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uniformly with the angular velocity (o, round S, the fixed attracting point. Let x, y, be
the co-ordinates of G with reference to these axes, and let XS, YS denote the components



ON THE STABILITY OF THE MOTION OF SATURN's RINGS. 375

of the whole force of attraction of S on the rigid body. Then since this force is in the
line through S, its moment round G is

SYx-SXy;
the components of the forces on the moving body being reckoned as positive when they
tend to diminish x and y respectively. Hence if k denote the radius of gyration of the
body round G, and if <f> denote the angle which OG makes with SX {i.e. the angle GOK),
the equations of motion are,

In the first place we see that one integral of these equations is

This is the "equation of angidar momentum."

In considering whether the motion round S with velocity co when coincides with
-S' is stable or unstable, we must find whether every possible motion with the same
" angular momentum " round S is such that it will never bring to more than an infinitely
small distance from S : that is to say, we must find whether, for every possible solution
in which H = M {ct" + k"") o), and for which the co-ordinates of are infinitely small at one
time, these co-ordinates remain infinitely small. Let these values at time t be denoted
thus: 8^ = ^, and NO='rj; let OG be at first infinitely nearly parallel to OX, i.e. let <f>
be infinitely small (the full solution will tell us whether or not <f) remains infinitely small) ;
then, as long as <f) is infinitely small, we have

x = a+ ^, y = v + ^<^>
and the equations of motion have the forms

31






and we may write the equation of angular momentum instead of the third equation.

If now we suppose f and rj to be infinitely small, the last of these equations becomes
{a' + k^)f^+2a>a^+af^=0 (a).



376 ON THE STABILITY OF THE MOTION OF SATUKN S RINGS.

If p and q denote the components parallel and perpendicular to OG of the attraction
of the body on a unit of matter at S, we have

X = pco?,^-q?,m4> = p, and F=psin^ + 5^003 ^=j3</> 4-^,
since q and ^ are each infinitely small ; and if we put V= potential at S, and

then p =/- a| - 777, q = -0v- 7^.

If we make these substitutions for X and Y, and take into account that

.f=co'a^ (*).

the first and second equations of motion become

g_2.^_„.f_2„af4(.f+„)=0 (0),

A,2„|_„., + „^4(^,+,« = W.

Combining equations (a), (c), and (tf), by the same method as that adopted in the text,
we find that the differential equation in ^, 7), or </>, is of the form



d*u ^d^u ^



where A = A;',



C = a>* (A;* - 3a*) + «" -^ {{a* + ^*) (a + yS) - 4a»y8} + {a' + Fj^^, (a'yS - 7).

In comparing this result with that obtained in the Essay, we must put

r^ for a,

R for M,

B+S for S,

L for o,

Nt: for y8,

Mr^ for 7.



Tv^ 7



Fi^ Z



VOL. L PLATE V.




Tig. ^.



Fi^ 6.



VOL. L PLATE V,




[From the Philosophical Magazine for January and July, I860.]



XX. Illustrations of the Dynamical Theory of Gases*.

PART L

On the Motions and Collisions of Perfectly Elastic Spheres.

So many of the properties of matter, especially when in the gaseous form,
can be deduced from the hypothesis that their minute parts are in rapid motion,
the velocity increasing with the temperature, that the precise nature of this
motion becomes a subject of rational curiosity. Daniel Bemouilli, Herapath,
Joule, Kronig, Clausius, &c. have shewn that the relations between pressure,
temperature, and density in a perfect gas can be explained by supposing the
particles to move with uniform velocity in straight lines, striking against the
sides of the containing vessel and thus producing pressure. It is not necessary
to suppose each particle to travel to any great distance in the same straight
line ; for the effect in producing pressure \vill be the same if the particles
strike against each other ; so that the straight line described may be very short .
M. Clausius has determined the mean length of path in terms of the average
distance of the particles, and the distance between the centres of two particles
when collision takes place. We have at present no means of ascertaining either
of these distances ; but certain phenomena, such as the internal friction of gases,
the conduction of heat through a gas, and the diffusion of one gas through
another, seem to indicate the possibility of determining accurately the mean
length of path which a particle describes between two successive collisions. In
order to lay the foundation of such investigations on strict mechanical principles,
I shall demonstrate the laws of motion of an indefinite number of small, hard,
and perfectly elastic spheres acting on one another only during impact.

* Read at the Meeting of the British Association at Aberdeen, Sei)tember 21, 1859.
VOL. I. 48



378 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

If the properties of such a system of bodies are found to correspond to
those of gases, an important physical analogy will be established, which may
lead to more accurate knowledge of the properties of matter. If experiments
on gases are inconsistent with the hypothesis of these propositions, then our
theory, though consistent w^th itself, is proved to be incapable of explaining
the phenomena of gases. In either case it is necessary to follow out the
consequences of the hypothesis.

Instead of saying that the particles are hard, spherical, and elastic, we may
if we please say that the particles are centres of force, of which the action is
insensible except at a certain small distance, when it suddenly appears as a
repulsive force of very great intensity. It is evident that either assumption
will lead to the same results. For the sake of avoiding the repetition of a
long phrase about these repulsive forces, I shall proceed upon the assumption
of perfectly elastic spherical bodies. If we suppose those aggregate molecules
which move together to have a bounding surface which is not spherical, then
the rotatory motion of the system will store up a certain proportion of the
whole vis viva, as has been shewn by Clausius, and in this way we may
accoimt for the value of the specific heat being greater than on the more
simple hypothesis.

On the Motion and Collision of Perfectly Elastic Spheres.

Prop. I. Two spheres moving in opposite directions with velocities* inversely
us their masses strike one another; to determine their motions after impact.

Let P and Q be the position of the centres at
impact; AP, BQ the directions and magnitudes of ^-V at

the velocities before impact; Pa, Qh the same after ^^^^^^^^ — j^

impact; then, resolving the velocities parallel and per- ^
pendicular to PQ the line of centres, we find that
tlie velocities parallel to the line of centres are exactly
reversed, while those perpendicular to that line are
luichanged. Compounding these velocities again, we find that the velocity of
each ball is the same before and after impact, and that the directions before
and after impact lie in the same plane with the line of centres, and make equal
angles with it.




ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 379

Prop. 11. To find the probability of the direction of the velocity after
impact lying between given limits.

In order that a collision may take place, the line of motion of one of the
balls must pass the centre of the other at a distance less than the sum of
their radii ; that is, it must pass through a circle whose centre is that of the
other ball, and radius (s) the sum of the radii of the balls. Within this circle
every position is equally probable, and therefore the probability of the distance
from the centre being between r and r + dr is

2rdr
~7~'

Now let <f> be the angle A Pa between the original direction and the directioii
after impact, then APN=^<f>, and 7- = 5 sin ^<^, and the probabihty becomes

^ sin 6d^.
The area of a spherical zone between the angles of polar distance <j> and <f) + d<f) is

27r sin (f)d<f> ;

therefore if a> be any small area on the surface of a sphere, radius unity, the
probability of the direction of rebound passing through this area is

to

4:ir *

so that the probability is independent of ^, that is, all directions of rebound
are equally likely.

Prop. III. Given the direction and magnitude of the velocities of two
spheres before impact, and the line of centres at impact ; to find the velocities
after impact.

Let OA, OB represent the velocities before impact, so that if there had been
no action between the bodies they would
have been at A and B at the end of a
second. Join AB, and let G be their centre
of gravity, the position of which is not
affected by their mutual action. Draw GN
parallel to the line of centres at impact (not
necessarily in the plane AOB). Draw aGh

in the plane AGN, making NGa = NGA, and Ga=GA and Gb = GB; then by

48—2




380 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

Prop. I. Ga and Gh will be the velocities relative to G ; and compounding
these with OG, we have Oa and Oh for the true velocities after impact.

By Prop. 11. all directions of the Une aGh are equally probable. It appears
therefore that the velocity after impact is compounded of the velocity of the
centre of gravity, and of a velocity equal to the velocity of the sphere relative
to the centre of gravity, which may with equal probability be in any direction
whatever.

If a great many equal spherical particles were in motion in a perfectly
elastic vessel, collisions would take place among the particles, and their velocities
would be altered at every collision; so that after a certain time the vis viva
will be divided among the particles according to some regular law, the average
number of particles whose velocity lies between certain Umits being ascertainable,
though the velocity of each particle changes at every colUsion.

Prop. IV. To find the average number of particles whose velocities he
between given limits, after a great number of collisions among a great number
of equal particles.

Let N be the whole number of particles. Let x, y, z be the components
of the velocity of each particle in three rectangular directions, and let the number
of particles for which x lies between x and x-hdx, be Nf{x)dx, where f{x) is
a function of x to be determined.

The number of particles for which y lies between y and y + dy wUl be
Nf{y)dy; and the number for which z Hes between z and z + dz will be Nf(z)dz,
where / always stands for the same function.

Now the existence of the velocity x does not in any way affect that of
the velocities y or z, since these are all at right angles to each other and
independent, so that the number of particles whose velocity lies between x and
x + dx, and also between y and y-{-dy, and also between z and z + dz, is

If we suppose the N particles to start from the origin at the same instant,
then this wil) be the number in the element of volume (dxdydz) after unit of
time, and the number referred to unit of volume will be



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 381

But the directions of the coordinates are perfectly arbitrary, and therefore this
number must depend on the distance from the origin alone, that is

f{x)f(y)f(z) = ^{^+y' + z%

Solving this functional equation, we find

f{x) = Ce^'', (^M = CV.

If we make A positive, the number of particles will increase with the
velocity, and we should find the whole number of particles infinite. We there-
fore make A negative and equal to — „ , so that the number between x and

x + dx is

NCe'^'dx.

Integrating from a:=— <» toa;=-foo,we find the whole number of particles,

aVTT

1 -?:

f[x) is therefore /-e " .

Whence we may draw the following conclusions : —

1st. The number of particles whose velocity, resolved in a certain direction,
lies between x and x + dx is

N^i'^'dx (1).

2nd. The number whose actual velocity lies between v and v + dv is

]Sf-^^^e~^'dv (2).

3rd. To find the mean value of v, add the velocities of all the particles
together and divide by the number of particles ; the result is

mean velocity = -p- (3).

Vtt

4th. To find the mean value of v; add all the values together and

divide by N,

mean value of t;' = |a- (4).

This is greater than the square of the mean velocity, as it ought to be.



382 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

It appears from this proposition that the velocities are distributed among
the particles according to the same law as the errors are distributed among
the observations in the theory of the " method of least squares." The velocities
i-ange from to oo , but the number of those having great velocities is com-
paratively small. In addition to these velocities, which are in all directions
equally, there may be a general motion of translation of the entire system of
particles which must be compounded with the motion of the particles relatively
to one another. We may call the one the motion of translation, and the other
the motion of agitation.

Prop. V. Two systems of particles move each according to the law stated
in Prop. IV. ; to find the number of pairs of particles, one of each system,
whose relative velocity lies between given limits.

Let there be N particles of the first system, and N' of the second, then
NN' is the whole number of such pairs. Let us consider the velocities in the
direction of x only ; then by Prop. IV. the number of the first kind, whose
velocities are between x and x + dx, is

1 -^
N — j=e '^ dx.

aV-Tr

The number of the second kind, whose velocity is between x + y and x + y + dy, is

1 (i±vl
N' — 7= e ^ dy,

where fi is the value of a for the second system.

The number of pairs which fulfil both conditions is

NN'^e'^^'^' dxdy.
apir

Now X may have any value from — qo to +cx> consistently with the difference
of velocities being between y and y + dy; therefore integrating between these
limits, we find

^^'7^^^"'^''^ ^'^

for the whole number of pairs whose difference of velocity lies between y and
y + dy.



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. ;}83

This expression, which is of the same form with (1) if we put XN' for
X, a' + ^ for a', and y for x, shews that the distribution of relative velocities
is regulated by the same law as that of the velocities themselves, and that
the mean relative velocity is the square root of the sum of the squares of tlie
mean velocities of the two systems.

Since the direction of motion of every particle in one of the systems may
be reversed without changing the distribution of velocities, it follows that the
velocities compounded of the velocities of two particles, one in each system, .-irr
distributed according to the same formula (5) as the relative velocities.

Prop. VI. Two systems of particles move in the same vessel ; to prove
that the mean vis viva of each particle will become the same in the two
systems.

Let P be the mass of each particle of the first system, Q that of each
particle of the second. Let p, q be the mean veloci-
ties in the two systems before impact, and let p', ((
be the mean velocities after one impact. Let OA = p
and OB = q, and let AOB be a right angle; then, by
Prop, v., AB will be the mean relative velocity, OG will
be the mean velocity of the centre of gravity ; and drawing
aGh at right angles to OG, and making aG = AG and
bG = BG, then Oa will be the mean velocity of P after
impact, compounded of OG and Ga, and Ob will be that of Q after impact.





^~ P+Q '

therefore p' = Oa = ^!^^±^Ipl±^: ,

^ P + Q

and q' = Ob = ^-^M±S±El±W,

P+Q

and Pp"-Qq" = {^)\Pp'-Qq') C^).



384 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

It appears therefore tKat the quantity Pp' — Qq^ is diminished at every impact
in the same ratio, so that after many impacts it will vanish, and then

Now the mean vis viva is f Pa'' = -^ Pp* for P, and ^ Qq^ for Q ; and it is

8 8

manifest that these quantities will be equal when Pp^ = Qq^.

If any number of different kinds of particles, having masses P, Q, R and
velocities jp, q, r respectively, move in the same vessel, then after many impacts

Pf^Q^ = m^, &c (7).

Prop. VII. A particle moves with velocity r relatively to a number of
particles of which there are N in imit of volume ; to find the number of these
which it approaches within a distance 5 in unit of time.

If we describe a tubular surface of which the axis is the path of the
particle, and the radius the distance s, the content of this surface generated
in unit of time will be irrs^, and the number of particles included in it will be

Nirrs' (8),

which is the number of particles to which the moving particle approaches within
a distance s.

Prop. VIII. A particle moves with velocity v in a system moving according
to the law of Prop. IV.; to find the number of particles which have a velocity
relative to the moving particle between r and r + dr.

Let u be the actual velocity of a particle of the system, v that of the
original particle, and r their relative velocity^ and 6 the angle between v and r,
then

u^z=v^ + 7^ — 2vr cos 0.

If we suppose, as in Prop. IV., all the particles to start from the origin, at
once, then after imit of time the "density" or number of particles to unit of
volume at distance u will be

1 -^
aM



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 385

From this we have to deduce the number of particles in a shell whose centre
is at distance v, radius = r, and thickness = dr,

^-n=l{^ *• -« *^ }^^' (9)>

which is the number required.

CoR. It is evident that if we integrate this expression from r = to
/• = oo , we ought to get the whole number of particles = iV, whence the following
mathematical result,

dx.x{e »' —e~ »' ) = V77-aa (lO).

Prop. IX. Two sets of particles move as in Prop. V.; to find the number
of pairs which approach within a distance s in unit of time.

The number of the second kind which have a velocity between v and v + dv ia

4 -^

The number of the first kind whose velocity relative to these is between r
and ri-dr is

iV — = - (e »' -e »* )dr = n,
and the number of pairs which approach within distance 5 in unit of time is

4 t. _ ("-»•)* (o^-r)*

^NN' -^.s'r've ^ {e »' -e" «" \drdv.
By the last proposition we are able to integrate with respect to v, and get

Integrating this again from r = to r = oo ,

2NN' J^ J'^FT^s' (11)

is the number of collisions in unit of time which take place in unit of volume

between particles of difierent kinds, s being the distance of centres at collision.

vol. I. 49



386 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

The number of collisions between two particles of the first kind, 5, being the
striking distance, is

and for the second system it is

The mean velocities in the two systems are -7= and -^ ; so that if l^ and l^

be the mean distances travelled by particles of the first and second systems
between each collision, then



ii a

Prop. X. To find the probability of a particle reaching a given distance
before striking any other.

Let us suppose that the probability of a particle being stopped while
passing through a distance dx, is adx ; that is, if iV particles arrived at a
distance x, Nadx of them would be stopped before getting to a distance x-^dx.
Putting this mathematically,

^=-Na, or N=Ce-'^.

Putting iV"=l when x = 0, we find e""* for the probability of a particle not
striking another before it reaches a distance x.

The mean distance travelled by each particle before striking is - = l. The

probability of a particle reaching a distance = 7i? without being struck is e"".
(See a paper by M. Clausius, Philosophical Magazine, February 1859.)

If all the particles are at rest but one, then the value of a is

a = Trs'N,
where s is the distance between the centres at collision, and N is the number
of particles in unit of volume. If v be the velocity of the moving particle
relatively to the rest, then the number of collisions in unit of time wiU be

virs W :



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 387

and if V, be the actual velocity, then the number will be r,a ; therefore



a = -7rsW,



where v, is the actual velocity of the striking particle, and v its velocity
relatively to those it strikes. If -y, be the actual velocity of the other particles,
then V — Jv* + v*. If i\ = i\ , then v = sl2i\ , and

a = j2TTS*N.
Note*. M. Clausius makes a = ^Trs^N,

Prop. XI. In a mixture of particles of two different kinds, to find the
mean path of each particle.

Let there be iV, of the first, and N^ of the second in unit of volume.
Let Si be the distance of centres for a collision between two particles of the
first set, 5j for the second set, and s for collision between one of each kind.
Let r, and i\ be the coefficients of velocity, M^, M^ the mass of each particle.

The probability of a particle M^ not being struck till after reaching a
distance x, by another particle of the same kind is

* [In the Philosophical Magazine of 1860, Vol I. pp. 434 — 6 Clausius explains the method by
which he found his value of the mean relative velocity. It is briefly as follows: If u, v be the
velocities of two particles their relative velocity is >Ju* + v* - 2uv cos 6 and the mean of this as
regards direction only, all directions of v being equally probable, is shewn to be

1 w* , ^ 1 V* ,

f + o — when u<v, and w + ^ — when u> v.
o V 3 w

If r = M these expressions coincide. Clausius in applying this result and putting u, v for the
mean velocities assumes that the mean relative velocity is given by expressions of the same form,
so that when the mean velocities are each equal to u the mean relative velocity would be ^u.
This step is, however, open to objection, and in fact if we take the expressions given above for the
mean velocity, treating u and v as the velocities of two particles which may have any values between
and 00 , to calculate the mean relative velocity we should proceed as follows : Since the number of

4 _*!

particles with velocities between u and w + rfu is N , , tt*g~«' du, the mean relative velocity is

2

This expression, when reduced, leads to -j= Ja* + /3', which is the result in the text. Ed.]

49—2



388 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES.

The probability of not being struck by a particle of the other kind in the same
distance is

Therefore the probability of not being struck by any particle before reaching a
distance x is

and if k be the mean distance for a particle of the first kind,

\ = j27rs-N, + 7: ^f^.s^N, (12).

Similarly, if k be the mean distance for a particle of the second kind,

l=^/27r5,W, + 7^ /l+^^/W, (13).

The mean density of the particles of the first kind is N,M, = p„ and that of
the second NJiI, = p,. If we put

i =Ap, + Bp,, l = Cp, + Dp, (15),

^^ C-Wr~< ^ ^

Prop. XII. To find the pressure on unit of area of the side of the vessel
due to the impact of the particles upon it.

Let iV= number of particles in unit of volume;
M= mass of each particle ;
V = velocity of each particle ;
I = mean path of each particle ;
then the number of particles in unit of area of a stratum dz thick is

Ndz (17).

The number of colHsions of these particles in unit of time is

Ndz J (18).



ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 389

The number of particles which after collision reach a distance between nl and
(n 4- dn) I is

Njc-^dzdn (19).



The proportion of these which strike on unit of area at distance z is

rd — z



,(20);



2nl

the mean velocity of these in the direction of 2 is

.'4±? (21).

Multiplying together (19), (20), and (21), and M, we find the momentum at
impact

MN-^j,(nn'-z')e-''dzdn.

Integrating with respect to z from to nl, we get

^MNi? nt"" dn.
Integrating with respect to n from to 00 , we get

for the momentum in the direction of z of the striking particles ; for the
momentum of the particles after impact is the same, but in the opposite
direction ; so that the whole pressure on unit of area is twice this quantity, or

This value of _p is independent of I the length of path. In applying this
result to the theory of gases, we put MN=p, and v- = 2>h, and then

which is Boyle and Mariotte's law. By (4) we have

^'^ = |a^ .-. o: = 2k (23).

We have seen that, on the hypothesis of elastic particles moving in straight
lines, the pressure of a gas can be explained by the assumption that the square
of the velocity is proportional directly to the absolute temperature, and inversely
to the specific gravity of the gas at constant temperature, so that at the same



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