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Charles Proteus Steinmetz.

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

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Factor, constant, with parabolic and

hyperbolic curves, 223.
Fan motor torque by potential series,

215.

Flat top wave, 246.
zero waves, 246.
Fourier series, see Trigonometric

series.
Fraction, 8.

as series, 52.
Fractional exponents, 11, 44.

expressions of general number, 49.
Full scale, 260.
Functions, theory of, 265.



Gamma function, 275.
Gen^caXnumber, 1, 16.

algebra, 2").

exponential forms, 50.

reduction, 48.

Graphical determination of extrema,
147, 150, 168.

H

Half angles in trigonometric func-

tions, 103.
Half waves, 117.
Half scale. 260.
Harmonics, even, 117.

odd, 117.

of trigonometric series, 114.

two, in wave, 246.



High harmonics in wave shape, 246.
Hunting of synchronous machines,

248.
Hyperbola, arc of, 61.

equilateral, 217.
Hyperbolic curves, 216.
functions, 275.

curve shape, 231.
Tables, 284, 285.

integrals and functions, 269.
Hyperelliptic integrals and functions,
272.



Imaginary number, 26.

quantity, see Quadrature number.
Incommensurable waves, 248.
Indeterminate coefficients, method,

71.
Indeterminate coefficients of infinite

series, 60.
Individuals, 8.
Inductance, 65.
Infinite series, 52.

values of curves, 211.

of empirical curves, 232.
Inflection points of curves, 153.
Impedance vector, 41.
Implicit analytic function, 265.
Integral function, 266.
Integation constant of series, 69, 79.

of differential equation, 65.

by infinite series, 60.

of trigonometric functions, 103.
Intelligibility of calculations, 256.
Intercepts, defining tangent and co-
tangent functions, 94.
Involution, 9.

of general numbers, 44.
Irrational numbers, 11.
Irrationality of representation by
potential series, 213.



Leastsquares, method of, 179, 186.
Limitation of mathematical represen-
tation, 40.

of method of least squares, 186.

of potential series, 216.



290



INDEX.



Limiting value of infinite series, 54.
Linear number, 33.

see Positive and negative number.
Line calculation, 249.

equations, approximated, 204.
Logarithm of exponential curve, 228.

as infinite series, 63.

of parabolic and hvperbolic curves,
225.

with small quantities, 197.
Logarithmation, 20.

of general numbers, 51.
Logarithmic curves, 226.

functions, 268.

paper, 232.
Loss of curve induction motor, 183.

M

Magnetite arc, volt-ampere character-
istic, 237.
Magnetite characteristic, evaluation,

244.
Magnitude of effect, determination,

253.

Maximum, see Extremum.
Maxima, 147.

McLaurin's series with small quan-
tities, 198.
Mechanism of calculating empirical

curves, 235.
Methods of calculation, 249.

of indeterminate coefficients, 71.
of least squares, 179, 186.
Minima, 147.

Minimum, see Extremum.
Multiple frequencies of waves, 248.
Multiplicand, 39.
Multiplication, 6.

of general numbers, 39.
with small quantities, 188.
of trigonometric functions, 102.
Multiplier, 39.

N

Negative angles in trigonometric
functions, 98.

exponents, 11.

number, 4.

Nodes in wave shape, 247.
Non-periodic curves, 212.
Nozzle efficiency, maximum, 150.
Number, general. 1.



Numerical calculations, 249.

values of trigonometric functions,
101.

O

Observation, errors, 180.
Odd functions, 81, 98, 276.

periodic, 122.
harmonics in symmetrical wave,

117.

separation, 120, 125, 134.
Omissions in calculations, 263.
Operator, 40.

Order of small quantity, 188.
Oscillating functions, 92.
Output, see power.



7i and added and subtracted in

trigonometric function, 100.
Parabola, common, 218.
Parabolic curves, 216.
Parallelogram law of general numbers

28.

of vectors, 29.
Peaked wave, 246.
Pendulum motion, 272.
Percentage change of parabolic and

hyperbolic curves, 223.
Periodic curves, 246.
decimal fraction, 12.
phenomena, 106.

Periodicity, double, of elliptic func-
tions, 270.

of trigonometric functions, 96.
Permeability maximum, 148, 170.
Phase angle of fractional expression,

49.

of general number, 28.
Plain number, 32.

see General number.
Plotting of curves, 212.

proper and improper, 259.
of empirical curve, 232.
Polar co-ordinates of general number,

25, 27.
expression of general number, 25,

27, 38, 43, 44, 48.

Polyphase relation, reducing trigo-
nometric series, 134.
of trigonometric functions, 104.
system of points or vectors, 46.



INDEX.



291



Positive number, 4.

Potential series, 52, 212.

Power factor maximum of induction

motor, 149.

maximum of alternating trans-
mission circuit, MS.
of generator, 161.
of shunted resistance, 155.
of storage battery, 172.
of transformer, 173.
of transmission line, 165.
not vector product, 42.
of shunt motor, approximated, 189.
with small quantities, 194.
Probability calculation, 181.
Product series, 274.

of trigonometric functions, 102.
Projection, defining cosine function,

94.
Projector, defining sine function, 94.

Q

Quadrants, sign of trigonometric

functions, 96.
Quadrature numbers, 13.
Quarter scale, 260.
Quaternions, 22.



R



Radius vector of general number, 28.

Range of convergency of series, 56.

Rational equation of curve, 210.
function, 266

Rationality of potential series, 214.

Real number, 26.

Rectangular co-ordinates of general
number, 25.

Reduction to absolute values, 48.

Relations of hyperbolic, trigono-
metric and exponential func-
tions, 280.

Relativeness of small quantities, 188.

Reliability of numerical calculations,
261.

Resistance, 65

Resolution of vectors, 29.

Reversal by negative unit, 14.

Reverse function, 265.

Right triangle defining trigonometric
functions, 94.

Ripples in wave, 45.



Roots of general numbers, 45.

with small quantities, 194.

of unit, 18, 19, 46.
Rotation by negative unit, 14.

by quadrature unit, 14.



Saw-tooth wave, 246.
Scalar, 26, 28, 30.

Scale in curve plotting, proper and
improper, 259, 212.

full, double, half, etc., 260.
Secant function, 98.
Secondary effects, 210.

phenomena, 233.
Series, exponential, 71.

infinite, 52.

trigonometric, 106.
Shape of curves, 212.

proper in plotting, 259.

of exponential curve, 226, 229.

of function, by curve, 257.

of hyperbolic functions, 231.

of parabolic and hvperbolic curves,

217.

Sharp zero wave, 246.
Short circuit current of alternator,

approximated, 195.
Sign error, 264.

of trigonometric functions, 95.
Simplification by approximation, 187.
Sine-amplitude, 270.

component of wave, 121, 125.

function, 94.

series, 82.

versus function, 98.
Sinh function, 276.
Slide rule accuracy, 254.
Small quantities, approximation, 187.
Squares, least, method of, 179, 186.
Special functions, 273.
Steam path of turbine, 33.
Subtraction, 1.

of general number, 28.

of trigonometric functions, 102.
Summation series, 274.
Surging of synchronous machines, 272.
Supplementary angles in trigonomet-
ric functions, 99.
Symmetrical curve maximum, 150.

periodic function, 117.

wave, 117.



292



INDEX.



Tabular form of calculation, 249.

Tangent function, 94.

Taylor's series with small quantities,

199. t

Temperature wave, 131.
Temporaryuse of potential series, 216.
Terminal conditions of problem, 69.
Terms, constant, of empirical curves,

232.

in exponential curve, 228.
with exponential curve, 227.
in parabolic and hyperbolic

curves, 225.
of infinite series, 53.
Theorem, binomial, infinite series,

59.

Thermomotive force wave, 133.
Theta functions, 271.
Third harmonic, separation, 136.
Torque of fan motor by potential

series, 215.
Transient current curve, evaluation,

239.

phenomena, 106.

Transmission line calculation, 249.
equations, approximated, 204.
Triangle, defining trigonometric func-
tions, 94.

trigonometric relations, 106.
Trigonometric and exponential func-
tions, relation, 83.
functions, 94, 275.
series, 82.

with small quantity, 198.
integrals and functions, 269.



Trigonometric series, 106.

calculation, 114, 116, 139.
Triple harmonic, separation, 136.
peaked wave, 246.
scale, 260.

Tungsten filament, volt-ampere char-
acteristic, 233.
Turbine, steam path, 33.



U

Univalent functions, 106.
Unsymmetric curve maximum, 151.
wave, 138.



Values of trigonometric functions,

101.

Vector analysis, 32.
multiplication, 39.
quantity, 32.

see General number,
representation by general number,

29.
Velocity diagram of turbine steam

path, 34.

functions of electric field, 275.
Versed sine and cosine functions, 98.
Volt-ampere characteristic of magnet-
ite arc, 237.
of tungsten filament. 233.



Zero values of curve, 211.
of empirical curves, 232.




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Online LibraryCharles Proteus SteinmetzEngineering mathematics; a series of lectures delivered at Union college → online text (page 17 of 17)