Mat 16, 1913.

______________________________________________________________________________________________________________________________

1012

THE COLLIERY GUARDIAN.

If we put a = b = 1, the equation becomes

___________
_______
.............

~ 1 /n-l +	/n-<L + ...............+

___Z” ,	, i
_______
j_n— 2 £2~*~ £n— 1

For different values of n, the binomial series has
different sets of coefficients. These were formed into a
triangle by Pascal.

(a + 5)° .........	1

(a + by .........	1 + 1

(a + b)2 ......... 1 + 2 1.........

(a 4“ 5)3 .........	1 4" 3 4- 3 4" 1

.........

(a 4- by ................ 1	+ 4 4“' 6 4 4 "F 1

(a 4- &)5 .........	1 4- 5 4- 10 4- 10 4- 5 4- 1

(a 4- 5)6 ......... 1 + 64- 15 4- 20 4- 15 4-64-1
Each figure in the triangle is the sum of the two others
which stand above it on the right and left hand respec-
tively. We might from this table construct a curve
for each of the horizontal rows of figures, by taking
abscissse to represent ^he successive numbers of the
terms and ordinates to represent the numbers. So that
the form of the curve would be as in Fig. 5. For successive
curves, as n becomes bigger, the curve if drawn to the
same scale becomes higher and higher, but the general
form remains the same.

As we are about to examine the general form of the
binomial curve apart from any special value of the
terms or the exponent, we may put it into the form most
convenient for examination provided we take care to
preserve its general characteristics.

To draw the curve, we may take along the axis of x
a series of equi-distant points representing, by their

20

Fig. 5.

15

15

6

6

distance from o, the 1st, 2nd and other terms of the
binomial. For the ordinate corresponding to any
abscissa x, we may take the arithmetical value of the a?th
term of the binomial series.

Thus when x = 1, y would be = a n b.

Z n

Z n — 1

When x = 2, y =

a n ~ 1 b; and so on.

If n is very large indeed, and a and b are relatively
small, then the first and last terms of the binomial will

be small, but towards some point dependent on the
relative values of a and b the terms will increase to a

maximum.

As it is this maximum value we wish to study, then, if
n is very large, the only feasible plan in drawing the
curve will (as is done in the case of all curves which
have branches to infinity) be to put the origin of
co-ordinates not at one end of the curve, but towards
the middle of it.

We shall therefore make the axis of y pass through
the highest point of the curve and select a symmetrical
curve to begin with.

To fulfil the condition of symmetry, the two members
of the binomial must be equal. We will make them
each =1. We will select a binomial with an even
number of terms. The exponent will therefore be an
odd number, say 2n - 1. The series will thus have 2n
terms, and the nth, and n + 1th will be the greatest, and
equal to one another.

(1 +1)2"-1 = l+(2n - 1) + (2w~J)(2n-2) +

1.2

(2n - 1) (2n - 2) (2n -3)

1.2.3	+

(2n — 1) (2n — 2)....(2n —(r —1)) p

—---------T.-----------F-l. +&c’

This last is obviously the rth term, not the r + 1th term.

The several terms have a regular order of progression,
and each is derived from the last, by multiplying it by a
certain factor.

Thus, for example, the r + ltb term is got from the

______

rth term by multiplying it by r9 and the r + 2th
r

term is got from the r + 1th by multiplying the r + 1th
___________

by	and so on. The terms go on

increasing, so long as the multiplier is greater than 1; after
this they begin to descend. Hence, in order to get the

______

greatest term, we put = 1 or n = r. The nth and

n + 1th terms are therefore the biggest. If n is very
large, the distinction between the nth and n + 1th term is
small, and we might if we had pleased use n instead of
2n — 1.

If we call this greatest value Q, and take the a?th term
from it—that is, the ccth from the middle point, or the
n + a?th from the commencement of the series—we shall
then have

(2n—r) (2n—(r + 1)) (2n-(-r + 2)) (2n —(r + 3))
__ p.	to x terms

y ~ r ; (r + 1) . (r + 2) . (r + 3)

to x terms

and since Q is the biggest term = 1—that is, r = n,
______

T

whence y = Q G - 1) (n - 2) (n - 3) to a terms

n \n + 1) (n + 2) (n + 3) to x terms

If we divide n + 1 by n — 1 the result is a series
2

= 1 + —F other terms involving n2 and higher powers
n

of n in the denominators. It is a well-known principle
(applicable in this instance) that if n is large, these may
be omitted. So that

____________________________1______________________
y	^1 + i 0. + £ to x terms

but if we may neglect the terms which have a higher
power of n than one in the denominators, then

and so on, whence then

_________________________________

V = Q ^ + 1	(2 + 4 + 6 + 8toa?terms)

1

y - Q L ) »+1)

But if x is large as compared with 1—that is, if we go
to a fairly good distance from the centre as compared
with n, which is infinite—then x (x 1) = sr2, and our
equation becomes

y = Q 0 + i )

But when n is very large + — ^ = e base of
the Naperian system of logarithms, where e = 2•71828.
Hence	«/ = Qe ~

The above operation involves several assumptions which,
though correct in fact, deteriorate from the exactitude
of the proof. A full proof involves Stirling’s theorem
(see ChrystaVs Algebra, vol. ii., p. 368.)

The form above given for y is perfectly general, and
the shape of the curve of which x and y are the
co-ordinates is quite independent of n. Hence we may
put y = Qe “ x2 as the general equation to the binomial
curve in its simplest form.

The shape of the curve is obviously symmetrical
about the axis of y. The maximum value of y is the
middle point where it intersects the axis of y and where
x = o. It slopes down towards each side, and y = o
when ± x = <x>. Hence the axis of x is an asymptote
to it.

The steepest part of the curve is some way down
where —= o—that is to say, where x =	.

dx*	V2

The general form of the curve is, therefore, that of
a typical hill such as one often sees in the Lake district,
with a rounded top and sloping gradually away to a
level plain.

To find the area of the curve we have to integrate the

• expression Q I e~x2dx. This is a difficult integral in its
J CO

general form, but presents no difficulty when taken
between limits only. Among other modes of solution is
that known as the use of temporary constants.

fo

e~x2dx. We may give other values

_ CO

to x if we please without altering the value of I. Thus

I = Q I e ~a2x2adx.

J 00

le ~ a2 = Q I e _ a2x2adx x e~a2 = Q I e~ a2{y+x2>)adx.

J 00	J co

I (°e-«2da = Q | | e ~a2^ + x2^ada dx.
J Ob	J CL J CO

ButQj' e~a2da = I.

Again, | e ~ ^da =. i ,
Joo	X

whence | e - a2 (1 + x2^a da =	—

Joe	2 i + a2

_____

I2 = Qa J i 1 —a?2 = Q2 £tan = Q2

I = Q

This area represents the sum of all the values of y on
one side of the axis of y.

If for simplicity we put this area

so as to make

the total area of the curve = 1, then on a corre-
' spending scale Q =	= O A. A few of the dimensions

I	a/-7T

of the curve are given in the following table to give an
1 idea of its shape. They can be calculated by means of
( a table of logarithms :—

	Value of y — MP	Areas A O P M
Value of x = O M.	= __L e-xZ * OM.	= | e ~ x2 d x.  J x
Bet ween 0*0 and 0’1	0-564 = ~  ^TT	0’056232
0*1 and 1*1	0*2	0-455157
1*5 and 1*6	0*029	0-49189
00	0	0*5

____________________________________________________

This ideal binomial curve has several interesting
properties, which will be demonstrated presently. It is to
be observed that when x has a value between 1*5 and 1*6
on a scale in which O A is represented by -J—=0*564,

V if

almost the whole area of the curve has been taken

Y

M

X

Fig. 6.

in, and y is only about ^th of O M, so that almost
the whole area of the binomial curve lies within a short
distance on each side of the axis of y. (Fig. 6.)

As we shall presently see, this represents the tendency
of Nature to keep the vast majority of her products
nearly to types, and only in rare instances to permit any
considerable departure from them.

(To be continued.}

_____________________________

The General Regulations.* The Signalling Code.—At
the annual conference of the National Federation of Colliery
Enginemen's and Boilerminders' Protective Associations,
held at Southport on Tuesday, the secretary (Mr. W. B.
Charlton, of Durham), in his report, referred in detail to
the proposed new signalling code. These new signals were,
Mr. Charlton said, apprehended with dread and fear by the
members. In the change of signals which comes to an
engineman with the change from one pit to another there
was less to fear than in the change with the same rapper,
in the same engine room, the same engine and the same
banksmen, for the fear of dropping into an old groove and
moving the engine in accordance with the old signals was
alarming. In consequence of the expressed opinions, the
committee had interviewed the executive of the Miners'
Federation and the Mining Association of Great Britain
with a view to inducing these bodies to join them in seeking
to allow signals to remain as at present. Mr. Charlton read
a letter which he sent to Sir Thomas R. Ratcliffe-Ellis on
May 6, in which he stated that they were formally advised
that the miners intended to press the uniformity of
signals into operation. This the enginemen deeply
regretted. It was most unfortunate that they, the
enginemen, clamoured for a uniform code, for the
miners respected their submission, and gave them their
support, but it had not then been with the enginemen
more than an abstract idea. Now, when they had
seen how they would be affected in practice, they
shuddered at the idea of its adoption, and at the grave
fear they had of injuring their fellow-comrades, the miners.
Sir Thomas, in reply, stated that the owners would meet
the Miners' Federation on the 21st inst., when they were
not without hope that they might be able to induce them
to modify their views with reference to the uniform code of
signals as proposed by the Home Office.—Mr. Charlton also
wrote to Mr. Thomas Ashton, secretary of the Miners'
Federation, expressing the hope that that body would not
press for the signals to be changed, and Mr. Ashton replied
stating that he would place the communication before his
officials. “ The repeal of the uniform signalling codes,"
the report concluded," “ alone from the general regulations
is no easy task to those of us who before important
audiences have expressed unbounded pleasure that the
Royal Commission on Safety in Mines had recommended
that the signalling codes should be uniform throughout the
coalfields; nevertheless, having seen what a positive source
of danger such a change will be, it is true manliness to
admit our mistake in advocating it, and we believe that
when the miners see that we have a genuine fear, they will,
too, assist us in having it repealed."