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IN THE GASEOUS STATE.

313

Throughout this investigation bars over the symbols indicate the mean
taken over some group of molecules; when no further indication as to the
particular group is given, it is to be understood that the mean is taken from
the entire group in a unit of volume at the same instant. Thus 2 , 7/ 2 , 2
indicate the mean squares of , t}, respectively for all the molecules in a
unit of volume of uniform gas which is in the same mean condition as the
gas at the point considered.

Q is used to represent any quantity belonging to a molecule, such as its
mass, momentum, energy, &c.

S(Q) is used to represent the aggregate value of Q for a group of
molecules as existing in a unit of volume ; and when no further indication
is given it will be understood that the aggregate is that of the entire group.

cr(Q) indicates the aggregate value of Q carried across a unit of plane
area in a unit of time by a group of molecules, which in the absence of
further indication will be understood to be the entire group which crosses
the plane.

<r x (Q), with the suffix, is used to express the direction of the plane as well
as the aggregate value carried across it.

&x(Q}, with the superimposed symbol, expresses the group over which the
summation extends; u+ indicates that the summation is taken over all
those molecules which are moving in the positive direction as regards the
axis of x. By varying the superimposed symbol, the general symbol may be
made to express the value of Q carried by a group of molecules having any
particular motion across the plane indicated by the suffix.

Fig. 9.

07. As indicated by the signs of the component velocities, the molecules
in a unit of volume, or the molecules which cross a surface at a point in a
unit of time, will be divided into eight groups.

These groups may be indicated by the eight corners of a cube, having its

314 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

edges parallel to the axes, circumscribed about the point considered. Thus
in fig. 9 the group which have u+, v+, w+, will approach from the
region indicated by the corner A, and similarly there will be a corner for
each group. The particular groups, therefore, may be distinguished by the
letters at the corners of the cube, fig. 10. And instead of

to+ w- w+ w- w+ w- w+- w

v+ v+ v~ u v + v+ v- v

u+ -u+ u+ + u- 11 u- u-

2<Q), 2(), 2(G), 2(0, 2(Q), 2(Q), 2(Q), 2<Q)
we have respectively,

2(Q), 2(Q), 2<Q), 2(Q), 2(Q), 2fo), 2(Q), 2(Q).

And in order still further to simplify the notation, instead of

*(), <r(Q), <r(Q), <r&), r(Q), crfe, <r (Q), r(Q)
we may write respectively the simple letters

, _ _ A, B, C, D, E, F, G, H.

68. The method of considering the value of Q carried across a plane by
groups of molecules distinguished by the directions in which they are moving,
constitutes the essential means by which the results of this investigation are
arrived at. And as it does not appear that this method has been resorted to
by any previous writer, it appears necessary for me to describe at some length
the preliminary steps.

The rate at which Q is carried across a plane.

69. Since the aggregate value of Q carried across any plane by the entire
group of molecules must be equal to the sum of the values of Q carried across
by all the various groups into which the gas may be divided, we have

I (1)

iH-

(2)

(3)

(4)

(5)

w+

(6)

(7)

33] IN THE GASEOUS STATE. 315

Gas in uniform condition.

70. When the gas is uniform, whether at rest or in motion, the value of
(r(Q) has already been determined by Professor Maxwell, but it is necessary
to transform the expressions to the notation of this paper.

We have by a well-known formula*

in which the suffixes x, y, z, indicate that it is the planes yz, zx, xy, that Q is
being carried across, and the superimposed symbols a a a indicate the group
of molecules over which the summation extends.

We have also

(9)

and similar expressions for the values of Q carried across each of the other
planes by all the other groups.

In the equation (8) and similar equations we may obviously substitute for
u, v, w their values

V = r + V

And since the gas is here supposed to be uniform, we shall have

U=u\
V=v (11)

W= w.
%, rj, being identically the same as if the gas were at rest.

71. For the purpose of this investigation it is necessary to express such

u+ it-
quantities as <r x (Q), <r x (Q) in terms of the groups distinguished by the signs

of , 7}, , instead of u, v, w ; and owing to the fact that in all the cases to be
* "On the Dynamical Theory of Gases," Maxwell; Phil. Trans. 1867, p. 69.

.316 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

considered U 2 , V 2 , W 2 are of the second order of small quantities compared
with u 2 , v 2 , w 2 this may be done. For we may put

u+ + + u+ and <U

U)Q] + 2 {(+U)Q} ...... (12)

+ and <U

\${(+ U)Q] ...... (13)

and when U is small compared with Ju\ the last term on the right in each
of these equations will be small to the second order as compared with the
first term. For the number of molecules over which the summation in these
terms extends is to the whole number of molecules in a unit of volume in

something less than the ratio of U to \/u 2 . Hence, as will subsequently
appear, in neglecting these last terms we shall be neglecting nothing within
the limits of our approximation. We have therefore

u+ +

and similarly for all other groups. Thus it appears that the letters a, b, c,
&c., may be used indifferently to indicate the groups as distinguished by the
signs of u, v, w or of , 77, .

Distribution of velocities amongst the molecules.

72. Although not actually essential to this investigation, as it will tend
greatly to simplify the results obtained, I shall adopt the conclusion arrived
at by Professor Maxwell* with respect to the distribution of velocities
amongst the molecules of a uniform gas, viz. :

where N is the whole number of molecules in a unit of volume, and dN the
number whose component velocities lie between and + dg, 77 and 77 + dy,
and f and +d.

From equation (15) we have for a uniform gas

VTT

V AW /
(17)

2

i 2

VTT

'

a 2

< 19 >

Phil. Trans. 1867, p. 65.

33] IN THE GASEOUS STATE. 317

Also if r is the absolute temperature of the gas. p the intensity of pressure,
M the mass of a molecule, and p the density of the gas, we have for uniform
gas

^=" 2 2 (20)

P = P5 (21)

p 2 T

p = *M (22)

in which /c 2 varies with the nature of the gas, and is otherwise constant.

73. The adoption of equations (15) to (22) restricts the application of
the results that may be arrived at to gases of uniform molecular texture such
as air and hydrogen. For these equations do not apply to a varying mixture
of gases. In order to render them applicable to such a mixture it would be
necessary to consider throughout the investigation the presence of at least
two systems of molecules. This would add greatly to the complication,
whereas none of the experimental results which it is my immediate object to
explain involve a varying mixture.

It will be seen, however, that at least one important result which has not
hitherto been explained could be fully explained in this way. This is the
transpiration of a varying mixture of two gases through a porous plate. The
possibility of such an explanation will be seen from the results obtained for a
simple gas.

74. Table XX. contains all the values of <r ( Q) carried across the axial
planes by the several groups of molecules in a uniform gas, for all the
quantities Q which are important in this investigation.

318

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

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33] IN THE GASEOUS STATE. 319

SECTION VII. THE MEAN RANGE.

75. So far the gas has been supposed to be in uniform condition as
regards space as well as time. When the condition varies from point to
point, the results given in Table XX. will not hold good, for the condition of
the molecules, arriving from any particular direction, which cross a plane at a
point A will not be determined by the mean condition of the gas at A, but
rather by the mean condition at the points at which the molecules receive
the direction and velocity with which they cross the plane.

These points will not necessarily be the points at which the molecules
last undergo encounter before crossing the plane, for one encounter may not
be sufficient completely to modify their motion. In order, therefore, to
determine from first principles the manner in which the molecules approach
the point A, we must know the law of action between the molecules, and
even then the complete solution would present difficulties which appear to be
insuperable.

Fortunately, however, for the purposes of the present investigation a
complete solution is not necessary. The point that has mainly to be con-
sidered is the effect of a solid surface on the mean condition of the molecules
which cross a plane in its immediate neighbourhood. And the principal
question is not how far such an effect would extend into gas in a particular
condition, but what would be the nature of the effect at points to which it
does extend, and what would be the comparative range of similar effects in
gases the condition of which differ with respect to density and variation of
temperature ? If it should be found that the number and mean condition of
the molecules which arrive at A from a given direction, partake in a definite
manner of the condition of the gas at a point in that direction whose distance
s from A is a definite function of the density of the gas and some function of
the temperature ; such a solution would be sufficient to allow of the deduction
of results corresponding to the experimental results.

Now it appears to follow from the view propounded at the commencement
of this article, that in the interior of the gas there must be some distance s
from a point A at which the mean condition of the gas must represent the
mean condition of the molecules which reach A from that direction. This
language is somewhat vague, but so must be the first idea. On closer
inspection the question naturally arises as to what is meant by the mean
condition of the gas, and by the mean condition of the molecules which reach
A ? Nor does this question at first sight appear to be difficult to answer.

320 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

The mean condition of the gas appears most naturally to resolve itself into
that which we can measure the density p, the mean pressure

(u? + & + w 2 )
o

and the mean component velocities u, v, w ; and with respect to the mean
condition of the arriving molecules why should not this be measured by their
density, their mean energy and their mean component velocities ? On com-
paring these with the corresponding quantities for the gas just mentioned,
one point of doubt presents itself: in the mean component velocities of the
molecules arriving from one direction we have a very different thing from the
mean component velocities of the gas. However, ignoring this caution, the
most obvious supposition appears to be that as an approximation towards the
condition of the molecules as they arrive at A, we may suppose them to come
from a uniform gas having the density, mean pressure and component
velocities of the gas at a point distant s from A in the direction from which
they arrive. Such an assumption can be worked out, and the results com-
pared with known experimental results. But we need go no farther than the
case of gas at equal pressure and varying temperature. As applied to this
case, our supposition leads to the inevitable conclusion that, unless s is zero,
such a gas must be in motion from the colder to the hotter part with a
velocity greater than its actual velocity, whatever this may be, which is
absurd. This brings us back to the caution already mentioned respecting
the difference between the component velocities of the group of molecules
approaching A, and the component velocities of the gas. Without attempting
to investigate this difference from first principles, we may follow the obvious
course of attributing certain arbitrary mean component velocities to the
uniform gas, as from which the molecules are supposed to arrive at A.

We now suppose the molecules to arrive at A as from a uniform gas
having the mean pressure and density at a distance s as before', but having
arbitrary component velocities U, V, W (where U, V, W are so small that
their squares may be neglected). This gets over the difficulty in the case
mentioned above, for U, V, W being arbitrary can be so determined that the
gas resulting from all the groups arriving at A shall have any mean velocity,
and hence the mean velocity of the gas. It is only one such case, however,
that we can meet in this way ; for having once determined U, V, W, they are
no longer arbitrary, and hence if the calculated results fit, to the same
degree of approximation, all other cases, it must be that the approximation is
a true one.

This test, however, can only be partially applied. As worked out in
the subsequent sections of this paper, it was found that the supposition
explained the phenomena of the radiometer and suggested the laws of

33] IN THE GASEOUS STATE. 321

transpiration and thermal transpiration exactly as they were afterwards
realised. And in so far as they can be compared there is a complete agree-
ment between the theoretical and experimental results.

Under these circumstances, the course which I first adopted in drawing
up this paper was to found the theoretical investigation on such an assumption
as has just been discussed.

The only other course was to look to first principles for the evidence
wanting to establish the truth of the assumption. This I had attempted.

Obviously the first step in this direction was to examine the values of U,
V, W, as determined by the case of gas at varying temperature and uniform
pressure. This showed that if a plane be supposed to be moving through the
gas with velocities U, V, W, then, measured with respect to the moving plane,
the aggregate momenta carried from opposite sides across the plane are equal.

This fact appeared pointed, but the exact point of it was not at once
obvious, nor did it fully occur to me until I had completed the investigation
founded on the assumption as already described.

Subsequently, however, working at the subject from the other end, so to
speak, I came to see that whatever might be the action between the molecules,
the probable effect of encounters in a varying gas would not tend to reduce
the molecules after encounter to the same state as those of a uniform gas
moving with the mean component velocities of the varying gas, but to a uniform
gas moving with the halves of the mean component velocities of all the
molecules which cross a unit of surface in a unit of time which pass through
an element in a unit of time.

I had not till then apprehended, nor do I know that it has anywhere
been pointed out, that the mean component velocities of the molecules which
pass through an element in a given time are not, in the case of a varying gas,
the doubles of the component velocities of the gas, as they would be in that
of a uniform gas (neglecting the squares of the mean component velocities).
But it turns out to be so (see Art. 77). And what is more, these mean
component velocities are the very velocities U, V, W, which had been found
to be necessary as already described.

The recognition of this fact therefore removed all fundamental difficulty
as regarded the velocities U, V, W.

There still, however, remained the question as to whether the molecules
might be considered to arrive in all respects, to the same degree of approxi-
mation, as from the same uniform gas whether the molecules would arrive
in respect to density from the same uniform gas as in respect to mean
velocity, &c. ; or whether severally in respect of density, mean velocity, &c.,
o. R. 21

322 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

the uniform gas would correspond to different values of s ? The answer to
this question depends on the law of action between the molecules, and hence
it is of necessity left for such light as accrues from the experiments and other
known properties of gas.

It is, however, now proved (not altogether from first principles, but on
certain elementary assumptions which might, it is thought, be deduced from
first principles) that as regards number the molecules will arrive at A as
from a uniform gas having the density, mean pressure and U, V, W, of the
actual gas at a certain distance s from A, and that as regards mean velocity,
mean square of velocity, and mean cube of velocity, the molecules will arrive
as from uniform gas corresponding in each respect with the same or another
point.

So that instead of having one value of s there are four ; the numerical
relations between which have not been determined from the elementary
assumptions, but which are all shown to be functions of the temperature and
inversely proportional to the density, and when the gas varies continuously
independent of the direction from which the molecules arrive.

On comparing the theoretical results with those of experiment it is
found

1. That the values of s for density and mean square of velocity are
equal ;

2. That s for the mean cube does not enter into any of the experimental
results of this investigation ;

3. That s for the mean velocity has a real value, but there are no data
for effecting a numerical comparison between this and the other value of s.

As this foundation of the theory on elementary assumptions renders it
more satisfactory, it is introduced at length into this section of the paper.
The argument, which is long and occupies Arts. (79 to 84), may be sketched
as follows :

Sketch of the method by which the fundamental theorems are deduced.

76. Upon certain elementary assumptions, which do not involve any
particular law of action between the molecules, it is first shown that, in
respect of density, mean velocity, &c., considered separately, any group of
molecules whose directions of approach differ by less than a given small angle
from any given direction A, will enter the element at A (within a sufficient
degree of approximation) as if the gas were uniform and had the same density
and mean pressure as at B, and had mean component velocities which,
although not the mean component velocities at B, are equal to one-half the

33] IN THE GASEOUS STATE. 323

mean component velocities of all the molecules which enter an indefinitely
small element at B in a unit of time. These component velocities, which are
written U, V, W, cannot in the first instance be expressed in terms of known
quantities, but they are shown to be functions of the position of B in the
gas.

The distance AB or s is shown to be a function of the pressure and
density of the gas, which function, although not completely expressed, as
such an expression would involve the law of action between the molecules, is
shown to be approximately independent of the variation of the density and
pressure, and hence of the direction of AB.

The relations between p, a, U, V, W for a uniform gas may thus be used
to express severally the density, mean velocity, &c., for each elementary group
of molecules arriving at A. And since p, a, U, V, W are functions of the
position of the point B (if x y z are the coordinates of A, and I m n are the
direction cosines of AB) they are functions of as + Is, y -\- ms, z + ns, s having
the value for the particular quantity to be represented. Therefore p, a, U,
V, W for B may, by expansion, be represented by p, a, U, V, W for A, and
their differential coefficients multiplied by powers of s. Thus the density,
mean velocity, &c., of the molecules of each group arriving at A may severally
be expressed in terms of p, a, U, V, W at A, and their differential coefficients
multiplied by a particular value of s.

Therefore as the elementary portions of a- (Q) for the group can always be
expressed in terms of the density, mean velocity, &c., and /, m, n, it can be
expressed in terms of p, a, U, V, W, for A, their differential coefficients mul-
tiplied by certain values of s and I, m, n. And, since all these quantities but
I, m, n are independent of the direction of the group, by integrating for all
values of I, m, n, cr (Q) is found in terms of p, a, U, V, W for A, and their
differential coefficients multiplied by s.

It also appears that within the limits of the necessary approximation,
terms multiplied by U 2 , F 2 , W 2 , or differentials of the second order, may be
neglected; so that a-(Q) is expressed in terms of p, a, U, V, W, and their
differential coefficients of the first order multiplied by some one or other of
the several values of s.

U, V, W, are then at once found by putting Q = M, so that cr x (M), <r y (M),
and a z (M), are respectively u, v, and w, which form the left sides of three
equations (48) in which U, V, and W respectively appear on the right side.

It is difficult to give an intelligible sketch of so complicated a series of
operations, but what has been stated above may serve to indicate the general
scheme of this section.

212

324 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Mean component velocities of the molecules which pass through an element.

77. It has been already pointed out that when the condition of the gas
varies, the mean component velocities of all the molecules which in a unit of
time pass through an element are not, to the same degree of approximation
as they would be if the gas were uniform, the doubles of the mean component
velocities of the molecules in the element at the same instant.

To express this, suppose that the condition of the gas varies only in the
direction of x, so that the mean momentum in any direction perpendicular to
x carried across all surfaces is zero.

Then taking a rectangular element, so that its edges are parallel to the
axes, and its edges parallel to x are indefinitely short compared with its edges
perpendicular to x, the only momentum carried through the element will be
by molecules entering and leaving the faces perpendicular to x ; and, since the
condition of the element remains unchanged, the aggregate momentum of the
molecules which enter must be equal to the aggregate momentum of the
molecules which leave.

u+

The aggregate momentum which enters at the face on the left is <r x (Mu),

u+

or as it may be written 2 (Mu*), while the aggregate momentum which enters

u- ii-

on the right is cr x (Mu) or 2 (Mu*).

Therefore the whole momentum in the direction of a; carried through the
element in a unit of time is

u+ u- u+ u-

<r x (Mu) - a- x (Mu) or 2 (Mu*} - 2 (Mu*).

And since the aggregate mass of the molecules which pass through the
element in the same time is

u + u- u+ u-

<r x (M) - <r x (M) or 2 (Mu) - 2 (Mu)

the mean component velocity of all the molecules which pass through the
element in a unit of time is

+ - + -

<r x (Mu) - <r x (Mu) ^(Mu 2

- 2 "(Mil)

which will not be, neglecting w 2 , the same as 2u, as it would be if the gas
were uniform and moving with the velocity u.

The same thing may be shown for faces parallel to x, and for variations

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