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Hi 



HEAT. 



HEAT 



BY 



P. G. 



, M.A., SEC. R.S.E. 



HONORARY FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE 

PROFESSOR OF NATURAL PHILOSOPHY IN THE 

UNIVERSITY OF EDINBURGH 




Eonfcon 
MACMILLAN AND CO. 

AND NEW YORK 
1892 

The Rights of Translation and Reproduction are Reserved 



3 



/rvPTtyc' r ' ^ 'OTi'T^ 



RICHARD CLAY AND SONS, LIMITED, 
LONDON AND BUNGAY. 

First Edition printed, 1884. 
Reprinted with corrections, 1892. 



PREFACE. 

IT was with considerable hesitation that I sanctioned the 
Publishers' proposal to print a very large number of copies 
of the first edition of this work. Some of my reasons 
will be found in the former Preface, reprinted below : 
especially those depending on the fact that the work was 
written, in occasional intervals of comparative leisure, during 
several very busy years. 

It is, therefore, with proportionate pleasure that I now 
find I can reissue it with a very moderate amount of alter- 
ation : the only serious error discovered (besides other 
pretty obvious typographical ones) being the unaccountable 
omission of some necessary words in the important quota- 
tion from Clerk-Maxwell given in 442. 

But Teachers, who have used the book, have kindly 
pointed out to me passages in which they or their pupils 
had found unexpected difficulty : and I have endeavoured 
to make these passages more clear. 

I have also rewritten the section which deals with the 
isothermals of gases from the point of view of the kinetic 
theory. Here temperature, of which so much was said 
in the former Preface, reappears in an even more formidable 
guise than before. 

P. G. TAIT. 

COLLEGE, EDINBURGH, 

November 2nd, 1891. 

673242 



PREFACE TO THE FIRST EDITION. 

THIS work originated in an article which I contributed in 
1876 to the Handbook to the Loan Collection of Scientific 
Apparatus (at South Kensington). As that article was 
based upon the system of teaching the subject of Heat 
which, after many years' experience, I have found best 
adapted to the wants of intelligent students taking up the 
subject for the first time, Mr. Macmillan asked me to 
develop it in the form of an elementary treatise. In 1876 7 
a large portion of the work was written ; the greater 
part of which was put in type, and had the advantage of 
Clerk- Maxwell's valuable criticism. Some of the earlier 
chapters were utilised when I had to give at short notice 
an evening discourse at the British Association Meeting 
in 1876. 

Pressing work of a very different character, such as the 
examination of the Challenger Thermometers, interfered 
from time to time with my writing by occupying most of my 
leisure. I have at last managed to finish the book, as 
nearly as possible on the lines laid down at starting ; but it 
cannot, under the circumstances, be expected to have the 
unity which it might have secured by being continuously 
written. 



PREFACE. * vii 

It may be asked Why publish another text-book on a 
subject which is already thoroughly treated in the excellent 
(and strictly scientific) works of Clerk-Maxwell and Balfour 
Stewart ? The only answer, and it may be a sufficient one, 
i s : Clerk-Maxwell's work is on the Theory of Heat, and is 
specially fitted for the Study ; that of Stewart is rather for 
the Physical Laboratory ; so that there still remains an 
opening for a work suited to the Lecture Room. By that 
expression I mean a work suitable for students who, without 
any intention of entering on a scientific career, whether 
theoretical or experimental, are yet desirous of knowing 
accurately the more prominent facts and theories of modern 
science to such an extent as to give them an intelligent 
interest in physical phenomena. 

In addition to what is stated in the text ( n) as to 
the special arrangement and division of the subject in 
the present work, it may be well to say a few words here 
as to the way in which the important subjects of tem- 
perature, and especially of absolute temperature, have 
been treated. 

Temperature is first introduced as a mere condition de- 
termining which of two bodies in contact shall part with 
heat to the other ( 6). In this sense it is compared to 
the pressure of the air in a receiver, the air itself being 
the analogue of heat ( 52). When two such receivers 
are made to communicate with one another, air passes 
from that in which the pressure is greater to that in which 
it is less. And this is altogether independent of the relative 
quantities of air in the two receivers. 

In 57 it is pointed out that there is an absolute method 



viii PREFACE. 

of defining temperature, which must therefore be the sole 
scientific method. But it is also stated that, in experi- 
mental work, few quantities are directly measured in 
terms of the strictly scientific units in which they are 
ultimately to be expressed. In 60 the reader is told 
that what is wanted, to prepare him for the absolute 
measurement of temperature, is "something which shall 
be at once easy to comprehend and easy to reproduce, 
and which shall afterwards require only very slight 
modifications to reduce it to the rigorous scale." 

Thus ( 61) a temporary centigrade scale is defined by 
means of mixtures of ice-cold water and boiling water. 
And this is stated to " accord so closely with the absolute 
scale, that careful experiment is required even to show 
that it does not exactly coincide with it." 

In terms of temperatures thus defined, the questions of 
Expansibility, Latent and Specific Heats, &c., are explained 
from the experimental point of view. 

But, before these are taken up, the first rapid resume of 
the whole subject introduces Carnot's Cycles, and his 
principle of Reversibility, with Thomson's application of 
them to the absolute measurement of temperature. It is 
pointed out ( 95) that Carnot's results still leave a certain 
option in the formal definition of absolute temperature, 
and that Thomson found it possible so to frame his 
definition (which is given at this stage) as to make a very 
close agreement between the new scale and that of an 
ordinary air-thermometer. Thus, the use of air-thermo- 
meter temperatures is justified by their close approximation 
to absolute temperatures, as well as by their comparative 



PREFACE. . ix 

simplicity and convenience, for the chapters which follow ; 
and they are thenceforth employed until (by means of 
the Indicator Diagram) it has been fully pointed out what 
is the scientific importance of absolute temperature. It is 
then shown, in 405, how to compare absolute tempera- 
ture with that given by the air-thermometer. 

All a priori notions as to strict logical order, however 
valuable, and in fact necessary, in a formal treatise, must 
be set aside in an elementary work, if they be found in 
practice to impede rather than assist the progress of the 
average student. And it is solely on this ground that 
the above system of exposition has been adopted in the 
present work. 

He who expects to find this work, elementary as it is, 
everywhere easy reading, will be deservedly disappointed. 
No branch of science is free from real and great difficulties, 
even in its elements. Any one who thinks otherwise has 
either not read at all, or has confined his reading to 
pseudo-science. 

The reader, who wishes to know more of the general 
science of Energy than was admissible into the present 
work, will find a connected historical sketch of it in my 
little book on Thermodynamics. There he will also find 
more of the purely analytical development of the subject, 
specially from W. Thomson's point of view. 

P. G. TAIT. 
COLLEGE, EDINBURGH, 

December isf, 1883. 



CONTENTS. 



CHAPTER I. 

PAGE 
FUNDAMENTAL PRINCIPLES I 

CHAPTER II. 

INTRODUCTORY 8 

CHAPTER III. 

DIGRESSION ON FORCE AND ENERGY 13 

CHAPTER IV. 

PRELIMINARY SKETCH OF THE SUBJECT 21 

CHAPTER V. 

DILATATION OF SOLIDS 71 

CHAPTER VI. 

DILATATION OF LIQUIDS AND GASES 88 

CHAPTER VII. 

THERMOMETERS 103 

b 



CONTENTS. 



CHAPTER VIII. 

J'AC'.B 

CHANGE OF MOLECULAR STATE. MELTING AND SOLIDIFICATION II& 



CHAPTER IX. 

CHANGE OF MOLECULAR STATE. VAPORISATION AND CON- 

DENSATION ...................... 131 

CHAPTER X. 

CHANGE OF TEMPERATURE. SPECIFIC HEAT ......... 149 

CHAPTER XI. 

THERMO-ELECTRICITY ................... 163 

CHAPTER XII. 

OTHER EFFECTS OF HEAT ................. 182 

CHAPTER XIII. 

COMBINATION AND DISSOCIATION .............. 193 

CHAPTER XIV. 

CONDUCTION OF HEAT ................... 2O 



CHAPTER XV. 

CONVECTION ....................... 230 



CHAPTER XVI. 



RADIATION 



239 



CHAPTER XVII. 

RADIATION AND ABSORPTION 250 



xii CONTENTS. 

CHAPTER XVIII. 

I'AGE 

RADIATION 276 

CHAPTER XIX. 

UNITS AND DIMENSIONS 2QO 

CHAPTER XX. 
WATT'S INDICATOR DIAGRAM 298 

CHAPTER XXI. 

ELEMENTS OF THERMODYNAMICS 324 

CHAPTER XXII. 

NATURE OF HEAT 3$2 




HEAT. 



CHAPTER I. 

FUNDAMENTAL PRINCIPLES. 

i. IN dealing with any branch of physical science it is 
absolutely necessary to keep well in view the fundamental 
and all-important principle that 

Nothi?ig can be learned as to the physical world save by 
observation and experiment, or by mathematical deductions 
from data so obtained. 

On such a text volumes might be written ; but they are 
unnecessary, for the student of physical science will feel at 
each successive stage of his progress more and more pro- 
found conviction of its truth. He must receive it, at starting, 
as the unanimous conclusion of all who have in a legitimate 
manner made true physical science the subject of their 
study ; and, as he gradually gains knowledge by this the 
only method, he will see more and more clearly the abso- 
lute impotence of all so-called metaphysics, or a priori 
reasoning, to help him to a single step in advance. 

2. Man has been left entirely to himself as regards the 
acquirement of physical knowledge. But he has been gifted 
with various senses (without which he could not even know 



2 HEAT. [CHAP. 

that the physical world exists) and with reason to enable him 
to control and understand their indications. 

Reason, unaided by the senses, is totally helpless in such 
matters. The indications given by the senses, unless inter- 
preted by reason, are in general utterly unmeaning. But 
>wr\en reasGii ;and the senses work harmoniously together 
ithey open <to l His - r an absolutely illimitable prospect of 
^mysteries 1 to ;be explored. This is the test of true science 
&her' te "na "resting-place each real advance discloses so 
much that is new and easily accessible, that the investigator 
has but scant time to co-ordinate and consolidate his know- 
ledge before he has additional materials poured into his 
store. 

3. To sight, without reason, the universe appears to be 
filled with light except, of course, in places surrounded by 
opaque bodies. 

Reason, controlling the indications of sense, shows us 
that the sensation of sight is our own property ; and that 
what we understand by brightness, &c., does not exist out- 
side our minds. It shows us also that the sensation of 
colour is purely -subjective, the only differences possible 
between different so-called rays of light outside the eye 
being merely in the extent, form, and rapidity of the vibra- 
tions of the luminiferous medium. 

To hearing, without reason, the air of a busy town seems 
to be filled with sounds. Reason, interpreting the indications 
of sense, tells us that if we could see the particles of air we 
should observe among them (superposed upon their rapid 
motions among one another) simply a comparatively slow 
agitation of the nature of alternate compressions and 
dilatations. And our classification of sounds as to loudness, 
pitch, and quality, is merely the subjective correlative of 
what in the air-particles is objectively the amount of com- 



i.J FUNDAMENTAL PRINCIPLES. 3 

pression, the rapidity of its alternations, and the greater or 
less complexity of the alternating motion. 

A blow from a stick or a stone produces pain and a 
bruise ; but the motion of the stick or stone before it 
reached the body is as different from the sensation produced 
by the blow as is the alternate compression and dilatation of 
the air from the sensation of sound, or the ethereal wave- 
motion from the sensation of light. 

Hence to speak of what we ordinarily mean by light, or 
sound, as existing outside ourselves, is as absurd as to speak 
of a swiftly-moving stick or stone as pain. But no incon- 
venience is occasioned if we announce the intention to use 
the terms light and sound for the objective phenomena, 
and to speak of their subjective effects as " luminous 
impressions " or " noise," as the case may be. In this sense 
there is outside us energy of motion of every kind, but in 
the mind mere corresponding impressions of brightness and 
colour, noise or harmony, pain, &c. &c. 

4. It would seem therefore that we must be extremely 
cautious in our own interpretation of the immediate evidence 
of our senses as to heat. And the very first instance that 
occurs to us fully justifies this caution. 

Touch, in succession, various objects on the table. A 
paper-weight, especially if it be metallic, is usually cold to 
the touch ; books, paper, and especially a woollen table- 
cover, comparatively warm. Test them, however, by means 
of a thermometer, not by the sense of touch, and in all proba- 
bility you will find little or no difference in what we call 
their temperatures. In fact, as we shall presently see, any 
number of bodies of any kind shut up in an enclosure 
(within which there is no fire or other source of heat) all 
tend to acquire ultimately the same temperature. Why 
then do some feel cold, others warm, to the touch ? 

B 2 



4 HEAT. [CHAP. 

The reason is simply this the sense of touch does noi 
inform us directly of temperature, but of the rate at which 
our finger gains or loses heat. As a rule, bodies in a room 
are colder than the hand, and heat always tends to pass from 
a warmer to a colder body. Of a number of bodies, all 
equally colder than the hand, that one will seem coldest to 
the touch which is able most rapidly to convey away heat 
from the hand. The question therefore is one of conduction 
of heat. And to assure ourselves that it is so, reverse the 
process : let us, in fact, try an experiment, though an exceed- 
ingly simple one; for the essence of experiment is to 
modify the circumstances of a physical phenomenon so as 
to increase its value as a test. Put the paper-weight, the 
books, and the woollen table-cloth into an oven, and raise 
them all to one and the same temperature, considerably 
above that of the hand. The woollen cloth will still be 
comparatively cool to the touch, while the metal paper- 
weight may be much too hot to hold. The order of these 
bodies, as to warm and cold in the popular sense, is in fact 
reversed ; and this is so, because the hand is now receiving 
heat from all the various bodies experimented on, and it 
receives most rapidly from those bodies which in their 
previous condition were capable of abstracting heat most 
rapidly. However it may be in the moral world, in the 
physical universe the giving and taking powers of one and 
the same body are strictly correlative and equal. 

5. Thus the direct indications of sense are in general 
utterly misleading as to the relative temperatures of different 
bodies. 

In a baker's or a sculptor's oven, at temperatures far above 
the boiling-point of water (on one occasion even 320 F.), 
so high indeed that a beef-steak was cooked in thirteen 
minutes, Tillet in France, and Blagden and Chantrey in 



i.j FUNDAMENTAL PRINCIPLES. * 5 

England, remained for nearly an hour in comparative com- 
fort. But, though their clothes gave them no great incon- 
venience, they could not hold a metallic pencil-case without 
being severely burned. 

On the other hand, great care has to be taken to cover 
with hemp, or wool, or other badly conducting substance, 
every piece of metal which has to be handled in the intense 
cold to which an Arctic expedition is subjected ; for contact 
with very cold metal produces sores almost undistinguish- 
able from burns, though due to a directly opposite cause. 
Both of these phenomena, however, ultimately depend on 
the comparative facility with which heat is conducted by 
metals. 

6. Even from the instance just given, the reader cannot 
fail to see that there is a profound distinction between heat 
and temperature. Heat, whatever it may be, is SOMETHING 
which can be transferred from one portion of matter to 
another; the consideration of temperatures is virtually that 
of the mere CONDITIONS which determine whether or not 
there shall be a transfer of heat, and in which direction the 
transfer is to take place. 

In fact, we may without risk of misleading the student 
tell him, even at the beginning of his work, that from one 
point of view the quantity of heat in a body bears a very 
close analogy to the water-power stored up in a cistern or 
reservoir, while the temperature of the body is as closely 
analogous to the elevation of the cistern. He must notice, 
however, that water-power does not depend upon mere 
quantity of water : to be capable of driving a mill the water 
must have " head/' or elevation. 

But, in another and quite different sense, heat in a body 
is analogous to the water, not to the water-power. Heat 
tends to pass from a hotter to a colder body, just as water 



6 HEAT. [CHAP. 

tends to flow from a cistern at a higher to another at a lower 
level. Thus heat in a hot body is in this property analo- 
gous to (or at least behaves like) water at a high level, and 
vice versa. 

Two such connected and yet different analogies can be 
safely presented to the student at an early stage ; for they 
will certainly help his conceptions, and their difference 
will prevent his being in any way misled by either. 

7. For all that, as will presently be seen, heat though 
not material has objective existence in as complete a sense as 
matter has. 

This may appear, at first sight, paradoxical ; but we must 
remember that so-called paradoxes are merely facts as yet 
unexplained, and therefore still apparently inconsistent with 
others, already understood in their full significance. 

When we say that matter has objective existence, we 
mean that it is something which exists altogether indepen- 
dently of the senses and brain-processes by which alone we 
are informed of its presence. An exact, or adequate con- 
ception of it, if it could be formed, would probably be 
something very different from any conception which our 
senses will ever enable us to form ; but the object of all 
pure physical science is to endeavour to grasp more and 
more perfectly the nature and laws of the external world, 
by using the imperfect means which are at our command 
reason acting as interpreter as well as judge ; while the 
senses are merely the witnesses, who may be more or less 
untrustworthy and incompetent, but are nevertheless of 
inconceivable value to us, because they are our only available 
ones. 

8. Without further discussion we may state once for all 
that our conviction of the objective reality of matter is based 
mainly upon the fact, discovered solely by experiment, that we 



I.] FUNDAMENTAL PRINCIPLES. 7 

cannot in the slightest degree alter its quantity. We cannot 
destroy, nor can we produce, even the smallest portion of 
matter. But reason requires us to be consistent in our 
logic ; and thus, if we find anything else in the physical 
world whose quantity we cannot alter, we are bound to 
admit it to have objective reality as truly as matter has, 
however strongly our senses may predispose us against the 
concession. Heat, therefore, as well as Light, Sound, 
Electric Currents. &c., though not forms of matter, must be 
looked upon as being as real as matter, simply because they 
have been found to be forms of Energy (Chap III. below), 
which in all its constant mutations satisfies the test which 
we adopt as conclusive of the reality of matter. 

But the student must here be again most carefully warned 
to distinguish between heat and the mere sensation of 
warmth; just as he distinguishes between the energy of 
motion of a cudgel and the pain produced by the blow. 
The one is the thing to be measured, the other is only the 
more or less imperfect reading or indication given by the 
instrument with which we attempt to measure it in terms of 
some one of its effects. 

9. There is one other point which must be insisted on as 
a necessary preliminary to all physical inquiries, to wit, the 
condition under which alone it is possible that physical 
science can exist We may enunciate it as follows : 

Under the same physical conditions the same physical results 
will always be produced, irrespective altogether of time or place. 

It requires no comment whatever, if the terms employed 
be fully understood and be interpreted in the strictest 
sense. It is, in fact, merely the assertion (based entirely on 
observation and experiment) of the existence of definite 
and unchanging laws to which all physical processes are 
found to be subject. 



CHAPTER II. 

INTRODUCTORY. 

i o. UNTIL all physical science is reduced to the deduction 
of the innumerable mathematical consequences of a few 
known and simple laws, it will be impossible completely 
to avoid some confusion and repetition, whatever be the 
arrangement of its various parts which we adopt in bringing 
them before a beginner. 

When we confine ourselves to one definite branch of the 
subject, all of whose fundamental laws can be distinctly 
formulated, there need be no such confusion. Here in fact 
the mathematician has it all in his own hands. He is the 
skilled artificer with his plan and his trowel, and the hodmen 
have handed up to him all the requisite bricks and mortar. 
This has long been known and recognised as a fact, but 
it has not often been put so neatly as in the following 
extract : 

" That which is properly called Physical Science is the 
knowledge of relations between natural phenomena and 
their physical antecedents, as necessary sequences of cause 
and effect ; these relations being investigated by the aid of 
mathematics that is, by a method in which processes of 
reasoning, on all questions that can be brought under the 



CHAP, ii.] INTRODUCTORY. 9 

categories of quantity and of space-conditions, are rendered 
perfectly exact, and simplified and made capable of general 
application to a degree almost inconceivable by the un- 
initiated, through the use of conventional symbols. There 
is no admission for any but a mathematician into this school 
of philosophy. But there is a lower department of natural 
science, most valuable as a precursor and auxiliary, which 
we may call Scientific Phenomenology ; the office of which 
is to observe and classify phenomena, and by induction to 
infer the laws that govern them. As, however, it is unable 
to determine these laws to be necessary results of the action 
of physical forces, they remain merely empirical until the 
higher science interprets them. But the inferior and 
auxiliary science has of late assumed a position to which it 
is by no means entitled. It gives itself airs, as if it were 
the mistress instead of the handmaid, and often conceals its 
own incapacity and want of scientific purity by high-sound- 
ing language as to the mysteries of nature. It may even 
complain of true science, the knowledge of causes, as 
merely mechanical. It will endue matter with mysterious 
qualities and occult powers, and imagines that it discerns 
in the physical atom 'the promise and potency of all 
terrestrial life.' " * 

Thus all who have even a slight acquaintance with the 
subject know that the laws of motion, and the law of gravita- 
tion, contain absolutely all of Physical Astronomy, in the 
sense in which that term is commonly employed : viz., the 
investigation of the motions and mutual perturbations of a 
number of masses (usually treated as mere points, or at 
least as rigid bodies) forming any system whatever of sun, 
planets, and satellites. 

But, as soon as physical science points out that we must 
* Church Quarterly Review, April 1876, p. 149. 



io HEAT. [CHAP. 

take account of the plasticity and elasticity of each mass of 
such a system, the amount and distribution of liquid on its 
surface, possibly of magnetic and other actions between 
them, and the resistance due to the medium in which they 
move ; the simplicity of the data of the mathematical 
problem is gone ; and physical astronomy, except in its 
grander outlines, becomes as much confused as any other 
branch of science. 

So it is with the Dynamics of Solid and Fluid bodies : 
so long as we are content to view solids as perfectly smooth 
and rigid, and fluids as incompressible and frictionless, the 
difficulties of Dynamics, though often enormously great, are 
entirely mathematical, it falls naturally into quite distinct 
and separate heads, and the classification of its various 
problems is comparatively simple. Introduce ideas of strain, 
and fluid friction, with consequent development of heat, and 
the confusion due to imperfect or impossible classification 
comes in at once. Each problem, instead of being treated 
by itself, has to borrow, sometimes over and over again, 
from others ; and the only fully satisfactory and uncom- 
plicated mode of attacking such a subject (were it con- 
ceivable) would be to work it all out at once. 

ii. Hence, in dealing with the general subject of heat 
we shall find it quite impossible to lay down definite lines 
of demarcation. Divide it as we choose, each part will be 
found to require for its development something borrowed 
from another. 

All that we can do under these conditions (the existence 
of which simply means that we do not yet know all about 
heat in the same sense as we may be said to know the laws 
of motion and of gravitation) is to make our classification 
confessedly somewhat indefinite, and freely to assume 
throughout, when needed, results of other parts of our 



n.] INTRODUCTORY. * n 

subject which we have not yet discussed. The advantages 
of this method, at least in my own experience, have been 
found much to outweigh its obvious but as yet inevitable 
disadvantages. But, to reduce the latter as far as possible, 
I shall first go over the whole subject briefly, so as to point 
out its main features and their mutual relations ; explaining 
some of the more important things, though not giving their 
experimental proof; and thus the student will be from the 
outset fairly prepared to take up in turn each of our divi- 
sions of the subject with as much detail as is consistent with 
the dimensions of an elementary treatise. And another 
advantage will be gained, inasmuch as such a rapid and 
general glance at the whole subject will admit of a number 
of useful and even important digressions which would 
seriously impair the consistency of the more detailed and 
definite part of the work. 

12. The classification I have found convenient is as 
follows : 

1. Nature of Heat. 

2. Effects of Heat. 

3. Measurement of Heat and of Temperature. 

4. Sources of Heat. 

5. Transference of Heat. 

6. Transformations of Heat. 

As already pointed out, there is no hard and fast line drawn 
between any two of these heads, in fact the explanations of 
many even among ordinary phenomena belong in part to 
more than one of them. 

It is well, however, to remark that there is a very intimate 
connection between the three heads (i), (4), and (6) above, 
which contain among them the chief recent advances of 
the Dynamical Theory of ffeaf y or Thermodynamics, as it is 
commonly called. Hence (i) will be more fully developed 



'12 HEAT. [CHAP. u. 

than the other heads in our first rapid resume of the whole 
subject, while its farther development will be wholly merged 
in that of (4) and (6), which may then profitably be studied 
together. Again (3) depends, at least in all its ordinary 
practical forms, on some application or other of one of the 
group (2). (5) stands to a great extent by itself, but na- 
turally divides itself into three perfectly distinct processes, 
all of which are of great scientific as well as practical 
importance. 



CHAPTER III. 

DIGRESSION ON FORCE AND ENERGY. 

13. WE must now take up briefly and in order these 
divisions of our subject ; but before we can do so intelligibly, a 
digression into the elements of Dynamics is absolutely essen- 
tial. This involves, in fact, the citation and explanation of 
a passage in Newton's Principia which, till very lately, seems 
to have altogether escaped the notice of scientific men. 
The reader will find that the consideration of this passage, 
especially when he sees it rendered into the terms used in 
modern science, will greatly facilitate his farther progress 
with regard to the nature of heat. 

Newton's Third Law of Motion is to the effect that 

" To every action there is always an equal and contrary re- 
action; or, the mutual actions of any two bodies are always 
equal and oppositely directed" 

This law Newton first shows to hold for ordinary pres- 
sures, tensions, attractions, &c., that is, for what we com- 
monly call forces exerted on one another by two bodies ; 
also for impacts, or impulses, which are merely the time- 
integrals of forces. 

But he proceeds to point out that the same law is true in 
another and much higher sense. He says : 

" If the acti'ity of an agent be measured by the product of its 



14 HEAT. [CHAP. 

force into its velocity ; and if, similarly p , the counter-activity of 
the resistance be measured by the velocities of its several parts 
into their several forces, whether these arise from friction, 
cohesion, weight, or acceleration ; activity and counter-activity, 
in all combinations of machines, will be equal and opposite" 

The actions and reactions which are here stated to be 
equal and opposite are no longer simple forces, but the 
products of forces into their velocities ; i.e. they are what 
are now called Rates of doing Work ; the time-rate of 
increase, or the increase per second, of a very tangible and 
real SOMETHING : for the measurement of which Watt intro- 
duced the practical unit of a horse-power, the rate at which 
an agent works when it lifts 33,000 pounds one foot high 
per minute, against the earth's attraction. 

14. Now let the reader think of the difference between 
raising a hundredweight and endeavouring to raise a ton. 
With a moderate exertion he can raise the hundredweight 
a few feet, and in its descent it might be employed to drive 
machinery, or to do some other species of work. Let him tug as 
he pleases at the ton, he will not be able to lift it; and 
therefore after all his exertion, it will not be capable of 
doing any work by descending again. 

In both cases the first interpretation of Newton's Third 
Law has been verified. With whatever force he pulled 
either of the masses, that mass reacted with an equal force. 
But the second interpretation cannot be applied to the ton ; 
for it did not acquire any velocity, it was not moved. Hence, 
as no work was spent upon it, it has not acquired the 
power of doing work. On the other hand, the hundred- 
weight was moved, work was done upon it, and that work 
was stored up in it in its raised position, ready for use 
at any future time. Newton's statement implies that in 
this case the work spent in raising the hundredweight is 



in.] DIGRESSION ON FORCE AND ENERGY. ^15 

stored up (without change of amount) in the mass when 
raised. 

15. Thus it appears that force is a mere name; but that 
the product of a force into the displacement of its point of 
application has an objective existence. [Even those who 
are so metaphysical as not to see that the product of a mere 
name into a displacement can have objective existence, may 
perhaps see that the quotient of a horse-power by a velocity 
is not likely to be more than a mere name.] In fact, 
modern science shows us that force is merely a convenient 
term employed for the present (very usefully) to shorten 
what would otherwise be cumbrous expressions; but it is 
not to be regarded as a thing ; any more than the bank rate 
of interest is to be looked upon as a sum of money, or than 
the birth-rate of a country is to be looked upon as the 
actual group of children born in a year. And a very simple 
mathematical operation shows us that it is precisely the 
same thing to say 

The horse-power of an agent, or the amount of work done 
by an agent in each second, is the product of the force into the 
average velocity of the agent : 
and to say 

Force is the rate at which an agent does work per unit 
of length* 

1 6. In the special illustration of Newton's words which 
we have just given, the resistance was a weight, that of a 
hundredweight or of a ton. When the resistance was 



* In symbols this is merely 
whence 



= fv = / , the first statement, 
at at 



- /, the second. 
dx 



1 6 HEAT. [CHAP. 

overcome, work was done, and it was stored up for use 
in the raised mass in a form which could be made use of 
at any future time. 

Following a hint given by Young, we now employ the 
term ENERGY to signify the power of doing work, in whatever 
that power may consist. The raised mass, then, we say, 
possesses in virtue of its elevation an amount of energy 
precisely equal to the work spent in raising it. This dormant, 
or passive, form, is called Potential Energy. Excellent 
instances of potential energy are supplied by water at a 
high level, or with a "head," as it is technically called, in 
virtue of which it can in its descent drive machinery ; by 
the wound-up "weights" of a clock, which in their descent 
keep it going for a week ; by gunpowder, the chemical 
affinities of whose constituents are called into play by a 
spark; &c. &c. 

Another example of it is suggested by the word "Cohe- 
sion" employed in Newton's statement ( 13), and which 
must be taken to include what are called molecular forces in 
general, such as for instance those upon which the elasticity 
of a solid depends. 

When we draw a bow we do work, because the force 
exerted has a velocity ; but the drawn bow (like the raised 
weight) has in potential energy the equivalent of the work 
so spent. That can in turn be expended upon the arrow ; 
and what then ? 

1 7 . Turn again to Newton's words ( 1 3) and we see that 
he speaks of one of the forms of resistance as arising from 
" acceleration" In fact, the arrow, by its inertia, resists being 
set in motion j work has to be spent in propelling it : but 
the moving arrow has that work in store in virtue of its 
motion. It appears from Newton's previous statements that 
the measure of the rate at which work is spent in producing 



in.] DIGRESSION ON FORCE AND ENERGY. -17 

acceleration * is the product of the momentum into the accele- 
ratioji in the direction of motion, and the energy produced 
is measured by half the product of the mass into the square 
of the velocity produced in it. This active form is called 
Kinetic Energy, and it is the double of this to which the 
term Vis Viva (erroneously translated Living Force) has 
been applied. 

As instances of ordinary kinetic energy, or of mixed 
kinetic and potential energies, take the following : A 
current of water capable of driving an undershot wheel ; 
winds, which also are used for driving machinery ; the energy 
of water-waves or of sound-waves ; the radiant energy which 
comes to ilk from the sun, whether it affect our nerves of 
touch or of sight (and therefore be called radiant heat or 
light) or produce chemical decomposition, as of carbonic 
acid and water in the leaves of plants, or of silver-salts in 
photography (and be therefore called actinism) ; the energy 
of motion of the particles of a gas, upon which its pressure 
depends, &c. [When the motion is vibratory the energy is 
generally half potential, half kinetic.] 

1 8. These explanations and definitions being premised, 
we can translate Newton's words (without alteration of their 
meaning) into the language of modern science, as follows : 

Work done on any system of bodies (in Newton's statement 
the parts of any machine) has its equivalent in work done 
against friction, molecular forces, or gravity, if there be no 
acceleration ; but if there be acceleration, part of the work is 
expended in overcoming the resistance to acceleration, and the 

* Let v be the acceleration, in the direction of motion, of a mass M 
whose velocity is v. Then Newton's expression for the rate of spending 
work against the resistance to acceleration is Afzr . v, or as above, Mv . v, 
and the whole work spent in giving the velocity v to the mass M, 
originally at rest, is \ Mv 1 . 

C 



1 8 HEAT. [CHAP. 

additional kinetic energy developed is equivalent to the work so 
spent. 

But we have just seen that when work is spent against 
molecular forces, as in drawing a bow or winding up a 
spring, it is stored up as potential energy. Also it is stored 
up in a similar form when done against gravity, as in raising 
a weight. 

Hence it appears that, according to Newton, whenever 
work is spent, it is stored up either as potential or as kinetic 
energy : except possibly in the case of work done against 
friction, about whose fate he gives us no information. Thus 
Newton expressly tells us that (except possibly when there 
is friction) energy is indestructible it is changed from one 
form to another, and so on, but never altered in quantity. 
To make this beautiful statement complete, all that is 
requisite is to know what becomes of work done against 
friction. 

19. Here, of course, experiment is requisite. Newton, 
unfortunately, seems to have forgotten that savage men had 
long since been in the habit of making it whenever they 
wished to procure fire. The patient rubbing of two dry 
sticks together, or (still better) the drilling of a soft piece of 
wood with the slightly blunted point of a hard piece, is 
known to all tribes of savages as a means of setting both 
pieces of wood on fire. Here, then, heat is undoubtedly 
produced, but it is produced by the expenditure of work. In 
fact, work done against friction has its equivalent in the heat 
produced. This Newton failed to see, and thus his grand 
generalization was left, though on one point only, incom- 
plete. The converse transformation, that of heat into work, 
dates back to the time of Hero at least. But the knowledge 
that a certain process will produce a certain result does not 
necessarily imply even a notion of the "why"; and Hero 



JJK] DIGRESSION ON FORCE AND ENERGY. - 19 

as little imagined that in his CEolipile heat was converted 
into work, as do savages that work can be converted into 
heat. 

Rumford and Davy, at the very end of last century, by 
totally different experimental processes, showed conclusively 
that the materiality of heat could not be maintained ; and 
thus gave the means of completing Newton's statement. 

20. One particular case of the Conservation of Energy 
had been formulated long before Newton's time, under the 
title of the Impossibility of the " Perpetual Motion." This 
was employed by Stevinus as the basis of Statics. In 1775 
the French Academy of Sciences refused to consider any 
scheme pretending to give work without corresponding and 
equivalent expenditure. The multiplied experiments of 
some of the most ingenious men who have ever lived have 
been directed to obtaining the Perpetual Motion, and their 
absolute failure in every case may be taken as proof of the 
impossibility. One mode of reasoning from this acknowledged 
impossibility may be here explained by a particular instance, 
as it will be found very useful in some of the more theoreti- 
cal parts of our subject. 

We employ it to show that in all cases of natural laws, 
such as the laws of gravitation and of magnetic attraction, 
the work spent in moving a body through a certain course 
in one direction is exactly restored by letting it return to its 
first position, not merely by the original path but by any 
other; always on the supposition that friction is avoided. 
Suppose there could be two courses, from A to B, by the 
one of which more work would be spent on the mass than 
by the other. Let these amounts be W and w. If such 
were the case the Perpetual Motion could be produced. 
Apply frictionless constraint to guide the mass, so that in its 
ascent it shall travel along the course A w B, and in its 

c 2 



20 HEAT. [CHAP. in. 

descent, along B W A. From A to B the amount w of 
work is spent against the forces of the system from B to 
A these forces refund the amount W. On the whole, after a 
complete cycle, the mass is restored to A with an amount 
W w of energy additional to what it possessed at starting. 




Every time the mass goes round the double course in the 
same direction it gains the difference between the larger 
quantity and the smaller one, and therefore at the end of 
each complete cycle, that amount may be drained off to 
turn some machine ; to do useful work. 

We here assume that the work spent in one course would 
be exactly recovered by letting the mass retrace its steps ; 
in other words, that the operation is reversible. This term 
will be fully discussed later. 

In general, if there were one way of doing a thing at less 
cost than another, and if the more costly operation were 
reversible, it would be possible to get unlimited amounts 
of useful work from nothing. 

We are now prepared to undertake the short general 
glance over our subject promised in n, according to the 
plan there laid down. 



CHAPTER IV. 

PRELIMINARY SKETCH OF THE SUBJECT. 

21. Nature of Heat. Heat, under the name of fire, 
which seems to have included everything either really or 
apparently of the nature of flame, such as sun and moon, 
stars, planets, and comets, lightning and Aurora, &c., as well 
as ordinary fire, was in old times regarded as one of the so- 
called Four Elements, of which, or of some of which, it was 
imagined that everything in the physical world was neces- 
sarily composed. 

The tendency of early experimental science, which took 
its first great impulse, if not its absolute origin, from Gilbert 
of Colchester (circa 1570), was to regard heat, light, elec- 
tricity, &c., as forms of matter excessively subtle and 
refined capable of freely pervading and combining with 
all ordinary gross matter. They were, in fact, classed as 
Imponderables, because a heated or electrified body, for 
instance, was not found to be increased in weight by the 
heat or electricity which it was supposed to have imbibed. 

22. In this sense heat was supposed to be an excessively 
light species of matter, and was called Caloric, or, in certain 
of its manifestations (especially in some chemical processes), 
Phlogiston : though, as has been recently shown,* the latter 

* Crum Brown, Proc. R.S.E., 1870. 



22 HEAT. [CHAP, 

term was also used to describe certain forms of what we now 
call chemical potential energy. (The meaning of this term 
will be obvious from the statement above ( 16) regarding 
the energy of gunpowder.) 

The notion of the materiality of heat or caloric, in spite 
of experiments proving the contrary, and of several shrewd 
guesses* as to their true nature, was all but universally 
accepted and taught till about 1840. Then commenced 
that rapid revolution which has entirely altered the generally 
accepted views of heat, just as a few years previously the 
Material or Corpuscular Theory of Light may be said to 
have received its death-blow; or, as in somewhat more 
recent times, the so-called Electric Fluids have been ban- 
ished from all British science except some portions of that 
spurious and worse than useless kind which is commonly 
called popular. 

23. One by one the Imponderables have been displaced 
from their old and proud position in science ; but they have 
all died hard, and among many of the continental schools 
of physicists, and especially of the German, the electric 

* " And tryall hath taught me that there are liquors, in which the 
bare admixture of milk, oyle, or other liquors nay, or of cold water, 
will presently occasion a notable heat : and I sometimes imploy a 
menstruum, in which nothing but a little flesh being put, though no 
visible Ebullition ensue, there will in a few minuts be excited a Heat, 
intense enough to be troublesome to him that holds the Glasse. And 
yet it seems not necessary that this should be ascrib'd to a true fermen- 
tation, which may rather proceed from the perturb'd motion of the 
Corpuscles of the menstruum, which being by the adventitious liquor 
or other body put out of their wonted motion, and into an inordinate 
one, there is produced in the menstruum a brisk confus'd Agitation of 
the small parts that compose it ; and in such an agitation (from what 
cause soever it proceeds), the nature of heat seems mainly to consist." 
Some Considerations touching the Usefulness of Experimental Naturall 
Philosophy. By Robert Boyle. Part II., Section I, p. 4=5. 1663. 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. : 23 

fluids still retain (thanks to an exceedingly ingenious idea 
of W. Weber, quite as good in its way as Newton's Fits of 
Easy Reflection and Transmission) their position as, in a 
certain sense, forms of imponderable matter, acting on one 
another according to a very peculiar law, quite different 
from anything met with in other branches of physics. But 
even there the end must soon come, when this last trace of 
the imponderables shall be handed over for preservation in 
the museum of the scientific antiquary. 

24. The experiments, which, as stated in 22, first 
proved that heat is not matter, are due to E.umford and 
Davy, and date from 1798 and 1799 respectively. 

The explanation of the heat produced by friction which 
was given by those who believed heat or caloric to be matter 
was simply this : The body in' its solid state, or rather in its 
massive state, before you began to abrade filings from it, 
possessed, at any particular temperature, a certain quantity 
of heat. It had a certain capacity for heat, as it was called ; 
in other words, it required so much heat to be mixed up 
with its particles in order to make the temperature of the 
whole that which was observed. But if you could make it 
more capacious if you could give it greater capacity for 
heat then it would hold more heat without becoming of a 
higher temperature. On the other hand, if by any process 
whatever you could diminish its capacity for heat, then, of 
course, it would become hotter in itself, and even give out 
heat to surrounding bodies ; so that, according to the notion 
of the supporters of this theory, the production of heat by 
friction or abrasion is due to the fact that you make the 
capacity of a body for heat smaller by reducing it to powder. 
For of course, when its capacity for heat is thus made 
smaller, it must part with some of the heat it had at first ; 
or if it retains it, it must necessarily show the effect of the 



24 HEAT. [CHAP. 

heat more than it did before, and must therefore rise in 
temperature. Now this reasoning is, so far, perfectly 
philosophical, and we can say nothing against it as a mode 
of reasoning. But it involves the fallacious assumption that 
heat is matter, and therefore indestructible. 

25. Next, see how well Rumford laid hold of that point, 
and how he proceeds by experiment to try if possible to 
satisfy his doubts about it. He says : 

"If this were the ca-:e, then, according to the modern doctrines of 
latent heat, and of caloric, the capacity for heat of the parts of the 
metal so reduced to chips, ought not only to be changed, lut the 
change undergone by them should be sufficiently great to account for 
#//the heat produced." 

Rumford found no difference, so far as his form of 
experiment enabled him to * test them, between the ca- 
pacities for heat of the abraded metal and of the metal 
before the abrasion had taken place : so that if this 
additional experiment had only been a satisfactory one : 
and Rumford did not see how to make it thoroughly satis- 
factory : the fact that heat is not matter would have been 
conclusively established in 1798. What Rumford really did 
want was a test to bring the abraded rnetal and the non- 
abraded metal, if possible, precisely to the same final state. 
He tried to do that by throwing them into water equal 
masses of the metal in lumps and filings, each raised to 
the same high temperature, into equal quantities of water 
at the same lower temperature, to see whether they would 
produce different changes of temperature, each in its own 
vessel of water. 

26. But then they were not in the same final state. The 
filings were not in the same state as the solid metal ; they 
might have been veiy considerably compressed, or they 
might have been distorted in shape, and in virtue of these 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 25 

they might have had a certain quantity of latent heat which 
Rumford could not discover by this process. The simplest 
legitimate process which we know of for completely answer- 
ing this question, which was Rumford's sole difficulty, is a 
chemical process. Dissolve the lumps and an equal weight 
of the filings in equal quantities of an acid. At the end of 
the operation, of course, there can be no doubt that the 
chemical substances produced will be precisely the same, 
whether you begin with lumps or with filings. If there be 
any mysterious difference as to the capacity for heat in 
them, that will be shown during the process of solution. In 
general, in dissolving a metal in an acid, there is a develop- 
ment of heat ; but if there were any difference in the 
quantity of heat which the lumps and an equal weight of 
filings contained when at the same temperature that is to 
say, if heat could by any possibility be matter there would 
necessarily have been a greater development of heat in one 
vessel than in the other. Had Rumford tried that one 
additional experiment, he would have had the sole credit of 
having established the non-materiality of heat. 

27. Rumford found that, in spite of inevitable loss of heat 
in his operations (which consisted in boring a cannon with 
a blunt borer), the work of a single horse for two hours and 
twenty minutes was sufficient to raise to the boiling-point 
about nineteen pounds of water, besides heating the cannon 
and all the machinery engaged in the process. Here is 
his final reasoning : 

" In reasoning on this subject, we must not forget to consider that 
most remarkable circumstance, that the source of heat generated by 
friction in these experiments appeared evidently to be inexhaustible. 

"It is hardly necessary to add, that anything which any insulated 
body or system of bodies can continue to furnish without limitation, 
cannot possibly be a material substance. It appears to me to be ex- 
tremely difficult, if not quite impossible, to form any distinct idea of 



26 HEAT. [CHAP. 

anything capable of being excited and communicated in the manner in 
which heat was excited and communicated in these experiments, except 
it be motion." 

28. When we make a calculation from the data furnished 
by Rumford's paper, we find that, supposing heat to be a 
form of energy, and taking 30,000 foot-pounds per minute 
as the work of a horse, the mechanical equivalent of heat is 
940 foot-pounds. The meaning of this statement is, that 
if you were to expend the amount of work designated as 
940 foot-pounds in stirring a single pound of water, that 
pound of water at the end of the operation would be i 
Fahrenheit hotter than before you commenced. We can put 
it in another form, which is perhaps still more striking. In 
the fall down a cascade or waterfall 940 feet high there would 
be 940 foot-pounds of work done by gravity upon each 
pound of water ; and therefore if all the energy which the 
moving water has as it reaches the bottom of the fall were 
spent simply in heating the water, the result would be that 
the water in the pool at the bottom of the fall would be i 
Fahrenheit hotter than the water at the top. 

29. Davy first showed that by rubbing two pieces of ice 
together by simply expending work in the friction of two 
pieces of ice ice could be melted. Now a believer in the 
caloric theory would have argued thus : two pieces of ice 
when rubbed together cannot possibly melt one another, 
because in order to melt them heat must be furnished to 
them. But if the heat can only come from themselves when 
they are rubbed together, if it cannot come from surround- 
ing bodies, they cannot possibly melt; because to melt 
one another they would have first to part with some of 
their heat. 

To Davy's experiment, which seems to decide the ques- 
tion against the calorists, there was therefore this possible 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. ! 27 

objection, that the heat might have come from some external 
source, so that he tried a second form of experiment. He 
rubbed two pieces of metal together, keeping them sur- 
rounded by ice, and in the exhausted receiver of an air- 
pump, so as to remove every possible disturbing cause, or 
even source of suspicion, from his experiment ; and still he 
found that these two pieces of metal when rubbed together 
produced heat and melted the ice, every precaution having 
been taken to prevent heat from getting at them from every 
side. 

30. It is curious that his reasoning upon the subject is 
extremely inconclusive, although his experiments themselves 
completely settle the question. He says : 

" From this experiment it is evident that ice by friction is converted 
into water, and according to the supposition its capacity is diminished ; 
but it is a well known fact that the capacity of water for heat is much 
greater than that of ice ; and ice must have an absolute quantity of 
heat added to it before it can be converted into water. Friction, conse- 
quently, does not diminish the capacities of bodies for heat." 

And there he stops. But some years afterwards he came to- 
this conclusion from these experiments : 

" Heat, then, or that power which prevents the actual contact of the 
corpuscles of bodies, and which is the cause of our own sensations of 
heat and cold, may be defined as a peculiar motion, probably a vibration, 
of the corpuscles of bodies tending to separate them. It may with 
propriety be called the repulsive motion. Bodies exist in different states, 
and these states depend upon the action of attraction and of the repul- 
sive power on their corpuscles, or, in other words, on their different 
quantities of repulsion and attraction. " 

31. Davy explains by these experiments the difference 
between a solid and a liquid, and that between a liquid 
and a gas. In general the melting of a solid is produced by 
communicating heat to it. In other words, according to 
Davy's explanation, the particles of the solid are set in 



28 HEAT. [CHAP. 

vibration, and thus, in consequence of repeated impacts 
upon one another, push one another aside. And as he 
.also says, you may consider this repulsive motion to 
have a complete analogy to the so-called centrifugal force 
in a planetary orbit, for the faster one particle is moving 
about another, the larger necessarily is the orbit into which 
it will be forced. The particles of a solid, then, are forced 
from one another by this repulsive action of heat, and it 
assumes what we call the liquid state. Increase still 
farther the amount of heat communicated to the body, the 
cohesive forces are at length wholly overcome, and you 
have free particles, as in a gas, flying about and impinging 
upon one another, but only for very brief periods coming 
near enough in the course of their gyrations to bring into 
play the molecular forces again. When, however, the 
molecular forces do come into play for a moment, you may 
have two particles adhering together, but they are soon 
knocked asunder by a blow from a third particle. 

32. There is one other sentence, however, which must 
be quoted from Davy, for it shows when and how he finally 
got over his difficulties and confusion of reasoning. In 
1812 he enunciated this proposition : 

" The immediate cause of the phenomenon of heat, then, is motion, 
and the laws of its communication are precisely the same as the laws of 
the communication of motion." 

When Davy was in a position to make that statement he 
had only to take it in addition to the second interpretation 
of Newton's third law (ante 18), and the whole dynamical 
theory of heat was in his possession. Still, that publication 
of Davy's in 1812, like the earlier ones of Rumford and of 
Davy himself, remained almost unnoticed looked upon, 
perhaps, as an ingenious guess, or something of that sort, but 



iv.J PRELIMINARY SKETCH OF THE SUBJECT. ; 29 

as something which it was not worth the trouble of philoso- 
phers to consider ; and it was not until Joule's time, some- 
where about 1840, that the subject was fairly taken up, and 
that justice was rendered to their real value. 

33. Notice, however, how distinctly these two great 
leaders were men who based their work directly upon 
experiment. There is no a priori guessing, or anything of 
that kind, about either Rumford's or Davy's work. They 
simply set to work to find out what heat is. They did not 
speculate on what it might be. But both before and after 
their time there have been numbers of philosophers who 
have, without trying a single experiment, or at best trying 
only the roughest forms of experiment, endeavoured to 
discover by a priori reasoning what heat is. 

34. The investigations of Colding and Joule, dating from 
about 1840, cannot here be treated in full; because, though 
it was to them that the final dethronement of caloric from 
its old position as an imponderable is unquestionably due, 
these exquisite experiments (especially those of Joule) 
extended to all forms of energy, and therefore included 
an immense range of subjects entirely beyond the scope of 
this work. 

In so far, however, as they bear upon our present ques- 
tion, these researches re-established the conclusions of 
Rumford and Davy ; and they supplied a much more exact 
determination of the dynamical (or, as it is commonly but 
inaccurately called, mechanical} equivalent of heat than that 
above deduced ( 28) from some of Rumford's experimental 
data. 

35. In fact the extensive, and exceedingly accurate, ex- 
periments of Joule led, in 1843 and subsequent years, by 
processes depending directly on friction, to numbers varying 
from 770 to 774 foot-pounds of energy as the equivalent 



30 HEAT. [CHAP. 

of one unit of heat (denned below, 37) on the Fahrenheit 
scale. The number finally assigned by Joule (for the lati- 
tude of Manchester) is 772, and it is almost certainly not 
in error by anything approaching to i per cent. 

In 1853 Joule verified this result by means of a very 
accurate determination of the specific heats of air, and a 
direct experimental proof (given in 1845) that the heat 
developed by the sudden compression of air is very nearly 
the equivalent of the work expended. 

36. Direct measurements of the heat produced by the 
expenditure of mechanical energy were made in various 
ways by Colding, in 1843, and have been repeated in many 
forms by Him, Regnault, &c., since the publication (in 
1849) of Joule's final result. 

The direct verification of the fact that heat disappears 
when work is done by a heat-engine, unsuccessfully at- 
tempted by Seguin in 1839, was first effected by Him in 

1857- 

A great variety of indirect methods of approximating to 
the mechanical equivalent of heat have been successfully 
applied within the last thirty years by different experi- 
menters. The earliest determinations of this kind are, of 
course, those of Joule, effected in 1843 an d subsequent 
years, by means of magneto-electricity. 

37. The results of all such experiments are briefly summed 
up in the exceedingly important statement known as the 

First Law of Thermodynamics. When equal quan- 
tities of mechanical effect are produced by any means from 
purely thermal sources, or lost in purely thermal effects, equal 
quantities of heat are put out of existence or are generated. 
And, in the latitude of Manchester, 772 foot-pounds of 
work are capable of raising the temperature of i Ib. of 
water from 50 F. to 51 F. 



UNIVERSITY OF CALIFORNI/ 

DEPARTMENT OF PHYSICS 

iv.] PRELIMINARY SKETCH OF THE SUBJECT. 31 

38. Though by experiments such as those we have 
mentioned (but not, as yet, described or explained) Heat 
has been proved to be a form of Energy whether kinetic 
or potential we have made no progress towards the dis- 
covery of the mechanism upon which it depends. 

And, referring again to 30, 31, it may be stated 
generally that we have as yet very little information about 
the nature of the internal heat-motions, &c., of the 
particles of solids and liquids. 

39. But we have obtained from several kinds of experi- 
ments, entirely different from these, what must be called a 
probable, rather than a plausible, explanation of the effects 
of heat on gases along with at least a general idea of the 
nature of the motions upon which heat depends in such 
bodies. This idea, hinted at in 31, is due originally to 
Hooke,* and (sixty years later) to D. Bernoulli ; t but it has 
been resuscitated and immensely developed of late years 
by Herapath, Joule, Clausius, Clerk-Maxwell, Boltzmann, 
and others. It has been found capable of explaining a very 
great number of the known physical properties of gases : 
and it seems destined, in a comparatively short time, to 
acquire all the claims to acceptance which can be demanded 
from a physical theory of the motions, collisions, &c., of 
particles of matter whose exceedingly minute dimensions 
put them altogether and for ever beyond the range of the 
most perfect possible microscope. 

40. From another absolutely distinct point of view we 
have obtained very remarkable information as to the nature 
of the motion upon which Radiant Heat depends. But 
here we are especially favoured, because we have complete 
experimental evidence that the nature and the mechanism 
of propagation of Radiant Heat are precisely the same as 

* De Potentia restitutivd, 1678. + Hydrodynamica, 1738. 



32 HEAT. (CHAP. 

those of Light '( 3). Thus we have two quite independent 
sense-organs adapted for the study of these allied pheno- 
mena. 

41. Resume of 20 40. Heat is now proved to be a 
form of energy. Hence the First Law of Thermodynamics, 
which merely states the equivalence of heat to work, with 
the requisite numerical datum called Joule's Equivalent. 
The mechanism upon which heat-energy depends is (pro- 
bably at least) approximately known so far as regards heat 
in a gas, and as regards radiant heat. Beyond these we 
have, as yet, little information on the subject. 

42. Effects of Heat. These are exceedingly varied 
and numerous, and in our present rapid sketch we cannot 
allude to any but the more common or more prominent of 
them. These we may classify as follows : 

a. Change of Dimensions, or of Stresses, in Solids, and 

of Volume, or of Pressure, in Fluids. 

b. Change of Molecular State. 

c. Change of Temperature. 

d. Electric Effects. 

e. Effects in starting Chemical Changes, 

43 (a). The change of dimensions and stresses of solid 
bodies by heat is known by experience, even of the com- 
monest kind, to every one. We mention a few instances 
taken at random, but it will be good exercise for the student 
to try to recollect others, and the same exercise will be 
found profitable in the other departments of this subject. 

The " shrinking on," as it is called, of a wheel-tire, and 
of coil after coil on the core of a wrought-iron gun, is 
accomplished by heating the tire or coil, slipping it on 
while expanded by heat, and then cooling it suddenly or 
gradually as may be necessary, 



iv.J PRELIMINARY SKETCH OF THE SUBJECT.* 33 

Rails are not laid down end to end, but with a small 
interval ; else on a summer day they might expand so much 
more than the ground supporting their bearings as to pucker, 
and displace one another. 

The huge metal tubes of the Menai bridge are not 
rigidly fixed at each end, else they would tear themselves 
or their supports ; they are free to expand and contract as 
their temperature changes, one end of each being supported 
on rollers. 

Uncompensated clocks and watches (if their average rate 
be exactly adjusted) go too slow in summer and too fast in 
winter ; the former simply, the latter mainly, on account of 
the change, by heat, of the dimensions of their moving 
parts. The touch of a finger on the graduated limb of a 
delicate meridian circle produces a perceptible change in 
the measured zenith-distance of a star. 

A harp tuned in a warm room rises notably in pitch 
when taken out into frosty air. 

Telegraph wires are seen to " sag " more and more as the 
temperature of the air rises. 

Massive walls pressed outwards (by overloading of the 
roofs or floors of buildings) have been forcibly restored to 
their vertical position by the contraction of iron rods passed 
through them ; nuts being screwed tight upon the rods up 
to the exterior surface, while the rods were in a state of 
expansion by heat. 

44 (a continuea). The grand circulations constantly going 
on in the ocean and in the atmosphere are mainly due to the 
expansion of water and of air by heat. Bulk for bulk, the 
heated portions are lighter than the colder, and rise above 
them in virtue of the ordinary hydrostatic laws, which 
explain the floating of oil on water or the rise of a balloon in 
the air. On a smaller scale the rise of what we call smoke 



34 HEAT. [CHAP. 

from a chimney, ventilation of mines by a fire at the bottom 
of an " upcast " shaft, &c., are examples of the same effect. 

45 (b and c). Take a piece of very cold ice. Though the 
assertion may appear a little startling at first, it is really a 
stone just as much as is a lump of rock-salt or galena 
only that its molecular or crystalline structure is somewhat 
more complex. It becomes warmer, just as other stones, by 
every fresh application of heat up to a certain point, which 
we call its melting-point but you cannot make it any hotter. 
Heat now does not change its temperature, but changes its 
molecular state. Precisely the same is true of the rock-salt 
only that the temperature of its melting-point is considerably 
higher than that of the ice. 

Suppose sufficient heat to have been applied just to melt 
all the ice. It is still the same substance from the chemical 
point of view, its temperature is still that of the melting- 
point, but it is a liquid instead of a solid. 

46 (b and c continued}. Apply more heat to the water. 
Its effect is now to make the water warmer : in scientific 
language, the temperature of the water rises. Every fresh 
application of heat raises the temperature more and more 
till it reaches what is called the boiling-point, but here the 
rise of temperature again stops. Farther application of heat 
produces a new alteration in the molecular state, and the 
liquid changes into steam or water vapour. 

47 (b and c continued] . Suppose heat to have been applied 
till the whole of the liquid has, without farther rise of tem- 
perature, been converted into vapour saturated steam, as it 
is technically called we can now, by applying more heat, 
raise the temperature of the steam, so that it becomes what 
is called superheated steam, and is practically a gas. But 
this gas cannot be heated indefinitely farther without the 
production of another molecular change this time what 



iv.J PRELIMINARY SKETCH OF THE SUBJECT.- 35 

is commonly called a chemical change dissociation : the 
analysis or separation of the water-gas into its constituents, 
oxygen and hydrogen. 

Experiment has not yet told us whether or not still farther 
applications of heat may be capable of altering the physical 
or chemical nature of either of these now merely mixed 
gases. 

48. Thus the successive effects, produced by continuous 
application of heat to a piece of very cold ice, are > 




2. Melting. 

3. Heating. 

4. Evaporation. 

5. Heating. 

6. Dissociation. 

7. Heating. 

Water-substance is familiar to all, and, as we have seen, 
gives an excellent and instructive example of the various 
successive changes of state and of temperature. Few other 
substances offer, at least with our present very limited ex- 
perimental facilities, so complete a series. On the other 
hand there are substances, such as carbon, which we cannot 
even melt; others, such as uncombined hydrogen, which, 
till the very end of 1877, we knew only in the gaseous 
form. 

49 (d). Certain crystals, such as tourmalines, when heated 
attract other bodies in the same way as do sealing-wax or 
glass which have been electrified by friction. This is, how- 
ever, still a very obscure branch of our subject, and has as 
yet yielded no results of great importance, though it may 
possibly become the source of enormous additions to our 
knowledge. 

D 2 



36 HEAT. [CHAP. 

49 (d continued}. But what is commonly called Thermo- 
electricity is already of immense practical as well as theo- 
retical importance, as the reader will see when we come to 
the question of measurement of heat and temperature. 

The fundamental phenomenon of Thermo-electricity is the 
following, discovered by Seebeck about 1820 : 

When one of the junctions of a closed circuit of two metals 
is raised to a higher temperature than the other, a current 
of electricity passes round the circuit and (in general) increases 
in intensity with increasing difference of temperature of the 
junctions. The direction of the current is, of course, reversed 
if the cold junction be now made the hotter. 

If, for instance, the ends of an iron and of a copper wire 
be joined by twisting, soldering, or otherwise, we form what 
is called a circuit or re-entrant path, round which an electric 
current can travel. 




Let A and B in the sketch be the (necessarily) two junc- 
tions. To a person going completely round the circuit in 
the direction indicated by the arrows, these junctions would 
be distinguished from one another by the fact that, while at 
B he is passing from iron to copper, and at A from copper 
to iron, if he reversed the direction of his motion round 
the circuit, these characteristics of the two junctions would 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 37 

be interchanged. Now suppose the whole to be at any 
ordinary temperature, and a lamp flame to be applied (for a 
moment only) at A. A current of electricity will pass round 
the circuit in the direction of the arrows, and will continue, 
though becoming gradually weaker, till A at length cools 
down again, so as to have the same temperature as B. This 
is Seebeck's discovery. What we have just said as to the 
characteristic difference between the two junctions shows us 
that, if the lamp be now applied for a moment to B, the 
current produced will run round the circuit in the opposite 
direction to that indicated by the arrows. In fact, under the 
conditions specified, the current passes from copper to iron 
through the warmer junction. (This statement applies to all 
cases in which neither junction is made very hot. It would 
only confuse the student at present to give him the general 
law, of which the above is merely a particular case. The 
whole subject will be thoroughly discussed later.) 

50 ( d continued]. Suppose now a closed circuit consisting 
of any number of pieces of iron and copper wire arranged 
alternately as below. It need not be longer (and therefore 
need not resist electricity more) than the circuit of two just 
described. These may be in fact cut up into shorter pieces 



7 /l'/f/l /I /I \ 




and re -arranged. Call the successive junctions in order 
A lf B 1? A 2 , B 2 , &c. Then it is obvious that a person going 



HEAT. 



[CHAP. 



round the circuit in the direction indicated by the arrows 
will at each A junction pass from copper to iron, and at 
each B junction from iron to copper. Hence all the A 
junctions, when heated, produce thermo-electric currents 
in the direction of the arrows. When the B junctions are 
all heated the current is the other way round. 

But in the simple circuit we had only one hot junction, 
while we may have here as many as we please ; it is obvious 
that by this arrangement we can increase the strength of the 
electric current (for the' same difference of temperatures of 
the hot and cold junctions) in the same proportion as we 
increase the total number of junctions. 








For convenience of application, the wires or rods Aj B lf 
A 2 , A 2 B 2 ,'&c., are made of equal length, and packed 



iv.] PRELIMINARY SKETCH OF THE SUBJECT: 39 

together as closely as possible into what is called a Thermo- 
electric Pile, so that all the A junctions are at one end and 
all the B junctions at the other. Thin paper, gutta percha, 
or other insulating material, is interposed between each two 
contiguous rods to prevent their touching one another, 
except where they are soldered together at the junctions. 
It has only to be added that as Seebeck (led by very curious 
reasoning) discovered that the currents produced by alter- 
nate bars of bismuth and antimony are far more powerful 
than those produced under the same circumstances of tem- 
perature and resistance by iron and copper the pile is 
generally built up of alternate bars of the former pair of 
metals. The invention of this valuable instrument is due to 
Nobili (1834). 

51 (e). There is little doubt that it is the heat developed 
by friction which inflames the particles of iron when the old 
" flint and steel " is used, just as it sets fire to amorphous 
phosphorus in contact with chlorate of potash in our modern 
" safety matches " ; or as the heat of a red-hot poker causes 
gunpowder to explode. In all such cases heat appears to 
induce or promote chemical combination ; to perform, in fact, 
the reverse of the operation which we have already attributed 
to it as dissociation (47). 

52. Measurement of Heat and of Temperature. 
The absolute distinction between the ideas of heat and of 
temperature cannot be too soon learned by the student. 
Heat, we have seen, is a REAL SOMETHING, a form of energy. 
Temperature be may be content at first to look on as a 
mere condition which determines which of two bodies put in 
contact shall part with heat to the other. That such a state- 
ment is consistent with observed facts is shown by this, that 
if A is at the same temperature as B and also at the same 
temperature as C no transfer of heat takes place between 



40 HEAT. [CHAP. 

B and C, whatever be these bodies : i.e. bodies which have 
the same temperature as another body have themselves the same 
temperature. 

To refer to one of the analogies formerly employed, Heat 
may be compared to the quantity of air in a receiver, tem- 
perature to the pressure of that air. When two receivers, 
each containing air, are connected by a pipe, air is forced to 
pass from the receiver in which the pressure is the greater to 
that in which it is less. And this is altogether independent 
of the quantities of air in the two receivers : that which 
parts with some of its air may be very small in comparison 
with the other, and contain far less air ; it is the difference of 
pressure alone which determines the direction of the transfer. 

53. But, just as one receiver may be more capacious than 
another, so as to contain more air at a given pressure so 
one body may have more capacity for heat than another, and 
therefore contain more heat even when their temperatures 
are the same. 

And this difference of capacity may be due in part either 
to mere quantity of matter, (as when we compare an ocean 
with a pond,) or to a specific property of the substance 
itself for, as we shall see later, mass for mass, water has 
thirtyfold the capacity for heat that mercury has. This 
specific property is known by the name of Specific Heat, a 
term derived from the old erroneous notions as to the nature 
of heat, but now fairly rooted in the language so as to be 
almos-t permanent, though as much calculated to mislead the 
student as is the celebrated "Centrifugal Fone" 

54. Thus the quantity of heat employed in the entire 
system of fires, furnaces, &c., of Great Britain during a 
whole year producing vivid incandescence of millions of 
tons of coal and of liquid iron may be a mere trifle 
compared v/ith the heat required to raise the average 



iv.J PRELIMINARY SKETCH OF THE SUBJECT ; 41 

temperature of the Atlantic Ocean by a single degree of 
the thermometer. 

55. Referring again to the general principle enunciated 
in 9, it is clear that we may measure heat in terms of 
a unit of heaf, which may be defined as the heat required 
to melt a pound of ice at the freezing-point, to raise to the 
boiling-point a pound of ice-cold water, or in general to 
produce any definite physical change in a given mass of a 
given substance. Or, knowing as we do, that heat is a 
form of energy, we may dynamically measure a quantity of 
it by the number of foot pounds of work to which it is 
equivalent ( 37). 

But the mode usually adopted in Britain is to define a 
unit of heat as the amount of heat required to raise a pound 
of water from the temperature called 50 F. to that called 

51* F. 

56. Of course we may equally well adopt what are called 
metrical units, and the scale of the Centigrade Thermometer 
instead of that of Fahrenheit, but the change from one of 
these systems to another is one not of principle but of 
convenience, and, at the worst, involves a mere arithmetical 
operation of multiplication or division by a definite number. 
Such questions rarely rise to scientific importance, though 
they may raise (often justly) discussions as to comparative 
convenience. 

[Thus, there can be no question about the fact that the 
metre is inconveniently long, and the kilogramme incon- 
veniently massive, for the ordinary affairs of life. 

The average length of the arms of shop-girls, and the 
average quantity of tea or sugar wanted at a time by a small 
purchaser, have no conceivable necessary relation to the 
ten-millionth part of the quadrant of the earth's meridian 
passing through Paris or the maximum density of water. 



42 HEAT. [CHAP. 

But the standard yard and pound were, no doubt, originally 
devised to suit these very requirements as regards the 
average dimensions of the shop-girl or the paying powers 
of the ordinary customer. Yet this invaluable superiority 
of our units over those of the metrical system is, with an 
almost over-refinement of barbarism, thrown away at once 
when we come to multiples or submultiples. The very 
lowest attempt at consistency should have rendered it 
impossible for any one who employs the decimal notation 
to use any but a decimal system of multiplication and sub- 
division of units. All the monstrosities of the old Logic 
with its Barbara celarent, &c., or of the Latin Grammar, 
with its As in presejiti, &c., seem almost natural and proper 
when compared with a statement like this : 

12 inches i foot, 3 feet = i yard, 220 yards = i 
furlong, 8 furlongs = i mile. 

And even this is nothing to the awful complex of poles 
or roods, grains Troy and Avoirdupois, drachms and fluid 
ounces.] 

57. The measurement of Temperature, upon which we 
have seen ( 52) that the measure of Heat ultimately 
depends, presents absolutely unlimited choice. 

Any of the effects of heat (which we have already briefly 
discussed) may be employed, and we may use any material 
substance whatever for the purpose. 

It will be seen, however, when we have passed these 
preliminaries, that the Second Law of Ther mo-dynamics 
enables us to lay down an ABSOLUTE definition of tem- 
perature : absolute in the sense that it is entirely inde- 
pendent of the physical properties of any particular kind 
of matter, and depends solely upon the laws of trans 
formation of energy. 

58. It will at once be obvious to the reader that, since 



IV.] PRELIMINARY SKETCH OF THE SUBJECT. 43 

there is such an absolute scale of temperature, it must be 
adopted in all really scientific reasoning from the results 
of experiments ; though it may happen to be very difficult 
to apply it directly, during the practical work of experiment 
and observation. 

59. Few experimental measurements, certainly very few 
in which great precision is sought, are made in terms of 
the scientific or other units to which they are ultimately 
reduced. Thus, in constructing a screw micrometer for 
delicate astronomical or other measurements, it is not 
necessary, even if it were practicable, to give the screw 
exactly 10 or 100 threads to the inch. What is necessary is 
a good, i.e. a uniform, screw : and the exact length of 
its step is of no consequence. When the observer has, 
once for all, measured with it a number of objects or angles 
of known magnitude, he knows the value of one whole turn, 
and thence the value of any reading whatever, by a simple 
arithmetical process. 

Very fine thermometers, for research, often have their 
tubes not only calibrated, but graduated before the bulbs are 
blown on them. The freezing and boiling-points are then 
determined with the utmost care in terms of the arbitrary 
scale of the instrument, which thus (by an arithmetical 
operation) becomes perfectly definite throughout. 

Far from being a drawback to the use of such an instru- 
ment, the arbitrary scale is often of positive advantage to 
the experimenter, for it prevents his being (as the most 
honest and careful experimenter is very liable to be) uncon- 
sciously influenced by a knowledge of the reading that is 
to be expected. And the mere arithmetical difficulty of 
passing from one scale to another is so trifling in com- 
parison with the difficulties of experiment that it would 
be wholly disregarded, even were there not the distinct 



44 HEAT. [CHAR 

gain thus secured, of absolute freedom from the bias of 
preconceived notions. 

60. What we want at present, in our preliminary sketch 
of the subject, is not the rigorous scientific measure of 
temperature, but something which shall be at once easy to 
comprehend and easy to reproduce, and which shall after- 
wards require only very slight modifications to reduce it 
to the rigorous scale. 

Approximately pure water is (practically) to be obtained 
with ease everywhere. Now our fundamental principle ( 9) 
assures us that the changes called melting and boiling will 
always take place, with the same substance, each under its 
own precise set of conditions. 

It will be found later that the only conditions which 
require to be insisted on here are those of temperature and 
pressure. Hence we merely assert for the present, that 

For a definite pressure there is a definite temperature at 
which pure ice melts, and another definite temperature at which 
pure water boils. 

6 1. The idea of using two such fixed temperatures for 
determining a scale is due to Newton. 

For our present purpose we may suppose we have one 
vessel in which ice is melting, and another in which water 
is boiling, the barometer standing at 30 inches.* 

If we call the temperature of the cold water zero, or o, 
that of the boiling water 100, we adopt what is called the 
Centigrade scale. 

Now it is obvious that we may obtain every possible 
intermediate temperature by mixing portions of water 
from the two vessels in different proportions. And we may 

* The student may here take for granted that it would only confuse 
him were we now to give the scientifically rigorous definition of the 
temperature called the "boiling-point." 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 45 

define the intermediate degrees by saying that the per- 
centage (by bulk or preferably) by weight of the hot water 
in each such mixture is its temperature in degrees. Thus 
10 C. would be the temperature of a mixture of 10 Ibs. of 
water at 100 C. with 90 Ibs. of water at o C., and so on. 
This is obviously but one of an infinite number of ways in 
which the intermediate degrees might have been defined- 
but it has the advantage of directness and simplicity, while it 
is near enough to the absolute scale for our present purposes. 
And it has the farther advantage that it is the only one 
upon which the temperature of a mixture of equal weights 
of water at any two temperatures has a value exactly half- 
way between these. 

Until we can point out clearly the necessity for, and the 
possibility of, making improvements on this scale, we shall 
employ it for our work ; for it accords so closely with the 
absolute scale that careful experiment is required even to 
show that it does not exactly coincide with it. 

62. The effect of heat which is most commonly em- 
ployed for measurements of heat and temperature is the 
change of volume of liquids ( 42). We may take for 
granted that every reader has some little familiarity with an 
ordinary Thermometer. The liquid employed is usually 
mercury or alcohol. The precautions necessary in making 
and using Thermometers will be discussed later. The 
other effect of Heat most commonly employed for these 
purposes is the Thermo-electric one already described. 

( 49). 

In two respects it is vastly superior to any of the others 
far, first, it is far more delicate; and, second, its indications 
can quite easily be made visible to the largest audience. 

This method is based on the fact that, at least for a small 
difference of temperatures of the junctions, the strength 



46 HEAT. [CHAP. 

of the electric current is directly proportional to that 
difference. 

Hence a galvanometer, an instrument which measures the 
strength of the electric current, measures at the same time 
the difference of temperatures of the junctions. 

63. Here a digression is necessary. The fundamental 
fact of Thermo-electricity ( 49) belongs directly to our 
subject ; but the fundamental fact of Electro-magnetism, 
on which the action of the galvanometer depends, is 
entirely foreign to it. 

The fact, discovered by Oersted in 1820, is simply that 
the position of a freely suspended magnet is, in general, altered 
by the passage of an elect rk current in its neighbourhood. 

The figure represents all the essential features of the only 
form of this experiment which we require for illustration. 




A long copper wire, covered with silk or gutta-percha, to 
prevent contact of the wire with itself or with other metal, 
is wound into a circular coil ot a considerable number of 
turns, and its free ends are connected one with a zinc, the 
other with a copper plate. The coil is placed vertically in the 
magnetic meridian, and a magnetized steel bar is suspended 
horizontally inside it by a fine silk thread. While the zinc 
and copper plates are dry, and not in metallic contact, the 



iv.l PRELIMINARY SKETCH OF THE SUBJECT. 47 

magnetized bar of course takes the line of magnetic north, 
and therefore settles in the plane of the coil. But the 
moment the zinc and copper plates are dipped into a vessel 
containing water slightly acidulated, so as to cause an 
electric current to pass round the coil, the magnet is 
deflected, and tends, so far as it can, to set itself at right 
angles to the plane of the coil As the figure is drawn 
above, the bar tends to turn one or other end towards the 
spectator. 

The actual position which it will finally take up depends 
upon the relative amounts of the forces exerted upon it 
by the earth's magnetism and by the electric current. If the 
current be very weak the magnet is very little deflected ; if 
the current be very strong the magnet is placed almost 
at right angles to the plane of the coil. Thus, from the 
amount of the deflection, the strength of the current may 
be calculated. The direction in which the current passes 
round the coil determines to which side of the coil the north 
pole of the magnet will be deflected. This can easily be 
shown by interchanging the copper and zinc plates, when we 
find the direction of the deflection reversed. The rule 
which is found to determine the direction of the deflection 
may be stated thus. Suppose that, in the figure, we are 
looking at the coil from the west side, then the left-hand 
end of the magnet is what is usually called its north pole. 
If, then, the current be so applied that positive electricity 
passes round the coils of wire in the direction in which the 
hands of a watch move, the north end of the magnet will 
move towards the east side of the coil, i.e. will move away 
from the spectator. 

64. The galvanometer is usually constructed on a small 
scale, because the action of the current on the magnetic poles 
within the coils is greater as their radius is smaller (provided 



48 HEAT. [CHAP. 

there be the same number of coils and the same strength 
of current). Also a small magnet has greater mobility ; and 
in general harder temper, so that it preserves its magnetiza- 
tion longer than a large one. Hence a double advantage 
is gained by diminishing, as far as possible, the dimensions 
of the apparatus. Another gain is thus insured : the number 
of coils, and with them the electro-magnetic action, may be 
greatly increased while the length of the wire, which by its 
resistance weakens the current, may actually be reduced. 
But all the more essential features of galvanometers are 
exhibited in the rude apparatus depicted in 63. 

65. Suppose, now, a small mirror to be fixed vertically 
to one side of the magnetized bar, and a ray of sunlight to 
be made to fall on it, and after reflection received on a 
white screen. Every motion of the magnet, would be at 
once indicated by the corresponding motion of the illumi- 
nated spot on the screen. And the farther off the screen is 
placed, the greater will be the motion of the spot for the 
same deflection of the mirror i.e. of the magnet. Thus 
deflections, however small, may be magnified so as to 
become visible even to a very large audience. This makes 
the thermo-electric method invaluable for lecture-illustration. 
And here a fourth important advantage is gained, for the 
smaller the deflections of the magnet, the more nearly are 
they proportional to the strength of the current. 

When accurate measurement is our object, the little mirror 
is made concave, and a wire is adjusted vertically between 
the mirror and the source of light at such a distance that a 
sharp image of it is formed on the screen. This, and not 
the ill-defined patch of light which it crosses, serves as an 
index, by which to read the deflections by the help of the 
scale of equal parts drawn on the screen. 

66. The various successful devices for rendering galvano- 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 49 

meters more and more sensitive, belong properly to the 
subject of electricity ; but it may be well to indicate here a 
few of the more important. 

The mirror, with a number of very small pieces of hard 
steel (magnetized to saturation), attached to it, need not 
weigh more than a fraction of a grain. It is supported by 
a single fibre of unspun silk. The coils of wire should be 
as numerous, and of as small diameter as possible, to secure 
the greatest effect ( 64) ; but the whole resistance of the 
wire should be. as small as possible since in the greater 
number of thermo-electric arrangements which we require 
for radiant heat, &c., the main resistance is in the galvano- 
meter coils. Thus the best copper should be employed, and 
the diameter of the wire should be so small as to allow 
of a great number of coils of small diameter, while not 
so small as to destroy this gain by the greater consequent 
resistance, and weakening of the current. 

The astatic principle may 
also be introduced with 
great advantage for very 
delicate instruments. Here 
two light needles, or sets of 
needles, are placed parallel 
to each other with their 
north poles oppositely di- 
rected, and their middle 
points connected by a thin 

aluminium wire, which is supported in a vertical position 
by a fibre of unspun silk. 

Each magnet has its own coil, and the ends of the wires 
are so connected, that (as in the figure) a current, passed 
through them, runs round them in opposite direction?. 
Thus, as the direction of the current, *s well as the direction 

E 




50 HEAT. [CHAP. 

of the north pole, are each reversed in passing from one 
coil to the other, the electro-magnetic effects tend in the 
same direction, while those of the earth's magnetism tend 
in opposite directions. 

If we could make the magnetic moments of the needles 
exactly equal, and place them with their magnetic axes 
exactly parallel, terrestrial magnetism would not affect the 
apparatus at all, and the electro-magnetic effect would be 
counteracted and balanced solely by the torsion of the silk 
fibre. Practically this cannot be done; but the effective 
directive force of the earth's magnetism can easily be 
reduced to yj^th of its whole amount, and even less ; thus 
securing at least a hundredfold greater delicacy by this 
arrangement. 

As an illustration of the delicacy thus attainable, it may 
be mentioned that it is easy to construct a galvanometer 
which will show the y^oo-th or even the To-Jo-o tn f a degree- 
Fahrenheit of difference of temperatures beween the two 
junctions of a copper-iron circuit ( 49) at ordinary tem- 
peratures. 

67. When a galvanometer is to be used sometimes for 
powerful, sometimes for weak currents, various devices may 
be employed ; such as making the coil in two or more 
nearly equal parts, one or more of which may be employed 
at a time. If two be employed, with the current running 
opposite ways in them, a much more powerful current will 
obviously be required to produce a given deflection than if 
it run the same way in both. 

Another device is to use external, adjustable, magnetized 
bars so as to increase or diminish at pleasure the intensity 
of the field of force due to the earth's magnetism. This 
method is satisfactory for powerful currents, but not for 
weak ones, because the earth's horizontal magnetic force 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 51 

is continually and rapidly changing, and when the greater 
part of it is neutralized, even a slight change may produce 
great percentage variation in the uncompensated part, on 
which alone the delicacy of the galvanometer depends. 

The great majority of those valuable improvements just de- 
scribed, were recently introduced by Sir W.Thomson (originally 
for practical telegraphic applications). Another, and by no 
means the least, consists in inclosing the magnets and mirror 
in a small, narrow, glass cell, with parallel faces. The vis- 
cosity of the air, and the large surface and small moment of 
inertia of the mirror, cause this instrument to pounce almost 
instantly upon its exact reading, without the long-continued 
and tantalizing oscillations which were unavoidable in the 
older instruments. This dead beat principle, as it is called, 
enables the experimenter to make accurate readings faster 
almost than he can record them ; and this not only effects 
an immense saving, both of time and trouble, but enables 
the observer to study a phenomenon in various ways before it 
has sensibly changed its conditions, a most desirable thing in 
itself, but only now made possible. 

68. Sources of Heat As heat is only one of the 
many forms of energy, and as any one form of energy can 
in general be transformed, in whole or in part, into another,, 
the sources of heat are as numerous as the forms of energy 
at our disposal. Our available sources of potential energy 
are mainly fuel. Under the head of fuel are included not 
merely coal, wood, and so on, but also all that may properly 
be called fuel the zinc used in a galvanic battery, for 
instance, and various other things of that kind, including 
the food of animals. 

We have also : 

Ordinary water-power. 
Tidal water-power. 

E 2 



52 HEAT. [CHAR 

In the kinetic form we have : 

Winds. 

Currents of water, especially ocean currents. 

Hot springs and volcanoes. 

There are other very small sources known to us, some 
exceedingly small, such as diamonds for instance j but 
those named include our principal resources. And every 
one of these, by proper processes, can be made a Source 
of Heat. 

69. Now comes the question, What are the sources of 
these supplies themselves ? They also can be classified under 
four heads. 

The first is primitive chemical affinity, which we may sup- 
pose to have existed between particles of matter from the 
earliest times, and still to exist between them, because these 
portions of matter have not combined with one another 
nor with other matter. If, for instance, while the materials 
of which the earth is composed were widely separated from 
one another, there were particles of meteoric iron and 
native sulphur which, when the materials fell together to 
form the earth, did not combine, but still remain separate 
from one another, the mutual chemical potential energy of 
the iron and sulphur xemains to us as a portion of energy 
primordially connected with the universe. But of that, so far 
as we know, at least near the surface of the earth, there is 
very little. There may be towards the interior enormous 
masses of as yet uncombined iron and uncombined sulphur, 
or various other materials ; but towards the surface, where 
they could be of any direct use to us, the quantities of these 
are excessively small. 

The second source is solar radiation, by far the most 
abundant source we have. Then we have two very 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 53 

instructive forms, tJie energy of the earth's rotation about its 
axis, and the internal heat of the earth. 

70. Now compare our available stock with the sources 
from which we derive it, and we easily see how the two- 
are connected. 

Our supplies of fuel are almost entirely due to the sun. 
In times long gone by, the sun's rays by their energy, as 
absorbed in the green leaves of plants, decomposed 
carbonic acid and stored up the carbon. That carbon, and 
various other things stored up ages ago along with it, we 
have still as an immense reserve fund of coal. 

For food we are mainly indebted to the sun again, because 
the food of animals must ultimately be vegetable food, even 
of the animals which live upon animal food. For ordinary 
water-power we are also indebted to the sun, because it is 
mainly the energy of the radiation from the sun which eva- 
porates water from the plains or seas, so as to be con- 
densed again at such an height that it has potential energy 
in virtue of its elevation. Ordinary currents of water are 
a mere transformation of this potential energy, because water 
on a height converts part of its potential energy into kinetic 
energy of visible motion as it flows down. 

But when we come to tidal water-power, we must look to 
another source. If we employ tidal power for the purpose 
of driving an engine, we take it in the rise of the water as the 
tide wave passes us. We secure a portion of water at a 
certain elevation, wait till the tide has gone back, and then 
take advantage of the descent of that portion of water. Now, 
if we were to go on doing this for any considerable period 
of time, and over large tracts of sea-coast, we should find that 
the effect would be to gradually slacken the rate of rotation 
of the earth. 

Winds and ocean currents are almost entirely due to 



54 HEAT. [CHAP. 

solar radiant heat. And hot springs and volcanoes, which 
have never been employed for any direct production of 
work, depend mainly, at least, upon the internal heat of 
the earth ; partly, perhaps, on potential energy of chemical 
affinity. 

71. It is obvious, then, that the sun is the great source 
of almost all our available energy ; and we can carry the 
investigation still one step farther back, so as to inquire 
into the source of rhe sun's enormous store of energy. It 
will be seen later, that no known chemical source can ac- 
count for more than a very small fraction of it ; and that the 
only adequate known source is the potential energy which 
its parts must have possessed, in virtue of their mutual 
gravitation, while they were widely dispersed throughout 
a space of dimensions at least equal to those of the known 
solar system. 

72. Transference of Heat. Under this head it is 
usual to classify as the three ways in which heat can be 
transferred from one place to another, the processes of 

Conduction. 

Radiation. 

Convection. 

As will be seen presently, this list involves some confu- 
sion, for as commonly understood it includes either too 
much or too little. 

73. By Conduction of Heat we usually mean that com- 
paratively slow transfer of heat from part to part of a 
body, or from one body to another with which it is in 
direct contact. It is mainly by conduction that any part of 
the earth's internal heat reaches the surface, and that the 
sun's heat penetrates into the crust. The study of the laws 
of conduction, and of the relative conducting powers of 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 55 

various substances, as well as of the conducting power of one 
and the same substance at different temperatures, is of very- 
high practical value, as well as of intense scientific interest. 

By Radiation of Heat we usually mean the process by 
which heat is transferred from one body to another with the 
velocity of light even through space altogether devoid of 
what is called tangible matter. It is entirely by radiation 
that we obtain heat from the sun, as also (in very small 
quantity) from other stars. The subject of radiation and 
the connected subject of absorption have, of late, received 
unexpectedly an enormous extension. The practical pro- 
cess of Spectrum Analysis, to which they have recently led, 
is one of the most important means at our command for 
the study alike of the most distant stellar and nebular 
systems, and of the almost inconceivably minute grained 
structure of matter. 

By Convection of Heat is usually meant the carriage of 
heat-energy from one locality to another along with the 
particular portion of matter with which it is associated. The 
Gulf-stream is a vast convection-current, whereby the solar 
heat of the tropics is carried into the North Atlantic. Every 
form of ventilation, whether of coal-mines or of private 
houses, at least if heat be directly the effective agent, is a 
mere case of convection. 

74. To avoid what the student might feel to be perplexing 
novelties, it may be well to adhere in this elementary book 
to the classification just given. But it may not be amiss to 
make here a remark or two upon it. 

In the first place we have absolutely no proof that radia- 
tion from the sun is in any of the forms of energy which we 
call heat, while it is passing through interplanetary space. 
That it is a form of energy, and that it depends upon some 
species of vibration of a medium, we have absolute proof. 



56 HEAT. [CHAP, 

But it seems probable that we are no more entitled to call 
it heat than to call an electric current heat ; for, though 
an electric current is a possible transformation of heat- 
energy and can again be frittered down into heat, it is 
not usually looked upon as being itself heat. Just so the 
energy of vibration al radiations is a transformation of the 
heat of a hot body, and can again be frittered down into 
heat but in the interval of its passage through space devoid 
of tangible matter, or even while passing (unabsorbed) 
through tangible matter, it is not necessarily Heat. 

Again, in dealing with Thermo-electricity, we shall find 
that a current of electricity has a convective power for 
heat: i.e. that when passed through a metal bar whose 
temperature is not the same at all points it may carry heat 
irom the hot to the colder parts, or vice versa; and that 
when passed through a circuit, made up partly of one 
metal, partly of another, and at the same temperature 
throughout, it carries heat from one of the two junctions 
to deposit it in the other. 

These facts render the usual hard and fast classification 
of 7 2 somewhat inadequate ; but without farther advert- 
ing to this we merely mention that the apparent exceptions 
can easily be treated under the head of Transformations of 
Heat, as will be seen when we go over the various divisions 
of the subject more fully. This is clearly a case of some- 
what illogical arrangement, but, as already explained ( ii), 
we have decided to adopt it, lest we should perplex rather 
than assist the beginner. The student who has got suc- 
cessfully over the earlier stages will easily surmount this 
difficulty also. 

75. Transformations of Heat. These are, of course, 
as numerous as the other forms of Energy, since we have 
already seen that any form of energy can be transformed (at 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 57 

least partially) into any other. On this point nothing farther 
need be said at present. But there is another point of the 
case which is of the utmost importance. We may state it 
thus : 

Given a certain quantity of energy in one form and 
under given conditions, how much of it can, by means 
of a given kind of apparatus, be converted into some 
other definitely assigned form, the rest being either untrans- 
formed, or transformed, in whole or in part, into some third 
form ? 

76. Now part at least of the enormous amount of waste 
which takes place in an ordinary steam-engine is familiar to- 
all. Not to speak of the unburned fuel which is allowed 
to escape as smoke, the very ascent of the column of smoke 
is due to wasted heat, and there is constant and large leak 
age from the furnaces, boilers, and cylinders. The very fact 
that the stoke-hole is so intolerably hot is due to the same 
waste. But there is unavoidable as well as unnecessary 
waste. It has been satisfactorily demonstrated that in the 
very best engine, even if it were theoretically perfect, and 
working at ordinary ranges of temperature, only somewhere 
about one-fourth very rarely so much, but at the best about 
one-fourth of the heat which is actually employed is con- 
verted into work ; that is to say, three-fourths of the coals r 
or three-fourths of the heat employed, are absolutely wasted 
under the most favourable circumstances. What is it that 
determines this ? Why is it that the whole of a quantity o 
work or of potential energy can be converted into heat r 
while the heat cannot be converted again, except in part, 
into the higher form of work or potential energy? 

77. The answer is included entirely in the word "higher," 
just used. When energy is to be converted from a higher 
form into a lower, the process can in general be carried out 



$8 HEAT. [CHAP. 

in its entirety ; but when it comes to be a question of reversal 
going up hill, as it were then it is only a fraction, in 
general (even under the most favourable circumstances) only 
a small fraction, of the lower kind of energy which can be 
raised up again into the higher form. All the rest usually 
sinks down still lower in the process. When it is low 
.already, and part of it has to be elevated and transformed 
into a higher order, a large part of it must inevitably be still 
farther degraded; in general the larger part of it. This is 
one of the most important advances ever made in science, 
and has most stupendous bearing on the future of the whole 
-visible universe. 

78. In all transformations of energy we find experiment- 
ally that there is a tendency for the useful energy to run 
down in the scale, so that, the quantity being unaltered, 
the quality becomes deteriorated, or the availability becomes 
less ; and thus we are entitled to enunciate, as Sir William 
Thomson did very early after the new ideas were brought 
into full development, the principle of Dissipation of Energy 
in Nature. 

79. The principle of dissipation, or degradation, as it may 
perhaps preferably be called, is simply this, that as every 
operation going on in nature involves a transformation of 
energy, and every transformation involves a certain amount 
of degradation (degraded energy meaning energy less capable 
of being transformed than before), energy is as a whole con- 
tinually becoming less and less transformable. 

80. Thus, as long as there are changes going on in 
nature, the energy of the universe is falling lower and lower 
in the scale, and we can at once see what its ultimate form 
must be, so far at all events as our knowledge yet extends. 
Its ultimate form must be that of heat so diffused as to give 
.all bodies the same temperature. Whether this be a high 



iv.J PRELIMINARY SKETCH OF THE SUBJECT. 59 

temperature or a low temperature does not matter, because 
whenever heat is so diffused as to produce uniformity of 
temperature, it is in a condition from which it cannot raise 
itself again. In order to procure any work from heat, it is 
absolutely necessary to have a hotter body and a colder one. 
When therefore all the energy in the universe is transformed 
into heat, and so distributed as to raise all bodies to the 
same temperature, it is impossible at all events by any 
process yet known, or even conceivable to raise any part 
of that energy into a more available form. Why it is so, 
and what (slight) exceptions there are to this general state- 
ment, will form an important subject for discussion farther 
on. For the present purpose we may hold it as a perfectly 
general law of nature. 

8 1. The grand question, therefore, with regard to all 
transformations of energy is To what extent can they be 
carried out ? This, of course, is a question to which nothing 
but experiment, or reasoning ultimately based on experi- 
ment, can possibly give an answer. And, so far at least as 
the transformation of heat into work, and various allied 
transformations, are concerned, it has been answered in the 
most brilliant and satisfactory manner. 

82. Perhaps no purely physical idea has done so much to 
simplify science, or led to so many singular and novel pre- 
dictions (subsequently verified by experiment) as has Carnot's 
idea of a Cycle, or his farther idea of a Reversible Cycle, of 
operations. 

It has supplied not merely the legitimate mode of finding 
the relation between the heat taken in and the work done 
by an engine, but also the test of perfection for a heat- 
engine, an absolute definition of temperature, the effect of 
pressure on the melting-points of solids, and innumerable 
important groups of associated properties of matter and 



60 HEAT. [CHAP. 

energy under various conditions. To a great extent these 
are included in the statement of the 

Second Law of Thermodynamics. If an engine be 
such that, when it is worked backwards, the physi:al and 
mechanical agencies in every part of its motions are all reversed 
(see 89), it produces as much mechanical effect as can be pio- 
duced by any thermodynamic engine, with the same tempera- 
tures of source and refrigerator, from a given quantity of heat. 

83. It is to be particularly observed here that Reversibility 
(see 88, 89) is the sole test of perfection of an engine. 
Also in the working of a reversible heat-engine nothing is 
said about the nature of the working substance ; the tempe- 
ratures of source and refrigerator, and the quantity of heat 
supplied, are the sole determining factors of the work which 
can be done. The importance of this proposition, as regards 
actual and proposed engines, cannot be over-estimated. 

84. Garnet's work is upon the Motive Power of Heat, and 
was published in 1824. It forms no inconsiderable portion 
of Sir W. Thomson's many scientific claims that he recog- 
nised at the right moment the full merits of this all but 
forgotten volume, and recalled the attention of scientific 
men to it in 1848; pointing out, among other things, that 
it enabled us to give, for the first time, an absolute definition 
of Temperature. Although Carnot (seemingly against his 
own convictions) reasons on the assumption that heat is 
matter, and therefore indestructible ; and although, in con- 
sequence, some of his investigations are net quite exact, his 
work is of inestimable value, because it has furnished us, 
not only with a correct basis on which to reason but, with 
a physical method of extraordinary novelty and power, 
which enables us at once to apply mathematical reasoning 
to all questions of this kind. These, then, are his two great 
claims first, the setting thermo-dynamics upon a proper 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 61 

physical and experimental basis; and, second, in the fur- 
nishing us with a means of reasoning upon it which was 
absolutely new in physical mathematics, and which has 
been, not merely in Carnot's hands, but in the hands of 
a great many of his successors, as fruitful in new discoveries 
as the idea of the conservation of energy itself. 

85. In order to reason upon the working of a heat- 
engine (suppose it for simplicity a steam-engine), we must 
imagine a set of operations, such that at the end of the 
series the steam or water is brought back to the exact 
state in which it was at starting. That is what Carnot 
calls a cycle of operations, and of it Carnot says, that only 
for such a cycle are we entitled to reason upon the rela- 
tion between the work done and the heat spent. If a 
quantity of steam were allowed merely to expand, losing 
heat in the process and doing work, we should have no 
right whatever to say that the quantity of heat which has 
disappeared is the equivalent of the work which is given 
out, because at the end of the operation the steam is in 
a different state as to pressure and temperature from 
that in which it was at the beginning. It was saturated 
steam at a certain temperature, let us say, to start with, 
and at the end of the operation it may still, if proper 
adjustments be made, be saturated steam, but it is neces- 
sarily at a different temperature, and therefore we have no 
right to assume that it possesses intrinsically the same 
amount of energy as it did in its former state. We have no 
right whatever to reason upon the quantity of heat which 
appears to have gone, as compared with the work which has 
been done, when the working substance begins in one state 
and ends in another. But if we can by any process bring 
the working substance back to its initial state, then we are 
entitled to assert that it must contain neither more nor less 



62 HEAT. [CHAP 

than it did at first, and therefore of course we are also 
entitled to reason upon all the external things that have 
taken place during the operation, and to determine the 
condition of equivalence among them. 

86. The hypothetical operation which Carnot introduced 
for the purpose of reasoning on this subject is, like 
most great ideas, excessively simple when found. Let us 
imagine two bodies, each maintained constantly at a definite 
temperature. These will be called the hot body and the 
cold body respectively. In addition to these suppose a 
body which, as regards other bodies, is neither cold nor 
hot, being incapable of absorbing heat or of giving it out 
a non-conductor of heat. Suppose the walls of the working 
cylinder and the piston to be non-conductors of heat, but 
the bottom of the cylinder a perfect conductor. Suppose 
water and steam (in proper proportions ( 391)) to be in 
the cylinder, both at the same temperature, that of the cold 
body. Place the cylinder on the non-conductor and expend 
work in pressing down the piston, the contents will become 
warmer, and some steam will be liquefied. Continue this 
process till the temperature rises to that of the hot body 
then transfer the cylinder to it. Now allow the piston to 
rise, the contents remain at the temperature of the hot body, 
fresh steam is generated, and work is done. Arrest this 
process at any stage and transfer the cylinder to the non- 
conducting body. If we now allow the contents farther to 
expand, more work is done, but the temperature gradually 
sinks. Continue this process till the temperature falls to 
that of the cold body, to which, therefore, without loss or 
gain of heat, it may now be transferred. Next apply work 
to compress it at the constant temperature of the cold body 
till (by condensation) the contents have become exactly as 
they were at starting. The cylinder must now be transferred 



iv.] PRELIMINARY SKETCH OF THE SUBJECT 63 

to the non-conducting stand, and everything is as it was at 
first save that some heat was taken from the hot body in 
the second operation, and heat was given to the cold body 
during the fourth. Also it is evident that more work has 
been done during the second and third operations than was 
spent in the first and fourth, for the temperature, and there- 
fore the pressure, of the contents was greater during the 
expansion than during the compression. Of course this 
operation may be repeated any number of times. 

87. Notice particularly what the peculiarity of the opera- 
tion is. The steam or expanding substance, whatever it is 
for air or anything else would do equally well must 
always be hi contact with bodies at its own temperature, or 
else with non-conducting bodies. If it were in contact with 
a body of a lower temperature, there would be a waste of 
heat. Heat would pass by conduction from the cylinder to 
external bodies, and would of course be wasted as regards 
work. The same would happen if the cylinder were to be 
removed from the non-conducting body and placed upon 
the cold body, before its contents had been allowed to 
expand sufficiently to cool down to the temperature of the 
cold body: some heat would then be conducted away at 
once, and be lost to the engine. So, throughout the whole 
of Carnot's operations, it is essential that there should be no 
direct transfer of heat at all except while heat is being taken 
in from the hot body or given out to the cold body : the 
temperature of the contents of the cylinder being in each of 
these cases (all but exactly) the same as that of the body 
with which they are for the moment in contact. 

88. We now come to another point, also perfectly novel, 
and of great importance. Suppose the operations in Carnot's 
cycle to be performed in exactly the reverse order. Begin, 
for instance, with the hot body, but do not allow the piston 



'64 HEAT. [CHAP. 

to rise there. Take the cylinder from the hot body when 
the water and the steam below the piston have acquired 
the higher temperature. Lift it to the non-conducting body, 
and then allow the piston to rise. Let it continue to rise, 
doing work all the time, till the temperature sinks to that of 
the cold body ; place it on the cold body ; allow the steam 
to expand still farther, it will be in that case giving out 
work but taking in heat. When it has risen to its former 
highest point, place it back again on the non-conducting 
body, force the piston back to the same extent as that to 
which it rose when (in Garnet's direct set of operations) it 
was first placed on that body. Everything has taken place 
in precisely the reverse order to that in which it took place 
before. Finish then upon the hot body, and press home. 
There is sent back in that final operation precisely the 
quantity of heat taken from the cold body ; but during the 
two first operations the piston was in contact with steam at 
a lower temperature, and therefore at a lower pressure 
than during the two last. And, therefore, in the reversed 
method of working the engine, work has on the whole to 
be spent in taking heat from the cold body and depositing 
it in the hot body. 

89. These are the grand ideas which Carnot introduced. 
Their two distinctive features are, first ', the idea of a com- 
plete cycle of operations, at the end of which the working 
substance, whatever it is, is brought back to precisely its 
primary condition ; this cycle can be repeated over and 
over again indefinitely. Secondly, the notion of making the 
cycle a reversible one, so that all the operations can be 
performed in the reverse order; i.e. instead of taking 
in heat at any stage, heat is given out ; instead of 
work being done by the engine at any stage, work is 
spent upon it. With these reversals of each part of the 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 65 

operation, the whole cycle can be gone over the reverse 
way. 

90. Now Carnot, considering heat as a material sub- 
stance, says that obviously it has done work in the direct 
series of operations by being let down from the higher 
temperature to the lower, just as water might do work by 
being let down through a turbine or other water-engine, 
doing work in proportion to the quantity that comes down 
and the height through which it is allowed to descend. In 
the reversed operations work is spent in pumping up the 
heat again from the cold body to the hot one. Here it is 
of course assumed that the quantity of heat which reaches 
the condenser is the same as that which leaves the boiler ; 
i.e. virtually that heat is matter, and therefore unchangeable 
in quantity. We now know that this notion of the nature 
of heat is erroneous, but Carnot's reasoning does not there r 
fore lose its value, because the change of a word or two 
only is required to render it perfectly applicable with our 
modern knowledge of the subject. 

91. One point which appears to show conclusively that 
Carnot's analogy (not his result) was incorrect, is that nothing 
is easier than to let heat down at once from the hot body to 
the cold, without the performance of any work. If the hot 
body be put into direct communication with the cold body, 
the same quantity of heat might be allowed to go down 
from one to the other, and yet give no work at all. Before 
we give the correct statement of the question here involved 
it may be well to complete this brief account of Carnot's 
contributions to the problem. For even with his false 
assumption he obtained much more than has yet been 
stated. We have yet to show why he introduced the notion 
of reversibility. And this is virtually what he said : If an 
engine be reversible (as this cycle of operations has been 



UN ,VERSrTY OF CALIFORNIA 



66 HEAT. [CHAP. 

shown to be), it does as much work as can be got from a 
given quantity of heat under the same given circumstances. 
So that, no matter what be the substance which is expand- 
ing and contracting, if a certain quantity of heat be let down 
from a source at a certain temperature through a reversible 
engine to a sink at another definite temperature, then the 
quantity of useful work which can be got from that heat 
will be absolutely the same. 

Reversibility is thus the sole necessary condition of equi- 
valence between two heat-engines. 

This is an enormous step in physical science. The 
reasoning is independent altogether of the properties of any 
particular substance. Whether steam, or air, or ether, &c., 
be the working substance, there is the same crucial test of 
the perfection of an engine. That test is, if a heat-engine be 
reversible it is perfect, perfect not in the popular sense, but 
in a scientific sense ; that is to say, // is as good as it is 
possible physically to make it. 

92. Carnot demonstrates this property by a simple reductio 
ad absurdum exactly analogous to that of 20. He says 
that a reversible heat-engine is a perfect one ; for, if not, 
suppose there could be one more perfect, and let these two 
engines be employed in conjunction. Let the more perfect 
engine be employed in taking a quantity of heat, conveying 
it to the condenser from the boiler, and giving from it a 
larger quantity of work than the reversible engine could 
do. Part only of the work thus gained will be required to 
enable the reversible engine to pump the same amount 
of heat back again. Every time the heat goes down, it is 
through the more perfect engine ; every time it comes up, 
it is through the less perfect engine, and therefore the 
double system necessarily gains more work than it spends. 
Thus we have an engine which will not merely go for ever, 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 67 

but which will for ever, without any drain on its sources, 
steadily do work on external bodies. 

Such a consequence, as we have seen ( 20), is incon- 
sistent with all experimental results, and therefore the 
supposition which led to it, viz. that there can be a more 
perfect engine than a reversible one, is necessarily false. 

Such is Garnet's proof that (on the assumption that heat 
is matter) a reversible heat-engine is a perfect engine. It 
requires very little indeed, as a moment's reflection shows, 
to make this reasoning consistent with modern knowledge 
of heat. 

93. We have now to consider the cycle in the light of the 
conservation of energy ; so that, if work be produced from 
heat at all, some of that heat must have disappeared in its 
production. Therefore, under no circumstances if the 
engine be doing external work at all can the quantity of 
heat which reaches the condenser ever be equal to that 
which leaves the boiler. If no heat has been wasted by 
conduction or in other unprofitable ways, the difference 
between the quantity which leaves the source and the 
quantity which reaches the condenser during a complete 
cycle must be precisely the equivalent of the external work 
which has been done. Taking that into account, suppose 
we could make an engine more perfect than a reversible 
one. Work the two together, as before. Make the re- 
versible engine continually restore to the source as much 
as the other takes from it. Then, as it is less perfect, it 
will require less work to be employed on it, when reversed 
(to restore to the source or boiler that quantity of heat), than 
is furnished by the other engine; and therefore, on the whole, 
while there is a pumping up of heat and letting it down 
which exactly compensate one another, at least so far as the 
source is concerned, there is a gain of work. But a heat- 

F 2 



68 HEAT. [CHAP. 

engine, however complex, can only work by expenditure of 
heat so that, as the arrangement in question takes no heat 
from the boiler, it must obtain it from the Condenser. The 
result amounts to this, that by taking, as the condenser for 
a compound engine such as that supposed, any limited 
portion of the available universe, the engine would con- 
tinue to give out work till it had removed all heat from that 
portion. It may safely be assumed as axiomatic that this 
cannot be done ; all experimental facts are against it. Thus 
we have it, ex absurdo, that there can be no engine more 
perfect than a reversible one. This question will be more 
carefully considered in a later chapter. 

94. Since all reversible engines are perfect, they are all 
of equal efficiency : that is, they all give precisely the same 
amount of work from the same quantity of heat, under the 
same conditions. It follows that these conditions alone 
determine how much work can be produced, by a perfect 
engine, from a given quantity of heat. Now, the tempera- 
tures of the boiler and condenser are the only particulars in 
which this set of perfect engines agree. Suppose each to be 
worked till it has employed a given quantity of heat, then 
each would do the same amount of work. That is to say, 
All perfect heat-engines under the same conditions convert 
into work the same fraction of the heat used, and the value 
of this fraction depends only upon the temperatures em- 
ployed. Hence follows immediately Thomson's absolute 
method of measuring temperature. (Refer again to 82.) 
For it is obvious that the relation between the temperatures 
of the boiler and the condenser can now be defined in terms 
of the fraction just spoken of. 

95. The terms of the definition are to a certain extent at 
our option. That which was finally fixed on by Thomson 
was chosen so as to make as near an agreement as possible 



iv.] PRELIMINARY SKETCH OF THE SUBJECT. 9 

between the new scale and that of the ordinary air- 
thermometer, and therefore to make the introduction of 
this method, the only scientific one, produce as small a 
dislocation of previous conventions as possible. The full 
reasons for this particular choice will be afterwards 
explained. Meanwhile we simply state the definition in 
Thomson's words : 

The temperatures of two bodies are proportional to tJie 
quantities of heat respectively taken in and given out in 
localities at one temperature and at the other, respectively, by 
a material system subjected to a complete cycle of perfectly 
reversible ther -mo-dynamic operations, and not allowed to part 
with or take in heat at any other temperature: or, the absolute 
values of two temperatures are to one another in the proportion 
of the heat taken in to the heat rejected in a perfect thermo- 
dytiamic engine, working with a source and refrigerator at the 
higher and lower of the temperatures respectively." * 

96. Suppose we keep a body at the temperature of 
boiling water, under the condition that the barometer shall 
be at a height of 30 inches. (See foot-note to 61.) Suppose 
we keep another body at the temperature of melting ice, 
with the barometer at the same height. Suppose we could 
measure what amount of heat is taken in, and what amount 
given out, by a perfect engine working between these two 
temperatures ; these amounts of heat, as will be shown later, 
would be found nearly in the proportion of 374 to 274. 
These particular numbers have been chosen for the terms of 
the ratio because their difference is 100. In the ordinary 
centigrade scale we call the freezing temperature zero, and 
we call the temperature of boiling water, under the 30 
inches of pressure of the atmosphere, 100. Thus, as we 
see by the experiment, in the case above mentioned, that 
* Trans. R. S. ., May 1854. 



70 HEAT. [CHAP. iv. 

for 374 taken in, 274 are given out, the temperature of 
boiling water will on this scale be represented by 374, and 
that of freezing water or melting ice by 274, the range 
between these being the ordinary 100 of the centigrade 
thermometer. Hence the curious and very important 
result, that a body cooled down 274 centigrade degrees 
below zero is absolutely deprived of heat. This limit, 
which is commonly called the absolute zero of temperature, 
is perhaps more correctly called the zero of absolute tem- 
perature. And it is obvious that the temperature of a 
body could not be reduced so as to be lower than this. 

97. The student is recommended fully to master the 
brief statement we have now given : for he will then be 
much better able to understand the relations of the various 
parts of our great subject than he would have been without 
such knowledge ; and he will also understand why, in the 
more detailed study of them which follows, it is constantly 
necessary to anticipate, by borrowing materials from a later 
chapter. 

98. Resume #/ 41 97. Temperature is a mere con- 
dition of a body as regards Heat. The utmost fraction of 
a given quantity of Heat which can be converted into useful 
work depends solely on the temperature of the body with 
which it is associated, and upon the lowest temperature 
available. Carnot's inestimable services to science consist in 
his suggestion of Cycles ; and especially of Reversible Cycles, 
for the criterion of a perfect engine. Second Law of Thermo- 
dynamics, and Absolute measurement of Temperature. 



CHAPTER V. 

DILATATION OF SOLIDS. 

99. Taking the process explained in 61 as a mode of 
defining temperatures between oand 100 of the Centigrade 
scale, we proceed to determine the dilatation of various bodies 
when they are raised to various temperatures within that range. 
One most important result of this inquiry will be found to be 
the (approximate) measurement of temperatures by means of 
these dilatations, so that we shall be enabled to get rid of 
the theoretically very simple but practically very cumbrous 
process which we first adopted for defining temperature. 

The order of simplicity of result would lead us to com- 
mence with gases such as air and its constituents, but as we 
cannot experiment on them except when they are enclosed 
in solid vessels, simplicity of process requires that we 
commence with SOLIDS. 

The importance of this inquiry from the merely popular 
point of view consists mainly in the common applications of 
the results to such matters as the regulation of watches and 
clocks, and the correction of measuring-rods for changes of 
temperature. Its pure scientific interest is of a far higher 
order. 

ico. A Solid which has the same properties at all points 
is called homogeneous, otherwise it is heterogeneous. But even 
a homogeneous solid has not necessarily the same properties 



72 HEAT. [CHAP. 

in all directions. The grain of wood, of fibrous iron, the 
crystalline forms and cleavage planes of minerals, the planes 
of deposition and of cleavage in slates and sandstone 
rocks, and many analogous phenomena, familiarize us with 
the idea of solids which are homogeneous but not isotropic. 
An isotropic body, then, is one from which if a small sphere 
were cut, it would be impossible to tell by any operation on 
it how it originally lay in the solid it has, in fact, precisely 
the same properties in all directions. Probably there does 
not exist any solid absolutely isotropic as regards all its 
physical properties, but such substances as non-crystalline 
metals (lead, gold, silver, copper,) and well-annealed glass 
may be taken as approximately isotropic as well as homo- 
geneous. We commence our work with a rod or bar of such 
a material, and we define thus : 

The co-efficient of linear dilatation at any temperature is the 
ratio which the increment of length of the bar when its 
temperature is raised one degree bears to the original length. 

101. As the bar is supposed to be homogeneous, and to 
be raised to the same temperature throughout, each inch of 
its length increases by the same amount ; so that the whole 
increase of length is directly proportional to the original 
length. To obtain a result independent of the dimensions of 
the specimen chosen, we must therefore take the ratio of the 
increase to the whole original length : which is the state- 
ment made in the definition. The linear dilatation might 
be put in the more formal terms of a specific property of the 
substance, by operating on a bar of unit length, and then 
defining its increase of length for one degree of temperature 
as the dilatation required. But, practically, no one operates on 
an exact unit, and therefore we must make our definition such 
as to be independent of the length of the specimen operated 
on ; and of course independent of the standard of length. 



v.] DILATATION OF SOLIDS. 73 

102. As this work is not designed to teach the details of 
experimental methods, which are in all cases much more 
readily and also more appropriately taught in the laboratory 
than in the lecture-room or by books, it is sufficient to say 
that the co-efficient of dilatation of a bar is usually obtained 
by actual measurement of its length, or of the distance 
between two marks upon it, first when it is surrounded 
by melting ice, and again when it has been for some time 
immersed in a bath of water or other suitable liquid at 
a definite temperature. The temperature of the bath is then 
altered, and another measurement made. The measurements 
may be made directly with a microscope and screw, or com- 
parisons may be made between the lengths of the heated 
bar and of another kept permanently in melting ice in a 
second bath placed alongside of the first. Methods in 
which one end of the bar is clamped, while the other in 
expanding moves a bent lever or a mirror, are not so trust- 
worthy as those in which the distance between two known 
points of the bar is directly measured. Their principal use, 
and it is often a very important one, is in lecture illustra- 
tions. The principle on which they depend is very simple. 
One end of the bar or rod is clamped in a pillar A ; the 
other end, passing freely through a hole in the pillar B, 
presses against the back of a plane mirror supported on a 
horizontal axis at C, perpendicular to the rod. (The reader 
may easily supply the diagram for himself.) Any advance 
of the free end of the rod must of course be due to expan- 
sion, and its amount can be at once determined by the 
change of direction of a ray of light reflected from the 
mirror, for that change is double the angle through which 
the mirror rotates. And the tangent of the angle through 
which the mirror rotates is the ratio of the increase of length 
of the bar to the height of the axis C above it. Thus by 



74 



HEAT. 



[CHAP. 



lowering C the sensibility of the arrangement admits of 
almost unlimited increase. The annexed cut represents the 
old form of this arrangement, where a bent lever is used 
instead of the mirror. 




A method of extreme delicacy has recently been intro- 
duced by Fizeau. It depends upon what are called 
Newton's rings, an optical phenomenon caused by the 
retardation of light in passing twice through the air-space 
between a lens and a flat plate of glass. The colour pro- 
duced varies with the distance of the plate from the lens : 
so that if one be fixed, and the other carried on the end of 
an expanding rod, the changes of colour enable us to 
measure with great accuracy the expansion of the rod. 

103. The general result of such measurements is found 
to be the very simplest possible : 

T/ie ratio of the increase of length to the length at zero is 
proportional to the rise of temperature. 



v.] DILATATION OF SOLIDS. 75 

Let /, represent the length of an assigned portion of the 
bar at temperature /C, then the above experimental result 
is at once stated in the form 



where k is the coefficient of proportionality. Hence, if we 
put /=i, we see by the definition ( 100) that k is the 
coefficient of linear dilatation at zero. 

The value of k is usually so small that to determine it 
with any accuracy we must use a considerable change of 
temperature. Taking then the extreme temperatures of our 
temporary scale i.e. measuring the length of the same 
bar, first in melting ice and then in boiling water we 
have 

/.oo /o 

- -, - = 100 k. 

^o 

The coefficient of linear dilatation is therefore the one- 
hundredth part of the ratio which the increase of length of 
the bar between the temperatures oC and iooC bears to 
the length at oC. 

104. It is rarely found that two specimens even of the 
same material possess any one specific property to exactly 
the same amount. This remark applies even to natural 
crystals, much more therefore to artificial substances such 
as glass, c., so that it is impossible to give any perfectly 
definite general statements of Dilatation-Coefficients. The 
following numbers must therefore be taken merely as an 
illustration of the sort of results arrived at ; quite sufficient 
as a basis for some general remarks upon the subject, but 
altogether unfit for any application in delicate calculations 
or constructions. When an optician (worthy of the name) 
has discs of flint and crown glass supplied him for the 



76 HEAT. [CHAP. 

purpose of making an achromatic lens, he does not look up 
tables of results of former experimenters to find the re- 
fractive index, he has to measure with great care for himself 
this particular constant for the material of each of the discs, 
and for more than one definite wave-length of light. The 
constructor of really high-class electrical apparatus, be it the 
smallest galvanometer or the longest submarine cable, most 
carefully chooses his copper by a determination of its specific 
resistance. And similar statements are, or ought to be, true 
in every practical branch of physical science. 

APPROXIMATE COEFFICIENTS OF LINEAR DILATATION OF 
ISOTROPIC SUBSTANCES. 

Glass -0000085 =r ~^ 

Platinum . . . . . . -0000085 i~^3 

Steel -000012 1 = -j~ 

Copper -0000191 =: -^~ 

Zinc -000029 = ^,000 

105. The foregoing table, meagre as it purposely has been 
made, shows two marked results. First, the Coefficient of 
Linear Dilatation is in general very small; secondly, its values 
are very considerably different in different solids. 

From the first result it follows that, unless very delicate 
operations or very high temperatures are involved, tempe- 
rature-change of dimensions in solids may be neglected 
except when we are dealing with great lengths such as the 
iron girders of a bridge, or the rails on a railway. 

From the second result, coupled with the experimental 
law of 103, it appears that we may by means of combina- 
tions of two different materials produce compensation, if we 



v.j DILATATION OF SOLIDS. 77 

arrange so that the distance between two assigned points of 
a complex system shall be increased by the expansion of 
some of its parts, and diminished to 
an equal amount by the expansion of 
others. 

Thus, to take the simplest con- 
ceivable arrangement, let us have two 
similar bars, of zinc and of copper, 
placed parallel to one another, and 
fastened at one end to a transverse 
piece. The condition that the dis- 
tance between two uprights, fixed to 
the free ends of these bars, shall 
remain constant at all ordinary tem- 
peratures is simply that the actual ex- 
pansion shall be the same for each 
bar. But, as the Coefficients for zinc 
and copper are to one another as 29 
to 19, this will be at once secured by 
making the lengths of the bars as 19 
to 29. Thus, by taking advantage of 
the different expansibilities of different 
substances, we can construct a measur- 
ing-rod whose length is independent 
of temperature. This is, practically, 
the same arrangement as that of 
Graham's mercurial Compensation Pen- 
dulum. A slightly more complex 
arrangement of the same kind gives 
the Gridiron Pendulum of Harrison. 
But the reader who has mastered the simple case given 
above can have little trouble with the more complex 
cases. 



HEAT. 



[CHAP. 



1 06. The balance-wheel and its spring perform for a 
watch or chronometer precisely the same function as the 
pendulum, assisted by gravity, performs for a clock. But 




here the application of the compensation process is con- 
siderably more complicated, because gravity is constant in 
the clock, while the coefficient of elasticity of the chron- 
ometer balance-spring depends on its temperature. How- 
ever, we may for the present consider only the question of 
compensating for the increased moment of inertia of the 
balance-wheel due to expansion. [The effect of heat on 
the balance-spring usually renders 0zw-correction necessary, 
but the principle is the same.] 







When a rod, or preferably a flat strip, of wood or metal is 
bent it is known that the layers on the convex side are 



v.J DILATATION OF SOLIDS. 79 

extended, those on the concave side compressed, relatively 
to the intermediate ones. Hence it is obvious that such a 
strip would bend itself if the layers on one side of its mesial 
plane were to be extended or contracted relatively to those 
on the other. Thus a strip formed by soldering or brazing 
together strips of two metals of different expansibilities will 
become curved by change of temperature, the more expan- 
sible metal being on the convex side when the whole is 
heated, and on the concave when it is cooled. 




Now suppose the rim of the balance-wheel to be con- 
structed of two concentric layers, the outer the more expan- 
sible. Also let it be cut into separate arcs sufficiently 
removed from one another to prevent interference. Then 
it is easy to see that the expansion of the radial bar tends to 
increase the moment of inertia of the whole, while the in- 
creased curvature of the arc tends to diminish it. A ribbon 
of two differently expansible metals similar to that just 
described was employed by Breguet for the purpose of con- 
structing an exceedingly delicate thermometer. The ribbon 
was usually coiled into a helix which was fixed vertically 



8o HEAT. [CHAP. 

in a support at its upper end, while the lower end carried 
a light index travelling on a horizontal dial. The ribbon 
is so thin that it almost instantly acquires the same tem- 
perature as the surrounding medium, and its capacity for 
heat is so small that it does so without sensibly affecting 
the temperature of the medium. 

107. One fact of exceeding great use in practice appears 
in the short table of 104. The coefficient is nearly the 
same for platinum as for glass, and farther experiment shows 
that this is at least approximately true for a very wide range 
of temperature. In consequence of this a platinum wire 
may be "fused in " to a glass vessel : the glass, melted 
under the blow-pipe, adhering to the hot wire, and the two 
materials contracting equally as they are allowed gradually 
to cool, so that the junction is perfectly air-tight. If the 
attempt be made with a wire of gold or of any other metal 
whose melting-point is high, it almost certainly fails however 
thin be the wire : either the glass cracks on cooling ; or the 
wire contracts more than the glass, so that the junction is 
not air-tight. 

1 08. The facts that platinum is practically infusible in 
any ordinary furnace, and that its coefficient of linear dilata- 
tion changes very little through great ranges of temperature, 
have led to its employment for the rough measurement 
of high temperatures. Instruments of this rude kind are 
usually called Pyrometers, The principle on which they 
work is to keep a record of the length which a bar of 
platinum had while plunged in the furnace, so that it may 
be compared accurately with the length of the bar when 
cold. 

The bar is put into a hole bored in a piece of graphite or 
plumbago, whose dilatation is exceedingly small. The bar 
rests against the bottom of the hole and is kept there by a 



v.j DILATATION OF SOLIDS. 81 

tightly-slipping plug of graphite or baked clay. As the 
platinum expands by heat it pushes this plug forward, but 
when it cools it does not draw it back. The principle is 
exactly that with which every one is familiar in the ordinary 
maximum thermometer. 

109. When a body is aeolotropic, i.e., non-isotropic, its 
properties are not the same in all directions. But, though 
the circumstances are now less simple than in the case of 
isotropic bodies, they are usually by no means very complex. 
For in general, even in the most complex cases, these 
properties can be referred to three definite directions at 
right angles to one another. When these directions are 
ascertained, and measurements of a particular property 
(as of coefficients of linear dilatation, with which we are at 
present engaged) are made parallel to them, the value 
of the property for any other assigned direction may be 
calculated by a simple process. 

[It would require a little more of mathematical reasoning 
than can well be introduced into an elementary work to 
show that, whereas the properties of an isotropic body are 
analogous to those of a sphere, those of an aeolotropic body 
have the same relation to an ellipsoid. This and what 
follows we take for granted. An ellipsoid may be regarded 
as formed from a sphere, by selecting three diameters at 
right angles to one another, and extending or compressing 
the whole sphere to different amounts in directions parallel 
to these three lines. An ellipsoid is known if the direc- 
tions and lengths of its three principal axes (which are the 
three lines at right angles to one another) are known. When 
the lengths of any two of its axes are equal it becomes an 
ellipsoid of revolution (prolateii the third axis is the longer, 
oblate if it is the shorter, of the three) ; and when all three 
are equal it becomes a sphere. Thus when the amounts of 

G 



82 HEAT. [CHAP. 

compression or extension parallel to two of the three 
diameters are equal, the sphere becomes an ellipsoid of 
revolution ; when all three amounts are equal, it remains a 
sphere. Every plane section of an ellipsoid is an ellipse. 
Every ellipse which has equal conjugate diameters in direc- 
tions perpendicular to one another is a circle. An ellipsoid 
which has two plane circular sections such that the line 
joining their centres is perpendicular to their planes is an 
ellipsoid of revolution, of which the line just mentioned is 
the axis. All lines through the centre of an ellipsoid of 
revolution, and equally inclined to the axis, are equal. If 
it have been formed from a sphere by one elongation and 
two equal contractions, or by one contraction and two equal 
elongations, there will be a set of diameters (all equally in- 
clined to the axis) which are each unaltered in length.] 

no. By far the most homogeneous substances we possess 
are natural crystals. None are absolutely isotropic, for all 
have planes of more or less perfect cleavage. But one 
great class of crystals, those of the first or regular system 
(with a few curious exceptions, probably only apparent], are 
isotropic as regards light, and are also isotropic as regards 
heat. Substances so very different as diamond, galena, 
rock-salt, &c. , belong to this system, and their coefficients of 
linear dilatation are the same in all directions. 

Crystals of the prismatic and rhombohedral systems, and 
in general bodies which have one optic axis as well as one 
axis of crystalline symmetry, have their greater or lesser 
coefficient of linear dilatation in the direction of that axis, 
and are equally expansible in all directions at right angles 
to that. 

All other crystalline bodies have three principal dilatation- 
coefficients, different from one another, and in directions at 
right angles to one another. 



v.] DILATATION OF SOLIDS. 83 

in. Optical instruments, such as the reflecting goniometer, 
enable us to measure with very great accuracy the angles 
between the plane faces of crystals ; and measurements of 
angle are in general much more exact and useful for such 
inquiries as those in which we are now engaged than direct 
measures of length. It was, in fact, almost entirely by 
measurements of angles that Mitscherlich and others arrived 
at the results briefly stated in no. 

As a simple illustration, suppose a square prism to be cut 
from an aeolotropic body, so that the diagonals of one of its 
ends are parallel to any two of the chief axes, and have in 
general different coefficients of linear dila- 
tation. An application of heat will neces- 
sarily change the square into a rhombus 
as in the cut, and the ratio of the lengths 
of the diagonals is very accurately found 
by measurement of one of the angles. 

The defect of this method is that it gives the difference of 
the linear dilatations in the directions of the two diagonals, 
but not the actual amount of either. For if the diagonals 
were of unit length at o C, their lengths will be i + kit 
and i -f kj at tC, and the tangent of half the measured 
angle of the rhombus is 

i + kj , . . . 

T^ , which is practically i + (k 2 #,)/, 

i -j- *,/ 

because k^ and k 2 are both exceedingly small. 

Suppose two such prisms cemented together as below, 




the edges brought into contact being those whose angle 
becomes more obtuse by heating. The upper surface will 

G 2 



84 HEAT. [CHAP. 

form a continuous plane mirror when the whole is cool, 
but will consist of two inclined mirrors after heating. An 
almost infinitesimal difference of expansibility in different 
-directions can be detected by observing with a small tele- 
scope the image of a distant object formed by reflection at 
the surfaces of the prisms. 

What are called twin-crystals are made up of two parts 
put together in a manner somewhat similar to that just 
described : and the phenomena are beautifully seen when a 
polished slab of "arrow-headed " selenite is slightly warmed. 

112. By the optical method the differences between the 
principal dilatation coefficients are very accurately deter- 
mined. All that remains to be done is to measure one of 
them directly, by the methods already referred to ( 102). 
Here Fizeau's method, already described, may be used with 
advantage. 

When this is done it is found that in some bodies, notably 
in Iceland spar, the difference of two dilatation coefficients is 
greater than either. In other words this substance, which 
has one axis of symmetry, expands parallel to its axis when 
heated, but simultaneously contracts equally in all directions 
perpendicular to the axis. 

It follows from the explanation in 109 that there are 
an infinite number of directions, equally inclined to the 
axis, in which Iceland spar neither expands nor contracts. As 
a similar property is found in certain masses of marble, 
Brewster long ago suggested the use of a cylinder of this 
substance, cut in a proper direction, as an invariable 
pendulum. 

But the apparent anomaly of contraction by heating is 
not confined to crystalline bodies. We shall presently have 
to discuss it as shown by water, but a very instructive 
instance is furnished by india-rubber. 



v.] DILATATION OF SOLIDS. 85. 

If the spiral wire be extracted from an ordinary vulcanised 
india-rubber gas-pipe, and the pipe be then suspended 
vertically, with a weight attached to its lower end, it con- 
tracts (in some specimens by five or even ten per cent, of 
its length) and raises the weight, when steam is blown 
through it from a little boiler. Thus we have a contrac- 
tion, easily visible to a large audience, without any of the 
artificial processes alluded to in 102 for the exhibition 
of expansion. 

113. The experimental result of 103 involves, as a 
consequence, the farther statement: 

The coefficient of cubical dilatation is in all cases tht sum of 
the three chief coefficients of linear dilatation. 

This is easily seen by supposing a body to be divided 
into a number of brick -shaped portions, with their edges 
parallel to the three chief axes. The lengths of the several 
edges of each brick are increased by heating from o to t C 
in the ratios 

i : i + M i : i + &/, i : i + /, 

and thus we nave for the final, in terms of the original, 
volume 



or, what comes to the same thing, since the quantities k are 
always exceedingly small 



This is precisely the same formula as (a) of 103, and thus 
the above statement is verified. 

When the body is isotropic the values of k are equal,. 
and the coefficient of cubical dilatation is three times that 
of linear dilatation. Thus, for such bodies, only one 



86 HEAT. [CHAP. 

determination (whichever may be the easier, or the more 
exact) need be made. 

In aeolotropic bodies there may be expansion, con- 
traction, or no change of volume, as a result of rise of 
temperature. 

Thus, just as there may be change of volume without 
change of form, we may have change of form without 
change of volume. 

114. Most of the modes of directly measuring change of 
volume depend on the use of expansible liquids. Hence, 
in describing their principle, we shall suppose first a non- 
expansible vessel and a non-expansible liquid, and then 
show how to correct for the expansion of the vessel ; 
leaving to a later section to show how the expansion of 
any actual liquid may be measured and allowed for. 

Let a vessel containing the solid whose cubical dilatation 
is to be measured be filled up to the end of its (narrow) 
neck with an inexpansible liquid, and let the whole be 
weighed. Then let it be raised to a known temperature, 
the expansion (if any) of the solid will drive out some of 
the liquid. Let it cool, and weigh it again. The difference 
of weight is the weight of the liquid driven out by the 
expansion of the solid. 

The correction for the expansion of the vessel, the 
material of which is usually glass or some other isotropic 
substance, is, by 113, 



where k is its coefficient of linear dilatation. 

115. The greater part of what precedes is only approxi- 
mately true, even for the small range of temperature (o 
to iooC.) to which our statements have been, in the 
main, confined. 



v.j DILATATION OF SOLIDS. 87 

Measurements carried out at much higher temperatures 
have shown that the coefficients both of linear and of cubical 
dilatation are not constant, but increase slowly with rise of 
temperature. We are not aware of any experiments made 
with the view of deciding whether, as is probable, these co- 
efficients become gradually less as the temperature is lowered 
below zero. 

It has been tacitly assumed, in what precedes, that heat- 
ing has not permanently altered the molecular state of the 
solid operated on. So long as this is the case the dimen- 
sions of the body are always practically the same at the 
same temperature, whether the body is being heated or is 
cooling. But it is familiar to every one that sudden cooling 
has often a marked effect on the properties of a body. The 
process of " annealing," as it is called, is devised essentially 
for the purpose of preventing such physical changes. The 
whole of this subject, with the exception of a few rules 
discovered tentatively, is still very obscure. Think, for 
instance, of the very different conditions of stress, &c., 
under which the different parts of a Rupert's drop are 
successively solidified. A recent great practical improve- 
ment in the manufacture of steel depends essentially upon 
causing it to solidify under considerable pressure. 

116. Resume of 99 115. The linear dilatation of a 
solid rod is approximately proportional to the rise of tem- 
perature. Hence, in isotropic solids, the cubical dilatation 
(being threefold the linear dilatation) is also proportional 
to the rise of temperature. Special exceptions, in the case 
of aeolotropic bodies. Applications to jneasuring-rods, 
compensation balances and pendulums, pyrometers, 
Breguet's metallic thermometers. Irregularities due to 
rapid cooling, &c. 



CHAPTER VI. 

DILATATION OF LIQUIDS AND GASES. 

117. In fluids this question is much more simple than 
in solids, for we have cubical dilatation only. Taking 
liquids first, we find that in general their cubical dilatation 
is, like that of solids, proportional directly to the rise of 
temperature, but greater. 

Exceptionally great labour and experimental skill have 
been devoted to this point, in the case especially of two 
common liquids : mercury, because of its importance in 
thermometers ; and water, because its expansion at ordinary 
ranges of temperature is anomalous, and because this 
anomaly is closely concerned with the laws of its circula- 
tion. We propose to devote several sections to each of 
these liquids, as they may be considered typical. 

Meanwhile we may merely remark that what we seek is 
the real dilatation ; not what is called apparent dilatation, 
which is affected by the dilatation of the vessel. 

1 1 8. Let, as in 114, a glass vessel with a very narrow 
neck be weighed when full of mercury at o C ; then let it 
be heated to t C, and weighed again after cooling. Let 
these weights be W , W t , respectively, and let w be the 
weight of the empty vessel. 



CH. vr.l DILATATION OF LIQUIDS AND GASES. 89 

Then, as the vessel at o holds an amount of mercury 
\vhose weight is 

W.-w, 

it would hold at temperature t an amount whose weight is 
(i + 3 &) ( W w\ 

provided mercury were not expansible. It actually holds 
an amount whose weight is only W t w. The ratio of 
these two numbers is therefore that of the densities of 
mercury at o and at /; and, if K be the mean coefficient of 
cubical dilatation of mercury between o and /, we have 

(i + AT/) ( W t w) = ( i + 3&) ( Ww\ 

from which K may at once be calculated. 

To verify this equation, suppose the cubical dilatation 
of the liquid to be the same as that of the vessel, i.e. 

K=& 
then we have at once 



showing that the vessel remains just filled at all tem- 
peratures. 

119. This process is a simple and effective one, although 
it involves the determination of the expansion of the vessel 
as well as of the liquid ; and it is so because the expansion 
of mercury is much greater than that of glass. But a very 
ingenious method of determining the expansion of a liquid, 
independently of that of the solid in which it is enclosed, 
was devised by Dulong and Petit, and applied with great 
skill by Regnault. The principle of the method is merely 
that of the simple hydrostatic equilibrium of two liquids, 
one in each of the branches of a U tube : viz., that the 
heights of the separate columns above the common surface 



HEAT. 



[CHAP. 



(where the liquids meet) are inversely as their specific 
gravities. 

In practice, to avoid direct contact between the hot and 
cold liquids, an air-space is made to intervene, so that the 
essential parts of the arrangement are as in the sketch. 
The tube is now a double U, one half of it being surrounded 
by a vessel containing ice-cold water, the other by a vessel 
whose liquid contents may be raised to any desired tem- 
perature. 




The columns a b, a' b', of mercury are each exposed to 
the pressure of the atmosphere at their upper ends, and at 
their lower ends to the pressure of the air in b b'. Hence, 
if the differences of level of the surfaces at a and b, and at 
.a and b', be measured with a cathetometer, the mercury in 
each tube has a density inversely as the corresponding 
measured number. 

120. Here are a few of Regnault's numbers for mer- 
cury : The second column contains numbers derived 
.directly from the experiment described ; the third is cal- 
culated from the second for a special purpose : 



Mean 
coefficient of 
dilatation 
from o. 


Coefficient 
referred to vol. 
at o. 


True 
Coefficient. 


0-OOOI803 . < 
0-000l8l5 . , 
O-OOOl828 . . 


0-OOOI79I . 
0-000l8l5 . 
0-0001841 . 

0*0001866 . 


. O-OOOI79 
. O'OOOlSo 
. O'OOOlSl 
. 0-OOOI82 


0*0001853 . , 
0*000l866 . , 
0-0001878 . 


0-0001891 . 
0-0001916 . 
0-0001941 . 
0-0001967 . 


. O*OOOl83 
. O*OOOl84 
. 0*000l85 

. 0*000186 



vj.] DILATATION OF LIQUIDS AND GASES. 91 
DILATATION OF MERCURY. 

CO 

Tempera- 
ture C. 

. 

50 
IOO 

150 . 

200 

250 . 

300 . 

350 

We have extended the list to temperatures far higher 
than those already temporarily defined, in fact up to the 
boiling point of mercury. As given in the table they are 
measured by the air-thermometer presently to be described. 

Meanwhile the numbers above give, as the reader may 
easily verify, the following very simple formula for the 
usually employed coefficient of cubical dilatation of mercury 
at any temperature t C : 

K t 0*0001791 + -00000005 1. 
Thus if / be 250 : 

A" 250 0*0001791 4-0-0000125=0*0001916, 

as given in the table. As this formula agrees with all the 
given numbers, it may safely be employed to calculate K 
for any intermediate temperature. 

It will be observed that the coefficient of dilatation 
increases steadily with rise of temperature. This is found 
to be the case with the great majority of liquids, through 
the whole range of temperature in which we can experiment 
upon them. But the rate of increase of the coefficient is 
generally greater as the temperature of the liquid rises 
towards its boiling point. 



92 HEAT. [CHAP 

The last column in the table gives the true coefficient 
of dilatation being (according to the analogy of the defini- 
tion in 100) the ratio which the increment of volume for 
one degree of rise of temperature bears to the original 
volume : not (as is tabulated in the third column in accord- 
ance with common usage) to the volume at zero C. With 
solids, in general, the dilatation is so small that this dis- 
tinction is of little importance but a comparison of the 
two last columns just given, shows that it cannot be 
neglected for a liquid like mercury. The formula for 
this true coefficient is very nearly 

K t = O'OOOiyQI H- O'OOOO0002/. 



In this case, and in that of water which follows, we 
abandon for a time the rule laid down in 104, and enter 
into details. The reason is that we are now dealing with 
a perfectly definite substance, whose properties (under the 
same conditions, 9) are the same at all times and in all 
places ; not a substance like glass or brass, &c., of 
which different specimens may differ seriously from one 
another. 

121. The behaviour of water between o and iooCis 
very different from that of mercury. From o upwards it 
contracts, more and more slowly, till it reaches its maximum 
density almost exactly at 4C. From 4 upwards it ex- 
pands, at first very slowly, then faster and faster to 100. 

The usual class-illustration of this phenomenon is known 
as Hope's experiment. It depends on the simple hydro- 
static law that, when a heterogeneous fluid is in stable 
equilibrium under the action of gravity, the density increases 
from above downwards. 

Hope merely applied what is called a freezing mixture (a 
mixture of common salt with snow or pounded ice is quite 



vi.] DILATATION OF LIQUIDS AND GASES. 93 

sufficient) to the middle of a cylindrical jar full of water. 
Thermometers were fixed by corks in holes in the side of 




the jar, so as to indicate the temperatures of the upper and 
lower stratum of water. 

Before the freezing mixture is applied, provided the tem- 
perature of the water is above 4 C, there is usually a slight 
excess of temperature indicated by the upper thermometer 
showing that the warmer water is less dense than the colder. 
The first effect of the freezing mixture is to reduce the tem- 
perature shown by the lower thermometer, without perceptibly 
affecting the higher. This goes on till the lower thermometer 
reaches 4C, for then it ceases to descend; very soon after- 
wards the temperature indicated by the higher thermometer 
begins to sink in its turn. But it does not stop at 4 C. It shows 
lower and lower temperatures till the water towards the top 
of the vessel begins to freeze, and is therefore at o. Thus 
water at 4 C is proved to be denser than at any other tem- 
perature from o to 100, because it remains persistently at 
the bottom of the vessel whatever be the temperature of the 
water above it. 

Such an experiment, however, is not adapted for the 
measurement of coefficients of dilatation. Some experi- 



94 HEAT. [CHAP, 

menters have for this purpose weighed a lump of glass in 
water at different temperatures, others have operated by the 
process indicated in 119. The following are from Kopp's 
paper (Poggendorff, 1 847 ) : the volume of water at 4 being 
taken as unit, the excess of any of the other numbers over 
unit is the whole expansion from 4 to the corresponding 
temperature : 

DILATATION OF WATER AT ORDINARY PRESSURE. 

Temp. C. Vol. of Water. Temp. C. Vol. of Water. 

I -00012 20 I 00169 

2 1-00003 30 I '00420 

3 i-ooooi 40 1-00766 

4 i -ooooo 50 1-01189 

5 i-ooooi 60 1*01672 

6 i -00003 70 1*02237 

8 1*00011 80 1-02871 

10 . ; . . . 1-00025 90 1*03553 

15 1-00082 ioo 1*04312 

The numbers in this table, from o to 20, are represented 
with a fair degree of approximation by the formula 

144,000* 

Hence the coefficient of dilatation of water (between 
these limits) is approximately 



72,000. 

Matthiessen, Pierre, and Hagen have given experimental 
results, the mean of which tends to show that the denomi- 
nator of this fraction should be more nearly 68,000. 

According to Despretz, the volume of water when cooled 
more than four degrees below its temperature of maximum 
density increases somewhat faster than if the water be heated 
to the same number of degrees above that point. Water 
can be cooled several degrees below o c C without freezing, 
provided it be not agitated. 



vi.] DILATATION OF LIQUIDS AND GASES. 95 

It was found by Canton, in 1764, that the compressibility 
of water is greater in winter than in summer. It follows 
from this that, as will be shown later, the maximum density 
point is lowered by increase of pressure. The experimental 
data do not yet enable us to obtain very definite information, 
but we may say (roughly) that a pressure of 50 atmospheres 
lowers the maximum density point by i C. The freezing 
point also is lowered, but not so much. 

122. The most expansible of all known liquids are those 
which require considerable pressure to keep them in the 
liquid state. The coefficients of dilatation of sulphurous 
acid, or carbonic acid, in hermetically sealed tubes, are 
considerably greater than that of air. 

The following are some of Thilorier's results for the 
density of liquid carbonic acid in presence of its vapour at 
different temperatures : 

Temperature. Density. 

2O C. . . . . O'QO 

o C . 0-83 

-i- 30 C . o - 6o. 

Thus the mean expansion for i between oC and 30 C 
is 0*013 nearly. 

As a rough general rule it may be stated that (at ordinary 
temperatures) the more volatile a liquid is the greater is its 
co-efficient of dilatation. Alcohol, however, (whose co- 
efficient of dilatation is 0-00105 at o c C.) is considerably 
more expansible than ether. 

123. When we come to consider the expansion of gases, 
we find a problem quite different from those we have already 
treated. For a given quantity of a gas cannot be said to 
have any particular volume, unless we specify the vessel in 
which it is confined, or the pressure to which it is subjected. 
The effects of moderate pressures upon the volumes of 



96 HEAT. [CHAP. 

solids and of the majority of liquids are so small that we 
have notfound it necessary to take them into account while 
discussing the expansion of such substances. In fact a 
solid or a liquid will expand very much as we have already 
described, even if it be confined in an envelop of consider- 
able strength simply bursting the envelop if it does not 
otherwise yield. But, practically, no superior limit has yet 
been assigned to the volume which a given quantity of air 
will occupy at ordinary temperatures \ and on the other 
hand a strong vessel containing a gas at atmospheric pres- 
sure would be melted or at least softened before the contained 
gas had been raised to a sufficient pressure to burst it. 

It follows, then, that we must measure either the dilata- 
tion (i.e. ratio of increase of volume to original volume) of 
a gas kept at a constant pressure ; or the ratio of increase of 
pressure to original pressure, in a gas kept at constant 
volume ; each for a given rise of temperature. 

The first rough experiments, due to Charles about 1787, 
and the subsequent more perfect measures of Gay Lussac, 
indicated that the quantities just defined are not only the 
same in any one gas, but have the same common numerical 
value for all gases, at least for the range from o to 100 C. 

Though these results are only approximately true, they are 
sufficiently exact for many very important applications, both 
theoretical and practical, and therefore we will devote a 
section or two to the study of their consequences before 
stating the more accurate results of modern experiments. 

124. In 1662 was published by the Hon. Robert Boyle 
the experimental fact now called 

BOYLE'S LAW. 

The volume of a given mass of gas, kept at a given temperature, 
is inversely as the pressure. 



vi.] DILATATION OF LIQUIDS AND GASES. 97 

Along with this we may take (in a somewhat extended 
form) the result stated in last section, which is called 

CHARLES' LAW. 

The volume of a given mass of gas, kept at a constant pressure, 
increases by a defi?iite fraction of its amount for a given rise of 
temperature. 

The two laws together may be stated in the simple 
form 

pv = C(i + a/), 

where C is a constant quantity for any one gas (depending 
in fact upon its quantity and quality), and a is the coefficient 
of its cubical dilatation. 

The first really good determination of the value of a was 
made by Rudberg. For our present purpose we may 
take 

a =: 0-003665. 

125. Three remarks must, at this stage, be made on the 
statement of last section. 

First. Note the remarkable fact that the coefficient of 
dilatation is (practically) the same for all gases. This points 
to a common simplicity of behaviour, and gives a hint that 
a gas-thermometer is probably to be preferred to a mercurial 
one. 

Second. Note that the two forms of statement ( 123, 
124), which we have used in giving Charles' result, are really 
deducible from one another by the help of Boyle's law. 
This is obvious from the equation written in last section ; 
because / and v are similarly involved in it. 

We have stated Charles' law as holding for all tempera- 
tures intermediate to o and 100 C. Practically this is 
true, though it seems to have been tacitly assumed. 

n 



98 HEAT. [CHAP. 

Third. If we assume the formula of 124 to hold for 
tf// temperatures, we arrive at the very important result that 
p or v must vanish at a particular temperature, which is the 
same for all gases : i.e. if any gas be cooled to the particular 
temperature 

* = - a = ~ 0^665 = - ^73 nearly, 

which makes the expression i -f / vanish, then either : it 
will cease to have any volume : or // will cease to press upon 
the walls of the containing vessel. Thus we obtain a first 
rough hint of the position of the absolute zero of temperature, 
i.e. the temperature of a body altogether deprived of heat. 

It may be well to recall the student's attention, in passing, 
to the hint (given in 96 above) that such an absolute zero 
does exist, and to state that the above rough approximation 
gives a fairly satisfactory notion of its position on the ordi- 
nary centigrade scale. The only lawful methods, however, of 
treating such questions are those based upon the general 
laws of Thermodynamics, which apply to all bodies : and 
not on the properties, simple in their statement though they 
be, of any particular class of substances. 

126. If we would shift our zero-point, which is ( 61) 
wholly at our option, to the absolute zero thus determined, 

we must put t - for ^and the expression for the laws of 
Boyle and Charles ( 124) becomes 
pv = Xf, 

where R is a definite constant for each gas. 

As already stated ( 123) this is only approximately true, 
and the divergences from it will be afterwards carefully con- 
sidered; but the degree of approximation is found to be 
closest in those gases which are least easily liquified ; 



VI.] DILATATION OF LIQUIDS AND GASES. 99 

and closer for any one gas the higher its temperature, is 
raised, and (with limitations) the more it is rarefied. 

The Kinetic theory of gases, presently to be described, 
shows us that the smaller the mutual action of the particles 
of a gas, when not in contact, in comparison with the 
action during collision, the more nearly should the above 
equation be satisfied. Hence we speak of the ideal perfect 
gas as a substance between whose particles there is no 
action except at the instants of collision and in which 
therefore (as it is proved in the Kinetic theory of gases) the 
product of the pressure and the volume is strictly propor- 
tional to the absolute temperature. 

The experiments of Joule and Thomson show that, 
under such circumstances, the absolute temperature, as 
indicated by the ideal perfect gas, is strictly identical with 
absolute temperature as defined ( 96) by the help of 
Carnot's reversible cycle. 

Hence the air-thermometer, when properly constructed, 
furnishes a measure of temperature much more in accord- 
ance with thermodynamic theory than does the mercurial 
thermometer. 

But it is necessary again to observe (as in 59) that it 
does not at all matter what instrument is used, provided it 
be a good one of its kind, and provided its indications can 
be translated into the corresponding absolute temperatures as 
denned in 95. 

127. Recurring to the statements of 123, we may now 
give an idea of the methods employed by Regnault in his 
classic investigations. The nature and use of the apparatus 
he employed are, in all the more essential features, suffi- 
ciently indicated by the rudimentary diagram on the next 
page, which represents a somewhat simplified form devised 
by Balfour Stewart. 

II 2 



HEAT. 



[CHAP. 




Two vertical glass tubes, of equal bore (to get rid as 
far as possible of the effects of capillary forces), are fastened 
vertically into the lid of a closed vessel 
full of mercury. The capacity of this 
vessel can be altered at will by screwing 
in, or unscrewing, a plug or piston. 
One of the tubes is open at the top, 
the other terminates in a glass balloon 
A, containing the gas to be experi- 
mented on. This balloon, and as 
much of the stem as is necessary, are 
surrounded alternately by melting ice, 
and by steam from water boiling at i 
atmosphere ( 134). 

When the change of volume at con- 
stant pressure is to be measured, the 
plug is screwed in or withdrawn till the 
mercury stands at the same level in 
each of the tubes at each of the tem- 
peratures selected. The pressure of 
the gas in A is then exactly that of the 
atmosphere for the time being, and the volume of the gas 
in A and its stem must be accurately measured at each 
temperature. A correction (small in comparison with the 
change of volume of the gas) must of course be made for 
the dilatation of the balloon by heat. 

When the change of pressure at constant volume is to be 
determined, the plug is adjusted at each temperature so as 
to bring the surface of the mercury to a definite point, a, on 
the tube carrying the balloon : and the difference of levels 
of the mercury in the two tubes is to be added to the 
column of mercury in the barometer for the estimation of 
the pressure of the gas. In addition to the correction for 




vi.] DILATATION OF LIQUIDS AND GASES. 101 

dilatation of the balloon by heat, there is also to be applied 
a correction (determined by experiment) for its dilatation by 
increase of pressure. 

Other methods were employed by P.egnault, such fo r 
instance as to fill with mercury, and weigh, <i glass vessc 1 
with a narrow neck. Thus the capacity of the vessel u. 
ascertained. Then, filling the vessel with dry air, to htit 
it to 100 C, to seal it hermetically while the contained 
air was thus rarefied ; open it when cold under mercury, and 
weigh the quantity of mercury which entered. But it 
is sufficient to say that the results of these experiments 
agreed satisfactorily with those obtained by the first- 
described apparatus. 

128. The following are a few of Regnault's results: 
The first column is the ratio of the volume of the gas at 
1 00 to that at o C. under atmospheric pressure ; the second 
the pressure at iooC. when the gas is taken at a pressure 
of i atmosphere at o C. and heated without change of 
volume : 

Hydrogen . . . 1-3661 .. . 1*3667 

Air .... 1-3670 . . 1-3665 

Nitrogen . . . 1-3670 ... . 1*3668 

Carbonic Oxide . . I -3669 . : . i -3667 

Carbonic Acid . . 1-3710 .* . 1*3688 

Sulphurous Acid . . I -3903 ... i '3845 

It is to be observed that, except in the case of Hydrogen, 
the numbers in the first column are larger than those in the 
second. Also that for the less easily liquifiable gases, such 
as the four first in the table, the numbers are all nearly equal 
while for the easily liquifiable gases they differ widely, 
and the more so the more easily liquifiable is the gas. 

129. The following numbers, also determined by Regnault, 
show the ratio in which the pressure is increased at constant 



102 HEAT [CHAP. \i. 

volume, for the same range of temperature, at initial pressures 
greater and less than an atmosphere : 

Pressure in Ratio of increase of 

Atm ^spheres at o. pressure from o to 100 C 

Any j ; : : i. :' . 0-1444 . - '36482 

' I -O3OO . . . '36650 

; . , 4'8i . . . '3709 1 

: < .GarlMnic Acid, < k J 0*998 . . . '36856 

4722 . . . -38598 



Thus, for a range of pressures from i to 4*8 atmospheres, 
the ratio for air changes by about -3 Jy-th, while in a range of 
but i to 47 atmospheres the ratio for carbonic acid changes 
by more than -g^-th. 

For dilatation under constant pressure, in the same range 
of temperature, we have 

Pressure in Ratio of volumes 

Atmospheres. at o" and 100 C. 

Air ... i ... 1*36706 

3-447 . . . 1-36964 

Hydrogen i .... 1-36613 

3-349 . . . 1-36616 

Carbonic Acid I .... I "37099 

3 '3i6 . . . 



We will return to this subject when we come to consider 
Thermodynamic principles as applied to the different states 
of matter. 

130. esumtof< 117 129. Dilatation of Liquids; 
especially Mercury and Water. Carbonic Acid. Dilatation 
of Gases. Laws of Boyle and Charles. Another hint of an 
Absolute Zero of Temperature. Regnault's results as to 
change of volume under constant pressure, and change of 
pressure at constant volume for different gases, 



CHAPTER VII. 

THERMOMETERS. 

131. IT seems now certain that the first inventor of the 
thermometer was Galileo, before 1597 (see Memoire sur la 
Determination de VEcheUe du Thermometre de V Academic del 
Cimento, par G. Libri, Ann. de C/iimie, xlv. 1830). His 
thermometer was an air-thermometer, consisting of a bulb 
with a tube dipping into a vessel of liquid. 
The first use to which it was applied was to 
ascertain the temperature of the human body. 
The patient took the bulb in his mouth, and 
the air, expanding, forced the liquid down the 
tube, the liquid descending as the temperature 
of the bulb rose. From the height at which 
the liquid finally stood in the tube, the physician 
could judge whether or not the disease was of 
the nature of a fever. 

A similar instrument was afterwards used, for 
a similar purpose, by the physician Sagredo, 
who, till recently, was regarded as the inventor 
of the thermometer. 

Air-thermometers, however, are affected by 
changes in the pressure of the atmosphere, as well as 
by changes in the temperature of the enclosed air, and 



104 HEAT. [CHAP. 

therefore, unless this disturbing cause is removed or ac- 
counted for, the reading of the thermometer is of no value. 

Thermometers, containing a liquid hermetically sealed up 
in glass, were first made under the direction of Rinieri (died 
1647), by Giuseppe Moriani, who, for his skill in glass- 
blowing, was surnamed II Gonfia. 

Many of the readings recorded by Rinieri are to be found 
in the Memoirs of the Academy del Cimento, but these were 
long supposed to have lost their value, as the instruments 
themselves could not be compared with our present ther- 
mometric scale. 

In 1829, however, a number of these very thermometers 
were found by Antinori, and their graduations were com- 
pared with those of Reaumur's scale, so that the readings 
of Rinieri can now be interpreted. 

One of the physical researches for which the Florentine 
Academy employed these thermometers, was to determine 
whether the melting of ice always takes place at the 
same temperature. This question they finally answered 
affirmatively. 

The next great step in thermometry was made by Newton, 
in his Scala graduum Caloris,\\\ the Philosophical Transactions 
for 1701, where he proposes the melting of ice and the 
boiling of water as standard temperatures. 

132. Fahrenheit of Dantzic, about 1714, first constructed 
thermometers of which the graduation was uniform. These 
thermometers were much used in England, and Fahrenheit's 
graduation is still the most common in English-speaking 
countries. In Fahrenheit's scale the temperature of melting 
ice is marked 32, and that of boiling water 212. 

The Centigrade scale was introduced by Celsius, of 
Upsala. In it the freezing-point is marked o, and the boil- 
ing point 100. The obvious simplicity of this mode of 



vn.l THERMOMETERS. 105 

dividing the space between the points of reference has 
caused it to be very generally adopted, along with the 
French decimal system of measurement, by scientific men, 
especially on the Continent of Europe. 

The scale of Reaumur, in which the freezing-point is 
marked o, and the boiling point 80, is still used for some 
medical and domestic purposes on the Continent of Europe. 

The existence of these three different thermometric scales 
furnishes an example of the inconvenience of the want of 
uniformity in systems of measurement. 

The division of the range from freezing to boiling points 
into 1 80, was probably in imitation of the division of a 
semicircle into 180 degrees of arc. This arose from the 
fact that 360 has a great number of divisors. 

The selection of Fahrenheit's zero probably arose from 
its being supposed that the ordinary freezing mixture of ice 
and salt gives the lowest attainable temperature. If we sup- 
pose the same thermometer to have these three separate scales 
adjusted to it, or (still better) engraved side by side upon 
the tube, we easily see how to reduce from one scale to 
the other. 



For, if/, ^, r be the various readings of one temperature, 
it is obvious that 

f 32 bears the same ratio to (212 32 or) 180, 
that c ' bears to J oo, 

and ;- bears to So. 



io6 HEAT. [CHAP. 

Hence 

/ 32 = __ _ r_ 
180 100 80 

[So far, all is legitimate and also necessary. But, in this 
latter half of the ipth century, the progress of science (!) 
seems to demand that, in Britain at least (for quasi-Chinese 
examination purposes), " problems " should be set, wherever 
a tempting opportunity like this presents itself. 

To prepare the student for such an ordeal, we append a 
couple of possible specimens, with their solutions. These 
specimens are not of more than average grotesqueness, and 
scarcely of average absurdity ; but we cannot devote more 
space to such things, useful as they may be. 

Thus, to find the temperature which has the same 
numerical expression in the Fahrenheit and Centigrade 
scales, put f for c or c for /, and we have 

/ -:-A 2 __ JL 

180 100' 

so that 

f - c = -_ 4 o. 

Again, to find the temperature whose Centigrade reading 
is the sum of its Fahrenheit and Reaumur readings, take 

c =/ + r, 

and we have 

180 80 

c = 32 + - - c + - - f, 

IOO IOO 

whence 

c = -20. Also, f 4, ;- 1 6. 

We do not exaggerate, when we say that practice at 
rubbish like this may turn the scale in some competitions for 
desirable appointments.] 

133. The glass tubes for liquid thermometers are drawn, 
not bored, and it is therefore necessary to go through the 



VII.] 



THERMOMETERS. 



107 



process of calibrating them : first, roughly, in order to 
reject all that are not nearly uniform in bore throughout ; 
then more carefully, so as to discover and tabulate the 
defects of the selected tube. 

This is a very simple process. It consists in introducing 
a little column of mercury an inch or two in length ; and, 
by slightly inclining the tube, causing it to slide from point 
to point, and measuring its length in each successive posi- 
tion. The narrower the bore at the place occupied by the 
mercury, the longer will be the column. These records are 
taken into account in graduating the tube by the Jielp of 
a dividing engine. 




The size of the bulb (which, in standard instruments, 
is usually cylindrical) is then to be estimated by three 
quantities : i, the section of the tube ; 2, the expansibility 
of the liquid to be used ; 3, the desired length of one 
degree of the scale. It varies of course directly as the 
first and third of these, and inversely as the second. 

The bulb, being blown, is gently heated in a spirit lamp 
to expel a little air, and the open end of the tube is 
plunged into the liquid to be used. As the bulb cools, 
some of the liquid is forced into the stem by atmospheric 
pressure : and, on erecting the instrument, part of it passes 
into the bulb. This portion of liquid is now made to boil 
freely so that its vapour expels the air, and the end of the 
tube is again suddenly immersed in the liquid. When the 
whole has cooled, both bulb and stem are filled with the 



loS HEAT. [CHAP. 

liquid, and the excess is expelled by again warming the 
bulb. The liquid is once more boiled so as to expel the air 
from the stem, and the end of the tube is then hermetically 
sealed. Thus, in a properly filled thermometer, when it 
lies horizontally, the contained liquid is subject to the 
pressure of its own vapour only. 

134. It is always desirable to have the range of a mer- 
curial thermometer made long enough to include both 
o and 100 C. When this cannot be done, or when instead 
of mercury there is a liquid which boils at a lower tempera- 
ture than water, definite points of the scale must be 
determined by careful comparison with a standard instru- 
ment. 




When the range is long enough, the cardinal points of the 
scale are determined as follows : 

Freezing -point. The bulb and part of the stem of the 
instrument are immersed, as in the sketch, in pounded ice, 
from which water is dripping away, and left for some time. 



VIL] THERMOMETERS. 



109 



till the upper surface of the mercury, which is just visible 
above the ice, is observed to take a definite position. This 
is carefully marked on the glass tube as the zero-point ; or, 
if the graduation has been previously effected according to 
An arbitrary scale (taking account of the calibration 133), 
the zero-point is carefully observed and registered. 

Boiling point. The instrument is now to be immersed, 
as carefully as possible, in steam freely escaping from water 
boiling under a pressure of i atmosphere. It is found that 
the temperature of boiling water is not steady, but that the 
temperature of the escaping steam is so. The annexed 
cut shows fully the arrangement usually adopted, and re- 
quires no explanation ; but we must show how to define 
accurately what is meant by a pressure of i atmosphere. 
(See 6 1, footnote.) 

The pressure at a given depth under the surface in the 
column of mercury of a well-filled barometer (whose 
tube is wide enough to prevent any sensible effect due to 
capillarity) depends solely upon two things the density of 
the mercury, and the intensity of the force of gravity. 

For a short column of mercury we may neglect the com- 
pression due to its own weight, so that the density (if the 
mercury be pure) depends practically upon the temperature 
alone. Hence, in defining barometric pressure, we must 
reduce the column to its length at o C. (See 120.) 

The intensity of the force of gravity depends upon the 
latitude, and upon the height above the mean sea-level. 
Hence, both of these must be taken into account in defining 
an " atmosphere." 

The definition, agreed on in this country, is the pressure 
when the barometer, reduced to o C., stands at 2 9 '90 5 
inches at the sea-level, in the latitude of London. 

The change of pressure, per degree of temperature of the 



HEAT. 



[CHAP, 




~_L I - - (5 "^^ 1 1 ^ : ;= " ^ :: - : ^ ; ~ : 



VI,.] 



THERMOMETERS. 



in 



boiling-point of water, in the neighbourhood of 100 C., is 
about 1-071 inch. It is rather less for temperatures lower 
than 100 C., and rather more for temperatures above that 
point. But the single number just given suffices for the 
practical determination of boiling-points near the sea-level. 
The further discussion of this question must be deferred 
till we are dealing with Changes of Molecular State due to 
heat. (See below, 165.) 

135. The mercurial thermometer is useful through a range 
of temperature from a little above the melting point 
( 40 C.) of mercury, to a few degrees below its boiling 
point (350 C.). When we desire to estimate temperatures 
about or below 40 C., alcohol is the liquid generally em- 
ployed, as its freezing-point is about i30C. For tempera- 
tures much over 300 C. recourse must be had to an air- ther- 
mometer ; or, for rough purposes, to Pyrometers ( 108), or 
Thermo-electric processes. When a thermometer of great 
sensitiveness is required for use at ordinary temperatures, 
carbonic acid is obviously ( 122) a suitable liquid. 

The contrivances for what are called maximum or mini- 
mum thermometers i.e. instruments to record the highest 
or lowest temperature during a given time are practically 
innumerable. The simplest, and most common, depend upon 
capillary forces. A column of mercury, pushing before it a 




little iron index (which it does not wet), is usually em- 
ployed in a maximum thermometer ; a column of alcohol 



ii2 HEAT. [CHAP. 

pulling back with it a little index of enamel (which it 
thoroughly wets), is usually employed for a minimum instru- 
ment. The force in either case is due to the tendency of 
the surface-film of the liquid to preserve the smallest area 
possible under the conditions to which it is subject Other 
principles are employed, such as for instance a maximum 
thermometer in which the mercury has to pass through a 
very narrow part of the tube near the bulb. While ex- 
panding, it passes freely : but as soon as it begins to con- 
tract the column breaks at the narrow neck, and remains 
in the tube as a record of the highest temperature reached. 

136. But all these devices are excessively imperfect, at 
least in comparison with continuous registration. This must, 
of course, be automatic. Many ingenious devices have 
been introduced for the purpose, but they are entirely super- 
seded by the introduction of photography. The photo- 
graphic process is so simple, and valuable, and at the same 
time so instructive, that we will devote a section or two to 
an explanation of it, and of some of its elementary 
consequences. 

It is applicable of course, not to temperature alone, but 
to every variable quantity which can be suitably indicated 
by an instrument such as (for instance) the barometer, the 
electrometer, or the variation compass. 

Suppose the stem of a mercurial thermometer to be 
placed close in front of a narrow slit, so that light can pass 
only where the slit is not blocked by the column of mercury. 
Then, if a gas flame be placed in front of the thermometer, 
and a sheet of photographic paper be drawn uniformly 
along behind the slit, a permanent record of the successive 
positions occupied by the mercury will be formed by the 
boundary between the blackened and the unaltered parts . 
of the paper. The clockwork by which the paper is drawn ; 



VII.] 



THERMOMETERS. 



along is easily made to mark it at every hour or minute, and 
the graduations of the thermometer, or some known marks 
made on its stem, may easily be recorded photographically. 
When such a sheet has been taken off, developed, and 
fixed, it presents an appearance somewhat like the sketch 
below : 




To find then what was the temperature at any instant, 
say 4 h .4i m , all we have to do is to draw (as in the figure) 
the line D E which at that time must have coincided with 
the slit. The point E, where this line meets the boundary 
of the blackened part of the paper, indicates where the end 
of the mercury column was at the given instant. In the 
figure, as drawn, the temperature is 5'2. 

137. If we look at the diagram above, we see that 
the portion of the curve drawn obviously contains two 
maxima and three minima ; and we see that the character- 
istic of either kind of point is simply that the tangent to 
the curve shall at that point be horizontal, and shall not 
pass through the curve. At a maximum, such as A, the 
tangent is horizontal, and (near A) wholly in the blackened 
space above the curve. Near a minimum, as B, it is wholly 
in the white space below the curve. 



ii4 HEAT. [CHAP. 

But a point such as C in the sketch is obviously neither 
a maximum nor a minimum, although the tangent is hori- 
zontal. At one side of C it is above, at the other below, 
the curve. 

Another way in which this may be put, and which is in 
fact quite obvious from the behaviour of the thermometer 
itself, is the statement that between two equal values of any 
varying quantity there always necessarily lies a maximum or 
a minimum. Therefore the nearer together (in point of 
time) are these equal values, the more nearly does either 
coincide (in time, and also in magnitude) with the maximum 
or minimum sought. 

[This simple consideration enables us to solve many 
problems which are usually thought to require the Differ- 
ential Calculus, or even the Calculus of Variations, for their 
proper treatment. Thus it suffices not merely for the in- 
vestigation of the law of refraction from the assumption 
that light takes the least time possible to pass from one 
point to another, but also for the investigation of brachisto- 
chrones, or lines of swiftest descent.] 

138. The general principle on which air-thermometers 
are constructed, is easily apprehended from the sketch of 
Balfour Stewart's apparatus in 127. If a metal ball be em- 
ployed, as in fact must be done if very high temperatures are 
to be measured, the gas must be preserved at atmospheric 
pressure, else there is danger of diffusion through the metal. 
Even in the case of a glass ball the constant pressure is 
desirable, not however on account of the permeability of the 
glass, but because of the alteration of its volume by in- 
creased pressure from within. There is not, as yet, any one 
particular form of air-thermometer which is in general use ; 
all the greater experimenters having devised special forms 
for the special inquiries they had on hand. 



ML] THERMOMETERS. H 5 

The Differential Thermometer of Leslie and Rumford is 
really a couple of air-thermometers working against one 
another. In its simplest form it consists merely of a tube 
with a ball blown at each end. These balls contain air, 




and the contents of the two are kept separate by a column 
of sulphuric acid, whose motions indicate differences of 
pressure, and therefore of temperature, in the two bulbs. 
This instrument is obviously unaffected by changes of baro- 
metric pressure ; it can be made very sensitive, and it is 
therefore of great use. Leslie made all his experiments 
on radiation of heat with this instrument ; and, by varying 
the materials of which the balls (or one of them) were 
made, he converted it into a hygrometer, a photometer, an 
aethrioscope, &c. 

I 2 



HEAT. 



[CHAP. 



139. The Mean or Average temperature during any 
assigned period can be obtained very accurately from the 
photographic record above described. Another method is 
to use a clock whose pendulum is not compensated ( 105) ; 
and, from the gain or loss of time which it shows as com- 
pared with the normal mean time clock, to calculate the 
average length of the pendulum, and thence the average 
temperature. 

Another method, originally suggested by Moseley's obser- 
vations on the way in which, in consequence of alterations 
of temperature, sheet lead gradually tears itself off from a 
sloping roof, is that of T. Stevenson's Creeper. It is simply 




a flat bar of an expansible metal, provided with equidistant 
rows of teeth along its upper and lower ends, and made to 
rest on an inclined slab of slightly expansible material, in 
which horizontal grooves are cut as close to one another as 
possible. 

The teeth and grooves are so shaped as to offer very 
slight resistance to motion upwards, while preventing all 
sliding down. When the instrument is raised in tempera- 
ture it expands ; and, as the lower end cannot move down- 
wards, the upper end must rise, so that its teeth reach a 



vii.] THERMOMETERS. 117 

higher groove. On the other hand, when it contracts by 
cooling, the lower teeth must be drawn up into a higher 
groove, because the upper end cannot slide down. It is 
obvious from this description that the " creeper " gives a 
record depending mainly upon the number and extent of 
the fluctuations of temperature. How it acquires the 
potential energy, which it has in its elevated position, is 
a question to be treated later. 

140. Resume of 131 139. Invention of the Ther- 
mometer. Newton's Fixed points. Scales of Fahrenheit, 
Celsius, and Reaumur. Calibration, Graduation, and Filling 
of Liquid Thermometers. Determination of Fixed points. 
Air Thermometer. Definition of "an Atmosphere." 
Ranges of different Thermometers. Maximum and Mini- 
mum Thermometers. Photographic Registering Thermo- 
meters. Properties of a Maximum, of a Minimum, and of 
a Maximum- Minimum. Differential Thermometer. Aver- 
aging Thermometers. The Creeper. 



CHAPTER VIII. 

CHANGE OF MOLECULAR STATE. MELTING AND 
SOLIDIFICATION. 

141. IN sections 45 and 48 above we have given a general 
sketch of the part of the subject to which this chapter is 
devoted. The student is recommended to re-read these 
sections before proceeding further. When a similar recom- 
mendation has to be made, it will be put simply in the 
form " Refer again to , ." 

142. Melting. In accordance with the principle of 9, 
and the fundamental principle of i, it is found that 
pressure and temperature alone require to be taken into 
account in the discussion of the melting of any definite 
solid. Hence the experimental law : 

The pressure remaining the same, there is a definite melting- 
point for every solid ; and (provided the mass be stirred} 
however much heat be slowly applied ', the temperature of the 
whole remains at the melting-point till the last particle is 
melted. 

This is one of the bases of Black's doctrine of Latent 
Heat. Our modern knowledge that heat is not matter leads 
us to regard the energy which escapes detection by the 
thermometer as being employed in tearing asunder the 
particles of the solid. This will not appear very startling 



CH. VIIL] CHANGE OF MOLECULAR STATE. 119 

if we think of the work required to reduce a mass to fine 
powder, every particle of which is still a portion of the 
solid. But the first clause of the statement leads to the 
important question of the influence of pressure upon the 
melting-point. This was first discovered by James Thomson, 
in 1849, and his calculations with regard to the lowering of 
the melting-point of ice by pressure were exactly verified 
experimentally by Sir W. Thomson in the same year. 
Hopkins, Bunsen, and others, have experimentally de- 
termined the elevation by pressure of the melting-points 
of substances which expand on becoming liquid. 

143. As an instance showing the nature of the experi- 
mental bases on which this statement rests, we refer again 
to the process of determining the zero-point of the thermo- 
meter ( 134). There nothing is said about the rate at 
which the ice is melting : i.e. about the quantity of heat 
supplied in a given time. See also 131. 

Water and mercury can be procured in very great purity, 
but the same statement cannot be made about the majority 
of other substances. Hence we give (as in 104) only a 
short table of examples of approximate 

MELTING-POINTS. 

Temperature C. 

Ice ....,..,. ..>.;>, i o 

Mercury . .,...- 40 

Sulphur in 

Lead 335 

Wrought Iron 1500 (?) 

Some metallic alloys, especially those containing bismuth 
(which have received the general name of " fusible 
metal "), melt at temperatures considerably under 100 C. 
The recently discovered metal, gallium, melts at about 
30 C. On the other hand, platinum cannot be fused in 



120 HEAT. [CHAP. 

any ordinary furnace ; and gas-coke (a form of carbon) has 
been softened only, not melted,, by the most intense heat 
yet produced artificially that of the electric arc. 

144. As already stated, the effect of pressure upon the 
melting-point of a body was deduced from theory, and 
subsequently verified by experiment. The reason of its 
having previously escaped experimenters is probably to be 
found in the extremely small amount of the effect even 
when great pressures are applied. This is not the place to 
enter upon the theory, which will be discussed later ; but we 
may mention the theoretical result in the form that 

Bodies which contract in the act of melting have their 
melting-points lowered by increase of pressure, and vice versa. 

145. The theoretical result for ice, exactly verified by 
experiment, is a lowering of the melting-point by 



for each additional atmosphere ( 134) of pressure. This 
may be roughly stated in the form that, under a pressure 
of one ton weight per square inch, ice melts at one degree 
Centigrade under its ordinary melting-point. 

[Along with this we have to remark that the density of 
ice is only about 0^92 of that of water, so that water-sub- 
stance contracts by eight per cent, in the act of melting.] 

Many of the consequences of this important fact were 
familiar to all before the fact itself was pointed out. 

Ons form in which it must have been well known for 
hundreds of years is the form in which we try the same 
experiment every time we make a snowball. Schoolboys 
know well that after a very frosty night the snow will not 
" make " : their hands cannot apply sufficient pressure* 
But, if the snow be held long enough in the hands to be 
warmed nearly to its melting-point, it recovers the power of 



MIL] CHANGE OF MOLECULAR STATE. iii 

" making," or rather of " being made." Every time we see 
a wheel-track in snow we see the snow is crushed, and even 
after one loaded cart has passed over it, certainly after 
two or three have passed, the snow has been crushed into 
clear transparent ice. The same thing takes place by degrees 
after people enough have walked over a snow-covered 
pavement; and in all these cases this minute lowering of 
the freezing-point has led to the result. And now we see 
how it is that the enormous mass of a glacier moves slowly 
on like a viscous body, because in consequence of this 
most extraordinary property it behaves under great pressure 
precisely as if it were a viscous body. The pressure down 
the mass of a glacier must of course be very great, and as 
the mass is especially in summer freely percolated 
throughout by water, its temperature can never (except on 
special occasions, and then near the free surface) fall 
notably below the freezing-point. Now, in the motion of the 
mass on its journey, there will be every instant places at 
which the pressure is greatest, where in fact a viscous 
body, if it were placed in the position of the glacier ice,, 
would give way. The ice, however, has no such power of 
yielding ; but it has what produces quite a similar result 
wherever there is concentration of pressure at one particular 
place it melts, and as water occupies less bulk than the 
ice from which it is formed, there is immediate relief, and 
the pressure is handed on to some other place or part of 
the mass. The water is thus relieved from the pressure by 
the yielding caused by its own diminution of bulk on 
melting. The pressure is handed on ; but the water 
remains still colder than the freezing-point, and therefore 
instantly becomes ice again. The only effect is that the 
glacier is melted for an instant at the place where there is 
the greatest pressure, and gives way there precisely as a 



122 HEAT. [CHAP. 

viscous body would have done. But the instant it has 
given way and shifted off the pressure from itself it becomes 
ice again, and that process goes on continually throughout 
the whole mass ; and thus it behaves, though for special 
reasons of its own, precisely as a viscous fluid would do 
under the same external circumstances. 

The first who seems to have realised this on a small scale 
t>y experiment was Dollfuss-Ausset, who showed that by 
compressing a number of fragments of ice in a Bramah 
press, it was possible to melt them ; and when pressure was 
taken off them, to allow them to revert again into a solid 
block. But he found that with very cold ice the experi- 
ment did not succeed. In fact, as we now see, even 
with his Bramah press, he could not apply pressure 
enough. 

By opening a hole in the end of a cylinder in which snow 
is compressed by a Bramah press, we obtain a cylinder or 
wire of solid ice gradually squeezed through the hole, just 
as wires are made of soft or crystalline metals. The 
mechanism of the process is different, but the results are 
exactly the same. 

Another simple but effective experiment may be made by 
passing a loop of wire round a bar of ice supported hori- 
zontally. A weight attached to the wire pulls it gradually 
through the ice, which melts before the wire and is imme- 
diately re-formed behind it : so that the wire passes entirely 
through (as in cutting cheese or soap) and yet leaves the bar 
as strong as ever. 

146. Every one knows that when melted bees-wax partially 
solidifies, the crust if broken sinks in the liquid. The same 
phenomenon is observed in lava-lakes such as that of 
Hawaii, where the crust " cracks in different directions, and 
first one half of the lake and then the other is covered with 



Yin.] CHANGE OF MOLECULAR STATE. 123 

a fresh coating of red-hot lava, the crust tumbling out of 
sight as it shrunk and cracked in cooling." * 

Just as the effect of pressure on the melting-point of ice 
enables us to account for the plasticity of glacier ice, so the 
effects now described enable us to explain with great pro- 
bability why it is that the materials forming the interior of 
the earth are practically rigid, and therefore solid, though at 
temperatures far above their ordinary melting-points. 

For, as will be seen when we consider (under the head of 
Conduction} underground temperatures, there can be little 
doubt that the temperature at a few hundred miles below 
the earth's surface must be very high certainly far above 
the usual melting-point of lava. Yet, as Sir W. Thomson 
has shown from the amount of the tides, there can be no 
doubt that, as a whole, the earth is nearly as rigid as a globe 
of steel of the same size. 

These two apparently inconsistent conditions are at once 
reconciled, if we suppose the average materials of the earth 
to be such as, like lava, to expand on melting : for the 
immense pressure to which they are subjected from super- 
incumbent strata is probably sufficient to raise their melting- 
point above their present temperature. Thus they may 
remain solid, even at a white- heat. But if by the cooling 
and shrinking of the lower strata within the solid crust 
this pressure 'should be anywhere considerably relieved, 
the mass affected would (almost explosively) melt with 
considerable expansion. This seems to have important 
bearings on earthquakes and upheavals. 

[It may be mentioned, in passing, that if the earth were 
liquid throughout, and of uniform density equal to its pre- 
sent mean density, the pressure within a few hundred miles 

* J. W. Nichol, Prcc. R. S. ., 1875-6, p. 117. 



124 HEAT. [CHAP. 

of the surface would increase at the rate of somewhere about 
800 atmospheres (or nearly 5^ tons weight per square inch) 
per mile of depth.] 

Hopkins has found that the melting-point of wax is raised 
about 10 C. by 500 atmospheres' pressure ; and Bunsen 
gives a rise of 3 '5 C. as the effect of 100 atmospheres on 
the melting-point of paraffin. 

147. We must now consider the second clause of the 
experimental statement of 142. 

That the temperature of the mixture of solid and liquid 
should remain at the melting-point, however much heat be 
supplied, till the last particle is melted, was explained by 
Black (on the hypothesis of the materiality of heat) by the 
supposition that the liquid differs from the solid simply by 
having taken into combination a certain proportion of 
caloric. Thus the liquid was regarded as a species of 
chemical combination of the solid with an equivalent of 
caloric. As no effect was produced on the temperature 
of the body by this admixture, Black introduced for this 
supposed equivalent of caloric the name of Latent Heat. 
Relatively to the knowledge of his time, the name was a 
felicitous one, because it was found that when a liquid 
solidified it gave out exactly as much heat as it had taken 
in on melting. 

Unfortunately, from our modern point of "view, the term 
is not a felicitous one and yet it is so ingrained in our 
language (like vacuum and centrifugal force} that we are 
not at all likely soon to get rid of it. But no difficulty 
will be found if we keep in mind that when we speak of 
latent heat we mean no more than this that a certain 
amount of heat is in every case required to change the 
molecular state of a substance even when there is no 
alteration of its temperature. 



vni j CHANGE OF MOLECULAR STATE. 125 

148. We are, as yet, almost wholly ignorant of the form 
in which energy exists in bodies generally ; and a great deal 
of mischief has been done (almost ever since the time of 
Black) by the assumption that bodies must possess a certain 
amount of what has been called " sensible," or " thermo- 
metric " heat. Until we know, at least partially, the nature 
of the internal mechanism connecting the particles of matter, 
it is altogether vain to discuss questions concerning the heat 
present in a body, farther than is involved in estimating the 
amounts of energy supplied to it, and given out by it, in the 
various stages of an operation. Refer again to 74. 

149. But if we consider for a moment the amount of work 
necessary to grind to fine powder even the most friable of 
solids, and, going farther, think what almost infinitely more 
perfect breaking up befalls such solids when they are melted, 
it is easy to see how a great part of the energy, supplied to 
a solid in the form of heat, must be applied to the mere 
mechanical work of pulling its particles into the position of 
comparatively slight constraint which they occupy in the 
liquid, against the molecular forces which originally main- 
tained them in the solid form. 

150. With these explanations, we retain the term Latent 
Heat as a general one applicable in all cases of change of 
molecular state where energy in the heat form can be 
supplied to a body without producing alteration of its 
temperature. 

[The student should carefully notice the words in italics, 
for the same amount of energy can be given to the ice in 
many other ways, without either melting it or changing its 
temperature e.g. by lifting it to the proper height against 
gravity, or by projecting it with the proper velocity.] 

Thus the latent heat of water is to be numerically mea- 
sured as the number of units of heat ( 55, 56) which must 



126 HEAT. [CHAP. 

be communicated to a pound of ice at o C. to convert it into 
a pound of water at o C. 

This is ( 9) evidently a definite quantity. For the 
temperature mentioned in the definition requires that the 
pressure shall be i atmosphere. 

151. The experimental determination of this quantity can 
be made in many ways : but all the usual ones depend 
upon the direct comparison of the effects of equal amounts 
of heat upon ice and upon water. 

The ordinary lecture-room form of the experiment con- 
sists in pouring over a pound of ice at o C. a pound of 
water at about 80 C., and carefully stirring the mixture. 
When proper precautions against loss of every kind are 
taken, it is found that under these circumstances the whole 
of the ice is just melted the resulting temperature being 
still o C. 

Assuming, what is very nearly the case, that the amount 
of heat required to raise the temperature of a pound of 
water one degree is the same throughout the whole range 
from o to 80 C., it is obvious that the pound of hot water 
has lost 80 units of heat, which must have gone to the pound 
of ice with the result of melting it without raising its tempera- 
ture. Hence it follows that (in accordance with the defini- 
tion of latent heat, 150), 80 is the latent heat of water. 
It appears from the experiments of Person and others that 
the correct value is more nearly 79*25. The experimental 
part of the work presents no very great difficulty, but the 
mode of reasoning from the results depends to some extent 
upon the opinion we may form as to whether the transition 
from ice to water is an abrupt or a gradual process. 

152. There can be no doubt that, in many substances at 
least, the transition from solid to liquid is gradual and not 
abrupt. Every one is familiar with the softening of wax and 



vin.] CHANGE OF MOLECULAR STATE. 127 

paraffin, as they are gradually raised in temperature. The 
welding of iron and of platinum, at high temperatures, is 
another case in point. And several excellent authorities 
state their conviction that something analogous occurs with 
ice. What farther we have to say on this question will be 
profitably deferred till we discuss the whole subject on 
thermodynamic principles. It is clear that a rapid increase 
of specific heat, just below the melting-point, might, if un- 
detected, lead to an over-estimate of the latent heat as we 
have defined it. But it is also clear that this explanation can 
only be valid if ice, from which water is trickling, is (all 
but its superficial layer) essentially a little colder than o C. 
Forbes and Balfour Stewart state that the temperature of a 
mass of ice, which has been rapidly pounded, is invariably 
found to be a little below the freezing-point. 

153. Water is almost exceptionally high among liquids as 
regards the amount of its latent heat. The following short 
table gives some general notions on the subject : 

LATENT HEAT OF FUSION. 

Water . 79 '25 

Phosphate of Soda (crystallised) .... 67*0 

Zinc 28' i 

Sulphur 9 '4 

Lead 5 '4 

Mercury 2 '8 

It is to be observed that the numbers here given denote the 
units of heat required just to melt one pound of each of 
these substances without change of temperature. 

154. We have already alluded to the abrupt changes of 
volume which usually take place on melting. In general, 
the change is of the nature of expansion ; but water, cast- 
iron, type-metal, and some other bodies, contract. As the 



128 



HEAT. 



[CHAP. 



process of solidification is exactly the converse of that of 
melting, it is accompanied by disengagement of latent heat 
and return to the former volume. Hence the nicety with 
which iron and type-metal adjust themselves to every little 
crevice in a mould. Hence also the bursting of water-pipes 
during (not, as the majority even of tl educated " people still 
think, after) a frost. The pressure requisite to prevent 
water from freezing in a closed vessel, nearly full, is enor- 
mous, provided the temperature be a few degrees under the 
usual freezing-point : sufficient, as has been found by trial, 
to burst the strongest bomb-shells. The amount of work 





which can be done by a pound of water, in freezing under 
given circumstances, can be at least approximately calcu- 
lated, as we will show when treating of the dynamical 
theory. In fact, while discussing the lowering of the 
freezing-point by pressure, we have pointed out ( 145) 
that a pressure of a ton weight per square inch is re- 
quisite to prevent ice from being formed at a temperature 
even one degree under zero. 



viii.] CHANGE OF MOLECULAR STATE. 129 

155. The process of solution of a solid in a liquid is, in 
so far at least as it is independent of chemical action, very 
closely analogous to melting ; and, as in 149, must obvi- 
ously require a supply of energy. This is usually taken 
from the body itself and from the menstruum, as well as 
from surrounding bodies, in the form of heat. 

Thus solution of a solid in a liquid is usually accom- 
panied by cooling. This is, in some cases, partially or wholly 
masked, sometimes even overcome, by the heat developed 
by chemical action. 

One of the commonest of these arrangements for pro- 
ducing cold (i.e. rendering heat latent), is to dissolve nitrate 
of ammonia in an equal weight of water. If both be taken 
at ordinary temperatures (say 10 C., or 50 F.), the tempera- 
ture of the solution falls to about 15 C. A somewhat 
greater cooling is produced by pouring commercial hydro- 
chloric acid upon snow. 

But by far the lowest temperatures yet attained by such 
means are procured by pouring ether upon solid carbonic 
acid and other bodies which are gaseous at ordinary tem- 
peratures and pressures. These temperatures are further 
considerably lowered by placing the mixture under a receiver, 
and exhausting the air. Natterer estimates the lowest tem- 
perature he has thus obtained at about i4oC., more than 
half-way from freezing-point to absolute zero (96, 125). 
Perhaps even more striking results may yet be obtained 
from solid hydrogen. 

156. When two solids, on being mixed as crystals or 
powder, melt one another, we have of course to supply 
from without the latent heat for each, except in so far as 
heat is chemically developed by the combination. Thus we 
explain the action of the well-known freezing mixture of 
snow and common salt, or of salt and pounded ice ( 132). 

K 



130 HEAT. [CHAP. via. 

If the salt has been previously cooled to o C., the tempera- 
ture of the mixture falls to about 20 C. 

157. Conversely, when a substance in solution crystallises 
out, we have in general a development of heat. This is 
very well shown by supersaturated solutions of sulphate or 
acetate of soda, which suddenly crystallise when the smallest 
fragment of the solid is dropped in. In the acetate of soda 
the water is almost entirely taken up by the solid as water 
of crystallisation. 

158. Resume of 141-157. Law of Melting. Effect of 
Pressure. Melting-points of some Solids. Behaviour of 
Ice and Lava. Consequences as regards Glaciers and the 
Strata under the Earth's Crust. Latent Heat of Fusion. 
Latent Heat in different Liquids. Solution. Freezing 
Mixtures. Heat developed in Solidification. 



CHAPTER IX. 

CHANGE OF MOLECULAR STATE. VAPORISATION AND 
CONDENSATION. 

159. Refer again to 46, 47. We have seen ( 152) 
that it is not yet settled whether the change from the solid 
to the liquid state takes place precisely at the definite tem- 
perature called the melting-point or not. There is no 
doubt, however, that the transition from the liquid (and, 
in some cases at least, from the solid) state to the state of 
vapour takes place at all temperatures, but more freely the 
higher the temperature, up to the so-called boiling-point. 

Every one must have noticed that even in the coldest 
winter day especially if it be windy the ice on the pave- 
ments gradually dwindles away, though constantly appear- 
ing solid and hard. It is, in fact, always evaporating though 
very slowly. Whether it passes or not through the liquid 
state in a thin film on the surface is not yet known. And 
distillation of liquids is often practised on a large scale, in 
manufacturing operations, at temperatures kept purposely 
below their boiling-points. Salt-pans form an excellent 
example. All but a very small fraction of the vapour con- 
stantly in the air, has been raised from the surface of 
oceans, lakes, or moist ground, at temperatures far below 
the boiling-point. 

K 2 



132 . HEAT. [CHAP. 

1 60. BOILING. Experiment enables us to state for this 
phenomenon a law precisely similar to that of 142, viz. 

The pressure remaining the same, there is a definite boiling- 
point for the free surface of every liquid ; and (provided the 
mass be stirred] however much heat be applied, the tempera- 
ture of the whole remains at the boiling-point till the last 
particle is evaporated. 

The effect of pressure upon the boiling-point can be cal- 
culated, as was that of pressure upon the melting-point ; but 
as we do not know of a substance which (at the same 
temperature and pressure) occupies less bulk in the form 
of vapour than in that of liquid, we may assert that the 
effect of pressure is in all cases to raise the boiling-point. 

So far as water is concerned, we have already ( 60) given 
some explanations and experimental data on this subject. 
We will now treat the whole experimental law more fully, 
and in a manner similar to that which we adopted for the 
analogous law of melting. 

161. We take water vapour as a type as it is the most 
important, and has been therefore the most carefully studied. 
The first even approximately accurate statement of its 
behaviour is due to Dalton. Numerical data of great pre- 
cision, and extending through a wide range of temperatures 
and pressures, have recently been furnished by Regnault. 
And an experimental result of startling novelty, and of the 
greatest theoretical importance, due to Andrews, revealed, so 
lately as 1869, what is the true distinction between a vapour 
and a gas. 

162. If a vessel of water be placed in an exhausted 
receiver, evaporation will immediately commence, and the 
process will go on with great rapidity until the pressure of 
the vapour in the receiver rises to a certain definite amount, . 
which is found to depend solely upon the temperature. 



ix.J VAPORISATION AND CONDENSATION. 133 
If the receiver is fitted with a piston, \yhich is pushed in 




or pulled out so as greatly to increase or to diminish its 
volume, still the pressure of the vapour remains unchanged ; 



134 HEAT. [CHAP. 

for fresh vapour is formed when the volume is increased, 
and part is condensed into liquid water when the volume is 
diminished. This is easily effected in practice by intro- 
ducing a drop of water into a tube filled with mercury, as 
for the Torricellian experiment. When the tube is inverted 
in a vessel of mercury, there is a definite amount of de- 
pression of the mercury column (as compared with the 
barometer), which is not altered by plunging the tube 
deeper into the mercury. [This is exhibited in the first 
of the above cuts, where a shows the level of the mercury 
in the barometer tube (with the Torricellian vacuum above 
it), while b is that in the tube with a drop of water inserted. 
The second of the above cuts represents the simultaneous 
behaviour in three barometer tubes of different lengths, into 
each of which a drop of water has been introduced.] 

The same result is ultimately arrived at, but not so rapidly, 
if the receiver contain air. The only effect of the air is to 
retard the production of vapour, but the process goes on 
as before until the pressure of vapour is that due to the 
particular temperature of the water and the denser the air 
the greater the retardation. 

163. Vapour, which is in equilibrium in contact with 
excess of liquid, is called saturated vapour, and Dalton's 
statements may be put in the form 

The ultimate pressure of the saturated vapour of any liquid 
depends only upon the temperature. 

[N.B. The student should remark that in many old 
books, and in too many modern ones, the word tension is 
improperly used in this connection instead of pressure. 
Chemists, especially, have been led to sanction this blunder, 
and almost invariably speak of vapour-tension.] 

164. While a liquid is thus in equilibrium with its own 
vapour, suppose the vapour to be removed by means of an 



IX.] VAPORISATION AND CONDENSATION. "135 



air-pump almost as fast as it is formed. Then, obviously, 
fresh vapour is furnished very rapidly from the liquid : and 
it is observed to come not merely from its surface, but in 
bubbles from the interior of the mass and specially from 
sharp points or edges of solids immersed in it. This free 
and rapid discharge of vapour is what is called boiling. 
And we thus see that 

The boiling-point of a liquid is the temperature at which its 
saturated vapour has a pressure equal to that to which the 
free surface of the liquid is subjected. 

PRESSURE OF SATURATED WATER VAPOUR REGNAULT. 



Temperature C. 



O 
10 
2O 
30 
4 
50 
60 
70 
80 
90 
TOO 
110 



'ressure in Atmo- 


Temperature C. 


spheres. 




o'oo6 


120 . 


0-012 


I 3 


0*023 


140 . 


0-042 


150 . 


0*072 


160 




O'I2I 


170 




0*196 


I8o 




0-306 


190 




0-466 


200 




0-691 


210 




I'OCO 


220 




I-4I5 


230 





Pressure in Atmo- 
spheres. 

1-962 



4712 

6*120 

7 -844 

9'929 

12-425 

i5-38o 

18-848 

22-882 



165. On this very instructive table (given by Regnault 
at p. 728 of his magnificent Relation des Experiences, etc., 
Mem. de VAc. des Sciences, 1847,) a few remarks may be 
made. 

(a.) We see how very rapidly the pressure of saturated 
water-vapour rises as the temperature is raised. 

Thus the rise of temperature from 100 to 180 increases 
the pressure from one to nearly ten atmospheres, while an 
additional rise of only .40 C. raises the pressure to nearly 
twenty-three atmospheres. 

Compare this with the behaviour of a gas such as air 



136 



HEAT. 



[CHAP. 



( 124), where the pressure (at constant volume) rises ap- 
proximately in proportion to the absolute temperature ; and 
we see how very much less dangerous, so far as the chances 
of ,an explosion are concerned, is an air-engine than a 
steam-engine working (with saturated steam) at the same 
high temperature. 

(b.) Water at ordinary temperatures may be made to boil 
by placing it in the receiver of an air-pump and producing 
a sufficient vacuum. Thus if it be taken at 10 C. (50 F.) 
it will boil when the pressure is reduced to 0*012 of an 




atmosphere (or, more accurately, to 9-16 millimetres of 
mercury). And a notable feature of this experiment is that 
the water is found to become gradually colder as the boiling 
proceeds, thus requiring farther exhaustion by the air-pump 



ix.J VAPORISATION AND CONDENSATION. 137 

to maintain the process. This will be explained in a 
subsequent section, when we are dealing with the Latent 
Heat of steam. 

A striking form of this experiment consists in making 
water boil in an open flask so furiously that the greater 
part of the air is expelled by the steam, then corking the 
flask and inverting it. When cold water is poured on the 
bottom of the inverted flask, it condenses the steam and 
thus diminishes the pressure on the water, so that it im- 
mediately begins to boil. Boiling water, poured on, at once 
stops the boiling. If the air has been very completely 
expelled, the boiling by the application of sufficiently cold 
water can be produced even when the contents of the flask 
have cooled to the temperature of the air of the room. 

(c.) The temperature at which water boils (in a small 
apparatus like that sketched in 134) may be used as a 
means of measuring the pressure to which it is subjected. 
Thus Wollaston made the thermometer take the place 
of the barometer in the measurement of the heights of 
mountains. It has, in fact, several advantages as regards 
portability, being far more compact and much less liable 
to breakage. The following roughly approximate formula, 
calculated from a table of Regnault's (p. 633 of the 
Relation, etc.) shows how the boiling-point varies with the 
ordinary fluctuations of the barometer near the sea-level. 

Pressure. 



Temperature C. Atmospheres. Inches. 

99 r. 0'9647 0*0348 T. 28*87 ^ I<0 4 T - 

[This formula must not be employed for values of r 
much exceeding 2. For greater ranges, or for more 
accurate values within this range, Regnault's full table 
must be consulted.] 



138 



HEAT. 



[CHAP. 



Roughly speaking, the boiling-point of water is lowered 
by i C. for 960 feet of vertical ascent above the sea-level. 
At the top of Mont Blanc the Hypsometric Thermometer 
was found by Bravais and Martins, in 1844, to stand at 
8 4 - 4 C. 

(d.} We see why a diminution of pressure reduces the 
solvent and cooking powers of boiling water, while an 
increase of pressure exalts them. Tea prepared at the top 
of Mont 'Blanc is poor stuff; and Papirfs digester, which is 




used for extracting everything soluble from bones, etc., is 
simply a very strong boiler, in which water, under the 
pressure of its saturated vapour, can be safely raised to 
temperatures far above its ordinary boiling-point. 



TX.] VAPORISATION AND CONDENSATION. 139 

1 6 6. Approximate expressions for the amount of heat 
rendered latent in the .evaporation of water at different 
temperatures were given by Watt and others. But the 
subject remained doubtful until Regnault cleared it up. 
He gives for what he calls the total heat of steam, at any 
temperature tC. the very simple formula 

606 '5 + 0*305 /. 

Thus a pound of saturated water vapour at oC. gives out 
606 '5 units of heat in condensing to water at o ; 'while if its 
temperature had been originally iooC. it would have given 
out 606*5 + 30*5 (or 637 units) in passing to water at o. 

In the next chapter the (very small) variation of specific 
heat of water with temperature will be treated. When it 
was taken into account along with the above formula, it 
was found by Regnault that the latent heat of steam at 
different temperatures falls 

from 606 *5 at oC., 
to 536'5 at 100, 

and to 464*3 at 200. 

The latent heat at any temperature is here the number of 
units of heat required simply to convert a pound of water 
into steam of the same temperature. 

For most practical purposes this may be taken as indi- 
cating a decrease simply in proportion to the rise of tempe- 
rature, and amounting to 11*5 per cent, for each hundred 
degrees above the freezing-point. Watt's hypothetical state- 
ment was to the effect that the heat required to change a 
pound of water at o C. into steam at any pressure whatever 
i.e., the total heat of steam is constant. If this were 
true the sums of the last three pairs of numbers should be 



140 HEAT. [CHAP. 

approximately equal. They are not so, and they show that 
Watt's statement involves an error in defect amounting to 
about five per cent, of the whole for every hundred degrees 
above o C. 

When we are dealing with the dynamical theory, we shall 
have occasion to discuss this question more fully. 

167. According to Andrews, the latent heat of a pound 
of vapour, produced from certain common liquids by 
boiling at the ordinary atmospheric pressure, is as follows : 

LATENT HEAT OF EVAPORATION AT i ATMOSPHERE PRESSURE. 

Water 536-0 

Alcohol 2O2'4 

Ether 90*5 

Bromine 45-6 

Here, again, water stands relatively very high. And, as 
we have already seen that the latent heat of steam increases 
as the temperature of evaporation is lower, we see what a 
very large amount of heat is required for the evaporation 
going on over the oceans ; and how great is the amount of 
heat set free when that vapour is condensed into fogs or 
clouds. This leads us to consider the phenomenon of 
condensation with some of its consequences. 

1 68. A striking illustration of the great latent heat of 
steam is given by a very simple arrangement. We have 
merely to lead steam from a boiler into a vessel containing 
a measured quantity of ice-cold water. At first, the steam 
is condensed as soon as it reaches the water ; but as the 
water becomes warmer the steam gradually advances from 
the end of the conducting pipe, forming an increasing 
bubble at whose surface condensation is steadily going 
on. As soon as the water is raised to the boiling-point, the 



ix.] VAPORISATION AND CONDENSATION. 141 

steam passes freely through it. When this stage is reached, 
remove the steam-pipe and measure the volume of the 
water. It is found to have increased by less than a fifth 
of its original amount. This increase is, of course, in a 
condensed form, the steam, whose latent heat has raised 
by 100 C. the temperature of the mass of water. 

169. The latent heat of vaporisation is utilised in many 
ways, especially for cooling and for freezing. 

Thus water, put into a vessel of unglazed clay, is kept 
permanently cool in warm dry air, by the evaporation from 
the surface of the vessel. A similar result is produced when 
a glass vessel is employed, if it be wrapped in a wet cloth 
and placed in a current of air. In some parts of India ice 
is procured by exposing water at night in shallow unglazed 
saucers, laid upon rice-straw. More rapid effects may, 
of course, be obtained by using instead of water highly- 
evaporable liquids such as sulphuric ether. A few drops of 
ether, sprinkled on the bulb of a thermometer, produce an 
immediate contraction of the contents, which is greater as 
the temperature of the air is higher. The cooling of water, 
when it is made to boil at low temperatures (by reducing 
the pressure, as in 165 (b) ) is due to the same cause. 
This process, with a quantity of dry oatmeal or a large sur- 
face of sulphuric acid (to absorb the vapour as it is formed) 
was employed by Sir John Leslie for the purpose of making 
ice ; and is still, with various modifications, the basis of 
some of the most convenient domestic ice machines. 

The Cryophorus (whose principle will be explained when 
we are dealing with Daniell's Hygrometer, 172) is another 
very curious illustration of the same fact. 

When a jet of carbonic acid, liquefied by pressure, is 
allowed to escape into the air, the outer layers of the jet 
vaporise at once ; taking the requisite latent heat from the 



1 42 HEAT. [CHAP. 

core of the jet, which is thus frozen into a solid and can.be 
collected in a proper receiver as a snow-like mass. It 
appears that hydrogen has in this way been obtained as a 
steel-grey powder. A very striking experiment of the same 
class is the freezing of water in a white-hot platinum dish. 
This is easily effected by the help of liquid sulphurous acid, 
which evaporates very freely from the water while it remains 
suspended above the dish in what is called the ''spheroidal 
state" ( 213). If the dish be not sufficiently heated the 
experiment fails. 

Faraday succeeded in freezing even mercury in a white- 
hot vessel by a process of this kind. The mercury was in a 
little capsule resting on a mixture of solid carbonic acid and 
ether in the spheroidal state in a highly heated platinum 
crucible. 

170. The reader will now easily understand why it is 
possible (as stated in 155) to procure very low tempera- 
tures by exposing solidified carbonic acid, mixed with ether, 
in an exhausted receiver. 

171. We must next consider the converse process, the 
condensation of vapour into liquid. And for the present, 
we confine our remarks to the behaviour of water-substance. 
In 162 we have already shown that condensation com- 
mences, in a vessel containing water-vapour alone, as soon 
as the pressure of the vapour exceeds that corresponding to 
saturation. We may state this, of course, in another form, 
viz. that condensation commences as soon as the tempera- 
ture falls below that corresponding, in Regnault's table. 
164, to the pressure of the water-vapour present. And it 
is found by experiment that the pressure of air or other gas 
does not modify this result except in more or less retarding 
it. Hence, if we present to air containing water-vapour a 
solid cooled below the temperature of saturation corres- 



ix.J VAPORISATION AND CONDENSATION, ! 43 

ponding to the vapour-pressure, condensation will take place 
in the layer of air immediately in contact with the solid, 
which will thus be covered with a film of Dew. This film 
will become thicker, until, by the latent heat given up by the 
vapour in condensing, the solid has been raised to the 
temperature of saturation corresponding to the vapour- 
pressure. If the solid be at a temperature below o C. the 
film freezes as it is deposited, and becomes what is called 
Hoar Frost. 

In Hope's experiment (121) the metal vessel containing, 
the freezing mixture is rapidly covered with a layer of hoar 
frost, however warm the surrounding air may be, provided 
it contains a moderate amount of aqueous vapour. 

172. The statements in last section show at once what is 
meant by the meteorological term, the Dew-point. It is 
the temperature at which saturated water-vapour would have 
the same pressure as that of the vapour present in the 
air at the time ; and it could therefore be found directly 
from Regnault's table ( 164) if we knew the quantity of 
vapour per cubic foot of air. But the converse process is 
that most commonly used, the dew-point being directly 
measured, and then employed for the purpose of estimating 
by Regnault's table the quantity of vapour present in the air 
at any time. 

Daniell's Hygrometer is an ingenious instrument by which 
a body is gradually cooled, so that we can measure its 
temperature at the instant when dew just begins to be 
deposited on it. It consists of a glass tube with a bulb at 
each end. One of these bulbs is made of black glass, to^ 
show at once the slightest trace of dew on its surface. In 
this bulb is a thermometer (whose indications can be read 
through the clear glass tube joining the two bulbs) and a 
quantity of sulphuric ether The instrument-maker boils 



144 HEAT. [CHAP. 

this ether ; and hermetically seals the second bulb when the 
ether-vapour has expelled the air from the whole apparatus 
(as in 165 (b) ). To use the instrument, pour a few drops 
of ether on a piece of cambric wrapped round the second 
bulb. The evaporation of the ether cools this bulb and 
.condenses the ether-vapour in it. More vapour is formed 
from the ether in the black bulb, and again condensed in 
the second bulb. Distillation in fact goes on between the 
two bulbs ; and the black bulb, with its contents, becomes 
gradually colder on account of the latent heat required for 
the persistent formation of vapour. The temperature of the 
black bulb is read by the inclosed thermometer at the 
instant of the first appearance of dew. The thermometer is 
again read at the instant that the dew disappears, when the 
.apparatus begins to be heated again to the temperature of 
the air. It is usual to assume the mean of these readings 
as the dew-point. 

Suppose water to be put in the instrument instead of 
ether, and snow and salt instead of ether to be applied to 
the empty bulb. The rapid evaporation of part of the 
water from the full bulb freezes what is left. This is the 
Cryophorus ( 169). 

Another method of finding the dew-point is by the Wet 
and Dry Bulb Thermometers. Two similar thermometers 
are placed side by side ; one having its bulb covered with 
cambric, connected by a few threads with a small vessel of 
water, so as to be kept constantly moist. The thermometer 
with the naked bulb shows the temperature of the air, the 
other shows a lower temperature, which differs from the first 
in consequence of the evaporation constantly going on from 
its moist covering. When the air is nearly saturated with 
moisture, the evaporation is very slow ; and the lowering of 
temperature small. But if the air be dry, and especially if 



ix.] VAPORISATION AND CONDENSATION. 145 

it be also warm, there is rapid evaporation and considerable 
lowering of temperature. Dr. Apjohn, who has studied this 
subject with particular care, gives for the determination of 
the dew-point by this arrangement the formula 



48 30 

In this formula 8 is the difference between the readings 
of the thermometers, p is the sought pressure of vapour in 
the air,/ that in Regnault's table corresponding to the 
temperature of the wet-bulb, and b is the height of the 
barometer in inches. To a certain extent this formula is, 
as yet, empirical. The full treatment of the question by 
theory remains to be discovered. 

173. The theory of the formation of dew was first 
correctly given in Dr. Wells's Essay of 1814. It cannot be 
fully stated till we have dealt with Radiation. But it may 
suffice for the present to say that the cooling of stones, 
blades of grass, &c., by radiation, takes place most rapidly 
in clear, calm nights. Clouds, in general, radiate back a 
great deal of the heat they receive, and so prevent objects 
on the ground from cooling sufficiently. Winds also, unless 
the air is itself very cold, constantly warm again bodies 
cooled by radiation. Hence it is after clear, calm nights 
that the dew is usually most copious ; and when the 
radiation has been sufficiently rapid we have it in the form 
of hoar-frost. The temperature of the air itself, near the 
ground, cannot at any time fall much below the dew-point ; 
for, as soon as it does so, condensation takes place, and 
latent heat of vapour is set free. 

174. What has been said in the preceding sections of 
the condensation of aqueous vapour is true, generally, of 



146 HEAT. [CHAP. 

the vapours of all substances which are liquid at ordinary 
temperatures and pressures. Some liquids, such as com- 
mercial sulphuric acid, have a scarcely measurable vapour- 
pressure under ordinary conditions; others, as sulphuric 
ether, evaporate with, great rapidity. 

But it is only within comparatively recent times that it 
has been shown experimentally that all gases are really 
vapours. This has been done by reducing them to the 
liquid, and sometimes even to the solid, state. Mere cold 
effects this with many substances, mere pressure with 
others ; but some (especially those bodies which used to be 
called permanent gases) require the combined application of 
cold and pressure. Faraday effected the liquefaction of a 
great number of gases, mainly by pressure. On the other 
hand Cagniard de la Tour rendered liquids, with little 
change of density, gaseous (or, at all events, not liquid), by 
heating them .under great pressure. It was Andrews, 
however, who first cleared up the whole subject by showing 
the nature of the distinction between what may now be 
called a true gas, and what may be called a true vapour. 
His experiments, on carbonic acid and other bodies, led to 
the experimental law that 

There is a Critical Temperature for every vaporous or 
gaseous substance; such that, only when its temperature is 
below this, can the substance be reduced to the liquid form by 
any pressure, however great* 

The critical temperature for sulphurous acid is high, and 
this substance is easily liquefied at ordinary temperatures 
by an atmosphere or two of pressure. That for carbonic 
acid is 3o.9C, and it requires, at ordinary temperatures, 
forty or fifty atmospheres for its condensation. Water vapour 
has to be raised to a temperature of about 41 2 C. before it 
ceases to be condensable by pressure. On the other hand. 



ix.] VAPORISATION AND CONDENSATION. 147 

the critical temperature for Hydrogen, Oxygen, Nitrogen, and 
Carbonic Oxide is so low that even considerable pressure 
requires to be assisted by the most powerful freezing mix- 
tures before the liquefaction of these substances is accom- 
plished. Cailletet and Pictet, separately but simultaneously, 
effected in the end of 1877 this complete verification of 
Andrews' s law. 

175. When a gaseous substance is at a temperature higher 
than its critical temperature, we may call it a true gas ; when 
at a lower temperature, a true vapour. The distinction is 
well shown by the following brief account of Andrews's 
experiments on carbonic acid. 

At 1 3. i C. carbonic acid gas was reduced by a pressure 
of forty-nine atmospheres, as measured by an air gauge, to 
y 1 ! of its original volume under one atmosphere, without 
undergoing any change of state. With a slight increase 
of pressure, liquefaction occurred, and the volume of the 
carbonic acid in the liquid state was T g- J of the original 
volume of the gas. During the process of liquefaction both 
liquid and gas were visible in the tube in which the experi- 
ment was made. After complete liquefaction, the carbonic 
acid continued sensibly to contract as the pressure was 
augmented. At 2i'5 C., similar results were obtained, but 
liquefaction did not occur till a pressure of sixty atmospheres 
was reached. At 3i'iC., which is o'2 above the critical 
temperature of carbonic acid, the gas behaved in the same 
way as at lower temperatures, till a pressure of seventy-four 
atmospheres was attained, when a further increase of 
pressure produced a very rapid, but nowhere abrupt, dimi- 
nution of volume, unaccompanied by any evidence of 
liquefaction or of the presence (at any stage of the process) 
of two different states of matter in the experimental tube. 
After this the carbonic acid, now reduced to 7 |^- of its 

L 2 



148 HEAT. [CHAP. ix. 

original volume, continued slowly to diminish in volume as 
the pressure was increased. At higher temperatures this 
rapid fall became less manifest, and at 48 it could no 
longer be observed. [This question will be more fully dis- 
cussed in Chapter XX. below.] 

176. For the formation of liquid water from ordinary 
vapour a nucleus of some kind must be present, else the 
vapour may remain in a state of saturation, sometimes even 
of supersaturation, without condensing. This has quite 
recently been shown by Aitken, who has traced the formation 
of clouds and fogs to the presence of excessively fine particles 
of dust in the air. These dust particles are found to be 
completely arrested by a plug of cotton-wool, through which 
air is made to pass. Each dust particle secures a share of 
the vapour, so that, when they are very numerous, the share 
of each is small, and a fog or mist is formed ; when they 
are few each gets more, and clouds or even rain-drops are 
produced. The smaller the number of nuclei the larger 
are the individual rain-drops. Aitken concludes from his 
experiments that, if the atmosphere were entirely free from 
such particles no mists or clouds could be formed, no rain 
would fall, and the air would get rid of its superfluous 
moisture only by direct .deposition upon bodies exposed 
to it. 

177. Resume of 159-176. Vaporisation. Dalton's 
Law. Regnault's Determination of the Pressure of Satu- 
rated Water-vapour at different temperatures. Hypsometric 
Thermometer. Total Heat of Steam. Freezing by Evapo- 
ration. Condensation. Dew-point. Hygrometers. Dis- 
tinction between Gas and Vapour. Andrews's Critical 
Point. Cause of Clouds and Fogs. 



CHAPTER X. 

CHANGE OF TEMPERATURE. SPECIFIC HEAT. 

178. Refer again to 45-48. In these sections we 
have spoken of the gradual rise of temperature of water 
substance, as heat is steadily applied to it, in the solid 
liquid, and gaseous forms, and the question immediately 
suggests itself : What amount of heat is required for each 
degree of rise of temperature by the substance in each of 
these three states ? In other words, does water substance 
become easier to heat or harder to heat by being changed 
from one of the molecular states to another? Again, if a 
pound of water and a pound of mercury, at the same tem- 
perature, have equal quantities of heat communicated to 
them, will the heating effect be the same for both ; or, if 
not, in what proportion will be their rise of temperature ? 

179. Experiment has answered these and other analogous 
questions, in a form which shows that bodies differ in a very 
marked manner as to this effect of heat ; and we therefore 
define, as a property of each particular substance (under 
assigned conditions) what is called its Specific Heat, as 
follows : 

The specific heat of a substance, under any specified con- 
ditions, is the number of units of heat required, under these 



150 HEAT. [CHAP. 

conditions, to raise the temperature of one pound of the substance 
by i C. 

We have already defined a unit of heat as the amount of 
heat required to raise the temperature of a pound of water 
by i C. ; it is therefore clear that, by our definition, the 
specific heat of water is i. It will be seen later ( 184) that 
the specific heat of water varies so slowly with temperature 
that it is practically the same at o C. and at 10 C., i.e. at 
32 F. and at 50 F. (see 55). The term Specific Heat was 
originally devised by the Calorists, but it is still used in 
science, like Latent Heat, Centrifugal Force, etc., etc., as it 
is convenient and quite harmless. 

1 80. There are many experimental processes for deter- 
mining Specific Heat, but we cannot spare space for more 
than two or three so far as solids and liquids, under 
ordinary pressures, are concerned. We commence with 
the most readily intelligible of them ; a process, in fact, 
long ago devised by Black. 

It will be found, when we are dealing with Radiation, 
that the quantity of heat radiated from the surface of a hot 
body, in a given time, depends (so far as the body itself is 
concerned) solely upon the temperature and the nature of 
the surface. Hence, if we take a thin shell of metal and fill 
it successively with different liquids, each at a temperature 
higher (or lower) than that of the surrounding bodies, we 
know that its rate of loss (or gain) of heat by radiation will 
be the same at any one temperature whatever be the sub- 
stance which fills it. Suppose, for simplicity, that the shell 
holds just one pound of hot water. Then, if the bulb of a 
thermometer be plunged in the water we know that, for 
every degiee the thermometer falls, one unit of heat has 
been removed from the water by radiation and convection. 
Let us note the number of seconds which elapse for each 



x.J CHANGE OF TEMPERATURE. '151 

degree of temperature as the thermometer falls, and we 
may then construct a table of the number of seconds 
required for the loss of one unit of heat by radiation from 
the shell at each successive degree of temperature. 

Now fill the shell with another hot liquid, mercury sup- 
pose. (To be thus employed the shell must be made of 
iion, for chemical reasons.) Note again the time required 
for each degree the thermometer sinks. It will be found to 
fall faster than when the contents were water ; although, in 
consequence of the great specific gravity of mercury, there 
are 13-6 pounds of that liquid in the shell. In fact, the 
time of cooling through any given range of temperature is 
now less than half of what it was when the shell was filled 
with water. Hence we conclude that the specific heat of 
water is more than 27-2 times that of mercury, because the 
same change of temperature is produced in one pound of 
water and in 13*6 pounds of mercury, while the water loses 
more than twice as much heat as the mercury. The correct 
specific heat of mercury is found in this way to be about 



1 8 1. .The process we have just described is liable to 
several objections. The most serious of these are (i) that 
the contents of the shell cannot, unless constantly stirred, 
have the same temperature throughout ; (2) that, unless the 
neck of the shell be closed, there is serious loss of heat by 
evaporation a loss which is of greatly different amounts 
with different liquids at the same temperature. 

But for the beginner in experimental science, the process 
is very instructive ; especially if he employs a graphic method 
in deducing the results. We may take this opportunity of 
giving a general notion of this most important process. 
Although from its very nature it cannot pretend to any 
great accuracy, it is always free from large error ; and is 



152 HEAT. [CHAP. 

therefore an almost indispensable auxiliary to numerical 
calculation, in which the most expert calculators may make 
serious mistakes. 

[Thus, suppose the results of the observations of tempe- 
rature in terms of the time, as furnished by the readings of 




the thermometer after the lapse of successive intervals, equal 
or not, be plotted (on mechanically ruled paper) as above ; 
intervals of time being measured horizontally, corresponding 
temperatures vertically. The figure represents, on a much 



x.] CHANGE OF TEMPERATURE. '153 

reduced scale, the original plotting of the data of part of an 
actual experiment ; the cooling of the short bar in Forbes's 
method for measuring thermal conductivity (Chapter XIV. 
below). The horizontal line is divided into minutes of 
time, the vertical line (on its left side) into degrees centi- 
grade. Suppose a smooth curve to be drawn libera manu 
(the full curve in the figure), so as to agree as nearly as 
possible with Ihe observed points. This curve will repre- 
sent an approximation (the more close the shorter are the 
intervals between the observations) to that which would have 
been obtained by the continuous photographic process of 
136. Then if rate of cooling at any temperature, say 
1 86, be required, all that has to be done is to draw a hori- 
zontal line through 186 on the scale of temperatures, and 
where this line meets the curve draw a tangent, producing it 
both ways to meet the two divided lines. As drawn, it cuts 
off 73 on the temperature scale, and i7 m '6 on the time scale. 
Hence the rate of cooling at 186 is 73 in i7 m> 6, or 4'i5 
per minute. 

Even the best mercury thermometers, when the column 
is rising or falling otherwise than very slowly, are found 
to move occasionally by sudden starts, the surface film 
sticking for a moment in the tube and thus subjecting 
the column to pressure or tension. It is possible also that 
the bulb recovers by starts from its state of dilatation. 
Hence for the purpose of finding rates of cooling, as in 
the case above, it is sometimes better to make the observa- 
tions of temperature at successive equal intervals taken as 
the unit of time, and to plot the differences of the successive 
readings midway between the beginning and end of the 
interval in question. In the figure the corresponding points 
are put in and connected by dotted lines. ' The time scale 
is the same as before, but the degrees on the temperature 



154 HEAT. [CHAP. 

scale are ten times as long as before. Its numbers are 
inserted on the right side of the temperature scale. We 
now find it impossible to draw a smooth curve through the 
various points, but if we draw a smooth curve (the dotted 
one in the figure) which (perhaps not passing through any 
of them) shall show on the whole as much divergence of 
observed points from itself on one side as on the other, the 
ordinates of this curve will be themselves the rates of cooling 
required, and no drawing of tangents will be necessary. In 
the figure, the ordinate of this dotted curve, corresponding 
to 1 86, is 4'i2, thus agreeing within one per cent, with the 
result of the former method. Experience alone can guide 
us, in any case, as to which of these methods is to be 
preferred.] 

182. Another method of determining specific heat is 
supplied by the ice-calorimeter, in which the amount of 
heat lost by a hot body in cooling to o C. is measured by the 
amount of ice which it melts. The general principle on 
which the calorimeter is constructed will be obvious from 
the figure. There are virtually three vessels, one within 
another, the two outermost containing ice and water at 
ordinary atmospheric pressure, and therefore always at o C. 
The middle vessel can part with no heat to the outer one 
unless its temperature changes; but this cannot be until 
all its contained ice is melted ( 142). We have thus a 
very simple and fairly effective method of preventing loss 
of heat. Into the interior vessel the substance whose 
specific heat is to be measured is dropped, at a known 
temperature, and the whole is closed up. After a short 
time the whole contents of the vessel are again at o C., but 
the heat disengaged from the substance has melted some of the 
ice in the middle vessel. In the older and rougher forms of 
the instrument, as designed by Lavoisier and Laplace (from 



X.] 



CHANGE OF TEMPERATURE. 



155 



one of which the sketch is taken), the amount of ice melted 
was given by the amount of water which drained away from 
the middle vessel. Of course there was considerable error, 
as some of the water was necessarily retained by capillary 




forces. In the improved form, as designed by Bunsen and 
others, the middle vessel is filled with solid ice to begin 
with, and the amount melted is calculated from the dimi- 
nution of the volume ( 145) of the contents, as shown by an 
external gauge containing mercury in contact with the ice. 



156 HEAT. [CHAP. 

The result of each experiment shows (151) the number 
of units of heat lost by a known mass of a substance in 
passing from a known temperature to o C. By dividing 
the number of units of heat lost by the number of pounds 
of the substance and by the number of degrees of change 
of temperature, we obtain the Average Specific If eat of the 
substance through the range of temperature employed. 

This method is very troublesome, and requires great 
care and skill; but it is much more trustworthy than any 
other, when properly conducted. 

183. The only other method we, need describe is the 
common one called the Method of Mixture. Here we 
determine the specific heat of a substance by dropping 
it, when hot, into a liquid (usually water) at a lower tem- 
perature, and observing the temperature which the mixture 
acquires. For rough determinations this process is sufficient, 
and it has the additional advantage of being very easy. 
But the corrections, which are absolutely necessary if we 
wish to obtain exact results, are exceedingly troublesome 
and require great experimental skill. These corrections are 
required chiefly on account of the loss of heat by radiation 
and evaporation, which do not affect the method last 
described. 

Suppose M pounds of one substance, at temperature 
T C., to be mixed with ;;/ pounds of another substance at 
a lower temperature, t C. ; and suppose the mixture (all 
corrections made) to have the temperature T C., which must 
obviously be intermediate between T and /. Then the 
heat lost by the one substance has gone to the other. If the 
specific heats of the two substances be *Sand s respectively, 
the loss of the first (in units of heat) is 

M S(T T). 



x.] CHANGE OF TEMPERATURE. '157 

The gain of the other is (in the same units) 

/// y (T /) 
Equating these quantities, we find at once 

S m (r /) 
s -~M(TT)' 

which gives the ratio of the specific heats ; or, if the second 
substance be water, the specific heat of the first substance. 
[It is to be observed that ^ is the average specific heat of 
J/from Tto T, s that of m from r to /.] 

It is well to observe that the temperature of the mixture, 
if no heat be lost by radiation or evaporation, is 

MS T + vi sf 



The quantities MS, MS, which appear in this expression, 
are called Water-equivalents of the substances. The water. 
equivalent of any mass is thus the number of pounds of 
water which would be altered in temperature to the same 
extent, by the same number of units of heat, as the given 
mass. 

[And we see that the temperature of a mixture is given 
in terms of the water-equivalents of the several substances 
and their several temperatures, by a formula exactly the 
same as that which gives the position of the centre of 
inertia of a number of particles in one line, in terms of 
their separate masses and positions.] 

Another term which is sometimes of use, is the thermal 
capacity of a substance. This may be regarded as the 
water-equivalent of unit volume of the substance, and its 
numerical value is obviously the product of the density by 
the specific heat of the substance. (See 249). 



153 HEAT. [CHAP. 

184. We will now give some general notions as to relative 
specific heats of solids and liquids. But before doing so, 
it is well to remark that, in general, specific heat rises with 
temperature ; and that (as above stated) our experimental 
determinations give directly the average specific heat of a 
substance through a certain range of temperature. From 
the results of such determinations, however, if they be 
sufficiently numerous and varied in their circumstances, it 
is not difficult to deduce the true specific heat as a function 
of the temperature. 

According to Regnault, a pound of water at o C. requires 
ioo'5 units of heat to raise it to iooC., and 203*2 units to 
raise it to 200 C. 

These are represented by the formula 

H / + 0*000,02 /'-' + 0*000,000,3 ^ 3 5 

which gives, for the specific heat of water at any temperature, 
/ C., the expression 

i + o'ooo,o4 t + o'ooo,ooo,9 / 2 . 

Thus the change for any ordinary ranges of temperature is 
extremely small. Refer again to 61. 

The specific heat of ice at oC. is almost exactly half 
that of water. Regnault has shown that it diminishes as 
the temperature is lowered. 

That of glass may be roughly assumed as about 0*2. 
That of platinum is about 0-0355, and varies very slightly 
even for large ranges of temperature. With other metals, 
such as copper, the increase of specific heat is (roughly) 
about io/. c. for iooC. It appears to be rather more in 
iron. But we have no very exact knowledge on this 
subject, as different specimens seem to give very different 
results. 



X.] CHANGE OF TEMPERATURE. i 59 



SPECIFIC HEAT OF ELEMENTARY SOLIDS. 

Lead 0x31 . . . 207 . . . 6'4 

Tin 0-056 . . . 118 ... 6*6 

Copper .... 0-095 . . . 63-5 ... 6-0 

Iron 0-114 ... 5 6 . . 6-4 

Sodium .... 0*293 . 23 ... 67 

Lithium .... 0-941 ... 7 . . . 6'6 

The first column of the table gives the specific* heat at 
ordinary temperatures, the second the atomic weight of the 
substance. The numbers in the third column are the 
products of those in the first and second. It will be seen 
that the numbers in this third column are within about 10 per 
cent, of one another. Hence it is concluded that the specific- 
heat of an elementary solid is inversely as its atomic weight. 
The small differences in the last column are attributed (in 
part at least) to the known fact that the specific heat of 
each substance increases as the temperature rises, and 
therefore that, as these specific heats are measured all at 
the same temperature, the different bodies compared are 
taken in different physical states, some being much more 
nearly at their melting-point than others. 

It is probable that very important information as to the 
nature of matter will be obtained from this experimental 
fact. Perhaps its value may be more easily seen if (by 
183) we put it in the form : 

The atomic water-equivalent is nearly the same for all 
elementary solids. 

A similar law has been found to hold for groups of 
compound bodies of similar atomic composition. But the 
product of the specific heat and the atomic weight varies in 
general from group to group of such compounds. 

185. In general, as we have seen, the specific heat of a 



160 HEAT. [CHAP. 

substance in the liquid state is greater than in the solid 
state. 

SPECIFIC HEATS OF ELEMENTARY LIQUIDS. 

Lead 0*040 400 C. 

Tin 0-064 3 C. 

Mercury 0*033 3C. 

The second column gives the mean temperature of the 
range through which the specific heat is measured. 

The specific heat of Alcohol, which it is important to 
keep in mind, is (at 30 C.) a little above 0*6. 

1 86. In the case of gases the problem of determination 
of specific heat is not only more difficult, from the experi- 
mental point of view, than in the case of a solid or of a 
liquid, but it is also more complex from a theoretical point 
of view. For, in consequence of the great expansibility of 
gases, we have to specify the conditions under which the 
measurement is to be made. Thus the gas may be main- 
tained at the same volume, or at the same pressure, through- 
out the range of temperature employed. And we therefore 
speak of the Specific Heat at constant volume, or of the 
Specific Heat at constant pressure. Thermo-dynamics gives 
us a simple relation between these quantities for the ideal 
perfect gas ( 126); and therefore for gases such as Air, 
Hydrogen, &c., it is only necessary to determine directly 
one of the two specific heats. This is a very happy cir- 
cumstance, because the experimental difficulties of deter- 
mining the specific heat at constant volume are extremely 
great. The mass of a gas at ordinary pressures is small 
compared with that of the containing vessel, unless the 
vessel be of unwieldy dimensions ; and we cannot get over 
this difficulty by compressing the gas, for we must then 
make the vessel strong (and therefore massive) in 
proportion. 



x.] CHANGE OF TEMPERATURE. 161 

The measurement of the specific heat at constant pressure 
is effected by passing the gas, at a uniform rate, through 
two spiral tubes in succession. In the first of these it is 
heated to a known temperature, and in the second it gives 
up its heat to a mass of water in a calorimeter in which 
that spiral is immersed. From the volume of the gas 
which has passed through the spirals, we can calculate 
its mass, and the observed rise of temperature in the 
water of the calorimeter supplies the requisite additional 
datum. 

This process was devised by De la Roche and Berard, 
but its details were greatly improved by Regnault. 

Further remarks on this subject, especially from the 
theoretical point of view, must be deferred for the 
present. 

187. Regnault' s experiments showed that, for gases like 
air, tne specific heat of a given mass of the gas is inde- 
pendent of the temperature and pressure, and therefore 
( 124) of the volume. 

Hence the specific heat of a given volume of such a gas 
varies directly as the density. 

Equal volumes of different gases of this class, at the 
same pressure, have approximately equal specific heats. 

SPECIFIC HEATS AT CONSTANT PRESSURE. 

Air ' 2 37 Oxygen .... 0*217 

Nitrogen . . . 0-244 Hydrogen .... 3'4CQ 

It will be seen that these numbers are very nearly in the 
inverse ratio of the densities of the various gases (see 
184). 

Experiments on sound ( 397) have shown that the ratio 
of the two specific heats of air is about 1-408, whence we 
see that the specific heat of air at constant volume is o'i68. 

II 



1 62 HEAT. [CHAP. x. 

The specific heat of water-vapour under the same con- 
dition is 0-48, a little less than that of ice ( 184). Hence 
water-substance has twice as great a specific heat in the 
liquid state as in the solid or in the vaporous state. 

iSS. Resume of 178-187. Change of temperature cf 
unit mass by a given quantity of heat. Specific Heal. 
Method of Cooling. [Digression on Graphic Methods.] 
Ice Calorimeter. Method of Mixtures. Water Equiva- 
lent. Thermal Capacity. Relation between Specific Heat 
and Atomic Weight. Specific Heat of Gases : (a) at 
constant volume, (b) at constant pressure. Specific Heat of 
Water- vapour. 



CHAPTER XI. 

THERMO-ELECTRICITY. 

189. We have already stated, in 49, the fundamental 
fact of Thermo-electricity as discovered by Seebeck ; and 
we now propose to examine the subject more closely from 
the experimental point of view. For this purpose we must 
use the term Electro-motive Force. So far as we* require its 
properties they may be enunciated as follows : 

The strength of a current in a given circuit is directly 
proportional to the Electro-motive Force, and inversely 
proportional to the resistance. The energy of the current 
is the product of the Electro-motive Force (contracted as 
E.M.F.} and the strength of the current. 

It will be advantageous to commence with the following 
statement of an experimental result : 

Jf e be the E.M.F. in a thermo-electric circuit when 
/ and /, are the temperatures of the junctions ; and e tJie 
E.M.F. when the temperatures are / x and / 2 / then when 
the temperatures are t and / 2 the E.M.F. is e + e'. 

[Consider this from the point of view given by the fol- 
lowing figure ; where the circuit is supposed to consist of 
Jour wires, alternately of two metals (say iron and copper). 
See. again, 49, where it is stated that the wires may be 

M 2 



164 



HEAT. 



[CHAP. 



soldered together at the junctions. No third substance, such 
as metal or solder, introduced into the solid circuit, is found 
to affect the result if it have the same temperature at the 
points where it meets each of the two metals. 

Let the temperatures of the four junctions be as in the 
figure. Then D and A together (if B and C had the same 
temperature with one another) would give E.M.F. e, 
and B and C together (if D and A had one common 
temperature) would give E.M.F. e. Hence as the figure 
is drawn, there would be total E.M.F. e + e', provided 
the assumptions made (in brackets) above were superfluous. 



Iron 




Iron 



But the E.M.F. may be looked upon as due entirely to 
the difference of temperatures of C and D, for the tem- 
peratures of A and B are equal, and therefore the copper 
wire A B produces no effect. Hence the contribution of 
each metal to the whole E.M.F. of such a circuit depends 
only on the temperatures of its ends, i.e. is independent of 
the temperature of the rest of the circuit.] 

190. It follows from the experimental fact above, that 
we may break up any interval of temperature, t to / ? ., 
into ranges : viz. t to , ^ to 4 &c., / M . x to t n ; and the 
E.M.F. in a simple circuit of two metals, when the junctions 



XI.] 



THERMO-ELECTRICITY 



'165 



are at t and t n will be the sum of its separate values when 
the junctions are successively at /, and /,, t l and / 2 , &c, and 
/., and 4. 

We shall now look upon these as successive equal ranges 
of temperature, and we shall define the thermo-electric power 
of a circuit of two metals, at mean temperature /, as the 
E.M.F. which is produced when one junction is kept half 
a degree above, the other half a degree below, /. 

We thus get the means of representing in a diagram the 
relative thermo-electric positions of any two metals at 
different temperatures. Thus we may erect little rectangles 
one degree in breadth, as. below, upon a line representing 
one of the metals. 




The area of each rectangle is the thermo-electric power 
corresponding to the temperature at the middle of its base ; 
and, by taking the intervals of temperature small enough, 
it is obvious that the final upper boundary of the group of 
rectangles will become a continuous line. This will repre- 
sent at every point the relative thermo-electric situation 
of the second metal with regard to the first, as the first is 
represented by the line on which temperatures are measured. 
And the sum of any number of the rectangles (giving the 
E.M.F. for any t\vo assigned temperatures of the June- 



1 66 HEAT. [CHAP. 

tions) is now represented by the area of the corresponding 
portion of the curve. 

1 9 1. But there is a much more general experimental 
result than that in 189. At every temperature the algebraic 
sum of the thermo-electric powers of metals a and /3, and 
p and y, is the thermo-electric power of a and y. 

[Suppose one of the iron wires in the diagram of 189 to be 
replaced by a third meial, say gold; and let the temperatures 
of the ends of one of the copper wires be t n and of the other 
/ T . Then A and D (if B and C had the same temperature) 



Iron, 

'Cold 



would give E. J/. F. depending on iron-copper alone at the 
temperatures /<,, /; ; B and C (if A and D had the same 
temperature), would give JE. M. F. due to a copper-gold 
circuit with its junctions at the same two temperatures. It 
is easy to see, as before, that the experimental result shows 
the assumptions (in brackets) to be superfluous. For, as 
the ends of each copper wire are at the same temperature, 
the circuit acts as one of iron-gold alone, its junctions also 
having the temperatures t , t^\ 

Thus if, by the process of 190, we form the lines repre- 
senting the positions of iron and of gold, each with regard 
to copper taken as the standard, the lines so drawn will 
show the position of iron and gold relatively to one 




XI.] THERMO-ELECTRICITY. 167 

another. Hence the possibility of constructing a. Thermo- 
electric Diagram containing a single line for each metal. 
The idea of such a representation was suggested by 
W. Thomson in 1855, and he gave a rough preliminary 
sketch of it. A "first approximation" to an accurate 
diagram was given by Tait in 1873. This is repro- 
duced on a small scale in 198, but it cannot be fully 
explained until some additional experimental facts are 
stated. It will then be found that a properly drawn thermo- 
electric diagram embodies all that is known about the 
subject of thermo-electricity. And thus, by the use of 
the diagram, we are now able to present the subject in 
a much more simple and connected manner than was 
formerly possible. 

192. The electric current in a thermo-electric circuit 
represents a certain (usually very small) amount of energy, 
whose measure as before stated is the product of the 
E.M.F. by the strength of the current. This can have 
no other source than the heat which has been given (as in 
49) to a part of the circuit. And the existence of the 
current implies a loss of heat by the circuit as a whole. 

It is a very remarkable fact in the history of science 
that, without any reference to the theory of energy, Peltier 
(1834) discovered by experiment that 

When a current of electricity from an external source 
passes through a junction of two metals, it causes an ab- 
sorption or a disengagement of heat. 

If the direction of the current be the same as that of the 
current which would be produced by heating the junction, the 
effect is absorption ; and vice versa. 

This is very easily proved by the help of a galvanometer. 
Two wires (say iron and copper) are brazed together at 
their middle points only ; one of the free ends of the iron 



168 HEAT. [CHAP. 

is maintained in connection with one pole of a single 
voltaic cell, the other end is in connection with one of the 
ends of a galvanometer coil. The copper wire is so 
arranged that by rocking it over to one side we close the 
circuit of the cell, leaving the galvanometer circuit open ; 
and by rocking to the other side we break the battery 
circuit and close the circuit of the galvanometer ; the iron- 
copper junction being thus alternately in one circuit and 
in the other. There is always a deflection of the galvano- 
meter after the voltaic current has traversed the junction. 
And it changes sign when the direction of that current 
is reversed. 

The wires used for this experiment should be stout, else 
the heat generated in the circuit by ordinary resistance to 
the battery current may be greater than the Peltier effect 
which is sought. 

When proper precautions of this kind are taken, a notable 
amount of heat may be produced or absorbed. Lenz, in 
1838, succeeded in freezing a little water by passing a 
current of electricity in the proper direction through a 
bismuth-antimony junction, the metals themselves being 
surrounded by melting snow. 

The direct quantitative measurement of the Peltier effect, 
at different temperatures, presents very great experimental 
difficulties. These have been only partially overcome as 
yet, mainly by the experiments of Naccari and Bellati. 
The results agree, as well as could be expected, with those 
stated below. There can be no doubt, however, that for 
any one pair of metals, kept at a given temperature, the 
Peltier effect is directly proportional to the strength of the 
current employed. 

193. It was Joule who first remarked that the Peltier 
phenomenon furnishes a clue to the source of the energy 



XL] THERMO-ELECTRICITY. '169 

of a thermo-electric current ; and it is quite possible that 
there may be pairs of metals (more probably alloys) for 
whose circuits it is the only source of the current. [This 
will be seen in 199 below.] 

But a phenomenon, noticed by dimming very soon after 
the publication of Seebeck's discovery, shows that the 
Peltier effect is in general a part only of the source of the 
current. This discovery of Cu mining's may be stated as 
follows : 

/// certain circuits, such as those of iron-copper, iron-silver, 
iron-gold, etc., if one junction be kept at ordinary tempera- 
tures, and the temperature of the other be steadily raised, the 
E.M.F. increases more and more slowly till it reaches a 
maximum, then gradually diminishes, and finally is 
reversed. 

Sir W. Thomson explained this effect as follows : At that 
temperature of the hot junction for which the E.M.F. is a 
maximum (in Cumming's experiment), the two metals are 
neutral to one another, i.e. their thermo-electric power 
vanishes, and the Peltier effect also. This occurs, as the 
figure in 190 shows, when the lines representing them 
intersect; for, after the intersection, the rectangles (from 
which the figure was composed) are turned the opposite 
way, and must therefore have the opposite algebraic sign. 

He proceeds to reason thus : Suppose a copper-iron 
circuit, in which the hot junction is at the neutral tem- 
perature, and the other at any lower temperature: we 
cannot suppose the energy of the current to come from the 
heat of the hot junction : for, as the metals are there 
neutral to one another, the Peltier effect must be nil. 
Also the cold junction is heated, not cooled, by the current. 
Hence the energy can only come from one or other, or 
both, of the wires themselves, and it must come from them 



i;o HEAT. [CHAP. 

in virtue of the differences of temperatures of their ends. 
The assumption that the Peltier effect vanishes at the 
neutral temperature requires experimental proof, which has 
as yet been only partially furnished. We merely mention 
this to show that the experimental basis of the reasoning is 
not quite complete. Campbell (Proc. JZ.S.E. 1882) has 
to a certain extent supplied a qualitative verification, by 
showing that the ratio of the Peltier effects at two tempera- 
tures is consistent with that deduced from the thermo- 
electric diagram. It is probable that by his method the 
vanishing of the Peltier effect at the neutral point may be 
experimentally established. 

Thus Thomson was led to the conclusion that : 

An electric current in an unequally heated conductor, of 
one at least of two metals which have a neutral point, must 
produce absorption, or disengagement, of heat, according as it 
passes from hot parts to cold, or vice versa. 

After a series of elaborate experiments (described in the 
Phil. Trans, for 1855) Thomson found that: 

When the current passes from cold to hot in copper there 
is absorption of heat, and vice versa. In iron the effects are 
the opposite. 

This "Thomson effect" is sometimes called Electric 
Convection of Heat. As will be seen later ( 199), the 
above statement, so far as iron is concerned, requires 
modification at very high temperatures. 

194. Thus, Thomson speaks of the specific heat of vitre- 
ous electricity in a metal ; and regards it as positive in 
copper and negative in iron, in consequence of the differ- 
ence of behaviour just explained. Thomson gave, along 
with these remarkable results, the thermo-dynamical theory 
of the thermo-electric circuit in terms of Peltier effect and 
electric convection, but without assigning a special expres- 



xi.] THERMO-ELECTRICITY. 171 

sion for the amount of convection or of the Peltier effect 
in terms of temperature. In fact, he says that the lines 
representing various metals, in the thermo-electric diagram, 
" will generally be curves." But he showed experimentally 
that the thermo-electric current, in a circuit of two metals, 
vanishes not only when the junctions have the same tem- 
perature (as in 190), but also when their temperatures 
are equidistant from that of their neutral point. When one 
of the metals is represented by a straight line (as in the 
diagram of 190) the other is therefore represented by a 
curve which crosses it at the neutral point, and has the same 
form and dimensions on opposite sides of that point. 

[The rest of the chapter is, for the most part, taken from 
the Rede Lecture for 1873 * ] 

It has been found, by measurements of the E. M. F. of 
numerous pairs of metals through the whole range of mer- 
cury thermometers, that if the thermo-electric position of 
any one metal be represented by a straight line on the 
diagram, the lines for other metals are (in general) also 
straight. This involves no assumption other than that just 
named. So that whatever may ultimately be found to be 
the rigorous expression for electric convection of heat in 
terms of temperature, and therefore the true form of the 
thermo-electric diagram, we may assert that if the diagram 
be so distorted (by shearing) that the line of any one metal 
(except iron, &c., 198, 199) becomes straight, those of 
the great majority of other metals will also become straight. 
From this it follows that the rate at which the thermo-electric 
power of two metals changes with temperature is constant. 

195. Thomson's theoretical expressions, just alluded to, 
are based on the supposition that there is no reversible 

* Nature, vol. viii. pp. 86 and 122. 



172 HEAT. LCHAP. 

thermal effect in the circuit but the Peltier and Thomson 
effects. If this be so, it follows from the theory of heat and 
the result just given (as will be shown in Chapter XXI.), 
that the Peltier effect is proportional to the thermo-electric 
power and to the absolute temperature. Also the (alge- 
braic) difference of the specific heats of electricity in the 
two metals is proportional to the absolute temperature. 

Hence, if there be a metal which has no specific heat of 
electricity at any temperature, it follows by theory from the 
above experimental result ( 194) that : 

The specific heat of electricity in a metal is in general 
directly proportional to the absolute temperature. 

Le Roux, in 1867, made a valuable series of direct 
measurements of the electric convection of heat, and 
showed that it is null, or at least exceedingly small, in 
lead. 

196. We are now in a position to explain the indications 
of the thermo-electric diagram, and to use it for the calcu- 
lation of the Peltier and Thomson effects, showing fully 
whence the energy of the thermo-electric current is derived. 

Let the cut represent a part of the diagram, two metals 
(copper and iron, suppose) being represented by the straight 
lines CC' and //'. The vertical axis O E corresponds to 
the absolute zero of temperature. Draw lines parallel to 
O E, corresponding to / and /, the (absolute) temperatures 
of the two junctions, and cutting the lines of the metals in 
C, /, and 6", 1', respectively. Also (for simplicity) suppose 
both of these temperatures to be lower than that of the 
neutral point N. Then the E.M.F. of the copper-iron 
circuit is ( 190) represented by the area C C I' /; and, as 
the figure is drawn, it produces a current going round the 
circuit in the positive direction as shown by the arrows, 
passing from copper to iron at the hotter junction. [The 



XL] 



THERMO-ELECTRICITY. 



reason why the line of copper has been drawn so as to rise 
towards the right, is the fact of its specific heat of electricity 
being positive ( 194).] This area C C' /' /will represent 
the amount of energy of the current, if it be made unit by 
properly adjusting the resistance of the circuit. 

Now (by 195) the areaZ>' B' C 1 I' represents the Peltier 
effect (absorption of heat) at the hot junction if unit current 
were passing through it ; and DICE the Peltier effect 
(development of heat) at the cold junction under the same 




condition. And therefore the area B! B C C' will represent 
under the same condition the Thomson effect in the copper, 
D Lf /' / that in the iron. Both of these represent absorp- 
tion, for the current passes from cold to hot in copper, and 
from hot to cold in iron. It will be easily seen (by Euclid) 
that G C is the specific heat of electricity in copper at t C. 
(positive) while K I (negative) is that in iron. 

For the whole area B C C' I' I D may be taken to re- 
present energy taken in, in the form of heat ; while the part 



174 



HEAT. 



[CHAP. 



D I C B represents energy given out in that form. The 
difference, C C' 1' f, remains for the production of the 
thermo-electric current. Thus the first law of thermo- 
dynamics is satisfied. In order to make these representa- 
tions exact, instead of merely proportional, each area must 
be multiplied by the strength of the current actually passing 
in the circuit. 

To show that the second law applies, the algebraic sum 
of all the portions of heat taken in and given out, each 
divided by the absolute temperature at which the operation 
takes place, must be proved to vanish. But we have 
obviously B' D' BD + BB' + D' D = o, which proves 
the proposition. 




[It will be good practice for the student to work out the 
corresponding diagrams, when one or both junctions are 



xi] THERMO-ELECTRICITY. '175 

above the neutral point ; and when both metals have 
positive specific heats of electricity. Also to treat the case 
of a circuit of three metals in which the three junctions are 
at different temperatures. We insert a sketch for this case. 
Here A and B have their specific heat of electricity posi- 
tive, C negative. The junction B, C, is at / C, A, at / 2 , 
and A, B, at /" 3 ; and the area bounded by the doubled . 
line (partly positive, partly negative) represents the^.J/!/!] 

197. Let us now consider how the strength of the current 
ought to vary with the temperatures of the junctions if these 
statements be correct. [First figure in last section]. 

In other words, if C/be fixed, how does the area CCI'I 
vary as the temperature /' increases? 

Without introducing symbols we may easily answer the 
question thus : Let Of represent velocity, and /' time. 
Then the area CC'J'I represents the space described in the 
time t't. But evidently the velocity diminishes uniformly 
as the time increases ; i.e. it is a case of uniform acceleration 
vertically downwards, associated with uniform velocity to 
the right ; exactly the case of an unresisted projectile. 
Hence : When one junction of a thermo-electric circuit is 
kept at a constant temperature, and the temperature of 
the other is gradually raised, the curve representing the 
strength of the current in terms of the variable temperature 
is a parabola, whose axis is parallel to that of E.M.F., and 
the abscissa of whose vertex is the temperature of the 
neutral point. 

This has been observed to be very accurately the case 
with the majority of thermo-electric pairs ; but it is not the 
case for all ranges of temperature Avhen iron, or nickel, is 
one of the two metals. 

[An erroneous explanation of thermo-electric currents, 
probably suggested by this experimental fact, but based 



i;6 HEAT. [CHAP. 

upon the assumption that the source of the E.M.F. of a 
circuit of two metals lies entirely in the junctions, was 
given by Akin in I863, 1 and shortly afterwards by 
Avenarius. 2 As this explanation has already found its way 
into some text -books, it is necessary to direct the reader's 
attention to the fact that it cannot possibly be correct. 
Among other defects it ignores entirely the existence of 
the Thomson effect, which has been thoroughly established 
as a vera causa by independent experimenters.] 

198. The thermo-electric diagram opposite (from Trans. 
R. S. ., 1873) contains the lines of a number of common 
metals for a range of temperatures such as can be measured 
by mercury thermometers. Bismuth and antimony cannot 
be shown in it. The line of the latter lies high above its 
limits, that of the former considerably below them. 

In consequence of the result obtained by Le Roux 
( 195) the line for lead has been adopted as the axis of 
abscissae along which temperatures are measured. 

An auxiliary circuit of two alloys of iridium and plati- 
num was employed in constructing the diagram. The 
specific heat of electricity is exceedingly small in each ; but 
it must exist in one of them at least, for they have a neutral 
point somewhere about a low white-heat. But the explana- 
tions which we have just given show that for an extended 
range of temperature the E.M.F. of a circuit of these 
alloys must vary very nearly as the difference of absolute 
temperatures of the junctions. Such a combination forms 
an extremely convenient substitute for a thermometer, when 
great accuracy is not required ; and it was indispensable for 
the investigation of the peculiar form of the iron line at 
high temperatures, which will presently be described. 

1 Proc. R. S., vol. xii., and Camb. Phil. Trans. 
* Pogg. Ann., 119. 



XL] THERMO-ELECTRICITY. 177 



100 Pl> 200 3pQ ^X. 40QC 




178 HEAT. [CHAP. 

The only line in the diagram which calls for special 
remark is that of Nickel. Its position and form are con- 
veniently studied (as, indeed, is obvious from the diagram), 
by making a nickel-palladium circuit. When this is done it 
is found that in nickel the specific heat of electricity, nega- 
tive at ordinary temperatures, changes sign somewhere about 
200 C. and again about 320 C. 

199. Iron shows, but at much higher temperatures, the 
same peculiarity as nickel. Its line in the diagram intersects 
that of N (one of the iridium-platinum alloys above men- 
tioned) at least three times below a low white-heat. In other 
words, the iron- A 7 " circuit has at least three neutral points. 

If we raise the junctions of such a circuit to the tem- 
peratures corresponding to two of these points, the current 
produced is due entirely to the Thomson effect, i.e. to the 
excess of the absorption of heat in one part of the iron, 
over the development of heat in the rest ; for N (as we 
have seen) has very little, if any, Thomson effect. 

On the other hand, in the circuit of the two alloys, what- 
ever the temperatures of its junctions, the current is due 
almost entirely to the Peltier effect. 

It is probable that the above peculiarities of nickel and 
iron are connected with the changes in their magnetic pro- 
perties, which take p!ace, as shown by Faraday, at about 
the same temperatures. If there be peculiarities of the 
same kind in cobalt, Faraday's results would lead us to look 
for them at a still higher temperature than even in iron. 

200. In practice it is very difficult to find the relative 
position of two metals on the diagram for the purpose of 
drawing the line of one (when that of the other is assumed, 
or already determined) unless, either they have a neutral 
point (intersection) within the limits of the diagram, or at 
least lie at no very great distance from one another. For 



XT.] THERMO-ELECTRICITY. 179 

when they are far apart, the changes of E.M.F., however 
large in themselves, are usually small in comparison with 
the whole quantity measured, and may thus to a con- 
siderable extent be lost sight of in the inevitable errors of 
observation. The lines shown in the diagram are very 
irregularly scattered. 

But it is possible, by a very simple process, to procure 
what is virtually a metal whose line shall lie anywhere 
between the lines of two given metals. 

For this purpose we use a double arc of these metals. 
This is made by soldering together at their ends a wire of 
gold (suppose) and a wire of platinum laid side by side 
without contact ; and this combination will (when treated as 
a single wire) have a line which passes through the neutral 
point of gold and platinum, and lies between them in 
thermo-electric position. Its actual position between them 
is determined by the relative electric resistances of the 
separate wires of the double arc. It would take us too 
much into pure electric science to give the calculation : 
so the following statement (from the Camb. Math. Tripos 
Exam. 1875), which contains all that is necessary for the 
experimental application of the process, must suffice. 

Let wires of three metals A, B, C y having resistances 
a, b, c, have their ends soldered together at two junctions 
which are maintained at (different) constant temperatures. 
Then if I a be the current when A is cut, T b the current 
when B is cut, the current in C when all the wires are 
continuous will be 

a (b + c) I a + b(c+a] I b 
bc+ca + ab 

In practice the resistance c includes that of the galvano- 
meter, which is usually large compared with either a or b; 

N 2 



i8o 



HEAT. 



[CHAP. 



and then the above formula takes the sufficiently approxi- 
mate and exceedingly simple form . 



a + b 

By the use of this artifice we can fill up all the gaps in 
the diagram. 

201. To conclude this part of the subject, we give a 
brief table of approximate thermo-electric data for some 
common metals. These show how to draw the lines on the 
diagram, between the limits o and 350 C. Different speci- 
mens of the same metal do not agree very closely, so that 
the numbers are given on the same terms as are those in 
104, 153, &c. 



NEUTRAL POINTS WITH LEAD. 



Ni 


(-424) 


Ag 


-144 


Sn 


+ 75 


Sb 


-156 


Au 


( - 276) 


Cu 


-132 


Ru 


+ 136 


Pb 




Arg 


-238 


Zu 


- 95 


Al 


+ 212 


Rd 


+ 132 


Co 


-228 


Cd 


- 59 


Mg 


+ 239 


Ir 


00 


Pd 


-172 


Ft 


- 56 


Fe 


+ 356 


Bi 


(-580) 



\The numbei's in parentheses are not, of course, temperatures. ,] 



SPECIFIC HEAT OF ELECTRICITY. 



Fe. . 
Pt (soft) 
Ir . . 
Mg . 
Arg . 



. - -00247 
. - '00056 

. 'OOOOO 

. - -00048 
. - '00260 



Cd + -00218 

Zn + -00122 

Ag + '00076 

Ru --00106 

Sb + '01140 

Co --00585 



Au + -00052 

Cu + -00048 

Pb -OOOGO 

Sn + -00028 

Pd - -00182 

Ni (to i/5C.) . .--00260 
Ni(25o 3ioC.) .-f -01225 
Ni (from 340 C.) . - -coiSo 

Rd - -00058 

Bi - -0055 

Al . . + '0002 



The quantities in this last table are to be multiplied by the 
absolute temperature C. The unit employed corresponds 



XI.] THERMO-ELECTRICITY. 181 

to about io~ 5 of the E.M.F. of a single Grove's cell. This 
is adopted because it enables the reader to form an 
idea of the extreme feebleness of the thermo-electric cur- 
rents in general. The mode of calculating the E.M.F. of 
a circuit of two metals will be given later, when we discuss 
the thermo-dynamic theory of the circuit. 

202. So far, we have considered the wires employed to 
be each of the same material and section throughout. 
Magnus long ago showed that differences of section in the 
circuit, or in any parts of it, had no effect on the E.M.F. 
[This is not necessarily true when the variation of section is 
so abrupt that there can be finite difference of temperature 
within a range comparable with that of molecular force.] 
But it has been found that hammering, knotting, twisting, 
and stress in general, applied to one part of a circuit of one 
metal only, makes it capable of giving thermo-electric cur- 
rents when irregularly heated. The altered parts of the 
circuit behave to the unaltered parts as if they were of a 
different metal. In iron, nickel, &c., such effects can be 
produced by magnetising part of the wires. Also it has 
been found that if the two ends of a wire are at different 
temperatures, and be brought into contact, a current (of very 
short duration) is produced. 

203. Resume of 189-202. General experimental pro- 
perties of thermo-electric circuits. Consequences. Thermo- 
electric power denned. Peltier Effect. Neutral Point. 
Thomson Effect. Specific heat of Electricity in a metal. 
Thermo-electric diagram. How it represents E.M.F., and 
the Peltier and Thomson effects. Mode of constructing 
the diagram experimentally. Anomalous behaviour of 
Nickel and Iron. Circuits in which there is (a) no Thomson 
effect, (b) no Peltier effect. Null effect of unequal thickness 
of wires. Effects of irregular stress and strain in a wire. 



CHAPTER XII. 

OTHER EFFECTS OF HEAT. 

204. IN this chapter we propose to collect a few odd 
remarks on some of the effects of heat, the discussion of 
which would have unduly interrupted the continuity of the 
more important divisions of the subject already treated ; 
and to take advantage of the opportunity to introduce others 
which cannot as yet be strictly classified. We shall thus 
be able, for instance, to say a few words about some of the 
chemical aspects of the subject, other than the heat of 
combination to which the next chapter is devoted. 

205. A curious example of mechanical motion, due 
directly to expansion, is afforded by what is commonly called 
the Trevelyan Experiment. A piece of heated metal is laid 
on a cold block, usually of lead, so as to touch it at a place 
which has been recently scraped or filed (to remove the 
oxide). If the heated mass be of such a form and so 
placed that it can rock easily on its supports, the rocking 
(once started) becomes very much more rapid, so much so 
as to produce a quasi-musical sound which often continues 
until the two bodies have acquired almost the same tempera- 
ture. The explanation of this phenomenon, given by 
Leslie and confirmed by Faraday, depends merely upon the 



CH. xn] OTHER EFFECTS OF HEAT. '183 

fact that at the point where the hot metal touches it the 
cold metal is so suddenly heated that it swells up, thus 
canting the rocker over to meet with similar treatment on 
the other side. Before it has come back again to its first 
position the swelling on the cold block has subsided, the 
heat having penetrated into the block by conduction, and 
thus the process is repeated until the temperatures of the 
two bodies are nearly equal. 

The effectiveness of the arrangement is oiten much 
increased by filing a notch in the lead, so that the rocker 
touches the block alternately on opposite sides of the 
notch. 

The rapidity of the vibrations, and therefore the pitch of 
the note produced, is usually much altered by pressing the 
rocker more or less forcibly on the cold block ; and, when 
this process is skilfully conducted, sounds of the most 
extraordinary character, sometimes almost vocal, are 
produced. 

206. An interesting example of the effect of heat, upon 
what are usually called "molecular" forces, is supplied by 
the modifications produced on capillary phenomena. 

In general the curvature of the surface of water and 
other liquids in a capillary tube becomes less as the tem- 
perature is raised ; and at the same time the surface 
tension also becomes less ; so that, on both accounts, the 
height to which the liquid rises, or the depth to which 
it is depressed, in a capillary tube, becomes less as the 
temperature rises. 

The former phenomenon is easily seen by careful in- 
spection, and the proof of the latter is established by a 
multitude of simple experiments. 

Thus if a thin layer of water of considerable surface be 
placed on a large metal plate, and a small lamp flame be 



UNIVERSITY OF CALIFORNIA 

DEPARTMENT OF PHYSICS 



1 84 HEAT. [CHAP. 

applied under the middle of the plate, the effect of the lamp 
is of exactly the same kind as that which it would produce 
on a tightly stretched sheet of India-rubber. The surface 
tension of the colder water overcomes the diminished 
surface tension of the hotter water, and thus we have 
the same sort of result as we get by putting a single drop 
of alcohol or ether on the middle of the water surface. In 
the course of a little time the water is dragged away on all 
sides from the heated part, in the one case, as it is in the 
other case from the point at which the alcohol was applied. 

Thus we see why a drop of solder seems to be repelled 
from the hotter to the colder parts of a soldering iron. 

207. Another curious fact, closely connected with this 
part of the subject, is the effect of the curvature of a liquid 
surface upon the pressure of saturated vapour which is in 
equilibrium in contact with it. 

When a number of narrow tubes, open at each end, dip 
into water in a receiver (free of air, let us suppose) the 
water surface is in each raised in proportion as its curva- 
ture is greater. Also the vapour pressure is less as the 
surface is more raised. Hence, the more concave is a 
water- surface, the less is the pressure of saturated vapour 
in contact with it. If the tubes be now closed at the 
bottom, with a little water in each, evaporation or conden- 
sation will take place according as the water level is now 
higher or lower than before, and will proceed until the 
heights above the external water-surface become the same 
as when the lower ends were open. W. Thomson, who 
first called attention to this phenomenon, considers that it 
accounts for the great amount of water absorbed from damp 
air by bodies with fine pores or interstices, as cotton-wool, 
&c. Clerk-Maxwell pointed out that it also accounts for 
the growth of the larger drops in a cloud at the expense of 



xn.J OTHER EFFECTS OF HEAT. '185 

the smaller ones ; and, at least in part, for that collapse of 
small bubbles of steam, which gives rise to the so-called 
" singing " of a kettle of water just before boiling commences. 

208. This leads us to make a few more remarks on the 
subject of boiling. 

Water boils at a lower temperature in a metallic vessel 
than in one made of glass. 

If water be carefully deprived of air, it is found that with 
care its temperature may be raised in a smooth vessel many 
degrees above the boiling point corresponding to the pres- 
sure to which the water is subjected. But if some chips or 
filings of metal, or any solid with sharp angular points or 
edges, be dropped in, it suddenly boils with violence. It 
is possible that this property may have something to do 
with boiler-explosions, at least when no fresh supply of 
water has been put in for some time before their occurrence. 

These facts have not yet been thoroughly explained, but 
it would appear that a nucleus of some kind is required for 
boiling, just as ( 176) it is required for condensation. 

209. A very singular example of explosive boiling is 
furnished by the geysers, or volcanic hot springs, of Iceland. 
The phenomena exhibited by these were carefully examined 
by Bunsen and Descloiseaux (Liebig's Annalen, 1847). 

They determined the temperature of the water at different 
depths in the shaft of the Great Geyser, at various intervals 
before and after a grand eruption ; and on these observa- 
tions Bunsen founded his theory of the action which 
takes place. 

Between two eruptions it was found that the temperature 
of the water in the shaft increased from the surface down- 
wards, but in no place rose so high as to reach the boiling- 
point corresponding to the sum of the pressures of the atmo- 
sphere and ,of the superincumbent water. The temperature 



186 HEAT. [CHAP. 

steadily rose at every point, steam and very hot water being 
supplied from below and probably also at the sides of the 
shaft until a small upward displacement of the water column 
into the basin lowered the pressure to the boiling-point 
at one part of the shaft. This usually occurred at a depth 
of somewhere about seventy feet under the surface. Then 
there was violent boiling, blowing out the upper part of the 
column of water ; and thus relieving the lower part from 
pressure, so that it also boiled explosively. Then colder 
water, from the geyser basin, ran back into the shaft, and 
the gradual heating from below recommenced. 

It is easy to exhibit the phenomena on a small scale, by 
artificially producing the observed state of temperatures, 
and thus to fully verify the sufficiency of Bunsen's theory. 

210. In 60, 134, while denning the fixed points of 
the Newtonian thermometer scale, we used the terms pure 
ice &c\&pure water. This implies, of course, that the freezing 
and boiling points of water (and of other liquids), are 
affected by impurities held in solution. [Mere mechanical 
suspension of impurities, as in the case of muddy water, 
produces in general no measurable effect.] 

The most important case, so far as the freezing point is 
concerned, is when water contains common salt in solution. 
It is found that the temperature of the freezing point is 
then notably lowered. Walker (in the Fox Expedition) 
found that the freezing temperature of ordinary sea-water is 
below 28'5 F. He observed, however, that a great deal of 
salt is extruded from the ice. So he melted the ice, and re- 
froze it, several times in the hope of thus obtaining drinkable 
water. Although the freezing-point was raised by each 
operation, the water was still brackish after four repetitions of 
the process. The lowest specific gravity of the liquid thus 
obtained was from 1*0025 to 1*002. He alsb tried the 



OTHER EFFECTS OF HEAT. ty 

converse process, removing in succession the various ice- 
crusts formed at lower and lower temperatures from a tub 
full of sea water, in order to find how strong a brine could 
thus be procured. 

211. Very extensive and careful measurements have 
been made of the boiling-points of aqueous solutions of 
various salts, of different degrees of concentration. In all 
cases the boiling-point is above 100 C., and the amount of 
rise is approximately proportional to the percentage of salt 
in solution : the co-efficient of proportionality being dif- 
ferent for different salts. Such solutions, when boiling, give 
off almost pure water-vapour; and it is sometimes stated 
that, however high be the boiling-point of the solution, 
the vapour comes off at the temperature corresponding (in 
Regnault's table) to the pressure. There can be no doubt, 
from special experiments made by Regnault and others, 
that a thermometer placed in the issuing vapour does 
indicate very nearly this temperature corresponding to the 
pressure. But it is to be remarked that the aqueous vapour 
just before it leaves the liquid is certainly superheated ; and 
that superheated vapour leaving the liquid freely in a par- 
tially closed vessel soon becomes saturated vapour, which 
deposits a layer of water on the bulb of the thermometer. 
The temperature of the bulb is therefore that at which pure 
water is in equilibrium with water-vapour at the pressure to 
which they are exposed, so that the method of observation 
is fallacious. We do not yet know, by any certain method, 
what is the exact temperature of vapour leaving a saline 
solution, boiling under ordinary pressure at a temperature 
above 100 C. 

212. The rise of the boiling-point produced by salts in 
solution is attributed to the molecular attraction between 
the salt and the water. Hence, when the circumstances of 



1 88 HEAT. [CHAP. 

the experiment are altered, so that pure steam at iooC. is 
made to pass into an aqueous solution of a salt, condensa- 
tion of steam (with its accompanying disengagement of 
latent heat) goes on until the mixture rises to its boiling- 
point. Thus, with steam at iooC. such solutions maybe 
raised to temperatures very considerably higher. 

In Leslie's mode of freezing water, the greater part of the 
heat developed in the sulphuric acid is the latent heat of 
the vapour absorbed. In the next chapter we shall see how 
to account for the rest. 

213. What is called the spheroidal state of liquids is a 
phenomenon which, in one at least of its many forms, has 
been known from remote times. When a laundress wishes 
to test whether a flat-iron is sufficiently hot, she dips her 
finger in water and allows a drop to fall gently on the iron. 
When the iron is not much above iooC. in temperature, 
the drop spreads over the surface, and rapidly boils away. 
But if it be considerably hotter, the drop glides off from the 
surface without wetting it, and without suffering much evapor- 
ation. [See Phil. Mag. 1850, I. 319, and 1832, II. 378.] 

The phenomenon is easily studied by dropping water 
cautiously from a pipette into a shallow platinum dish 
heated from below by a Bunsen lamp. When the dish is 
sufficiently hot, the water appears to behave on it very much 
as it does on a cabbage-leaf or on an oiled or greasy surface. 
The water is, of course, not in contact with the metal. This 
can be verified in many ways. For instance, if the dish be 
slightly convex instead of concave, a ray of light can be 
made to pass between it and the drop. Poggendorff con- 
nected one pole of a battery with the water and the other 
with the dish, and found that no current passed. The 
temperature of water in the spheroidal state is found to be 
about 95 C. only. 



xn.J OTHER EFFECTS OF HEAT. i3y 

The force required to support the drop is easily calculated. 
Thus, suppose it to be a square inch in lower surface, and a 
quarter of an inch thick. A water barometer stands at about 
thirty-three and a half feet at the mean atmospheric pressure. 
Hence the additional pressure required to support the drop 
is only about 1/33*5 X 12 x 4, or roughly, 1/1600 of the at- 
mospheric pressure. This is supplied, as will be easily seen 
when we are dealing with the Kinetic theory of gases, by the 
momentum acquired by air and vapour particles which have 
come in contact with the hot surface. On leaving it they 
move in directions more nearly perpendicular to the surface 
than those in which they impinged ; i.e. more nearly verti- 
cal : and thus, in the very thin layer between the water 
and the metal, the gaseous medium exerts a somewhat greater 
pressure in a vertical than in a horizontal direction. To 
produce a similar result in a thicker layer the gas or vapour 
must be rarefied, so that the mean free path (Chap. XXII.) 
may be much increased in length. 

214. This simple consideration gives the explanation of 
the chief phenomena shown by the Radiometer. This instru- 
ment consists essentially of a set of very light vanes attached 
to an axle, about which they can freely turn. One side of 
each vane is blackened (so as to absorb heat), the other 
side is polished. The whole is fixed in a partial vacuum. 
When it is exposed to radiation there is greater gaseous 
pressure on the blackened than on the polished sides of the 
vanes, and the apparatus consequently rotates. 1 The first 
experimental results which seem to have had a direct bearing 
on this explanation are due to Fresnel, but they were left 
unnoticed. 

215. With care water may be kept in the spheroidal state in 
a glass vessel, such as a watch-glass. But to insure success 

1 Dewar and Tait, Nature, xii. 217. 



190 HEAT. [CHAP. 

in this experiment the water must be nearly at its boiling- 
point when it is dropped on the glass. We may even have 
a spheroid of water above the surface of very hot oil. But 
here great caution is requisite, as explosions often occur, 
scattering the hot oil in all directions. 

216. Another striking form of the same experiment is to 
dip under water a solid ball of silver heated almost to its 
melting-point. The ball is seen to glow in the middle of 
the water for a few seconds (cooling, however, more rapidly 
than it would have done in air), when suddenly the water 
comes in contact with it, a slight explosion occurs, and all 
is dark. The experiment is considerably facilitated by 
previously adding some ammonia to the water. 

217. Brewster long ago discovered that in many speci- 
mens of topaz and other crystals there are cavities, usually 
microscopic, which are partially filled with liquid. Sang 
observed that little bubbles of gas in the liquid in the 
cavities of Iceland spar seem to move towards the side of 
the cavity at which the temperature of the crystal is slightly 
raised. If part of the walls of the bubble be formed by the 
solid, the explanation of this curious fact probably involves 
the capillary phenomena described in 206 above. But 
when the bubble is wholly surrounded by liquid, it would 
seem that the motion is only apparent, the liquid distilling 
from the warmer side of the bubble and condensing on the 
colder. As the liquid is usually carbonic acid, and under 
considerable pressure, this explanation seems to accord with 
known facts. Very similar phenomena can be produced on 
a comparatively large scale by holding a warm body near 
one end of a small free bubble in a horizontal sealed tube 
containing liquid sulphurous acid. 

218. Davy's ingenious safety-lamp is merely an ordinary 
lamp surrounded by wire-gauze. He found by trial that 



XII.] 



OTHER EFFECTS OF HEAT. 



what is called flame cannot, except in extreme cases, pass 
through such gauze. If we lower a piece of wire-gauze, 




whose meshes are not too wide, over the flame of a Bunsen 
lamp, the flame is arrested by the gauze, although the un- 





consumed gaseous mixture which passes through the meshes 
is highly inflammable. This may be proved by applying a 



192 HEAT. [CH. xii. 

lighted match to it. Or we may operate with the lamp 
unlit, when it will be found that the mixed gases can be 
inflamed above the gauze without igniting the explosive 
mixture below it. It will be seen, in the next chapter, that 
the ignition of such a mixture begins at a definite tempera- 
ture. The conducting and radiating powers of the gauze 
(Chaps. XIV. and XVI. below) prevent the heated part 
from being raised to this temperature. Our further remarks 
on flame will come more suitably in next chapter. 

219. Resume of 204-218. Trevelyan experiment. 
Effects of heat on molecular forces. Vapour pressure over 
curved surface of liquid. Boiling under abnormal condi- 
tions. Geysers. Freezing and boiling points of aqueous 
solutions. Spheroidal state. Radiometer. Motion of 
bubbles. Safety-lamp. 



CHAPTER XIII. 

COMBINATION AND DISSOCIATION. 

220. REFER again to 47, 51, 155 7. As we have 
already remarked ( 38), we know almost nothing as to the 
state in which energy exists in a body. We can measure, in 
general, the amount which goes in, or which comes out ; and 
we can tell in what forms it does so. But we have seen that 
molecular changes such as melting, solution, solidification, 
crystallisation, &c., are usually associated with absorption 
or evolution of heat. We attribute these thermal phenomena 
to the work done against, or by, the so-called molecular 
forces. This is, of course, merely an hypothesis ; so framed, 
however, as to fit in with the law of conservation of energy. 
The actual nature of the process is wholly unknown to us. 
It is not to be expected, then, that in the present state of 
science we should have arrived at anything more satis- 
factory than this in connection with the profound changes 
which take place in what is called chemical combination. 
Still, the little we do know is of great importance from the 
physical, as well as from the more purely chemical, point of 
view. And, from the purely practical point of view, it is one 
of the most important parts of our subject. For it is 
directly concerned with almost all our artificial processes for 
producing heat. 



194 HEAT. [CHAP. 

221. It might at first sight appear, from some well-known 
experiments, that no very definite information is to be had 
on such subjects. For we can make the same two bodies 
combine in many different ways. Thus, to take a simple 
example, we may burn a jet of hydrogen in an atmosphere of 
oxygen, both originally at any one ordinary temperature ; or 
we may mix together, once for all, in the proper proportions, 
and at the same temperature, our oxygen and hydrogen, and 
apply a lighted match or an electric spark to the mixture. 
The one process may be made to take place almost as 
slowly and as quietly as we please ; the other (in a vessel 
of moderate size) is practically instantaneous, 1 and is ac- 
companied by all the physical concomitants of a violent 
explosion. But, if we consider the two processes in the 
light of Carnot's fundamental principle ( 85), we see that 
when the water produced in each case is of the same 
amount, and has reached the same final state, the amounts 
of energy set free must also be the same. In the case of 
slow combustion it is given out almost entirely in the form 
of heat ; in the explosion a great part appears at first as 
sound and ordinary mechanical energy : but both of these 
ultimately become heat. And the quantity of heat thus pro- 
duced must be the same in the two cases, when the products 
have arrived at the same final state. So far, well. But suppose 
that the mixed gases, just before the explosion, had been 
raised to a higher temperature than that at which the slow 
combustion took place, would the whole amount of energy 
involved in the explosion be the same as in the former case ? 
We simply mention this difficulty for the moment. 

222. It would take us far beyond our imposed limits to 

1 There is a definite rate at which a surface of combination runs 
along in each explosive mixture of gases, but it is usually considerable 
in comparison with the dimensions of any ordinary apparatus. 



xiii.j COMBINATION AND DISSOCIATION. '195 

give even a complete sketch, without details, of a subject 
like this, which has been developed in many directions by 
careful experimenters. We therefore content ourselves with 
a mere mention (in addition to what has been said in 69, 
155, 156 above) of such facts as the following : 

Heat is developed, often in considerable amount, when 
gases such as chlorine, hydrochloric or hydriodic acid, 
&c., are dissolved in water. Part of this heat is, of course, 
due to the change to the liquid from the gaseous form. But 
the rest is to be accounted for by the same process as are 
the facts which follow. 

When liquids, e.g. sulphuric acid of commerce, alcohol, 
&c., are diluted with water, there is generally evolution of 
heat accompanied by diminution of volume. 

But it is not always the case that there is evolution of 
heat in the mere mixture of liquids. Thus, although when 
27 parts (by weight) of water and 23 of alcohol are mixed 
at the same ordinary temperature, the temperature of the 
mixture rises about 8'3 C.; if we mix 31 parts (by weight) 
of bisulphide of carbon with 19 of alcohol at the same 
temperature, the temperature of the mixture is lou'ered by 
5'9 C. Now it has been found by direct experiment that, 
in the former case, the water-equivalent ( 183) of the mix- 
ture is about \\\\ greater than the sum of those of the con- 
stituents ; while in the latter case the excess is- only 
about T Vth. On the other hand, the volume of the first 
mixture is 3-6 per cent, less than the sum of the component 
volumes ; with the second mixture it is 17 per cent. 
greater. The first of these supplementary facts would tell 
against the observed results, the second in favour of them. 
The subject is thus easily seen to be one of considerable 
complexity, and we cannot farther develop it here. 

We may mention, however, that experiments have been 

o 2 



196 HEAT. [CHAP. 

carried out with mixtures of the same pair of liquids at 
various (common) initial temperatures. And it appears 
probable (according to Berthelot) that the specific heats of 
mixtures vary as much with temperature as do those of 
simple liquids so much so, in fact, that the mixture of 
bisulphide of carbon and alcohol mentioned above would 
give rise to evolution of heat if both components were 
originally at a temperature lower than o C. 

223. The application of the great laws of thermodynamics 
( 37? 82 ) to chemical combination is, in its elements at 
least, a matter of no great difficulty. The experimental 
part of the work has been carried out with great ability by 
many investigators, prominent among whom are Andrews, 
Favre and Silbermann, and (more recently) Berthelot, 
Deville, Thomsen, and Stohmann. We must confine our- 
selves to the more salient features of the subject. 

It follows from the first law that, if any mixture undergo 
a definite chemical change without taking in or giving out 
work, the heat developed or absorbed depends solely upon 
the initial and final states of the system, and in no way upon 
the intermediate transformations. This is, in one respect, a 
great gain ; for it much simplifies the reasoning from the 
results of experiment ; but it is also a great loss, inasmuch 
as it prevents our obtaining (by this means alone) any in- 
formation as to the nature and order of the successive 
steps of a complex reaction. 

Again, in any transformation which takes place without 
the application or the giving out of work, the heat developed 
is the equivalent of the excess of the original over the 
final potential energy due to the chemical affinities involved. 
[We have spoken here only of heat developed. The reason 
for this restriction will appear at once from the next 
statement] 



xin.] COMBINATION AND DISSOCIATION. 197 

In accordance with the second law of thermodynamics (or 
its consequence, the degradation of energy), the final state 
of every combination is that in which the potential energy 
of chemical affinity is a minimum ; or, what comes prac- 
tically to the same thing, the reactions which take place 
are such as to develop the greatest amount of heat. And, 
conversely, compounds which are formed in this way require 
for their decomposition a supply of energy from without, 

224. Thus, to take a simple case, suppose equal weights 
of hydrogen to enter into combination with oxygen no 
matter how. There are but two compounds which (so far 
as we know) can be formed, water and peroxide of hydro- 
gen. The heat of combination is found to be nearly 50 
per cent, greater for the former product than for the latter. 
Hence, when hydrogen and oxygen act directly on one 
another, water alone is formed, and the excess of either 
gas is simply unacted on. And, by the first of the three 
statements of 223, if we wished to cause water to be 
oxidised into peroxide of hydrogen, we should have to 
supply energy equal to the excess of the heat of combina- 
tion of two atoms of hydrogen with one of oxygen, over 
that of two atoms of hydrogen with two of oxygen. 

Hence we should expect to find, as in fact we do find, 
that peroxide of hydrogen is essentially an unstable com- 
pound. The energy required for the formation of such 
compounds is usually supplied from some direct reaction 
(forming a stable compound) which takes place at the same 
time. Thus peroxide of hydrogen is obtained during the 
formation of chloride of barium from peroxide of barium 
and dilute hydrochloric acid. 

225. We have seen that a compound, which is formed by 
the direct action of its components with the evolution of 
heat, is essentially stable. To decompose it, external energy 



198 HEAT. [CHAP. 

is required. Thus, in the case just cited, peroxide of barium 
and hydrochloric acid are each essentially stable, for heat is 
given out when either is formed directly from its elements. 
But a mixture of the two is not stable, for a rearrangement 
of elements is possible by which farther heat can be 
developed. Here the energy required for the decom- 
positions is supplied by a chemical process. 

But there are many other ways in which it can be 
supplied. Thus heat itself, as in the formation of quick- 
lime from limestone, may be directly the agent. Light also 
is effective under certain conditions, as in the decom- 
position of carbonic acid in the leaves of plants, or of salts 
of silver on a photographic plate. And the discovery of the 
alkaline and earthy metals, Davy's grandest contribution to 
chemistry, was effected by the electric decomposition of 
their hydrated oxides. 

On the other hand, compounds which are formed with 
absorption of heat are often liable to spontaneous decom- 
position. This is the case with many highly explosive 
bodies, such as chloride of nitrogen and the oxides of 
chlorine. 

226. The displacement of one element by another from 
a compound gives an excellent instance of dissipation of 
energy. Thus the heat of combination of iodine with 
hydrogen or other metal is less than that of chlorine with 
the same metal and chlorine has, in consequence, the 
power of decomposing such iodides, setting the iodine free. 
But the whole of this part of the subject is much more 
easily studied by measuring the energy, given out during a 
transformation, in its electrical forms than in the form of 
heat ; and therefore, so far as it is not chemical, it belongs 
more properly to Electricity than to Heat. 

Although all combinations take place in accordance with 



XIIL] COMBINATION AND DISSOCIATION. 1*99 

the laws of energy, these laws alone do not enable us 
to determine the result in any particular case. We must 
have special data with reference to each pair of the sub- 
stances involved. These, of course, can be found only by 
experiment. As they belong much more directly to 
Chemistry than to Heat, we cannot enter minutely into 
what is known of them, but will quote, as a specimen, a few 
general results of one branch of this extensive subject as 
given by Berthelot. 

Strong acids and strong bases, when made to combine in 
equivalent proportions, each being previously dissolved in a 
sufficient quantity of water, evolve nearly equal amounts of 
heat in the formation of stable neutral salts, whatever be 
the acid or the base. 

The heat disengaged is but slightly altered by the presence 
of greater quantities of water, or of additional quantities 
either of the same, or of another, base. 

Examples of this are furnished by the neutral salts of 
soda, potash, baryta, &c., with hydrochloric, nitric, sulphuric, 
<Scc., acids. 

Even in the formation of the alkaline salts of strong acids 
more heat is disengaged than in the combinations of feeble 
acids with the same alkaline bases. 

The feeble acids form, even with the strongest bases, 
salts which are decomposable by water to an extent in- 
creasing with the amount of water, but less as there is 
greater excess of base or acid present. 

The increase of decomposition goes on when water is 
added, without limit in the case of some feeble acids, but 
tends to a definite limit with others. But there are instances 
in which a moderate amount of water wholly effects the de- 
composition, so that the effect on the thermometer is almost 
exactly the reverse of that due to the formation of the salt. 



200 HEAT. [CHAP. 

227. To illustrate some of the above, and a few further, 
remarks, we give a little table of roughly approximate 
values of 

HEAT OF COMBINATION. 

H 2 with O 68,000 34,000 

H 2 with O 2 45,ooj 22,500 

C with O 30,000 2,500 

C with O 2 97>coo 8,100 

CO with O 67,000 2,400 

The first column indicates the substances combining, and 
the proportions in which they combine. Here H, O, C, 
stand for "atoms" (not "molecules") of hydrogen, oxygen, 
and carbon, in the chemical sense of the term ; so that their 
relative masses are as i, 16, and 12. The second column 
gives the number of units of heat ( 179) produced when 
2 Ibs. of hydrogen, 12 Ibs. of carbon, or 281bs of carbonic 
oxide, respectively, enter into the oxygen combination indi- 
cated. The numbers in the third column give the units of 
heat set free per Ib. of the substance to be oxidised. The 
processes by which these numbers were obtained were all 
essentially such as to reduce the compound to the same 
(ordinary) temperature as that of the components, and to 
measure the amount of heat given out during the reduction 
of temperature. 

Thus, when a pound of hydrogen is oxidised into 9 Ibs. 
of water, 34,000 units of heat are given out. If it could be 
directly peroxidised, only 22,500 units would be given out. 
Hence, to effect the farther oxidation of 9 Ibs of water, 
energy equivalent to the difference of these, or 11,500 units 
of heat, must be supplied. 

And we also see that the number of units of heat set free 
by oxidising a quantity of carbon at once into carbonic 
acid, is the sum of those corresponding to the two stages 



xni.] COMBINATION AND DISSOCIATION. ioi 

formation of carbonic oxide, and its subsequent farther 
oxidation. 

228. Many combinations, even of the most vigorous kind, 
do not take place at ordinary temperatures. Even the ex- 
plosive mixture of one volume of oxygen with two of hydro- 
gen requires a start, as it were. If the smallest portion of 
the mixture be raised to the requisite temperature (as by a 
match or an electric spark) the heat developed by the 
combination suffices to raise other portions, in turn, to the 
proper temperature. It is not yet certainly known how a 
jet of hydrogen is inflamed by contact (in air) with spongy 
platinum. It is possible that this may be due to the heat 
developed by its sudden condensation, in which case this 
would be a phenomenon of the same class as that just 
mentioned. But it may be due to the mere approximation 
of oxygen and hydrogen particles by surface condensation. 

229. We are now prepared to consider some of the chief 
phenomena of ordinary combustion. For the table, meagre 
as it is, shows how very large are the quantities of heat 
developed in some of the commonest cases of combination. 
When the gaseous products of a combination are raised to 
a sufficiently high temperature to become self-luminous, 
they form what is called a flame. We will take one case 
with a little detail. 

230. An excellent and typical instance, which has been 
very carefully studied by Deville, is that of a blowpipe 
flame fed with a mixture of oxygen and carbonic oxide, 
escaping from an orifice of i^th of a square inch in 
area under a pressure of from half an inch to an inch 
of water. The flame proper consists of an exterior cone, 
three to four inches long, vividly blue in its lower parts and 
of a very faint yellow at the apex. Within this there is an 
interior cone, barely half an inch long, which consists of as 



202 HEAT. [CHAP. 

yet unignited gas. Just at the apex of this interior cone, 
where the gases have just begun to burn, the flame has the 
highest temperature, and it is there capable of readily melt- 
ing even a stoutish platinum wire. The temperature of the 
flame falls off rapidly from this point upwards. [The 
velocity of propagation of the combining temperature (foot- 
note to 221 ) is obviously that of the jet of gas at the apex 
of the interior cone.] 

To find the composition of the luminous matter at any 
assigned point of the axis of the flame, Deville employed a 
thin tube of silver, about T \t ns f an mcn m diameter, 
which was kept cool by a current of cold water passing 
through it. This tube was fixed transversely in the flame, 
in such a way that a small hole ( T |^th of a inch in 
diameter) in its side occupied the assigned position in the 
axis of the flame, and faced the orifice of the jet. The cold 
water was made to pass so rapidly as to produce suction 
through this small hole. Thus a sample of the gases of the 
flame was carried along in the current of water. This was 
led, after escaping from the water, through caustic potash 
to remove the undissolved carbonic acid, and then carefully 
analysed. In order to determine from what amount of the 
unignited mixture this combustion product had been 
furnished, the experimenter had previously added to the 
mixture a known (small) percentage of nitrogen. This gas, 
of course, passed almost entirely to the collecting vessel, 
the only loss being due to slight diffusion through the 
flame. 

A great number of concordant determinations made by 
this method showed that at the apex of the flame there is 
practically nothing but carbonic acid (provided, of course, 
that the original combustibles were mixed in the proper pro- 
portion). But, from the apex downwards, the proportion of 



xiii.J COMBINATION AND DISSOCIATION. 263 

uncombined to combined gases steadily rose, till at the apex 
of the interior cone (where the temperature is highest) the 
uncombined gases formed about ^rd of the whole. This 
'result is, at first sight, somewhat surprising, but its explana- 
tion is not far to seek. We must take account of the dis- 
sociation ( 47) due to the high temperature. 

231. On the theoretical aspects of dissociation we intend 
to make some remarks later. Meanwhile we may, roughly, 
assimilate its laws to those of evaporation. We have seen 
that a liquid, at any definite temperature, is in kinetic 
equilibrium at its free surface with its own vapour at a 
definite pressure ( 162), so that, at the common boundary, 
there is constant condensation accompanied by an equal 
amount of evaporation, with equal continuous disengage- 
ment and absorption of latent heat. 

Thus, in a compound gas, at least at any one temperature 
within certain limits (which now probably depend on the 
pressure), there is kinetic equilibrium maintained by a 
constant amount of dissociation, accompanied by an equal 
amount of recombination, and also with equal continuous 
disengagement and absorption of heat, the heat of combina- 
tion. The percentage of the whole, which is dissociated, 
rises from zero at the lower limit of temperature (at, and 
below, which there is no dissociation) faster and faster till 
about half way, and then slower and slower till the upper 
limit (at, and above, which the whole is dissociated) is 
reached. This half-way temperature, at which one half of 
the gas is dissociated, is called the temperature of dis- 
sociation. 

When the mixed gases are gradually heated, they cannot 
begin to combine till a certain temperature, depending on 
the constituents and their relative proportions (in a way not 
yet ascertained), is reached. Thus we see how the explosive 



204 HEAT. [CH. xill. 

mixture (228) requires a "start." And, if the constituents be 
such as to combine directly with evolution of heat, this com- 
mencement of combination is sufficient of itself to raise 
the temperature, with the effect of producing farther and 
farther combination. This goes on (provided the gas lose 
no energy to external bodies) until kinetic equilibrium is 
established, i.e. until the percentage of the mixture which 
is in the combined state corresponds to the temperature 
reached by the whole in consequence of the heat of com- 
bination set free. If, however, the gas be permitted to lose 
energy, as by contact with a cold vessel or by expanding 
and doing work, the process of combination will go on 
farther and may become complete, in which case the final 
temperature must be under the lower limit. 

Similar consequences may be traced if we suppose that 
the gases be first individually heated above the higher 
limit, then mixed, and the temperature gradually reduced. 

232. Resume of 220-231. Different modes of com- 
bination. Heat or cold produced by mixing. Direct con- 
sequences of the Laws of Thermodynamics. Stability of 
compounds. Displacement. Heat of combination. Com- 
bustion. Composition of flame. Dissociation. 




CHAPTER XIV. 

CONDUCTION OF HEAT. 

233. REFER to 4, 73. We have already seen that 
bodies in contact ultimately acquire the same temperature. 
This experimental fact is the necessary basis of almost all 
our methods of measuring temperature. The same must, of 
course, be true of contiguous parts of the same body. This 
can only occur by transference of heat from part to part of 
the body. For the present we need not inquire whether 
this equalisation is effected by the mere passage of heat from 
the warmer to the colder parts, or whether there is a mutual 
interchange in which the warmer part gives more to the 
colder part than it receives from it. So far, indeed, as the 
ultimate result is concerned, it does not matter which of 
these is the correct view, for we cannot identify any portion 
of energy ; though the question is theoretically of great 
interest and importance, as we shall find when we come to 
Radiation. To this latter head also we will refer all cases 
in which the transfer of heat in a body appears to take place 
directly between parts at a finite distance from one another. 

234. That bodies differ very greatly in conducting- 
power for heat, is matter of common observation. We 
hold without inconvenience one end of a short piece of 



20b HEAT. [CHAP. 

glass, even when the other end is melting in a blowpipe 
flame. No one would try the same experiment with a 
similar piece of copper or even of iron. The pieces of 
wood or bone which are inserted between the handle and 
the body of a metal tea-pot owe their presence to a recog- 
nition of the same fact. So does the packing of the interior 
of a Norwegian cooking-stove. 

Two of the ordinary forms of lecture experiment may 
also be mentioned here. 

First, that of Ingenhouz, in which a metallic trough, from 
the front of which a number of rods of the same diameter, 
but of different materials, project. These rods are covered 
with a thin film of bees-wax, which melts whenever it is 




raised to the proper temperature. Suppose the whole to be 
at the temperature of the air, and hot water to be suddenly 
poured into the trough. By watching the limits of the 
melted wax we can judge of the relative thermometric 
conducting powers of the various materials. 

A less objectionable form of the experiment consists in 
placing end to end two bars of iron and copper, of equal 
sectional area, having small bullets attached by bees-wax 
to their lower sides, and placing a lamp flame impartially 
between them. The temperature at which the wax softens 
sufficiently to let the bullets fall is found to travel much 



xiv] CONDUCTION OF HEAT. 107 

faster along the copper than along the iron. And here it 
does really indicate superior conducting power on the part 




of the copper, for the thermal capacities ( 183) of iron 
and copper are not very different 

235. Some creditable work was done on this subject 
last century by Lambert, But the first to give a thoroughly 
satisfactory definition of conducting power, or conductivity 
(not, in English at least, conductibility or conducibility) was 
Fourier. The entire subject of the present chapter, with 
all but its later experimental developments, may be said to 
have been created by his Theorie Analytique de la Chaleur, 
first published in a complete form in 1822 ; and these later 
developments may be said to have been suggested, or ren- 
dered possible, by Fourier's work. Its exquisitely original 
methods have been the source of inspiration of some of the 
greatest mathematicians ; and the mere application of one 
of its simplest portions, to the conduction of electricity, 
has made the name of Ohm famous. And more, in this 
work of Fourier's, attention was first specially called to 
the extremely important subject of Dimensions of physical 
quantities in terms of the fundamental units. [See 
Chapter XIX.] 

236. Conducting power must, ceteris paribus, be mea- 
sured by the quantity of heat which passes. We must 
therefore find how to estimate the amount of heat passing 



208 HEAT. [CHAP. 

from part to part of any one body, and what are the cir- 
cumstances which have influence on it. We can then place 
two different bodies under precisely the same determining 
circumstances, and thus compare their conductivities by 
comparing the quantities of heat transferred. 

237. The simplest mode of passage is that in which the 
transfer of heat takes place, throughout a body, in parallel 
lines. Thus, for instance, when a lake is covered with a 
uniform sheet of ice, the ice grows gradually thicker if the 
temperature of the air be below the freezing point. This is 
caused by the passage of heat through the ice from the 
water immediately below it, the upper surface of the ice 
being kept colder than the lower. Thus the layer of water 
next the ice freezes. And as the temperature is the same 
throughout any horizontal layer, whether of water or ice, 
the transference of heat is wholly perpendicular to such 
layers, i.e. it is vertically upwards. 

Hence we choose, as our typical form of conducting 
solid, a plate of uniform thickness and of (practically) 
infinite surface, whose sides are kept permanently at two 
different temperatures. After a lapse of time, theoretically 
infinite but in general practically short, a permanent state 
of distribution of temperature is arrived at throughout the 
slab, and therefore every layer parts with as much heat as 
it receives. If, under these circumstances, we measure the 
quantity of heat which passes through a square inch, foot, 
or yard of any layer of the plate in a minute, we may take 
this as representing numerically the conducting power. To 
compare plates of different substances, we must of course 
take them all of the same thickness, and use the same pair 
of temperatures for their sides. 

Thus we are led to define as follows : The thermal con- 
ductivity of a body at any temperature is the number of units 



xiv.J CONDUCTION OF HEAT. 269 

of heat which pass, per unit of time, per unit of surface, 
through an infinite plate (or layer} of the substance, of unit 
thickness, w/ien its sides are kept at temperatures respectively 
half a degree above, and half a degree below, that temperature. 
The units which we employ are, as hitherto, the foot, the 
minute, the pound, and the degree centigrade. In a 
special chapter we will show how to pass to any other 
system of units. [It will be observed that conductivity 
may, so far as the above definition goes, depend upon 
the temperature of the body. This will be considered 
later.] 

238. The methods chiefly employed for measuring thermal 
conductivity depend ultimately upon observations of tem- 
perature of the conducting body at different parts of its 
mass. Thus the amount of heat which passes across any 
surface in the body, one of the quantities in terms of which 
the conductivity is defined, is deduced from the change of 
temperature which it produces in the body itself. Now the 
temperature effects of a given quantity of heat are inversely 
as the thermal capacity ( 183) of the body. Hence what 
we directly deduce from such experiments is not the con 
ductivity itself, as defined above, but its ratio to the thermal 
capacity. [This ratio is called by Maxwell the Thermometric 
Conductivity, and by Thomson the Thermal Diffusiuity^\ 
Thus it follows that the determination of conductivity re- 
quires that these methods be supplemented by a separate 
set of experiments for the determination of specific gravity 
and of specific heat. 

239. Let us now think of the state of things in the interior 
of the slab while a steady passage of heat is going on. The 
only reason why it goes on is that every layer differs in 
temperature from those adjacent to it. And, because it 
goes on to the same amount across each (thin) layer, the 

p 



210 



HEAT. [CHAP. 



difference of temperatures ot the sides of such layer must be 
proportional to its thickness. [This is easily seen by sup- 
posing the layer to be divided into a great number of sub- 
layers of equal thickness, and then considering two, three, 
or more adjacent ones as a single layer.] Thus the amount 
of heat passing across any layer depends ceteris paribus on 
the gradient of temperature across that layer. This may 
usefully be substituted for the difference of temperatures of 
the sides of a layer of unit thickness. For we thus have 
the means of calculating the amount of heat which passes 
across any layer, even when there is a variable state of 
temperature : as for instance when one side of the slab is 
kept at a constant temperature while the other is alternately 
heated and cooled. 

240. All experimental results hitherto obtained are con- 
sistent with the assumption, which must evidently be true 
for small gradients, that the amount of heat which passes 
across any layer, kept at a given temperature, is directly 
proportional to the gradient of temperature perpendicular 
to that layer. [It is necessary to point out that the exten- 
sion to high gradients is an assumption found to be consistent 
with facts, because conductivity is generally measured under 
circumstances in which the gradient is very much greater 
than iC. per foot, which is that implied in our definition. 
And we cannot yet positively assert that the quantity of 
heat which would pass through a slab a millionth of a foot 
in thickness is exactly a million times as great as that passing 
through a slab one foot thick, when the surface tempera- 
tures are the same, even if the conductivity of the material 
were the same at all temperatures. The corresponding pro- 
position in electric conduction (which is called Ohm's Law, 
though directly based on the splendid work of Fourier) in- 
volves the same assumption, but here we can much more 



xiv.] CONDUCTION OF HEAT. 211 

easily verify it, and the verification has been carried to 
extreme lengths by Maxwell and Chrystal.~| 

241. It is practically impossible to realise experimentally 
the simple conditions which are assumed in the above 
definition of conductivity. Hence, in order to measure 
the value of this quantity in different substances, experi- 
mental arrangements of a somewhat more complex character 
must be adopted. 

Two chief principles have been employed as the bases of 
experiment. The first requires a steady state of tempera- 
ture, and a consequent steady flow of heat. The second 
requires a variable, but essentially periodic, state of tem- 
perature, maintained by external causes in one part of a 
body and thence propagated as a species of wave. In 
both we calculate, in terms of the (known) gradients of 
temperature and the (unknown) conductivity, how much 
more heat enters an assigned portion of a body by con- 
duction than leaves it by the same process. If the tem- 
perature remain constant, this part of the body must lose 
heat otherwise than by conduction ; if it do not so lose 
heat, its temperature must rise. In the first method, as 
each part of the body maintains a steady temperature, we 
must measure by some independent process how much 
heat the assigned portion loses otherwise than by conduc- 
tion. In the second, we must equate the gain of heat hy 
conduction to the quantity of heat indicated by the thermal 
capacity of the element and its change of temperature. 
A third method of some celebrity involves both of these 
considerations. 

242. The first method was employed by Lambert, in an 
imperfect form : but was afterwards greatly improved by 
Forbes, who obtained by means of it the first absolute 
determination of conductivity of any real value. The 

p 2 



212 



HEAT. 



[CHAP. 



principle of the method is extremely simple, but the 
experimental work and the details of the calculation from 
it are both very tedious. 

A long bar of uniform cross section has one end raised 
to, and maintained at, a definite high temperature, while 
the rest is exposed directly to the air of the laboratory. 
Small holes are bored in the bar at regular intervals ; these 
(when the bar is of any other metal than iron) are lined 
with thin iron shells, and in them the cylindrical bulbs of 




accurate ;thermometers are inserted, each surrounded by a 
few drops of mercury. [It is found by calculation based 
on the results of the experiment itself, that these holes do 
not, to any perceptible extent, interfere with the transference 
of heat along the bar.] 

A second bar, of the same material and of equal cross 
section, but of much smaller length, also provided with an 
inserted thermometer, is placed near to the first. 

The bars have been left all night, let us say, in the 
laboratory, no heat being applied. They will then be of the 
same temperature throughout, viz., that of the air, and all 



CONDUCTION OF HEAT. 313 

the thermometers will show that temperature. Now suppose 
heat to be applied to one end of the long bar. [This is 
usually done by inserting it in a bath of melted solder, or 
lead, which is maintained by a proper gas-regulator at a 
steady temperature.] After a short time the thermometer 
nearest to the source of heat begins to rise, then rises 
faster and faster. Meanwhile, the next thermometer, in 
turn, begins to rise. And so on. In Forbes's iron bar, 
which is 8 feet long by about i inch square section, the 
final steady state of temperatures was not reached till after a 
lapse of from six to eight hours. When this steady state 
was reached (i.e. when, for half an hour or more, no change 
of reading was observed in any of the thermometers) it was 
found that even the thermometer farthest from the source 
indicated a temperature perceptibly higher than that of the 
air of the room, as shown by the thermometer in the short 
bar. The others showed each a rise of temperature, increas- 
ing more and more rapidly as they were placed nearer the 
source. [The conductivity of copper is so much greater 
than that of iron, that the farther end of the eight-foot bar 
has to be kept cool by immersion in a vessel, into which a 
steady supply of water is admitted from below.] From this* 
experiment, repeated if necessary with different tempera- 
tures of the source, it is obvious that we can obtain a very 
accurate determination of the permanent distribution of 
temperature along the axis of the bar. [And calculation, by 
Fourier's methods, shows that the temperature is practically 
the same throughout any cross section of the bar, unless the 
section be very great, or the conductivity of the material 
very small.] 

243. So far, the investigation has dealt with temperatures 
only : we must now consider heat. The quantity of heat 
which passes, per minute, across any transverse section of 



214 HEAT. [CHAP. 

the bar, must obviously be the product of the area of the 
section, the conductivity ( 237), and the gradient of 
temperature ( 239) at that section. The gradient can, of 
course, be calculated from the observed distribution of 
temperature, and the area of the section is known. Hence, 
the heat passing is expressed by a definite multiple of the 
unknown conductivity. But that heat does not raise the 
temperature of the part of the bar to which it passes, for we 
have supposed that the stationary state has been reached. 
Hence the rest of the bar, beyond the section in question, 
must lose by cooling in the air (and, in the case of copper, 
the water- bath at the end) precisely as much as it gains 
by conduction. It is only necessary, then, to find what 
amount of heat is thus lost, and we have at once the 
determination of the conductivity of the bar at the tempera- 
ture of the particular cross section considered. [The words in 
italics are of special importance, as they indicate one of the 
most valuable consequences of the improvements due to 
Forbes, which we now give.] 

244. Starting afresh, we once for all heat the short bar to 
a high temperature, insert its thermometer, and leave it to 
cool in the neighbourhood of the long bar (which is, in 
this experiment, left at the temperature of the air). The 
thermometer in the short bar is now read at exactly equal 
intervals of time, say every half minute while it is very hot 
and cooling fast, then every minute, every five minutes, 
and finally every half hour, till it has acquired practically 
the temperature of the air. Some of the thermometers in 
the long bar are read at intervals during this process. 
From what we have said above it will be obvious that, 
from this form of experiment, we can calculate how much 
heat is lost by the short bar, per minute, per unit of 
length, at each temperature within the range employed ; 



xiv.J CONDUCTION OF HEAT. ^15 

and hence we can calculate what was lost in the former 
experiment at any assigned part of the long bar, since 
we know what was the stationary distribution of temperature 
along it. 

245. The loss of heat by cooling depends mainly upon 
the excess of temperature of the bar above that of the 
air, and it is for the purpose of finding this that we use 
the two bars in each part of the experiment. But it also 
depends, to some extent, upon the actual temperature and 
pressure of the air ; and thus the two parts of the experi- 
ment should be conducted as nearly as possible at the 
same air-temperature and pressure. (See 336.) 

246. The unit of heat, in terms of which the conductivity 
is given by direct comparison of the results of 242, 244, 
is of course that which raises by i C. the temperature of 
unit volume of the substance of the bar. The results of 
one of Forbes's experiments on iron, in terms of this unit, 
are as follows : 

THERMOMETRIC CONDUCTIVITY OF IRON. 

Temperature C. o 100 200* 

0*01506 . . . 0*01140 . . . 0*00987 

Thus it would at first sight appear that the conduc- 
tivity of iron for heat, like its electric conductivity, is 
diminished by rise of temperature. Forbes had expected 
to find it so, having remarked that the order of the metals 
as conductors of heat is the same as their order as electric 
conductors, while the electric conductivity of every metal is 
diminished by rise of temperature. This remark was fully 
confirmed by the well-planned experiments of Wiedemann 
and Franz, but Forbes's farther expectation was not, as we 
shall see, realised. 

The following results, in terms of the same or similar 



216 HEAT. [CHAP. 

units, were obtained by Tait in a repetition and extension 
of Forbes's experiments. The iron bar employed was that 
of Forbes. 

THERMOMETRIC CONDUCTIVITY. 

Temperature C. o 100 200 300 

Iron ................................. 0*0149 o'oi28 ... 0-0114 ... 0*0105 52-9(1+0-0014*) 

Copper, electrically good ...0-076 ...0-079 0*082 ...0-085 53-4(1+0-000880 

Copper, electrically bad ...... 0*054 ...0-057 0-060 ...0-063 52-5(1+0-0009*) 

German Silver .................. o "0088 ... 0*009 0-0092 ... 0*0094 46-6(1+0-0009*) 

In this list, all the substances but iron seem to improve 
in therm ometric conductivity by rise of temperature. 

To convert these numbers into thermal conductivities 
as defined above ( 237) they must each be multiplied by 
the numerical value of the water equivalent of a cubic foot 
of the corresponding substance, i.e. by its thermal capacity 
( 183). These water equivalents are given in the last 
column of the table. The temperature changes indicated 
by the factors in brackets are very uncertain. 

The following table contains the results of Forbes, with 
some more recent determinations. Mitchell employed Tait's 
bars, nickelized to prevent oxidation. 

THERMAL CONDUCTIVITY OF IRON AND COPPER. 

To foot, minute, and degree C. 
Iron. Copper. 

0-835 (l-:O*COI47/) .............. Forbes. 

0796 (, -o.oo.870. . ' 



0-666(1-0-000230. .2-88(1+0-000040. - . Lorenz. 
0-677 (I-0-002/) . . . . 2-04 (1+0-00570 . . . Kirchhoff. 

0-6,9 (,+0-00070 . .*, .Mitche,,. 



247. It is to be noted, however, that in deducing his 
final numbers from the data of experiment, Forbes did 
not allow for the increase of specific heat with rise of 



xiv.] CONDUCTION OF HEAT. ^17 

temperature ( 184). Angstrom unfortunately says expressly 
that this cause can affect the change of conductivity with tem- 
perature only to an inconsiderable extent, so that his num- 
bers also are not corrected for this cause. But the specific 
heat of iron increases by about i per cent, for every 7 C., 
and the introduction of this consideration would reduce to 
about -|th of their amount the changes of conductivity given 
by Forbes, and they would then be (in a matter of such 
delicacy and difficulty) within the limits of experimental 
error. This change of specific heat has been taken into 
account by the other experimenters whose results are given 
above. Thus, it would appear that, though the order of 
the metals as heat conductors is practically the same as 
their order in electric conductivity, the farther analogy 
sought by Forbes does not exist. The diminution of elec- 
tric conductivity by rise of temperature holds for all metals, 
and nearly to the same extent in all. On the contrary, the 
thermal conductivity seems (at least in the majority of 
metals) to improve with rise of temperature. 

But it has been shown that copper of good electric quality 
has higher thermal conductivity than that of bad electric 
quality. The whole subject, so far as experimental details 
are concerned, is still in a very crude state, as may be 
judged from the preceding tables. Of course, a good deal 
of the discrepancy arises from the fact that the materials 
operated on differed not only in chemical constitution, 
but also physically, i.e. having been cast, rolled, drawn, 
annealed, &c. 

248. Angstrom's process is the mixed method referred to 
in 241. It consists in alternately heating and cooling, 
for fixed periods and to a fixed amount, one end of a bar, 
which need not be nearly so long as that employed in 
Forbes's method. This periodic change is steadily carried 



2i8 HEAT. [CHAP. 

out until the indications of each of the thermometers at 
different parts of the bar have also become strictly periodic, 
or at least practically so. Fourier's method, applied to 
this problem, shows that (at least if the conductivity and 
specific heat do not vary with temperature) the conductivity 
can be calculated directly from the rate of diminution of 
ranges of the successive thermometers, and the postpone- 
ment of their dates of maximum temperature, per unit of 
length along the bar ; altogether independently of the rate 
of loss of heat by the surface (provided this loss be every- 
where proportional to the excess of temperature over that of 
the air). The details of the calculation cannot be given 
here, but a general idea of its nature will be gleaned from 
the solution of a more restricted problem to be given in the 
next section, the mathematical part of which may be omitted 
by any reader without interruption of the continuity of our 
exposition. 

249. Two extremely important practical questions con- 
nected with this part of our subject are 

(i.) How does the internal heat of the earth affect (by 
conduction) the temperature near the surface ? 

(2.) How far, and according to what law, do the 
fluctuations of surface temperature, from day to night or 
from summer to winter, penetrate into the crust of the 
earth ? 

The second of these questions obviously involves a 
problem somewhat similar to that presented by Angstrom's 
method which we have just discussed. And both questions 
afford us very simple instances of the application of 
Fourier's beautiful method. 

So long as we confine ourselves to the upper strata of the 
earth's crust, we may treat them as planes, the temperature 
throughout depending on the depth only, and the passage 



xiv.J CONDUCTION OF HEAT. 219 

of heat being therefore in parallel (vertical) lines. This 
greatly simplifies the investigation. 

Let v be the temperature at a depth x under the earth's 
surface. Then the temperature gradient is dvjdx. Thus 
the rate at which heat passes upwards through a horizontal 
area of one square foot, at depth x, is kdv/dx, where k 
is the conductivity at temperature v. For a depth x + So- 
under the surface this becomes 

k d + 
dx dx 

The excess of the latter over the former denotes the rate at 
which heat has been (on the whole) communicated to a slab 
of crust a square foot in surface and of thickness S_v. 

The rate at which its temperature rises is dv-dt\ whence, 
if c be the thermal capacity ( 183) of the crust, we have as 
another expression for the rate at which heat is communi- 
cated to the slab, per square foot of surface, and thick- 
ness 8x, 

; - ^ \.:'v : :.'. .,, 

Equating these independently obtained expressions for the 
same quantity of heat, we have finally 

dr d i r </?'\ 
dt dx I dxl 

This equation is merely the translation into symbols of the 
final statement of 241. 

250. It is entirely inconsistent with our plan to attempt a 
full discussion of the varied possible consequences of this 
equation of Fourier's. We will therefore restrict ourselves to 
the two particular questions given above. And in each case 
we will suppose that the conductivity and thermal capacity 
are constant throughout the range of temperature involved . 



220 HEAT. [CHAP. 

251. When there is a steady state of temperature we 

have 

dv 

7/ = ' 

whence 

L 

dx 

Here A is the surface temperature, and J3jk is the rate at 
which temperature rises per foot as we descend. This 
might have been obtained at once from the statements of 
239. 

Hence a stationary state of temperature near the earths 
surface, maintained by the internal heat, implies a uniform 
rise of temperature per foot of descent, if the strata are all 
of the same conducting power. 

If the conducting power be different in different strata, 
the rise of temperature per foot is greater in each in pro- 
portion as the conductivity is less. It is carefully to be 
observed that, as we are dealing with a steady state of 
temperature, the thermal capacity of the crust does not 
appear in our equation. 

The quantity of heat which, in one minute, passes through 
a square foot of area of any stratum, is numerically the 
product of the conductivity by the rate of increase of 
temperature per foot of depth. 

Very numerous observations connected with this subject 
have been collected in late years by a committee of the 
British Association. The results vary greatly in different 
localities, but on the average the rise of temperature in 
mines, artesian bores, &c., is somewhere about o c> oi C. per 
foot. The average value of k may be taken as about 0*024, 
so that the yearly supply of heat to a square foot of surface 
from the interior, averages somewhere about 126 units of 



xiv.J CONDUCTION OF HEAT. 221 

heat. This is of no consequence in tropical or temperate 
regions, but towards the poles it forms a minute one among 
the causes which prevent the formation of an ice-crust of 
more than some 400 or 500 feet in thickness. 

252. Particular stationary periodic solutions of the equa- 
tion, such as are due to surface conditions (any number of 
which solutions may exist simultaneously because the 
equation is linear), are necessarily of the form 



where T is the period of the disturbance, and 2z- its 
range at the surface. If we substitute this value in the 
equation of 249 we find the condition 



Here, of course, we assume k, &c., to be constant; i.e. 
independent alike of temperature and of depth. To 
introduce temperature changes of these quantities is un- 
necessary, because the range of temperature in any of these 
periodic solutions is usually small. And the problem 
becomes too complex for such a work as this when the 
strata are supposed to vary (with depth) in conductivity 
and capacity. 

253. -Hence, when the earth's surface is subjected for a 
sufficiently long time to a simple harmonic change of tem- 
perature, all layers of the crust have simple harmonic 
changes of temperature of the same period. 

But tJie ranges of temperature at successive equal increments 
of depth diminish in geometrical progression. The factor for 
unit of depth is 

c-' or 



222 HEAT. [CHAP. 

Thus, the range diminishes more slowly with depth if the 
conductivity be higher, or if the period be longer ; more 
quickly if the thermal capacity of the crust be greater. And, 
in a precise form, the depth at which the surface disturbance 
is reduced to a definite fraction of its amount, is directly as 
the square roots of the conductivity and of the period, and 
inversely as the square root of the thermal capacity. Thus, 
as the square root of 365 is about 19, a surface disturbance 
of a year period will be sensible at nineteen times as great 
a depth as a disturbance of a day period provided their 
original ranges are the same. 

Again, the date at which the maximum temperature arrives 
at any depth is later in simple proportion as the depth is 
greater. 

The rate at which the crest of the periodic heat wave 
travels is 



It is therefore directly as the square root of the conductivity, 
and inversely as the square roots of the period and of the 
thermal capacity. A rough representation of what takes 
place in this case may be obtained by imagining waves at 
sea to run from shallower to deeper water, i.e. in the 
opposite direction to that in which we usually see them 
coming up a shelving beach, so that their heights shall 
steadily dimmish, instead of increasing, as they progress. 

254. Observations of temperature at different depths 
have been made in various places. The most extensive 
series is that which, commenced in 1837 by Forbes, has 
been carried on ever since (with the exception of the years 
1876 79, which were occupied in replacing the instruments, 
which had been destroyed by a madman). These observa- 
tions are made in the grounds of the Edinburgh Observatory ? 



xiv.] CONDUCTION OF HEAT. 223 

where four gigantic thermometers, with their scales above 
ground, have their bulbs sunk to depths of 3, 6, 12, and 
24 French feet respectively, in the porphyritic rock of the 
Calton Hill. It is found sufficient, so slow are the usual 
changes, to read these thermometers once a week. 

From the sketch, just given, of the theoretical results, 
we should expect to find the indications of the four thermo- 
meters very different in general character: that nearest 
the surface being considerably affected by disturbances of 
short period, which do not penetrate far into the crust ; 
while those of the 24 feet thermometer should depend only 
upon the surface disturbances of longer periods. This is 
found to be the case. 

255. Fourier has given us a method of decomposing any 
strictly periodical disturbance, however complicated, into 
its separate simple harmonic elements. By the aid of this 
method we can separate from the recorded indications of 
each thermometer that part which belongs to the yearly 
change from winter, through summer, to winter again ; and 
to this, as by far the most important, we will confine our 
attention. We give the approximate average numbers for 
the years 1837 42. 

The mean temperature is about 7 -5 C. at the highest 
thermometer, and rises steadily to about 8'i at the lowest 
(251). The ranges of the annual wave are, for the various 
instruments in order 

8- 2 , 5 -6, 2 -7, and o 7 C. 

These do not very accurately follow the law of geometrical 
progression ( 253), but from them we obtain, a year being 
our unit of time, the average value 



V 



0-115 P er French foot, nearly. 
k 



224 HEAT. [CHAP, 

The epochs of maximum temperature are, in order- 
August 19, September 8, October 19, January 6. 
These follow very closely the arithmetical law ( 253), and 
give for the velocity of propagation of the heat-wave 

2 y 7 = 54-8 in French feet per annum. 

Multiplying together the right hand sides, and the left hand 
sides, of these two equations, we have, as a test of their 
accordance, 

2 TT = 6-302 ; 

which, being within Jrd per cent, of the truth, gives us some 
confidence in the general accuracy of the work. 

Taking, therefore, the quotients of corresponding sides 
of the equations, we have 

k 27-4 

= -*~ = 238 nearly. 
c 0-115 

The value of c for this rock, found directly by Regnault, 
is about 0*528. 

Hence, to the system of units employed, we have 

k 125*6. 

To reduce to British feet we must multiply by 1*136, the 
square of the ratio of the lengths of the French and the 
British foot. And, further, to express the conductivity in 
minutes instead of years, and in terms of the usual unit of 
heat ( 56), we must divide by the number of minutes in 
a year, and multiply by the number of pounds in a cubic 
foot of water. (See Chap. XIX.) After these reductions we 
finally find - 

Conductivity of Calton Hill porphyry = 0-017, 
which may be compared with the numbers in the table of 
246. Forbes observed in the same way the changes of 



XIV.] CONDUCTION OF HEAT. 235 

underground temperature at two other stations near Edin- 
burgh, where the materials of the crust are very different. 
The following are the approximate results : 

Conductivity of Sandy Soil (Experimental Gardens), O'oii. 
Conductivity of Sandstone (Craigleith Quarry), 0-043. 

It was the average (o'024) of the three numbers just 
given, which we employed in 251 to form a rough estimate 
of the heat passing annually from the interior of the earth. 
We may calculate from these results, of course only in the 
roughest way, that the heat lost at present per annum from 
the interior of the earth is capable of raising the tempera- 
ture of its whole mass somewhere about one ten-millionth 
of a degree centigrade. 

256. The records of these deeply buried thermometers, 
virtually purified as they are from the effects of the more 
trivial surface disturbances, are extremely valuable as 
indications of the periodic changes in solar activity. A very 
remarkable and suggestive collation of them with the 
corresponding observed development of sun-spots, by the 
Scottish Astronomer Royal, is to be found in vol. xiv. plate 
17, of the Edinburgh Observations. 

257. Heat conduction is essentially a dissipation of 
energy : for, whatever be the actual mechanism of the 
process (see 233 again), and however short be its dura- 
tion, there is always necessarily loss of availability incurred 
by the falling of part at least of the heat from a higher to 
a lower temperature. 

It would lead us far beyond the scope of such a work as 
this to treat of the Secular cooling of the Earth, or the 
Age of the Sun's Heat subjects which have been recently 
discussed by Sir W. Thomson. Their importance, especially 
as regards questions of geological time, is so great, that 

Q 



226 HEAT. [CHAP. 

it is to be hoped that before many years have passed, 
experimental methods may be devised and carried out,, 
capable of giving really satisfactory determinations of the 
conductivity and thermal capacity of rocks and lavas 
through wide ranges of temperature. This branch of our 
subject is especially one in which experiment is as yet far 
behind theory and mathematical methods. 

258. There is, however, one important remark to be 
made. A discontinuity of temperature is impossible in 
a conducting body, if there be no sudden generation of 
heat. Hence if, in tracing back by the help of Fourier's 
methods from a present to the various necessarily precedent 
distributions of heat, we find that at some definite epoch 
there was such a discontinuity, we have a new problem ; we 
are forced to conclude that this state of matters did not 
arise by mere conduction, and that there must have existed 
at that epoch some source of heat not taken account of in 
our equations. But questions like this are of a higher 
order of difficulty than those to which an elementary treatise 
must be confined. 

259. Hitherto, the conducting solids we have dealt with 
have been assumed to be isotropic ( 100). But special 
experiments had of course to be made to determine whether 
bodies which are obviously not isotropic in their molecular 
structure, such as wood, fibrous or rolled iron, crystals (of 
any but the first system), &c., are also non-isotropic as 
regards thermal conductivity. This question was answered 
in the affirmative : chiefly by the elaborate researches of 
De Senarmont, who devised the following extremely simple 
and elegant experimental method. 

A thin plate, with parallel faces, is cut from the substance 
to be examined, the direction of its normal making known 
angles with the chief axes of structure. The surfaces of 



xiv.j CONDUCTION OF HEAT. 227 

this plate are covered with a very thin film of beeswax. 
A stout wire of copper or silver is passed tightly through a 
hole bored perpendicularly to the faces of the plate ; and is 
heated to a definite temperature by a lamp (the radiant 
heat from which is prevented by screens from reaching the 
plate), or. preferably by passing a voltaic current through 
it. If the plate be of glass, or other isotropic material, the 
wax melts over a circular portion of the surface (surround- 
ing the hole as centre) which increases more and more 
slowly till it acquires a definite diameter, which depends on- 
the conductivity, the thermal capacity, and the rate of 
surface loss, of the plate. But, if the plate be cut from a 
non-isotropic substance, the boundary of the melted wax is 
usually an ellipse (the centre still being at the hole). The 
cuts represent, roughly, the effect when the plate is of quartz 




cut (i) perpendicular, (2) parallel, to the axis of the hex- 
agonal prism ; along which the conductivity is the greatest. 
If the substance (as in this case) be homogeneous, the 
thermal capacity is the same at all points, and thus the 
results of this experiment are conclusive as to the different 
conductivities in different directions. By measuring the 
ratio of the axes of this ellipse and observing their 
directions, in slices cut with various orientations, we can 

Q 2 



228 HEAT. [CHAP. 

approximate to the form and position of the ellipsoid at the 
.surface of which a given temperature would be produced 
in the interior of a mass of the substance by a single 
point-source of heat. 

260. From the mathematical theory of this experiment 
(which has been admirably given by Stokes in the Cambridge 
-and Dublin Mathematical Journal, vol. vi.) it appears that 
the conductivities parallel to the axes of this ellipsoid 
(which are the principal axes of thermal conductivity) are 
proportional to the squares of the lengths of these axes. 
It also appears from theory that the dimensions only, not 
the form, of the ellipse on the plate are altered by the surface 
loss of heat. 

When the plate is of moderate thickness, the curves on 
the two surfaces are usually ovoid rather than elliptical, the 
smaller ends being turned opposite ways on the two. This 
arises from the fact that the wire has been passed perpen- 
dicularly through the plate. When the hole is bored in the 
direction of the diameter of the conduction-ellipsoid 
conjugate to plane sections parallel to the plate, the curves 
on the two surfaces of the plate are similar, equal, and 
similarly situated ellipses, with their centres at the ends of 
the hole. 

261. The thermal conductivity in liquids (except mercury 
and other melted metals) is in general extremely small. 
We know that the upper layers of water in a vessel may be 
made to boil, by cautious introduction of steam, while the 
temperature an inch or two below is not sensibly affected. 
To the units already employed, the conductivity of water, 
which is one of the liquids highest on the list, is probably 
less than o'oi. 

The conductivity of gases is still smaller, and therefore 
very difficult to measure. But it can be calculated from the 



Xiv.] CONDUCTION OF HEAT. 2*29 

kinetic theory ; and we therefore defer its consideration for 
the present. 

262. Resume of 233-261. Mechanism of Conduction. 
Preliminary illustrations. Fourier's great work. Definition 
of Thermal Conductivity. Thermometric Conductivity. 
Gradient of temperature. Fundamental assumption. Ex- 
perimental method of Forbes. Results of Forbes, 
Angstrom, and others. Angstrom's method. Effects of 
internal heat, and of periodical surface heating, on the 
earth's crust. Underground Thermometers. Conductivity 
of crust, Conduction in non-isotropic bodies. In liquids 
and in gases. 



CHAPTER XV. 

CONVECTION. 

263. Refer to 73, 74. From what has been said 
above it will be clear that a slight inversion of subjects is 
now advisable so that Convection, as being more inti- 
mately allied with the matters lately discussed, should take 
precedence of Radiation, which involves a somewhat 
different class of ideas. 

264. It is a subject of transcendent importance, but 
also, except in its elements, one of excessive difficulty. 
And in consequence it has been, and still is at least in its 
more practical branches too much in the hands of mere 
empirics, and far too little in those of strictly scientific 
men. 

Of its importance, it need only be said that it includes, 
directly or indirectly, by far the greater part of meteorology, 
as well as the more humble subject of ventilation. Its 
difficulties arise principally from the great complexity of 
almost every one of its problems, not from any doubts as to 
its fundamental principles, for these (in the main at least) 
are exceedingly simple. But in applying them, we must 
know and take account of all the circumstances ; for in this 



en. xv.] CONVECTION. 231 

subject, far more than in almost any other branch of 
natural philosophy, causes which appear at first sight the 
most trifling and unimportant often completely alter the 
very character of the result. 

265. As convection takes place entirely by means of 
the relative motion of the parts of a fluid, it is properly a 
branch of hydrokinetics, and therefore depends for its full 
discussion upon principles which we cannot introduce here. 
But, as all accelerated motion of fluids is due to differences 
of pressure, we must consider how such differences can be 
produced by heat. A body essentially lighter than its bulk 
of water tends to rise in that liquid because the excess of 
vertical pressure on its lower surface, above that on its upper 
surface, is greater than the weight of the whole body. In the 
same way, because a portion of air, in the free atmosphere, 
when its temperature is altered, behaves practically as if it 
were expanding or contracting at constant pressure ( 127), 
it will tend to rise if its temperature be raised, and to sink 
if that be lowered. This is well shown by an ordinary fire- 
balloon, whose gaseous contents are considerably rarefied 
by a flame applied at the lower end of the bag, which is 
open. If it be not inclosed, the rarefied air rises still faster, 
as may be beautifully shown by gently wafting a cloud of 
smoke towards a flame burning free in the air. In this 
case the motion of the smoke indicates exactly the form 
and rapidity of the currents produced. There is a powerful 
vertical current, or updraught, over the flame, and a (nearly 
horizontal) inflow of cold air from all sides towards the 
flame. 

266. The effect is considerably intensified if the lower 
end of a glass or metal tube, held vertically, be slipped over 
the flame. This trivial experiment gives the basis of the 
whole system of ventilation by a shaft or chimney with an 



232 HEAT. [CHAF. 

open fire at the bottom. As already stated, the actual cal- 
culation of the effect even in this simple way, except to a 
first rough estimate, is an exceedingly difficult problem : 
which, having hitherto defied rigorous treatment, has been 
solved approximately with the aid of hypotheses suggested 
by experiment. 

In fine summer weather near the sea coast we have 
constant examples of this process on a large scale, due to 
the large changes of temperature of the land, while the sea 
remains nearly constant in temperature during the whole 
twenty-four hours. 

When the land is heated by direct sunshine, it heats the 
air above it, which rises, and has its place supplied by a 
horizontal inrush of colder air. This is the Sea-breeze. 
When towards, or after, sunset the land grows cold by 
uncompensated radiation, the air above it becomes colder 
than that over the sea. The latter rises in its turn, while the 
former flows outwards to take its place, and we have the 
Land-breeze. 

267. The same thing is well shown by water heated in a 
flask by a small flame placed below. If a little bran be 
mixed with the water, we see at a glance the nature of the 
currents produced. A column of heated water rises 
steadily up along the axis of the flask, and its place is 
supplied by colder water flowing downwards near the sides. 
A constant circulation can thus be maintained if the sides 
of the flask be kept permanently at a low temperature. 

The converse is easily shown if the sides and bottom 
of the flask be kept at a moderate temperature, and some 
fragments of ice be put in to float on the water. But, if 
the sides be not kept warm, and if the ice-supply be kept 
up, this process fails (as in 121) so soon as the water 
attains the temperature of 4. When things are in this state, 



xv.] CONVECTION. 235 

the application of hydrostatic pressure restores the cir- 
culation, in consequence of the lowering of the maximum 
density point thus produced ( 121). Continuing the 
pressure till the ice is all melted, a distribution of tem- 
perature will be arrived at which will prevent currents. 
These, however, will start afresh if the pressure be 
relaxed. But the pressure required for this will be of the 
order of tons-weight on the square inch. 

Further experimental data are required for the complete 
treatment of this subject; but enough has been said to 
show that it is one of curious interest, especially as regards 
the currents in deep fresh-water lakes at a season when 
the temperature at any particular depth fluctuates about 
the maximum density point for that depth. These diffi- 
culties do not present themselves in the case of sea-water ; 
for, even under atmospheric pressure, it freezes before the 
temperature falls to the probable maximum density point. 

268. We cannot, in a work like this, enter upon the 
general questions of atmospheric or of ocean circulation, 
though they belong most definitely to our subject. Both 
problems would be sufficiently formidable even if there 
were no continents, and if the solid earth were truly 
spherical. For, not to speak of the earth's rotation, whose 
influence is noteworthy (as regards, for instance, the produc- 
tion of the trade-winds, and the opposite sense of rotation 
of cyclones in the northern and southern hemispheres), we 
have to consider not merely the varying amount and 
direction of the sun's radiation and its consequences on 
the sea and air, but also the way in which it is occasionally 
cut off for a time from large regions by the formation of 
cloud. This one element alone, though of course ( 9), 
like every physical phenomenon, absolutely determinate 
to one who knew all the conditions and could completely 



234 HEAT. [CHAP, 

work out their consequences, presents itself to our igno- 
rance in a way which can only be called " capricious " ; 
i.e. apparently subject to no law. It is found, however, by 
trial that, in this as in many other apparently hopeless 
cases, approximate results may be obtained by using mean 
values deduced from long series of observations. Such 
results must be sought in works devoted to these special 
problems. 

269. Another illustration of the " capriciousness " of such 
phenomena will occur to any one who thinks of two strata 
of liquids of different densities, and which do not mix, in 
the same vessel. As long as the denser liquid is the lower 
the equilibrium is stable, and any moderate disturbance 
(produced, for instance, by raising artificially a portion of 
the lower liquid into the upper, or by pressing part of the 
upper into the lower) at once rights itself, the energy 
applied being dissipated in the form of waves at the 
surface of separation, spreading gradually from the centre 
of disturbance. 

But when the upper liquid is the denser, the equilibrium 
is essentially unstable; so that this arrangement is, of 
course, not fully realisable in practice. But, in every case 
in which such an arrangement has been nearly realised, 
the inevitable up-and-down-rushes have taken place, not 
over the whole surface of separation, but at two or more 
places which are perfectly determined though we do not 
possess the data for predicting them. 

At some of these places columns of the denser liquid 
pour downwards, at the others columns of the lighter liquid 
ascend. There is, no doubt, an important difference be- 
tween this case and that of two layers of the same liquid 
at different temperatures; for, in the latter, conduction 
(small as it is) very quickly does away with discontinuity 



XV.J CONVECTION. 235 

at the surface of separation ; while in the former, there is 
usually a capillary surface-tension in the separating film. 
But the results, so far as the nature of the motion is 
concerned, are very similar in the two cases. 

270. Vertical currents at definite places may thus be at 
once produced either by heating the requisite part of the 
lower portions of a fluid mass, or by cooling that of the 
upper portions. But the effects of cooling part of the 
lower portions, or heating some of the upper, are usually 
much less important. Hence the grander phases of Ocean 
circulation (except in so far as they depend on winds, and 
therefore on Atmospheric circulation) are much more de- 
pendent upon polar cold than upon tropical heat. On the 
other hand, those of atmospheric circulation depend more 
upon tropical heat than on polar cold. For the great 
temperature effects are produced mainly at the upper 
surface of the ocean, and at the lower surface of the atmo- 
sphere. Hence, if there were no great modifying causes, 
we should expect to find (on the whole) the lower water, 
as well as the lower air, coming from both sides towards 
the equator, and the upper currents of each flowing to 
the poles. 

Similarly, if at any locality in the northern temperate 
zone there should be a large expanse of heated surface, a 
column of hot air would rise from it and colder air would 
flow in from all sides. But, as that coming from the south 
would have, in consequence of the earth's rotation, a 
relatively greater eastward motion, and that from the north 
a relatively less, than the central column, the result would 
be that the inflow would take place spirally, and the whole 
would form a vortex (cyclone) rotating in the sense in 
which the earth turns about the north pole. A down-flow 
of cold air from above would also give rise to a vortex. 



236 HEAT. 

but its direction of rotation would be the opposite (anti- 
cyclone) ; and, because it takes place from above down- 
wards, it should usually be a less violent phenomenon. 
These rotations would each be reversed in direction in 
the Southern Hemisphere. It is to be observed, however, 
that such results as these would only necessarily follow in 
the case of an extensive disturbance. Smaller vortices, 
from waterspouts, dust-pillars, c., up to some forms of 
local thunderstorms, in general owe their rotation to fluid 
friction ; and with this we are not now concerned, farther 
than to say that the sense of the rotation produced depends 
upon merely local circumstances. 

271. So far, we have taken no account of aqueous 
vapour, though the atmosphere usually contains it in 
considerable amount. It is the effective agent in many of 
the most important meteorological phenomena; but we 
must confine our attention to it only in so far as its efficacy 
is connected with heat. 

Every one knows by experience how thin a layer of 
cloud or mist suffices to arrest the greater part of the sun's 
heat. It can only do so by absorbing the heat rays (except 
in so far as it reflects or scatters them), and thus indirectly 
heating the air in which it is suspended. Thus the cloud, 
instead of the earth's surface, becomes the intermediary by 
which the sun's radiations are given to the air. 

But water-substance acts powerfully in quite another way. 
We have seen how great is the latent heat of steam ; we 
know also that an inch per hour is not an unusual fall of 
rain ; while the greater part of it, at least, comes from the 
lowest mile of the atmosphere. The "homogeneous atmos- 
phere" is about 5 miles in height : and the water barometer 
( 213) stands at about 34 feet. Hence the mass of lower 
air, in a column a mile high and a square foot in section, is 



XV. CONVECTION. 237 

considerably less than that of 7 cubic feet of water it is in 
fact about 360 Ibs. only. Its specific heat is ( 187) about 
0*25, so that its thermal capacity is about 90. An inch of 
rain corresponds to about 5 Ibs. of water per square foot, 
giving out about 3,000 units of heat when condensed from 
vapour. Hence, if all the vapour were condensed at once, 
the lower air would be raised in temperature by somewhere 
about 33C. ! We get a still more striking notion of the 
magnitude of this result by expressing it in terms of work. 
It is more than four millions of foot-pounds, or the work of 
one horse for two hours ! In a cubic mile of the lower 
atmosphere this would correspond to the work of a 
million horses for fifty hours. [If this number of horses 
were placed on a square mile they would barely have 
standing room each would have just three square yards.] 
This is, of course, an extreme case, but it shows that there 
can be no difficulty in explaining, by the mere latent heat 
of the vapour in the air, the energy displayed in the most 
violent hurricanes. And this energy came, of course, mainly 
in a radiant form from the sun. This points directly to 
the subject of the next chapter. 

272. We cannot here discuss how the warm, moist, air 
-cools as it expands on rising, becomes partially warm again 
by precipitation of water, and so on. Even the convective 
equilibrium of a moist atmosphere is not a very simple 
question, 1 though almost incomparably easier than that of 
its motions. Nor can we do more than mention the 
extraordinary convection currents in the sun's atmosphere ; 
where speeds of one hundred miles per second, and up- 
wards, have been actually measured by perfectly trustworthy 
methods. 

1 See a short discussion of it by Sir W. Thomson (Ma)uhester 
Memoirs, 1862). 



238 HEAT. |CH. xv. 

273. A very beautiful practical application of the prin- 
ciples of this chapter has been made by Joule, who employed 
it to determine the maximum density point of water, and 
also to measure the heat-radiation from the moon. The 
details of his apparatus need not be given, the essential 
feature being a closed circuit in a vertical plane, filled with 
fluid. A current sets in when, in consequence of difference 
of temperature or otherwise, there is a difference of density 
in the contents of the two vertical parts of this circuit. 

274. Resume of 263-273. Mechanism of Convection. 
Basis of Ventilation. Land and Sea breezes. Causes of 
Atmospheric and Ocean circulation. Cyclones and Anti- 
cyclones. Waterspouts and Dust-pillars. Effects of 
Aqueous Vapour. Convective Equilibrium. Currents in 
the Sun's atmosphere. Joule's Convection Apparatus. 



CHAPTER XVI. 

RADIATION. 

275. Refer to 40, 73, 74. The term "Radiant," 
as applied to Heat, though we cannot hope to get rid of 
it, is even less defensible than "Specific" or "Latent," 
whose claims we have already examined ( 147, 179). 
Radiations, like Sounds, are all of one class or family, 
so that the so-called Radiant Heat is simply Light. 

The following passage is Query 18 of the Third Book of 
Newton's Optics: 

"If in two large cylindrical vessels of glass inverted, 
two little thermometers be suspended so as not to touch 
the vessels, and the air be drawn out of one of these 
vessels, and these vessels, thus prepared, be carried out 
of a cold place into a warm one ; the thermometer in 
vacua will grow warm as much, and almost as soon, as the 
thermometer which is not in vacua ; and when the vessel 
is carried back into the cold place, the thermometer in 
vacua will grow cold almost as soon as the other ther- 
mometer. Is not the heat of the warm room conveyed 
through the vacuum by the vibrations of a much subtiler 
medium than air, which, after the air was drawn out, 
remained in the vacuum ? And is not this medium the 



240 HEAT. [CHAP. 

same by which light is refracted or reflected, and by whose 
vibrations light communicates heat to bodies, and is put 
into fits of easy Reflexion and easy Transmission ? And 
do not the vibrations of this medium, in hot bodies, 
contribute to the intenseness and duration of their heat ? 
And do not hot bodies communicate their heat to con- 
tiguous cold ones, by the vibrations of this medium pro- 
pagated from them into the cold ones ? And is not this 
medium exceedingly more rare and subtile than the air, 
and exceedingly more elastick and active ? And doth it 
not readily pervade all bodies ? And is it not (by its 
elastic force) expanded through the heavens ? " 

276. We know by actual trial that there are sounds which, 
while painfully affecting certain people, are absolutely in- 
audible to others whose ordinary hearing is quite as good, 
and who are as favourably situated with regard to the source 
of sound. But the cause is not far to seek. Such sounds are 
always either very grave or very shrill. In other words, 
there is, for each human ear, a continuous range of 
audible sounds. With the majority of individuals this 
range is somewhere about eleven octaves, the middle being 
a note corresponding to about 1,500 vibrations per second. 
If, therefore, we were to trust to our ears only, we might 
suppose that periodic compressions and dilatations of air 
must necessarily be confined to this range, a statement alto- 
gether inconsistent with the known dynamical properties 
of air. But here we have the advantage of knowing (to 
a great degree of accuracy at least) the nature of the ob- 
jective phenomenon ; and we are thus prepared easily to 
understand that the limits of our range of hearing may 
depend to a notable extent upon the mechanism of the 
organs of hearing. In this sense the range may be re- 
garded as wholly subjective, and we can easily conceive 



xvi. ] RADIATION. 241 

that animals might exist of greater dimensions than man, 
whose coarser audition-mechanism would enable them to 
hear, as continuous sounds, periodic disturbances of air 
occurring twice or thrice per second. Again, we might con- 
ceive of insects whose range of hearing should lie entirely 
among sounds too shrill to affect any human ear. This is still 
farther impressed upon us by another well-known result. 
If a large bell or tuning-fork be struck sharply with a hard 
body, such as a hammer, the sound produced is at first ex- 
tremely harsh and unmusical ; but, as it gradually dies away, 
it becomes the pure, proper musical note of the instrument. 
The metallic clang which is first heard is made up of a great 
number of notes having no necessary musical relation to one 
another. But these die out at very different rates, the 
shriller the faster, so that after a short time nothing is 
audible but the grave, fundamental note. 

But we may go even farther. Very low notes, such as 
those of the longest pedal-pipes of an organ, are ^//almost as 
much as heard ; and the same thing is often presented to 
us in the working of massive machinery, as in factories and 
engine-shops, where a singularly oppressive and even 
painful effect is produced on the brain, though we can 
scarcely call the impression by the name of " sound." It 
seems, in fact, to be almost devoid of what we commonly 
call loudness ; and sometimes we are only conscious of its 
having existed, by the exceedingly pleasant feeling of relief 
which \ve experience when it stops. 

277. Analogy at once leads us to inquire whether there 
is not something similar in the case of our subjective 
perception of Light. And the very simplest observations 
at once suggest that such an idea is at least extremely 
plausible. 

The existence in all civilised languages of terms equivalent 



242 HEAT. [CHAP. 

to our own red-hot, white-hot, &c., is one of the most 
striking of these. It shows that men have long been accus- 
tomed, as the result of experience, to associate htat with 
luminosity. When we think of this, in connection with the 
fundamental fact of observation, that heat always passes from 
warmer to colder bodies, we are led to look upon the emis- 
sion of light by a hot body as at least part of this process. 
When a body is gradually heated in the dark, as for instance 
a stout iron wire by the passage of a powerful electric cur- 
rent, it first becomes visible by feeble rays of long wave- 
length, or low refrangibility, only; then, as it becomes 
hotter, it gives out more of these lower rays, along with 
others of higher refrangibility. This process continues, if 
the current be sufficiently powerful, until the wire becomes 
white-hot, i.e. until it gives off all kinds of visible light in 
something like the same relative proportions as those in 
which they come to us from the sun. If the current be 
now interrupted, the wire presents exactly the same con- 
tinuous series of appearances in the reverse order. But, 
even when it has ceased to be visible, we feel (on holding 
the hand below it, so as to avoid convection currents) that 
it is still giving off something, but at gradually-diminishing 
rate, which produces the sensation of heat. Bearing in 
mind the analogy of sounds, and also the fact that the 
shriller sounds correspond, like the more refrangible rays, to 
the shorter periods or the shorter waves, we are led to the 
conclusion that radiation is all one continuous phenome- 
non, of which only a certain range affects the eye, while 
below that range it can be detected directly only by the far 
inferior sense of touch. 

278. This, however, is presumption only: though on 
very strong grounds. To make the matter certain we must 
investigate the behaviour of the invisible radiations, and 



xvi.j RADIATION. 343 

compare it under conditions as varied as possible with that 
of the visible. Thanks to the thermo-electric pile, and the 
galvanometer ( 50, 62), we can now treat obscure radia- 
tions with an amount of accuracy which, though far inferior 
in many respects to that usually attainable in optical 
processes, is sufficient for our purpose. 

[Some of the fundamental facts in Optics must here be 
assumed. We cannot digress to indicate the nature of the 
evidence for their truth.] 

279. In free space, or in air of uniform density, light 
moves in straight lines. So does radiant heat, whether from 
the sun or from a terrestrial source. For an opaque obstacle 
which intercepts the light intercepts the heat also, whatever 
be the source. 

Radiant heat is propagated with a speed practically the 
same as that of light. After a total eclipse of the sun, the 
heat rays reappear (as nearly as can be tested) simultaneously 
with those of light. They are therefore propagated with a 
speed of somewhere about 186,000 miles per second. 
This great common speed alone is almost sufficient proof 
of identity in mechanism of propagation. 

Burning-mirrors and burning-glasses show that the laws 
of reflection and of refraction are the same for light and 
for obscure radiations. For they are adjusted so as to focus 
the light rays, and the heat rays are also found focussed at 
the same place. This is rigorously the case with a mirror, 
but not so with a lens. To obtain the most powerful effect 
with a burning lens, we must place it a little farther from 
the screen than if we wished to obtain the sharpest luminous 
image of the sun. But the fact that it is so affords a still 
stronger argument in favour of our proposition, for the focal 
distance for red rays is greater than that for blue rays (unless 
the lens be achromatic for these two sets of rays), and we 

R 2 



244 



HEAT. 



[CHAR 



should expect the focal distance to be still greater for 
lower radiations. 

Experiments with a burning-glass are, of course, of a 
very rude character; but Lord Rosse, and others, have 
found that, to throw the obscure rays from a star upon a 
small thermo-electric pile, by means of the mirror or the 
object-glass of the largest telescope, it is only necessary 
that the luminous image should be formed on the pile. 

280. The intensity of light proceeding from a small source 
through a transparent medium is inversely as the square of 
the distance from the source. The thermo-electric pile 
shows us that the same is true of radiant heat. But this 
is no additional analogy, for it is necessarily true of any 
form of energy which spreads rectilinearly in all directions 
from a small source ', always provided that none is absorbed 
by the way. 

A luminous surface appears of the same brightness what- 
ever be its distance from the eye (provided there be no 
absorption of light by the intervening medium). Similarly,. 



s' 




when the conical collecting mirror of the thermo-electric 
pile is turned towards a uniformly hot surface, the effect on 
the galvanometer is the same at all distances, so long at 



xvi.] RADIATION. 245 

least as the surface appears from the pile to fill the entire 
aperture of the cone. This, again, is a mere consequence 
of rectilinear propagation, without absorption by the way. 

A uniformly luminous surface appears equally bright all 
over, however various may be its inclinations at various 
points to the line of sight Similarly, when a large hot 
surface of uniform texture radiates to the pile through an 
aperture, the amount of heat received is not altered by 
altering the orientation of the surface, so long as (seen from 
the pile) it appears completely to fill the aperture. Hence 
it follows that the intensity of radiation, whether luminous 
or obscure, from a hot surface, in any direction, is propor- 
tioned to the cosine of the angle which that direction 
makes with the perpendicular to the surface. [We shall 
find presently that this is a necessary consequence of the 
fundamental tendency of heat to uniformity of temperature.] 

281. Bodies differ extremely from one another in the 
amount of light which they absorb. The same is true as 
regards radiant heat. But, just as there are bodies extremely 
transparent to blue rays and practically opaque to red, or 
vice versd, so we have bodies extremely transparent to light 
and practically opaque to obscure radiations, and vice versd. 

A polished plate of rock salt is, when dry, almost equally 
transparent to obscure radiations and to light. If it be 
thickly covered with lamp-black it becomes practically 
opaque to light, while still transmitting a considerable frac- 
tion of the obscure radiations. A strong aqueous solution 
of alum is almost perfectly transparent to light, while very 
nearly opaque to obscure radiations. The same is true of 
a thick plate of glass : as is nowadays practically recognised 
when we use such a plate as a fire-screen, which permits 
us to enjoy the cheerful light of the fire without suffering 
unduly from its heat. 



246 



HEAT. 



[CHAP. 



282. But there is a farther analogy connected with this. 
Light, which has passed through one plate of coloured 

glass, and has thereby been weakened, is much less weak- 
ened by passing through a second plate of the same glass, 
still less by a third, and so on. 

So if, by using a sufficiently powerful radiating source, we 
force a considerable quantity of obscure radiations to pass 
through a sheet of ordinary plate-glass, the ray so sifted is 
capable of passing in much greater per-centage through a 
second plate, and so on. 

283. All the facts above described ( 279-282) can 
now be established without the least trouble by the use of 
the pile and galvanometer. There is no need of describing 
the obvious necessary adjustments. All this part of our 
subject was excellently worked out by Melloni, who 
extended in various directions the valuable work of 
Leslie. A sketch of the arrangement of Melloni's apparatus 
is subjoined. It requires no explanation. 




284. We might mention a great many more of these less 
perfect analogies, but the above must suffice. We now 



xvi. ] RADIATION. 247 

come to others which are practically conclusive of the 
identity of mechanism in all kinds of radiation. They are, 
in fact, simply those which established the Undulatory 
Theory of Light. When they were found true of radiant 
heat also, the identity of the phenomena could no longer 
be doubted. For this part of the subject we are again 
indebted to Melloni, and also, in a special manner, to 
Forbes. It is curious to notice that the original speculations of 
Mohr, of date 1837, as to the true nature of heat were mainly 
based upon these discoveries. We will content ourselves 
with a very brief statement, because the experiments are 
far more striking when their results are directly exhibited 
to the eye by luminous rays, than when these have to be 
groped for, as it were, in darkness by the help of the thermo- 
electric pile. 

285. Every one knows that the establishment of the 
undulatory theory of light was aided to a wonderful degree 
by Young, mainly by his doctrine of Interference and its 
consequences. Exact measurements of these phenomena 
gave the means of determining the wave-length of any 
particular ray; and a very simple adjustment gave the 
means of proving that light moves faster in air than in 
glass or in water, and must therefore be due to a trans- 
ference of energy and not to a transference of material 
particles. But all this gave no information as to the nature 
of the wave-motion. It proved merely that, along the path 
of a ray, some kind of displacement occurs which is exactly 
and periodically reversed every half wave-length This 
might be of the nature of compression and dilatation merely, 
as in a sound wave ; it might be wholly transverse, as in a 
pianoforte wire; or it might be partly longitudinal and 
partly transverse, as in ripples or waves at the surface of 
water. 



248 HEAT. [CHAP. 

But the phenomena of Polarization at once gave further 
information. They showed that the displacement, what- 
ever be its nature, must be transverse to the ray ; and 
they enabled Fresnel to give a consistent theory of 
Double Refraction, and of Circular and Elliptical Polari- 
zation. 

All the details of these splendid investigations properly 
belong to Optics. We confine ourselves to the mere 
mention of the simplest proofs of Interference and of Polari- 
zation in the case of obscure rays, for these are the keys of 
the position. 

286. Fizeau and Foucault found that all the ordinary 
interference experiments, such as that with Fresnel's in- 
clined mirrors and that of ordinary diffraction, succeed 
with obscure rays ; and, as was to be expected, indicate 
greater wave-lengths than those of visible light. 

Forbes was the first to obtain positive proof of the polari- 
zation of obscure radiations. The form of apparatus which 
he employed is still amongst the best for that purpose. It 
consists merely of a stout plate of mica, split by cautious 
heating into a great number of exceedingly thin parallel 
films. This is placed inside a tube, at the proper inclina- 
tion to the axis, and polarizes by transmission just as a 
parallel bundle of " microscope glass " polarizes light. 
When two such tubes are placed end to end, between a 
non-luminous source of heat and the end of the pile, it is 
found that, if the mica plates in the two are parallel, a 
considerable amount of obscure radiation is transmitted. 
This is greatly reduced by making one of the tubes rotate 
through a right angle ; and is restored to its original value 
by a farther rotation of one right angle. Similar results 
though not quite so well marked, are obtained with sources 
hot enough to be self-luminous. But the mica bundles 



xvi.j RADIATION. 249 

require to be placed at a special angle for each class of 
radiations so as to give the greatest effect. 

We may merely add that the electro-magnetic rotation 
of the plane of polarization of light, discovered by 
Faraday, has been found to hold good for obscure rays 
also. 

In comparatively recent years it has been conclusively 
shown by Clerk-Maxwell, mainly by theoretical deductions 
from Faraday's wonderful series of experiments, that Radia- 
tion generally is an electromagnetic phenomenon. But to 
discuss his theory, or even to describe the ingenious experi- 
ments by which Hertz and others have recently illustrated 
it, would involve a digression of a much more extensive 
character than any which have been admitted into this very 
discursive volume. 

287. Resume of 275-286. Radiant Heat is merely 
Light. Analogy from Sound. Comparison of visible and 
invisible Radiations, as to speed of propagation, reflection, 
refraction, absorption, &c. Proofs of identity furnished by 
Interference and by Polarization. Clerk- Maxwell's Electro- 
magnetic Theory. 



CHAPTER XVII. 

RADIATION AND ABSORPTION. 

288. THE remarks in the preceding chapter, coupled with 
a mere reference to the radiations of shorter wave-length than 
those of light (which are studied either by means of fluor- 
escent bodies, or by their action on photographic plates), 
entitle us to speak of radiation as one phenomenon. 
Different modes of experimenting are specially adapted for 
special properties of rays of different wave-lengths, that is 
all. But the indications furnished by the pile, being inde- 
pendent of particular nerves, or particular chemical com- 
pounds (on which sight and photographic processes depend), 
are obviously fitted, in a special manner, to give us informa- 
tion of one standard kind for all species of radiation alike. 

289. If we think again of the phenomena presented by 
a hot wire ( 277), we see that we may now state that such a 
body gives off radiations of all wave-lengths, from the 
longest up to a certain limit, which depends upon its tem- 
perature. The higher the temperature, the higher is the 
limit, and also the greater is the amount radiated corres- 
ponding to each wave-length longer than the limiting one. 

290. This prepares us to see that the radiation from a 
body depends upon itself alone (i.e. upon its constitution, its 



CH. XVIL] RADIATION AND ABSCRPTIOX. 251 

temperature, the nature of its surface, &c.), and therefore 
that the equilibrium of temperature which ultimately obtains 
among bodies within an inclosure which contains no source 
of heat ( 4), is arrived at, not by the warmer bodies alone 
radiating to the colder, but by all the bodies simultaneously 
radiating, each to an amount depending on its own nature, 
surface-condition, and temperature. Also that equilibrium, 
once attained, is maintained by the same process. This is 
called the Theory of Exchanges, and was propounded by 
Prevost last century. He was led to it in trying to explain 
what had been called radiation of Cold ; as when, for instance, 
a thermometer in a warm room falls if a piece of ice be 
held in the neighbourhood of its bulb. 

In 1858 and 1859, respectively, Stewart and Kirchhoff 
greatly extended our knowledge of this subject, both by 
experiment and by theory. Their chief theoretical result, 
however, had been to a great extent anticipated by Stokes 
about 1850, by the help of a dynamical analogy. 

[In the immediate farther discussion of this subject we 
will suppose that our experiments are all carried out in 
racuo. This practically restricts our reasoning to solids, 
which is of course a great limitation ; but, per contra, it frees 
us from all the complications of convection. In another 
chapter we will consider the effects of simultaneous radia- 
tion and convection.] 

291. It was well known to Leslie and others that the 
amount of radiation from different bodies, at the same 
temperature, varies greatly with the nature of the radiating 
surface. Leslie, for instance, constructed a large cube of 
tinned iron, one vertical face of which was left bright, a 
second dimmed by scratching with emery, the third covered 
with lamp-black, and the fourth with white enamel. When 
the cube was filled with hot water, it was found that there 



-252 HEAT. [CHAP. 

was very little radiation from the polished surface, consider- 
ably more from the scratched one, but very much more from 
either of the others, which (in spite of their contrast of 
colour) behaved nearly alike. The amounts of radiation 
from the polished and the black surfaces are so different 
that this observation can be made by the sense of touch 
merely by holding the hand parallel to these sides in 
succession, at a distance of a few inches. 

292. Leslie, and afterwards more fully De la Provostaye 
and Desains, made numerous experimental determinations 
of the radiating, absorbing, and reflecting powers of bodies 
for the groups of obscure rays from each of many 
different sources. These experiments cannot lay claim to 
great exactness ; but they tend to show that, on the 
whole, the radiating and absorbing powers of one and the 
same body are proportioned to one another, while the 
reflecting power diminishes as the other two properties 
simultaneously increase. The apparently anomalous beha- 
viour of some bodies, especially as regards their comparative 
reflective powers for obscure and for luminous radiations, 
prevented any grand generalisation from these experiments. 

The whole subject, as we now know, is so intimately 
connected with Carnot's Principle of Reversibility ( 83) that 
every experimental development which it has received must 
be looked at from this great theoretical point of view. And 
this principle, whether an author was aware of it or not, 
has been the basis of all sound theoretical reasoning on the 
subject. But, just as Carnot's principle itself is (as will be 
shown later) exact only in a statistical sense, due to the 
extreme minuteness of the particles of matter in com- 
parison with our instruments for measuring temperature; 
so any conclusions we may deduce from it are subject to 
the same limitations. It is therefore vain, - at least in the 



xvii.] RADIATION AND ABSORPTION. 253 

present state of science, to look for a truly rigorous inves- 
tigation of the relation between radiating, absorbing, and 
reflecting powers. In all the professedly rigorous investiga- 
tions which have been given, the careful reader will detect 
one or more steps which are to be justified only by the 
statistical process of averages. Why not, therefore, boldly 
begin by assuming its validity, and thus acknowledging that 
the demonstration is not rigorous ? To avoid unnecessary 
complication, without in any way losing accuracy, we adopt 
a method which has been found of great value in older 
branches of physics. 

293. When, in elementary dynamics, we consider the 
equilibrium of a lever under various forces, we simplify the 
mathematical treatment of the question in a very marked 
degree by assuming the lever to be rigid. And we find that 
we thus obtain a solution of a practically accurate character. 
If the lever be very long, or the forces very great, such a 
solution is not sufficient, for we find by trial that flexure 
takes place. But, unless the flexure be great, we can 
determine its amount and consequences by means of a 
simple hypothesis as to the law of bending, and this in 
turn enables us to obtain in the more complex problem 
results accurate enough for all practical purposes. Hooke's 
Law, Coulomb's Laws of Friction, &c., are all mere 
approximations to the truth, based on experiment, and 
employed to enable us to avoid difficulties of theory as 
well as labour of calculation. So with the assumptions of 
perfect fluids, and of gases which exactly follow Boyle's 
Law. No such thing as a rigid body, or a perfect fluid, 
has yet been met with: but the conception of either 
presents no difficulty, while it affords valuable facilities for 
simplifying our investigations. And a similar set of assump- 
tions will greatly help our present work. 



254 HEAT. [CHAP. 

294. (i.) Though no substance is perfectly transparent 
all the most transparent bodies being found to be more or 
less coloured when taken in sufficient thickness, and thus 
to absorb some of the light which falls upon them, while 
they also reflect or scatter a portion of it : we can imagine, 
and use for reasoning, a perfectly transparent non-reflecting 
body. Every ray, whether luminous or obscure, which falls 
on such a body, passes through it without loss. 

(2.) A well-polished surface of metallic silver suggests by 
its behaviour the theoretical conception of z. perfectly reflect- 
ing surface. No ray can penetrate into a body provided 
with such a surface. 

(3.) Similarly, a lump of gas-coke, or a body covered with 
lamp-black, suggests the conception of a perfectly black body. 
Every ray which falls on such a body is at once absorbed. 
Hence such bodies can neither reflect nor transmit. 

Bodies belonging to one and the same of these three 
classes have properties (so far as we are concerned with 
them) absolutely identical. 

(4.) A thick plate of cobalt-blue glass, which transmits 
only one class of rays (red), suggests the conception of a body 
partially transparent for one definite ray and perfectly 
absorbent of all others. 

(5.) Certain compounds of didymium, &c., suggest a body 
partially absorbent of one definite ray and transparent to all 
others. 

(6.) And, finally, a thin film of metallic silver, such as is 
now used for the glass mirrors of reflecting telescopes, sug- 
gests an ideal body partially transparent to a certain definite 
ray and a perfect reflector of all others. 

Bodies belonging to any one of these three classes may 
have properties entirely different from one another. 

Various other combinations easily suggest themselves, 



xvji.] RADIATION AND ABSORPTION. ^55 

but those named above are the most important for our 
object. 

295. Recurring to our fundamental experimental fact, 
viz. that all bodies in an impervious inclosure, which contains 
no source of heat, ultimately acquire and maintain the same 
temperature ( 4), we see that the state of each is altogether 
independent of the others. Any number of them may be 
removed, or any number of new ones introduced, without 
disturbing the equilibrium, always provided they be at the 
common temperature. [It might be thought that a perfectly 
reflecting body would form an exception. But if it, and its 
contents, were originally at the assigned temperature, they 
would remain so indefinitely. In fact its surface becomes 
part of the boundary of the original inclosure, while it is 
also the boundary of an inclosure within an inclosure.] 
Hence the total radiation, either way, passing perpen- 
dicularly across an elementary unit of surface inside the 
inclosure, must be the same, to the quality and quantity of 
each of its components, for all positions and for all orienta- 
tions of the element. For it must remain the same what- 
ever additional bodies at the same temperature are inserted. 
As one of these bodies may be a black body, the total 
radiation as above defined must obviously be that of a black 
body at the particular temperature. This is the aggregate 
of an infinite number of components, each belonging to a 
definite wave-length. 

This statement virtually contains the whole theory. To 
make its consequences more clear, we must now fix our 
attention on one of the components only. In front of a 
black body in the inclosure, this is supplied entirely by 
emission, because such a body can neither reflect nor 
transmit. But if we now suppose a plate of the substance 
(5) of 294, at the proper temperature, to be interposed 



UNIVERSITY OF CALIFORNIA 



256 HEAT. [CHAP. 

between our place of observation and the black body, the 
radiation of each particular wave-length will remain un- 
changed, even of that to which alone the plate is 
impervious. Hence what the plate absorbs on one side it 
must, under the circumstances supposed, radiate on the 
other, exact both as to quality and quantity. 

The student may exercise himself profitably in obtaining 
the same result by supposing the plate to be of either of the 
ideal kinds (4) or (6) of 294. And he may then make use 
of other suppositions of a less restricted character. From 
all, however, he will find the same common result. We 
must have proper terms in which to express it. 

Let us then define as follows : 

The Emissivity of a body, for a particular wave-length, is 
the ratio of that part of its radiation to the corresponding part 
of the radiation of a black body at the same temperature. 

The Absorptive Power, for a particular wave-length, is the 
fraction expressing the portion of incident radiation of that 
wave-length which is absorbed at a given temperature, the 
rest being reflected, scattered, or transmitted. [Strictly, the 
radiation here referred to is that from a black body at the 
same temperature as the absorbing body ; but we assume, 
with probability, that these restrictions are unnecessary.] 

With these definitions we see that for all bodies : 

The emissivity, and the absorptive power, at any one tem- 
perature, and for any definite wave-length, are equal to one 
another. 

296. This is the general proposition, undoubtedly true, 
as already stated, in a statistical sense. The proof which 
we have given, or indicated, above was (in all essential 
particulars) communicated by Stewart to the Royal Society 
of Edinbicrgh in March 1858. His process, so far as theory 
is concerned, is apparently much more elaborate than that 



xvii.] RADIATION AND ABSORPTION. 257 

just given, and was applied chiefly to obscure radiations. 1 
Still more elaborate is that given by Kirchhoffin 1859, and 
since developed by others with the help of an imposing array 
of symbols. But all these methods really involve no more 
than we have given above, and the unnecessary complica- 
tion, which adds nothing to the soundness of the proofs, 
tends to prevent the reader from appreciating the compara- 
tively simple, though not rigorous, foundations on which all 
of them are ultimately built. 

297. The dynamical analogy, given by Stokes about 1850, 
was called forth (like the investigations of Kirchhoff) by 
the phenomena which gave the first hint of the modern 
process of Spectrum Analysis. Here we must make a 
slight digression. 

Fraunhofer observed in 1817 that the flame of a candle, 
when examined by what we now call a spectroscopic pro- 
cess, showed a double bright line in the orange region 
exactly where the solar spectrum showed the double 
dark line which he called D. Hallowes Miller, with 
improved optical means, found the coincidence abso- 
lute. But Foucault, in 1849, went farther. Finding this 
double bright line in the electric arc, he examined the 
spectrum ,of sunlight which had passed through the arc, and 
to his surprise observed that the double dark line appeared 
darker than before. He then, by means of a mirror, 
reflected through the arc the light from one of the incan- 
descent carbon points. This light (from a nearly black 
body) gives of itself a continuous spectrum, parts of which 
are somewhat brighter than others. But when it was 
analysed, after passing through the arc, it was found to 
have dark lines exactly where the arc itself gave bright 
ones. 

1 See his own statement in Phil. Mag. 1863, i. 354. 

S 



258 HEAT. [CHAP. 

298. Stokes explained the phenomena by an analogy 
drawn from sound. Suppose a space to contain a very 
large number of tuning-forks or pianoforte wires all tuned 
to the same note. If they were set in vibration, that 
particular note alone would be given out. But if a listener 
were placed at one side of the space, while a cornopean 
was played at the other, the listener would hear (except 
so far as mere obstruction is concerned) any note played, 
except the particular note to which the strings are tuned. 
The strings, in fact, would all be set in vibration by that 
note, whose energy would thus be greatly diminished. 
Thus the strings form a medium which, when agitated, 
gives one definite sound ; and which is, at the same time, 
specially opaque to that particular sound when it originates 
from an external source. 

This is precisely the equivalence of radiating and absorbing 
power for one definite wave-length. And it may be the 
very sort of mechanism on which all radiation depends. 

299. Now #11 flames, especially in places near the sea, 
contain small quantities of common salt, and Swan had de- 
finitely proved that the double bright line observed by 
Fraunhofer in a candle-flame is due to the. presence of 
this substance. Hence it followed from Stokes's reason- 
ing that the production of the double dark line D in the 
solar spectrum must be due to salt, or rather to metallic 
sodium in a vaporous form, somewhere between the body of 
the sun and the earth. 

This part of the subject, so far as light is concerned, 
was very ably worked out by Kirchhoff ; but the more 
close consideration of it belongs to treatises on Light, and 
specially to those on Spectrum Analysis. 

300. There is, however, one part of the question to 
which, though it may easily be deduced from Stokes's 



xvn.J RADIATION AND ABSORPTION. 259 

dynamical analogy, Stewart and Kirchhoff first called 
attention. That there may be a reversal as it is called, i.e. 
a dark line in a spectrum instead of a bright one, the 
absorbing medium must be at a temperature so much lower 
than the source that a black body, at that temperature, 
radiates less of the particular ray than does the source. 
This follows at once from the main proposition ( 295), 
coupled with the fact that the emission of a black body, 
for any one wave-length, increases as the temperature is 
raised ( 277). This also can be derived from the general 
theory of 295 by supposing two bodies only to fae 
in our inclosure, both of the same material, of the species 
(5) of 294, but originally at different temperatures. 

KirchhofFs investigation was direct, and specially limited 
to single, definite wave-lengths. And it was experimentally 
verified by the fact that when the source was an incandes- 
cent lime-ball, and the absorbing medium the flame of a 
Bunsen lamp, no reversal of the sodium line took place ; 
while, when a spirit-flame was substituted for the Bunsen, it 
was obtained at once. 

Stewart also operated with light, but did not confine him- 
self (as he might easily have done) to one particular wave- 
length. He showed that red glass, for instance, loses all 
colour in the fire when it is at the same temperature as the 
coal behind it, appears red when a hotter coal is behind it, 
and green (the complementary colour) when the coal 
behind it is not so hot as it is. 

301. When giving the general demonstration in 295, we 
did not allude to polarization as one of the characteristics of 
a ray. This we purposely left to a subsequent section, to 
avoid confusion to the reader. But, if we now revert to the 
demonstration above, we easily see that a few additional 
words will meet this consideration also. For, among the 

S 2 



260 HEAT. [CHAP. 

bodies in the inclosure, some may polarise by reflection, 
some by transmission, as we know they do with light-rays. 
But the radiation must be throughout of the identical char- 
acter of that from a black body, and thus cannot be polar- 
ized. (If this were not so, the mere turning of the black 
body into a new position would alter the character of the 
radiation in the inclosure.) Hence a body which polarizes 
the reflected radiation in one plane, must give off by emis- 
sion in the same direction light polarized in a plane perpen- 
dicular to the first. And similarly with a tourmaline plate 
which polarizes by transmission. It does so because it 
absorbs one of the polarized rays produced by its double 
refraction ; and, absorbing, it must also radiate, that 
definite polarized ray. 

This very beautiful and important instance of the truth of 
the general theory was given, independently, by Kirchhoff 
and by Stewart. 

302. Other instances, but not so simply or directly con- 
nected with the theory, may be given in great numbers. 
We take only a few common ones, easily tested. 

If we write with ink on a piece of polished platinum 
foil, and heat the foil in a non -luminous flame, such as that 
of a large coal-gas blowpipe, it becomes covered with a 
thin dark coating of oxide of iron where the ink was applied. 
When the whole is very hot, the writing appears bright on 
a dark ground. It does so by radiating more than does the 
polished surface. But, because it radiates more, that part of 
the foil is permanently colder, as is seen when the writing is 
turned to the flame, for the back of the foil now shows 
dark letters on a light ground. 

If pieces of china-ware or crockery, with a dark pattern 
on a light ground, are strongly heated in a bright fire, 
we find, on examining them in a dark room, the pattern 



xvn.] RADIATION AND ABSORPTION. 261 

reversed. It is very instructive to flash a ray of sunlight on 
them at intervals, and then withdraw it, while they are still 




very hot The reversed and direct patterns are thus seen 
alternately in rapid succession. 

When we heat a piece of gas-coke to a high temperature 
it becomes brilliantly luminous. But rock-salt melted, and 
at the same temperature, is scarcely visible in the dark. 
The one is very nearly a black body, the other almost a 
transparent one. 

When an incandescent body, in an inclosure, is separated 
from a colder body by a screen, the ultimate result (unless 
the screen is a perfectly reflecting body, in which case there 
are two separate inclosures) must of course be equalisation 
of temperature, whatever be the particular classes of rays 
absorbed by the screen. Hence if the screen be very 
transparent for the obscure rays (which have by far the 
greater share of the whole energy emitted) while practically 
opaque to the luminous rays, the colder body may be 
raised to incandescence before the screen gives off any 
visible rays. Thus a burning lens of rock-salt is but 
slightly heated by direct sunlight even when its surfaces 



262 HEAT. [CHAP. 

are smoked : and, therefore, if of sufficient aperture, performs 
its proper function of raising bodies to incandescence at the 
focus, though it is practically opaque to visible rays. Instead 
of smoking the lens, we may absorb most of the luminous 
rays by means of a strong solution of iodine in bisulphide 
of carbon, as suggested by Debus. Caution is required if 
this solution is interposed between the lens and the body to 
be heated. Akin and others have imagined that this ex- 
periment gives the opposite of Fluorescence i.e. that here 
the radiation is elevated in character, while in Fluorescence 
it is always degraded. But there is no analogy, for the 
phenomenon we have just described is a consequence of 
the general proposition ( 295), while Fluorescence is, as 
we shall see, at least apparently an exception to that theory. 
Of course it is to be remarked that the theory does not 
exactly apply to this phenomenon, for it is essentially con- 
nected with a state in which thermal equilibrium is not yet 
attained; while the theory is based entirely upon the 
assumption of thermal equilibrium. 

303. In all that precedes, we have for simplicity assumed 
that emission is a mere surface-operation, like reflection or 
scattering. For these last never take place in the interior 
of a body except where there is heterogeneity, and in that 
case the interface between different portions may be looked 
upon as a netv surface. And (except in a passing hint in 
the preceding section) we have entirely omitted all reference 
to fluorescence. 

But the first inspection of a wedge of coloured glass 
(when the colour is not applied as a mere surface layer) 
suffices to show that absorption is not confined to the mere 
surface ; and we should consequently expect to find that 
emission also takes place from below the surface as well as 
directly from it. Stewart proved experimentally that the 



xvii.J RADIATION AND ABSORPTION. 263 

radiation from a thick plate (of a body neither black nor 
transparent) is greater than that from a thin plate of the 
same substance,- both being at the same temperature. 

304. A very simple calculation enables us to trace the 
effect of thickness on the direct radiation from a plate. 
Let R represent the direct radiation of a definite kind, 
when the plate is of unit thickness (the unit may be chosen 
as small as we please), and a the corresponding absorptive 
power for the same radiation at the same definite tempera- 
ture. Then a second unit plate of the same substance, 
at the same temperature, placed in front of the first, and 
so close as to be virtually one with it, radiates also R, 
but stops aR of that emitted by the first. The whole 
radiation from the double plate is thus : 

R(l + (I -a)). 

A third unit plate placed in front of these contributes R 
and stops aR (i -f (i a) ). Hence the whole effect of 
three plates is 

R(i +(i -a) + (i -a) 2 ). 

The law is now obvious ; and, by summing a geometric 
series, we find for the radiation of a plate n units thick 



However small may be the value of a, provided it do not 
vanish, (i - a)" vanishes when n is infinite. This gives 
the utmost amount which a mass of the substance can 
emit, and the value is then 

R_ 
a 

This is, of course, equal to its absorptive power for the 
corresponding radiation from a black body of the same 



264 HEAT. [CHAP. 

temperature. Hence if a stand for this perfectly definite 
quantity, we have 

R = aa; 

and the preceding expression for the direct emission from a 
plate of n units thickness becomes 

a (i - (i - a) -)- 1 

The following little table will be useful in showing the 
consequences of this expression. It gives the values of 
the quantity i-(i-a)*, the corresponding values of a 
being in the upper line of the table (i.e. the values of the 
expression for =i), and those of ;/ in the left-hand 
column. The numbers in the table may of course be taken 
as representing the relative absorptions by different thick- 
nesses of the plate. Their defects from unity give the 
corresponding transmissions : 



I 


O'l 


O'OI 


O'OOI 


O'OOOI 


10 


0-651 


0-096 


O'OIO 


O'OOI 


100 


0-99997 


0-634 


0-095 


O'OI 


IOOO 


I'O 


0-99996 


0*632 


0-095 


1 0000 


I'O 


I'O 


0-99995 


0-632 


lOOOOO 


I'O 


I'O 


I'O 


0-99995 



305. One very remarkable conclusion from the above 
expression is that a stratum of any substance, however 
slight its emissivity for particular radiations, will, if only 

1 The above is not rigorous, Lut is sufficient for our purpose. To 
obtain a rigorous investigation we may put a/m for o, and mn for n, 
and then make m infinite. Or we may proceed as follows : Let 
R, a, be such that a plate of thickness 5 x, at the given temperature, 
emits RSx of a particular wave-length, absorbing a8x of that of the 
same wave-length, from a black body. Then, if <>(jr) be the whole 
radiation of that kind from a plate of thickness x, we have at once 
<J> (x + Sx) = (i - a8x) <f>(x) + fiSx, so that </>' (x) = R - a<f)(x'), and 

f(*)=(l--* )=(!--* ). 

a 



XV! i] RADIATION AND ABSORPTION. 265 

thick enough, behave exactly as a black body. For, in all 
probability, all actual substances emit, when hot enough, 
though in very different amounts, every ray. Thus we can 
understand why the body of the sun, though probably 
gaseous (the greater part of its materials at all events being 
at temperatures far above their critical points), radiates as if 
it were a black body. 

Another very important conclusion is that a body, whose 
absorption is specially and sharply selective, may stop a 
large percentage of the incident radiation from a black 
body very early in its career, while the rest may pass 
through considerable additional thickness with compara- 
tively little loss. 

It is quite otherwise with a body whose absorption is not 
selective, even though it be nearly transparent ; such a 
body, in great thickness, is nearly opaque. 

306. From the general statements above, it is clear that 
experimental determinations connected with this subject 
may be made either directly on the radiation from a sub- 
stance, when it is hotter than the surrounding bodies, or 
indirectly on the radiations which it allows to pass through 
it from a hotter body. 

Such experiments, direct or indirect, are comparatively 
easy when the radiation is visible, although even then there is 
one insuperable difficulty, viz., the eye estimation of relative 
brightness of light of different wave-lengths. This, however, 
belongs properly to Photometry r , a branch of practical Optics. 
We may merely mention that the principle usually employed 
is that of the law of diminution of intensity of radiation 
with increase of distance from a point-source ( 280). 
The two sources to be compared illuminate each a portion 
of a screen, and their relative distances from the screen are 
adjusted till the eye judges these portions to be equally 



266 HEAT. [CHAP. 

illuminated. But it is quite clear that rays belonging to the 
middle portion of the range of visible radiation will, by 
this process, be favoured at the expense of those belonging 
to the ends of the range. 

The same is true of all photographic processes. Each of 
these, like the eye, has its particular range (sometimes 
above, sometimes in, and sometimes altogether below, the 
visible spectrum), the middle part of which it especially 
favours. In fact, the true use of any photographic process 
is to detect special regions of very great absorption. The 
only true method of comparing radiations of very different 
wave-lengths is to convert them into heat, and measure the 
amount produced in a given time. This could be perfectly 
effected if we had bodies such as (6) of 294, each trans- 
parent for one definite group of wave-lengths only, and 
perfectly reflecting all others ; and if we could keep the 
normal temperature of our pile or thermometer close to 
absolute zero. These conditions are alike unattainable, so 
we must employ the least faulty method at our disposal. 

307. One such method depends on one of the early 
results obtained by Young, when he revived Huyghens' 
Undulatory Theory of Light, and developed it by the h'elp 
of his own doctrine of Interference. He showed that if a 
parallel beam of white light passed from a distant source 
perpendicularly through the plane of a grating of equi- 
distant parallel wires, and were afterwards received on a 
screen, the white spot (which would have been formed on 
the screen had no grating been interposed) would be 
weakened, and accompanied by a series of coloured spectra. 
These are arranged symmetrically on each side of the white 
spot, in a line perpendicular to the wires of the grating. 
The amount of displacement of any particular colour from 
the central white spot is very nearly in direct proportion to 



xvn J RADIATION AND ABSORPTION. 267 

the corresponding wave-length. It is also directly as the 
distance of the screen from the grating, and inversely as the 
distance between the axes of the successive wires. By 
adding, after Newton, the refinement of a narrow slit 
(parallel to the wires) through which the light falls on the 
grating, and an achromatic lens of long focus between the 
grating and the screen, the various wave-lengths can be so 
perfectly separated from one another that the chief 
Fraunhofer lines are distinctly shown when sunlight is 
employed. 

The distribution of energy in such a spectrum can now 
be studied directly by means of a thermo-electric pile, 
whose junctions are all arranged in one line parallel to the 
wires of the grating. Placed in any position on the screen 
it gives, by the deflection of the galvanometer, a- measure 
of the amount of energy of the original radiation which is 
comprised between two wave-lengths differing by an amount 
proportional to the breadth of the face of the pile. The 
pile can be made to travel, keeping its length parallel to 
the wires of the grating, all along the spectrum, by means 
of a screw motion. 

308. The objections to this method are very numerous. 

(i.) The greatest intensity of the spectral radiation is 
necessarily only a very small fraction of the whole radiation. 

(2.) The lengths of the spectra, so far as visible radiations 
are concerned, are necessarily very small, because, for our 
present purpose, we must use actual wires or something 
equivalent, and we cannot well have more than a hundred 
or so per inch. 

In such a case, even if the slit and the screen were each 
twenty feet from the grating, the utmost deflection of the 
red of the first spectrum from the central line would be 
two thirds of an inch. 



268 HEAT. [CHAP. 

[Gratings, such as are employed for optical purposes, may 
have many thousand lines per inch; but their heat indications 
must be received with caution, as they are usually formed 
of lines ruled with a diamond on glass or speculum metal ; 
and the various classes of obscure radiations may be treated 
very differently (one from another) by passage through 
glass, or by reflection from metal. Thus, such gratings 
give perfect indication of intense special absorption (such 
as the Fraunhofer lines), but require supplementary evi- 
dence when they appear to weaken a large region of the 
spectrum.] 

(3.) As the pile deals with all radiations, and not merely 
with those which are visible, the effect upon it in any 
position is due to a particular wave-length of the first 
spectrum, radiation of half that wave-length from the 
second spectrum, &c. 

(4.) The statements made above hold only for waves 
whose length is considerably less than the distance between 
the axes of successive wires. With longer waves the 
phenomenon is of a different character. 

(5.) The comparison, at different parts of the spectrum, 
ought to be between groups of wave-lengths whose extreme 
ratio in each group is the same ; whereas this method deals 
with groups in which the extreme difference is the same, 
thus giving the shorter waves an undue advantage. 

309. When we try a refraction spectrum, instead of a 
diffraction one, we are met by a somewhat different, but 
quite as formidable, array of difficulties. Some of these 
may probably be got over by the recent improvements in 
photography, which enable us to utilise the obscure rays of 
longer wave-lengths than red, just as the old methods 
employed chiefly obscure rays of wave-lengths shorter than 
violet. 



xvii.] RADIATION AND ABSORPTION. -^69 

(i.) We must have independent means of discovering the 
absorptive power of the material of our prism for each 
obscure wave-length. We cannot trust even such a sub- 
stance as rock-salt, without direct proof that it has not 
some special absorption for a definite region of the 
spectrum. 

(2.) Even supposing we had this, the crowding together, 
according to an as yet unknown law, of the less refrangible 
rays, offers a very serious difficulty. With light we know 
that, to a rough degree of approximation, the refractive 
index of a transparent body can be expressed by the 
formula 



where X is the wave-length, and A and B are special con- 
stants for each medium. If this law holds, even approx- 
imately, for wave-lengths much exceeding those of visible 
light, there must be an immense assemblage of radiations 
for which the refractive index is practically the same ; and 
the intensity of the whole radiation in a given fraction of 
the length of the spectrum must increase (provided there is 
no special absorption) steadily to the limit, for which p = A. 
But, in practice it has been found impossible to prevent 
scattering, however carefully the substance used as a prism 
has been selected, and however truly its surfaces have been 
worked into polished planes. 

Again, it must be noticed that what the pile indicates 
is the excess of the radiation reaching it over that which is 
leaving it. Thus, all its indications are in defect, and to a 
greater amount the higher is its mean temperature. 

Many more difficulties might be urged, but those specified 
are sufficient. 

310. The remarks just made must have prepared the 



270 HEAT. [CHAP. 

reader to find that our positive knowledge on this subject, 
so far at least as obscure rays are concerned, is of very 
slender amount ; and that even our general information is 
of a somewhat doubtful character. 

Attempts have been made to exhibit graphically the 
intensity of visible rays at each point of the spectrum. 
These are all, however, doubtful on account of the difficulty 
already hinted at ( 306) of estimating the apparent relative 
brightness of rays of different colours. There is a general 
consensus (except with some of the colour-blind), that the 
maximum is somewhere in the yellow, but (as might be 
expected) not much farther agreement. 

W. Herschel was the first to point out that the hottest 
part of the solar spectrum (as determined by a thermometer) 
was, not only not coincident with the brightest -part but, 
situated differently with regard to the visible part according 
to the material employed as a prism. It is in fact within 
the visible part when a prism of water is used, but outside 
it and beyond the red end when the prism is of rock-salt. 
The reason is obviously the transparency of rock-salt, and 
the opacity of water, to the obscure radiations. 

311. Melloni, Forbes, and others have tried to meet 
some of the difficulties of this problem in another way, 
viz. by measuring directly the amount of absorption of any 
one body for radiations from a series of sources at different 
temperatures ; e.g. a Leslie's cube ( 291), blackened copper 
at 200 C., 400 C, &c., a Locatelli lamp, a white-hot 
platinum spiral, &c. But all such observations are seen to 
be of comparatively little interest when we look at them 
from the point of view suggested by the behaviour of 
coloured glass. For there we find that an apparently trivial 
quantity of foreign matter (cobalt, manganese, gold, &c.). 
entirely alters the behaviour of the glass as regards visible 



xvii.] RADIATION AND ABSORPTION. -27* 

radiations. There can be little doubt that this effect 
extends, to at least as formidable an amount, to the 
behaviour of bodies with regard to obscure radiations. And 
this prepares us to expect, what in fact we find, that there 
can be no close agreement between the results obtained 
by different experimenters unless they have operated, not 
only on the same substance, but on the same specimen of 
the substance, so long at least as it is a natural and not 
an artificial product. 

312. When substances can be obtained chemically pure, 
we naturally expect a much closer agreement between the 
independent results of different experimenters. Thus all 
experimenters seem to agree that water, and aqueous 
solutions (especially that of alum), are singularly opaque 
to the lower obscure radiations. There is also a fair agree- 
ment as to the behaviour of gases ; some (as dry air for 
instance) being found to exert very slight absorption on 
obscure rays in general, others (like olefiant gas) being 
powerful absorbers of such rays. This subject has been 
elaborately investigated by various experimenters. We are 
prepared of course to find such differences by our knowledge 
of the existence of coloured gases ; such as chlorine, and 
especially nitrous acid which (as Brewster showed) can 
be rendered practically opaque to sunlight, even when in 
small thickness, by being sufficiently heated. 

313. But when we come to vapours the question is not 
by any means so definitely answered. There can be no 
doubt that even a small amount of water-substance in air 
considerably increases the absorption of obscure radiations. 
But we do not yet know whether' this effect is general or 
selective. And more, it has not yet been decisively shown 
that this effect is wholly due to vapour of water. We all 
know how slight a cloud or fog is sufficient greatly to mitigate 



272 HEAT. [CHAP. 

the glare of the sun, while stopping his heat as well as his 
light. This can be accounted for, to some extent, by the 
mere reflection, &c., from the small drops of water. But if 
we were to attempt a rough approximation to the brightness 
of a cloud in full sunshine by assuming that it sends back to 
a hemisphere the amount of light which it receives from the 
sun (a disc of o-25 in radius); i.e. its brightness would, 
on this hypothesis, be to that of the sun as i : 120,000 
nearly. This would be nearly equivalent to assuming that, 
if the whole sky were covered with cloud as bright as it is 
in full sunshine, the amount of light we should receive would 
be the same as that from the direct sun alone. This estimate 
must be very considerably above the truth. For though 
clouds in sunshine appear brighter than the moon, yet the ratio 
of full moonlight to sunlight is given by Herschel (after 
Wollaston) as about i : 800,000, only. Thus the amount of 
sunlight reflected by a cloud must be only about \ or \ of 
the whole. The greater part of the remainder is absorbed. 

The direct heat-radiation from the sun is felt powerfully 
even when the air appears to be saturated with moisture ; 
as in the intervals between heavy showers, the " clear 
shining after rain." But the showers have extracted, 
at least in great part, the dust nuclei which are required for 
condensation ( 176); and it would appear that such 
saturated air, if only supplied with a little dust, would at 
once become practically opaque to sun-heat by the forma- 
tion of mist or clouds. Still, we must also bear in mind that 
a rain-cloud is the summit of an ascending column of 
warm, moist air, and therefore the interval between two 
clouds is in great part a descending column of drier air. 

314. The absorption due to true water-vapour, both as to 
quality and as to quantity, has been a specially vexed 
question for the last twenty years, and still remains 



xvii.] RADIATION AND ABSORPTION. 273 

unsettled. All that can yet be definitely said on the subject 
is that, for obscure radiations as a whole, true water-vapour 
is certainly much more transparent than olefiant gas, and is 
possibly much less transparent than dry air. Its main 
absorption is probably for certain definite obscure radiations. 
This question may ppssibly admit of solution by the recent 
invention of Abney, which gives the means of photographing 
the obscure radiation. Had steam an absorptive (and therefore 
a radiative) power at all approaching to that of olefiant gas, 
there would be practically no difference in the amount of 
dew after a clear night and after a cloudy one : for, if the 
night were clear and the air nearly dry, there would be 
considerable cooling of the ground, but little dew to fall 
and, if the air were moist, there would be material for dew 
but little cooling to precipitate it. Practically, the results 
of Wells are independent of the absorption due to true 
water-vapour. 1 The great difference between the effects of 
true water- vapour and cloud or mist is seen at once by any 
one who watches the puffs of steam from a locomotive. 
These are practically transparent for the first few feet above 
the funnel, but become almost absolutely opaque when 
condensation takes place. 

315. There is yet another class of phenomena, already 
referred to, which claims mention here, though (from its 
very nature) it has hitherto been observed only in connec- 
tion with visible radiations, and therefore is more properly 
treated in works on Light. This is designated by the terms 
Fluorescence and Phosphorescence, which are probably mere 
varieties of one phenomenon. 

By Phosphorescence is meant something very different 
from the luminosity (in the dark) of a stick of phosphorus 
or a line drawn with it on a wall, of decaying wood, or of 

1 W. Thomson, Proc. R.S.E., v. 203 (1864). 

T 



274 HEAT. [CHAP. 

dried fish. These probably depend upon slow oxidation, or 
other chemical combination, and therefore belong to com- 
bustion ; though, in some special cases, they are known tx> 
be due to colonies of bacteria. But it was known to the 
ancients that certain gems, after being heated, continue 
luminous in the dark long after they have cooled down to 
the temperature of the air. The properties of the Bologna 
stone (sulphide of barium) were discovered in the seven- 
teenth century, those of Canton's phosphorus (sulphide of 
calcium) in the eighteenth. And now we have many varieties 
of chlorides, sulphides, &c., which, like these preparations, 
shine brilliantly in the dark for hours after they have been 
exposed to sunlight or to a burning magnesium wire. 
Recently these bodies have been applied to various purposes 
as the bases of Luminous Paints. The curious property 
exhibited by them is correctly called Phosphorescence. 

Certain crystals of greenish Fluor-spar, a piece of canary 
glass (coloured with oxide of uranium), a decoction of 
horse-chestnut-bark (esculine), and a slightly acidulated 
solution of sulphate of quinine, show each a peculiar surface 
colour of its own even in ordinary daylight, much more 
strongly in sunlight or electric light. A brilliant investi- 
gation by Stokes, in 1852, showed that this was due to the 
giving out, as altered light, of light absorbed by the body : 
in fact, Stokes's paper has the title On the Change of 
Refrangibility of Light. This is correctly called Fluores- 
cence. It appears that in all cases the wave-length of 
the emitted rays is greater than that of the rays absorbed. 

Fluorescence appears to differ from true Phosphorescence 
mainly in being of very much shorter duration. But we 
must refer to works on Light for the description of the 
Phosphoroscope, and the results obtained by its use. 

316. The mechanism by which this apparent storage 
and subsequent doling out of light are produced is still 



xvii.] RADIATION AND ABSORPTION. - 275 

somewhat obscure, and the hypotheses which have been' 
advanced by Stokes and others are too difficult to be 
treated here, especially from the point of view of the 
general proposition ( 295). 

We merely refer again to the serious limitations ( 292) 
under which the whole theoretical part of this subject 
admittedly still lies; and point out the apparent incon- 
sistency which appears between the two statements : (a) that 
the emission of any particular radiation by a given body 
depends on its temperature only, increasing in intensity 
as the temperature rises ; and (b) that certain fluorescent 
bodies give out visible radiations at temperatures at which 
even a black body would emit obscure radiations only. 

317. Resume of 288-316. Pre vest's Theory of Ex- 
changes. Radiation as depending on Nature of Surface. 
Extension of Prevost's Theory. General relation between 
Emissivity and Absorptive Power. Physical Analogy. 
Spectrum Analysis. Internal Radiation. Absorption by 
plates of different thickness. Distribution of Energy in 
the Spectrum. Absorption by gases and vapours. Fluor- 
escence and Phosphorescence. Defects in the Theory. 



T 2 



CHAPTER XVIII. 

RADIATION. 

318. So far we have been dealing mainly with the 
relations between the absorptive and emissive powers of a 
body, or bodies, at one definite temperature. And we 
have had, to some extent at least, the help of theoretical 
considerations. 

But we must appeal directly to experiment for informa- 
tion as to the comparative amounts of emission or absorp- 
tion by the same body at different temperatures, whether 
these refer to the radiations as a whole or are confined to 
one particular wave-length. Mechanical analogies, which 
were of considerable assistance in the former part of the 
work, fail us here because of our ignorance of the nature of 
the vibrations of the particles of a hot body, and of the 
mode in which their energy is transferred to the ether. All 
that we have yet arrived at, in this connection, is that a rise 
of temperature in the radiating body is accompanied by 
an increase of emissive power for each kind of radiation. 
Whether this is nearly the same for different wave-lengths 
or not, we have had as yet no information. 

319. It must be remembered that we are now dealing 
with emission alone ; not with the whole radiation, which, 
as we have seen, is the same for all bodies in our imaginary 



CH. xvili.] RADIATION. 277 

inclosure. With black bodies this is wholly emission, but 
with other bodies it is partly emission, partly transmission, 
and partly reflection. For simplicity we will, therefore, at 
first confine ourselves to black bodies. And, for comparison 
with Pre vest's Theory of Exchanges ( 290) we will suppose 
that our inclosure contains two black bodies at different 
temperatures, and these only. Farther, to approximate as 
nearly as may be to a case realisable in actual experiment, 
\ve will suppose that the colder black body lines the inclo- 
sure, w4iile the hotter is suspended within it. This is nearly 
realised in the case of a thermometer, with a very large 
blackened bulb, inclosed in a receiver blackened internally 
and exhausted of air a form of experiment which is easily 
carried out. To avoid the difficulties of convection cur- 
rents, &c., in the liquid filling the bulb, we may suppose 
the blackened body to be a large solid ball made of a good 
metallic conductor, with a hole in it containing mercury, 
into which the bulb of an ordinary thermometer is inserted ; 
the stem of the thermometer passing air-tight through the 
inclosure. If the inclosure be also of good conducting 
metal, we may suppose it to be kept at a constant tempe- 
rature by immersion in a large vessel full of water, which 
is constantly renewed from a cistern. The ball is heated, 
once for all, inserted with the thermometer in the inclosure, 
and the vacuum made as speedily as possible. The read- 
ings of the thermometer are then taken (as in 180) at 
intervals of a minute. The water-equivalent of the ball, 
mercury, and thermometer-bulb together are found by the 
proper methods, and enable us to calculate the number 
of units of heat lost for the fall of temperature observed. 
This number, divided by the number of square units in the 
surface of the ball, gives the loss of heat by emission per 
unit of surface, per minute, at each temperature. 



2?S HEAT. [CHAP. 

320. If the absolute temperature of the ball be /, and 
that of the inclosure &, and if /(/) represent the rate oi 
surface-loss per unit surface at temperature /, the quantities 
thus calculated from the results of experiment are the 
successive values of 



321. The earliest speculations on the subject, including 
those of Newton, assumed that this quantity is proportional 
simply to the difference of temperatures 

'-'o, 

so that, for all temperatures, 

where A and B are constants. This is the assumption 
made (even when convection also comes into play) in all 
the theoretical writings of Fourier. And it is found to be 
approximately true so long as the temperature-differences 
are not above a few degrees. 

322. Dulong and Petit, by a very careful series of ex- 
periments, found that this approximation errs in defect, 
even for moderate temperature-differences, and that the 
percentage of error rapidly increases as these differences 
increase. But they found that the results of all their 
experiments, up to temperature-differences of 200 C. at 
least, when the inclosure was at temperature from o up 
to even 80 C., could be well represented by an exponential 
formula of the form 

/(/) = A a< + B. 

The quantity A was found to depend upon the nature of 
the radiating surface onjy; while a = 1*0077, is, according 
to these experimenters, practically the same for all bodies. 
\B, of course, cannot be found experimentally : but should 



XVIIL] RADIATION. 279 

be equal to A if this formula were true for all tempera- 
tures down to absolute zero, for there obviously can be no 
radiation when t o.] 

323. De la Provostaye and Desains have verified the 
consequences of this formula from o C. to 200 C. ; but 
they state that above the latter temperature it is no longer 
applicable. 

The simplicity of this expression, especially as a was 
found sensibly the same whatever was the nature of the 
emitting surface, might lead us to fancy that it might be 
obtained as a consequence of theory, but we have as yet no 
information about the mechanism of emission on which to 
base a theory. 

324. Accepting it, however, as a fair approximation to 
experimental results between the limits assigned above, we 
find for the rate of surface-loss 

/(/)-/(/=)== .4 a'o(a'-'o-l). 

Thus we have the approximate statements : 

For a given temperature of the indosure, the rate of surface 
loss is proportional to 

(1-0077)* I 

where is the difference of temperatures in degrees C. 

For a given difference of temperatures the rate of surf ace loss 
is proportional to 



where t<> is the absolute temperature of the indosure. 

It therefore rises, in geometrical progression for equal 
increments of temperature. 

325. The first of these results shows that the rate of 
cooling, for moderate temperature- differences, is propor- 
tional to 

(i +0-0038 6/-&C.). 



23o HEAT. [CHAP. 

from which we see the nature of the approximation in what 
is called Newton's Law. The error is about 4 per cent., 
in defect, for a temperature difference of 10 C. 

We have no very definite information as to the actual 
value of the quantity A. According to the experiments of 
Hopkins, it is about th of a unit of heat per minute, per 
square foot, from a surface of glass. From this it follows 
that the radiation from glass at iooC. to an inclosure at 
oC. is about one unit of heat per square foot per minute. 

326. Though the law of Dulong and Petit is found to 
hold approximately for a considerable range of temperature, 
and for surfaces of all kinds, it does not appear to hold 
even approximately for each separate wave-length. If it 
did so, the character of the radiation from a black body 
(i.e. the proportions in which its whole energy is distributed 
among the various separate wave-lengths) would be the 
same throughout that range of temperature. Thus the 
percentage absorption by one and the same body would 
be the same for radiations from all black sources at tem- 
peratures within that range. This is at variance with 
the great majority of Melloni's results. Thus he found for 
the percentage absorption of heat from different sources by 
plates of equal thickness at ordinary temperatures : 





Blackened 


Copper 


Platinum 


Locatelli 




at 100 C. 


at 400 C. 


incandesced 


lamp. 


Rock-salt . 


. . 8 . . 


. 8 . 


. . 8 . 


. 8 


Fluor spar . 


. . 67 . . 


. 58 


. . 31 


. 22 


Iceland spar 


. . IOO . . 


94 . 


.' '. 72 . 


. 61 


Alum . . 


. . IOO . . 


. IOO . 


. . 98 . 


91 



We have added, for comparison, two other sources, of 
which one at least is certainly not a black body. 

327. Even a brief table like this seems to prove beyond 
all doubt (at least so far as the theoretical conclusions of 



XVIIL] RADIATION. 281 

Chapter XVII. may be trusted), that the amount of emis- 
sion of any particular radiation by a black body increases, 
after being first perceptible, more and more rapidly as the 
temperature is raised, thenceforward more and more slowly 
as it is still farther raised. 

[A desideratum, which could easily be supplied with 
modern apparatus, is a careful series of experiments like 
those of Melloni, made with the same blackened body as 
source, but taken successively at temperatures 50, 100^ 
150. . . 500 C higher than the absorbing plate and the 
pile. Such a series, for each of a well-selected set of 
absorbing plates, would be of high scientific value.] 

328. We must here mention a very curious speculation 
(due to Balfour Stewart) which, possibly from some inaccur- 
acies involved in the first statement of it, does not seem to 
have met with the consideration it certainly merits. It is 
based on what is called Doppler's Principle, which is merely 
an application of the reasoning by which Romer first 
measured the velocity of light. 

329. If a periodic disturbance of any kind (producing 
water-waves, sound-waves, light-waves, c., propagated at 
a uniform rate), take place at a centre, the number of these 
disturbances which reach a spectator in a given time 
depends on his relative velocity with regard to the centre. 
When he is approaching it they are more, and when he is 
receding from it less numerous, than if he were at relative 
rest We are not, however, entitled (without proof) to 
assume that the result will necessarily in all these cases be 
precisely the same as if the spectator and the centre had 
been at relative rest, while the period of the original dis- 
turbance was shortened or lengthened. To show the 
necessity for caution in this matter, let us suppose both 
centre and spectator to be at rest, but the wave -medium to 



:282 HEAT. [CHAP. 

be moving from one to the other. In this case the waves 
will be lengthened or shortened, but their rate of propaga- 
tion will be proportionally increased or diminished, so that 
the number received in a given time remains unaltered. 

But experiments on musical sounds produced by a trum- 
peter on a passing train have verified the principle for sound ; 
.and it has also been, to a certain extent, verified for light 
by the observed difference of refrangibility of definite lines 
in the solar spectrum according as the light comes from the 
-east or west limb of the sun. [The east side is approaching 
us, and the west side receding from us, in consequence of 
the sun's rotation, at such a rate that while 150,000 light 
waves of any species are given off, there reach us about 
150,001 of the same species from the east side, and 149,999 
from the west. Thus, if the spectra of these two sources 
of light be formed side by side by the same prism, the 
Fraunhofer lines in the two will not coincide. The relative 
displacement of each in the two spectra will be as if one 
had a wave-length about y^^th greater than the other.] 
A similar agreement with theory has been obtained with 
the light reflected from Venus. This principle has been 
successfully applied by Huggins and others to find the 
proper motions of the stars in the direction of the line of 
vision. Thus it is (so far) proved that radiation from one 
body to another depends upon their relative motion only, 
and not upon the motion of the ether relative to either of 
them. 

330. Assuming, then, the principle to be accurate (as it 
is at least partially verified by experiment) if in our inclo- 
sure ( 295) one of the bodies is moving relatively to the 
others, the only way in which we could conceive equality of 
temperature to be maintained would be by supposing the 
energy corresponding to each of the infinite variety of 



XVIIL] RADIATION. 283 

wave-lengths in the radiation from a black body to be of 
the same amount at any one temperature. This is certainly 
not the case ; and therefore, if our assumption is correct, 
it follows that relative motion of radiating bodies in an 
indosure is inconsistent with ultimate equality of temperature 
amongst them. 

Now differences of temperature involve Motivity (see 
Chap. XX). Kence it would appear, from the Second Law 
( 82), that the energy of relative motion among radiating 
bodies in an inclosure must gradually be diminished, with 
the result of raising the temperature of the whole contents. 
Compare the roughly analogous case ( 70) of the effect of 
tides on the earth's energy of rotation. 

[Other trains of thought lead to similar conclusions. 
Thus, if there be resistance by the ether to the motion of 
ordinary matter, it is clear that part at least of the energy of 
motion lost must be transferred to the ether. This may 
be given off (as we gather from Maxwell's beautiful theory) 
in various electrical forms, or wholly in the form of trans- 
verse vibrations or radiations (which, in Maxwell's theory, 
are a special case of magnetic and electric disturbances). 
But it would lead us altogether away from the proper business 
of a text-book to pursue this highly interesting digression.] 

331. The absolute amount of energy radiated from the 
sun is to us a matter of truly vital importance. It could be 
determined with great ease, and with considerable accuracy, 
were it not for reflection, absorption, and scattering, by the 
various constituents of the atmospheres of sun and earth. 
But it is not probable that we shall soon be able to estimate 
even roughly the effect of the sun's atmosphere ; nor is it, 
for our present purpose, of much consequence that we 
should. What we are most concerned with is how much 
radiant energy reaches the earth in a given time. 



284 



HEAT. 



[CHAP. 



Pouillet was the first who made a successful attempt to 
answer this question ; and his results have not been much 
altered by farther research. The ingenious and simple 
instrument he devised for the purpose is called the 
Pyrheliometer. 




332. It consists essentially of a short but wide cylindrical 
vessel constructed of highly-polished metal, one end only 
being covered with lamp-black. This is filled with water 
or mercury, and immersed in the liquid is the bulb of a 



A viii.] RADIATION. 285 

thermometer, whose stem protrudes along the axis of the 
cylinder. A circular metallic disc, equal in diameter to the 
cylinder, is supported at the other extremity of the stem. 
When the instrument is turned so that the shadows of the 
cylinder and disc coincide, we know that the blackened end 
of the cylinder must be turned exactly to face the sun. 
The water-equivalent of the cylinder and its contents, and 
the area of the blackened end, must both be carefully 
determined. 

Suppose the instrument, at the temperature of the air, 
to be shaded from the sun, and turned towards the sky. 
The thermometer will after five minutes show a definite 
fall of temperature, say / - 

Now turn the instrument towards the sun for five minutes, 
let us say. The thermometer rises through /. Turn it imme- 
diately to the sky, shaded as before, for five minutes. The 
temperature falls through t t . 

During the time of its exposure to the sun, its tempera- 
ture was steadily rising by solar radiation, but as steadily 
falling by its own radiation. Hence we conclude approxi- 
mately that it lost in temperature by cooling to the amount 



In fact, we take the mean of the two coolings, before and 
after exposure to the sun, as the cooling during that 
exposure. 

Thus the full gain from the sun would have raised it in 
five minutes by 



From this, and the numerical data for the instrument (which 
we have specified above), we calculate at once how many 



286 HEAT. [CHAP. 

units of heat, per unit of surface, have been received from 
the sun per minute. 

333. From a great number of experiments with this instru- 
ment, Pouillet concluded that the rate of rise of temperature 
due to direct solar radiation in a given time, at different 
hours of the same day, could be represented very closely 
by an expression of the form 



where A and e are constants, and r represents the length of 
the course of the rays through the atmosphere, as depending 
on the sun's altitude. 

A was found to be sensibly the same for all the days of 
observation, but e (which is of course less than unity) varied 
in a marked manner from day to day. 

Pouillet considered that these facts warranted his as- 
suming that the formula would hold good even if there 
were no atmosphere ; in which case we should have T = o, 
and the expression for the rate of rise of temperature would 
be A simply. 

Thus he calculated from the data of his instrument that 
the amount of heat which in one minute would reach 
directly a square centimetre of the earth's surface, if there 
were no atmosphere, would raise the temperature of 176 
grammes of water by i C. In the units we have hitherto 
employed this is about 3*6 units of heat per minute, per 
square foot of surface. From this it can be easily calculated 
that the total heat received from the sun in the course of a 
year would, if distributed uniformly over the earth, be 
capable of melting a crust of ice somewhere about 95 feet 
thick. 

334. From the average values obtained for the quantity 
e in his formula, Pouillet concluded that, when the sun is 



xviii.] RADIATION. 287" 

vertical and the sky clear, from eighteen to twenty-five per 
cent, of the solar heat is absorbed before reaching the 
earth ; and that (taking obliquity into account) half of the 
entire ;heat which comes from the sun to the illuminated 
hemisphere of the earth is absorbed by the atmosphere. 
Forbes' s experiments on the Faulhorn showed that on the 
clearest days the lowest 6,000 feet of the atmosphere 
absorbs one-fifth of the radiant energy received from the 
sun. 

335. The earth, as seen from the sun, is a disc occupying 
only about a3 oovbooT7g' ^ ^ ie sur f ac e of the celestial sphere. 
Hence we can calculate (assuming it to be uniform) the 
amount of heat lost by each square foot of the sun's surface 
in a given time. W. Thomson, using (along with Pouillet's 
just given) the data of Herschel, finds that the sun's 
radiation is at the rate of about 7,000 horse-power per 
square foot, about thirtyfold that of the same area of the 
furnace ^of a locomotive. Also that the whole heat which 
leaves the sun per annum is about 6 x io 30 units C. 

All these estimates, however, though they are almost 
certainly of the proper order of magnitude, must be looked 
on as mere approximations. 

336. The great majority of the experimental determina- 
tions of absolute rate of loss of heat have been made in 
air, so that convection as well as emission has a share in 
the effect. 

Here, again, Dulong and Petit furnish us with an em- 
pirical law. They find that the convection effect is entirely 
independent of the radiation ; and is the same at the same 
temperature for a particular body, whether the surface be 
polished or blackened. It depends, of course, upon the 
nature of the surrounding gas, and upon its pressure (p) as 
well as upon the temperature excess of the body. . This 



. 2 88 HEAT. [CHAR 

part of the loss of heat can, they find, be expressed in the 
form 

A p'(t **,)***. 

Here A is a special constant ; and ft varies from one gas 
to another, being about 0-45 for air. 

These experiments were made with an apparatus such as 
that described in 319 above, which could be rilled, at 
pleasure, with any gas at any desired pressure. The law 
to which they led has, however, been shown by De la 
Provostaye and Desains to be only roughly approximate. 
Especially is this the case for low pressures, when 'the 
inclosing vessel is small. 

337. Determinations of absolute amount of loss by ra- 
diation and convection together are not numerous. We 
may cite the following, due to Nicol : 

Loss in heat units, per square foot, per minute, from 

Lright Copper . . 1-09 . . 0*51 . . 0*42 
Blackened. . . . 2*03 . . 1-46 . . 1*35 

Here the hot body was 50 C. above the inclosure at 
8 C., and the pressure of the contained air was, in the 
three columns, about 30, 4, and 0-4 inches of mercury 
respectively. 

According to Macfarlane's experiments, the numbers in 
the first column should be 1*38 and 2*01 respectively. But 
in this case the temperature of the inclosure was 14 C., 
and the air was saturated with water vapour. 

The agreement is very satisfactory, since there can 
be no doubt that differences in the amount of polish for 
the bright surfaces, and in the quantity and quality of 
the material used to blacken them, are capable of 
accounting for much larger discrepancies. 



xviii.] RADIATION. 289 

338. Experiments with the short bars as in Forbes's con- 
duction method ( 244), give us valuable information of this 
kind through a much wider range of temperature ; though 
the bars are simply exposed in a room not in an internally 
blackened inclosure ; while the circumstances must be 
different for the vertical and the horizontal faces of the bar. 
The following rough numerical values have been arrived at 
in this way. The numbers are in the same units as those 
just given, the air temperature being ij z C. 

Temperature Excess. 50 C. 100. 150. 200. 250. 

Polished iron . . 1*13 2*76 4 76 7^33 IO'55 
]ron smoked . . 1^40 364 6*75 H'25 i~'2i 

In reducing the experiments, it has been assumed (as in 
247), that the specific heat of iron increases by about i per 
cent, for 7 C. And we observe the curious fact that the 
ratios of these numbers at each step, of the arithmetical 
series of temperatures, form almost an exact geometric 
series. 

Though not very concordant, these various estimates 
give us at least a general idea of the order of magnitude of 
the loss in heat units of a cooling body. 

339. Resume of 318-338. Radiation from the same 
body at different temperatures. Dulong and Petit's Law. 
Melloni's measures of absorption. Pouillet on Solar 
Radiation. Pyrheliometer. Loss of heat by radiation and 
convection jointly. 



CHAPTER XIX. 

UNITS AND DIMENSIONS. 

340. No one has the least difficulty in apprehending that 
half-a-crown is thirty pence, or that a furlong is 220 yards, 
or 660 feet. A process precisely similar to that which is. 
really involved in such every-day transformations would 
enable us to express a given sum of British money in its 
equivalent in francs, or a number of yards in the equiva- 
lent in centimetres. We have already ( 132), in showing 
how to transform from one thermometer scale to another, 
said all that is necessary on the principle of such simple 
transformations, where we reduce from a mere multiple 
of one unit to the corresponding multiple of another of 
the same kind. When two or more units have to be 
simultaneously changed, the operation is (numerically) 
somewhat more complex, but the principle is the same. 
In fact, we might apply it directly by the more laborious 
process of changing one unit at a time till we had got 
through the list Thus a speed of 10 feet per second is 
5,280 feet in 528 seconds, i.e. a mile in 528 seconds, or 
(finally) a mile in 8m. 485. Or we might put it as 600 feet 
per minute ; that is, 36,000 feet, or about 6*82 miles, per 
hour. Such things present only possible labour, but no 
difficulty of principle. 



CH. xix.] UNITS AND DIMENSIONS. . 291 

341. The case is somewhat less easy when we take such 
a question as this : The intensity of gravity is represented 
by 32-2 in feet and seconds, what is it in miles and hours ? 
[\Ve might have desired it in centimetres and minutes, but 
that would only have altered the necessary numerical 
factors; leaving the principle of the process unchanged.] 
In attacking such a question, we must begin by carefully 
investigating how the various units are involved. Once this 
is done, the rest is mere ordinary arithmetic. Now the 
meaning of the above statement is that gravity produces in 
a falling body, in each second, an additional speed of 32-2 
feet per second. Notice that the foot is mentioned only 
once, while the second occurs twice, in this amplified state- 
ment. A little thought will show that, so far as the unit of 
length is concerned, the numerical expression for the value 
of gravity must be 5,280 times greater in terms of feet than 
in terms of miles. Again, the effect of gravity in an hour 
must be 3,600 fold of what it is in a second ; and if we 
measure this by the space passed over in an hour, instead 
of that passed over in a second, the result will again be in- 
creased 3,600 fold. Altogether, then, we must multiply the 
given number, 32*2, twice over by 3,600, and divide the 
result by 5,280. We thus obtain for gravity, in miles and 
hours, 

^ 79^36 nearly. 



That is, a body falling under [constant] gravity acquires, 
in each hour, an additional speed of 79,036 miles per 
hour. 

-42. Look on these two examples from another point of 
view. Speed is greater as the space passed over is 
greater, and less as the time employed is greater. Hence 

u 2 



292 HEAT. [CHAP. 

it involves length directly, and time inversely ; or as it is 
the custom to write it, 

L 



But acceleration is greater as the additional speed produced 
is greater, and less as the time employed in producing it is 
greater : thus 

M -CM*} " , ;: 

Now, when a unit is increased in any proportion, a con- 
crete quantity, expressed in terms of it has its numerical 
value diminished in the same proportion. [Thus when we 
increase the unit twentyfold, as in passing from shillings 
to pounds, we find 480^. = -f^ 480!. = 24!.] Every definite 
quantity is homogeneous in terms of each fundamental unit 
it involves. Thus in the expression for the space described 
under constant acceleration in the line of motion, we have 

s --= a + bt -f \ ct 2 . 

Here s and a are each of dimensions [L] 

b, a speed, is , ' L f J 

and c^ an acceleration . . . - I -jfi I 

Thus each term of the value of s is, like s itself, of the 
dimensions [L]. 

Hence it appears that, to determine the numerical factor 
required to pass from any one system of units to another, 
all that is required is to find the dimensions of the quantity 
we are measuring, in terms of the fundamental quantities, to 
one or more of which every other measurable quantity can 



xix.] UNITS AND DIMENSIONS. 293 

always be referred. As already remarked, the theory of 
dimensions is due to Fourier. 

343. The fundamental quantities are length [L], mass 
[M], and time [T], and it is a matter of mere convention 
\vhat amounts of these we shall assume as our units. 

Thus we may employ, as we have hitherto done, a foot- 
pound-minute system; we might adopt a mile-ton-day system ; 
or, what seems to be in a fair way towards adoption in 
science, a centimetre-gramme-second (C. G. S.) system. 

The student must remember that the choice of units is in 
no way whatever of scientific importance, being a matter 
mainly of convenience : what is really wanted for science is 
a general system, even if it be inconvenient for mere busi- 
ness purposes. (See again, 56.) 

344. But the question of dimensions is of the utmost scien- 
tific importance, and is too apt to be lost sight of in the 
contest about units. It would be, perhaps, too much to say 
that an ill-chosen system of units, which should force a man 
to t/rin/c, would be preferable to a well-chosen system, likely 
to cause error by inspiring a blind, mechanical confidence. 
Still, there is some force in such arguments. Spelling and 
composition are altogether independent of the form of 
handwriting one employs. But one must know spelling 
and grammar before he can write correctly, even in the 
best of "hands." And the unfortunate advocating of the 
C. G. S. system, under the specious denomination of Abso- 
lute Units, is very apt to mislead the beginner, by giving 
him the impression that this choice of units has some 
mysterious connection with the truths of science. If a pro- 
posed system of units were more handy for general purposes 
than those which they are designed to supersede, every one 
would cry out for a change. It is undoubtedly on the 
score of general convenience that our present standards 



294 HEAT. [CHAP. 

have come into use. Every one knows what is meant by 
a man of five feet eight, who " scales " twelve stone. In 
the C. G. S. system this advantage is wholly sacrificed for 
the sake of another advantage (felt especially in electrical 
measurements). The average height of a man, and his 
ordinary walking pace, are here each expressed by about 
170 of the proper units; his mass is somewhere about 
70,000 units; while his weight, also in the proper unit 
(dyne), approaches the gigantic figure of 70,000,000. A 
horse-power is about 7,500,000,000 ergs per second. Thus 
the system is not likely to be employed for any but strictly 
scientific purposes. 

[A much more imperative want, of the same kind, is a 
common language of science. If original works were now, 
as they used to be, written in Latin how much more 
rapidly would not science progress ? No doubt most of the 
new and valuable scientific work of the day is published in 
English, French, or German ; or at least given in abstract 
in some one of these languages. But there is much of it 
ieft imbedded in Czech, Danish, Dutch. Italian, Russian, 
Swedish, &c., and thus practically lost to the great majority 
of those to whom it might be of the utmost importance. 
The wise conservatism of the botanists, much as it is ridi- 
culed by " advanced " science, has preserved to them this 
invaluable system of freemasonry.] 

345. The following statements of dimensions are self- 
evident : 

Volume [v] = [f]. Speed [V] =[ ]. 

TUT "TWTT 

Density [p] = [p]' Momentum [^] = | J. 

Force [F ] 



XIX.] UNITS AND DIMENSIONS. 295 

Pressure [p] = force per unit surface = |->J LT 
Energy [E] = [PL] = [MV*] = [pv] 



346. We now come to complex dimensions specially con- 
nected with our subject. And here a new fundamental 
unit, temperature [0], is required. This, again, is optional, 
but we may take it for illustration as i C. on the absolute 
scale ( 96). 

A few instances, fully worked out and explained, will 
show the reader the principles of this subject ; and he may 
then easily work out the other cases for himself. 

The co-efficient of dilatation ( 100, 113) is the ratio of 
the fractional change of length or volume to the corre- 
sponding change of temperature. The fractional change 
is a mere number, so that the dimensions of expansibility 
are simply 

[e-]. 

Heat, [H], as we have seen, may be measured in many 
ways. If it be measured in dynamical units, its dimensions 
are those of [E] above. 

If it be measured in thermal units, i.e. by the rise of 
temperature it produces in a mass of some standard sub- 
stance (as of water, 55), its quantity is proportional to the 
mass and to the rise of temperature, and its dimensions are 
therefore 

[M0]. 

But if it be measured in thermometric units ( 246) it is of 
the order 



296 HEAT. [CHAP. 

Now by the definition of thermal conductivity ( 237), 
we see that 

Conductivity x Gradient of temperature x Surface x Time 
is the measure of the heat which has passed. Thus if k 
denote thermal conductivity 



[k] [L 2 ] [T] = [H], 



or 



w = [ft] 

ML 



. 

m energy ' 



= =r= in thermal units, 

rI 2 l . 
= I ^j-j in thermometric units. 

Similarly we find that 



Rate of Emission = m 



as energy, 
= I 7901 i n thermal units, 
= f~ J in thermometric units. 

347. Thus, to turn the results of 246, which are in 
thermal foot-pound-minute units, into the corresponding 
expressions in C. G. S. units, we have as above 



\ix.j UNITS AND DIMENSIONS. 297 

But 

One pound = 45 3 "6 grammes, nearly, 
One foot = 30-48 centimetres, 
One minute = 60 seconds. 

Hence the factor required is 



453-6 



- nearly. 



60 x 30-48 4-03 

Thus in C. G. S. units the thermal conductivity of iron is 
about 0-2, and that of copper from i to 0-5. 

Again for rate of emission, as in 337, we have as 
the reducing factor 



rM0 ~i 
Ll 2 TJ 



Here [0] is a centigrade degree in both systems, so that 
the requisite factor is 



- nearly. 



60 x (30-48)- 123 

Similarly, and with equal ease, the reducing factor from 
any one system to another can be found for the other 
experimental data of our subject. 

348. Resume of 340-347. Change of units is a 
mere arithmetical operation. The real difficulty lies in 
ascertaining how the various units are involved. Objection 
to the term Absolute Units. Dimensions of Volume, 
Pressure, Energy, &c. Of Heat, Conductivity, Rate of 
Emission, &c. Examples of reduction to C. G. S. units. 



CHAPTER XX. 

WATT'S INDICATOR DIAGRAM. 

349. Besides the many capital improvements which Watt 
introduced into the steam engine [some, such as the 
separate condenser or the expansive action, being appli- 
cations of physical knowledge, others, such as the parallel 
motion, being applications of mechanical ingenuity] we owe 
to him what is called the Indicator Diagram, which is of 
the utmost importance to the elementary exposition of the 
fundamental principles of Thermodynamics. Watt devised 
it for the purpose of determining the work done by a steam 
engine ; and it is still employed for that and similar pur- 
poses. But in the hands of Clapeyron, and more recently 
of Rankine, its properties have been so fully developed 
that we can represent by means of it not merely the work 
done by an engine, but the various stages of the process ; 
the thermal properties of the working substance itself; and 
their connection with the laws of Thermodynamics. The 
germs of the method indeed are to be found in various 
parts of the Prindpia, wherever Newton had to represent 
graphically what we now call an integral. 

350. It would be inconsistent with our plan to enter into 
the practical details of construction of the Indicator itself, 
of which many ingenious forms are in use. These, as well 
as the details of construction of steam engines, &c., belong 



CH. xx.] WATTS INDICATOR DIAGRAM. 299 

rather to engineering than to physics proper ; and can be 
far more successfully studied by careful examination of the 
working instrument than by reading descriptions ever so 
minute, or inspecting drawings ever so accurate. All we 
need do is to explain the principle involved. And it is 
simply this : A pencil is so attached to the piston-rod of 
the engine, that it shares the to-and-fro motion of the piston, 
and its consequent position at any instant thus indicates the 
volume of the contents of the cylinder. The pencil, how- 
ever, has another motion, in a direction perpendicular to 
the first, such that its displacement in the new direction is, 
at every instant, proportional to the pressure of the contents 
of the cylinder. Thus, as the pencil-point moves over a 
fixed sheet of paper, it traces a line, every point of which 
represents a pair of simultaneous values of volume and 
pressure of the working substance. [In some forms of the 
instrument, the pencil has one of the two motions, and the 
paper the other. Also, it is usual to make the adjustments 
so that the volumes and pressures are represented on a 
conveniently reduced scale. But the final result is the 
same : the mode of attainment being mere matter of 
ingenuity or convenience.] 

351. The figure PP 'Q'Q represents one of these dia- 
grams. The various values of OM represent the volume 
of the working substance, the corresponding values of MP 
its pressure. 

From Watt's point of view, the diagram gives the work 
done during a stroke of the engine. In fact if S be the 
area of the piston, and / the pressure (understood as 
pressure on unit surface) the whole force exerted is pS. 
If then the piston move under the action of this force 
through a space //, the work done is ( 13) 
fS. h or /. S/i. 



3 co 



HEAT. 



[CHAP. 



But S/i is the increase of volume of the working substance. 
Hence, when the pressure is constant, the work done is the 
product of the pressure by the increase of volume. This 
would be the case in the figure, if PP' were a straight line 
parallel to Ov, and then the woik during the expansion from 
OM to OM' would be represented by the area of the 
rectangle MPP' M' . 

When the pressure is (as in the . figure) not uniform, we 
must break up the expansion into separate stages, each 




corresponding to an infinitesimal change of volume. We 
thus obtain (as in 190) a number of narrow rectangles, 
the sum of whose areas is ultimately the curvilinear space 
MPP'M* . Similar constructions must be made for the 
expansion from P' to Q f and the contractions from Q to 
<2 and from Q to P. 

If the volume diminish instead of increasing, the work, 
estimated as before, must be regarded as spent upon the 
working substance, and therefore reckoned as negative. 



xx.] WATT'S INDICATOR DIAGRAM. 301 

Bearing this in mind, we see at once that if the closed 
figure PP'QQ be the diagram of an engine, its area repre- 
sents the work given out during a complete cycle. For the 
work is positive from P to /*, and from P' to Q' : but 
negative from Q to Q, and from Q to P. 

The work done on the whole, in any such cycle, is there- 
fore positive if the pencil run round the diagram in the 
direction of tire hands of a watch, negative if in the opposite 
direction. 

If the diagram intersect itself, some parts of its area will 
be positive, others negative ; but the statement above applies 
separately to each of the parts. 

352. Were this all that the diagram affords, its value 
^great as it is) would be mainly practical, as Watt originally 
designed it to be. But we must now examine it from a 
higher point of view. 

We assume for the sake of reasoning that there is a definite 
amount, say unit of mass, of the working substance, and 
that it does not leave the cylinder ; also that it has, through- 
out (at each instant), the same temperature and also the 
same hydrostatic pressure. By this last consideration our 
reasoning is practically restricted to fluids, whether they be 
liquids, vapours, or gases, or even a complex arrangement 
such as a liquid in the presence of its saturated vapour. 

In the last of these cases there is, between the limits of 
volume at which the whole is liquid, or the whole is vapour, 
a definite relation between temperature and pressure alone. 
The volume, when assigned, gives us in this case the farther 
information how much of the substance is in the liquid 
state. 

But in the first three of these cases we have seen 
( 121, 124) that there is a definite relation between the 
volume, pressure, and temperature ; a relation whose form 



302 HEAT. [CHAP. 

depends upon the particular substance treated, but -which 
is sufficient to determine any one of the three quantities 
above when the other two are assigned. That relation we 
assume to have been experimentally obtained for the par- 
ticular substance whose behaviour we are for the time 
discussing. 

353. But this is not all. The physical state of the sub- 
stance is entirely defined when any two of these quantities 
are assigned. [The reader must be reminded that we are 
dealing with a definite quantity of the substance.] Hence, 
as a particular case, when the volume and pressure are 
assigned, the temperature can be definitely calculated. 
Every point, P, on the diagram thus corresponds to one 
temperature ; and, by drawing lines, each through all the 
points corresponding to one particular temperature, we may 
cover the diagram with Isothermal^ or lines of equal tem- 
perature. Portions of two of these, PP' and QQ', are roughly 
indicated in the diagram of section 351. 

Each of these lines gives a graphic representation of the 
relation between the pressure and volume of the substance 
so long as its temperature is unchanged. This corresponds 
to the second &&& fourth of the operations in Carnot's Cycle 
( 86). Thejirst and third operations of that cycle involve 
the behaviour of the working substance when it is sur- 
rounded by non-conducting bodies, and therefore cannot 
gain or lose heat directly. From any point in the diagram 
(which, as we have seen, represents a definite state of the 
body) we may suppose a line drawn representing the 
relation between pressure and volume under this new 
limitation. Thus the whole diagram may be covered with 
a new set of curves, called by Rankine Adiabatic lines. In 
the rough diagram above, PQ and P Q' represent portions 
of two such lines. Any other definite condition will, in 



xx.] WATT'S INDICATOR DIAGRAM. 303 

general, give rise to its own particular class of lines ; but 
the two classes we have mentioned are by far the most 
important for our present purpose. We must discuss their 
properties with some care. 

354. So far, temperature may be considered as being 
measured on any scale, no matter how denned. But one 
of the great results to be developed in this chapter is the 
absolute measurement suggested to Thomson by the remark- 
able investigation of Carnot. Once we have got this mode 
of measurement, every other method must give place to it. 

355. Meanwhile we make the general remark that any 
class of lines on the plane diagram may be regarded as suc- 
cessive parallel sections of a surface, which represents the 
general relation between volume, pressure, and the quantity 
characteristic of the class of lines. Thus the lines of equal 
temperature are sections, perpendicular to the axis on which 
temperature is measured, of the surface which gives the 
relation between volume, pressure, and temperature. Sec- 
tions of this surface perpendicular to the axis of volumes- 
would be a set of curves of equal volume in terms of pres- 
sure and temperature as co-ordinates. The surfaces them- 
selves may in fact be regarded as portions of a hilly 
country, while the parallel sections play the part of con- 
tour lines. And all the properties of contour lines find here 
new and interesting applications. 

Any .number of such surfaces and corresponding curves 
of section can be devised ; we will refer to those only 
which are of paramount importance. 

356. The isothermals or lines of equal temperature 
might be conceived as being drawn by the indicator it- 
self, the contents of the cylinder being kept successively 
at temperatures rising step by step, while at each tempera- 
ture the piston is made to go to and fro in the cylinder. 



304 HEAT. [CHAP. 

We may for the moment assume these steps to be each 
i C, such as we have hitherto employed, or a definite mul- 
tiple or fraction of such a degree. But one of the great 
objects which we have now irt view is the absolute measure- 
ment of temperature. When we have secured this, we shall 
have an obviously appropriate rule suggested to us for 
drawing the successive isothermals. 

357. The isothermals of the ideal perfect gas ( 126) may 
be very briefly treated. Since the product of the pressure 
and volume is constant at any one temperature, the lines 
PP 1 , QQ', in the fig. of 351 are equilateral hyperbolas 
of which the lines Ov and Op are the asymptotes. These 
curves are all similar, and similarly situated, and the linear 
dimensions of each are as the square root of the correspond- 
ing absolute temperature. Thus, if drawn at successive 
equal intervals of temperature, they approach more and 
more closely to one another as the temperature is higher. 

358. The isothermals of the more permanent gases, such 
as hydrogen, air and its constituents, &c., do not, within 
ordinary ranges of temperature and pressure, differ much 
from equilateral hyperbolas. For pressures less than about 
150 atmospheres, the air isothermals lie a very little below 
the hyperbolas of the ideal perfect gas. 

What happens at exceedingly small pressures is not cer- 
tainly known. In fact, if the kinetic gas theory be true, 
a gas whose volume is immensely increased, cannot in any 
strict sense be said to have one definite pressure through- 
out. At any instant there would be here and there isolated 
impacts on widely different portions of the walls of the con- 
taining vessel ; instead of that close and continuous bom- 
bardment which (to our coarse senses) appears as uniform 
and constant pressure. 

At ordinary temperatures, and about 1 50 atmospheres, the 



XX.] WATT'S INDICATOR DIAGRAM. 305 

air isothermals cross the hyperbolas, and for 'higher pres- 
sures show volumes in constantly increasing ratio to those 
of the ideal gas under the same pressure. 

They appear, in fact, to have an asymptote parallel to the 
line of no volume (Op in the fig. of 351), but at a finite 
distance from it. This, of course, we are prepared to 
expect for we have absolutely no reason to think that any 
finite portion of matter can be deprived altogether of volume 
be the pressure what it may. 

359. The isothermals of several gases, within ordinary 
ranges of temperature, have been recently determined up to 
very high pressures with great care by Amagat. The dis- 
tinctive merit of his process lay in measuring the pressure 
directly by means of a column of mercury ; which some- 
times exceeded 1,000 feet in height. His results are of 
very great value, but they are only remotely connected with 
the main features of our present inquiry. 

360. The distinction between a true^ gas and a vapour 
appears very clearly from the isothermals of carbonic acid, 
obtained by Andrews in the classical investigation already 
mentioned ( 174). Andrews' diagram is reproduced on 
next page, with some slight modifications which he pointed 
out and accounted for in his paper. According to a sug- 
gestion of Clerk-Maxwell, one new line (dotted) has been 
introduced. Part of this line is to some extent conjectural, 
from the want of experimental data : but, imperfect as it 
is, it gives much novel and valuable information. 

As the reader is now supposed to understand fully the 
immediate teachings of the indicator diagram, he is referred 
again to 175, which he should peruse attentively with the 
aid of the figure on the next page. 

x 



CH. xx.] WATT'S INDICATOR DIAGRAM. 307 

This figure contains two groups of curved lines which are 
the isothermals for carbonic acid and air, respectively, at 
temperatures between i3 3 . i C. and 48'.! C. 

[The masses of the two quantities of gas were not equal, 
so that in that respect the comparison is not of the kind 
which has been hitherto assumed in this chapter. For the 
air was used merely as a manometer, to give (by its changes 
of volume) the pressure of the carbonic acid at each stage 
of the process. The isothermals, therefore, are those of a 
mass of air which at i atmosphere and 13.! C. had the same 
volume as the carbonic acid.] 

The new dotted line is drawn so as to pass through all the 
pairs of points on each isothermal (under the critical tem- 
perature) at which the carbonic acid either just ceases to be 
wholly vapour, or just becomes wholly liquid. The region 
included by it is therefore that in which the vapour can be 
in thermal equilibrium with the liquid. A glance at the figure 
shows that the limits of volume corresponding to this region 
gradually approach one another as the temperature is raised ; 
the left-hand branch of the dotted curve leaning towards 
the right, and the right-hand branch towards the left. Thus, 
as the temperature is gradually raised, the smallest volume 
at which the carbonic acid is wholly vaporous becomes 
less, while the greatest volume at which it is wholly liquid 
becomes greater. Close to the critical temperature (but 
under it) these volumes are practically equal. 

Also, as is obvious from the figure, the compressibility of 
the carbonic acid just before it is partially liquefied becomes 
less and less as the temperature is raised, while that of the 
liquid (when just completely formed) becomes greater and 
greater. 

Thus in volume, and in compressibility, at one tempera- 
ture, these two states gradually approach one another, until 

X 2 



3o8 HEAT. [CHAP. 

at and above the critical temperature they can no longer be 
distinguished from one another. 

Thilorier's result ( 122) as to the great expansibility of 
liquid carbonic acid, is obvious from the figure : as is also 
its great compressibility (discovered by Andrews). 

If we imagine a broken line to be drawn in the figure, 
formed of the left-hand branch of the dotted curve, and the 
critical isothermal (for the higher range of pressures) it is 
obvious that, for any condition represented by a point to the 
left of this line, the carbonic acid is wholly liquid. Another 
broken line, consisting of the right- hand branch of the 
dotted curve and the same portion of the critical isothermal, 
has to the right of it all points expressing conditions at 
which the substance is wholly non-liquid. Now, by properly 
applying heat and pressure, we can bring the substance by 
any path we choose (on the diagram) from one of these 
states to the other. Choose two such paths, one (A) wholly 
free of the dotted curve, the other (B) intersecting it (twice). 
Operate on the gas according to the B path, and we see it at 
one part of the course partly vapour and partly liquid. Return 
from the undoubtedly liquid state to the undoubtedly non- 
liquid state, by the path A. At no stage of the operation 
is there any indication that the substance is partly in one 
molecular state, partly in another. 

This, however, is on the supposition, which we have 
hitherto made for the indicator diagram, viz., that the 
pressure and temperature shall be uniform at every instant 
throughout the whole mass of the substance operated on. 
If carbonic acid be in a state represented by a point near 
the apex of the dotted curve, very slight differences of tem- 
perature or pressure at different points of the mass give rise 
to extraordinary differences of optical properties ; and the 
whole presents, in an exaggerated form, the appearances. 



xx.] WATT'S INDICATOR DIAGRAM. 309 

seen when we look through a column of air ascending 
from a hot body, or through a vessel in which water and 
strong brine have been suddenly mixed. 

361. The relative densities of the liquid carbonic acid 
and its saturated vapour are, as the diagram shows, about 
57 : i at 13.! C. At 2i.5 C. the ratio is only about half 
as great. Thus there is no difficulty in representing the 
relative volumes graphically. 

But when we deal with water and saturated steam at any 
ordinary temperature, the ratio of densities is (roughly) 
1600: i. To give anything approaching this we must take 
carbonic acid at very low temperatures. Hence no diagram 
of moderate size can be constructed so as to represent fully 
the isothermals of water-substance, at the temperatures for 
which Andrews has given us those of carbonic acid. On 
the other hand, a diagram, somewhat resembling that 
of Andrews, would represent the isothermals of water for 
temperatures over 400 C. 

362. It will be observed that in all these cases, as a rule, 
the isothermal lines are inclined downwards towards the 
right, i.e., when the substance is kept at constant tempera- 
ture, increase of volume implies diminution of pressure ; or 
increase of pressure implies diminution of volume. This 
merely signifies that every known fluid, in whatever state it 
be, is compressed by the application of greater pressure, its 
temperature being kept unaltered. 

Apparent exceptions are necessarily collapsible or explosive 
bodies, which suddenly and abruptly change volume when 
pressure is applied. Such exceptions are only apparent be- 
cause in them the isothermal condition is necessarily violated. 

A real exception is in the case of saturated vapour in 
presence of the liquid, for here the pressure remains un- 
changed as the volume varies, whether by diminution or by 



UNIVERSITY OF CALIFORNIA 



3io HEAT. [CHAP. 

increase. This, of course, is due to the fact that part of the 
substance undergoes a change of molecular state, involving 
abrupt change of volume with considerable absorption or 
evolution of heat, and the proper realisation of the iso- 
thermal requires that the volume be made to alter so slowly 
that the change of temperature which would thus be caused 
can be guarded against by external applications. 

363. The adiabatic lines 'cannot conveniently be drawn 
for any substance as the result of direct experiment, 
simply because it is impossible to make an absolutely non- 
conducting vessel in which to conduct the experiments. It 
is, however, possible to calculate their form for any class of 
substances by the help of theory, from the results of experi- 
ments which can be carried out. This point must be 
deferred for the present. 

364. Meanwhile we may, but only for the purpose of 
reasoning ( 86), suppose that we have the substance in- 
closed in a cylinder which can be made, at will, either a 
perfect conductor of heat or an absolute non-conductor. 
Let this cylinder be supposed to be surrounded by a mass 
of perfectly conducting liquid, whose specific heat is so 
great that its temperature remains practically unaltered by 
any transference of heat, either way, between it and the 
contents of the cylinder. Then, if the indicator be attached 
to the piston, it will trace the isothermal, or the adiabatic, 
according as the walls of the cylinder conduct heat or not. 

365. Now suppose the substance to give out heat under 
compression. Let the piston be forced inwards. If the 
cylinder conduct, the heat developed is at once removed ; 
but for all that the pressure in general rises, as we saw in 
362. But, if the cylinder do not conduct, the effect will 
be the same as if the substance, with its pressure already 
increased by (isothermal) compression, had farther heat 



XX.] WATT'S INDICATOR DIAGRAM. 311 

supplied to it without being permitted to change its volume. 
In such bodies as we are now considering, the effect will be 
to still further increase the pressure. Thus the adiabatic 
line through any point of the diagram is more inclined to the 
axis of volume than is the corresponding isothermal. 

366. The same thing is true if the substance be, like 
water between o C. and 4 C., one of those which are cooled 
by the application of pressure, because they contract when 
heated. For if the cylinder be a conductor, heat passes 
through it into the substance, and thus the pressure becomes 
less than if, as in the adiabatic, heat be not allowed to enter. 

The statement is still true when we are dealing with 
a liquid, and its saturated vapour, in presence of one 
another. For compression liquefies some of the vapour, 
and sets free its latent heat. When this is allowed to 
escape as it is developed, the pressure remains unchanged. 
But in the adiabatic the pressure must, in consequence of 
the heating, increase with diminution of volume. 

It holds also when we are dealing with a mixture of ice 
and water. For here it is the ice which melts, because it is 
bulkier than the water produced from it, and the whole 
becomes colder in consequence of the latent heat required 
for the water which is formed. If heat be allowed to enter, 
so as to restore the original temperature, more ice is melted, 
and the pressure sinks in consequence. 

367. These phenomena are instances of a general law 
which has been formulated independently by different 
physicists. Thus Helmholtz says, as to the effect of 
pressure on a mixture of ice and water : " Here mechanical 
pressure, as happens in the majority of cases of interaction 
of different natural forces, favours the production of the 
change, melting, which is favourable to the development of 
its own action." 



312 HEAT. [CHAP. 

Clerk-Maxwell, speaking of the greater steepness of the 
adiabatics than of the isothermals, says : " This is an 
illustration of the general principle that, when the state of a 
body is changed in any way by the application of force in 
any form, and if in one case the body is subjected to some 
constraint, while in another case it is free from this con- 
straint but similarly circumstanced in all other respects, 
then if during the change the 'body takes advantage of this 
freedom, less force will be required to produce the change 
than when the body is subjected to constraint." 

368. In general, any one point of the diagram cor- 
responds to one perfectly definite state of the working 
substance, and therefore there can be drawn through it 
only one isothermal and one adiabatic The adiabatic (as 
we see by 365, 366) crosses the isothermal from above 
downwards, and towards the right of the diagram. 

But, in certain special cases, a point of the diagram may 
correspond to more than one essentially different state of 
the substance. Each of these states has its own isothermal 
and its own adiabatic ; and thus we find it sometimes said 
that two or more isothermals, or adiabatics, may intersect 
one another. 

The proper view to take of such cases is to look on them 
as instances in which part of the diagram overlies another 
part, so that (as in the contour lines of an overhanging cliff), 
though designated in the diagram by the same rectangular 
co-ordinates, they lie in regions which must be regarded as 
perfectly distinct from one another. (See, again, 355.) 

This will appear clearly enough if we consider the rela- 
tion between volume and temperature in water at the 
ordinary atmospheric pressure. Thus ( 121) we know 
that the volume of water is the same at 2 C. and at about 
6 C. Hence the isothermals for water at 2 C. and at 6 C. 



XX.] WATTS INDICATOR DIAGRAM. 313 

intersect in a point given by one atmosphere pressure and 
volume 1-00003. But if we think of a water thermometer, 
we see that the scale of such an instrument would be, as it 
were, doubled back on itself, the lowest point being at 4 C. 
(the maximum density point), and the scale reading 
upwards from this point to 5, 6, c., for increase of tem- 
perature, but also upwards to 3, 2, c., for diminution of 
temperature. These are not to be regarded as one scale, 
but as two distinct parts of a scale doubled back on itself. 
And in a similar way we must regard the corresponding 
part of the indicator diagram above mentioned. 

369. Hence, in the reasonings which follow, we shall 
consider the systems of isothermals and adiabatics as in 
themselves groups of non-intersecting curves ; but such that 
each curve of one group intersects once, and once only, 
each curve of the other group. Two of the same group 
which appear to intersect will be regarded as lying in 
essentially different regions, though depicted on the same 
part of our diagram. 

370. Another remark must be made here. We have 
seen ( 358) that our direct knowledge of the form of iso- 
thermals is limited to their middle regions, where the volume 
of the substance is neither very great nor very small. The 
same is true as regards the adiabatics, of which our know- 
ledge is considerably less complete because less direct. 
But we must presently consider the area (which will be 
proved to be finite), included between a finite portion of one 
isothermal and two adiabatics passing towards the right 
through its extremities. Remark that all that is wanted is 
a mode of completing our diagram, however imperfectly or 
even (as subsequent experiments may show) erroneously, 
provided that it can lead us into no error in the special 
reasoning for which it is devised, and for which alone it is 



HEAT. 



[CHAP. 



to be employed. Clerk- Maxwell suggests the following 
method which, while convenient and sufficient for our 
solitary object, is so obviously incorrect as to details that no 
one can run any risk of being misled by it. 

Let RR'R" (in the fig. of 374 below) be the isothermal 
of lowest temperature whose form we know, QR, QR* ', 
Q'R", . . . adiabatic lines. Draw any line SS S" and call 
it (for our temporary purpose) the isothermal of absolute 
zero. Then it is clear that we may draw (each in an infinite 
number of different ways) lines RS, R' S , R'S", . . . such 
that the curvilinear areas RR' S' S, R'R"S"S' &c., shall have 
any assignable finite values. 

371. We now recur to Carnot's Reversible Cycle, in order 
that we may be able to interpret the diagram in the light of 
the two Laws of Thermodynamics. 




Let PP be the isothermal of the working substance at the 
temperature / r , of the hot body; QQ' that at the tempera- 
ture f of the cold body; while QP and P'Q' are the adia- 
batics of the first and third operations respectively. Also 



xx.l WATT'S INDICATOR DIAGRAM. 315 

let HI, H be the quantities of heat taken in and given out 
respectively in the direct working of the cycle. Then, by 
the definition of absolute temperature ( 95), we have 



t, t* 

As the cycle is reversible, no heat-transaction takes 
place except these, and therefore the work done is the 
equivalent of the excess of the heat supplied from the 
source over that given out to the condenser. If we 
choose, for convenience, to measure heat in dynamical 
units, we must use the word "equal" instead of "equiva- 
lent." [Joule's equivalent, as originally given (37), was 
772 footlbs. for the unit of heat to the Fahrenheit degree. 
This is, of course, about 1,390 foot Ibs. for the centigrade 
degree.] Hence, with this convention and the proposition 
of 351, we have 



372. As the extent of the isothermal expansion, P to P r . 
may be what we choose, we will for simplicity suppose it 
so taken that we have numerically (the unit of heat being 
now the foot-pound) 

HI = A- 
The definition of absolute temperature then gives 

^ = 4; 

so that, in our present system of units we have 
/, - / = area PP' Q'Q. 

373. Xow we are prepared to choose scientifically our 
system of isothermals and adiabatics, and thus to settle the 
values of the several degrees of the scale of absolute 
temperature. 



3i6 HEAT. [CHAP. 

Proceeding along the isothermal PP' in the diagram 
opposite, let us mark successive additional points /"', P'" , 
&c., so that ti heat units are taken in from P' to /"', /"' to 
P" f , &c., as well as from P to />'. Through the points P", 
P'", &c., draw adiabatics meeting <2<2' i n <2", <?'": &c - 
Then it is clear that the heat given out in passing from 
<2" to <2', or from Q'" to <2", &c., is (like that from Q' 
to represented by / . Thus, /, -/ = area PP'Q'Q 
-- area P'PQTQ' = area P"P'"Q"Q" = & c . 
And this holds whatever be the value of / . 

Hence, if we assume t to be one degree lower than / x , so 
that A - f = i, each of the areas PP'Q'Q, P'P"Q'Q', 
c., will be one unit. And a third isothermal, two degrees 
under / will be one degree under / , and will thus cut off 
a new set of unit areas from the series of adiabatics. 

The whole explored part of the field may thus be divided 
into unit areas by the system of adiabatics just described, 
and a set of isothermals for successive degrees of tempera- 
ture. But the length of a degree, so far, is perfectly 
arbitrary, though when its value is assigned at any part of 
the scale the whole becomes definite. 

374. The area contained between two successive adi- 
abatics of this series, and any two isothermals, has therefore 
as its measure the number expressed by the difference of 
the absolute temperatures of the isothermals. 

[Here we see, at once, one of the great merits of Carnot's 
process. For the statement just made is altogether inde- 
pendent of the nature of the working substance.] 

This area also represents, as we have seen, the excess of 
the heat supplied over that given out. Its utmost value, 
therefore, when the lower isothermal is that of absolute 
zero, is a finite quantity representing the whole heat sup- 
plied. Thus if RRR'R". ... be the lowest isothermal 



XX.] 



WATT'S INDICATOR DIAGRAM. 



317 



whose form is known, we may (as in 370) take any line 
SS'S"S'" .... as the isothermal of absolute zero; and 
the lines XS, 'S, "S" c., must be so drawn that the 
several areas PP'QR'S'SRQ, P'P'Q'tf'S'S'R'Q, .... 
may each be equal to /,. 

From what has been already said we see at once 
that whatever isothermal is represented by QQ'Q"Q'", - - 
the areas QQ'R'SSR, QQ'"S"S'J?, .... must also be 




equal to one another. Thus the absolute temperatures in any 
two isotliermals PP and QQ' are to one another as the com- 
plete areas PP'Q'R'S SR Q and QQ'R'S'SR. The determina- 
tion of the ratio of these areas, when PP' and QQ' belong 
to any two definite temperatures, such as those of water 
boiling, and of ice melting, under one atmosphere of 
pressure, is a matter entirely for experiment. (How such 



318 HEAT. [CHAP. 

experiments have been conducted we will afterwards show.) 
The result for these two temperatures was found by Joule 
and Thomson to be nearly i'365 : i. (Phil. Trans. 1854.) 

375. Hence, if we adopt the centigrade scale, but merely 
in so far as to divide the interval between the freezing and 
boiling points into 100 degrees, whether these be the same 
degrees as those of our earlier scale ( 61) or not : and \ix be, 
on our new scale,, the absolute temperature of melting ice, 
A + 100 will be that of boiling water : so that 

x+ 100 : x : : 1*365 : i. 

Thus we have, as already stated, #=274 very nearly. 

376. This may be stated in the easily intelligible form : 
If a reversible engine work between the boiling and the 

freezing points of water, it gives to the condenser 274 out of 
every 374 units of heat which it takes from the boiler. 

Or, to introduce a practical term, Efficiency, whose mea- 
sure is the fraction of the whole heat taken in which is 
converted into work : The efficiency of a perfect engine, 
working between the boiling point and the freezing point of 
water is |4J. The efficiency, of course, rises with the ratio 
of the higher to the lower absolute temperature. 

In practice, the very best engines fall far short of this. 
Joule gives as a fair instance of the data for a good high- 
pressure steam-engine the following : 

If it work at 3^ atmospheres' pressure (about 53 Ibs. 
weight per square inch) the temperature of the boiler must 
be about 300 F., and it is found practically impossible to 
keep the condenser at a lower temperature than about 
uo c F. nearly. Absolute zero on the Fahrenheit scale is 
-274 l + 32 = 461 F. nearly. Hence even the theo- 
retical efficiency is only y|~J, very nearly \. The actual 
efficiency is rarely more than about half as much. 



XX.] WATT'S INDICATOR DIAGRAM. 319 

377. To recur to our diagram, 374. The absolute 
temperature, /, completely defines a particular isothermal, 
when we know the working substance. Let, now, <j> be the 
corresponding characteristic of an adiabatic, i.e. the quantity 
which has the same value at all points of such a line. 
Rankine originally called it the Thermodynamic Function, 
and Clausius has since called it Entropy. It is obvious 
that < depends in some way on the heat given to or taken 
from the substance, for it is constant only when there is no 
direct gain or loss of heat. 

And we see at once from the equation 



which is true for all values of / x and / , that the amount by 
which < increases, in passing from one adiabatic to another 
along an isothermal, may be defined as simply the common 
value of these equal quantities. Thus since in our standard 
method of drawing a group of adiabatics ( 372) we took 
H t numerically equal to / the value of <f> increases by 
unity from any one to the next of the group. 

378. By working backwards through the group of adia- 
batics, along the isothermal /, we remove / units of heat for 
each unit by which <f> diminishes. This suggests the measure- 
ment of < from a zero at which the substance has no heat 
to part with. Practically, however, we measure <f> (as, in 
dynamics, we measure a potential) from some assumed 
origin. For it is with its changes alone, and not with its 
actual value, that we are mainly concerned. 

Suppose, then, that we assume for this purpose a definite 
point in the diagram as the origin. Draw the corresponding 
isothermal, say / > and produce it to cut the adiabatic for 
which < is to be measured. Let the substance expand or 



320 HEAT. [CHAP. 

contract adiabatically till its temperature is / , and let its 
volume then change isothermally till its state is that of the 
assumed origin. If H be the heat given out during this 
last operation, we have 

- * 



Here //"may be negative, in which case < is also negative. 
It follows that if a substance change its state isothermally 
at temperature /, from < to <, it takes in an amount of 
heat denoted by 

/(4>-< ). 

If it be restored to < along the isothermal / , it gives out 
heat to the amount 

4 (4> - &) 

In a Carnot's cycle, bounded by /, </>, / , < , the excess of 
heat taken in over that given out, i.e. the work done, or the 
diagram area of the cycle, is thus 



379. The changes in the total energy of a substance may 
also be exhibited on the diagram. [The whole amount of 
energy in a substance is a quantity which we have no means 
of ascertaining. We can tell ( 220) how much goes in, 
and how much comes out, and in what forms it goes in or 
comes out, but no more. Fortunately this is all we require 
for practical applications.] 

Let the point P in the diagram represent the initial state 
of the substance, PM the line of volume, PQ the adiabatic 
corresponding to P. Then, if the state be altered (by any 
path) to one represented by a point P r , there will be work 
done by the body, (i.e. loss of energy,) if P is to the right 
of MP ; also heat will be given out (again loss of energy) 



XX.] 



WATT'S INDICATOR DIAGRAM. 



if P' lie below PQ. [\Ve will take this as the standard case. 
A little consideration will show what modifications are re- 
quired if P is to the left of MP, or above PQ.] Draw the 
volume line PM', and the adiabatic P'Q', and let QQ be 
the (arbitrary) line corresponding to S'SS' S'" in 370. 
Then the area PPM'Mvs. the work done, and PQQP the 
heat given out ( 374), both represented in dynamical units. 
Hence the area PQQ'P'M'M represents the whole loss of 




M 



energy from the state P to the state P' ; and is evidently 
independent of the form of the path PP'. 

But it is necessary to remark that each of its parts, the 
work done, and the heat given out, has in general a value 
which depends on the form of the path. This consideration 
is one of the utmost importance, as it enables us to repre- 
sent graphically the amounts of heat which are either 
necessarily, or needlessly, wasted in any particular course. 
If. for instance, we suppose that no heat is supplied to the 
working substance, the course PP' cannot anywhere pass 

v 



322 HEAT. [CHAP. 

above PQ. Again, if F' has the lowest available tempera- 
ture, PP' cannot anywhere lie below the isothermal PR. 
Thus the maximum amount of work which can be obtained, 
in passing from one given state to the other, will correspond 
to expansion adiabatically along PQ, followed by compres- 
sion along the isothermal to P'. In the course PP', the 
area P'RQQ' represents the heat necessarily wasted, 
PRP' that wasted unnecessarily. 

380. Hence we see that the improved diagram, with the 
proper groups of isothermals and adiabatics as recently 
explained, enables us to trace simultaneously the changes 
of volume, pressure, and temperature, and the amounts of 
heat taken in or given out ; as well as the amount of work 
done, and the changes of the total energy. Thus we fully 
justify the remark in 349 as to the development of the 
powers of the diagram. 

We might, after Rankinc and Clerk-Maxwell, develop 
much farther the applications of the diagram (see especially 
Rankine's paper, Phil. Trans., 1854; and chaps, ix. to 
xiii. of Maxwell's Theory of Heat} so as to deduce from 
the properties already given, and by geometrical methods 
only, the relations between different physical properties of 
a substance. These are all ingeniously obtained by Max- 
well from different expressions for the unit area of the 
elementary parallelogram formed by two consecutive adia- 
batics and two consecutive isothermals, as defined in 373. 
Bat experience has taught us that, elegant, simple, and 
powerful as these methods are, they are found (by such 
students as are able to understand them) considerably 
more difficult to follow than the analytical methods. 

No one who wishes thoroughly to realise Thermo- 
dynamic Theory, can safely omit the study of Clerk-Max- 
well's book ; but he will learn something of the subject, 



xx.] WATTS INDICATOR DIAGRAM. 323 

and be better prepared for further study, if he begins by 
looking at the elements of the theory from a simple 
analytical point of view. 

381. There is, unfortunately, a wide-spread notion (of 
course only among those who have not taken the little 
trouble required to know better), that the differential cal- 
culus, even in its elements, is something terribly profound. 
When one of this class, as sometimes happens, can be per- 
suaded to listen, he is made conscious in a very short time 
that many of the mental processes to which (unless he is 
simply an idiot) he has been accustomed since his childhood, 
are based on these very elements of the dreaded calculus, 
sometimes upon more recondite parts of it. For such 
people there should be no consideration, and there shall be 
none here. 

382. Resume of 349-381. Watt's Indicator. Indicator 
Diagram. How it expresses work. Isothermal and Adia- 
batic Lines. Regarded as successive parallel sections of a 
surface. Isothermals of air, carbonic acid, water. Adia- 
batics steeper than isothermals. Carnot's cycle on the 
diagram. Absolute temperature. Efficiency of an engine. 
Thermodynamic Function. En'tropy. Total Energy. 



Y 2 



CHAPTER XXI. 

ELEMENTS OF THERMODYNAMICS. 

383. IN the preceding chapter we have exhibited, by 
means of the indicator diagram, many of the thermal pro- 
perties of individual substances, and some of the relations 
which the application of the Laws of Thermodynamics has 
shown to exist among these. The quantities, in terms of 
which we there expressed the state of a substance, were at 
first v (the volume) and/ (the pressure) of unit mass of the 
substance. 

Then we saw that the whole explored region of the 
diagram could be divided into equal areas (of any size) by 
properly grouped isothermals and adiabatics, so that the 
state of the substance might equally well be expressed in 
terms of / (absolute temperature) and < (thermodynamic 
function, or entropy). 

But we can also easily see that we might have expressed 
the position of a point (i.e. the state of the substance) by 
any other pair of these four quantities : -i.e. in terms of 
v and /, v and </>, p and /, or p and <f>. 

And we also saw that (E) the whole energy (to a constant 
pres) could be expressed in terms of p and v, and therefore 
in terms of any other pair of the four quantities. Hence 



CH. XXL] ELEMENTS OF THERMODYNAMICS. 325 

we see that there exist three necessary relations among the 
five quantities 

/, v, /, <, E, 

and therefore that any three of them can be expressed in 
terms of the remaining two. 

We proceed to develop, analytically, the more ele- 
mentary results of the application of the Laws of 
Thermodynamics : and we will choose sometimes one pair, 
sometimes another, of these five quantities (as shall best 
suit our immediate purpose) for the independetit variables 
in terms of which the others are to be expressed. 

384. We commence with the expressions for the Energy. 
This subject was attacked by W. Thomson in 1851. 

If, under mean pressure /, the volume of the substance 
change from v to v + dv, the work done (corresponding to 
so much loss of energy) is 

pdv. 

If at the same time the substance, at mean temperature /, 
pass from the adiabatic < to the proximate one <f> + d <, 
it must take in (i.e. gain in energy by) the amount of heat 



represented ( 371, 378) in dynamical units. 
Thus we have, for the change of energy, 

dE = td<$> -pdv . . . . (i) 

Hence E may be regarded as a function of the 
two independent variables v and <f>, such that its partial 
differential coefficients are 



> 



326 HEAT. [CHAP. 

385. These equations tell us, respectively, that 

(a) The loss of energy per unit increase of volume in 
adiabatic expansion is measured by the pressure. 

(ft) If the substance be kept at constant volume, the gain 
of energy, per unit increase of entropy, is measured by 
the absolute temperature. This is merely the same as : 
At constant volume, the increase of energy is measured by 
the heat supplied. 

386. From equations (2), by partial differentiation, we 
find 



= = 

^</>\ dv ' dv \ d$)\ \ dv )' 

To interpret this equation, multiply and divide the left 
hand member by /, and we have : -The fall of temperature 
per unit increase of volume in adiabatic expansion, is equal 
to the increase of pressure per (dynamical) unit of heat 
taken in at constant volume, multiplied by the absolute 
temperature. 

387. The expression for dE [(i) of 384] consists of two 
parts, neither of which is (separately) a complete differen- 
tial, though their (algebraic) sum is necessarily so. Com- 
pare 379, where the reason is made obvious. Both 
the work done, and the heat supplied, depend on the 
form of the path (i.e. on the succession of states through 
which the substance passes) and this is wholly arbitrary. 
The change of energy, on the other hand, depends only 
on the initial and final states of the substance. 

388. By adding various complete differentials to both 
sides of (i), we may change the independent variables to any 
of the other pairs v and t,p and <, p and /. And from the 
results we can, as before, draw definite conclusions as to 
relations between the thermal properties of the substance. 



xxi.J ELEMENTS OF THERMODYNAMICS. 3^7 

389. Thus, for v and / as independent variables, 

d (E tty] = < dt - pdr. 

This gives, by a process precisely similar to that employed 
in 386, 

/ dp \ / d$ \ 

\di) ~\av)' 

Multiplying both sides by /, this reads : The latent heat 
of (isothermal) expansion is measured by the product of 
the absolute temperature and the increase of pressure per 
unit rise of temperature at constant volume. 

Thus if we have steam in presence of water, or water in 
presence of ice (i.e. the same substance in two states in 
which the first differs thermally from the second by the 
latent heat, Z, per unit mass) ; let v , v t be the volumes of 
unit mass of each at/, /; e : i - e the ratio in which the 
unit of substance operated on is made up of these forms, 

Hence 

dp\ _ 



(p\ = i / 
V dt ' z' - v, \ de 



The first member is the rate at which pressure must 
change with temperature so that there shall be no change 
of volume, i.e. no alteration of the relative proportions of 
the parts of the substance in the two different states. 

The second, multiplied by /, is the heat which must be 
supplied per unit increase of volume, the temperature 
remaining unchanged. Hence Lde, the latent heat required 
for the change of state represented by de, must be equal to 
the heat supplied. 

Thus the preceding equation takes the form 



. 

dt 



328 HEAT. [CHAP. 

For small simultaneous changes of pressure and tempera- 
ture, this becomes 



,, 



t(v - z 



- z', 
-- 8/, 



since under the assigned conditions, p is a function of / 
only, not of v ( 362, 366). 

Thus, when v > z\ (as with steam and water), the tem- 
perature is raised by increase of pressure : but when v < v^ 
(as with water and ice), the temperature of the mixture 
falls when the pressure is raised. 

As a test let us consider the dimensions ( 345) of the 
above expression. Term for term they are 



[FL] 

390. The formula above is equivalent to that given by 
J. Thomson in 1849 ( 142) from Carnot's principle alone. 
Let us deduce a numerical result. A cubic foot of water 
at o C. weighs about 62*5 pounds. Hence the volume of a 
pound of water is about 0*016 cubic feet. The density of ice 
is ( 145) 0*92, so that the volume of a pound of ice is 
[o'oi6-^o'92 = ] 0-0174 cubic feet. The latent heat of water 
is ( 151) 79*25 units of heat, which must be multiplied by 
Joule's equivalent, 1390 ( 371), to reduce it to foot-pounds. 
One atmosphere of pressure is about 2117 pounds weight 
per square foot. Hence, calculating from the formula, we 
find that the freezing point is lowered by 



for each atmosphere of pressure. 

391. We cannot directly apply this process to a numerical 
calculation in the case of water and steam, for experimental 
difficulties of a formidable character lie in the way of the 



xxi.] ELEMENTS OF THERMODYNAMICS. 329 

determination of the density of saturated steam at different 
temperatures. Still, if we take the corresponding values of 
8/ and S/ from Regnault's table ( 164), and L from 166, 
we may use the formula to calculate the volume of one 
pound of saturated steam at any temperature. 

But a curious result, due to Rankine and Clausius, 
may easily be deduced. Let r , c t (functions of / alone) 
be the specific heats of any substance in its two states of 
saturated vapour, and liquid, in the same vessel. Then 
remembering that the condition of unit mass of the sub- 
stance is fully characterised by the quantities / and e, we 
have for the heat required to change these to / 4- df, and 
e + de, respectively, 

td$ = (c e + <r x (i - e))dt+ Lde . . . . (i). 
d*<> d 



dL L 
*--*-? 

Now by 166, dL\dt for steam is (roughly) 07 ; so that 
as d = i for water, dLldt + d = 0-3. But Lit has the 
values 2 '2, i '4, 0*98 at o, 100, and 200 C. respectively. 
Hence, we find that C Q is negative. 

To investigate the results of adiabatic compression on 
steam in the presence of water, note that ( 393) the 
mixture is heated, so that dt does not vanish, but has a 
positive value. Now the left-hand member of (i) is zero. 
Hence in order that de may vanish, i.e. that there may be 
neither evaporation nor liquefaction, we must have 



[Since c is negative, the value of e is a proper fraction.] 



330 HEAT. [CHAP. 

If the steam be in excess of this ratio, the value of de is 
positive, so that compression leads to further evaporation ; 
if in defect, some steam is liquefied. (See, again, 86.) 

Saturated steam, when no water is present, becomes 
superheated by adiabatic compression. 

Also, if saturated steam is allowed to expand in a vessel 
impervious to heat, it cools so as to keep at the temperature 
of saturation ; and, besides, a portion of it liquefies. 

This result appears at first sight inconsistent with the 
paradoxical experiment long known, that high-pressure 
steam escaping into the air through a small orifice does 
not scald the hand, or even the face, of a person exposed 
to it ; while, on the contrary, low-pressure steam inflicts 
fearful burns. W. Thomson has explained the difficulty 
thus : The steam rushing through the orifice produces 
mechanical effect, immediately wasted in fluid friction, and 
consequently recotiverted into heat, from which, by Regnault's 
numerical data, it follows that the issuing steam (in the 
case of the high-pressure, but not of the low-pressure, 
boiler) must be over 100 C. in temperature, and dry. 

392. Again we have from our fundamental equation, 

d (E + pv) - td$ + vdp, 
whence, as before, 

*L 

dp 

Introducing t as a divisor of each side, we read : The 
rate of increase of volume, per unit of heat supplied at 
constant pressure, is the ratio of the adiabatic rate of 
change of temperature with pressure to the absolute 
temperature. 

393. Finally 

d(E t$ + pv) = - $dt + vdp, 



xxi.] ELEMENTS OF THERMODYNAMICS. 
whence 



^dpJ \dt, 

Introducing / as a factor on both sides, we have : The heat 
given out per unit increase of pressure at constant tempera- 
ture is equal to the product of the absolute temperature by 
the rate of change of volume per unit rise of temperature 
at constant pressure, i.e. to the continued product of the 
absolute temperature, the volume, and the expansibility. 

Thus we see that bodies which expand by heating are 
heated by compression, while those which contract by 
heating (as water under its maximum density point) are 
cooled by compression. 

394. We have seen ( 121) that, for temperatures near to 
the maximum density point, we have for water 

Ldv _ t 278 
v dt 72,000 

Hence, for one additional atmosphere of pressure, the heat 
developed in unit mass of water (about this region of 
temperature) is, in dynamical units (see 390), 

2117 /(/- 278). 
62*5 x 72,000 

and it therefore produces a rise of temperature 

2117 /(/- 278) = /(/- 278) 

62-5 x 1390 ' 72,000 2,950,000 

nearly. Thus the change of temperature produced, in water 
at o C., by a sudden increase of pressure to the amount of 
150 atmospheres (roughly i mile of sea water, or a ton 
weight per square inch), would seem to be about 

- o'o 5 5 C. 



332 HEAT. [CHAP. 

This result, however, is considerably too large, because 
we have not taken account of the fact that the maximum 
density point is notably lowered by so large a pressure as 
150 atm., and in consequence the expansibility of water at 
o C. is correspondingly diminished. To take account of 
this, let 

i tdv\ i idv\ 

e = - ( -7- ) and K = ( ) 

v \dt) v \dp) 

represent respectively the expansibility and the compres- 
sibility ; p and t being the independent variables. We see 
at once that 



\dp) \dtr 



dp 

or : The rate at which the expansibility of a substance is 
raised by increase of pressure is equal to the rate at which 
its compressibility is diminished by rise of temperature. 
Thus Canton's early result ( 121) shows that water, at 
ordinary temperatures, becomes more expansible (or, as the 
case may be, less contractile] under pressure. 

Now the relation between small simultaneous changes of 
pressure and temperature which leave the expansibility un- 
changed is given by 



By means of the above relation we see that this may be 
put in the form 

' 



But experiment gives, at ordinary temperatures and pressures, 
the approximate expression 

K = Q'000052 - O'OOOOOO^ T. 



xxi.] ELEMENTS OF THERMODYNAMICS. 333 

where T is temperature C. Hence, as the maximum density 
point is subject to the present conditions, being that of 
zero expansibility, it is lowered through 



by one additional atmosphere. 

\Ye may now write the more accurate expression for the 
expansibility 3t ordinary temperatures 

I dv _ t 278 + 0'02/ 

v dt 72,000 

I 
and we find that the change of temperature produced by 

150 atmospheres in water at o C. is only 

150 x 274(4 15 x o'oi) 

= o 'o i s C. 



or about two-thirds of the former estimate. 

In the case of \vater at 8 C., the effect of 150 atmo- 
spheres is to raise the temperature about one-thirteenth of a 
degree. 

395. If/' be the specific heat of a substance at constant 
pressure, we have for the unit mass 

kdt = td$ 
with the condition 

dp = o. 

If we choose, as is usually done, v and / as independent 
variables, these equations may be written 

M = &# + && 

dt dv 

dp dp 
= dt + - dv. 



334 HEAT. [CHAP. 

Now / -5* is obviously the specific heat at constant volume, 

at 

c, suppose. For it is the rate at which heat is supplied per 
degree of rise of temperature, when the volume of unit mass 
of a substance is kept constant. Hence these equations 
give, by elimination of the special ratio of dt to dv enforced 
by our condition, 

dp < 

k.^--*#^.* 

dv dp dp 

dv dv 

which (since by 362, -f- is negative) is obviously always 

dv 

positive. This applies to all substances in which there can 
be uniform hydrostatic pressure, but it takes an exceedingly 
simple form for the ideal perfect gas. 

396. The thermodynamic relations of the ideal perfect 
gas may next be taken, as they are all excessively simple 
and form at least fair approximations (within ordinary 
ranges of temperature and pressure) to those of gases or 
gaseous mixtures such as hydrogen, oxygen, air, &c. 

Here ( 126) our fundamental condition is 

pv = fit. 
To find the form of the adiabatics, we have ( 389) 

^ .._ dp = R _ 
dv dt v 
We have also 

d t= - t 

dv v ' 

so that (395) 



p 



xxi.] ELEMENTS OF THERMODYNAMICS. 335 

Hence, as 



we have 



t v 

= <Ldp + -dv. 
p* v 

Thus 



gives, for various assigned values of <, the form of the 
adiabatics for the ideal perfect gas. 

397. As Laplace first pointed out, the compressions and 
dilatations of air, in the passage of sound, take place so 
rapidly that there is no time for much more than a tendency 
towards equalisation of temperature, and the changes of 
volume therefore take place adiabatically. 

Thus we have, as the relation between pressure and 
volume during the passage of sound, the relation 

ptf lc = constant, 
instead of 

pv = constant, 

i.e. Boyle's Law, which was employed by Xewton for the 
solution of this most important problem. 

The ordinary process for the investigation of the motion 
of plane waves (which would be out of place here), gives 
for the velocity of sound, with the above relations between 
pressure and volume, the respective expressions 



V - Rt (Laplace). 
\/~Rt (Newton). 



336 HEAT. [CHAP. 

R and /, as well as the velocity of sound, can be directly 
measured ; and thus from Laplace's result the ratio kjc can be 
determined with considerable accuracy. (See, again, 187.) 
Rankine was the first to give a theoretical determination 
of the specific heats of air, which rivalled in accuracy the 
subsequent experimental determinations of Joule and Reg- 
nault. His method was founded on this value of the ratio 
of k to <r, combined with the value of the difference k c, 
as calculated from the formula of 395. 

398. To discover how ordinary gases deviate in their 
behaviour from the ideal perfect gas, many methods have 
been employed, of which that devised by Thomson (as an 
improvement of one of Joule's methods) is the most 
interesting and instructive in its consequences. We will 
first describe Joule's experiment of 1844, already referred to, 
and then consider Thomson's improvement of it. 

Temperature C, on the air-thermometer bears, by defini- 
tion, the same ratio to 100 that the expansion of air at 
constant pressure [or the increase of pressure at constant 
volume], from the freezing-point to that temperature, bears 
to the corresponding expansion [or increase of pressure] 
from the freezing-point to the boiling-point. 

399. By his experiment, presently to be described, Joule 
proved that, to a considerable degree of accuracy, the heat 
developed in the compression of air is the equivalent of the 
work spent. 

This is a truth, but not a truism. And the following 
words of Clerk -Maxwell, whether or not they shall ever 
come to be carefully considered by the reckless partisans at 
whom they were aimed, are at least calculated to convey a 
much needed and salutary lesson : 

" This is by no means a self-evident proposition. In fact, 
it is not true in the case of substances which are not in the 



xxi. J ELEMENTS OF THERMODYNAMICS. 337 

gaseous state, and even in the case of the more imperfect 
gases it deviates from the truth. Hence the calculation of 
the dynamical equivalent of heat, which Mayer founded on 
this proposition, at a time when its truth had not been 
experimentally proved, cannot be regarded as legitimate." 

British science has not as yet been so warmly cherished 
by British statesmen that it can afford to have some of its 
very best achievements wrongly attributed to foreigners. 

400. Joule took a strong vessel containing compressed 
air, and connected it with another equal vessel which was 
exhausted of air. These two vessels were immersed each in 
a tank of water. After the water in the tanks had been 
stirred carefully, so as to bring everything to a perfectly 
uniform state of temperature, a stop-cock in the pipe con- 
necting the two vessels was suddenly opened. The com- 
pressed air immediately began to rush violently into the 
empty vessel,, and continued to do so till the pressure 
became the same in both ; and the result was, as every one 
might have expected, that the vessel from which the air had 
been forcibly extruded fell in temperature in consequence of 
that operation. It had expended some of its energy in 
forcing the air into the other vessel. But that air, being 
violently forced into the other vessel, impinged against the 
sides of that vessel, and thus the energy with which it was 
forced in through the tap was again converted into heat 
Thus the air which was forced into the vacuum became 
hotter than before, while the air which was left behind 
became colder than before. But, on stirring the water 
round these vessels, after the transmission of air had been 
completed, and the stop-cock closed, Joule found that the 
number of units of heat lost by the vessel and the water on 
the one side was almost precisely equal to the quantity of 
heat which had been gained on the other side. 

z 



338 HEAT. [CHAP. 

He then repeated the experiment, employing instead of two 
tanks of water, each holding one of the two strong vessels, 
one larger tank also filled with water, with both vessels 
buried side by side in it : then, on allowing part of the air 
to escape, as before, from the one into the other, and 
stirring till everything had acquired exactly a common tem- 
perature, he found that there was scarcely any measurable 
change in temperature. 

These experimental methods proved indisputably that 
the quantity of heat lost by the one part of the air was : 
at least as nearly as so rough an experiment enabled 
him to test it : equal to the quantity of heat gained by 
the other. 

401. Now the compressed air had at first a certain capa- 
bility of doing work. It might have been used to drive a 
compressed-air engine for instance ; but in its final state, 
when it had expanded to double its original bulk, it had 
not so much available working power stored up in it as it 
had before. There was, therefore, dissipation of part of 
the energy, originally present ; and yet the apparatus and 
its contents had not lost any heat. 

There was, on the whole, no heat lost, because what was 
lost to the one vessel was gained by the other. No heat 
was given out to external bodies, and no available work 
was done. The air was simply allowed to expand to 
change its bulk without driving out pistons or doing 
anything by which it could convey work to external bodies. 
It had, therefore, at last precisely the same amount of 
energy as at first ; and yet of that not nearly so much was 
available. The air had seized at once the chance given 
it of dissipating part of its energy, and did dissipate it, 
as far as was compatible with the circumstances of the 
arrangement. 



XXI.] ELEMENTS OF THERMODYNAMICS. 339 

402. The really curious point about this is, that in order to 
restore the lost availability to the energy of the air to get 
the air back into its former condition, so as to be capable 
of doing as much work as it was capable of doing at first 
it would be necessary to spend work upon it, pumping half 
of it back from the second vessel into the first ; but the 
amount of work which would be spent in pumping it 
back goes to heat the whole mass of air ; and, when work 
enough to force back the air into the first vessel from the 
second has been expended, the amount of heat which is 
given out during the process which can be measured with 
great exactness is almost precisely equivalent to the work 
which is spent in forcing the air back. 

Thus, to restore to the energy its former availability, no 
supply of energy is required, but some high-class energy 
must be degraded. Work has been spent, and we have got 
instead its less useful heat-equivalent. We must waste a 
certain amount of energy, or rather get a lower form of 
energy in place of it, in order to restore to the mass of air 
the availability of the energy which it possessed originally, 
and which it had been allowed to lose during its sudden 
expansion. 

403. Thomson's modification of this experiment resembles 
in some respects the process of Regnault described in 
1 86. The gas to be experimented on is made to pass, as 
uniformly and noiselessly as possible, through a tube in 
which there is an obstruction, in the form of a porous plug, 
such as a pellet of cotton-wool or the like. The tempera- 
ture of the stream is carefully measured on each side of the 
plug, and at such distances from the plug as to avoid the 
local irregularities produced by it. 

Since the motion is uniform, equal masses of the gas pass 
in equal times through the various cross sections of the tube. 

z 2 



34o HEAT. [CHAP. 

Let the pressures before and after passing the plug be/ and 
p' t the corresponding volumes of unit mass v and v. 

Then pv is the work done on unit mass of the gas as it 
passes a cross section of the tube before reaching the plug, 
while p'v is the work it gives out as it passes a section 
after leaving the plug. Their difference is the gain of 
energy, provided no heat be supplied from without, and no 
energy lost as sound. Hence, since the motion is regarded 
as uniform, if E and E be the intrinsic energy of unit mass 
before and after passing the plug, 

E' - E = / v - p'v', 
or the conditions of the experiment are such that 

E + p v 
is constant. 
Therefore ( 392), 

v = o, 



or, taking / and / as independent variables, 



dt 
Now / is here evidently the specific heat at constant 

pressure, which ( 395) we called k. Also (393) we have 

d<$> _ dv _ 
~dp ' ~di 

if e be the expansibility at constant pressure. With these 
values our equation becomes 

kt = - v (i - et) S/ . . . . (i) 

Here / represents absolute temperature. To compare 
this with the scale of a gas-thermometer, let T be the tern- 



xxi.J ELEMENTS OF THERMODYNAMICS. 34* 

perature Centigrade on such a thermometer, corresponding 
to the absolute temperature / / the lengths of the degrees 
on the two being assumed as equal throughout the (very 
small) range of the experiment. Then we have, by 124, 

pV = C(l + aT) 

as an approximate expression, to be rectified. Thus, as the 
degrees are practically equal on the two scales, throughout 
the small range of change, 

i dv a. 

= vdT ^ I + af 

and therefore 



Thus, finally, 



In the experiments of Joule and Thomson, the changes 
of pressure and temperature are not infinitesimal. To adapt 
the formula to such a case we may lawfully integrate (2), 
through the small range of the experiment, neglecting the 
variation of / T. Thus if the observed change of tempera- 
ture be 0. and the pressures/ and/', the formula becomes. 



. 

a A' (log./ - log./')' 

404. The first two terms of this expression give the re- 
sult of 126, directly suggested by Charles' Law. The 
third term was found by experiment to be small for bodies 
at temperatures above their critical points, i.e. for true 



342 HEAT. [CHAP. 

gases ; and considerably larger for vapours. All the true 
gases, except hydrogen, were slightly colder after than 
before passing the plug ; hydrogen slightly warmer. It 
was by these experiments that the temperature of absolute 
zero was determined ( 374) to be very nearly - 2737 on 
the Centigrade scale. 

405. The following little table exhibits the main features 
of Joule's and Thomson's results, so far as the air-thermo- 
meter is concerned. The numbers in the first column (each 
with 2737 added) are absolute temperatures. Those in 
the second column are the quantities to be added to the 
corresponding absolute temperatures to give temperatures 
by the air-thermometer, when the air is kept at the density 
corresponding to o C. and i atmosphere of pressure. The 
third column gives the corresponding numbers for the air- 
thermometer when the pressure is maintained throughout at 
one atmosphere. (Phil. Trans. 1854.) 

DIFFERENCE BETWEEN AIR-THERMOMETERS AND ABSOLUTE 
SCALE. 



Absolute Scale. Constant Volume. Constant Pressure. 


2 73 7 + .... 


+0' . , . . 


+ 0* 


+ 20 .... 


+ 0-0298 .... 


+ 0-0404 


+40 .... 


+ 0-0403 .... 


+ 0*0477 


+60 .... 


+ 0*0366 . . . . 


+ 0-0467 


+80 .... 


+ 0-0223 .... 


+ 0-0277 


+ 100 .... 


+ O'OOOO .... 


+ O'COOO 


+ 120 . . . . 


- 0-0284 .... 


- 0-0339 


+ 140 . . . . 


- 0-0615 ...... 


- 0*0721 


+ 160 .... 


- 0-0983 .... 


-0-1134 


+ 180 . . . . 


- 0-1382 .... 


-0-1571 


+ 200 .... 


- 0*1798 .... 


- 0*20lS 


+ 220 .... 


- 0-2232 .... 


- 0*2478 


+ 240 .... 


- 0-2663 .... 


- 0-2932 


+ 260 .... 


- 0-3141 .... 


- 0*3420 


+ 280 . . . . 


- 0-3610 .... 


- 0*3897 


+ 300 . . . . 


- 0-4085 . . - 


- 0-4377 



XXI.] ELEMENTS OF THERMODYNAMK.t .. Sft Tj-r 

r* F*IJ . 

406. Joule's earliest determinations of the vaUie of the 
dynamical equivalent of -heat were made by working a 
porous plug or piston up and down in a cylinder full of 
water, and measuring the change of temperature produced 
for a given difference of pressure above and below the plug. 
Equation (i) of section 403 gives us the means of making 
the necessary calculation. For 8 t is measured ; S/ is 
known ; v is the volume of water present multiplied by 
the number of up and down strokes made by the piston ; 
and the remaining quantities, e and /, may be estimated 
with sufficient accuracy by the air-thermometer. The result 
is k in dynamical units ; i.e. the value of Joule's equivalent 
itself, because the value of k in thermal units is simply 
unity. 

407. In 78-81 a first notion was given as to the dissi- 
pation of energy. Closely connected with this is the 
Restoration of energy, a question also first treated by W. 
Thomson. The tendency of heat (whether by conduction, 
radiation, or convection) towards equalisation of tempera- 
ture, i.e. to loss of availability, gave the first hint of dissipa- 
tion or degradation. It becomes, then, an interesting 
problem to seek what amount of work can be obtained, 
by perfect engines, from an assigned distribution of heat. 

408. If we continue to measure heat in dynamical units, 
the dynamical value of the quantity H is simply H itself, 
whatever be the temperature of the body which contains it. 
But the utmost realisable value, unless we have a body at 
absolute zero to act as the condenser of a perfect engine, is 
always less. In fact, if / be the absolute temperature of the 
hot body, and 4 the lowest available temperature of the 
condenser, the realisable value is ( 376) only 



344 HEAT. [CHAP. 

/r-ftf. 

Suppose that we operate upon a number of bodies at 
different temperatures ; some being used to supply heat, 
others to have heat supplied to them ; then the work will 
be simply 



the excess of the heat taken from some of the bodies over 
that given to others. This must always, except when per- 
fect engines are employed, be less than the realisable 
value 

S (O) - t, 2 (f). 
Hence we see that the expression 

is necessarily negative ; except when perfect engines only 
are used, in which case alone its value is zero. This is 
Thomson's expression for the heat dissipated during the 
cycle of operations. 1 

We have already shown, in 379, how to treat such a 
question when the operations are not cyclical. The repre- 
sentation in symbols presents no difficulty. 

409. When this mode of investigation is applied to any 
distribution of heat in a body, or system of bodies, it leads 
to the measurement of the available energy, or, as W. 
Thomson proposes to call it, the Thermodynamic Motivity 
of the system. This is "the possession, the waste of which 
is called dissipation." 

410. Motivity may be regarded from without, or from 
within the system. 

1 See Phil. Mag. May 1879. 



xxi.] ELEMENTS OF THERMODYNAMICS 545 

In the former case it is the utmost amount of work 
which can be obtained from the system by reducing the 
temperatures of all its parts to some assigned temperature ; 
suppose that of an infinite medium by which the system 
is surrounded. 

In the latter case it is the utmost amount of work which 
can be obtained by equalising the temperatures of the 
various parts of the system among themselves. 

In either case the expression for its value has the form 



* m i 



t / 



where m is any element of mass, c its specific heat in dynam- 
ical units, / its absolute temperature, and / the final 
temperature to which the whole system is to be reduced. 
For mcdt is an element of heat, in a body at temperature 

/. Of this, the amount me Y^ dt can be realised by a 

perfect engine with condenser at t . The integration sums 
up all the heat thus realised till /// is cooled to / . And the 
sign 2 collects these quantities for the separate masses of 
the system. 

When the system is regarded from without, / is given ; 
and the value of this expression can be calculated at once. 

When the system is regarded from within, f must be 
determined by the condition that if a new body at that 
temperature had been used as a condenser for the engines 
using the heat of the hotter parts of the body, and as a 
source to which the colder parts of the body acted as 
condensers, it should on the whole have neither gained nor 
lost heat. This condition is expressed by 



o. 



346 HEAT. [CHAP. 

[For, when the temperature, /, of one of the masses is 
higher than / , the quantity H leaving that mass deposits 

/ ff 

2 in the body at / . But when / is lower than / , the 

supply of H to the mass at / necessitates the taking of 

/ J-T 
' from the body at /. The form of the expression is 

therefore the same in either case; and the distinction 
between giving and taking is provided for by the fixed 
order of the limits in the integral, which is taken from a 
lower to a higher temperature in the former case, and from 
a higher to a lower in the latter.] 

From this / can be calculated, and the expression for 
the motivity then takes the very simple form 



411. One very curious consequence of this is, that if 
the system consist of two equal masses, m, of the same 
substance, at temperatures ^ and / 2 , and if we assume the 
specific heat to be independent of the temperature, the 
common temperature when the internal motivity has been 
entirely realised is 

V'/t/a, 

and the motivity itself is 



Thus the internal motivity of a system consisting of a 
pound of ice-cold water and a pound of boiling water is, 
in foot-pounds, 

'39 (i9'339 - 16-553)--= 1390 x 776. 



xxi.] ELEMENTS OF THERMODYNAMICS. 347 

The absolute temperature to which the system is reduced, 
when its internal motivity is thus exhausted, is 

320-12, corresponding to 46-! 2 C. 

But, if the two parts of the system had been simply mixed, 
the resulting temperature would have been 324, or 50 C. 
Hence the energy would have been greater by that of 2 Ibs. 
of water raised 3'88, i.e. by the quantity of work 
1390 x 7-76 obtained in the former process. But this 
excess of energy is only in part available. With an un- 
limited external system at temperature 46-! 2 C. we can 
realise only about 

1390 x 0-048 foot-pounds. 

Thus when the water is, in each case, brought to the 
uniform temperature of 46-! 2 C., we realise more than 
1 60 times as much work by the first process as by the 
second. 

412. The entropy of a system changes along with its 
motivity; but the two things are quite distinct, as the 
following simple case shows. 

When the element, 77, of heat is in a body at tempera- 
ture /,, its motivity is (as we have seen) 

A-/o rr 

~7T 

where t is the lowest available temperature. Hence, if this 
heat be transferred to another body at a lower tempera- 
ture /,, the loss of motivity is 



A 



On the other hand, when the element H of heat passes from 
a body at temperature / to another at temperature / a , the first 



348 HEAT. [CHAP. 

TJ 

body loses entropy to the amount , and the second gains 

TT 

to the amount -_ ( 378) ; so that the whole entropy of the 
system increases by the amount 



Thus the loss of motivity is simultaneous with gain of 
entropy. But the loss of motivity by the passage of heat 
from a warmer to a colder body is less as the lowest avail- 
able temperature is lower ; while the corresponding gain of 
entropy is the same whatever be this lowest temperature. 

413. The only effect of a limit of temperature, on the 
entropy, is to limit its final amount. But if the external 
universe were at the temperature of absolute zero, there 
need be (theoretically) no loss of motivity, i.e. no dissipation 
of thermal energy ; while the entropy would go on increasing 
without limit as the heat gradually passed to colder bodies. 

Thus we see that Clausius' theorem, " The entropy of 
the universe tends to a maximum" is by no means iden- 
tical with, though it is closely connected with, Thomson's 
previously published theory of dissipation. 

414. In 195 we promised to give Thomson's investiga- 
tion of the phenomena of the Thermo-electric circuit. We 
can now do so, by the help of the formulas of this chapter. 

Let / , /be the absolute temperatures of the junctions in 
a circuit of two metals, in which the specific heat of elec- 
tricity has the values o- 1} o- 2 respectively. Let T be the 
temperature of the neutral point, and II the Peltier effect of 
unit current passing through the junction at temperature t. 
Then the change of E, the electromotive force, caused by 



XXI.] ELEMENTS OF THERMODYNAMICS. 349 

raising the temperature of the hot junction from t to 
will be 



dt 

for it must be remembered that the direction from hot to 
cold is necessarily reversed in passing from one to the other 
of the two metals. 

But the change of the expression 

$( } 

which ( 408) is always zero for a set of reversible opera- 
tions, such as this is ( 195) supposed to be, gives 



Without any assumption as to the expression for o^ cr a , we 
may eliminate it from these equations, and we obtain the 
very interesting result 

dE J 
dt t ' 

[Hence we see at once that II vanishes at the neutral 
point, for then ( 193) E is a maximum, and therefore 



This equation shows that the value of II may be com- 
pletely determined from measurements of the current in 
the circuit as depending on the temperature of the hotter 
junction, the colder being kept at constant temperature. 
Conversely the comparison of the observed currents with 
corresponding measures of the Peltier effect, would enable 



350 HEAT. [CHAP. 

us to test the admissibility of the assumptions we have 
made. 

415. So far, we have been following Thomson. But if 
we now introduce the experimental result ( 197) that the 
specific heat of electricity is proportional to the absolute 
temperature, we have 



and our equations become 

dE d 
dt -' ~d 



The second gives 

3 = c - (*, - k,)t, 

= (k, - k a ) (T - /). 

because IT vanishes at the neutral point. 
With this, the first gives finally 



Thus we see that the expression for the electromotive 
force has no other variable factors than the two ; the first of 
which was of course known to Seebeck, while the second 
w r as discovered experimentally by Thomson. 

The constant factor, (/ r - / 2 ), is the same in the expres- 
sion for the E.M.F., and in that for the Peltier effect. Hence 
we cannot, by measuring either of these quantities, find 
more than the algebraic difference between the specific heats 
<of electricity in the two metals. 



XXL] ELEMENTS OF THERMODYNAMICS. 351 

The measurement of the actual amount of the Thomson 
effect in some one metal must therefore be carried out 
directly. 

All the results of this section are, however, quite easily 
obtained by mere inspection of the diagram ( 196). 

A general notion of the first effect produced by an 
electric current on the distribution of temperature in a wire 
will be found in Nature, xxxiv. 121. 

416. Resume of 383, 415. The state of unit mass 
of the working substance can be expressed in terms of any 
two of the five quantities, energy, volume, pressure, temper- 
ature, and entropy. Hence various thermodynamic relations 
among these. Relation between specific heats. Develop- 
ment of heat by compression. Determination of absolute 
temperatures in terms of air-thermometer. Restoration of 
energy. Thermodynamic motivity. Comparison with 
entropy. Theory of the thermoelectric circuit. 



CHAPTER XXII. 

NATURE OF HEAT. 

417. EVERY ONE has heard of the celebrated phrase 
" Ohne Phosphor, kein Gedanke" Be what there may of 
truth in this, the facts stated in the preceding chapters 
entitle us to say " Without motion, no heat." But just 
as from the first statement, even if it were proved to be 
absolutely true, we should not be entitled to say that 
" Thought is phosphorus " : so the second by no means im- 
plies that " Heat is motion." It is therefore much to be 
regretted that some of the highest modern authorities on the 
subject, such as Clerk-Maxwell and Sir W. Thomson, have 
occasionally been betrayed into the use of this phrase in the 
form in which it was originally employed by Rumford and 
Davy, before clear ideas had been obtained about energy ; 
and that they have thus given an unintentional support to a 
practice altogether inconsistent alike with their usual mode 
of speaking of heat, and with their knowledge of its true 
nature. For, in their less unguarded statements, such men 
invariably speak of heat as energy. The word Motion has 
two different uses ; the one, popular as well as scientific, 
referring to mere change of position : the other, purely 
scientific, meaning (after Newton) momentum. It is, of 
course, in the former of these senses that the word is used 



CH. xxil.] NATURE OF HEAT. 353 

with regard to heat : for its use in the latter sense would not 
be merely slipshod, it would be untrue. In the posthumous 
MSS. of Sadi Carnot, published for the first time in 1878, 
we find that he carefully guards himself in his statements 
on this matter. Thus he says: Un mouvement (celui de 
la chaleur rayonnante) pourrait-il produire un corps (le 
calorique)? Non, sans doute, -il ne pent produire qu'un 
mouvement. La chaleur est done le resultat (Tun mouvement. 
Alors il est tout simple qu'elle puisse se produire par la con- 
sommation de puissance motrice, et qu'elle peut produire 
cette puissance. [This is one of the very striking passages 
to which we referred in 84 above.] 

418. Heat, like all other forms of energy, can be per- 
ceived by us only while it is being transformed or trans- 
ferred. We cannot individualise any portion of it so as to 
be able to recognise it again, any more than we can dis- 
tinguish, in a vessel of water, the portions of the contents 
which were put in first from those which were afterwards 
added. As we have already seen ( 220) we can tell how 
much heat enters a body, and how much leaves it ; what it 
is in the body is quite another question. That some of it, 
at least, is in the form of kinetic energy we know from the 
facts of radiation. For we have seen that all bodies radiate, 
i.e. communicate vibratory energy to the luminiferous 
medium. This can only be in consequence of motions of 
the particles of the body, for it is not motion of the body as 
a whole. And we have seen that radiation is not confined 
to the surface of a body. Hence all the particles (by which, 
for the moment, we mean literally " little parts," parts too 
small to be distinguished by the microscope) of a hot body 
must be in motion. But there are two quite distinct kinds 
of motion which these particles may have : motion of each 
particle as a whole, relative to the others, and motion of the 

A A 



354 HEAT. [CHAP, 

parts of a particle relative to its centre of inertia. This con- 
sideration, as we shall see later, leads to some very important 
consequences. 

419. How much of the energy in a body is that of 
invisible motions of its small parts is, for the majority of 
bodies, still an extremely obscure question. 

Before entering on a short discussion of the subject, it is 
necessary that we should briefly consider what is known as 
to the ultimate structure of matter. This leads, of course, 
to a considerable digression. 

420. That even apparently homogeneous liquid or solid 
matter, such as a mass of water, of glass, or of gold, though 
in our most powerful miscroscopes it still preserves the ap- 
pearance of homogeneity, has ultimately a certain coarse- 
grainedness or heterogeneity, is made certain by a great 
variety of facts. Thus, as water is a compound body, it is- 
quite clear that the division of a drop of it cannot, even in 
thought, be carried farther than a certain limit, if the parts 
are still to be portions of water. This, it must be remarked,, 
is absolutely independent of the question between atoms and 
the infinite divisibility of matter. The only mode of escape 
from the conclusion would be to suppose the doctrine of 
Impenetrability untrue : i.e. to suppose that the same (in- 
definitely small) portion of space could be simultaneously 
occupied by hydrogen and by oxygen. Such a form of argu- 
ment, however, is too metaphysical to be admitted into 
physics. 

421. Fortunately, we can dispense with it. Hetero- 
geneity of ultimate structure is proved "by (i) the difference 
of the refractive indices of every substance for light of 
different wave-lengths ; (2) the phenomena of contact 
electricity ; (3) the behaviour of liquid films, as in a soap- 
bubble ; and (4) the behaviour of gases. To the last of 



xxii.] NATURE OF HEAT. 355 

these, as more intimately connected with our subject than 
the others, we must here confine ourselves. The results of 
the kinetic theory will be given, and compared one by one 
with the known properties of gases. 

422. The atomic speculations of the old Greek philoso- 
sophers, which have been so lucidly expounded by 
Lucretius, have always more or less affected more recent 
thought \ and probably had a share in suggesting to Hooke 
and Bernoulli ( 39) their notion of the cause of pressure in 
a gas. That notion was revived by Lesage and Prevost ; and 
again in this country by Herapath, in a very curious work : 
but the first precise calculations based on it were made by 
Joule. He showed that, if the pressure of a mass of 
hydrogen be due to the continual impacts of its particles on 
the walls of the containing vessel, the speed (assumed to be 
the same for all the particles) must be about 6055 feet per 
second, at o C., and i atmosphere pressure. Also that, the 
volume being supposed to remain constant, this speed in- 
creases in proportion to the square root of the pressure. 
These calculations were of a very simple character, as they 
took no account of the number or size of the particles, or of 
their mutual impacts. [The more complete investigations 
which have since been made have shown that this result is 
correct if, for the supposed uniform speed of the particles, we 
substitute the (so-called) velocity of mean square.] 

423. We may easily see how Boyle's Law can be 
arrived at by this elementary hypothesis. For, suppose a 
gas to be inclosed in a cylinder, and suppose that we could, 
without altering the common speed of its particles, compress it 
into half its former bulk by pushing in one end of the 
cylinder. Then the number of impacts, per second, on the 
ends of the cylinder would be doubled ; because the lengths 
of the paths of the particles from end to end have been 

AA 2 



356 HEAT. [CHAP. 

reduced to half, while the speed of the particles remains 
the same. Also the impacts on the curved sides of the 
cylinder are just as numerous and as intense as before, but 
they are applied to only half as great a surface. Thus the 
pressure is everywhere doubled. 

Joule's calculation, referred to above, may be made as 
follows. Let M be the whole mass of the gas in a cube 
whose side is /, u the speed of each particle, n the number 
of particles. Then, if we suppose one-third of them to be 
moving perpendicular to each pair of faces, the number 
of impacts per second made by any one particle on any one 
face is uJ2l. Hence the whole number of impacts per 
second on a face is nuj6l. Also each impact reverses the 
speed of a mass M/n, so that its measure is 2.Mu\n. And 
the pressure per unit surface of the face is found by 
multiplying the value of each impact by their number, and 
dividing by the area of the face. It is therefore J/w 2 /3# or 
one-third of the product of the density of the gas by the 
square of the common speed of its particles. Thus we may 
write, as the result of this approximate investigation, 

pv = \Mu z , 

and the Law of Boyle and Charles at once suggests that 
the absolute temperature is proportional to the whole kinetic 
energy of the particles. 

424. An easy extension shows how to estimate the 
specific heats, if we assume that the whole kinetic energy 
of the particles is increased by the energy of the heat com- 
municated. But here Clausius found it necessary to take 
into account the internal energy of each particle, whether 
due to rotation or to relative motion of its parts. If we 
assume that this bears on the average a constant ratio to its 
energy of translation, we find that the experimental facts of 
this part of the subject can be accounted for. 



xxn.] NATURE OF HEAT. 357 

425. The words put in italics in section 423 merit 
special remarks, which can only in part be made as yet : 
so we must recur to them. In pushing down the piston 
work is done against the pressure of the gas, and we know 
that the result is that the gas is heated, and ( 365) that 
its pressure increases adiabatically, i.e. more rapidly than 
according to Boyle's Law. To prevent this, the heat due 
to compression must be removed as it is communicated. 
Thus work is spent, and its equivalent in heat communi- 
cated to the surroundings of the cylinder. In other words, 
there is dissipation of energy. But if the italicised state- 
ment could be realised in practice, there would be restora- 
tion of energy without any expenditure of work. For the 
motivity of the compressed gas would be increased, without 
any expenditure of work ; though the energy of motion of 
its particles would not be altered. 

426. It is to Clausius that we are indebted for the 
earliest approach to an adequate treatment of this question. 
He was the first to take into account the collisions between 
the particles, and to show that these did not alter the pre- 
viously obtained results. He has also the great credit of 
introducing the statistical methods of the theory of proba- 
bilities, and of thus giving at least approximate ideas as to 
the probable length of the mean free path, i.e. the average 
distance travelled over by a particle before it impinges on 
another, and thus has its course changed. He thus ex- 
plained also the slowness of diffusion of gases, and their 
very small conductivity for heat. Clerk-Maxwell shortly 
afterwards improved the theory by introducing, also from 
the statistical point of view, the consideration of the variety 
of speeds at which the different particles are moving; 
Clausius having expressly limited his investigations by 
assuming for simplicity that all moved with equal speed. 
Maxwell explained gaseous friction, and gave a more 



358 HEAT. [CHAP. 

definite determination of the length of the mean free 
path. 

These investigators, with the able concurrence of Boltz- 
mann and others, have since still farther extended the 
theory. All that we can attempt here is to give a general 
account of their more prominent results. This will be 
done, as far as possible, in such a way as to show how far 
the theory is capable of accounting for known facts : for 
it is by this, and this alone, that the validity and adequacy 
of a theory can be judged. Our plan forbids the introduc- 
tion of the more complex mathematical processes, so that 
we can (as a rule) merely give the assumptions, with their 
chief theoretical consequences. But we must make an 
exception in favour of a method which has practically 
superseded those previously employed in the treatment of 
this question. 

427. Assuming that the gaseous contents of a vessel 
consist of independent particles which exert certain forces 
on one another, while the average distribution of the 
particles, and of energy among them, is subject only to 
slight and very rapid periodic changes, Clausius in 1870 
established a perfectly general theorem which is symbolically 
expressed as follows : 



Here the left-hand side is the whole kinetic energy of the 
system, half the sum of the products of each mass m, by 
the square of its speed u. If the parts of a particle move 
relatively to the centre of inertia of the whole, the kinetic 
energy of this motion must be included. The right-hand 
side consists of two terms, the first of which is a mere 
numerical multiple of the product of the volume and the 
pressure on the containing walls (supposed uniform). The 
second term, which Clausius names Virial, depends on the 



xxii.] NATURE OF HEAT. 359 

mutual action between pairs of particles (which is assumed 
to be exerted in the direction of the line joining them). 
Each element of it is the product of this force R (positive 
when there is attraction) into the distance r between the 
two particles. In this term must be included the part due 
to distance between portions of the same particle. 

428. If we suppose for a moment that the particles are 
small hard bodies, exerting no action on one another except 
when they are in collision, the term 



depends merely upon the impulses, multiplied by the 
common diameter of the particles. 

When the diameter is small in comparison with the 
average distance of any particle from its nearest neighbours 
(i.e. when the whole space occupied by the particles is small 
in comparison with the volume of the gas) this term is 
negligible : so that the equation practically reduces itself 
to the form 

S(/ a ) =pv. 

This is, in a slightly different form, the relation given by 
Joule ( 423) ; and, as already stated, it leads us (by com- 
parison with the Law of Boyle and Charles) to regard the 
absolute temperature of a gas in this condition as directly 
proportional to the kinetic energy. 

When the particles are assumed to be equal, smooth, 
hard spheres, whose diameters are comparable with the 
average distance between contiguous ones, the equation is 
found to take the approximate form 



where ft is four-fold the sum of the volumes of the particles. 
For a given total kinetic energy (i.e. for an isothermal) the 



360 HEAT. [CHAP. 

gas now shows greater resistance to compression than before, 
and this effect increases rapidly as the volume is diminished. 
The equation just written shows a fair agreement with the 
behaviour of hydrogen, from ordinary temperatures upwards. 
But the deviations, from the Law of Boyle and Charles, 
which are exhibited under ordinary circumstances of pressure 
and temperature by the more easily liquinable gases (and by 
the so-called permanent gases at temperatures sufficiently 
low, and pressures sufficiently high) are to be accounted for 
by molecular attraction between the particles. In this case 
the virial equation takes some such form as 

A 



r + a 

where Afv 2 is approximately the molecular pressure of 
Laplace. This shows at once how, for a given amount of 
kinetic energy, the molecular force tends to increase the 
compressibility of the gas : or, what comes to the same 
thing, to reduce its resistance to compression. 

The first on the whole soundly-based suggestion of an 
equation of this species, viz. 

J 2 (mu 12 ) = (p + 

is due to Van der Waals ; who seems to have been led to it 
by endeavouring to extract from the virial the part due to 
Laplace's molecular pressure. Though objections may be 
freely taken to this particular equation, as well as to parts 
of the process by which it was obtained, there can be no 
doubt that the mere suggestion of it formed an extremely 
important step towards the solution of this very interesting 
and difficult problem. 

But so soon as we introduce the idea of a molecular 
action between the particles of a gas, and the consequent 
increase of kinetic energy which they acquire in merely 



xxii.] NATURE OF HEAT. 361 

approaching one another, we are brought face to face with a 
new and excessively serious difficulty : viz. What now 
represents the absolute temperature of the group of particles? 
Until this point is settled, we cannot put the virial equation- 
into the desired form of a relation among/, z\ and /, which 
is to take the place of the Laws of Boyle and Charles. 
Van der Waals and Clausius assume that the whole kinetic 
energy is still proportional to the absolute temperature. 
Clerk-Maxwell is more guarded : for he says " we have no 
evidence that any other law holds for gases, even near their 
liquefying point " ; while Sir W. Thomson and others have 
gone at least so far as to express doubt on the subject. 
The question cannot be even provisionally answered, at least 
with any feeling of confidence, until we have the solution of 
the purely dynamical problem : Find the ultimate average 
distribution of kinetic energy in two systems of impinging 
particles, which are separated from one another by a flexible^ 
inextensible, massless diaphragm ; when one of the systems, 
only, has molecular forces among its particles. 

Further remarks on this subject are thus, for the present 
at all events, unsuited for an elementary treatise : and it 
happens, fortunately, that the great majority of the results 
now to be given were obtained without the introduction of 
molecular forces : so that the temperature difficulty was 
cither not raised, or raised only in an indirect -iianner. 

429. Clerk-Maxwell showed that if there be a mixture of 
two kinds of particles, they will interchange energy until the 
average kinetic energy is the same for each species of 
particle. Hence it follows that when masses of different 
gases are at the same temperature and pressure the number 
of particles per unit volume is the same in each. This is 
what chemists call Avogadro's Law (also pointed out by 
Ampere in 1814). 

He also demonstrated that, in a mixture of gases in 



362 HEAT. [CHAP. 

hydrostatic equilibrium under the action of gravity, each 
gas behaves as if the others were absent. Further, that in 
equilibrium, under gravity alone, conduction would lead to 
uniform temperature throughout a gaseous column. [From 
this, by a simple application of Carnot's process, we show 
at once that gravity can have no influence on the thermal 
equilibrium of any substance.] 

430. A remarkable relation, deuuced from the kinetic 
theory, enables us to approximate to the length of the mean 
free path of the particles of a gas. For this is shown to 
bear, to the diameter of any one particle, the same ratio as 
the volume of the gas bears to about 8^ times the whole 
bulk of the particles. If we assume that, in the liquid 
state,' the particles are very nearly in contact, we can find 
approximately the size of a particle from the relative den- 
sities of the gas and the liquid. Thus it is found that the 
number of particles of gas in a cubic inch, at ordinary 
temperature and pressure, is somewhere about 3 x io 20 ; 
while if it be oxygen, the mean free path is about - T fa ru 
inch, and the velocity of mean square about 1550 feet per 
second. It follows that, on the average, each particle has 
about 7^600 million collisions with others per second. 

431. Now we are prepared to understand why it is that 
the oxygen and nitrogen of our atmosphere are so tho- 
roughly mixed that all analyses of different portions of the 
same mass of air give exactly the same ratio of the oxygen 
to the nitrogen. If each cubic inch contained only a dozen 
or two of particles, the theory of probabilities shows that in 
a large extent of air there would be, at any instant, certain 
cubic inches in which the relative proportions of the con- 
stituents might be any whatever. Even then this state of 
things could last only for an exceedingly small fraction of a 
second. Biit e in consequence of the enormous number of 



xxii.] NATURE OF HEAT. 363 

the particles in each cubic inch, the probability of one single 
cubic inch of the lower atmosphere being (even for a 
moment) exclusively rilled with oxygen or nitrogen alone, 
becomes practically a vanishing quantity. Spaces less than 
the trillionth part of a cubic inch will, however, occasion- 
ally (but only for periods less than the billionth part of a 
second) contain oxygen or nitrogen particles alone. [These 
numbers are purposely stated in the roughest terms, but 
they give all that we require : a notion of the relative 
magnitudes involved.] 

432. The following experiment, due to Graham, is one of 
the most beautiful and suggestive in the whole range of 
physical science : while it requires, for its successful 
performance, no training whatever. 

Take a hollow ball of unglazed clay, and lute into its 
neck a glass tube of moderate bore, so as to form a vessel 
resembling a large thermometer. Invert it, full of air, and 
plunge the open end of the tube into water. The difference 
of levels of the water inside and out settles in a short time 
into that due to capillary forces, and there remains. [Had 
the ball been made of glazed clay, or of glass, the water 
inside would not have risen so high : in fact, by push- 
ing the tube farther down we should obtain a permanent 
depression of the level of the inner surface.] This state of 
things is maintained by the absolute equality of the numbers 
of particles continually passing opposite ways through the 
fine pores of the clay. Unfortunately we cannot mark the 
particles, so as to be able to see directly that the contents 
of the apparatus are steadily changing. We must use 
another method. If we could arrange matters so that the 
number passing in should be greater than those passing out, 
iras would bubble out from the end of the tube : if fewer 
entered than escaped, atmospheric pressure would cause 



364 HEAT. [CHAP. 

the water to rise in the tube. Now this is precisely what 
happens when an atmosphere of coal-gas (collected in an 
inverted beaker) is made to surround the ball. The 
moment it is applied, air begins to bubble rapidly out from 
the tube. If this be continued for a little, till the contents 
of the bulb are mainly coal-gas, the moment the beaker is 
removed the water rapidly rushes up : showing in either 
case how much more rapidly coal-gas passes through the 
pores than does air. 

The rates of passage of different gases under similar 
conditions are inversely as the square-roots of their densities. 

But the main purpose for which we have described the 
experiment here is to show how thoroughly independent 
of one another are the particles of a gas, and how fast they 
must be moving, even at ordinary temperatures ; and thus 
to give an additional and very strong argument in favour of 
the kinetic theory. 

Here, incidentally, we have a first method of testing the 
equality of the various particles of any one gaseous 
substance. For a gas consisting of particles of different 
sizes would be partially sifted (as it were), by allowing half 
of it to diffuse through the clay ball. Half of the remainder 
may then be allowed to diffuse, and so on. It is clear that 
we should thus have a process somewhat analogous to what 
the chemists call Levigation ; and by repeating it sufficiently 
often, we might obtain two specimens of the same gas of 
different densities (at the same temperature and pressure). 

433. Additional, and equally strong, arguments in favour 
of the theory are furnished by the diffusion of one gas into 
another, without the use of a porous septum as in the ex- 
periment above described. The general laws of this 
diffusion are found to be precisely such as are indicated by 
the theory ; and the careful measures of Loschmidt have 



XXII.J NATURE OF HEAT. 365 

enabled us to arrive at much more definite ideas of the 
actual size and number of the particles in various gases than 
could be obtained by the rough assumption indicated in 
430 above. The following approximate numbers are 
taken from Clerk-Maxwell's (1873) Bradford Lecture on 
3 1 oU ailes : 

Carbonic Carbonic 
Hydrogen. Oxygen. Oxide. Acid. 

Velocity of mean-square at o c C. 6190 1550 1656 1320 

Mean path 386 224 193 151 

Collisions 1775 7646 9489 9720 

Diameter 2-3 3 3-5 3*7 

The collisions are here given (as in 430) in millions per 
second ; the mean path and the diameter in hundred 
millionths of an inch ; and the speed in feet per second. 

434. Another valuable confirmation of the theory is 
afforded by the satisfactory manner in which it accounts for 
what would othenvise be an extremely puzzling fact the 
existence of viscosity in gases. This, however, is at once 
seen to be a consequence of inter-diffusion between layers of 
gas moving with different velocities, and thus, as it were, 
sliding over one another. For this process obviously tends 
to do away with the relative velocity of the two layers, and 
thus produces precisely the same kind of effect as the 
ordinary friction between solids. 

435. The conduction of heat in a gas is obviously to be 
.attributed to the transference of energy by impacts of the 
more quickly moving particles on the others. The kinetic 
theory gives very simple relations among the numerical 
coefficients of diffusion, viscosity, and heat-conduction in 
a gas. But these could not easily be made intelligible 
without a more careful examination of the theory than our 
limits permit Nor can we do more than point out that it 
has also been successfully applied to the explanation of the 



3 66 HEAT. [CHAP.. 

processes of evaporation and dissociation, which (as we have 
seen in 231) have a considerable resemblance to one 
another. In the one case we have a particle of the liquid 
moving so last that it tears itself away from the rest in spite 
ot the molecular forces. In the other case a compound 
particle is struck so sharply by another that its constituents 
are separated in spite of their chemical affinity. 

436. We saw in 418 that, besides the motions of the 
particles of a gas as wholes, there are generally relative 
motions of the different parts of a particle. These motions 
are at least in part communicated to the luminiferous 
medium in the form of transverse vibrations, and probably 
account for the main loss of energy in the radiant form. Of 
this, however, we cannot be absolutely certain, for we are 
altogether ignorant of the mode in which energy is trans- 
ferred from ordinary matter to the luminiferous medium : 
and it is possible that there may be a direct transfer even 
when the particle is moving merely as a whole. In such a 
case there would be something producing effects similar to 
those of a resisting medium ; but in what form the energy 
thus transferred would be stored or propagated in the 
luminiferous medium we have no knowledge. 

437. So long as a particle is pursuing its free path, we 
should expect its internal motions to be of perfectly definite 
periodic character, like those of a bell for instance. Hence, 
if the particles are all equal and similar, we are prepared to 
find that rarefied gases, when their temperature is sufficiently 
raised, give out one or more radiations each of perfectly 
definite wave-length. This corresponds to one or more 
bright lines in the spectrum. But even with absolutely 
equal particles these lines cannot be perfectly sharp ; for, 
though all the particles are vibrating in precisely the same 
periods, some are approaching the spectator, and others 



xxn.J NATURE OF HEAT 3^7 

receding from him, with velocities not negligible in com- 
parison with that of light. Hence, from this cause, which 
tells more as the temperature is higher, the lines are of 
finite (though small) breadth. At the instants of collision, 
the vibrating body is under constraint, and the periods are 
necessarily altered. The result of this action also tends in 
the direction of broadening the lines, but it is practically 
insensible when the gas is very rare ; unless it be in ex- 
tremely great thicknesses, for then (as we saw in 305) it 
behaves as a black body. 

But the effect of compression, the temperature being 
supposed to be maintained unaltered, is to reduce the time 
during which a particle is free from collisions : i.e. to 
make the constrained radiation a larger fraction of the 
whole. And this effect steadily increases with farther com- 
pression, the lines gradually broadening out until we have 
a practically continuous spectrum. Such is necessarily the 
case when the body is in the liquid or the solid state. 

438. Another point of importance, in which also experi- 
ment has fully verified the previsions of theory, is that the 
more sharp the impact the greater will be the number of 
vibrations of shorter periods called into play; as well as the 
greater the intensity of the lower forms. Hence the spectral 
lines increase in intensity as the temperature rises, and new 
ones of higher refrangibility come into view. But we must 
not pursue this discussion, as we have not, in the preceding 
pages, entered with any detail into the methods and results 
of spectrum analysis. (See Nature, xxiv. 582.) 

439. There remains, however, one point of fundamental 
importance on which our present subject enables us to 
obtain more light : the nature of the evidence for the 
Second Law of Thermodynamics ( 82). The proof which 
we gave in passing, in 93, rests entirely on the assumption 



368 HEAT. [CHAP. 

(which is essentially that of W. Thomson) that we cannot 
convert into work heat derived wholly from the colder of 
the two bodies employed in the process. If we look on 
this statement in the light of the definition in 410, it 
will be seen to amount to this : that it is impossible to 
obtain work by a process which at the same time increases 
the intrinsic motivity of the system we employ. 

Clausius, who was the first to state Carnot's principle in 
a manner consistent with the true theory of heat, offered 
in support of it merely the following argument, with which 
he concludes a statement analogous to that of 93 : 
" This contradicts the general deportment of heat, which 
everywhere exhibits the tendency to annul differences of 
temperature, and therefore to pass from a warmer body to 
a colder one." 

As will be seen presently, heat does pass (though on an 
almost infinitesimal scale) from colder to hotter bodies. 
And a thermo-electric current, from a sufficient pile of 
elements, can raise a fine wire to a higher temperature 
than that of any of the junctions. 

Several years later, Clausius put his argument into the 
following form : " Heat cannot, of itself, pass from a colder 
to a warmer body." The words "of itself" he afterwards 
interpreted into "without compensation." This form is 
not liable to the thermo-electric objection above. Clausius' 
basis thus ultimately becomes, " In all uncompensated 
transformations there is increase of entropy." 

440. The true basis of the second law is to be found in 
the extreme number and minuteness of the particles of 
matter : and the consequent impossibility of dealing with 
them individually. For, by mere frictionless constraint, 
we can transfer energy of translatory motion from one body 
to another ; so that, if we could operate on each particle 



xxii.] NATURE OF HEAT. 369 

of a gas separately, we might transfer at least the greater 
part of their motion to one and the same body ; and thus 
the motivity of the system might be notably increased 
without any external application of work. 

441. If, in the illustration of 423, we suppose the 
piston to move in, bit by bit, those parts only advancing 
upon which (for the moment) there is no impact, the con- 
sequences there mentioned would be realised : no work 
would be spent, the energy of the gas would remain 
unaltered, but its motivity would be increased. The 
multitude and minuteness of the particles of the gas render 
this impossible ; and thus we are forced to conclude that, 
to increase the motivity of the gas, without increasing its 
energy, we must degrade a portion of energy by applying 
it as work to compress the gas, and then removing its 
equivalent in the form of heat. 

And in fact, as the statement of 43 1 shows, heat does 
of itself pass from colder to hotter bodies. For the same 
argument which applies to the relative amounts of two 
different gases at different parts of a mixture applies also 
to the relative amounts of slow and fast moving particles 
of any one gas. But this, of course, is only on the same 
exceedingly minute scale as in the former case. 

442. Clerk-Maxwell has treated the question from this 
point of view with great clearness. He says : 

" One of the best-established facts in thermodynamics 
is that it is impossible in a system inclosed in an envelope 
which permits neither change of volume nor passage of 
heat, and in which both the temperature and the pressure 
are everywhere the same, to produce any inequality of 
temperature or of pressure without the expenditure of work. 
This is the second law of thermodynamics, and it is un- 
doubtedly true as long as we can deal with bodies only in 

B B 



37 HEAT. [CHAP. 

mass, and have no power of procuring or handling the 
separate molecules of which they are made up. But if we 
conceive a being whose faculties are so sharpened that he 
can follow every molecule in its course, such a being, whose 
attributes are still as essentially finite as our own, would be 
able to do what is at present impossible to us. For we 
have seen that the molecules in a vessel full of air at 
uniform temperature are moving with velocities by no 
means uniform, though the mean velocity of any great 
number of them, arbitrarily selected, is almost exactly 
uniform. Now let us suppose that such a vessel is divided 
into two portions, A and B, by a division in which there is 
a small hole, and that a being, who can see the individual 
molecules, opens and closes this hole, so as to allow only 
the swifter molecules to pass from A to B, and only the 
slower ones from B to A. He will thus, without expendi- 
ture of work, raise the temperature of B and lower that of 
A, in contradiction to the second law of thermodynamics. 

" This is only one of the instances in which conclusions, 
which we have drawn from our experience of bodies con- 
sisting of an immense number of molecules, may be found 
not to be applicable to the more delicate observations and 
experiments which we may suppose made by one who can 
perceive and handle the individual molecules which we 
deal with only in large masses." 

443. It is to be hoped that, in time, the mathematical 
treatment of this statistical (not dynamical) problem may 
be brought within the range of students in general. This 
appears to be quite possible, though no attempt of any 
value seems yet to have been recorded. 

But the real difficulty of the whole theory (beside, of course, 
the tremendous one of temperature, 428) and one which, 
even when overcome, will probably be capable of treatment 
only by methods far beyond the grasp of the ordinary reader, 



xxii.] NATURE OF HEAT. 37 1 

is that (so far as it has yet been developed) it has not been 
reconciled with the observed values of the specific heats of 
gases in general. To account for the large number of lines 
in the spectrum of a glowing gas, it is necessary to assume 
that the constituents of each particle have a great many 
degrees of 'freedom -, i.e. of perfectly distinct kinds of relative 
motion. But, at least in the present mode of applying the 
theory, every additional complexity of the gaseous particle 
seems to require that the ratio of its intrinsic energy to its 
energy of translation shall be made larger. Thus the 
calculated values of the specific heats are as a rule much 
too large. But it would appear that the difficulty has been 
unnecessarily increased by the assumption that, on the 
average, the whole energy is shared in equal proportions 
by each degree of freedom. This is not likely, so far at 
least as the character of the radiation of a gas at different 
temperatures can afford us any information on the subject. 
But such outstanding difficulties, grave as they certainly 
are, must not prevent our acceptance of a theory alike 
elegant and simple, and one which has already explained 
in an unexceptionable manner a whole series of apparently 
unconnected facts. (See, again, 438.) 

444. In the present state of science we are not likely to 
take the course of Clairaut, D'Alembert, and others who, 
when they found the calculated motion of the moon's nodes 
to differ considerably from that given by observation, pro- 
posed to modify the gravitation law of the inverse square of 
the distance, in spite of the almost innumerable series of facts 
which had been exactly accounted for by means of it : 
forgetting altogether that, in forcing the law to suit the 
results of their calculations of this special phenomenon, 
they were thereby rendering it totally unfit for the purposes 
to which it had already so fully adapted itself. 



372 HEAT. [CH. xxii. 

445. It is mainly the existence of difficulties like those 
above referred to, which are felt throughout every portion of 
the now extensive range of physical science, that gives such 
a zest to the struggles of its true votaries. If science were 
all reduced to a matter of certainty, it could be embodied 
in one gigantic encyclopaedia, and too many of its parts 
would then have for such men little more than the com- 
paratively tranquil or, rather, languid interest which we feel 
in looking up in a good gazetteer such places as Bangkok, 
Ak-Hissar, or Tortuga. A science which could be com- 
pleted would have far less interest than a dead language ; 
it would sink to the level of a puzzle of which we had 
discovered the key. 

But we have merely to think of the ideas which we try to 
express by such words as Time, Space, and Matter, to see that, 
however far discovery may be pushed, our little " clearing " 
can never form more than an infinitesimal fraction of the 
" boundless prairie." No part of this, however, can strictly 
be called inaccessible to unaided human reason, if time 
and patience fail not. But far beyond in one sense, though 
in another sense ever intimately present with us, are thd* 
higher mysteries of the true Metaphysic, of which our 
senses and our reason, unaided, are alike unable to gain us 
any information. 

446. Resume of 417-445. Nature of heat in a 
body. Grained structure of matter. Kinetic gas theory. 
Calculation of speed of hydrogen particles. Statistical 
methods. Virial. Relation of Kinetic Energy to Absolute 
Temperature. Law of Avogadro. Mean free path. Number 
of collisions. Transpiration of gases. Viscosity, diffusion, 
thermal conduction. Real basis of Second Law. Present 
difficulties of the kinetic theory. 

RICHARD CLAY AND SONS S LIMITED, LONDON AND BUNGAY. V >, 



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