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Fundamentals of Physics I
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the open yale courses series is designed to bring the depth and
breadth of a Yale education to a wide variety of readers. Based on Yale’s
Open Yale Courses program (http://oyc.yale.edu), these books bring out-
standing lectures by Yale faculty to the curious reader, whether student or
adult. Covering a wide variety of topics across disciplines in the social sci-
ences, physical sciences, and humanities, Open Yale Courses books offer
accessible introductions at affordable prices.
The production of Open Yale Courses for the Internet was made possible
by a grant from the William and Flora Hewlett Foundation.
books in the open yale courses series
Paul H. Fry,Theory of Literature
Roberto Gonz´alez Echevarr´ıa,Cervantes’ “Don Quixote”
Christine Hayes,Introduction to the Bible
Shelly Kagan,Death
Dale B. Martin,New Testament History and Literature
Giuseppe Mazzotta,Reading Dante
R. Shankar,Fundamentals of Physics I: Mechanics, Relativity,
and Thermodynamics, Expanded Edition
R. Shankar,Fundamentals of Physics II: Electromagnetism, Optics,
and Quantum Mechanics
Ian Shapiro,The Moral Foundations of Politics
Steven B. Smith,Political Philosophy
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Fundamentals
ofPhysicsI
Mechanics, Relativity,
and Thermodynamics
Expanded Edition
r. shankar
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First edition 2014. Expanded edition 2019.
Copyrightc2014, 2019 by Yale University.
All rights reserved.
This book may not be reproduced, in whole or in part, including
illustrations, in any form (beyond that copying permitted by Sections
107 and 108 of the U.S. Copyright Law and except by reviewers for the
public press), without written permission from the publishers.
Yale University Press books may be purchased in quantity for educational,
business, or promotional use. For information, please e-mail sales.press@
yale.edu (U.S. ofﬁce) or [email protected] (U.K. ofﬁce).
Set in Minion type by Newgen North America.
Printed in the United States of America.
ISBN: 978-0-300-24377-2
Library of Congress Control Number: 2019931271
A catalogue record for this book is available from the British Library.
This paper meets the requirements of ANSI/NISO Z39.48-1992
(Permanence of Paper).
10987654321
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To my students
for their friendship and inspiration
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Deep and original, but also humble and generous, the physicist
Josiah Willard Gibbs spent much of his life at Yale University. His father was
a professor of sacred languages at Yale, and Gibbs received his bachelor’s
and doctorate degrees from the university before teaching there until his death in
1903. The sculptor Lee Lawrie created the memorial bronze tablet pictured
above, which was installed in Yale’s Sloane Physics Laboratory in 1912. It now
resides in the entrance to the J. W. Gibbs Laboratories, Yale University.
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Contents
Preface to the Expanded Edition xiii
Preface to the First Edition xiv
1. The Structure of Mechanics 1
1.1 Introduction and some useful tips 1
1.2 Kinematics and dynamics 2
1.3 Average and instantaneous quantities 4
1.4 Motion at constant acceleration 6
1.5 Sample problem 10
1.6 Derivingv
2
−v
2
0
=2a(x−x 0)using calculus 13
2. Motion in Higher Dimensions 15
2.1 Review 15
2.2 Vectors ind=216
2.3 Unit vectors 19
2.4 Choice of axes and basis vectors 22
2.5 Derivatives of the position vectorr 26
2.6 Application to circular motion 29
2.7 Projectile motion 32
3. Newton’s Laws I 36
3.1 Introduction to Newton’s laws of motion 36
3.2 Newton’s second law 38
3.3 Two halves of the second law 41
3.4 Newton’s third law 45
3.5 Weight and weightlessness 49
4. Newton’s Laws II 51
4.1 A solved example 51
4.2 Never the whole story 54
4.3 Motion ind=255
4.4 Friction: static and kinetic 56
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viii Contents
4.5 Inclined plane 57
4.6 Coupled masses 61
4.7 Circular motion, loop-the-loop 64
5. Law of Conservation of Energy 70
5.1 Introduction to energy 70
5.2 The work-energy theorem and power 71
5.3 Conservation of energy:K
2+U2=K1+U1 75
5.4 Friction and the work-energy theorem 78
6. Conservation of Energy ind=282
6.1 Calculus review 82
6.2 Work done ind=284
6.3 Work done ind=2 and the dot product 88
6.4 Conservative and non-conservative forces 92
6.5 Conservative forces 95
6.6 Application to gravitational potential energy 98
7. The Kepler Problem 101
7.1 Kepler’s laws 101
7.2 The law of universal gravity 104
7.3 Details of the orbits 108
7.4 Law of conservation of energy far from the earth 112
7.5 Choosing the constant inU 114
8. Multi-particle Dynamics 118
8.1 The two-body problem 118
8.2 The center of mass 119
8.3 Law of conservation of momentum 128
8.4 Rocket science 134
8.5 Elastic and inelastic collisions 136
8.6 Scattering in higher dimensions 140
9. Rotational Dynamics I 143
9.1 Introduction to rigid bodies 143
9.2 Angle of rotation, the radian 145
9.3 Rotation at constant angular acceleration 147
9.4 Rotational inertia, momentum, and energy 148
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Contents ix
9.5 Torque and the work-energy theorem 154
9.6 Calculating the moment of inertia 156
10. Rotational Dynamics II 159
10.1 The parallel axis theorem 159
10.2 Kinetic energy for a general N-body system 163
10.3 Simultaneous translations and rotations 165
10.4 Conservation of energy 167
10.5 Rotational dynamics usingτ=
dL
dt
168
10.6 Advanced rotations 169
10.7 Conservation of angular momentum 171
10.8 Angular momentum of the ﬁgure skater 172
11. Rotational Dynamics III 175
11.1 Static equilibrium 175
11.2 The seesaw 176
11.3 A hanging sign 178
11.4 The leaning ladder 180
11.5 Rigid-body dynamics in 3d 182
11.6 The gyroscope 191
12. Special Relativity I: The Lorentz Transformation 194
12.1 Galilean and Newtonian relativity 195
12.2 Proof of Galilean relativity 196
12.3 Enter Einstein 200
12.4 The postulates 203
12.5 The Lorentz transformation 204
13. Special Relativity II: Some Consequences 209
13.1 Summary of the Lorentz transformation 209
13.2 The velocity transformation law 212
13.3 Relativity of simultaneity 214
13.4 Time dilatation 216
13.4.1 Twin paradox 219
13.4.2 Length contraction 220
13.5 More paradoxes 222
13.5.1 Too big to fall 222
13.5.2 Muons in ﬂight 226
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xContents
14. Special Relativity III: Past, Present, and Future 227
14.1 Past, present, and future in relativity 227
14.2 Geometry of spacetime 232
14.3 Rapidity 235
14.4 Four-vectors 238
14.5 Proper time 239
15. Four-momentum 241
15.1 Relativistic scattering 249
15.1.1 Compton effect 249
15.1.2 Pair production 251
15.1.3 Photon absorption 252
16. Mathematical Methods 255
16.1 Taylor series of a function 255
16.2 Examples and issues with the Taylor series 261
16.3 Taylor series of some popular functions 263
16.4 Trigonometric and exponential functions 265
16.5 Properties of complex numbers 267
16.6 Polar form of complex numbers 272
17. Simple Harmonic Motion 275
17.1 More examples of oscillations 280
17.2 Superposition of solutions 283
17.3 Conditions on solutions to the harmonic oscillator 288
17.4 Exponential functions as generic solutions 290
17.5 Damped oscillations: a classiﬁcation 291
17.5.1 Over-damped oscillations 291
17.5.2 Under-damped oscillations 292
17.5.3 Critically damped oscillations 294
17.6 Driven oscillator 294
18. Waves I 303
18.1 The wave equation 306
18.2 Solutions of the wave equation 310
18.3 Frequency and period 313
19. Waves II 316
19.1 Wave energy and power transmitted 316
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Contents xi
19.2 Doppler effect 320
19.3 Superposition of waves 323
19.4 Interference: the double-slit experiment 326
19.5 Standing waves and musical instruments 330
20. Fluids 335
20.1 Introduction to ﬂuid dynamics and statics 335
20.1.1 Density and pressure 335
20.1.2 Pressure as a function of depth 336
20.2 The hydraulic press 341
20.3 Archimedes’ principle 343
20.4 Bernoulli’s equation 346
20.4.1 Continuity equation 346
20.5 Applications of Bernoulli’s equation 349
21. Heat 352
21.1 Equilibrium and the zeroth law: temperature 352
21.2 Calibrating temperature 354
21.3 Absolute zero and the Kelvin scale 360
21.4 Heat and speciﬁc heat 361
21.5 Phase change 365
21.6 Radiation, convection, and conduction 368
21.7 Heat as molecular kinetic energy 371
22. Thermodynamics I 375
22.1 Recap 375
22.2 Boltzmann’s constant and Avogadro’s number 376
22.3 Microscopic deﬁnition of absolute temperature 379
22.4 Statistical properties of matter and radiation 382
22.5 Thermodynamic processes 384
22.6 Quasi-static processes 386
22.7 The ﬁrst law of thermodynamics 387
22.8 Speciﬁc heats:c
vandc p 391
23. Thermodynamics II 394
23.1 Cycles and state variables 394
23.2 Adiabatic processes 396
23.3 The second law of thermodynamics 399
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xii Contents
23.4 The Carnot engine 403
23.4.1 DeﬁningTusing Carnot engines 409
24. Entropy and Irreversibility 411
24.1 Entropy 411
24.2 The second law: law of increasing entropy 418
24.3 Statistical mechanics and entropy 423
24.4 Entropy of an ideal gas: full microscopic analysis 430
24.5 Maximum entropy principle illustrated 434
24.6 The Gibbs formalism 437
24.7 The third law of thermodynamics 441
Exercises 443
Problem Set 1, for Chapter 1 443
Problem Set 2, for Chapter 2 446
Problem Set 3, for Chapters 3 and 4 449
Problem Set 4, for Chapters 5, 6, and 7 455
Problem Set 5, for Chapter 8 458
Problem Set 6, for Chapters 9, 10, and 11 461
Problem Set 7, for Chapters 12, 13, 14, and 15 466
Problem Set 8, for Chapters 16 and 17 470
Problem Set 9, for Chapters 18 and 19 475
Problem Set 10, for Chapter 20 478
Problem Set 11, for Chapters 21, 22, 23, and 24 481
Answers to Exercises 487
Problem Set 1, for Chapter 1 487
Problem Set 2, for Chapter 2 488
Problem Set 3, for Chapters 3 and 4 489
Problem Set 4, for Chapters 5, 6, and 7 491
Problem Set 5, for Chapter 8 491
Problem Set 6, for Chapters 9, 10, and 11 492
Problem Set 7, for Chapters 12, 13, 14, and 15 494
Problem Set 8, for Chapters 16 and 17 495
Problem Set 9, for Chapters 18 and 19 497
Problem Set 10, for Chapter 20 498
Problem Set 11, for Chapters 21, 22, 23, and 24 498
Constants and Other Data 501
Index 503
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Preface to the Expanded Edition
Besides the correction of typos, the most important change is the inclusion
of over 300 exercises to go with these 24 chapters. Answers are given to
all exercises, but not the solutions. They were chosen to test the material
covered in this book. This was done in response to calls from students
and instructors. A similar inclusion will be made in the next edition of the
companion volume,Fundamentals of Physics II.
xiii
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Preface to the First Edition
Given that the size of textbooks has nearly tripled during my own career,
without a corresponding increase in the cranial dimensions of my students,
I have always found it necessary, like my colleagues elsewhere, to cull the
essentials into a manageable size. I did that in the course Fundamentals
of Physics I taught at Yale, and this book preserves that feature. It covers
the fundamental ideas of Newtonian mechanics, relativity, ﬂuids, waves,
oscillations, and thermodynamics without compromise. It requires only
the basic notions of differentiation and integration, which I often review
as part of the lectures. It is aimed at college students in physics, chemistry,
and engineering as well as advanced high school students and independent
self-taught learners at various stages in life, in various careers.
The chapters in the book more or less follow my 24 lectures,
with a few minor modiﬁcations. The style preserves the classroom
atmosphere. Often I introduce the questions asked by the students
or the answers they give when I believe they will be of value to
the reader. The simple ﬁgures serve to communicate the point with-
out driving up the price. The equations have been typeset and are
a lot easier to read than in the videos. The problem sets and ex-
ams, without which one cannot learn or be sure one has learned the
physics, may be found along with their solutions at the Yale website,
http://oyc.yale.edu/physics, free and open to all. The lectures may also be
found at venues such as YouTube, iTunes (https://itunes.apple.com/us/
itunes-u/physics-video/id341651848?mt=10), and Academic Earth, to
name a few.
The book, along with the material available at the Yale website, may
be used as a stand-alone resource for a course or self-study, though some
instructors may prescribe it as a supplement to another one adapted for
the class, so as to provide a wider choice of problems or more worked
examples.
To my online viewers I say, “You have seen the movie; now read the
book!” The advantage of having the printed version is that you can read it
during take-off and landing.
xiv
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Preface to the First Edition xv
In the lectures I sometimes refer to myBasic Training in Mathematics,
published by Springer and intended for anyone who wants to master the
undergraduate mathematics needed for the physical sciences.
This book owes its existence to many people. It all began when Peter
Salovey, now President, then Dean of Yale College, asked me if I minded
having cameras in my Physics 200 lectures to make them part of the ﬁrst
batch of Open Yale Courses, funded by the Hewlett Foundation. Since my
answer was that I had yet to meet a camera I did not like, the taping be-
gan. The key person hereafter was Diana E. E. Kleiner, Dunham Professor,
History of Art and Classics, who encouraged and guided me in many ways.
She was also the one who persuaded me to write this book. Initially reluc-
tant, I soon found myself thoroughly enjoying proselytizing my favorite
subject in this new format. At Yale University Press, Joe Calamia was my
friend, philosopher, and guide. Liz Casey did some very skilled editing. Be-
sides correcting errors in style (such as a long sentence that began in ﬁrst
person past tense and ended in third person future tense) and matters of
grammar and punctuation (which I sprinkle pretty much randomly), she
also made sure my intent was clear in every sentence.
Barry Bradlyn and Alexey Shkarin were two graduate students and
Qiwei Claire Xue and Dennis Mou were two undergraduates who proof-
read earlier versions.
My family, from my wife, Uma, down to little Stella, have encouraged
me in various ways.
I take this opportunity to acknowledge my debt to the students at Yale
who, over nearly four decades, have been the reason I jump out of bed on
two or three days a week. I am grateful for their friendship and curiosity.
In recent years, they were often non-majors, willing to be persuaded that
physics was a fascinating subject. This I never got tired of doing, thanks to
the nature of the subject and the students.
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chapter 1
The Structure of Mechanics
1.1 Introduction and some useful tips
This book is based on the ﬁrst half of a year-long course that introduces
you to all the major ideas in physics, starting from Galileo and Newton,
right up to the big revolutions of the twentieth century: relativity and
quantum mechanics. The target audience for this course and book is really
very broad. In fact, I have always been surprised by the breadth of interests
of my students. I don’t know what you are going to do later in life, so I have
picked the topics that all of us in physics ﬁnd fascinating. Some may not
be useful, but you just don’t know. Some of you are probably going to be
doctors, and you don’t know why I’m going to cover special relativity or
quantum mechanics. Well, if you’re a doctor and you have a patient who’s
running away from you at the speed of light, you’ll know what to do. Or, if
you’re a pediatrician, you will understand why your patient will not sit still:
the laws of quantum mechanics don’t allow a very small object to have a
deﬁnite position and momentum. Whether or not you become a physicist,
you should certainly learn about these great strides in the human attempt
to understand the physical world.
Most textbooks are about 1,200 pages long, but when I learned
physics they were around 400 pages long. When I look around, I don’t
see any student whose head is three times as big as mine, so I know that
you cannot digest everything the books have. I take what I think are the
really essential parts and cover them in these lectures. So you need the lec-
tures to ﬁnd out what’s in the syllabus and what’s not. If you don’t do that,
1
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2 The Structure of Mechanics
there’s a danger you will learn something you don’t have to, and we don’t
want that, right?
To learn physics well, you have to do the problems. If you watch me
online doing things on the blackboard or working through derivations in
the book, it all looks very reasonable. It looks like you can do it yourself
and that you understand what is going on, but the only way you’re go-
ing to ﬁnd out is by actually doing problems. A fair number are available,
with their solutions, at http://oyc.yale.edu/physics/phys-200, and over 300
(with answers but not solutions) in the exercises and answers sections at
the back of this book. You don’t have to do them by yourself. That’s not
how physics is done. I am now writing a paper with two other people. My
experimental colleagues write papers with four hundred or even a thou-
sand other people when engaged in the big collider experiments like the
ones in Geneva or Fermilab. It’s perfectly okay to be part of a collaboration,
but you have to make sure that you’re pulling your weight, that everybody
makes contributions to ﬁnding the solution and understands it.
This calculus-based course assumes you know the rudiments of dif-
ferential and integral calculus, such as functions, derivatives, derivatives
of elementary functions, elementary integrals, changing variables in inte-
grals, and so on. Later, I will deal with functions of more than one variable,
which I will brieﬂy introduce to you, because that is not a prerequisite. You
have to know your trigonometry, to know what’s a sine and what’s a co-
sine and some simple identities. You cannot say, “I will look it up.” Your
birthday and social security number are things you look up; trigonometric
functions and identities are what you know all the time.
1.2 Kinematics and dynamics
We are going to be studying Newtonian mechanics. Standing on the shoul-
ders of his predecessors, notably Galileo, Isaac Newton placed us on the
road to understanding all the mechanical phenomena for centuries until
the laws of electromagnetism were discovered, culminating in Maxwell’s
equations. Our concern here is mechanics, which is the motion of billiard
balls and trucks and marbles and whatnot. You will ﬁnd out that the laws
of physics for this entire semester can be written down on the back of
an envelope. A central purpose of this course is to show you repeatedly
that starting with those few laws, you can deduce everything. I would en-
courage you to think the way physicists do, even if you don’t plan to be a
physicist. The easiest way to master this subject is to follow the reasoning I
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The Structure of Mechanics 3
give you. That way, you don’t have to store too many things in your head.
Early on, when there are four or ﬁve formulas, you can memorize all of
them and you can try every one of them until something works, but, after
a couple of weeks, you will have hundreds of formulas, and you cannot
memorize all of them. You cannot resort to trial and error. You have to
know the logic.
The goal of physics is to predict the future given the present. We will
pick some part of the universe that we want to study and call it “the sys-
tem,” and we will ask, “What information do we need to know about that
system at the initial time, like right now, in order to be able to predict its
future evolution?” If I throw a piece of candy at you and you catch it, that’s
an example of Newtonian mechanics at work. What did I do? I threw a
piece of candy from my hand, and the initial conditions are where I re-
leased it and with what velocity. That’s what you see with your eyes. You
know it’s going to go up, it’s going to follow some kind of parabola, and
your hands get to the right place at the right time to receive it. That is an
example of Newtonian mechanics at work, and your brain performed the
necessary calculations effortlessly.
You only have to know the candy’s initial location and the initial ve-
locity. The fact that it was blue or red is not relevant. If I threw a gorilla
at you, its color and mood would not matter. These are things that do not
affect the physics. If a guy jumps off a tall building, we want to know when,
and with what speed, he will land. We don’t ask why this guy is ending it
all today; that is a question for the psych department. So we don’t answer
everything. We ask very limited questions about inanimate objects, and we
brag about how accurately we can predict the future.
The Newtonian procedure for predicting the future, given the
present, has two parts,kinematicsanddynamics. Kinematics is a complete
description of the present. It’s a list of what you have to know about a sys-
tem right now. For example, if you’re talking about a piece of chalk, you
will want to know where it is and how fast it’s moving. Dynamics then
tells you why the chalk goes up, why it goes down, and so on. It comes
down due to the force of gravity. In kinematics, you don’t ask for the rea-
son behind anything. You simply want to describe things the way they are,
and then dynamics tells you how and why that description changes with
time.
I’m going to illustrate the idea of kinematics by following my pre-
ferred approach: starting with the simplest possible example and slowly
adding bells and whistles to make it more and more complicated. In the
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4 The Structure of Mechanics
initial stages, some of you might say, “Well, I have seen this before, so
maybe there is nothing new here.” That may well be true. I don’t know how
much you have seen, but it is likely that the way you learned physics in high
school is different from the way professional physicists think about it. Our
priorities, and the things that we get excited about, are often different; and
theproblemswillbemoredifﬁcult.
1.3 Average and instantaneous quantities
We are going to study an object that is a mathematical point. It has no
size. If you rotate it, it will look the same, unlike a potato, which will look
different upon rotation. It is not enough to just say where the potato is;
you have to say which way its nose is pointing. The study of such extended
bodies comes later. Right now, we want to study an entity that has no spa-
tial extent, a dot. It can move around all over space. We’re going to simplify
that too. We’re going to take an entity that moves only along thex-axis. So
you can imagine a bead with a straight wire going through it, which allows
it to only slide back and forth. This is about the simplest thing. I cannot
reduce the number of dimensions. I cannot make the object simpler than
a mathematical point.
To describe what the point is doing, we pick an origin, call itx=0,
and put some markers along thex-axis to measure distance. Then we will
say this guy is sitting atx=5. Now, of course, we have to have units and the
unit for length is going to be the meter. The unit for time will be a second.
Sometimes I might not write the units, but I have earned the right to do
that and you haven’t. Everything has got to be in the right units. If you
don’t have the units, and if you say the answer is 42, then we don’t know if
you are right or wrong.
Back to the object. At a given instant, it’s got a location. We would like
to describe the object’s motion by plotting a graph of space versus time. A
typical graph would be something like Figure 1.1. Even though the plot is
going up and down, the object is moving horizontally, back and forth along
the spatialx-axis. When it is atA, it’s crossing the origin from the left and
going to the right. Later, atB, it is crossing back to the left. In the language
of calculus,xis a function of time,x=x(t), and the graph corresponds
to some generic function that doesn’t have a name. We will also encounter
functions that do have a name, likex(t)=t,x(t)=t
2
,x(t)=sint,cost,
and so on.
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The Structure of Mechanics 5
Figure 1.1 Trajectory of a particle. The position,x(t),ismeasuredverticallyand
the time,t, is measured horizontally.
Consider¯v,theaverage velocityof an object, given by
¯v=
x(t
2)−x(t 1)
t2−t1
(1.1)
wheret
2>t1are two times between which we have chosen to average the
velocity. In the example in Figure 1.1,¯v<0 for the indicated choice oft
1
andt 2since the ﬁnalx(t 2)is less than the initialx(t 1).
The average velocity may not tell you the whole story. For example,
if you started atx(t
1)and at timet 1ended up at pointCwith the same
coordinate, the average velocity would be zero, which is the average you
would get if the particle had never moved!
Theaverage acceleration,¯a, involves a similar difference of velocities:
¯a=
v(t
2)−v(t 1)
t2−t1
. (1.2)
Now for an important concept, thevelocity at a given timeorinstan-
taneous velocity, v(t). Figure 1.1 shows some particle moving a distance
→xbetween timestandt+→t. The average velocity in that interval is
→x
→t
. What you want is the velocityattimet. We all have an intuitive no-
tion of velocity right now. When you’re driving your car, if the needle
says 60 miles per hour, that’s your velocity at that instant. Though veloc-
ity seems to involve two different times in its very deﬁnition—the initial
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6 The Structure of Mechanics
time and the ﬁnal time—we want to talk about the velocity right now. That
is obtained by examining the position now and the position slightly later,
and taking the ratio of the change in position to the time elapsed between
the two events, while bringing the two points closer and closer in time. We
see in the ﬁgure that when we do this, both→x→0 and→t→0, but their
ratio becomes the tangent of the angleθ, shown in Figure 1.1. Thus the
velocity at the instanttis:
v(t)=lim
→t→0
→x
→t
=
dx
dt
. (1.3)
Once you take one derivative, you can take any number of deriva-
tives. The derivative of the velocity is theacceleration, and we write it as
the second derivative of position:
a(t)=
dv
dt
=
d
2
x
dt
2
. (1.4)
You are supposed to know the derivatives of simple functions likex(t)=
t
n
(
dx
dt
=nt
n−1
), as well as derivatives of sines, cosines, logarithms, and
exponentials. If you don’t know them, you should ﬁx that weakness before
proceeding.
1.4 Motion at constant acceleration
We are now going to focus on problems in which the accelerationa(t)is
just a constant denoted bya, with no time argument. This is not the most
general motion, but a very relevant one. When things fall near the surface
of the earth, they all have the same acceleration,a=−9.8ms
−2
=−g.
If I tell you that a particle has a constant accelerationa,canyoutellme
what the positionx(t)is? Your job is to guess a functionx(t)whose second
derivative isa. This is called integration, which is the opposite of differenti-
ation. Integration is not an algorithmic process like differentiation, though
it is governed by many rules that allow us to map a given problem into
others with a known solution. If I give you a function, you know how to
take the derivative: change the independent variable, ﬁnd the change in
the function, divide by the change in the independent variable, take the
ratio as all changes approach zero. The opposite has to be done here. The
way we do that is we guess, and such guessing has been going on for three
hundred years, and we have become very good at it. The successful guesses
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The Structure of Mechanics 7
are published as the Table of Integrals. I have a copy of such a table at
home, at work, and even in my car in case there is a breakdown.
So, let me guess aloud. I want to ﬁnd a function that reduces to the
numberawhen I take two derivatives. I know that each time I take a
derivative, I lose a power oft. In the end, I don’t want any powers oft.
It’s clear I have to start with a function that looks liket
2
. Well, unfortu-
nately, we knowt
2
is not the right answer, because the second derivative is
2, while I want to geta. So I multiply the original guess by
1
2
aand I know
x(t)=
1
2
at
2
will have a second derivativea.
This certainly describes a particle with an accelerationa. But is this
the most general answer? You all know that it is just one of many: for ex-
ample, I can add to this answer some number, say 96, and the answer will
still have the property that if you take two derivatives, you get the same ac-
celeration. Now 96 is a typical constant, so I’m going to give the namecto
that constant. We know from basic calculus that in ﬁnding a function with
a given derivative, you can always add a constant to any one answer to get
another answer. But if you only ﬁx thesecondderivative, you can also add
anything with one power oftin it, because the extra part will get wiped
out when you take two derivatives. If you ﬁxed only the third derivative of
the function, you can also add something quadratic intwithout changing
the outcome.
So the most general expression for the position of a particle with
constant accelerationais
x(t)=
1
2
at
2
+bt+c (1.5)
whereb, likec, is a constant that can be anything.
Remember thatx(t)in the ﬁgure describes a particle going from side
to side. I can also describe a particle going up and down. If I do that, I
would like to call the vertical coordinatey(t). You have to realize that in
calculus, the symbols that you callxandyare arbitrary. If you know the
second derivative ofyto bea, then the answer is
y(t)=
1
2
at
2
+bt+c. (1.6)
Let me go back now to Eqn. 1.5. It is true, mathematically, you can add
bt+cas we did, but you have to ask yourself, “What am I doing as a
physicist when I add these two terms?” What am I supposed to do with
bandc? What value should I pick? Simply knowing that the particle has
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8 The Structure of Mechanics
an accelerationais not enough to tell you where the particle will be. Take
the case of a particle falling under gravity with acceleration−g. Then
y(t)=−
1
2
gt
2
+bt+c. (1.7)
The formula describes every object falling under gravity, and each has its
own history. What’s different between one object and another object is the
initial height,y(0)≡y
0, and the initial velocityv(0)≡v 0. That’s what these
numbersbandcare going to tell us. To ﬁndcin Eqn. 1.7 put timet=0
on the right and the initial height ofy
0on the left:
y
0=0+0+c (1.8)
which tells uscis just theinitial coordinate. Feeding this into Eqn. 1.7 we
obtain
y(t)=−
1
2
gt
2
+bt+y 0. (1.9)
To use the information on the initial velocity, let us ﬁrst ﬁnd the velocity
associated with this trajectory:
v(t)=
dy
dt
=−gt+b (1.10)
and compare both sides att=0
v
0=b. (1.11)
Thusbis theinitial velocity. Tradingbandcforv
0andy 0, which makes
their physical signiﬁcance more transparent, we now write
y(t)=−
1
2
gt
2
+v0t+y0. (1.12)
Likewise for the trajectoryx(t)when the acceleration is some constanta,
the answer with speciﬁc initial positionx
0and initial velocityv 0is
x(t)=
1
2
at
2
+v0t+x0. (1.13)
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The Structure of Mechanics 9
In every situation where the body has an accelerationa,thelocation
has to have this form. So when I throw a candy and you catch it, you are
mentally estimating the initial position and velocity and computing the
trajectory and intercepting it with your hands. (The candy moves in three
spatial dimensions, but the idea is the same.)
Now, there is one other celebrated formula that relatesv(t), the ﬁnal
velocity at some time, to the initial velocityv
0and the distance traveled,
with no reference to time. The trick is to eliminate time from Eqn. 1.13.
Letusrewriteitas
x(t)−x
0=
1
2
at
2
+v0t. (1.14)
Upon taking the time-derivative of both sides we get
v(t)=at+v
0 (1.15)
which may be solved fort:
t=
v(t)−v
0
a
. (1.16)
Feeding this into Eqn. 1.14 we ﬁnd
x(t)−x
0=
1
2
a
τ
v(t)−v
0
a
→
2
+v0
τ
v(t)−v
0
a
→
(1.17)
=
v
2
(t)−v
2
0
2a
(1.18)
which is usually written as
v
2
−v
2
0
=2a(x−x 0) (1.19)
wherevandxare assumed to be the values at some common generic
timet.
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10 The Structure of Mechanics
1.5 Sample problem
We will work through one standard problem to convince ourselves that we
know how to apply these formulas and predict the future given the present.
Figure 1.2 shows a building of heighty
0=15m.
I am going to throw a rock with an initial velocityv
0=10m/sfrom
the top. Notice I am measuringyfrom the ground. The rock is going to
go up to pointTand come down as shown in Figure 1.2. You can ask me
any question you want about this rock, and I can give the answer. You can
ask me where it will be 9 seconds from now, how fast will it be moving 8
seconds from now, and so on. All I need are the two initial conditionsy
0
andv 0that are given. To make life simple, I will usea=−g=−10ms
−2
.
The positiony(t)is known for all future times:
y=15+10t−5t
2
. (1.20)
Of course, you must be a little careful when you use this result. Say you
puttequal to 10,000 years. What are you going to get? You’re going to
ﬁndyis some huge negative number. That reasoning is ﬂawed because you
cannot use the formula once the rock hits the ground and the fundamental
premise thata=−10ms
−2
becomes invalid. Now, if you had dug a hole of
Figure 1.2 From the top of a building of heighty 0=15m, I throw a rock with
an initial upward velocity ofv
0=10m/s. The dotted line represents the trajectory
continued back to earlier times.
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The Structure of Mechanics 11
depthdwhere the rock was going to land,ycould go down to−d.The
moral is that when applying a formula, you must bear in mind the terms
under which it was derived.
If you want to know the velocity at any timet,justtakethederivative
of Eqn. 1.20:
v(t)=10−10t. (1.21)
Let me pick a few more trivial questions. What is the heighty
maxof the
turning pointTin the ﬁgure? Eqn. 1.20 tells youy if you know t,butwe
don’t know the timet
∗
when it turns around. So you have to put in some-
thing else that you know, which is that the highest point occurs when it’s
neither going up nor coming down. So at the highest pointv(t
∗
)=0. From
Eqn. 1.21
0=10−10t
∗
which meanst
∗
=1s. (1.22)
So we know that it will go up for one second and then turn around
and come back. Now we can ﬁndy
max:
y
max=y(t
∗
)=y(1)=15+10−5=20m. (1.23)
When does it hit the ground? That is the same as asking wheny=0,
which is our origin. Wheny=0,
0=15+10t−5t
2
. (1.24)
The solutions to this quadratic equation are
t=3sort=−1s. (1.25)
Why is it giving me a second solution? Cantbe negative? First of all,
negative times should not bother anybody;t=0 is when I set the clock
to zero, and I measured time forward, but yesterday would bet=−1day,
right? So we don’t have any trouble with negative time; it is like the year
300 BC. The point is that this equation does not know that I went to a
building and launched a rock or anything. What does it know? It knows
that this particle had a height ofy=15mand velocityv=10m/sat time
t=0, and it is falling under gravity with an acceleration of−10ms
−2
.
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12 The Structure of Mechanics
That’s all it knows. If that’s all it knows, then in that scenario there is no
building or anything else; it continues a trajectory both forward in time
and backward in time, and it says that one second before I set my clock to
0, this particle would have been on the ground. What it means is that if you
had released a rock aty=0 one second before I did with a certain speed
that we can calculate (v(−1)=20m/sfrom Eqn. 1.21), your rock would
have ended up at the top of the building when I began my experiment,
with the same heighty=15m, and velocityv
0=10m/s. So sometimes
the extra solution is very interesting, and you should always listen to the
mathematics when you get extra solutions.
When Paul Dirac was looking for the energy of a particle in rela-
tivistic quantum mechanics, he found the energyEwas connected to its
momentump,massm, and velocity of light,c,by
E
2
=p
2
c
2
+m
2
c
4
, (1.26)
in accord with a relation we will encounter in relativity. Now, this quadratic
equation has two solutions:
E=±
≡
c
2
p
2
+m
2
c
4
. (1.27)
You may be tempted to keep the plus sign because you know energy
is not going to be negative. The particle’s moving, it’s got some energy
and that’s it. This is correct in classical mechanics, but in quantum me-
chanics the mathematicians told Dirac, “You cannot ignore the negative
energy solution in quantum theory; the mathematics tells you it is there.”
It turns out the second solution, with negative energy, was telling us that
if there are particles, then there must be anti-particles, and the nega-
tive energy particles, when properly interpreted, describe anti-particles of
positive energy.
So the equations are very smart. When you ﬁnd some laws in mathe-
matical form, you have to follow the mathematical consequences; you have
no choice. Here was Dirac, who was not looking for anti-particles. He was
trying to describe electrons, but the theory said there are two roots to the
quadratic equation and the second root is mathematically as signiﬁcant as
the ﬁrst one. In trying to accommodate and interpret it, Dirac was led to
the positron, the electron’s anti-particle.
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The Structure of Mechanics 13
Returning to our problem, if you were only asking for the maximum
heighty
max, and not the timet
∗
when it got there, there is a shortcut using
v
2
=v
2
0
+2a(y−y 0). (1.28)
Usingv=0,v
0=10m/s, anda=−10ms
−2
we ﬁnd
y
max−y0=5m (1.29)
—that is, the rock reached a maximum height of 20mfrom the ground.
You can ﬁnd the speed when it hits the ground (y=0) using
v
2
=10
2
+2·(−10)(0−15)=400 which meansv=±20m/s.
(1.30)
The root we should take for when it hits the ground is of course−20m/s.
As mentioned earlier, the other root+20m/sis the speed with which it
should have been launched upward, fromy=0att=−1, to follow the
dotted trajectory in the ﬁgure.
1.6 Derivingv
2
−v
2
0
=2a(x−x 0)using calculus
I want to derive Eqn. 1.19,v
2
−v
2
0
=2a(x−x 0), in another way that
illustrates the judicious use of calculus.
Start with
dv
dt
=a (1.31)
and multiply both sides byvand writev=
dx
dt
in the right-hand side:
v
dv
dt
=a
dx
dt
. (1.32)
Now I’m going to do something that is viewed with suspicion, which is just
to cancel thedton both sides. Although I agree that you’re not supposed
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14 The Structure of Mechanics
to cancel thatdin
dy
dx
, canceling thedton both sides gives valid results if
interpreted carefully. Doing so here gives us
vdv=adx. (1.33)
This equation tells us that in an inﬁnitesimal time interval[t,t+dt],the
variablesvandxchange bydvanddx, and these changes are related as
abovein the limit dx,dv,dt→0.Now the limit ofdx→0ordv→0(as
compared to their ratio) is of course trivial, and Eqn. 1.33 reduces to 0=0.
However, the way we interpret and use Eqn. 1.33 is as follows. Suppose in
the ﬁnite time interval[t
1,t2], the variablevchanges fromv 1tov2, and
xchanges fromx
1tox2. Let us divide the interval[t 1,t2]into a very large
numberNof equal sub-intervals of widthdt, and letdxanddvbe the
changes inxandvin the interval[t,t+dt]. The relation between these
changes is given in Eqn. 1.33. If we sum up theNchanges on both sides of
Eqn. 1.33 asN→∞, the sums converge to nontrivial limits, namely the
corresponding integrals:
∗
v2
v1
vdv=a
∗
x2
x1
dx (1.34)
1
2
v
2
2
−
1
2
v
2
1
=a(x 2−x1). (1.35)
Thus it must be understood that the two sides of a relation like Eqn.
1.33 are to be ultimately integrated between some limits to obtain a useful
equality.
Eqn. 1.19 follows upon setting
v
2=v,v 1=v0,x2=x,x 1=x0. (1.36)
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chapter 2
Motion in Higher Dimensions
2.1 Review
In the last chapter we took the simplest case, of a point particle moving
along thex-axis with a constant accelerationa.Whatisthefateofthis
particle? The answer is that at any timet, the location of the particle is
given by
x(t)=x
0+v0t+
1
2
at
2
, (2.1)
wherex
0andv 0are its initial position and velocity. If you took the
derivative of this, you would get
v(t)=v
0+at. (2.2)
You can easily check, by taking one more derivative, that this particle does
indeed have a constant accelerationa. This equation, which gives the ve-
locity of the object at timet, in terms of its initial velocity and acceleration
can be inverted to givetin terms ofv:
t=
v−v
0
a
. (2.3)
15
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16 Motion in Higher Dimensions
Feeding this into Eqn. 2.1 we obtain the result that makes no reference to
time:
v
2
=v
2
0
+2a(x−x 0). (2.4)
It is understoodvandxcorrespond to some common time.
I showed you in the end how we can use calculus to derive this re-
sult. It is important to brush up on your calculus. When a student says,
“I know calculus,” sometimes that means the student knows it, and some-
times that means he or she once met someone who did. One solution for
that is to get a copy of a textbook I wrote calledBasic Training in Mathe-
matics.Thisisalittleawkward;Idon’twanttofoistmybookonyou.On
the other hand, I don’t want to withhold relevant information. If you’re
going into any science that uses mathematics—chemistry, engineering, or
even economics—you should ﬁnd the contents of that book useful. Don’t
wait for the movie: it is not coming.
2.2 Vectors ind=2
The next difﬁcult thing is to consider motion in higher dimensions. Ev-
erything moves around ind=3. However, I’m going to use only two
dimensions for most of the time. Whereas the difference between one di-
mension and two is very great, that between two and higher dimensions
is not. Later we will encounter a few concepts that make sense ind=3
but notd<3. String theorists will tell you that actually we need 9 spatial
dimensions plus time to describe superstrings, which will be discussed in
depth in Chapter 3,498 of this book.
Picture some particle that’s traveling in thex−yplane as shown in
Figure 2.1. This is not anxversustplot or ayversustplot. It’s the actual
path the particle traces out on thex−yplane. You might say “Where is
time?” One way to mark time is to imagine the particle carries a clock
with it, and put markers every second. Four representative markers att=
1,2,33, and 34 are shown. It obviously is going much slower between 33
and 34 than between 1 and 2.
The kinematics of this particle requires a pair of numbersxandy.
It’s more convenient to lump these into a single entity, called avector.The
simplest context in which one can motivate a vector and the rules for deal-
ing with vectors is to look at movements in the plane. Let’s imagine that
when I went camping I walked for 5kmfrom the base camp on the ﬁrst
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Motion in Higher Dimensions 17
Figure 2.1 Path of a particle ind=2. Equal intervals in time are indicated by
markersonthepathnumbered1,2,...,33,and34.
day and another 5kmon the second day. How far am I from the base camp?
You cannot answer that, even if I promised to move only along thex-axis.
It’s not enough to say I went 5km. I have to tell you whether I went to the
rightortotheleft.SoIcouldbe10km,0km,or−10kmfrom base. If I say
not just that I walked 5km, but speciﬁed whether it was±5km,thattakes
care of all ambiguity in one dimension.
But ind=2 the options are not just left and right, but an inﬁnity
of possible directions. For example, on the ﬁrst day I could leave the base
camp at the origin and move along the arrow labeledAto arrive at the
point labeled 1 in Figure 2.2(a). The second hike is described by the arrow
Figure 2.2 Adding vectors. Part (a) shows how to add vectors and that
A+B=B+A. Part (b) illustrates the meaning of multiplying a vector by a
number (2 in this example) and the null vector0.
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18 Motion in Higher Dimensions
B, which starts whereAended and brings me to 2. These two arrows are
examples ofvectorsand I use them here for describingdisplacement,or
changes in position. Vectors can be used to describe many other physical
quantities, as we will see.
A vector is an arrow that has got a beginning and an end. This is why
one says a vector has a magnitude and a direction. The magnitude is how
long it is, and direction is its angle relative to some ﬁxed direction, usually
thex-axis. When you refer to a vectorAin your notes, you’re supposed to
put a little arrow on top like this:A. In textbooks, vectors are in boldface:
A. If you don’t put an arrow on top or do not use boldface, you’re talk-
ing about just a numberA. When applied to a vectorA,Astands for its
length.
From Figure 2.2(a), we see that there is a very natural quantity that
you can callA+B. One day I moved byAand on the next byB.IfIwant
to do it all in one shot, what is the equivalent step I should take from the
start? It’s obvious that the bottom line of my two-day trip is this objectC.
We will call thatA+B. It does represent the sum, in the same sense that if
I gave you 4 bucks and then I gave you 5 bucks, you have the equivalent of
a single payment of 9 bucks. Here, we are not talking about a single num-
ber, but a displacement in the plane, andCindeed represents an effective
displacement due toAandB.
So here is the rule for adding two vectors that comes from a study of
displacements: you draw the ﬁrst one and at the end of that ﬁrst one, you
begin the second one, and their sum starts at the beginning of the ﬁrst and
ends at the end of the second.
You can verify, as illustrated in the ﬁgure, thatA+Bis the same as
B+Awhere you ﬁrst drawBand from whereBends you drawA.Youwill
end up with the same point, 2, as shown by the sum of the dotted arrows.
The next thing I want to do is to deﬁne the vector that plays the role
of the number 0, which has the property that when you add it to anynum-
ber,itgivesthesamenumber.Thevector0that I want to call the zero
ornull vectorshould have the property that when I add it to anyvector,
I should get the same vector. So you can guess who it is: a vector of no
length. I cannot show you the0. If you can see it, I’m doing something
wrong.
Look at part (b) of Figure 2.2. What if I drawA, then I add to it
another
Ato getA+A. You have to agree that if there’s any vector that
deserves to be called 2A,itisthisguy,A, stretched to twice its length. Now
we have discovered a notion of multiplying a vector by a number. If you
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Motion in Higher Dimensions 19
multiply it by 2, you get a vector twice as long and in the same direction.
Then we’re able to generalize that and say, if you multiply it by 2.6, you get
a vector 2.6 times as long. So multiplying a vector by a positive number
means to stretch it (or shrink it) by that factor.
Let us keep going. I want to think of a vector that I can call−A. What
do I expect of−A? I expect that if I add−AtoA, I should get0,which
plays the role of 0 among vectors. What should I add toAso I get the null
vector? It’s clear that you want to add a vector that looks like−Ain part(b)
of Figure 2.2, because, if you go from the start ofAto the ﬁnish of−A,you
end up where you started and you get this invisible0vector. So the minus
vector is the same vector ﬂipped over, pointing the opposite way. That’s
like−1timesavector.Onceyouhavegotthat,youcando−7.3 times a
vector: just take the vector, rescale it by 7.3 and ﬂip it over. Multiplying a
vector by a number is calledscalar multiplication, and ordinary numbers
are calledscalars. You can do more complicated things. You can take one
vector, multiply it by one scalar, take another vector, multiply that by an-
other scalar, and add the two of them. We know what all those operations
mean now. You don’t have to memorize the rules for all this. The only rule
is: “Do what comes naturally.” Do what you normally do with ordinary
numbers.
2.3 Unit vectors
Letusgobacktothesamex−yplane. I’m going to introduce two very
special vectors. They are theunit vectors:iandj, pointing along thexand
yaxes and of unit length, as shown in Figure 2.3. If I had a third axis per-
pendicular to the page, I would draw ak,butwedon’tneedthatyet.I
claim I can write any vector you give me as a numberA
xtimesi,plusa
numberA
ytimesj. There’s nothing you can throw at me that lies in the
plane that I cannot describe as some multiple ofiplus some multiple of
j. It’s intuitively clear, but I will just prove it beyond any doubt. Here is
some vectorA. It is clear from the ﬁgure that it is the sum of the dot-
ted horizontal vector and the dotted vertical vector, by the rules of vector
addition. The horizontal part, parallel toi, has to be a multiple ofi.We
knowthatbecausewecanstretchiby whatever factor we like. Call that
factorA
x, which happens to be positive in this example. It is called the
x-component ofAor theprojectionofAalongior along thex-axis. The ver-
tical part is likewisejA
ywhereA yis they-component ofA, or the projection
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20 Motion in Higher Dimensions
Figure 2.3 The unit vectorsiandjand an arbitrary vectorA=iA x+jAybuilt
out of them.
ofAalongjor along the y-axis.Therefore I have managed to write
Aas
A=iA
x+jAy. (2.5)
We refer to the pairiandj, in terms of which any vector can be
expressed, asbasis vectorsor as thebasis.
If you gave me a particular vectorAas an arrow of some lengthAand
orientationθrelative to thex-axis, what do I use forA
xandA y?Youcan
see from trigonometry that
A
x=Acosθ (2.6)
A
y=Asinθ. (2.7)
Conversely, given the components, the length and angle are
A=
≡
A
2
x
+A
2
y
(2.8)
θ=tan
−1
Ay
Ax
. (2.9)
Eqns. 2.6 to 2.9 will be invoked often. So please commit them to memory.
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Motion in Higher Dimensions 21
If you give me a pair of numbers,(A x,Ay), that’s as good as giving
me this arrow, because I can ﬁnd the length of the arrow by Pythagoras’s
theorem and I can ﬁnd the orientation from tanθ=
Ay
Ax
.Youhavetheop-
tion of either working with the two components ofAor with the arrow.
Inpractice,mostofthetimeweworkwiththesetwonumbers,(A
x,Ay).
In particular, if we are describing a particle whose location is theposition
vectorr, then we write it in terms of its components as
r=ix+jy. (2.10)
Thechangesinrare thedisplacement vectorsand examples areAandBin
Figure 2.2 that described the two hikes.
I have not given you any other example of vectors besides the dis-
placement vector, but at the moment, we’ll deﬁne a vector to be any object
that looks like some multiple ofiplus some multiple ofj.IfItellyouto
add two vectorsAandB,youhavegottwooptions.Youcandrawthear-
row corresponding toAand attach to its end an arrow corresponding toB,
and then add them, as in Figure 2.2. But you can also do the bookkeeping
without drawing any pictures as follows:
A+B=iA
x+jAy+iBx+jBy (2.11)
=i(A
x+Bx)+j(A y+By) (2.12)
so that the sumCis the vector with components(A
x+Bx,Ay+By).
Intheabove,Ihaveusedthefactthatvectorscanbeaddedinany
order. So I grouped the things involving justiand likewisej. Then I argued
that sinceiA
xandiB xare vectors alongi, their sum is a vector of length
A
x+Bxalso alongi. I did the same forj.
In summary if
A+B=C (2.13)
then
C
x=Ax+Bx (2.14)
C
y=Ay+By (2.15)
which can be summarized as follows:
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22 Motion in Higher Dimensions
To add two vectors, add their respective components.
An important result is thatA=Bis possible only ifA
x=BxandA y=
B
y. You cannot have two vectors equal without having exactly the samex
component and exactly the sameycomponent. If two arrows are equal,
one cannot be longer in thexdirection and correspondingly shorter in
theydirection. Everything has to match completely. The vector equation
A=Bis actually a shorthand for two equations:A
x=BxandA y=By.
2.4 Choice of axes and basis vectors
Ihaveinmindavectorwhosecomponentsare3and5.Canyoudraw
the vector for me? If you immediately said, “It is 3i+5j,” you’re making
the assumption that I am writing the vector in terms ofiandj.Iagree
iandjpoint along two natural directions. For most of us, given that the
blackboard or notebook is oriented this way, it is very natural to line up
our axes with it. But there is no reason why somebody else couldn’t come
along and say, “I want to use a different set of axes. Thexandyaxes or
iandjare not nailed in absolute space. They are human constructs and
we’re not wedded to any of them.”
Quite often, it’s natural to pick the axes in a certain way to suit the
problem. If you are studying a cannon ball launched from the earth, it
makes sense to pick the horizontal as thex-axis and the vertical as they-
axis, but, mathematically, you don’t have to. Another set of rotated but
mutually perpendicular unit vectorsi
∂
andj
∂
that form another basis can
also be rescaled and added to form any given vectorAin the plane. For
example, when we study objects sliding down an inclined plane, we will
choose our axes parallel and perpendicular to the incline.
If I draw an arrowAon a blank sheet of paper, it has life of its own
without reference to any axes. ThesamevectorAcan be written either in
terms ofiandj, which is the old basis, or in terms ofi
∂
andj
∂
, the new basis.
How do the components(A
∂
x
,A
∂
y
)in the new basis relate to the components
of the old basis? It’s a simple problem, but I just want to do it so you get
used to working with vectors.
ForthisweneedtheverybusyFigure2.4.Itshowstheoldxandyaxes
and thex
∂
andy
∂
axes obtained by rotating thex−yaxes counterclockwise
by an angleφ. The unit vectorsi
∂
andj
∂
are likewise rotated versions of
iandj. The components in the two bases are shown by dotted lines and
are simply the projections ofAalong the various axes. We want to relate
(A
∂
x
,A
∂
y
)to(A x,Ay).
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Motion in Higher Dimensions 23
Figure 2.4 The same vectorAis written asiA x+jAyin one frame and as
i
∂
A
∂
x
+j
∂
A
∂
y
in the other. The dotted lines indicate the components in the two
frames.
First we expressi
∂
andj
∂
in terms ofiandjusing the ﬁgure:
i
∂
=icosφ+jsinφ (2.16)
j
∂
=jcosφ−isinφ. (2.17)
Here are the details. The vectori
∂
has got a horizontal part, which is its
length, namely, 1, times cosφ, and a vertical part that is 1 times sinφ.How
aboutj
∂
?Itisatanangleφrelative toj.Soitsycomponent is cosφ. Finally,
itsxor horizontal component is(−sinφ), where the minus sign comes
because it is pointing to the left, along the negativex-axis. All that remains
now is to eliminatei
∂
andj
∂
in favor ofiandjinA=i
∂
A
∂
x
+j
∂
A
∂
y
and equate
it toAwritten in terms ofiandj:
A=i
∂
A
∂
x
+j
∂
A
∂
y
(2.18)
=(icosφ+jsinφ)A
∂
x
+(jcosφ−isinφ)A
∂
y
(2.19)
=i(A
∂
x
cosφ−A
∂
y
sinφ)+j(A
∂
x
sinφ+A
∂
y
cosφ) (2.20)
=iA
x+jAy. (2.21)
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24 Motion in Higher Dimensions
When we equate the coefﬁcients ofiandjon the right-hand sides of Eqns.
2.20 and 2.21, we obtain the desired expression forA
xandA yin terms of
A
∂
x
andA
∂
y
:
A
x=A
∂
x
cosφ−A
∂
y
sinφ (2.22)
A
y=A
∂
x
sinφ+A
∂
y
cosφ. (2.23)
So, you can pick your basis vectors any way you like and so can I. Your
basis is obtained from mine by a counterclockwise rotation by an angle
φ.ThesameentityA, the same arrow which has an existence of its own,
independent of axes, can be described by you and by me using different
components. Your components with primes on them are related to mine
by Eqns. 2.22 and 2.23. This is called thetransformation lawfor the vector
components under rotation of basis vectors.
Now, you can ask the opposite question. How do I getA
∂
x
andA
∂
y
in
terms ofA
xandA y? The quickest way is to replaceφby−φ:ifwegofrom
the unprimed to the primed system by a rotationφ, then rotation by−φ
is the way to go from the primed to the unprimed basis. The result, using
cos(−φ)=cosφand sin(−φ)=−sinφ,is
A
∂
x
=Axcosφ+A ysinφ (2.24)
A
∂
y
=−A xsinφ+A ycosφ. (2.25)
That turns out to be the correct answer. But I want you to think about
another way to show this, which often seems to bother some students. If I
told you
3x+5y=21 (2.26)
4x+6y=26, (2.27)
you certainly know how to solve forxandy, right? You have got to juggle
the two equations, multiply the ﬁrst by 6, the second by 5, and subtract to
isolatexand so on. Why is it when some of you see Eqns. 2.22 and 2.23,
you don’t realize it’s the same kind of problem, where you can multiply
Eqn. 2.22 by cosφ,Eqn.2.23bysinφand add to isolateA
∂
x
, for example?
For any particular value ofφ,sinφand cosφare just some numbers. For
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Motion in Higher Dimensions 25
example, if I pickφ=
π
3
=60
◦
,cosφ=
1
2
and sinφ=
√
3
2
. The equations
become (for this angle)
A
x=
1
2
A
∂
x
−
√
3
2
A
∂
y
(2.28)
A
y=
√
3
2
A
∂
x
+
1
2
A
∂
y
. (2.29)
If you multiply the second by
√
3 and add it to the ﬁrst you obtain
A
x+
√
3Ay=2A
∂
x
which means (2.30)
A
∂
x
=
1
2
A
x+
√
3
2
A
y (2.31)
=A
xcos
π 3
+A
ysin
π
3
(2.32)
in accordance with Eqn. 2.24. So go forth and treat sinφand cosφ
as plain numbers and juggle Eqns. 2.22 and 2.23 to derive Eqns. 2.24
and 2.25. Along the way of course you will have to use identities like
sin
2
φ+cos
2
φ=1.
The components of the vector depend on who is looking at the vector.
However, there’s one quantity that’s going to come out the same, no matter
who is looking at the vector. It is the length of the vector. It is unaffected
by the rotation of axes. It is aninvariantunder rotations. You may verify
from Eqns. 2.24 and 2.25 that
(A
∂
x
)
2
+(A
∂
y
)
2
=(Axcosφ+A ysinφ)
2
+(−A xsinφ+A ycosφ)
2
(2.33)
=A
2
x
(cos
2
φ+sin
2
φ)+A
2
y
(sin
2
φ+cos
2
φ)
(2.34)
=A
2
x
+A
2
y
. (2.35)
TheA
xAyterm is gone since its coefﬁcient is 2(cosφsinφ−sinφcosφ).
I want to conclude with one important point. We learned that a vec-
tor is a quantity that has a magnitude and a direction. A more advanced
view of vectors is that they are a pair of numbers (ind=2) which, un-
der rotation of axes, transform as per Eqns. 2.24 and 2.25. Anything that
transforms this way is called a vector. We already know about the position
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26 Motion in Higher Dimensions
vectorrand the changes in it, the displacement vectors (used in describing
the hike). How about more vectors? There turns out to be a very nice way
to produce vectors, given one vector like the position vector. And that’s the
following.
2.5 Derivatives of the position vectorr
Let’s take a particle in thex−yplane that moves fromrat timettor+→r
at timet+→tas in Figure 2.5. At timetits location is
r=ix(t)+jy(t) (2.36)
and att+→tit is
r+→r=i(x(t)+→x)+j(y(t)+→y)so that, (2.37)
→r=i→x+j→yand by the usual limiting process, (2.38)
v=lim
→t→0
→r
→t
=
dr
dt
=i
dx
dt
+j
dy
dt
. (2.39)
Figure 2.5 The particle moving along some curved path goes fromrat timetto
r+→rat timet+→t. The velocityvis the limit of the ratio
→r
→t
as→t→0, and
thus parallel to→r, which eventually becomes tangent to the curve.
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Motion in Higher Dimensions 27
When you move just along thex-axis, you wait a small time→tand
you move by an amount→x, and their ratio gives the velocity in the ap-
propriate limit. When you move in the plane, your position and its change
are both vectors.
Can you see why the derivative of a vector is also a vector? Because
→r, the difference in the vector between two times, is itself a vector. Di-
viding it by→tis like multiplying by 1/→t, but we know that when we
multiply a vector by a number, we simply rescale the vector. So the limit
will be some arrow that we call theinstantaneous velocity vector.Itwillbe
tangential to the curver(t)and point toward the instantaneous direction
of travel.
If I gave you the location of a particle as a function of time, you
can ﬁnd its velocity by taking derivatives. For example, if I say a particle’s
location is
r=t
2
i+9t
3
j (2.40)
then its velocity at timetis
v=2ti+27t
2
j. (2.41)
You can take a derivative of the velocity or the second derivative of
the position to get the acceleration vector
a(t)=
dv
dt
=
d
2
r
dt
2
=2i+54tj(in our example). (2.42)
You can then also multiplyaby the massm, which is ascalarun-
affected by rotations, to get a vectorma, which Newton’s law equates to
another vector, the forceF.
Even though we started with one example of a vectorr,we’renow
ﬁnding out that its derivative has to be a vector and the derivative of the
derivative is also a vector. When you learn relativity, you will ﬁnd out
there’s again one vector that’s staring at you, the analog of the position vec-
tor, but with four components. But more vectors can be manufactured by
multiplying vectors by scalars (like mass) or taking derivatives with respect
to a parameter that plays the role analogous to time.
Here is an illustration of vector addition and differentiation. Imagine
an airplane in ﬂight, as depicted in Figure 2.6. Letr
pgbe the location of a
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28 Motion in Higher Dimensions
Figure 2.6 The position of the ball relative to (some origin on) the groundr bgis
the vector sum of the position of the ball relative to the (tail of the) plane,r
bp,
and the position of (the tail of) the planer
pgrelative to the ground.
ﬁxed point in the airplane, say the tail, with respect to a ﬁxed point on the
ground. Imagine that in the airplane there is a ball located atr
bpas mea-
sured from this ﬁxed point in the airplane. By vector addition the location
of the ball with respect to the ground is
r
bg=rbp+rpg. (2.43)
Upon taking a time derivative and in the same notation, the law of
composition of velocities follows:
v
bg=vbp+vpg, (2.44)
which says the velocity of the ball as seen by a person on the ground is
the velocity of the ball relative to the airplane plus the velocity of the air-
plane relative to the ground. Taking yet another derivative we may relate
the accelerations:
a
bg=abp+apg. (2.45)
In the special case of an airplane moving at constant velocity,a
pg=0. Then
we ﬁnd
a
bg=apg, (2.46)
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Motion in Higher Dimensions 29
which means, in this case, the acceleration of the ball is the same as mea-
sured by an observer on the ground and an observer on the airplane. These
results will be recalled in our study of relativity.
2.6 Application to circular motion
Now we’ll take a concrete problem where you will see how to take deriva-
tives to obtain very useful results. I’m going to write a particular case of
r(t):
r(t)=R(icosωt+jsinωt) (2.47)
whereRandωare constants. What is going on as a function of time?
What’s this particle doing? Look at the length squared of this vector:
r
2
x
+r
2
y
=R
2
(cos
2
ωt+sin
2
ωt)=R
2
. (2.48)
That means the particle is going around in a circle of radiusRas shown in
Figure 2.7. Thexcomponent isRcosωtand theycomponent isRsinωt
Figure 2.7 The particle moves along a circle of radiusRwith an angular
velocityω.
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30 Motion in Higher Dimensions
whereωis a ﬁxed number. Astincreases, this angleωtincreases and the
particle goes round and round. Let’s get a feeling forω. As time increases,
the angle increases and we can ask how long it will take the particle to come
back to the starting point. Suppose the starting point was on thex-axis. As
tincreases,ωtincreases, and the particle will come back at a timeTsuch
that
ωT=2π. (2.49)
Thusωis related to the time periodTby
ω=
2π
T
=2πf (2.50)
wheref=
1
T
is the frequency or number of cycles per second. It is mea-
sured inHz, which stands for Hertz. Since in every cycle the particle rotates
by 2π, and it completesfrevolutions per second,ω=2πfis called the
angular velocityand measures the radians swept out per second.
Notice that in equating a full cycle to 2πI am usingradiansand not
degrees to measure angles. For those who have not seen a radian, it’s just
another way to measure angles, wherein a full circle, which we used to
think was worth 360
◦
, now equals 2πradians. Since 2π6.3, a radian is
roughly 60
◦
. You will see the advantages of using radians later. For now just
remember that a half circle, instead of being 180
◦
,willnowbeπradians,
and a quarter circle will be
π
2
, and so forth.
How fast is this particle moving? It’s going around a circle, the angle
is increasing at a steady rateω, and so we know it’s going at a steady speed.
Let us verify that by computing the velocity
v(t)=
dr(t)
dt
(2.51)
=R
∞
i
dcosωt
dt
+j
dsinωt
dt
ω
(2.52)
=Rω(−isinωt+jcosωt). (2.53)
Att=0, the velocity isv=Rωcos0j=Rωj, so it is moving straight up
at speedv=ωR. You may verify that it has the velocity as shown in the
ﬁgure at later times. The magnitude of the velocity is alwaysωRalthough
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Motion in Higher Dimensions 31
the direction is changing. From the ﬁgure we see it remains tangential to
the circle. The constancy of the speedvat an arbitrary time may also be
established by computing
v
2
=(ωR)
2
(sin
2
ωt+cos
2
ωt)=(ωR)
2
(2.54)
v=ωR. (2.55)
Remember the tangential velocity isv=ωR.
Let’s take the derivative of the derivative to ﬁnd the accelerationaand
its magnitude:
a=−ω
2
R(icosωt+jsinωt)=−ω
2
r (2.56)
a=ω
2
R. (2.57)
That’s a very important result. It tells you thatwhen a particle moves
in a circle of radius R at constant speed v, it has an acceleration, called the
centripetal acceleration, directed toward the center and of magnitude
a=ω
2
R=
(ωR)
2
R
=
v
2
R
. (2.58)
This acceleration at constant speed reﬂects the fact that velocity is a vector
and you can change the velocity vector by changing its direction. For ex-
ample, if a car is going on a racetrack and the speedometer says 60 miles
perhour,thelayperson’sviewisthatthecarisnotaccelerating.Butyou
will say from now on that it indeed has an acceleration equal to
v
2
R
even
though no one’s stepping on the accelerator or the brake.
Suppose the particle is not moving fully around a circle but travers-
ing just a quarter of the circle. When it is traveling the quarter of a circle, it
has the same acceleration directed toward the center of that quarter circle.
In other words, you don’t have to be moving actually in a circle to have the
acceleration
v
2
R
. At any instant, the curve you are following can be locally
approximated as part of some circle, and, in the formulaa=
v
2
R
, the accel-
eration is directed toward the center of that circle,Ris its radius andvthe
instantaneous tangential velocity.
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32 Motion in Higher Dimensions
2.7 Projectile motion
I want to consider a particle for whichr 0andv 0are the position and
velocity att=0 and which has a constant vector accelerationa. What
is its location at all future times? By analogy with what I did in one
dimension
r(t)=r
0+v0t+
1
2
at
2
. (2.59)
Once you knowr
0andv 0, you can ﬁnd the position of the object at all
future times. Let’s take one simple example. Somebody in a car has decided
to drive off a cliff as shown in Figure 2.8(a). We want to know when and
where the car hits the ground.
We pick our origin(0,0)at the foot of the cliff. Let the height of
the cliff beh. The car is traveling with some initial speedv
0xin the hor-
izontal direction. Equation 2.59 is really a pair of equations, one alongx
and one alongywitha=−jg,v
0=v0xi, andr 0=hj. Separating out the
components
x(t)=0+v
0xt+0 (2.60)
y(t)=h+0−
1
2
gt
2
. (2.61)
Notice that the evolution of the two coordinates is completely indepen-
dent. The timet
∗
when the car hits the ground (y=0) satisﬁes the
Figure 2.8 (a) A car ﬂies off the cliff at(0,h)and lands at(d,0).(b)Aprojectile
is launched with initial velocityv
0=(i+
√
3j)m/s. The range isR=0.35mand
the maximum height reached isy
max=0.15m.
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Motion in Higher Dimensions 33
equation
0=h−
1
2
gt
∗2
(2.62)
t
∗
=
∂
2h
g
. (2.63)
This is exactly how long it would take to hit the ground had it simply top-
pled over the edge from rest. The horizontal velocity does not delay the
crash one bit (unless you take into account the curvature of the earth). As
to where the car lands, the location is given by(x(t
∗
),0)=(d,0)where
d=v
0xt
∗
=v0x
∂
2h
g
. (2.64)
Finally the problem of projectile motion is depicted in Figure 2.8(b).
You ﬁre a projectile from(0,0)with some velocityv
0at some angleθ.It
will go up and then come down, moving horizontally at the same time.
Where is it going to land? What is the maximum heighty
maxto which it
rises? With what speed will it hit the ground? At what angle should you ﬁre
your projectile so it will go the furthest?
Here are the equations that contain all the answers, namely Eqn. 2.59
written out in component form:
x(t)=0+v
0xt=v0cosθ·t (2.65)
y(t)=0+v
0yt−
1
2
gt
2
=v0sinθ·t−
1
2
gt
2
. (2.66)
You can solve them but it is good to have an idea of what’s coming.
Imagine you have this monster cannon to ﬁre things. It has a ﬁxed muzzle
speed,v
0, but allows you to ﬁre at any angle. How do you aim it so the ball
goes as far as possible? There are two schools of thought. One says, aim at
your enemy and ﬁre horizontally. Then the ball lands on your foot because
it has zero time of ﬂight (assuming the cannon is at zero height). The other
school says, maximize the time of ﬂight and point the cannon vertically. It
goes up, stays in the air for a very long time, and lands on your head. Then
it hits you: the correct answer is somewhere between 0 and 90
◦
=
π
2
.The
naive guess 45
◦
=
π
4
turns out to be correct.
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uniform olive; first primary, with the outer half of the outer web, nearly white. Female with the gray of
the crown glossed with olive; the chin and throat paler centrally, and tinged with fulvous; a dull
whitish ring round the eye. Length, 5.50; wing, 2.45; tail, 2.25. Young not seen.
Hab. Eastern Province of United States to British America; Greenland; Southeastern Mexico, Panama
R. R., and Colombia. Not recorded from West Indies or Guatemala. Costa Rica (Lawr.).
Specimens vary in the amount of black on the jugulum, and the purity of the ash of
the throat. The species is often confounded with Oporornis agilis, to which the
resemblance is quite close. They may, however, be distinguished by the much longer
and more pointed wings, and more even tail, shorter legs, etc., of agilis. The white
ring round the eye in the female philadelphia increases the difficulty of separation.
The adult male in autumn is scarcely different from the spring bird, there being
merely a faint olive-tinge to the ash on top of the head, and the black jugular patch
more restricted, being more concealed by the ashy borders to the feathers; the
yellow beneath somewhat deeper.
Habits. The Mourning Warbler was first discovered and described by Wilson, who
captured it in the early part of June, on the borders of a marsh, within a few miles of
Philadelphia. This was the only specimen he ever met with. He found it flitting from
one low bush to another in search of insects. It had a sprightly and pleasant
warbling song, the novelty of which first attracted his attention. For a long while
Wilson’s single bird remained unique, and from its excessive rarity Bonaparte
conjectured that it might be an accidental variety of the Yellow-Throat. At present,
though still of unfrequent occurrence, it is by no means a doubtful, though generally
a comparatively rare species. Audubon mentions having received several specimens
of this Warbler, procured in the neighborhood of Philadelphia, New York, and
Vermont, all of which were obtained in the spring or summer months. He met with a
single specimen in Louisiana, and thinks its habits closely resemble those of the
Maryland Yellow-Throat.
Nuttall met with what he presumes to have been one of these birds in the Botanical
Garden at Cambridge. It had all the manners of the Yellow-Throat, was busy in the
search of insects in the low bushes, and, at intervals, warbled out some very
pleasant notes, which partly resembled the lively chant of the Trichas, and in some
degree the song of the Summer Yellow-Bird.
Professor Reinhardt states that two individuals of this species have been taken in
Greenland,—one in Fiskenæsset, in 1846, and the other at Julianhaab, in 1853.
Mr. Turnbull gives it as still quite rare in Eastern Pennsylvania, arriving there in the
middle of May on its way farther north. Mr. Lawrence includes it in his list of the birds

of New York. Mr. Dresser obtained five specimens early in May, in Southern Texas.
It has been met with as far to the north as Greenland by Reinhardt, and in Selkirk
Settlement by Donald Gunn. It has been procured in Eastern Mexico, in Panama, in
Carlisle, Penn., Southern Illinois, Missouri, Nova Scotia, and various other places. It
has been known to breed in Waterville, Me., and is not uncommon in Northwestern
and Northern New York. A single specimen of this bird was obtained at Ocana, in
Colombia, South America, by Mr. C. W. Wyatt.
Late in May, 1838, I have a note of having met with this species in Mount Auburn.
The bird was fearless and unsuspecting, busily engaged, among some low shrubbery,
in search of insects. It suffered our near presence, was often within a few feet, and
was so readily distinguishable that my companion, with no acquaintance with birds,
at once recognized it from Audubon’s plates. Its habits were the exact counterpart of
those of the Yellow-Throat. We did not notice its song.
Mr. Maynard states that, May 21, 1866, Mr. William Brewster shot a male of this
species in Cambridge, on the top of a tall tree. Another specimen was taken at
Franconia Mountains, New Hampshire, August 3, 1867. It was in company with four
fully fledged young, which it was feeding. The young were shy, and could not be
procured. The old bird was catching flies, after the manner of Flycatchers. Mr.
Maynard has met this species but once in Massachusetts, and then in May, among
low bushes and in a swampy place. He has since found it rather common at Lake
Umbagog, Maine, in June, where it breeds. He states that it frequents the bushes
along fences, stone walls, and the edges of woods. The male often perches and
sings in the early morning on the top rail of a fence, or the dead branch of a tree. Its
song he speaks of as loud and clear, somewhat resembling that of the Seiurus
noveboracensis.
Mr. Paine considers this Warbler to be very rare in Vermont. He once observed a pair,
with their young, at Randolph. The male was singing a quite pleasing, though
somewhat monotonous song.
Mr. George Welch met with these birds in the Adirondack region, New York, in June,
1870. They seemed rather abundant, and were evidently breeding there. He
obtained a single specimen.
Mr. John Burroughs, of Washington, was so fortunate as to obtain the nest and eggs
of this Warbler near the head-waters of the Delaware River, in Roxbury, Delaware
County, N. Y. “The nest,” he writes me, “was in the edge of an old bark-peeling, in a
hemlock wood, and was placed in some ferns about one foot from the ground. The
nest was quite massive, its outer portions being composed of small dry stalks and
leaves. The cavity was very deep, and was lined with fine black roots. I have
frequently observed this Warbler in that section. About the head of the Neversink
and Esopus, in the northwest part of Ulster County, New York, they are the prevailing
Warbler, and their song may be heard all day long. Their song suggests that of the

Kentucky Ground Warbler, but is not so loud and fine.” Mr. Burroughs states
elsewhere that “the eggs, three in number, were of light flesh-color, uniformly
speckled with fine brown specks. The cavity of the nest was so deep that the back of
the sitting bird sank below the edge.”
Their eggs are of an oblong-oval shape, pointed at one end. They measure .75 by
.55 of an inch. Their ground-color is a pinkish-white, and they are marked with dots
and blotches, of varying size, of dark purplish-brown.
Geothlypis macgillivrayi, Baird.
MACGILLIVRAY’S GROUND WARBLER.
Sylvia macgillivrayi, Aud. Orn. Biog. V, 1839, 75, pl. cccxcix. Trichas macg. Aud. Geothlypis macg.
Baird, Birds N. Am. 1858, 244, pl. lxxix, fig. 4; Rev. 227.—SclatÉr, Catal. 1861, 27 (Jalapa and
Guat.).—Ib. P. Z. S. 1859, 363, 373 (Xalapa, Oaxaca).—Cab. Jour. 1861, 84 (Costa Rica).—CoopÉr
& SucklÉy , P. R. R. Rep. XII, II, 1859, 177.—CoopÉr, Orn. Cal. 1, 1870, 96. Sylvicola macg. Mañ.
Cab. Jour. VI, 1858, 118. Sylvia tolmiæi, Towns. J. A. N. Sc. 1839. Trichas tolmiæi, Nutt. Man. I.
Trichas vegeta (Licht.), Bp. Consp. 1850, 310; fide Cab. Jour. 1861, 84 (Mexico).
Sp. Char. Adult male. Head and neck all round, throat and forepart of the breast, dark ash-color; a
narrow frontlet, loral region, and space round the eye (scarcely complete behind), black. The eyelids
above and below the eye (not in a continuous ring) white. The feathers of the chin, throat, and fore
breast really black, with ashy-gray tips more or less concealing the black. Rest of upper parts dark
olive-green (sides under the wings paler); of lower, bright yellow. Female with the throat paler and
without any black. Length of male, 5 inches; wing, 2.45; tail, 2.45. Young not seen.
Hab. Western and Middle Provinces of United States, to northern boundary; east to Fort Laramie;
south to Costa Rica.
The white eyelids of this species distinguish its males from those of G. philadelphia,
in which there is a black jugular patch not seen in the present species. The females
can only be known by the slenderer bill and more rounded wing, the first quill being
intermediate between the fifth and sixth, instead of being considerably longer than
the fifth.
The autumnal adult male is as described above, except that there is a faint tinge of
green on the crown, and the ashy borders to feathers of throat and jugulum broader,
concealing more the black. The adult female in autumn is considerably more dully
colored than in spring.
Habits. This comparatively new Warbler was first met with by Townsend, and
described by Audubon in the last volume of his Ornithological Biography. It has since
been found to have a wide range throughout the western portion of North America,
from Cape St. Lucas to British America, and from the Plains to the Pacific. It has also
been obtained at Choapan in the State of Orizaba, Mexico, by Mr. Boucard, and in
Guatemala by Mr. Salvin, who states that throughout the district between the
volcanoes of Agua and Fuego this was a common species, frequenting the outskirts
of the forests and the edges of the clearings. It breeds in abundance in Utah,

Montana, Idaho, Oregon, Washington Territory, and probably also in Northern
California.
Townsend first met with it on the banks of the Columbia. He states that it was mostly
solitary and extremely wary, keeping chiefly in the most impenetrable thickets, and
gliding through them in a cautious and suspicious manner. Sometimes it might be
seen, at midday, perched upon a dead twig, over its favorite places of concealment,
at such times warbling a very sprightly and pleasant little song, raising its head until
its bill is nearly vertical.
Mr. Nuttall informed Mr. Audubon that this Warbler is one of the most common
summer residents of the woods and plains of the Columbia, where it appears early in
May, and remains until the approach of winter. It keeps near the ground, and gleans
its subsistence among the low bushes. It is shy, and when surprised or closely
watched it immediately skulks off, often uttering a loud click. Its notes, he states,
resemble those of the Seiurus aurocapillus. On the 12th of June a nest was brought
to Mr. Nuttall, containing two young birds quite fledged, in the plumage of the
mother. The nest was chiefly made of strips of the inner bark of the Thuja
occidentalis, lined with slender wiry stalks. It was built near the ground in the dead,
moss-covered limbs of a fallen oak, and was partly hidden by long tufts of usnea. It
was less artificial than the Yellow-Throat’s nest, but was of the same general
appearance. On his restoring the nest to its place, the parents immediately
approached to feed their charge.
Dr. Suckley found this Warbler very abundant between the Cascade Mountains and
the Pacific coast. Like all Ground Warblers it was entirely insectivorous, all the
stomachs examined containing coleoptera and other insects. He did not find them
shy, but as they frequented thick brush they were very difficult to procure.
Dr. Cooper found this species very common about Puget Sound, frequenting the
underbrush in dry woods, occasionally singing a song from a low tree, similar to that
of the Yellow-Throat. He found its nest built in a bush, a foot from the ground. It was
of straw, loosely made, and without any soft lining. Dr. Cooper found this species as
far east as Fort Laramie, in Wyoming. They reach the Columbia River by the 3d of
May.
The same writer noticed the first of this species at Fort Mojave, April 24. He regarded
their habits as varying in some respects from those of the Trichas, as they prefer dry
localities, and hunt for insects not only in low bushes but also in trees, like the
Dendroicæ. Dr. Cooper twice describes their eggs as white, which is inaccurate. He
thinks that some of them winter in the warmer portions of California. He regards
them as shy, if watched, seeking the densest thickets, but brought out again by their
curiosity if a person waits for them, and the birds will approach within a few feet,
keeping up a scolding chirp.

The nests of this species obtained by Dr. Kennerly from Puget Sound were all built on
the ground, and were constructed almost exclusively of beautifully delicate mosses,
peculiar to that country. They are shallow nests, with a diameter of four and a height
of two inches, the cavity occupying a large proportion of the nest. Its walls and base
are of uniform thickness, averaging about one inch. The nests are lined with finer
mosses and a few slender stems and fibres.
Mr. Ridgway found these Warblers breeding in great numbers, June 23, 1869, at
Parley’s Park, Utah, among the Wahsatch Mountains. One of these nests (S. I.,
15,238) was in a bunch of weeds, among the underbrush of a willow-thicket along a
cañon stream. It was situated about eight inches from the ground, is cuplike in
shape, two inches in height, three in diameter, and somewhat loosely constructed of
slender strips of bark, decayed stalks of plants, dry grasses, intermixed with a few
fine roots, and lined with finer materials of the same. The cavity is one and a half
inches in depth, and two in diameter at the rim.
The eggs, four in number, are .75 of an inch in length and .50 in breadth. Their
ground-color is a pinkish-white, marbled and spotted with purple, lilac, reddish-
brown, and dark brown, approaching black. The blotches of the last color vary much
in size, in one instance having a length of .21 of an inch, and having the appearance
of hieroglyphics. When these spots are large, they are very sparse.
“This species,” Mr. Ridgway writes, “inhabits exclusively the brushwood along the
streams of the mountain cañons and ravines. Among the weeds in such localities
numerous nests were found. In no case were they on the ground, though they were
always near it; being fixed between upright stalks of herbs, occasionally, perhaps, in
a brier, from about one to two feet above the ground. The note of the parent bird,
when a nest was disturbed, was a strong chip, much like that of the Cyanospiza
amæna or C. cyanea.” He also states that it was abundant in the East Humboldt
Mountains in August and in September, and also throughout the summer. A pair of
fully fledged young was caught on the 21st of July.
Subfamily ICTERIANÆ.
SÉction ICTERIEÆ.
In this section there are two American genera; one found in the United States, the
other not. The diagnoses are as follows:—
Size large (about 8 inches). Lower jaw not deeper than upper anterior to
nostrils. Tail moderate. Partly yellow beneath, olive-green above … Icteria.
Size smaller (about 6 inches). Lower jaw deeper than upper. Tail almost fan-
shaped. Partly red beneath, plumbeous-blue above … Granatellus.
[58]

Icteria virens.
2260
GÉnus ICTERIA, ViÉill.
Icteria, ViÉillot, Ois. Am. Sept. I, 1807, iii and 85. (Type, Muscicapa viridis , Gm. Turdus virens, Linn.)
GÉn. Char. Bill broad at base, but contracting rapidly and
becoming attenuated when viewed from above; high at the base
(higher than broad opposite the nostrils); the culmen and
commissure much curved from base; the gonys straight. Upper
jaw deeper than the lower; bill without notch or rictal bristles.
Nostrils circular, edged above with membrane, the feathers close
to their borders. Wings shorter than tail, considerably rounded;
first quill rather shorter than the sixth. Tail moderately graduated;
the feathers rounded, but narrow. Middle toe without claw about
two thirds the length of tarsus, which has the scutellæ fused
externally in part into one plate.
The precise systematic position of the genus Icteria is
a matter of much contrariety of opinion among
ornithologists; but we have little hesitation in including
it among the Sylvicolidæ. It has been most frequently
assigned to the Vireonidæ, but differs essentially in
the deeply cleft inner toe (not half united as in Vireo),
the partially booted tarsi, the lengthened middle toe,
the slightly curved claws, the entire absence of notch or hook in the bill, and the
short, rounded wing with only nine primaries. The wing of Vireo, when much
rounded, has ten primaries,—nine only being met with when the wing is very long
and pointed.
Of this genus only one species is known, although two races are recognized by
naturalists, differing in the length of the tail.
I. virens. Above olive-green; beneath gamboge-yellow for the anterior half, and white for the
posterior. A white stripe over the eye.
Length of tail, 3.30 inches. Hab. Eastern United States to the Plains; in winter through
Eastern Mexico to Guatemala … var. virens.
Length of tail, 3.70 inches. Hab. Western United States from the Plains to the Pacific;
Western Mexico in winter … var. longicauda.
Icteria virens, Baird.
YELLOW-BREASTED CHAT.
Turdus virens, Linn. Syst. Nat. 10th ed. 1758, 171, No. 16.(based on Œnanthe americana, pectore
luteo, Yellow-breasted Chat, CatÉsby, Carol. I, tab. 50). Icteria virens, Baird, Rev. Am. B. 1864,
228. Muscicapa viridis, GmÉlin, Syst. Nat. I, 1788, 936. Icteria viridis, Bon.; Aud. Orn. Biog. II, pl.
cxxxvii.—Baird, Birds N. Am. 1858, 248. Icteria dumecola, ViÉill. Pipra polyglotta, Wils. ? Icteria
velasquezi, Bon. P. Z. S. 1837, 117 (Mexico).—SclatÉr & Salv. Ibis, I, 1859, 12 (Guatemala).
Localities quoted: Costa Rica, Caban., Orizaba (winter), Sum. Yucatan, Lawr.

Icteria virens.
Sp. Char. Third and fourth quills longest; second and fifth little shorter; first nearly equal to the sixth.
Tail graduated. Upper parts uniform olive-green; under parts, including the inside of wing, gamboge-
yellow as far as nearly half-way from the point of the bill to the tip of the tail; rest of under parts
white, tinged with brown on the sides; the outer side of the tibiæ plumbeous; a slight tinge of orange
across the breast. Forehead and sides of the head ash, the lores and region below the eye blackish. A
white stripe from the nostrils over the eye and involving the upper eyelid; a patch on the lower lid,
and a short stripe from the side of the lower mandible, and running to a point opposite the hinder
border of the eye, white. Bill black; feet brown. Female like the male, but smaller; the markings
indistinct; the lower mandible not pure black. Length, 7.40; wing, 3.25; tail, 3.30. Nest in thickets,
near the ground. Eggs white, spotted with reddish.
Hab. Eastern United States, west to Arkansas; rare north of Pennsylvania; south to Eastern Mexico
and Guatemala. Not noticed in West Indies.
Both sexes in winter apparently have the base of
lower mandible light-colored, the olive more brown,
the sides and crissum with a strong ochraceous tinge.
It is this plumage that has been recognized as I.
velasquezi.
Habits. The Yellow-breasted Chat is found throughout
the Eastern United States, from Massachusetts to
Florida, and as far to the west as Fort Riley and
Eastern Kansas. Mr. Say met with it among the Rocky
Mountains as far north as the sources of the Arkansas.
It is not very rare in Massachusetts, but a few breed in
that State as far north as Lynn. It has been found in
Mexico and Guatemala, but not, so far as I am aware,
in the West Indies.
Probably no one of our birds has more distinctly
marked or greater peculiarities of voice, manners, and
habits than this very singular bird. It is somewhat terrestrial in its life, frequenting
tangled thickets of vines, briers, and brambles, and keeping itself very carefully
concealed. It is noisy and vociferous, constantly changing its position and moving
from place to place.
It is not abundant north of Pennsylvania, where it arrives early in May and leaves the
last of August. The males are said always to arrive three or four days before their
mates.
This species is described by Wilson as very much attached to certain localities where
they have once taken up their residence, appearing very jealous, and offended at the
least intrusion. They scold vehemently at every one who approaches or even passes
by their places of retreat, giving utterance to a great variety of odd and uncouth
sounds. Wilson states that these sounds may be easily imitated, so as to deceive the
bird itself, and to draw it after one; the bird following repeating its cries, but never
permitting itself to be seen. Such responses he describes as constant and rapid, and

strongly expressive both of anger and anxiety, their voice, as it shifts, unseen, from
place to place, seeming to be more like that of a spirit than a bird. These sounds
Wilson compares to the whistling of the wings of a duck, being repetitions of short
notes, beginning loud and rapid, and falling lower and lower. Again a succession of
other notes, said to closely resemble the barking of young puppies, is followed by a
variety of hollow, guttural sounds, each eight or ten times repeated, at times
resembling the mewing of a cat, only hoarser,—all of these, as he states, uttered
with great vehemence, in different keys and with peculiar modulations, now as if at a
considerable distance, and the next moment as if close by your side; so that, by
these tricks of ventriloquism, one is utterly at a loss to ascertain from what particular
quarter they proceed. In mild weather this strange melody of sounds is kept up
throughout the night during the first of the pairing-season, but ceases as soon as
incubation commences.
They construct their nest about the middle of May. These are placed within a few
feet of the ground, in the midst of low brambles, vines, and bushes, generally in a
tangled thicket. They build a rude but strongly woven nest, the outer portions more
loosely made of dry leaves; within these are interwoven thin strips of the bark of the
wild grape, fibrous roots, and fine dry grasses.
The eggs, four or five in number, are usually hatched out within twelve days, and in
about as many more the young are ready to leave their nest.
While the female is sitting, and still more after the young are hatched, the cries of
the male are loud and incessant when his nest is approached. He no longer seeks to
conceal himself, but rises in the air, his legs dangling in a peculiar manner, ascending
and descending in sudden jerks that betray his great irritation.
The food of this bird consists chiefly of beetles and other insects, and of different
kinds of berries and small fruit, and it said to be especially fond of wild strawberries.
Audubon states that in their migrations they move from bush to bush by day, and
frequently continue their march by night. Their flight at all times is short and
irregular. He also states that when on the ground they squat, jerk their tails, spring
on their legs, and are ever in a state of great activity. Although the existence of this
bird north of Pennsylvania is generally disputed, I have no doubt that it has always
been, and still is, a constant visitor of Massachusetts, and has been found to within a
score of miles of the New Hampshire line. Among my notes I find that a nest was
found in Brookline, in 1852, by Mr. Theodore Lyman; in Danvers, by Mr. Byron
Goodale; in Lynn, by Messrs. Vickary and Welch; and in many other parts of the
State. It certainly breeds as far south as Georgia on the coast, and in Louisiana and
Texas in the southwest. On the Pacific coast it is replaced by the long-tailed variety,
longicauda.
A nest of this species from Concord, Mass., obtained by Mr. B. P. Mann, and now in
the collection of the Boston Natural History Society, has a diameter of four inches

and a height of three and a half. The cavity has a depth of two and a quarter inches,
and is two and a half wide. This is built upon a base of coarse skeleton leaves, and is
made of coarse sedges, dried grasses, and stems of plants, and lined with long, dry,
and wiry stems of plants, resembling pine-needles. Another from Pomfret, Conn.,
obtained by Mr. Sessions, is a much larger nest, measuring five inches in diameter
and three and three quarters in height. The cup is two and a half inches deep by
three in width. It is made of an interweaving of leaves, bark of the grapevine, and
stems of plants, and is lined with fine, long wiry stems and pine-needles.
Their eggs are of a slightly rounded oval shape, vary in length from .85 to .95 of an
inch, and in breadth from .65 to .70. They have a white ground with a very slight
tinge of yellow, and are marked with reddish-brown and a few fainter purplish and
lilac spots.
Icteria virens, var. longicauda, Lawr.
LONG-TAILED CHAT.
Icteria longicauda, LawrÉncÉ , Ann. N. Y. Lyc. VI, April, 1853, 4.—Baird, Birds N. Am. 1858, 249, pl.
xxxiv, fig. 2; Rev. 230.—SclatÉr, Catal. 42, No. 253.—Finsch, Abh. Nat. Brem. 1870, 331
(Mazatlan).—CoopÉr, Orn. Cal. 1, 1870, 98. ? Icteria auricollis (Licht. Mus. Berl.), Bon. Consp.
1850, 331.
Sp. Char. Similar to var. virens. Fourth quill longest; third and fifth shorter; first shorter than the
seventh. Above ash-color, tinged with olive on the back and neck; the outer surface of the wings and
tail olive. The under parts as far as the middle of the belly bright gamboge-yellow, with a tinge of
orange; the remaining portions white. The superciliary and maxillary white stripes extend some
distance behind the eye. Outer edge of the first primary white. Length, 7 inches; wing, 3.20; tail,
3.70.
Young (8,841, Loup Fork of Platte, August 5; F. V. Hayden). Above light grayish-brown; beneath yellow
on anterior half as in adult, but yellow less pure; rest of under parts (except abdomen) ochraceous;
markings on head obsolete, the eyelids only being distinctly white.
Hab. Western and Middle Provinces of United States, east to Missouri River and Texas; Cape St. Lucas
and Western Mexico.
The most tangible difference between this bird and typical virens consists in the
longer tail. In addition, the upper plumage is grayish, with hardly any olive tinge, and
the white maxillary stripe extends farther back; the bill is not so deep as that of the
Eastern bird. All these differences, however, are in strict accordance with various
laws; the more grayish cast of plumage is what we should expect in birds from the
Middle Province, while the restriction of the yellow from the maxillæ we see also in
Western specimens of Helminthophaga ruficapilla; the longer tail, also, is a well-
known characteristic of Western birds, as distinguished from Eastern of the same
species.
Upon the whole, therefore, taking into consideration the absolute identity of their
habits and notes, we can only consider the I. longicauda and I. virens as restricted,

as being merely geographical races of one species.
This variety, as well as the Eastern, has in autumn and winter a slightly different
plumage. A pair (53,348 ♂, and 53,347 ♀, West Humboldt Mountains, Nevada)
obtained September 4 differ in the following respects from spring adults: the upper
plumage is decidedly brown, with even a russet tinge,—not gray, with a greenish
wash; the lores are less purely black, and the sides and crissum are deep cream-
color, instead of pure white; the female has a shade of olive across the jugulum;
both male and female have the lower mandible almost wholly white, and the
commissure broadly edged with the same.
No. 38,402 ♂, Laramie Peak, June, has the throat and jugulum strongly stained with
deep cadmium-orange.
Habits. The Western or Long-tailed Chat has an exclusively Western distribution, and
has been found from Mexico and Cape St. Lucas to Oregon, on the Pacific coast, and
as far to the east as the Upper Missouri.
According to Dr. Cooper, these birds appear in San Diego and at Fort Mojave in the
latter part of April. They are said to inhabit chiefly the warmer valleys near streams
and marshes, rarely on the coast. At Fort Mojave, Dr. Cooper found a nest of this bird
May 19, built in a dense thicket of algarobia. It contained three eggs, and one of the
Molothrus. The nest was built of slender green twigs and leaves, lined with grass and
hair. The eggs were white, sprinkled with cinnamon, somewhat in the form of a ring
near the larger end, and measured .75 by .64 of an inch.
These nests were usually very closely concealed, but one that he found at Santa
Cruz, near the coast, was in a very open situation, only two feet above the ground.
When the nest is approached, the old birds are very bold, keeping up a constant
scolding, and almost flying in the face of an intruder. At other times they are very
shy. The notes and sounds uttered by the Western bird Dr. Cooper states to be the
same as those of the Eastern species, and with the same grotesqueness. They leave
the State of California on or before the first of September.
Dr. Gambel states that the Chat appears in California about the middle of April,
resorting to the hedges, vineyards, and bushy portions of gardens to breed.
Mr. Xantus found a nest of this bird (S. I., 896) at Fort Tejon, California, in May. It is
a very symmetrical and exactly circular nest, six inches wide and three in height. The
cavity has a diameter of three inches at the brim, and a depth of two. It is built of
soft strips of bark, large stems, and branches of dry plants, leaves, twigs, and other
vegetable substances. These are very neatly and compactly interwoven. The nest is
elaborately lined with finer stems and flexible grasses. Another nest (S. I., 1816),
obtained at Neosho Falls, Kansas, by Mr. B. F. Goss, is of irregular shape. Its height is
four inches, and its diameter varies from three and three quarters to five inches. It
was built in a depression in the ground, and its shape adapted to its location. The

base is composed entirely of leaves, impacted when in a moist and decaying
condition. Within these is interwoven a strong basket-like structure, made of long
and slender stems, strips of bark, and fine rootlets, lined with finer grasses and
stems of plants.
A nest of this species from Sacramento is composed, externally, of fine strips of inner
bark of the grape and of deciduous trees, coarse straws, stems of plants, twigs, and
dried remains of weeds, etc. It is lined with finer stems and long wiry roots,
resembling hair. This nest has a diameter of four inches and a height of three. The
cavity has a diameter of three inches at the rim, and a depth of two.
In regard to this variety Mr. Ridgway writes: “In no respect that I could discover does
this Western bird differ from the Eastern in habits, manners, or notes. The nesting-
habits are exactly the same.”
The eggs of this species are, for the most part, larger than are those of the virens.
They vary in length from .95 to 1.00 of an inch, and have an average breadth of .70
of an inch. Their markings do not differ essentially in shadings from those of the
common species.
Subfamily SETOPHAGINÆ.
GÉn. Char. Sylvicoline birds with the characters of Flycatchers; the bill notched at tip, depressed and
broad at the base, though quite deep; the rictus with well-developed bristles reaching beyond the
nostrils, sometimes to the end of the bill. First quill rather less than the fourth, or still shorter. Size of
the species rarely exceeding six inches. Colors red, yellow, and olive.
The species of this section resemble the small Flycatchers of the family Tyrannidæ in
the structure of the bill, etc., and in the habit of capturing insects more or less on the
wing, though they are more restless in their movements, seeking their prey among
trees or in bushes, rapidly changing their place, instead of occupying a perch and
returning to it after pursuing an insect through the air. The yellow or orange crown
found in many species also carries out the analogy; but the strictly Oscine characters
of the tarsal scutellæ and the nine primaries will serve to distinguish them.
The Setophaginæ have their greatest development in Middle and South America, no
less than nine genera and subgenera being on record, of which only two extend into
the United States. Of one of these, Setophaga, we have only a single species of the
many described; the other, Myiodioctes, has no members other than those found in
the United States.
The following diagnosis is prepared to distinguish our genera from the South
American:—
A. Wings pointed; the first quill longer than the fifth; the third as long as or longer than the
fourth. Tail nearly even, or slightly rounded (the difference of the feathers less than .20); the

feathers broad and firm; the outer webs of exterior feathers narrow at base, but widening to
nearly double the width near the end.
1. Bill from gape nearly as long as skull, broad at base and much depressed; rictal bristles
reaching half-way from nostrils to tip. Culmen and commissure nearly straight. Wings equal
to the tail. Tarsi long; toes short; middle toe without claw, about half the
tarsus … Setophaga.
2. Bill from gape nearly as long as skull, broad at base, but deep and more sylvicoline; rictal
bristles reaching but little beyond nostrils. Culmen and commissure straight to the tip. Wings
longer than the almost even tail. Middle toe without claw, three fifths the
tarsus … Myiodioctes.
3. Bill from gape much shorter than head, wide at base, but compressed and high; the
culmen and commissure much curved from base, scarcely notched at tip; rictal bristles
reaching nearly half-way from nostrils to tip. Wings about equal to the almost even tail.
Middle toe without claw, about three fifths the rather short tarsus … Cardellina.
B. Wings rounded; the first quill shorter than in the preceding section; always less than the
fifth. South American genera.
[59]
Several species of Setophaginæ have, on not very well established grounds, been
assigned to the southern borders of the United States. They are as follows:—
Cardellina rubra, Baird, Rev. Am. Birds, 1865, 264. (Setophaga rubra, Swainson .) Parus
leucotis, Giraud, Birds Texas. Hab. Mexico. Rich carmine-red. Wing and tail-feathers brown.
Ear-coverts silvery white. Length, 4.70; wing, 2.40; tail, 2.55.
Basileuterus culicivorus, Baird, Rev. Am. Birds, 1865, 246. (Sylvia culicivora, Licht. )
Muscicapa brasieri, Giraud, Texas Birds. Hab. Southern Mexico; Guatemala and Costa Rica.
Top of head with two black stripes enclosing a median of yellow. Back olivaceous-ash.
Beneath entirely yellow. No rufous on side of head. Length, 4.90; wing, 2.40; tail, 2.25.
Basileuterus belli, Baird, Rev. Am. Birds, 1865, 247. Muscicapa belli, Giraud, Texas Birds.
Hab. Mexico and Guatemala. Top of head and face chestnut. A yellow superciliary stripe
bordered above by dusky. Back olive; beneath yellow. Length, 5.10; wing, 2.28; tail, 2.50.
GÉnus MYIODIOCTES, Aud.
Myiodioctes, Audubon, Synopsis, 1839, 48. (Type, Motacilla mitrata, Gm.)—Baird, Birds N. Am. 1858,
291.
Wilsonia, Bonap. List. 1838 (preoccupied in botany).
Myioctonus, Cabanis , Mus. Hein. 1850, 18. (Type, Motacilla mitrata.)
GÉn. Char. Bill broad, depressed; the lateral outlines a little concave; the bristles reaching not quite
half-way from nostrils to tip. Culmen and commissure nearly straight to near the tip. Nostrils oval, with
membrane above. Wings pointed, rather longer than the nearly even but slightly rounded tail; first
quill shorter than the fourth, much longer than the fifth; the second and third quills longest. Tarsi
rather lengthened, the scutellar divisions rather indistinct; the middle toe without claw, about three
fifths the tarsus.
This genus is distinguished from Setophaga, mainly by stouter feet and longer toes;
shorter and more even tail, narrower bill, etc. The species are decidedly muscicapine
in general appearance, as shown by the depressed bill with bristly rictus. The type M.

Myiodioctes mitratus.
2226
mitratus is very similar in character of bill to Dendroica
castanea, but the wings are much shorter; the tail
longer and more graduated; the legs and hind toe
longer, and the first primary shorter than the fourth
(.15 of an inch less than the longest), not almost
equal to the longest. The species are plain olive or
plumbeous above, and yellow beneath. They may be
grouped as follows:—
A. Tail with white patches on the inner feathers.
1. M. mitratus. Head and neck black. Front, cheeks, and
under parts yellow. Back olive-green. Hab. Eastern Province of
United States, south to Panama and West Indies.
2. M. minutus. Olive above; yellowish beneath. Two white
bands on the wings. Hab. Eastern United States.
B. Tail without white patch on the outer feathers.
3. M. pusillus. Crown black. Forehead, cheeks, and under
parts yellow. Back olive.
Yellow of forehead without an orange tinge; upper parts dull olive-green; pileum with very
dull steel-blue lustre. Hab. Eastern Province and Rocky Mountains of North America, south
to Costa Rica. … var. pusillus.
Yellow of forehead with an orange cast; upper parts bright yellowish-green; pileum with a
bright steel-blue lustre. Hab. Pacific Province of North America, from Sitka to Costa
Rica … var. pileolata.
4. M. canadensis. Streaks on the crown, stripes on sides of head and neck, with pectoral
collar of streaks, black. Rest of under parts, and line to and around the eye, yellow. Back
bluish. Hab. Eastern Province of United States, south to Ecuador.
Myiodioctes mitratus, Aud.
HOODED WARBLER.
Motacilla mitrata, GmÉlin, S. N. I, 1788, 293. Sylvia m. Lath.; ViÉill.; Bon.; Nutt.; Aud. Orn. Biog. II,
pl. cx. Sylvicola m. Mañ. Sylvania m. Nuttall, Man. I, 1840, 333. Setophaga m. Jard. Wilsonia m.
Bon. 1838.—AllÉn, Pr. Essex Inst. 1864. Myiodioctes m. Aud. Syn. 1839, 48.—Ib. Birds Am. II, pl.
lxxi.—SclatÉr, P. Z. S. 1856, 291 (Cordova); 1858, 358 (Honduras).—Baird, Birds N. Am. 1858,
292; Rev. 239.—JonÉs, Nat. Bermuda, 1859, 26 (March).—SclatÉr & Salvin, Ibis, 1859, 11
(Guatemala).—LawrÉncÉ , Ann. N. Y. Lyc. VIII, 63 (Panama R. R.).—Gundlach, Cab. Jour. 1861, 326
(Cuba).—SamuÉls , 245. Myioctonus m. Cab. Mus. Hein. 1851.—Ib. Jour. Orn. III, 1855, 472 (Cuba).
Muscicapa cucullata, Wilson, III, pl. xxvi, fig. 3. Muscicapa selbyi, Aud. Orn. Biog. I, pl. ix.
Sp. Char. Male. Bill black; feet pale yellow. Head and neck all round and forepart of the breast black. A
broad patch on the forehead extending round on the entire cheeks and ear-coverts, with the under
parts, bright yellow. Upper parts and sides of the body olive-green. Greater portion of inner web of
outer three tail-feathers white.

Female similar, but without the black; the crown like the back; the forehead yellowish; the sides of the
head yellow, tinged with olive on the lores and ear-coverts. Throat bright yellow.
Length, 5.00; wing, 2.75; tail, 2.55. (Skin.)
Hab. Eastern Province of United States, rather southern; Bermuda; Cuba; Jamaica; Eastern Mexico;
Honduras and Guatemala to Panama R. R. Orizaba (autumn, Sumichrast); Yucatan (LawrÉncÉ ).
A young male in second year (2,245, Carlisle, Penn., May) is similar to the female,
but the hood is sharply defined anteriorly, though only bordered with black, the olive-
green reaching forward almost to the yellow; there are only very slight indications of
black on the throat. Apparently the male of this species does not attain the full
plumage until at least the third year, as is the case with Setophaga ruticilla.

Myiodioctes pusillus.
Habits. This beautiful and singularly marked
Warbler is a Southern species, though not
exclusively so. It is more abundant in South
Carolina than any other State, so far as I am
aware. It is, however, found as far to the
north as Northern New Jersey and
Pennsylvania, and Southern New York, and,
farther west, as far north as the shores of
Lake Erie. It has also been found in
Bermuda, Cuba, Jamaica, Eastern Mexico,
Honduras, and Guatemala. Throughout
Central America it appears to be abundant
during the winter.
Mr. Audubon also states that it abounds in Louisiana and along the banks of
the Mississippi and the Ohio. It occurs on the Hudson to some distance
above New York. It appears from the South early in March, and has young
already hatched, in Louisiana, early in May.
It is said to be one of the liveliest of its tribe, and to be almost constantly in
motion. It is fond of secluded places, and is equally common in the thick
canebrakes, both of the high and the low lands, and in the tangled
undergrowth of impenetrable swamps. It has a peculiarly graceful manner
of closing and opening its broad tail, that at once distinguishes it from
every other bird, as it gambols from tree to tree, now in sight, and now hid
from the eye, but ever within hearing.
Mr. Audubon adds that its call-note so closely resembles that of the Spiza
ciris that it requires a practised ear to distinguish them. But its song is very
different. This consists of three notes, and is loud, lively, and pleasing. This
song is said to be made of sounds resembling the syllables weet, weet,
weetēē. Extremely vocal in the early spring, it becomes nearly silent as
soon as its brood is hatched. It resumes its song when its mate is again
sitting on her eggs, as they have more than one brood in a season.
They are described as expert flycatchers, full of activity and spirit, flying
swiftly after their insect prey; and catching the greater part on the wing.
Their flight is low, gliding, and often protracted.

Mr. Bachman narrates a striking instance of its courage and conjugal
devotion. While a pair of these Warblers were constructing a nest, a Sharp-
shinned Hawk pounced upon and bore off the female. The male followed
close after the Hawk, flying within a few inches and darting at him in all
directions, and so continued until quite out of sight.
Wilson states that it builds a very neat and compact nest, generally in the
fork of a small bush. It is formed of moss and flaxen fibres of plants, and
lined with hair or feathers. The eggs, five in number, he describes as of a
grayish-white, with red spots at the larger end. He noticed its arrival at
Savannah as early as the 20th of March. Mr. Audubon adds that these nests
are always placed in low situations, a few feet from the ground.
The late Dr. Gerhardt, of Varnell’s Station, Georgia, informed me, by letter,
that the Hooded Warbler deposits her eggs about the middle of May, laying
four. The nest is not unlike that of the Spiza cyanea, but is larger. It is
constructed of dry leaves and coarse grass on the outside, and within of dry
pine-needles, interwoven with long yellow grasses and sometimes with
horsehair. They are built, for the most part, in the neighborhood of brooks
and creeks, in oak bushes, four or five feet from the ground. The female
sits so closely, and is so fearless, that Dr. Gerhardt states he has sometimes
nearly caught her in his hand.
In another letter Dr. Gerhardt describes a nest of this species as measuring
three inches in height, three in external diameter, and an inch and a quarter
in the depth of its cavity. Externally it was built of dry leaves and coarse
grasses, lined inside with horsehair, fine leaves of pine, and dry slender
grasses. It was constructed on a small oak growing in low bottom-land, and
was three feet from the ground. The complement of eggs is four.
Mr. Ridgway states that this species is a common summer resident in the
bottom-lands along the Lower Wabash, in Southern Illinois, inhabiting the
cane-brakes and the margins of bushy swamps.
The eggs of this Warbler are oval in shape, with one end quite pointed.
They measure .70 by .50 of an inch. Their ground-color is a beautiful bright
white, when the egg is fresh, strongly tinged with flesh-color. The spots are
of a fine red, with a few markings of a subdued purple.
Myiodioctes minutus, Baird.

SMALL-HEADED FLYCATCHER.
Muscicapa minuta, Wilson, Am. Orn. VI, 1812, 62, pl. 1, fig. 5.—Aud. Orn. Biog. V, pl.
ccccxxxiv, fig. 3.—Ib. Birds Am. I, pl. lxvii. Sylvia minuta, Bon. Wilsonia m. Bon. List,
1838. Myiodioctes minutus, Baird, Rev. Am. Birds, 1864, 241. Sylvania pumilia, Nutt.
Man. I, 1840, 334.
Sp. Char. Wings short, the second quills longest. Tail of moderate-length, even. General
color of upper parts light greenish-brown; wings and tail dark olive-brown, the outer
feathers of the latter with a terminal white spot on the inner web; a narrow white ring
surrounding the eye; two bands of dull white on the wings; sides of the head and neck
greenish-yellow; the rest of the lower parts pale yellow, gradually fading into white behind.
Male, 5 inches long; extent, 8.25 inches.
Hab. Eastern United States.
Habits. All that is known in regard to this species we receive from Wilson
and Audubon, and there is a decided discrepancy in their several
statements. Wilson states that his figure was taken from a young male shot
on the 24th of April, but in what locality he does not mention. He adds that
he afterwards shot several individuals in various parts of New Jersey,
particularly in swamps. He found these in June, and has no doubt they
breed there.
Audubon claims that Wilson’s drawing was a copy from his own of a bird
shot by him in Kentucky on the margin of a pond. He throws a doubt as to
the correctness of Wilson’s statement that they have been found in New
Jersey, as no one else has ever met with any there. That may be, however,
and Wilson’s statement yet be correct. The same argument carried out
would reject the very existence of the bird itself, as no well-authenticated
records of its occurrence since then can be found. They are at least too
doubtful to be received as unquestionable until the genuine bird can be
produced. Mr. Nuttall, it is true, states that Mr. Charles Pickering obtained a
specimen of this bird many years ago, near Salem, Mass., and that he had
himself also seen it in the same State, at the approach of winter. In the fall
of 1836, when the writer resided in Roxbury, a cat caught and brought into
the house a small Flycatcher, which was supposed to be of this species. It
was given to Mr. Audubon, who assented to its correct identification, but
afterwards made no mention of it. The presumption, therefore, is that we
may have been mistaken.
In regard to its habits, Wilson represents it as “remarkably active, running,
climbing, and darting about among the opening buds and blossoms with
extraordinary agility.” Audubon states that in its habits it is closely allied

with the pusillus and the mitratus, being fond of low thick coverts in
swamps and by the margin of pools. He also attributes to it a song of rather
pleasing notes, enunciated at regular intervals, loud enough to be heard at
the distance of sixty yards. These peculiarities seem to separate it from the
true Flycatchers and to place it among the Warblers.
Myiodioctes pusillus, Bonap.
GREEN BLACK-CAPPED FLYCATCHER.
Muscicapa pusilla, Wilson, Am. Orn. III, 1811, 103, pl. xxvi, fig. 4. Wilsonia pus. Bon.
Sylvania pus. Nutt. Myiodioctes pus. Bon. Consp. 1850, 315.—SclatÉr, P. Z. S. 1856,
291 (Cordova); 1858, 299 (Oaxaca Mts.; Dec.); 1859, 363 (Xalapa); 373.—Ib. Catal.
1861, 34, No. 203.—Baird, Birds N. Am. 1858, 293 (in part); Rev. 240 (in part).—
SclatÉr & Salvin, Ibis, 1859, 11 (Guatemala).—SamuÉls , 246. Myioctonus pus. Cab. M.
H. 1851, 18.—Ib. Jour. 1860, 325 (Costa Rica). Sylvia wilsoni, Bon.; Nutt. Muscicapa
wilsoni, Aud. Orn. Biog. II, pl. cxxiv. Setophaga wilsoni, Jard. Myiodioctes wilsoni, Aud.
Birds Am. II, pl. lxxv. Sylvia petasodes, Licht. Preis-Verz. 1830.
Sp. Char. Forehead, line over and around the eye, and under parts generally, bright yellow.
Upper part olive-green; a square patch on the crown lustrous-black. Sides of body and
cheeks tinged with olive. No white on wings or tail. Female similar, the black of the crown
replaced by olive-green. Length, 4.75; wing, 2.25; tail, 2.30.
Hab. Eastern portions of United States, west to the Snake and Humboldt Rivers; north to
Alaska, south through Eastern Mexico and Guatemala to Costa Rica; Chiriqui (Salvin).
Habits. Wilson’s Black-Cap is found throughout the United States from ocean
to ocean, and as far to the north as Alaska and the Arctic shores, where,
however, it is not common. Mr. Dall shot a specimen, May 30, on the Yukon
River, where it was breeding. Mr. Bischoff obtained others with nests and
eggs at Sitka, and afterwards found it more abundant at Kodiak. On the
Pacific coast Dr. Suckley found it very abundant in the neighborhood of Fort
Steilacoom, where it frequented thickets and small scrub-oak groves, in its
habits resembling the Helminthophaga celata, flitting about among the
dense foliage of bushes and low trees in a busy, restless manner. He
describes its cry as a short chit-chat call. In California, Dr. Cooper notes
their first arrival early in May, and states that they migrate along the coast,
up at least to the Straits of Fuca. At Santa Cruz he noted their arrival, in
1866, about the 20th of April. They were then gathering materials for a
nest, the male bird singing merrily during his employment. As they have
been observed in Oregon as early as this, it has been conjectured that
some may remain all winter among the dense shrubbery of the forests.

This bird winters in large numbers in Central America, where it is apparently
very generally distributed. Mr. Salvin found it very common at Duenas. It
was taken at Totontepec, among the mountains of Oaxaca, Mexico, by Mr.
Boucard.
Mr. Ridgway found it very common during the summer and autumn months
among the willows of the fertile river valleys, and among the rank
shrubbery bordering upon the streams of the cañons of the higher interior
range of mountains. It was found in similar situations with the Dendroica
æstiva, but it was much more numerous. During September it was most
abundant among the thickets and copses of the East Humboldt Mountains,
and in Ruby Valley, at all altitudes, frequenting the bushes along the
streams, from their sources in the snow to the valleys.
Wilson first met with and described this species from specimens obtained in
Delaware and New Jersey. He regarded it as an inhabitant of the swamps of
the Southern States, and characterized its song as “a sharp, squeaking
note, in no wise musical.” It is said by him to leave the Southern States in
October.
Audubon states that it is never found in the Southern States in the summer
months, but passes rapidly through them on its way to the northern
districts, where it breeds, reaching Labrador early in June and returning by
the middle of August. He describes it as having all the habits of a true
Flycatcher, feeding on small insects, which it catches on the wing, snapping
its bill with a sharp clicking sound. It frequents the borders of lakes and
streams fringed with low bushes.
Mr. Nuttall observed this species in Oregon, where it arrived early in May.
He calls it a “little cheerful songster, the very counterpart of our brilliant and
cheerful Yellow-Bird.” Their song he describes as like ’tsh-’tsh-’tsh-tshea.
Their call is brief, and not so loud. It appeared familiar and unsuspicious,
kept in bushes busily collecting its insect fare, and only varied its
employment by an occasional and earnest warble. By the 12th of May some
were already feeding their full-fledged young. Yet on the 16th of the same
month he found a nest containing four eggs with incubation only just
commenced. This nest was in a branch of a small service-bush, laid very
adroitly, as to concealment, upon a mass of Usnea. It was built chiefly of
hypnum mosses, with a thick lining of dry, wiry, slender grasses. The
female, when approached, slipped off the nest, and ran along the ground

like a mouse. The eggs were very similar to those of Dendroica æstiva, with
spots of a pale olive-brown, confluent at the greater end.
A nest found by Audubon in Labrador was placed on the extremity of a
small horizontal branch, among the thick foliage of a dwarf fir, a few feet
from the ground and in the very centre of a thicket. It was made of bits of
dry mosses and delicate pine twigs, agglutinated together and to the
branches and leaves around it, from which it was suspended. It was lined
with fine vegetable fibres. The diameter of the nest was three and a half
and the depth one and a half inches. He describes the eggs, which were
four, as white; spotted with reddish and brown dots, the markings being
principally around the larger end, forming a circle, leaving the extremity
plain.
In this instance the parents showed much uneasiness at the approach of
intruders, moving about among the twigs, snapping their bills, and uttering
a plaintive note. In Newfoundland these birds had already begun to migrate
on the 20th of August. He met with them in considerable numbers in
Northern Maine in October, 1832. Mr. Turnbull mentions it as a rather
abundant bird of Eastern Pennsylvania, appearing there early in May, in
transitu, and again in October.
Mr. T. M. Trippe has observed this species at Orange, N. J., from the 19th to
the 30th of May. It is said to keep low down in the trees, and is fond of
haunting thickets and open brush fields. Occasionally he has heard it utter a
loud chattering song, which it repeats at short intervals.
A nest of this species from Fort Yukon (Smith. Coll., 13,346), obtained May
20, by Mr. McDougal, contained four eggs. These varied from .60 to .63 of
an inch in length, and from .45 to .49 in breadth. They were obovate in
shape, their ground-color was a pure white; this was finely sprinkled round
the larger end with brownish-red and lilac. No mention is made of the
position of the nest, but it is probable this bird builds on the ground.
Myiodioctes pusillus, var. pileolatus, Ridgway.
Motacilla pileolata, Pallas, Zoög. Rosso Asiat. I, 1831, 497 (Russian America).
Myiodioctes pusillus, var. pileolata, Ridgway, Report U. S. Geol. Expl. 40th Par.
Myiodioctes pusillus, Auct. (all citations from Pacific coast of North and Middle
America).—Lord, Pr. R. Art. Inst. Woolw. IV, 1864, 115 (Br. Col.).—Dall & BannistÉr
(Alaska).—CoopÉr, Orn. Cal. 1, 1870, 101.

Sp. Char. Similar to var. pusillus, but much richer yellow, scarcely tinged with olive laterally,
and deepened into an almost orange shade on the front and chin. Above much brighter
and more yellowish olive-green. The black pileum with a brighter steel-blue gloss. Bill
much narrower, and deep, light brown above, instead of nearly black. Measures (4,222 ♂,
San Francisco, Cal.), wing, 2.15; tail, 2.00.
Hab. Pacific coast region of North America, from Kodiak (Alaska); south through Western
Mexico (and Lower California) to Costa Rica.
This is an appreciably different race from that inhabiting the eastern
division of the continent; the differences, tested by a large series of
specimens, being very constant.
A Costa-Rican specimen before me is almost exactly like specimens from
California.
Habits. The remarks, in the preceding article relative to specimens from the
Pacific coast belong to this variety.
Myiodioctes canadensis, Aud.
CANADA FLYCATCHER.
Muscicapa canadensis, Linn. Syst. Nat. I, 1766, 327. (Muscicapa canadensis cinerea,
Brisson, II, 406, tab. 39, fig. 4.)—GmÉlin.—Wilson, III, pl. xxvi, fig. 2.—Aud. Orn. Biog.
II, pl. ciii. Setophaga can. Swains.; Rich.; Gray. Myiodioctes can. Aud. Birds Am. II, pl.
ciii.—BrÉwÉr, Pr. Bost. Soc. VI, 5 (nest and eggs).—SclatÉr, P. Z. S. 1854, 111
(Ecuador; winter); 1855, 143 (Bogota); 1858, 451 (Ecuador).—Ib. Catal. 1861, 34, No.
204.—SclatÉr & Salvin, Ibis, 1859, 11 (Guatemala).—LawrÉncÉ , Ann. N. Y. Lyc. VI, 1862.
—Baird, Birds N. Am. 1858, 294; Rev. 239.—SamuÉls , 247. Euthlypis can. Cab. Mus.
Hein. 1850, 1851, 18; Jour. Orn. 1860, 326 (Costa Rica). Sylvia pardalina, Bon.; Nutt.
Sylvicola pardalina, Bon. Myiodioctes pardalina, Bon. ? Muscicapa bonapartei, Aud. Orn.
Biog. I, 1831, 27, pl. v. Setophaga bon. Rich. Wilsonia bon. Bon. Sylvania bon. Nutt. ?
Myiodioctes bon. Aud. Syn.—Ib. Birds Am. II, 1841, 17, pl. xvii.—Baird, Birds N. Am.
1858, 295. Setophaga nigricincta, Lafr. Rev. Zoöl. 1843, 292; 1844, 79.
Sp. Char. Upper part bluish-ash; a ring around the eye, with a line running to the nostrils,
and the whole under part (except the tail-coverts, which are white), bright yellow. Centres
of the feathers in the anterior half of the crown, the cheeks, continuous with a line on the
side of the neck to the breast, and a series of spots across the forepart of the breast,
black. Tail-feathers unspotted. Female similar, with the black of the head and breast less
distinct. In the Young obsolete. Length, 5.34; wing, 2.67; tail, 2.50.
Hab. Whole Eastern Province of United States, west to the Missouri; north to Lake
Winnipeg; Eastern Mexico to Guatemala, and south to Bogota and Ecuador (SclatÉr). Not
noted from West Indies.

Habits. This is a migratory species, abundant during its passage, in most of
the Atlantic States. It breeds, though not abundantly, in New York and
Massachusetts, and in the regions north of latitude 42°. How far northward
it is found is not well ascertained, probably as far, however, as the wooded
country extends. It was met with on Winnepeg River, by Mr. Kennicott, the
second of June. It winters in Central and in Northern South America, having
been procured at Bogota, in Guatemala, and in Costa Rica, in large
numbers.
Mr. Audubon states that he found this bird breeding in the mountainous
regions of Pennsylvania, and afterwards in Maine, New Brunswick, Nova
Scotia, Newfoundland, and Labrador. Although he describes with some
minuteness its nests, yet his description of their position and structure is so
entirely different in all respects from those that have been found in
Massachusetts, that I am constrained to believe he has been mistaken in
his identifications, and that those he supposed to belong to this species
were really the nests of a different bird.
“In Vermont,” Mr. Charles S. Paine, of Randolph, informs me, “the Canada
Flycatcher is a summer visitant, and is first seen about the 18th of May.
They do not spread themselves over the woods, like most of our small fly-
catching birds, but keep near the borders, where there is a low growth of
bushes, and where they may be heard throughout the day singing their
regular chant. A few pairs may occasionally be found in the same
neighborhood. At other times only a single pair can be found in quite a
wide extent of territory of similar character. They build their nests, as well
as I can judge, about the first of June, as the young are hatched out and
on the wing about the last of that month, or the first of July. I have never
found a nest, but I think they are built on the ground. They are silent after
the first of July, and are rarely to be seen after that period.” The song of
this bird is a very pleasing one, though heard but seldom, and only in a few
localities in Massachusetts.
Near Washington Dr. Coues found the Canada Flycatcher only a spring and
autumnal visitant, at which seasons they were abundant. They frequented
high open woods, and kept mostly in the lower branches of the trees, and
also in the more open undergrowth of marshy places. They arrive the last
week in April and remain about two weeks, arriving in fall the first week in
September, and remaining until the last of that month.

The first well-identified nest of this bird that came to my knowledge was
obtained in Lynn, Mass., by Mr. George O. Welch, in June, 1856. It was built
in a tussock of grass, in swampy woods, concealed by the surrounding rank
vegetation, in the midst of which it was placed. It was constructed entirely
of pine-needles and a few fragments of decayed leaves, grapevine bark,
fine stems, and rootlets. These were so loosely interwoven that the nest
could not be removed without great care to keep its several portions
together. Its diameter was three and a half inches, and it was very nearly
flat. Its greatest depth, at the centre of its depression, was hardly half an
inch. It contained four young, and an unhatched egg.
Another nest found in June, 1864, by the same observing naturalist, was
also obtained in the neighborhood. This was built in a tussock of meadow-
grass, in the midst of a small boggy piece of swamp, in which were a few
scattered trees and bushes. The ground was so marshy that it could be
crossed only with difficulty, and by stepping from one tussock of reedy
herbage to another. In the centre of one of these bunches the nest was
concealed. It measures six inches in its larger diameter, and has a height of
two and a quarter inches. The cavity of this nest is two and three quarters
inches wide, and one and three quarters deep. It is very strongly
constructed of pine-needles, interwoven with fine strips of bark, dry
deciduous leaves, stems of dry grasses, sedges, etc. The whole is firmly
and compactly interwoven with and strengthened around the rim of the
cavity by strong, wiry, and fibrous roots. The nest is very carefully and
elaborately lined with the black fibrous roots of some plant. The eggs,
which were five in number, measure .72 of an inch in length by .56 in
breadth. Their ground-color is a clear and brilliant white, and this is
beautifully marked with dots and small blotches of blended brown, purple,
and violet, varying in shades and tints, and grouped in a wreath around the
larger end.
GÉnus SETOPHAGA, Swains.
Setophaga, Swainson , Zoöl. Jour. III, Dec. 1827, 360. (Type, Muscicapa ruticilla, L.)—
Baird, Birds N. Am. 1858, 297. Sylvania, Nuttall, Man. Orn. I, 1832. (Same type.)
GÉn. Char. Bill much depressed, the lateral outlines straight towards tip. Bristles reach half-
way from nostril to tip. Culmen almost straight to near the tip; commissure very slightly
curved. Nostrils oval, with membrane above them. Wings rather longer than tail, pointed;
second, third, and fourth quills nearly equal; first intermediate between fourth and fifth.
Tail rather long, rather rounded; the feathers broad, and widening at ends, the outer web

Setophaga ruticilla, Sw.
984
narrow. Tarsi with scutellar divisions indistinct
externally. Legs slender; toes short, inner cleft nearly
to base of first joint, outer with first joint adherent;
middle toe without claw, not quite half the tarsus.
The genus Setophaga is very largely
represented in America, although of the
many species scarcely any agree exactly in
form with the type. In the following diagnosis
I give several species, referred to, perhaps
erroneously, as occurring in Texas.
Belly white. End of lateral tail-feathers black.
Sexes dissimilar.
Ground-color black, without vertex spot. Sides
of breast and bases of quills and tail-feathers
reddish-orange in male, yellowish in
female … ruticilla.
Belly vermilion or carmine red. Lateral tail-
feathers, including their tips, white. Sexes
similar.
Entirely lustrous black, including head and
neck. No vertex spot. A white patch on the
wings … picta.
[60]
Plumbeous-ash, including head and neck. A chestnut-brown vertex spot. No white
on wings … miniata.
[61]
Setophaga ruticilla, Swains.
AMERICAN REDSTART.
Motacilla ruticilla, Linn. Syst. Nat. 10th ed. 1758, 186 (Catesby, Car. tab. 67). Muscicapa
ruticilla, Linn.; GmÉlin; ViÉillot; Wils.; Bon.; Aud. Orn. Biog. I, pl. xl. Setophaga rut.
Swains. Zoöl. Jour. III, 1827, 358.—Bon.; Aud. Birds Am.—SclatÉr, P. Z. S. (Ecuador,
Bogota, Cordova, Oaxaca, City of Mexico).—SclatÉr & Salvin, Ibis, 1859, 12
(Guatemala).—Baird, Birds N. Am. 1858, 297; Rev. 256.—Mañ.; Sallé, P. Z. S. 1857 (St.
Domingo).—NÉwton, Ibis, 1859, 143 (St. Croix; winter).—Cab. Jour. 1856, 472 (Cuba);
1860, 325 (Costa Rica).—Gundlach, Ib. 1861, 326 (Cuba).—Bryant, Pr. Bost. Soc. VII,
1859 (Bahamas).—LawrÉncÉ , Ann. N. Y. Lyc. 1861, 322 (Panama R. R.).—SamuÉls , 249.
Sylvania rut. Nuttall, Man. I, 1832, 291 (type of genus). Motacilla flavicauda, GmÉlin, I,
1788, 997 (♀).

PLATE XVI.
1. Setophaga ruticilla, Linn. ♂ Pa.,
984.

2. Myiodioctes minutus, Aud. (Copied
from Aud.)
3. Myiodioctes pusillus, Wils. ♂ Cal.,
7683.

4. Myiodioctes pusillus, Wils. ♀ Pa.,
2325.
5. Setophaga ruticilla, Linn. ♀ Pa.,
2281.

6. Myiodioctes canadensis, Linn. ♂
Pa., 945.
7. Progne subis, Linn. ♀ 40704.

8. Tachycineta bicolor, Vieill. ♂ Pa.,
2896.
9. Hirundo horreorum, Bart. ♂ Pa.,
1452.

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