Copyright
Theodore R. (Theodore Rudolph) Running.

Empirical formulas online

. (page 4 of 8)
Online LibraryTheodore R. (Theodore Rudolph) RunningEmpirical formulas → online text (page 4 of 8)
Font size
QR-code for this ebook


/




















/
















^'" = 1.175,








/






















/


















log cos i^ = 9.9370— 10,






/


















5 = 30° 10' ap-
proximately,




/






















/





















A .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
Values o£ —,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;



-^ — ^ — _^ = c:+J tan 30^^°^



Y-5 is plotted to tan



which is a straight line when —^^ —

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



DETERMINATION 0¥ 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;° — .496 sin 3oiit:°).



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






























"■


~~"




^


"^




































"V




























».










""v


"^






































^


















a.




















^














m


























s








>! -0.5


























^


\


































N



-1.5 -1.0 -0.5 0.5

Values of tan 30i/^x°

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:»: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




yk


yk


^.00


x"^^


x''


puted y


•OS


.283


•05


283






.192


.078


1.470


.283


.10


.402


.10


402






.282


.141


1.426


.402


.15


.488






,




-352


.199


1-385


.488


.20


• 556


.20


556


1.420


1.965


-413


•255


1-347


.556


• 25


.613










.466


.308


I-315


.612


• 30


.658










.516


.359


1.276


.658


■35


• 695










-561


.410


1.238


.697


.40


• 730


• 40


730


1-383


1. 816


.609


•459


1.208


.730


•45


•757










.645


•507


1. 174


.757


.50


.780










.683


-555


1. 142


.780


•55


.800










.720


.602


1. 114


.799


.60


.814










• 755


.648


1.078


.814


•65


.826










• 789


•693


1.047


.826


.70


.835










.822


.738


1. 016


•83s


.75


.840










• 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^±ld^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
—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,
^ = •55-



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
derivatives may be made use of to advantage in evaluating the
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 = e°'(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' — 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 —{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


-.


■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


— . 04 39


-.0066





.1976


-.0319


13


45 -06


I. 1972


.9713


.2447


+


.1776


4


+


.1622


-.0481


-.0078





. 29b5


— .0481


18


20.08


I. 2712


.9492


.3314


+


.1344


.■)


+


.1102


- OS57


- 0075





.5054


-.0681


22


55- 10


1-3499


.9211


.4228


4-


.0886


6


+


.0506


- 063s


-.0086


— I


.2549


— .1700


27


30.12


1.4333


.8870


.5206


+


.0398


7





.0175


-.0721


- . 0080


+4


.1200


+.4571


32


05-14


1.5220


.8472


.6270




.0136


8





.0937


-.0805


-.0087


+


.8591


+ .0928


36


40.16


1.6161


.8021


.7446





.0723


9





.178b


-.0894


— .0091


+


• SOU


+ .0510


41


15.18


I. 7160


.7519


.8771





.1384


10





.2726


-.0985


— .0091


+


.3247


+ .0334


45


50.20


I. 8221


.6967


1.0296





.2147


II





■^iv


-.1078


-.0085


+


.2869


+ .0226


50


25.22


1.9348


.6372


I . 2097





.3047


12





.4881


-.1168


-.0087


+


.2393


+ .0178


55


00.24


2.0544


.5735


1.4284





.4143


13





.6093


-.1257


— .0091


+


.2063


+ .0149


59


35.26


2.1815


.5062


I . 7036





.5518


14





.739b


-.1348


-.0089


+


.1823


+ .0120


64


10.28


2.3164


• 4357


2.0659





.7328


15





.8788


-•1435


-.0084


+


. Ib33


+.0096


68


45.30


2.4596


.3624


2.5722





.9859


Tft
















73
77


20.32
55-34


2.6117
2.7732


.2867
.2098


3-3414
4-6735




• 3707
.0306




— I


.1814












—2







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

y'n =—r {yn-2-Sy„-i+ Syn+i-yn+2),
i2n

y"n= -^^ (>-2-i6y„-i+30>'n-i6y„+i+y„-f2),

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

Plotting the points represented by ( — ,—), Fig. 24, it is

\y y I

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

and intercept —.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 ^ — ^ = ^+^ tan .oSx.
e"®* 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.»


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 — i and the inter-



oc^y . XV

when —^— is plotted to —

X V



cept is —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


-•447




35


.695


0. 720


—0.400


.363


-.071




40


.730


0.623


-3. 533


.341


-.774




45


•757


0.492


— 1.500


.292


-.401




50


. .780


0.433


-1.067


• 277


-•342




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


• 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 — — ) 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 y„ = log a+6 log Xn-\-Xn log c,
log y„+i =log a-\-b log Xn^rrxn log c-^b log r.



DETERMINATION OF CONSTANTS 73

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

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

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



-.1








°


















.














°




°










°




o












g-6












-s8

-.9

-1.0








o


























°









.1 .2

Values of

Fig. 26.



From the first equation

log Jn - Xn log € = 1) log X„ + log a.

If then log jn—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 (r— 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





60°


120°


180°


240°


300°


360°


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-\ — ^h-] ^^>2,

2 2

"X/o \/-3

2.O = fl0-§fll-|«2 + a3-^| ^^1 -b2f

2 2

74



DETERMINATION OF CONSTANTS 75

i.S=ao-\ai-\a2+az ^61 H — ^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 = ^—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





60°


120°


i8o^


240°


300°


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-§ai-§fl^2+a3H — ^61 -h2i

2 2

>'3=fl0 — fll + fl2— fl3, _ ■

y4. = ao-\ai-\a2-\-az ^hi-\ ^62,

2 2



3'5 = ^ + Jfl^l — 1^2 — ^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- §3'2+>'3- b4-j3'5,

6a3=>'o- yi-\- >'2-3'3+ 3^4- 3'5,

2



3^>i



+ — 3'iH— ^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 — \pi,
6a3 = ro-si,

2

3^2 = — ^^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

•3 ^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
-•9



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 —ci-2 +^4+ ^1 — ^3,




'v/2 V^2 V 2 V 2

y3 = cio fli H ^3—^4 4 ^1—^2 4 263,

^2 2 2.2




y4 = flo- a\-\-a2— <i3+<^4,



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

y5 = a0 «1 H ^3 — ^4 61+^2 ^3,

2 2 2 2

y6=flk) — «2 +^4 — ^1 + ^3,

-\/2 ^f 2 V 2 V 2

y7 = floH (i\ az — a\ b\ — 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

• 4^2 =yo — y2 +y4 —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= — y\^y2^ yz y^-y^ y-i,

2222

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

y/ 2 v'2 'v/2 V2

4^3= — >'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


•vs




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:=po—pi,










^^ V2 -
4h=r2-\ riy

2 '


^\-fv^


, •+ -' 1






4h = qi,










463 =-^2+ n.


'-^^


^^^ f.^3



The process will be made clear by an example :






45°


90°


135°


180°


225°


270°


315°


360°


4


— 2


— I


2


3


3


— I


2


4



80 EMPIRICAL FORMULAS

For computation the arrangement is as follows:





4




-2


— I


2






3_




3


— I


2




vo


7




I


— 2


4




Wo


I




-5










7


I






I


-5





— 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-


■.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






■ 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 — Cip2,
5a3=qo-C2qi—Ciq2,

5^4 = pO — Cipl + C2p2y
10^5 = ^0 — ^1 + ^2,

561=51/1+52/2,

5&2=52Wl+5lW2,

563=52/1—51/2,
5^4 = 51^1— 52M2.



In the above equations

Ci = cos 36°,
C2 = cos 72°,



5i=sin36°,
52 = sin 72°.



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 • 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=Po—p3+h(j>i—P2)f

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

6a5=qo -qi+^q2,

2

i2a6=/o— /i,

2

V~z

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

663=/!,

6^4=— ^(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

•W7 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 — Ciga +C2^i,

2

8^4=^0,

8a5 = go ^2+Cig3— C2^i,

2

8a6 = wo Wl,

2

V2

8^7 = go H q2 — Ciqi — C2qs ,

2

i6a^ — to—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 = ^1— -^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 :


221° =


sin


67r,










C2 = sin :


22^ =


cos


671°.












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











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 j2°-\-q2 sin 54°+g3 sin 36°+?4 sin i8°,

ioa2=mo+mi sin 54°+W2 sin i8°,

ioa3 = qo — q3 sin 72° — g4 sin 54°+gi sin 36° — 5^2 sin 18°,

10^4 = /o — ^2 sin 54°+/i sin 18°,

10^5 = ^0 — ^2,


1 2 4 6 7 8

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