8.960
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. ya\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+RhR^+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
iR 'Â°Â«'*
iR
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
3125
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
....
y2.72
1.49
2.53
3.68
4.93
6
24
7.64
9.09
log(y2.72)
.1732
.4031
.5658
.6928
7952
.8831
.9586
Computed y
4.21
525
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
from the water head
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
35
4.0
y
231
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 (ybx<^)
.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)ioÂ»"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.
\ yyk/
If two variables, x and y, are so related that the points repre
xâ€”Xk, ) lie on a straight line, the relation between
yyJ
the variables is expressed by the equation
XVI {x^a){y^b)=c.
'LQtX â€” Xu=X,
yy, = 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
/ XXk\
[xx , )
\ yy>^
yyic
= p{xx,)+q.
Clearing of fractions
xx, = p{xx.){yyk){q{yyK).
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
5975
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
425
7.25
8.50
â€” 11.00
13.50
â€” 15.00
i56o
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
5394
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=Ai.g6,
7=767.00.
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)Xf(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)(744.78) =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.
\xxt 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
\xXk yt yt/
between the variables is expressed by the equation
b
XVIa y = aio'^'.
By the condition stated
log^ = w â€” llog^+^Â»,
yk xxt 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 ylog yk){xXk) = w(log ylog yk)+b{xXk),
or
log y{xXkm) = (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 yklog a)(xk\c) =b,
since the point (xk, yk) lies on the curve.
X log F+log Y(xk+c)^X(log yklog a) =o.
Dividing this equation by X
log Y=(xk+c) ^^+logalog>'A;.
Replacing log F and X by their values
log ^ =  (xk\c)^ log â€” +log alog yk.
yk X Xk yk
From this it is seen that if log â€” be plotted to â€” â€” log â€”
yic xxt 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
35
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
+1577
+ 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€” , lL1 \
\ 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 Hce^""^^ 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
eÂ«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!fix.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
tog2aAx^ Since Ax is
equal to unity we have
/
1 n
/
/
;^<
/
^
/
/
1 '
/
> â€¢*
/
/
26Â° cos 5 = 1.875,