividing by F'tdt, 

N 
e= -j^\ogv = Tlogv. 

JV 
Calling T the fraction -^ which is a function of t 

only, the equation 

e = T log v 

is the analytical expression of the law stated pp. 80, 
81. It is common to all gases, since the laws ot 
which we have made use are common to all. 

If we call s the quantity of heat necessary to 
change the air that we have employed from the 
volume 1 and from the temperature zero to the 
volume v and to the temperature t, the difference 
between s and e will be the quantity of heat re- 
quired to bring the air at the volume 1 from zero 
to t. This quantity depends on t alone; we will 
call it U. It will be any function whatever of t. 
We shall have 

s = e + U= Tlogv + U. 

If we differentiate this equation with relation to t 
alone, and if we represent it by T' and U', the dif- 
ferential coefficients of T and U, we shall get 

//<? 

g=ZMogt;+Z7'; ... (5) 



APPENDIX B. 249 



-j2 is simply the specific heat of the gas under 
cl t 

constant volume, and our equation (1) is the an- 
alytical expression of the law stated on page 86. 

If we suppose the specific heat constant at all 
temperatures (hypothesis discussed above, page 92), 

ds 

the quantity '- will be independent of t', and in 
dt 

order to satisfy equation (5) for two particular 
values of v, it will be necessary that T' and U' be 
independent of t; we shall then have T' = C, a 
constant quantity. Multiplying T' and C by dt, 
and taking the integral of both, we find 



but as T = =- , we have 



- T ~ Ct + C; 
Multiplying both by dt and integrating, we have 

& = log (01 + C,) + C,; 

or changing arbitrary constants, and remarking 
further that Ft is when t = 0, 

Ft 
The nature of the function Ft would be thus 



= A log (l + |) . . . . (6) 



250 APPENDIX B. 

determined, and we would thus be able to estimate 
the motive power developed by any fall of heat. 
But this latter conclusion is founded on the hy- 
pothesis of the constancy of the specific heat of a 
gas which does not change in volume an hypoth- 
esis which has not yet been sufficiently verified by 
experiment. Until there is fresh proof, our equa- 
tion (6) can be admitted only throughout a limited 
portion of the thermometric scale. 

In equation (5), the first member represents, as 
we have remarked, the specific heat of the air oc- 
cupying the volume v. Experiment having shown 
that this heat varies little in spite of the quite con- 
siderable changes of volume, it is necessary that 
the coefficient T' of log v should be a very small 
quantity. If we consider it nothing, and, after 
having multiplied by dt the equation 

Z"=0, 
we take the integral of it, we find 

T= C, constant quantity; 
but 



~ F't' 

whence 

_,, N N 

Ft = -?=-{? = A; 

whence we deduce finally, by a second integration, 
Ft = At B. 



APPENDIX B. 251 

As Ft = when t = 0, B is 0; thus 



that is, the motive power produced would be found 
to be exactly proportional to the fall of the caloric. 
This is the analytical translation of what was 
stated on page 98. 

NOTE F. M. Dalton believed that he had dis- 
covered that the vapors of different liquids at equal 
thermometric distances from the boiling-point 
possess equal tensions; but this law is not pre- 
cisely exact; it is only approximate. It is the 
same with the law of the proportionality of the 
latent heat of vapors with their densities (see Ex- 
tracts from a Memoire of M. C. Despretz, Annales 
de CMmie et de Physique, t. xvi. p. 105, and t. 
xxiv. p. 323). Questions of this nature are closely 
connected with those of the motive power of heat. 
Quite recently MM. H. Davy and Faraday, after 
having conducted a series of elegant experiments 
on the liquefaction of gases by means of consider- 
able pressure, have tried to observe the changes of 
tension of these liquefied gases on account of slight 
changes of temperature. They have in view the 
application of the new liquids to the production 
of motive power (see Annales de CMmie et de 
Physique, January, 1824, p. 80). 



252 APPENDIX B. 

According to the above-mentioned theory, we 
can foresee that the use of these liquids would 
present no advantages relatively to the economy 
of heat. The advantages would be found only in 
the lower temperature at which it would be possi- 
ble to work, and in the sources whence, for this 
reason, it would become possible to obtain caloric. 

NOTE G. This principle, the real foundation 
of the theory of steam-engines, was very clearly 
developed by M. Clement in a memoir presented 
to the Academy of Sciences several years ago. 
This Memoir has never been printed, and I owe 
the knowledge of it to the kindness of the author. 
Not only is the principle established therein, but 
it is applied to the different systems of steam- 
engines actually in use. The motive power of 
each of them is estimated therein by the aid of 
the law cited page 92, and compared with the re- 
sults of experiment. 

The principle in question is so little known or 
so poorly appreciated, that recently Mr. Perkins, a 
celebrated mechanician of London, constructed a 
machine in which steam produced under the pres- 
sure of 35 atmospheres a pressure never before 
used is subjected to very little expansion of vol- 
ume, as any one with the least knowledge of this 
machine can understand. It consists of a single 
cylinder of very small dimensions, which at ench 



APPENDIX B. 253 

stroke is entirely filled with steam, formed under 
the pressure of 35 atmospheres. The steam pro- 
duces no effect by the expansion of its volume,, for 
no space is provided in which the expansion can 
take place. It is condensed as soon as it leaves 
the small cylinder. It works therefore only under 
a pressure of 35 atmospheres, and not, as its use- 
ful employment would require, under progressively 
decreasing pressures. The machine of Mr. Per- 
kins seems not to realize the hopes which it at 
first awakened. It has been asserted that the 
economy of coal in this engine was j\ above the 
best engines of Watt, and that it possessed still 
other advantages (see Annales de Chimie et de 
Physique, April, 1823, p. 429). These assertions 
have not been verified. The engine of Mr. Per- 
kins is nevertheless a valuable invention, in that 
it has proved the possibility of making use of 
steam under much higher pressure than previously, 
and because, being easily modified, it may lead to 
very useful results. 

Watt, to whom we owe almost all the great im- 
provements in steam-engines, and who brought 
these engines to a state of perfection difficult 
even now to surpass, was also the first who em- 
ployed steam under progressively decreasing pres- 
sures. In many cases he suppressed the introduc- 
tion of the steam into the cylinder at a half, a 



254 



APPENDIX B. 



third, or a quarter of the stroke. The piston com- 
pletes its stroke, therefore, under a constantly 
diminishing pressure. The first engines working on 
this principle date from 1778. Watt conceived the 
idea of them in 1769, and took out a patent in 1782. 
We give here the Table appended to Watt's 
patent. He supposed the steam introduced into 
the cylinder during the first quarter of the stroke of 
the piston; then, dividing this stroke into twenty 
parts, he calculated the mean pressure as follows: 



Portions of the descent from the 
top of the cylinder. 



Decreasing pressure of the 
steam, the entire pressure 
being 1. 





0.05 


f 1.0001 




0.10 
0.15 


Steam arriving 1.000 | T , 
- freely from ^ 1.000 [ Lo L l 




0.20 


the boiler. 


1.000 


Quarter. . . 


..0.25 




,1.000 J 




0.30 




r 0.830 




0.35 




0.714 




0.40 




0.625 


Half ... 


0.45 
0.50 
55 


The steam be- 


5'Sgj Half original 
0454^ pressure. 






o.'eo 

0.65 
0.70 
0.75 

0.80 


ing cut off 
and the de- 
scent taking 
place only by 
expansion. 


0'.417 
0.385 
0.375 
0.333 One third. 
0.312 




0.85 




0.294 




0.90 




0.277 


Bottom of 


0.95 




0.262 


cylinder. 


..1.00 


[0.025 Quarter. 




Total, 11.583 



11 583 
Mean pressure - -- = 0.579. 



APPENDIX R 255 

On which he remarked,, that the mean pressure is 
more than half the original pressure; also that in 
employing a quantity of steam equal to a quarter, 
it would produce an effect more than half, 

Watt here supposed that steam follows in its ex- 
pansion the law of Mariotte, which should not be 
considered exact, because, in the first place, the 
elastic fluid in dilating falls in temperature, and 
in the second plac3 there is nothing to prove that 
a part of this fluid is not condensed by its expan- 
sion. Watt should also have taken into considera- 
tion the force necessary to expel the steam which 
remains after condensation, and which is found in 
quantity as much greater as the expansion of the 
volume has been carried further. Dr. Robinson 
has supplemented the work of Watt by a simple 
formula to calculate the effect of the expansion of 
steam, but this formula is found to have the same 
faults that we have just noticed. It has neverthe- 
less been useful to constructors by furnishing them 
approximate data practically quite satisfactory. 
We have considered it useful to recall these facts 
because they are little known, especially in 
France. These engines have been built after the 
models of the inventors, but the ideas by which 
the inventors were originally influenced have been 
but little understood. Ignorance of these ideas 



256 APPENDIX B. 

has often led to grave errors. Engines originally 
well conceived have deteriorated in the hands of 
unskilful' builders, who, wishing to introduce in 
them improvements of little value, have neglected 
the capital considerations which they did not know 
enough to appreciate. 

NOTE H. The advantage in substituting two 
cylinders for one is evident. In a single cylinder 
the impulsion of the piston would be extremely 
variable from the beginning to the end of the 
stroke. It would be necessary for all the parts by 
which the motion is transmitted to be of sufficient 
strength to resist the first impulsion, and perfectly 
fitted to avoid the abrupt movements which would 
greatly injure and soon destroy them. It would 
be especially on the working beam, on the 
supports, on the crank, on the connecting-rod, 
and on the first gear-wheels that the unequal 
effort would be felt, and would produce the 
most injurious effects. It would be necessary 
that the steam-cylinder should be both sufficiently 
strong to sustain the highest pressure, and with 
a large enough capacity to contain the steam 
after its expansion of volume, while in using two 
successive cylinders it is only necessary to have 
the first sufficiently strong and of medium ca- 
pacity, which is not at all difficult, and to have 



APPENDIX B. 257 

the second of ample dimensions, with moderate 
strength. 

Double-cylinder engines, although founded on 
correct principles, often fail to secure the advan- 
tages expected from them. This is due principally 
to the fact that the dimensions of the different 
parts of these engines are difficult to adjust, and 
that they are rarely found to be in correct propor- 
tion. Good models for the construction of double- 
cylinder engines are wanting, while excellent de- 
signs exist for the construction of engines on the 
plan of Watt. From this arises the diversity that 
we see in the results of the former, and the great 
uniformity that we have observed in the results of 
the latter. 

NOTE I. Among the attempts made to develop \ 
the motive power of heat by means of atmospheric 
air, we should mention those of MM. Niepce, which 
were made in France several years ago, by means 
of an apparatus called by the inventors a pyre- 
olophore. The apparatus was made thus: There 
was a cylinder furnished with a piston, into which 
the atmospheric air was introduced at ordinary 
density. A very combustible material, reduced to 
a condition of extreme tenuity, was thrown into it, 
remained a moment in suspension in the air, and 
then flame was applied. The inflammation pro- 



258 APPENDIX B. 

duced very nearly the same effect as if the elastic, 
fluid had been a mixture of air and combustible 
gas, of air and carburetted hydrogen gas,, for ex- 
ample. There was a sort of explosion, and a sud- 
den dilatation of the elastic fluid & dilatation that 
was utilized by making it act upon the piston. 
The latter may have a motion of any amplitude 
whatever, and the motive power is thus realized. 
The air is next renewed, and the operation re- 
peated. 

This machine, very ingenious and interesting, 
especially on account of the novelty of its princi- 
ple, fails in an essential point. The material used 
as a combustible (it was the dust of Lycopodium, 
used to produce flame in our theatres) was so ex- 
pensive, that all the advantage was lost through 
that cause; and unfortunately it was difficult to 
employ a combustible of moderate price, since a 
very finely powdered substance was required which 
would burn quickly, spread rapidly, and leave little 
or no ash. 

Instead of working as did MM. Niepce, it would 
seem to us preferable to compress the air by means 
of pumps, to make it traverse a perfectly closed 
furnace into which the combustible had been in- 
troduced in small portions by a mechanism easy of 
conception, to make it develop its action in a cylin- 



APPENDIX B. 259 

der with a piston, or in any other variable space; 
finally, to throw it out again into the atmosphere, 
or even to make it pass under a steam-boiler in 
order to utilize the temperature remaining. 

The principal difficulties that we should meet in 
this mode of operation would be to enclose the fur- 
nace in a sufficiently strong envelope, to keep the 
combustion meanwhile in the requisite state, to 
maintain the different parts of the apparatus at a 
moderate temperature, and to prevent rapid abra- 
sion of the cylinder and of the piston. These dif- 
ficulties do not appear to be insurmountable. 

There have been made, it is said, recently in 
England, successful attempts to develop motive 
power through the action of heat on atmospheric 
air. We are entirely ignorant in what these at- 
tempts have consisted if indeed they have really 
been made. 

NOTE J. The result given here was furnished by 
an engine whose large cylinder was 45 inches in 
diameter and 7 feet stroke. It is used in one of the 
mines of Cornwall called Wheal Abraham. This 
result should be considered as somewhat excep- 
tional, for it was only temporary, continuing but a 
single month. Thirty millions of Ibs. raised one 
English foot per bushel of coal of 88 Ibs. is generally 
regarded as an excellent result for steam-engines. 



260 APPENDIX B. 

It is sometimes attained by engines of the Watt 
type, but very rarely surpassed. This latter prod- 
uct amounts, in French measures, to 104,000 kilo- 
grams raised one metre per kilogram of coal con- 
sumed. 

According to what is generally understood by 
one horse-power, in estimating the duty of steam- 
engines, an engine of ten horse-power should raise 
per second 10 X 75 kilograms, or 750 kilograms, to 
a height of one metre, or more, per hour; 750 X 
3600 = 2,700,000 kilograms to one metre. If we 
suppose that each kilogram of coal raised to this 
height 104,000 kilograms, it will be necessary, in 
order to ascertain how much coal is burnt in one 
hour by our ten-horse-power engine, to divide 
2,700,000 by 104,000, which gives *fiff- = 26 kilo- 
grams. Now it is seldom that a ten -horse-power 
engine consumes less than 26 kilograms of coal per 
hour. 



APPENDIX C. 

NOTE BY THE EDITOR. 

ALL the preceding data are to-day subject to 
modification. 

Thus a duty of 150,000,000 ft.-lbs. per 100 Ibs. 
good coal is to-day attainable, and two thirds that 
figure is extremely common. With engines of 
large size the coal-consumption has fallen to one 
half, sometimes even to one fourth, the figure in 
the text. 

Hot-air engines are superseded by the gas- 
engine and the oil-vapor engine ; which even 
threaten, in the opinion of many engineers, to 
ultimately displace the steam-engine. 

Compound and other multiple-cylinder engines, 
with two, three, and even four cylinders in series, 
are now always employed where fuel is costly. The 
reason of their success is, in part, that given in 
Note H; but in only small part. The real cause 
of their general adoption is the fact that the in- 
ternal thermal waste by "cylinder-condensation" 
which in simple engines ordinarily amounts, 
according to size, to from 25 to 50 per cent, or 

261 



262 APPENDIX C. 

more, of all heat supplied by the boiler is reduced 
nearly in proportion to the number of steam -cylin- 
ders in series. 

For the applied thermodynamics of the steam- 
engine, following Carnot and Thomson, see the 
pages of Kankine and of Clausius of 1850 to 1860, 
and especially the treatise of Rankine on the 
Steam-engine. The editor has adopted the methods 
of these great successors of Carnot in his " Manual 
of the Steam-engine" (2 vols. 8vo; N. Y., J. Wiley 
& Sons), which may be consulted in this connec- 
tion, and especially for details of the theory and 
the structure of this prime mover. 




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