Electricity and magnetism for engineers

Front matter

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Part I Electric And Magnetic Circuits By Habold Pender

380 pages, 6X9, Illustrated.

Pabt II—- Elbctbostatics and Altbrnatinq Cubbbnts. 221 pages, 6X0, Illustrated.

ELECTBICITT ASD MAGNETISM

FOR

ENGINEERS

PART I

ELECTRIC AND MAGNETIC CIRCUITS

FiitsT Edition

FOUBTH ImPBEBSION

McGRAW-HILL BOOK COMPANY, Inc.

NEW YORK: 370 SEVENTH AVENUE

LONDON : 6 A 8 BOU VERIE ST., E. C. 4

Copyright, 1918, by thb McGraw-Hill Book Company, iNa

PRINTED IN THE UNITED STATES OF AMERICA

THE MAPLE PRESS COMPANY, YORK, PA.

Preface

In the following pages is given, from an engineering point of view, (1) a description of the more important eflFects conmionly described as electric and magnetic phenomena, (2) a statement of the fundamental laws in accord with which these phenomena have been foimd to occur, and (3) the application of these laws to some of the simpler problems which arise in connection with the generation, transmission and utilization of electric energy.

Particular emphasis is laid upon exact qiuintiiative statements of the fundamental laws or principles. Both safety and economy demand that the engineer be able to answer not only ''how," but also ''how much. " To this end, the student of engineering • should be taught to analyze, not only qualitatively, but also quantitatively, each problem which may be presented to him.

Most of the simpler formulas used by scientists and engineers are special cases of certain general relations, and these special formulas are applicable only under certain specific conditions. One of the most common causes of confusion on the part of the beginner arises from his attempt to apply such special formulas to cases to which they are not applicable. This is due in part to the failure in many text-books to state the limitations of such formulas. Particular care is therefore taken in these pages to state specifically the exact conditions under which each formula is applicable.

The procedure adopted throughout the book is to pass from simple phenomena, known to practically every school-boy, to the more complex phenomena and principles with which the engineer has to deal.

For convenience the book has been divided into two parts. Part I deals with the electric and the magnetic circuits, and Part II with electrostatics and alternating currents. Each part of the book can readily be covered in four hours of classroom work per week for a term. Part I may be looked upon as an introduction to the study of direct-current machinery, and Part II as an introduction to the study of alternating-current machinery.

vi PREFACE

At the end of each important section are given one or more problemsi illustrating the principles developed in the text. The answers to these problems are also given. The student should be required to solve each problem, and when time is available additional problems, without answers, should be assigned. It is only by the solution of numerical problems that the student can imderstand the full significance of the relations developed in the text.

This book covers substantially the same ground as that of the author's '^ Principles of Electrical Engineering/' McGraw- Hill Book Company, 1911. The method of treatment, however, is distinctly different, the various laws and relations are more fully discussed, and a greater number of practical applications is given. Habold Pbndbb.

Philadelphia, Pa., Od. 3, 101&

CHAPTER I Paqb

1. Introduction 1

8. Scalar and Vector Quantities 1

8. Linear Displacement, Velocity and Acceleration 6

4. Angular Displacement, Velocity and Acceleration 8

5. Mass

6. Systems of Units 10

7. Density and Specific Gravity 11

8. Center of Mass 12

9. Moment of Inertia and Radius of Gyration 12

10. Linear Momentum and Angular Momentum 13

11. Force and Weight 13

12. Units of Force 16

18. Pressure 17

14. Torque or Moment of a Force 17

15. Work and Energy 19

16. Units of Work and Energy 21

17. Kinetic and Potential Energy 22

18. Frictional and Elastic Forces 24

19. Power 26

80. Efficiency and Losses 28

81. Flow of Water Through Hydraulic Motors and Pumps .... 30

CHAPTER II Electricity, Electric Currents and Electric Energy

82. Introduction 36

23. Voltaic Cells *. 36

24. Electricity and Magnetism 38

26. Electricity and Matter 40

26. Conductors and Insulators 42

27. Electrolytes and Electrolysis 43

28. Quantity of electricity and Current of Electricity 45

29. Kirchhoff's First Law for Conducting Networks 48

80. Series and Parallel Connections 50

81. Systems of Electric Units 52

32. The International Ampere and the International Coulomb ... 53

88. Galvanometers, Ammeters, and Electrodynamometers 55

84. Electric Energy and Electric Power 60

86. Electric Resistance 61

86. Units of Electric Reststance, Power and Energy 62

87. Sources and Receivers of Electric Energy 66

vii viii TABLE OF CONTENTS

Pagb

88. Electric Pressure and Electromotive Force 70

89. Units of Electromotive Force 73

40. Difference of Electric Potential and Resistance Drop 75

41. Direction of a Potential Difference, of an Electromotive Force, and

of a Resistance Drop 80

42. Kirchhoff's Second Law for Networks 82

48. The Voltmeter 84

44. Millivoltmeter and Shunt for Measuring Large Currents .... 85

46. The Potentiometer 87

46. The Wheatstone Bridge 89

47. The Wattmeter 91

CHAPTER III Direct-current Circuit Calculations

48. Fundamental Principles Involved in Circuit Calculations .... 93

49. Resistances and Electromotive Forces in Series 95

60. Voltage Drop and Power Loss in a Transmission Line Supplying a

Single Load 98

61. Resistances and Electromotive Forces in Parallel 101

62. Single Transmission line Supplying Two or More Loads .... 107 68. Three-wire Systems of Distribution 109

CHAPTER IV Their Dependence on the Material, Dimensions, Shape and Temperature of

the Conductor

66. Resistivity or Specific Resistance 113

66. Ohms per Meter-gram 117

67. Direct-current Conductance 119

68. Conductivity or Specific Conductance 120

69. Percent Conductivity — Annealed Copper Standard 120

60. American Wire Gage 121

61. Size of Wire Required for Direct-current Distribution Circuits . 123

62. Temperature Coefficient of Electric Resistance 124

68. Determination of the Temperature Rise of Conductors by the

Change in Their Resistance 127

64. Resistance of Non-linear Conductors 128

66. Insulation Resistance of a Single-conductor Cable 133

CHAPTER V Electrolytic Conduction

67. Faraday's Laws of Electrolysis 138

68. Dissociation Theory of Electrolytes 140

TABLE OF CONTENTS ix

CHAPTER VI Sources of Electromotive Force

Paqb

70. Contact Electromotive Forces at Junctions Between Metals. . . 144

71. Thermal Electromotive Force 145

72. Electromotive Force of Voltaic Cells 146

78. Standard Cells 148

CHAPTER VII Magnetic Flux and the Magnetic Circuit

74. Electromagnetic Induction 150

76. Magnetic Fields 151

76. lines of Magnetic Force 153

77. Right-handed and Left-handed Screw Relations 154

78. Continuity of the Lines of Magnetic Force 155

79. Magnetic Flux 156

80. Units of Magnetic Flux 160

81. Induced Electromotive Force and Rate of Change of Flux . . .161 88. Direction of the Induced Electromotive Force — Resultant Flux

Through a Surface 163

83. The Direct-current Dynamo 165

84, Electromotive Force Induced in the Armature of a Direct-current

Dynamo 171

86. Flux Linkages 173

86. Measurement of Magnetic Flux 175

87. Magnetic Flux Density 177

88. Magnetomotive Force and Magnetic Reluctance 180

89. Magnetic Permeability and Reluctivity 184

90. Magnetizing Force 188

91. Practical Calculation of the Ampere-turns Required to Produce a

Given Flux 196

92. Difference of Magnetic Potential 199

98. Mutual and Leakage Fluxes 202

94. Kirchhoff's Laws for the Magnetic Circuit 206

96. Refraction of Lines of Force at the Surface of a Magnetic Medium . 207

CHAPTER VIII Calculation of Flux Density

97. Mux density Due to a Current in an Elementary Length of its Cir-

cuit 211

98. Flux Density Due to a Current in a Finite Length of a Straight

Wire 213

99. Flux Density Expressed in Terms of a Solid Angle 220

100. Flux Density Due to a Current in a Circular Coil 225

X TABLE OF CONTENTS

Pagb

101. Flux Density Due to a Current in a Solenoid 228

102. Intensity of Magnetization and Magnetic Susceptibility .... 236

108. Magnetic Poles and Magnetic Pole Strength 240

104. Magnetizing Force and Flux Density When Magnetic Substances

are Pr^ent 246

100. Distribution of the Magnetic Poles on the Surface of a Magnetic

Body 249

106. Solenoid with Magnetic Core 252

107. Magnetic Potential Due to a Magnetic Pole 255

108. Flux Density due to a Current in a Toroid 258

109. The Earth's Magnetic Field 262

CHAPTER IX Hysteresis, Magnetic Energy and Eddy Currents

110. Determination of the Relation Between the Flux Density B and

the Magnetizing Force H in a Magnetic Substance 264

111. Magnetic Hysteresis 268

112. B-H Curves 272

118. Magnetic Saturation 277

114. Magnetic Energy and Hysteresis Loss 278

115. Total Energy Required to Establish a Magnetic Flux 281

116. Magnetic Energy in a Circuit of Constant Reluctance 283

117. Hysteresis Loss and Magnetic Energy in Ferromagnetic Sub-

stances 287

118. Eddy-currents and Eddy-current Loss 296

119. Total Core Loss 306

CHAPTER X Self- and Mutual Inductance

121. Self -inductance of a Coil 312

122. Mutual Inductance of Two Coils 316

128. Mutual Inductance and Mechanical Forces 322

124. Energy of the Magnetic Field Due to One or More Electric Cur-

rents in Terms of the Induction Coefficients of Their Circuits . 325

125. Leakage Inductance and Exciting Current 328

126. Establishment and Decay of a Current in a Circuit Containing '

Resistance and Inductance Only 336

127. Alternating-current Transformer 341

128. Alternating-current Generator 346

CHAPTER XI Mechanical Forces in Magnetic Fields

130. Force exerted on a Straight Wire by a Uniform Magnetic Field . 357

TABLE OF CONTENTS xi

Paqb

181. Force of Attraction or Repulsion Between Two Coilfl 359

132. Torque Exerted on a Coil by a Magnetic Field 360

188. Average Torque Exerted by a Magnetic Field on a Rotating Coil 363

184. Forces on Magnetic Poles in a Non-magnetic Medium 366

186. Practical Calculation of the Mechanical Forces Exerted on Magnetic Bodies by a Magnetic Held 370 l^ART I ELECTRIC AND MAGNETIC CIRCUITS

1. Introduction. — Electricity manifests itself in the production of forces between material bodies and in the production of various forms of energy. A knowledge of the terms employed in describing the effects of mechanical forces and energy, and of the fundamental relations between matter, force and energy, is therefore essential to an imderstanding of the facts and "laws" of electricity and magnetism. For the convenience of the reader a brief review of these terms and principles is here given.

2. Scalar and Vector Quantities. — ^A quantity which can be completely specified by a magnitude, such as mass, work, temperature, etc., is called a "scalar" quantity. A scalar quantity may be either positive or negative. In the first case it is represented by a positive number and in the latter case by a negative number. Two or more scalar quantities are added and subtracted according to the ordinary laws of algebra.

A quantity, such as a force, which requires for its complete representation not only a magnitude but a definite direction in space, is called a "vector" quantity. A vector quantity can always be represented by a straight line of definite length drawn in a definite direction in space. The line representing a vector quantity is called a "vector."

A vector may be conveniently represented by two letters which designate its ends. The convention universally adopted is to write as the first letter that letter which designates the point /row which the vector is drawn. For example, by the vector AB is meant the straight line drawn from AtoB. Similarly, by the vector bZ is meant the straight line drawn from B to A.

When a vector is represented by a line AB, the direction from ii to fi is called the ''positive sense" of this line.

The length of a line may also be represented by a single letter or number, and the positive sense of this line by an arrow placed on the line pointing in it s positive sense. Hence if 7 is used to designate the vector AB, then the length and sense of the vector drawn from Aio B may be represented by the algebraic expression +V and the length and sense of the vector drawn from B to il by the algebraic expression — V,

A J^ B

Fig. 1.

The relative direction of two vectors, such as AB and AG in Fig. 2, is conveniently specified as follows: When the angle measured around from AB to AC in the courUer-dockwiae direction is less than 180 degrees, the vector AC is said to "lead" AB by this angle, and conversely, the vector AB is said to "lag" AC by this same angle. When the angle measured around from AB to AC in the counter-clockwise direction is greater than 180 degrees, then the angle measured around from AB to AC in the clockwise direction must be less than 180 degrees, and therefore AB leads 'AC by this latter angle, or AC lags AB by this same angle. A

Fig. 2.

lead of <t> degrees is therefore equivalent to a lag of ^ = 360 — ^ degrees, and a lag of ^ degrees is equivalent to a lead of = 360 — if> degrees. Hence it is always possible, by employing the conceptions of lead and lag, to specify the relative direction of two vectors by an angle less than 180 degrees.

By the "vector sum" of two or more vectors, such asAB and

AC in Fig. 2, is meant vector equal to the diagonal ZS of the parallelogram formed by drawing through C a line CS parallel to AB and through B a line BS parallel to AC. If the angle by which AC leads AB is $, then the vector sum of AB and AC results in a vector of length

AS = VaB* + AC' + 2 ABAC cos $ (1)

leading AB by the angle

<r = tan M= = (la)

VaB+AChobBI

From these relations it follows that the vector sum of two vectors which are equal in length and opposite in sense, such as AS and Ba (Fig. 1) is zero, for the difference in direction between these two vectors is 180 degrees, and therefore cos ^ = — 1. This fact may be expressed mathematically by the relation

= lB + M

or

BA^ - AB

By the "vector subtraction" of a vector AC from a vector AB is meant the vector addition of — AC to AS, which, from the

Fig. 3.

relation just deduced, is equivalent to turning the vector AC through 180 degrees and adding it to AB, as shown in Fig. 3. Hence the vector subtraction of AC from any vector AB results in a vector of length

AD = A/AB* + AC' - 2 AB,AC cos e (2)

lagging 'AB by the angle

\AB - AC cob tf J

It is also of interest to note that the length and direction of this vector difference are respectively the same as the length and direction of diagonal CB of the parallelogram in Fig. 2.

A special case of vector subtraction of particular importance is that of the vector difference of two vector quantities which have equal magnitudes and which differ in direction by an infinitely small angle. Let V be the magnitude of each of the two vectors AB and AC in Fig. 4, let d$ be the angle between them, expressed in radians, and let dV represent the vector difference BC. Since the angle between AB and AC ia infinitely small, the ratio of the length of the line BC to the length of the line AB is equal to the angle dd in radians.^ Hence

dV = Vdd (26)

Also, since d$ is infinitely small, the line BC is perpendicular to AB. Hence, from Fig. 4, the vector difference dV is perpendicular to the vector V and is in the direction of the positive sense of the angle d$.

Fio 5.

It may be readily shown that any vector AB may be considered as the sum or "resultant" of two or more component vectors, provided these component vectors have such lengths and directions that when placed end to end they form a continuous path from AtoB, such that a point in passing from A to B over this path moves along each component in its positive sense. In particular, it is frequently convenient to consider a vector AB

*The measure of an angle in radians is by definition the ratio of the arc which subtends this angle to the radius of this arc. When the angle is infinitely small the chord of this arc is equal to the arc itself.

as the sum of two mutually perpendicular components, such as AP and PB in Fig. 5. If tf» is the angle by which AB leads the component AP, then the two components AP and PB are

AP PB

AB cos 0k AB sin 0i

Conversely, when the two mutually perpendicular components AP and PB are known, then the resultant AB has the length

AB = \AP' + PB^ and leads the component AC by the angle

fPB\ <AP)

Sb = tan-^ I

These relations lead to another way of expressing the sum of two vectors. Let AB and AC be the two vectors. At the end of AB lay off £iS equal in length to AC and in the same direction

Pro. 6.

as AC> and resolve AB and BS into their horizontal and vertical components. Then, using the notation indicated in Fig. 6, the vector sum of AB and BS is a vector of length

AS = \Af^ + TS^ which leads AP by the angle 6„ where tan St = == AT

The values of TS and AT are

TS^PB + QS = AB8inei, + ACyin 6^ AT ^ AP + BQ = AB COB Ob + AC cos Op

That is, the component, hi any dkection, of the resultant of two or more vectors is the sum of the components, in this same direction, of the individual vectors*

By calculating first the angle B, from equation (5a), the evaluation of the radical expression may be avoided by calculating the vector siun AS from the formula cos ^,

A very convenient table of trigonometric functions for use in such calculations is given in Pender's Handbook for Electrical Engineers, pages 1709 to 1715.

Problem 1. — ^Two vectors A and B, 5 and 10 inches in length respectively, make an angle of 30 degrees with each other, the shorter vector A leading the longer one.

(a) What is the vector sum of these two vectors? (6) By what angle does this vector sum, or resultant, lead the longer vector? (c) What is the vector difference of A minus B? (d) By what angle does this vector difference lead A ? {e) By what angle does the vector difference B minus A lag A ? if) What is the domponent of A in the direction of B? (jg) What is the component of B perpendicular to A, and does it lead or lag A? (A) If the angle by which A leads B is changing at the rate of 157 radians per second, what is the length and direction, with respect to A, of the vector which is equal to the rate of change of the vector A with respect to time? Draw a vector diagram to illustrate the answer to each of the above questions.

Anaum. — (a) 14.55 inches. (&) 9.9 degrees, (c) 6.20 inches, (d) 156.2 degrees, (e) 23.8 degrees, if) 4.33 inches, {g) 5 inches, lagging. (A) 785 inches, leading A by 90 degrees.

Problem 2. — A vector A which has a length of 12 inches leads a reference line OiX by 40 degrees. Another vector B has a component in the direction of OX equal to 0.6 inch, and a component 90 degrees ahead of OX equal to 1.5 inches.

(a) What is the component in the direction of OX of the vector sum of A and B? (6) What is the component of this resultant 90 degrees ahead of OX? (c) What is the angle by which this resultant leads OX? (d) What is the magnitude of this resultant?

Answer,— {a) 9.79 inches. (6) 9.21 inches, (c) 43.3 degrees, (d) 13.45 inches.

3. Linear Displacement, Velocity and Acceleration. — When a particle moves from a point A to some other point B, fixed relative to the point A, the straight line drawn from A to fi is called the "linear displacement" of this particle relative to A.

The rate of change with respect to time in the linear displacement of a particle relative to any point A is called the ''linear velocity" of this particle with respect to the point A. When this rate of change is uniform, the vdocity is where I is the displacement in time t. When the rate of change in the displacement of the particle is not uniform, then the velocity at any instant t is

where dl represents the change in the displacement in an infinitesimal interval of time dt measured from this instant.

The rate of change with respect to time in the linear velocity of a particle, relative to any point Ay is called the ''linear acceleration" of this particle with respect to the point A. When the rate of change of velocity is uniform, then the acceleration is where vi — vtia the increase in velocity in time t. When this rate of change is not uniform the acceleration is

dv d*l ,_ ,

« = ^ = ^« ^7«>

Linear displacement, velocity and acceleration are all vector quantities, i.e., each has a direction as well as a magnitude. The term "speed" is conunonly used to designate the magnitude of a velocity.

Displacements, velocities and acceleration therefore cannot be added and subtracted algebraically, but must be added and subtracted vectoriaUy. In particular, when the direction of the velocity of a point changes, Vi — t;2 in equation (7) and dv in equation (7a) must be interpreted as vector differences. For example, when a point is moving in a circular path of radius r at a constant speed F, the direction of its velocity will change during any infinitesimal interval of time dt by the angle dd =

Vdt

— . Hence, from equation (26), the change in the linear veloc-

V^dt ity of this particle in time (ft is dF = Vdd = . From equation (7a) the linear acceleration of this particle at this instant is therefore and, flince dV is perpendicular to the direction of the velocity V and in the positive sense of the angle dS, the direction of this acceleration is toward the center of the circle. In this formula V and r must be expressed in the same units of length; for example, V in miles per hour and r in miles, in which case a will be in miles per hour per hour.

Problem S. — ^A locomotive is traveling at the rate of 60 miles an hour around a curve of 1000 feet radius.

(a) What ia the speed of this locomotive in meters per second? (h) In feet per second? (c) What is the direction and amount of its acceleration in miles per hour per second? (d) In feet per second per second?

Ansioer. — (a) 26.8 meters per second, (h) 88.0 feet per second, (c) 5.28 miles per hour per second, toward the center of the curve, (jd) 7.75 feet per second per second.

4. Angular Displacement, Velocity and Displacement — ^Let OX be any arbitrarily chosen axis, and imagine a plane drawn through this axis and a moving particle. The angle through which this plane turns when the particle moves from any point A to some other point B, fixed with respect to the point A and the axis OX, is called the ''angular displacement" of the particle about this axis relative to the point A.

The rate of change with respect to time in the angular displacement of a particle about a given axis OX is called the angular velocity of the particle about this axis, and the rate of change in this angular velocity with respect to time is called its angular acceleration about this axis.

When the angular displacement is changing at a constant rate, the angular velocity is where $ is angular displacement in time t When the rate of change in the angular displacement is not uniform, then the angular velocity is

" - 1' (*•)

Similarly, when the angular velocity is changing at a constant rate, the angular acceleration is where «i — wj is the increase in the angular velocity in time t.

When the rate of change in the angular velocity is not uniform, then the angular acceleration is

The fundamental unit of angular displacement is the radian, and the corresponding units of angular velocity and angular acceleration are respectively 1 radian per second and 1 radian per second per second. In the case of a rigid body, rotating about a fixed axis, the unit of angular velocity commonly employed is one complete revolution per minute. One complete revolution is equal to 2ir radians, and hence one revolution per

minute is equal to xtt radians per second.

Problem 4. — ^The armature (i.e., the rotating member) of an electric motor, starting from rest, reaches a speed of 750 revolutions per minute (abbreviated r.p.m.) in 10 seconds, the change in velocity being at a uniform rate.

(a) What is the ang^ular acceleration of the armature in radians per second per second? (b) How many revolutions wiU the armature make before reaching the speed of 750 r.p.m.?

Answer, — (a) 7.854 radians per second per second. (6) 62.5 revolutions.

6. Mass. — ^The quantity of matter in a body, called its "mass," can be defined only in terms of some effect produced on the body by some other body or bodies external to it. It is a fact of experiment that two bodies, which appear to our senses to be identical in every respect, will exactly counterbalance each other when suspended one from each end of an equal-armed balance in a vacuum. . As an arbitrary assimiption, the mass of any two bodies may then be defined as equal, irrespective of their volume, shape or chemical composition, if, when suspended in a vacuum, one from each end of an equal-armed balance, and each free from all external influences other than that due to the earth, there is no tipping of the beam of the balance from its original I>osition of equilibrium.

This is an entirely arbitrary definition, but it has been found that mass as thus defined is a fundamental property of matter. Any arbitrary portion of matter may be taken as the unit of mass; the mass of any given portion of matter may then be expressed as the number of such equal units, which taken together, and suspended from one arm of an equal-armed balance in a vacuum, will just counterbalance the given body suspended from the other ann. Note that mass is a scalar quantity.

6. Systems of Units. — The standard unit of mass to which all other units of mass are referred is a certain platinum-iridium cylinder, known as the International Kilogram, preserved at the International Bureau of Weights and Measures near Paris. The standard unit of length is similarly the length, at a temperature of O^C, of a certain platinum-iridium bar, known as the International Meter, preserved at the same place.

As the standard unit of time is taken the mean solar day. By a solar day is meant the interval of time between two successive transits of the sun across the meridian of the earth at the point of observation. This interval varies in length at different times during the year, but the mean solar day, i.e., the average length of this interval for each year, remains constant; or more precisely, may be assumed to remain constant.

Submultiples of the meter, kilogram and solar year are generally employed as more convenient units. The three standard submultiples are:

1 gram = j^ kilogram.

1 second = oaAQo ^^^^ solar day.

In terms of these three units, namely, the centimeter, the gram and the second, a unit may be specified for every measurable quantity. These three units are therefore called the three "fundamental" units, and any unit specified in terms of these three fundamental units is called a '^ derived" unit.

The system of units specified in terms of the centimeter, gram and second in such a manner that the fundamental units appear in the specification only as unity or as integral multiples of 10, 100, 1000, etc., or their reciprocals, is called the "c.g.s. metric system" of units.

Multiple and submultiple units in the c.g.s. system are designated by the following prefixes:

^^^ = 150 = ^°""

deka == 10 hecto = 100 = 10* kflo = 1000 = 10» myria = 10,000 = 10* mega = 1,000,000 = 10« For example, a millimeter is one-thousandth of a meter or one-tenth of a centimeter; a kilometer is 1000 meters; a decigram is one-tenth of a gram or one ten-thousandth of a kilogram, etc. The English, or foot-pound-second, system of units was originally based upon arbitrary units of mass and length, independent of the kilogram and meter, namely, the standard poimd and the standard yard. However, the present U. 3. legal definitions (Act of Congress, July 28, 1866) are and