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log X. . .

0.9908

1.3014

1.6072

I . 7847

I. 9104

2.0078

2.3096

log y. . .

â€”0.4815

-0.1580

0.1847

0.3820

0.5178

0.6217

0.9523

Comp.

y....

0.324

0.714

I -541

2.416

3323

4.252

9.132

408.57
19.490

2.6II3

I . 2898

19.600

Selecting the values of x which form a geometrical series,
or nearly so, it is seen that the corresponding values of y form
approximately a geometrical series, and, therefore, the relation
between the variables is expressed by the equation

or

y = ax ,
log 3; = log a-\-b logx.

If now logy be plotted to logx the value of b will be the
slope of the line and the intercept will be the value log a. Fig.
12 gives b = i.i. In computing the slope it must be remembered
that the horizontal unit is twice as long as the vertical unit.
The intercept is â€”1.5800 or 8.4200 â€” 10, which is equal to
log 0.0263. The formula is

y = .0263:i;''^.

The values of y computed from this equation are written
in the last line of the table. They agree quite well with the
observed values.

44

EMPIRICAL FORMULAS

XIII. y-a-\-b\ogx-\-c\og^x
Values of log x form an arithmetical series and A^y constant.

// two variables, x and y, are so related that when values of
log X are taken in an arithmetical series the second differences of
the corresponding values of y are constant the relation between the
variables is expressed by the equation

XIII

y = a-\-b\ogx-{-c\o^x.

This becomes evident from I by replacing x by log x. The
law can also be stated as follows: If the values of x form a geo-

1.4
1.2
1.0

.8

i:

It

-.2
-.4

y

,X

y

_y

/

/

y

y"

x'

^

i^

-

,x

y

,^^

^

^

y

y

X'

-.8

X

.8 J9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1,8 l.'J 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
Values of log a:

Fig. 12.

metrical series and the second differences of the corresponding
values of y are constant the relation between the variables is
expressed by the equation

y=^a-\-b\ogx-\-c\o^x.
If c is zero the formula becomes

y = a-\-b\Qgx,
which is V with x and y interchanged.

DETERMINATION OF CONSTANTS 45

Formula XIII represents a parabola when y is plotted to
\ogx. The constants are determined in the same way as the
constants in I.

XIV. y = a^hx\.

Values of x form a geometrical series and 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 first differences of the cor-
responding values of y form a geometrical series, the relation between
the variables is expressed by the equation

XIV y = a+bx\

As in XII the wth value of x is

Xn=xir''~'^ {c)

The series of first differences of y may be written

A3;i, ^yiR, AyiR^y AyiR^ . . . AyiR''-^

and the values of y are

yi, yi-^Ayi, yi+Ayi-\-AyiR, yi+Ayi+AyiR+AyiR^ . . .

yi-\-Ayi+AyiR+AyiR2-\-AyiR^+ . . . ^AyiR^'-K

That is the nth. value of y will be

yn = yi+Ayi+AyiR+AyiR^+AyiR3+ . . . +AyiR''-''
=yi-\-Ayi(i+R-hR^+R^+ . . . +R"-')

= yi-\-Ayi- â€” (d)

I â€” K

Taking the logarithm of each member of (c),
log Xn = log xi + (n â€” i) log r

log r

46

KMPIRICAL FORMULAS

Substituting this value of n â€” i in the wth value of y given
in W,

>'Â» = >'i+A7

log xn -log Xi

i-R 'Â°Â«'*

i-R

log XI / 1 \ log Xn

. . log Xl / 1 \

Let it be required to find the law connecting x and y having
given the values in the first two lines of the table.

*

2

3

4

s

6

7

8

y

4.21

5.2s

6.40

7.65

8.96

10.36

II.81

log*

.3010

.4771

.6021

.6990

7782

â€˘8451

.9031

X

2

2.5

3-125

3 906

4

883

6. 104

7.630

y

4.210

4.720

5.388

6.290

7

515

9.110

11.275

logx

.3010

.3979

.4948

.5918

6887

.7856

Ay

.510

.668

.902

I. 225

I

595

2.165

....

log Ay

- .2924

-.1752

-.0448

.0881

2028

.3358

....

y-2.72

1.49

2.53

3.68

4.93

6

24

7.64

9.09

log(y-2.72)

.1732

.4031

.5658

.6928

7952

.8831

.9586

Computed y

4.21

5-25

6.41

7.65

8

98

10.36

II. 81

In the fourth line values of x are given in a geometrical
series with the ratio 1.25. In the fifth line are given the cor-
responding values of y read from Fig. 13. The first differences
of the values of y are written in the seventh line. These differ-
ences form very nearly a geometrical series with the ratio 1.336.
Since the ratio is nearly constant the law connecting x and y
is fairly well represented by the equation

y = a-\-bx''.

There are two methods which may be employed for deter-
mining the values of the constants, either one of which may
serve as a check on the other.

DETERMINATION OF CONSTANTS

47

First Method. Select three points, A, P, and Q on the
curve, Fig. 13, such that their abscissas form a geometrical
series and two other
points, R and S, such
that R has the same
ordinate as A and the
same abscissa as P, S
the same ordinate as P
and the same abscissa
as Q. The points may
be represented as fol-
lows:

A = (xo, a + bxo");
P={xor, a+bxoY);
Q^{xor^a-\-bxo'r^'):
R={xor, a-\-bxo');
S^{xor^a+bxoY).

The equation of the
line passing through P
and Q is

bxoY{r'

A

^

V

/

/

/

/

r

;

/

^

/

/

^

/^

A,

V

^

'p

x^

/

X

A

3

i
V

alues of

'

8

Fig. 13.

xor(râ€” i)

The equation of the line passing through the points R and
5 is

^_b x,'{r'-i) ^^^ bxQ^r'-r)

xor{râ€”i)

r â€” i

These two lines intersect in a point whose ordinate is a. In
Fig. 13 xo is taken equal to 2 and r equal to 2. The value of
a is found to be 2.72. The formula then becomes

or

y â€” 2.'j2 = bx'',
log (3^â€”2. 72)= log b-\-c\ogx.

48

EMPIRICAL FORMULAS

In Fig. 14 log (yâ€” 2. 72) is plotted to log x and h and c
determined as in XII. It is seen that the points lie very nearly
on a straight line. The values of c and h are read from Fig. 14.

c = i.3;

log 6 = 9.7840 â€” 10;

The law, connecting x and y then is

>' = 2.72-|-.6ir*;^-^.

.5
.4
jt
.2
5* 1

-i3
i4
t6

-.6

/ /

X y

-7 V

Z Z

/ / ._. ,

ji y V

~X- -.^

it z ^z

/ 7

^ ^

7^ r

z z

1 /

^ J^

X .2 .3 .4 .5 .
Values of

6 .7 .8 .9 1.0
log X

Fig. 14.

The values of y computed from this formula are written in
the last line of the table.

Second Method. From the equation

y = a-\-hx'''

DETERMINATION OF CONSTANTS 49

we have

y-\-^y = a-\-hx'^r^\

Ly = bx''{r''-i)\
log A>; = log h{r''-i) +c log x.

This is the equation of a straight line when log ^y is plotted
to log^i:. Fig. 14 shows the points so plotted and from the
line drawn through them the values of h and c are obtained.

c = i.3,

a is found by taking the average of all the values obtained

from the equation

a=y â€” .6i^-^,

a is equal to 2.72.

XV. y = aid'''\

Values of X form a geometrical series and A log y form a geometrical
series.

// two variables, x and y, are so related that when values of
X are taken in a geometrical series the first differences of the cor-
responding values of log y form a geometrical series, the relation
between the variables is expressed by the equation

XV y = aid>^\

This equation written in the logarithmic form is

\ogy = \oga-\-bx'^.

Comparing this with XIV it is evident that if the values of x
form a geometrical series the first differences of the corre-
sponding values of log y also form a geometrical series.

In an experiment to determine the upward pressure of water
seeping through sand a tank in the form shown in Fig. 15 was
filled with sand of a given porosity and a constant head of

50

EMPIRICAL FORMULAS

water of four feet maintained.* The water was allowed to flow
freely from the tank at A. The height of the column of water

in each glass tube,
six inches apart,
was measured. In
the table below x
represents the dis-
tance of the tube
In feet, and y the
height of the column
of water in the tube,
also in feet. It is
required to find the
law connecting x
Fig. 15. and y. .

Tube

I

2

3

4

S

6

7

8

9

X

â€˘ s

I.O

1.5

2.0

2.5

3.0

3-5

4.0

y

2-31

2.30

2.20

2.00

1.66

1.24

0.84

0.54

0.28

logy

.3636

â€˘3617

.3424

.3010

.2201

.0934

- .0757

- .2676

- .5528

X

â€˘s

1.0

2.0

4.0

y

2.30

2.20

1.66

.28

logy

.3617

.3424

.2201

- .5528

A log y

- .0193

- .1223

- .7729

log*

â€” .3010

.0000

.3010

log (-A log y)

-1.7144

â€” .9126

â€” .1119

. **' c

.0000

â€” .0036

â€” .0228

- .0673

- .1449

â€” .2627

- .4272

- .6445

- .9201

log (y-bx<^)

.3b3b

â–  3653

â€˘ 3652

.3683

â€˘ 3650

.3561

.3515

.3769

â€˘ 3673

Computed y

2.314

2.29s

2.19s

1.982

1.658

1.264

.865

.525

.278

In the fifth line values of x are selected in a geometrical
series and the corresponding values of y written in the next
line. In Fig. 16 log (â€”A log y) is plotted to logic. The
points lie on a straight line. On account of the small number
of points used in the test we select formula XV on trial.
From the formula

it follows that

y = aid^
log y = \og a-^hxf

* Coleman's Thesis, University of Michigan.

DETERMINATION OF CONSTANTS 51

logyk = loga-\-bxt
log yk+ 1 = log a + bxtY
^logyk = bxt'(r'-i)
log (A log y) =log b(r'-i)-^c log x.

If A log y is negative b is negative, in which case it is only
necessary to divide the equation by â€” i before taking the
logarithms of the two members of the equation.

-1 .1

Values of log x

Fig. I 6.

The last equation above represents a straight line when
log (a log y) is plotted to log x. The slope gives the value of
c and the intercept gives \ogb(r''-i). From Fig. i6 values
of b and c are readily obtained.

c = 2%,

b= â€”.02282.

In the next to the last line the value of a is computed for
each value of x from the equation

log a = \ogy-\-.o22^x^^.

52 EMPIRICAL FORMULAS

The average of these values of a gives

= 2.314.
The formula obtained is

y = (2.3i4)io-Â»"Â«'".

The values of y computed from this equation are written
in the last line of the table. The agreement is not a bad one.

CHAPTER IV

XVL {x+a){y-^h)=c.
Points represented by [xâ€”xt, ) He on a straight line.

\ y-yk/

If two variables, x and y, are so related that the points repre-
xâ€”Xk, ) lie on a straight line, the relation between

y-yJ

the variables is expressed by the equation
XVI {x^a){y-^b)=c.

'LQtX â€” Xu=X,

y-y, = Y,

where Xk and yk are any two corresponding values of x and y.
From the above equations

x = X+Xk,

y = Y-\-yt.

Substituting these values of x and y in equation XVI we
have

(X-\-Xk+a){Y-\-y,-\-b)=c,
or

XY+{yk-\-b)X+{x,+a)Y-}-(x,c+a)(yt+b)=c.

Since (iCt, yt) is a point on the curve

{xt+a)(yk+b)=c,
and

XYi-{y>c-^b)X+{xk+a)Y=o.

53

54

EMPIRICAL FORMULAS

Dividing the last equation by Y

A'+0'*+^)^+^*+Â«=o,

or

F

I _j^_^ -\-o>

yk-\-h y.,-\-h

This represents a straight line when X is plotted to â€” .
The theorem is proved directly as follows: If the points
â– ** ' lie on a straight line its equation will be

X â€” Xk

/ X-Xk\
[x-x , )

\ y-y>^

y-yic

= p{x-x,)+q.

Clearing of fractions

x-x, = p{x-x.){y-yk)-{-q{y-yK).
This is plainly of the form

{x+a){y+h)=c.

The following tables of values is taken from Ex. i8, page 138
of Saxelby's Practical Mathematics. It represents the results
of experiments to find the relation between the potential differ-
ence V and the current A in the electric arc. The length of
the arc was 3 mm.

A (am-

peres)

i.9fa

2.46

2.97

3.4s

3.96

4.97

5.97

6.97

7.97

V (volts

67.00

62.7s

59-75

58.50

56.00

53.50

52.00

51.40

50.60

X

0.50

l.OI

1.49

2.00

3.01

4.01

50I

6.01

Y
X
Y

-4-25

-7.25

-8.50

â€” 11.00

-13.50

â€” 15.00

-i5-6o

-16.40

- .1176

- .1393

- .1752

- .1817

- .2228

â€” .2670

â€” .3210

- .3665

Com-

puted F

66.99

62.74

59.80

57.80

56.19

53-94

52.44

SI. 36

50. 55

Let A be taken as abscissa and V as ordinate and transfer
the origin to the point (1.96, 67.00) by the substitution

X=A-i.g6,

7=7-67.00.

DETERMINATION OF CONSTANTS

55

The values of X and Y are given in the third and fourth

X

lines of the table. The values of â€” are plotted to X in Fig. 17

3 i

Yalues of X

Fig. 17.

and are seen to lie nearly on a straight line. It is therefore
concluded that the formula is

{V^-h){A-^a)=c.

By the equations of substitution this becomes

(X+i.96+a)(F+67.oo+6) =c,
or

XF+(67.oo+6)X-f-(i.96+6)F=o.

Dividing by 7(67.00+/^)

1.96+d^

' X^_ _j_

Y 67.00+6

The slope of this line is â€”

â€” â€” -. From Fig. 17

67.00 -fo

X-

67.00+6*

Solving these equations

From formula

67.00+6

1.96+a
67.00+6

67.00+6

= â€˘045;
= .095.

and the intercept is

^ = 0.151,
6= -44.78,

c = 46.89.

56 EMPIRICAL FORMULAS

These values give

(A +o.isi)(7-44.78) =46.89.

In the last line of the table are written the values of V com-
puted from the above formula

XVIa. y = aio'-^'.

Points represented by ( log -â€” , log â€” ) lie on a straight line.

\x-xt yt ytj

If two variables, x and y, are so related that the points repre-
sented by I log â€” , log â€” ) lie on a straight line, the relation

\x-Xk yt yt/

between the variables is expressed by the equation

b
XVIa y = aio'-^'.

By the condition stated

log^ = w â€” l-log^+^Â»,

yk x-xt yt

where Xk and yk represent any two corresponding values of
X and y. m is the slope of the line and b its intercept. Clear-
ing the equation of fractions

(log y-log yk){x-Xk) = w(log y-log yk)+b{x-Xk),
or

log y{x-Xk-m) = (b-^log y*)a;-log yk(xk-\-m)-bxt.

log ,.- (^+lQg yt)x-\og yk(xk-\-m) -bxt
xâ€”Xtâ€”tn

^ Ax+B

x+C

^+ x+C

=logaH â€” â€” .

x-\-c

DETERMINATION OF CONSTANTS 57

Therefore

6

For the purpose of determining the constants the equation
is written in the form

log>; = logaH â€” â€” ,

x-\-c

{log y -log a){x+c)=b,
Let

log3; = logF+log3;;fc,
and

x = X-\-Xk.
Then follows

(log F+log y,-log a){X+x,-\-c)=b,

X log F +log Y(xic +c) +X(log yic -log a) + (log yk -log a) {xk +c) = b.

But

(log yk-log a)(xk-\-c) =b,

since the point (xk, yk) lies on the curve.

X log F+log Y(xk+c)-^X(log yk-log a) =o.

Dividing this equation by X

log Y=-(xk+c) -^^+loga-log>'A;.

Replacing log F and X by their values

log ^ = - (xk-\-c)-^ log â€” +log a-log yk.

yk X Xk yk

From this it is seen that if log â€” be plotted to â€” â€” log â€”

yic x-xt yt

a straight line is obtained whose slope is â€”{xk-\-c) and whose
intercept is log aâ€” log y^. If the slope of the line is represented
by M and the intercept by B

c= â€”Mâ€”Xkj
loga = B-\-logyk.

58 EMPIRICAL FORMULAS

By writing XVla in the logarithmic form

\ogy = bâ€” - hloga
a line is obtained whose slope is b.

XVII. y = ae"-{-be''.
Values of x form an arithmetical series and the points

ht+i yt+2 \
\ yk ' yt /

lie on a straight line whose slope, M, is positive and intercept, B, is negative,
and M^-^4B positive.

If two variables, x and y, are so related that when values of x
are taken in an arithmetical series the points represented by

(yk+i^yk+2\ 11^ ^^^ ^ straight line whose slope, M, is positive
yt yt I
and whose intercept, B, is negative and also M^-\-4B is positive
the relation between the variables is expressed by the equation

XVII y^ae'^'+be^^

Let fe, >'/.), (x+Ax, yk+\), {xkâ– \-2^x, yk+2) be three sets of
corresponding values of x and y where the values of x are taken
in an arithmetical series. We can then write the three equations,
provided these values satisfy XVII.

y.^ae^'+be'^^^ ........ (i)

y.+i^ae^'^V^^+fte^V^^ (2)

Multiplying (i) by e^^^ and subtracting the resulting
equation from (2)

>'.+i-^^^^3'^ = 6e^^K^^^^-^'^^). .... (4)

Multiplying (2) by e^^^ and subtracting from (3)

Multiplying (4) by e'^^^ and subtracting from (5) there
results

DETERMINATION OF CONSTANTS

59

> = o, . . . (6)

or

>^A:+2 ^ fc^X X ^d^X\ Jk+l _^{c +d) Ax

Jk

yk

The values of c and d are fixed for any tabulated function
which can be represented by XVII, and therefore, the last

equation represents a straight line when ^^^ is plotted to ^â€” i.

yk, yk

The slope of the Hne is

and the intercept is

It is seen that M is positive, B negative, and M^-{-^ posi-
tive, for

and

In the first two lines of the table are given corresponding
values of x and y. It is desired to find a formula which will
express ^he relations between them.

X

I.O

1.5

2.0

2.5

3.0

3-5

4.0

4.5

5.0

y

+ .3762

+ .0906

-.1826

- .4463

- .7039

- .9582

â€” 1.2119

-1.4677

-1.7280

Jk + l

yk

+ .241

â€” 2. 015

+2.444

+1-577

+ I.361

+ 1.265

+I.211

yk+2
yk

-.485

-4.926

+3.855

+2.147

+1.722

+I.S32

+ 1.426

,-.412.

+ .662
+ â€˘319

+ .539
+ .071

+ â€˘439
-â– 131

+ .359
- .295

+ .290
- .429

+ .236
- .538

+ .192
- .626

+ .157
- .698

+ .127
- .757

Computed y

+ .371

+ .087

-.185

- .447

- .704

- -957

â€” 1. 210

-1.464

-1.723

Plotting the points represented by (^^, ^^), Fig. 18,

\ yk yt /

a straight line is obtained whose equation is

60

EMPIRICAL FORMULAS

M=1.97,

B=- .96.

Since M is positive, B negative, and M^-\-^ positive, it
follows tiiat the relation between the variables is expressed
approximately by XVII. It has been shown that the slope

i
s

2

1

/

/

/

/

/

/

-2
-3
-4

/

/

/

J

^

-i>

/

>-A

Values ol^^YT

Fig. 18.

.3 .4 .5 .6
Values of â‚¬~-^*2x

Fig. 19.

of the line is equal to e'^'^+e'^^'', and the intercept is equal to
_^(c+<f)A^^ Since A:*: is .5

n^ ^ e''V'^ = i.97,

gKc+d)^ .96.

From these equations are obtained the values of c and J,
c=-.247,
^ = .165.

DETERMINATION OF CONSTANTS 61

The formula is now

Dividing both sides of this equation by e-^^^* gives the
equation

which represents a straight line when ye~-^^^' is plotted to
^-.4121 jjjg values of these quantities taken from the table
are plotted in Fig. 19 and are seen to lie very nearly on a straight
line.

This line has the slope 2.00 and intercept â€” i.oi. Sub-
stituting these values of a and h in the formula it becomes

It is seen that the errors in the values of y computed from
this formula are in the third decimal place. The values are as
good as could be expected from a formula in which the con-
stants are determined graphically. For a better determination
of the constants the method of Chapter VI must be employed.

XVIII. y = e\c cos hx-\-d sin hx).
Values of x form an arithmetical series, and the points ( â€” â€” , lL-1 \

\ yt yt /

lie on a straight hne. Also M^-^4B is negative.

// two variables, x and y, are so related that when values of
X are taken in an arithmetical series the points represented by

iy]^Â±l^ ^VH^j ^^ ^^ ^ straight line whose slope M and intercept

B have such values that M'^-^-^B is negative, the relation between
the variables is expressed by the equation

XVIII y = e''^(cQ,o^bx+dsmbx),

Let X and y^ be any two corresponding values of the variables.
We have the three equations

yk = e'^''{c cosbx-\-d?>\jibx), (i)

yic+i =eÂ°V^^[c cos {bx+b^x)-^d sin {bx+b^x)]

62 EMPIRICAL FORMULAS

=ef"ef^[c(cos bx cos b Ax â€” sin bx sin bAx)

+rf(sin bx cos bAx-\-cos bx sin bAx)]

= e"'6Â«^[(c cos bAx+d sin 6Aii;)cos bx

-\- {d cos bAx â€” c sin 6A:r)sin bx]. . (2)

The value yk+2 can be written directly from the value of
y..+ 1 by replacing Ax by 2 Aic.

3'.;+2 = eÂ°'c^'^'[(c cos 26Aa:+<^ sin 2&Aii;)cos bx

H-((/cos 2i5)Aii;â€” c sin 26A;i(;)sin&jt:] (3)

Subtracting (i) multiplied by e'^''{c cos bAx+d sin ftAic)
from (2) multiplied by c we have

cyt+i â€” e'^^(c cos 5Aa;+f/ sin bAx)yk

= ceÂ°^e'^^(d cos ftArrâ€” c sin bAx)sm bx

â€” f/e^V^'^Cc cos bAx+d sin &Aa;)sin ftjc
= -(c2+i/2)ga^gaAxgin^,Aa;sin^>a; (4)

Similarly

cyk+2 â€” e^'^^'^(c cos 2bAx+d sin 2^>Aa:)y&

= - (c2 + C?2)ga^g2aAx gj^ 2^>AX sin Jo:. .... (5)

Multiplying equation (4) by e"^"" sin 2b Ax and subtracting
it from (5) multiplied by sin bAx

c sin bAxyk+2 â€” e'^''^(c cos 2bAx sin 6Ax+c? sin 2bAx sin &Ax)yA:

â€” ce^' sin 26Aa;>'t+ 1 + e^"^"" (c cos Z>Ait; sin 2 6Ax

+(/ sin ftAit; sin 2bAx)y;c = o.
Simplifying

c sin bAxyk+2â€”ce'^^^ sin 2&A:r>'A+i H-ce^""^^ sin ftArc^'t =0.

Dividing by c sin JArry^,

yic yt

DETERMINATION OF CONSTANTS

63

The values of a and b will be fixed for any tabulated function
which can be represented by XVIII, and therefore, the last

equation represents a straight line when ^^ is plotted to
^-^. The slope of the line is

and the intercept

M = 2e''^^ cos JAiu,

It is evident that M^-\-4.B is negative.

It is possible that in a special case M^-^-^B might be zero,
but then h would be zero and hence

y = ce'''y
which is formula V.

Corresponding values of x and y are given in the first two
columns of the table below. It is required to find a formula
which will represent approximately the relation between them.

y

yk+x
yic

yk+2
yk

^.OSx

cosbx

tan bx

cosbx

y

Com-

X

e-Â«Â«^os bx

puted y

+ .300

I. 0000

I

.0000

.0000

I. 0000

+ .300

+ .308

I

+ .oil

1.0833

+

.8646

+ .5812

+ .9366

+ .012

+ .018

2

- .332

+ .04

- I. II

1.1735

+

.4950

+ 1.7556

+ .5809

- .571

- .327

3

- .636

-30.2

-57.8

I. 2712

â€”

.0087

-114.59

â€” .0111

+57.3

- .634

4

â€” .803

+ 1.92

+ 2.42

I. 3771

-

.5100

- 1.6864

- .7023

+ I. 143

â€” .804

5

- .761

+1.26

+ 1.20

I. 4918

â€”

.8732

- .5581

â€” 1.3026

+ .584

- .761

6

- .485

+ .95

+ .60

1.6161

â€”

.9998

+ .0175

-1. 6159

+ .300

- .485

7

- .017

+ .64

+ .02

1.7507

-

.8557

+ .6048

-1. 4981

+ .011

â€” .012

8

+ .537

+ .04

- I. II

1.8965

â€”

.4797

+ I. 8291

- .9098

- .590

+ .545

9

+ 1.027

-31.6

-60.4

2.0544

+

.0262

â€” 38.1880

+ .0538

+19.08

+ 1.035

10

+ 1.298

+ I.91

+ 2.42

2.2255

+

.5250

â€” I. 6212

+I.I684

+ 1 .III

+ 1.299

In Fig. 20 the points represented by ( ^^, ^^ ) are plotted.

\ yt yt I

They lie very nearly on the straight line whose equation is

>^ = i.875 2!fi-x.i7S.

yk jk

64

EMPIRICAL FORMULAS

Since (1.878)2â€”4(1.18) is negative the relation between the

variables is expressed

/

approximately by the

/

equation

A

/

/

y = e'^'^ic cos bx-\-d sin bx).

/

1.6

/

It was shown that
the slope of the line is
equal to 2 (cos bAx)e''^''
and the intercept equal
to-g2aAx^ Since Ax is
equal to unity we have

/

1 n

/

/

;^<

/

^

/

/

1 -'

/

> â€˘*

/

/

26Â° cos 5 = 1.875,

Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 3 of 8)