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Empirical formulas online

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/

^'" = 1.175,

/

/

log cos i^ = 9.9370â 10,

/

5 = 30ÂḞ 10' ap-
proximately,

/

/

A .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
Values oÂ£ â,r^-

FlG. 20.

a = .08.
The formula is now

y = e-^^^(c cos ^Oyx+d sin 3oiic),

where 30^: is expressed in degrees.

Dividing the equation by c-^"^ cos 30^0;

-^ â ^ â _^ = c:+J tan 30^^ÂḞ^

Y-5 is plotted to tan

which is a straight line when â^^ â

Zok^^' In Fig. 21 these points are plotted and are seen to
lie nearly on a straight line whose slope is â .496 and intercept

DETERMINATION 0Âċ CONSTANTS

65

.308. Two of the points are omitted in the figure on account
of the magnitude of the coordinates. Substituting the values
of constants just found in the formula the equation expressing
the relation between x and y is

y = g08^(.3o8 cos 3oiii;ÂḞ â .496 sin 3oiit:ÂḞ).

The last column
in the table gives
the values of y
computed from the
equation. The
agreement with the
original values is
fairly good.

In case c is zero
XVIII becomes the
equation for damped vibrations, y = ^e"^ sin bx.

1.5

1^1.0

S

"â

~~"

^

"^

"V

ÂṠ.

""v

"^

^

a.

^

m

s

>! -0.5

^

\

N

-1.5 -1.0 -0.5 0.5

Values of tan 30i/^xÂḞ

Fig. 21.

2.0

XIX. y = ax'^-i-bx^.

(yt+i yft+2 \
yt yk I

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

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

/ j^+i >^fc+2 \ ^^ ^^ ^ straight line whose slope, M, is positive,
\ jk yjc I

and intercept, B, negative, and also M^-\-4B positive, the relation
between the variables is expressed by the equation

XIX y = ax'~^bx''.

Let X and y^ be any two corresponding values of the variables.
The following equations are evident:

yt = ax^-\-bx^, (i)

yt+i=ax'r'+bx'^r^, (2)

66 EMPIRICAL FORMULAS

>'^h2 = a:ÂṠ:V2'^^-6:*:V^ (3)

yt^-i-fyt ^bx'if^-r'), (4)

yt+2-r'yt+i =bxfr^{r^-r') (5)

Multiplying equation (4) by y'^ and subtracting it from
equation (5) there results

yt+2-r'yt+i-r^yk+i-^r'-^'^yt = o,
or

yt yt

It is seen that the slope of this line is positive and the inter-
cept negative, and M^+4B positive.

In the table* below, the values of x and y from a; = .05 to
ii; = .55 are taken from Peddle's Construction of Graphical
Charts.

yk+i

yfc+i

.55

85

y

Com-

X

y

X

y

âḃ

yk

yk

^.00

x"^^

x''

puted y

âḃOS

.283

âḃ05

283

.192

.078

1.470

.283

.10

.402

.10

402

.282

.141

1.426

.402

.15

.488

,

-352

.199

1-385

.488

.20

âḃ 556

.20

556

1.420

1.965

-413

âḃ255

1-347

.556

âḃ 25

.613

.466

.308

I-315

.612

âḃ 30

.658

.516

.359

1.276

.658

â 35

âḃ 695

-561

.410

1.238

.697

.40

âḃ 730

âḃ 40

730

1-383

1. 816

.609

âḃ459

1.208

.730

âḃ45

âḃ757

.645

âḃ507

1. 174

.757

.50

.780

.683

-555

1. 142

.780

âḃ55

.800

.720

.602

1. 114

.799

.60

.814

âḃ 755

.648

1.078

.814

âḃ65

.826

âḃ 789

âḃ693

1.047

.826

.70

.835

.822

.738

1. 016

âḃ83s

.75

.840

âḃ 854

.783

0.984

.840

.80

.845

.80

845

1-313

1.520

.885

.829

0.955

.846

In column 3 the values of x are selected in geometrical ratio
and the corresponding values of y are given in column 4. The

points /Z^Âḟld^j are plotted in Fig. 22, and although the

\ yt yt /

* See Rateau's " Flow of Steam Through Nozzles."

DETERMINATION OF CONSTANTS

67

three points do not lie exactly on a straight line the approxima-
tion is good. The slope of the line is 4.10 and the intercept
â3.86 which give the equations

1.50

^1.30
O

1 1.20
^UO

LOO

.90

2^^4-2'^ =4.10,

3-86.

Values of X'^^

.10 .20 .30 .40 .50 .60 .70 .80 .90

\

N

\

N

X

H

\

Si

s

N

N

V

N

\

y

N

v/

/

/

^s

N,

/

/

/

/

y

/

/

/

y

y

1.30

1.32

1.3J

Values of

2.0O

1.80^

1.70

1.60

1.40 1.^

Fig. 22 AND Fig. 23.

Solving these equations the values of c and d are found to be
6 = 1.40,
^ = âḃ55-

The formula now is

68 EMPIRICAL FORMULAS

Dividing both members of this equation by x-^^

which represents a straight line when -^ is plotted to r^*.

The slope of the line is equal to a and the intercept equal to b.
From Fig. 23

a=- .685,

6 = 1.522.

The formula after the constants have been replaced by their
numerical values is

>' = i.522r"-.685x^'*^

The last column of the table shows that the fit is quite
good.

If the errors of observation are so small that the values
of the dependent variable can be relied on to the last figure
constants in empirical formulas. But when the values can not
be so relied on, or when the data must first be leveled graphically
or otherwise, the employment of derivatives may lead to very
erroneous results. This will be illustrated by two examples
worked out in detail.

The first step in the process is to write the differential equa-
tion of the formula used and then from this equation find the
values of the constants.

Consider the formula

y = eÂḞ'(c cos bx+d sin bx).

Looking upon a and b as known constants and c and d
as constants of integration, the corresponding differential
equation is

y' â 2ay-]-{a^-\-b^)y = o.

Dividing this equation by y

DETERMINATION OF CONSTANTS

69

which, if the data can be represented by XVIII, represents a
straight Hne whose slope is 2a and whose intercept is â{a?-^-})^).
Corresponding values of x and y are given in the table.

y

y'

y"

/

yl

X

g.06z

cos .o?>x

tan .080;

y

X

g.OBx

y

y

De-

Min-

cos .08a;

grees

utes

+

.3000

I. 0000

I. 0000

.0000

+

.3000

-.

â 4-

.2750

4
9

35-02
10.04

I 0618

.9968
.9872

0802

+
+

.2598
.2193

2

1

.2441

-.0342

-.0068

-

.1401

-.0279

1. 1275

.1614

3

+

.2065

â . 04 39

-.0066

â

.1976

-.0319

13

45 -06

I. 1972

.9713

.2447

+

.1776

4

+

.1622

-.0481

-.0078

â

. 29b5

â .0481

18

20.08

I. 2712

.9492

.3314

+

.1344

.â )

+

.1102

- OS57

- 0075

â

.5054

-.0681

22

55- 10

1-3499

.9211

.4228

4-

.0886

6

+

.0506

- 063s

-.0086

â I

.2549

â .1700

27

30.12

1.4333

.8870

.5206

+

.0398

7

â

.0175

-.0721

- . 0080

+4

.1200

+.4571

32

05-14

1.5220

.8472

.6270

.0136

8

â

.0937

-.0805

-.0087

+

.8591

+ .0928

36

40.16

1.6161

.8021

.7446

â

.0723

9

â

.178b

-.0894

â .0091

+

âḃ SOU

+ .0510

41

15.18

I. 7160

.7519

.8771

â

.1384

10

â

.2726

-.0985

â .0091

+

.3247

+ .0334

45

50.20

I. 8221

.6967

1.0296

â

.2147

II

â

â ^iv

-.1078

-.0085

+

.2869

+ .0226

50

25.22

1.9348

.6372

I . 2097

â

.3047

12

â

.4881

-.1168

-.0087

+

.2393

+ .0178

55

00.24

2.0544

.5735

1.4284

â

.4143

13

â

.6093

-.1257

â .0091

+

.2063

+ .0149

59

35.26

2.1815

.5062

I . 7036

â

.5518

14

â

.739b

-.1348

-.0089

+

.1823

+ .0120

64

10.28

2.3164

âḃ 4357

2.0659

â

.7328

15

â

.8788

-âḃ1435

-.0084

+

. Ib33

+.0096

68

45.30

2.4596

.3624

2.5722

â

.9859

Tft

73
77

20.32
55-34

2.6117
2.7732

.2867
.2098

3-3414
4-6735

âḃ 3707
.0306

â I

.1814

â2

The values of y' and y" are obtained by the formulas

y'n =âr {yn-2-Syâ-i+ Syn+i-yn+2),
i2n

y"n= -^^ (>-2-i6yâ-i+30>'n-i6yâ+i+yâ-f2),

where /? = Ax = i . These formulas are derived in Chapter VI.

Plotting the points represented by ( â ,â), Fig. 24, it is

\y y I

seen that they lie nearly on a straight line whose slope is .12

and intercept â.01. Therefore

We have then

2a = .i2,-
a = .06,

y = e-^^^{c cos .o2>x-\-d sin .o8x).

70 EMPIRICAL FORMULAS

Dividing this equation by c^^ cos .oSx

oto ^ â ^ = ^+^ tan .oSx.
e"Â®* cos .o8jc

-1.2 -1

-.

8 -.

6 -i4 -.

2 -

\jii

2 .

4

6 .8 1

/

/

/

/

/

/

/

/

A

T

/

/

/

r

/

/

/

/

/

/

/

/

A

Fig. 24.

This represents a straight line when

e-^^^ cos .o8x

is plotted

to tan ,Q%x. The slope is d and intercept c. From Fig. 25

c= .3.
The law connecting the variables is represented by

^ = e06a;(^_2 (.Qg q3^_^^ sin .080;).

DETERMINATION OF CONSTANTS

71

The values of y computed from this formula agree with
those given in the second column of the table.
Consider formula XIX

y =^ ax'' -\-hy^ .

The corresponding differential equation is

y

= (c+^-]

y

-cd^

M

e0.cx

cos

A

V

.2

\

k.

i

\

\

Ji
-.2 o

\

\

\

\

P

f

\

\

-A

\

\

1

\

\

l.ÂṠ

1

\

\

N

\

\

\

\

n A

\

1

\a

uesc

f tan

3
MX

^

Fig. 25.

where c and d are known constants and a and h constants of
integration. The differential equation represents a straight line

The slope is c-\-d â i and the inter-

oc^y . XV

when â^â is plotted to â

X V

cept is âcd.

The values of x and y in the table below are the same as
those given in the discussion of XIX.

72

EMPIRICAL FORMULAS

/

//

xy

*y'

X

y

y

'

y

y

OS

.283

lO

.402

IS

.488

1.503

-6.933

.462

-.320

20

.556

1.240

-4.133

.446

-.297

2S

.613

i.ois

-4.967

.414

-.506

30

.6s8

0.803

-3.267

.366

-âḃ447

35

.695

0. 720

â0.400

.363

-.071

40

.730

0.623

-3. 533

.341

-.774

45

âḃ757

0.492

â 1.500

.292

-.401

50

. .780

0.433

-1.067

âḃ 277

-âḃ342

55

.800

0.338

-2.633

.232

-.996

60

.814

0.255

-0.633

.188

-.280

65

.826

0.213

-1.200

.167

-.614

70

.835

0.135

-1.767

âḃ 113

-1.037

75

.840

80

.845

The values of y' and y" were computed by means of the
formulas used in the preceding table. In Fig. 26 the points

( _Z. _^_ I are plotted and, as is seen, the points do not deter-

\y y /

mine a line. It is clear that the constants can not be determined
by this method.

XlXa. y==dx"(f.

Points represented by ixn, log â â ) lie on a straight line.

\ yn /

If two variables, x and y, are so related that when values of x
are taken in a geometrical series the corresponding values of y are

such that the points represented by ( Xn, log ^^^^ ) lie on a straight

\ yn I

line, the relation between the variables is expressed by the equation
XlXa y = aa;V. ' .

Using logarithms:

log yâ = log a+6 log Xn-\-Xn log c,
log yâ+i =log a-\-b log Xn^rrxn log c-^b log r.

DETERMINATION OF CONSTANTS 73

Subtracting the first equation from the second
log =^^^â = {r- i)a:n \og c-\-h log r.

By plotting log ^^ to it^n a line is obtained whose slope is

* . ^" .
(r â i)log c, and since r is known c can be determined.

-.1

ÂḞ

.

ÂḞ

ÂḞ

ÂḞ

o

g-6

-s8

-.9

-1.0

o

ÂḞ

.1 .2

Values of

Fig. 26.

From the first equation

log Jn - Xn log âỲ = 1) log Xâ + log a.

If then log jnâXn log c be plotted to log Xn a line is obtained
whose slope is h and whose intercept is log a.

CHAPTER V

XX. y=oo+ai cosar+flacos 2x4-a3C0S3x+ . . . -\-aTCosrx

-\-bisinx-]rb2sm 2X-{-bz^n$x-\- . . . +6r_iSm (râ i)x. Values of y periodic. The right-hand member of XX is called a Fourier Series when the number of terms is infinite. In the application of the formula the practical problem is to obtain a Fourier Series, of a limited number of terms, which will represent to a sufficiently close approximation a given set of data. The values of y are given as the ordinates on a curve or the ordinates of isolated points. In what follows it is assumed that the values of y are periodic and that the period is known. We will determine the constants in the equation y = ao+ai cos x-\-a2 cos 2x-\-a3 cos sx,+bi sin x+^2 sin 2x, so that the curve represented by it passes through the points given by the values in the table. X oÂḞ 60ÂḞ 120ÂḞ 180ÂḞ 240ÂḞ 300ÂḞ 360ÂḞ y I.O 1-7 2.0 1.8 1-5 0.9 1.0 Substituting these values in the equation we have the fol- lowing six linear relations from which the values of the six constants can be determined: I.O = flO+ fll+ (l2-\-(l3, i'7 = ao-\-iai-la2-as-\ â ^h-] ^^>2, 2 2 "X/o \/-3 2.O = fl0-Â§fll-|Âḋ2 + a3-^| ^^1 -b2f 2 2 74 DETERMINATION OF CONSTANTS 75 i.S=ao-\ai-\a2+az ^61 H â ^62, 2 2 \/-2 \/o 2 2 Multiplying each of the above equations by the coefficient of ao (in this case unity) in that equation and adding the result- ing equations we obtain (i) below. Multiplying each equation by the coefficient of ai in that equation and adding we obtain (2). Proceeding in this rnanner with each of the constants a new set of six equations is obtained. 6^0 = 8.9. (i) 3^1 =-1.25. ....... (2) 3(Z2=-.25 (3) 6a3 = .io (4) 36i = .65V3. (S) 3^2 = .i5V'3 (6) The equation sought is y = ^âT2Cosx-^2 cos 2:r+-^V cos sx+M^3 sin x+^o^s sin 2X. It reproduces exactly each one of the six given values. The solution of a large number of equations becomes tedious and the probability of error is great. It is, therefore, very desirable to have a short and convenient method for com- puting the numerical values of the coefficients.* *The scheme here used is based upon the 12 -ordinate scheme of Runge. For a fuller discussion see "A Course in Fourier's Analysis and Periodogram Analysis " by Carse and Shearer. 76 EMPIRICAL FORMULAS Take the table of six sets of values X oÂḞ 60ÂḞ 120ÂḞ i8o^ 240ÂḞ 300ÂḞ y yo yi y2 ^3 y^ ys where the period is 27r. For the determination of the coefficients the following six equations are obtained: 2 2 >'2 = ao-Â§ai-Â§fl^2+a3H â ^61 -h2i 2 2 >'3=fl0 â fll + fl2â fl3, _ â y4. = ao-\ai-\a2-\-az ^hi-\ ^62, 2 2 3'5 = ^ + Jfl^l â 1^2 â ^3 V ^^1 2 Proceeding in the same way as was done with numerical equations the following relations are obtained: 6^0=3^0+ 3^1+ 3'2+3'3+ >'4+ yby 2,ai=yo+ \yi- \y2-yz- ly4.-\-hyb, 2,a2=yo- hyi- Â§3'2+>'3- b4-j3'5, 6a3=>'o- yi-\- >'2-3'3+ 3^4- 3'5, 2 3^>i + â 3'iHâ ^3'2 2 2 ^^3 '>'5, 3^2= H ^>'l ^>'2 2 2 H-^^3'4 2 >'5. (a) For convenience in computation the values of y are arranged according to the following scheme: yo yi y2 n 3^4 3^ Sum vq vi V2 Difference wo wi W2 DETERMINATION OF CONSTANTS 77 vo Vl V2 Wo Wi W2 Sum Difference pQ Pi Sum Difference ro Sl (b) 6ao = po-^pi, 1 Sai=roi-isi, 2,(^2= po â \pi, 6a3 = ro-si, 2 3^2 = â ^^1. 2 It is evident that the equations in set (ft) are the same as those in set (a). For the numerical example the arrangement would be as follows : i.o 1.8 1-7 1-5 2.0 0.9 VO Wo 2.8 2.8 -.8 3-2 2.0 3-2 .2 2.9 I.I -.8 .2 I.I po 2.8 6.1 To âḃ3 ^1 6ao=+8.90, 3^1 = -1.25, 3^2=- .55, 6^3 = + .10, 3Z>i = + .65 V3, 3^2 = + .i5V'3- -.8 1-3 -âḃ9 78 EMPIRICAL FORMULAS It is seen that the computation is made comparatively simple. The values of the z;'s are indicated by vo, the first one. The values of the p\ etc., are indicated in the same way. 8-ORDiNATE SCHEME. The formula for eight ordinates which lends itself to easy computation is y = ao+ai cos 6-\-a2 cos ad+az cos 3^4-^4 cos 4^ +61 sin 0+62 sin 2d-\-b3 sin$d.

For determining the values of the constants eight equations
are written from the table:

d

o<-iO 00

45 90 135 180 225 270 315

y

yo yi y2 ys y* ys ye yi

yo = ao+ ai+a2+ ^3+^4,

\/2 \/2 \^2 V 2

^2222

y2 = ao âci-2 +^4+ ^1 â ^3,

'v/2 V^2 V 2 V 2

y3 = cio fli H ^3â^4 4 ^1â^2 4 263,

^2 2 2.2

y4 = flo- a\-\-a2â <i3+<^4,

'\/2 ^^2 V'2 'V^2

y5 = a0 Âḋ1 H ^3 â ^4 61+^2 ^3,

2 2 2 2

y6=flk) â Âḋ2 +^4 â ^1 + ^3,

-\/2 ^f 2 V 2 V 2

y7 = floH (i\ az â a\ b\ â b2 63.

22 22

From which are obtained the following eight equations:

8ao=yo+ yi+y2+ y3+y4+ y5+y6+ y?,

\/2 V2 V2 , V'2

4^i=>'oH yi y3-y4 ys H y?,

2222

âḃ 4^2 =yo â y2 +y4 âye,

\/2 . V2 , V2 V2

4^3 = yo y 1 H y3 - y4 H ys y?,

2222

DETERMINATION OF CONSTANTS 79

Sa^=yo- yi+3'2- >'3+3'4- y5-\-y6- yi,

V'2 '\/2 \^2 ^2

461= â y\^y2^ yz y^-y^ y-i,

2222

4^2= y\ - yz -V yb - yi,

y/ 2 v'2 'v/2 V2

4^3= â >'i-3'2H >'3 y5+y& yi-

2222

For the purpose of computation the values of y will be
arranged as follows:

yo yi y2 y^

yA yo yo, yi

Sum

Vo

Vi V2

^^3

Difference

Wo

Wl W2

W3

Vq

Vl

Wo

Wl W2

V2

âḃvs

Ws

Sum

Po

pi Sum

ro

ri n

Difference

Qo

qi Difference

Sao = po+ pi, .

s\ ^0

V~2

4ai=ro;i ^i, '

2

L^C' ^

4a2 = qo,

V-2

4(^3 =ro, si,

2

^a "

w,^.^>

Sa4:=poâpi,

^^ V2 -
4h=r2-\ riy

2 '

^\-fv^

, âḃ+ -' 1

4h = qi,

463 =-^2+ n.

'-^^

^^^ f.^3

The process will be made clear by an example :

45ÂḞ

90ÂḞ

135ÂḞ

180ÂḞ

225ÂḞ

270ÂḞ

315ÂḞ

360ÂḞ

4

â 2

â I

2

3

3

â I

2

4

80 EMPIRICAL FORMULAS

For computation the arrangement is as follows:

4

-2

â I

2

3_

3

â I

2

vo

7

I

â 2

4

Wo

I

-5

7

I

I

-5

â 2

4
5

ro

Po 5

I

-5

^0 9

-3

Sao =
4^1 =

4^2 =
4^3 =

8a4 =

4*1 =
4*2 =

4*3 =

Sl

10,
9,

I+fV2,
0,

-3,

-5

The formula becomes

y = i-25-

â .634.

cos d-\-

2.25 cos 2

^+1

.134 cos 3^

J

-.8841

sin^-

.75 sin 21

9- .

884 sin 3(9.

10

-Ordinate Scheme

yo

yi

>'2

>'3

3'4

:vo

3^8

>'7

3^0

y^

Sum Vo

Vl

V2

?'3

V4:

V5

Difference

Wi

W2

ws

W^

Vo

Vl

V2

â  Wi

W2

V5

va

vz

Sum

W4

Wz

Sum po

pi

P2

h

h

Difference ^o

Qi

Q2

Difference wi

m2

DETERMINATION OF CONSTANTS

81

ioao = po+pi+p2,
5ai=qo-\-Ciqi+C2q2,
Sa2= pQ-^C2pi â Cip2,
5a3=qo-C2qiâCiq2,

5^4 = pO â Cipl + C2p2y
10^5 = ^0 â ^1 + ^2,

561=51/1+52/2,

5&2=52Wl+5lW2,

563=52/1â51/2,
5^4 = 51^1â 52M2.

In the above equations

Ci = cos 36ÂḞ,
C2 = cos 72ÂḞ,

5i=sin36ÂḞ,
52 = sin 72ÂḞ.

In the schemes that follow, as in the lo-ordinate one, only
the results will be given.

12-Ordinate Scheme

3'c

)

y\
yn

y2
yiQ

ys

yQ

y^ y^

ys âḃ y7

ye

Sum

Vq

Vl

V2

V3

Va V5

VQ

Difference

Wi

W2

W3

W4 W5

vo

n

V2

n

Wi

W2 Wz

Vq

V5

^4

Sum

W5

Wa

Sum

Po

Pl

P2

pz

r\

r2 n

Difference

qo

qi

q2

Difference si

S2

p^

)

pl

n qo

ll

I

Ps

rs q2

Sum

Difference

/2

82 EMPIRICAL FORMULAS

I2flo = /o+/l,

2

6a2=Poâp3+h(j>iâP2)f

6a4^po+p3-i{pi-\-p2)f

6a5=qo -qi+^q2,

2

i2a6=/oâ /i,

2

V~z

6b2=-^(Si+S2),
2

663=/!,

6^4=â ^(51-^2),

2

2

16-ORDINATE Scheme

yo yi y2 ys y^. y^ y^ yi y^,
y\b yw y\z 712 yw 3^10 y^

Sum z'o

1^1

V2

vs

^4

V5

^6 V7 Vi

Difference

W\

W2

W3

W4:

W5

We wj

vo

Vl

V2

n

V4:

V8

V7

Vq

V5

Sum

po

Pl

P2

ps

P4.

Difference

qo

qi

q2

qs

DETERMINATION OF CONSTANTS 83

Wl W2. Wz Wa

âḃW7 Wq W5

Sum ri r2 rs ^4

Difference ^i S2 S3

po pi p2 lo h

pA p3 I2

Sum lo h I2 Sum to

Difference mo mi Difference xo

^ai=qo-\ g2+Cigi+C25'3,

2

V2

d>a2 = mo-\ Wl,

2

V2

8^3 = ^0 ?2 â Ciga +C2^i,

2

8^4=^0,

8a5 = go ^2+Cig3â C2^i,

2

8a6 = wo Wl,

2

V2

8^7 = go H q2 â Ciqi â C2qs ,

2

i6a^ â toâhj

V2

8&i = r4H r2+Cir3-^C2rij

2

V2
8^>2=^2H (51+^3),

2

8&3= -^4H ^2 + Cm -^2^3,

2

8^4 = ^1â -^3 J

84

EMPIRICAL FORMULAS

V2

865 = fA r2 +Ciri - C2r3,

2

V2

860= -^2 H (^1+53),

2

8^7= -r4 r24-Cir3+C2ri.

Ci = cos :

221ÂḞ =

sin

67r,

C2 = sin :

22^ =

cos

671ÂḞ.

20-Ordinate

Scheme

yo

yi

y2 ys

y^

y5

y6 yi

ys

y^ yxz

yi9

yi8 yi7 yi6

yi5

3^14 3^13

y\2

y\\

Sum

1^0

Vl

V2 V3

V4:

V5

Vo Vl

V8

Vo V\o

Difference

Wi

W2 W3

; W4.

W5

Wo W7

Ws

Wo

vo

n

V2

V3

V4. \

V5

^10

Vd

V8

V7

Vo

Sum

Po

pi

P2

P3

P^ .

P5

Difference

qo

Qi

q2

?3

^4

âḃ

Wi

W2

Ws

'

W4 W5

.

W9

Wg

Wi

Wo

Sum

ri

r2

rs

ta r5

Difference

Sl

S2

S3

S4:

po

Pi

P2

qo qi

?2

P5

P^

ps

qA gs

Sum

/o

h

I2

Sum

ko ki

T2

Difference

Wo

mi

W2

/o

Wo

Wl

ri rz

/l

W2

rs

/2

Sum

/o

Sum

no

m

Sum

Ol 03

DETERMINATION OF CONSTANTS 85

Si

S2

S4

S3

gl

g2

hi

h2

Sum
Difference

20^0 = /Oj

loai =^0+^1 sin j2ÂḞ-\-q2 sin 54ÂḞ+g3 sin 36ÂḞ+?4 sin i8ÂḞ,

ioa2=mo+mi sin 54ÂḞ+W2 sin i8ÂḞ,

ioa3 = qo â q3 sin 72ÂḞ â g4 sin 54ÂḞ+gi sin 36ÂḞ â 5^2 sin 18ÂḞ,

10^4 = /o â ^2 sin 54ÂḞ+/i sin 18ÂḞ,

10^5 = ^0 â ^2,

Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 4 of 8)