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accordance with the kinetic theory.

Theoretical Difference of Temperature.

Whatever may be the nature of the action by which heat is communicated
from a surface to a gas, the result, according to the kinetic theory, is to
increase the mean square of the velocity with which the molecules move, in
the ratio of the temperature : thus, if v be the initial velocity, and r the
initial absolute temperature, and if

V 2 = AT,
where A is a constant depending on the nature of the gas, then

(V + dvf = A (r + dr\
or, neglecting dv 2 as a small quantity,

Adr = 2vdv,

rf

dr = 2dv - .
v

Now, if we assume that each molecule comes up to the surface with
a velocity v, and leaves with a velocity v + dv, we shall have the greatest
reactionary force which it is possible that the heat could produce. That the
force produced is as large as this is not probable. We know that at ordinary
densities the molecules communicate the heat to each other, so that they do
not come up to the surface with so small a velocity as v. The smaller the
tension of the air, however, the less will be the difference ; so that the force
which we have assumed is the limit towards which the force tends as the



23] HEAT BETWEEN A SURFACE AND A GAS, ETC. 177

vacuum improves, so long as the conditions of a perfect gas are fulfilled*. The
increase dv in the velocity with which the molecules leave the surface would
increase the pressure in the ratio



or



p



2v '

dp dv

= 1 +



dp _ .dv
and by the foregoing



p * v



dv _ .dr

ir = *v

dp _ . dr
p r '

Therefore if, as we have calculated,

^ = 0008,
P

^ = 0032,

T

and taking r = 520 F.,

.'. dr = 1-6640.

If, therefore, the difference of temperature caused by the light were not
greater than 1 0> 7 R, it would appear from these measurements that the
forces arising from the communication of heat would not be adequate to
cause the effect produced. That is to say, 1 0> 7 is the lowest limit that the
theory admits for the heat reaction to have caused the effects in this
particular case. The theory points to the probability, however, that the
difference was considerably greater than l - 7.

To put this to the test it was necessary to obtain some measure of the

Actual Difference of Temperature on the Black and Bright sides of the

Plates.

. '\

So far as I am aware there is no recognized means of measuring this
difference ; and although it is admitted that a black surface exposed to light
will attain a higher degree of temperature than a white or bright surface, no
comparative experiments have been made.

* Proc. Boy. Soc. 1874, vol. xxii. p. 407.
O. R. 12



178



ON THE FORCES CAUSED BY THE COMMUNICATION OF



[23



While taking part with Dr Schuster in his experiments, I held an
ordinary thermometer containing some dark red fluid, in the place which the
mill had occupied, exposed to the light. This came from a lime-light, and
was condensed by an ordinary lantern.

The thermometer rose to 130 F., and was still rising when the experiment
had to be discontinued.

This measure, great as it was, was not satisfactory, for it was not com-
parative, and a white-bulbed thermometer would obviously have risen to
some extent. I therefore took two similar mercurial thermometers, blackened
the bulb of one and whitened that of the other, and exposed them to similar
intensities of light. Under all circumstances the black bulb was the most
affected, for however long a time the exposure was continued; the light
of a candle which caused the light-mill to make 30 turns per minute made
a difference of 2^ in the thermometers, whereas a feeble sun, which gave
the mill about 60 turns, caused a difference of 5. These results showed a
close agreement with the action of the light-mill ; but whereas the light
acted instantaneously on the mill, the thermometers did not show signs of
moving for some time. It also seemed probable that the immediate surface
which was exposed to the light, besides coming to its temperature almost
instantaneously, would probably assume a higher temperature than that which
would be communicated through the material. In order to show this it
occurred to me to construct



A New Photometer.

This instrument consists of two very thin hollow glass globes,
in diameter, connected by a siphon-tube ^ inch internal diameter.



inches





FRONT END

One of the globes was blackened on the inside with lampblack over one



23] HEAT BETWEEN A SURFACE AND A GAS, ETC. 179

hemisphere, and the other was whitened with chalk in a similar manner ; the
siphon-tube was filled with oil, the air within the globes was carefully dried,
and they were sealed. The two clean sides of the globes are turned in the
same direction, so that any light entering through these clean sides falls
equally on the blackened and whitened surfaces within. The air within
instantly commences to receive heat in proportion to the temperature of
these surfaces, and, expanding, moves the liquid in the tube.

By comparing the volume of a certain length of the tube with the
volume of the globes, the distance which the liquid moves for 1 degree
difference of temperature has been found : 1 inch means 2'2 degrees. A scale
having been fixed to the tube, the effect of light to cause a difference of
temperature in the air can be read off.

There is. however, still one difficulty : the air within the globes does not
arrive at the temperature of the surfaces, as these do not entirely enclose it.
All that can be said is that it is proportional, probably about ^ or rather
more.

This difference may, however, be set off against the difference which
must exist in the mean temperature of the vanes of the mill, and what it
would be if they remained steadily perpendicular to the light. As it is, each
part of the surface of the vane is only exposed to the light for half its
time, and then at varying angles ; so that the light that it receives bears to
the light which would fall on it, if fixed and perpendicular, the ratio of the
diameter to the circumference of a circle, i.e. the ratio 1/vr. In the case of
the photometer the ratio of the section of the intercepted beam to the whole
surface of the sphere is that of the area of a great circle to that of the
sphere, or \ ; so that it is probable the photometer only registers f the
difference of temperature which similar surfaces would acquire on the mill.

The white surfaces on the mill, however, are not similar to those of the
photometer, and they probably absorb considerably more light, and con-
sequently diminish the difference of temperature; so that, on the whole,
it is probable that the differences recorded by the photometer are quite as
great, if not greater than those which exist in the mill.

The instrument is very sensitive, and begins to move as soon as the light
falls on it. Its indications agree surprisingly with those of the light-mill :
1 on the photometer corresponds with 11 revolutions per minute of the
mill. When the mill made 200 revolutions per minute, the reading on the
photometer was 21, which is the highest it will record. Differences to ^ of
a degree can be read on the photometer, or the effect of light which will turn
the mill at 1 revolution per minute. It can be used, therefore, for all

122



180 ON THE FORCES CAUSED BY THE COMMUNICATION OF [23

purposes of photometry for which the mill may be useful. It is much
more convenient, as it requires no counting, and it can be made with
much less trouble.

Measured by this photometer, the difference of temperature in Dr Schuster's
experiment would have been 24. This, which must be looked on as an
outside measure, leaves ample room for allowance for the inaccuracy of the
calculation. We have, on the one hand, the least estimated heat 1'7, and
the greatest limit of the measured heat 24, and the probability that both
these quantities tend towards each other.



Conclusion.

The investigation of which this paper gives an account was undertaken
with a view to settle the only point respecting my previous explanation of
the motion caused by heat which appeared to me to remain doubtful, after
I had discovered that, according to the kinetic theory, the communication of
heat to a gas was attended by a reaction on the surface, viz. whether this
reaction was adequate in amount to produce the motion. This point has
now been cleared up. We have :

1. The remarkable agreement between the law of the resistance ex-
perienced by the mill and the peculiar law of the resistance which air offers
at small tension.

2. Dr Schuster's positive proof that the force which acts on the vanes
arises within the mill itself.

3. The exceedingly small magnitude of the actual force, as shown by
quantitative measurements.

4. The fact, that the estimated difference of temperature necessary to
produce heat-reactions, of equal magnitude with the forces which act, is
well within the difference of temperature actually found to exist.

Taking all these facts into consideration, it seems to me that the evidence
is conclusive as regards the nature of the forces which cause the motion in
light-mills, and that we may now look upon the motion caused by light and
heat as a direct proof of the kinetic or molecular theory of gas.

A new Light-Mill.

Although the proofs against the forces in the light-mills being directly
referable to radiation are already more than sufficient, I will venture to
suggest one more test, which the difficulty of obtaining the instrument has



23] HEAT BETWEEN A SURFACE AND A GAS, ETC. 181

as yet prevented me applying. If a " light-mill " were made unlike those
which have hitherto been constructed, inasmuch that, instead of its vanes
being perpendicular to the direction of motion, and having one side black
and the other white, it has vanes arranged like the sails of a wind-mill or
the screw of a ship, all inclined to the direction of motion, and of the same
colour on both sides ; then if this mill turned, it would show that the force
is not influenced by the direction from which the light and heat come, but
that, like the wind on a wind-mill, it acts perpendicularly to the surface of
the vanes*.

It seems to me that, inasmuch as the vanes of such a mill would be
continuously acted upon, and would experience the full and not merely the
differentiated effect of light, it would be much more sensitive than those at
present constructed.



APPENDIX (March 7, 1877).

Vanes fixed in the Envelope.

In the discussion which followed the reading of this paper, it was stated
by Mr Crookes that he had suspended his instruments upside down by a
single fibre, and floated them upside down in water, and had then found,
when the vanes could not turn in the envelope, that the whole envelope
rotated very slowly under the action of light, steadily and continuously in
the same direction as that in which the vanes would have turned had they
been free. And at the Meeting on March 30th, subsequent to the reading
of this paper, Mr Crookes described how the case of one of his instruments,
floating in water, revolved at a rate of about 1 revolution an hour when the
vanes were free to turn. Comparing this effect with that which was caused
when the vanes were fixed by the magnet one revolution in 2 minutes, it
appears that the force turning the envelope with the vanes free was ^th
that turning the vanes; for the resistance of the water at such small
velocities would be proportional to the velocity.

As no such effect to turn the envelope had been observed during
Dr Schuster's experiment, in which I took part, and as it was difficult to
conceive any method of suspension more delicate than that then adopted, I
was forced to believe that the effect found by Mr Crookes was due to some

* In the discussion which followed the reading of the paper, Mr Crookes mentioned that
he had already constructed mills with inclined vanes, and found them answer ; and I am
informed that he exhibited one at the next meeting of the Society. I may mention here
that I have received a mill from Dr Geissler, which I had previously ordered. This instru-
ment, although damaged in transit, is sufficiently sensitive to prove that the action of heat
is altogether independent of the direction from which the heat comes. July 31, 1876.



182 ON THE FORCES CAUSED BY THE COMMUNICATION OF HEAT. [23

accidental cause, such as air-currents, about the outside of the case of his
mill. I therefore repeated Mr Crook es's experiment; first, by floating the
mill, as he describes, in a beaker of water, and simply covering the whole
with a glass shade. I then found that it was impossible to bring sufficient
light to bear on the mill to cause the vanes to revolve without causing the
case to turn ; although this turning was irregular, and such as might be
caused by air-currents. Dr Schuster and myself then suspended the same
light-mill we had previously used in a manner in all respects similar to that
of his former experiments, except that the mill was upside down, so that
the vanes could not turn in the envelope. On the light being turned on
a certain amount of disturbance was always consequent so long as the
receiver was not exhausted ; but when the receiver was exhausted to about
^ inch of mercury, no motion at all could be observed. At the soiree given
by the Royal Society on the 14th of June, 1876, I had two mills suspended,
the one upright and the other reversed. The envelope of the upright mill
moved when the light was turned on through a distance represented by
several hundred divisions of the scale; but the reversed mill showed no
motion at all, although a motion of two divisions must have been perceived.
The mills were suspended in vessels from which the air had been pumped
until the pressure was about half an inch of mercury. In these experiments,
therefore, there was no residual force tending to turn the envelope with the
mill so great as -^ of the force on the mill.



24.



ON VARIOUS FORMS OF VORTEX MOTION.



[From the " Proceedings of the Literary and Philosophical Society of
Manchester," Feb. 1877.]

(Read February 6, 1877.)

PROFESSOR OSBORNE REYNOLDS exhibited various forms of vortex motion
in a large glass tank by means of colour, or bubbles of air, the vortex lines
behind an oblique vane, the vortex ring behind a circular disc, the vortex
rings caused by raindrops, and the vortex rings caused by a puff of water.
The various ways in which these vortices move were also shown. But Pro-
fessor Reynolds' object in showing these experiments was to illustrate the
importance of the method of study rather than the intrinsic importance
of the results already obtained, which are not as yet sufficiently complete for
publication.

(For continuation see p. 184.)



25.



ON VORTEX MOTION.

[From the "Proceedings of the Royal Institution of Great Britain,"

Feb. 1877.]

(Read February 2, 1877.)

IN commencing this discourse the author said : whatever interest or
significance the facts I hope to set before you may have, is in no small
degree owing to their having, as it were, eluded the close mathematical
search which has been made for them, and to their having in the end been
discovered in a simple, not to say commonplace, manner. In this room you
are accustomed to have set before you the latest triumphs of mind over
matter, the secrets last wrested from nature by gigantic efforts of reason,
imagination, and the most skilful manipulation. To-night, however, after
you have seen what I shall endeavour to show you, I think you will readily
admit that for once the case is reversed, and that the triumph rests with
nature, in having for so long concealed what has been so eagerly sought, and
what is at last found to have been so thinly covered.

The various motions which may be caused in a homogeneous fluid like
water, present one of the most tempting fields for mathematical research.
For not only are the conditions of the simplest, but the student or philosopher
has on all hands the object of his research, which, whether in the form of
the Atlantic waves or of the eddies in his teacup, constantly claims his
attention. And, besides this, the exigencies of our existence render a know-
ledge of these motions of the greatest value to us in overcoming the limitations
to which our actions are otherwise subject.

Accordingly we find that the study of fluid motion formed one of the
very earliest branches of philosophy, and has ever since held its place, no
subject having occupied the attention of mathematicians more closely. The
results have been, in one sense, very successful ; most important methods of



25] ON VORTEX MOTION. 185

reasoning have been developed, mathematical methods, which have helped
to reveal numberless truths in other departments of science, and have taught
us many things about fluids which most certainly we should not otherwise
have found out, and of which we may some day find the application. But
as regards the direct object in view, the revelation of the actual motion of
fluids, the research has completely failed. And now that generations of
mathematicians have passed away, now that the mysteries of the motions
of the heavenly bodies, of the earth itself, and almost of every piece of
solid matter on the earth have been explained by mathematicians, the
simplest problems of fluid motion are yet unsolved.

If we draw a disc flatwise through the water, we know by a process of
unconscious geometrical reasoning that the water must move round the
disc ; but by no known mathematical process could the motion be ascertained
from the laws of motion. If we draw the plate obliquely through the water
we experience a greater pressure on the one side than on the other. Now
this case, representing as it does the principle of action of the screw propeller,
is of the very highest importance to us; and yet, great as has been the
research, it has revealed no law by which we may in a given case calculate
the resistance to be obtained, or indeed tell from elementary principles in
what way the water moves to let the plate pass. Again, the determination
of the resistance which solid bodies, such as ships, encounter, is of such
exceeding economic importance, that theory, as shipbuilders call it, having
failed to inform them what to expect, efforts have been, and are still being
made to ascertain the laws by direct experiment. Instances might be
multiplied, but one other must suffice. If we send a puff of fluid into other
fluid we know that it will travel to a considerable distance, but the manner
in which it will travel and the motion it will cause in the surrounding fluid,
mathematics has not revealed to us.

Now the reasons why mathematicians have thus been baffled by the
internal motions of fluids appear to be very simple. Of the internal
motions of water or air we can see nothing. On drawing the disc through
the water there is no evidence of the water being in motion at all, so that
those who have tried to explain these results have had no clue ; they have
had not only to determine the degree arid direction of the motion, but also
its character.

But although the want of a clue to the character of the motion may
explain why so little has been done, it is not so easy to understand how it
is that no attempts were made to obtain such a clue. It would seem that
a certain pride in mathematics has prevented those engaged in these in-
vestigations from availing themselves of methods which might reflect on the
infallibility of reason.



186 ON VORTEX MOTION. [25

Suggestions as to the means have been plentiful. In other cases where
it has been necessary to trace a particular portion of matter in its wanderings
amongst other exactly similar portions, ways have been found to do it. It
may be argued that the influences which determine the path of a particular
portion of water are slight, subtle, and uncertain, but not so much so as
those which determine the path of a sheep. And yet thousands of sheep
belonging to different owners, have been from time immemorial turned loose
on the mountains, and although it probably never occurred to anyone to
reason out the paths of his particular sheep, they have been easily identified
by the aid of a little colour. And that the same plan might be pursued
with fluids, every column of smoke has been evidence.

But these hints appear to have been entirely neglected, and it was left
for nature herself, when, as it were, fully satisfied with having maintained
her secret so long, and tired of throwing out hints which were not taken, at
last to divulge the secret completely in the beautiful phenomenon of the
smoke ring. At last; for the smoke ring is probably a phenomenon of
modern times. The curls of smoke, as they ascend in an open space, present
to the eye a hopeless entanglement ; and although, when we know what to
look for, we can see as it were imperfect rings in almost every smoke cloud,
it is rarely that anything sufficiently definite is formed to attract attention,
or suggest anything more important than an accidental curl. The accidental
rings, when they are formed in a systematic manner, come either from the
mouth of a gun, the puff of a steam engine, or the mouth of a smoker, none
of which circumstances existed in ancient times.

Although, however, mathematicians can in no sense be said to have
discovered the smoke ring, or the form of motion which it reveals, they
were undoubtedly the first to invest it with importance. Had not Professor
Helmholtz some twenty years ago called attention to the smoke ring by the
beautiful mathematical explanation which he gave of its motion, it would
in all probability still be regarded as a casual phenomenon, chiefly interesting
from its beauty and rarity. Following close on Helmholtz came Sir William
Thomson, who invested these rings with a transcendental interest by his
suggestions that they are the type after which the molecules of solid matter
are constituted.

The next thing to enhance the interest which these rings excited, was
Professor Tait's simple and perfect process of producing them at will, and
thus rendering them subjects for lecture-room experiments. Considering
that this method will probably play a great part in perfecting our notions
of fluid motion, it is an interesting question how Professor Tait came to hit
upon it. There is only one of the accidental sources of these rings which
bears even a faint resemblance to this box, and that is the mouth of a



25] ON VORTEX MOTION. 187

smoker as he produces these rings. This might have suggested the box
to Professor Tait. But since this supposition involves the assumption that
Professor Tait sometimes indulges in a bad habit, and as we all know that
Professor Tait is an eminent mathematician, perhaps we ought rather to
suppose that he was led to his discovery by some occult process of reasoning
which his modesty has hitherto kept him from propounding.

But however this may be, his discovery was a most important one, and
by its means the study of the actual motion of these rings has been carried
far beyond what would otherwise have been possible.

But it has been for their own sake, and for such light as they might
throw on the constitution of matter, that these rings were studied. The
most important lesson which they were capable of teaching still remained
unlearned. It does not appear to have occurred to anyone that they were
evidence of a general form of fluid motion, or that the means by which
these had been revealed, would reveal other forms of motion.

There was, however, at least one exception, which will not be forgotten
in this room : the use of smoke to show the effect of sound upon jets of air.

Also, the late Mr Henry Deacon, in 1871, showed that minute vortex
rings might be produced in water by projecting a drop of coloured water
from a small tube. And his experiments, in spite of their small scale,
excited considerable interest.

Four years ago, being engaged in investigating the action of the screw
propeller, and being very much struck by the difference between some of
the results he obtained and what he had been led to expect, the author
made use of coiour to try and explain the anomalies, when he found that
the vortex played a part in fluid motion which he had never dreamt of;
that, in fact, it was the key to almost all the problems of internal fluid
motion. That these results were equally new to those who had considered
the subject much more deeply than he had, did not occur to him until after
some conversation with Mr Froude and Sir William Thomson.

Having noticed that the action of the screw propeller was greatly



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