May 16, 1913

THU COLLIERY GUARDIAN.

1011

ON THE KINETIC THEORY OF CASES.

[Rights of Reproduction Reserved.~\

By Sir Henry Cunynghame, Chairman of the Royal
Commission upon Accidents in Mines.

(Continued from page 955.)

Section III.

The previous investigation has enabled us to compute
the velocity which molecules of oxygen must have at
0 deg. Cent, and standard pressure of 760 mm. if those
velocities are assumed to be equal to one another. But
a very little consideration will show that even if the
molecules all started with fixed equal velocities, collisions
would soon produce inequalities. Thus, for instance (fig. 3),
if an elastic molecule moving with a velocity v were struck
on one side by an elastic molecule of equal size with a
velocity v, it would bound away at an angle of 45 degs.
with an increased velocity = u ^2. Other forms of
collision might diminish its velocity. All, therefore,
that our investigation has enabled us to say is, that if G
be the general average velocity, then that G2 X M
=	9i2 +	+ ^3 032.....This general value of G

of course limits the individual velocities of the various
molecules.

For instance, if one molecule had a velocity v9 such
that J v2 m = J G2 M, then it would in itself contain
the whole energy of the gas and all the others would be
at rest—an absurd conclusion. For if a gas in such a
state as this were in a bag, the sides would partially
collapse. For it would obviously be improbable that
the molecule, by darting about here, there and every-
where, could keep all the sides distended. On the
contrary, the speed would be so enormous that it would

Fig. 3.

go right through the bag like an electric spark. No
fabric could retain a molecule with such a velocity.
Again, if the velocities of the molecules in a confined
gas were constantly changing in groups, the sides of the
bag would move about in undulations. But we see
nothing of this. A gas, though in violent motion, may
appear at rest. The still air of the evening hardly
moves a thistledown, and yet every molecule of that
air is in violent motion—whence, then, the velocities of
molecules must be fairly uniform.

We have now to see whether a glimpse can be
obtained of the manner in which a vast mass of
molecules governed by the law of the conservation of
energy distribute themselves as to position and velocity.

To enable us to do this we must treat the whole mass
as subject to a law of distribution depending on the law
of conservation of energy, and also on a general tendency
towards what may be called mediocrity or uniformity of
speed. This tendency resembles closely the tendency
of what is called chance to produce equalities—to make,
for instance, a whist-player who plays many games,
neither on the whole to win or lose much. Accordingly
the mathematical process by which the distribution of
velocity is investigated is the same in form as that
employed in the investigation of the laws of chance and
of error.

The “ chance ” that governs the velocities of various
molecules is not simply “ luck,” but is only that sort of
uniformity which is produced by the combined operation
of an enormous number of small causes.

As an example of such chances, let us consider the
following problem:—

Suppose we wished to determine the chances that, if
we took the sexes of the children in any family, there
would be a given number of boys and a given number
of girls.

For example, suppose we wished to calculate what the
odds are that in a family taken at random in which
there are five children, two of them would be boys, and
the remainder girls.

On the assumption that it is equally probable that a
a child will be born a boy or a girl, one would be inclined
to say that if the family be taken at random, it would be
equally probable that all the children would be boys, or
that there would be one boy and four girls or two boys
and three girls, or, in fact, that any sex distribution of
the selected family would be equally probable with any
other sex distribution.

But reflection will dispel this impression. For it is
obvious that if all sex distributions are equally probable,
the chance in favour of any given sex distribution will
depend on the number of the different possible arrange-
ments in order of birth of the five different children of a

family; just as the probability of a given number of
trumps in a hand at whist depends on the number of
different hands which it is possible to deal, the order of
the cards being taken into account as well as their
character.

For it is only by considering the order as well as the
suits that we can give proper effect to the competition
of chances.

If, therefore, a schedule be drawn up showing all the
possible orders in which five boys^ or girls might be
born, it will be found that the probability of a given
total number of boys is not uniform.

The schedule will be in the form given below, where
b stands for boy, g for girl, and each line represents the
composition of a family. There are 32 families, with a
total of 160 children (80 boys and 80 girls):—

bbbbb^
bbbb
b b b g b
bb g b b }>
b g b b b
gbbbb^
bbb g g"
bb g g b
b g g b b
g g b bb
gb bb g >
g b g b b r
b g b b g
gbb gb
bbgb g
bgbgbj

One family,
all boys.

Five families
with four
boys and
one girl.

Ten families
with three
boys and
two girls.

ggggg{
ggggV'
gg gbg
ggbgg >
gbg gg
bg q g gj
g g gbb"
g gbbg
gbbgg
bb gg g
b g g gb
b gbg g
gbg gb
b g g bg
g gbgb
gbgbgj

One family,
all girls.

Five families
with four
girls and
one boy.

Ten families
with three
girls and
two boys.

The above table is complete; no other arrangements
of boys and girls in a family are possible, It is there-
fore certain that whatever number of families we

consider, any one of them taken at random must have
one or other of the arrangements of children set forth
above, and as all are by hypothesis equally likely to
occur, the chance of lighting on a family of 3 boys and
2 girls is 10 out of 32 families or = if. Whereas the
chance of lighting upon a family of all boys is only 1
out of 32, or gL. Hence, if people marry, the probability
that they will have 3 boys and 2 girls is twice as great
as that they will have 4 boys and 1 girl; and ten times
as great as the probability that they will have boys
only. If we take other cases we shall see that this law

prevails.

Thus if the average height of man is 5 ft. 8 in., far
the greatest number of men will have about this stature;
very tall men or very short men will be rare. The reason
is, that abnormal shortness or tallness must be caused
by the co-operation of various factors which have to
coincide, such as a short father who has short parents,
and a short mother who has also short parents. Hence
inasmuch as a union between parents who are not only
short, but who also have short progenitors is more rare
than one in which there is not such a marked family
divergence on both sides from average ideals, it follows
that mediocrity in height is more probable than
exceptional tallness or shortness.

This tendency to mediocrity prevails throughout
nature. It is seen no less in leaves on the trees, than in
grains of sand. It makes the yearly number of deaths
approximately uniform, it makes men’s talents, their
religious opinions, their morals, their earnings, their
strength, their energies, all oscillate about a mean
position, and far the greatest number of men are
mediocrities, rarely diverging to any considerable
degree from a “ type.”

The problem of the children’s sex might be set out in
a figure in which each small square denoted a family,
and the black portion of each square represented the
proportion of girls in the family (fig. 4).

A consideration of the table of families previously
given and of this figure, which is only a method of dis-
playing the same facts, will no doubt lead the reader to
think that possibly the theory of permutations may
have something to do with the matter.

Moreover, the way in which the numbers of families
and the proportion of boys and girls in them will remind
him of the binomial theorem,

(b + g)6 = 165 + 56*0 + 1063 g* + 106a g* 4-

5 bg41 + g5.

Here the large figures in the right-hand member of the
equation indicate the numbers of the families having
the proportions of children shown in the indices to
b and g.

The co-efficients are the numbers of permutations
that can be made by taking five things, one of which is
a boy and four are girls ; or again two of which are boys
and three girls, and so on. For it is obvious that we
can only have five differently arranged families in
which there are one girl and four boys. And again, if
there are three boys and two girls, the numbers of
5x4

families will be ; for trial will easily show that

the various age positions the three boys can occupy is

ten; and in general the permutations which can be
made out of n things, of which r are of one sort, as, for
instance, boys, and the remainder n — r are of another
sort, as, for instance, girls, is

n. n — 1 . . . n—r 4 1	£n

Lr	[_r £_n — r

A curve may be drawn round the squares in fig. 4 given
above, the ordinates of which represent the numbers of
families having certain sexual proportions of children.
The area of the figure is the total of all the individuals
in the case = 160.

Thus, then, we have seen that the average sexual
composition of a family can be considered as governed
by laws of probability the expression of which in
mathematical language depends upon the laws of
permutations, and is in some way also connected with
the binomial theorem.

The law of the average distribution of the velocities
of gaseous molecules depends also on similar considera
tions.	•

For suppose that there is in a cubic centimetre an
enormous number of molecules each of which has its
own velocity of motion. The velocity of any one
individual molecule is the result of a very great number
of impacts with other molecules. Hence this velocity
may be regarded as the result of a very large number
of small independent causes, some of which act to
diminish the velocity, some to increase it. And we may
regard all these small causes as having, on the average,
equal efficacy and being equally likely to occur.

>5

ig 3
sgseK®
COOQJsi
CQ QQ QQ CQ M

Fig. 4.

The resulting velocity may therefore be compared to
the condition of a family of boy* and girls which has
been produced by the fortuitous birth of boys and girls
in all sorts of orders, each cause that comes in to retard
or accelerate the velocity being analogous to the birth of
a boy or girl to modify the characteristic condition of
the family.

And if it can be shown that families tend to mediocrity
of conditions—that is, that on the average the tendency
is to produce families in which neither boys or girls
are greatly in excess of one another, but, so to speak,
gravitate round equality in numbers of the sexes, so
that the enormously greatest number of families will be
those in which the sexes are equally, or nearly equally,
divided—so we shall be prepared to find that the
velocities of molecules, each of which has been produced
in an analogous way by the co-operation of an enormous
number of small causes, also exhibits a scheme of
distribution in which the various velocities, instead of
being chaotically disparate, tend to cluster thickly
round some given standard velocity.

This investigation is one which applies not merely to
the velocities of gas molecules. In its general form it
applies to the whole theory of statistics, and as the
statistical method of dealing with problems comes more
and more into use, so will the use of the theory of
statistical distribution become of increasing importance.

Before, however, we proceed to develop that theory, it
will be desirable to examine the theory of the binomial
theorem more closely, our object being to discover a
formula which shall, so to speak, exhibit the quintessence
of the binomial theorem in its most general and essential
form. The ordinary form is as follows:—

(«+&)” — an + naH'1b + ”' ” - o'* ~ 8	+

X • Al

n.n-1.n-2aB-8&3+&c

1.2.3

It is to be observed that the several terms consist each

of a coefficient and of powers of a and b.

The coefficients are symmetrically arranged, rising to
the middle term and falling again, those equidistant
from the centre being equal. Unless a is equal to 5, the
biggest term is not in the middle of the series, and the
whole series will not be symmetrical. The binomial
may also be put in the more symmetrical and general
form :—

+	=	■ Ln , .a*-,b' +

[_n	£i Ln — 1

--- an -2 b*+.....