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in the directions y arid z.

IN THE GASEOUS STATE. 325

In all the phenomena considered, the velocity

<rl + (Mu) - ff~(Mu)

~, u

<r x (M)-<r x (M)

is very small compared with the mean velocity of a molecule ; but the relation

is of the same order as that of the unhindered rate of thermal transpiration

and the mean velocity of a molecule.

78. The following limitations and definitions will tend to the simpli-

fication of subsequent expressions.

The condition of the gas.

All the assumptions and theorems, as indeed the entire investigation,

with the exception of Arts. 108 A and 109, relate to a simple gas in which

the diameters of the molecules may be neglected in comparison with the

mean distance which separates them, the condition of which gas is at all

points steady as regards time, and the molecules of which are subjected to no

external forces, such as gravity and electric attractions ; and the term gas is

to be understood in this sense unless otherwise defined.

The small quantities neglected.

As a first approximation, i.e., in theorems (I.) and (II.) no account is

taken of variations of the second order, such as are expressed by

d 2 p d 2 a

the effects of such variations being too small to make any difference in the

results of the first approximation.

Also throughout the investigation the velocity of the gas is assumed to be

so small that such quantities 'as -u 2 and

,' M-f M \ 2

/<r x (Mu)-(r x (Mu)\

u+

may be neglected.

Definitions.

An elementary group of molecules. In addition to the separation of the

molecules into the groups A, B, C, D, E, F, G, H, as explained in Art. 67, a

further subdivision is necessary in order to render the reasoning of this

section definite.

326 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

From any one of the eight groups are selected all the molecules having

directions of motion which differ by less than certain small angles from a

given direction, or, in other words, those molecules of which the directions of

motion are parallel to some line which may be included within a pyramidal

surface having indefinitely small angles at the apex. Such a group will be

called an elementary group, and in this sense only will the term elementary

group be used. The mean ray or axis of the pyramid is the mean direction

of the group. And it is to be noticed that only those molecules that are

moving in the same direction parallel to the axis of the pyramid are included

in the same group, those with opposite motion constituting another elementary

group.

The distinguishing features of an elementary group, apart from the

direction of the group, are the number of molecules at any instant in a unit

of volume the symbol A r will be used to signify this number ; their mean

velocity, mean square of velocity, &c., will be indicated without regard to

direction by the symbols v, v 2 ; and to avoid confusion, instead of using Q to

indicate the two latter quantities the letter G will be used to represent

severally y, v, F 2 , &c.

The resultant uniform gas. It has been already pointed out (Art. 75)

that if the encounters within an element of volume resulted in the molecules

leaving the element in the same manner as they would leave if the gas about

and within the element were uniform, this uniform gas must have component

velocities which are one-half the mean component velocities of all the

molecules of the varying gas which in a unit of time pass through the

element. This uniform gas, which would also have approximately the mean

pressure and density of the actual gas in the element, is called the resultant

uniform gas of the gas within the element. U, V, W are used to designate

pa 2

its component velocities, p to express its density, and * to express its

2

pressure. U, V, W are functions of u, v, w and of the variations of p, at, or,

in other words, they are functions of the condition of the gas at the point

considered, but they cannot be completely determined in the first stage of

the investigation.

The inequalities in elementary groups. All the elementary groups relating

to a unit of volume in a varying gas are compared with corresponding

elementary groups in the resultant uniform gas for the element, and the

differences in respect of the density and velocities of the molecules are spoken

of as the inequalities of the group. There are only four quantities in respect

to which the groups can be compared, namely : the numbers of molecules, the

mean velocity, the mean square, and the mean cube of the velocity ; essentially,

therefore, the differences in these constitute the inequalities of the group.

33] J& THE GASEOUS STATE. 327

Thus, if G standing for .v, v, v 3 or F a refers to an elementary group of

the resultant uniform gas for an indefinitely small element, and G + 1 refers

to the corresponding elementary group of the varying gas, then 1 represents

the inequality in an elementary group at a point as compared with the

resultant uniform gas at that point.

When the element has small but definite dimensions (Sr) the inequalities

ot the elementary groups entering or leaving will be

,dGSr dISr

dr 2 4 l dr 2

r\

for the inequality has reference to the uniform gas at a point distant J from

2

the point at which I represents the inequalities, and therefore the change in

G + I must be added to or subtracted from the inequality, as the case may

be.

79. The following assumptions may all be deduced from first principles,

but the necessary reasoning is long, and it is thought that the assumptions

are sufficiently obvious.

Assumptions.

I. That the condition of the gas, as already defined (Art. 78), at any

instant within an element of volume depends entirely on the numbers and

component velocities of the molecules which, in a unit of time, enter at each

part of the surface of the element ; and hence if the molecules enter one element

in exactly the same manner as the molecules enter a geometrically similar

element, the condition of the gas within the elements must be similar.

II. That the number and component velocities of the molecules which leave

each elementary portion of the surface of an element, depend only on the

condition of the gas within ^the element and the manner in which the molecules

enter ; and therefore by (I.) depend only on the manner in which molecules

enter. Also since the gas immediately outside the element consists of the

molecules entering and leaving, its condition depends only on the molecules

entering. So that if molecules enter corresponding portions of the surfaces

of two geometrically similar elements in exactly the same manner the gas

about the elements must be exactly similar.

III. That whatever be the nature of the action between the molecules, the

effect of encounters within an element must always tend to produce or maintain

the same relative motion amongst the molecules, which relative motion is that of

a uniform gas ; and hence the encounters must render the manner in which

the molecules leave the element, as compared with that in which they enter,

more nearly similar to the manner of a uniform gas.

328 ON CERTAIN DIMENSIONAL PROPERTIES OP MATTER [33

That is to say, if A, B, C, D, &c., be a series of geometrically similar

spherical elements, and the gas about B is such that the molecules enter B

in exactly the same manner as they leave the opposite sides of A, and the gas

about C such that the molecules enter as they leave the opposite side of B

and so on, the gas about each element being such that the molecules enter

the element exactly as they leave the opposite side of the preceding element,

then, according to the assumption, the gas about each element will be more

nearly uniform than that about the preceding element, so that eventually

about the wth element the gas would be uniform, n being indefinitely great.

This may be expressed algebraically. Putting h for the number of

encounters necessary to obliterate the inequalities in the groups which pass

through A in a unit of time, h will be infinite, and as / is so small that it

may be considered as taking no part in the distribution, the rate of dis-

tribution will depend on the number of encounters in a unit of volume, and

on some function /(a) of a, -= being the mean velocity of the molecules.

VTT

Therefore approximately

dl

So that if /' is the initial value of /, then after h encounters we have

integrating

//>-/*

and if h is infinite

1 = (24).

/(a) is a positive function of a, and is not a function of /; but both

as regards form and coefficients /(a) may depend on the nature of the

quantity G.

The question whether /(a) is different for any or all of the quantities

N, F, F 2 , &c., must depend on the nature of the action between the molecules

during encounters.

If therefore by comparing the mathematical results with those from

experiments the several values of /(a) can be compared, a certain amount of

light would be thrown on the action between the molecules. So far, however,

the conditions of equilibrium in the interior of gas of which the temperature

varies, form the only instance in which the values of /(a) are brought into

direct comparison. This instance affords means of comparing the values of

/(a) for N and F 2 , and shows that these values must be equal. As regards

/(a) for F or F 3 , there are no experimental results which furnish any further

light than that/ (a) has real positive values.

33] IN THE GASEOUS STATE. 329

These questions do not rise in this investigation, since /(a) for v 3 does not

appear in the results, and should /(a) have a different value for v from that

which it has for N and F 2 , the only result would be a numerical difference in

certain coefficients as to the comparative value of which the experiment

affords no approximate evidence.

IV. That when the molecules which enter or leave an element of volume in

a unit of time are considered separately, the proportion of the molecules (N v)

entering in a unit of time, in each entering group, which will subsequently

undergo encounters within the element, and the proportion of the molecules

leaving in a unit of time, in each leaving group, which have undergone

encounters within the element, are approximately proportional to the mean

distance (8r) through the element in direction of the group, and to the number

of molecules in each unit of volume of the element.

V. That the mean effect of encounters in distributing the several inequalities

of the molecules which, entering in a unit of time, encounter within the element,

is a function (f(a)) of the mean velocity of the molecules within the element at

the instant.

Fundamental Theorems.

80. On the assumptions I. to V., remembering the fact pointed out in

Arts. 75 and 79 with respect to the component velocities of the resultant

uniform gas, the following theorems are established :

Theorem (I.). Each of the several inequalities, as defined in Art. 78, in evert/

elementary group of molecules which in a unit of time leave an element of

volume of small but definite size, will severally be less than in the cor-

responding elementary group, which in the same time enter the element in

the same direction, by, quantities which bear approximately the same

relation to the mean inequalities of the two groups, as the distance through

the element in direction of the group bears to a distance (s) which is a

function of the density of the gas, and the mean square of the velocity of

the molecules only.

To express this theorem algebraically, let G and /, as explained in the

last article, refer to the point in the middle of the element. Then the

inequality in the entering group is expressed by

dG Br 7 dl 8r

~~dr"2 + ~"dr ~2 '

and for the leaving group by

dG Br dl &r

dr 2 dr 2

OF THE

_.Mvf

UNIVERSITY

OF

330 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

And what the theorem asserts is

(25)

. .

wherein s is a function of p and a 3 only.

Proof of Theorem (/.).

(a) From assumptions I. and II., Art. 7.9, it follows at once that when

the condition of the gas varies from point to point, the molecules cannot

enter an element of volume in the same manner as they would from any

uniform gas.

(6) From (a) and assumption III. it follows that the effect of encounters

within an element in a varying gas is to render the manner in which the

molecules leave as compared with that in which they enter more nearly

similar to that of some uniform gas.

(c) The uniform gas referred to in (6) must, as has been already pointed

out, have component velocities equal to half the mean component velocities

of all the molecules which in a unit of time pass through the element.

This at once follows from the illustration appended to assumption III.,

Art. 79. For the molecules which leave an element in a unit of time must

have the same mean component velocities as those which enter, their aggregate

mass being the same and the momentum within the element remaining

unaltered, and as the molecules enter each successive element in the same

manner as they left the preceding, the molecules which enter the nth element

in a unit of time must have the same mean component velocities as those

which enter the first ; but in the ?ith element the gas is uniform. Therefore,

if U, V, W are the component velocities of the uniform gas, when these are

small so that we may neglect U' 2 , V'\ W 2

u+ u- v+ v- w+

<r x (Mu) - a- x (Mu)

- x (Mu) v _ <r y (Mv) - a-y (Mv} w _

- "' ' ~* v+ v- ' r ~~

w + w -

<r x (M)-<r x (M) <r y (M)-<r y (M) <r z (M) - <r z (M)

...... (26).

The number of molecules which enter the nth element will also be equal

to the number which enter the 1st.

_ 32

Therefore putting w 2 + v 2 + w 2 = ~- , and using the dash to indicate the

Z

first element

pa = //a' .................................... (27).

33] IN THE GASEOUS STATE. 331

And the energy carried into the >ith is equal to the energy carried into the

first element. Therefore

pa? = p'a' 3 (28).

From which equations

a a = a' s and p = p (29).

Or the density and pressure of the uniform gas is approximately the same

as the density and mean pressure of the actual gas. This uniform gas is,

therefore, the resultant uniform gas according to the definition Art. 78.

(d) From assumptions IV. and V. it follows directly that the several

changes in the inequalities, considered separately, of each elementary group

which enters the element in a unit of time, will be proportional to the mean

inequalities of the group as it enters and leaves, multiplied by /(a) and by

the product of ^ and the mean distance through the element traversed by

the group.

Or, as before, putting 7 for the mean inequality of the group as it enters

and leaves in respect of G, the separate inequalities are

Whence from assumptions IV. and V. it follows that

-i((? + /)r/()8ftf (30).

dr ^ ' M

M

And from the dimensions of this equation it follows that ... . represents a

/()P

distance. Therefore putting 6- for this distance

4-(G + I)8r = /.. ..(31).

dr s

[Q. E. D.]

Corollary to Theorem (/.).

When -= - is nearly constant, so that we may neglect s ~j- 2 as compared

with -5- , then integrating equation (31) we have

or

^n

=s-j- + Ce

dr

dG I C .L

= e *

dr s s

(32).

332 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Near a solid surface.

Equation (32) shows the nature of the inequalities as affected by dis-

continuity such as may arise at a solid surface. The last term on the right

gives the effect of discontinuity for an element at a distance r from the

surface, r being measured in the direction of the group. This effect

diminishes as r increases.

In the first of equations (32) we may obviously put

dG , dG n -L

Si -T- for s i h Ue .

dr dr

! being a function of the position of the element and of the direction of the

group.

Theorem (II.). When the variation in the condition of the gas is approximately

constant, then in respect of any one of the quantities N, r, v 2 , &c., each

elementary group of molecules entering a small element of volume at any

point will enter approximately as if from the resultant uniform gas at a

point in the direction from which the group arrives, the distance of which

point from the element is a function of the mean velocity of a molecule, and

inversely proportional to the number of molecules within a unit of volume,

and is independent of the variation of the gas and the direction of the

elementary group.

To illustrate this, supposing a small spherical element at A, and con-

sidering the group arriving in the direction BA, then if the gas varies in the

Fig. 10.

direction BA the resultant uniform gas for points along BA will differ, and if

A were to be surrounded by a gas identical with the resultant uniform gas at

a point P, the elementary group in the direction BA or PA would arrive at

the element with different values as to density, mean velocity, &c., from a

similar group if the gas were identical with the resultant uniform gas at

another point in AB.

Now what the theorem asserts is, that there is some point Pj at which

the resultant uniform gas is such that the elementary group in direction BA

33] IN THE GASEOUS STATE. 333

would arrive with approximately the same value of N as the actual group, and

that the distance PI A is independent of the direction of BA, i.e., would be

the same for all directions from A. In the same way there is some point P 2

at which the resultant uniform gas is such that the group of molecules BA

would have the same value of v as for the actual group, and so for F 2 and F 3 .

It is not however asserted that AP lt AP Z , &c., either are or are not

identical.

Proof of Theorem (II.).

This follows directly from theorem (I.).

Taking a series of elements bounded by a cylindrical surface described

about the element at A and having its axis in the direction of the group,

then all the molecules of the group leaving one element may be supposed to

enter the next.

In entering the first element at B there will be a difference / between

the value of G for the actual group and the value of G for the resultant

uniform gas. If G B is taken for the resultant uniform gas, G B + I B will

represent the corresponding value for the actual gas at B.

On emerging from the first element G + 1 for the group will, by theorem

Sr

(I.), have been diminished by /, Sr being the thickness of the element ; on

emerging from the next element, G + I will be still further diminished by

Sr r

I, and

s

equal to

Sr

I, and so on through all the elements, the total diminution of G + I being

S

'/

dr.

*

And by the corollary to theorem (I.), since the variation in the condition

of the gas is approximately constant, I is approximately constant through the

distance s, and s will be approximately constant through this distance ;

therefore

~dr = I B (33).

Hence, having traversed the distance s, the group will emerge having

= G B (34).

That is, on arriving at A, the molecules will, in respect of G, enter the

element as if from the resultant uniform gas at B, a point in the direction of

334 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

the group, the distance s of which from A is a function of a, is inversely

proportional to ^ , and is independent of the variation of the gas and of the

direction of the group. [Q. E. D.]

Corollary to Theorem (II.}. The effect of a solid surface.

If in the neighbourhood of A there is a solid surface such that, if B is a

point on this surface, BA is of the same order of magnitude as s, then putting

r = BA for the group arriving at A from the direction BA, equation (33)

gives

(35)

and substituting for - from equation (32) and integrating

S

dG _j T n _ n dG

dr

n /- a j r r """" 4.1, r

or since G B = G. + r -=- , and /. (7= s -;- , therefore

* * ^7^. ' a -*

- (36)

G will be a function of I, m, n, and it may be written f(lmri) s -^- ; therefore

(37).

The mean range.

M

81. The distance s, or . (equation 30) is thus shown to be the

distance at which the elementary groups radiating outwards from a point

have the mean value of G for the molecules which, in a unit of time, pass the

central point. And hence it is proposed to call 6' the mean range of the

quantity G.

The mean range is thus seen to be approximately independent of the

space variations of the gas, but since s involves^ a), which, as pointed out in

assumption III , Art. 78, may, so far as is yet known, have different values

for v and r 3 from its values for N and F 2 , which latter are equal, so the values

of s for the mean velocity and mean cube of the velocity may be different

from the values of s for the density and mean square of the velocity, which

latter are equal.

33] IN THE GASEOUS STATE. 335

Such a difference in the values of s, however, is not important as regards

the immediate results of this investigation, and in the absence of any evidence

to the contrary all values of s will be treated as equal.

The mean component values of s and general expression of <r(Q). Gas

continuous.

82. When the gas is continuous, by theorem (II.) all values of a-(Q) for

the groups A, B, C, &c , at any point may severally be expressed as functions

of p, a, U y V, W for this point, their space variations, and s.

The first step is to express as a function of p, a, U, V, W, the elementary

portion of <r (Q) belonging to an elementary group of molecules, and then to

integrate for the larger groups.

Putting a-' (Q) for the value of <r (Q), which would result from the resultant

uniform gas at a point, and cr (Q) for the elementary portion of a (Q) belonging

to an elementary group whose directions are I, m, n, then since p, a, U, V, W,

for any point x, y, z, are functions of x, y, z, &(r(Q) is a function of x, y, z, and

for the point x + Is, y + ms, z + ns we have

' ......... (38)

~

together with terms which are neglected as small.

Whence integrating for all the groups in A, and putting A for a-' (Q)

...... (39)

where cos = I, m = sin 6 cos <j>, n sin 6 sin <j>, and similarly for the other

seven groups, B, G, &c.

The values of A, &c., are given in Table XX.

The integrals of s \(l -5- +m-j- + n-j-}SA[ sin 0d6d<p will involve terms

I \ CLOG ^y Ct2 J

which will be the differentials of the corresponding terms in Table XX.,

multiplied by s and by certain numerical coefficients which are the mean

values of I, of P divided by /, and so on, and which may be written I,

, , &c. The values of s multiplied by these coefficients are the mean

i L

component values of s for p, a, a 2 , &c., for the groups A, B, C, &c. As it is

these component values which come into comparison with the distances from

a solid surface, it is important to preserve the coefficients, therefore instead of

using the numerical values they will be indicated by the letters L 1} L 2 , &c.,

as about to be assigned.

336

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

Putting i for unity or any power of a or U, V, W, the coefficients by

which the differentials of the corresponding terms in Table XX. must be

multiplied, are as follows:

.

dx

by

I-J

dpgi

dprji

dpty

~(H

&c.

I*

2

dx '

dy '

" 1

lm

3

4

dy '

dz '

dprfi

" 1

Z 3

3?r

3

dx '

dplj*i

dy

' dz

. <foc.

" I 2

^m

" Z 2

4

3

dy

<*

8

3

dp^rji

expressed by L 1} .

> (40).

L 4>

dx

dx

lm

The coefficients L lt L 2 , &c., all occur in some one or other of the

expressions for A, B, G, &c. ; but when these expressions come to be added

together it is found that L 2 is the only coefficient of s which has to be

considered. This being the case, instead of L 2 s, the simple s will be used,

so that in all subsequent expressions

2

s = K (mean range) ........................... (41).

o

The signs of the products of s and the differentials of the several terms

in the table may, as will be seen from Art. 69, be expressed in the following

manner, ignoring the numerical coefficients L lt L 2 , &c.

-..(42),

with similar expressions for'o- 3/ (Q) and <r z (Q\ the suffix to the letters being

the same as the suffix to a (Q) on the Feft.

The following are the resulting values of <r(Q) which are required for

this investigation, terms of the order U 2 having been neglected.

33]

IN THE GASEOUS STATE.

IN THE GASEOUS STATE. 325

In all the phenomena considered, the velocity

<rl + (Mu) - ff~(Mu)

~, u

<r x (M)-<r x (M)

is very small compared with the mean velocity of a molecule ; but the relation

is of the same order as that of the unhindered rate of thermal transpiration

and the mean velocity of a molecule.

78. The following limitations and definitions will tend to the simpli-

fication of subsequent expressions.

The condition of the gas.

All the assumptions and theorems, as indeed the entire investigation,

with the exception of Arts. 108 A and 109, relate to a simple gas in which

the diameters of the molecules may be neglected in comparison with the

mean distance which separates them, the condition of which gas is at all

points steady as regards time, and the molecules of which are subjected to no

external forces, such as gravity and electric attractions ; and the term gas is

to be understood in this sense unless otherwise defined.

The small quantities neglected.

As a first approximation, i.e., in theorems (I.) and (II.) no account is

taken of variations of the second order, such as are expressed by

d 2 p d 2 a

the effects of such variations being too small to make any difference in the

results of the first approximation.

Also throughout the investigation the velocity of the gas is assumed to be

so small that such quantities 'as -u 2 and

,' M-f M \ 2

/<r x (Mu)-(r x (Mu)\

u+

may be neglected.

Definitions.

An elementary group of molecules. In addition to the separation of the

molecules into the groups A, B, C, D, E, F, G, H, as explained in Art. 67, a

further subdivision is necessary in order to render the reasoning of this

section definite.

326 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

From any one of the eight groups are selected all the molecules having

directions of motion which differ by less than certain small angles from a

given direction, or, in other words, those molecules of which the directions of

motion are parallel to some line which may be included within a pyramidal

surface having indefinitely small angles at the apex. Such a group will be

called an elementary group, and in this sense only will the term elementary

group be used. The mean ray or axis of the pyramid is the mean direction

of the group. And it is to be noticed that only those molecules that are

moving in the same direction parallel to the axis of the pyramid are included

in the same group, those with opposite motion constituting another elementary

group.

The distinguishing features of an elementary group, apart from the

direction of the group, are the number of molecules at any instant in a unit

of volume the symbol A r will be used to signify this number ; their mean

velocity, mean square of velocity, &c., will be indicated without regard to

direction by the symbols v, v 2 ; and to avoid confusion, instead of using Q to

indicate the two latter quantities the letter G will be used to represent

severally y, v, F 2 , &c.

The resultant uniform gas. It has been already pointed out (Art. 75)

that if the encounters within an element of volume resulted in the molecules

leaving the element in the same manner as they would leave if the gas about

and within the element were uniform, this uniform gas must have component

velocities which are one-half the mean component velocities of all the

molecules of the varying gas which in a unit of time pass through the

element. This uniform gas, which would also have approximately the mean

pressure and density of the actual gas in the element, is called the resultant

uniform gas of the gas within the element. U, V, W are used to designate

pa 2

its component velocities, p to express its density, and * to express its

2

pressure. U, V, W are functions of u, v, w and of the variations of p, at, or,

in other words, they are functions of the condition of the gas at the point

considered, but they cannot be completely determined in the first stage of

the investigation.

The inequalities in elementary groups. All the elementary groups relating

to a unit of volume in a varying gas are compared with corresponding

elementary groups in the resultant uniform gas for the element, and the

differences in respect of the density and velocities of the molecules are spoken

of as the inequalities of the group. There are only four quantities in respect

to which the groups can be compared, namely : the numbers of molecules, the

mean velocity, the mean square, and the mean cube of the velocity ; essentially,

therefore, the differences in these constitute the inequalities of the group.

33] J& THE GASEOUS STATE. 327

Thus, if G standing for .v, v, v 3 or F a refers to an elementary group of

the resultant uniform gas for an indefinitely small element, and G + 1 refers

to the corresponding elementary group of the varying gas, then 1 represents

the inequality in an elementary group at a point as compared with the

resultant uniform gas at that point.

When the element has small but definite dimensions (Sr) the inequalities

ot the elementary groups entering or leaving will be

,dGSr dISr

dr 2 4 l dr 2

r\

for the inequality has reference to the uniform gas at a point distant J from

2

the point at which I represents the inequalities, and therefore the change in

G + I must be added to or subtracted from the inequality, as the case may

be.

79. The following assumptions may all be deduced from first principles,

but the necessary reasoning is long, and it is thought that the assumptions

are sufficiently obvious.

Assumptions.

I. That the condition of the gas, as already defined (Art. 78), at any

instant within an element of volume depends entirely on the numbers and

component velocities of the molecules which, in a unit of time, enter at each

part of the surface of the element ; and hence if the molecules enter one element

in exactly the same manner as the molecules enter a geometrically similar

element, the condition of the gas within the elements must be similar.

II. That the number and component velocities of the molecules which leave

each elementary portion of the surface of an element, depend only on the

condition of the gas within ^the element and the manner in which the molecules

enter ; and therefore by (I.) depend only on the manner in which molecules

enter. Also since the gas immediately outside the element consists of the

molecules entering and leaving, its condition depends only on the molecules

entering. So that if molecules enter corresponding portions of the surfaces

of two geometrically similar elements in exactly the same manner the gas

about the elements must be exactly similar.

III. That whatever be the nature of the action between the molecules, the

effect of encounters within an element must always tend to produce or maintain

the same relative motion amongst the molecules, which relative motion is that of

a uniform gas ; and hence the encounters must render the manner in which

the molecules leave the element, as compared with that in which they enter,

more nearly similar to the manner of a uniform gas.

328 ON CERTAIN DIMENSIONAL PROPERTIES OP MATTER [33

That is to say, if A, B, C, D, &c., be a series of geometrically similar

spherical elements, and the gas about B is such that the molecules enter B

in exactly the same manner as they leave the opposite sides of A, and the gas

about C such that the molecules enter as they leave the opposite side of B

and so on, the gas about each element being such that the molecules enter

the element exactly as they leave the opposite side of the preceding element,

then, according to the assumption, the gas about each element will be more

nearly uniform than that about the preceding element, so that eventually

about the wth element the gas would be uniform, n being indefinitely great.

This may be expressed algebraically. Putting h for the number of

encounters necessary to obliterate the inequalities in the groups which pass

through A in a unit of time, h will be infinite, and as / is so small that it

may be considered as taking no part in the distribution, the rate of dis-

tribution will depend on the number of encounters in a unit of volume, and

on some function /(a) of a, -= being the mean velocity of the molecules.

VTT

Therefore approximately

dl

So that if /' is the initial value of /, then after h encounters we have

integrating

//>-/*

and if h is infinite

1 = (24).

/(a) is a positive function of a, and is not a function of /; but both

as regards form and coefficients /(a) may depend on the nature of the

quantity G.

The question whether /(a) is different for any or all of the quantities

N, F, F 2 , &c., must depend on the nature of the action between the molecules

during encounters.

If therefore by comparing the mathematical results with those from

experiments the several values of /(a) can be compared, a certain amount of

light would be thrown on the action between the molecules. So far, however,

the conditions of equilibrium in the interior of gas of which the temperature

varies, form the only instance in which the values of /(a) are brought into

direct comparison. This instance affords means of comparing the values of

/(a) for N and F 2 , and shows that these values must be equal. As regards

/(a) for F or F 3 , there are no experimental results which furnish any further

light than that/ (a) has real positive values.

33] IN THE GASEOUS STATE. 329

These questions do not rise in this investigation, since /(a) for v 3 does not

appear in the results, and should /(a) have a different value for v from that

which it has for N and F 2 , the only result would be a numerical difference in

certain coefficients as to the comparative value of which the experiment

affords no approximate evidence.

IV. That when the molecules which enter or leave an element of volume in

a unit of time are considered separately, the proportion of the molecules (N v)

entering in a unit of time, in each entering group, which will subsequently

undergo encounters within the element, and the proportion of the molecules

leaving in a unit of time, in each leaving group, which have undergone

encounters within the element, are approximately proportional to the mean

distance (8r) through the element in direction of the group, and to the number

of molecules in each unit of volume of the element.

V. That the mean effect of encounters in distributing the several inequalities

of the molecules which, entering in a unit of time, encounter within the element,

is a function (f(a)) of the mean velocity of the molecules within the element at

the instant.

Fundamental Theorems.

80. On the assumptions I. to V., remembering the fact pointed out in

Arts. 75 and 79 with respect to the component velocities of the resultant

uniform gas, the following theorems are established :

Theorem (I.). Each of the several inequalities, as defined in Art. 78, in evert/

elementary group of molecules which in a unit of time leave an element of

volume of small but definite size, will severally be less than in the cor-

responding elementary group, which in the same time enter the element in

the same direction, by, quantities which bear approximately the same

relation to the mean inequalities of the two groups, as the distance through

the element in direction of the group bears to a distance (s) which is a

function of the density of the gas, and the mean square of the velocity of

the molecules only.

To express this theorem algebraically, let G and /, as explained in the

last article, refer to the point in the middle of the element. Then the

inequality in the entering group is expressed by

dG Br 7 dl 8r

~~dr"2 + ~"dr ~2 '

and for the leaving group by

dG Br dl &r

dr 2 dr 2

OF THE

_.Mvf

UNIVERSITY

OF

330 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

And what the theorem asserts is

(25)

. .

wherein s is a function of p and a 3 only.

Proof of Theorem (/.).

(a) From assumptions I. and II., Art. 7.9, it follows at once that when

the condition of the gas varies from point to point, the molecules cannot

enter an element of volume in the same manner as they would from any

uniform gas.

(6) From (a) and assumption III. it follows that the effect of encounters

within an element in a varying gas is to render the manner in which the

molecules leave as compared with that in which they enter more nearly

similar to that of some uniform gas.

(c) The uniform gas referred to in (6) must, as has been already pointed

out, have component velocities equal to half the mean component velocities

of all the molecules which in a unit of time pass through the element.

This at once follows from the illustration appended to assumption III.,

Art. 79. For the molecules which leave an element in a unit of time must

have the same mean component velocities as those which enter, their aggregate

mass being the same and the momentum within the element remaining

unaltered, and as the molecules enter each successive element in the same

manner as they left the preceding, the molecules which enter the nth element

in a unit of time must have the same mean component velocities as those

which enter the first ; but in the ?ith element the gas is uniform. Therefore,

if U, V, W are the component velocities of the uniform gas, when these are

small so that we may neglect U' 2 , V'\ W 2

u+ u- v+ v- w+

<r x (Mu) - a- x (Mu)

- x (Mu) v _ <r y (Mv) - a-y (Mv} w _

- "' ' ~* v+ v- ' r ~~

w + w -

<r x (M)-<r x (M) <r y (M)-<r y (M) <r z (M) - <r z (M)

...... (26).

The number of molecules which enter the nth element will also be equal

to the number which enter the 1st.

_ 32

Therefore putting w 2 + v 2 + w 2 = ~- , and using the dash to indicate the

Z

first element

pa = //a' .................................... (27).

33] IN THE GASEOUS STATE. 331

And the energy carried into the >ith is equal to the energy carried into the

first element. Therefore

pa? = p'a' 3 (28).

From which equations

a a = a' s and p = p (29).

Or the density and pressure of the uniform gas is approximately the same

as the density and mean pressure of the actual gas. This uniform gas is,

therefore, the resultant uniform gas according to the definition Art. 78.

(d) From assumptions IV. and V. it follows directly that the several

changes in the inequalities, considered separately, of each elementary group

which enters the element in a unit of time, will be proportional to the mean

inequalities of the group as it enters and leaves, multiplied by /(a) and by

the product of ^ and the mean distance through the element traversed by

the group.

Or, as before, putting 7 for the mean inequality of the group as it enters

and leaves in respect of G, the separate inequalities are

Whence from assumptions IV. and V. it follows that

-i((? + /)r/()8ftf (30).

dr ^ ' M

M

And from the dimensions of this equation it follows that ... . represents a

/()P

distance. Therefore putting 6- for this distance

4-(G + I)8r = /.. ..(31).

dr s

[Q. E. D.]

Corollary to Theorem (/.).

When -= - is nearly constant, so that we may neglect s ~j- 2 as compared

with -5- , then integrating equation (31) we have

or

^n

=s-j- + Ce

dr

dG I C .L

= e *

dr s s

(32).

332 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Near a solid surface.

Equation (32) shows the nature of the inequalities as affected by dis-

continuity such as may arise at a solid surface. The last term on the right

gives the effect of discontinuity for an element at a distance r from the

surface, r being measured in the direction of the group. This effect

diminishes as r increases.

In the first of equations (32) we may obviously put

dG , dG n -L

Si -T- for s i h Ue .

dr dr

! being a function of the position of the element and of the direction of the

group.

Theorem (II.). When the variation in the condition of the gas is approximately

constant, then in respect of any one of the quantities N, r, v 2 , &c., each

elementary group of molecules entering a small element of volume at any

point will enter approximately as if from the resultant uniform gas at a

point in the direction from which the group arrives, the distance of which

point from the element is a function of the mean velocity of a molecule, and

inversely proportional to the number of molecules within a unit of volume,

and is independent of the variation of the gas and the direction of the

elementary group.

To illustrate this, supposing a small spherical element at A, and con-

sidering the group arriving in the direction BA, then if the gas varies in the

Fig. 10.

direction BA the resultant uniform gas for points along BA will differ, and if

A were to be surrounded by a gas identical with the resultant uniform gas at

a point P, the elementary group in the direction BA or PA would arrive at

the element with different values as to density, mean velocity, &c., from a

similar group if the gas were identical with the resultant uniform gas at

another point in AB.

Now what the theorem asserts is, that there is some point Pj at which

the resultant uniform gas is such that the elementary group in direction BA

33] IN THE GASEOUS STATE. 333

would arrive with approximately the same value of N as the actual group, and

that the distance PI A is independent of the direction of BA, i.e., would be

the same for all directions from A. In the same way there is some point P 2

at which the resultant uniform gas is such that the group of molecules BA

would have the same value of v as for the actual group, and so for F 2 and F 3 .

It is not however asserted that AP lt AP Z , &c., either are or are not

identical.

Proof of Theorem (II.).

This follows directly from theorem (I.).

Taking a series of elements bounded by a cylindrical surface described

about the element at A and having its axis in the direction of the group,

then all the molecules of the group leaving one element may be supposed to

enter the next.

In entering the first element at B there will be a difference / between

the value of G for the actual group and the value of G for the resultant

uniform gas. If G B is taken for the resultant uniform gas, G B + I B will

represent the corresponding value for the actual gas at B.

On emerging from the first element G + 1 for the group will, by theorem

Sr

(I.), have been diminished by /, Sr being the thickness of the element ; on

emerging from the next element, G + I will be still further diminished by

Sr r

I, and

s

equal to

Sr

I, and so on through all the elements, the total diminution of G + I being

S

'/

dr.

*

And by the corollary to theorem (I.), since the variation in the condition

of the gas is approximately constant, I is approximately constant through the

distance s, and s will be approximately constant through this distance ;

therefore

~dr = I B (33).

Hence, having traversed the distance s, the group will emerge having

= G B (34).

That is, on arriving at A, the molecules will, in respect of G, enter the

element as if from the resultant uniform gas at B, a point in the direction of

334 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

the group, the distance s of which from A is a function of a, is inversely

proportional to ^ , and is independent of the variation of the gas and of the

direction of the group. [Q. E. D.]

Corollary to Theorem (II.}. The effect of a solid surface.

If in the neighbourhood of A there is a solid surface such that, if B is a

point on this surface, BA is of the same order of magnitude as s, then putting

r = BA for the group arriving at A from the direction BA, equation (33)

gives

(35)

and substituting for - from equation (32) and integrating

S

dG _j T n _ n dG

dr

n /- a j r r """" 4.1, r

or since G B = G. + r -=- , and /. (7= s -;- , therefore

* * ^7^. ' a -*

- (36)

G will be a function of I, m, n, and it may be written f(lmri) s -^- ; therefore

(37).

The mean range.

M

81. The distance s, or . (equation 30) is thus shown to be the

distance at which the elementary groups radiating outwards from a point

have the mean value of G for the molecules which, in a unit of time, pass the

central point. And hence it is proposed to call 6' the mean range of the

quantity G.

The mean range is thus seen to be approximately independent of the

space variations of the gas, but since s involves^ a), which, as pointed out in

assumption III , Art. 78, may, so far as is yet known, have different values

for v and r 3 from its values for N and F 2 , which latter are equal, so the values

of s for the mean velocity and mean cube of the velocity may be different

from the values of s for the density and mean square of the velocity, which

latter are equal.

33] IN THE GASEOUS STATE. 335

Such a difference in the values of s, however, is not important as regards

the immediate results of this investigation, and in the absence of any evidence

to the contrary all values of s will be treated as equal.

The mean component values of s and general expression of <r(Q). Gas

continuous.

82. When the gas is continuous, by theorem (II.) all values of a-(Q) for

the groups A, B, C, &c , at any point may severally be expressed as functions

of p, a, U y V, W for this point, their space variations, and s.

The first step is to express as a function of p, a, U, V, W, the elementary

portion of <r (Q) belonging to an elementary group of molecules, and then to

integrate for the larger groups.

Putting a-' (Q) for the value of <r (Q), which would result from the resultant

uniform gas at a point, and cr (Q) for the elementary portion of a (Q) belonging

to an elementary group whose directions are I, m, n, then since p, a, U, V, W,

for any point x, y, z, are functions of x, y, z, &(r(Q) is a function of x, y, z, and

for the point x + Is, y + ms, z + ns we have

' ......... (38)

~

together with terms which are neglected as small.

Whence integrating for all the groups in A, and putting A for a-' (Q)

...... (39)

where cos = I, m = sin 6 cos <j>, n sin 6 sin <j>, and similarly for the other

seven groups, B, G, &c.

The values of A, &c., are given in Table XX.

The integrals of s \(l -5- +m-j- + n-j-}SA[ sin 0d6d<p will involve terms

I \ CLOG ^y Ct2 J

which will be the differentials of the corresponding terms in Table XX.,

multiplied by s and by certain numerical coefficients which are the mean

values of I, of P divided by /, and so on, and which may be written I,

, , &c. The values of s multiplied by these coefficients are the mean

i L

component values of s for p, a, a 2 , &c., for the groups A, B, C, &c. As it is

these component values which come into comparison with the distances from

a solid surface, it is important to preserve the coefficients, therefore instead of

using the numerical values they will be indicated by the letters L 1} L 2 , &c.,

as about to be assigned.

336

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

Putting i for unity or any power of a or U, V, W, the coefficients by

which the differentials of the corresponding terms in Table XX. must be

multiplied, are as follows:

.

dx

by

I-J

dpgi

dprji

dpty

~(H

&c.

I*

2

dx '

dy '

" 1

lm

3

4

dy '

dz '

dprfi

" 1

Z 3

3?r

3

dx '

dplj*i

dy

' dz

. <foc.

" I 2

^m

" Z 2

4

3

dy

<*

8

3

dp^rji

expressed by L 1} .

> (40).

L 4>

dx

dx

lm

The coefficients L lt L 2 , &c., all occur in some one or other of the

expressions for A, B, G, &c. ; but when these expressions come to be added

together it is found that L 2 is the only coefficient of s which has to be

considered. This being the case, instead of L 2 s, the simple s will be used,

so that in all subsequent expressions

2

s = K (mean range) ........................... (41).

o

The signs of the products of s and the differentials of the several terms

in the table may, as will be seen from Art. 69, be expressed in the following

manner, ignoring the numerical coefficients L lt L 2 , &c.

-..(42),

with similar expressions for'o- 3/ (Q) and <r z (Q\ the suffix to the letters being

the same as the suffix to a (Q) on the Feft.

The following are the resulting values of <r(Q) which are required for

this investigation, terms of the order U 2 having been neglected.

33]

IN THE GASEOUS STATE.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

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