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heater as far as its frame would allow.

From the fact that the fibre of silk had shown positive motion so nearly

up to the pressure of the atmosphere it might have been anticipated that

the spider line, on account of its much greater thinness, would have shown

positive motion even at pressures considerably above that of the atmosphere.

But the reasoning of Art. 46 respecting the differences of temperature to be

maintained and the effect of the air currents, obviously applies with greater

force to the spider line than to the fibre of silk, and at once accounts for the

observed fact that the positive motion with the spider line was not obtained

until the pressures were somewhat lower than those necessary for the fibre

of silk.

304 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

52. Both with the fibre of silk and the spider line, the phenomena of

impulsion (the excess of pressure against warm surfaces) were apparent and

consistent at densities many hundred times greater than the highest densities

at which like results are obtained with vanes several hundred times broader

than the fibre of silk ; this verifies the theoretical conclusion on which this

part of the investigation was based. The results in this case are not so

definite as is the agreement of the logarithmic homologues in the instances

of transpiration ; but the one fact supports the other, and we may consider

the law of impulsion Law VIII., Art. 9 to have been sufficiently proved.

This concludes the experimental investigation.

PART II. (THEOEETICAL).

SECTION V. INTRODUCTION TO THE THEORY.

53. In suggesting in a former paper that the results discovered by

Mr Crookes were due to the communication of heat from the surface of the

solid bodies to the gas surrounding them, I pointed out as the fundamental

fact on which I based my explanation, that when heat is communicated from

a solid surface to a gas, the mean velocity of the molecules which rebound

from the surface must be greater as they rebound than as they approach, and

hence the momentum which these particular molecules communicate to the

surface must be greater than it would be if the surface were at the same

temperature as the gas.

So far the reasoning is incontrovertible. But in order to explain the ex-

perimental results, it was necessary to assume that the number of cold

molecules which approached the hot surface would be the same as if the

surface were at the same temperature as the gas, or at any rate, if reduced,

the number would not be sufficiently reduced to counteract the effect of

increased velocity of rebound.

Although at that time I could not see any definite proof of this, nor any

way of definitely examining the question, yet I had a strong impression that

the assumption was legitimate ; and although I hoped at some future time

to be able to complete the theoretical explanation, I was content for the time

to rest the evidence of the truth of the assumptions involved on the adequacy

of the reasoning to explain the experimental results obtained.

As other suggestions respecting the cause of the phenomena, widely

different in character from mine, had found supporters, and a good deal of

scepticism was expressed as to the fitness of the cause which I had suggested,

my attention was occupied in deducing the actions which must result from

such a force, and comparing them with experimental ^results. Having, how-

ever, at length satisfied myself, and seeing that a conviction was spreading

33] IN THE GASEOUS STATE. 305

that what I suggested contained the germ of the explanation, I set to work

in earnest to complete the explanation, and ascertain by an extension of the

dynamical theory of gases what effect the hot molecules receding from

the surface should produce on the number arid temperature of those

approaching.

My first attempts to accomplish this were altogether unsuccessful. When

contemplating the phenomena it seemed to me that I could perceive a

glimmering of the method of reasoning for which I was in search, but as

soon as ever I attempted to give definite expression to it this glimmering

vanished.

The reason for this I now perceive clearly. When contemplating the

phenomena, I had a surface of limited extent before me, and I considered the

effect on such a surface without recognising the fundamental importance of

the limit to size.

On the other hand, when I came to definite reasoning, for the sake of

what appeared to be a simplification of the conditions of the problem, I

assumed the surface to be without limit, thus introducing a fundamental

alteration into the conditions of the problem without perceiving it.

The importance of this limit only became apparent to me when I found,

by simple dynamical reasoning, that with surfaces of unlimited extent such

results as those actually obtained would be impossible. This appeared as

follows :

No force on unlimited surface.

54. If we had two plane plates of unlimited extent, H and C, the surface

of H opposite to C being hotter than the surface of C which was opposite to

H, the outside surfaces of both plates being at the same temperature, then

in order to produce results similar to those obtained with limited plates, the

gas between the two plates must maintain a greater steady pressure on the

plate H, than that which it exerts on the colder plate C. Whereas it is at

once obvious that such a condition is contrary to the laws of motion, which

require that the gas between the two surfaces should exert an equal and

opposite pressure on both surfaces.

Having once perceived the force of this reasoning, it became clear to me

that if, as I had supposed, the results obtained in the experiments were due

to gaseous pressure, then they must depend on the limited extent of the

surfaces.

This gave me the clue, in following which I have not only had the satis-

faction of finding the explanation complete as regards the phenomena from

o. R. 20

306 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

which it originated, but I have also found that the theory indicated the

phenomena of thermal transpiration, and explains much that hitherto has

been considered anomalous respecting the laws of transpiration of gases

through small channels suggesting the experiments by which might be

established the relation between these actions.

The manner in which the force arises in the case of a limited surface was

at first rendered much clearer to me by considering an illustration, which

I introduce here, although it forms no part of the proof which will follow.

55. Instead of H and C being plates with gas between them, let them

be earthen batteries of unlimited length, and suppose that guns are distri-

buted at uniform intervals along those batteries ; suppose, also, that all the

shot fired from H bury themselves in the earth of C, and vice versd.

Then, in the first place, it is obvious that since on firing a shot the

momentum imparted to the gun is equal and opposite to the momentum given

to the shot, every shot fired from H will exercise the same force to move the

battery H away from C as the shot will exercise to move C away from H ;

and in the same way the recoil of the guns on C will exercise the same

tendency to move C away from H as the shot will exercise to move H away

from G. And this will be the case whether the guns are supposed to be

pointed straight across the interval between the batteries, or, as / shall

suppose, are pointed with various degrees of obliquity.

Since, then, the result of every shot, whether fired from H or G, causes

equal and opposite forces on the two batteries, the result of all the firing, no

matter how much harder one battery may bombard than the other, must be

to cause an equal force on each battery, the batteries being of unlimited

length.

This case will be seen to be strictly analogous to the effect of the gas

between two plates of unlimited extent to cause equal pressures on the

plates, no matter what may be the differences in the temperature of the

plates.

If now we consider the batteries of limited extent, then, owing to the

obliquity of the guns, some of the shot from H may pass beyond the ends

of C, and vice versd ; and in this case the force of recoil on the battery which

fires will no longer be balanced by the stopping of the shot on the other

battery. So that supposing the directions of firing to be similar, that

battery which fires the hardest will be subject to the greatest tendency to

move back.

The battery which fires the hardest corresponds with the hottest plate ;

and hence we perceive by analogy that, if of limited extent, the hottest

plate will experience the greatest pressure from the gas between the plates.

33] IN THE GASEOUS STATE. 307

56. The analogy between the batteries and the plates is rendered more

strict if we suppose the batteries H and C to be two limited batteries, each

placed in front of a battery of unlimited extent, and that these unlimited

batteries are pounding away in an exactly similar manner.

The effect of the shot from these unlimited batteries on H and G will be

analogous to the effect of the gas outside and beyond the plates. And it is

at once seen that these unlimited batteries will produce similar effects on H

and G respectively, and that the effect of the firing between H and G will

be uninfluenced by the batteries behind, and therefore, as before, that battery

will be subject to the greatest tendency to move back which fires the hardest.

To make the analogy between the two cases complete, suppose that H

and G, in addition to pounding away at each other, are exactly returning the

fire of the batteries from behind, and that the mean rate at which H fires at

G and G at H are exactly the same as the rate at which the other firing goes

on, but that the velocity of the shot from H is just as much greater than

the mean velocity, as the velocity of the shot from G is below the mean.

Then it is at once seen that the total tendency on H is to move back, while

the total tendency on G is to move forward.

It obviously follows from the foregoing that the inequality in the forces

on H and C could only occur at a certain distance from their ends, which

distance would depend on the distance between the batteries ; and hence

that the ratio which this inequality (due to any particular rate of firing)

would bear to the whole reaction on either battery would increase as the

length of the batteries diminished ; or in other words, the inequality of force

would be proportional to the distance between the batteries, and would be

constant whatever might be the length of the batteries beyond a certain

point.

At first sight it may appear that the distance between the batteries H

and G should be analogous to the distance between the hot and cold plates ;

but it is necessary to remember that it is only in case of the gas being

extremely rare, as compared with the distance between the plates, that the

molecules can be supposed to go straight from the one plate to the other.

In ordinary cases the molecules encounter other molecules, and the effect of

such encounters is to reduce the motion to a mean. Hence it appears that

the distance between the batteries as affecting the equality in the reactions

is somewhat analogous to the distance which a molecule may be supposed to

travel without losing its characteristic motion. And hence it would appear

that in the case of gas the inequalities of force on the two plates would be

proportional to the inverse density of the gas and the extent of the boundaries

of plates.

57. The shot from H which miss G, and those from G which miss //,

202

308 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

must be stopped by the outside batteries. Therefore the inequalities in the

forces on H and C will be balanced by inequalities in the forces on the

batteries behind, and the sum of the forces on // and the battery behind will

be equal to the sum of the forces on C and the battery behind.

And this is strictly analogous to the result of Schuster's experiment, viz. :

that the effect upon the vanes of the light mill is exactly balanced by the

effect on the containing vessel.

58. The batteries also serve to illustrate the action of thermal tran-

spiration. In the case already considered (Art. 57) the inequality between

the shot from H which miss C and those from C which miss H is transferred

to the outside batteries, or in the case of the gas. to the containing vessel.

The better to illustrate the present point, suppose that the outside batteries

are ranged across the ends of the open space between H and C, This will

make no difference to the result. The inequality of the action of the shot

which miss H and C must now cause a force parallel to the end batteries,

tending to cause these batteries to move end-wise in the direction of C.

Suppose that the two batteries H and C were free to move together in

the direction from C vo H (suppose them on a truck). The inequality in the

force would set them in motion in this direction, which motion would increase

until the actual velocity of the shot from C equalled the actual velocity of

the shot from H; then all inequalities in the reactions would cease, and there

would be no reactions on the limiting batteries.

In this case the limiting batteries are obviously analogous to the sides of

a tube, and the interval between the planes H and C corresponds with a layer

of gas at equal pressures, but across which the heat is being conducted by the

greater velocity of the molecules which move from H to C ; and the con-

clusion is that such a layer of gas when maintained at rest exerts a tangential

force on the sides of the tube tending to move the tube in the direction of

the flow of heat, whereas if the gas were free to move it would flow towards

the hottest end ; arid this is the phenomenon of thermal transpiration.

59. The foregoing illustration, with the exception that the action is con-

fined to a plane instead of being distributed through a space, is more than

analogous : it is strictly parallel to the case of gas as long as the gas is so

rare that the molecules proceed straight across the intervals between the

plates or sides of a tube. When this is the case, therefore, the example of

the batteries explains the phenomena of thermal transpiration as well as the

phenomena of the radiometer. But when the gas is so dense that in crossing

the interval between the surfaces the molecules undergo several encounters,

the parallelism no longer holds. Even then, however, the analogy holds, for

33] IN THE GASEOUS STATE. 309

the gas at any point may be considered as consisting of two sets of molecules

which are moving across a plane from opposite sides. And by examining the

difference in the velocity of these two sets of molecules a general explanation

of many of the phenomena may be obtained without recourse being had to a

strict analytical investigation. The analogy has, however, been pursued far

enough to serve the purpose of an introduction.

Before proceeding to the mathematical investigation, which is novel and

somewhat intricate, I have thought it advisable to further introduce it by a

short description of the method used and the assumptions involved.

Prefatory description of the mathematical method.

60. The characteristic as well as the novelty of this investigation consists

in the method by which not only the mean of the motions of the molecules

at the point under consideration is taken into account, but also the manner

in which this mean motion may vary from point to point in any direction

across the point under consideration. It appears that such a variation gives

rise to certain stresses in the gas (tangential and normal), and it is of these

stresses that the phenomena of transpiration and impulsion afford evidence.

Instead of considering only the condition of the molecules comprised

within an elementary unit of volume of the gas, what is chiefly considered in

this investigation is the condition of the molecules which cross a plane sup-

posed to be drawn through the point, which plane may or may not be in

motion along its normal.

The molecules which cross this plane are considered as consisting of two

groups, one crossing from the positive to the negative side of the plane, and

the other crossing from the negative to the positive side. Considered in

opposite directions, the mean characteristics (the number, mass, velocity,

momentum, energy, &c.) of these two groups are not necessarily equal : they

may differ in consequence of the motion of the gas, the motion of the plane

through the gas, or a varying condition of the gas. And the determination

of the effects of these causes on the mass, momentum, and energy that may

be carried across by either group is the more general result of the in-

vestigation.

61. As a preliminary step, it is shown that whatever may be the nature

of the encounters between the molecules within a small element, the

encounters can produce no change on the mean component velocities of the

molecules which in a definite time pass through the element; and hence,

whatever may be the state towards which the encounters tend to reduce the

gas, this state must be such that the mean component velocities of the

310 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

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

These mean component velocities, it is to be noticed, are not the mean

component velocities of the molecules within an element at any instant.

Certain assumptions are then made. These do not involve any law of

action between the molecules. They are equivalent to assuming that the

tendency of the encounters within an element is to reduce the gas to a

uniform state.

From these assumptions two theorems (I. and II.) are deduced. From

theorem I. it follows that the rate of approximation to a uniform gas is

inversely proportional to a certain distance s, which distance is inversely

proportional to the density, and is some unknown function of the mean

velocity of the molecules. From theorem II. it follows that the molecules

which enter a small element from any particular direction, arrive as if from

the uniform gas, to which the actual gas tends at a point distant s in the

direction from which the molecules come.

When the gas is continuous about the element for distances large com-

pared with s, then s is independent of the direction from which the molecules

come ; but near a solid surface s is a function of this direction and of the

position of the element with respect to the solid surface.

These theorems are fundamental to all the reasoning which follows ; and

the distance s enters as a quantity of primary importance into all the results

obtained.

It is proposed to call this distance the mean range of the characteristics

of the molecules. Thus we have the mean range of the mass, the mean range

of momentum, and the mean range of energy. By qualifying the term

" mean range " by the name of the quantity carried, instead of considering

it as a general characteristic of the condition of the gas, two things are

avoided

(1) It is not implied that the mean range is the same for all the

quantities which may be considered;

(2) There is no fear of confusing the mean range with the mean path of

a molecule.

The mean range, whatever may be the nature of the quantity considered,

is obviously a function of the mean path of the molecules, and is a small

quantity of the same order as the mean path, but it also depends on the

nature of the impacts between the molecules.

The symbol s is used to express the mean range of any particular

quantity Q.

33] IN THE GASEOUS STATE. 311

62. Assuming that the mean value of Q for the molecules in an elementary

unit of volume at a point is a function of the position of the point, the

aggregate value of Q carried across the plane at a point is obtained in a series

of ascending powers of s. And by neglecting the terms which involve the

higher powers of s, which terms also involve differentials of Q of orders and

degrees higher than the first, equations are obtained between s and the

aggregate value of Q carried across the plane.

63. The dynamical conditions of steady momentum, steady density, and

steady pressure are next considered. General equations are obtained for

these conditions, which general equations involve s, the motion of the plane

and other quantities depending on the condition of the gas.

The condition that there may be no tangential stress in the gas is also

considered.

It is found that when there is no tangential stress on a solid surface

wherever it may be in the gas, the mean component velocities of all the

molecules which pass through the element in a definite time must be zero at

all points in the gas.

64. The equations of motion are then applied to the particular cases

which it is the object of this investigation to explain. Two cases are

considered. The first, that of a gas in which the temperature and pressure

only vary along one particular direction, so that the isothermal surfaces and

surfaces of equal pressure are parallel planes ; this is the case of transpiration.

The second case is that in which the isothermal surfaces and the surfaces of

equal pressure are curved surfaces (whether of single or double curvature) ;

this is the case of impulsion and the radiometer.

As regards the first case, the condition of steady pressure proves to be of

no importance ; but from the conditions of steady momentum and steady

density an equation is obtained between the velocity of the gas, the rate at

which the temperature varies, and the rate at which the pressure varies ; the

coefficients being functions of the absolute temperature of the gas, the

diameters of the apertures, and the ratio of the diameters of the apertures to

the mean range. These coefficients are determined in the limiting conditions

of the gas, when the density is small and large, and as they vary continuously

with the condition of the gas, the limiting values afford indications of what

must be the intermediate values.

From this equation, which is the general equation of transpiration, the

experimental results, both as regards thermal transpiration and transpiration

under pressure, are deduced.

In dealing with the second case, that in which the isothermal surfaces

312 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

are curved, the three conditions steady momentum, density, and pressure

are all of them important. These conditions reduced to an equation between

the motion of the gas, the variation in the absolute temperature, and the

variation in pressure, in which, as in the equation of transpiration, the

coefficients are functions of .the absolute temperature, the diameters of the

apertures, and the ratios of the diameters of the apertures to the mean

range.

The reduction of the conditions of equilibrium to this equation, however,

involves the assumption that the gas should not be extremely rarefied. In

order to take this case into account a particular example is examined, and

the equation so obtained, together with the equation obtained from the

conditions of steady motion, is shown to lead to the results of impulsion and

the phenomena of the radiometer.

SECTION VI. NOTATION AND PRELIMINARY EXPRESSIONS.

65. In arranging the notation I have endeavoured as far as possible to

make it similar to the notation already adapted to the kinetic theory of gases

by previous writers. With this object I have adopted almost entirely, both

as regards symbols and expressions, the notation used by Professor Maxwell

in his paper "On the Dynamical Theory of Gases*." But his notation,

copious as it is, has fallen far short of my requirements. I have had to take

under consideration certain quantities which have not hitherto been

recognised ; and what has particularly taxed my resources in symbolising,

is that I have had, according to my method, to devise symbols to express

each of twenty-four partial or component quantities which spring from any

one of certain quantities, which have hitherto been dealt with as simple

quantities.

Explanation of the symbols.

66. u, v, w, are used to represent the component velocities of a molecule

with reference to the fixed axes x, y, z.

%, i), are used to represent the component velocities of a molecule with

reference to axes parallel to x, y, z, but which move with the halves of the

mean component velocities of the molecules which pass through an element

in a definite time.

U, V, W are used to represent the component velocities of the moving

axes.

* Phil. Trans. 1867.

From the fact that the fibre of silk had shown positive motion so nearly

up to the pressure of the atmosphere it might have been anticipated that

the spider line, on account of its much greater thinness, would have shown

positive motion even at pressures considerably above that of the atmosphere.

But the reasoning of Art. 46 respecting the differences of temperature to be

maintained and the effect of the air currents, obviously applies with greater

force to the spider line than to the fibre of silk, and at once accounts for the

observed fact that the positive motion with the spider line was not obtained

until the pressures were somewhat lower than those necessary for the fibre

of silk.

304 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

52. Both with the fibre of silk and the spider line, the phenomena of

impulsion (the excess of pressure against warm surfaces) were apparent and

consistent at densities many hundred times greater than the highest densities

at which like results are obtained with vanes several hundred times broader

than the fibre of silk ; this verifies the theoretical conclusion on which this

part of the investigation was based. The results in this case are not so

definite as is the agreement of the logarithmic homologues in the instances

of transpiration ; but the one fact supports the other, and we may consider

the law of impulsion Law VIII., Art. 9 to have been sufficiently proved.

This concludes the experimental investigation.

PART II. (THEOEETICAL).

SECTION V. INTRODUCTION TO THE THEORY.

53. In suggesting in a former paper that the results discovered by

Mr Crookes were due to the communication of heat from the surface of the

solid bodies to the gas surrounding them, I pointed out as the fundamental

fact on which I based my explanation, that when heat is communicated from

a solid surface to a gas, the mean velocity of the molecules which rebound

from the surface must be greater as they rebound than as they approach, and

hence the momentum which these particular molecules communicate to the

surface must be greater than it would be if the surface were at the same

temperature as the gas.

So far the reasoning is incontrovertible. But in order to explain the ex-

perimental results, it was necessary to assume that the number of cold

molecules which approached the hot surface would be the same as if the

surface were at the same temperature as the gas, or at any rate, if reduced,

the number would not be sufficiently reduced to counteract the effect of

increased velocity of rebound.

Although at that time I could not see any definite proof of this, nor any

way of definitely examining the question, yet I had a strong impression that

the assumption was legitimate ; and although I hoped at some future time

to be able to complete the theoretical explanation, I was content for the time

to rest the evidence of the truth of the assumptions involved on the adequacy

of the reasoning to explain the experimental results obtained.

As other suggestions respecting the cause of the phenomena, widely

different in character from mine, had found supporters, and a good deal of

scepticism was expressed as to the fitness of the cause which I had suggested,

my attention was occupied in deducing the actions which must result from

such a force, and comparing them with experimental ^results. Having, how-

ever, at length satisfied myself, and seeing that a conviction was spreading

33] IN THE GASEOUS STATE. 305

that what I suggested contained the germ of the explanation, I set to work

in earnest to complete the explanation, and ascertain by an extension of the

dynamical theory of gases what effect the hot molecules receding from

the surface should produce on the number arid temperature of those

approaching.

My first attempts to accomplish this were altogether unsuccessful. When

contemplating the phenomena it seemed to me that I could perceive a

glimmering of the method of reasoning for which I was in search, but as

soon as ever I attempted to give definite expression to it this glimmering

vanished.

The reason for this I now perceive clearly. When contemplating the

phenomena, I had a surface of limited extent before me, and I considered the

effect on such a surface without recognising the fundamental importance of

the limit to size.

On the other hand, when I came to definite reasoning, for the sake of

what appeared to be a simplification of the conditions of the problem, I

assumed the surface to be without limit, thus introducing a fundamental

alteration into the conditions of the problem without perceiving it.

The importance of this limit only became apparent to me when I found,

by simple dynamical reasoning, that with surfaces of unlimited extent such

results as those actually obtained would be impossible. This appeared as

follows :

No force on unlimited surface.

54. If we had two plane plates of unlimited extent, H and C, the surface

of H opposite to C being hotter than the surface of C which was opposite to

H, the outside surfaces of both plates being at the same temperature, then

in order to produce results similar to those obtained with limited plates, the

gas between the two plates must maintain a greater steady pressure on the

plate H, than that which it exerts on the colder plate C. Whereas it is at

once obvious that such a condition is contrary to the laws of motion, which

require that the gas between the two surfaces should exert an equal and

opposite pressure on both surfaces.

Having once perceived the force of this reasoning, it became clear to me

that if, as I had supposed, the results obtained in the experiments were due

to gaseous pressure, then they must depend on the limited extent of the

surfaces.

This gave me the clue, in following which I have not only had the satis-

faction of finding the explanation complete as regards the phenomena from

o. R. 20

306 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

which it originated, but I have also found that the theory indicated the

phenomena of thermal transpiration, and explains much that hitherto has

been considered anomalous respecting the laws of transpiration of gases

through small channels suggesting the experiments by which might be

established the relation between these actions.

The manner in which the force arises in the case of a limited surface was

at first rendered much clearer to me by considering an illustration, which

I introduce here, although it forms no part of the proof which will follow.

55. Instead of H and C being plates with gas between them, let them

be earthen batteries of unlimited length, and suppose that guns are distri-

buted at uniform intervals along those batteries ; suppose, also, that all the

shot fired from H bury themselves in the earth of C, and vice versd.

Then, in the first place, it is obvious that since on firing a shot the

momentum imparted to the gun is equal and opposite to the momentum given

to the shot, every shot fired from H will exercise the same force to move the

battery H away from C as the shot will exercise to move C away from H ;

and in the same way the recoil of the guns on C will exercise the same

tendency to move C away from H as the shot will exercise to move H away

from G. And this will be the case whether the guns are supposed to be

pointed straight across the interval between the batteries, or, as / shall

suppose, are pointed with various degrees of obliquity.

Since, then, the result of every shot, whether fired from H or G, causes

equal and opposite forces on the two batteries, the result of all the firing, no

matter how much harder one battery may bombard than the other, must be

to cause an equal force on each battery, the batteries being of unlimited

length.

This case will be seen to be strictly analogous to the effect of the gas

between two plates of unlimited extent to cause equal pressures on the

plates, no matter what may be the differences in the temperature of the

plates.

If now we consider the batteries of limited extent, then, owing to the

obliquity of the guns, some of the shot from H may pass beyond the ends

of C, and vice versd ; and in this case the force of recoil on the battery which

fires will no longer be balanced by the stopping of the shot on the other

battery. So that supposing the directions of firing to be similar, that

battery which fires the hardest will be subject to the greatest tendency to

move back.

The battery which fires the hardest corresponds with the hottest plate ;

and hence we perceive by analogy that, if of limited extent, the hottest

plate will experience the greatest pressure from the gas between the plates.

33] IN THE GASEOUS STATE. 307

56. The analogy between the batteries and the plates is rendered more

strict if we suppose the batteries H and C to be two limited batteries, each

placed in front of a battery of unlimited extent, and that these unlimited

batteries are pounding away in an exactly similar manner.

The effect of the shot from these unlimited batteries on H and G will be

analogous to the effect of the gas outside and beyond the plates. And it is

at once seen that these unlimited batteries will produce similar effects on H

and G respectively, and that the effect of the firing between H and G will

be uninfluenced by the batteries behind, and therefore, as before, that battery

will be subject to the greatest tendency to move back which fires the hardest.

To make the analogy between the two cases complete, suppose that H

and G, in addition to pounding away at each other, are exactly returning the

fire of the batteries from behind, and that the mean rate at which H fires at

G and G at H are exactly the same as the rate at which the other firing goes

on, but that the velocity of the shot from H is just as much greater than

the mean velocity, as the velocity of the shot from G is below the mean.

Then it is at once seen that the total tendency on H is to move back, while

the total tendency on G is to move forward.

It obviously follows from the foregoing that the inequality in the forces

on H and C could only occur at a certain distance from their ends, which

distance would depend on the distance between the batteries ; and hence

that the ratio which this inequality (due to any particular rate of firing)

would bear to the whole reaction on either battery would increase as the

length of the batteries diminished ; or in other words, the inequality of force

would be proportional to the distance between the batteries, and would be

constant whatever might be the length of the batteries beyond a certain

point.

At first sight it may appear that the distance between the batteries H

and G should be analogous to the distance between the hot and cold plates ;

but it is necessary to remember that it is only in case of the gas being

extremely rare, as compared with the distance between the plates, that the

molecules can be supposed to go straight from the one plate to the other.

In ordinary cases the molecules encounter other molecules, and the effect of

such encounters is to reduce the motion to a mean. Hence it appears that

the distance between the batteries as affecting the equality in the reactions

is somewhat analogous to the distance which a molecule may be supposed to

travel without losing its characteristic motion. And hence it would appear

that in the case of gas the inequalities of force on the two plates would be

proportional to the inverse density of the gas and the extent of the boundaries

of plates.

57. The shot from H which miss G, and those from G which miss //,

202

308 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

must be stopped by the outside batteries. Therefore the inequalities in the

forces on H and C will be balanced by inequalities in the forces on the

batteries behind, and the sum of the forces on // and the battery behind will

be equal to the sum of the forces on C and the battery behind.

And this is strictly analogous to the result of Schuster's experiment, viz. :

that the effect upon the vanes of the light mill is exactly balanced by the

effect on the containing vessel.

58. The batteries also serve to illustrate the action of thermal tran-

spiration. In the case already considered (Art. 57) the inequality between

the shot from H which miss C and those from C which miss H is transferred

to the outside batteries, or in the case of the gas. to the containing vessel.

The better to illustrate the present point, suppose that the outside batteries

are ranged across the ends of the open space between H and C, This will

make no difference to the result. The inequality of the action of the shot

which miss H and C must now cause a force parallel to the end batteries,

tending to cause these batteries to move end-wise in the direction of C.

Suppose that the two batteries H and C were free to move together in

the direction from C vo H (suppose them on a truck). The inequality in the

force would set them in motion in this direction, which motion would increase

until the actual velocity of the shot from C equalled the actual velocity of

the shot from H; then all inequalities in the reactions would cease, and there

would be no reactions on the limiting batteries.

In this case the limiting batteries are obviously analogous to the sides of

a tube, and the interval between the planes H and C corresponds with a layer

of gas at equal pressures, but across which the heat is being conducted by the

greater velocity of the molecules which move from H to C ; and the con-

clusion is that such a layer of gas when maintained at rest exerts a tangential

force on the sides of the tube tending to move the tube in the direction of

the flow of heat, whereas if the gas were free to move it would flow towards

the hottest end ; arid this is the phenomenon of thermal transpiration.

59. The foregoing illustration, with the exception that the action is con-

fined to a plane instead of being distributed through a space, is more than

analogous : it is strictly parallel to the case of gas as long as the gas is so

rare that the molecules proceed straight across the intervals between the

plates or sides of a tube. When this is the case, therefore, the example of

the batteries explains the phenomena of thermal transpiration as well as the

phenomena of the radiometer. But when the gas is so dense that in crossing

the interval between the surfaces the molecules undergo several encounters,

the parallelism no longer holds. Even then, however, the analogy holds, for

33] IN THE GASEOUS STATE. 309

the gas at any point may be considered as consisting of two sets of molecules

which are moving across a plane from opposite sides. And by examining the

difference in the velocity of these two sets of molecules a general explanation

of many of the phenomena may be obtained without recourse being had to a

strict analytical investigation. The analogy has, however, been pursued far

enough to serve the purpose of an introduction.

Before proceeding to the mathematical investigation, which is novel and

somewhat intricate, I have thought it advisable to further introduce it by a

short description of the method used and the assumptions involved.

Prefatory description of the mathematical method.

60. The characteristic as well as the novelty of this investigation consists

in the method by which not only the mean of the motions of the molecules

at the point under consideration is taken into account, but also the manner

in which this mean motion may vary from point to point in any direction

across the point under consideration. It appears that such a variation gives

rise to certain stresses in the gas (tangential and normal), and it is of these

stresses that the phenomena of transpiration and impulsion afford evidence.

Instead of considering only the condition of the molecules comprised

within an elementary unit of volume of the gas, what is chiefly considered in

this investigation is the condition of the molecules which cross a plane sup-

posed to be drawn through the point, which plane may or may not be in

motion along its normal.

The molecules which cross this plane are considered as consisting of two

groups, one crossing from the positive to the negative side of the plane, and

the other crossing from the negative to the positive side. Considered in

opposite directions, the mean characteristics (the number, mass, velocity,

momentum, energy, &c.) of these two groups are not necessarily equal : they

may differ in consequence of the motion of the gas, the motion of the plane

through the gas, or a varying condition of the gas. And the determination

of the effects of these causes on the mass, momentum, and energy that may

be carried across by either group is the more general result of the in-

vestigation.

61. As a preliminary step, it is shown that whatever may be the nature

of the encounters between the molecules within a small element, the

encounters can produce no change on the mean component velocities of the

molecules which in a definite time pass through the element; and hence,

whatever may be the state towards which the encounters tend to reduce the

gas, this state must be such that the mean component velocities of the

310 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

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

These mean component velocities, it is to be noticed, are not the mean

component velocities of the molecules within an element at any instant.

Certain assumptions are then made. These do not involve any law of

action between the molecules. They are equivalent to assuming that the

tendency of the encounters within an element is to reduce the gas to a

uniform state.

From these assumptions two theorems (I. and II.) are deduced. From

theorem I. it follows that the rate of approximation to a uniform gas is

inversely proportional to a certain distance s, which distance is inversely

proportional to the density, and is some unknown function of the mean

velocity of the molecules. From theorem II. it follows that the molecules

which enter a small element from any particular direction, arrive as if from

the uniform gas, to which the actual gas tends at a point distant s in the

direction from which the molecules come.

When the gas is continuous about the element for distances large com-

pared with s, then s is independent of the direction from which the molecules

come ; but near a solid surface s is a function of this direction and of the

position of the element with respect to the solid surface.

These theorems are fundamental to all the reasoning which follows ; and

the distance s enters as a quantity of primary importance into all the results

obtained.

It is proposed to call this distance the mean range of the characteristics

of the molecules. Thus we have the mean range of the mass, the mean range

of momentum, and the mean range of energy. By qualifying the term

" mean range " by the name of the quantity carried, instead of considering

it as a general characteristic of the condition of the gas, two things are

avoided

(1) It is not implied that the mean range is the same for all the

quantities which may be considered;

(2) There is no fear of confusing the mean range with the mean path of

a molecule.

The mean range, whatever may be the nature of the quantity considered,

is obviously a function of the mean path of the molecules, and is a small

quantity of the same order as the mean path, but it also depends on the

nature of the impacts between the molecules.

The symbol s is used to express the mean range of any particular

quantity Q.

33] IN THE GASEOUS STATE. 311

62. Assuming that the mean value of Q for the molecules in an elementary

unit of volume at a point is a function of the position of the point, the

aggregate value of Q carried across the plane at a point is obtained in a series

of ascending powers of s. And by neglecting the terms which involve the

higher powers of s, which terms also involve differentials of Q of orders and

degrees higher than the first, equations are obtained between s and the

aggregate value of Q carried across the plane.

63. The dynamical conditions of steady momentum, steady density, and

steady pressure are next considered. General equations are obtained for

these conditions, which general equations involve s, the motion of the plane

and other quantities depending on the condition of the gas.

The condition that there may be no tangential stress in the gas is also

considered.

It is found that when there is no tangential stress on a solid surface

wherever it may be in the gas, the mean component velocities of all the

molecules which pass through the element in a definite time must be zero at

all points in the gas.

64. The equations of motion are then applied to the particular cases

which it is the object of this investigation to explain. Two cases are

considered. The first, that of a gas in which the temperature and pressure

only vary along one particular direction, so that the isothermal surfaces and

surfaces of equal pressure are parallel planes ; this is the case of transpiration.

The second case is that in which the isothermal surfaces and the surfaces of

equal pressure are curved surfaces (whether of single or double curvature) ;

this is the case of impulsion and the radiometer.

As regards the first case, the condition of steady pressure proves to be of

no importance ; but from the conditions of steady momentum and steady

density an equation is obtained between the velocity of the gas, the rate at

which the temperature varies, and the rate at which the pressure varies ; the

coefficients being functions of the absolute temperature of the gas, the

diameters of the apertures, and the ratio of the diameters of the apertures to

the mean range. These coefficients are determined in the limiting conditions

of the gas, when the density is small and large, and as they vary continuously

with the condition of the gas, the limiting values afford indications of what

must be the intermediate values.

From this equation, which is the general equation of transpiration, the

experimental results, both as regards thermal transpiration and transpiration

under pressure, are deduced.

In dealing with the second case, that in which the isothermal surfaces

312 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

are curved, the three conditions steady momentum, density, and pressure

are all of them important. These conditions reduced to an equation between

the motion of the gas, the variation in the absolute temperature, and the

variation in pressure, in which, as in the equation of transpiration, the

coefficients are functions of .the absolute temperature, the diameters of the

apertures, and the ratios of the diameters of the apertures to the mean

range.

The reduction of the conditions of equilibrium to this equation, however,

involves the assumption that the gas should not be extremely rarefied. In

order to take this case into account a particular example is examined, and

the equation so obtained, together with the equation obtained from the

conditions of steady motion, is shown to lead to the results of impulsion and

the phenomena of the radiometer.

SECTION VI. NOTATION AND PRELIMINARY EXPRESSIONS.

65. In arranging the notation I have endeavoured as far as possible to

make it similar to the notation already adapted to the kinetic theory of gases

by previous writers. With this object I have adopted almost entirely, both

as regards symbols and expressions, the notation used by Professor Maxwell

in his paper "On the Dynamical Theory of Gases*." But his notation,

copious as it is, has fallen far short of my requirements. I have had to take

under consideration certain quantities which have not hitherto been

recognised ; and what has particularly taxed my resources in symbolising,

is that I have had, according to my method, to devise symbols to express

each of twenty-four partial or component quantities which spring from any

one of certain quantities, which have hitherto been dealt with as simple

quantities.

Explanation of the symbols.

66. u, v, w, are used to represent the component velocities of a molecule

with reference to the fixed axes x, y, z.

%, i), are used to represent the component velocities of a molecule with

reference to axes parallel to x, y, z, but which move with the halves of the

mean component velocities of the molecules which pass through an element

in a definite time.

U, V, W are used to represent the component velocities of the moving

axes.

* Phil. Trans. 1867.

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