Charles Proteus Steinmetz.

# Engineering mathematics; a series of lectures delivered at Union college online

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(21)

as seen in Chapter II, paragraph 58.

The hyperbolic functions, by substituting for k +z and
the series (13), can be expressed by the series:

smh

z 3 z 5 z 7

7^+7=-+F7 + - ;
\- l_ II

. . . . (22)

174. In the next case, p = 3 or 4,

_ rVbv

dx, . . (23)

cannot be integrated by rational, logarithmic or arc functions,

270 ENGINEERING MATHEMATICS.

but gives a new class of functions, the elliptic integrals, and
their reverse functions, the elliptic functions, so called, because
they bear to the ellipse a relation similar to that, which the
trigonometric functions bear to the circle and the hyperbolic
functions to the equilateral hyperbola.

The integral (23) can be resolved into elementary functions,
and the three classes of elliptic integrals:

dx

Vx(l-x)(l-c 2 xY
xdx

dx

:-b)Vx(l-x)(l-c 2 x)

. . . . (24)

(These three classes of integrals may be expressed in several
different forms.)

The reverse functions of the elliptic integrals are given by
the elliptic functions :

am(u, c) ;

(25)

known, respectively, as sine-amplitude, cosine-amplitude, delta-
amplitude.

Elliptic functions are in some respects similar to trigo-
nometric functions, as is seen, but they are more general,
depending, as they do, not only on the variable x t but also on
the constant c. They have the interesting property of being
doubly periodic. The trigonometric functions are periodic, with
the periodicity 2;r, that is, repeat the same values after every
change of the angle by 2n. The elliptic functions have two
periods pi and p 2 , that is,

sin am(u +npi +mp 2 , c) =sin am(u, c), etc. ; . (26)

hence, increasing the variable u by any multiple of either
period p\ and p 2 , repeats the same values.

APPENDIX A.

271

The two periods are given by the equations,

dx

I 2\ / x(l-x)(l-c 2 x) )

i

dx

(27)

2\ / x(l-x)(l-c 2 x)

175. Elliptic functions can be expressed as ratios of two
infinite series, and these series, which form the numerator and
the denominator of the elliptic function, are called theta func-
tions and expressed by the symbol d, thus

sin am (u, c) =

.

'()'

cos am(u } c) =

J

. (28)

and the four functions may be expressed by the series :
(x) = 1 -2g cos 2x + 2q* cos 4x -2q g cos 6x +-...;

25

sin x
cos

sn

cos

where

=e a and a = ^

t sin bx -

25

4 COS

, (29)

(30)

In the case of integral function (14), where p>4, similar
integrals and their reverse functions appear, more complex

272 ENGINEERING MATHEMATICS.

than the elliptic functions, and of a greater number of periodici-
ties. They are called hyperelliptic integrals and hyperelliptic
functions, and the latter are again expressed by means of auxil-
iary functions, the hyperelliptic 6 functions.

176. Many problems of physics and of engineering lead to
elliptic functions, and these functions thus are of considerable 4
importance. For instance, the motion of the pendulum is
expressed by elliptic functions of time, and its period thereby
is a function of the amplitude, increasing with increasing ampli-
tude; that is, in the so-called " second pendulum," the time of
one swing is not constant and equal to one second, but only
approximately so. This approximation is very close, as long
as the amplitude of the swing is very small and constant, but
if the amplitude of the swing of the pendulum varies and
reaches large values, the time of the swing, or the period or
the pendulum, can no longer be assumed as constant and an
exact calculation of the motion of the pendulum by elliptic
functions becomes necessary.

In electrical engineering, one has frequently to deal with
oscillations similar to those of the pendulum, for instance,
in the hunting or surging of synchronous machines. In
general, the frequency of oscillation is assumed as constant,
but where, as in cumulative hunting of synchronous machines,
the amplitude of the swing reaches large values, an appreciable
change of the period must be expected, and where the hunting
is a resonance effect with some other periodic motion, as the
engine rotation, the change of frequency with increase of
amplitude of the oscillation breaks the complete resonance and
thereby tends to limit the amplitude of the swing.

177. As example of the application of elliptic integrals, may
be considered the determination of the length of the arc of an
ellipse.

Let the ellipse of equation

be represented in Fig. 93, with the circumscribed circle,

a 2 m . (32)

APPENDIX A.

273

To every point P = x, y of the ellipse then corresponds a
point PI = X, y\ on the circle, which has the same abscissa JT,
and an angle = AOP\.

The arc of the ellipse, from A to P, then is given by the
integral,

f* (l-c^dz
L = a I - , .. , TT, .... (33)

(34)

where

z = sin 2 6 = ( ) and c ^
\a/

is the eccentricity of the ellipse.

FIG. 93. Rectification of Ellipse.

Thus the problem leads to an elliptic integral of the first
and of the second class.

For more complete discussion of the elliptic integrals and
the elliptic functions, rererence must be made to the text-books
of mathematics.

B. Special Functions.

178. Numerous special functions have been derived by the
exigencies of mathematical problems, mainly of astronomy, but
in the latter decades also of physics and of engineering. Some
of them have already been discussed as special cases of the
general Abelian integral and its reverse function, as the expo-
nential, trigonometric, hyperbolic, etc., functions.

274 ENGINEERING MATHEMATICS.

Functions may be represented by an infinite series of terms;
that is, as a .sum of an infinite number of terms, which pro-
gressively decrease, that is, approach zero. The denotation of
the terms is commonly represented by the summation sign 2.

Thus the exponential functions may be written, when
defining,

|0 = 1; |n = lX2X3x4x. . . Xn,
as

/v2 -r3 oo /vn

e,-i +a!+ - f _ + ... = |._ .... (35)

X n *

which means, that terms , are to be added for all values of n

|n

from n = to n=cc .

The trigonometric and hyperbolic functions may be written
in the form :

X 3 X 5 X 7 oo X 2n + l

sina;=x __ + ___ + . = 2.(-iy. ; . (36)

(37)

X 3 X^ X^ 3 X 2n "*"^

sinh x-=x+T7r+r+^+. . . = 2nT- r ; . . . C38)
|3 |5 |/ o [2n-fl '

(39)

Functions also may be expressed by a series of factors;
that is, as a product of an infinite series of factors, which pro-
gressively approach unity. The product series is commonly
represented by the symbol ~[~j~.

Thus, for instance, the sine function can be expressed in the
form,

2 2 2 2

. (40)

179. Integration of known functions frequently leads to new
functions. Thus from the general algebraic functions were

APPENDIX A. 275

derived the Abelian functions. In physics and in engineering,
integration of special functions in this manner frequently leads
to new special functions.

For instance, in the study of the propagation through space,
of the magnetic field of a conductor, in wireless telegraphy,
lightning protection, etc., we get new functions. If i=f (t}
is the current in the conductor, as function of the time t, at a
distance x from the conductor the magnetic field lags by the

time ti=-, where S is the speed of propagation (velocity of
o

light). Since the field intensity decreases inversely propor-
tional to the distance x, it thus is proportional to

and the total magnetic flux then is /

r

z= \ ydx

If the current is an alternating current, that is, f (t) a
trigonometric function of time, equation (42) leads to the
functions,

"sin x

dx',

(43)

COS X ,

dx.

If the current is a direct current, rising as exponential
function of the time, equation (42) leads to the function,

w -

/e*dx

(44)

27G

ENGINEERING MATHEMATICS.

Substituting in (43) and (44), for sin x, cos x, e x their
infinite series (21) and (13), and then integrating, gives the
following :

'sin x x 3 x 5 x 7

dx = x-^+^-^+-. .. ;

Axwz, x 2 x 4 x 6

J -x- dx = l Z X -^ + 4fi-^

, (45)

For further discussion of these functions see "Theory and
Calculation of Transient Electric Phenomena and Oscillations,"
Section III, Chapter VIII.

180. If y"=f(x] is a function of x, and 2 = I f (x)dx = 6(x)

fb
f(x)dx, is no longer

a function of x but a constant,

For instance, if y = c(x ri) 2 , then

c(x-n)*

C
I

,

c(x n) 2 dx =

and the definite integral is

Z= C b c(x-n) 2 dx = ^

Ja O

This definite integral does not contain x, but it contains
all the constants of the function / (x}, thus is a function of
these constants c and n, as it varies with a variation of these
constants.

In this manner new functions may be derived by definite
integrals.

Thus, if

y=f(x;u,v...) ....... (46)

is a function of x, containing the constants u, v . . .

APPENDIX A. 277

The definite integral,

Z= C b f(x,u,v...)dx, . .... (47)

Ja

is not a function of x, but still is a function of w, v . . . , and
may be a new function.

181. For instance, let *

y== -x x n-l. m ...... (48)

then the integral,

f(u)= (*e-*x u ~ l dx, ...... (49)

is a new function of u, called the gamma function.

Some properties of this function may be derived by partial
integration, thus :

; ..... .... (50)

if n is an integer number,

r(u) = (u-l)(u-2\..(u-n)r(u-n), . . (51)
and since

r(i)=i, ........... (52)

if u is an integer number, then,

r(u) = \u-l. .......... (53)

C. Exponential, Trigonometric and Hyperbolic Functions.
(a) FUNCTIONS OF REAL VARIABLES.

182. The exponential, trigonometric, and hyperbolic func-
tions are defined as the reverse functions of the integrals,

a. u= - = logz. ...... .... (54)

and: x=P ... ..... . . (55)

/dx
- arc sin x; ...... (56)
vl-z 2

278 ENGINEERING MATHEMATICS.

and: x=\$mu, .......... (57)

Vl-z 2 = cosu, .......... (58)

/J
^=~=-log{vT+^-a;!; . . . . (59)

/ _ U

and # = - ~ =sinhw; .... (60)

-f- -

..... (61)

From (57) and (58) it follows that

sin 2 M+cos 2 u = l ....... (62)

From (60) and (61) it follows that

cos 2 /m-sin 2 /m = l ...... (63)

Substituting (x) for x in (56), gives (u) instead of u,
and therefrom,

sin ( u) = sin u ...... (64)

Substituting (u) for u in (60), reverses the sign of x,
that is,

sinh ( u) = sinh u. . . . (65)

Substituting ( x) for x in (58) and (61), does not change
the value of the square root, that is,

cos ( w)=cos u, ...... (66)

cosh (u)= cosh u, ..... (67)

Which signifies that cos u and cosh u are even functions, while
sin u and sinh u are odd functions.

Adding and subtracting (60) and (61), gives

g* = cosh u sinh u ...... (68)

APPENDIX A.

279

(56.)

(6) FUNCTIONS OF IMAGINARY VARIABLES.
183. Substituting, in (56) and (59), a; = jy, thus t/ = ,

dx

gves

x = sinh u

1 + x 2 = cosh u

hence, ju= = , hence, j u

Vl-y*'
=sin/u; . . . (69)

Resubstituting x in both
sinh ju J '" e~ J ' u

. . (70)

sin m

T-; (71)

1 x 2 = cos u = cosh

2

= cos/w. . (72)

u
and s u = cosh u sinh u = cos /M =F j sin /w. . . (73)

(c) FUNCTIONS OF COMPLEX VARIABLES

184. It is :

. . (74)

280 ENGINEERING MATHEMATICS.

sin (ujv) =sin u cos jv cos u sin jv

v +~ v . , v e~ v

= sin w cosh v db? 60S tt smh ?; = sin w j ~ cos u ;

cos(w p) = cos u cos pT sin u sin jv

fi f+ e - je-er v .

= cos u cosh v T J sm it smh v = cos w =F ] ^ sm u ;

= sinh it cos vj cosh w sin v ;
cosh(ttjv) = -

(76)

(77)

(78)

= cosh u cos v / sinh u sin v;
etc.

(d) RELATIONS.

185. From the preceding equations it thus follows that the
three functions, exponential, trigonometric, and hyperbolic,
are so related to each other, that any one of them can be
expressed by any other one, so- that when allowing imaginary
and complex imaginary variables, one function is sufficient.
As such, naturally, the exponential function would generally
be chosen.

Furthermore, it follows, that all functions with imaginary
and complex imaginary variables can be reduced to functions
of real variables by the use of only two of the three classes
of functions. In this case, the exponential and the trigono-
metric functions would usually be chosen.

Since functions with complex imaginary variables can be
resolved into functions with real variables, for their calculation
tables of the functions of real variables are sufficient.

The relations, by which any function can be expressed by
any other, are calculated from the preceding paragraph, by
the following equations :

APPENDIX A.

e " = cosh u sinh u = cos ju =F j sin ju ;
jr = cos v j sin v = cosh j v j sinh j v ;
= M (cos vj sin v),

sinh ju e* u s~ Jn
sinu= <p ;

^ . v

sin ? v = ? sinh v = ? ^ :

sin (w- ;'?;) = sin u cosh v j cos w sinh v

= - sin u j' -^ cos w ;

^I*_L . 3

cos w = cosh ju =
cos ? v = cosh v =

cos (u jv) = cos u cosh v^ j sin w sinh i?

sinh 'a

v -~ r

2

- " sin

sm

_
sinh jv = j sin v = ^ - ;

sinh (ujv) =sinh w cos vj cosh w sin v

cos

Sln

coshw =

cosh jv = cos v = ;
z

cosh (ii jv) =cosh w cos v j sinh u sin v

7^ cos vj ^ sin v.

281

(a)

(c)

(d)

282 ENGINEERING MATHEMATICS.

And from (b) and (d), respectively (c) and (c), it follows that

sinh (u + jv) = j sin (v fa) = j sin (v ju) ;
cosh (t*/v)=cos

Tables of the exponential functions and their logarithms,
and of the hyperbolic functions with real variables, are given
in the following Appendix B.

APPENDIX B.

TWO TABLES OF EXPONENTIAL AND HYPERBOLIC

FUNCTIONS.

TABLE I.

= 2.7183,

log = 0.4343.

X

XlO-a

X10- 2

xio-i

XI

1.0

0.999

0.990

0.905

0.368

1.2

0.988

0.887

0.301

1.4

0.986

0.869

0.247

1.6

0.984

0.852

0.202

1.8

0.982

0.835

0.165

2.0

0.998

0.980

0.819

0.135

2.5

0.975

0.779

0.082

3.0

0.997

0.970

0.741

0.050

3.5

0.966

0.705

0.030

4.0

0.996

0.961

0.670

0.018

4.5

0.956

0.638

0.011

5.0

0.995

0.951

0.607

0.007

6

0.994

0.942

0.549

0.002

7

0.993

0.932

0.497

0.001

8

0.992

0.923

0.449

0.000

9

0.991

0.914

0.407

10

0.990

0.905

0.368

283

284

ENGINEERING MATHEMATICS.

TABLE II.

EXPONENTIAL AND HYPERBOLIC FUNCTIONS.

= 2.718282 2.7183, log e = 0.4342945 ~ 0.4343.

cosh x =

p.p.

434

435

0.1

43

43

0.2

87

87

0.3

130

130

0.4

174

174

0.5

217

217

0.6

261

261

0.7

304

304

0.8.
0.9

347
391

348
391

1.0

434

435

X

log-+*

^log

*

log e-*

+ x

g-X

cosh x

sinh x

x

b

434
435
434
434

434

435

434
434
435

434

1

1

1

0.001
0.002
0.003
0.004

0.000434
0.000869
0.001303
0.001737

9.999566
9.999131
9.998697
9.998263

1.00100
1.00200
1.00301
1.00401

0.99900
0.99800
0.99700
0.99601

1.00000
1.00000
1.00000
1.00001

0.00100
0.00200
0.00300
0.00400

0.001
0.002
0.003
0.004

0.005

0.002171

9.997829

1.00501

0.99501

1.00001

0.00500

0.005

0.006
0.007
0.008
0.009

0.002606
0.003040
. 003474
. 003909

9.997394
9.996960
9.996526
9.996091

1 . 00602
1.00702
1.00803
1.00904

0.99402
0.99302
0.99203
0.99104

1.00002
1.00002
1.00003
1 . 00004

0.00600
0.00700
0.00800
0.00900

0.01000

0.006
0.007
0.008
0.009

0.010

0.004343

9.995657

1.01005

0.99005

1.00005

0.010

0.012
0.014
0.016
0.018

0.005212
. 006080
0.006949
0.007817

9.994788
9.993920
9.993051
9.992183

1.01207
1.01410
1.01613
1.01816

. 98807
0.98610
0. 984 13
0.98216

1.00007
1.00010
1.00013
1.00016

0.01200
0.01400
0.01600
0.01800

0.012
0.014
0.016
0.018

0.020

0.008686

9.991314

1.02020

0.98020

1.00020

0.02000

0.020

0.025
0.030
0.035
0.040
0.045

0.050

0.010857
0.013029
0.015200
0.017372
0.019543

9.989143
9.986971
9.984800
9.982628
9.980457

1.02531
1.03046
1.03562
1.04081
1.04603

0.97531
0.97045
0.96561
0.96079
0.95600

1.00031
1 . 00046
1.00062
1.00080
1.00102

0.02500
0.03000
0.03500
0.04001
0.04502

0.025
0.030
0.035
0.040
0.045

0.021715

9.978285

1.05127

0.95123

1.00125

0.05003

0.050

0.06
0.07
0.08
0.09

0.026058
0.030401
0.034744
0.039086

9.973942
9.969599
9.965256
9.960914

1.06184
1.07251
1.08329
1.09417

0.94176
. 93239
0.92312
0.91393

1.001800.06004
1.002450.07006
1.00321; 0.08008
1.004050.09011

0.06

0.07
0.08
. 09

0.10

0-043429

9.956571

1.10516

0.90484

1. 00500 'o. 10016

0.10

0.12
0.14
0.16
0.18

0.052115
0.060801
0.069487
0.078173

9.947885
9.939199
9.930513
9.921827

1.12750
1 . 15027
1.17351
1 . 19721

. 88692
. 86936
0.85214
0.83527

1.00721 0.12028
1.009820.14046
1.012830.16069
1.016240.18097

0.12
ii. 14
0.16
0.18

0.20

0.086859

9.913141

1 . 22140

0.81873

1. 02006 : 0.20 134

0.20

+ o-ooi = i.oo 1000494,

e - 0-001 = 0.99900049.

APPENDIX B.

TABLE II Continued.

EXPONENTIAL AND HYPERBOLIC FUNCTIONS.

285

X

log e + *

log -*

+ *

-*

cosh x

sinh x

x

0.20

0.086859

9.913141

1 . 22140

0.81873

1.02006

0.20134

0.20

0.25
0.30
0.35
0.40
0.45

0.25
0.30
0.35
0.40
0.45

0.108574
0.130288
0.152003
0.173718
. 195433

9.891426
9.869712
9.847997
9.826282
9 . 804567

1 . 28403
1.34986
1.41907
1.49183
1.56831

0.77880
. 74082
0.70469
0.67032
0.63763

1.03142
1.04534
1.06188
1.08108
1 . 10297

0.25261
0.30457
0.35719
0.41076
0.46534

0.50

0.217147

9.782853

1.64870

0.60653

1.12761

0.52108

0.50

0.6
0.7
0.8
0.9

. 260577
0.304006
0.347436
0.390865

9.739423
9.695994
9.652564
9.609135

1.82212
2.01375
2 . 22554
2.45960

0.54881
0.49659
0.44933
0.40657

1 . 19546
1.25517
1.33744
1.43309

0.63666

. 75858
0.88811
1.02657

0.6
0.7
0.8
0.9

1.0

0.434294

9.565706

2.71828

0.36788

1.54308

1 . 17520

1.0

1.2
1.4
1.6
1.8

0.521153
0.608012
0.694871
0.781730

9.478847
9.391988
9.305129
9.218270

3.32011
4.05520
4.95304
6.04965

0.30119
. 24660
0.20190
. 16530

1.81065
2.15090
2.57745
3.10745

1 . 50946
1 . 90430
2 . 37557
3.44218

1.2
1.4
1.6
1.8

2.0

0.868589

9.131411

7 . 38906

. 13534

3.76220

3 . 62686

2.0

2.5
3.0
3.5
4.0
4.5

1.085736
1.302883
1 . 520030
1.737178
1.954325

8.914264
8.694117
8.479970
8.262822
8.045675

12.1825
20.0855
33.1154
54.5983
90.0170

0.082085
0.049797
0.030197
0.018316
0.011109

6.1323
10.0677
16.5728
27.3083
45.0141

6 . 0002
10.0178
16 . 5426
27 . 2900
45.0030

2.5
3.0
3.5
4.0
4.5

5.0

2.171472

7.828528

148.413

0.006738

74.210

74 . 203

5.0

6

7
8
9

2.605767
3.040061
3.474356
3.908650

7.394233
6.959939
6.525644
6.091350

403.428
1096.63
2980.96
8103.08

0.002479
0.000912
0.000335
0.000123

201.715 1201.713

6

7
8
9

10

= * + *
for x> 6

10

4.342945

5.657055

22026 . 5

0.0000454

12
14
16

18

5.211534
6.080123
6.948712

7.817301

4.788466
3.919877
3.051288
2.182699

162755
1202610
8886120
65660000

0.0000061
0.00000083
0.00000011
0.00000002

12
14
16
18

20

20

8.685890

1.314110

485166000

0.00000000

INDEX.

Abelian integrals and functions, 276.
Absolute number, 4.

value of fractional expression, 49.

of general number, 30.
Accuracy of approximation estimated

200.
of transmission line equations,

208.

of calculation, 252.
of curve equation, 210.

of general number, 28.

and subtraction of trigonometric

functions, 102.

Algebra of general number or com-
plex quantity, 25.
Algebraic expression, 265.

function, 265.

Alternating current and voltage vec-
tor, 41.

functions, 117, 125.
waves, 117, 125.
Alternations, 117.
Alternator short circuit current,

approximated, 195.
Analytical calculation of extrema,

152.

function, 265.

Approximate calculation, 254.
Approximations giving (1 + s) and

(1-s), 201.
of infinite series, 53.
methods of, 187.

Arbitrary constants of series, 69, 79.
Area of triangle, 106.
Arrangement of numerical calcula-
tions, 249.

B

Base of logarithm, 21.
Binomial series with small quantities,
193.

theorem, infinite series, 59.

of trigonometric function, 104.

Calculation, accuracy, 252.
checking of, 262.
numerical, 249.
reliability, 261.
Capacity, 65.

Change of curve law, 211, 233.
Characteristics of exponential curves,

227.
of parabolic and hyperbolic curves,

223.

Charging current maximum of con-
denser, 176.

Checking calculations, 262.
Ciphers, number of, in calculations,

255.

Circle defining trigonometric func-
tions, 94.
Coefficients, unknown, of infinite

series, 60.
Combination of exponential functions,

229.

of general numbers, 28.
of vectors, 29.

Comparison of exponential and hyper-
bolic curves, 228.
Complex imaginary quantities, see

General number,
quantity, 17.
algebra, 27.
see General number.

287

288

INDEX.

Complementary angles in trigono-
metric functions, 99.
Conjugate numbers, 31.
Constant, arbitrary of series, 69, 79.
errors, 186.

factor with parabolic and hyper-
bolic curves, 223.
phenomena, 106.
terms of curve equation, 211.
of empirical curves, 232.
in exponential curves, 228.
with exponential curves, 227.
in parabolic and hyperbolic

curves, 225.
Convergency determinations of series,

57.

of potential series, 215.
Convergent series, 56.
Coreloss by potential series, 213.

curve evaluation, 242.
Cosecant function, 98.
Cosh function, 276.
Cosine-amplitude, 270.
function, 94.

components of wave, 121, 125.
series, 82.

versus function, 98.
Cotangent function, 94.
Counting, 1.

Current change curve evaluation, 239.
input of induction motor, approxi-

imated, 191.

maximum of alternating trans-
mission circuit, 159.
of distorted voltage wave, 169.
Curves, checking calculations, 263.
empirical, 209.
law, change, 233.
rational equation, 210.
use of, 257.

D

Data on calculations and curves, 261.

derived from curve, 258.
Decimal error, 263.

number of, in calculations, 255.
Definite integrals of trignometric

functions, 103.
Degrees of accuracy, 253.
Delta-amplitude, 270.
Differential equations, 64.

of electrical engineering, 65, 78,
86.

Differential equations of second order,

78.

Differentiation of trigonometric func-
tions, 103.

Distorted electric waves, 108.
Distortion of wave, 139.
Divergent series, 56.
Division, 6.

of general number, 42.

with small quantities, 190.
Double angles in trigonometric func-
tions, 103.

peaked wave, 246.

periodicity of elliptic functions, 270.

scale, 260.

E

, 21.

Efficiency maximum of alternator,

162.

of impulse turbine, 154.
of induction generator, 177.
of transformer, 155, 174.
Electrical engineering, differential

equations, 65, 78, 86.
Ellipse, length of arc, 272.
Elliptic integrals and functions, 270.
Empirical curves, 209.

evaluation, 232.
equation of curve, 210.
Engineering, defferential equations,

65, 78, 86.

Equilateral hyperbola, 217.
Errors, constant, 186.
numerical, 263.
of observation, 180.
Estimate of accuracy of approxima-
tion, 200.

Evaluation of empirical curves, 232.
Even functions, 81, 98, 276.

periodic, 122.
harmonics, 117.

separation, 120, 125, 134.
Evolution, 9.

of general number, 44.
of series, 70.
Exact calculation, 254.
Exciting current of transformer,

resolution, 137.

Explicit analytic function, 265.
Exponent, 9.
Exponential curves, 226.

forms of general number, 50.

INDEX.

289

Exponential functions, 52, 268, 275.
with small quantities, 196.
tables, 283, 284, 285.
series, 71.

ami trignometric functions, rela-
tion, 83.
Extrapolation on curve, limitation,

210.
Extrema, 147.

analytic determination, 152.
graphical construction of differen-
tial function, 170.
graphical determination, 147, 150,

168.

with intermediate variables, 155.
with several variables, 163.
simplification of function, 157.

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