curve approximately follows a simple law, as a straight line,
to represent by small terms the deviation from this simple law,
that is, the secondary effects, etc. Its use, thus, is often
temporary, giving an empirical approximation pending the
derivation of a more rational law.
The Parabolic and the Hyperbolic Curves.
146. One of the most useful classes of curves in engineering
are those represented by the equation,
y = ax n  ......... (15)
or, the more general equation,
yb = a(xc) n ........ (16)
Equation (16) differs from (15) only by the constant terms b
and c; that is, it gives a different location to the coordinate
EMPIRICAL CURVES. 217
center, but the curve shape is the same, so that in discussing
the general shapes, only equation (15) need be considered.
If n is positive, the curves y = ax n are parabolic curves,
passing through the origin and increasing with increasing x.
If n>l, y increases with increasing rapidity, if n<l, y increases
with decreasing rapidity.
If the exponent is negative, the curves y=ax~ n = are
x n
hyperbolic curves, starting from 2/=oo for z=0, and decreasing
to 2/=0 for x oo.
n^l gives the straight line through the origin, n=0 and
n=oo give, respectively, straight horizontal and vertical lines.
Figs. 61 to 71 give various curve shapes, corresponding to
different values of n.
Parabolic Curves.
Fig. 61. n = 2; y = x 2 ; the common parabola.
Fig. 62. n = 4' } y = x 4 ', the biquadratic parabola.
Fig. 63. n = 8; y = x 8 .
Fig. 64. n = J; y=Vx\ again the common parabola.
Fig. 65. n = i; y= fix; the biquadratic parabola,
Fig. 66. n = i; y = Vx.
Hyperbolic Curves.
Fig. 67. n= 1; y=', the equilateral hyperbola.
Fig. 68.
Fig. 69. n=4; y =  4 .
Fig. 70. n=~> y
Fig. 71. n=; y==.
218
ENGINEERING MATHEMATICS.
EMPIRICAL CURVES.
219
r* 1
, '
^
^
^
x^
x^
^
^
X
^
x
X
x""
X
X
/
X
/
/
/
/
/
J
1
2
4
p
6
8
1
J
a
i
4
1
6
i
8
2
FIG. 64. Parabolic Curve.
FIG. 65. Parabolic Curve. y= jjx.
220
ENGINEERING MATHEMATICS.
1.0
^*~

_ i
~
_
_

_
^.
 
; == w
^
^
"^
/
/
2
4
6
o
8
1
1
2
1
4
1
6
.1
8
2
FIG. 66. Parabolic Curve.
O.I;
.
1
E
\
l\
\
irtJ
1 p
\
V
\
\
^v,
^x.
^^^


.
1 .
^
SS^
*~
* 
^ .
_
1 !!

OJ4
8
1
2
1
6
2
2
4
2
8
3
2
3
6
4,
FIG. 67. Hyperbolic Curve (Equilateral Hyperbola). y= .
EMPIRICAL CURVES.
221
*z
28
9 A.
I
9(\
\
\
\
\
tae
\
\
\
s
(V4
\
x
I

^
^.

* _
*
*

.^.

ii
.
0.4 0.8 1.2 1.6 .2.0 2.4 2.8 3.2 3.6 4.0 4.
FIG. 68. Hyperbolic Curve. y= 5.
1
rLR_
1 2
\
\
08
\
\
\
s
4
0.8
1
]
2
?IG.
"^
1
69
*
6
I
^ .
2
iyp
erb

2
olic
4
Cu
2.
rve.
=
8
3
1
X
^^=5
2
=sss
3
=s^m
6
mi^mm
4.
i..
4.^
222 ENGINEERING MATHEMATICS.
o A \
\
l\
\
x
s
>^
x
^v.
***.
0_^

*****
.
~ _
** .
 .
__
4
8
1
2
1
6
2
2
4
2
8
3
2
3
6
4
4
FIG. 70. Hyperbolic Curve. y==.
Vx
I
lr :(
1
\
L
\
^^
^
^
Or8
^~
~
 __
?=9M_

0,
i
8
1
2
1
6
2
2.
4
2
8
3
2
3
6
4
A
FIG. 71. Hyperbolic Curve. y==.
EMPIRICAL CURVES. 223
In Fig. 72, sixteen different parabolic and hyperbolic curves
are drawn together on the same sheet, for the following values :
n = l, 2, 4, 8, oo; i, i, i, 0; 1, 2, 4, 8; J, J, f
147. Parabolic arid hyperbolic curves may easily be recog
nized by the fact that if x is changed by a constant factor, y also
changes by a constant factor.
Thus, in the curve y = x 2 , doubling the x increases the y
fourfold; in the curve y = x 1 ' 59 , doubling the x increases the y
threefold, etc.; that is, if in a curve,
*/(*),
f?\ = constant, for constant > q, . . . (17)
the curve is a parabolic or hyperbolic curve, y = ax n , and
/^^M!_,n
f(jr) ~ ax ~ q
If q is nearly 1, that is, the x is changed 'onjy by a small
value, substituting q = l+s, where s is a small quantity, from
equation (18),
hence,
f(x+sx)f(x)=ns; (19)
that is, changing x by a small percentage sx, y changes by a pro
portional small percentage nsy.
Thus, parabolic and hyperbolic curves can be recognized by
a small percentage change of x, giving a proportional small
percentage change of y, and the proportionality factor is the
exponent n; or, they can be recognized by doubling x and
seeing whether y hereby changes by a constant factor.
As illustration are shown in Fig. 73 the parabolic curves,
which, for a doubling of re, increase y: 2, 3, 4, 5, 6, and 8 fold.
Unfortunately, this convenient way of recognizing parabolic
and hyperbolic curves applies only if the curve passes through
the origin, that is, has no constant term. If constant terms
exist, as in equation (16), not x and y, but (rcc) and (yb)
follow the law of proportionate increases, and the recognition
224
ENGINEERING MATHEMATICS.
becomes more difficult; that is, various values of c and of 6
are to be tried to find one which gives the proportionality.
2.2 1
2.0
r=a?
1.8
1.6
1.4
\
\
%
1.2
1.0
Hyperbolic Curves
Parabolic Curves
^
0.6
0.4
0.2
0,2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
FIG. 72. Parabolic and Hyperbolic Curves, y xn.
148. Taking the logarithm of equation (15) gives
lg 2/ = lg o>+n log X] (20)
EMPIRICAL CURVED
225
that is, a straight line; hence, a parabolic or hyperbolic curve can
be recognized by plotting the logarithm of y against the loga
rithm of x. If this gives a straight line, the curve is parabolic
or hyperbolic, and the slope of the logarithmic curve, tan = n,
is the exponent.
0.2 0.4
FIG. 73.
0.6 0.8 1.0 1.2 1.4 1.6
Parabolic Curves. y=xn.
This again applies only if the curve contain no constant
term. If constant terms exist, the logarithmic line is curved.
Therefore, by trying different constants c and 5, the curvature
of the logarithmic line changes, and by interpolation such
constants can be found, which make the logarithmic line straight,
and in this way, the constants c and b may be evaluated. If
only one constant exist, that is, only b or only c, the process is
relatively simple, but it becomes rather complicated with both
226
ENGINEERING MATHEMATICS.
constants. This fact makes it all the more desirable to get
from the physical nature of the problem some idea on the
existence and the value of the constant terms.
Exponential and Logarithmic Curves.
149. A function, which is very frequently met in electrical
engineering, and in engineering and physics in general, is the
exponential function,
y = af; . (21)
which may be written in the more general form, .
(22)
Usually, it appears with negative exponent, that is, in the
form,
y = ae* (23)
Fig. 74 shows the curve given by the exponential function
(23) for o = l; w = l; that is,
y=e*, (24)
as seen, with increasing positive x, y decreases to at x= 4 oo,
and with increasing negative x, y increases to oo at x= <x.
The curve, y=z +x , has the same shape, except that the
positive and the negative side (right and left) are interchanged.
Inverted these equations (21) to (24) may also be written
thus,
n(x c) = log
yb.
(25)
that is, as logarithmic curves.
EMPIRICAL CURVES.
227
150. The characteristic of the exponential function (21) is,
that an increase of x by a constant term increases (or, in (23)
and (24), decreases) y by a constant factor.
Thus, if an empirical curve, y=f(x), has such characteristic
that
= constant, for constant q, . . . (26)
A
V
1T4
.o\
\
b;U
\
\
1.2
\
\
onj
\
\
1.0
4.0
\
\
\
\
D_/\
\
\
\
\
\
s
\
o.fl
\
\
\
\
_A.J
"\,
\
^s,
\
i n
\
^
^
v^
^
^.
=s=
^=
~


"  ^.
~^
8.0 1.6 1.2 0.8 0,4 0.4 0.8
Fi G. 74 . Exponential Function . y
1.2
1.6
2.0
the curve is an exponential function, y=ae nx , and the following
equation may be written :
f(x+q) _
f(x) as
_ ng
(27)
Hereby the exponential function can easily be recognized,
and distinguished from the parabolic curve; in the former a
constant term,, in the latter a constant factor of x causes a
change of y by a constant factor.
As result hereof, the exponential curve with negative
exponent vanishes, that is, becomes negligibly small, with far
greater rapidity than the hyperbolic curve, and the exponential
228
ENGINEERING MATHEMATICS.
function with positive exponent reaches practically infinite
values far more rapidly than the parabolic curve. This is
illustrated in Fig. 75, in which are shown superimposed
the exponential curve, y=s~ x , and the hyperbolic curve,
2.4
y = 7. , rrxo? which coincides with the exponential curve
\X +1.OOJ
at x=0 and at x = l.
Taking the logarithm of equation (21) gives logi/=loga +
nx log e, that is, log y is a linear function of x, and plotting
log y against x gives a straight line. This is characteristic of
0;8
0.6
0:4
2.4
0:2
0,4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6
FIG. 75. Hyperbolic and Exponential Curved Comparison.
4.0
the exponential functions, and a convenient method of recog
nizing them.
However, both of these characteristics apply only if x and y
contain no constant terms. With a single exponential function,
only the constant term of y needs consideration, as the constant
term of x may be eliminated. Equation (22) may be written
thus:
= Ae*, ....... (28)
where A = as~ c is a constant.
An exponential function which contains a constant term b
would not give a straight line when plotting logy against x,
EMPIRICAL CURVES.
229
but would give a curve. In this case then log (yb) would be
plotted against x for various values of 6, and by interpolation
that value of b found which makes the logarithmic curve a
straight line.
151. While, the exponential function, when appearing singly,
is easily recognized, this becomes more difficult with com
FIG. 76. Exponential Functions.
binations of two exponential functions of different coefficients
in the exponent, thus,
y = a l s Clx a 2 e C2X , ..... (29)
since for the various values of i, a 2 , ci, c 2 , quite a number of
various forms of the function appear.
As such a combination of two exponential functions fre
quently appears in engineering, some of the characteristic forms
are plotted in Figs. 76 to 78.
230
ENGINEERING MATHEMATICS.
j
L2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
1
;\
(1) 2/=
(2^ y=
(3) y=
(4) y=
(5) y=
(6) y=
'^+0.5
X
x nt
10 a?
VI
 lc
l(
C
)
\
^0.5
""*" i^
\ 2 \
LOC
lo a?
(3^
\
J(4^
^
/5/
r^
^
f //
\
V.
//((
)
x
x^
//
^
\
II
^
^
1
^
\
^~^.
~~
4
8
1
2
1
6
2
2
4
2
8
1
FIG. 77. Exponential Functions.
Fig. 76 gives the following combinations of e~ x and z~ 2x ;
(1) 2/=*+0.5e 2 *;
(2) y=*+Q.2 2x ',
(3) rr;
(4) y=*Q.22*' }
(5) y=  0.5 2 ;
(6) 7/=  O.S2
(7) y= X 2x.
(8) 2/= a5 1.5 2 ^.
EMPIRICAL CURVES.
231
cosh x =
 x
02 04 06 08 10 1.2 14
7 !7
2.0
fc8
1.&
1i
1.2
1.0
0^
0;6
0:4
0:2^
FIG. 78. Hyperbolic Functions.
Fig. 77 gives the following combination of e~ x and s~ 10x
(1) ?/=*+0.5 10 *;
(3) 7/=  0.l 10 
(4) i/=*0.5 10 *;
(5) yC'f 1 *';
(6) y= *1.5fi 10a: .
Fig. 78 gives the hyperbolic functions as combinations of
e +ar and ~ x thus,
232 ENGINEERING MATHEMATICS.
C. Evaluation of Empirical Curves.
152. In attempting to solve the problem of finding a mathe
matical equation, y=f(x), for a series of observations or tests,
the corresponding values of x and y are first tabulated and
plotted as a curve.
From the nature of the physical problem, which is repre
sented by the numerical values, there are derived as many
data as possible concerning the nature of the curve and of the
function which represents it, especially at the zero values and
the values at infinity. Frequently hereby the existence or
absence of constant terms in the equation is indicated.
The log x and log y are tabulated and curves plotted between
x, y, log x, log y, and seen, whether some of these curves is a
straight line and thereby indicates the exponential function, or
the parabolic or hyperbolic function.
If crosssection paper is available, having both coordinates
divided in logarithmic scale, and also crosssection paper having
one coordinate divided in logarithmic, the other in common
scale, the tabulation of log x and log y can be saved and x
and y directly plotted on these two forms of logarithmic cross
section paper.
If neither of the four curves: x,y\ x, log y; log x, y; log x,
logy is a straight line, and from the physical condition the
absence of a constant term is assured, the function is neither
an exponential nor a parabolic or hyperbolic. If a constant
term is probable or possible, curves are plotted between x,
yb, log x, log (y b) for various values of 6, and if hereby
one of the curves straightens out, then, by interpolation,
that value of b is found, which makes one of the curves a straight
line, and thereby gives the curve law. In the same manner,
if a constant term is suspected in the x } the value (xc) is
used and curves plotted for various values of c. Frequently the
existence and the character of a constant term is indicated by
the shape of the curve ; for instance, if one of the curves plotted
between x, y } log x } log y approaches straightness for high, or for
low values of the abscissas, but curves considerably at the
other end, a constant term may be suspected, which becomes
less appreciable at one end of the range. For instance, the
effect of the constant c in (xc) decreases with increase of x.
EMPIRICAL CURVES. 233
Sometimes one of the curves may be a straight line at one
end, but curve at the other end. This may indicate the presence
of a term, which vanishes for a part of the observations. In
this case only the observations of the range which gives a
straight line are used for deriving the curve law, the curve
calculated therefrom, and then the difference between the
calculated curve and the observations further investigated.
Such a deviation of the curve from a straight line may also
indicate a change of the curve law, by the appearance of
secondary phenomena, as magnetic saturation, and in this case,
an equation may exist only for that part of the curve where the
secondary phenomena are not yet appreciable.
If neither the exponential functions nor the parabolic and
hyperbolic curves satisfactorily represent the observations,
x
further trials may be made by calculating and tabulating 
*j
fj X U
and , and plotting curves between x, y,, . Also expressions
x y x
as x 2 +by 2 , and (x a) 2 +b(y c) 2 , etc., may be studied.
Theoretical reasoning based on the nature of the phenomenon
represented by the numerical data frequently gives an indi
cation of the form of the equation, which is to be expected,
and inversely, after a mathematical equation has been derived
a trial may be made to relate the equation to known laws and
thereby reduce it to a rational equation.
In general, the resolution of empirical data into a mathe
matical expression largely depends on trial, directed by judg
ment based on the shape of the curve and on a knowledge of
the curve shapes of various functions, and only general rules
can thus be given.
A number of examples may illustrate the general methods of
reduction of empirical data into mathematical functions.
153. Example 1. In a 118volt tungsten filament incan
descent lamp, corresponding values of the terminal voltage e
and the current i are observed, that is, the socalled " volt
ampere characteristic " is taken, and therefrom an equation for
the voltampere characteristic is to be found.
The corresponding values of e and i are tabulated in the
first two columns of Table III and plotted as curve I in Fig. 79.
In the third and fourth column of Table III are given loge
234
ENGINEERING MATHEMATICS.
and logl In Fig. 79 then are plotted e, logi, as curve II;
log e, i, as curve III; log e, log i, as curve IV.
As seen from Fig. 79, curve IV is a straight line, that is,
log i = A+n log e ; or, i = ae n ,
which is a parabolic curve.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2:0 2.2 2A=log e
2
3 4
) e
) 8
) 1(
K V
1,
1
50 1
40 2
)0 2!
^r**
k)=e
 r/
log i
9.6
as
WA
9.3
9.2
9.1
i
0.45 8.9
0.40 ^.8
0.35 8.7
0.30 8.6
0.25 8.5
0.20 8".4
0.15 8".3
0.10 8.2
0.05 8.1
in
^^
j^^
$
/
fliV*
^
^
c
/
/
^
[/
/
/
IV
/
/
/
/
/
?
/
$
y
7
f
^*
* /
/
_^
Ix
x^
/
/
/
S*
X"
s*^
/
f
/
\
C2xd
X
s
/
/
X
X
/
x
I /
7
/
4
Y_
/
s
X
&
1 /
^**
^
^
T_
<
*^*^"
FIG. 79. Investigation of Voltampere Characteristic of Tungsten Lamp
Filament.
The constants a and n may now be calculated from the
numerical data of Table III by the method of least squares,
as discussed in Chapter IV, paragraph 120. While this method
gives the most accurate results, it is so laborious as to be seldom
EMPIRICAL CURVES.
235
used in engineering; generally, values of the constants a
and n, sufficiently accurate for most practical purposes, are
derived by the following method :
TABLE III.
VOLTAMPERE CHARACTERISTIC OF 118VOLT TUNGSTEN LAMP.
e
i
log e
log i
8.211 +0.6 log e
4
2
4
8
00245
0037
00568
0301
0602
0903
8392
8568
8754
8389
5572
8753
0003
0004
+ 0001
16
25
32
00855
01125
01295
1204
1398
1505
8932
9051
9112
8933
9050
9114
0001
+ 0001
0002
50
64
100
01715
0200
02605
1.699
.234
9230
9295
9411
+ 0004
+ 0006
+ 0005
1.806
2000
9.301
9416
125
150
180
02965
03295
0.3635
2097
2176
2255
9472
9518
9561
9469
9518
9564
+ 0003
0003
200
218
3865
407
2301
2338
9587
9610
9592
9614
0 005
0004
^7= 7612 2040
27= 14973 6465
avg. . 003 = 07 per cent
4= 7361 4425
r
4425
0.6011~0.6
7361
214 =22 585 8505
06X22585 = 13 551
4 = 8.50513551 = 4.954
495414= 8211
log t= 8211+ 06 log e and t = 001625e 6
The fourteen sets of observations are divided into two
groups of seven each, and the sums of log e and log i formed.
They are indicated as 7 in Table III.
Then subtracting the two groups 27 from each other,
eliminates A, and dividing the two differences J, gives the
exponent, n = 0.6011; this is so near to 0.6 that it is reasonable
to assume that ft = 0.6, and this value then is used.