Online Library → Theodore R. (Theodore Rudolph) Running → Empirical formulas → online text (page 2 of 8)

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-0.0977

â€” . 0644

3

1910

3.1304

8

3

0594

8

-0.1356

-0.1695

-0.1344

3

1938

3.0604

9

2.9759

9

â€” 0. 2191

-0.2434

â€” 0. 2196

3.1955

2.9752

I

3)31.9476

a

= 3

1948

The numbers in column 6 were found after b and c were deter-

mined in Fig. 3. The sum of the numbers in the seventh column

divided by ten gives the value of a. In the last column are

written the values of y computed from the formula

>' = 3.i948 + .44X-76x2

(4)

22

EMPIRICAL FORMULAS

II.

Values of - form an arithmetical series and A^y are constant.

Another method of determining the constants is illustrated

in the following example: Let it be required to find an equation

which shall express approximately the relation between x and y

having given the corresponding values in the first two columns

of the table below.

I

2

3

4

s

6

7

8

<J

*

y

I _

X

X

y

Ay

aV

2

Com-

puted y

I.O

1.2

1-4

1.6

1.8

2.0

2.2

2.4

4.000

2.889

2.163

1.656

1.284

1. 000

0.777

O.S97

1.0

0.9

0.8

0.7

0.6

0.5

0.4

03

1. 000

i.m

1.250

1.429

1.667

2.000

2.500

3-333

4.00

3-32

2.68

2.07

I-5I

1. 00

0.52

0.08

-0.68

â€” 0.64

-0.61

-0.56

-0.51

-0.48

-0.44

0.04

0.03

0.05

0.05

0.03

0.04

2.00

1.50

1. 14

0.87

0.67

0.50

0.36

0.25

4.000

2.889

2.163

1.656

1.284

1. 000

0.777

O.S97

In column 3 are given values of - in arithmetical series

and the corresponding values of x and y are written in columns

4 and 5 of the table. The values of y were read from Fig. 4.

It is seen that the second differences of the values of y given

in column 7 are nearly constant, and therefore the relation

between the variables is represented approximately by the

equation

^-+*Â©+<^^)- â€¢ â€¢ â€¢ â€¢ (5)

This becomes evident if x be replaced by - in I. The law

X

may then be stated :

If two variables, x and y, are so related that when values of -

X

are taken in arithmetical series the nih. diferences of the corre-

DETERMINATION OF CONSTANTS

23

sponding values of y are constant, the law connecting the variables

is expressed by the equation

II

= +-+-+â€” +

X X^ 0(^

Values of -^

.3 .4 .5 .6 .7

1 1.2 1.4 L6

1.8 2.0 2.

Values of x

Fig. 4.

If in equation (5) - be replaced by X, then

and

.9 1

K

â€”

â€”

\

>

y

N

\

/

/

\

s

/

\

\,

/

^

N

^

^

^

V,

y

^

"^

â€¢ >

r

/

^

^

^-^

/

/

/

_j

lA 2.6 2.8 3.0

/y = a+6Z+cX2,

y^Ly = a^b{X^LX) +c(X+AX)2.

By subtracting (6) from this equation

A3; = tAX+2c(AX)Z+c(AX)2; . . .

and from (7)

A3;+A2>; = MX + 2c(AX)(X+AZ)+c(AX)2. .

Subtracting (7) from (8)

A23; = 2c(AX)2;

A2y

"" 2(AZ)2-

(6)

. (7)

. (8)

24

EMPIRICAL FORMULAS

From column 7 it is seen that the average value of t^y is

0.04, and as AX was taken â€”.1,

_ 0.04

Writing the equation in the form

y-h 7

/

^

_/

t

Jl

1

7

i^

\ 17

7

j2

1

7

t

1

t

J2

17

I

A .5 .6 .7 .S .9 1.0

Fig. 5.

it is seen that it represents a

straight line when - and y â€”

x"

are the coordinates. From

Fig. 5 & is found to be 3

and a to be â€” i. The for-

mula is

x2/'

>'=-i+3(-)+2(-

The last column gives the

values of y computed from

this equation.

The following, taken from

Saxelby's Practical Mathe-

matics, page 134, gives the

relation between the poten-

tial difference V and the cur-

rent ".4 in the electric arc.

Length of arc =2 mm., A is

given in amperes, F in volts.

A

Observed V . .

z

I

Computed V .

1.96

50.25

.5102

50.52

2.46

48.70

.4065

48.79

2.97

47-90

â€¢ 3367

47.62

3-45

47-50

.2899

46.84

3-96

46.80

â€¢2525

46.22

4-97

45-70

45 36

5-97

45-00

â€¢1675

44.80

6.97

44.00

â€¢1435

44-40

7-97

43.60

-1255

44.10

9.00

43-50

.iiii

43-85

DETERMINATION OF CONSTANTS

25

Fig. 6 shows V plotted to ^ as abscissa. The slope of this

line is 12.5 divided by .75 on 6. 7. The intercept on the V â€” axis

is 42. This gives for the relation between V and A

F = 42-+

16.7

Although the

points in Fig. 6 do

not follow the straight

line very closely the

agreement between

the observed and the

computed values of V

is fairly good.

â€” I 1 â€” I â€” I â€” I â€” I â€” 1 â€” I 1 1 â€” >

.3

Fig. 6.

III. - = a-^bx-\-cx^+do(^+ . . . +gx".

y

nl

Values of x form an arithmetical series and A - constant

// two variables, x and y, are so related that when values of

X are taken in an arithmetical series the n\h differences of the cor-

responding values of - are constant, the law connecting the variables

y

is expressed by the equation

III

= a+bx-\-cx'^ -\-dx^ -{-

+gx".

This becomes evident by replacing >> in I by -. The con-

stants in III may be determined in the same way as they were

in I.

IV.

i2_

a-\-bx-\-cx^+dx^-{- . . . -\-qx^.

Values of x form an arithmetical series and A" y^- constant.

If two variables, x and y, are so related that when values of

X are taken in an arithmetical series the nih. differences of the cor-

26 EMPIRICAL FORMULAS

responding values of y^ are constant, the law connecting the variables

is expressed by the equation

IV 'f = a-^bx-{-co(^^-dx^-\- . . . -^qo^.

This also becomes evident from I by replacing y by y'^.

The method of obtaining the values of the constants in

formulas III and IV is similar to that employed in formulas I

and II and needs no particular discussion.

CHAPTER II

V. y = ah\

Values of x form an arithmetical series and the values of y a geometrical

senes.

// two variables, x and y, are so related that when values oj

X are taken in an arithmetical series the corresponding values of

y form a geometrical series, the relation between the variables is

expressed by the equation

V y = ah'.

If the equation be written in the form

\ogy = \oga^-([ogb)x,

it is seen at once that if the values of x form an arithmetical

series the corresponding values of log y will also form an arith-

metical series, and, hence, the values of y form a geometrical

series.

The law expressed by equation V has been called the com-

pound interest law. If a represents the principal invested, b the

amount of one dollar for one year, y will represent the amount

at the end of x years.

The following example is an illustration under formula V.

In an experiment to determine the coefficient of friction, /x,

for a belt passing round a pulley, a load of W lb. was hung

from one end of the belt, and a pull of P lb. applied to the other

end in order to raise the weight W. The table below gives cor-

responding values of a and /z, when a is the angle of contact

between the belt and pulley measured in radians.

a

2

27r

3

6

TT

77r

6

47r

3

3![

2

5^

3

IITT

~6~

P

5.62

6.93

8.52

10.50

12.90

15.96

19.67

24.24

29.94

27

28

EMPIRICAL FORMULAS

The values of a form an arithmetical series and the values

of P form very nearly a geometrical series, the ratio being 1.23.

The law connecting the variables is

The constants are determined graphically by first writing

the equation in the form

log P = \oga -\-a log h

and plotting the values of a and P on semi-logarithmic paper;

or, using ordinary cross-section paper and plotting the values

of a as abscissas and the values of log P as ordinates. Fig. 7

gives the points so located. The straight line which most

nearly passes through all of the points has the slope .1733 and

the intercept .4750. The slope is the value of log h and the

intercept the value of log a.

2J0

an

^â– ^^

log a =0.4750,

^

â– ^

'^

log ^>= 0.1733;

i

-â€”

l^

'

Â« = 3,

\

^TT

F

[G.

TT

alue

7.

sof

a

r^ir

raT

^Ke^

& = i.49-

or

The formula expressing the relation between the variables is

i^ = 3(i49r,

VI. y = a-Vhe.

Values of x form an arithmetical series and the values of Ay form a

geometrical series.

// two variables^ x and y, are so related that when values of x

are taken in an arithmetical series the first differences of the values

DETERMINATION OF CONSTANTS

29

oj y form a geometrical series, the relation between the variables

is expressed by the equation

VI y = a^-bc^.

By the conditions stated the wth value of x will be

Xn = Xi+{n-l) ^x,

and the series of first differences of the values of y will be

Lyij Lyir, Ayir^, Ayir^ ^^- "^ a.. .,n-2

The values of y will form the series

yu >'i+A>'i, yi+Ayi+rAyi,

yi-{-Ayi-YrAyi-\-r'^Ayi-\-r^Ayi-\- . . . +r""^A>'i.

The wth value of y will be represented by

Ayir^ . . . Ayir"

yi+rAyi-\-r^Ayi . . .

yn=yi-{-Ayi

iâ€”r

From the nth. value of x

n â€” i=-

Xn X\

Ax

Substituting this value in the above equation there is ob-

tained

Xn â€”XI

yn=yi-\-Ayi-

= a+bc\

iâ€”r

where a stands for 71 H â€” â€”, b for ~â€” r ^ , and c for r^x.

Iâ€”r iâ€”r

Let it be required to find the law connecting x and y having

given the corresponding values in the first two lines of the

table.

X

. I

. 2

â– 3

-4

â€¢5

.6

â€¢ 7

.8

â€¢9

I.O

y

Ay

y

1.300

0. 140

1.300

1.440

O.IS7

1-439

1-597

0.177

1-597

1-774

0. 200

1-774

1-974

0. 224

1-973

2.198

0.254

2.198

2.452

0.285

2.452

2.737

0.323

2.738

3.060

0.363

3 . 059

3-423

0.407

3-421

3-830

3 830

30 EMPIRICAL FORMULAS

Since the values of A^ form very nearly a geometrical series

the relation between the variables is expressed approximately

by

y â€” a'\-hc'.

The constants in this formula can be determined graphically

in either of two ways. First determine a and then subtract

this value from each of the values of y giving a new relation

yâ€”a = hc'\

which may be written in the logarithmic form

log (;y-a) =log h^x log c,

and h and c determined as in Fig. 7; or, determine c first and

plot c' as abscissas to y as ordinate giving the straight line

y = a^h{c'),

whose slope is h and whose intercept is a.

First Method. The determination of a is very simple.

Select three points P, Q, and R on the curve drawn through

the points represented by the data such that their abscissas

form an arithmetical series. Fig. 8 shows the construction.

R={xQ^-2^x, a+h(f'c^^).

Select also two more points S and T such that

S^{xQ-\-^x,a-\-h(f')\

T={xQ-\-2^x, a-\-h(f'c^).

The equation of the line passing through Q and R is

y= ^ Lx ^ -{xo+Axj+a-^bcfc^. (i)

Ax Ax

DETERMINATION OF CONSTANTS

31

The equation of the Hne through the points S and T is

y==

Ax

A:*:

{x,^+^x)+a+hc'\ , (2)

These Hnes intersect in a point whose ordinate is a. For,

multiplying equation (2) by c^"" and subtracting the resulting

equation from (i) gives

y = a.

Fig. 8 gives the value of a equal to 0.2. The formula now

becomes

log (y â€” .2)=\ogb-\-x\ogc.

In Fig. 9 log {y â€” .2) is plotted to x as abscissa. The slcpe of

the line is 0.5185 which is the value of logc, hence c is equal to

S-T,. The intercept is the ordinate of the first point or 0.0414,

which is the logarithm of b, hence b is equal to i.i.

The formula is

v = o.2 + i.i(3.3)^

32

EMPIRICAL FORMULAS

or

The last line in the table gives the values of y computed from

this formula.

Second Method. For any point {x^y) the relation between

X and y is expressed by

^ y^a-^-hc',

and for any other point (jc+Aic, y+^y) by

y-^^y = a-\-h(fc^.

From these two equations is obtained

^y^h(f{c^-'l)

log A3; = log h{c^* â€” i)-\-x log c,

6

If now log i^y be

plotted to X as

abscissa a straight

line is obtained

whose slope is log c.

The value of c hav-

ing been determined,

the relation

y=^a+h{(f)

will represent a

straight line pro-

vided y is plotted

to c* as abscissa.

Fig. 9.

The slope of this line is h and its intercept a,

VII. \ogy = a^hc\

Values of x form an arithmetical series and the values of A log y form a

geometrical series.

If two variables, x and y, are so related that when values of x

are taken in an arithmetical series the first differences of the cor-

DETERMINATION OF CONSTANTS 33

responding values of log y form a geometrical series, the relation

between the variables is expressed by the equation

VII log y = a-^bc\

This is at once evident from VI when y is replaced by log y.

The only difference in the proof is that instead of the series

of differences of y the series of differences of log y is taken.

VIII. y = a+bx+cd\

Values of x form an arithmetical series and the values of i^'^y form a

geometrical series.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the values of the second differ-

ences of the corresponding values of y form a geometrical series,

the relation between the variables is expressed by the equation

VIII y = a-\-bx+cd\

The nih. value of x is represented by

Xn=xi + {n â€” i)d^x.

The values of y and the first and second differences may be

m colu

imns

yi

^yi

y2

Ay2

A2yi

^3

Ays

A2y2

y^

Ay4

A2y3

y^

Ays

A2y4

y^

etc.

etc.

etc.

34 EMPIRICAL FORMULAS

Since the second difTerences of y are to form a geometrical

series they may be written

A^yi, r^^yu r^^^yl, r^^^yl . . . r^-^A^yi.

The series of first differences will then be

A>'i , Aji +d?yi , A^'i +^^yl â– ^r^^yl , ^yl -\-L^yi â– \-rd?yi -\-r^^^yi

^yl+^^yl-{-r^^yl-\-r^^^yl-\- . . . -{â– r^'-^L^yi.

The Â«th value of y will be equal to the first value plus all

the first differences. For convenience the Â«th value of y is

written in the table below.

yn=yi

-\â€¢^yl

+^yl-\-^^yl

+A>'i -{-d^yi -{-rA^yi

+A3'i + A2;yi 4-rA2;yi +r2A2;yi

+ A^/i + A^yi + r^^yi + r^^^yi + r^^^yi

â– \-^yi-\-^^yi-\-rL^yi-\-r'^^^y\+r^i^^yi-\- . . . -{-r'^'^^^yi.

Adding gives

I / \a I a9 fiâ€” ^ I Iâ€” ^^ , Iâ€” ^ , Iâ€” r^

yn=yl + {n-l)^yl-{-^^yl\ \ 1 \-

Liâ€” r iâ€”r iâ€”r iâ€”r

+lz^+ . . . +l^I^\

Iâ€”r iâ€”r J

The first two terms on the right-hand side represent the sum

of all the terms in the first column of the value of yn. The

remaining terms contain the common factor A^^yi. The terms

inside the bracket are easily obtained when it is remembered

that each line, omitting the first term, in the value of y form a

geometrical series. It is easily seen that the value of yâ€ž may be

written

DETERMINATION OF CONSTANTS

35

yn^yi-{-(n-i)Ayi+^n-2)-'^^(r+r^+r^+ . . . +r - ')

iâ€”r

iâ€”r

=yi-\-(n-i)Ayi

A2y,

-{n-i) ^

iâ€”r iâ€”r Iâ€”r

=.4+5(w-i)+CrÂ»-';

where

A^y

A=y, - ^^^,B=^Ay,+^,^ndC =

A^y,

A^yi

(i-r)

Iâ€”r

(i-r)'-

From the value of Xn is obtained

Xnâ€”Xl

nâ€”i =-

Ax

Substituting this in the value of yn it is found

xn-xi

yâ€ž=A+E^^^^^^^+Cr ^'

Ax

= a-\-hXn-\-cd'''.

Since Xn and yn stand for any set of corresponding values

of X and y the resulting formula is

YIII y = a-\-hx+cd\

In the first two columns of the following table are given

corresponding values of x and y from which it is required to

find a formula representing the law connecting them.

*

y

Ay

A^y

log A^y

(2.00)* y â€” I.

01(2.00)^

Computed y

.0

1.500

048

023

-1.6383

1. 000

490

1.492

.2

1.548

071

026

â€” I

5850

1. 149

388

1.550

.4

1. 619

097

028

-I

5528

1.320

286

1.620

.6

1.716

125

034

-I

4685

1.517

184

I.71S

.8

1. 841

159

039

-I

4089

1.742

082

1.841

I.O

2.000

198

043

-I

3665

2.000 â€”

020

1.999

1.2

2.198

241

051

-I

2924

2.300 â€”

125

2.196

1.4

2.439

292

059

â€” I

2291

2 . 640 â€”

227

2.440

1.6

2.731

351

067

â€” I

1739

3.032 -

331

2.735

1.8

3.082

418

3.482

4.000 â€”

435

540

3.085

2.0

3.500

3 506

36

EMPIRICAL FORMULAS

Since the values of x form an arithmetical series and the

second differences of the values of y form approximately a

geometrical series, it is evident that the relation between the

variables is fairly well represented by

y = a-{-bx-\-cd'.

Taking the second difference

log A^y = log c(d^ - 1)2+ (log d)x.

or

Plotting the logarithms of the second differences of y from

the table to the values of x, Fig. lo, it is found that log d = .3000

-1.0

-1.1

o

Â§4.4

^

^

^

C4

^

^

^

^

X

^

y

[^

00

^

X

J

H

x

n â€”â€¢

^^

X.

X

V

^

[y

X

J.

^

hN

^

>^'

\^

y

^

^ ^

1.0 1.2 L4 1.6 1-8 2jO

Yaluea of <Â»

Fig.

or (/ = 1.995, approximately 2. The intercept of this line,

â€” 1.6500, is equal to log c(d^' â€” iy.

Since

d = 2,

.02239 = ^(2-^-1)2,

C = I.OII.

DETERMINATION OF CONSTANTS 37

Plotting 3;â€” (1.01)2* to X, Fig. lo, the values of a and b

are found to be

a= 0.5,

J= -0.515.

The formula derived from the data is

3; = o.5-o.5i5:j;+(i.oi)2^

In the last column of the table the values of y computed

from the formula are written down. Comparing these values

with the given values of y it is seen that the formula reproduces

the values of ^^ to a fair approximation.

IX. y^id^^'^^'^'.

Values of x form an arithmetical series and A^ log y constant.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

values of log y are constant, the relation between the variables is

expressed by the equation

IX y = id'+^^+'='\

This becomes evident from I when y is replaced by log y.

\o%y = a-\-bx-\-cx^,

which represents a parabola when log>' is plotted to x. The

constants are determined in the same way as they were in

formula I.

X. y = ks'g^.

Values of -x form an arithmetical series and values of A^ log y form a

geometrical series.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

corresponding values of log y form a geometrical series, the relation

between the variables is expressed by the equation

38 EMPIRICAL FORMULAS

This becomes evident by taking the logarithms of both sides

and comparing the equations thus obtained with VIII. X

becomes

log >; = log ^ + (log s)x + (log g)d''.

This is the same as VIII when y is replaced by log y, a by

log k, b by log 5, and c by log g*

XL y =

a-\-bx-\-cx^

Values of x form an arithmetical series and A*- are constant.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

corresponding values oj - are constant, the relation between the

y

variables is expressed by the equation

XI

a-\-bx-{-cx^

Clearing equation XI of fractions and dividing by y

- = a-\-bx-\-cx^.

y

X

This is of the same form as I, and when - is replaced by y

y

the law stated above becomes evident.

If a is zero XI becomes

_ I

^~b-\-cx

which, by clearing of fractions and dividing by y, reduces to

- = b-{-cx,

y

a special case of III.

* For an extended discussion of X see Chapter VI of the Institute of

Actuaries' Text Book by George King.

DETERMINATION OF CONSTANTS

39

If c is zero XI becomes a special case of XVI, or

y

X

which is a straight line when - is plotted to x.

y

Corresponding values of x and y are given in the table below,

find a formula which will express approximately the relation

between them.

X

X

X ,,

Y X

Com-

X

y

Aâ€”

A2- X

Y

-2.5^2

y

y

y

X y

puted y

o

o.ooo

...

0.000

.1

1-333

0.075

100

.050 â€”

9

- 2 . 703

3

003

050

1.329

2

I -143

0^175

150

.050 â€”

8

â€” 2 . 603

3

254

075

1. 140

3

0.923

0.325

200

â€¢ 050 â€”

7

-2.453

3

504

100

0.929

4

0.762

0.52s

250

.050 â€”

6

-2.253

3

755

125

0.760

5

0.645

0.775

300

-051 -

5

â€” 2.003

4

006

150

0.644

6

0.558

I -075

351

-049 -

4

- 1 . 703

4

257

175

0.558

7

0.491

1.426

400

.047 -

3

-1352

4

507

201

0.491

8

0.438

1.826

447

.058 â€”

2

-0.952

4

760

226

0.438

9

0.396

2.273

505

.040 â€”

I

-0.503

5

030

248

0.395

Â°

0.360

2.778

545

â€¢054

0.000

0.360

I

0.331

3 323

599

.056

I

0.545

5

450

298

0.331

2

0.306

3.922

655

-051

2

1. 144

5

720

332

0.305

3

0.284

4.577

706

-035

3

1.799

5

997

352

0.284

4

0.265

5-283

741

4

2.505

6

262

383

0.265

5

0.249

6.024

5

3.246

6

492

399

0.249

The values of x form an arithmetical series and since the

X

second differences of - are nearly constant the values of y will

y

be fairly well represented by

y=

a-\-bx-^cx^^

or

= a-\-bx-\-cx^.

This represents a parabola when - is plotted to x.

y

Let X = xâ€”i,

Y = -

y

.778.

40 EMPIRICAL FORMULAS

From these equations are obtained

- = F+2.778.

y

The formula becomes

Since the new origin lies on the curve

a-\-h-\-c = 2.^^2>,

the equation reduces to

Y={h+2c)X+cX^,

Y

or

X

= b+2c-{-cX.

Y .

This represents a straight line when â€” is plotted to X. The

value obtained for c from P'ig. ii is 2.5. The value of b could

be obtained from the intercept of this line but the approximation

DETERMINATION OF CONSTANTS 41

will be better by plotting â€” 2.5ii;2 to x. In this way is obtained

the line

â€” 2.<x^ = a+bx.

y

From the lower part of Fig. ii the values of a and b are

found to be

a = .o2S,

^ = â€¢2525.

Substituting the values of the constants in XI the formula

becomes

X

y=

.025 + .25250:+ 2. 5x2"

In the last column of the table the values of y computed

from this equation are given and are seen to agree very well

with the given values.

yc

c \

-M

^\c^

CHAPTER ra

XII. y^a^.

Values of x form a geometrical series and the values of y form a

geometrical series.

// two variables^ x and y, are so related that when the values of

X are taken in a geometrical series the corresponding values of y

also form a geometrical series, the relation between the variables is

expressed by the equation

XII y = ax^.

From the conditions stated equations {a) and {b) are obtained.

Xn=xir''-'^, {a)

â– yn=yiR!'-\ {b)

where r is the ratio of any value of x to the preceding one

and R is the ratio of any value of y to the preceding one.

Taking the logarithm of each member of (a)

log Xn = log xi + in-i) log r,

logr

Also by substituting this value of n â€” i in the value of yn in

equation (b),

log XT â€”log XI

yn=yiR 'Â°^^

_ log XI / 1 \ log Xn

DETERMINATION OF CONSTANTS

43

where

and

log xn

a = yiR 'Â°er

Io^=i^log^

The following data (Bach, Elastizitat und Festigkeit) refer

to a hollow cast-iron tube subject to a tensile stress; x represents

the stress in kilogrammes per square centimeter of cross-section

and y the elongation in terms of e^o cm. as unit.

X

9-79

20.02

40.47

60.92

81.37

101.82

204 . 00

y

0.33

0.695

I 530

2.410

3.29s

4.185

â€” . 0644

3

1910

3.1304

8

3

0594

8

-0.1356

-0.1695

-0.1344

3

1938

3.0604

9

2.9759

9

â€” 0. 2191

-0.2434

â€” 0. 2196

3.1955

2.9752

I

3)31.9476

a

= 3

1948

The numbers in column 6 were found after b and c were deter-

mined in Fig. 3. The sum of the numbers in the seventh column

divided by ten gives the value of a. In the last column are

written the values of y computed from the formula

>' = 3.i948 + .44X-76x2

(4)

22

EMPIRICAL FORMULAS

II.

Values of - form an arithmetical series and A^y are constant.

Another method of determining the constants is illustrated

in the following example: Let it be required to find an equation

which shall express approximately the relation between x and y

having given the corresponding values in the first two columns

of the table below.

I

2

3

4

s

6

7

8

<J

*

y

I _

X

X

y

Ay

aV

2

Com-

puted y

I.O

1.2

1-4

1.6

1.8

2.0

2.2

2.4

4.000

2.889

2.163

1.656

1.284

1. 000

0.777

O.S97

1.0

0.9

0.8

0.7

0.6

0.5

0.4

03

1. 000

i.m

1.250

1.429

1.667

2.000

2.500

3-333

4.00

3-32

2.68

2.07

I-5I

1. 00

0.52

0.08

-0.68

â€” 0.64

-0.61

-0.56

-0.51

-0.48

-0.44

0.04

0.03

0.05

0.05

0.03

0.04

2.00

1.50

1. 14

0.87

0.67

0.50

0.36

0.25

4.000

2.889

2.163

1.656

1.284

1. 000

0.777

O.S97

In column 3 are given values of - in arithmetical series

and the corresponding values of x and y are written in columns

4 and 5 of the table. The values of y were read from Fig. 4.

It is seen that the second differences of the values of y given

in column 7 are nearly constant, and therefore the relation

between the variables is represented approximately by the

equation

^-+*Â©+<^^)- â€¢ â€¢ â€¢ â€¢ (5)

This becomes evident if x be replaced by - in I. The law

X

may then be stated :

If two variables, x and y, are so related that when values of -

X

are taken in arithmetical series the nih. diferences of the corre-

DETERMINATION OF CONSTANTS

23

sponding values of y are constant, the law connecting the variables

is expressed by the equation

II

= +-+-+â€” +

X X^ 0(^

Values of -^

.3 .4 .5 .6 .7

1 1.2 1.4 L6

1.8 2.0 2.

Values of x

Fig. 4.

If in equation (5) - be replaced by X, then

and

.9 1

K

â€”

â€”

\

>

y

N

\

/

/

\

s

/

\

\,

/

^

N

^

^

^

V,

y

^

"^

â€¢ >

r

/

^

^

^-^

/

/

/

_j

lA 2.6 2.8 3.0

/y = a+6Z+cX2,

y^Ly = a^b{X^LX) +c(X+AX)2.

By subtracting (6) from this equation

A3; = tAX+2c(AX)Z+c(AX)2; . . .

and from (7)

A3;+A2>; = MX + 2c(AX)(X+AZ)+c(AX)2. .

Subtracting (7) from (8)

A23; = 2c(AX)2;

A2y

"" 2(AZ)2-

(6)

. (7)

. (8)

24

EMPIRICAL FORMULAS

From column 7 it is seen that the average value of t^y is

0.04, and as AX was taken â€”.1,

_ 0.04

Writing the equation in the form

y-h 7

/

^

_/

t

Jl

1

7

i^

\ 17

7

j2

1

7

t

1

t

J2

17

I

A .5 .6 .7 .S .9 1.0

Fig. 5.

it is seen that it represents a

straight line when - and y â€”

x"

are the coordinates. From

Fig. 5 & is found to be 3

and a to be â€” i. The for-

mula is

x2/'

>'=-i+3(-)+2(-

The last column gives the

values of y computed from

this equation.

The following, taken from

Saxelby's Practical Mathe-

matics, page 134, gives the

relation between the poten-

tial difference V and the cur-

rent ".4 in the electric arc.

Length of arc =2 mm., A is

given in amperes, F in volts.

A

Observed V . .

z

I

Computed V .

1.96

50.25

.5102

50.52

2.46

48.70

.4065

48.79

2.97

47-90

â€¢ 3367

47.62

3-45

47-50

.2899

46.84

3-96

46.80

â€¢2525

46.22

4-97

45-70

45 36

5-97

45-00

â€¢1675

44.80

6.97

44.00

â€¢1435

44-40

7-97

43.60

-1255

44.10

9.00

43-50

.iiii

43-85

DETERMINATION OF CONSTANTS

25

Fig. 6 shows V plotted to ^ as abscissa. The slope of this

line is 12.5 divided by .75 on 6. 7. The intercept on the V â€” axis

is 42. This gives for the relation between V and A

F = 42-+

16.7

Although the

points in Fig. 6 do

not follow the straight

line very closely the

agreement between

the observed and the

computed values of V

is fairly good.

â€” I 1 â€” I â€” I â€” I â€” I â€” 1 â€” I 1 1 â€” >

.3

Fig. 6.

III. - = a-^bx-\-cx^+do(^+ . . . +gx".

y

nl

Values of x form an arithmetical series and A - constant

// two variables, x and y, are so related that when values of

X are taken in an arithmetical series the n\h differences of the cor-

responding values of - are constant, the law connecting the variables

y

is expressed by the equation

III

= a+bx-\-cx'^ -\-dx^ -{-

+gx".

This becomes evident by replacing >> in I by -. The con-

stants in III may be determined in the same way as they were

in I.

IV.

i2_

a-\-bx-\-cx^+dx^-{- . . . -\-qx^.

Values of x form an arithmetical series and A" y^- constant.

If two variables, x and y, are so related that when values of

X are taken in an arithmetical series the nih. differences of the cor-

26 EMPIRICAL FORMULAS

responding values of y^ are constant, the law connecting the variables

is expressed by the equation

IV 'f = a-^bx-{-co(^^-dx^-\- . . . -^qo^.

This also becomes evident from I by replacing y by y'^.

The method of obtaining the values of the constants in

formulas III and IV is similar to that employed in formulas I

and II and needs no particular discussion.

CHAPTER II

V. y = ah\

Values of x form an arithmetical series and the values of y a geometrical

senes.

// two variables, x and y, are so related that when values oj

X are taken in an arithmetical series the corresponding values of

y form a geometrical series, the relation between the variables is

expressed by the equation

V y = ah'.

If the equation be written in the form

\ogy = \oga^-([ogb)x,

it is seen at once that if the values of x form an arithmetical

series the corresponding values of log y will also form an arith-

metical series, and, hence, the values of y form a geometrical

series.

The law expressed by equation V has been called the com-

pound interest law. If a represents the principal invested, b the

amount of one dollar for one year, y will represent the amount

at the end of x years.

The following example is an illustration under formula V.

In an experiment to determine the coefficient of friction, /x,

for a belt passing round a pulley, a load of W lb. was hung

from one end of the belt, and a pull of P lb. applied to the other

end in order to raise the weight W. The table below gives cor-

responding values of a and /z, when a is the angle of contact

between the belt and pulley measured in radians.

a

2

27r

3

6

TT

77r

6

47r

3

3![

2

5^

3

IITT

~6~

P

5.62

6.93

8.52

10.50

12.90

15.96

19.67

24.24

29.94

27

28

EMPIRICAL FORMULAS

The values of a form an arithmetical series and the values

of P form very nearly a geometrical series, the ratio being 1.23.

The law connecting the variables is

The constants are determined graphically by first writing

the equation in the form

log P = \oga -\-a log h

and plotting the values of a and P on semi-logarithmic paper;

or, using ordinary cross-section paper and plotting the values

of a as abscissas and the values of log P as ordinates. Fig. 7

gives the points so located. The straight line which most

nearly passes through all of the points has the slope .1733 and

the intercept .4750. The slope is the value of log h and the

intercept the value of log a.

2J0

an

^â– ^^

log a =0.4750,

^

â– ^

'^

log ^>= 0.1733;

i

-â€”

l^

'

Â« = 3,

\

^TT

F

[G.

TT

alue

7.

sof

a

r^ir

raT

^Ke^

& = i.49-

or

The formula expressing the relation between the variables is

i^ = 3(i49r,

VI. y = a-Vhe.

Values of x form an arithmetical series and the values of Ay form a

geometrical series.

// two variables^ x and y, are so related that when values of x

are taken in an arithmetical series the first differences of the values

DETERMINATION OF CONSTANTS

29

oj y form a geometrical series, the relation between the variables

is expressed by the equation

VI y = a^-bc^.

By the conditions stated the wth value of x will be

Xn = Xi+{n-l) ^x,

and the series of first differences of the values of y will be

Lyij Lyir, Ayir^, Ayir^ ^^- "^ a.. .,n-2

The values of y will form the series

yu >'i+A>'i, yi+Ayi+rAyi,

yi-{-Ayi-YrAyi-\-r'^Ayi-\-r^Ayi-\- . . . +r""^A>'i.

The wth value of y will be represented by

Ayir^ . . . Ayir"

yi+rAyi-\-r^Ayi . . .

yn=yi-{-Ayi

iâ€”r

From the nth. value of x

n â€” i=-

Xn X\

Ax

Substituting this value in the above equation there is ob-

tained

Xn â€”XI

yn=yi-\-Ayi-

= a+bc\

iâ€”r

where a stands for 71 H â€” â€”, b for ~â€” r ^ , and c for r^x.

Iâ€”r iâ€”r

Let it be required to find the law connecting x and y having

given the corresponding values in the first two lines of the

table.

X

. I

. 2

â– 3

-4

â€¢5

.6

â€¢ 7

.8

â€¢9

I.O

y

Ay

y

1.300

0. 140

1.300

1.440

O.IS7

1-439

1-597

0.177

1-597

1-774

0. 200

1-774

1-974

0. 224

1-973

2.198

0.254

2.198

2.452

0.285

2.452

2.737

0.323

2.738

3.060

0.363

3 . 059

3-423

0.407

3-421

3-830

3 830

30 EMPIRICAL FORMULAS

Since the values of A^ form very nearly a geometrical series

the relation between the variables is expressed approximately

by

y â€” a'\-hc'.

The constants in this formula can be determined graphically

in either of two ways. First determine a and then subtract

this value from each of the values of y giving a new relation

yâ€”a = hc'\

which may be written in the logarithmic form

log (;y-a) =log h^x log c,

and h and c determined as in Fig. 7; or, determine c first and

plot c' as abscissas to y as ordinate giving the straight line

y = a^h{c'),

whose slope is h and whose intercept is a.

First Method. The determination of a is very simple.

Select three points P, Q, and R on the curve drawn through

the points represented by the data such that their abscissas

form an arithmetical series. Fig. 8 shows the construction.

R={xQ^-2^x, a+h(f'c^^).

Select also two more points S and T such that

S^{xQ-\-^x,a-\-h(f')\

T={xQ-\-2^x, a-\-h(f'c^).

The equation of the line passing through Q and R is

y= ^ Lx ^ -{xo+Axj+a-^bcfc^. (i)

Ax Ax

DETERMINATION OF CONSTANTS

31

The equation of the Hne through the points S and T is

y==

Ax

A:*:

{x,^+^x)+a+hc'\ , (2)

These Hnes intersect in a point whose ordinate is a. For,

multiplying equation (2) by c^"" and subtracting the resulting

equation from (i) gives

y = a.

Fig. 8 gives the value of a equal to 0.2. The formula now

becomes

log (y â€” .2)=\ogb-\-x\ogc.

In Fig. 9 log {y â€” .2) is plotted to x as abscissa. The slcpe of

the line is 0.5185 which is the value of logc, hence c is equal to

S-T,. The intercept is the ordinate of the first point or 0.0414,

which is the logarithm of b, hence b is equal to i.i.

The formula is

v = o.2 + i.i(3.3)^

32

EMPIRICAL FORMULAS

or

The last line in the table gives the values of y computed from

this formula.

Second Method. For any point {x^y) the relation between

X and y is expressed by

^ y^a-^-hc',

and for any other point (jc+Aic, y+^y) by

y-^^y = a-\-h(fc^.

From these two equations is obtained

^y^h(f{c^-'l)

log A3; = log h{c^* â€” i)-\-x log c,

6

If now log i^y be

plotted to X as

abscissa a straight

line is obtained

whose slope is log c.

The value of c hav-

ing been determined,

the relation

y=^a+h{(f)

will represent a

straight line pro-

vided y is plotted

to c* as abscissa.

Fig. 9.

The slope of this line is h and its intercept a,

VII. \ogy = a^hc\

Values of x form an arithmetical series and the values of A log y form a

geometrical series.

If two variables, x and y, are so related that when values of x

are taken in an arithmetical series the first differences of the cor-

DETERMINATION OF CONSTANTS 33

responding values of log y form a geometrical series, the relation

between the variables is expressed by the equation

VII log y = a-^bc\

This is at once evident from VI when y is replaced by log y.

The only difference in the proof is that instead of the series

of differences of y the series of differences of log y is taken.

VIII. y = a+bx+cd\

Values of x form an arithmetical series and the values of i^'^y form a

geometrical series.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the values of the second differ-

ences of the corresponding values of y form a geometrical series,

the relation between the variables is expressed by the equation

VIII y = a-\-bx+cd\

The nih. value of x is represented by

Xn=xi + {n â€” i)d^x.

The values of y and the first and second differences may be

m colu

imns

yi

^yi

y2

Ay2

A2yi

^3

Ays

A2y2

y^

Ay4

A2y3

y^

Ays

A2y4

y^

etc.

etc.

etc.

34 EMPIRICAL FORMULAS

Since the second difTerences of y are to form a geometrical

series they may be written

A^yi, r^^yu r^^^yl, r^^^yl . . . r^-^A^yi.

The series of first differences will then be

A>'i , Aji +d?yi , A^'i +^^yl â– ^r^^yl , ^yl -\-L^yi â– \-rd?yi -\-r^^^yi

^yl+^^yl-{-r^^yl-\-r^^^yl-\- . . . -{â– r^'-^L^yi.

The Â«th value of y will be equal to the first value plus all

the first differences. For convenience the Â«th value of y is

written in the table below.

yn=yi

-\â€¢^yl

+^yl-\-^^yl

+A>'i -{-d^yi -{-rA^yi

+A3'i + A2;yi 4-rA2;yi +r2A2;yi

+ A^/i + A^yi + r^^yi + r^^^yi + r^^^yi

â– \-^yi-\-^^yi-\-rL^yi-\-r'^^^y\+r^i^^yi-\- . . . -{-r'^'^^^yi.

Adding gives

I / \a I a9 fiâ€” ^ I Iâ€” ^^ , Iâ€” ^ , Iâ€” r^

yn=yl + {n-l)^yl-{-^^yl\ \ 1 \-

Liâ€” r iâ€”r iâ€”r iâ€”r

+lz^+ . . . +l^I^\

Iâ€”r iâ€”r J

The first two terms on the right-hand side represent the sum

of all the terms in the first column of the value of yn. The

remaining terms contain the common factor A^^yi. The terms

inside the bracket are easily obtained when it is remembered

that each line, omitting the first term, in the value of y form a

geometrical series. It is easily seen that the value of yâ€ž may be

written

DETERMINATION OF CONSTANTS

35

yn^yi-{-(n-i)Ayi+^n-2)-'^^(r+r^+r^+ . . . +r - ')

iâ€”r

iâ€”r

=yi-\-(n-i)Ayi

A2y,

-{n-i) ^

iâ€”r iâ€”r Iâ€”r

=.4+5(w-i)+CrÂ»-';

where

A^y

A=y, - ^^^,B=^Ay,+^,^ndC =

A^y,

A^yi

(i-r)

Iâ€”r

(i-r)'-

From the value of Xn is obtained

Xnâ€”Xl

nâ€”i =-

Ax

Substituting this in the value of yn it is found

xn-xi

yâ€ž=A+E^^^^^^^+Cr ^'

Ax

= a-\-hXn-\-cd'''.

Since Xn and yn stand for any set of corresponding values

of X and y the resulting formula is

YIII y = a-\-hx+cd\

In the first two columns of the following table are given

corresponding values of x and y from which it is required to

find a formula representing the law connecting them.

*

y

Ay

A^y

log A^y

(2.00)* y â€” I.

01(2.00)^

Computed y

.0

1.500

048

023

-1.6383

1. 000

490

1.492

.2

1.548

071

026

â€” I

5850

1. 149

388

1.550

.4

1. 619

097

028

-I

5528

1.320

286

1.620

.6

1.716

125

034

-I

4685

1.517

184

I.71S

.8

1. 841

159

039

-I

4089

1.742

082

1.841

I.O

2.000

198

043

-I

3665

2.000 â€”

020

1.999

1.2

2.198

241

051

-I

2924

2.300 â€”

125

2.196

1.4

2.439

292

059

â€” I

2291

2 . 640 â€”

227

2.440

1.6

2.731

351

067

â€” I

1739

3.032 -

331

2.735

1.8

3.082

418

3.482

4.000 â€”

435

540

3.085

2.0

3.500

3 506

36

EMPIRICAL FORMULAS

Since the values of x form an arithmetical series and the

second differences of the values of y form approximately a

geometrical series, it is evident that the relation between the

variables is fairly well represented by

y = a-{-bx-\-cd'.

Taking the second difference

log A^y = log c(d^ - 1)2+ (log d)x.

or

Plotting the logarithms of the second differences of y from

the table to the values of x, Fig. lo, it is found that log d = .3000

-1.0

-1.1

o

Â§4.4

^

^

^

C4

^

^

^

^

X

^

y

[^

00

^

X

J

H

x

n â€”â€¢

^^

X.

X

V

^

[y

X

J.

^

hN

^

>^'

\^

y

^

^ ^

1.0 1.2 L4 1.6 1-8 2jO

Yaluea of <Â»

Fig.

or (/ = 1.995, approximately 2. The intercept of this line,

â€” 1.6500, is equal to log c(d^' â€” iy.

Since

d = 2,

.02239 = ^(2-^-1)2,

C = I.OII.

DETERMINATION OF CONSTANTS 37

Plotting 3;â€” (1.01)2* to X, Fig. lo, the values of a and b

are found to be

a= 0.5,

J= -0.515.

The formula derived from the data is

3; = o.5-o.5i5:j;+(i.oi)2^

In the last column of the table the values of y computed

from the formula are written down. Comparing these values

with the given values of y it is seen that the formula reproduces

the values of ^^ to a fair approximation.

IX. y^id^^'^^'^'.

Values of x form an arithmetical series and A^ log y constant.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

values of log y are constant, the relation between the variables is

expressed by the equation

IX y = id'+^^+'='\

This becomes evident from I when y is replaced by log y.

\o%y = a-\-bx-\-cx^,

which represents a parabola when log>' is plotted to x. The

constants are determined in the same way as they were in

formula I.

X. y = ks'g^.

Values of -x form an arithmetical series and values of A^ log y form a

geometrical series.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

corresponding values of log y form a geometrical series, the relation

between the variables is expressed by the equation

38 EMPIRICAL FORMULAS

This becomes evident by taking the logarithms of both sides

and comparing the equations thus obtained with VIII. X

becomes

log >; = log ^ + (log s)x + (log g)d''.

This is the same as VIII when y is replaced by log y, a by

log k, b by log 5, and c by log g*

XL y =

a-\-bx-\-cx^

Values of x form an arithmetical series and A*- are constant.

// two variables, x and y, are so related that when values of x

are taken in an arithmetical series the second differences of the

corresponding values oj - are constant, the relation between the

y

variables is expressed by the equation

XI

a-\-bx-{-cx^

Clearing equation XI of fractions and dividing by y

- = a-\-bx-\-cx^.

y

X

This is of the same form as I, and when - is replaced by y

y

the law stated above becomes evident.

If a is zero XI becomes

_ I

^~b-\-cx

which, by clearing of fractions and dividing by y, reduces to

- = b-{-cx,

y

a special case of III.

* For an extended discussion of X see Chapter VI of the Institute of

Actuaries' Text Book by George King.

DETERMINATION OF CONSTANTS

39

If c is zero XI becomes a special case of XVI, or

y

X

which is a straight line when - is plotted to x.

y

Corresponding values of x and y are given in the table below,

find a formula which will express approximately the relation

between them.

X

X

X ,,

Y X

Com-

X

y

Aâ€”

A2- X

Y

-2.5^2

y

y

y

X y

puted y

o

o.ooo

...

0.000

.1

1-333

0.075

100

.050 â€”

9

- 2 . 703

3

003

050

1.329

2

I -143

0^175

150

.050 â€”

8

â€” 2 . 603

3

254

075

1. 140

3

0.923

0.325

200

â€¢ 050 â€”

7

-2.453

3

504

100

0.929

4

0.762

0.52s

250

.050 â€”

6

-2.253

3

755

125

0.760

5

0.645

0.775

300

-051 -

5

â€” 2.003

4

006

150

0.644

6

0.558

I -075

351

-049 -

4

- 1 . 703

4

257

175

0.558

7

0.491

1.426

400

.047 -

3

-1352

4

507

201

0.491

8

0.438

1.826

447

.058 â€”

2

-0.952

4

760

226

0.438

9

0.396

2.273

505

.040 â€”

I

-0.503

5

030

248

0.395

Â°

0.360

2.778

545

â€¢054

0.000

0.360

I

0.331

3 323

599

.056

I

0.545

5

450

298

0.331

2

0.306

3.922

655

-051

2

1. 144

5

720

332

0.305

3

0.284

4.577

706

-035

3

1.799

5

997

352

0.284

4

0.265

5-283

741

4

2.505

6

262

383

0.265

5

0.249

6.024

5

3.246

6

492

399

0.249

The values of x form an arithmetical series and since the

X

second differences of - are nearly constant the values of y will

y

be fairly well represented by

y=

a-\-bx-^cx^^

or

= a-\-bx-\-cx^.

This represents a parabola when - is plotted to x.

y

Let X = xâ€”i,

Y = -

y

.778.

40 EMPIRICAL FORMULAS

From these equations are obtained

- = F+2.778.

y

The formula becomes

Since the new origin lies on the curve

a-\-h-\-c = 2.^^2>,

the equation reduces to

Y={h+2c)X+cX^,

Y

or

X

= b+2c-{-cX.

Y .

This represents a straight line when â€” is plotted to X. The

value obtained for c from P'ig. ii is 2.5. The value of b could

be obtained from the intercept of this line but the approximation

DETERMINATION OF CONSTANTS 41

will be better by plotting â€” 2.5ii;2 to x. In this way is obtained

the line

â€” 2.<x^ = a+bx.

y

From the lower part of Fig. ii the values of a and b are

found to be

a = .o2S,

^ = â€¢2525.

Substituting the values of the constants in XI the formula

becomes

X

y=

.025 + .25250:+ 2. 5x2"

In the last column of the table the values of y computed

from this equation are given and are seen to agree very well

with the given values.

yc

c \

-M

^\c^

CHAPTER ra

XII. y^a^.

Values of x form a geometrical series and the values of y form a

geometrical series.

// two variables^ x and y, are so related that when the values of

X are taken in a geometrical series the corresponding values of y

also form a geometrical series, the relation between the variables is

expressed by the equation

XII y = ax^.

From the conditions stated equations {a) and {b) are obtained.

Xn=xir''-'^, {a)

â– yn=yiR!'-\ {b)

where r is the ratio of any value of x to the preceding one

and R is the ratio of any value of y to the preceding one.

Taking the logarithm of each member of (a)

log Xn = log xi + in-i) log r,

logr

Also by substituting this value of n â€” i in the value of yn in

equation (b),

log XT â€”log XI

yn=yiR 'Â°^^

_ log XI / 1 \ log Xn

DETERMINATION OF CONSTANTS

43

where

and

log xn

a = yiR 'Â°er

Io^=i^log^

The following data (Bach, Elastizitat und Festigkeit) refer

to a hollow cast-iron tube subject to a tensile stress; x represents

the stress in kilogrammes per square centimeter of cross-section

and y the elongation in terms of e^o cm. as unit.

X

9-79

20.02

40.47

60.92

81.37

101.82

204 . 00

y

0.33

0.695

I 530

2.410

3.29s

4.185

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