Theodore R. (Theodore Rudolph) Running.

Empirical formulas online

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Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 5 of 8)
Font size ioaQ = mo â€” m2 sin 54Â° â€” mi sin 18Â°,

10^7 = ^0+^3 sin 72Â° â€” ^4 sin 54Â° â€” ?i sin 36Â°â€” ^2 sin 18Â°,

ioa8 = /oâ€” ^1 sin 54Â°+/2 sin 18Â°,

10^9 = ^0â€”^1 sin 72Â°+g2 sin 54Â° â€” ^3 sin 36^+^4 sin 18Â°,

2oaio=woâ€” wi,

1061 =f5+^4 sin 72Â°+r3 sin 54.Â°+/'2 sin 36Â°+ri sin 18Â°,

io&2=g2 sin 72Â°+gi sin 36Â°,

1063= â€” f5+^2 sin 72Â°+ri sin 54Â° â€” r4 sin s^Â°+^3 sin 18Â°,

io&4 = /?i sin 72Â°+/j2 sin 36Â°,

I0b5 = 0iâ€”03,

loh^gi sin 72Â°â€” g2 sin 36Â°,

ioh=â€”r5â€”r2 sin 72Â°+^ sin 54Â°+r4 sin 36Â°+f3 sin 18Â°,

10&8 = â€”h2 sin 72Â°+/ji sin 36Â°,

iob<d = r5 â€” r4. sin 72Â°+r3 sin 54Â°â€” /'2 sin 36Â°+ri sin 18Â°.

24-OKDiNATE Scheme

yo yi y2 y3 y4. ys y^ yi y^ jg >'io y\\ yi2
y23 y22 y2i y2o yi^ yis yn ym yib yi4: yi3

Sum Vq Vi V2 V3 V4. V5 Vq V7 Vs Vq 2^10 Z'll Z'12

Difference wi W2 ws ^4 w^ wq wi ws wq wio wn

86

EMPIRICAL FORMULAS

vo

Vl

V2

V3

V4

V5

VQ

1'12 V]

11 fio

Vo

V8

Vl

Sum

Po

Pl p2

P3

P^

P5

po

Difference

qo

Qi 92

^3

94

95

Wi

W2

W3

W4,

W5

w&

wn

Wio

Wq

Ws

W7

Sum

ri

r2

rs

n

rr,

To

Difference

Sl

S2

S3

S4:

S5

Po

pi

p2 p3

Sl

S2

S3

p6

P5

P^

S5

S4

Sum

/o

h

k h

Sum

Vl

k2

k3

Difference

nio

mi

m2

Difference

ni

W2

h

h

Wc

)

mi

h
go

l2

gl

Sum

m:

>

Sum

Co

ei

Difference

ho

hi

Difference

/o

24ao=go-\-gi,

12^1 =^0 + ^^4 + 1 V2g3 + |V^392+Cl^l+C2g5,

i2a2 = wo+iw2+|V3mi,

12^3 = 90-94 + 2^^2(^1 -gs-^s),

I2a4 = /f0 + pl,

i2a5 = qo+C2qi-W3Q2-W2qs + 2Q4:-\-Ciq5,

12^6 =/o,

i2a7 = qo-C2qi â€” ^^3q2 + 2"^2qs-\-^q^â€”Ciq5y
i2a8 = go-kij

I2a9 = go-94 + |V2(-gi+g3+g5),

i2aio = mo-\-^m2-iy^Smij

i2aii=qo-Ciqi+^y^q2-iV2q3+^q4,~C2q5j
24ai2=ho-hil

DETERMINATION OF CONSTANTS

i26i=CVi+ir2+|v'2r3+Â§v'3r4+Cir5+r6,

1 2^>2 = Â§^1+1^3^2 + ^3,
I2^>3=r2-r6 + |V2(n+^3-^5)>

I2b4: = ^^s{ni+n2),

i2b5 = Ciri-^lr2-l^ 2n-lV y4,-\-C2r5+rQ,

I2bQ = kiâ€”k3,

i2b7 = Ciri â€” ^r2-W2^3+Ws^4.+C2r5-rQ,

I2&8 = |V^3(^l-^2),

i2b9=rG-r2+i^2{ri+r3-r5),

I2Z>io = 53+K^l+^5)-|V3(52+54),

i2bn = C2ri-^r2+iV^f'3-2^3^4.+Cir5-rQ,

V3 + 1

87

Ci

V^

â€˘96593.

C2=^^^V^ = . 25882.

2V 2

As an illustration let it be required to find a Fourier series
of 24 terms to fit the data given in the table below.

x"

y

xÂ°

y

â–  xÂ°-

y

xÂ°

y

00

149

90

159

180

178

270

179

15

137

105

178

195

170

285

185

30

128

120

189

210

177

300

182

45

126

13s

191

225

183

315

176

60

128

150

189

240

181

330

166

75

135

i6s

187

255

179

345

160

149 137 128 126 128 135 159 178 189 191 189 187 178

160 166' 176 182 185 179 179 181 183 177 170

1^0 149297 294 302 310 320 338 357370374366357178

^1 -23 -38 -50 -54 -50 -20 -I 8 8 12 17

88 EMPIRICAL FORMULAS

149

297

294

302

310

320

338

178

357

366

374

370

357

Po

327

654

660

676

680

677

338

^

-29

-60

-72

-72

-60

-37

-23

-38

-50

-54

-50

â€” 20

17

12

8

8

â€” I

n

- 6

-26

-42

-46

-51

â€” 20

Sl

-40

-50

-58

-62

-49

327

654 660 676

-

-40 -

50 -58

338-
665 ]

677 â€˘ 680
[331 1340 676

^1

-

-49 -

62

h

-89 -

112 -58

wa

â€” II -

-23 -

20

ni

9

12

665

1331

â€” II

-23

^0

676
1341

1340
2671

eo

â€” 20

-31

-23

h

â€” II

-9

/o

9

The formula becomes

3^ = 167.167 â€” 19.983 cos ic â€” 3. 410 cos 2X+5.470 cos 3a;

â€” 1.292 cos 40;+. 249 cos 5::t:+.75 cos 6:x: + .3io cos yx
+ .458 cos 8a: â€” .304 cos 9:^- .090 cos lorc- .243 cos iirc

â€” .083 cos 12:^-12.779 sin X â€” 16.624 sin 2x â€” .323 sin 3^1;
+ 1.516 sin4x+i.46i sin 50; â€” 2.583 sin 6X+.321 sin yx

â€” .216 sin 8X+.676 sin 9X â€” .459 sin loic â€” .639 sin iiic.

In what precedes the period was taken as 2t. This is not
necessary; it may be any multiple of 2t. The process of finding
a Fourier series of a limited number of terms which represent
data whose period is not 2ir will be best set forth by an example.
In the table below the period is t/^ and the values of y are
given at intervals of 7r/i8. The 12-ordinate scheme can be
used by first making the substitution

x=^9 or d = ^x.

DETERMINATION OF CONSTANTS

xÂ°

0Â°

y

xÂ°

0Â°

y

:Â»;Â°

eÂ°

y

oo

00

+27.2

40

120

+9-8 â–

80

240

-ii-S

lO

30

+34-5

SO

150

+ 8.5

90

270

-17s

20

60

+ 21.5

60

180

+0.2

100

300

-17.2

30

90

+ 10. 1

70

210

-7.1

no

330

+ 1. 5

27.2

34-5

21.5

10

I 9.8

8.5 0.2

1-5 -

-17.2 -

17

â– 5 -II-5

-7-1

vo

27.2

56.0

4-3 â–

-7

4 -1.7

1.4 0.2

Wi

33-0

38-7

27

6 21.3

15.6

27.2

36.0

4-3

7.4

Po

0.2

^â€˘4

-1-7

27.4

37-4

2.6

7-4

Qo

27.0

34-6

6.0

33 -o

38.7

27.6

27.4

37.4

15.6

21.3

2.6

-7-4

ri

48.6

60.0

27.6

lo 30.0

30.0

Sl

17-4

17-4

48.6
27.6

27.0
6.0

21.0

21.0

The formula is

3^ = 5+9.994 cos ^+8.7 cos 2^+3.5 cos 3^+.oo6 cos 5^
+ 17.31 sin ^+5.023 sin 2^+3.5 sin 3^ â€” .01 sin 5^.

Replacing 6 by its value t^x,

3' = 5+9.994 cos 3ii;+8.7 cos 6X+3.5 cos 9ii;+.oo6 cos 15^1;
+ 17.31 sin 3a;+5.o23 sin 6a-+3.5 sin 9X â€” .01 sin 15:31;.

CHAPTER VI

EMPIRICAL FORMULAS DEDUCED BY THE METHOD
OF LEAST SQUARES

In the preceding chapters we computed approximately the
values of constants in empirical formulas. The methods em-
ployed were almost wholly graphical, and although the results
so obtained are satisfactory for most observational data, other
methods must be employed when dealing with data of greater
precision.

It is not the purpose of this chapter to develop the method
of least squares, but only to show how to apply the method to
observation equations so as to obtain the best values of the
constants. For a discussion of the subject recourse must be
had to one of the numerous books dealing with the method of
least squares.*

It was found in Chapter I that the equation

y = a-\-bx-{-cx^ (i)

represents to a close approximation the relation between the
values of x and y given by the data

X

y

X

y

o

3-I950

â€˘5

3.2282

.1

3.2299

.6

. 3-1807

.2

3-2532

â€˘7

3.1266

â€˘3

3.2611

.8

3-0594

4

3-2516

.9

2-9759

* Three well-known books are: Merriman, Method of Least Squares;
Johnson, Theory of Errors and Method of Least Squares; Comstock,
Method of Least Squares.

90

DEDUCED BY THE METHOD OF LEAST SQUARES 91

where x represents distance below the surface and y represents
velocity in feet per second.

Substituting the above values of x and ym (i), the following
ten linear observation equations are found :

a-\- oh-\- 0^ = 3.1950,
a+. lb + .01^ = 3.2299,
a+.2&+.o4c = 3.2532,
Â«+.3^+.09^ = 3-26ii,
^+.46+. 16^ = 3.2516,
(I+.55+. 25^=3. 2282,
a+.65+.36c = 3.i8o7,
Â«+.7^+49^ = 3-i266,
fl+.8J+.64c = 3.0594,
a+.96+.8ic = 2.9759.

Here is presented the problem of the solution of a set of
simultaneous equations in which the number of equations is
greater than the number of unknown quantities. Any set of
three equations selected from the ten will suffice for finding
values of the unknown quantities. But the values so found
will not satisfy any of the remaining seven equations. Since all
of the equations are entitled to an equal amount of confidence
it would manifestly be wrong to disregard or throw out any one
of the equations. Any solution of the above set must include
each one of the equations.

The problem is to combine the ten equations so as to obtain
three equations which will yield the most probable values of
the three unknown quantities a, &, and c. It is shown in works
on the method of least squares that the first of such a set of
equations is obtained by multiplying each one of the ten equa-
tions by the coefficient of a in that equation and adding the result-
ing equations. The second is obtained by multiplying each one
of the ten equations by the coefficient of h in that equation
and adding the equations so obtained. The third is obtained
by multiplying .each of the ten equations by the coefficient of

92 EMPIRICAL FORMULAS

c in that equation and adding the equation so obtained. The
process of computing the coefficients in the three equations is
shown in the table. The coefficients of a, 6, and c are represented
by i4, 5, and C respectively, and the right-hand members are
designated by iV. The number 5, which stands for the numeri-
cal sum oi A,B,C and iV, is introduced as a check on the work.
It must be remembered that this method of finding the values
of the -constants holds only for linear equations.

The sum of the numbers in the column headed AA=ZAA
= io. The sum of the numbers in the column headed AB =
2i4B=4.5. The sum of the numbers in the column headed
-4C = 2^C = 2.85. Also the sum of the numbers in the column
headed AN ^llAN = ^i.'j6i6. These sums give the coefficients
in the first equation.* The second and third equations are
obtained in the same way.

The three equations from which we obtain the most probable
values of the constants are:

10 a -\- 4.56 +2.85C =31.7616;
4.5a -f2.85^> -f 2.025c =14.08957;
2.85a -1-2.0256+1.5333^= 8.828813.

These are called normal equations. From them are obtained

^=+3-19513;
b=+ .44254;
c=- .76531-
I'he check for the first equation is

i:AA-\-lAB+i:AC+XAN = ZAS = 4g.iii6;
lor the second equation

2^5-f255 + 25C+25iV = 2^5 = 23.46457;
for the third equation

2^C-|-25C-[-2CC+2CA^=2C5=i5.237ii3.
* Cf. Wright and Hayford, Adjustment of Observations.

DEDUCED BY THE METHOD OF LEAST SQUARES

93

AA

AB

^C

.liV

AS

o

I

2

3
4
5
6

7
8

9

o

OI

04
09
16

25

36

49
64
81

3-1950
3.2299

3-2532
3.2611
3-2516
3.2282
3-1807
3.1266
3-0594
2.9759

5
5
5

5

1950
3399
4932
6511
8116
9782
1407
3166
4994
6859

lO

4-5

2.85

31.7616

49. I I 16

AB

BB

BC

BN

BS

.01

.001

.32299

â€˘43399

04

.008

.65064

89864

09

.027

.97833

I

39533

16

.064

1.30064

I

92464

25

.125

1.61410

2

48910

36

.216

I . 90842

3

08442

49

â€˘343

2.18862

3

72162

64

.512

2.44752

4

39952

81

â€˘ 729

2.67831

5

11731

4-5

2.85

2.025

14.08957

23-46457

AC

BC

CC

CN

CS

.0001

.032299

â€˘043399

.0016

.130128

.179728

.0081

â€˘ 293499

.418599

.0256

.520256

- 769856

.0625

. 807050

I . 244550

.1296

I. 145052

1.850652

.2401

1.532034

2.605134

.4096

I. 958016

3.519616

.6561

2.410479

4-605579

2.85

2.C

25

1-5333

8.828813

15-

237113

94

EMPIRICAL FORMULAS

The formula is

3; = 3.i9Si34-.44254^-. 76531^-
For the purpose of comparison the observed values and the
computed values are written in the table, v (called residual)
stands for the observed value minus the value computed from
the formula.

X

Observed

y

Ccmputed

y

V

Â»2

3 1950

3-1951

â€” .0001

.00000001

.1

3-2299

3-2317

â€” .cx>i8

.00000324

2

3-2532

3-2530

+ .0002

.00000004

3

3.2611

3-2590

+ .0021

.00000441

4

3-2516

3-2497

+ .0019

.00000361

S

3.2282

3-2251

+ .0031

.00000961

6

3-1807

3-1851

â€” .0044

.00001936

7

3.1266

3.1299

- 0033

.00001089

8

3 -0594

3-0594

.0000

.00000000

9

2-9759

2-9735

+ .0024

.00000576

+ 0001 .00005493

This method derives its name from the fact that the sum of
the squares of the residuals is a minimum. A discussion of
this will be found in the books referred to above.

In case the formula selected to express the relation between
the variables is not linear the method of least squares cannot
be applied directly. In order to apply the method the formula
must be expanded by means of Taylor's Theorem. Even when
the formula is linear in the constants it may be advantageous
to make use of Taylor's Theorem. In order to make this trans-
formation clear we will apply it to the formula just considered.

Suppose that there have been found approximate values of
a, b, and c, ao, bo and cq, say, then it is evident that corrections
must be added in order to obtain the most probable values of
the constants. Let the corrections be represented by Aa, Ab,

and Ac. And let

a = ao-\-Aa,

b = bo+Ab,

c = Co-{-Ac.

DEDUCED BY THE METHOD OF LEAST SQUARES 95

The formula was

y=a+bx+cx^.

This may be written

y=f{a, b, c) =f(ao+Aa, 60+A&, co+Ac).

Expanding the right-hand member

/(ao+Aa, bo+Ab, co-\-Ac)=f{ao,bo,Co)+-^^Aa-\-^lAb+^Ai

ddo doo dco

+-^^(AaAc)+^i-{AbAc)]+ . . .
8^0 9co doodco J

where -^ stands for the value of the partial derivative of
doo

f(a, b, c) with respect to a and ao substituted for a, ^^-^ stands

for the value of the second partial derivative of /(a, b, c) with
respect to a and ao substituted for a, etc. If qq, bo, and co have
been found to a sufficiently close approximation the second and
higher powers of the corrections may be neglected.

dao
dbo '

The formula becomes

y-f{ao,bo,Co)=^Aa+^fAb+^Ac,

9Â«o dOo dco

or

yâ€”(oo-j-box+ cox'^) = Aa + xAb + x~Ac.

96 EMPIRICAL FORMULAS

Selecting for the values of ao, bo, and co those found in
Chapter I, the new set of observation equations are

Aa-\- oAb-\- oAc= .0002,

Aa+.iAb+.oiAc= â€”.0013,

Aa+.2Ab-\-.04Ac= .0008,

Aa+.:^Ab+.ogAc= .0027,

Aa-\-.4Ab-\-.i6Ac= .0024,

Aa-\-.sAb-\-.2sAc= .0034,

Aa-^.6Ab-^.\$6Ac= â€”.0045,

Aa+.jAb+.4gAc= â€”.0038,

Aa+.8A6+.64Ac= â€” .0010,

Aa + .gAb+.SiAc= .0007.

From these are obtained the three normal equations

ioAa+4.5 A& + 2.85 Ac=â€” .0004,

4.5Aa+2.85 A6+2.025 Ac=â€” .00203,

2.85Aa+2.025A6+i.5333A6:= -.002059.
Solving

Aa= +.00033,

A6= +.00254,

Ac =-.00531,

which added to the values of ao, bo, and co, give

a= 3-19513,

b= .44254,

^=-â€˘76531-
the same as just found.

The above process may be applied to linear equations con-
taining more than three constants. But as the method of pro-
cedure is quite evident from the above the general statement
of the process will be made with reference to equations con-
taining only three constants.

DEDUCED BY THE METHOD OF LEAST SQUARES 97

Let the observation equations be represented by

. aix-\rbiy-\rciz = ni pi,
a2X+b2y-\rC2Z = n2 p2,
asx+b3y-{-C3Z=n3 p3,

amX-\-hmy-\-CmZ = nTn pm-

The normal equations will then be

Upa^ ' X + i:pah â€˘ y + i:pac â–  z = Zpan,
i:pab'X+i:pb^-y+i:pbc'Z = i:pbn,
Xpac-x+Upbc-y-h^pc^ 'Z^llpcn,

where a, b, c, and n are observed quantities, and x, y, and z
are to be determined, pi, p2, p3 - - â€˘ pmSire the weights assigned
to the observation equations. In the problem treated at the
beginning of the chapter the weight of each equation was taken
as unity.

It was stated on a preceding page that when a formula to be
fitted to a set of observations is not linear in the constants it
must be expanded by Taylor's Theorem.

Take as an illustration a problem considered in Chapter IV.

The formula considered was

y.

=f{A,B,

m,n)

= Ax"'+Bx'',

a/.

dAa

=x^\

9/.

= x''\

9/.
9wo

= ^ox""

' log X,

a/.

dno

= BoX^''

logo:;

EMPIRICAL FORMULAS

y ^f(A , B, m, n) =/Uo, Bo, m, Â«o) 4-:^A^ -^~w^

df

df

dAo

dBo

dm) 'dm
y-f(Ao, Bo, m, no) =^A^ +^aJ5+|^Aw4-|^Aw.

The observation equations will be of the form

^^A^^^-^B+-^^m+^^n = y-yo.
dAo dBo dmo dm

Assume the approximate values found in Chapter IV.
A= 1.522,
5= -.685,

w= .55,
n= 1.4.

X

^0

Bo

logx

5ox"" log X.

OS

.10

â€˘15

.20

â€˘25

19

.28

â€˘35

â€˘ 41

â€˘47

02

.04

.07

.10

â€˘14

I

522

-

685

â€” 2

996

-2.303

-1.897

â€” 1 . 609

-1.386

-

88

- -99

â€” 1.02

â€” 1. 01

- .98

03

.06

.09

.12

â€˘ 14

.30
â€˘ 52
.19

â€” 1 . 204

- .94
â€˘ 15

X-

log a;

.4 o*"*" logic.
50*"Â° log X.

â€˘35
.56
.23
1.050
.90
.i6

.40

.45

SO

.60

.64

.68

.28

â– 3d,

.38

â€”0.916

-0.799

-0.693

â€” 0.

- .84

- .78

- .72

- .

â€˘ 17

.18

.18

The new observation equations become

.I9A^ + -02A5â€” .88Aw+.03A;i:
.28A^+.04A5â€” .99AW+.06AW
.3 5 A^ + .07 A5 â€” 1 .02 Aw + .09 Aw

.0004,
.0002,
.0001,

55

72

43

598

66

18

DEDUCED BY THE METHOD OF LEAST SQUARES 99

.4iA^+.ioA5 â€” i.oiAm+.i2Aw= .0000,
.4'jAA-{-.i4ABâ€” .98Aw+.i4Aw= .0013,
.52A^+.i9A5â€” .94Aw4 - i5Aw= â€” .0001,
.S6AA-{-.2^ABâ€” .9oAw+.i6A;z= â€” .0019,
.6oA^ + .28A5â€” .84Aw+.i7A;z= â€” .0016,
.64AA-\-.^^ABâ€” .78Aw+,i8Aw= â€” .0001,
.68A^+.38A5â€” .72Aw+.i8A;^= -.0001.
.72A^+.43A5â€” .66Aw + .i8Aw= .0011.

From these the four normal equations are obtained

2.96oA^ + i.32iA5â€” 4.637AW+ .8o6An= â€”.00071,

i.32iA^+ . 642 A5 â€” 1. 802 Aw 4- .359Aw= â€”.00031,

â€” 4.637A^ â€” 1.802A5+8.737AW â€” i.253A7^= +.00085,

.8o6A^4- .359AJ5 â€” 1.253AW+ .22iAw= â€”.00023.

From which

A^ = -.oo68,

Aj5 = +.oii2,
Aw =â€”.0022,
An= â€” .0070.

These corrections being applied the final formula becomes

3; = i.5i52a;*^478_ (5y^3^i.393^

CHAPTER VII

INTERPOLATION.â€” DIFFERENTIATION OF TABULATED
FUNCTIONS

Interpolation ^ '

In Chapter II we found that the formula

XL y=

.025+. 2525^4-2. 5ac2

represents to a fair degree of approximation the values of y
given by the data. Any other value of 3;, within the range of
values given, can be obtained in the same way. This rests on
the assumption that the formula derived expresses the law con-
necting X and y. For example, the value of y corresponding
to x = \.o^ will be

^".025 + .2525(i.05) + 2.5(i.05)2~Â°-^45.

When a formula is used for the purpose of obtaining values
of y, within the range of the data given it is called an inter-
polation formula. Interpolation denotes the process of calcu-
lating under some assumed law, any term of a series from values
of any other terms supposed given.* It is evident that empirical
formulas cannot safely be used for obtaining values outside
of the range of the data from which they were derived.

* For a more extended discussion of the subject the reader is referred
to Text-book of the Institute of Actuaries, part II (ist ed. 1887, 2nd ed.
1902), p. 434; Encyklopadie der Mathematischen Wissenchaften, Vol. I,
pp. 799-820; Encyclopedia Britannica; T. N. Thiele, Interpolationsrechnung.

As to relative accuracy of different formulas, see Proceedings London
Mathematical Society (2) Vol. IV., p. 320.

100

INTERPOLATION 101

There are two convenient formulas for interpolation which
will be developed.*

The first one of these requires the expression for yx+n in
terms of yx and its successive differences, yx represents the
value of a function of % for any chosen value of x, and yx+n
represents the value of that function when %-\-n has been sub-
stituted for X.

yx+\-=yx-^^yx\
yx^2-=yx-^i^yx^^{yx-V^y^

^yx-\r'2.^yx-^t^yx\
yx+ 3=yx+2Ayx+ A^yx +A(yx+ 2Ayx + A^yx)

=yx+3^y^+3^^y=^+^^y^'^

yx+4: = yx+3^yx+3^^yx+^^yx+A(yx+3Ayx+sA^yx+A^yx)
= yx+4^yx+6A^yx+4^^yx+A'^yx.

These results suggest, by their resemblance to the binomial
expression, the general formula

yx+n=yx+nAyx+^ -A^yx-\ â€” ^ r^ -A^yx+etc.

If we suppose this theorem true for a particular value of n,
then for the next greater value we have

, . .n(n â€” i).^ . n(n â€” i)(n â€” 2) ^o , ,
yx+n+i=yx+nAyx+ I ' A^yx+â€” p -A^yx+etc,

+Ayx+nA^yx+â€”, â€” ^A^^x+etc,

=yx+{n+i)Ayx-\- ^ I A^yx+- y-^ -A^yx+etc

The form of the last result shows that the theorem remains
true for the next greater value of n, and therefore for the next

* See Chapter III, Boole's Finite Dififerences.

102

EMPIRICAL FORMULAS

greater value. But it is true when w = 4, therefore it is true
when Â« = 5. Since it is true forn = 5 it is true when Â« = 6, etc.
If now o is substituted for x and x for n, it follows that

, ^ ,x(xâ€”l)^^ , x(xâ€”l)(xâ€”2)

|3

A^yo+etc.

If A'*^'. =0, the right-hand member of the above equation
is a rational integral function of x of degree n â€” i. The formula
becomes

, ^ ,x{xâ€”l)^^ , x{xâ€”l){x â€” 2) ^^

yx=yo+xAyo+-^, â€” -A^yo-\-â€” /^ -A^yo-\- . .

^ x(x-i){x-2) . â–  . (x-n4-2) ^â€ž_i
\n â€” i

(i)

Formula (i) will now be applied to problems. // must not
be forgotten that in applying this formula x is taken to represent
the distance of the term required from the first term in the series,
the common distance of the terms given being taken as unity.

I. Required to find the value of y corresponding to a; = 42 5
having given the values under XIX. In the interpolation
formula a; = .5.

yo y\ yi yz

â€˘730

'1S1

.780

.800

AyQ

.027

.023

.020

A^y^....

â€” .004

-.003

^y^

.001

y=yo^-hAyQ-{A^yQ-{-^A^yo

= .730+.oi35 + .ooo5 + .oooi

= .744.

This is the same as given by XIX.

2. Find the value of y corresponding to 0^ = 2.3. x in the
formula will have the value f if we take >'o=â€” .1826 when

X = 2.

INTERPOLATION

103

yo

yi

y2

. y^

J4

y5

-.1826 -

4463

-â€˘7039

-.9582

â€” I.2II9

-1.4677

-.2637 -

2576

-â€˘2543

-â€˘2537

- .2558

.0061

0033

.0006

â€”.0021

-.0028 -

0027

â€” .0027

.0001

.0000

â€”.0001

Ayo
A'^yo

-.i826+f(-.2637) + ^^^(.oo6i) + ^

(-Â«(-Â«

(-.0028)

i( i)( n( x)(.oooi)

24

-â€˘3417-

3. The following example is taken from Boole's Finite Differ-
ences. Given log 3.14 = .4969296, log 3.15 = .4983106, log 3.16 =
.4996871, log 3.17 = .5010593; required an approximate value of
log 3-I4I59-

>'o yi y2 y?,

^^yQ

.4969296

.4983106

.4996871

â€˘5010593

.0013810"

.0013765

.0013722

â€” .0000045

-.0000043

. 0000002

Here the value of x in the formula is equal to 0.159.

>;;.= .4969296+ (.159) (.0013810) +-!^^^^^i!^-^^^^^( - .0000045)

, â€˘l59(-l59-l)Cl59-2) ^ r.r.r.r^.\

i (^.0000002 )

6

= .4971495.

104 EMPIRICAL FORMULAS

This is correct to the last decimal place. If only two terms
had been used in the right-hand member of the formula, which
is equivalent to the rule of proportional parts, there would
have been an error of 3 in the last decimal place. The rapid
decrease in the value of the differences enables us to judge
quite well of the accuracy of the results. The above formula
can be appHed only wh en the values of x form an ar ithmetical
series.

In case the series of values given are not equidistant, that is,
the values of the independent variable do not form an arithmetical
series, it becomes necessary to apply another formula.

Let ya, y*, yc^ ya, - - â€˘ y* be the given values corresponding
to a, b, c, d, . . . k respectively as values of x. It is required
to find an approximate expression for y^, an unknown term
corresponding to a value of x between x = a and x = k.

Since there are n conditions to be satisfied the expression
which is to represent all of the values must contain n constants.
Assume as the general expression

y,=A+Bx+Cx^+Dx^+ . . . -^Nx^'-K

Geometrically this is equivalent to drawing through the n
points represented by the n sets of corresponding values a
parabola of degree n â€” i.

Substituting the sets of values given by the data in the
equation above n equations are obtained from which to determine
the values oi A, B, C, etc.,

ya=A+Ba-^Ca^+Da^+ . . . Na""'^;
yi,=A-\-Bb+Cb^+Db^+ . . . W"^;

yt=A-\-Bk-\-Ck^+Dk^+ . . . Nk'^-K

But the solution of these equations would require a great
deal of work which can be avoided by using another but equiva-

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