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the spider- line in place of the plate- vanes.

(4) In Section XII. it is also shown that while the phenomena of the
radiometer result from the communication of heat from a surface to a gas, as
explained in my former paper, these phenomena also depend on the divergence
of the lines of flow ; whence it is shown that all the peculiar facts that have
been observed may be explained.

(5) In Section X. it is also shown that the phenomena of transpiration,
resulting from a variation in the molecular constitution of the gas (investigated
by Graham), are also to be explained by the equation of transpiration.

(6) Section II. (Part I.) contains a description of the experiments under-
taken to verify the revelations of Section X. respecting thermal transpiration ;
which experiments establish not only the existence of the phenomena, but
also an exact correspondence between the results for different plates at
corresponding densities of the gas.

(7) Section III. contains a description of the experiments on transpiration
under pressure, undertaken to verify the revelations of Section X. with respect
to the correspondence between the results to be obtained with plates of
different coarseness at certain corresponding densities of the gas ; which
experiments proved, not only the existence of this correspondence, but also
that the ratio of the corresponding densities in these experiments are the
same as the ratio of the corresponding densities with the same plates for
thermal transpiration a fact which proves that the ratio depends on the
relative coarseness of the plates.

(8) Section IV. contains a description of the experiments with the fibre
of silk and with the spider-line, undertaken to verify the revelations of
Section XII. ; from which experiments it appears that, with these small
surfaces, phenomena of impulsion similar to those of the radiometer occur at
pressures but little less than that of the atmosphere.

126. As regards transpiration and impulsion, the investigation appears
to be complete. Most, if not all, the phenomena previously known have been
shown to be such as must result from the tangential and normal stresses
consequent on a varying condition of molecularly constituted gas ; while the
previously unsuspected phenomena to which it was found that a variation in
the condition of a molecular gas must give rise, have, on trial, been found to
exist.

The results of the investigation lead to certain general conclusions which
lie outside the immediate object for which it was undertaken. The most



380 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

important of these, viz. that gas is not a continuous plenum, has already
been noticed in Art. 5, Part I.

The dimensional properties of gas.

127. The experimental results, considered by themselves, bring to light
the dependence of a class of phenomena on the relation between the density
of the gas and the dimensions of objects, owing to the presence of which the
phenomena occur. As long as the density of the gas is inversely proportional
to the coarseness of the plate, the transpiration results correspond ; and in
the same way, although not so fully investigated, corresponding phenomena
of impulsion are obtained as long as the density of the gas is inversely pro-
portional to the linear size of the objects exposed to its action. In fact, the
same correspondence appears with all the phenomena investigated.

We may examine this result in various ways, but, in whichever way we
look at it, it can have but one meaning. If in a gas we had to do with a
continuous plenum such that any portion must possess the same properties,
we should only find the same properties, however small might be the quantity
of gas operated upon. Hence, in the fact that we find properties of a gas
depending on the size of the space in which it is enclosed, and of the quantity
of the gas enclosed in this space, we have proof that gas is not continuous
or, in other words, that gas possesses a dimensional structure.

In virtue of their depending on this dimensional structure, and having
afforded us proof thereof, I propose to call the general properties of gas on
which the phenomena of transpiration and impulsion depend, the Dimensional
Properties of Gases.

This name is also indicative of the nature of these properties as deduced
from the molecular theory; for by this it appears that these properties
depend on the mean range a linear quantity which, cceteris paribus, depends
on the distance between the molecules.

In forming a conception of a molecular constitution of gas, there is no
difficulty in realizing that such dimensional properties exist ; there is,
perhaps, greater difficulty in conceiving molecules so minute and so numerous
that, in the resulting phenomena, all evidence of the individual action is lost.
But the real difficulty is to conceive such a range of observational power as
shall embrace, on the one hand, a sufficient number of molecules for their
individualities to be entirely lost, while, on the other hand, it can be so far
localized as regards time and space that, if not the action of individuals, the
actions of certain groups or classes of individuals become distinguishable
from the action of the entire mass. Yet this is what we have in the
phenomena of transpiration and impulsion.



33] IN THE GASEOUS STATE. 381

Although the results of the dimensional properties of gases are so minute
that it has required our utmost powers to detect them, it does not follow that
the actions which they reveal are of philosophical importance only. The
actions only become considerable within extremely small spaces, but then
the work of construction in the animal and vegetable world, and the work of
destruction in the mineral world, are carried on within such spaces. The
varying action of the sun must be to cause alternate inspiration and expiration
of air, promoting continual change of air within the interstices of the soil as
well as within the tissue of plants. What may be the effects of such changes
we do not know, but the changes go on ; and we may fairly assume that in
the processes of nature the dimensional properties of gas play no unimportant
part.

Nor is this all. It is by aid of the analogy which gas affords us that we
must look forward to solve the mystery of the luminiferous ether. And
although all attempts to frame a satisfactory hypothesis as to the molecular
constitution of ether have hitherto failed, in none of these hypotheses have
the tangential and normal stresses arising from a varying condition been
taken into account ; whereas the recognition of the part which these stresses
play in the properties of gases shows, or at least suggests, the possibility that
the phenomena of ether which we observe may depend largely upon analogous
stresses.



APPENDIX.

(Added December, 1879.)
NOTE 1.

Since the reading of this paper I have had my attention called to a paper by
W. Feddersen (" Uber Thermodiffusion von Gasen," Pogg. Ann., 1873). Feddersen made
some experiments, and seems to have thought that he had discovered some such pheno-
menon. But the results he obtained were attributed by M. J. Violle to the presence of
the vapour of water, against which no precautions appear to have been taken (Journal de
Physique, 1875, p. 90). That M. J. Violle was right there can be no doubt, for the results
obtained are now seen to be much too large for the true results, and are similar to those
which I obtained before I had succeeded in sufficiently drying the air.

NOTE 2.

Graham applied the term " transpiration " to the passage of gases through capillary
tubes as distinguished from the passage of gases through larger tubes and through
apertures in thin plates, and applied the term " effusion " to the passage of gases through
minute apertures in thin plates.



382 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

He did not apply either of these terms to the passage of gases through porous plates,
because his experiments led him to conclude that the phenomena attending such passage
were not the same as the phenomena attending either of the former, but were somewhere
between the two.

By the fuller light thrown on to the subject by this investigation it appears that in the
limit, when the tubes and holes are small enough according to the condition of the gas, the
laws of transpiration are strictly the same as those of effusion, the theory of the phenomena
being the same. Hence the continued use of two names appears to be unadvisable.

The term " transpiration " has been chosen in preference to " effusion," because it is
found that as the passages become coarser, according to the condition of the gas, the law of
the passage of gas through porous plates is still in strict accordance with the law of the
passage through tubes, showing that the passages are of the nature of tubes rather than
thin plates.



NOTE 3, ART. 7.

It will be observed that this dependence of the phenomena on a relation between the
size of the surfaces and the mean path of a molecule is essentially different from what has
been a common, but as is herein shown, erroneous supposition, that the phenomena
essentially depend on distance separating the opposite surfaces. The one supposition
makes the action of the radiometer depend on the size of the vanes, but leaves it
independent of the size of the envelope, while the other makes the action depend on the
size of the envelope, but leaves it so far independent of the size of the vanes.



NOTE 4, ARTS. 41 AND 104.

The assumption that the coefficients A, TO, X 1} and X 2 , also TO' and X 3 , equation (99),
are the same for stucco as for meerschaum, is equivalent to assuming that the only respect
in which the interstices of these plates differ is that of coarseness. There is no a priori
ground for making this assumption. The fact that the logarithmic homologues for stucco
fit those for meerschaum through such a considerable range of densities proves the
approximate truth of the assumption ; but it is possible, since c s and c m are arbitrary
dimensions, that the curves for transpiration under pressure depending on A, m, X 1? and
X 2 may approximately fit for one value of c g /c m , and the curves for thermal transpiration
depending on A lt m, X 1? X 2 , m', and X 3 may approximately fit for another value of c s /c m . If
this were so log. (c s /c m ) ; the shift necessary to bring the curves into coincidence would not
be the same for transpiration under pressure as for thermal transpiration, and as has been
pointed out (Art. 41), this is to a certain extent the case, this ratio having the values 6*5
and 5 -6 a difference which was sufficiently decided to call for notice, but which is not so
large but that, as pointed out (Art. 41), it may possibly be due to the plates being hot in
the one case and cold in the other. In any case the smallness of the difference is an
additional proof that the interstices do not greatly differ as passages in any respect except
that of size.



33] IN THE GASEOUS STATE. 383

NOTE 5, ART. 119.

Whence at the surface when is sufficiently small we have by eqiiation (18)



Ci 2 2 Ji

fa a'\ 2
neglecting ( -~- \



jsr



ft

And when is large, we have by equation (128)



H 9p' s
-=-7= -(-)



Therefore substituting for "' , a in equation (140)



I?



(6) In my former paper (Proc. Roy. Soc. 1874, p. 407)



1 d , rf

^=3-^and-

Therefore, since



a 2 /3 /3 .

^ = ^2~' V= V 2' 8v = V 2 '



and putting 8a = a c - a',

/3^H

f ~ V 2 18 c 2 '



384 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33



CERTAIN DIMENSIONAL PROPERTIES OF MATTER IN THE

GASEOUS STATE.

(An Answer to Mr GEORGE FRANCIS FITZGERALD.)
[From the " Philosophical Magazine " for May, 1881.]

IN the February number of the Philosophical Magazine there appeared
a paper by Mr Fitzgerald, in which he criticised my paper " On Certain
Dimensional Properties of Matter in the Gaseous State," Philosophical
Transactions of the Royal Society, 1879. Mr Fitzgerald courteously put his
remarks in the form of questions, expressing the hope that I would answer
them. I was prevented by other work from preparing anything in time
for insertion in the April number; but I now ask your space for a few
remarks.

The objections taken by Mr Fitzgerald to my work may be summed up
as three :

(1) That by dividing space into eight regions I have adopted a method
which is at once inelegant and unnecessarily elaborate.

(2) That I have omitted terms which, if retained, would have altered
the results.

(3) That I have changed my views and adopted the theory which I had
previously combated.

To all these accusations I would most emphatically plead not guilty. And
I would further suggest, in explanation of Mr Fitzgerald's difficulty, (1) that
he has not paid equal attention to all parts of my paper, but has rather
confined his attention to those parts which relate to the phenomena of
impulsion, in which he seems to be especially interested, and that thus he has
failed to see that, in order to obtain any results whatever for transpiration,
the division of space into regions is necessary ; and (2) that in his anxiety to
find a different result in the case of impulsion from that which I had obtained,



33] IN THE GASEOUS STATE. 385

he has failed to perceive that the terms which I have neglected, and of which
he instances one as disproving my conclusion, are of a distinctly smaller order
of magnitude than those which appear in my result.

As regards, then, the charge of inelegance, I am sure that Mr Fitzgerald
would not for one moment have urged it had he not thought that the par-
ticular step to which he objects might be replaced by some other known
method. One might as well abuse David because he used a stone and sling,
as object to the inelegance of a mathematical method by which alone true
results have been obtained. Of course I do not for one moment defend my
method as being elegant, nor should I have noticed this remark were it not
that, taken together with the more definite criticism to the same effect, it
shows conclusively that Mr Fitzgerald has failed to notice the gist of the
greater portion of my paper that he has failed to notice one of the most
important terms in the equation of transpiration and the manner in which
this term enters. In the paragraph beginning at the bottom of page 104 he
says, " With the symbols and notation I have no fault to find ; but I must
enter a protest against his elaborate and totally unnecessary division of space
into eight regions. He might have perfectly well calculated equations (43)
to (47) without rendering a difficult subject tenfold as elaborate as was
necessary." And then he goes on to show how I might have obtained
equations for the aggregate results at one integration. Clearly, then, he has
seen no object in my division of space into regions, and is at a loss to account
for it except as mere clumsiness in the integrations. Had he, however,
looked closer, or even been careful to be accurate in his statement, he would
have seen that the two equations (44), which are among those to which he
refers, only apply to the partial groups for which u is respectively positive
and negative, and that they contain a term w^hich apparently disappears if
the respective members of the two equations be added ; and he would have
seen that the same thing is true of equations (45)*, which hold only for
groups for which v is respectively positive and negative, and from which two
terms disappear when the results are added. Now these terms, which are
the first and second, are sufficiently obvious in the partial equations, whereas
they do not appear at all if the integration be extended to both groups ; and
if Mr Fitzgerald had followed the next articles (83) and (84), he would have

* The partial equations (45) :

^ pa.U s dpaU s dpa? _ s dpaV

2*J* 2*Jir dy 27r dx 2^/ir dx

paU s dpaU s dpa? s
< rr(M M )=-^p-^-^- + - -^p

The equation obtained by complete integration :

s dpaU s dpaV
-



25



386 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

seen why these terms are important. To ignore these two articles is to ignore
the method by which the results for transpiration are obtained ; and these
results were the main purpose of the preliminary work in the paper.

To obtain any results at all for transpiration, it is necessary to divide
space into two regions, or else to consider the mean range s as a function of the
position of the point, and discontinuous at the solid boundaries ; and by the
latter method the determination of the form of the function requires that space
should be divided. The results depend entirely on the terms which, when
s is constant, disappear in the complete integration, but which, if different
arbitrary values are assigned to s for the different regions, do not cancel
when the partial integrals are added. No result whatever is obtained by
complete integration if s be constant ; and although Mr Fitzgerald does not
seem to have noticed it, the late Professor Maxwell followed me in dividing
space into two regions at the bounding surfaces, calling the two groups the
absorbed and evaporated cjas. But without the use of arbitrary coefficients
he had no means of dealing with the variable condition of his gas, except by
assuming that the same distribution holds in both groups at all points. To
meet this assumption (which, he points out at the top of page 253*, is
improbable) he had further to assume a highly complex and improbable
condition of surface; and the result is that the equation he obtains (77) is
short of the most important term. This term is that which gives the result
when the tubes are small compared with s ; and as this is the only case in
which the results are appreciable, when Maxwell came to apply his equation
to an actual case there was no sensible result.

In the first instance, I also began by considering space as divided only at
the bounding surface, and, assuming the distribution in the two groups the
same, integrated for the complete space ; and the result I then obtained was
precisely the same in form as that subsequently obtained by Maxwell.
These results correspond with the experimental results for a tube whose
diameter is large compared with s called by Graham transpiration; but
they do not at all correspond with the law "which Graham found to hold when
he used a fine graphite plug, and which I have shown to hold also with
coarse stucco plugs when the gas is sufficiently rare, viz. that the times of
transpiration of equal volumes of different gases are proportional to the
square roots of the atomic weights. Graham had considered this law as
depending on the fineness of the pores of the plug, and had suggested that
the action then resembled that of effusion through a small aperture in a thin
plate, rather than transpiration through a tube of uniform bore; and this is
the assumption which Maxwell falls back upon to account for the difference
between his calculated results and those of experiment. That I did not do

* " On Stresses in Rarefied Gases," Appendix, p. 249, Phil. Trans. 1879.



33] IN THE GASEOUS STATE. 387

the same was owing to my having, by reasoning ab initio, after the manner
explained in the analogy of the batteries, in the very first instance found that
the law of the square roots of the atomic weights must hold in a tube when-
ever the gas was so rare that the molecules ranged from side to side without
encounter, and to my having proved by experiment that both laws might be
obtained with the same plug by changing the density of the gas. It was
thus clear to me that some term had been omitted in my equation ; and after
a long search it was found that, though the term vanished in the complete
integral, it appeared in the partial integrals when space was divided into
regions, and that, as the values of s were obviously different in the different
regions, the assumptions on which the complete integral had been obtained
were clearly at fault. The further division into eight regions was not only
for the sake of symmetry, but that all the other terms which enter into the
partial integrals might be examined, and as being necessary in particular
cases as, for instance, in that of a round tube, which is also treated of in
the paper.

Having thus shown that, however elaborate and inelegant, the division of
space into regions is essential, it is unnecessary to defend it on other grounds.
But I may remark, by the way, that such a division does tend greatly to
simplify the consideration of motion. This, I think, is proved by the universal
adoption of north, east, south, west, zenith, and nadir.

I have dwelt at considerable length on the foregoing point, as the mis-
conception of this point is fundamental to all Mr Fitzgerald's criticism. The
rest I may answer shortly.

With regard to Professor Maxwell's remarks on my paper, and his own
work on the same problem, of course the sad circumstance of his death
occurring, so that this was about the last-work he did, renders it very difficult
to approach the subject ; but with reference to what I have already said, and
in explanation of the apparently imperfect idea at which he arrived as to the
scope and purpose of my method, it may be stated that, before writing his
own paper, Professor Maxwell had only seen my paper in manuscript in
the condition in which it was first sent in to the Royal Society, when the
preliminary part was very much compressed, and, as I fear, somewhat vaguely
stated, besides being founded on different assumptions from the present.
Without entering further upon this now, I may refer to a letter which I
addressed to Prof. Stokes after seeing an early copy of Prof. Maxwell's paper,
and before I was aware of his illness, which letter was subsequently published
in the Proceedings of the Royal Society for April 1880, p. 300.

Mr Fitzgerald has asked me for an explanation of the system on which
certain terms are retained and others neglected. This is difficult to give in
a few words ; but I was under the impression that it is sufficiently explained

252



388 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

in the paper. It seems to me that the difficulty which Mr Fitzgerald has
found must have arisen from his having adopted the hitherto vague way of
looking at the mean path of a particle (or in this case the mean range) as a
small quantity, without strictly inquiring as compared with what it is small.
In my paper, s is nowhere to be regarded as small except in cases where it
comes into direct comparison with some definitely larger quantity. The

small factors are - , - - , and - j- ; the squares of such quantities being
a a. dx a. dx

' s 2 d?d

consistently neglected. Such factors as - ,- 2 and variations of higher order

a doc

are zero in the case of transpiration, but in the case of impulsion they are of
the same order as the results. But the retention of such terms in equations
(42) to (48), or in the fundamental theorem, would only give rise in the

s 3 d^d

results to such terms as 5 ; so that as long as s is small compared with

ra dx 2

r no error can have arisen from the neglect of these terms. And this is the
only case to which these results have been applied, the extreme case where
s is large compared with r having been dealt with by a special method which
gives rigorous results. In the first instance, all terms of the second order

s 2 d 2 a.

such as T were retained ; and it was only after it was found that these
a. da?

did not in any way affect the results as a first approximation that they were
neglected. The terms I have neglected are, as far as I perceive, the same as
those neglected by Professor Maxwell ; and such was the care taken in this
matter (which is of fundamental importance) that I am very confident that
there is no mistake. On the other hand, it is difficult for me to see how
Mr Fitzgerald can have failed to see that the residual term, which he
instances as showing that I am wrong in saying that my equations show that
there is no force in the case of parallel flow, is distinctly of the second order
of small quantities. But even to this term he has no right ; for in order to
obtain results to such an order the variations of s would have to be considered.
It seems that Mr Fitzgerald is of opinion that the parallel flow of heat does
cause stresses in the gas, and that he has been trying to find that I have not
disproved the possibility of such stresses. If he confines his attention to
stresses of the same order of magnitude as those now shown to exist in the
case of converging or diverging flow, he will find that both Professor Maxwell



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