/
/
^'" = 1.175,
/
/
log cos i^ = 9.9370â€” 10,
/
5 = 30Â° 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^^Â°^
Y5 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 0Â¥ 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;Â° â€” .496 sin 3oiit:Â°).
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
Â».
""v
"^
^
a.
^
m
s
>! 0.5
^
\
N
1.5 1.0 0.5 0.5
Values of tan 30i/^xÂ°
Fig. 21.
2.0
XIX. y = ax'^ibx^.
(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:Â»:V2'^^6:*:V^ (3)
yt^ifyt ^bx'if^r'), (4)
yt+2r'yt+i =bxfr^{r^r') (5)
Multiplying equation (4) by y'^ and subtracting it from
equation (5) there results
yt+2r'yt+ir^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€¢
yk
yk
^.00
x"^^
x''
puted y
â€¢OS
.283
â€¢05
283
.192
.078
1.470
.283
.10
.402
.10
402
.282
.141
1.426
.402
.15
.488
,
352
.199
1385
.488
.20
â€¢ 556
.20
556
1.420
1.965
413
â€¢255
1347
.556
â€¢ 25
.613
.466
.308
I315
.612
â€¢ 30
.658
.516
.359
1.276
.658
â– 35
â€¢ 695
561
.410
1.238
.697
.40
â€¢ 730
â€¢ 40
730
1383
1. 816
.609
â€¢459
1.208
.730
â€¢45
â€¢757
.645
â€¢507
1. 174
.757
.50
.780
.683
555
1. 142
.780
â€¢55
.800
.720
.602
1. 114
.799
.60
.814
â€¢ 755
.648
1.078
.814
â€¢65
.826
â€¢ 789
â€¢693
1.047
.826
.70
.835
.822
.738
1. 016
â€¢83s
.75
.840
â€¢ 854
.783
0.984
.840
.80
.845
.80
845
1313
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^Â±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
â€”3.86 which give the equations
1.50
^1.30
O
1 1.20
^UO
LOO
.90
2^^42'^ =4.10,
386.
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€¢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 = eÂ°'(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
3502
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
13499
.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
0514
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€¢ 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€¢ 4357
2.0659
â€”
.7328
15
â€”
.8788
â€¢1435
.0084
+
. Ib33
+.0096
68
45.30
2.4596
.3624
2.5722
â€”
.9859
Tft
73
77
20.32
5534
2.6117
2.7732
.2867
.2098
33414
46735
â€¢ 3707
.0306
â€” I
.1814
â€”2
The values of y' and y" are obtained by the formulas
y'n =â€”r {yn2Syâ€ži+ Syn+iyn+2),
i2n
y"n= ^^ (>2i6yâ€ži+30>'ni6yâ€ž+i+yâ€žf2),
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.Â»
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€¢447
35
.695
0. 720
â€”0.400
.363
.071
40
.730
0.623
3. 533
.341
.774
45
â€¢757
0.492
â€” 1.500
.292
.401
50
. .780
0.433
1.067
â€¢ 277
â€¢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
â€¢ 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€ž = log a+6 log Xn\Xn log c,
log yâ€ž+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
Â°
.
Â°
Â°
Â°
o
g6
s8
.9
1.0
o
Â°
.1 .2
Values of
Fig. 26.
From the first equation
log Jn  Xn log â‚¬ = 1) log Xâ€ž + 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 2x4a3C0S3x+ . . . \aTCosrx
\bisinx]rb2sm 2X{bz^n$x\ . . . +6r_iSm (râ€” i)x.
Values of y periodic.
The righthand 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
oÂ°
60Â°
120Â°
180Â°
240Â°
300Â°
360Â°
y
I.O
17
2.0
1.8
15
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\iaila2as\ â€” ^h] ^^>2,
2 2
"X/o \/3
2.O = fl0Â§fllÂ«2 + 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
oÂ°
60Â°
120Â°
i8o^
240Â°
300Â°
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 = aoÂ§aiÂ§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 \y2yz ly4.\hyb,
2,a2=yo hyi Â§3'2+>'3 b4j3'5,
6a3=>'o yi\ >'23'3+ 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=roiisi,
2,(^2= po â€” \pi,
6a3 = rosi,
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
17
15
2.0
0.9
VO
Wo
2.8
2.8
.8
32
2.0
32
.2
2.9
I.I
.8
.2
I.I
po
2.8 6.1 To
â€¢3 ^1
6ao=+8.90,
3^1 = 1.25,
3^2= .55,
6^3 = + .10,
3Z>i = + .65 V3,
3^2 = + .i5V'3
.8
13
â€¢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.
8ORDiNATE 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€”ci2 +^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 Â«1 H ^3 â€” ^4 61+^2 ^3,
2 2 2 2
y6=flk) â€” Â«2 +^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 y3y4 ys H y?,
2222
â€¢ 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^ yi,
2222
4^2= y\  yz V yb  yi,
y/ 2 v'2 'v/2 V2
4^3= â€” >'i3'2H >'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€¢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,
V2
4(^3 =ro, si,
2
^a "
w,^.^>
Sa4:=poâ€”pi,
^^ V2 
4h=r2\ riy
2 '
^\fv^
, â€¢+ ' 1
4h = qi,
463 =^2+ n.
'^^
^^^ f.^3
The process will be made clear by an example :
45Â°
90Â°
135Â°
180Â°
225Â°
270Â°
315Â°
360Â°
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 = i25
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=qoC2qiâ€”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 36Â°,
C2 = cos 72Â°,
5i=sin36Â°,
52 = sin 72Â°.
In the schemes that follow, as in the loordinate one, only
the results will be given.
12Ordinate Scheme
3'c
)
y\
yn
y2
yiQ
ys
yQ
y^ y^
ys â€¢ 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+p3i{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
16ORDINATE 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€¢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 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 r24Cir3+C2ri.
Ci = cos :
221Â° =
sin
67r,
C2 = sin :
22^ =
cos
671Â°.
20Ordinate
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€¢
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 j2Â°\q2 sin 54Â°+g3 sin 36Â°+?4 sin i8Â°,
ioa2=mo+mi sin 54Â°+W2 sin i8Â°,
ioa3 = qo â€” q3 sin 72Â° â€” g4 sin 54Â°+gi sin 36Â° â€” 5^2 sin 18Â°,
10^4 = /o â€” ^2 sin 54Â°+/i sin 18Â°,
10^5 = ^0 â€” ^2,