May 23, 1913.

THE COLLIERY GUARDIAN.

1075

ON THE KINETIC THEORY OF GASES.

[Rights of Reproduction Reserved.']
By Sir Henry Cunynghame, Chairman of the Royal
Commission upon Accidents in Mines.

Section III.

(Continued from page 1012).

We have now to show the application of the binomial
curve to the theory of probabilities and deviations.

Suppose that an enormous number of balls were put
into a bag. Then if one were drawn out, the probability
of its being a white ball would depend on the relative
number of white and black balls in the bag. Let the
probability of its being a white ball be p. If a second
ball were drawn the probability of its being a white ball
would also be p ; supposing always that the number of
balls in the bag were so large that the abstraction of any
finite number of balls made no appreciable difference in
the probability of drawing a white ball. Hence the
probability that both balls would be white would be
p x p, and so on for any number of drawings. Similarly
the probability that a black ball would be drawn would
be 1 — p. This is, of course, evident, for if the odds in
favour of a horse are, say, 1 to 9—that is to say, 1 in 10,
or T^th, the odds against him must be T9oths—that is,
1 — Jq. Hence then, speaking generally, if n balls are
drawn, the probability that r of them will be white and
that the remainder, viz., n — r will be black is pr x
(1 — p)n~r. But hitherto we have said nothing as to
the order in which the white and black balls came. If
the order is to be taken into account then we might
have a very great number of different arrangements of
balls of which the totals were r white and n — r black
balls. For we might have a white ball first, then two black?
then a white, and so on. And the number of arrange-
ments would be the permutations that could be made out
of groups of r white and n — r black balls. The number
of arrangements that can be made of n things, when r of
them are alike, and the remainder n — r are also alike, is

[_n

£r j^n-T

This is easily proved, for if the balls were all of
different colours, you could have Z n permutations or
arrangements out of them. As, however, r are white
the number of arrangements is reduced to and as

Z r

n — r are black the number of arrangements is further

reduced to

Ln

£r j_n — r

Whence, then, it follows that if an enormous number of
black and white balls are in a bag, the proportion being
such that the probability of drawing a white ball is p,
then if n balls are drawn the chances of any given
arrangement containing r white balls is pr (1 — p)w“r.
And, therefore, the chance that some arrangement is
drawn such as to contain r balls is

—— pr (1 - »)” -r.

l_n — r

But our previous investigations will have prepared us to
see that this term is the r + 1th term in the expansion
——	/ %

(1-p+2J)»=(1-p)”+—(i-pyr'p
----------------------------

+—— pra-p)n~r
f_r j_n — r

+................

From whence it follows that the various terms of the
expansion of the above binomial are the chances that a
drawing of n balls out of a bag will contain one, two,
three or more white balls. Hence, then, if any given
abscissa of the binomial curve represent the number of
white balls in a drawing of n balls, the corresponding
ordinate of the curve will represent the probability of
making a draw of n balls containing the number of
white balls indicated by the abscissa.

The ordinate y corresponding to any abscissa x of the
curve is the ajth term of the binomial series, taking Q
the biggest term as the first.

The r + 1th term is got by multiplying the rth term by

V and the r 4- 2th term by multiplying
r \X-pJ

the r + rh term by n ~ x and 80 on-

Thus the a?th term from Q

(n - r}(n — r — 1) (n - r - 2) to a? terms Z p \ ®
r * (r + 1) (r 4- 2) to x terms \ 1 — p)

If, however, Q is the greatest term, then —— .	—

= 1. Putting the value of r thus obtained into the last
equation, we have

o (^(1 —P)) (h(1 -p) - 1) (n (l-p)-2) Z p \*
np(np+l) (np 4- 2)	\1— p/

M

0

Fig. 7.

of causes any one of which is equally likely to be
present or absent, we have then a case exactly analogous
to that of the white and black balls. Thus, if there
were an absolutely fortuitous rush of atoms of two
classes, one of which would push a thing one way?
and the other class of which would push it the contrary
way, then the resultant probable motion would obviously
be governed by a binomial curve.

So again, supposing a man were casting bowls along a
green at a peg, the deviation from the peg would be
produced by a consensus of a vast number of small
causes. If we suppose these causes of error to be
divided up into individual causes of error each of which
is equally likely to occur, and each of which has a
definite chance p of occurrence, then we should be
prepared to find that on trial the deviations from
the peg (counting a deviation in the same class, whether
it was to the left or right) would arrange themselves
according to the equation to the binomial curve. This
by actual experimental trial is found to be so. In
all cases in which deviations vary according to what we
call the law of chance, the binomial curve is applicable.

Thus the total number of casts would be all the values
of y added together. Any strip of the curve between
the ordinates y and y 4- dy, and the abscissae x and
x + dx would indicate the number of casts which had
a deviation between x and x 4- dx. The total number
of casts would be equal to the total area of the curve

— (1 — p p)n = 1, and 0 A = Q = —0’564, as has
nJ 7T

been already proved, y is of course not the number of
casts which have the deviation x, but the proportion
which that number bears to the whole number of casts
made. (Fig. 7.)

But we may here ask, What will be the average error ?
This can be got in the following way. Let us suppose
that it = X ; then, if all the casts had this error, the sum
of the errors thus obtained would be equal to the sum
of all the errors that are actually shown by the curve.
That is to say, if we take the sum of x y d x between the

and on the same method as previously employed, if n is

very large, this

= Q e2w(1“p)

e 2~np

— Q e 2« k p 1 - P/

- x2

y = Q . e

Since n and p are constants depending on the number
of balls drawn, and the probability in any one case of
drawing a white ball, we may put

2 n p (1 — p) = c2, and by adopting a suitable
scale in drawing the curve we may put c = 1. We
shall thus have as the equation to the curve, y =
Qe-*2. In this case x would represent the number of
white balls in any given drawing of balls, and y the
probable number of drawings which would be found to
contain x white balls. Here Q is taken as the maximum
value of y, that is the greatest term in the binomial
expansion. If we take the origin of co-ordinates at the
point where y has the value Q, then x will equal the
excess of white balls over black balls in any drawing.
As has been said, Q is the maximum value of y, at
which the numbers of white and black balls drawn will
be in the ratio of —. For in this case x = o. and

1 - P

Q is the centre of the curve, which will be of the form
already shown in fig. 6.

Thus the application of the binomial curve to a
simple case of the theory of probabilities has been
demonstrated. It can be also shown that on certain
assumptions the binomial curve can be used as the basis
of a theory of error, or deviation.

For suppose that an error or deviation of any sort is
produced by the presence or absence of a whole group

limits of oo and — oo on each side, it will equal X multi-
plied by the total area. But

f°°	f°°	1	-#2	1

xydx = I	x.~-= e	dx =~= = X X 1

J ~ 00	J - 00 V 7F	7T

wherefore the mean deviation is given by

X = O M = -i= = 0’56418

v/7T

and P M = y = ~==* e -C564i8)2= 0’4097.

v/tt

The curve also gives the arrangement that the other
casts will take. For instance, out of 1,000 casts, about
41 would have a mean deviation between 0’4 and 0 5, and
the numbers corresponding to any given deviation would
be proportional to the ordinates of the curve.

By experimental investigation it has been shown how
closely this theory accords with the facts—so much so,
that tables founded on the binomial curve are used for
actuarial computation of ages at death.

Thus, then, we have investigated the theory of simple
deviations, and shown the regularity with which the
deviations from a type arrange themselves.

It must not, however, be imagined that every chaotic
mass of things can be arranged in this orderly way.

Where there is no common similarity of origin or
cause of coming into existence, things would not exhibit
this uniformity, so that what we call equal chances is
only really average equality of the effects of like causes.
And therefore the more that things are produced and
governed by laws, however complicated, the more do they
tend to arrange themselves according to the law of
statistical distribution.

The above law has been determined for deviations
along one line only, and with a view to one effect only
viz., the magnitude of a horizontal distance from a
mark.

For any simple deviation the law above given would
hold true, as for age at death, deviation from standard
height, and similar cases.

But we can conceive cases in which it is necessary to
consider a classification dependent on two different
simultaneous deviations, each of which is quite indepen-
dent of the other. Thus we might consider the case of
the magnitude of deviation of a bowl from the line of a
peg, combined with the magnitude of the quantity by
which it came to rest either beyond or short of the peg

0 x

Fig. 8.

These two groups of quantities would be quite indepen
dent of one another, one depending on accuracy of aim,
the other on the force of the throw, but it is possible to
arrange a classification of deviation dependent on both
of them.

It will be necessary, before we apply the theory of
deviation to the distribution of the velocities of gaseous
molecules, to consider a simple case of a double
compound deviation.

Let us commence by extending the case of simple
derivation which we have been considering, of a bowl
aimed at a peg, to rifle practice at a target. As before,
we shall simply consider the distances of the shots from
the bull’s eye, taking no account of whether they go
high or low, or to the right or left. It is distance only
from the bull’s eye that is to be recorded and reckoned.

In this case, as we are dealing with a surface, the
deviation may be treated as the combination of two
groups of deviations, viz., of a horizontal x and a
vertical x' (in each case independently of sign), so that
the deviations are x and x'. We may have then
deviations combined in any way, so that x may be small
while x' is large, or vice versa. (Fig. 8.)

The distance f from the bull’s eye is of course deter-
mined by the relation

ys — a.2 +

but the magnitude of x has no influence on the mag-
nitude of x', because the motion of a body in any given
direction may vary without producing any change of
velocity in directions at right-angles to the variation.

The chance that a bullet will have a horizontal
deviation x is, as before, y= —The chance that

y/ it

it will have a vertical deviation x' is y = —-—e-x>i.

J 7T

Whence, then, the probability that it will at the same
time have a vertical deviation xf and a horizontal
deviation x is

J_ e-^ X 1 e-«'« = _1 e -(»»+»'!) — _1_ e-/> .

yir	TT	IT