THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
DEC 11 1976
THEORETICAL PHYSICS
ELEMENTS
OF
THEORETICAL PHYSICS
BY
DR. C. CHRISTIANSEN
PROFESSOR OF PHYSICS IN THE UNIVERSITY OF COPENHAGEN
TRANSLATED INTO ENGLISH BY
W. F. MAGIE, PH.D.
PROFESSOR OF PHYSICS IN PRINCETON UNIVERSITY
MACMILLAN AND CO, LIMITED
NEW YORK : THE MACMILLAN COMPANY
1897
All rights resei~eed
GLASGOW : PRINTED AT THE UNIVERSITY PRESS
BY ROBERT MACLEHOSE AND CO.
Physii
TRANSLATOR'S PREFACE.
THE treatise of Professor Christiansen, of which a translation is here
given, presents the fundamental principles of Theoretical Physics,
and develops them so far as to bring the reader in touch with
much of the new work that is being done in that subject. It is
not in every respect exhaustive, but it is stimulating and informing,
and furnishes a view of the whole field, which will facilitate the
reader's subsequent progress in special parts of it. The need of such
a book, in which the various branches of the subject are developed
in connection with one another and in a consistent notation, has
been long felt by both teachers and students.
The thanks of the translator are due to Professor Christiansen for
his courtesy in permitting the use of his book. The translation was
made from the German of Miiller. The first draft of it was
prepared by the translator's wife, without whose aid the task might
never have been accomplished.
W. F. MAGIE.
PRINCETON UNIVERSITY, September, 1896.
TABLE OF CONTENTS.
PAGE
INTRODUCTION, - 1
CHAPTER I.
GENERAL THEORY OF MOTION.
SECTION
I. Freely Falling Bodies, - 5
II. The Motion of Projectiles, 7
III. Equations of Motion for a Material Point, - 8
IV. The Tangential and Normal Forces, - 13
V. Work and Kinetic Energy, - 14
VI. The Work Done on a Body during its Motion in a Closed
Path, - 16
VII. The Potential, .... - - 21
VIII. Constrained Motion, 24
IX. Kepler's Laws, - - 27
X. Universal Attraction, - 30
XI. Universal Attraction (continued}, - 31
XII. The Potential of a System of Masses, - ... 34
XIII. Examples. Calculation of Potentials, - - 36
XIV. Gauss's Theorem. The Equations of Laplace and Poisson, - 41
XV. Examples of the Application of Laplace's and Poisson's
Equations, ------- ---46
XVI. Action and Reaction. On the Molecular and Atomic Structure
of Bodies,. - - - 48
XVII. The Centre of Gravity, - -.- * . - - - 50
CONTENTS.
SECTION
PAGE
XVIII.
A Material System,
53
XIX.
Moment of Momentum, - - - - - -
55
XX.
The Energy of a System of Masses,
56
XXI.
Conditions of Equilibrium. Rigid Bodies, - - -
58
XXII.
Rotation of a Rigid Body. The Pendulum,
60
CHAPTER II.
THE THEORY OF ELASTICITY.
XXIII.
62
XXIV.
Components of Stress, - - -
64
XXV.
Relations among the Components of Stress,
67
XXVI.
The Principal Stresses, -
69
XXVII.
Faraday's Views on the Nature of Forces Acting at a
Distance, -
72
XXVIII.
Deformation, ---------
74
XXIX.
Relations between Stresses and Deformations,
79
XXX.
Conditions of Equilibrium of an Elastic Body,
82
XXXI.
Stresses in a Spherical Shell,
83
XXXII.
Torsion,
85
XXXIII.
Flexure, ----------
87
XXXIV.
Equations of Motion of an Elastic Body, -
89
XXXV.
Plane Waves in an Infinitely Extended Body,
90
XXXVI.
Other Wave Motions, - - - -
93
XXXVII.
Vibrating Strings, - - - ....
95
XXXVIII.
Potential Energy of an Elastic Body, - - - -
96
CHAPTER III.
EQUILIBRIUM OF FLUIDS.
XXXIX. Conditions of Equilibrium, ...
XL. Examples of the Equilibrium of Fluids,
101
CHAPTER IV.
MOTION OF FLUIDS.
XLI. Euler's Equations of Motion,
XLII. Transformation of Euler's Equations,
103
106
CONTENTS.
IX
SECTION
XLIII.
XLIV.
XLV.
XL VI.
Vortex Motions and Currents in a Fluid,
Steady Motion with Velocity-Potential,
Lagrange's Equations of Motion, -
Wave Motions,
PAGE
107
109
111
112
CHAPTER V.
INTERNAL FRICTION.
XL VII. Internal Forces,
XLVIII. Equations of Motion of a Viscous Fluid,
XLIX. Flow through a Tube of Circular Cross Section,
115
118
119
CHAPTER VI.
CAPILLARITY.
L. Surface Energy, -
LI. Conditions of Equilibrium, -
LII. Capillary Tubes, -
121
123
125
CHAPTER VII.
ELECTROSTATICS.
LIIL Fundamental Phenomena of Electricity, - - - 127
LIV. Electrical Potential, 128
LV. The Distribution of Electricity on a Good Conductor. - 130
LVI. The Distribution of Electricity on a Sphere and on an
Ellipsoid, - - - 132
LVII. Electrical Distribution, - ..... 135
LVI 1 1. Complete Distribution, - - 139
LIX. Mechanical Force Acting on a Charged Body, - - 141
LX. Lines of Electrical Force, 143
LXI. Electrical Energy, - - - - - - - - 145
LXII. A System of Conductors, 147
LXIII. Mechanical Forces, 150
LXIV. The Condenser and Electrometer, 151
LXV. The Dielectric, 155-
CONTENTS.
SECTION
PAGE
LXVI. Conditions of Equilibrium, 157
LXVII. Mechanical Force and Electrical Energy in the Dielectric, 158
CHAPTER VIII.
MAGNETISM.
LXVIII. General Properties of Magnets, 163
LXIX. The Magnetic Potential, ------ 166
LXX. The Potential of a Magnetized Sphere, - - - 168
LXXL The Forces which Act on a Magnet, - - - - 169
LXXII. Potential Energy of a Magnet, - - - - - 171
. LXXIII. Magnetic Distribution, - - . _ .' - 173
LXXIV. Lines of Magnetic Force, 174
LXXV. The Equation of Lines of Force, - - - - 178
LXXVI. Magnetic Induction, - 179
LXXVII. Magnetic Shells, - - . . fc . . 180
CHAPTEE IX.
ELECTEO-MAGNETISM.
LXX VIII. Biot and Savart's Law, - . „ _ . j84
LXXIX. Systems of Currents, - - . , . . - 186
LXXX. The Fundamental Equations of Electro-Magnetism, - 188
LXXXI. Systems of Currents in General, - - . - 190
LXXXII. The Action of Electrical Currents on each other, - 192
LXXXIII. The Measurement of Current-Strength on the Quantity
of Electricity, - .. . . . . _ '-194
LXXXI V. Ohm's Law and Joule's Law, . „ . . - 197
CHAPTER X.
INDUCTION.
LXXXV. Induction, - - ....
LXXXVI. Coefficients of Induction, - ' .
LXXXVII. Measurement of Resistance,
LXXXVIII. Fundamental Equations of Induction,
LXXXIX. Electro-Kinetic Energy,
XC. Absolute Units, -
202
205
- 210
- 211
CONTENTS.
CHAPTER XL
ELECTRICAL OSCILLATIONS.
SECTION
XCI.
Oscillations in a Conductor, ------
XCII. Calculation of the Period, ------
XCI II. The Fundamental Equations for Electrical Insulators
or Dielectrics, --------
XCIV. Plane Waves in the Dielectric, -
XCV. The Hertzian Oscillations,
XCVI. Poynting's Theorem, 224
215
217
219
221
223
CHAPTER XII.
REFRACTION OF LIGHT IN ISOTROPIC AND TRANSPARENT
BODIES.
XCVII. Introduction, - -
XCVIII. Fresnel's Formulas,
XCIX. The Electro-Magnetic Theory of Light,
Equations of the Electro-Magnetic Theory of Light,
Refraction in a Plate, 242
Double Refraction, ----.... 246
Discussion of the Velocities of Propagation, - - - 249
CIV. The Wave Surface, 251
The Wave Surface (continued), - - ... 254
The Direction of the Rays, 256
Uniaxial Crystals, - - - 259
C.
or.
GIL
GUI.
CV.
CVI.
CVII.
229
231
235
237
CVIII. Double Refraction at the Surface of a Crystal, - - 261
CIX. Double Refraction in Uniaxial Crystals, - - - 264
CHAPTER XIII.
THERMODYNAMICS.
CX. The State of a Body, ------- 266
CXI. Ideal Gases, . 270
CXII. Cyclic Processes, - - 272
CXIII. Carnot's and Clausius' Theorem, 274
CXIV. Application of the Second Law, 279
Xll
CONTENTS.
SECTION PAGE
CXV. The Differential Coefficients, - 280
CXVI. Liquids and Solids, - - 281
CXVII. The Development of Heat by Change of Length, - 282
CXVITI. Van der Waal's Equation of State, - - 283
CXIX. Saturated Vapours, 290
CXX. The Entropy, - - 292
CXXI. Dissociation, --------- 295
CHAPTER XIV.
CONDUCTION OF HEAT.
CXXII. Fourier's Equation, - 298
CXXIII. Steady State, - - 300
CXXIV. The Periodic Flow of Heat in a given Direction, - 301
CXXV. A Heated Surface, - - 303
CXXVI. The Flow of Heat from a Point, 304
CXXVII. The Flow of Heat in an Infinitely Extended Body, - 305
CXXVIII. The Formation of Ice, - - 307
CXXIX. The Flow of Heat in a Plate whose Surface is kept at
a Constant Temperature, 308
CXXX. The Development of Functions in Series of Sines and
Cosines, - 312
CXXXI. The Application of Fourier's Theorem to the Conduction
of Heat, - - 315
CXXXII. The Cooling of a Sphere, 318
CXXXIII. The Motion of Heat in an Infinitely Long Cylinder, - 322
CXXXIV. On the Conduction of Heat in Fluids, - - - - 325
CXX XV. The Influence of the Conduction of Heat on the In-
tensity and Velocity of Sound in Gases, - 330
INTRODUCTION.
IN the Science of Physics it is assumed that all phenomena are
capable of ultimate representation by motions, that is, by changes
of place considered with reference to the time required for their
accomplishment. We therefore begin with a brief discussion of the
theory of pure motion (Kinematics). We will treat first the motion
of a point. The continuous line traced out by the successive positions
which a moving point occupies in space is called its path. The
symbol s represents the distance which the point traverses along its
path in the time t. In measuring these quantities the second is
used as the unit of time; the centimetre, as the unit of length.
The measures of all the magnitudes which occur in the discussion
of motions may be stated in terms of these two units.
Motions are distinguished by the form of the path, as rectilinear,
curvilinear, or periodic. Eectilinear and curvilinear motions are
sufficiently defined by their names. A periodic motion is one in
which the same condition of motion recurs after a definite interval
of time; that is, one in which the moving point returns after a
definite time to the same position with the same velocity and direction
of motion.
Rectilinear motion may be either uniform or variable. It is uniform
if the moving point traverses equal distances in equal times. In this
case the point traverses the same distance in each unit of time, and
the distance traversed in the unit of time measures its velocity. If
the point traverses the distance 5 in the time t with a uniform motion,
the velocity c is the ratio of s to t, or (a) c = s/t. A velocity is
therefore a length divided by a time.
If a point moves on the circumference of a circle with a constant
velocity, the radius vector drawn to this point sweeps out equal
sectors in equal times. In this case the angle which is swept out by
this radius vector in the unit of time measures the angular velocity.
4 INTRODUCTION.
is xl + x2. The component of velocity in the direction of the a--axis
is ac = sc1 + «2.
Similar expressions hold for motions in the directions of the other
axes. The resultant velocity is represented by the diagonal of the
parallelepiped, whose edges are x, y, z, or by s = *]d? + y- + z2.
Since an acceleration is the increment of a velocity, the resultant
acceleration will be determined in a similar manner. Let ajj, a%
represent the .r-components of the increments of velocity due to
the two motions. We then have for the total acceleration in the
direction of the .r-axis, x = xl + x.2, and for the acceleration of the
point, 3 = ^/(iBj + «2)2 + (yl + i/^f + (^ + z2)2, by which s is expressed as
the diagonal of the parallelepiped whose edges are a;, y, z.
If the coordinates of the moving point are given as functions of
the time, the equation of the path is obtained by determining the
values of x and y which hold for the same time L If, for example,
x=fl(t)a,ndy=f2(t), the relation between x and y is found by
eliminating the variable t from the equations by any appropriate
method.
From this brief discussion of these purely kinematic questions we
turn to the consideration of the causes of motion, taking as our
starting point the researches of Galileo on freely falling bodies.
CHAPTER I.
GENERAL THEORY OF MOTION.
SECTION I. FREELY FALLING BODIES.
THE investigation by Galileo of the motion of freely falling bodies
was the first step in the development of modern physics. It is
advantageous to start from the same point in our study of the
subject. Galileo concluded from his experiments that all bodies falling
freely in vacuo will fall at the same rate. This is one of the most
important discoveries in natural science, since it shows that all bodies,
independent of their condition in other respects, have one property
in common. No parallel to this has been found in Nature. It points
to a unity in the constitution of matter, of which we certainly do
not as yet appreciate the full significance.
Galileo's conclusions have been confirmed by the careful experi-
ments of Newton, Bessel, and others. Galileo concluded further,
that the distance s traversed by a falling body in the time t is proportional
to the square of the time, so that (a) s = ^gt2, where g is a constant.
The constant g is called the acceleration of gravity. The falling body
has a uniformly accelerated motion, since
= s = t and
Its acceleration is therefore constant. This second law of falling
bodies is not to be considered a fundamental law in the sense in
which the first is.* In the time r immediately following the time /,
the body traverses the distance <r, which is determined from the
equation s + a- = ^(t + r)2g. By the use of equation (a) we obtain
* Later researches have shown that the value of the force of gravity depends
on the distance of the falling body from the centre of the earth, and therefore
<j varies during the fall. However, the variation of // is so slight that it has
not yet been detected by direct experiment on falling bodies.
6 GENERAL THEORY OF MOTION. [CHAP. i.
(b) a- = gtr + ^r2. During the time r the velocity is variable, but if
v is the mean velocity during that time, we will have v = <r/T = gt + ^gT.
If r is infinitely small and equal to dt, we have <r = ds, and neglecting
\gdt in comparison with gt, (c) v = ds/dt='s = gt. The velocity therefore
increases proportionally to the time, and g represents the increment of
relocity in the unit of time.
The body falls through the space s in the time t, which, from (a),
is determined by (d) t = <j2s/g.
The velocity at the time t is obtained by substituting this value
of t in (c) ; making this substitution, we have (e) v = J2sg. We
reach the same result by eliminating t between equations (a) and (c).
From the laws of falling bodies we deduce the law of inertia.
In order to explain the fact that the velocity of a falling body
increases uniformly with the time, we make the assumption, that
a body retains a velocity once imparted to it unchanged in magnitude and
direction; any change of its velocity is due to external causes. This law
is called the principle of inertia.
At the time (t + T) the velocity v' is v' = gt + gr. The initial velocity
is here gt, to which, in consequence of an external cause, namely,
the force of gravity, the velocity gr is added. Under the action of
gravity the body traverses the space cr = gtT + ^gT2 in the time T,
immediately following the time t. The falling body traverses the
space gtr during the time T, with the velocity gt attained at the
end of the time t ; the additional distance \grl traversed by the
falling body is due to the action of gravity during the time T.
The principle of inertia holds not only when the increment of
velocity is in the same direction as the original velocity, but also
when it makes any angle whatever with the original velocity. This
principle justifies the application of the methods of geometrical
addition to the motions and accelerations of bodies.
The laws of falling bodies lead also to an answer to the question,
how forces are to be measured. It is evident that the gain in
velocity of a body, and its pressure on a support, that is, its weight,
are properly regarded as actions of one and the same force. The
increasing velocity of a body in its fall is an evidence of that force,
and the increment of velocity in the unit of time gives a new measure
of it. This definition shows what before Galileo's time was not
clearly understood, how the combined action of several forces may be
measured. The increments of velocity corresponding to the separate
forces are combined by the method previously described, and the re-
sulting increment gives a measure" of the combined action of the forces.
SECT. II.]
THE MOTION OF PBOJECTILES.
SECTION II. THE MOTION OF PROJECTILES.
We will apply the foregoing principles to the motion of projectiles,
which is closely connected with that of freely falling bodies. We
consider
1. Vertical projection, both downward and upward.
Galileo, in his study of the motion of projectiles, proceeded on
the assumption that a body which is given an initial motion in any
direction retains this motion, which is combined with that imparted
to it by gravity in accordance with the laws of freely falling bodies.
If, at the time t = 0, the velocity u is given to a body, directed
vertically downward, its velocity r, after the lapse of the time t,
is (a) r = u + gt, and the distance traversed is (b) s = ut + ^gt2. If
the body is given the initial velocity u, directed vertically upward,
the corresponding formulas are (c) (d) v = u- gt and s = ut- ^gt~.
2. Projection in a direction inclined to the vertical.
Let a body be projected in the direction OA, making an angle a
with the horizontal. Let OA represent the initial velocity u (Fig. 1).
The space which the body
would traverse in the time
t if gravity did not act on
it is OB = ut. The body,
however, does not reach B,
but, at the end of the time
/, is beneath B at the point
C, so that BC=\gP. Let
the x-axis Ox and the #-axis
Oy lie in the vertical plane
containing OB ; then the
coordinates of the point C
at the time t are
(e) x= OD =
By these equations the posi-
tion of the body at any
time is determined. During
FIG. 1.
the time-element dt the coordinates x and y increase by
(f) dx = ucosadt and dy = u sin a dt - gtdt.
The distance ds traversed in the time dt is determined by
ds* = dx* + df = [(u cos a)2 + (« sin a - gt)2]dt*.
8 GENERAL THEORY OF MOTION. [CHAP. i.
The velocity v is given by
(g) (h) v = s and v* = s2 = & + f = u?- Zugtsin a + gW.
The components of the velocity parallel to the axes Ox and Oy
are respectively v . dx/ds and v . dy/ds, or, by (g), are equal to x and y.
From equation (f ) we have (i) x = u cos a ; y = u sin a - gt. Hence
the horizontal velocity is constant, while the vertical velocity
diminishes uniformly; this follows because the only force acting is
directed vertically downward.
If t is eliminated from the equations (e) we obtain for the equation
of the path (k) y = x tan a - x2/4h . (1 + tan'2a), where h = %u2/g is the
distance through which the body must fall under the action of gravity
to attain the velocity u. Equation (k) shows that the path is a
parabola. The range, or the distance reckoned from 0, at which
the path cuts the z-axis, is given by (k) if we set y = 0. We have
0 = tan « -\gx(\ +tan2a)/u2. The range W, or the particular value
of x given by this equation, is W= u2 sin 2a/</ ; the maximum range
is attained when o. = \ir.
If the velocity u is given, we may determine from equation (k)
the direction in which a body must be projected in order to reach
a prescribed point. Transposing, we obtain
tan a = (2h ± V4A2 - 4hy - x2)/x.
This equation shows that there are in general two directions in
which the body may be projected with the initial velocity u so as
to reach a prescribed point. If the expression under the radical
is zero, there is only one possible direction. If the point to be
reached by the body is so situated that 4/i2 - 4hy - x2<0, tan a will
be imaginary, and the body will not reach the prescribed point.
SECTION III. EQUATIONS OF MOTION FOR A MATERIAL POINT.
In the theory of motion we use the word force to designate the
causes, known or unknown, of a change in the motion of a body.
If a body at rest is set in motion, or if a moving body comes to
rest, these changes are ascribed to the action of a force. If the
change is sudden, the force acting on the body is called an instantaneous
force, or impulse. Close examination shows, however, that finite
changes in the motion of a body are never instantaneous, but occur
only in a finite time. This time may, in many cases, be very small.
The motion of a body, which is measured by its velocity, may vary
•SECT, in.] EQUATIONS OF MOTION. 9
both in amount and in direction. The velocity of a freely falling
body varies only in amount ; the velocity of a body revolving round
a centre varies in direction, and sometimes also in amount. Experi-
ment shows that all changes in the direction, as well as in the
amount of velocity, are due to external causes, which act during a
longer or shorter time, but never instantaneously.
We may set aside all questions as to the origin of force, and
measure the amount of a force by its action. We may take as a
measure of a force either the space which a body, starting from
rest, traverses under the action of the force, or the velocity which
the force imparts to the body in a given time. There is no essential
difference between these two modes of measurement, but generally
the velocity produced, or better, the change in velocity, is used for
the purpose. We measure the amount of an impulse by the change in
velocity imparted to the body by the impulse, and the amount of a constant
force acting continually upon the body, by the change in velocity which
occurs in a second. Newton assumed further, that the force is proportional
to the quantity of that which is set in motion, that is, to the mass m of
the body. He therefore set F=f.m. b where b is the acceleration of
the body, and / is a factor dependent on the units of force, mass,
and acceleration, or on the units of mass, time, and length. If we
set/=l, then F=m.b, and we obtain the following definition for
the unit of force : The unit of force is that foi~ce which imparts the unit
of acceleration to the unit of mass, or which imparts to a body in a second
the unit of momentum (cf. XVI.). This unit of force, called a dyne,
is therefore that force which, acting for one second, imparts to a
mass of one gram the velocity of one centimetre per second. Hence
the dimensions of force are MLT~2 (cf. Introduction).
The force with which a body is attracted by the earth is called
its weight, and is measured by the product of its mass and the
acceleration which it would have if it were falling freely. If a body
is prevented from falling by a support, it exerts a pressure on the
support which is equal to its weight. Conversely, the support exerts
the same pressure on the body, in accordance with the law of action
and reaction. This pressure may be determined by the balance, by
the elasticity of a spring, etc.
Since the velocity which is caused by a force F may be resolved
into components in the directions of the three axes of a system of
rectangular coordinates, so, in the same way, the force F may be
resolved into components along the three coordinate axes. If these
components are represented by X, Y, and Z, we have F2 = X'2 + Y'2 + Z2.
10 GENERAL THEORY OF MOTION. [CHAP. i.
We may also resolve forces into their components in other ways.
These will be treated later.
If a body moves with the velocity v in the direction AB (Fig. 2)»
and if a force acts on it in the direction AC,
the path of the body may be determined by
the method used by Galileo to obtain the law
of the motion of projectiles. Consider the motion
in the time T. In that time the body will tra-
verse the distance AM = VT, in consequence of
its initial velocity ; in the same time it will
traverse the distance AN= |yr2 under the action
of the force F, if y represents the acceleration
due to the force F. If the parallelogram AMDN
^ is constructed, D will be the position of the body
at the end of the time r.
Let the direction of motion make the angles a, /8, y with the axes
OX, OY, OZ of a system of rectangular coordinates ; let the direction
of the force make the angles A, /j., v with the same axes. If the
coordinates of the point A are a*, y, and z, the x coordinate of the
point D is (b) .r + AM cos a + DM cos A = x + VT cos a + £yr2 cos A. Since
the coordinates are functions of the time, we may obtain an expression
for the ^-coordinate of D by the use of Taylor's theorem. Applying
this theorem, we obtain (c) x + XT + |arr2 + . . . . By comparing (b) and
(c) it follows that (d) (e) v cos a = x and y cos A = .r. In a similar
way we obtain v cos jK = y, y cos n — i/ ; v cos y = z, y cos v = z. Tb e
symbols x, y, z represent the velocities along the coordinate axes ;
this appeai-s also if we write v = ds/dt, and notice that cosa = dx/ds,
etc., so that v cos a = dxjds . s = x. From (e) it follows further that
ray cos \ = mx. Since my is the force F=JX2+ Y* + Z* and my cos A
represents the x component X of the force F, we have (f) X=mx;
similarly Y=my, Z = mz. These equations (f) are the equations of
motion of the particle m. If X, Y, and Z are given functions of the
coordinates, of the time, and sometimes of the velocity, equations
(f) will determine the motion of the mass m, if its position and
velocity are given at the beginning of the motion. To determine
the motion, however, it is necessary to integrate equations (f), which
can be done in only a very few cases. If the motion is known,
that is, if .r, y, and z are given as functions of the time #, these
equations may be more easily applied to find the force which causes
the motion.
We will now consider some examples.
SECT. III.]
EQUATIONS OF MOTION.
11
1. Motion in a Circle.
Let a body of mass in move with constant velocity in the circle
ABC, whose centre lies at the origin of coordinates, and whose
radius is R (Fig. 3). Let T represent the time of revolution, or
period. If w represents the angular velocity of the body, and if the
,«-axis is drawn through the point occupied by the body at the
time t = 0, we have x = Bcos (at), y = E sin (wi). It then follows
from (f) that X=mx= -raw2/i?cos ((at), Y=my = -moPRsm ((at) or
X= —m<a?x, Y= — rridPy. The force acting on the body is, therefore,
F= *JX'2 + 1' 2 = maPll. The cosines of the angles made by the direction
FIG. 3.
of the force with the x- and y-axes are respectively -x/ft&nd — y/R.
The force is therefore directed toward the centre of the circle. If
v represents the velocity of the body in the circle, we have
v = /fw = 27r£/r and F=mv*/R = 4Tr*mfi/T2. The acceleration directed
toward the centre, the so-called centripetal acceleration, is equal tov2/R = Ii(a2.
F is called the centripetal force. This result was first obtained by
Huygens.
2. The Motion of Projectiles.
Let a body be projected from the origin of coordinates with the
velocity u in a direction which makes the angle a with the horizontal
#-axis ; let the positive ?/-axis be directed upward. Then
A"=0, Y=-mg.
12
GENERAL THEORY OF MOTION.
[CHAP.
The equations of motion are mx = Q, mi/ = -ing. By integration
we have x = a + a^, y = b + b1t- \gfi, where a, av b, bl are constants.
Since the body is at the origin at the time t = 0, we have a = 0 and
6 = 0. The components of velocity at the time t are x = av y = \-gi
From the value of the velocity at the time t = Q we have
al = u cos a, l)l = u sin a.
We thus obtain again the equations given in II. (e).
3. Oscillatory Motion.
If an elastic cylindrical rod, whose weight is so small as to be
negligible, and which carries on one end a heavy sphere, is clamped
firmly by the other end, and if it is then bent, it will be urged back
toward its position of equilibrium by a force which is proportional
to its displacement from that position. If r (Fig. 4) is the distance
FIG. 4.
from the position of equilibrium 0 to the point P, which the body
occupies at the time t, the force which urges it toward 0 may be
set equal to - m&V, where k is a constant. The components of this
force are X= -mk'2x, Y= -mkzy, and the equations of motion are
SECT. III.]
EQUATIONS OF MOTION.
13
The integrals of these equations are
x = a: cos let + Jx sin kt, y = a2 cos let + b.2 sin kt,
where av bv a2, b2 are constants. If the coordinates of the point P
at the time t = 0 are x0, y0, and if the components of the velocity v
at the same time are UQ and #0, we have
z0 = ai' ^0 = %; «o = &A v0 = bjc.
With these values of the constants, the equations become
x = x0 cos kt + u0/k . sin kt, y = yQ cos kt + vjk . sin K
The components of the velocity are
x= -kx0sinkt + u0coskt, y = - ky0sin kt + v0cos kt.
If, when / = 0, the point P lies on the axis Oy, and if its initial
velocity v is parallel to the or-axis, V = UQ and v0 = 0; and
x = MO/^ . sin kt, y = y0cos kt.
For two values of t which differ by 2ir/k, the values of x and y
are the same. The motion is therefore periodic. The period is
T=2ir/k. If we divide the first equation by ujk, the second by y0,
and add the squares of the right and left sides of both equations,
we eliminate t, and obtain the equation of an ellipse as the equation
of the path of the body.
SECTION IV. THE TANGENTIAL AND NORMAL FORCES.
Let MAD (Fig. 5) be a part of the path of a body whose mass
FIG. 5.
is m, let AB be the tangent to the path at the point A, AC, the
14 GENERAL THEORY OF MOTION. [CHAP. i.
direction of the force F acting on the body. The directions of the
motion and of the force lie in the plane* of the path. We choose
this plane for the ay-plane of a system of rectangular coordinates
whose .vaxis lies in the direction AB. The normal AH, drawn to
the same side as the force AC, is taken as the positive #-axis. The
equations of motion are mx = T, my — N.
T and N are the components of the force in the direction of the
tangent and of the normal, and are called in consequence tangential
and normal forces. If the small arc AD is represented by s, if H
is the centre of curvature of the curve at the point A, and if the
radius of curvature AH is represented by It, the coordinates of D are
x = E . sin (s/R), y = R-Rcos (s/R).
Hence x = s. cos (s/R) - sin (s/R) . £/R,
i/ = s.sin (s/R) + cos (s/R) . ^,'R.
If s is very small, we may set cos (s/R) = 1 and sin (s/R) = 0. We
have then x = s, y = s2/R = v*/R,
and therefore T=ms and N=m^jR,
that is, the tangential force is proportional to the acceleration in the path.
The normal force is proportional directly to the square of the velocity, and
inversely to the radius of curvature.
SECTION V. WORK AND KINETIC ENERGY.*
If a particle, under the action of a force S, moves along a path ds,
whose direction is that of the force S, the force is said to do work
equal to Sds. If the direction of motion and the direction of the
force make the angle 6 with each other, we must use the component
of the force in the direction of motion, instead of the total force S ;
the work done is Sds cos 0. If the body moves in a given path s0s
under the action of the tangential force T, the work done by motion
through the element ds is Tds, and the work done in the path s0s
is given by the integral I Tds. If the velocity of the particle is
represented by v, v = ds/dt and T=mii — mv. Hence
(a) ('Tds = fmvvdt = \m$ - \mi-*,
where v0 represents the velocity of the body in its initial position s0.
The quantity £mt'2, or the product of one half the mass and the
* Kinetic energy is also called actual energy or vis viva.
SECT, v.] WORK AND KINETIC ENERGY. 15
square of the velocity, is called the kinetic energy of the body.
From equation (a) the gain in kinetic energy is equal to the work done
by the tangential foi'ce, or is equal to the work done by the total force,
since in the calculation of the work as it has been defined it is
necessary to consider only the component of the total force which
acts in the direction of the path. If a, (3, y are the angles which ds
makes with the coordinate axes, and X, Y, Z the components of the
force T, then the following equations hold :
T = JTcos a + Fcos ft + Zcos y,
ds cos a = dx, ds cos /3 = dy, ds cos y = dz.
The work done by the force T in the infinitely small distance ds is
Xdx + Ydy + Zdz.
Equation (a) then takes the form
(b) \(Xdx + Ydy + Zdz) = %mv2 - \mv*.
This equatipn may be applied to advantage in many cases, especially
if the force is a function of the coordinates only. If the path is
also given, we may use this equation to determine the velocity at
any point in the path.
1. Example. — Let the a^-plane of a system of rectangular co-
ordinates be horizontal, f and let the y-axis be directed vertically
upward. Let a body of mass m be situated on the «/-axis, and let
the only force acting on it be gravity. Its components are
X=Q, Y= -mg, Z=0.
We have therefore { (Xdx + Ydy + Zdz) = - mg(y - b), if the body
begins to move at the point y = b. From (b) we obtain
(c) 0 = v<*-2g(y-b).
Hence the velocity is determined by the ^-coordinate alone. This
example is discussed in II.
2. Example. — The force is a function of the distance of the particle
from a fixed point. Let the force be a repulsion and a central force,
that is, one whose direction passes always through a fixed point 0.
We take this point as the origin. The components of the force
which acts at the point (x, y, z) are
X=f(r).x/r, Y=f(r).ylr, Z=f(r).z/r.
Using these values, we have
I (Xdx 4- Ydy + Zdz) - f^(xdx + ydy + zdz).
Since r2 = x2 + f + z2, and therefore rdr — xdx + ydy + zdz, the work
16 GENERAL THEORY OF MOTION. [CHAP, u
which is done by the force during the movement of the body from
the point A to the point B is I f(r)dr, if r0 and r are respectively
the distances from the point 0 to the points A and B. Let the
velocities at the points A and B be. respectively v0 and r, then
The gain in kinetic energy depends only on r0 and r, and is con-
sequently independent of the form of the path. The general con-
dition which must be fulfilled that the work done may depend
only on the initial and final positions of the body, and be inde-
pendent of ther path traversed, will be examined in the next section.
SECTION VI. THE WORK DONE ON A BODY DURING ITS MOTION
IN A CLOSED PATH.
If a body describes a closed path A BCD (Fig. 6) under the action
of a force whose components are X, Y, Z, the work done upon it is
determined by taking the integral (a) ^(Xdx+ Ydy + Zdz) over the
whole path. If the body, moving from A with the velocity r0,
FIG. 6.
traverses the closed path in the direction indicated by the arrow,
and returns to A with the velocity v, the work done is equal to
Supposing v>v0, kinetic energy is produced during the motion, and
will increase continuously if the motion is continued. On the other
hand, supposing v<v0, kinetic energy will be produced if the body
SECT, vi.] MOTION IN- A CLOSED PATH. 17
traverses the path A BCD in the Opposite direction. Now, we know
by experience that a body, under the action of forces proceeding
from fixed points, after traversing a closed path, returns to the
starting point with the same kinetic energy which it had when it
started. It is therefore important to investigate to what conditions
the components of the force must conform in order that the integral
(a), taken over a closed path, shall be zero; that is,. the conditions
which must hold in order that a body, moving through a closed
path, shall return to its original position with the same kinetic
energy with which it started.
If the integral taken over the closed path A BCD equals zero, that
{•ABC rCDA
is, if / (Xdx + Ydy + Zdz) + J (Xdx + Ydy + Zdz) = 0,
where the letters connected with the integral signs indicate that the
first integral is to be taken over the line ABC, the second, over
CD A, we have
/ABC fADC
(Xdx + Ydy + Zdz) = J (Xdx + Ydy + Zdz).
If the work done during the passage of the body from one point
to another is independent of the path and dependent only on the
initial and final points of its path, the components X, Y, Z are
single valued and continuous functions of the position of the point.
Before we deduce the general conditions which must hold in order
that the work performed by a force shall be dependent only on the
initial and final points of the path, we will determine the work
done in the case in which the area enclosed by the path is
infinitely small. Through the point 0 (Fig. 7), whose coordinates
are x, y, z, we draw the lines Ox, Oy, Oz parallel to the coordinate
axes, whose positive directions are determined in the following way.
If the right hand is stretched out in the direction of the positive
£-axis, a line drawn from the palm will give the direction of the
positive y-axis, and the thumb that of the positive ?-axis. A
positive rotation around the re-axis is that by which the +«/-axis
is brought by a rotation through a right angle
into coincidence with the +2-axis. This rule,
by cyclic* interchange of the letters x, y, z,
gives the directions of the positive rotations
about the y- and 2-axes. If OBDC is a rect- *~
angle in the i/2-plane, and if its perimeter is
* That is, if y is replaced by x, z will be replaced
by y, and x by z.
B
18
GENERAL THEOEY OF MOTION.
[CHAP.
traversed in the direction OBDCO, the motion by which it is traversed
is said to be in the positive direction. This convention as to the
sign of the direction of rotation shall hold in all our subsequent
work. If we set OB = dy, the work done by the transfer of the
body from 0 to B equals Ydy. If the body moves from B to D,
the work done is (Z+'dZ/'dy . dy)dz. The work done in the path
DC is -(Y+dYfd2.dz)dy, and that done in the path CO is - Zdz.
FIG. 7.
Hence the total work done is (c>Zpy>-'dYj'dz)dydz. In general, the
work done by a force during the movement of a body around a
surface element dS^ which is parallel to the y^-plane, is
(b) F.dSx
In the same way we obtain
G . dS, = (dX/'dz -
SECT. VI.]
MOTION IN A CLOSED PATH.
19
F, G, and H are the quantities of work done during the movement
of the body around a unit area at the point 0, when perpendicular
to the x-, y-, and s-axes respectively.
If OABC (Fig. 8) is an infinitely small tetrahedron, whose three
edges OA, OB, OC are parallel to the coordinate axes, and if the
body moves on the boundary of the surface ABC in the direction
given by the order of the letters, the work done is equal to that
which is done by moving the body in succession about OAB, OBC,
and OCA. By this set of motions, the distances AB, EC, CA will
each be traversed once in the positive direction, while the distances
FIG. 8.
OA, OB, OC will each be traversed twice and in opposite directions,
so that the work done in them is zero. The work done during the
movement of the body about the surface ABC = ds is, therefore,
(c) J .dS=F .dS.l + G.dS .m + H.dS.n, where /, m, n are the
cosines of the angles which the normal to the surface dS drawn
outward from the tetrahedron makes with the coordinate axes.
Hence the work J done during the movement of the body around
unit area is (d) J=Fl + Gm + Hn, where Z, m, and n determine the
position of the unit area.
20 GENERAL THEORY OF MOTION. [CHAP. i.
If the work done during the movement of a body about an infinitely
small surface is zero, we must have .7=0 for all positions of the
surface, or F=G = H=Q,
( -dZfdy - 'dY/'dz = 0, ax/a? - VZ/Vx = 0,
\ 9F/aB-aX/<ty = 0.
When the equations of condition (e) are satisfied, the expression under
the integral sign in (a) is the complete differential of a function V
of x, y, z, whence X = rdVj'dx, Y='dF/'dy, Z='dF/'dz. The equations
of condition (e) are satisfied by this assumption. The function V is
the potential of the acting forces ; we here obtain for the first time
the mathematical definition of this function, whose differential co-
efficients with respect to x, y, z are the components of force X, Y, Z.
If V is a value of the potential of the acting forces, V is also a
value, if V=V+C, where C is a constant; since
.Y=3F/3z = 3F7ar, etc.
The value of the potential therefore involves an unknown or arbitrary
constant. We will return to the consideration of this point in the
next section.
If the equations of condition (e) are everywhere satisfied, the
work done during the movement of the body about a surface is also
zero when the surface is finite.
// >^ The surface may be divided into
'- — -t -i L*~*il — /\ surface-elements, as shown in Fig. 9.
/ / //, I I \ If the body moves about these
~7 7 7 7 7 7| elements one after another in the
v — 4- •+• 4- + -r -f — 7 same direction, the total work done
X^ / / / / I/ will equal zero. It is here assumed
"x^,/ 7 1^^^ ^Dat ^e f°rces X, F, Z are con-
•^~^^_ " J tinuous and single valued func-
tions of the coordinates. Every
line-element thus introduced will be traversed twice in opposite
directions, with the exception of those which form the boundary of
the finite surface.
Those forces or systems of forces, which are such that the work
done by them is independent of the path in which the body is trans-
ferred from its initial to its final position, are called conservative forces.
The most important examples of such forces are those which act
from a fixed point, and have values which depend only on their
distance from it. If the force acting at the point P depends only on
the distance of that point from the origin of coordinates 0, that is,
SECT, vi.] MOTION IN A CLOSED PATH. 21
if it equals /(r), then X=f(r) . x/r, since x/r is the cosine of the angle
which the line OP makes with the #-axis. We have similarly
X=f(r).x/r, r=/(r).y/r, Z=f(r).z/r.
If we set f(r)/r = R, then X=Ex, Y=Ry, Z=Rz. We have then
-dZfiy = dR/dr . yz/r, 3F/3* = dR/dr . yz/r.
The equation of condition ~dZ/c)y -?>Y/'d2 = 0 is therefore satisfied.
The same is true of the other equations of condition (e).
The work done during the movement of the body about a surface
is given by the integral \(Xdx + Ydy + Zdz). This work is also done
if the body moves in succession about all the surface-elements into
which the finite surface is divided (Fig. 9). In this process the
motion must be uniformly carried out in the same sense. From (c)
this work is equal to \(Fl + Gm + Hn)dS. If we substitute the ex-
pressions for F, Gf, H formerly obtained, we have, by the use of
(a) and (b),
{\(X . dx/ds + Y. dyjds + Z . dz/ds)ds
= I j [(dZfdy - -dY/3z)l + (dX/Vz - -dZ/3x)m
+ (dY/3x-'dXI'dy)n]dS,
where s is the perimeter of the surface S, and I, m, n are the direction
cosines of the normal to each surface-element. Equation (f) shows
that the line integral along a closed curve may be replaced by a
surface- integral over a surface bounded by this curve. The only
conditions which the surface S must fulfil are that it shall be bounded
by the curve and have no singular points. The theorem contained
in (f) was discovered by Stokes.
SECTION VII. THE POTENTIAL.
The only applications of the potential that we will discuss are
those like the foregoing, in which the work done during the motion
is completely determined by the initial and final points of the path.
That this may be the case, we must have
We exclude from the discussion all cases in which these equations
do not hold.
Let the components of the force in the field be X, Y, Z, Let
there be a unit of mass at the point 0 (Fig. 10), whose rectangular
22
GENERAL THEORY OF MOTION.
[CHAP. i.
coordinates are a, b, c, and let it move from 0 to P along the path s.
The work V done by the force during this motion is
(a)
V= f'(Xdx + Ydy + Zdz) = VP -
it being assumed that A', Y, Z are the partial derivatives of a single
function V, which itself is a function only of x, y, z. The work
required to transfer the unit of mass from any point 0 to P is
equal to the difference of the potentials VP and V0 at those points,
or is equal to the difference of potential. Such differences of potential
are all that can be directly measured. The value of the potential itself
FIG. 10.
involves an unknown constant, and therefore cannot be completely
determined. If we assume that the potential is zero at the point 0,
then VP is the potential at P. Hence the potential at any point is the
work required to transfer the unit of mass to that point from a point
where the potential is zero.
The potential V is a function of the coordinates. The equation
(b) y(x, y, z) = C, when C is constant, represents a surface which is
the locus of points, such that the amount of work required to
SECT. VII.]
THE POTENTIAL.
23
transfer the unit of mass from a point where the potential is zero
to any one of them is the same. If different values of C are taken,
we obtain a system of surfaces, which are called level or equipotential
surfaces. Let PF and QQ' (Fig. 11) be two infinitely near surfaces
of this system ; let the potential on PP be V, and on QQ' be V+dV.
Let ds be the element of an arbitrary curve crossing these surfaces,
which is cut off by them. If the force acting in the direction of
ds is T, the quantity of work Tds will be done by the transfer
of the unit of mass from P to
Q ; this work is also equal to
V^-V^dV. We have, there-
fore, (c) T .ds = dV or T=dVjds.
Hence the force in any direction
at a point is determined from the
potential ; the relation between
force and potential being given by
equation (c). Since the direction
of the element ds is arbitrary, we
may substitute for ds the elements
dx, dy, dz, and obtain for the com-
ponents of the force X='dV/'d.r,
Y= dF/c)y, Z^'dF/^z. From equa-
tion (c) the force is inversely pro-
portional to the element ds drawn
between the two equipotential sur- "
faces V and V '+ dV. If the
direction of ds is that of the normal to the surface PP', the force
has its greatest value. If a series of lines is drawn which cut the
equipotential surfaces orthogonally, their directions are the directions
of the force at the points of intersection. Such lines are conse-
quently called lines of force. The tangent to the line of force at a
point gives the direction of the force at that point.
If Pl and P2 are two infinitely near points in an equipotential
surface, no work need be done to transfer a body from Pl to P2,
for VPl - VP^ = §\ the force acting on the body is perpendicular to
the direction of motion.
1. Example. — Gravity. — If at a place near the earth's surface we
set up a system of rectangular coordinates, so that the .r^-plane is
horizontal, and the positive ?y-axis directed vertically upward, then
^=0, Y= -mg, Z=0. Hence we have V= -mgy, that is, the
equipotential surfaces are horizontal planes.
FIG. 11.
24
GENERAL THEOEY OF MOTION.
[CHAP. i.
2. Example. — In the case discussed in V., Ex. 2, the work
F=fjf(r)dr
is needed to move the body from its position at the distance r0
from a fixed point to another position at the distance r. Hence
we have F"= F(r) - jP(r0), and the equipotential surfaces are spheres
whose centres are at the centre of attraction 0.
SECTION VIII. CONSTRAINED MOTION.
Galileo investigated not only freely falling bodies and the motion
of projectiles, but also motion on an inclined plane and the motion
of a pendulum, and thus made the first step in the investigation of
constrained motion.
If a body is compelled by any cause to move in a given path,
which is not that which it would follow if free to yield to the
action of the forces applied to it, its motion is said to be constrained.
1. Example.— The Inclined Plane.— Let the body D (Fig. 12), acted
on by gravity, slide down an inclined plane AB, which makes the
FIG. 12.
angle a with the horizontal plane BC. We neglect any resistance
which may arise from friction. The force or reaction exerted by
the inclined plane acts perpendicularly to the plane AB, and does
not affect the motion of the body. The expression sought may be
best obtained by using the relation between kinetic energy and
work. If in represents the mass of the body, v the velocity acquired
at B, g the acceleration of gravity, and I the length AB of the
inclined plane, we have fymv2 = mg sin a . /, assuming that the motion
begins at A, so that the initial velocity is zero. If a represents the
height AC of the inclined plane, we have I sin a = a, and the work
-SECT VIII.]
CONSTEAINED MOTION.
25
<lone equals mga. Hence the velocity of the body at the foot of
the inclined plane B is v = .j2ga, and is the same as that which it
would have at C, if it were to fall freely through the distance AC.
If a body moves on the curve AB (Fig. 13) under the action of gravity,
we determine the velocity at B in a similar way, from (the initial
velocity v0 at A and the distance of the fall AC. That is, we have
FIG. 13.
%mv2 - %mv0* = mga, and therefore (a) #2 = v02 + 2ga. The time t required
for the movement of the body from A to B is
<"> -jft
where ds is an element of the path AB.
2. Example. — The Pendulum. — If we suspend a body A (Fig: 14)
at the end of a weightless rod which can swing freely about the
point 0, it is compelled to move on the surface of a sphere whose
radius is equal to the length I of the rod. We will treat only the
simple case in which the departure of the pendulum from its position
of equilibrium is small. If, at the time t = 0, the body starts from
rest at A, it will move in the arc A BCD through the point C lying
perpendicularly under 0. If we set OA = l, ^AOC = a, i_£OC=0,
and if A A' and BB' are drawn perpendicular to OC, then the velocity
which the body gains in moving from A to B equals that which would
be gained if it were to fall freely from A' to B'. The distance from
A to B' is A'B' = l(cos 0-cosa), and therefore the velocity at B is
v = *j2gl (cos 0 - cos a).
26
GENERAL THEORY OF MOTION.
[CHAP. i.
For v = 0, or for 6= ±a, the pendulum bob will be at rest, and is
then at A or D, if i.DOC = LAOC. The time t taken by the
body to move from A to B is found by substituting this value of v
in (b). We thus obtain
(c) t= - Ieid6/J2gl(cos 6 - cos a),
•'a
0
This expression is easily integrated if a, and therefore 0, are so
small that we may set cos0=l-£02 and cosa = l-£a2. The ex-
pansion of the cosine in a series is of the form
and if x is very small we may neglect terms of higher orders than
the second. Making this restriction, we have
and by integration (d) 6 = acos(t>JgJl). If t^/g/l = ^Tr, we have 0 = 0;
the body moving from A reaches the lowest point of its path in
the time t = £ir . *Jl/g. The time T required for the movement of
the body from A to D is twice this, or (e) T=irjl/g. T is called the
period of oscillation.* The period of oscillation is directly proportional
* In the case of the pendulum here treated, which swings in a plane, it must
be clearly understood that by the period of oscillation T only one advancing
or returning beat is meant. In other periodic motions, the period of oscilla-
tion is the time between two instants, at which the motion of the body is
precisely similar, that is, at which the body has the same velocity and direction
of motion ; or, it is the time required for both the advancing and returning
l>eats.
SECT. VIIL] CONSTEAINED MOTION. 27
to the square root of the length of the pendulum, and' is inversely proportional
to the square root of the acceleration of gravity.
The equation (e) holds only for very small arcs. In the case of
finite values of a, we use the formula
(f)
It is only for very small arcs that the oscillations of the pendulum
are isochronous, that is, independent of the size of the arcs. If the
arc is not infinitesimal, the period will increase rather rapidly with
the length of the arc.
The pendulum may also be studied in the following way. Let
an oscillating body of mass m be at the point J5, and be acted on
by the force mg. We may represent this force by the line OE
(Fig. 14). Draw EF perpendicular to OB; then OF and FE are
components of the force OE. The magnitude of the tangential force
is mgsind. If we set BC = s, and reckon the tangential force posi-
tive, when it tends to increase st we have P= -mgsin(s/l), and if
we assume s to be very small, P= -mgs/L Hence the equation -of
motion is (g) ms = P or »= -gs/l. By integration we obtain, by
a suitable choice of constants, (h) s = a cos (t\/g/l). This equation
corresponds to (d).
If a body is compelled to move on a given surface, the deter-
mination of its motion is in general very difficult. We will not
enter into the discussion of the general case, but will consider only
the motion of an infinitely small body on a spherical surface, when
the body during the motion always remains near the lowest point
C of this surface, and when gravity is the only force acting on it.
We may then assume that the component of gravity which moves
the body is directed toward the point C, and that it is equal to
mgsjl, when I represents the radius of the sphere. This assumption
gives the motion treated in III., Ex. 3. The path is an ellipse and
the time of oscillation is T=2ir*Jl/g. The time of oscillation is
therefore independent of the form and dimensions of the path.
SECTION IX. KEPLER'S LAWS.
In our deduction of the principal theorems of the general theory
of motion, we proceeded from Galileo's laws of falling bodies. We
turn now to that force of which gravity is a special example, and
28
GENERAL THEORY OF MOTION.
[CHAP. i.
from whose properties the laws of planetary motion may be deduced.
Starting with the hypothesis of Copernicus, that the sun is stationary
and that the earth rotates on its own axis and also revolves round
the sun, Kepler announced the following laws :
1. A radius vector drawn from the sun to a planet describes equal
sectors in equal times.
2. The orbits of the planets are ellipses with the sun at one of the foci.
3. The squares of the, periodic, times of two planets are proportional to
the cubes of the semi-major axes of their orbits.
These laws may be expressed" analytically in the following way.
Let S be the centre of the sun (Fig. 15) and APQ a part of the
orbit of a planet. Let the
planet be at A at the time
= 0, and at P at the time t.
In the next time element dt
the planet moves from P to
Q, and its radius vector drawn
from the sun describes the
sector PSQ. Let -ASP = Q,
LPSQ = dQ, and SP = r. The
surface PSQ is equal to \r-dQ.
Since by Kepler's first law
the surface described by the
radius vector increases pro-
portionally to the time, we
have r2dQ = Mt, where k is
constant, or, writing the equa-
tion in another form,
(a) 7-2.0 = &.
Kepler's first law is a special
case of a general law which is
FIG. 15.
If the force which acts upon a
the surface described by the
called the law of areas. This law is
moving body pi'oceeds from a fixt
radius vector drawn from that point to the body increases at a constant
rate. Hence Kepler's first law holds for all central forces.
From (a) it appears that the angular velocity 0 is inversely
proportional to the square of the distance of the planet from the
sun.
We represent the velocity of the planet at P by v, and the
perpendicular from S upon the tangent to the orbit at the point P
by SN=p.
SECT, ix.] KEPLER'S LAWS. 29
If we set PQ = ds, the area of the sector PSQ is equal ,to
^pds = fypvdt. But it is also equal to ^r2dQ = \k . dt. Hence we have
pvdt = Mt, or pv = Tc, that is, the velocities of the planet at different
points in its orbit are inversely proportional to the distances of the
tangents at those points from the sun, the centre of attraction.
Let BPC be the elliptical
orbit of the planet (Fig. 16),
with the sun situated at the
focus S. Let the major axis
be BC=2a, and let SA be a
fixed radius vector which makes
the angle a with the major axis.
We set SP = r, ±_ASP = Q. If
n i ci if- v. FIG. 16.
F and S are the foci; we have
PF+PS=2a, PF=2a-r,
and hence (2a-?-)2--=4a2e2 + r2 + 4aercos(6-a),
if e is the eccentricity, and if, therefore, FS=2ae. From this equation
we obtain for the equation of the path in polar coordinates
(b) 1/r = [1 + e cos (6 - o)]/a(l - e2).
From equation (a) we have J^r2 . dO = |& . T, if the integration is
taken over the whole orbit and, if T is the periodic time. The
integral is equal to the area of the ellipse, or to a . b . TT, if b represents
the minor axis. Hence we have %Trab = k.T. If we notice that
a2 = b2 + a?e2, we obtain (c) 27r«2V 1 - e2 = JcT, and squaring,
By Kepler's third law T2/a3 is constant for all the planets. We
must therefore have (d) /* = k2/a(l - e2) = 47r2a3/T2, a constant.
The velocity v may be determined in the following way. Let
S (Fig. 16) be the origin of a system of rectangular coordinates,
and let SA be the a-axis. We have x = rcosQ and y = rsin6, and
v2 = z? 4- if2. From the equations
(e) x = r cos 0 - r sin 0 . 0 ; y = r sin 0 + r cos 0 . 9,
we obtain (f) vz = r2 + r262.
If we substitute for r6 its value given by equations (a) and (b),
and for r the value got by differentiating equation (b), we have
v2 = (l + 2e cos (0 - a) + e2) . &2/a2(l - e2)2.
Noticing that 1 +2ecos (Q - a) + e2 = 2(l + e cos (6 - a)J - (1 -e2), we
obtain, by the help of equation (b), «2 = (2/r- I/a) . &2/a(l -e2), or,
introducing the quantity p. defined by (d), (g) v2 = 2p,/r — p/a.
30
GENERAL THEORY OF MOTION.
[CHAP. i.
SECTION X. UNIVERSAL ATTRACTION.
We owe to Newton the determination of the law of the force which
must act on a planet in order that its motions may conform to
Kepler's laws. To determine this force, we use equation (g) IX.
Let the centre of the sun be the origin of a system of rectangular
coordinates, and let the planet be situated at the point (z, y)
Represent the components of the unknown force by X and Y. From
the law of kinetic energy (V.) we have |#2 - £t'02 = { Xdx + Ydy. If v0
is the velocity at the distance r0, we obtain, by the help of equation
(g) IX., fJL/r-fjL/rQ = \Xdx + Ydy. If Xdx+Ydy is a complete differ-
ential d<f>, we will have
X = 'tyfdx**'d(plrydx and
or X = - /x/V2 - 3r/ac = - /iz/r8, Y
The force R with which the sun acts on the planet is R= - /*/r2,
that is, is a force which is inversely proportional to the square of the
distance of the planet from the sun. It is evident from equation (d) IX.
that the quantity p. has the same value for all the planets.
FIG. 17.
We may also obtain these results from the general equations
x = X and y=Y. The unknown components of force X and Y may
be represented by the lines PA and PB (Fig. 17), and resolved
into a component R in the direction SP = r and a component T
perpendicular to SP. Setting -PSX = Qt we have
T= - XsinQ+ Fcos6.
SECT, x.] UNIVERSAL ATTRACTION. 31
By the help of equation (e) IX., this becomes
(d) 72 = r-r62 and T=2fQ + re=l/r . d(r*6)ldt.
But since r26 = constant by Kepler's first law, we have T=Q. The
attractive force is therefore directed toward the sun. Using equations
(a) and (b) IX., we obtain (e) R= -£2/a(l -e2>'2 = - f*,'r2.
We apply this result to the motion of the moon. By reference
to (d) IX., where the value of p. is given, we find that
R= -4irV/ZV.
The orbit of the moon is approximately a circle with a radius
60,27 times as great as that of the earth. Setting
r=a=4. 109.60,27/27rcm,
we have the acceleration 7 of the moon toward the earth,
7 = 47r2a/r2 = 877 . 60,27 . 109/2 360 6002 cm,
since the period of the moon's rotation is 27,322 days or 2 360 600
seconds. Hence we have 7 = 0,27183 cm. If the centre of the
moon were situated at the distance of the radius of the earth from
the earth's centre, it would have an acceleration equal to
0,27183.60,272cm = 987cm,
assuming that the force is inversely proportional to the square of
the distance. This value accords so well with that of the accelera-
tion at the surface of the earth, that we are justified in assuming
that the motion of a falling body is an action of the same force as
that which keeps the moon and the planets in their orbits. The
final proof of the validity of Newton's law of mass attraction is
obtained from the complete agreement of the theoretical conclusions
drawn from it with the results of observations on the heavenly
bodies.
SECTION XI. UNIVERSAL ATTRACTION (continued).
We will now use a method precisely the opposite of our former
one. We will assume the law of attraction known, and determine
the path of a planet whose position and velocity at the time t = 0 are
given. Let S be the centre of the sun (Fig. 18), let the attracted
body be situated at A at the time £ = 0, and let AC represent the
velocity r0, whose direction makes the angle CAD = </> Avith SA = r0
produced. If the acceleration which the sun imparts to the planet
is set equal to p./r2, then, using a system of polar coordinates whose
32 GENERAL THEORY OF MOTION. [CHAP. i_
origin is at S, and writing the force with the minus sign because
it is directed toward the sun, we obtain
(a) (b) f - r62 = - fM/r2 and 1/r . d(r*Q)/dt = 0.
It follows from (b) that (c) r29 = &, where k is a constant. This for-
mula was obtained from Kepler's first law, (a) IX. Since ?-6 is the
component of velocity perpendicular to the direction of r, we obtain
for t = Q, (d) k/r0 = vQsm(f>. By the help of equation (c), (a) take&
the form r - F/r3 = - /*/r2. If this equation is multiplied by 2i-dt,
we have d(f3)+d(k2/r2) = 2d(p/r), and by integration
r2 + F/r2 = 2/i/r + Const.
FIG. 18.
In the initial point A, we have v = v() and r = v0cos<f>, therefore for
/ = 0, we obtain ?'02cos2<£ + F/r02 = 2fi/r0 + Const., from which by the
help of (d) it follows that ?;02 = 2/*/r0 + Const. Hence we have
(e) r2 = V-2f0o + 2/Vr-F/r>.
Since the velocity v, from (f) IX., may be expressed generally by
2j we obtain by the use of (c) and (e)
This equation agrees with (g) IX.
The same result may be derived from the theorem connectin
kinetic energy and work. From (e) we have
(g) r=±x/V-2ft
where the upper sign is to be taken, if r and t increase or diminish
together. It follows from equation (c) that Q = k/r2. Since r and 0
depend only on t, we obtain from (c) and (g)
ft . d(l/r)/dB = + VV-2/
SECT, xi.] UNIVERSAL ATTRACTION. 33
This is the differential equation of the orbit. By adding and sub-
tracting fj.2/k2 under the radical sign, and by noticing that p/k is a
constant, and that therefore its differential is zero, this equation
may be written
(h) dQ = d(k/r -
If we set M2 = v0z - 2/x/r0 + p?/k2, we get by integration
0 = arc cos (k/ur - p/uk) + a,
where a is a constant.
Hence the equation of the path is
(i) l/r=(l+Jb//tco8(e-o))/<^//*), .
In this equation u may always be considered positive, since a is
arbitrary.
The polar equation of a conic is (k) l/r= (l +e cos (0 - a))/a(l -e2),
which represents an ellipse, a parabola, or a branch of an hyperbola
respectively, according as e<l, e = l or e>l. If e = 0 we have the
equation of the circle. From the equation e = ku/p, by introducing
the value of «, we obtain (1) 1 -e2 = (2/»/r0-i>02) . &2//*2. If a body
approaches the sun from infinity to the distance r0, its velocity will
be vv determining by the following equation
Therefore we have (m) e2 = 1 - (v^ - v02) . F//*2. Hence the path is
either an ellipse, a parabola, or an hyperbola, according as
«o<*'i» vo = vi or vo>vi'>
that is, the path of the body is an ellipse, parabola, or hyperbola,
according as the vis viva imparted to the planet at the first instant
is too small to send it to infinity against the attraction of the sun,
or exactly sufficient, or more than is sufficient, to accomplish that
result.
By comparison of the formulas (i) and (k), we obtain
(n) o(l -#) = *»//*.
This corresponds to (d) IX. From (m) and (n) it follows, moreover,
that (o) yu. = ± (flj2 - v02) . a. The upper sign is used when v1>v0,
the lower when v1<v0.
In the first case, if the value for #x2 is substituted in (o), we
have #02 = 2/i/7-0 - p-ja. In conjunction with (f) this equation becomes
v2 = 2/x/r - p/a, which corresponds to (g) IX.
34
GENEEAL THEOEY OF MOTION.
[CHAP. i.
SECTION XII. THE POTENTIAL OF A SYSTEM OF MASSES.
In the previous discussion Newton's law of gravitation was derived
from Kepler's laws by the assumption that the attraction proceeds
from the centre of the sun, or, what is the same thing, that the
whole mass of the sun is concentrated at its centre. A similar
assumption was made in the case of the planets. These assumptions
might be made without further demonstration if the radius of the
sun were infinitely small in comparison with the orbits of the planets ;
since this is not the case, it is necessary to investigate with what
force a mass distributed throughout a given space acts on a body.
This problem in the simplest cases was solved by Newton. His
researches, and those of other distinguished mathematicians, have
led to results of the greatest importance, both in physics and
mathematics. The method by which such problems are treated is
due to Laplace, and the theory was developed by Poisson, Green,
Gauss, and others.
Let the masses mv m2, mB be situated at the points A, B, C (Fig. 19),
and let a unit of mass be concentrated at the point whose coordinates
are x, y, z. The force with which the unit of mass is attracted by
ml is -fmjr^ where 1\ is the
distance AP, and / a constant
dependent on the units of mass,
force, and length. Call the co-
ordinates of A, £v rjv fr The
components Xv Yv Zl of the force
by which P is attracted to A are
evidently X, = -fmjr,2. (x - ^)/rv
etc. In the same way we calcu-
late the components of the forces
which originate at the other
points, B, C, etc. If the sum of all the X components is repre-
sented by X, we obtain
(a) X= -/K(z - fJW + m.(x - &)lr* +...}.
We set (b) F= mj^ + w2/r2 + m3/r3 + . . . .
Now V = (x - &)» + (y-^Y +(*- &)2,
and therefore r^dr^fdx =(x- gj, etc.
Hence we have
and (c)
X=f.
SECT, xii.] POTENTIAL OF MASSES. 35
In a similar way we derive the equations
(d) (e) F=/.3r/3y and Z =f . VP'I'dz.
The quantity V defined by equation (b) is (VII.) the potential at
the point P of the given system of masses. If the potential is given,
the equations (c) (d) and (e) determine the forces acting in the
directions of the coordinate axes. Since the position of the system
of coordinates is arbitrary, the force acting in any direction may be
derived from V. This has been already shown in VII. The force
acting in the direction s is therefore
d F/ds = 3 F/3z . dx/ds + 3F/fy . dy/ds + 9F/3* . dz/ds.
The work A performed by the force in moving a unit of mass
along an arbitrarily chosen path is given by
A = l\Xdx + Ydy + Zdz),
where o and s are respectively the initial and final points of the
path. If the values given in the formulas (c) (d) (e) are substituted
for X, Y, and Z, and the element of the path whose projections on
the coordinate axes are dx, dy, dz is designated by ds, then
A =ff(d Vj'dx . dx/ds + 3 FJ-dy . dy/ds+ 3 F/3z . dz{ds)ds =fj'd V,
and hence we have A=f(Tt-F0). If the body traverses a closed
path, the work done by the forces equals zero (cf. VI., VIL).
Therefore, if we let a body traverse a closed path under the action
of gravity, the work which gravity performs in moving the body
forward is equal in absolute value to the work which must be
performed against gravity in order to bring the body back to the
starting point. There is no surplus work performed. Hence it is
evidently impossible to produce a perpetuum mobile, that is, an
arrangement which continuously creates work out of nothing.
We have assumed that the masses considered are concentrated at
points; this, however, does not occur in nature. Matter is more
or less continuously distributed throughout space or on surfaces. If
it is uniformly distributed in space, the mass p contained in the
unit of volume is called the density. If it is not uniformly
distributed, let a sphere of infinitely small radius be constructed
about the point P ; the ratio of the mass contained in the sphere
to its volume is the volume density p at the point P. If the
mass is distributed over a surface, the surface density a- at the
point P is defined by the ratio of the mass contained within a
circle of infinitely small radius drawn about the point P as centre
to the area of the circle.
36 GENERAL THEORY OF MOTION. [CHAP. i.
If the mass contained in the unit of volume is p, the element
of volume dw will contain the mass pe?w. The potential of a mass
which is distributed in space is therefore, from equation (b),
(g) F=JJJ^o,/r.
This integral is extended over the whole volume occupied by the
mass, r is the distance between dt» and the point for which the
potential is to be determined.
It is sometimes necessary to consider the mass as distributed in
an infinitely thin sheet over a surface. If the mass on the unit of
surface is a-, the surface-element dS will contain the mass crdS.
The potential takes the form (h) F=\\a-dS/r.
The potential cannot be determined without some further informa-
tion ; in the next paragraph we will discuss some of the simplest cases.
SECTION XIII. EXAMPLES. CALCULATION OF POTENTIALS.
The sun and planets are approximately spherical. On the sup-
position that they are spheres, their potential can be easily calculated,
if the density p is given, and if we assume that it is a function of
the radius, and therefore has the same value for all parts of the
concentric spherical layers which compose the sphere.
FIG. 20.
1. The Potential of an Infinitely Thin Spherical Shell of Constant Surface
Density <r.
Let ABD (Fig. 20) be a sphere, whose centre is the point C and
whose radius is R. The potential at the point 0 is to be deter-
mined. If we set OC = r, L.OCB = <j> and OB = u, we have
?= r-2-n-R sin <£ . Rd<i> . <r/u.
SECT, xin.] CALCULATION OF POTENTIALS. 37
Since u2 = r'2 + R2- 2Rr cos $ and udu = Rr sin <f>d(j>, the integral takes
the form V= faR/r . udu/u . <r = 2TrRa-/r. \du. If 0 lies outside the
sphere, we have \du = (r + R) - (r - R) = 2 R ; while if it lies within
the sphere, we have \du = (R + r)-(R- r) = 2r. If we designate the
potential outside the sphere by Va and that within the sphere by
V-t, we have (a) Vi = IvRa- • Va = ^irR^a-jr. The potential is therefore
constant inside the spherical shell; and at points outside the spherical shell
it is inversely proportional to the distance from its centre. Hence the
potential for the whole region is given by the two different ex-
pressions Va and V^ It is not discontinuous at the surface, since
for r = R we have Va=Vt = iirRo-. On the other hand, its differential
coefficient is discontinuous at the surface. For we have
dVJdr = -±irR?<r/r2 and
and therefore
(b) [dVJdr]r=Jt = -4™ and
Hence an infinitely thin spherical shell exerts no force at a point
lying within it. The sphere acts only on' points outside of it, as if
its whole mass were concentrated at its centre. Hence a solid
sphere made up of homogeneous concentric spherical shells acts on
outside points in a similar manner. If the attracted point is situated
within the mass of a spherical shell, it will be attracted to the centre
by the portion of the mass which lies within a sphere described about
the centre with the distance of the point from the centre as radius.
The portion lying outside this surface exerts no action.
2. The Potential of a Solid 'Sphere.
We will now calculate the potential of a solid sphere of constant
density p. We have for points outside the sphere
and for points inside the sphere
V, = f 47T.R2 . dR . P/r + f iirR . dR . P = ~r*P + 2irp(R2 - r2),
• 0 ~r O
or (d) F—Zirp^2-^). In this case also, the potentials within
and without the sphere are represented by two different expressions
Vt and Va. Both values, however, coincide at the surface, since for
r = R we have for the potential Vi=Va = ^irR^p.
The function Vt which represents the potential of a mass dis-
tributed through space, is everywhere continuous. Its differential
coefficients with respect to r are
(e) dVt/dr= - frrp, dVJdr= -
38 GENERAL THEORY OF MOTION. [CHAP. i.
and these values at the surface, where r = R, are also equal, that is,
Hence the first derivatives of the potential of a mass distributed in
space are nowhere discontinuous, but are continuous throughout all
space. On the other hand, its second derivatives vary continuously
in the interior and the exterior regions, but on passage through the
spherical surface a discontinuity occurs, That is, d2F'/dr2 at the
surface has two values, since
[cVFJdr*]^^ -$irp and [dzFa/dr2]r=J! = +fay>.
From equation (e) the force outside the sphere is inversely pro-
portional to the square of the distance of the unit of mass from
the centre of the sphere. We thus justify the assumption that the
planets and the sun may be treated as points in which their re-
spective masses are concentrated. In the interior of the sphere the
force is proportional to the distance of the attracted point from the
centre. If we transform equation (e) to dFt/dr = -^m^p.l/r2, we
see that the force proceeds from that portion of the sphere whose
distance from the centre is less than r. This only holds on the
assumption made about p. The earth's density very probably increases
toward the centre; hence the force of gravity will not have its
greatest value at the surface, but at some point beneath it. This
corresponds with the results of experiments on the time of vibration
of a pendulum in a deep mine.
3. The Potential of a Circular Plate.
Let AE (Fig. 21) be a circular plate of surface density a-; the
centre of the plate is 0 and the axis OP.
The point P, for which the potential is
to be determined, lies on the axis at the
distance x from the plate. The potential
V is then
f*
F= I 2irr) . drj . o-/<u,
where R is the radius of the plate and
7; and u are the distances of a point on
the plate from 0 and P respectively.
We have v? = rf + x2, therefore ndu»i)d^t
and hence
IG' V= fadu .<r=27ra-(p- x),
if p is the distance of the point P from the edge of the plate. If
SECT. XIII.]
CALCULATION OF POTENTIALS.
39
(f)
the x drawn from one face of the plate is considered positive, the
potential for negative values of x is Pr=2Trcr(p + x). Hence
Jfora;>0, Vl = 2ir(r(p-x)
If the radius of the plate is infinitely great in comparison with x,
we may set Prl = C-2Tra-x and F2—C+2iro-x, where C is an infinitely
great constant, since p is infinitely great and o- remains finite. We
have for x>Q, dVljdx = -2ircr,
and for x<0, d F2/dx = + 27nr.
By passage through the surface, dF/dx, that is, the force, changes dis-
continuously by 47rcr.
4. The Potential of an infinitely long straight line.
Suppose each unit of length of the line AB (Fig. 22) to have the
. Let G be a point at the
distance a from AB, and CD the "B
perpendicular let fall from C upon
AB. The potential V, at the point
(7, is
V= 2^ log (z'/a
Since z' is infinitely great in com-
parison with a, we may neglect 1
under the radical and write
(k) V= 2p log ( 2z'/a) = C.-p log a2, .
where C is an infinitely great con-
stant, if z' is infinitely great.
Further, we obtain
dF/da= -2/*/a,
that is, the force is inversely pro- FlG' 22>
portioned to the distance of the point from the straight line.
5. The Potential of a Circular
Represent the surface density of a circular cylinder by a-. Through
the point P, for which the potential is to be determined, pass a plane
40 GENERAL THEOEY OF MOTION. [CHAP. i.
perpendicular to the axis of the cylinder (Fig. 23). Let R be the
radius of the cross section of the cylinder and r the distance of the
point P from its centre.
FIG. 23.
We then have from (k)
v= c-^JJme.a-. log a2.
We now find the value of the integral
in which a = *Jf2 - 2rE cos 6 + If2. First consider the case in which
r>R. The integral may then be written, if we set a = E/r,
rir _
^ = 2 Jo Rd6(\og r + log Vl - 2a cos 0 + a2).
Since cos 6 = \(&9 + «-**), we have 1 - 2a cos 0 + a2 = (1 - a.ei6}( 1 - ae~ie),
and jf
Now developing the terms in this integral in series, and carrying
out the integration, we find that the integral is equal to zero,
and hence that A = Z-n-R log r. Thus the mean value of log a for
all points of the circumference of the circle is equal to log r or to
the logarithm of the mean distance from P to the circumference of the
circle.
If now V< R, that is, if P lies between 0 and the circumference,
and if we set a = rjR, we have
= 2 Rd6(\0g R + log N/l-2aCOS0+a2),
and hence A = 2-rrE log R. In this case also the mean value of log a
SECT, xiii.] CALCULATION OF POTENTIALS. 41
is equal to the logarithm of the mean distance from P to the circum-
ference of the circle.
Now setting Vn and Vi for the potentials of points outside and
inside the cylinder respectively, we have from these values,
ra=C- 47r^o- log r, Vi = c- lirEo- log R.
The potential is therefore constant, and the force zero within the
cylinder. Outside the cylinder the force is given by
(n) d7Jdr= - 4irftr/r,
that is, the force is inversely proportional to the distance of the
point from the axis of the cylinder.
SECTION XIV. GAUSS'S THEOREM. THE EQUATIONS or LAPLACE
AND POISSON.
Let ABF (Fig. 24) be a closed surface, of which AB = dS is a
surface-element, and at the point 0 within the surface, let the mass
m be concentrated. On the element ds at C, construct the normal
CE. Let the length of the line connecting 0 and C be r, and let
the normal CE make the angle DCE = Q with 00 produced. If the
potential at C due to ml is Vv then Prl = ml/r, and the force ^
acting at the point C in the direction CE is Nl = 'dVl[dn) while the
total force in the direction of CO is mjr2. We have
(a) JVt = wij/r2 . cos («• - 6) = - mjr2 . cos 0.
42
GENEEAL THEORY OF MOTION.
[CHAP. i.
If a sphere of unit radius is described about the point 0 as
centre, the straight lines drawn to the contour of (IS mark out on
this unit sphere a surface-element whose magnitude is equal to
(b) du = dS.cose/r*.
From (a) and (b) we obtain
NjdS= - mjr2 . cos QdS= - m^w and 'dPJ'dn . dS= - mfa.
If there are still other masses, m^ my etc., within the closed
surface, we obtain similarly
3 fy3» . dS = - m^u, "d V^^n, . dS = - msdo>, ....
Vv Vy Fs are the potentials at the point C due to the masses
mv m2, ms respectively. For the total potential at the point C we
~ ..., and therefore
.dS= -
If we designate the mass enclosed by the surface by 2m, integration
over the whole surface gives (c) foV/'dn. dS= - 47r2m. The force
acting in the direction of the normal to the surface S is 'dF'/'dn ; we
call 'dF'/'dn. dS the flux of force which passes through the element dS.
Hence the total flux of force passing through a finite closed surface equals
the sum of the acting masses contained within the surface multiplied
by - 4:ir. Hence, if the entire acting mass is enclosed by the surface,
and 'dVfdn is given for all points on the surface, the sum of the
masses may be determined by the help of equation (c).
SECT, xiv.] GAUSS'S THEOREM. 43
The theorem expressed in (c) also holds in case the acting mass lies
outside the closed surface. Let the mass m' be situated at the point
0' (Fig. 25) outside the surface ABB' A. If the surface-element do>
is taken on the surface of the unit sphere described about 0' as
centre, the straight lines drawn from 0' through the boundary of this
surface-element mark out on the closed surface the surface-elements
AB = dS and A'B' = dS'. Let the normals to AB and A'ff directed
outward from the closed surface be n and n' respectively, and let
the forces 'dF'/'dn and 'dF'/'dn' act in the direction of these normals.
The V in these expressions represents the potential due to m'. If
the angles made by the normals directed outward and the straight
line drawn from 0' are designated by 6 and 6' respectively, and if
we set 0'A=r, 0'A' = r', we then obtain
-dF'fdn = m'/r2 . cos (a- - 9) ; 'dF'/^n' = m'/r'2 . cos (a- - 6'),
dS . cos (TT - 6) = r2du ; dS' cos 6' = r'2 . du,
and therefore Wpn.dS + W/dri .dS' = 0. We therefore have (d)
\?)V'/'dn.dS=Q, if the integral is extended over the whole surface.
The flux of force proceeding from a point outside a closed surface, and
passing through the surface, is equal to zero. Therefore the value of the
integral is independent of the mass outside the surface. We have
then, generally, (e) foF'fdn.dS** -4-jrM, where S is a closed surface,
V the potential, n the normal directed outward, and M the sum of
all the masses within the surface. This theorem is due to Gauss.
Equation (e) may be put into another form. We have
3 Ffdn = 3 Vfdx . dx/dn + 3 F/^ . dy/dn + 3 V\"bz . dz/dn,
and dx/dn = A, dyjdn = p, dzldn = v, where A, /*, and v are the cosines of
the angles which the normal to the surface makes with the axes.
We have then 3F/3tt = \X + ^Y+ vZ. X, Y, and Z are the com-
ponents of the force, and / is set equal to 1. We then obtain from
Gauss's theorem (f) J(ZX + Yp + Zv)dS= - 1-rrM.
Let x, y, z (Fig. 26) be the coordinates of the point 0, Ox, Oy, and Oz
be parallel to the coordinate axes, and 00' be a parallelepiped whose
edges are parallel to these axes. Let X, Y, Z be the components of
the force acting at 0. The components of the force at the point A,
whose coordinates are x + dx, y, z, will be
X+VXfdx.dx, Y+'dY/'dx.dx, Z+-dZ/Vx.dx.
We apply Gauss's theorem to the surface of this parallelepiped. The
force acting normal to the surface OA' is - X, that acting normal
to AO' is +X+~dX/c)x.dx. In the same way the force acting normal
44
GENERAL THEORY OF MOTION.
[CHAP. i.
to OB1 is -Y, and that acting normal to BO' is Y+'dY/'dy.d
similar statement holds for the z coordinate. We have therefore
\-dVfdn, . dS = Jf J[ - Xdydz + (X+ VX/Vx . dx)dydz]
+ [ - Ydxdz + (Y+ VY/oy . dy)dzdx]
We suppose the volume-element 00' to contain the mass M of density
/>, so that M=pdxdydz. We then have from (e)
(g)
or (h)
On account of the frequent use made of this equation in mathe-
matical physics, we use for the sum of the first derivatives of a
function / with respect to the three coordinates the symbol
and for the sum of the second derivatives with respect to the same
variables the symbol V2/= 32//a«2 + 32//c)?/2 + 32//3«2. With this notation
SECT, xiv.] GAUSS'S THEOREM. 45
equation (h) may be written (i) V2 F+ 4ny> = 0. By the help of this
equation, which was first used by Poisson, we can determine the
density when the potential is known. If no matter is present in
the region under consideration, that is, if p = 0, we have
(k) 32r/9a;2 + 32F/3y2 + 32r/o^ = V2r=0.
This equation was first derived by Laplace. It may be obtained
more simply in the following way. We start from
where £, i], and £ are constant, and obtain
3(l/r)/a* = - (x - £)/r», ^(l/r)/^ = - l/r» + 3(a? -
Analogous expressions hold for 38(l/r)/8jy2 and 32(l/r)<322. Adding
these equations, we have
3«2 = 0.
Since the potential F=2w/r [(b) XII.], this is equivalent to V2F=0.
Poisson's equation may also be obtained in the following way.
Let the density at the point P be p. Describe a sphere of infinitely
small radius R so as to contain the point P, and suppose the density
in the interior of the sphere to be constant. The potential P at
the point P consists of two parts, Fj and F~a ; Va is due to the mass
outside the sphere, and Vi to the mass within the sphere. The
potential at P is V=Va+Vi. If P is at the distance r from the
centre of the sphere, we have from (d) XIII.
(k') Ft = 2aX#» - £r2) ; V= Va + **p(& - &*)>
If £, rj, £ and x, y, z are the coordinates of the centre of the sphere
and of P respectively, we will have r2 = (x - £)2 + (y - r))2 + (z - £)2.
By differentiation with respect to x, we obtain
3(r2)/ae = 2(x - £) and 92(r2)/ae2 = 2,
therefore W = 6, and from equation (k') V*V=V*Va-lvp. Now Va
is the potential due to the mass lying outside of the sphere, and
therefore V2F"a = 0 and V2F+47r/a = 0. This is Poisson's equation.
In the parts of the region where p is infinitely great, Poisson's
equation loses its meaning. In this case we return to the fundamental
equation (e). For example, let a mass be distributed on a surface S
with surface density a-. Draw the normals vt and va to the element
dS on both sides of the surface, and construct right cylinders on
both sides of the surface on dS as base, and with the heights dvi
and dva ; the linear elements of these cylinders are lines of force. By
applying equation (e) to the volume included in the cylinders, we
46 GENERAL THEORY OF MOTION. [CHAP. i.
obtain ?>V^vi . dS+'dFJ'dva. dS= -4WS, where Vt and Va represent
the values of the potential on both sides of the surface. Hence
(1) 3r«/3v, + ^Val^va + lira- = 0.
This equation finds an application in the theory of electricity.
Comparing formulas (e) and (h), and noticing that M= ^^pdxdydz,
we obtain the relation (m) \\\VzVdxdydz = Jf3F/3». dS. The triple
integral in (m) must be extended over the volume bounded by the
surface S, and the double integral over the surface S. This theorem
may also be proved by integration by parts.
SECTION XV. EXAMPLES OF THE APPLICATION OF LAPLACE'S AND
POISSON'S EQUATIONS.
The potential V at the point x, yt z is a function of the three
coordinates, and, from the previous discussion, has the form
(a) V= Mpd&vdt/J(z - & + (y- ^ + (* - 02>
where the density p at the point (£, rj, £) is a function of the
coordinates.
We may, however, use the differential equation (b) V2F+47r/> = 0
as the starting point for the determination of the potential; we
thus often obtain the desired result by a more convenient method.
The density p must be given as a function of x, y, and z. The
integral of (b) is always given by (a), but V may often be found
more conveniently by direct integration of Poisson's equation.
In the solution of problems in potential, special attention must be
paid to the boundary conditions which serve to determine the functions
which are obtained by integration. We shall investigate the equations
of condition to which the potential Vt within a closed surface S and
the potential Va outside that surface must conform, if the surface S
encloses all the masses which are present in the field, and if no
mass is present outside the surface. Applying Poisson's equation
to the region enclosed by #, we have (c) V2J^ + 47r/> = 0. Outside
the surface S we have (d) V2F"a = 0.
If 0 is any point within S, P a, point outside S, and if we set
OP = r, we have, when r is very great, (e) Fa = M/r. M represents
the whole mass enclosed by S. Hence, for r = <x>, the potential
Ftt = 0 and (f) lim(rFa)r=00 =M, that is, the product rVa approaches
the finite limit M if the point P moves off to infinity.
SECT, xv.] LAPLACE'S AND POISSON'S EQUATIONS. 47
If Pl and P2 are two points which lie infinitely near each other
on different sides of the surface S, the potentials at both points are
equal, and we have for all points on the surface S, (g) Vi = Fa. The
dash drawn over V is used to denote the value of V at the surface.
From (1) XIV. it follows further that for the points on the surface
where o- = 0, we have (h) 'dFi/'dv = 'dFa/?)v, where the normal to S is
designated by v = - v; = va. The potential is therefore everywhere
finite.
It is here assumed that p is everywhere finite. For the places
where p = so we obtain other equations of condition, which may
readily be derived from those already given. For example, if o- is
the surface density on a surface S, in which, therefore, p is infinitely
great, and if p = Q for all other points in the region, then, in our
former notation, we have V2F"j = 0 and V2F"tt = 0, but
(i) Vi=V« Wtl'dvl + Wjdva+4:ira- = Q
for all points on the surface S.
By these equations we may determine the potential if the density
p within a sphere of radius R is constant. Outside the sphere p
is supposed to be zero. The potential within the sphere is Ft, and
outside of it Fa. We have then W, + 4ir/o = 0, V2Fft = 0. Now
we have 'dF/'dx = dFfdr . x/r,
&Ffda? = d*Fjdr* . x2/r2 + dFjdr . 1/r -dF/dr . x2/^.
Similar equations hold for the derivatives of V with respect to y
and z. We thus obtain V2F=d?F/dr2 + 2/r . dFjdr. Since, however,
d(r F){dr = rd F/dr + V and d2(rF)/dr2 = rd2 F/dr2 + 2d F/dr,
we have (1) V2F= 1/r . d2(rF)/dr2.
The differential equations which Vi and Fa must satisfy are therefore
(m) d2(rFt)ldr2 + l-rpr = 0, d2(r Fa)/dr* = 0.
From these we obtain by integration
For r = ao it is assumed that Fa = 0, so that Fa = Cl'/r. Since Ft
cannot become infinite for r = 0, we have (7 = 0, and hence
Since the force is a continuous function of the coordinates, arid
since, therefore, for all points on the surface, dFi/dr = dFJdr, we
will also have, when r = B, ^TrpR = Cl'/R2. Hence C^^-n-pB3, and
therefore (n) Fa = £irEsp/r. Since F"4=Fa when r = R, we have
C=27rR2p, and therefore (o) F^^^R2-^-2). These formulas are
the same as (c) and (d) XIII.
48 GENERAL THEORY OF MOTION. [CHAP. i.
If the potential depends on the distance of the point under con-
sideration from a straight line, we choose this line as the £-axis of
a system of rectangular coordinates. Let the distance from the
2-axis of the point for which the potential is to be determined
be r. We have then r~ = x2 + y-, and further
x*/r2 . dtF/dr* + l/r . dFfdr - x2/^ . dF/dr, etc.
Therefore
(p) ^Y^^YI^ + ijr . dp/fir = i IT t d(rdF/dr)/dr.
If we are dealing with an infinitely long circular cylinder of
radius E and surface-density a-, the axis of which is taken as the
z-axis, we have (q) VsFi = 0 and V2F"0 = 0, while for r = R we have
(r) Ft=FM d FJdr - dFt/dr = - 47ro-.
It follows from equations (p) and (q) that
d(rdFt/dr)/dr = Q and d(rdfra/dr)ldr = 0.
Hence dFt/dr=Cl/r and dFa/dr = C2/r,
ri = (71logr+(71' and Fa=C2\ogr+C2.
GI must be equal to zero, since no force acts at points in the axis
of the cylinder. Further, for r = R, we have C^ = C.2 log E + C2.
From equation (r) we have C2= -47rKo-, and therefore
V. = C2 - lirRo- log R; Fa = C2- iirRo- log r.
These equations are the same as those given in (m) XIII.
SECTION XVI. ACTION AND REACTION. ON THE MOLECULAR AND
ATOMIC STRUCTURE OF BODIES.
In our discussions up to this point we have considered the motion
of a body under the action of given forces ; but nothing has yet
been said as to the origin of these forces. A body upon which no
forces act moves forward, by the principle of inertia, in a straight
line with a uniform velocity. A change in the motion can arise
only from outside causes. We learn from experience that the motion
of one body in the presence of another undergoes a change, and we
are therefore led to assume that in the mutual action of these bodies
is to be found the reason for the change of motion. We will first
consider the mutual action of two bodies. We thus obtain the means
of investigating the more general case in which three or more bodies
SECT, xvi.] ACTION AND REACTION. 49
act on one another. The mutual action may be of different kinds.
If two bodies collide their motion changes. A similar change occurs
when the bodies slide over each other. In both cases the bodies
are at least momentarily in contact. Bodies also act on each other
without contact; thus, for example, a magnet attracts a piece of
iron, or a piece of amber when rubbed attracts a feather. The
first serious eifort to explain these mutual actions or so-called actions
at a distance was made by Descartes. His explanation was based
on the assumption that all space is filled with very small particles
in motion, and that all observed motions of bodies are due to
collisions between them and these invisible particles.
Hence the discovery of the laws of collision became one of the
most important tasks in the study of physics. Descartes investigated
this question, but without success. It was not until the close of
the 17th century that Huygens, Wallis, and Wren contemporaneously
succeeded in solving it. A sphere in motion can set in motion a
sphere at rest; the moving sphere, therefore, possesses energy of
itself. Let the collision be central, that is, let the direction of motion
coincide with the line joining the centres of the spheres. An iron
sphere produces a greater effect on the sphere at rest than a wooden
sphere of equal size moving with the same velocity. Of two equally
large spheres whose mass is the same, the one produces the greater
effect which has the greater velocity. Hence the force which the
moving sphere possesses increases with its mass and with its velocity
jointly. The product of the mass and the velocity gives a measure
for the force residing in the body, and is called its momentum, or
quantity of motion.
The principal result which Huygens, Wallis, and others obtained
was the following : If two bodies collide, they undergo changes of
momentum which are equally great and in opposite directions, or,
they act on each other with equal but oppositely directed forces.
The action and reaction are therefore equal and oppositely directed.
This is one of the most important laws of natural philosophy,
and we will discuss the grounds upon which it is founded. It was
first derived from observations on collision, without its thereby
becoming apparent how far it holds for other interactions between
bodies. Xewton first recognized in this law a universal law of
nature, which always applies when bodies act on one another. By
careful investigation of the collisions of different bodies (steel, glass,
wool, cork) he found that the action and reaction are equal, if
allowance is made for the resistance of the air. In order to examine
D
50 GENEKAL THEORY OF MOTION. [CHAP. i.
whether the same law holds for actions at a distance, he mounted
a magnet and a piece of iron on corks and floated them on water.
The iron and the magnet approached each other and remained at
rest after they had come in contact, so that the forces by which
the iron and the magnet were mutually attracted were oppositely
directed, and equal. He showed further by the following argument
that action and reaction are equal in the case of attraction or
repulsion : If two bodies acting on each other are rigidly connected,
they should both move in the direction of the greater force if action
and reaction were not equal ; this would contradict the principle of
inertia.
Since Newton's time this law has been established in many ways,
and many discoveries in physics have furnished proofs of its correct-
ness. It has led in many cases to new discoveries, and there is no
longer any doubt of its universal applicability.
The simplest conception of the structure of bodies is that, according
to which bodies are composed of discrete particles, for whose mutual
action the law of action and reaction holds. Starting from this
view, Newton calculated the action of gravity. Gravity is a function
of distance alone ; its value is therefore the same so long as the
distance is unchanged. This conception of the structure of bodies
has led to important results in other branches of physics. There
are, however, many cases in which it seems inadequate. Chemistry
teaches that bodies are composed of molecules, which themselves
may be groups of smaller particles or atoms. These molecules have
certainly a very complex structure, and the mutual actions among
them, especially if the distances between them are great in comparison
with their size, must therefore be of a very complicated nature. As
yet we have little knowledge on this subject. In what follows
we will confine ourselves to the treatment of the motions of particles
acting on each other with forces which are functions only of the
distances between them.
SECTION XVII. THE CENTRE OF GRAVITY.
Gravity acts on all parts of a body; the forces thus arising may
be considered parallel for all parts of the same body. The action
of gravity on all the particles of a body may be combined in a
resultant whose point of application is at the centre of gravity. If
the centre of gravity is rigidly connected with the body and rests
SECT. XVII.]
THE CENTRE OF GRAVITY.
51
on a support, the body is in equilibrium in any position. Since
gravity is proportional to the mass, the centre of gravity coincides
with the centre of mass. The resultant applied at the centre of gravity
is the weight of the body. The straight lever is in equilibrium if there
is applied to its centre of gravity a force equal to its weight and
acting in the opposite direction ; the particles on the one side of
the centre of gravity tend by their weight to produce rotation in
one sense which is equal to that produced in the opposite sense by
the particles on the other side.
Let the masses rax and m2, whose velocities are represented by A A'
and BB' (Fig. 27), be situated at the points A and B. Let the point
C be so determined on the line joining A and B that mlAC = m.2BC.
The point C is then .called the centre of gravity of the two masses
ml and m2. If the point C' is so determined on the line joining
m2B'C', we may consider CC' the velocity of
FIG. 28.
tJie centre of gravity. If AD and BE are equal and parallel to CO',
the velocity AA of the mass ml may be resolved into the components
AD and DA', and similarly the velocity BB' may be resolved into
the components BE and EB'. Now, since ml/m.2 = BC/AC=B'C'/A'C',
the triangles A'CfD and B'C'E are similar, the sides A'D and B'E are
parallel, and hence ml/m2 = B'EIA'D^ The velocities of the masses
may be considered as compounded of the velocity of the common
centre of gravity and two velocities vl and v2, which are parallel
to each other and inversely proportional to the masses ; so that
If, therefore, oa and ob (Fig. 28) represent the velocities of the
masses m-^ and m2, and if ab is divided by the point c into the
parts ac and be, which are inversely proportional to the masses, then
oc represents the velocity of the centre of gravity, and ca and cb
52 GENERAL THEORY OF MOTION. [CHAP. i.
represent the velocities of the masses ml and m.2 relative to the centre
of gravity. It is convenient to resolve the velocity in this way,
because the velocity of the centre of gravity is changed by external
forces only.
If momenta are resolved and compounded like forces, then from
Fig. 28 the momentum of the centre of gravity, in which we may consider
both masses united, equals the resultant of the momenta of the
separate masses ml and m2. The momentum ml.oa may be resolved
into m1oc + m1c«, the momentum m2ob into m2oc + m2cb. Now, m^ca
and m.ycb are equal but opposite in direction. Hence we have for
the resultant momentum m^c + m.2oc = (ml + m2)oc.
The velocity of the centre of gravity remains unchanged if the
bodies m^ and m2 act on each other according to the law of action
and reaction. In this case both bodies receive momenta which are
equal but oppositely directed, and which annul each other. This
result may be derived analytically in the following way. If xl and x2
are the coordinates of the particles m^ and m.2, the line joining which is
taken for the a-axis, and if the ,r-components of the forces with which
the masses act on each other are X^ and X>, the equations of motion
are (a) m1xl = Xl and m^cz = X2. Adding these equations, we have
(b) d*(mlxl+m<p2)/dt- = Xl + X2. Since Xl and X2 arise from the
mutual action of the masses on each other, they are equal but opposite
in direction, and hence Xl + X2 = Q. Setting
(c) mfa + mjK2 = (TWj + ro2)£,
we have £ = 0, £ = Const. Hence the point determined by the
^-coordinate moves with the constant velocity £. The .r-coordinate
of the centre of gravity is £, because m1(.r1 - £) = w2(£ - x2). Differ-
entiating equation (c) with respect to t, we have
That is, the momentum of the centre of gravity equals the sum of the
momenta of the separate masses.
By Newton's law of universal attraction two masses ml and m.2
act on each other with a force -fm^m^f2, where r is the distance
between the two masses. Their motion may be determined in the
following way. The velocity of the centre of gravity and the
velocities of the masses relative to the centre of gravity are deter-
mined from the velocities of the masses. These act on each other
with forces directed toward the centre of gravity, and we can
therefore consider this as the attracting point. If 1\ is the distance
of the mass m^ from the centre of gravity, then m1rl = m2(r-r1), and
SECT, xvii.] THE CENTRE OF GRAVITY. 53
therefore m.2r = (m1 + m.2)rl. By substitution of this value of r we
obtain for the force the expression fmlm.2Al(ml + w2)2ri2- The mass
% therefore moves round the centre of gravity as if the force acting
on it were due to a mass Ml = m23j(ml + m.->)2 situated at that point.
SECTION XVIII. A MATERIAL SYSTEM.
We will now consider a system of separate masses in vacuo, which
act on each other with forces which are functions of the distances
of the masses from each other, and obey the law of action and
reaction. The forces which act in such a way within the system
are called internal forces. External forces, proceeding from bodies
which do not belong to the system, may also act on it. The masses
are designated by %, m2, m3, etc., and the positions of the masses are
determined by the coordinates x, y, z, with appropriate indices. We
may determine the position of the system by supposing each mass
to be made up of different numbers of units of mass ; the mean
values £, y, £ of the x-, y-, ^-coordinates will then be
(a) £ = (mft + m^ + myx3 + ...)/(ml + m2+ ...). etc.
£, -i], £ are the coordinates of the centre of gravity of the system of
masses. If the equation (a) is differentiated with respect to the
time t, it appears that the velocity of the centre of gravity depends
on the velocities of the particles. That is,
(b) £ = (m^ + m^c2 + m^bs + . . . )/(•/», + m2 + m3+ ... ), etc.
The internal forces cannot change the motion
of the centre of gravity, since by the law of
action and reaction two masses impart to each a,
other equal and opposite momenta, the sum
of whose projections on any axis is equal to
zero.
This result may be represented geometrically
in the following way. From any point 0 (Fig.
29) draw the lines Oa, Ob, Oc, etc., which repre-
sent the velocities of the masses mv m2, m3, etc.
Then if the masses mv mv ms, etc., are placed
at the points a, b, c, etc., respectively, and if
p is the centre of gravity of the masses, Op is the velocity of the
centre of gravity.
54 GENERAL THEORY OF MOTION. [CHAP. i.
In order to determine the motion of the separate particles we
must know the separate forces which act on them. If the com-
ponents of the external forces acting on the mass ma are designated
by XM Ya, Za, if Fab is the force with which ma is attracted by
mM and if r,^ is the distance between rna and mb, the ,r-component
of the forces acting on ma is
Xa + Fab.(xa-xb)/rab + Fac(x<l-xr)lrae+....
We obtain similar expressions for mb, mc, etc. Hence we have
(c) m^a = Xa + F^ . (xa - xb)/rab + FM . (xa - xe)/rM + . . . , etc.
Since by the law of action and reaction Fn6 = Fba, Fac = Fca, we have
f
\
If we now introduce the coordinates of the centre of gravity, we
have, using equation (a),
This equation contains the law of the motion of the centre of gravity,
which may be thus stated : The centre of gravity of a system of masses
moves like a material point in which all the masses of the system are
united, and at which all the forces are applied.
The momentum of the whole system is compounded of the
momenta of the separate masses. From a point 0 (Fig. 30) draw
OA = mava parallel to the direction of the velocity
va. In the same way draw AB = mbrb, BC=mevc,
etc. Taking account of all particles we reach
2?/ / a point D. The line OD then represents the
momentum of the system. The sums of the pro-
jections of the momenta on the coordinate axes are
By equation (b) these sums equal the components
of the momentum of the centre of gravity. In
the time-element dt the motions of the separate
masses are changed by the forces which act on
them; nevertheless, the internal forces do not
change the momentum, since the resultant of the
momenta which these forces occasion is zero, by the law of action
and reaction. On the other hand, changes of momentum are occa-
sioned by the action of the external forces. A force K produces
the momentum K .dt in the time dt. If all the momenta which
the external forces produce are determined in this way, and com-
SECT, xviii.] A MATERIAL SYSTEM. 55
bined with those originally given, the actual momentum is obtained.
This also appears from equation (d), which may be thus written :
(f ) d(maxa + mbxb + m,xe +...) = (Xa + Xb + Xc +. ..)dt.
Sir William Rowan Hamilton introduced the word vector to repre-
sent magnitudes which have direction, and which may be compounded
like motions, velocities, forces, etc. The sum of vectors is called
their resultant. If we consider momentum and force as vectors, the
increase of the momentum which a system receives in the time dt
equals the product of the resultant of the external forces and the
time dt. Since the momentum of the system equals the momentum
of the centre of gravity, the law just stated holds also for this latter.
SECTION XIX. MOMENT OF MOMENTUM.
If the mass m at the point A (Fig. 31) moves in the direction AB
with the velocity v, its momentum is mi: If 0 is an arbitrary
fixed point, and 00 =p a line perpendicular to ^
AB, the product mvp is called the moment of
momentum with respect to 0. The value of the
moment depends on the position of the point 0.
If we erect a perpendicular on the plane deter-
mined by 0 and AB, and lay off on it from 0
a length proportional to mvp, the vector deter-
mined in this way is called the moment of !
momentum. This vector is to be so constructed
that it points in the direction of the thumb, ;
if the right hand points in the direction OC,
and the palm is turned toward the direction of the force.
In the same way the vectors corresponding to all parts of the
system can be determined, and compounded by the method given
in Fig 30. If neither external nor internal forces act on the parts
of the system, the moment of momentum of the entire system is
invariable, since the separate moments remain invariable. The
moment of momentum of the system is also not changed by the
action of internal forces. If, for example, A and B (Fig. 32) are
the points occupied by two masses m^ and m^ which repel each
other with the force K, then A receives in the time dt the momentum
K.dt in the direction A A', and B receives the same momentum in
the opposite direction. The moments of momentum of A and B
56
GENERAL THEORY OF MOTION.
[CHAP. i.
annul each other. On the other hand, the moment of momentum
of the system will in general be changed by external forces, but it
will remain constant in case the directions of all the external forces
always pass through the fixed point 0.
Hence, if the moments of momentum and the moments of the
external forces are considered as vectors, the increment of the moment
of momentum of the system in the time dt equals the resultant of
the moments of the external forces multiplied by dt.
X
FIG. 32.
FIG. 33.
This may be represented analytically in the following way. Let
AB — x and AC=y be the velocity components of the particle M
situated at A (Fig. 33). The distance of the moving mass from the
x-axis is y, and from the ?/-axis is x. Hence the moments of momentum
with respect to the £-axis are mxy and myx. These being oppositely
directed, their difference mxy -myx is the moment of momentum of
m with respect to the z-axis. This moment receives in the time dt
the increment md(xy - yx) = m(xy - yx)dt. Hence we have
(a) 2m(a# - yx) = 2(zF- yX), or (b) d2m(xy - yd) = dfS,(xY- yX),
that is, the increment which the moment of momentum about any axis
receives in the time dt is equal to the pi'oduct of the sum of the moments
of the external forces about the same axis and the time-element dt.
SECTION XX. THE ENERGY OF A SYSTEM OF MASSES.
If a particle m moves with a velocity v = ds/dt, its kinetic energy
[V.] is £/nfl2 = £m(<fe/<ft)2 = £ws2. Since ds2 = dx* + df + dz2, this may
be written %mvz = \m(xl + y2 + 22). The kinetic energy of the system
SECT, xx.] ENERGY OF SYSTEM OF MASSES. 57
is determined from the velocities of the separate particles of the
system. It is expressed by T = % . *2m(x* + yz + z2). If xt y, z are the
coordinates of a particle, and £, rj, £ the coordinates of the centre of
gravity, the coordinates of the particle with respect to the centre
of gravity are x — £ = x', y — ri = y', z — £ = z1. Using these new
coordinates, we obtain
Swfce2 = 2m(£ + x)'2 = £22m + Zmx'2 + 2£2m£', etc.
From XVIII. (a) we may set ^mx' = 0, if the centre of gravity is
chosen as the origin of coordinates. Then
T= \ . (¥+ if + C2) . 2m + 1 . ^m(x"1 + y"2 + z2).
The kinetic energy of the system is equal to the sum of the kinetic energy
of the masses due to the motion of the centre of gravity, and the kinetic
energy of the masses due to their motion relative to the centre of gravity.
The increment of the kinetic energy of the system in the time-
element dt equals the work done by the forces during that time.
This is divisible into two parts, that of the external and that of
the internal forces. If we designate the components of the motion
of a particle parallel to the axes by dx, dy, dz, the work done by
the external forces is ^(Xdx+Ydy + Zdz).
If /• is the distance between two particles and the repulsive force
F acts between them, the work done by the internal forces is *2.Fdr.
Hence we have dT=^(Xdx+Ydy + Zdz) + ^Fdr. If the force with
which the masses act on each other is a function of the distance r
only, we can set Fdr = d^, where ^ is a function of r only. Now
setting ^d$ = dU, we have finally
(d ) dT = 2 (Xdx + Ydy + Zdz) + d U.
The function U depends only on the distance between the particles
or on the configuration of the system. U is the potential of the system
on itself or the internal potential energy of the system. Further, U is
the work which would be done by the internal forces if the particles
were to move from their positions at any instant into other positions
in which their mutual actions are zero.
If, for example, the given masses act on each other according to
Newton's law, we have F= -fmlm2/r'2, and therefore
Fdr= -fmlm2dr/r'2 = +fd(mlm2/r).
If several masses mv m.2, m3, ... are present, whose distances from
each other are r12, r13, r23, ... respectively, we have
(e) d U =.fd(mlm2lrl2 + w»1»»8/rI8 + m2m3/r23 +...).
58 GENERAL THEORY OF MOTION. [CHAP. i..
If the system passes from one configuration to another, the work
done by the internal forces is determined only by the initial and
final positions of the particles, and does not depend on the paths
traversed by them. If no external forces act on the system, we have
(f) clT= dU or T - T0 = U - U0.
Now, from the discussion in VII., we may set E/"0 = 0, and so obtain
T=U+T0, that is, the kinetic energy of the system equals the
original kinetic energy T0 increased by the work U done by the
forces. In case of a change in the relative positions of the particles,
supposing no external forces to act, a transformation of the one
form of energy into the other occurs without causing a change in
the total energy of the system, that is, the sum of the kinetic and
potential energies of such a system is constant.
SECTION XXI. CONDITIONS OF EQUILIBRIUM. RIGID BODIES.
We will now consider the conditions of equilibrium of a system.
If the positions of the separate masses at a definite instant, and
also the internal and external forces are given, the system is in
equilibrium, when the resultant of all the forces acting on each
particle is zero. If the internal forces are in equilibrium, no change
occurs in the motion of the system, so long as no external forces
act on it. External forces will ' as a rule set the system in motion ;
but it is also possible that they will not change the equilibrium of
the system as a whole even if its separate parts are set in motion.
If the resultant of the external forces is zero, the motion of the
centre of gravity remains unchanged [XVIII. ] ; so that, for example,
if the centre of gravity, is at rest, it remains at rest. But even
when this is the case, the external forces may set the separate
masses in motion ; the relative positions of the particles may be
changed, and changes of form or rotations may occur. The con-
ditions for such changes are developed in the theory of elasticity
and in hydrodynamics. At present we will consider only the
behaviour of rigid bodies.
The particles of such bodies are so conditioned that the distances
between them are constant or nearly so. If the positions of three
particles of the body are given, the positions of all the other particles
are also given, and the position of the body is determined. If the
SECT. XXI.]
CONDITIONS OF EQUILIBRIUM.
body (Fig. 34) is moved from its position so that the points A, B, C
are brought to the points A', B', C' respectively, it can be brought
back to its original position by a series of simple operations. The
body may first be displaced parallel with itself through the distance
AA', so that the point A' coincides with A, and the points B' and
C' are brought to b and c. C'c, B'b, and A' A are equal and parallel.
The body may then be turned about an
axis passing through A, perpendicular
to the plane determined by BA and
bA, through the angle BA £>, so that b
coincides with B and c is brought to
c'. By a second rotation about the
axis AB, c' may be made to coincide
with C. The motion of the body is
thus reduced to a translation and two
rotations. A rigid body, therefore,
cannot be moved, if it can neither be
displaced nor rotated.
In order that a body acted on by
external forces shall be in equilibrium,
its centre of gravity must remain at
rest. The necessary condition for this
is that the resultant of the external forces is zero. It must also
have no rotation about any axis. If such rotations exist, its particles
receive a certain momentum, which has a moment with respect to
the axis. Since each of the elements in this moment is positive,
because the parts of the body all move in the same sense, and,
therefore, the momenta of the separate particles have the same
sign, the sum of the momenta can vanish only when each one of
them is separately zero. Now the sum of the moments of momentum
is [XIX.] equal to the product of the moment of force and the time
during which the force acts. Hence it is required for equilibrium
that the forces which act on the body have no moment with respect
to the axis. This must hold for each axis about which the body
can turn. Now, if the moment of the forces equals zero, the sum
of the moments of momentum equals zero, therefore the moment of
momentum of each particle equals zero; that is, each particle is in
equilibrium. Furthermore, since moments can be compounded like
forces, equilibrium will exist if the moments with respect to three
arbitrary axes are zero.
FIG. 34.
60 GENERAL THEORY OF MOTION. [CHAP. i.
SECTION XXII. ROTATION OF A RIGID BODY. THE PENDULUM.
Let a solid body revolve around an invariable axis, which is chosen
as the 0-axis of a system of rectangular coordinates. Let the angular
velocity of the body be w. If r represents the distance of any particle
m from the 2-axis, the velocity of this particle is ro> and its kinetic
energy ^mr-wr. Since tu has the same value for all particles, the
kinetic energy T equals T=W22mr2. The factor 2mr2 is called the
moment of inertia J of the body with respect to the 3-axis ; the moment
of inertia is equal to the sum of the products of the particles into
the squares of their respective distances from the 0-axis. Hence we
have T=\(»-J, that is, the kinetic energy of a rotating body is equal to
its moment of inertia multiplied by half tJie square of its angular velocity.
A length K may always be found, such that 2»w2 = K'2-m. This
length is called the radius of gyration of the body. It is the distance
from the axis at which a mass equal to the mass of the body would
have the same moment of inertia with respect to the axis as that
of the body.
If the only external forces which act on the body pass through
the axis, the work done by them is zero, since the axis does not
move. Since the internal forces also do no Avork, the kinetic energy
and therefore also the angular velocity o> must remain constant.
Since [XVIII.] the centre of gravity moves as if the resultant of all
the forces acted on the mass of the body concentrated at the centre
of gravity, this resultant R can be determined. If we represent
the distance OP (Fig. 35) of the centre of gravity from the r-axis
by a, we have [IV. (b)] E = ^ma-(a-/a = ^mao)-. E is the resultant
of the forces with which the body acts
on the axis of rotation. In general, the
forces applied to the body so act that
they have no resultant; they tend only
to produce rotation about the axis. In
order to determine them, the theorem in
XIX. concerning moments of momentum
must be used.
If external forces act on the body, its
angular velocity changes. The amount
of this change is determined from XIX.
The momentum of a particle m is represented by mro>, and its
moment of momentum by mrtar. Hence the moment for all particles
of the body is <o2wr2 = wJ. If the moment of the forces with respect
SKCT. xxii.] ROTATION OF A RIGID BODY. . 61
to the i-axis is represented by M, we have, from XIX., d(<oj) = Melt
or (c) Jdwfdt = M.
If, for example, the moment is constant, the angular velocity
increases in direct ratio with the time.
If the moment is due to gravity, the body under certain conditions
performs oscillations. We suppose the a-axis taken parallel to the
direction of gravity and represent the force of gravity by g. The
position of the centre of gravity P (Fig. 35) is determined by the
angle POX=Q and the angular velocity o> by dS = <adt. The moment of
force with respect to the s-axis is - (m^ + rn.^ +...)#= -?/. g?m,
where -/; is the y-coordinate of the centre of gravity. 2m denotes the
sum of all the masses. Since t] = a sin 0, we have from (c)
/. 0 = - a sin 0 . g~2m.
If 6 is very small, so that we may set sin 9 = 9, this becomes
(d) 9= -aeg'Sm/J.
Comparing this equation with that given in VIII. (g), we find that
they are identical when we set l/7 = «2m//. The period of oscillation
of the physical pendulum is therefore (e) t = vxjlfg = irJjjgaSm. Since
J=2m(x- + y2 + z2), we get by transferring the origin of the system
of coordinates to the centre of gravity (£, 77, £), if #', y', z' are the
coordinates with respect to the new origin,
(f ) J = 2m{(z' + £)2 + (y' + ,)2 + (z' + O2} = 2ma* + ?m(yf* + y"> + ^\
since the terms £2w.?', rj^my' and £2mtf vanish.
Now, if we set /= a22w + &22m, where k is the radius of gyration,
we obtain from (e) (g) t = irj(a2 + k*)/ga. We call l = — the
reduced length of the pendulum or the length of the equivalent simple
pendulum. The point S which is at the extremity of the line OS = l
(Fig. 35), drawn through 0 and P, is called the centre of oscillation.
If an axis is passed through S parallel to the £-axis, and the body
oscillates about it, the reduced length of the pendulum I' is
Since, however, l-a = k2/a, we have I' = (a2 + k2)/a = I. The reduced
length of the pendulum and therefore the time of oscillation are
the same for this new axis as for the former one.
CHAPTER II.
THE THEOEY OF ELASTICITY.
SECTION XXIII. INTERNAL FORCES.
IF all parts of a body are in equilibrium and if no tensions or pressures
act on them, yet internal forces must be present acting between the
separate parts of the body. Every action produces changes of form
in the body, and thus develops forces in its interior, which act in a
sense opposite to the external forces. These internal forces con-
dition the nature of the body, determining, for example, the difference
between solids and fluids. No sharp distinction can be drawn, how-
ever, between these two classes of bodies. Viscous fluids and jelly-
like solids are bodies which seem to be transition forms between
true solids and fluids.
If a pressure acts on the surface of a fluid, it must be equally
great on equal areas of the surface at all points, and it must be
perpendicular to the surface, if the fluid is to be in equilibrium.
This pressure is exerted throughout the whole mass ; all equal
surface-elements at a point are subjected to equal pressures, which
are always perpendicular to the surface-elements. We call such a
pressure hydrostatic pressure. A similar pressure may also be present
in solids. If a solid, a piece of glass, for example, which fills the
volume enclosed by its external surface, is immersed in a fluid on
which a pressure is exerted, the same pressure exists at every point
in the surface of the glass as in the fluid. The pressure is everywhere
the same, and perpendicular to the surface-elements. We may there-
fore speak of hydrostatic pressure in solids also.
Yet, in general, internal forces in solids are very different from
those in fluids. Let a cylindrical rod be fastened at one end, and
let the force V be applied at the other end so as to lengthen the
62
CH. II. SECT. XXIII.]
INTERNAL FORCES.
63
rod. In a cross section perpendicular to the axis of the cylinder
the internal forces are everywhere equal. Let the area of the cross
section be A (Fig. 36), then the force V\A acts on unit of area in A.
This quotient represents the stress S in the rod. If
another plane cross section B is taken in the rod,
which makes the angle <£ with A, the force & acts
on each unit of area of B, so that
& . AJ ^ *b . ^4 = o . Jj cos <PJ
and hence (a) S' = Scos<J>. The stress S' is no longer
perpendicular to the surface B on which it acts ; its
magnitude decreases with cos <f> and vanishes for
<£ = |TT. A surface-element within the cylinder and
. parallel to its axis is therefore subjected neither to
pressure nor to tension ; this conclusion holds for
an element of the surface of the cylinder. We may
resolve S' into two components, one of which, T,
is tangent, and the other, N, normal to B, and have
(b) N=Scos-<f), T = S cos <f> sin <J>.
If internal forces of this type exist within a body, we call the stresses
axial. In the direction of the axis the stress is S; a unit of surface
whose normal makes the angle <£ with the axis is acted on by a
force S cos <£ in the direction of the axis.
We will consider a rectangular parallelepiped (Fig. 37), of which
the lines OA, OB, and OC are adjacent edges. The stresses which
act on each unit of area of the faces
which are perpendicular to OA, OB, and
OC are Sa, Sb, Sc respectively. If the
normal to an arbitrarily situated unit
of surface / makes the angles a, (3, y,
with the edges OA, OB, OC respectively,
the force acting on / is the resultant
of the forces Sacos a, Sbcos /3, Sccos y,
which are parallel to OA, OB, OC re- /
spectively. If the stresses Sa, Sb, Sc
have the same value S, this resultant is &/ cos2a + cosL'/3 + cos'-'y = S.
Hence three equal stresses which are perpendicular to each other
cause a hydrostatic stress, since their resultant has the same value
whatever may be the position of the surface. Since the components
of this stress are S cos a, S cos ft, and S cos y, it is perpendicular to
the unit area /.
y
B
,/
'
A x
o
/
/
J3'
FIG. 37.
64
THE THEOEY OF ELASTICITY.
[CHAP n.
On the other hand, if Sc = 0 and Sa = Sb = S, that is, if two stresses
act at right angles to each other, while the stress perpendicular to
them both is zero, the components in the directions OA, OB, OC
respectively are S cos a, S cos /3, 0. Hence the force acting on / is
Sj cos-a + cos2/? = &Jl - cos-y = S sin y,
and is perpendicular to OC. Such a state of stress in a body may
be called equatorial. The plane which contains OA and OB, or rather,
every plane parallel to both these lines, may be called an equatorial
plane. The same stress S acts on each unit area perpendicular to
the equatorial plane. If the normal to the surface / makes the angle
(f) with the equatorial plane, the stress on it is proportional to cos <£.
SECTION XXIV. COMPONENTS OF STRESS.
Let the surface F (Fig. 38) divide a body into two parts, A and
B. If the portion of A which touches the element dF of the surface F
is removed, a force must act on dF to keep B in equilibrium. This
force SdF is not, as a rule, perpendicular to the element dF. The
forces acting at the various points
of F are, in general, different.
If the force tends to move the
element dF into the space occu-
pied by B, it is called a pressure
on the surface dF; if it tends to
move the element dF into the
space occupied by A, it is called
a tension. In all cases we call the
force S a stress; if this acts as a
tension, it is a positive stress, if as
a pressure, it is a negative stress.
If the part of B which touches
dF is removed, then to maintain
equilibrium in A a force SdF
must act on dF, since action and
reaction are equal Hence both forces which act on an element of
surface within a body are equal, but oppositely directed. It is
characteristic of a stress that it may be looked on as made up of
two equal and opposite forces.
FIG. 38.
SECT, xxiv.] COMPONENTS OF STRESS. 65
If the surface-element dF remains in its original place in the body,
but is turned about one of its points, a particular value of the stress
corresponds to every one of its positions; for special positions the
stress may be zero. When the body in which the surface is drawn
is a fluid, the stress is independent of the position of the surface.
We assume in the body a system of rectangular coordinates. The
stresses in the surface-elements, which are perpendicular to the
directions of the axes, are determined by their components.
Let the surface-element dF be perpendicular to the z-axis, and let
dF = dy . dz. If that part of the body is removed which lies on
the positive side of the surface-element dydz, the positive side being
determined by the positive direction of the z-axis, then, to maintain
equilibrium, a force Sdydz must act on the surface dydz. The
force AS' is resolved into the components Xa Ya Z^ which are
respectively parallel to the coordinate axes. The index indicates
that the forces act on an element which is perpendicular to the
a--axis. Xx is perpendicular to the surface-element; it is therefore
called the normal force ; Yx and Zx are tangential forces. Xow, let
the element dF remain in the same place, but be turned so that
it is perpendicular to the y-axis. We may then set dF=dzdx.
As before, there are three components of force X^ Yy, Zy acting on
the surface-element dzdx, of which Yy is the normal force, Xy and Zy
are the tangential forces. If the surface-element dF is turned so as
to be perpendicular to the z-axis, we have as components Xa Ya Z,,
of which Zt is the normal force and Xz and Yz are the tangential
forces. There are therefore, in all, nine components,
Xa Y^Z*; X0Y,,Z,; Xa Yz, Zz.
By these components the stress on any surface is determined. Let
OA, OB, 00 (Fig. 39) represent line-elements, parallel respectively
to the x-, y-, 2-axes. Let a plane be passed
through A, B, and (7, so as to form the
tetrahedron OABC. Let P, Q, and R be
the components of the stress in the direc-
tions of the coordinate axes at a point
in the base ABC of the tetrahedron. We
now form the equation of condition, which
must hold that the tetrahedron shall not ' FIG.
move in the direction of the z-axis. The
forces which tend to move the tetrahedron in that direction are
P. ABC acting on its base, and -XX.OBC, -Xy.OAC, -XZ.OAB
acting on its faces. Hence the force which urges the tetrahedron in
66 THE THEORY OF ELASTICITY. [CHAP. n.
the direction of the o-axis is P. ABC - Xx . OBC-Xy .OAC- X: . OAB.
Designate by a, /3, y the angles made with the axes by the normal
to the surface ABC drawn outward from the tetrahedron; then the
expression for the force in the direction of the ,r-axis becomes
(P - Xx cos a - Xy cos (3 - Xz cos y) . ABC.
Now, if no external attractions or repulsions act on any part of the
body, the conditions of equilibrium, obtained by setting this, and
the two similar expressions which hold for the other axes, equal to
zero, are
f P = Xx cos a + Xy cos ft + Xz cos y,
(a) I Q=Yxcosa+ Fycosp+Yscosy,
[^=^cosa+ Zy cos /3 + Zt cos y.
If other forces besides the stresses act on the parts of the body,
these must be taken into account in equations (a). If the force X
acts on the unit of mass in the direction of the a;-axis, the force
acting in that direction on the tetrahedron is Xpdv, if dv represents
its volume, and p its density. The condition of equilibrium in the
direction of the z-axis then becomes
(P - Xx cos a - X, cos j3 - X, cos y) . ABC + Xpdv = 0.
Now, since dv = %h . ABC, where h is the height of the tetrahedron,
this equation is equivalent to
P - Xx cos a - Xy cos (3 - Xz cos y + $hpX = 0.
Since the height h of the tetrahedron is infinitely small, we may
neglect the term containing it, and again obtain the first of equations
(a), which hold generally.
In order to exhibit the meaning of equations (a), we will consider
the following case. Suppose a tension S to act in the direction of the
re-axis, and a pressure of the same value to act in the direction of the
y-axis. Then XX = S, Yy = - S, and all other components of stress are
equal to zero. Hence P = S cos a, Q = -Scosfl, E = 0.
The resultant A of these components is ^=$siny. If A, //,, v
are the angles between A and the axes, we have cos A
B
cos M = — -. — — . cos v = 0. The angle e between A and the normal
smT o . , •>/}
to the surface-element considered is determined by cos e = — ' —. - -.
smy
If the surface - element is parallel to the ^axis, y = a A=S,
COS e = COS 2a, € = ± 2a.
SECT. XXIV.]
COMPONENTS OF STEESS.
67
If a= then e = ^, the resultant is a tangential force. Thus the
surface of a prism whose axis is parallel to the s-axis, and whose
o
1
FIG. 39 a.
make angles of 45° with the xz- and y^-planes, is acted on only
by tangential forces, each equal to S.
SECTION XXV. RELATIONS AMONG THE COMPONENTS OF STRESS.
The force which acts on the volume-element dxdydz (Fig. 40) is
determined from the components of stress. Let the components
acting at the point 0 be given, and let the force which acts on OA'
in the direction of the x-axis be equal
to - Xjlydz. By development by Mac-
laurin's theorem we obtain for the force
acting on AO' the expression
(Xx + 'dXx/'dx.dx)dydz.
The resultant of these two forces is
'dXJ'dx . dxdydz. The forces - Xydxdz
and ( Xu + 'dXJ'dy . dy}dxdz, whose re-
sultant is 'dXj'dy . dxdydz, act on the
surfaces OB' and O'B respectively in
the direction of the z-axis. The resultant of the forces acting in
the same direction on the surfaces O'C and OC' is 'dXJ'dz . dxdydz.
Hence the total force acting on the parallelepiped dxdydz in the
direction of the £-axis is
68 THE THEORY OF ELASTICITY. [CHAP. n.
If (X), (Y), and (Z) represent the components of the force with
which the stresses act on unit of volume, we have
(X) = 'dXJ
(Y) = VYJ-dx + 3 YJ-dy + 'd YJ-dz,
(Z) = c)ZJ3x + 'dZyfdy + 'dZJ'dz.
If the body is acted on only by stresses, equilibrium will exist
if the three components (X), (Y), (Z) are each equal to zero. The
equations (b) in this case are three differential equations which the
components of stress must satisfy. If a force whose components
are X, Y, Z acts on each unit of mass, and if the density of the
body is p, we obtain the conditions of equilibrium,
[ VXJ'dx + VXJdy + 'dXJ^z + PX= 0,
(c) I 'dYJ'dx + 'dYJ-dy + 'dYJ-dz + pY^O,
( -d Zx/*dx + 3 ZJdy + ~d ZJ-dz + PZ=0.
Internal forces produce both translations and rotations in the
The tangential components tend to rotate the parallelepiped 00'
about the £-axis. The tangential force Xy acts on the surface OB'
in the .negative direction, while the tangential force Xy + 3 XJ'dy . dy
acts on the opposite surface O'B in the positive direction. These
two forces form a couple acting on the parallelepiped with a moment
Xy . dxdz . dy, if terms of an order higher than the third are neglected.
This moment tends to turn the parallelepiped about the 2-axis in
the negative direction. The tangential forces acting on the surfaces
OA' and O'A have the moment Yz . dydz . dx, which tends to turn
the parallelepiped in the positive direction. The total moment which
tends to rotate the parallelepiped about the z-axis is (Yx- Xy)dxdydz.
If the body is in equilibrium under the action of the stresses con-
sidered, this moment must be zero, that is, (d) Yx = Xy, and similarly
Zy = Ya X,= Z^ The last two equations are derived in the same
way as the first. If attractive forces, such as gravity, or in general,
if any forces acting at a distance act on the body, equations (d)
will still be applicable. The point of application of such forces, in
infinitely small bodies, coincides with the centre of gravity ; such
forces, therefore, cannot produce rotations, and, therefore, cannot
make equilibrium with the forces which tend to rotate the body.
It appears from equations (d) that six quantities are sufficient to
determine the stress at a point in a body, namely, Xx, I7,,, Zz ;
Zy=Ya X2 = ZX, Yx = Xy. The first three are normal forces, the other
three tangential farces. It is possible to express these forces by a
SECT, xxv.] COMPONENTS OF STEESS. 69
simpler notation, but we will retain the above, which has the advantage
that it exhibits more clearly than any other the true significance of
the quantities involved. It must be borne in mind that the value
of a component of stress remains unchanged if the direction of the
force and the direction of the normal to the surface-element, on
which the stress acts, are interchanged.
SECTION XXVI. THE PRINCIPAL STRESSES.
In order to obtain a better understanding of the nature of internal
forces, we will examine if it is possible to pass a surface through a
given point in a body in such a position that no tangential force
acts on it. We may anticipate our conclusion by the statement that
three such surfaces may be drawn through any point and that they
are perpendicular to each other. To show this, we proceed from
the equations [XXIV. (a)]
( P = Xx cos a + Xy cos (3 + Xz cos y,
(a) \ Q= Yxcosa+Yycos/3 + F.cosy,
[ R = Zx cos a + Zy cos (3 + Zt cos 7,
in which a, /3, y are the angles between the normal to the surface
and the axes, and determine the position of the surface on which
the components of stress P. Q, R act. It is to be shown that this
surface may have such a position in the body that the stress acts
perpendicularly to it ; we will call the stress in this case the principal
stress S. The angles which the direction of S makes with the axes
are as before, a, /3, y, and
(b) P = Scosa, Q = Scos(3,
Introducing these values in (a), we have
f (Xx - S) cos a + Xy cos ft + X, cos y = 0,
J
(c)
If cos a, cos (3, cosy are eliminated from these equations, we obtain
(S*-(Xt+Y,, + Zt)S* + (XxY,+ YJt + ZtX,-Z*-X*-Y1?)S
\ - (XxYJt + 2ZyXz Yx - XJ? - YyX* - ZZYX2) = 0.
This equation has always one real root A, and we can find the cor-
responding values of a, /?, y from equations (c) and the relation
70 THE THEOEY OF ELASTICITY. [CHAP. n.
cos2a + cos2/? + cos2y = 1. Therefore, through any point in the body
there may be passed at least one plane having the property that no
tangential forces act on it. We call such a plane a principal plant.
Let the system of coordinates be so rotated that this principal
plane is parallel to the 7/2-plane. On this supposition, we have
XX = A, Yx = 0, ZX^Q. The equations (c) then become
(A - S) cos a = 0 ; (Y,,- S) cos p+Yzcosy = Q;
These equations are satisfied when we set
S=A, cosa = l, cos [3 = cos y = 0.
We thus return to the principal plane already found, with its appro-
priate normal stress A. The same equations are also satisfied if we
set cos a = 0 ; cos /3/cos y = - Yj(Yy - S) = - (Zt - S)/Z,r
Since cos a = 0, and a = |TT, the new principal planes are perpen-
dicular to the first one. We have further,
and cos /3/cos y=\(Y!l-Z2± J( Yy - Z
These equations present two values of S and two values each of
(3 and y. If we represent the values of P and y by /5' and /3", y'
and y" respectively, we have
cos (3' cos P" / cos y' cos y" = -1,
and hence cos ft' cos /?" + cos y' cos y" = 0.
Since the corresponding values of a are equal to JTT, it follows that
the two new principal planes are perpendicular to each other.
It is thus proved that, in general, through any point in a body,
there may be drawn three surface-elements, and only three, on
which only normal forces act, and that they are perpendicular to
one another. The normal stresses corresponding to the three planes
may be designated by A, B, and C. From (d) the following relations
hold among these normal stresses and the components of stress,
(e)
( ABC= XJ^i + 1ZVX.YX - XtZ* - YJt? - ZY?.
The first of these equations should be especially noticed; it shows,
that the sum of the normal forces for three planes perpendicular to each
other is constant.
SECT. XXVI.]
THE PRINCIPAL STRESSES.
71
If the axes of the system of coordinates are parallel to the directions
of the principal stresses A, B, and C, equations (a) become
P = Aco$a, Q = Bcos[3, E=Ccosj.
IfA>E>C, and we set A = B+SV C=B-S2, the principal stresses
can be replaced by a hydrostatic stress B and two axial stresses Sl
and S.2, the first of which is a tension, the second a pressure.
This investigation shows that through any point in a body three
planes can always be passed which are acted on only by normal
stresses, equal to the principal stresses A, B, and C. A, B, and C
are the three roots of equation (d) ; their directions may be determined
by the help of equations (c). A makes the angles a, (3, y, with the
coordinate axes. We write cos ax = /1} cos f3l = mv and cos yl = nv The
corresponding notation for B and C is exhibited in the following
table :
(g)
From equations (c) the following relations hold among these
quantities :
Am1 = YJ1 +
C13
Em. = Yxlz + Yym., + Yzn2 :
Bnz = ZJZ + Zum.2 + Zpz
A>3,
J3 + Zym3 + Ztna.
These equations can be solved for the components of stress Xx, Yy, etc.
These quantities may, however, be determined more easily in the
following way. Through a point P draw the lines PA', PB', and PC'
parallel to the directions of the principal stresses A, B, and C. These
three lines, together with a plane F parallel to the p-plane, deter-
mine a tetrahedron. The plane F is so placed that the tetrahedron
is infinitely small ; its base is dF. The areas of the faces which meet
at P are I^dF, l2dF, and lsdF. The force acting on unit area in
1-^dF in the direction of the o>axis is Al^ ; the forces acting on unit
area in the two other faces are B12, Cls, respectively, and the force
72 THE THEORY OF ELASTICITY. [CHAP. n.
acting on unit area in dF is Xx. That the tetrahedron shall not move
in the direction of the ar-axis we must have
2B . l2dF+ 13C . l3dF= XxdF or X, = Al* + Bl.22 + C732.
By a similar process we obtain for the other components the following
equations :
2 + Cl3n5,
Cl3m3.
It may easily be seen that these values of the components of stress
satisfy equations (h), if the known relations among the quantities
given in (g) are taken into account.
SECTION XXVII. FARADAY'S VIEWS ON THE NATURE OF FORCES
ACTING AT A DISTANCE.
Newton considered the action between two masses as an action at
a distance which is not propagated from particle to particle of the
medium surrounding the masses. Faraday, on the other hand, in
discussing electrical action, held that the intervening medium is the
seat of the action between two charged bodies, and that the action
is transferred from particle to particle. In each of these particles
electricity is displaced in the direction of a line of force, one end
of which becomes positively and the other negatively electrified.
In a body thus polarized the particles are so arranged that poles
of opposite name are contiguous. Hence the lines of force tend
to contract, and a state of stress arises in the medium. This stress
is similar to the elastic stress, and was called by Maxwell electrical
elasticity. In Chapter V. of his Treatise on Electricity, Maxwell,
using Faraday's hypothesis, developed a theory which we will now
proceed to discuss. Since electrical and magnetic forces conform to
the same law as that of universal attraction, the discussion may be
made perfectly general, and applicable to all forces between bodies
which are inversely proportional to the squares of the distances
separating the bodies.
Let the potential $ be given for all points of the region. The
density p is determined from the potential by Poisson's equation
(a) 32^/?.r2 + 3-Y/3y2 + ^/dz2 + ±TTP = 0.
SECT, xxvii.] FORCES ACTING AT A DISTANCE. 73
The mass pdv contained in the volume-element d-v is acted on by
a force whose components are
The upper sign holds for magnetic or electrical attractions, the lower
for mass attractions. Introducing the value of p given in (a) the
component acting in the direction of the .r-axis becomes
This quantity must be capable of representation as the sum of three
differential coefficients with respect to x, y, and z. We have
Hence the force which acts in the direction of the a-axis on the
volume-element dv is
If we designate the components of force which act on the unit
of volume by (X), (Y), and (Z) [XXV.], and if, for brevity, we set
X = - -d^px, Y=-
we obtain
f (X) = ±
(b) | (F)
( (Z) = ± I/STT . [Zd(XZ)/-dx + Zd(YZ)l*dy + 3(Z2 -X2-
Since these equations are perfectly analogous to those which deter-
mine the force with which stresses act on the unit of volume, we
may consider forces acting at a distance as arising from stresses in
the medium. If we are dealing with universal mass attraction, the
ether may be assumed to be the intervening medium; if we are
discussing electrical actions, the dependence of the stress in the
ether on the matter which fills the region, air, water, etc., must be
taken into account. It is not necessary to enter upon this question
in our treatment of the subject.
A comparison of equation (b) with equation XXV. (b) shows that
Xx = ± (X2 - Y2 - Z2)/Sir, Y
(c) -I Yy=±(Y2-X2-Z2)/87r, Zx
Zz=± (Z2 -X2- FO/STT, X,
74 THE THEORY OF ELASTICITY. [CHAP. 11.
To determine the principal stresses in the medium, we use equations
XXVI. (e), Avhich give
A + B + C = + (X- + F- + Z2)/8ir,
BC+AC+AB= - ((A'2+ F2 + Z2)/87r)2,
ABC = ± ((X'2 + Y1 + Z-')/8ir)8,
If we set (d) (X'2+Y* + Z'2)/8ir = S, A, B, and C are the roots of
the equation D3 ± SD'2 - SW + Ss = 0 or (D + S) (D ± S)'2 = 0. We
have therefore either (e) A = + S, B=C = -S or A =- S, B=C = + S.
Hence two principal stresses are always equal. In order to deter-
mine their directions, a, (3, and y must be calculated from XXVI. (c).
It is easiest to determine the directions of the equal stresses B and
C. If the values of ±S, given in (d), are substituted for S in the
equations referred to [XXVI. (c)], using the negative value of S in
combination with the positive value of Xx, etc., and vice versa, we
obtain (f) A" cos a + Ycos (3 + Zcosy = Q. Hence both of the equal
principal stresses are perpendicular to the direction of the force ; the
third principal stress is in the direction of the force, and is equal
to the square of the force divided by STT.
It has thus been shown that all forces acting at a distance may
be explained by a state of stress in an intervening medium. From
this point of view universal mass attraction is replaced by a negative
stress, that is, a pressure, in the direction of the lines of force, and
a positive stress, that is, a tension in all directions perpendicular to
the force. A surface-element which lies perpendicular to the direction
of the force is acted on by a tension which is equal to the force.
In the case of magnetic and electrical attractions the opposite holds
true. There is no independent evidence for the existence of such
stresses in the case of gravity ; but several phenomena in electricity
indicate that the medium between two electrified bodies is in a state
of stress, and no facts are known that are inconsistent with the
assumption that this stress is the cause of the forces acting on the
bodies.
SECTION XXVIII. DEFORMATION.
If a body changes its shape or its position in space, one of its
points, whose coordinates are originally x, y, .:, may be so displaced
that its coordinates become x + £, y + ??, z+ £ £, ->;, £ are the pro-
jections of the path which P has traversed or the components of
SECT. XXVIII.]
DEFORMATION.
the displacement. If £, 77, £ are given as functions of the time, the
position of the point P at any instant is determined. The motions
of the separate points of the body are in general different, that is,
£, ?/, £ are functions of x, ?/, z. We will first consider some simple
motions of the body.
If £, 77, £ are equal for all points of the body, the points all
move through equal distances and in the same direction ; the motion
is a translation. In this motion all parts of the body remain at fixed
distances from each other, and there are no internal forces developed.
This holds also in the case of a rotation of the body about an axis.
Let the axis of rotation be parallel to the o;-axis, and pass through
the point P (Fig. 41), whose coordinates are x, ?/, s. Let a point Q,
whose coordinates are x', y , z, traverse the path Qli = lix . r, where
r=QS is the distance of the point Q from the axis, aud hx is the
angle of rotation. By this rotation the ^/-coordinate is diminished
by BB' = QR(z' -z}/r = hx(z'- z), and the .^-coordinate is increased by
CO' = QR(y' -y}lr = hx(ij -y). If the body rotates at the same time
about two other axes, which are parallel to the y- and z-axes, and
if the angles of rotation are designated by liy and hz respectively,
the coordinates of Q are increased by £, 17, £, which have the following
values :
* W --9)*.- &-*)*,-
We may now proceed to the discussion of the general case, in
which the points of the body change their relative positions. Let
the point P, whose coordinates are .T, y, z, pass during this motion
to the point P', whose coordinates are x + g, y + r,, z + £; let another
76 THE THEORY OF ELASTICITY. [CHAP. n.
point Q, whose coordinates are originally x\ y', z1, pass to the point
Q, whose coordinates are x' + £', y' + i/, z' + £ '. If £ is a known func-
tion of x, y, z, we will have
£' = £ + (x' - x^j-dx + (y - y)3£/3// + (z' - *)3£/3* + . .* .
We may assume that P and Q are infinitely near, so that
x' — x = dx, y' — y = dy, z' — z = dz.
Neglecting terms of the second order we obtain the following relations,
£' = £ + 3£/d.r . dx + ^I'dy . dy + V£/3z . dz,
77' = ?; + 'drf/'dx . dx + 'drj/'dy . dy + 'dq/'dz . dz,
(' = £ + 3£/3.r . dx + 'dtfdy . dy + *dtfdz . dz.
By introducing the following notation,
yx = xy
we obtain
(£' = £ 4- x,dx + xvdy + x.dz - hzdy + hjlz,
rj' = r] + yjlx + yydy + yjlz - hjz + hzdx,
T = C + M* + z^y + zzdz - hydx + h,dy.
These equations determine the motion of a point in the neighbour-
hood of P. This motion is compounded of a translation, whose
components are £, »;, £ a rotation, whose components are h^ ky, h:,
and two motions, determined by x^ y^ zz and zf xa yf If we confine
our attention to the way in which the form of the body changes,
we need only consider the motion whose components e?£, d^, d£ are
determined by the following equations :
dr) = yrd
d£ = ztdz + z,dx + zydy.
To interpret the coefficients xz, y^ sz and z^ a;,, y^ we assume that
all except xx are equal to zero. Then dg = xx.dx and dr) = d£=Q. The
change of form corresponding to this is a dilatation of the body in
the direction of the ar-axis, by which dx increases by d£. The co-
efficient xx therefore represents the dilatation of a unit of length
parallel to the x-axis, or is the dilatation in the direction of the
;e-axis. Hence yv and zt are the dilatations in the directions of the
y- and 2-axis respectively.
SECT, xxvin.] DEFORMATION. 77
If, on the other hand, all the coefficients vanish with the exception
of zy, we have dg = Q, d^=zy.dz, d£=zy.dy. The particles are dis-
placed in a plane parallel to the yz-plane, and their distances from
the yz-plane remain unchanged. Let the original coordinates of the
point P (Fig. 42) be x, y, z; let ABCD be a square, the length of
whose sides is 2a. The point A, whose original coordinates were
x, y + a, z + a, referred to the axes PY and PZ, is displaced to A\
whose coordinates are a + zya, a + zya.
A therefore lies on PA produced. The
points B and D are displaced to B' and
Z>', which lie on BD ; C is displaced along
AC produced to C". The square ABCD
becomes the rhombus A'B'C'D. This
change of form is called a shear; the
quantities zy, xa yx are called components
of shear.
In the theory of elasticity we consider ^ FIG 49
only very small deformations of the body ;
the components xx, yy, etc., are consequently small quantities, whose
second and higher powers may be neglected. The volume of the
body is not changed by a shear; the square whose area is 4a2 will
become a rhombus A'B'C'D' whose area is
2PA' . PB' = 2(a 4 zya)j2(a - zya)j2 = 4a2(l - z?).
If we neglect zy-, the area of the square is equal to that of the
rhombus; hence the volume will not be changed by the shear.
From Fig. 42 it is evident that the infinitely small angle between
AB and A'B' is equal to azy/a = zy; hence the right angle DAB is
diminished by the shear by 2zy, so that
As the result of a dilatation determined by xa yy, zz, the volume
of the parallelepiped dxdydz becomes dxdydz(\ + xx)(l +yy)(l+zz). If
the components of dilatation are supposed infinitely small, we may
neglect their second and higher powers. Hence the increase in
unit volume is 0 = xx + yy + zz. 0 is called the volume dilatation. Sub-
stituting the values of x^ y^ za we have also
(e) Q = 'd£fdx + 'fr)l'dy + 'dtrdz.
Let dr be an element of a straight line which makes the angles
a, /3, y, with the coordinate axes ; then
dx = dr cos a, dy = dr cos (3, dz = dr cos y.
78 THE THEORY OF ELASTICITY. [CHAP. n.
By the deformation dr becomes dr', and makes the angles a, (3', y'
with the axis, so that
dx + dg = dr'cosa ; dy + djj = dr' cos (B' ; dz + d£ =dr' cosy',
from which dg, dt], d£ may be determined by equations (d). If the
direction of the line dr remains unchanged, we have a = a, /3 = /3',
and 7 = 7', and hence dg = dpcosa, drj = dp cos (3, d£=dpcosy, where
dp = d(r'-r). The length dp is the elongation of dr, and dp/dr is
the dilatation s in the direction of the line dr. Hence we have
.9 = dp/dr.
Equations (d) then assume the following form:
{(xx - s)cos a + xycos (3 + xzcos 7 = 0,
yxcos a- + (yy- s) cos (3 + yzcos y = 0,
2^ cos a + zy cos (3 + (zz- s) cos 7 = 0.
A comparison of these relations with those of XXVI. (c) shows that
they both may be interpreted in a similar way.
There are therefore three directions perpendicular to each other,
called the principal axes of dilatation, in which only dilatations occur ;
every line-element which is parallel to one of these three directions
contains after deformation the particles which were in it before the
deformation. This conclusion holds only on the supposition that
the body does not rotate, a supposition which has been made in
deducing equations (d). If the principal dilatations thus deter-
mined are called a, b, c, we have, as in XXVI. (e),
f a + b + c = xx + yy + zz,
(g) j be + ac + ab = z$, + x^ + ypx - z* - x* - y*,
{ abc = xji^t + 2ZfCjff - x^f - yp* - zjy*.
The first of these equations shows that the volume dilatation does not
depend on the position of the system of coordinates.
In the same way as that in which the components of stress are ex-
pressed in terms of the principal stresses [XXVI. (i)] xa xy,... may be
expressed in terms of the principal dilatations a, b, and c. Denoting
the cosines of the angles which the direction of a makes with the axes
by lv mv nv and the cosines of the angles which b and c make with
the axes by 12, m.2, n2; 13, m3, %, we obtain
f xx = all2 +bl.22 +c/32; 2,«om1n
(h) | yy = a™,l2 + bm22 + cffl./ ; xt-=alln1
I zt = an^ + bn.2- + en./ ; yx = al-pi^
SECT. XXIX.
STRESSES AND DEFORMATIONS.
SECTION XXIX.
RELATIONS BETWEEN STRESSES AND
DEFORMATIONS.
The study of the deformations of an elastic body has shown that
a parallelepiped which is stretched by forces applied to its ends, is
increased in length and diminished in cross section. If we only
consider forces which are so small that the limits of elasticity are
not exceeded, the elongation s per unit of length is s = S/E, where
E is the coefficient of elasticity and S the force acting on the unit of
surface. The contraction s' per unit of length parallel to the end
surfaces, is given by s' = k . S/E, where k is a constant. It is assumed
that the body is isotropic, that is, equally elastic in all directions
and at all points.
We will first consider a rectangular parallelepiped, whose edges
are parallel with the coordinate axes. The normal forces are denoted
by Xx, Yy, Za and a unit of length which is parallel to the a-axis
increases by xx; the units of length which are parallel to the y- and
2-axes respectively increase by yy and zz. We then have
-k(Xx+Yy)/E.
From XXVIII. (e) the volume dilatation 0 is
From these equations we deduce
Xx = kEQ/(l + k)(l- 2k) + Exx/(l + k).
Setting A. = kE/( I + k) (1 - 2k), p. = i£/(l + k),
we obtain (a) X, = XO + 2/tr, ; Yy = XQ + 2p.yy;
and by addition
(b) Xx + Yy + Zz = (3X + 2^)9.
To investigate the relation between the
shears and the tangential forces we may use
the following method, due to V. v. Lang.*
If the prism ABCD (Fig. 43) is stretched by
the tension S applied to each unit of surface
of its ends AB and CD, it takes the form
AB'C'D'. Four plane sections EF, FG, GH,
and HE are passed through the prism, which
mark out the rectangle EFGH on a plane
parallel to the axis; the rectangle EFGH becomes by deformation
* V. v. Lang, Theoretisc.he Physik; § 411.
80 THE THEOEY OF ELASTICITY. [CHAP. n.
the parallelogram E'F'G'H'. The angle AFE is represented by </>.
The tangential stress T, which acts on the surface EF in the
direction EF, is given [XXIII. (b)] by T = S sin <f> cos <£.
Since < BFG = |?r - <£, the same tangential stress T acts on GF
in the direction GF. On deformation the angle AFE becomes
AF'E' = <f> + d<f>, and we have
Now since s=--SjE and s' = kS/E are infinitely small, we have
( 1 + s)/(l - s') = 1 + s + s' = 1 + (1 + k)S/E.
Further, we have
and tg (<£ + dfy = tg <f> + d(f>/co$2<j>,
so that d<l> = (l+k)S sin <f> cos <£/ E = ( 1
Hence the change of the angle <£ is proportional to the tangential
stress T. Since the same tangential stress acts on GF as on EF, the
angle BFG increases by d<f>, the angle EFG diminishes by 2d(f>, and
the angle FGH increases by 2d<f). The shear is thus equal to 2e?<£,
and we have 2d<f>=2(l+k)T/E. But 2d<f> is the quantity 2zy pre-
viously introduced, when the rectangle EFGH is parallel to the
2/3-plane; and hence T=Zy and zy = (l+k)ZJE. If we set
we have (c) ^ = 2^, A>2/^z, Yx = 2^yx.
The equations (a) and (c) are the solution of the problem, to find
the components of stress, when the deformations are given, and
conversely. They contain only two constants, A and /*, which involve
the deformations caused by simple dilatation in the following way :
f X = *£/(l +*)(!- 2*); ,* =
t£-<8V+4i*V(*+p); *
Since A and /A are positive, & must be less than \.
The relations between the elastic forces and the deformations may
also be derived by another method. Let the principal stresses A, B,
and C, at the point P, be known in magnitude and direction [cf. XXVI.
(g)]. An infinitely small parallelepiped, whose edges are parallel
to the directions of the stresses A, B, and C, is extended in those
three directions. The increments a, b, and c of the unit of length
are parallel to A, B, and (7, and as in (a), we have
(e) A = \e + -2iM, B = XQ + 2pb,
when 0 = a + b + c, or [XXVIII. (g)], Q =
SECT, xxix.] STRESSES AND DEFORMATIONS. 81
By applying the formula XXVI. (i), we obtain the equation
Xx = X0 + 2/x(a/12 + bl/ + c/32),
which, from XXVIIL (h), becomes Xx = X9 + 2/xa^ The expressions for
Yy and Zt are obtained in a similar way.
From XXVI. (i), we have
and hence [XXVIIL (h)] Zy = 2fj.zy. We obtain the expressions for A",
and Yx in a similar way.
The coefficients E and k depend on the nature of the body. It
was at one time believed that k had the same value for all bodies.
This opinion was first expressed by Navier. He assumed that bodies
are made up of material points which repel one another, and on this
assumption concluded that k = £. Poisson also had the same opinion.
While k is a mere number, the coefficient of elasticity E is deter-
mined by E = S/s; the fraction \\E is called the modulus of elasticity.
Sis the force which acts on the unit of surface, and [III.] its dimensions
are LT~2M/L2 = L~1T~'2M. Since s is the ratio between the elongation
and the original length, it is also a mere number. Hence the dimen-
sions of E are L~1T-'2M.
In practical units E denotes the number of kilograms which would
produce an elongation in a rod of one square millimetre cross section,
such that its length is doubled. In order to transform it into absolute
measure, we notice that the weight of one gram is about equal to
981 dynes, and that therefore the weight of one kilogram is equal
to 981 000 dynes. The cross section must be taken equal to 1 sq. cm.,
and the number must therefore be multiplied by 100, so that the
factor of transformation becomes 98,100,000. According to Wertheim,
E equals 17278 in practical units for English steel ; therefore, in
absolute units it equals 17278. 981 . 105= 1,695 . 1012.
In the case of fluids, the discussion is simplified by the condition
that a fluid always yields to tangential forces, so that, when it is in
equilibrium, there are no tangential forces acting in it. This condition,
from (c), enables us to set /^ = 0. If the fluid is subjected to the
pressure p, and if its volume v is thereby diminished by dv, we
have from (b)
(f) -{ or, since /^ = 0,
dv =pv/X.
82 THE THEORY OF ELASTICITY. [CHAP. n.
If, for example, the unit volume of water is diminished by 0,000 046,
when the pressure is increased by 1 atmosphere, we have
X=pv/dv = 7& . 13,596 . 981/0,000,046 = 2,204 . 1010.
In the case of gases, if we represent the original pressure by P,
and its increase by p, Mariotte's law gives the equation
Pv = (P+p)(v-dv).
Assuming that p is very small in comparison with P, we obtain
dv=pv/P,
and therefore, for gases, we have from (f), (g) P=X.
SECTION XXX. CONDITIONS OF EQUILIBRIUM OF AN ELASTIC BODY.
If a force whose components are X, Y, Z acts on the unit of
mass of a body, we have from XXV. (c)
(a) VX,J'dx + 'dX,l'dy + 'dXJ'dz + pX=Ot etc.
Further [XXIX. (a) and (c)] we have
(b) Xx = XQ + 2fj..^px, Zy=Yi = p.(dil'dy + 'd-ql^z\ etc.
If the values for XM etc., are substituted in (a), it follows that
c (X + p) . ae/a-e + /*V2£ + pX = 0,
(c)
I
By the use of our former symbols for the components of rotation, viz.,
| 2A
\
equations (c) become
? -
<e) (X + 2/t)
I (x + 2/*)
If the first equation is differentiated with respect to .r, the second
with respect to y, and the third with respect to ,r, we have by
addition when p is constant,
(X + 2/x)y26 + P(dXfix + VY^ij + -dZI-dz) = 0.
If X, Y, and Z are the derivatives of a potential ^, and if y'-^ = 0
everywhere within the body, then (f) y20 = 0.
This result must be supplemented by the conditions of equilibrium
of the surface of the body. The force acting on the surface-element
dS, whose components are FQ'R, is in equilibrium with the elastic
SECT, xxx.] STRESSES IN A SPHERICAL SHELL. 83
forces which act on the parts of the body contiguous to dS. If
.Xx, Yy'} etc., are the components of the elastic forces, we have
(g) F = XX COS a + Xy COS ft + Xi COS 7.
There are similar values for Q' and R'. The symbols a, (3, 7
represent the angles which the normal to the surface directed out-
ward makes with the coordinate axes.
We assume (h) £ = ax, r) = by, {=cz, where a, b, and c are constants.
.£, ij, and £ are therefore linear functions of x, y, and z, and £ depends
only on x, 77 only on y, and £ only on z. On this supposition
[XXVIII. (b)] the deformation of the body is made up of dilatations
•only. The volume dilatation is 0 = a + i + c; hence the values
assumed for £, r/, £ satisfy equations (c), if we neglect the action of
external forces. We have further
If we set XX = S, Yy = 0,
we have b = c and
S=X(a + 2b) + '2pa; 0 =
The last equation gives
b/a= -£A/(X + /*)= -k,
and the first # = a(3A/* + 2/*2)/(A + p) = Ea.
The equations thus obtained give the law of the expansion of an
elastic prism.
SECTION XXXI. STRESSES IN A SPHERICAL SHELL. '
Suppose a spherical shell, bounded by two concentric spheres,
whose radii are rl and r2. Of these we assume r2 > rr Suppose a
constant hydrostatic pressure pl applied to the inner surface, and
a similar pressure p.2 applied to the outer surface. The pressures p^
and p2 are perpendicular to the surfaces. Let the centre 0 of the
sphere be the origin of coordinates, and let the distance from 0 of
any point in the shell be r. On the hypothesis that has been made
with respect to the pressures, all points lying in the same spherical
surface having the centre 0, receive equal displacements from the
centre. Let the displacement of the point considered be er, where e
is a very small quantity. We then have
84 THE THEORY OF ELASTICITY. [CHAP. n.
Since e is a function of r only, we may set
£ = fr . x/r = d<j>/dr . "drfdx = 'd^j'dx,
where <f> is a new function of r. We may represent ij and £ in a
similar way, so that
(b) £ = 3.£/3.£, ^30/fy, t=3</>/32.
Hence we have (c) 6 = V2<£.
The equations XXX. (c), if the action of gravity is neglected,
become
(A + 2fi).V23<£/9.c = 0, (A. + 2/z). V23</>/3y = 0, (A + 2/*) . V^fdz = 0,
so that (d) 0 = V-<£ = a, where a is a constant.
From XXX. (b) the components of stress are
c Xx = Aa + 2p .
(e) J Yy = Xa + 2 p.
I Z, = Xa + 2/i.32^/?38; F, = 2/t . 32</>/3o%.
The stress in a surface-element perpendicular to r is given by
XXIV. (a), if we set
cos a = z/'r, cos (3 = y/r, cos y = z/r.
If the components of stress are P, Q, and R, we have
P = Xa . a-/r + 2/*(.r/r . 32<^>/3a;2 + y/r . ^fdxdy + zlr . &<t>/'dzdz).
Using the equations
Pt/dr* - x*/i* . d<f>/dr +l/r. dj/dr,
xy/f* . d*<j>/dr* - xy/r* . d<j>/dr,
xz/r* . d^/dr* - xzji* . d<f>/dr,
we have P = (\a + 2p. . d^jdr*) . xfr.
Similar expressions may be obtained for Q and It. Hence a prin-
cipal stress (f) A = Aa + 2/* . d*<j>/dr2 acts on the surface-element
considered.
For a surface-element which contains r, the components are obtained
in the same way. If a, /?, y are the angles which the normal to the
surface-element makes with the axes, we have
P = Aa cos a + 2/*(92<£/3;e2 . cos a + 32<£/3.c3y . cos ft + 32</>/3x3^ . cos y).
If we notice that in this case
cos a . x/r + cos (3 . y/r + cos 7 . z/r = 0,
and use the expressions given above for the differential coefficients,
we have P = (\a + 2/*/r . d<J>/dr) cos a.
We may obtain Q and R by replacing a by ft and y respectively.
SECT, xxxi.] TOESION. 85
Hence the principal stress B acting on the element is
From (d) and XV. (1) we have
and therefore
(h) d<j>/dr = *ar + b/r* ; d2<j>/dr2 = ^a- 26/r3.
From (f) and (g) it follows that
A = (X + |/i)a - 4/xJ/r3 ; B = ( A. + |/z)a + fyb/r3.
For r = rv A= -pv and for r = rz, A= -p2, therefore
a = 3/(3X + 2/x) . (pfi
and A-
B =
SECTION XXXII. TORSION.
Let us consider a circular cylinder whose axis coincides with the
£-axis; and let the circle in which the xy -plane cuts the cylinder
be the end of the cylinder and be fixed in position. If torsion is
applied to the cylinder, a point at the distance r from the axis describes
an arc r<f>, parallel to the xy-plane, whose centre lies on the z-axis.
This angle, in the case of pure torsion, is proportional to the distance
of the point from the zy-plane, so that <f> = kz, where k is a constant.
The displacement of this point is krz, and its components £, 77, £ are
(a) £=-kyz, r, = kxz, f=0.
Using these values, we find that the volume-dilatation 0 is zero, that
is, pure torsion dots not cause a change of volume. We have further
[XXX. (b)], Xx = 0, Yy = 0, ZZ = Q,
and hence no normal forces act on the surfaces which are parallel
to the coordinate planes. On the other hand, we have
Z^fjJcx, Xt=-fdy, Yx = 0.
A surface-element perpendicular to the £-axis is acted on by the
tangential forces Y,= + fj.kx and Jfz= - pJcy, whose resultant pkr is
perpendicular to the radius r and to the z-axis.
By XXIV. (a) we reach the same result. That is, we get
f P = - ftky cos y, Q = pkx cos 7,
I R = -nkycosa
86 THE THEORY OF ELASTICITY. [CHAP. n.
For the stress on the surface of the cylinder we must set
cos a = x/r, cos f3 = y/r, cos y = 0.
We will then have P = 0, Q = 0, ^ = 0.
Hence a surface-element perpendicular to the radius, or which is part of
the surface of a circular cylinder whose axis is the z-axis, is not acted on-
by a force.
To find the surface-elements on which the only forces which act
are normal forces, we use equation XXVI. (d), which, in the case
before us, becomes
If A, B, and C are the roots of this equation, we can set
A = 0, B = pkr, C=-ph\
If the angles between the axes and the normal to one of these surface-
elements are represented by a, /3, y,
S cos a = — p.ky cos y, S cos f3 = pJcx cosy, Scosy = — //.£?/ cos a + //Xvr cos /£.
If we substitute in these equations the particular values of S given
by A, B, and (7, the values of a, J3, y thus obtained show that the
stress A = 0 acts on a surface-element perpendicular to r ; and that
B and C act in directions perpendicular to the radius r, and making
angles of 45 degrees with the 2-axis. B acts in the same direction
as the torsion, C in the opposite direction.
For example, considering a point which lies in the surface of the
cylinder and in the .s-plane, and setting therefore Y=0 and A" = r,
we have S cos a = 0, S cos /? = pkr cos y, S cos y = pier cos p.
When S = Q we have y = /3 = ^7r; when S— ±pkr we have a = ^7r,
cos (3 = cosy. Since cos2a + cos2/? + cos2y = 1 , we have eos/? = ±s/|.
The moment of force M to which the torsion of the cylinder is
due is
M= (f&r. Zirrdr.r.
The upper limit R of the integral is the radius of the cylinder,
Integrating, we have M=^TrfjJcR* = ir<f>ii.R4j2l, where I is the length
of the cylinder and </> the angle of torsion.
The factor T = Trp.R*/2l is called the moment of torsion of the cylinder.
It depends only on the dimensions of the cylinder and the constant
of elasticity /x. For this reason //, is called the coefficient of torsion.
SECT. XXXIII.
FLEXURE.
87
0.
a
SECTION XXXIII. FLEXURE.
It is not possible to give a rigorous discussion of the flexure of
a prism. We will, therefore, confine ourselves to an approximate
calculation in one very simple case*.
Let ABCD (Fig. 44) be the prism considered. Its length is supposed
horizontal and coincident with the axis Ox. The axis Oz is directed
perpendicularly upward, and the axis Oy is therefore horizontal.
After flexure, the cross section AB is
displaced to A'£', which may lie in the
same plane as AB. Another plane cross
section FG, also perpendicular to the
axis, is displaced by the flexure to F'G' ;
we assume that the section F'G' is also
plane, and that the plane F'G' cuts the
plane A'B' in a horizontal line passing y
through P. This line of intersection is
supposed to be common to the planes
of all sections perpendicular to the axis.
The parts of the prism which originally
lay in OQ lie after the flexure in OQ', FlG 44
which we will consider as the arc of a
circle whose centre is P. Such a flexure is called circular. All the
lines in the prism which were originally parallel to the x-axis become
circles, whose centres lie on the straight line passing through P.
Represent the original coordinates of a point M in the section
AB by 0, y, z, and its coordinates after flexure by 0, y + ^0, 2+f0*
The same changes occur in the other cross sections, for example in
FG. If the coordinates of a point M' in FG are originally x, y, z,
they will become by flexure x + g, y + t], z + £ We set L. OPQ' = <f>,
OP = pa,nd OQ = OQ'. This last assumption is admissible, since there
is always one line whose length does not change by flexure, and
since we have as yet made no assumption as to the position of the
z-axis. We therefore obtain
* + 1 = * + to -(/> + * + &)(! - cos <£).
If p is very great in comparison with x, z, and £0, we may set
sin <£ = x/p ; 1 — cos <f> — a^/2p8
and obtain (a) £ = xz/p, n = ^ t=to-x2/2P-
* Barre de Saint- Venant, Mem. pres. par div. Savants. T. 14. Paris, 1856.
88 THE THEORY OF ELASTICITY. [CHAP. n.
We may so determine % and £0 that all the components of stress
except Xx vanish ; hence we may write
(1 ) X, = A6 + 2^/p = S, (4) Z, =
(2) y,=Ae+2/u3v^=0> (5) xz
(3) Zt = \Q + 2iidf0fdz = Q, (6) Y,
Further, we have Q = z/p + 'dr]
From (2) and (3) it follows that (b)
Comparing (b) with XXIX. (d) it appears that the contraction of the
cross section is to the increase in length in the ratio of |A. to A + /z,
or of k to 1. Further, since T?O and £0 do not involve x, we have
from (b) rj0= - kyz/p +/(*), £0 = - kz*/2p + g(y),
when / and g designate two unknown functions. From (4) we have
-ty/p+f'(z) + g'(y) = 0, and hence f\z) = c, where c is an unknown
constant. It follows that
f(z) = cz + c', g(y) = kf/ 2p-cy + c"
and T,O = - kyz/p + cz + c', £0 = k(V2 ~ Z<2)/%P ~W + c".
At the point 0, where y = z = Q, we have Vo = ^> & = 0> and hence
c' = 0 and c" = 0. Since the prism does not turn about the ;r-axis
during flexure, it follows that for y = Q, ^ = 0 also, and consequently
c = 0. We obtain therefore
and further, from (a),
(d) t = xzl* ri=- kyz/p, t=k(y*-
These values for ^, rj, and ^ satisfy the equations XXX. (c), since
by hypothesis X=Y=Z=0. The equations 1-6 show that the
conditions of equilibrium are fulfilled. From (1) and (b) we get
X, --=S = (3X/A + 2/*2)/(A. + p.) . zfp. If we introduce the general coefficient
of elasticity E [XXIX. (d)], we have (e) S = Ez/P.
The resultant E of the forces S is (f ) R = Ejp . \z . dydz, and is equal
to zero, if the a;-axis passes through the centre of gravity of the
prism. If we assume this and then determine the moment M of
the forces S with respect to a horizontal line passing through the
centre of gravity, we will have M = \Szdydz = Ejp . \zzdydz = EJjp,
where J is the moment of inertia of the cross section. In order to
bend the prism so that an axis passing through the centre of gravity
of the prism becomes a circle of radius p, a rotating force of moment
M must act on each end surface ; the axes of the rotating forces
are perpendicular to the plane of the circle, and are oppositely
directed.
SECT, xxxin.] MOTION OF AN ELASTIC BODY. 89
The cross section of the prism is noticeably altered by the flexure.
Since the parts on the convex side of the prism are extended, and
the parts on the concave side compressed, the former tend to contract
in the directions of the y and .s-axes, the
latter to expand. If, for example, the
cross section is a rectangle ABCD, as in
Fig. 45, ABCD takes the form A'FC'D'.
The two plane surfaces whose projections
are represented in the figure by AB and ' r"~
CD are transformed into surfaces of double
curvature. We may consider A'B' and
C'D' as arcs with the centre E, while A'D'
and B'C' are straight lines which intersect
at E. The lines A' I? and B'C' are not
changed in length, AB is shortened and FlG- 45-
CD lengthened. If z = %BC, it follows from the definition of k [cf.
XXIX.] that
A'B' = AB(\ - kz/p), C'D' = CD(l + kz/p).
UOE = p, then
A'B' /C'D' = (p' - z)I(p +z) = (l- kz/P)/(l + kz/p),
from which it follows that p = kp. This relation has been applied
to the determination of k for glass prisms.
SECTION XXXIV. EQUATIONS OF MOTION OF AN ELASTIC BODY.
The resultant with which the elastic forces act on an infinitely
small volume-element dv of an elastic body in the direction of the
3-axis is [XXV.], (dXx/^x + 'dXy!dy + 'dXipz)dv. If the body is acted
on besides by attractions or repulsions, whose component in the
direction of the z-axis is X, the element dv is also acted on by the
component of force X.dv.p, where p is the density of the body.
Hence the .r-component of the acting forces is
(dXjVx + VXJVy + VXfiz + pX)dv.
If this resultant is not equal to zero, motion occurs in the direction
of the .T-axis, and the momentum imparted to the part of the body
under consideration in unit time is pdvd-(x + g)/dt? — pdvd'2g/dt2, where t
denotes the time. Hence we have
= VXJdx + VXJciy + "dXJdz + PX.
90 THE THEOEY OF ELASTICITY. [CHAP. ir.
If the components of stress are expressed by £, ry, and £ as in
XXX. (b), we obtain the equation
(a) p£ = (* + /*)• 39/3* + )"V2£ + pX
The equations for V) and £ are similar. «
As in XXX. (e) the equations (a) take the form
(b) p£= (A + 2/*) . 36/3x + 2/i(3A,/3« - 3/*2/3y) + /oA'.
If the force whose components are AT, F, and Z has a potential, and
if therefore X= -3*/3a:, F= -3¥/3y, 2T = -3¥/3*,
by differentiation of equation (b) with respect to #, y, 2 respectively,
and by addition, we have
(c) Pe = (A + 2/*)V2e-/>V2¥.
In what follows we assume that no external forces act, so that
the components X, Y, Z are zero. Therefore V2^ drops out of
equation (c).
SECTION XXXV. PLANE WAVES IN AN INFINITELY
EXTENDED BODY.
Lame* treated this form of motion in the following way. Suppose
a plane wave propagated in a direction which makes the angles a, J3, y
with the axes; let the velocity of propagation be V, and let the
direction of vibration make with the axes the angles a, b, c. If u
represents the distance of a point from its position of equilibrium,
U the amplitude, and T the period of vibration, the vibration at the
origin may be expressed by u= Ucos(2trt/T). At any other point,
whose coordinates are x, y, z, we have
(a) «= tfcosW.
We have further
(b) £ = u cos a, rj = u cos b, £ = ti cos c.
If the angle between the direction of propagation and the direction
of vibration is represented by <£, we have
cos <f> = cos a cos a + cos bcos/3 + cos c cos y.
For brevity we set
- tfsin jar/
**, t -
* Lame, Theorie de 1'elasticit^, p. 138. Paris, 1866.
SECT, xxxv.] PLANE WAVES. 91
and obtain
9 = 2KS/T V. cos </>, 36/ar - - ^2u/Tz V- .cos a. cos &
V2£ = - 47r2M/r2 V1 . cos a, £ = - 47r2M/r2 . cos a.
By the help of these relations and corresponding ones for t\ and
£ we obtain from XXXIV. (a)
r (A + p.) cos a cos <£ + (p. - pV2) cos a = 0,
(c)
I (A + /*) cos y cos <£ + (/x - pV2) cos c = 0.
If these equations are multiplied by cos a, cos/2, cosy respectively
and then added, we have (X + 2/X-/3/72) cos <£ = 0. We therefore have
either (d) (e) pV'2 = \ + 2fj. or cos<£ = 0. In the first case, equations (c)
become
cos a = cos a cos <£, cos b = cos (3 cos <£, cos c = cos 7 cos <£.
If the right and left sides of these equations are squared and added,
we obtain (f) cos2(/>=l, so that either <£ = 0 or <£ = TT. The vibrations
therefore occur in the direction of propagation ; they are called
longitudinal vibrations. In the second case <£ = JTT, that is, the vibra-
tions are perpendicular to the direction of propagation ; they are
called transverse vibrations.
Longitudinal Vibrations. — The velocity of propagation 12 of these
vibrations is determined by (d), (g) tt = <J(\ + '2/j.)lp. Hence con-
densations and rarefactions occur, since
To determine the stresses we assume that the waves are propagated
in the direction of one of the coordinate axes, say the 0-axis. In this
case we have
£ = 0, 17 = 0, C=Ucos{27r/T.(t-z/^)}.
From XXX. (b) the tangential forces are zero ; the normal forces are
(h) Xx = Y, = 27rX/7'I2 . U sin { 2^/T . (t - g/Q)}.
(i) Zt = 2v(\ + 2p)/TQ. Usin{2Tr/T.(t-z/tt)}.
Transverse Vibrations. — The velocity of propagation w, from (c), equals
(k) w = x//Vp. Since, for these vibrations, we have cos<£ = 0, we
also have 9 = 0, that is, neither condensations nor rarefactions occur.
If the wave is propagated in the direction of the 2-axis, and if the
vibrations are parallel to the .T-axis,
£= Ucos{2ir/T.(t-z/^)}, 77 = 0, £=0.
All components of stress vanish with the exception of the tangential
force Za (1) Zx = 2^/T^ .Usin{ 2v/T. (t - z/w) }.
•92 THE THEORY OF ELASTICITY. [CHAP. n.
In a solid, therefore, two different wave motions may exist, which
^re propagated with different velocities 12 and u>. From formulas
(g) and (k) the velocity 12 of the longitudinal vibrations is always
greater than the velocity w of the transverse vibrations. In liquids
and gases the only vibrations which can occur are longitudinal, since
for these bodies p. = 0.
For gases we have A. =JPJXXIX. (g)], and hence the velocity of
sound in air is (m) 12 = *JP/p. P must here be expressed in absolute
units. According to Regnault the density of atmospheric air at
Paris equals 0,0012932 under a pressure of 76 cm. of mercury, and
At a temperature of 0°C. Since the acceleration of gravity at Paris
is 980,94, the pressure of the air on a square centimetre equals
76.13,596.980,94 in absolute units. Hence the density p of the
air under a pressure P in absolute units is
<n) P = 0,001 293 P/76 . 13,596. 980,94 = P. 1,2759 .-lO'9.
Using this value of P/p we obtain 12 = 27 996 cm, or approximately
280 metres per second at 0°C. Since the density of the air at f C
is p = P. 1,2759. 10~9/(l+af) the velocity of sound at f is
where a is the coefficient of expansion of air 0,00 366. The result
obtained from this form of the theory does not agree with that found
by observation. Observation shows that 12 is about 330. The
reason why theory and observation are not in accord will be discussed
later in the theory of heat.
The velocity of sound in water is obtained in a similiar manner.
For water at 15° C. we have A = 2,22. 1010. At the same tempera-
ture we have P = 0,999 173, whence 12 =149 060 cm.
In a research carried out on the Lake of Geneva, Colladon and
Sturm found that the velocity of sound in water at 8,1°C. is 12 = 143 500
centimetres; the difference between the observed and calculated
values is explained by the difference in temperature, since A. increases
rapidly for water as the temperature rises.*
No observations have been made on wave motions in large masses
of metal ; but the velocity of sound has been determined in a metallic
wire. In such a body, however, sound is propagated with a different
velocity from that which it would have in an extended body. If
the wire is parallel to the z-axis, and if we consider only the motion
of the particles in the direction of this axis, the stress Z, at the
•distance z from the .r//-plane is, in our usual notation, Zt =
* Fogliani und Vicentini, Wied. Beibl. Bd. 8. S. 794.
SECT, xxxv.] WAVE MOTIONS. 93
At the distance (z + dz) the stress is
Zt + dZJdz . dz = E(dffd3 + 32£/322 . dz).
Hence a portion of the wire whose length is dz and whose cross
section is A, is acted on by a force given by AE&ydz2 . dz. The
equation of motion is pA . dz. ^=AE^i/dz2 . dz or (o) £= F232t/9.s2,
where V=jE/p. The integral of the differential equation (o) is
(p) £=cos {2ir/T. (t-zlV)} ; the velocity of propagation Vis obtained
from equation (p).
According to the researches of Wertheim the velocity calculated
from (p) agrees fairly well with the results of observation.
SECTION XXXVI. OTHER WAVE MOTIONS.
Spherical Waves. — We will investigate the circumstances of the
propagation of spherical waves in an infinitely extended elastic body,
when the direction of vibration of every particle passes through the
same point. We take this point as the origin of coordinates. As
in XXXI. (b) we set
(a) £ = 30/3a, i7 = 30/3y, £=30/3*,
where e£ is an unknown function of t and of the distance r from the
origin. The equations of motion [XXXIV. (b)] give (b) 0 = Q2V2<£.
In this case [XV. (1)] we may set V2<£ = !/?• . 32(r0)/3r2, and hence
(c) 32(r<£)/3<2 = fi232(r<£)/3r2. This equation is satisfied by
(d) 0 = a/r.cos{27r/r.(*-r/fl)},
when a is a constant and T the period of vibration.
The distance u of a point from its position of equilibrium is
u = 30/3r = - fl/r2 . cos { ^JT . (t - r/fl)} + ^irafBr . sin {2-n-fT. (t - r/Q)},
where B = £lT. If r is very much greater than the wave length we
can neglect the first term on the right, and have
u = A/r.sin{-27r/T.(t-rltt)}.
The wave motion is therefore one in which the wave surfaces are
spheres propagated with the velocit}^ 12.
Since the expressions (a) satisfy the equations of motion, if <f> has
the value given in (d), these equations are also satisfied if </> is replaced
by 30/3x, or by another differential coefficient taken with respect
to one or more coordinates.
Vibrations Due to Torsion.— Let the axis of a circular cylinder
coincide with the 2-axis, and its separate parts oscillate in arcs
94 THE THEORY OF ELASTICITY. [CHAP. n.
about the same axis. The components of displacement of a particle
from its position of equilibrium may be expressed [XXXII. (a)] by
{e) £= -<^y, fi = <t>x, £=0, where <£ is a function of z.
From XXVIII. (e) we have 0 = 0; therefore condensation and
rarefaction do not occur. The equations of motion are [XXXIV. (a)
and XXXV. (k)], £ = w2V2£, 77=0,^,,
whence we again obtain (f) <j> = w?'d2<t>/'dz'2. This equation is satisfied
by (g) <£ = asin {27T/T. (t-zjw)}. Hence w = v//x//j is the velocity
with which a wave motion is propagated in the direction of the
axis of the cylinder. From XXX. (b) the components of stress are
Zt = - Apx, Xz = + AM where A = Zira/Tta . cos {2ir/T. (t - z/u)}.
The other components of stress are zero.
If the cylinder is of finite length, stationary waves can exist in
it, that is, waves such that certain definite points of the cylinder
called nodal points are at rest, while on both sides of a nodal point
the vibrations are in opposite phase. The amplitude of the vibration
is greatest half way between two nodal points, at the ventral segments.
Stationary waves are formed when waves which have passed over a
certain point return to that point again in the opposite direction.
To find the period T of these vibrations, we notice that equation (f)
will be satisfied not only by (g), but also by <}> = b sin{2?r/T. (t + z /«>)},
and in general by
<£ = B sin 27rt/T. cos (ZirzjTu) + Ccos (2irt/T) . sin (2vz/T<o),
where B, C, and T are constants. If the points for which ^ = 0 are
fixed, the constant B will be zero and (i) <j>=Ccos (2vtjT) . sin (2irz/Tw).
If I represents the length of the cylinder, and if the points for which
.z = l are also fixed, we will have <£ = 0 when z = l, and therefore
pTr, where p is a whole number. Hence
If, on the other hand, one end of the rod is free, Yz = X2 = 0 when z = I.
.Since Xt = - w . 3<£/a?, YZ=+IM. 3<f>/'dz,
we have 9<£/3.z = 0 when z = l.
In this case we obtain from equation (i) 2irl/T<a = ^(2p+ 1). TT, where
j? is a whole number; and hence T=M/(2p+ 1). v//>//x. If both
ends of the rod are free, T=2l/
SECT, xxxvii.] VIBRATING STRINGS. 95
SECTION XXXVII. VIBRATING STRINGS.
Although the problem of the motion of vibrating strings is only
slightly connected with the theory of elasticity, a simple example
of this form of motion will be considered here. We suppose a
perfectly flexible string stretched between two fixed points A and
B. If P is the stress in the string, /0 the length of the string before
the application of the stress, and / its length while the stress is
applied, p the cross section of the string, E the coefficient of elasticity,
we have 1-10 = P10/FE. Let the string be slightly moved from its
position of equilibrium, that is, the straight line which joins A and
B, and let the new form of the string be designated by ACDB. By
this deformation the length of the string is increased by dl = dP . 10'FE.
It is here assumed that dP is infinitely small in comparison with P,
so that we may set the stress in the string everywhere equal to P.
For the sake of simplicity we suppose that the motion of the
string is always in one plane, say the z#-plane. Let A be the origin
of coordinates, and let B lie on the a>axis at the distance I from A.
The distance of any point C of the string from A may be represented
by 5, and that of the infinitely near point D by s + ds. The com-
ponents of stress at C in the directions of the x- and y-axis respectively
are Pdx/'ds and P'dy/'ds. For the point D the similar components are
PCdx/'ds + 'd-x/^s2 . ds) and P(byl^s + 32y/3s2 . ds).
The infinitely short portion CD of the string is therefore acted
on by the force P^x/cts- . ds in the direction of the z-axis, and by
the force P&y/'ds2 . ds in the direction of the y-axis. If the string
is displaced only very slightly from its position of equilibrium, we
can set s = x; the z-component then vanishes and the particles of
the string oscillate perpendicularly to the z-axis. If m represents the
mass of unit length of the string, the equation of motion is
or, if we set ma? = P, (a) ij = a2d*y/3a:2. The integral of this differential
equation is (b) y = AH cos (mrat/l) . sin (mrx/l), where n is a whole
number. When 2 = 0 and x = l we have y = 0, and when / = 0,
y = Ansin(mrx/l) ;
this is the equation of a sinusoid.
If, in the general case, the form of the string when t = 0 is given
by the equation y=f(x), then
f(x) = Al sin (TTX/I) + AZ sin (2irx/l) + A3 sin (3vx/l) + ... .
96 THE THEORY OF ELASTICITY. [CHAP. H.
The coefficients Av A2, A3 ... are determined in the following way.
Let the general term of the series be Ansinn<f>, where <t> = -xjl. If
both sides of the last equation are multiplied by sin «<£, we will have
/(/^>/7r)sinn^) = ^1 sin <£sin?t<£ + ^2sin2(£sin72<£ + ... + Ansin-n<f> + ....
If this equation is multiplied by dfa and integrated between the
limits 0 and IT, we will have
ff(l<f>/Tr) sin n<f> . d<f> = An / s
For if in and n are different numbers, we have
~w. »»
/ sin m<f> sin n<f>d(f> = \ I (cos (m - w)<£ - cos (m -f n)$)e?<£ = 0.
But when they are equal, / sin%<£d<£ = ITT. Hence
(c) A „ = 2/7T . /W«-) sin n<f> d<f> = 2 // . f(x) sin (n«c//) . rfz.
If, for example, the string is so displaced from its position of
equilibrium that a point in it at the distance p from the end A is
moved through the distance h in the direction of the y-axis, we have
f(x) = hxjp for 0 < x <p, but f(x) = h(l - x)j(l -p) for p < x < /. Hence
.. . .
p I JP l-p I
and therefore
An = 2hF/p(l -p) . (sin ^\ /»V2.
We obtain for y,
y = 2a%/(a - l);r2 . fl/12 . sin - . sin ^ . cos
[_ a / /
+ 1/22. sin 27r. sin ^-.cos^+
at I
where a = Z/p. If the string is struck in the middle, we have a = 2 and
SECTION XXXVIII. POTENTIAL ENERGY OF AN ELASTIC BODY.
When an elastic body changes its form work will be done. This is
stored up in the body as potential energy if the body is perfectly
elastic, which we will assume to be the case. The work necessary
to bring about a particular change is equal to the potential energy
gained by the body, and may be determined in the following way.
SECT, xxxvni.] VIBRATING STRINGS. 97
Let A', B1 and C" represent the principal stresses at a point in the
body ; about this point construct the infinitely small parallelepiped,
whose edges u, v and ic are parallel to the principal stresses. When
the stresses are applied, the edges of the parallelepiped are extended,
u becoming u(l+a'), v becoming v(l+b'), and iv becoming iv(l+c').
From XXIX. (e) we have
A' = A9' + 2/iw', B" = A6' + 2/aJ', C' = A9' + 2/xc'.
If a', b' and c' change by the increments da', db' and dc respectively,
the edges of the parallelepiped are increased by uda, vdb' and wdc',
and the parallelepiped undergoes an infinitely small change of form.
The forces which act in the directions of the edges are vwA', uwB'
and uvC". Hence the work done by the stresses during the change
of form is
(A'da + B'dV + C'dc')uvw = ( A6W + ^(a'da + b'db' + c'dc'))uvw,
since 0' = a' + b' + c'.
To change the form of the parallelepiped by an amount which is
determined by the elongations a, b, c, the work
4( A62 + 2/n(a2 + b2 + c2))uvw
must be done. If we set dv for the volume uvw of the parallelepiped,
the potential energy Ep of the whole body is given by
(a) ^ = |J(Ae2 + 2/>i(a2 + &2 + c2))cfo.
If we introduce the principal stresses A, B, C [XXIX. (e)], we have
(b) Ep = ^{(A + £ + C)2/E-(AS + £C+CA)lfi}dv.
By this equation (b) the potential energy is determined, the com-
ponents of stress and of elongation are known. We confine ourselves
to the statement of the following relation,
(c) Er = i Jfe + Yjy, + Z:zz + *Zjs, + 2A>, + 2YjJdr,
from which the others can easily be deduced.
Galileo was the first to study the properties of elastic bodies; he
failed, however, to reach correct results. The physical basis for the
theory of elasticity was given by Eobert Hooke, Avho in 1678 published
a treatise, De potential restitutira, in which he showed by experiment
that the changes of form of an elastic body are proportional to the
forces applied to it. Among earlier investigations those of Mariotte
and Coulomb deserve especial mention. More recently the theory
of elasticity has been developed principally by the French mathe-
maticians, Cauchy, Poisson, Lame, Barr6 de Saint-Venant, and others.
98 THE THEORY OF ELASTICITY. [CH. u. SECT, xxxvin.
We owe to Cauchy the theory of the components of stress in the
form here given. For more extended accounts of the theory of
elasticity we may mention Lame, Theorie Mathe'matique de V Elasticity
des Carps Bolides. Paris, 1866. Clebsch, Theorie der Elasticitdt fester
Korper. Leipzig, 1862. Among the more important recent treatises
on the theory of elasticity we mention : Boussinesq, Application des
Potentiel a I'Etude de I'Equilibre et du mouvement des Solides Elastiques.
Paris, 1885. Barr6 de Saint- Venant, Me'moire sur la Torsion des Prismes.
M6m. d. sav. Mr. T. XIV. Paris, 1856 ; Mtmoire sur la Flexion des
Prismes. Liouville I., 1856. William Thomson, Elements of a
Mathem. Theory of Elasticity. Phil. Tr. London, 1856; Dynamical
Problems on Elastic Spheroids. Phil. Tr. London, 1864.
Further researches on the theory of elasticity have been carried
out in recent years by W. Voigt.
CHAPTER III.
EQUILIBRIUM OF FLUIDS.
SECTION XXXIX. CONDITIONS OF EQUILIBRIUM.
THE principal difference between solids on the one hand and liquids
and gases on the other consists in the fact that the latter do not,
like the former, offer a great resistance to change of form. A force
is always needed to change the form of a fluid mass, but the resistance
offered by the fluid is determined by the rate at which the change
of form proceeds, and will be infinitely small if it proceeds very
slowly. We assume that the motion by which the condition of
equilibrium is attained proceeds very slowly, and we may therefore
assume, in hydrostatics, that a fluid offers no resistance to change of
form, so long as this does not involve change of volume.
Each infinitely small change of form of an infinitely small part of the
body may [XXVIII.] be treated as if it were produced by the dilatations
«, b, c in three directions perpendicular to each other. The lengths
u, v, w drawn in these three directions become u(\ + a), v(l +b), w(\ +c).
If A, B, C are the corresponding normal forces per unit of surface
which act on the surfaces vw, uw, iw respectively, the work done
by the normal forces in this change of form is
Avwua 4- Buwvb + Cuvwc or (Aa + Bb + Cc)u .v.w.
The change of form considered will, in general, involve an increase
of volume, given by uvw(\ + a}(\ +b)(l +c) - uvw. Since a, b, c are
infinitely small, the increment of the volume equals (a + b + c)u. v. w,
if we neglect infinitely small quantities of a higher order.
If we start from the assumption that the work done by the forces
equals zero if the volume is not changed, we have at the same
time Aa + Bb + C'c = Q and a + b + c = 0. These equations can both be
true only if A = B = C.
100
EQUILIBRIUM OF FLUIDS.
[CHAP. in.
The equations for the components of stress [XXVI. (i)] give
Xt=Y, = Z, and Z, = 0, X, = 0, Yx = 0.
There are, therefore, no tangential forces in a fluid in equilibrium.
If we start from the condition that the only forces which act on
fluids in equilibrium are perpendicular to their surfaces, we reach the
same result, namely, that the normal stresses are all equal. To
show this we set Zy = Q, A'2 = 0, 1^ = 0,
and have, from XXVI. (a),
P=Xxcosa, Q=Y,cos/3, fi = Z;Cosy.
P, Q and R are the components of stress for a surface whose normal
makes the angles a, (3, y with the axes. The stress acting on this
surface is <JP- + Q- + R\ the normal force N is determined by
N= Pcosa + Qcosfi + R cos y.
The tangential force T is
T- = (P2 + Q* + R2} - (P cos a + Q cos P + R cos y)2.
Introducing the values of P, Q and R, we have
(X, - F,)2 cos2a cos2j3 + (Yy- Zy- cos2£ cos'y + (Z2 - Xx)* cos2« cos2y = 0,
and hence Xx=Yy = Zz.
From the expression given above for N, it follows that N=XX,
that is, the normal force acting on a surface-element in the interior of
the fluid is independent of the position of the element.
If we neglect the force of cohesion of the fluid, which will be
treated later, the normal force will be a pressure ; if this is designated
by p, we have
(a) X,= Y, = Z,= -p; Zv = 0, X, = Q, Yx = 0.
If p is the density of the fluid, we obtain from XXV. (c) the condi-
tions of equilibrium
(b) "dp/fa = pX, dp/'dy = pY, oppz = PZ.
The components of the force acting on
the unit of mass are A", Y, Z; we may
consider p as constant in liquids ; in gases
p is a function of the pressure.
Equations (b) may also be developed in
/C ~B' the following way. Represent the sides
of the parallelepiped 00' (Fig. 46) by
OA=dx, OB = dy, OC = dz.
The pressure on OA' ispdydz, the pressure on O'A is (p + "dp/dx . dx)dydz.
The resultant of the pressures is the pressure - "dp/*dx .dx.dy.dz in
SECT, xxxix.] EQUILIBRIUM OF FLUIDS. 101
the direction of the a-axis. The force pXdxdydz also acts on the
parallelepiped in the same direction. The condition of equilibrium
is therefore ( - 'dp/'dx + pX)dxdydz = 0, from which we obtain the first
of equations (b).
If equilibrium obtains, p must satisfy equations (b) ; the conditions
for this are
(
1
These equations will hold if a function 4> exists, such that
(d) -d3>px = pX, -d3>py = pY, ?&fiz = f>Z.
Equations (c) are the essential conditions of equilibrium ; if they
are satisfied, p may be determined from the equation
dp = p(Xdx + Ydy + Zdz).
If the forces have a potential i/s, so that
we will have (e) dp= - pd^.
In gases p is a function of p • in liquids p may be considered
constant. In the latter case we obtain (f) p = c - pif>, where c is
constant.
SECTION XL. EXAMPLES OF THE EQUILIBRIUM OF FLUIDS.
The conditions of equilibrium of a liquid mass contained in a
vessel, and acted on by gravity only, may be determined in the
following way : Suppose the position of the particles of the liquid
referred to a system of rectangular coordinates, whose 2-axis is directed
perpendicularly upward ; we then have
X = 0, F-0, Z=-g,
and therefore $ = gz. Since the density p is considered constant,
equilibrium can obtain under the action of gravity. From XXXIX. (f)
we have p = c-gpz. Hence the pressure at the same level is every-
where the same.
We now determine the pressure in a liquid contained in a vessel, which
rotates about a perpendicular axis A with constant angular velocity w.
The fluid will turn, like a solid, about the axis A with the same
angular velocity as that of the vessel.
A particle at the distance r from the axis A is acted on both by
gravity and by a centrifugal force whose acceleration is w2r. We
102 EQUILIBEIUM OF FLUIDS. [CHAP. HI. SECT. XL.
refer it to a system of rectangular coordinates whose 2-axis is directed
perpendicularly upward, and coincides with the axis of rotation.
We then have
X=tfx, Y=^y, Z=-g,
and the potential ^ is ^= -%<azr- + gz. From XXXIX. (f) the pres-
sure is p = c + p( Jw'V2 - gz). The surfaces of constant pressure are
paraboloids of revolution with the common axis A.
We will make a third application of the conditions of equilibrium
to the determination of the pressure in the atmosphere. We suppose
gravity directed toward the centre of the earth, its acceleration y
may then be expressed by 7= - ga?/f2, where g is the acceleration
at the surface of the earth, a the earth's radius, and r the distance
of the point considered from the centre of the earth. We then have
\}/= -ga2/r. If the temperature is constant p = k.p, where k is con-
stant. We have then [XXXIX. (e)]
dp= -k.pdif* or log p = c - k^.
If, at the earth's surface, the pressure is p0, and the potential «/-0, we
have logpQ = c-k$0 and Iog(pjp) = k(\f'-^0) = kg(ar-a2)/r. If the
difference r-a is very small in comparison with a, we can set
Iog(p0/p) = kgh, where h = r — a is the height of the point considered
above the earth's surface, and k is equal to 1,2759 . 10~9 for dry air
at 0°C.
CHAPTER IV.
MOTION OF FLUIDS.
SECTION XLI. EULER'S EQUATIONS or MOTION.
IN the study of the motion of fluids very serious difficulties are
encountered, and thus far only a few problems have been completely
solved. In the following chapter we will consider only the so-called
ideal fluids, and therefore neglect the friction between their moving
particles and the forces of adhesion and cohesion, which will be
treated later. We further assume that the fluids considered are in-
compressible, and thus limit our discussion to liquids, which are only
slightly compressible. On these assumptions several characteristics
of the motion of liquids may be derived. As the study of the motion
of gases is extremely difficult, and as little success has so far been
obtained in it, we will not enter upon it here.
For a complete determination of the motion of a fluid the path of
each separate particle, as well as the position of the particle in the
path at any instant, may be given. The coordinates x, y, z of the
particle M may be given as functions of the time t.
An easier method is one in which the motion is expressed in
terms of the components of velocity, which according to circumstances
shall be designated by U, V, W, or u, v, w. Suppose U, F, W to
be the components of velocity of a particular particle of the fluid ;
if the particle at the time t is situated at P and at the time / + dt at P',
U, V, W will be the components of velocity at the point P, and
TT dUlt r dVlt ,„ d,W,.
U+lltdt> V+lTtdt> W+-^
the components of velocity at P. The quantities dU, dF, dW repre-
sent the increments which U, V, W respectively receive during the
time dt, if our attention is confined to the motion of a particular
particle.
103
104 MOTION OF FLUIDS. [CHAP. iv.
The other symbols, u, v, iv, represent the components of velocity
at a definite point in space, where one particle replaces another in
the course of the motion. If x, y, z are the coordinates of the point
considered, u, v, w are the components of velocity of a particle situated
at that point at the time t. After the lapse of the time dt the same
point is occupied by another particle, whose components of velocity
are u + du/'dt.dt, v + 'dv/'dt.dt, w + 'dw/^t.dt.
A particle situated, at the time t, at a point whose coordinates are
x + dz, y + dy, z + dz, has a velocity whose projection on the ,r-axis is
u + 'du/'dx . dx + "dufdy . dy + "dufdz . dz.
The velocities u, v, w are everywhere functions of x, y, z and /. If
u, v, w are the components of velocity, at the time /, at the point P,
whose coordinates are x, y, z, the components at the time t + dt at
another point P, whose coordinates are x + dx, y + dy, z + dz, will be
u + 'du/'dt . dt + 'du/'dx . dx + "dufdy . dy + 'du/'dz . dz, etc.
If the fluid particle is situated at the time t at P and at the time
t + dt at P', then we have U=u and
U+ d U/dt .dt = u+ oupt . dt + 'duj'dx . dx + ^fdy . dy + 'du/'dz . dz,
or (a) d U/dt = Vu/?)t + Vu/'dx . dx/dt + 'du/'dy . dyfdt + 'dufdz . dz/dt.
The particle considered traverses the distance PP' in the time dt,
hence its velocity is PF/dt, the projections of which on the coordinate
axis are evidently dxfdt = u, dy/dt = v, dz/dt = w. Thus we obtain
d U/dt = 'du/'dt + u . du/'dx + v . 'du/'dy + w . 'du/'dz.
The equations for dVjdt and dlVjdt are similar.
To find the equations of motion of a fluid let us cut from it a
parallelepiped dw, whose edges are dx, dy, dz, and on which a force
acts whose components are X, Y, Z. In the time dt the parallelepiped
receives an increase of momentum, whose components are
pXdudt, pYdwdt, pZdwdt,
when p denotes the density of the fluid.
The pressure p acting at the point .r, y, z, as has been shown in
a former chapter, imparts to dto the components of momentum
- 'dp/'dx'. d<adt, - 'dp/'dy . dwdt, - ty/'dz . dwdt.
Under the action of these forces the body receives, in unit time, an
increment of velocity whose components are dU/dt, dVjdt, dWjdt,
and hence we have
(b)
SECT. XLL] MOTION OF FLUIDS. 105
By the help of equations (a) we then obtain
f "duj'dt + udul'dx + v'dufiy + wdttfiz = X- l/p.'dp/'dx,
(c) i.'dv fit + u*v fix + vdv fiy + wdv fiz=Y-l/p. Vpfiy,
\ "divfit + tidwfiz + vdwfiy + wdwj'dz = Z -\jp. *dpfiz.
These equations are due to Euler, and are known as Euler's Equations
of Motion. To them must be added the so-called equation of continuity,
which is found in the following way : The parallelepiped receives in
the time dt, through the face dydz, the quantity of fluid pudydzdt ; and
loses, through the opposite face, the quantity (pu + 'd(pu)j'dx.dx)dydzdt.
The difference between the quantities flowing through the two surfaces,
which indicates a loss of fluid, if ~d(pu)fix . dx is positive, will be
3(ptt)/3a; . do> . dt.
By a similar argument applied to the two other pairs of faces it
appears that the total difference between the quantities of fluid which
leave and enter the parallelepiped is
(d(pu)fix + ^(pv)fiy + -d(pw)fiz)d<» . dt.
The parallelepiped at first contained the quantity p . do> ; after the
lapse of the time dt it contains the quantity (p + ~dpfit . d£)d<a ; the
difference of these two quantities is -~dpfit.dw.dt. By equating
these two expressions for the same quantity we get the equation of
continuity (d) "dp fit + ^pujfix + ?>(pv)fiy + ^(pw^fiz = 0. If the density
p of the fluid is constant, the equation of continuity becomes
(e) 'dU'/'dx + "do fiy + Vwfiz = 0.
Euler's equations are specially suited to investigations of the motion
in fluid masses with fixed boundaries. If the surface of the fluid
changes there will be points which will lie sometimes within and
sometimes without the fluid ; the velocity at such a point cannot be
determined by the method here given. Lagrange's method is the one
then employed. To this we will return later.
In equations (c) and (e) there are contained four unknown quan-
tities u, v, 10 and p, for whose determination we have four equations
given. To determine the constants of integration the conditions of
the motion of the fluid must be given at a definite time. If the
fluid is bounded by a fixed surface, the components of velocity in
the direction of the normals to the bounding surface are zero. If
u, v, w are the components of velocity of a particle at the boundary of
the fluid, and if the normal to the bounding surface makes the angles
a, ft, 7 with the axes, we have
(f ) u cos a + v cos ft + w cos 7 = 0.
106 MOTION OF FLUIDS. [CHAP. iv.
SECTION XLIL TRANSFORMATION OF EULER'S EQUATIONS.
In k fluid in motion an elementary parallelepiped, whose edges
are originally dx, dy, dz, not only changes its position in space but
may also rotate and change its form at the same time. Its motion
at any instant is determined by the components of velocity u, r, v: ;
the rotations and changes of form may be determined in the following
way : In the theory of elasticity the component of rotation hx of such
an element is expressed by hx = ^(dC'/'dy - ty'/'dz), if £' and rf are the
infinitely small changes of the coordinates z and y introduced by the
motion. We may set £' = iv.dt, rj' = v.dt, and obtain
If £ is the corresponding angular velocity, we will have hx = %. dt
and hence
(a) £ = £(3tp/By-a0/az); 77 = $
The equations for t] and £ may be derived in the same way as the
first; £, »/, £ are the components of angular velocity in a rotation
about the three coordinate axes.
If no rotation exists in the fluid, we have £ = y = {=Q or
These equations are the conditions for the existence of a function 4>
of x, y, z and t which has the property that
it = - 3^/Da;, v = - 3<£/9y, tr = - "d^fdz.
This function </> is called by v. Helmholtz the velocity potential, since
the components of velocity U, v, w, are related to each other in the
same way as the components of a force if it has a potential.
The equation of continuity [XLI. (e)], on the assumption that a
velocity potential exists and that the fluid is incompressible, becomes
* = V2<t> = Q. The velocity h of a particle is
From equations a it follows that 'du['dy = 'dv/'dx - 2£ 'du/'dz = 'dtv/'dx + 2r).
The first of equations XLI. (c) becomes
dw/3* + 2(ui) - vQ + u . 'du/'dx + v . dr/da: 4- w . 'dwj'dx = X-l/p. 'dpjox.
This equation may be written
f 3tt fit + 2(v») - rfl = X - 1 Ip .
\ We have similarly
I -dv fit + 2« -w£)=Y-\lp.
SECT. XLII.] VORTEX MOTIONS. 107
where h is the velocity of a particle. "We may eliminate p from
equations (b) by differentiating the second of those equations with
respect to z, and the third with respect to y and subtracting. We
thus obtain
If we use the equation of continuity 'du/'dx + 'dv/'dy + 'dw/'dz = 0, and
the relation following from (a) 3£/3z + cfy/3?/ + 3£/?z = 0, we obtain
3£/3< + u . 3£/3z + v . 9£/3y + w . m'dz - £ . 'du/ox - rj . 'duf'dy -
If & 7/' £ represent the components of rotation at a point in the
region containing the fluid at the time /, we may use &, H, Z to
represent the components of rotation of a particle at the time t + dt,
whose components at the time t were £, -rj, £.
The connection between the components £, 77, £ and &, H, Z may
be established in the same way as that previously used to find the
relation between the velocity at a point in space and the velocity
of a particle of the fluid. We have
£ = S, dS/dt = 3£/3rf +
Using this equation we obtain
(c) dS/dt = £ . 'du/'dx + rj . -duj-dy + £. 'du/'dz + tfdZ/'dy -
If at any time no rotation exists in the fluid, and if therefore £ = y = £=0
at any point in the fluid, a rotation may still be set up if Z and Y have
no potential. If, on the other hand, Z and Y have a potential so
that Z= -W/oz and F=-3¥/3y, we will have dSjdt = 0. If
besides X= -o¥/'dx, we have dU/dt = Q and dZ/dt = Q. Hence no
rotation can be set up in an ideal fluid if the fwces have a potential In
this case, the particles which rotate already continue to rotate, but the particles
which do not rotate from the beginning will never rotate. This theorem
was first given by v. Helmholtz.
SECTION XLIII. VORTEX MOTIONS AND CURRENTS IN A FLUID.
In researches on the motion of fluids it is important to observe
whether the particles rotate or not. If there is rotation it is called
cortex motion. We then have
(a) £ = $
From this it follows at once that (b) 3£/3z + 3^/3y + 3f/3^ = 0. The
equation of continuity is (c) 'du/'dx + 'dv/'dy + 'dw/'dz = 0.
108 MOTION OF FLUIDS. [CHAP. iv.
If the forces have a potential it follows from equations XLIL (c) that
C dS/dt = £ . 3n / cte + 17 . 'dw I'dy + £ .
(d)
{ dZ fdt = £ . Sw/dx + 77 . dw?/3# + £ . 'dwj'dz.
In these equations £, 77, £ are the components of rotation at the
point x, y, z; S, H, Z are the same components for a particle which
at the .time t is situated at the point x, y, z, but which at the time
t + dt is situated at the point x + dx, y + dy, z + dz.
On the other hand, if the components £, 77, £ are zero at every point
in the fluid at a definite instant they are equal to zero at any time,
from equations (d). In this case we call the motion a flow. It is
characterized by the equations
(e) 'dw/'dy = Zv/'dz, ?m/dz = 'dw/'dx, Vvj'dx = 'du/'dy.
From XLII. u, v and w then have a velocity potential <f>, which depends
in general on x, y, z and t. The equation of continuity is
(g) 32<£/ar2 + a^/a^8 + a^/a?2 = v-<£ = o.
Euler's equations XLI. (c) take the form
r X = -
J Y = -
(h) Y = - ^pfdy + ^W/Vy + 1 fp .
Hence such a motion can exist only when the forces have a poten-
tial ¥. In case this condition holds, we obtain from (h) by integration
(i) ¥ + T= 'd^/'dt - %h? -p/p, where T is a function of the time only.
In order to have a simple example of the two classes of motion
just described, we consider the case of an infinite fluid mass, all
particles of which move in circles parallel to the zy-plane whose
centres lie on the z-axis. All particles at the same distance from the
.z-axis move with the same velocity and in the same sense. We
have then from XXXVI. (e) u= -wy, v= +ux, w = 0. w depends
only on the distance r of the particle from the z-axis. Since we have
'du/'dx = - xy/r . dw/dr and 'dvj'dy = + xy/r . dw/dr. the equation of con-
tinuity is satisfied, because 'du/'dx + "dv/^y — 0.
In general there is rotation of the separate particles, since
'duj'dy = - (a - y2/r . dw/dr ; 'dv/'dx = w + x2/r . dw/dr,
and therefore £=<» + £?• .do>/dr. Since £ = 0, 77 = 0 and u, r, w are
independent of z, the equations of motion (d) are satisfied.
We assume f = £0 for r <r0 and £=0 for r > r0, where £0 is a constant.
In the first case, we have w = f0 + Cjr2, where C is a new constant.
SECT. XLIII.] STEADY MOTION. 109
C must vanish, because otherwise the particles at the axis would have
an infinitely great velocity. Hence o> = £0 for the part of the fluid
lying within a circular cylinder whose radius is ?-(), and whose axis
coincides with the 2-axis. These fluid particles therefore rotate about
the £-axis, just as if they formed a solid body. If, on the other hand,
r>r0, and hence {=0, the angular velocity w' will be ot'=C'/rz. The
linear velocity is rw' or C'/r, and therefore inversely proportional to the
distance of the particle from the axis. On the condition that there is
no discontinuity in the motion of the fluid, we have for r = r0, £0 = C'/r02.
Hence for r > r0, we have rw' = r02f0/r. If r0 is infinitely small and £0
infinitely great, we obtain a so-called vortex filament.
The action of the vortex filament on the surrounding fluid depends
on its cross-section and its angular velocity. If we set m = irr0^0,
the velocity h of a fluid particle which does not belong to the vortex
is h = rw = m/Trr.
Vortex filaments may have other forms ; they were first investigated
by v. Helmholtz,* and afterwards by William Thomson, and several
others. We see from this example, that the separate parts of the
fluid do not need to turn about themselves as their centres of gravity
describe circles : although the fluid surrounding the vortex filament
revolves about the 2-axis, the separate drops, into which the mass
may be divided, do not rotate about themselves.
SECTION XLIV. STEADY MOTION WITH VELOCITY-POTENTIAL.
If the components of velocity are independent of the time, or if
the condition of motion at any definite point in the fluid does not
change, the motion is called steady. If a velocity-potential <£ exists,
we have (a) u= — 'd<f>/'dz, v— — c)(£/3y, w= -'dtfij'dz, where <f> is a
function of x, y, z only. The same holds for the potential M*1 of the
forces ; hence the function T in XLIII. (i) must be constant. If we
set T= -C, we have (b) V+p/p + $h2 = C. If the only forces which
act are pressures within the fluid, we may set ^ = 0 and conclude
that the velocity of the particles increases as they pass from places
of higher pressure to places of lower pressure, and inversely.
For a motion for which there is a velocity-potential, the equation
of continuity is (c) V2<£ = 0.
As an example of such a motion, we will consider a sphere at
rest in an infinitely extended fluid. The particles of the fluid which
* Helmholtz, Crelle's Journal, Bel. 55, S. 25, 1858.
110 MOTION OF FLUIDS. [CHAP. iv.
are at a great distance from the centre of the sphere move with
equal velocities in the same direction.
Let the sphere be placed so that its centre is at the origin of
coordinates 0, and let the radius of the sphere be R. The particles
whose distance r from 0 is infinite are supposed to move in a
direction parallel to the positive 2-axis with the velocity w0. We
set the velocity-potential (d) $=V- u-^z, in Avhich V = 0 when
r = oo. Then (e) u = -'dF/'dx, v= -'dF/'dy, 10 = -'dPrjdz + tc0. Using
equation (c) we have (f) V2F=0. If we set P=l/r, or equal to a
differential coefficient of 1/r taken with respect to x, y, or z, the
equation (f) will be satisfied. Since the arrangement around the
.r-axis is symmetrical, we will consider if the assumption
will satisfy the given conditions.
The particles of the fluid move over the surface of the sphere,
and hence the component of velocity in the direction of the radius
is equal to zero, that is (3</>/<3r)r=Jj = 0. If we set z/r = cosy, we
have
</>=-(? cos y/r2 - ra'0 cos y and d<f>/dr = 2C cos y/r3 - w?0 cos y.
From this it follows that (h) C=±wQR*.
From equations (d), (g), and (h), it follows that
Using equations (e) we obtain
u = - %wQR3zx/r>, v = - f w0R?zyli*t w= - ^R^Sz^/r5 - 1/r3) + zr0.
If we set tt,-2 + t>2 = s2 and x2 + y2 = q-, we will have s= -^w^qz/r5.
If q and z are the coordinates of the path of a particle, the equation
of the path will be (k) dq/dz = s/w.
Remembering that r2 = q2 + z2, we may integrate equation (k) and
obtain q~(\ -R?/r3) = c. If c is constant, this is the equation of a
stream line. If c = 0, we will have either r = R or <? = 0; in the first
case, we get the equation of a great circle, in the second, the equa-
tion of the z-axis.
The pressure p may be determined by the help of equation (b).
Since ^P = 0, we have p = p(C- ^h'2). Now h2 = u2 + v* + w2, and hence
for a point on the surface of the sphere we have h = %w(>q/E.
Hence the pressure p on the part of the sphere which lies toward
the positive side of the 0-axis, is as great as that on that part of
the sphere which lies on the negative side of the 2-axis ; the moving
mass of fluid will therefore impart no motion to the sphere. And
SECT. XLIV.] LAGRANGE'S EQUATIONS. Ill
further, a sphere which moves with constant velocity in any direc-
tion in an infinite mass of fluid experiences no resistance during its
motion. This result, which is at first sight so startling, is explained
by the fact that the resistance offered by friction is not taken into
account.
SECTION XLV. LAGRANGE'S EQUATIONS OF MOTION.
Suppose a particle P of a fluid to be originally situated at the
point whose coordinates are a, b, c, and after the lapse of the time
dt, to have reached the point x, y, z. The general coordinates x, y, z,
are functions of t, a, b, c; if t alone varies in these functions, we
obtain the path of a particular particle. If, on the other hand, we
give to the coordinates a, b, c, all possible values, and keep t constant,
we have the positions of all the particles of the fluid at the same
time. If the pressure is designated by p, and the density of the
fluid by p, and if we set U=x, V=y, JF=z, we obtain from
XLI. (b)
(a) x = X-l/P.dp/dx, ij=Y -Ijp.dpldy, z = Z-l/p.dp/dz.
In order to eliminate the differential coefficients with respect to
x, y, z, we multiply these equations respectively by dx/da, dy/da,
dz/da, by dx/db, dy/db, dz/db, and finally by dx/dc, dy/dc, dz/dc.
By addition we then get the following equations :
f (x - X) . dx/da + (y-Y). dy/da + (z-Z). dz/'da +l/p. dp/da = 0,
(b) - (x-X). dx/db + (y-Y). dy/'db + (z-Z). dzjdb + l/p . dp/db = 0,
( (x - X] . dx/dc + 0/-Y). dy/dc + (z-Z). dz/'dc + l(p. dp/dc = 0,
These equations are due to Lagrange.
To these equations there must be added a relation which expresses
the fact that the volume of the fluid does not change. The particles
originally situated in a rectangular parallelepiped with the edges
da, db, dc, are at the time t contained in a parallelepiped, the pro-
jections of whose edges are
dx/da . da, dy/da . da, dz/da . da ;
dx/db . db, dy/db . db, dz/db . db ;
dx/dc . dc, dy/dc . dc, dz/dc . dc.
The volume of the parallelepiped at the time t will therefore be
dx/da, dy/da, dz/da
dx/db, dy/db, dz/db
dx/dc, dy/dc, dz/dc
.dadbdc.
112 MOTION OF FLUIDS. [CHAP. iv.
Since the fluid is assumed incompressible, the equation of con-
tinuity is
'dxj'da, 'dy/da, 'dz/'da
(c) -dxj-db, -dy/cib, -dz/Zb =1.
'dx/'dc, 'dy/'dc, 'dzj'dc
To apply Lagrange's equations, we will consider a fluid mass,
which turns with a constant angular velocity <u about the z-axis,
directed vertically downward. We then have X=0, F=0, Z = g,
and we set z = c, a = r cos (f>, b = r sin </>, and further,
x = r cos (<f> + (at) = a cos ut -b sin W,
y = r sin (<£ + W) = b cos tat + a sin wt.
From these relations it follows that
dx/'da = cos (at, ~dx/db = - sin (at, 'dx/'dc = 0,
~dy/da = sin (at, ~dy/db = cos (at, 'dy/'dc = 0 ;
i1 = — o>2a;, y—— (a2y, z = 0.
The equation of continuity (c) is satisfied and the equations of
motion (b) are 'dp/'da = /ooj%, 'dp/'db = p(a2b, 'dp/'dc = gp. Hence we
obtain by integration p= C + p(iw2(a- + J2)+^c). This solution agrees
with that given in XL.
SECTION XLVI. WAVE MOTIONS.
Lagrange's equations may be used to advantage in investigations
on wave motions in a fluid acted on by gravity. All the particles
of the fluid may be assumed to move in plane curves parallel to
the x^-plane ; let the a;-axis be horizontal, and the £-axis be directed
perpendicularly downward. Then, if we set y = b, we obtain
'dx/'db = 0, 9y/3a = 0, dy/3& = l, 3y/3c = 0, "dz[db = 0 and y = 0.
If we further set p = pP, the equations of motion given in XLV.
(b) become
,. ( x. 'dx/da + (z-g). 'dz/'da + ?>P/da = 0,
t x . dx/'dc + (z-g). dz/dc + 'dPj'dc = 0,
and the equation of continuity XLV. (c) takes the form
(b) 'dx/'da . 'dz/dc - 'dzj'da . "dx/dc = 1 .
Suppose the particle B (Fig. 47), having, while in its position of
equilibrium, the coordinates OA=a and AB = c, to move in a circle
SECT. XLVI.]
WAVE MOTIONS.
113
DFE, whose centre is at C. Let D be the position of the particle,
represent the angle between CD and the perpendicular CE by 0,
and set BC=s, CD = r. We then have
(c) x
Here m and n are constants and r and s are functions of c. We
therefore obtain
"dzfda = 1 + nr cos 0, 'dz/'da = - nr sin 0, 'dxj'dc = 'dr/'dc . sin 0,
"dz/dc = 1 + 'dsj'dc + 9r/3c . cos 0.
By these relations, equation (b) takes the form
3s/3e + nr . 'dr/'dc + {nr(l + 3s/3c) + 3r/3c} cos 0 = 0.
Since this equation holds for all values of t or 0, we have
(d) 3s/9e + nr.3r/dc = 0 and 3r/3c + »r(l+a*/3c) = 0.
We obtain further from equations (a) the relations
(e)
-mh:'drl'dc-
the last of which is transformed by the help of equations (d) into
(f ) - (w2 - gnfrCdrfdc + ( 1 + ^s/'dc) cos 0 } - g + 'dP/'dc = 0.
If the pressure depends on c only, it follows from (e) and (f) that
(g) (h) m2 = gn, P = gc, if the constant is set equal to zero; the pres-
sure therefore disappears for c = 0. This condition must hold at the
free surface of the fluid.
The paths of the particles are circles. If the time required by the
particle to traverse its path is T, that is, if T is the period of oscilla-
tion, we have m = 27r/77 and 6 = 2ir/T.
114 MOTION OF FLUIDS. [CHAP. iv. SECT. XLVI.
If X is the wave length and h the velocity of the wave, we will
have h = Tg/2Tr and h = \/T, from which it follows that
(i) h = *Jg\j'2ir and n = 2ir/X.
The motion of the particle is such that it describes a circle whose
centre lies a little above the position of equilibrium of the particle.
From the first of equations (d) we have s= -^nr2, where the con-
stant disappears, since s and r vanish simultaneously. Hence also
(k) s= -7rr2/A. From the second of equations (d) it follows that
d log r + nd(c + s) = 0, and hence, by integration, log r + n(c + s) = k, when
k is a constant. For a particle on the surface we have c = 0 ; if the
values of r and s for this particle are designated by E and S, we
have log B + nS=k. We have further log (r/E) + n(c + s-S) = Q. The
factor c + s-S=H is the perpendicular distance between the centre
of the path of the particle considered and the centre of the path of
a particle in the surface. We have therefore (1) r = Ee~Zirai'x.
If ds/dc is eliminated from equations (d) we obtain
dr + nr(dc - nrdr) = 0,
and by integration 1 ,'n . log r + c - |nr2 = Tc'. For the particles on the
surface we have 1/n. log E - ^nE2 = k' . Hence
(m) c = A/27T . log (E/r) - ir/A . (E2 - r2).
The free surface can be thought of as formed by the rolling of a
circular cylinder on the under side of a horizontal surface AB (Fig. 48),
FIG. 48.
which lies at the height OA = X/2ir over the centres of the paths
which the particles in the surface describe. The free surface is then
represented by a straight line whose distance from the axis of the
cylinder is E,
CHAPTER V.
INTERNAL FRICTION.
SECTION XLVII. INTERNAL FORCES.
IN the discussion of the motion of fluids, the friction among the
fluid particles has not been considered. Friction is excited in different
degrees between the particles of the fluid when they move among
themselves at different rates. In consequence of friction the viscosity
of the fluid is more or less great. We will try to determine the
friction caused by the motion of the fluid.
We will suppose that the particles of a fluid mass are moving in
a direction parallel to the ar-axis, and that those situated at the same
distance from the ar^-plane have the same velocity. The velocity
increases in proportion to the distance from the xy-pl&ne. One sheet
of the fluid glides over another and thereby gives rise to a definite
frictional resistance, which, according to Newton, may be assumed
proportional to the rate of change of velocity with respect to the
distance from the ^-plane, so that du/dy = e. The friction between
two contiguous sheets is then propor-
tional to the difference of their velocities,
and inversely proportional to the distance
between them. We therefore set the
velocity u = <uQ + ey. Let 00' (Fig. 49)
be a part of the fluid mass ; on each
unit of surface of O'B there acts, in the
direction O.r, a tangential force
(a) T= p. . du/dy = /*«,
where //, is the coefficient of friction. A FlG- 49-
force - T acts on .OB' in the direction Ox. Further, the tangential
forces T [XXV. (d)] must act on O'A and OA', of which the one
acting on O'A is in the direction Oy, and the one acting on OA' is
in the direction yO.
115
116
INTERNAL FRICTION.
[CHAP. v.
If the fluid mass moves in the direction of the ^/-axis with a velocity
v = v0 + e'x, the tangential force necessary to produce this motion
is T' = p . dv/dx. If both motions exist simultaneously, a tangential
force Xy acts on the fluid, such that we have
X, = T+T' = iJ.(dujdy + dv/dx).
In order to examine the physical meaning of this expression, we
will consider a fluid particle, originally situated at the point x, y, z,
which has moved through an infinitely small distance, whose pro-
jections on the axes are £, 7;, £ We then have u = 3£/9/, fl=cty/^,
and X=p.'dl'dt('dgfiy + 'dii/'dx) = 2p.'dz,rdt. This expression gives
the tangential force which arises from motion in a fluid in terms of
the friction. "We may therefore set
Zy = 2[j. . "dzj'dt
2/t . 'dyj'dt
v fis),
+ 'div/'dx),
/*(9t> fix + *dufiy).
"We know by experiment that ft is independent of the pressure. The
meaning of the other quantities in (b) is clear without explanation.
By the help of the formulas given in (b), we can determine the
tangential forces which must act in the fluid to overcome the frictional
resistances. We will now determine the magnitudes of the normal
forces which are necessary for
G d' D J)' the extension of a viscous fluid
in a given direction. Let the
fluid move in a direction parallel
to AB (Fig. 50), and let the velo-
city of a particle situated at the
distance i/ from this line be equal
to u. As before, we may set
u = ?/o + *y-
After the lapse of the time dty
A has traversed the distance u0dt,
and G the distance (u0 + fAC)dt.
CC' represents the motion of the
point C relative to A, and we have CC' = e . AC . dt. If we designate
the angle GAG' by d<f>, we have (c) d<f> = f.dt.
The rectangle EFGH, described in the rectangle ABCD, transforms
into the parallelogram EF'G'H', and we determine the increments
which the sides EH and EF receive by this transformation. Eepre-
senting the angle HEB by ^, and noticing that HH' and FF are
SECT. XLVII.] INTERNAL FORCES. H7
parallel to AB, we have EH' = EH '+ HH1 'cos & EF' = EF- FF'sin +.
We further have
HH' = BH. d<j> = EHsint.d<t>, FF' = AF. d<j> = EFcos $ . d<t>,
and hence (EH' - EH)/EH= sin ^ cos $d<j> ;
(£P" - EF)/EF = - sin ^ cos ^ty.
If the increment of length per unit-length of EH is designated by ds,
we have (d) ds = sin ^ cos \W<£ ; ds is also the diminution of length
per unit-length of EF.
To bring about the deformation considered, a tangential force T
must act on ABCD, which is, from (a), (e) T=p.e. This force acts
on the surface corresponding to CD in the direction CD, and on that
corresponding to AB in the direction BA ; on the other two surfaces
the forces act in the directions CA and BD. To determine the
normal force N acting on the surface EF, we 'set in XXIV. (a) a = ^,
/3 = |TT - ^, y = £TT, and Xy=Yx = T. Since all the other components
of stress are equal to zero, we obtain
(f) N=Pcost + Qsinf = 2TsiniI>cost.
From (c) and (d) we have ds = sin ^ cos ^ . e . dt, and from (e) and (f )
JV=2/xesin ^cos \j/. Hence, we have (g) N=2p..ds/dL The stress
acting on the surface EH is -N. It has been shown that a unit of
length, in the direction EF, is increased by - ds. If a normal stress
were to act on the surface of ABCD, it would have no influence on
the deformation; but a normal force S + N would act on EF, and a
normal force S-N on EH.
If the normal stresses XM Y^ Zt act on a rectangular parallelepiped
whose edges are parallel to the coordinate axes, when the fluid is in
motion, they cause deformations and a change of volume. If, as in
the theory of elasticity, we set the volume dilatation Q = xx + y, + za
then xx - JO is the part of the increase in the direction of the ic-axis
which is here considered. Similarly, we set 3S = Xx + Yy -f Za and
Xx - S is the part of the normal force in the direction of the a;-axis
which causes the deformation. By the help of (g), we obtain
If, finally, we set for - S a quantity p, which may be considered a
pressure, on account of its analogy with the pressure in ideal fluids
and gases, and remember that xx = 'd£l'dx, 'dxt/'dt = 'du/'dx, etc., we will
have (h) Xx = -p + 2/j. . 'duf'dx - ^('du/'dx + dv/3y + "dwfdz). Analogous
expressions hold for Yy and Zf
118 INTERNAL FRICTION. [CHAP. v.
From equation (h) the dimensions of ju, are ML~lT~l. The co-
efficient p. has been determined for many fluids and gases. It changes
very much with the temperature. The following values hold for 0° C. :
Water, 0,01775; Alcohol, 0,01838; Air, 0,000182.
SECTION XLVIII. EQUATIONS OF MOTION OF A Viscous FLUID.
We will now present the equations of motion of a fluid exhibiting
internal friction. From XXV. the components of stress act on the
unit of volume in the direction of the x-axis with the force
(X) = -dXJ-dx + aX,/3y + *dXJVz.
If U is the velocity of a single particle of the fluid in the direction
of the z-axis, then pU=(X) +pX, where, in the usual notation, A"
denotes the component of force in the direction of the x-axis. From
XL VII. (b) and (h), we have
(a) pU=pX- dpfdx + p-V2u + £/x . 3(3tt/3z + dv/dy + 'dw/'dz)/'dx.
This equation, and those analogous to it, which hold for the velocities
V and W in the directions of the y- and £-axes, are due to Stokes.*
They hold in connection with the equation of continuity
(b) 3/0/3* + 3{pw)/3a; + 'd(pv)/Vy + 3(pw')/^ = 0.
We assume that the fluid is incompressible, and have
(c) 'du/'dx + 'dv/'dy + 'dw/'dz = 0.
( p(u+u. 'du/'dx + v . 'duj'dy + w . 'du/'dz) = pV2u + pX - "dpfdx,
(d) J p(v + u . ?>v fdx + v . 'dv fdy + w.d>v fdz) = /*V2? +pY- 3p/3y,
{ p(w + u . 'dw/'dx + v . 'dw/'dy + w . 'dw/'dz) = p.V2w + pZ - 'dp/'dz.
The equations are simplified if the motion is steady, that is, if
ft = 0, v = 0, w = 0. If the velocity is very small, the terms u . 'du/'dx,
v . 'du/'dy, etc., may be neglected ; we then have
(e) nV2u + PX-'dpl'dz = 0, fj.V2v + pY-'dp/'dy = 0, p&w + pZ - 'dp/'dz = 0.
If the forces have a potential ¥, (f) V2p + pV2^ = 0. If we introduce
the components of rotation
£ = ±(dw/dy - 'dv/dz), y = $(dufdz - dw/dx),
and if the forces have a potential, we have from (e)
(g) V2£ = 0, V2»/ = 0, V2C=0.
Further we have (h)
* Stokes, Cambridge Phil. Tr., Vol. vin., p. 297, 1845.
SECT. XLVIII.] MOTION OF A VISCOUS FLUID. 119
With reference to the boundary conditions, it is assumed that the
particles of the fluid which are in contact with solid boundaries have
no relative motion with respect to them ; at the boundary of the fluid
we have therefore w = 0, v = 0, w = 0, if u, v, and w represent the
components of velocity at the bounding surface. If solids are present
in the moving fluid, we may generally assume that each particle in
the surface of the solid has the same velocity as the particle of the
fluid which is in contact with it.
SECTION XLIX. FLOW THROUGH A TUBE OF CIRCULAR
CROSS SECTION.
We consider a viscous fluid moving slowly through a narrow tube,
which is set horizontal, so that gravity does not influence the
motion. Let the axis of the tube be taken as the z-axis, and suppose
that the particles of the fluid move parallel to it. We then have
Equations XLVIII. (e), (c) and (f) then become
(a) 9p/a» = 0, 3p/3y = 0, ^w^'dp/'dz; (b), (c) dw/'dz = 0 ; \72p = 0.
From (c) it follows that (d) d2p/dzz = Q and p=fa+p(), where / and
pQ are constant.
It follows further from (a) that ju,y%=/. Since w depends on
the distance r of the particle from the axis of the tube, we have,
since r2 = x1 + y2,
V2w = d2w/dr2 + l/r . dw/dr.
Hence we obtain d^/dr^+l/r.dw/dr^f/fjL. By integration
w = c log r +fr2/4:fj. + WQ.
Since w has a finite value for r = 0, the constant c must, equal 0.
Therefore (e) w = wl)+fr2/4/JL, where WQ is the velocity in the axis
of the tube. If the pressure is equal to p0 when z = 0 and to pl
when z = l, we have from (d) f=(pi-p0)/l- If we substitute this
value of/ in equation (e), we have w = w0-rz. (p0-pi)/4fd- For
all particles of the fluid which are in contact with the wall of the
tube, we have w = 0. Representing by R the radius of the tube, we
will therefore have 0=»W0-^.(p0-j^)/4/tZ. We obtain finally
120 INTERNAL FRICTION. [CHAP. v. SECT. XLIX.
The volume m of the fluid which flows in one second through a
cross-section of the tube is given by
(f) m = I 2irrdr . w = 7r(pQ -p^W/Spl,
that is, the volume of the fluid is directly proportional to the fourth power
of the radius of the tube, inversely proportional to its length, and inversely
proportional to the constant /*.
Poiseuille was the first who investigated the flow of a fluid through
narrow tubes ; he was led to results which agree with the above
formulas.*
* Among recent works on hydrodynamics are to be mentioned : Lamb, Treatise
on the Motion of Fluids. Cambridge, 1879. Auerbach, Die Theoretische Hydro-
dynamik. Braunschweig, 1881.
CHAPTER VI.
CAPILLARITY.
SECTION L.— SURFACE ENERGY.
THE form of a fluid mass on which no external forces act is determined
by the forces with which its particles act on one another. If the
mass is very great, it will take the spherical form, in consequence of
the gravitational attraction of its parts ; if, on the other hand, the
mass is small, the force of gravitation between the particles will
have no perceptible influence. If the force of gravitation can be
neglected, the force of cohesion, which acts in every fluid mass, tends
to bring it into the same spherical form. From researches which have
been made on the mode of action of this force, it appears that it
acts only between particles which are at very small distances from
one another. The law of its dependence on the distance between the
particles is not yet known. We may nevertheless develop the laws
of capillarity by the use of a method which does not require a
knowledge of that law.
If the form of a fluid mass is originally a sphere, work must be
done to change it into any other form. If the fluid offers no frictional
resistance, this work can be due only to the fluid particles situated
in or near the surface ; since the only particles which can act on a
particle at a greater distance from the surface are those which
immediately surround it; these either remain in their positions or
are replaced by others which act in the same way as those replaced.
The work done is therefore expended in adding new particles to
those already present in the surface, or, what is the same thing, in
enlarging the surface. To increase the surface S of the fluid by the
infinitely small quantity dS, the work CdS is necessary ; C is constant
and may be called the capillary constant.
Two bodies in general meet in a surface, and C depends on the
character of these two bodies. In the case of a falling raindrop the
121
122 CAPILLARITY. [CHAP. vi.
two bodies in contact are water and air. At the surface of a drop
of oil which floats in a mixture of water and alcohol, as in the
well-known experiment of Plateau, two liquids are in contact. Even
when a fluid is in contact with a solid, or when two solids are in
contact, the common surface possesses a definite surface energy.
Let the capillary constant of two bodies a and b be Cab, and let
S be the surface in which the two bodies meet. The potential energy
Ep of the surface S is (a) Ep = Cab . S. Since Cab = EP/S, and since
the dimensions of Ep and S are L2T~2M and L- respectively, the
dimensions of the capillary constant are T~'2M.
The surface of the fluid is under a definite tension somewhat
analogous to that of an elastic membrane. If a rectangle DEFG
is described in a plane surface of a fluid, and if three sides of it
retain their positions unchanged while the fourth side FG, along with
the particles of the fluid present in it, is moved through the distance
FH in the direction EF, the surface is enlarged by the area FG . FH,
and the surface energy is increased by C . FG . FH, where C is the
capillary constant. To produce the motion considered, a force K must
act on FG ; the work done is therefore equal to K . FG . FH. It
follows that (b) K=C, or the tension per unit-length of the fluid surface
is numerically equal to the capillary constant.
This tension existing in the surface exerts a pressure in the fluid.
Let P (Fig. 51) be a point in the surface which is supposed convex
in the neighbourhood of P. Suppose two plane
sections erected at P, which contain the normal
to the surface at that point. One of these
planes cuts the surface in the curve PA, the
other in the curve PR PA and PB shall
intersect at right angles and their radii of curva-
ture shall be the principal radii of curvature of
the surface at the point P. A third plane
containing the normal to P cuts the surface
in the curve PF, whose radius of curvature R is determined from
Euler's theorem by the equation (c) l/E = cos2<f>/fil + sin2(f)/R2, where (f>
is the angle between PA and PF. About the point P as a centre we
suppose described a sphere of infinitely small radius which cuts the
surface in the curve AFBDE. The element FG of this curve is acted
on by the tension C . FG, proceeding from the adjacent parts of the
surface. FG may be set equal to rd<f> and the tension to Crd<f). Its
direction makes an angle with the normal to the surface, whose
cosine is r/R; hence the force acting in the direction of the normal
SECT. L.] SURFACE ENERGY. 123
is Cr2/E . d<f>. The surface tension therefore draws the surface-element
ABDE toward the interior of the fluid with a force
Cr2 [*'d<f>/lt.
Jo
Therefore if P represents the pressure on unit of surface we have
PTTT- = Cr2 l^d^jR and P = C/ir . f^'d^/B.
If for R we introduce the value given in (c), we obtain
(d) P
It is probable that in addition to the pressure here found, which arises
from the curvature of the surface, there also exists a constant pressure
M which acts in the fluid when its surface is plane. The total
pressure due to capillary forces is therefore M+C(l/Rl + l/R2), where
M and C depend on the character of the two bodies which are in
contact in the surface. Since the phenomena of capillarity do not
permit of the measurement of this quantity M, it need not be further
considered.
At 20° C. the value of C for the surface of contact between water
and air is 81, between mercury and air 540, and between mercury
and water 418.
SECTION LI. CONDITIONS OF EQUILIBRIUM.
Equilibrium exists in a fluid mass if its potential energy remains
unchanged when the position and form of the mass are changed by
an infinitely small amount. Since the energy depends on the extent
of surface we must obtain an expression for the increment SS of the
surface. Suppose a fluid A surrounded by another fluid B, the two
fluids being such that they do not mix. If no external forces act on
them, A will assume the spherical form.
Suppose the surface S to be concave toward A and to move toward
B so that it undergoes an infinitely small change of form. Let s
be the contour of the surface S, and S' represent that surface after
the change of form has occurred. The contour of S' may be repre-
sented by s'. Erect at all points of s normals to S which cut the
surface S' in a new curve o-, which may be supposed to lie within s'.
If we designate the infinitely small distance between a- and s' by 81,
the part of S' which lies between o- and s' will be given by (b) j£/ . ds.
124 CAPILLARITY. [CHAP. vi.
We now erect at a point P in S the normal PP which cuts «S"
at F ; set PP' = 8t>, and draw through P on the surface S two curves
PE and PF, one of which corresponds to the maximum curvature
of the surface at P, the other to the minimum. These principal curves
and two others infinitely near them will bound a rectangle PEQF,
whose sides PE = a and PF=b are infinitely small. If P^ and 722
are the principal radii of curvature, there will always be two angles
a and (3 such that a = Rla, b = R.2(3, and therefore dS = a.b = filR2a/3.
The normals to S erected at E, Q and F, intersect S' at E', Q\ F'.
We set FE' = a', P'F = b' and obtain a' = (R1 + 8v)a, b' = (Rz + 8v)/3
If Si is the part of S' bounded by <r, we will have
dSi = a'b' = (R^ + (Rl + R2)8v)a{3 and d(Sl' -S) = (l JP^ + I JR2)8vdS.
We have therefore (c) £/ -S=\(l/Rl + \/R2)8vdS. The total incre-
ment 8S which S receives in consequence of the change of form is
therefore (d) 8S= f (l/^ + l/R2)8vdS + \8l . ds. This expression remains
valid even if 81 and 8v at particular points or at all points of the
surface S are negative. If the fluid mass is bounded by a single
surface, the contour s will be zero; if the contour is fixed we have
$ = 0. In both cases the condition of equilibrium is
(e) \(I/E1 + l/R2)8vdS=0.
Since the space occupied by the fluid mass is supposed constant, we
have (f) ftvdS = 0, since ftvdS represents the increment of volume.
From (e) and (f) it follows that (g) l/Rl + ljRz = ct where c is a
constant. The same result is also given by L. (d) if we notice that
the pressure in the fluid mass must be constant.
If three fluids which do not mix meet in a line, the three angles
which the surfaces of the fluids make with one another may be
determined. Such relations occur if a drop of oil lies on the surface
of water. In this case the three fluids which meet are water, oil,
and air. We shall designate these fluids, for greater generality, by
a, b, and c; let the energy of a unit area of the surface separating
a and b be Cab ; let €„. and (76c have similar meanings. It is sufficient
to examine the case in which the edge is a straight line. The
directions of the three surface tensions Gab, C^, and (7^, determine
the inclination of the surfaces to each other ; equilibrium exists when
the three forces Cab, C^., C^ are in equilibrium. Let a, /?, y be the
three angles sought, belonging respectively to the three fluids a. b,
and c. We then have as the condition of equilibrium,
(h) CJ sin a = CJ sin (3 = CJ sin y.
SECT. LI.]
CONDITIONS OF EQUILIBRIUM.
125
From these equations we may in general determine a, (3, y. If,
however, one of the tensions, say C^, is greater than the sum of
the other two, equilibrium cannot exist. In this case, the fluid
spreads out into a very thin sheet which separates the fluids b and
c ; we have as an example the behaviour of a drop of oil of turpen-
tine on water.
The theorem given in (h) may be also obtained by the following
method : If the edge is displaced by an infinitely small distance
from its original position, the sum of the surface energies
increased by Cbc8S1 + Coc8S2 + Cab8S3. This increment must be equal
to zero, if equilibrium exists; we thus obtain equations (h).
If a solid c (Fig. 51) is in contact with two fluids a and b, the
edges may be displaced infinitely little along the surface of the
solid. In this case we have 8S: = - 8S2, 8S3 = - S^j . cos a, and further,
@ic - Cac = CM - cos a- Hence we have (i) cos a = (C^ - C^)ICaM where
a is the so-called contact angle.
SECTION LII. CAPILLARY TUBES.
To make an application of the foregoing principles, we will consider
a cylindrical tube c (Fig. 51 A) placed perpendicularly, the lower end
of which is immersed in a fluid b ; the upper part of the tube is
surrounded by air, which may be repre-
sented by a. The bounding surfaces may
be represented as before by Sv S2, and
S3. We may call the surface of contact
between the fluid and the tube Sv that
between the air and the tube S2, and
that between the air and the fluid S3. ~M N
The fluid surface MM outside the tube
may be infinitely great; it may also be
considered as at rest, even when the surface in the tube is in motion.
Take the surface MM for the xy-plsuue, and let the z-axis be directed
perpendicularly upward. If g represents the acceleration of gravity,
and p the density of the fluid, the potential energy of a particle of
the fluid pdv, will be g . pdv . z. Hence the potential energy of the
fluid mass lying above the xy-plane is
i b
FIG. 51 A.
M
126 CAPILLARITY. [CHAP. vi. SECT. LIT.
if x, y, z are from now on considered as belonging to S3. The part
of the potential energy Ep whose variations are to be considered is
EP = &P\ \Mxdy + CabS3 + CJ3, + CJ3V
In the case of equilibrium we have 8EP = Q, or
<a) 0 - gp\ \z8zdxdy + Cab8S3 + CM8S2 + CJS^
If s is the length of the line of section of the surface S3 with
the inner surface of the tube, if </> represents the angle between ds and
the xy-pla.ne, and if all points of the surface S3 are elevated by the
same infinitely small amount Sz, where 8z is constant, we will have
8S3 = Q, &§!= -8S2 = \cos<j>8zds. The equation (a) then becomes
(b) gp\ \zdxdy = (CM- C7fc) J cos <frds.
Hence the fluid will be displaced by the difference of the tensions
in the surfaces S.2 and Sr
If, on the other hand, the line of section s retains its position,
and if the only change is the change in the shape of the surface S3,
we will have from LI. (d) 8S3 = ^(l/El + l/Sa)8vdSs and 8Sl = 8S2 = Q,
in which 8v is an element of a normal lying between the surfaces
S and -ST.
We may replace Szdxdy by 8vdS3, and obtain from equation (a)
f {gpz+ C^l/E, + l/R2)}8vdS3 = 0.
Since 8v is arbitrary, we must have (c) gpz + Cab(l/Rl + 1/A>2) = 0. If
the curvature of the surface is' expressed by the differential coefficients
of z with respect to x and y, we will obtain from (c) a differential
equation for the determination of the form of the surface. If the
contact angle is also given, the surface is completely determined.
If the cross-section of the tube is circular and very narrow, we
may assume that approximately B^ = R.2 = r/ cos a, where r is the
radius of the tube and a the contact angle. The height z to which
the fluid rises is then z = - 2 cos a/gpr . C^. We obtain the same
result from equation (b) if we set ^zdxdy = Trr2z, Jc08<kfca*2rr, and
use equation LI. (i).
The theory of capillarity was discussed by Laplace in a supple-
ment to the tenth book of the Mtcanique Celeste. Poisson wrote a
larger work on the subject, called Nmivelle Thdorie de I' Action
Capillaire : Paris, 1831. Finally Gauss made an epoch-making in-
vestigation on the theory of capillarity, published in the Commentationes
Soc. Scient. Gottingensis. Vol. VII. 1830. (Works., Vol. V., p. 29.)
The most elaborate recent publication on the subject is that of Mathieu,
Tktorie de la Capillarity : Paris, 1883.
CHAPTER VII.
ELECTEOSTATICS.
SECTION LIII. FUNDAMENTAL PHENOMENA OF ELECTRICITY.
THE theory of electricity is founded upon the observation that
amber and other bodies obtain by friction the property of attracting
light bodies. Gray showed that this property may be transferred
from one body to another. The conception was thus suggested that
this property depends upon the presence of a fluid which is formed or
set free by friction in the body, and which, under certain conditions,
can pass from body to body. Dufay first showed that there are two
so-called electrical conditions, or according to the conception just
stated, two fluids, which Franklin named positive and negative, since
they can completely neutralize each other.
The hypothesis of two fluids has had an extraordinary influence
on the development of the theory of electricity. Poisson proceeded
from this conception in his researches on electrical distribution,
and W. Weber founded on it his theory of electrical currents.
In opposition to this theory, which, in its mathematical treatment,
proceeds from the conception that electrical action is, like gravity,
a force acting at a distance, Faraday adopted the view that the
electrical forces propagate themselves from particle to particle, not
immediately, therefore, but by the action of an intervening medium.
On this view it is not possible to explain the phenomena of electricity
completely without introducing a hypothetical medium, the ether,
and we thus meet with peculiar difficulties, which have not yet been
entirely overcome. Fruitful as Faraday's conceptions have been,
we still cannot explain many phenomena, even by their help, and a
complete systematic discussion of electricity cannot yet be given.
In the following presentation it has not been possible to consider the
whole subject from one point of view ; we have only endeavoured
to present the most important results which have been obtained.
127
128 ELECTROSTATICS. [CHAP. vn.
The starting point of our study is Coulomb's investigation of the
mechanical force which two electrified bodies exert on each other.
Two bodies which carry the charges el and e2, measured in any manner
and separated from each other by the distance r, will then act on
each other, according to Coulomb's law, with a force ^ = c . e^/r2,
where c is a constant. According as the two charges are similar or
dissimilar the bodies repel or attract each other. If the distribution
of electricity on extended bodies is known, the mechanical force
with which the bodies act on each other may be calculated. As a
rule, however, the distribution of electricity on a body cannot be
considered given. If electricity is developed by friction on a glass
rod or a stick of sealing wax, that is, on relatively poor conductors,
a slow discharge of the same occurs with lapse of time. The distribution
of electricity on good conductors depends on their form, on the char-
acter of the body surrounding them, and on the charge. In deter-
mining the distribution, we start from the assumption that the same
force acts between two quantities of electricity as between two bodies
which are charged with these quantities of electricity. If a definite
charge is imparted to an insulated conductor its separate parts will
act on each other, and there will be a definite distribution.
Charged bodies excite an electrical distribution in neighbouring
bodies. The attraction which a charged body exerts on an uncharged
body is explained by the assumption that positive and negative
electricities in equal quantities are present in the latter body, and
that under the influence of the charged body a separation of the
opposite electricities occurs, the force which proceeds from the charged
body acting as an electromotive force. The distribution thus pro-
duced acts against the external electromotive force, and a condition
of equilibrium is brought about if the electromotive force, which arises
partly from the force acting from without which causes the distri-
bution, and partly from the electricity separated in the body itself
and therefore free, is everywhere zero within the conductor. In poor
conductors also as, for example, the air, an electromotive force must
arise under similar circumstances, whose action we will at present
not consider.
SECTION LIV. ELECTRICAL POTENTIAL.
Suppose that electricity of density p is contained within the body
L (Fig. 52), and that electricity of density a- is present on its surface.
A volume-element dr then contains the quantity of electricity p . dr,
SECT. LIV.] ELECTRICAL POTENTIAL. 129
and a surface-element dS contains the quantity o- . dS. Suppose that
unit quantity of electricity is pre-
sent at the point P, whose co-
ordinates are x, y, z, and that the
charges within L and on its surface ( L ] "jT Q
act upon it. Let the coordinates
of any point in the body L be £,
»7, £ and let X, Y, Z be the com- FlG 52
ponents of the force which acts at
the point P. We then have from Coulomb's law (cf. XII.),
A' = \(x - £)/»« . pdr + \(x - £)/r» . a-dS,
y= (y - 1* •
where r* = (x - £)2 + (y- r,f + (z- ()2.
If we set (b) ¥ = J/o . dr/r + Jo- . dS/r, it follows that
(c) X=-Wrdx, Y= -'d^py, Z= -W/Vz.
*¥ is the electrical potential. If the unit charge moves from P to Q,
and if r' is the distance of the point Q from the point (£, ?/, £) of
the body L, we will have W = ^p. dr/r' + Jo- . dS/rr The work done
by the electrical forces during the motion is
\(Xdx + Ydy + Zdz) = ¥-¥'.
If the point Q is so far distant from the body L that ¥' = 0, the
work done will equal M*. We can therefore say : the electrical potential
at a point is equal to the work done by the electrical forces when a unit
of electricity moves from the point considered to a point at an infinite dis-
tance from the charged body.
If the point P is within the body L, we describe a sphere K of
radius E about the point P as centre, so small that the density p
within it may be considered constant. The force due to the charge
in K is then zero (cf. XIII.) ; the potential within L due to the
electricity present in it cannot, therefore, be infinite.
The potential has the same value on loth sides of a surface charged
with electricity. If the surface density is o-, the potential ^ which
arises from the surface distribution is ^F = Jo- . dS/r. We will consider
the values "^ and W.2 of the potential at two points Pl and P2
(Fig. 53) which lie on each side of the surface AB, and are separated
by an infinitely small distance. Let the line P^PZ be a normal to
the surface. We suppose an infinitely small portion of the surface
i
130 ELECTROSTATICS. [CHAP. VH.
cut off by a circular cylinder, whose axis is the line /^P., and whose
radius is R. The potential ^ is made up of two parts, one of which
*Py arises from the part of the surface which is cut off by
„ the cylinder, and the other ^rl - ^Py from the remaining
7 part of the surface. The latter value, ^ - "*?/, is neither
discontinuous nor infinite in the distance from Pl to P0.
The value ¥2 is likewise made up of two parts. The
radius of the infinitely small circle cut out of the surface
is R. Let n be the distance of Pl from the surface ; we
will then have
It therefore appears that >Py vanishes if R and n are
infinitely small.
It has been shown [XIV.] that
a2(i/r)/az2 + a2(i/r)/a#2 + a2(i/r)/a^2 = o.
FIG 53 ^e ^ave tneref°re> from (b), for a point outside of L,
(d) a^pyaa;2 + a^/a^2 + a^/a^2 = v'2^ = o.
On the other hand, if P lies within the body we have [XIV. (h)]
(e) V21? + 4irp = 0. If we designate the normals to the surface drawn
inward and outward by vs and va, we have for the surface density
o- [XIV. (1)], (f) W,/3v, + a^0/ava + 47TO- = 0. The horizontal lines over
the differential coefficients indicate that their values are to be taken
at the surface. Hence if v represents the electrical density on a surface,
the sum of the forces acting in the direction of the normals drawn outward
from both faces equals lira-.
These properties of the potential hold for every system of bodies
charged in any way with electricity. If the distribution is given, the
potential can be determined either from (b) or from (e) and (f). If
the potential is given, the densities p and o- are determined from
(e) and (f), while the components of the electrical force are given
from (c). The force F, acting in any direction ds, is F= -
SECTION LV. THE DISTRIBUTION OF ELECTRICITY ON A
GOOD CONDUCTOR.
If a charge e is communicated to a good conductor, it distributes
itself over the conductor. \Ve will determine its volume density p
SECT. LV.] DISTRIBUTION ON A CONDUCTOR. 131
and the surface density <r. If M^ is the potential inside, and *Pa the
potential outside the conductor, we have from LIV. (d) and (e)
(a) V2^,. + 47rp = 0 and V*P,, = 0.
After equilibrium is attained there is no electrical separation in the
interior, that is, we have (b) 3¥,/ae = 0, 3¥y3z/ = 0, 3¥,/a? = 0 for
all points in the interior of the conductor. Hence (c) V2^i==0, and
therefore from (a) /> = 0. Hence the electricity is distributed only on
the surface of the conductor.
From (b) ^ is constant in the interior of the conductor and equal
to ¥", where ¥ is the value of the potential on the surface. We may
determine ¥rt from the equations y2^fa = 0 and '¥„ = *¥ for all points
of the surface. The surface density is given by
3*V9i'« + 'd¥m[dvil + 47TO- = 0.
Since ¥, is constant we obtain (d) 4iro- = -^J?>va.
If we represent the force acting outward at the surface of the
conductor by F, we have (e) F— - 'dj¥a/dva = 47ro-, that is, the force
acting at a point on the surface of the conductor in the direction of the
normal is equal to the surface density a- at that point multiplied by 47r.
If ds is an element of a curve drawn on the surface of the con-
ductor, we have 3^/3$ = 0, since "*"„ is equal to ^f everywhere on
the surface, as has just been shown. Hence F has no components
in the surface. The surface of the conductor is a surface of constant
potential, and the direction of the force F is everywhere perpendicular
to it.
To determine the potential of the conductor its charge e must be
calculated ; we have e = J Jo- . dS, and therefore
<f) 'e = 1/47T . JJ F. dS= - 1 ,/47T . J j3*V3v0 . dS.
Since there is no charge in the interior of the conductor, we have
*Pa = \\<r.dSlr. If the electricity on the conductor, whose density
is <r, is in equilibrium, it will remain so if the density becomes wr,
where n is a number. If one distribution which is in equilibrium is
superposed on another, the new distribution is still in equilibrium.
If the density is everywhere rw, the potential has the value ri*Fa.
The potential is therefore proportional to the charge. If the charge C will
bring the conductor to the potential 1, the charge (g) Q = C^r must
be imparted to the conductor in order to bring the potential from
0 to "V. We call C the capacity of the conductor. The capacity C
is the ratio of the charge Q of a conductor to its potential ¥. In
order to give a means of representing the magnitude and direction
132
ELECTROSTATICS.
[CHAP. vn.
of the electrical force in the region around the conductor, we determine
the position of the surfaces of constant potential which surround
it. Their equation is ^ = c, where c is a constant. The first equi-
potential surface is the surface of the conductor, for which ¥ = XK
At a distance which is very great in comparison with the dimensions
of the conductor, the equi-potential surfaces will be spheres, since in
that case the expression reduces to ^ = ejr.
From VII. the surfaces of constant potential are perpendicular to
the direction of the force. If two such surfaces are considered which
are infinitely near each other, it appears from VII. that the force
at every point in one of them is inversely proportional to the distance
between the surfaces. If a line of force PP1P2PB (Fig. 54) is drawn
from a point on the surface, and if the points Pv P2, P3, etc., are
so chosen that F. PPl = Fl. P1P2 = F.2. P2P3, etc., where Flt F^ are
the electrical forces at the points Pv P2, we
have a relation which may be carried out to
any distance. If we draw the equi-potential
surfaces in such a way that the potential, as
we pass from one to the next, increases or
diminishes regularly by the same amount, the
product Fn.(PJPn+1) will be constant. The
further we pass from the acting quantity of
electricity the greater will be the distance
between the successive equi-potential surfaces.
If the magnitude of the force at one of the
equi-potential surfaces is given, its magnitude
at another point of the figure may be calculated from the distance
between the successive equi-potential surfaces, and if the magnitude
of the electrical force F at the surface of the body is given, its
magnitude at every other point of the figure may be determined.
FIG. 54.
SECTION LVI. THE DISTRIBUTION OF ELECTRICITY ON A
SPHERE AND ON AN ELLIPSOID.
1. The Sphere, Suppose the charge Q given to an insulated sphere
of radius R. We are to determine the potential "*P0 at a point outside
the sphere. Let the centre of the sphere be taken as the origin of
coordinates. We have V2^a = 0. Since ¥„ is a function of the dis-
tance from the centre 0, we have [XV. (1)] 1/r .d'2(rVn)/dr* = 0, and
hence "(Pa = cl + c<Jr, where cx and c2 are constants.
SECT. LVL] DISTEIBUTION ON AN ELLIPSOID. 133
We assume that no charged bodies are present besides the sphere,
so that ¥„ = 0 when r = <x> , and hence cx = 0. The electrical force
at the distance r from the centre is F= -d^rjdr = c.2/r2.
From LV. (e) we have further
The potential ¥„ and the capacity C are therefore
¥a=0/r, C=QI^ = R.
The dimensions of capacity are therefore those of a length.
2. The Ellipsoid. Eepresent the semi-axes of the ellipsoid by a, b, c,
and its charge by Q. It is most natural to assume that the surfaces
of constant potential are confocal ellipsoids. The equation of a
system of such surfaces is
(a) E = 3? I (a? + A) + f/(b2 + A) + *2/(c2 + A) = 1 .
On our assumption the potential must be a function of A, so that
we will write ¥ =/(A). To find this function / we proceed from
and the analogous expressions for y and z. These give
<b) V2* = d'WJdX* . ( (3A/3.C)2 + (3A/ay)2 + (3A/3z)2) + dV/dX . V2A.
We will set for brevity,
A = .«2/(o2 + A)2 + f/(b* + A)2 + ^/(c2 + A)2
B = ./:2/(«2 + A)3
We then have
(c)
(d)
Analogous expressions hold for the differential coeflBcients \vith respect
to y and z. Hence we have
c>2£/a*;2 = 2/(tt2 + A) - ^ . 92 A/3.7:2 - 2ar/(a2 + A)2 .
or, using equation (c),
(e) 32£/ac2 = 2/(a2 + A) - A . 32A/3x2 + '2B.
From equations (c) and (a) it follows that
(f) A2((d\px)* + (3A/3y)2 + (9A/o?)2) = 4A,
and from (e) and (f) that
(g) A . V2 A = 2/(a2 + A) + 2/(ft2 + A) + 2/(c2 + A).
By the help of equations (f) and (g) it follows from (b) that
<h) A . V2*" = 4d-^/dX2 + 2d¥jdX . (l/(fl2 + A) + l/(62 + A) + l/(c2 + A)).
134 ELECTEOSTATICS. [CHAP. vn.
Outside the ellipsoid we have y2^ = 0. If C is a constant, we
have from (h), (i) dV/d\ = - Cjjfa* + A)(62 + A)(c2 + A) and
A)(c2 + A) + <72.
At an infinitely distant point, for which x = y = z = QC , ¥" is assumed
equal to zero ; and for such a point equation (a) shows that A. = x> .
The potential ¥ at any point (x, y, z) is therefore given by
+ A)(62 + A)(c2 + A),
where A is known from the equation
z2/(a2 + A) + ?/2/(&2 + A) + s2/(c2 + A) = 1 .
If the charge on the ellipsoid is Q, the potential at a point at a
great distance from the ellipsoid is Qj^JX ; by comparison with (k)
we then obtain
(1) ¥ = Q/2 . d\lJ(a* + A) (ft2 + A)(c2 + A).
The electrical force F and its components X, Y, Z are determined
from the equations
X = - dVdX . 3Aar Y= - dVdX . 9A3 Z = -
F = -
From (f) and (1) we have
F= -
If we represent the perpendicular let fall from the origin on the
plane tangent to the ellipsoid at the point x, y, z of its surface by Nt
we have JA-\IN> and hence F=Q. JV/>/(^+A)(6* + A)(c2 + A).
The surface density o- on the ellipsoid itself is determined by the
equation 47ro- = F, and A = 0 for this ellipsoid, so that (m) a- = N. Q/lirabc.
Hence the electrical density at a point on the ellipsoid is proportional io
the perpendicular let fall from the centre on the plane tangent to the
ellipsoid at that point.
We will now consider several special cases. In the case of an
ellipsoid of rotation a = b, and therefore from (1), if ^ is the potential
of the ellipsoid, we have
% = Q/2 . J°°fU/(a2 + AjVc^+A:
Hence for a > c, (n) ¥0 = Qf-Ja* - c2 . (\TT - arctg c/V«2 - c2) ; for « = c,
(o)^P0 = Q/a; and for a<c,
(P) ^o = £/2v/c^2 . log [(c + N/^^)/(C - Vc2^^)].
SECT. LVI.]
DISTEIBUTION ON AN ELLIPSOID.
135
If in (n) we set c = 0 the ellipsoid becomes a circular plate, and its
capacity is C= Q/yir0 = a/(^Tr). For an ellipsoid of rotation whose length
is great in comparison with the equatorial diameter, we have from (p)
¥0 = Q/c . log (2c/o) and C'=c/log(2c/o).
The surface density a-, from equation (m), is
a- = Q/l-rrabc . l/Na:2/
If z is eliminated by the help of the equation of the ellipsoid, and
if c is infinitely small, we have for the density on an elliptical
plate whose semi-axes are a and b, a-^Q/^Trab. !/>/! -x2/a2-y'2/b'2.
If the plate is circular, that is, if a = b, and if we set x2 + y'2 = r2,
we have <r = Qjlira . l/>Ja? - r*. At a point whose distance u from
the edge is very small, we have o- = Q/4:Tra. l/*/2au. In this case,
therefore, the density is inversely proportional to the square root
of the distance of the point from the
SECTION LVII. ELECTRICAL DISTRIBUTION.
If several charged conductors are present in a region, the distribu-
tion of electricity on the conductors is determined not only by their
form and magnitude, but also by their mutual action. The deter-
mination of the conditions of electrical equilibrium is, as a rule, very
difficult. The most important work on this subject has been done
by Poisson and William Thomson. We will here make use of the
method of electrical images given by Thomson.
(a) Distribution on a Plane Surface. — Suppose the quantity of
electricity e present at the point 0 (Fig. 55) ; let AS be the plane
surface of a very large conductor L,
which is in conducting contact with
the earth. The potential ^ of L is
therefore zero, since we assume the
potential of the earth equal to zero
(cf. VII.). We are to determine the
surface density o- of the distribution
on the surface. Let the potential at
an arbitrary point in space due to the
conductor L be "*",., so that ^ is the
work done by the electrical forces of FlG- 55-
the conductor if unit quantity of electricity, which is supposed to be
merely a test charge and to have no effect on the electrical distribution
136
ELECTROSTATICS.
[CHAP. vu.
on L, is transferred from the point P to infinity. If OP = r, the
potential ^F at P will be ^ = e/r + V,.
We now suppose a quantity of electricity - e situated at the point
0' (Fig. 55), which is the image of the point 0 with respect to the
plane AB. This imaginary quantity at 0' would act on all points
lying on the same side of the plane AB as 0, in the same way as
the quantity of electricity which is distributed on AB ; for the
potential which arises from the quantities at 0 and 0' satisfies
Laplace's equation at all points which lie on the same side of the
FIG. 56.
plane AB as 0, except at the point 0 itself. Further, the potential
vanishes at all points of the plane AB, since all points of that plane,
which passes perpendicularly through the middle point of the line
00', are equally distant from the points 0 and 0', and hence for
all points of the plane AB we have e/r - e/r' = 0, where r and r'
represent the distances of a point in the plane from 0 and 0'. Now
if a function satisfies Laplace's equation and assumes assigned values
over a given surface, and if the function itself and its differential
coefficients are continuous, it is single-valued and determinate. This
theorem is known as Dirichlet's Principle.
SECT. LVII.] DISTRIBUTION. ELECTRICAL IMAGES. 137
It should be noticed that "9~e = e/r', and that the potential ¥ at the
point P is V = e/r -e/r'.
A unit quantity of positive electricity lying at a point P (Fig. 56)
in the plane AB is acted on by two forces, K=e/r2 and K = e/r''2,
whose directions coincide with OP and PO' respectively. Hence the
direction of the resultant force F is parallel to 00' and equal to
F= - 2eOI»/OP3, if we consider the force positive when it is directed
toward the region in which 0 lies. Now since 4:Tra- = F we obtain
<T= - e/2ir . OB/OPS. The surface density at the point P (Fig. 56)
is therefore inversely proportional to the cube of the distance of the point P
from the point 0, at which the quantity of electricity +e is situated.
The potential and the surface density are calculated in the same
way when several points carrying charges are present in the region.
(b) The Sphere. — Suppose that the quantities of electricity e and e'
are situated at the points 0 and 0' (Fig. 57). The equi-potential
FIG. 57.
surface for which the potential vanishes is given by the equation
4/r + e'/r' = Q if r and r represent the distances of a point on the
equi-potential surface from 0 and 0' respectively. If e and e' have
the same signs, this equation represents a surface lying at infinity ;
if e and e' have opposite signs it represents a sphere and, in the
limiting case, a plane.
The centre C of the sphere lies on the line 00' and we have
00 : 0'C= e2 : e"2 and CO'/CB = CBJCO = e'/e. The triangles CO'B and
CBO are similar, and the radius CB of the sphere is the mean
proportional between the distances of the points 0 and 0' from the
centre of the sphere.
If a hollow sphere of very thin sheet metal, in conducting connection
with the earth, is brought into the place occupied by this spherical
surface of zero-potential, the potential of points in the region will
not be changed, either within or without the sphere ; the electrical
action depends only on the quantities of electricity e and e'.
138 ELECTROSTATICS. [CHAP. vn.
If the sphere remains in conducting contact with the earth, and if
we remove the quantity of electricity e' from the interior of the
sphere, the potential within the sphere will become zero, while
outside the sphere it retains its former value, since the quantity of
electricity e does not change its position and the potential of the
sphere still remains zero. Hence, the quantity e lying outside the
sphere, kept at potential zero, together with the electricity induced
on the sphere, exerts on points outside the sphere the same action
as if the induced electricity were replaced by the mass e' lying inside
the sphere. We call the point 0', where the mass e' is situated, the
electrical image of the point 0. The quantity of electricity e' at that
point will exert the same action as the quantity of electricity actually
present on the sphere. In optics a point which appears to emit
light from behind a mirror or lens which, if it were self-luminous,
would emit rays in the same direction as those which proceed from
the mirror or lens, is called a virtual image. Hence, we may consider
0' the electrical image.
If we set C0'=f, (70 = a, and CB = R, we have
e'/e = 0'B/OB=f/E = R/a;
and hence e' = Be/a. We set OB = r and 0'B = r'. The force e/r2
acts at B in the direction OB, and the force e'/r'2 acts at the same
point in the direction BO'. The former of these may be resolved
into the components e/r'2. a/r along OC and e/r2. Rjr along CB, the
latter into -e'/r'2.f/Y along OC and -e'/r'2.R/r along CB. The
two components in the direction OC are equal but oppositely directed,
and therefore annul each other. The other two combine to give the
force efi/r3 - e'R/r'3 = - (a2 - R?)/E . e/r5, which acts in the direction
CB. The sphere is there/we an equi-potential surface, since the direction of
the force, at any point of its surface, coincides tvith the direction of the
normal to that surface.
The density <r, as obtained from F= 4™, is cr= - (a- - R^/^irR.e/r3,
and hence is inversely p-oportional to the third power of the distance
from the charged point 0. The quantity of electricity on the sphere
is - e' = - Re I 'a, since this charge produces the same potential in the
region as the actual charge on the sphere, and therefore [LV. (f)] must
be equal to it.
The sphere is attracted by the point 0 with the force
If the distance (70 = a is very great, this force becomes Re^/a3.
SECT. LVII.] ELECTRICAL IMAGES. 139
(c) If the sphere is originally insulated and uncharged, we can
find the electrical distribution on it by assuming that, besides carrying
a charge, distributed as above described, it also carries a uniformly
distributed charge whose surface-density is e'/4:-!rR2 = e/4:TrRa. The
sum of these two charges or the charge of the sphere is equal to
zero. The surface-density is then o- = e/4;rj?.(l/a- (a2 - R2)/^). The
surface-density cr is zero on a circle whose periphery is distant
r = af/1 - ft? /a2 from the point 0. The plane of this circle lies nearer
to 0 than the centre of the sphere. In order to find the potential^
of the sphere, we determine it for the centre. Since the charge on
the sphere is zero, the potential due to that charge is also zero ; the
potential at the centre is therefore ^ = eja. This follows from the
remark that the induced charge - e' and the charge e at 0 together
have no effect on the potential of the sphere ; the potential is due to
the additional charge + e', which makes the potential e /R = e/a.
The force with which the sphere is attracted by 0 is in this case
very much smaller than if the sphere were in conducting connection
with the earth. It is
ee'/(a -/)2 - ee'/a2 = Re2/as. R2('2a2 - R2)/(a2 - R2)2.
When R is very small in comparison with a, the force is approximately
Re2/a3. 2R2fa2. In this case we have a simpler expression for the
surface-density a-. Designating the angle BCO by 6, we have
If a is so great that the higher powers of R/a can be neglected, we
have r~3 = a~3(l + 3R/a . cos 6). If we designate the inducing force
e/a? which proceeds from 0 by X, we have a- = - 3 cos 6/47r . X, if,
in the formula for o-, we consider the radius R as infinitely small in
comparison with a and substitute the value just given for r~3.
SECTION LVIII. COMPLETE DISTRIBUTION.
If a charged body A (Fig. 58) is situated in the interior of a metallic
shell BC, there will be a distribution of electricity on the shell. If
A is charged positively, the inner surface B will be negatively, and
the outer surface C positively electrified. Let the charge on A be
e, that on B - e', and that on C + e'. We will show that the
quantity of induced electricity e' is equal to the quantity of induc-
ing electricity e. Let us suppose a closed surface D drawn in the
interior of BC. If ¥ is the potential in the shell BC, and v the
140
ELECTROSTATICS.
[CHAP. vii.
normal to the element dS of the surface D, we have from XIV.
(c), 4ir(e -e')=- \(dy/Vv)dS. Since the potential "*~ in the shell is
constant the integral vanishes, and hence e = e'. The charge e on C
can be conducted off, or another charge can be conducted to it
without causing any change in the charges A and B. If the inducing
body is entirely surrounded by the body in which electricity is
induced, we may say that the induction is complete ; the inducing and
induced quantities of electricity are equal.
C
FIG. 58.
FIG. 59.
This theorem may be used in the comparison of the charges of
different conductors. If the outer surface of BC is connected with
an instrument which will measure potentials, and if the charged
bodies to be tested are brought successively into the hollow within
BC, the potentials indicated by the instrument are proportional to
the magnitudes of the charges.
Suppose the sphere A, whose radius is R, to be charged with the
quantity e (Fig. 59). Let BC be a spherical shell concentric with A,
whose radii are R2 and Ry The inner surface of BC is then
charged with the quantity - e. If there is no electricity on the
outer surface (7, the potential ¥ at A is * = ejRl - e/R2. The potential
at a point within the shell BC or outside the surface C is zero, since
both charges act with respect to an external point, as if they were
concentrated at the common centre of the spheres. From the
definition of the capacity (7, we have [cf. LV. (g)]
(a) C=e/V = RiRJ(Rz - A).
The induction can, in many cases, be almost complete, even when
the one conductor is not completely surrounded by the other. Let
ABC and DEF (Fig. 60) be two conductors whose surfaces BC and
SECT. LVIII.]
COMPLETE DISTRIBUTION.
141
DF lie very near each other. On the surface BC describe a closed
curve GH, and from all points of it draw lines of force, which cut
out on DF the curve KJ. Now draw a closed
surface G'GJJ'K'KHH' in such a manner that the
two curves GH and JK lie in it. The surfaces
bounded by the curves G'H' and J'K' lie inside
the conductors and are congruent to the surfaces
GH and JK respectively. Let dS be a surface-
element of the closed surface G'H' J'K', and let e
and e' be the charges of the surfaces GH and JK.
The potential is represented by ¥ and the normal
by v. We then have from LV.
(b) ±Tr(e + e')= - \\Wfiv . dS.
The integral vanishes, since ^ is constant within the conductors, and
since between the conductors the force is parallel to the closed surface.
Hence we have e = -e'. If the surfaces BC and DF lie very near each
other, we also have <r= -a-', that is, the densities on the two surfaces
are equal but of opposite sign.
If a is the distance between the surfaces BC and DF, and if ^
is the potential of ABC, and ^2 the potential of DEF, the electrical
force F in the intervening space is [VII. (e)]
¥j = ¥2 + Fa, F=(¥1- ¥2)/a
- ¥2)/47
The
The surface-density o- is [cf. LV. (e)] a-= - <r' =
charge on the surface S is e = (^ - ¥2)/47ra . S.
If the conductor DF is connected with the earth, that is, if ^ = 0,
the capacity C will be C* = $/47ra, that is, the capacity is inversely
proportional to the distance between the conductors,
This formula is used in air-condensers, when a is very small.
SECTION LIX. MECHANICAL FORCE ACTING ON A CHARGED BODY.
If an element of volume contains the quantity of electricity pdv,
it is acted on by a force whose components are Xpdv, Ypdv, and Zpdv.
From LV. (a) the ,r-component can be expressed by
1/47T . Wrdx . ^dv = + 1/47T . X(dX/*dx + VY/dy + dZfdz)dr.
In the interior of a good conductor the force and electrical density
are zero ; but a force acts on each element of its surface, where the
density is not zero. This force is determined in the following way :
p+-
1 E
142 ELECTROSTATICS. [CHAP. vn.
Let AD (Fig. 61) be the electrified surface, and BC = dS the surface-
element, which is cut out by an infinitely small sphere described
- about the point P as centre, 'with the radius PB = PC.
Let P2 lie on the normal PP2 to BC, and let P.2P be
infinitely small in comparison with PB. A unit of
electricity at the point P2 is acted on by the force
27ro- arising from the distribution on the surface BC
[cf. XIII. (3)]. At the corresponding point Pl on
the opposite side of BC the force acting is - 27nr. If
/, m, n are the cosines of the angles which PjP., makes
with the axes, and X, Y, Z are the components of
force which arise from all the electricity present except
FIG. 61. that on BC, we have
X., = - 3¥2/ae = X + 2Tro-l, Xl = - 3¥,/az = X - 2Tro-l.
X2, Xl and ¥2, ^x are the components of force and the potentials
at the points P2 and Pl respectively. We have (c) X=^(X<, + Xl).
Analogous expressions hold for the other components of force. X
represents the force Avhich acts on a unit of electricity on dS in the
direction of the a:-axis. The element dS is therefore moved in the
direction of the #-axis by the force (d) Xa-dS •= %(X.2 + XJo-dS.
If the element dS is part of the surface of a good conductor, and
if Pl lies within the conductor, the force at Pl is equal to zero. If
we represent the force acting at P.2 by F, the force which acts on
(IS is, from (c) and (d), (e) ^Fa-dS. Since, from LV. (e), we have
F= +47ro-, the force sought will be (f) 2Tro-'2dS=l/8ir.F2.dS.
W. Thomson has made an interesting application of this equation
in the construction of his absolute electrometer. This consists of an
insulated metal plate EF and a smaller circular plate CD, which is
parallel with EF. CD forms a part of the base of a metallic cylinder
AB (Fig. 62). ' If the potential of EF
is ¥, and that of CD and AB is zero,
CD will be attracted to EF by a force
which is determined in the following
way : Since AB and CD are almost
like a single continuous body, there
is no perceptible surface distribution
on the inner surface of A BCD. Re-
FIG. 62. present the density on the external
surface of CD by o-, the distance between CD and EF by a, and let
the surface CD = S. The force K which attracts CD toward EF is,
C D
SECT. L1X.]
FORCE ON CHARGED BODY.
143
from (e), equal to K=$F<rS. We have Fa = ^ [cf. VII.], ^<r = F,
and therefore K=S^'2l%iraz. We determine the weight which is
necessary to counterbalance the electrical attraction. If M grams are
necessary for this purpose, we have ^ = a»j8TrMg/St where g is the
acceleration of gravity.
A B
FIG. 63.
FIG. 64.
SECTION LX. LINES OF ELECTRICAL FORCE.
All actions by which electricity is produced, such as friction, induc-
tion, etc., produce equal quantities of positive and negative electricity ;
for this reason we are led to assume that in every unelectrified body
equal quantities of positive and negative electricity are present. Let
A and B (Fig. 63) be two bodies electrified
by friction which are gradually separated
further and further from each other, as in
Fig. 64 ; during this separation they retain
equal but opposite charges. Suppose A and
B to be good conductors and to be insulated. "
Now, construct lines of force from all points
of the contour of a surface-element dS on A.
They will determine a surface-element dS' on
B. The region bounded by the lines of force
and the two elements dS and dS' may be
called a sphondyloid. If the density is o- on dS and a-' on dS', we may
prove, as in the discussion following LVIII. (b), that a-dS - a-'dS' = 0.
The surface-densities on the two charged conductors A and B are
therefore inversely proportional to the surfaces limited by the
sphondyloid.
If the conductor A is charged with a quantity Q of positive elec-
tricity, and if the surface of A is cut into Q parts, each one of which
is charged with unit quantity, the lines of force drawn from the
contours of the Q parts on A cut the surface B also into Q parts,
each one of which is charged with unit quantity of negative elec-
tricity.
If a conductor A BCD is charged in any manner to the potential
""P, the equi-potential surfaces about it are spheres, at all distances
which are great in comparison with its dimensions. If we construct
the equi-potential surface, whose potential is (¥ - 1), it lies nearest
the conductor at the points A, B, and D (Fig. 65). At these points
[LV.] the electrical force and the surface-density are greatest. The
144
ELECTROSTATICS.
[CHAP. vn.
lines of force also lie in closest proximity to one another at these
points. If the conductor has edges or points projecting outward,
the density on them is very great; it will be infinitely great on a
perfectly sharp edge. On this depends the so-called action of points ;
the density of the electricity is greatest at these points, and therefore
the electricity flows out from them with especial ease.
FIG. 66.
FIG. 65.
If A (Fig. 66) is a conductor charged with positive electricity,
and B an insulated conductor without charge, negative electricity
will be present at all parts of B which are met by the lines of force
proceeding from A. Lines of force also proceed from the other points
of B ; the number of the lines which fall upon B is equal to the
number of those proceeding from B. Hence, the surface of B is
divided into two parts with opposite charges. The parts are separated
by a curve encircling the body B, along which the surface-density
a- and therefore also the electrical force are zero. This curve is the
line of intersection of B and an equi-potential surface around A.
Let ABC (Fig. 67) be a conductor on whose surface the potential
is constant and equal to ¥, and let A'B'C" be an equi-potential
surface at which the potential is *Pr We
suppose that the charge of each surface-
element, for example of AB = dS, is moved
outward in the direction of the lines of
force and transferred to the surface A'B'C'.
If this surface is a conductor, the electricity
transferred to it is in equilibrium. If the
potential within the surface A'B'C' is ¥j,
and if it retains its former values outside
of that surface, the condition V^a = 0 is
fulfilled for all external points. If the electrical forces at AB and
A'B' are F and F' respectively, we have from LV. (f), since AB and
SECT. LX.] FORCE ON CHARGED BODY. 145
A'B' are bounded by lines of force, AB . F= A'B' . F'. If o- and a-'
represent the densities at AB and A'B' respectively, we have further
AB.a- = A'B'.a-'. We therefore obtain F/<r = F'/(r'. We have, how-
ever, F=±irv and, therefore, also F' = 4Tra-'.
Hence, if we divide the surface of the conductor into elements, each
of which contains unit quantity of electricity, and if we draw lines
of force outward from the boundary of the element, these lines bound
a tube. The tube cuts the equi-potential surfaces surrounding the
conductor in such a way that for all of them we have
F'/<r' = F"/<r" = ...F/v.
From the form of the tubes of force we obtain a representation of
the distribution of electrical force in the region. And also from the
distribution of the lines of force we may distinguish between the
attractive and repulsive forces. Two lines of force proceeding in the
same sense repel each other, so that the repulsion maintains equili-
brium with the tension which acts along the lines of force [cf. XXVII].
SECTION LXI. ELECTRICAL ENERGY.
A conductor charged with the quantity of electricity e can do
work in consequence of that charge ; it possesses electrical energy. If
the charged surface ABC (Fig. 67) is extended so that it assumes in
succession the form of its equi-potential surfaces, the forces acting
on the electrified surface do an amount of work which can be deter-
mined. If the surface of the conductor has the potential ^, and
if it is extended until it coincides with the equi-potential surface
whose potential is ^r + d^r, the work done is calculated in the follow-
ing way : A surface-element dS carrying the charge crdS is acted
on by the force i . Fa-dS. We represent by dv the distance of the
equi-potential surface V + cW from the conductor. The work done
on the element dS during its motion is ^Fa-dSdv. The total work
done is therefore ^^Fo-dSdv.
From the definition of the equi-potential surface [cf. LV. (e)] we
have Fdv = -d^f • we therefore have for the total work done
If the surface of the body is extended until it coincides with the
equi-potential surface at which the potential is zero, the work W
done is
(a) W=
146 ELECTEOSTATICS. [CHAP. vn.
All the work which can be done on the given conditions is represented
"by W, and hence it is called the potential energy of the conductor.
The electrical energy of the conductor can be geometrically repre-
sented in the following way : About the conductor L (Fig. 68),
whose potential is M*", we construct the
equi-potential surfaces whose potentials are
successively IP-I, ¥-2, ¥-3, etc., and
we divide the surface of the body L in
such a way that each part carries unit
charge. The space surrounding the body
L is divided into e^f parts by the lines
of force starting from the boundaries of
the separate parts and by the equi-potential
surfaces ; the number of these parts is
double the value of the electrical energy.
If we designate the electrical energy by W and the capacity by
C, we have e^CY, and thus (b) JF-^IVjC^-l^/a We have
seen before that the energy is also given by
W= \\ \FvdSdv = I / Sir . J \F\lSclv.
If X, Y, Z are the components of the electrical force, and if dxdydz
is a volume-element, we will have
(c) W = 1 /Sir . J J \(X2 + T- + Z*)dxdydz.
If the two conductors A EG and A'B'C' (Fig. 67) are charged with the
quantities of electricity + e and - e respectively, and have the potentials
M^ and *Fy we obtain their potential energy in the same way by
supposing the body ABC carrying the charge +e to be gradually
extended so as to coincide with the equi-potential surfaces which
surround it. In this way the charge e is finally transferred to A'B'C'.
The integral in (a) then becomes
(d) W= - \ l**edV = \e(^ - ¥2).
This method of treatment may be applied in all cases. A system
of conductors whose charges are ev e2, e3, ..., and whose potentials are
"*?!, ^2, ^Pg, ..., have, with respect to a conductor whose potential is
^PO, the potential energy
<e) W= K(*i - %) + &(*« - *0) + - -
If the sum of all the charges is zero, that is, if el + e.2 + . . . = 0
we have (f) ir=%el¥l + ie2¥2+ ... . Therefore, if all charged con-
ductors are brought to the same potential, the electrical energy is inde-
pendent of the value of the common potential.
SECT. LXI.] ELECTRICAL ENERGY. 147
The expression for the electrical energy may be derived in still
another way. In a system of conductors the electrical distribution
is determined if the density p is given at all points in terms of the
coordinates x, y, z. The potential at any point is then, in the usual
notation, * = \pdr/r. The potential increases and diminishes pro-
portionally to p. If the density of the electricity is doubled at all
points, the value of the potential also is doubled.
If the charge l/n.pdr is removed from every volume-element, the
potential becomes (n — 1 )/n . "*F. In order to transfer the quantity of
electricity l/n. \pdr to a distant and very large body whose potential
is •*•„, for instance to the earth, the work l/n. f/orfrC'P - ¥•„) must be
done, if n is a very large number. If the quantity l/n . \pdr is again
removed, the work required is
If the whole charge is at last transferred to the earth, the work
W which is done is
W= l/n . \pdr( I + (n - I )/n + (n - 2)/n + ... + l/rt)* - *0f pdr.
Now, we have 1 +(n- l)/n + (n- 2)/n+ ... + l/n = n(n+ l)/2n, and
if n is very great (g) W= ^^pdr-^Q . \pdr. If the sum of the
quantities of electricity present is zero, we will have (h) W= ^^pdr.
Since V2* = 32*/3x2 + VW/'dy2 + 32*/os2 = -4^, we have
Now, I HV.'&VI'da?. dxdydz
= f J(* . 3*/ae) . dydz - J { f (3*/9z)2 . dxdydz.
Hence, by partial integration extended over the whole volume, we
obtain
<i) W= 1 /STT . {((3*/3z)2 + (d^j'dyf + (3*/^)2)^r = I /Sir . \F*dr.
This result has already been derived for the energy in a good
conductor.
SECTION LXII. A SYSTEM OF CONDUCTORS.
If several insulated conductors Av Ay A3 are given, and if a unit
of electricity is imparted to one of them, say to Av while the others
have no charge, then the potential of A^ becomes plv while the potentials
of A 2 and Ay etc., become pn and pl3 respectively. If A.2 were to
become charged with unit quantity while the other conductors were
to remain uncharged, the potential of Az would equal p22, and the
148 ELECTEOSTATICS. [CHAP. vn.
potentials of Av As, A4, . . . would be p.2v p.2.A, p.2v . . . respectively. Now,
if the conductor Al is charged with the quantity e^, the conductor
A 2 with the quantity e2, etc., the potentials M-^, ^2, ... of the con-
ductors Av A2, A5, ... respectively will be expressed by
The total energy of the electrical system is
(b) W=
Hence,
(c)
If an infinitely small quantity of electricity Sel is communicated to
one of the conductors, for example to Av the energy of the system
will be increased by 8/iT= ^Er18e1. This increment of the energy may
also be expressed by the help of (c), since we have
(d) 8fT= (p^ + |(p12 +p^e2 + $(pl3 +p3l)e3 + ... )&r
Now, since §W is in this case equal to ^f18e1, it follows from the
first of equations (a) that (e) S?F= (^>11e1 +^21e2 +^)31e3 + ...)Ser Com-
paring the formulas (d) and (e), we obtain
Therefore, (f) p.2l =plv psl =^13, and in general pmn =pnm.
The electrical energy W, of the system may therefore be expressed
as a homogeneous quadratic function of the charges by
(g) W* = &nei2 + \Pve* + fce32 + • • • +Pi2eie2 +Pi
where the coefficients pmn are called coefficients of potential.
If equations (a) are solved for ev e.2, es, we obtain
(h)
If the charge 8el is communicated to the conductor Al and the charge
8e2 to the conductor A^ etc., so that the potential ^ is increased
by 8¥v while the other potentials retain their original values, we have
The increment of the energy is therefore
(i) 8/
SECT. LXII.] A SYSTEM OF CONDUCTORS. 149
From equation (h) the energy W is given by
(k)
If ^ is increased by 8WV we have
Comparing equations (i) and (1), it follows that (m) qmn = qnm. The
energy JF^,, expressed in terms of the potentials, is therefore given by
•" +028*^8+ ••••
The coefficients <?„„, in which the indices are the same, are the capacities
of the different conductors ; the coefficients qmia in which the indices
differ from each other, are the coefficients of induction. The energy
can therefore be expressed by the charges as well as by the potentials ;
in the former case it is represented by We, in the latter by W^,.
The significance of the coefficients qlv q^, ... is shown as follows:
If Al (Fig. 69) is an insulated conductor having the charge ev and
if A^ Ay etc., are connected with the earth, we have
ei = fti*!. e2 = qlz^rv e3 = ql3¥v ...,
since "*P2, ^F3, etc.. are equal to zero. The coefficient ^n is the
capacity of the conductor Al under these conditions. Hence, the capacity
of a conductor is the quantity of electricity
which it must contain in order that its potential
shall be unity, while the potential of all other
conductors is zero. The quantity of elec-
tricity induced on the conductors A^ A3
when connected with the earth is given
by <?12, <?13, ..., if A^ is electrified to unit
potential. The coefficients <?12, ql3 are
negative. The lines of force proceeding
from Al may either pass to the earth or terminate on the conductors
Ac,, Ay etc. Since a positive charge is present at the points at
which they leave Av the points at which they fall upon the other
conductors must have a negative charge.
On the other hand, the coefficients pl2, pl3 ...pmn are positive. If the
charges of the conductors A2, A3, . . . are e2 = e3 = e± = . . . = 0, we have
^i ~P\\e\i ^2 =Pi2ev ••• • -^s many lines of force enter the uncharged
conductors A2, A3 as pass out from them. Since lines of force pass
from points of higher to points of lower potential, the potential of
an uncharged conductor in an electrical field cannot be a maximum ;
150 ELECTROSTATICS. [CHAP. VH.
it lies between the greatest and least values of the potential in the
field. If the conductor Al is charged with the unit of electricity,
its potential is pn. At a point infinitely distant the potential is zero.
Hence, we have pu >J912, in general pnn >pmn and pmm >pnm. Further,
the value of pnm lies between pnn and zero, and since pnn is positive,
pnm is also positive. The potentials of the two conductors are equal
only when the charged conductor encloses the uncharged conductor.
If one conductor does not enclose the other, we will always have
Pnn>Pmn an(i Pmm>Pmn-
SECTION LXI1I. MECHANICAL FORCES.
Let us suppose a set of insulated conductors ; their charges will
remain unchanged in quantity when the conductors are displaced.
Their potentials depend on the charges in the manner given in LXII.
The forces acting on the charged surfaces tend to set the conductors
in motion. We assume that all the conductors except Al retain
their relative positions; that Al can move in the direction of the
z-axis ; and we then determine the force which tends to move Al in
this direction. Let the displacement of A^ be 8.1: The energy We
of the system will be diminished in consequence of this displacement
by X8x. At the end of the motion the energy is We + §Wn and
hence we have W.-X. 8x= W. + 8JT., and (a) X= -8J7f/8.v. Now,
from LXII. (g) we have
(b) X = |e128/>11/&c + %e2~8p22/8x + . . . + «1e28p12/Sj: + . . . ,
because the charges do not change during the motion, and are inde-
pendent of the displacement 8x. This method may be always applied
if the motion of the conductor is.
one for which the mechanical work
done by the system may be repre-
sented in the form X8x.
We now determine the force
with which one of the conductors
will move in the direction of the
.r-axis if the potentials remain con-
stant. Let Av A2, A3, ... be the
FIG. 70. given conductors (Fig. 70). They
are supposed to be connected by very thin wires with the very large
conductors Bv B2, B3, ..., whose potentials are ^f1, ^o, ^3, ... respec-
tively, and which are so remote from the system of conductors A
SECT. Lxiii.j MECHANICAL FORCES. 151
that they have no influence upon it by induction. If the conductor
Al is displaced by Sx, the charges ev e2, es, ... increase by 8elt 8e2, 8e3, ...,
and we have from LXII. (h)
The electrical energy of the system thus increases by
SJr= ^tej + ¥2&2 + ¥3Se3 + . . . ,
, + . . . .
But from LXII. (n) the energy is equal to W^, + Sf^ in the new
configuration of the system, where
The work done is JTSa;. The sum of the energy 7F^, originally present
and the energy 8fF supplied is equal to the sum of the energy in
the new position and the work done. We therefore have
tf* + 877=17^, + SIT* + X . 8x, X.8x = 8fT - 8!^.
If we substitute the expressions found for 8JF and 8JF^,, we have
v + ^2^3^23 + «. .
Hence, we obtain (d) X.8x = 8JF^, or X=8fTyjSx, and further
8JF= '2X . 8x.
The electrical energy supplied to the system is therefore twice as great
as the mechanical work done. Now, if during the displacement of
the conductors their potentials do not change, energy must flow
from Blt By B3, etc., to the conductors Av A%, A3, etc. One-half
of the energy SJF supplied is expended in doing the mechanical
work, the other half in increasing the electrical energy.
SECTION LXIY. THE CONDENSER AND ELECTROMETER.
1. Parallel Plates.
If two bodies at different potentials are placed near each other,
a relatively great quantity of electricity can be collected on the
surfaces which face each other. If A and B are two such bodies,
152 ELECTROSTATICS. [CHAP. vn.
whose potentials are ^ and Mfg respectively, and if the opposing
surfaces of the bodies are planes, the electrical force in the inter-
vening space is everywhere constant, except near the edges of the
plane surfaces. If a represents the distance between the planes,
and if ^fl is greater than ^fy so that this force is directed from
A to B, we have from VII. (c), (a) ¥1 = ¥2 + JFa and F=(¥1-¥2)la.
The surface-density on Al is determined from
471-0- = ^ or <r = (*1-'4r2)/4ira.
If S represents the surface of the conductor A which faces B, we
have for the charge el on S, (b) e1 = S<r = (>¥l-V2) . S/faa. The
charge e2 on B is equal to - er The electrical energy W^ of the
system is
or (c) JT^ = I/Sir. (^-^^.S/a, where the energy is expressed
in terms of the potentials. From (b) and (c) it follows that
(d) JFe = 2irael2/S.
If the z-axis is perpendicular to the plane surfaces of the conductors,
and if it is directed from A to B, we have, representing the x-co-
ordinate of the plane face of A by xv and that of the plane face
of B by x2, a = x2-xl, and
W* = 1/87T . ppj - ¥/ . S/(x2 - xj ; W. = 2we^(x2 - xJ/S.
The mechanical force which acts on A is [LXIIL]
Xl = - SWJSa^ = 2^/S ; Xl = 2™^ = $Fev
This corresponds to LIX. (e). We further have [LXIIL (d)]
X1 = Sfr+ISXi = 1/87T . (^ - ¥2)2 . S/(X2 - Xrf- = 1 1 Sir . F* . S.
This agrees with LIX. (f). From the expressions which have been
given for W^,, we also have W^,= l/8-n-. F2 . S(x<, - xj, which agrees
with LXI. (i).
The capacity C is C=e1/(^rl-^r2)-) if ^2 = 0. we have C=S/4ira.
2. Concentric Spherical Surfaces.
If a sphere Av whose radius is R, is enclosed by the concentric
spherical shell, whose internal and external radii are 7?2 and Rz,
and if Al is given the charge ev and A2 the charge e2, the inner
surface of
the charge
SECT. LXIV.] CONDENSER AND ELECTROMETER. 153
The potentials within the sphere Al and within the spherical
shell As are therefore "¥l = eJP^ - el/R2 + (el + e2)/E3 ; ¥2 = (e2 + el)jR3 ;
and hence ex = R^(RZ - RJ . ^ - JR2. Rlf(Ra - RJ . ¥/ and
These equations agree with those given in LXII. (h). The potential
^ in the space between the two spheres is
where /• is the distance of the point considered from the common
centre of the spheres. The potential outside the spherical shell is
^P0 = (e1 + e2)/r. The capacity C of the inner sphere is determined
by el = C^¥v when ¥2 is set equal to zero ; we have therefore C = R^RJat
if we represent by a the distance between the surface of the inner
sphere and the inner surface of the spherical shell.
3. Coaxial Cylinders.
Suppose two coaxial cylindrical surfaces, Al and A^ confronting
each other. Let their potentials be ^ and ^2, and their radii Rl
and R0 respectively. Let a point in the space between Al and A2
be at the distance r from the common axis of the cylinders. The
potential ^f must satisfy the equation V2xF = 0 for this space. Since
the equi-potential surfaces in the space considered are cylindrical
surfaces coaxial with Alt the equation V2^" = 0 may be given the
form [cf. XV.] d*P/dr2 + 2lr.dV/dr = (), and we obtain by integration
* = c log r + cr For r = R^ we have ¥ = ¥lf and for r = R^ ¥ = ¥„,
therefore
*. = *!• (log R2 - log r)/(log E2 - log RJ
+ V, . (log r - log ^)/(log R, - log Rj).
For a point outside the outer cylinder, the potential is
where r is the distance of the point considered from the axis of
the cylinder. The constant c cannot be determined from the
potentials alone. The electrical force F in the intervening space is
F = - <t9tjdr = - OP2 - "¥"!)/ (log R2 - log RJ . 1/r. The surface-densities
o-j and o-2 on Al and A2 are [LV. (e)]
2 - log RJ .
2 - log .Bj) .
The charge on a portion of the cylinder Av whose length is /, is
or e1 = //2.(Y1-'»P2)/(log^2-log^1). The charge on
154
ELECTEOSTATICS.
[CHAP. vn.
the inner surface of A.2 is equal and opposite to this. The capacity
C of a portion of the inner cylinder, whose length is /, is
4. Tlie Quadrant Electrometer.
Suppose AlAl to be two metal plates whose potential is 1Ir1, and
A2A2 to be two similar plates whose potential is ^2 (Fig. 71).
In the middle between the two pairs of plates there is placed a
plate A3 whose potential is ¥3. We assume that ^1<->P2<'»Ir3. If
A3 is displaced by the distance 8x in the direction from Az to Av
and therefore in the direction of its length, the area b8x is
displaced from right to left, if b represents the width of the
I
1
Jt
Az
\
plate Ay If the distance of As from Al and A2 is a, the force
acting between Al and A3 is ^ = ("*F3 — "ty^/a, and that acting between
A.2 and ^43 is ^ = (¥3 - ^2)/a, on the assumption that the points
considered do not lie near the edges of As, or in the space between
Al and Ay From LXI. (i) the electrical energy is 7F=l/87r. \F-d-r.
During the motion considered, the energy on the left side of the pair
of plates is increased by
that on the right side is diminished by l/8ir. (M/3-'*J/2)2/a2. 2ab8x.
The gain of energy is therefore
This expression does not fully represent the gain of energy, since
no account is taken of the relations at the edges. We therefore set
SECT, LX.IV.] CONDENSES AND ELECTEOMETEE.
155
The force X which tends to move A3 from right to left is then
[LXIIL (d)] X=k(Vt-Vl).(Vi-Mp1 + VJ). We may apply this
result to the quadrant electrometer. If, in this instrument, the
movable aluminium plate turns through the angle 6, we may set
approximately
6 = aC*, - Yj) . (^3 - £C*-j + *2))»
where ^ and ^ are the potentials of the quadrants, ¥3 is the
potential of the aluminium plate, and a is a constant whose value
depends on the form and dimensions of the apparatus.
SECTION LXV. THE DIELECTRIC.
We have assumed until now that the bodies considered were
either good conductors or perfect insulators, on which the charge
was immovable. Experiment shows, however, that there are no
perfect insulators.
Electricity on insulators is often lost by conduction, which for
the most part is due to the film of fluid deposited on them by
the air. But even if this film is removed by careful drying, con-
duction still persists. If a charge of electricity is communicated
to one part of an insulator, it is distributed after a considerable
time in the insulator in the same way as it would be in a good
conductor. Besides this, another action also exists which is instan-
taneous. When a movable insulator is brought into the neighbour-
hood of a charged conductor, the insulator sets itself
in the same way as a good conductor, from which it
follows that an instantaneous distribution of electricity
takes place in it. According to Faraday, insulators con-
sist of very small conductors which are separated by
an insulating medium. The capacity of a condenser
is increased by replacing the air which serves as the
insulator between its surfaces by other insulators, such
as glass, shellac, calc spar, etc.
Let A and B (Fig. 72) be two conducting plates which (^ j
are separated by the insulator CD. Let A be brought
to the potential ^fv and B to the potential ^2; let the
surface-density on A be a-, that on B is then — <r.
In order to explain this, Faraday assumed that a peculiar electri-
fication exists in the insulator CD, by which each of the conducting
particles contained in it acquires negative electricity on its right
C
D
FIG. 72.
156 ELECTROSTATICS. [CHAP. vn.
and positive electricity on its left. Just as a mechanical force can
give rise to an elastic displacement, the forces proceeding from
the plates of the condenser produce an electromotive action by setting
up a current of electricity in the particles of the dielectric. The
positive electricity flows in the particles toward the left, the negative
toward the right.
By this process, which we may call dielectric displacement, there
arises a polarization of all the particles.
The condition in the dielectric can be compared with the polarity
of the particles of a permanent magnet.
The quantity $) which flows through a unit of area parallel to
A and B must be equal to a-. A unit of area of the surface of the
insulator at A receives the charge -a-, and that at B the charge + 0-.
Let ft be the distance between A and B, whose difference of potential
is ¥x - ¥2. The force in the intervening space is (a) F= (M^ - ^.2)/a.
If the quantity of electricity £) which flows through the unit of area
is proportional to the force acting in the insulator, we can set
(b) £) = #/47r..F,
where K is a constant. Hence, the surface of A, whose area is S,
will have the charge (c) S3) = K/±ir . (¥\ - ¥2)/a . S. A comparison
of this equation with LXIV. (b) shows how many times greater the
capacity of the condenser becomes if another insulator is used in
place of air. We call K the dielectric constant ; for air, which is chosen
as the standard medium, we set K= 1.
The dielectric constant of an insulating medium is the ratio of the
capacity of a condenser having that medium as an insulator to the capacity
of the same condenser when air is the insulator. It is, however, more
correct to set K= 1 for a vacuum ; it has been shown that K for
gases is a little greater than 1. As examples of the values of the
dielectric constant, we have for
Glass, - #=5,83—6,34,
Paraffin, - - - #=2—2,32,
Sulphur, - - #=3,84,
Shellac, - - - #=3—3,7,
Bi-sulphide of carbon, #=2,6,
Oil of turpentine, - #=2,2.
On the whole, the results obtained by different observers are not
very consistent.
Let us, as in LVIIL, suppose that the body A is brought
into the space enclosed by the metallic shell BC. Suppose that
SECT. LXV.] THE DIELECTEIC. 157
B receives the charge - Q, and C the charge Q. The quantity
Q of positive electricity flows outward through the closed surface 7>
taken within BC. Hence, when the quantity Q is introduced through
the closed surface D, the same quantity flows out through the same
surface. We are therefore justified in assuming that the quantity
enclosed by the surface D is always zero. This holds for the closed
surface E, which may be drawn in the insulator surrounding BC,
and thus we obtain the general theorem that the total quantity of
electricity contained within a closed surface is equal to zero.
If the quantity of electricity £>, which flows through a unit of area
perpendicular to the direction of electrical force, is proportional to
that force at every point, we will have (d) £) = Kjkir . F. If /, g,
and h are the quantities which pass through three units of area in
an isotropic body taken perpendicular to the three coordinate axes,
that is, if they are the rectangular components of the displacement £), and
if X, Y, Z are the components of the electromotive force F, we have
(e) f=K/4:Tr. X, g=K/4ir.Y, h = K/4:Tr.Z. These expressions are
consistent with the relations
(f) Z=-3*/3ar, Y= -3*"/3y, Z= -
SECTION LXVL CONDITIONS OF EQUILIBRIUM.
Suppose the closed surface AS' to enclose a portion of the electrical
system, passing partly through the dielectric, partly through the
conductors. We will make use of our previous conclusion, that the
total quantity of electricity within S is zero. If we represent by
e the total quantity enclosed by S, and by £)' the quantity which
flows out through a unit of area in consequence of the displacement
in the dielectric, we have e—fo'.dS, The normal to the surface
S directed outward makes angles with the axes whose cosines are
I, TO, 7i. We have ^)' = K/^ir. Fcose, if e is the angle between the
normal to dS and the direction of the electromotive force F. Since
F cos e = XI + Ym + Zn, we obtain
(b) (c) e = 1/47T . \K(Xl + Ym + Zn)dS = \(fl + gm + hn}dS.
If we apply equation (c) to an infinitely small parallelepiped, whose
edges are dx, dy, and dz, we find that the quantity which enters,
it is fdydz + gdxdz + hdxdy, and that which leaves it
'/+ dydz + g + dydxdz + h + dz
158 ELECTROSTATICS. [CHAP. vn.
if p is the volume-density, the charge contained in the parallelepiped
is pdxdydz; now the total quantity of electricity within it equals
zero, and therefore
fdydz + gdxdz + hdxdy + pdxdydz
= (f + ?fdx}dydz + (g + ^dy}dxdz + (h + fdz}dxdy.
\ ox J \ oy J \ oz J
From this it follows that the volume-density p of the free electricity
within the body [cf. LXVIIL] is given by (d) p = 'd/J'd
-and from LXV. (e) we obtain
In terms of the potential, this becomes
<f ) -o(K . 3¥/az)/ae + -d(K. 3¥,%)/3y + 3(K. 3¥/3*)/3,? + 4rp = 0.
If we consider a surface on which the surface-density is o-, we
obtain by the same method as that used in LIV. (g) a- = 2)1 + $)2, if
<£)l and £)2 are the polarizations in the directions of the normals
to the surface drawn outward. If the forces along the same normals
are NI and JV2, we have (h) a- = KJiir . ^ + K^v . N.2, where K^
And K.2 are the dielectric constants on the opposite sides of the
surface.
By means of this equation questions on electrical distribution and
•on the relations between density and potential may be solved. If
K is constant, it follows from (f ) that K V2^ + kirp = 0. In a region
where the dielectric constant is K, the potential which arises from
.a given charge p is equal to only the Ktb part of that which arises
from the same charge in a region where K= 1. In the latter case
ithe potential Hf' is determined by V2¥ ' + 4ny) = 0, so that ¥ = V/K.
The electrical force is also diminished in the same ratio. If the
•charges el and e.2 are placed at the points A and B respectively, and
if the distance AB = r, the charges repel each other with the force E,
B-l/K.qiJA
SECTION LXVII. MECHANICAL FORCE AND ELECTRICAL ENERGY IN
THE DIELECTRIC.
Suppose that the dielectric constant is constant in the region con-
sidered. The forces acting in the directions of the axes on the
parallelepiped dxdydz = dr, in which the density is p, are
SECT. LXVII.] MECHANICAL FORCE IN DIELECTRIC. 159
Using LXVI. (e), we have
(X) = l/4ir(X . -d(KX)l^x + X . V(KY)rdy + X . d(KZ)pz).
Since [LXV. (f)] the forces have a potential, we obtain from
XXVII. (b)
(X) = K/8ir . (d(X* -Y2- Z2)/cx + 2^(XY)py + 23(A^)/3.?).
The force which acts on the volume-element dr may be considered
as due to the stresses Xx, Yy, etc. [cf. XXVII. (c)], where
f Xx = K/STT . (X2 - Y 2 - Z2), YZ=ZU= A747T . YZ,
[ Zt = K/8ir . (Z2 - A'2 - 72), Xy = Yx = A747T . YX.
If the direction of the a;-axis is that of the electrical force, we have
and the tangential components vanish. In the direction of the electro-
motive force F, there is a tension S, and in all directions perpen-
dicular to the force F, a pressure T, such that (d) S=T=KF'2/8Tr.
If A and B (Fig. 73) are two conducting surfaces which are
separated by an insulator whose dielectric constant is K, and if AB
is a line of force, a tension S acts along that line,
and a pressure T = S acts perpendicularly to it.
A surface-element at A is under a tension
1/Sir.KF2,
acting in the direction of the normal drawn
outward. When K= I this reduces to the result
reached in LIX.
For example, let A B be a hollow sphere of glass whose inner and
outer radii are rl and r2 respectively. Let the potential of the surface
A be ^v and that of B be 0. The potential in the interior of the
spherical shell is determined from LXVI. (f ) ; when p = 0, we have
V2*P" = 0. Since the potential depends only on the distance r from
the centre, we have from XV. d*¥/dr* + 2/r .d^/dr=0, and hence
M*1 = A + Bfr. Having regard to the boundary conditions, we obtain
W = rl(r2-r)/(r2-rl).yi/r. The forces F1 and F2 which act at the
inner and outer surfaces are
Representing the stresses on these surfaces by pl and pv we obtain
These stresses may be regarded as pressures which act on the surfaces.
160 ELECTROSTATICS. [CHAP, vn
If d<f>/dr represents the increment of r which results from the pressure
on the surface, we have from XXXI. (h) d<f>/dr = ^ar+b/r2, where
a = 3/(3 A + 2/0 . (prf -pff)/(r* - r*) ; b = 1 /4/» . (Pl -pjr* . r2*/(rs» - rf).
From this it follows that
If we set K^^/Sir . fjf j/[(*j — **i)(^23 ~ ri3)] = -^r» we w^ have
The volume contained by the hollow sphere will be increased by
the action of the electrical force.
If we represent by 60 the increment of the unit of volume, we have
47T/3.(r + d<£/e?r)3 = 47r/3.r3(l+60), so that 00 = 3d(^/rdr. For the
volume of the sphere, for which r = rv we obtain
60 = 3(l/(3A + 2/0 + (r2* - r-*)i(r, - rj . l/iprflN.
If we set r2- ^==8, and if 5 is very small in comparison with rv
we have 00 = 9JV. (A + /*)//*(3A + 2^), where H=K'¥l*/8ir. 1/332. It
thus follows, using XXIX. (d), that (e) 60 = 3/E8* . K^/Sv. The
increase of volume here considered has been observed for various
condensers.
If a region, in which the dielectric constant K is a function of
the coordinates, contains electrical charges, whose density is p, and
which give rise to the potential ¥, the energy W is determined
as in LXI. by (f) W=\\f&dr.
If p is expressed in terms of the potential by
we have for the energy W>
JF=-1/87T.
By integration by parts, we obtain
W= 1 1 Sir . J J f tf((3¥/aB)2 + Cd¥ pyy- +
where the integration is extended over the entire region, and it is
assumed that the force and the potential vanish at infinity. If F
represents electrical force, we have (g) JF=lj8ir.
Electrical Double Sheets. We have seen in LVIII. that two conductors
which are very near each other, and are kept at the two different
potentials ^ifl and W^ are oppositely charged. The surface-densities
<r and -a- of their charges are given by o- = (^r1 — ¥2)/47r«, where a
SECT. LXVII.] MECHANICAL FORCE IN DIELECTRIC. 161
is the distance between the surfaces. If a is taken infinitely small,
there is still a finite difference of potential between the surfaces ;
calling this difference V, we have (a) a- = F/4?ra.
Now, there are several ways by which such finite potential differ-
ences may be established across a surface ; for example, it may be
done by friction or, what amounts to the same thing, by contact.
This being so, we must necessarily assume, as was first remarked by
v. Helmholtz, that a double sheet of electricity is formed on the two
surfaces which are near each other, in which, if the distance a is
extremely small, the density o- must be extremely great if the potential
difference V is to be finite. When two such bodies are separated
from each other, provided they still retain the electricity thus disposed
on them, they will both be very strongly charged. It is usually not
possible to separate them without discharging them, but if one or
both of them are insulators a very considerable charge remains.
V. Helmholtz thus explains the action of the rubber of the frictional
electrical machine.
In the same way v. Helmholtz explained several remarkable pheno-
mena, for example the phenomena of electrical convection, studied
by Quincke and Wiedemann. Let us consider a capillary tube,
of circular cross-section, whose inner radius is R and whose
length is /. A liquid is supposed to be flowing through this tube.
If there is a difference of potential V between the liquid and the
wall of the tube, a layer of electricity forms around the liquid, whose
density a- is determined by equation (a). Setting the radius of this
cylindrical electrical layer equal to r, so that a = E-r, we have (b)
o- = VI±ir(R - r). Let the velocity of the flow at the place at which
this layer is present be w. The quantity of electricity which is carried
on by the current in unit time through any cross-section of the tube is
Now, we have found [XLIX.] that
')t and hence
We may in this equation set E = r, and obtain
(0
where S is the area of the cross-section of the tube.
If the pressure at the two ends of the tube is the same, while a
difference of potential ^ - M*^ exists between them, an electrical
162 ELECTROSTATICS. [CHAP. vn.
_
force F= — ~ — 2 will act within the tube, and therefore a force Fa-
will act on every unit of area of the electrical layer. The liquid will
thus be set in motion. The velocity increases from the wall of the
tube, where it is 0, to the layer a-, where it may be called u. By
the definition of internal friction [XLVIL], we then have F<r = p.~.
If or is expressed in terms of the potential difference, we have
VF
(d) u = . —
CHAPTER VIII.
MAGNETISM.
SECTION LXVIII. GENERAL PROPERTIES OF MAGNETS.
IT was very early known to the Greeks that in the city of Magnesia,
in Asia Minor, stones were to be found which had the power of attract-
ing iron. If such a stone, which consisted mostly of magnetic oxide
of iron, were thrown into iron filings, they would adhere with special
strength to it at certain points. A long magnet is especially active
in the vicinity of its ends, which are called poles. If a bar magnet is
suspended horizontally, so that it can turn about a vertical axis passing
through its centre, it assumes of itself a definite direction which
approximately coincides with the meridian of the place. The end
of the bar which points toward the north is called the north pole, the
other end the south pole. Following the indication of this experiment,
we assume the existence of two magnetic fluids, which are separated
in each particle of iron by the influence of a magnetic force. If a
bar magnet is floated on a fluid at rest, it assumes a definite direction
when exposed to the influence of a magnetic force, but the force
does not set the floating magnet in motion if the dimensions of the
magnet are very small in comparison with its distance from the seat
of the magnetic force. We conclude from this that the north and
south magnetic fluids are present in a magnet in equal quantities.
Let us represent the quantity of one fluid by + m, that of the other
by — m, the north magnetism being conventionally taken as positive.
Coulomb proved that the poles of two magnets repel each other with
a force F, which is given by (a) F^m^^r^ where m1 and m2 are
the quantities of magnetism at the poles and r is the distance
between the poles.
If a magnet is broken into many small pieces, each part is still a
163
164
MAGNETISM.
[CHAP. vni.
magnet. Because of this fact, we conclude that every magnet is
made up of a very great number of very small magnets.
If a magnet is broken, positive magnetism appears on one, and
negative magnetism on the other, of the surfaces formed by the
fracture, and their quantities are equal unless the magnetization is
changed by the jar given to the magnet when it is broken. We will
represent by + 0- the quantity of magnetism that is present on a unit
of area of one of the newly formed faces. For each point on this
face a- has a definite value, dependent on the position of the point
on the face and of the face in the original magnet. Construct a normal
to this face drawn outward ; <r then depends on the coordinates x, y, z
of the point and on the direction of the normal, which makes angles
with the axes whose cosines are /, m, n. Let ABC = dS (Fig. 74)
be an element of the positive face
and 0 a point in the magnet infinitely
near it. A, B, C are points in which
the surface dS is met by the lines
Ox, Oy, Oz parallel to the axes drawn
through 0. The surface OBC, one of
the surfaces of the tetrahedron OABC,
may be considered as a negative face.
Suppose that the magnet is mag-
netized with the components A, B,
C in the directions OA, OB, and
OC, the surface ABC of the tetrahedron exhibits positive magnetism,
the surfaces OBC, etc., negative magnetism. Represent the surface-
density perpendicular to the z-axis by A ; each unit of surface of
OBC then exhibits the quantity of magnetism - A. The unit of
surface of OA C and OB A will, in a similar notation, exhibit the
quantities of magnetism - B and - C respectively. The position of
the surface ABC is determined by the cosines /, m, n of the angles
which the normal to the surface makes with the axes ; we have
OBC = l.dS, OAC=m. dS, OB A =n.dS. If h represents the perpen-
dicular let fall from 0 to the surface ABC, and if the quantity of
magnetism in the unit of volume is represented by p, the total quantity
of magnetism contained in the tetrahedron is
(<r-lA-mB- nC)dS+ ^hpdS.
Now, since the total quantity of magnetism in a magnet is zero, and
the altitude h of the tetrahedron is assumed to be infinitely small, we
have (b) o- = Al + Bm + Cn. Hence, if the surface-density on three
perpendicular surface-elements passed through a point is known, the
FIG. 74.
SECT. LXVIII.]
PROPERTIES OF MAGNETS.
165
density on any other surface passed through the same point is deter-
mined from (b). If we set
(c) J* = A* + B* + C* and A=JX, B = Jp., C = Jv,
where A2 + /*2 + v-= 1, it follows from (b) that a- = J(l\ + mp, + nv). J
is the intensity or strength of magnetization. The direction of the inten-
sity makes angles with the coordinate axes, whose cosines are A., /*, v.
If we set l\ + mp. + nv = cos e, where e is the angle between the
intensity of magnetization and the normal to the surface-element,
we have 0- = J cose, that is, a- is the component of the intensity of
magnetization along the normal to the surface-element. The greatest
value of o- is reached when « = 0, that is, when the direction of the
intensity of magnetization coincides with the normal to the surface-
element on which the density is o-. A surface may be passed through
any point in a magnet for which the surface-density is a maximum.
The direction of the intensity of magnetization / lies in the normal
to this surface. For such a surface, whose normal coincides with the
direction of magnetization, cr = t/; that is, the quantity of magnetism
on the unit of area of this surface is /. We construct a parallelepiped,
one of whose ends, dS, lies in the surface considered, and whose edges
perpendicular to the surface are ds ; J.dS.ds is called the magnetic
moment of this parallelepiped. The intensity of magnetization J is
equal to the ratio of the magnetic moment of the magnet to its volume.
The magnetic condition of a magnet is defined by the components
of magnetization A, B, C. The density o- on the surface of the magnet
is determined by (b) from these
components. Free magnetism
may be present in the interior
of the magnet also. If 00' (Fig.
75) is a rectangular parallel-
epiped, whose edges are dx, dy,
dz, taken within the magnet, and
if A, B, C are the components
of the intensity of magnetization
at the point 0, OA' will con-
tain the quantity of magnetism
-Adydz, and O'A the quantity
(A + 'dA/'dx . dx)dydz. Analogous expressions hold for the other
surfaces. Representing by p the quantity of magnetism contained
in the unit of volume, we have
('dA/'dx + "dBfdy + 'dC/'dz + p)dxdydz = 0,
FIG. 75.
166 MAGNETISM. [CHAP. vin.
since the total magnetism in the parallelepiped is zero. Hence (e)
P= -(dA/'dx + 'dBfdy + 'dCfdz). While we may consider A, B, C as
the natural and direct expressions for the condition of a magnet, we
determine its free magnetism from the derived magnitudes p and or.
SECTION LXIX. THE MAGNETIC POTENTIAL.
The force with which a magnet acts on a pole at which unit
quantity of magnetism is concentrated is called the magnetic force.
Its components are represented by a, ft, y. These are determined
from the potential in the same way as the components of the electrical
force. Let the quantities of magnetism m, m', m", ... be present at
given points; let the pole P, for which the potential is to be deter-
mined, be distant r, r', r", ... respectively from those points. The
potential is then given by
F=m/r + m'/r'+m"/r"+..., and m + m' + m"+ ... = 0.
If N is a north pole with magnetism +m (Fig. 76) and S a south
pole with magnetism - m, and if P is the point for which the potential
^p is to be determined, and whose distances
from the north and south poles are r and
fS/ r respectively, we have
V= m/r - m/r' = m(r - r)/rr'.
If the length / of the magnet is very small
in comparison with r and /, and if we set
2, we have (a) r=
90} = lm is called the moment of the magnet ;
SNM is called the magnetic axis. Its positive direction is that from
the south to the north pole.
We now determine the potential of a magnet whose components of
magnetization A, B, and C are given. Let the coordinates with respect
to any point taken as origin be £, ?/, £; A, B, C are then functions
of these three coordinates. A parallelepiped whose edges are d£,
drj, d£ will have the magnetic moment Adijd{.d£, if, for the present,
we consider only the magnetization determined by A. If the co-
ordinates of the point P, for which the potential is to be determined,
are x, y, z, and if its distance from the point £, 77, £ is r, we have
r2 = (x - £)2 + (y - rj)2 + (z- C)2, cos 9 = (x - £)/>-.
The potential due to the element d% . d^ . d£ arising from the com-
ponent of magnetization A is, from (a), Ad£dr]d£/r2 . (x - £)>. B and C
SECT. LXIX.] THE MAGNETIC POTENTIAL. 167
give rise to the potentials Bdgdrjdflr2 . (y - rf)jr, Cdgdrjdflr2 . (z - £)/r.
The sum of these three potentials, integrated over the whole magnet,
gives the total potential V,
(b) V= \\\\A(x - £) + B(y - r,) + G(z - fl^W-
Since r'dr/'dx = x-g, rdrj'dy = y—% rdrfdz = z-£, we have
(c) V= - \\\(A .3(l/r)/3a; + £.3(l
If we set
^ = Jff^/r.d£eMt *,= \\lBfr.
we have (d) F"= -(3</'1/3a; + 3i/'2/3y + 3^3/32), since the components of
magnetization ^4, -6, (7 are independent of the coordinates x, y, z of
the point P.
Another more general transformation may be made by help of the
equations, r . 3r/3£ = - (x - £), r . 3r/3?/ = - (y - 17), r . 3r/3£= - (« - 0,
by the use of which we obtain from (b),
(e) V= \\\[A . 3(l/r)/3£ + B . 3(l/r)/3, + (7. 3(l/r)/3f]^^df.
Let the normal to the surface-element make angles with the axes
whose cosines are /, m, n. By integration by parts, it follows that
(f) V= \\(Al + Bm + Cn)/r . dS - J J J(3^/3^ + 3£/3ry + 3(7/30 . dgdr)d{/r,
and using LXVIII. (b) and (e), (f) becomes
(g) r=JforfS/r+Jffpd^<*0r.
The correctness of the last equation is immediately evident from
the meaning of <r and p.
The components a, /3, y of the magnetic force are expressed, as
in the theory of electricity, by
(h) a=-3F/3z, /3=-3F/3z/, y=-3F"/3z.
The force N, which acts in the direction of the element of length dv,
is JV= -3F/3^. If the potential is F] inside the magnet and Va
outside the magnet, we have at the surface Vi=Va. If vt is the
normal to a surface-element on which the density is cr drawn into
the magnet, and va the normal to the same element drawn outward,
we have from the general laws of the potential [cf. XIV. (1)], which
are applicable here, (i) 3Fi/3vi + 3F0/3v(l+ 47r<r = 0. For every point
within the magnet, we have (k) V2F|- + top = 0, or introducing the value
for P given in LXVIII. (e), (1) V2Fi = ^(dA/'dx + dB/'dy+aC/^z), if
p and A, J5, C are functions of x, y, and z. Outside the magnet we
have, on the other hand, (m) V2F"ra = 0. If S is a closed surface, v
its normal drawn outward, and M the total magnetism enclosed by
168 MAGNETISM. [CHAP. vm.
the surface, we have from XIV. (c) 4irM = - JJdF/3v . dS, or designating
by ^)n, the magnetic force in the direction of the normal to the surface,
(n) 4irM=l\$ndS.
The equations (i) and (k) are special cases of this equation.
SECTION LXX. THE POTENTIAL OF A MAGNETIZED SPHERE.
If the components of magnetization are given functions of the
coordinates, and £, 77, £ are the coordinates of a point within the
magnet, we obtain the potential most easily by using the formula
LXIX. (d). If A, B, C are constant, the problem is to determine the
potential of a body of constant volume-density. Hence we set
(a) t
and obtain (b) V= - (A . -d^fdx + B . 3</y cty + C . tyfiz). The potential
of a sphere whose components of magnetization are A, B, C is to be
determined by means of this equation. Take the origin of the system
of coordinates at the centre of the sphere. The potential $ has
different values, according as the point for which the potential is to
be determined lies inside or outside the sphere. In the usual notation
we have, from XIII. (c) and (d), ^ = 47r^/3r; i}>t = 2ir(It2-r2/3),
where R is the radius of the sphere.
If we represent the magnetic potential for points outside the sphere
by ^a> an(l for points inside it by Viy we have
(c), (d) Va = 47T/3 . ^/r3 . (Ax + By + Cz) • Vi = 47r/3 . (Ax + By + Cz).
Let / be the intensity of magnetization, and let its direction make
angles with the axes whose cosines are A, /*, v. Let 6 be the angle
between the direction of J and the line r. We then have
Xx/r + fjiy/r + vz/'r = cos 0
and (e) Va = 47r/3 . 1PJ cos 0/r2 ; F; = 47r/3./rcose.
Hence, the potential outside the magnet is the same as that which
is set up by an infinitely small magnet, whose magnetic moment is
3K = 4ir/3.W [LXIX (a)].
If the a-axis lies in the direction of magnetization, the potential in
the interior of the magnet is Vi = 4?r/3 . Jx. The magnetic force ^)
inside the sphere is therefore constant, and is expressed by
(f) $=- -Air/3. J.
Outside the sphere we divide the force into two components, one of
SECT. LXX.] POTENTIAL OF A MAGNETIZED SPHERE. 169
which, P, acts in the direction of the line r, the other, Q, perpendicularly
to that line. We then have P= -Vrjdr, Q= -l/r .'dTJ'dQ, or
P = 87T/3 . Wcos 0/7-3 = 2%R/r3 . cos 0 ;
Q = 47T/3 . E3Jsin 0/r3 = Wfr3 . sin 0.
From LXVIII. the surface-density is determined by cr = /cos9. The
resultant forced is F^^ft/r3.^/! + 3cos26. We have further tgQ = 2Q/P.
If </> is the angle between the direction of the force F and the direc-
tion of r, we have tg<f) = Q/P = \tgQ.
SECTION LXXI. THE FORCES WHICH ACT ON A MAGNET.
Let us suppose that the magnetic forces of a magnet, whose com-
ponents we may represent by a, /?, y, are functions of the coordinates.
Let us suppose also another magnet,
whose components of magnetization are
A, B, C. its action on the first magnet
is to be determined. Consider the in-
finitely small parallelepiped 00' within
the second magnet (Fig. 77) ; on the
face OA' there is a quantity of magnet-
ism present equal to - Adydz, which is
acted on by the force - Adydza in the ' Flo
direction of the positive z-axis. The face
AO', on which is the quantity of magnetism Adydz, is acted on by
the force (a + 'da/'dx.dx)Adydz in the same direction. The resultant
of these two forces is A . cta/dx . dxdydz. On the surface-element
OB' there is the quantity of magnetism - Bdxdz, and on BO' the
quantity Bdxdz. The former is acted on by the force —Bdxdz.a in
the direction of the positive .r-axis, and the latter by the force
+ B(a + 'da/'dy . dy)dxdz in the same direction. The resultant of these
two forces is B . 'da/'dy . dxdydz. For the surface-elements 00' and O'C,
we obtain the resultant C. 'daj'dz. dxdydz. We form the sum of these
three resultants, integrate over the whole volume occupied by the
magnet, and obtain for the force X, which tends to move the magnet
in the direction of the z-axis,
(a) X= \H(A . -dafdx + B . 3a/3y + C . Va/c)z)dxdydz.
Analogous expressions hold for the forces Y and Z.
If the magnetic force whose components are a, /?, y is due to
a system of magnets which give rise to the potential Fat the point
170 MAGNETISM. [CHAP. vm.
z, y, z, we have a = -3F/3x, /3= -3F/3y, 7= -^Vfdz. We then
have also 9a/3y = 'dfi/'dx, "dafdz = 7tyfdx, and hence
(b) X= \\\(A . -dafdx + B . Vpfix + C . Vy/-dx)dxd>/dz.
We now determine the moment of the forces which tend to turn
the magnet about one of the coordinate axis, say the z-axis, on the
assumption that the magnetic forces are constant. Let the coordinates
of the point 0 (Fig. 77) be x, y, z. The force which acts on the surface
BO' in the direction of the z-axis has a moment with respect to the
x-axis equal to Bdxdz . y. (y + dy). The force acting on OB' has
a moment with respect to the same axis equal to - Bdxdz .y.y.
Neglecting small terms of higher order, the resultant moment is
Bydxdz , dy. The forces acting on the surfaces O'C and OC' give
rise to the moment - Cftdxdy . dz. The moment L, which tends to
turn the magnet about the z-axis, is therefore
(c) L = \\\(By-C($)dxdydz.
The moments of rotation M and N, with respect to the two other
axes, are determined from analogous expressions.
If the magnet is subjected to the action of the earth's magnetism
only, the magnetic force may be considered as constant both in
magnitude and direction. The components a, (3, y are then inde-
pendent of x, y, z, and therefore X=F=Z=Q. The centre of gravity
of a magnet does not move under the action of the earth's magnetism.
The magnet is, however, acted on by a moment of rotation, which
may be determined in the following way :
Let the magnetic moment of the magnet be 9)?, and suppose its
direction with respect to the coordinate axes to be determined by
the angles whose cosines are I, m, n. We then have
Wl = \\\A.dT, mm = \\\B.dr, Wn = \\lC.dr,
and hence L = Wl(ym - /3n) ; M=$tt(an - yl) ; N= Wl(/3l - am). From
these equations, it follows that La + Mtf + Ny = 0 and LI + Mm + Nn = 0,
that is, the resultant moment is perpendicular to the magnetic force and
also to the magnetic axis of the magnet. If the direction of the force
is parallel to the or-axis, and if the magnetic axis lies in the xy- plane
and forms with the ,T-axis the angle 6, we will have
(d) Z = 0, M=Q, N= -2tta.sine.
If a magnet can turn about a vertical axis, the moment which tends
to increase the angle 0 between the magnetic axis and the magnetic
meridian is -~9RlT.dll 6, where H denotes the horizontal component
SECT. LXXI.]
FORCES ON A MAGNET.
171
of the earth's magnetism. If w is the angular velocity of the magnet
and / its moment of inertia, we have, from XXII. (c),
d(Ja>) = - WH .s'mQ.dt, or since w = dQ/dt = 0,
(e) je= -WlH.sinQ.
If the angle 6 is very small, the period of oscillation of the magnet
is given by XXII. (e), (f) T
SECTION LXXII. POTENTIAL ENERGY OF A MAGNET.
By the potential energy of a magnet is meant the work which is
needed to transfer the magnet from a position in which no magnetic
forces act on it to the position
in which the magnetic potential
is V. We will first consider an
infinitely small parallelepiped
(Fig. 78), whose components of
magnetization are A, B, C. In
order to bring the magnetic
surface OA' to the position in
which the potential is V, the
work - A . dydz . V must be
done. The opposite surface O'A
is brought to the position in
which the potential is F'+'dV/'dx.dx, and the work done on it is
A . dydz. (P'+'dT/'dx.dx). The work done on these two surfaces
therefore amounts to A . 'dVj'dx.. dxdydz. If we obtain in like
manner the work done on the two other pairs of surfaces, we
find that the whole work W done in transporting the magnet is
(a) W= \\\(A.^ Vfdx +B.3 Vj'dy + C . 9 Vfdz)dxdydz,
or since a, /?, y, the components of the magnetic force, are
FlG 78
we have (b) JF= -\\\(Aa + B(3 + Cy)dxdydz. We will apply this
equation to the case of a magnet subjected to the action of the earth's
magnetism only. Let its magnetic moment be 3ft, and let the
direction of its magnetic axis make angles with the coordinate axes
whose cosines are /, m, n. We then have
. dxdydz = m, \\\B .dxdydz = nm, \\\C .dxdydz = nW,
172 MAGNETISM. [CHAP. vui.
Representing the magnetic force by §, and supposing its direction
to make angles with the coordinate axes whose cosines are A, p, i>,
we obtain W= - Wl^lX + mp + nv). Letting 6 represent the angle
between the magnetic axis of the magnet and the magnetic force,
we have (c) W= - 3J?^) . cos0. If the direction of the force is
parallel to the z-axis, as in LXXf., and if the magnetic axis lies
in the ay-plane, the work done in turning the magnet through the
angle dQ is dW= + 9tf£ . sin 6 . dQ. This agrees with LXXI. (d).
We will now consider a very small magnet situated near a very
strong magnet. If the small magnet has sufficient freedom of motion,
it will turn so that its magnetic axis is parallel to the direction of
the magnetic force. In this case we have 6 = 0 and its potential
energy W is W= - 3ft$.
Since the motion of the small magnet involves the loss of potential
energy, it moves in such a way that W diminishes. This occurs by
the last equation, when ^) increases ; the magnet therefore moves in
the direction in which the magnetic force increases. A particle of
a paramagnetic substance therefore tends to move towards the place
where the magnetic force is greatest. On the other hand, diamagnetic
bodies move toward the place where the magnetic force is a
minimum.
In order to find the magnetic energy residing in a system of magnets,
we proceed in the following way : The potential at every point within
the system varies proportionally with the values of the components
of magnetization. We assume that the components of magnetization
change only in such a way that, in successive instants, they always
increase by the same fraction of their final values. On these con-
ditions the potential increases in the same proportion. If the com-
ponents of magnetization arc originally zero, the potential is also
originally equal to zero. Let the final values of the components of
magnetization be A, B, C. At a particular instant during the increase
of the components of magnetization, let these be represented by nA,
nB, nC, where n is a proper fraction. At the same time the potential
at any point is equal to nV. If the components of magnetization
increase by A.dn, B.dn, G.dn respectively, the potential at the
point considered increases by Fdn. If A, B, C increase by A . dn,
B.dn, G.dn respectively, the work needed to accomplish this is,
\>y (a),
\\\(A . dn . ridF/dx + B.dn. ndVj'dy +C.dn. n
= n.dn\\\(A .
SECT. LXXII.] POTENTIAL ENERGY OF A MAGNET. 173
Now, if n increases from 0 to 1, we have / ndn = ^, and the work
done is
(d) W= \\\\(A . -dVftx + B . -dFj-dy + C . VFj?)z)dxdydz,
or, by introducing the components of the magnetic force,
(e) W= - \\\\(A* + Bfl + Cy)dxdydz.
The energy of a magnetic system may be expressed in another
way. The same method by which we before obtained an expression
for the energy, shows that (f) W=*\\<r. V. dS+\\\\p. F.dxdydz,
where cr and p represent the surface and volume-densities respectively.
If V and its first differential coefficient vary continuously, we have
o- = 0 and W=\\\\p.V.dxdydz. Now, V2F+4«/> = 0, and hence
W = - 1 /STT . J f J V( 32 Vfdy? + 92 Vfif + 32 Vfiz^dxdydz.
By integration by parts over the whole infinite region, we obtain
or (g) W=-\l&ir. ^\(az + /3'2 + y2)dxdydz. Similar expressions hold for
dielectric polarization [cf. LXI.].
SECTION LXXIII. MAGNETIC DISTRIBUTION.
A piece of soft iron brought into a magnetic field becomes magnetized
by induction. "We assume that the intensity of magnetization at
any point is a function of the total magnetic force acting at that
point. We assume that the intensity of magnetization is proportional
to the magnetic force, or that (a) A=ka, B = kf3, C=ky, where k is
a constant. The magnetizing force proceeds partly from the per-
manent magnets present in the field, and partly from the quantities
of magnetism induced in the soft iron. The potential due to the
former may be designated by V, that due to the latter by U,
so that
A = - k . 3( F+ U)fix, B = - k . 3( F+ U)fdy, C = - k . 3( F+ U)fiz.
Now, in the space not occupied by permanent magnets, we have V2F"= 0,
and therefore 'dA/'dx + 'dB/'dy + 'dC/'dz^ -kV2U, or since, from LXVIII.
(e), p= -(dAj'dx+oBj'dy + 'dCI'dz}, we have finally V2U-p/k = 0.
Since the potential W is due to the components of magnetization
A, B, C, the equation that holds within the soft iron is V2 U + 4-rp = 0.
From the last two equations we obtain (b) (1 + ±irk)p = Q, and hence
p = 0 ; that is, there is no free magnetism present within the soft iron. The
174 MAGNETISM. [CHAP. vm.
magnetism present is therefore situated on the surface of the iron.
We will now determine the surface-density cr of this distribution.
For this purpose we use the equation
where va and vt are the normals drawn from any point on the surface
of the iron inward and outward respectively. Ut and Ua are the
values of the potential due to the induced magnetism inside and
outside the iron mass. Now we have
3 VfiVl = - 3 F/3vrt, and hence 4™ + 3 £7,/3v, + 3 UJdva = 0.
The magnetizing force just outside the surface of the soft iron is
-3(P"+ £7,)/3v4 in the direction of vt. The free magnetism on the
corresponding surface-element is therefore cr .dS = k.'d(V+ Ui)/'dvi.dS.
Hence we have
(c) 4vk . 3 F/'dvt + ( 1 + 47r£)3 U^v, + 3 Ujova = 0
The relation (c) in connection with the equations (d) V2Ut = Q, V2f/0 = 0
serves to determine the potentials Ut and Ua.
As an example of the theory here presented, we will consider
the magnetization of a sphere subjected to the action of a constant
magnetizing force ^) which acts in the direction of the £-axis. Let
the intensity of magnetization of the sphere in the direction of the
a-axis be A ; the force due to the magnetization and acting in the
direction of the z-axis is [LXX. (f)] equal to -4TT/3.A. From
equation (a) we have, therefore, A = k(5g> - 4ir/3 . A), and hence
^=&<p/(l+47r/3.&). We have in this case [LXX. (e)]
Ut = 47T/3 . Ax ; Ua = 4;r/3 . R3A . cos 0/r2.
These values satisfy equation (c), since V= -$gx.
SECTION LXXIV. LINES OF MAGNETIC FORCE.
If M represents the free magnetism within a closed surface, and
<^)n the component of magnetic force in the direction of the normal
to the surface, we have, from LXIX., (a) 4jr.3/=f ]"£)„ .dS.
If we lay a sheet of paper on a magnet which lies horizontally,
and scatter iron filings over it, they arrange themselves in curves
which are called lines of magnetic force. Let DE (Fig. 79) be a small
surface of area dS; lines of force proceed from its perimeter which
bound a tube of force. If D'E' represents another section cut through
the tube of force, equation (a) can be applied to the part of the
tube of force thus bounded. Since the direction of the magnetic
N
SECT. LXXIV.] LINES OF MAGNETIC FOKCE. 175
force coincides with the direction of the lines of force, the normal
force over the surface of the tube is everywhere zero except at the
ends DE and D'E'. Let <p be the force acting at DE and £)' that
acting at D'E'; let 0 and 6' be
the angles between the direction of
the force and the directions of the
normals to DE and D'E' respectively. ™
Since there is no free magnetism
present in the interior of the tube,
we have
- £) . (IS . cos 6 + £)' . dS' . cos 9' = 0.
If the sections dS and dS' are perpendicular to the lines of force,
we have ^)/^)' = dS'/dS. The force is therefore inversely proportional to
the cross-section of the tube of force. The lines of force may be so
drawn that their distances from one another furnish a representation
of the magnitude of the magnetic force in the field.
A tube of magnetic force cannot return into itself or form a hollow ring.
If this were not so, the work which is done by the magnetic
forces during the transfer of a unit quantity of magnetism from any
point over a closed path back to the same point again would not
be zero. If ds is an element of the tube of force, the work done
would be J^).cfc>>0, if the direction of motion coincides with the
direction of the force. If the magnetic potential is V, we have,
however, ^)= -dF/ds, and therefore, for a closed path,
since the potential is a single-valued function of the position of the
point.
Any tube of magnetic force must begin and end on the surface of a
magnet. If the tube ends with the cross-section PQ (Fig. 80), so
that a magnetic force is present in the tube TUQP, while
it is zero outside the tube at R and S, we may apply Jt,---,S
equation (a) to the region TUQSRP. Since a magnetic
force acts at the surface TU, but not in the region
PQSB, the surface integral taken over TUQSRP cannot
be zero. Magnetism must therefore be present within
the closed surface, which contradicts our assumption.
Therefore, any tube of magnetic force ends at the surface FlG
of a magnet.
In order to represent the magnitude and direction of magnetic
forces, Faraday used lines of magnetic force ; he assumed that the lines
176
MAGNETISM.
[CHAP. vm.
of force are continued in the body of the magnet. His mode of repre-
sentation has become of very great importance. If a magnet is broken,
and the surfaces exposed by the fracture are placed so as to face
each other and separated by only a small distance, a strong magnetic
T
FIG. 81.
force acts in the region PQEU (Fig. 81). This force is due partly
to the free magnetism in the interior of the magnet and on its original
surface, and partly to the free magnetism on the newly-formed sur-
faces. The force due to the former cause is directed from the north
pole n to the south pole s, that due to the latter from s to n. The
latter force is in practice the stronger, so that we may say with a
certain propriety that the magnetic tubes of force are produced through
the interior of the magnet along the path D'F'FD (Fig. 81).
FIG. 82.
If S (Fig. 82) is a closed surface lying outside all the magnets in
the field, and therefore containing no magnetism, we have from XIV.,
It is customary to express this result in the following way : The
integral J^)n . dS which is extended over a part of the surface may be
SECT. LXXIV.] LINES OF MAGNETIC FORCE. 177
divided into the parts $£>nl . dSlt ^)n2 . dS2, etc. Let them be so taken
that they are all equal, and let their common value be taken as unity.
Since the product §£)n.dS is constant for the same tube or line of
force, the integral J.£)n . dS gives the number of lines of force which traverse
the surface. If this integral is zero, as many lines of force entei' the sur-
face as leave it.
This holds for a surface which contains one or more magnets, for
the sum of the magnetism in every magnet is zero. On the other
hand, the theorem does not hold if the surface cuts through a magnet.
Nevertheless, if the magnet is divided into two parts, MNQP (Fig. 81)
and RSTU, and if they are situated infinitely near each other, the
theorem holds for either of them if the surface considered contains
one part, but excludes the other. Now, if this mode of division of
the magnet produces no disturbance in its magnetization, the theorem
can be expressed in the following way :
We represent the components of the magnetic force by a, (3, y.
In the part of the surface lying outside the cleft, no other magnetic
force is acting; on the other hand, the free magnetism +0- on the
element dS of PQ (Fig. 81), and — o- on the corresponding element
of RU, produce a force which can be determined in the following
way : From XIII. a surface on which the surface-density is o- exerts
an attractive force 27ro- on a unit of mass lying very near it. In
the case of magnetism, the force lira- is a repulsion. If there are
two parallel surfaces, on one of which the density of the magnetic
distribution is o-, while on the other it is - o-, the magnetic force
acting between the surfaces is 47r<r. If the normal to the surface-
element dS directed outward makes angles with the axes whose cosines
are /, m, n, we have, if A, B, C are the components of magnetization,
a- = lA + mB + nC. The magnetic force in the direction of the normal
is la + m/3 + ny. Since, in the case considered, the surface-integral must
be zero, we have {](Ai+Jit0+ny+4xv)d9«0, or, from LXVIII. (b),
\[l(a + lirA) + m(p + ±irB) + n(y + 4irC)]dS= 0.
We set (b) a = a + 4vA, b = P + ±irB, c = y + lirC, and obtain
(c) f (al + bm + cn)dS = 0.
The quantities a, b, c are the components of magnetic induction. If,
therefore, the directions of the lines of force are determined by the
directions of the resultants of the magnetic induction, it follows that
the lines of force may be considered as continued through the magnet itself,
and that they therefore return into themselves. Now, equation (c) shows
M
178
MAGNETISM.
[CHAP. vin.
that as many lines enter a surface as leave it. If we consider an
arbitrary surface S so drawn as to pass through every point of a closed
curve 5, and determine the magnetic induction whose components
are a, b, c, the magnitude N= \(al + bm + cn)dS is determined by the
boundary s of the surface S. We may say, the curve s encloses N
lines of magnetic force.
From equations (b) it follows that
'da/'dx + 'db/'dy + 'dc/'dz
= Vapz + Vppy + V7rdz + tor(dAfdx + 'dBj'dy + Wj'dz) = - V2 V- 4ir/o.
Hence, we have
SECTION LXXV. THE EQUATION OF LINES OF FORCE.
We will develop the equation of the lines of force of a small
straight magnet NS (Fig. 83) which is magnetized in the direction
9
fl
S
FIG. 83.
of its length with the intensity of magnetization J. Let N and S
represent the north and south poles respectively. Free magnetism
is present on the end surfaces S and N, supposed to be plane and to
have the area dA ; the quantity at the north end is /. dA, that at
the south end - .7 . dA. Let the centre of the magnet be the origin
of coordinates, and the x-axis coincide with the length of the magnet.
We are to determine the components a and (3 of the magnetic force
which acts at the point P, whose coordinates are x and y. If 21 is
the length of the magnet, and if PS = rv PN=r2, J.dA = q, we have
a = q.(z- Z)/r./ - q . (x + J)/V, ft = q . y/r* - q . y/rf.
SECT. LXXV.] EQUATION OF LINES OF FOECE. 179
If dx and dy are the projections of an element of the line of force,
we have dy/dx = f3/a, or
(a) (x - l)tr/ . dy - (x + l)/r^ . dy = y/r/ . dx - y/r^ . dx.
If we set ^PSx = Qv ^PNx = Q.2, we have
cosei = (x + l)/rl and cos 0, = (x - T)lrr
If x and y increase by dx and dy respectively, cos0 will increase by
or
In the same way d cos 02 = y2/rzs . dx - (x - l)y/r/ . dy. From equation
(a) we obtain d(cosSl - cos62) = 0, or, if c is a constant, cos61 - cos02 = c.
This is the equation of the lines of force.
SECTION LXXVI. MAGNETIC INDUCTION.
The components of magnetic induction are
If we consider a, b, c as components of flux, they have a property
similar to that of the components of flow of an incompressible fluid
[cf. XLI. (e)], for 'dal'dx + 'db/'dy + 'dcj'dz = 0, or, what amounts to the
same thing, ^(al + bm + cn)dS = 0. If 23 represents the resultant of
a, b, c, and € the angle between 23 and the normal to the surface-element
dS of the closed surface S, we have (53. cose. ^=0.
Let EF=dS (Fig. 84) be an element of the surface of the magnet,
and let the quantity of magnetism ar.dS be present on dS. Let the
induction outside the surface EF, gr p'
in the direction EE', be 230, and __ E ' ~4^L_
inside that surface, in the same -^""^ ' ------ - £
f" f"
direction, be 23;. Suppose perpen-
diculars erected on the perimeter of
the element EF and the surfaces E'F' and E"F" drawn parallel
to EF. We then have
(S3i-S3re).f^=0, or 23, = 23rt.
Outside the surface, 93a is the same as the magnetic force in the direc-
tion EE' ; it will be 23rt= -3F0/3va, if the potential outside the sur-
face is Va and the normal EE' is equal to dvn. Let V. be the potential
within the surface. The induction 23; is then [LXXIV.]
and we have, therefore, 3Fi/3v< + 3F"a/3va + 4jrcr = 0. This is the same
180
MAGNETISM.
[CHAP. vui.
equation as LXIX. (i). If the body considered is a mass of iron
whose coefficient of magnetization is k, we have
A=ka, JB = kP, C = ky, fJ.= l+±irk, a = pa, b = pP, C = /xy.
It follows that 'da/'dx + 'dbl'dy + 30/^ = 1*. V2Vi = 0 within the mass of
iron. Within that mass, therefore, no free magnetism is present.
The magnetic induction perpendicular to the surface has the same
value on both sides of it, that is [cf. LXXIII. (c)],
The magnitude /*, which is the ratio of the magnetic induction to
the magnetic force, may be called the magnetic inductive capacity (mag-
netic permeability}. The coefficient of induction or the permeability p is
equal to unity in vacuo, where k = Q ; in paramagnetic bodies /x > 1, in
diamagnetic bodies /*< 1.
SECTION LXXVII. MAGNETIC SHELLS.
Suppose a thin steel plate to be magnetized so that one face is
covered with north magnetism and the other with south magnetism.
At any point A in the face N (Fig. 85) draw a normal to the plate
which cuts the surface S at B. Let the plate be so magnetized
FIG. 85.
that -o- represents the magnetic surface-density at B, and +0- that
at A. We set AB = e, and call o-e = 4> the strength of the shell* at
the point under consideration. If the plate is infinitely thin and the
surface-density infinitely great, <f> has a finite value. Such a plate is
called a magnetic "shell"
The potential of such a shell may be expressed in the following
way: Let LM (Fig. 86) be the shell, dS a surface-element on its
positive face, BC the . normal to this surface-element, and P the point
* In the original, the moment of the surface. — TB.
SECT. LXXVII.] MAGNETIC SHELLS. 181
at which the potential is to be determined. We represent the angle
between BC and BP by e. The potential at the point P, due to
that part of the shell whose end-surface is dS, is [LXIX.]
dF=(r.dS.e.cosf/rz.
Hence, if the strength of the shell is constant, (a) V= <£ . J{ cos e . dS/r2,
where the integral is to be extended over the whole surface. If
the solid angle subtended by dS at the point P is called dw, and if
we set BP = r, we have dS . cos e = r^dw.
Therefore dpr=or. e. e?w = <J?. dw, and hence (b) F=<&.<a, where w
is the solid angle subtended by the shell at the point P. We may
call W the apparent magnitude of the shell seen from the point P.
If the point for which the potential is to be determined lies on
the opposite side of the surface, say at P', and if w' is the solid
angle subtended by the shell at the point P', we have V = - * . w'.
If the points P and P' approach each other until they are infinitely
near, but on opposite sides of the shell, we have (c) V = — & . (47r- w),
since 4?r is the total solid angle about a point. Hence, finally,
V-V' = \TT.<$>.
If PQP' (Fig. 87) is a curve which does not cut the shell, and
whose ends lie infinitely near each other on opposite sides of it, the
work done by the magnetic forces in moving a unit „---,.
magnet pole over the path PQP' is equal to 4ir«i».
This theorem holds even if other magnets are present
in the field. They act on the pole with forces which
have a single-valued potential, and the work done by
them during the motion of the unit pole in the curve
PQP' is equal to zero ; for this curve may be con-
sidered as a closed curve, since P and P' are infinitely
near each other.
After obtaining an expression for the potential of a
magnetic shell, we determine the force with which
the shell acts on a magnet pole of unit strength. The normal to
the shell makes angles with the axes whose cosines are I, m, n;
let £ */> £ ke the coordinates of a point in the shell, and x, y, z
the coordinates of the point outside the shell for which the potential
is to be determined. We then have
cos e = I. (x - £)/r + m.(y- ^)/r + n . (z - f)/r,
where r'2 = (x - £)2 + (y- rjf + (z- £)2. From equation (a) the potential
is
r= & . J J [(,- _ g) .1 + (y _ rj)m + (z - f )«]/r» . dS.
182 MAGNETISM. [CHAP. vm.
Since 3r~1/c3£ = (as-£)/rsi we obtain
V= *f J(/ . Br-1/^ + wi . Br"1/^ + n . 'dr~lj'dt)dS.
Represent the components of the magnetic force in the direction of
the .r-axis by a, we then have (d) a= -'dP'/'dx, and
because 3r~1/3«= -Br"1/^. If the shell does not pass through the
point .T, y, 2, 7- will never become zero, and we have
3V-i/3£2 + 3V- W + BV-'/Bf2 = 0,
(e) a = + *. J J[TO.32i-V3i/3£ + «. 32r-Y3£3£ - l(Wr~ll'dr
From the theorem of VI. (f), we have
f \(X. d£/ds + Y . drj/ds + Z . dC/ds)ds
+ n
WesetJf=0, Y= +*.3r~1/9t 2= -^.'dr~1/'dri, by which the right
sides of equations (e) and (f) become identical, and then obtain
(g) a = * . {Br-VBf - *?/<** - Sr-1/^ . dC/ds)ds.
Analogous expressions hold for (3, •/. By carrying out the differen-
tiation, we obtain (h) a = $ . J[(z - fl/r8 . diy/rfe - (y - 1?)/?-3 . d{/ds]ds.
The force is therefore determined by the contour and the strength of the
magnetic shell. This result follows from the fact that the potential
is determined by the solid angle and the strength of the shell.
In order to find the geometrical meaning of equation (h), we use
the following method : Let £, 77, £ be the coordinates of the point 0
(Fig. 88) ; Oy and Oz represent the directions of
the y- and s-axes respectively. Let the element
ds be parallel to the 2-axis, and represented by
OA=d£; we then have drj = Q. Let the point P,
for which the potential is to be determined, lie
in the y^-plane, and let OP = r. We set
and have y -rj = r . sin 6. The magnetic force
due to ds = OA is, (i) a = - * . ds . sin 6/r3 ; it
is perpendicular to the y.^-plane. Its direction may be determined
in the following way : If the right hand is held so that the fingers
point in the direction of ds, and the palm is turned toward the pole P,
the thumb gives the direction of the force.
Finally, we determine the work which must be done to bring a
magnetic shell from an infinite distance to a place where the magnetic
potential is equal to V. Let the shell be divided into elements dS.
In order to bring the surface-element which carries the quantity
SECT. LXXVIL] MAGNETIC SHELLS. 183
o- . dS of south magnetism to its final position, work equal to - o- . dS . V
must be done. In order to bring the corresponding surface-element
carrying the same quantity of north magnetism to its place, work
equal to (F'+dF'/dv. e)ar . dS must be done, if v represents the normal
to the surface-element dS. Hence the total work done is
A = f \dVldv .&r.dS=3>. \\dVldv . dS.
Now, since dVjdv = - (la + mfi + ny), we obtain for the work done
A= -3>.l\(loi + mp + ny)dS.
If JV represents the number of lines of force contained by the contour
of the shell, we have A = - & . N.
CHAPTER IX.
ELECTRO-MAGNETISM.
SECTION LXXVIII. BIOT AND SAVART'S LAW.
OERSTED discovered that the electrical current exerts an action on
magnets ; the law of the magnetic force which is due to an electrical
current was discovered by Biot and Savart. Let AB (Fig. 89) be
a conductor traversed by a current which is
S*P measured by the quantity of electricity flowing
/* in unit time through any cross-section. Let
D
0,-'
the quantity of magnetism p. be situated at the
point P, and let the conductor AB be divided
into infinitely small parts ds. If CD = ds is an
infinitely small part of the conductor CP = r, and
6 the angle between r and the direction of the
current in CD, the direction of the force will be
perpendicular to the plane determined by r and ds.
If the right hand points in the direction of the current and the palm is
turned toward the magnet pole, the direction of the force exerted by the
current-element on the pole is given by the direction of the thumb. The
magnitude of the force K is (a) K=fi . i.ds/r2. sin0.
The magnetic force which is due to any system of electrical currents,
whose direction, strength, and position in space are known, may
be calculated from (a).
If the current forms a closed circuit, and if the intensity of the
current is the same at all points in the conductor, we may determine
the force due to the current and also the potential which the current
produces. The force due to any current-element is equal to that
exerted by a line-element of the same length, which forms part of
the contour of a magnetic shell, whose strength is equal to the current-
184
CH. ix. SECT. Lxxviii.JBIOT AND SAV ART'S LAW. 185
strength. This follows from a comparison of equation (a) with
LXXVII. (i).
From LXXVII. (b), the potential V of a closed circuit of strength i,
at the point P, is (b) V=iw, where w is the solid angle subtended
by the circuit at P. If a, /?, y are the components of the magnetic
force acting at P, we have, from LXXVII. (h),
( a = i . f ((* - {)/rS . d-nlds -(y- n)li* . dtfds)ds,
(c) J8 = i . \((x - £)/r3 . dflds -(z- f)/»
7 = * •
If ABC (Fig. 90) is a conductor through which a current i flows
in the direction indicated by the arrow, and if a unit magnet pole
moves around the current in the direction
found by using the right hand in the manner
before described, the work done by the mag-
netic forces during the movement over the /j
path DFED is, from (b), equal to 4iri. If the
path of the pole encircles several currents i, i',
i" etc., the magnetic forces due to these
currents do upon it, during its motion, the
work A, given by (d) A = 4ir(i + i' + i" + ...),
in which the currents which flow in one FlG
direction are to be reckoned positive, and
those which flow in the opposite direction, negative. Hence, the
potential which an electrical current produces at the point F is
not determined only by the position of that point. If we bring a
unit pole (Fig. 90) to F over the path GF from an infinite distance,
the work which is done will be equal to V, the potential at the
point F. If the pole then passes around the current over the path
FEDF, the work k-n-i will be done, and the potential at F becomes
V+±iri. If the pole passes n times around the current in the
same way, the potential at F becomes V+ 47rni. Hence, the potential
at the point F has an infinite number of values. The differential
coefficients of the potential with respect to x, y, z are nevertheless
completely determined.
If a pole of strength ^ passes once around the current, the work
done on it is iTrip; if a magnet passes once around the current and
returns to its original position, the work done is 47ri2/*, when 2/*
represents the sum of the quantities of magnetism in the magnet.
But since for any magnet 2/* = 0, the work done is in this case equal
to zero.
186 ELECTRO-MAGNETISM. [CHAP. ix.
Since an electrical current may be replaced by a magnetic shell,
we can obtain the magnetic moment of an infinitely small closed
current. If i is the current-strength, the strength of the equivalent
magnetic shell is a-e=--i. If dS is the surface of the shell, we have
i.dS = <re.dS. The quantity of magnetism on one side of the shell
is <rdS, and the thickness of the shell is e. Hence, the magnetic
moment of the current equals the product of the current-strength
and the area enclosed by the circuit.
SECTION LXXIX. SYSTEMS OF CURRENTS.
Let a conductor be wound around a cylinder, so that the distances
of the separate turns from each other are equal. We may approxi-
mately determine the magnetic action of the system in the following
way : If L is the length of the cylinder, N the number of turns,
and i the current-strength, the current flowing in unit length of
the cylinder is Ni/L, and that flowing in the length dx is Ni.dxjL.
A portion of the cylinder whose length is dx may be replaced by
a magnetic shell, whose thickness is dx and whose surface-density
is a-, if (a) <r.dx = Ni.dx/L and o- = Ni/L. If this substitution is
carried out for the whole length of the cylinder, the actions of the
positive and negative faces of the substituted magnetic shells annul
each other everywhere except on the ends of the cylinder. If the
current flows in the way shown in Fig. 91, A exhibits negative and
B positive magnetism. Such a system of currents is called a solenoid.
Outside the cylinder, the only magnetic forces which act are those
which proceed from the poles A and B. If the length of the solenoid
is great in comparison- with its diameter, the magnetic force outside
of it vanishes near its middle point. The force in the interior of
the solenoid may be determined in the following way : Let the
line CD be parallel to the axis of the solenoid (Fig. 91), let CF and
DE be perpendicular to its sur-
face, and let the line FE be
parallel to CD. We assume that
the magnetic force y is parallel
FIG 91 to the axis of the solenoid in its
interior, and that no magnetic
force acts outside of it. Suppose a unit pole to traverse the closed
path CDEF. The work done by the magnetic force is y . d:c, if we
set CD = dx. From LXXVIII. (d), we have (b) y.dx=4ir.Ni.dxlL,
SECT. LXXIX.] SYSTEMS OF CURRENTS. 187
y = ^irNijL. If we represent the number of turns in unit length by
N, we have y = ±irNli, that is, the magnetic force (number of lines
of force per square centimetre) in the interior of the solenoid and
\ecir its middle point is given by the product of the current-strength i
into the number N of turns per unit length of the solenoid.
We will now determine the magnetic force of a sphere on whose
surface a conductor is wound. Suppose that a conductor is wound
on a sphere of radius It, so that the planes of
the turns are parallel and separated from each
other by the distance a. Suppose that ABCD
and EFGH (Fig. 92) are two of the turns. If
the current-strength is i, a single turn may be
replaced by a magnetic shell whose surface-density
is s, if as = i. For points outside the sphere,
the action of the positive magnetism on the
surface EG will be nearly annulled by the action
of the negative magnetism on the surface EH;
the only effective part of the two surfaces is the " FlG 92
circular ring, whose width is BE. Suppose the
magnetism on this ring to be distributed with the density <r, over
the zone BFGC. We have then
BE .s = BF.v, or <r/s = BE/BF= cos 9,
if 6 is the angle between the radius OB, and the line OP perpendicular
to the plane of the coils. Using the relation as = i, we obtain
a- = i/a. cos 6. From LXX. (e), (f) the magnetic potential for points
outside the sphere is given by Va = 47r/3 . RHja . cos 6/r2, since i/a is
equivalent to the intensity of magnetization /.
In determining the magnetic force in the interior of the sphere,
we must remember that the magnetic lines of force due to the currents
are continuous, and that the magnetic force in this case is the same as
the magnetic induction of the equivalent magnetized sphere [LXXIV.].
To find it, we suppose, as in that section, that the magnetized sphere
is divided into two parts by cutting out an infinitely thin section
perpendicular to the lines of magnetic force, and that the unit pole
is placed in the opening between these two parts. It will then be
subjected to the force '2-n-s directed toward the north end of the
magnetized sphere due to the repulsion of the north magnetism
exposed on one face of the cut, to the force 2-n-s due to the attraction
of the south face of the cut and in the same direction, and to the
force - f Tri/a, [LXX. (f)] due to the distribution on the outside of the
188 ELECTRO-MAGNETISM. [CHAP. ix.
sphere and in the opposite direction. The total force F acting on
the pole is therefore F = 4iri/a - £iri/a = ^ni/a.
If the whole number of turns is N, we have F=§ir.Ni/R, that is,
the force in the interior of the sphere is proportional directly to the current-
strength i and to the number of turns A7, and inversely to the radius It
of the sphere. We get the same result if we consider that this sphere
behaves like a magnetized iron sphere whose intensity of magnet-
ization is J=i/a.
In this way we can set up an almost constant magnetic field, which
may be applied in the construction of instruments used to determine
current-strength.
If the solenoid forms a closed ring, we obtain a system of currents
which has many applications. Let AB (Fig. 93) be a circle whose
centre is 0 and whose radius is r, and
let R be the distance between the
centre 0 and a straight line CD,
which lies in the plane of the circle.
If the circle rotates about the axis
CD, it describes a circular ring. Sup-
~j) pose that on this ring there are JV
turns of wire, through which the
current i flows. We replace the separate turns by magnetic shells,
and determine the magnetic force in the interior of the ring. We
reach this result most simply if we suppose a unit pole to move
on a circle of radius R about the axis CD. The work done by the
magnetic force ^), which acts in the interior of the ring, when the
unit pole has completed one revolution, is 2-n-IiQ. This work is also
equal to ItrNi if the path of the pole is in the interior of the ring ;
it is equal to zero if the path is outside the ring.
Hence we have, in the former case, 2irB§£> = faNi and ^) = INifR ;
in the latter case, ^p = 0.
SECTION LXXX. THE FUNDAMENTAL EQUATIONS OF
ELECTRO-MAGNETISM.
Up to this point we have considered the path of the electrical current
as a geometrical line. In reality the current always occupies space,
and is determined by its components along the coordinate axes. For
example, if dy . dz is a surface-element perpendicular to the z-axis,
and if the quantity of electricity u.dy.dz. dt passes through it in
SECT. LXXX.] EQUATIONS OF ELECTRO-MAGNETISM. 189
the positive direction in the time dt, u is the component of current
in the direction of the re-axis. The components of current in the
directions of the two other axes are represented by v and w. If Oy
and Oz are drawn through the point 0 (Fig. 94),
whose coordinates are x, y, z, parallel to the cor-
responding coordinate axes, and if the rectangle
OBDC is constructed with the sides dy and dz,
the current u.dy . dz flows through the element
OBDC. If the components of the magnetic force
are represented by a, p, y, and if a unit pole o
moves about the rectangle in the direction OBDCO,
the work done by the magnetic forces will be
(3.dy + (y + 'dy/'dy . dy) .dz-(p + 3/3/3z ,dz).dy-y.dz
This is [LXXVIII. (d)] equal to 4?r . u . dy . dz. Hence, we obtain
the equations
(a) 4iru = (dy[dy-'dpfdz), 4m = (da[dz-'dy[dx)i 4irw = (d(J/'dx-'da/'dy).
These equations express the current in terms of the magnetic force. In a
region where there is no current we have u = 0, v = 0, w — 0, and
therefore oy/3y = 3/3/32, da/32 = 3y/3z, 'dfij'dx = 3a/3y,
or a. dx + p.dy + y.dz= -dV. Therefore, in a region where there is no
current the magnetic forces have a potential. In this case the forces arise
from magnets.
From equations (a) the magnetic force is not determined only by
the components of current. If u, v, w are given and a, p, y so deter-
mined that equations (a) are satisfied, these equations will also be
satisfied if we replace a, p, y by
where V is an arbitrary function. The potential due to the magnets
present in the region is V.
We will now consider a few simple examples :
(a) Suppose the direction of the magnetic force to be parallel to
the 0-axis, and its magnitude to be a function of the distance r from.
this axis (Fig. 95). We then obtain from equations (a)
47TM = +dyjdr.y/r, 4irv = -dy/dr.x/r, w = 0.
The current is parallel to the zy-plane and perpendicular to r. The
current-strength J is
J= u cos (uJ) + v cos (vJ), J— -u. y/r + v . x/r = — l/4?r . dy/dr.
190
ELECTRO-MAGNETISM.
[CHAP. ix.
If y is constant in the interior of a cylinder whose radius is OA=rl
(Fig. 95), and equal to zero outside a cylinder of radius r.2, the current
in the unit length of the cylinder is
j . dr = -
. dr
FIG. 95.
which agrees with LXXIX. (b).
(b) If the current-strength is given,
we can find the magnetic force by
integrating equations (a). Let u and
v be zero, and w be a function of the
distance r from the ^-axis. We then
have from (a)
These equations will be satisfied if we assume that y = 0 and that
a and (3 are functions of x and y only. Suppose that the magnetic
force is resolved into two components, one of which, R, acts in the
direction of the prolongation of r, and the other, S, is perpendicular
to r. We then obtain a = R . x/r - S . y/r, fl --= R . yjr + S . .r/r, and
therefore 47rw = dS/dr + S/r =l/r. d(Sr)/dr.
If the conductor is a tube bounded by two coaxial cylinders whose
radii are Rl and R2, and if w is constant in the conductor we have,
if Cv C2, C3 are constants, Slr=Cv 2Trwr2 + C.2 = S.2r, S3r=C3. The
first of these equations holds for the interior, the second for the con-
ductor, the third for the space outside the conductor. From the nature
of the problem, Sl must have a finite value in the axis ; we therefore
have C\ = 0. Since the magnetic force changes continuously, we have
£, = 0 when r = Rly and therefore C.2 = - 2-n-wR^, S.2 = Airier -
Since irw(R.22 - R^) is equal to the current-strength i in the conductor,
we have S3 = 2i/r.
Therefore, an infinitely long straight linear current exerts a magnetic
force at a given point, which is inversely pi-oportional to the distance of that
point from the current.
SECTION LXXXI. SYSTEMS OF CURRENTS IN GENERAL.
The components of current and the components of magnetic force
are connected by the equations [LXXX. (a)]
(a) 4?rM = 'dyj'dy - 'dfi'fdz, ITTV = 'da/'dz - 'dy/'dx, lirw = 'dft/'dx - 'da/ay.
SECT. LXXXI.] SYSTEMS OF CUREENTS. 191
From these equations it follows that (b)
This equation corresponds with the equation of continuity in mechanics,
and asserts that the total quantity of electricity contained in a closed region is
constant. It thus appears that the current, whose components are u, v, w,
moves like an incompressible fluid. There is never any accumulation
of electricity, but only a displacement of it. This apparently con-
tradicts experience ; in order to be consistent with our method of
treatment we assume with Faraday that an electrical polarization or
an electrical displacement occurs. We represent the components of
this displacement by /, g, h. If one of the components, say /, increases
by the increment df in the time dt, df/dt=f represents the quantity
of electricity which passes in unit time through a unit of area per-
pendicular to the z-axis, in consequence of the change of polarization.
If p, q, r represent the components of the electrical current which is
due to the flow of electricity through the body, we have
(c) u=p + df/dt, v = q + dg/dt, w = r + dhjdt.
These quantities, u, v, iv, are the components of the actual current,
which is made up of the current conducted by the body and the current
arising from the change of polarization or the electrical displacement.
If the components of the current are finite, the components of
magnetic force vary continuously when no magnets are present in the
region. The components of force perpendicular to the surfaces of the
magnets, if any are present, is in general discontinuous. We assume
that currents of infinite strength do not occur in practice ; however
we sometimes consider the flow in a surface, in which case we must
assume that the components of current in the surface are infinite.
In this case the components of force parallel to the surface vary dis-
continuously on passage from one side of the surface to the other.
If a: and a2 represent these components of force, and J the quantity
of electricity which flows through a unit of length perpendicular to
the components, we have from LXXVIII. (d) 47r/=a2-a1. We may
obtain the same result from (a) as follows : We consider two surfaces
whose equations are z = cl and z = cy We obtain from the first two
equations (a)
477 . Fu . dz = 33 .dz- + fi ±- . /"*» . dz = a -<L- f*33fc . dz.
Pi and p.2 are the components of the magnetic force in the direction
of the y-axis on both sides of the plane surface ; ^ and a.2 have similar
meanings. If c.2 - cl = c is infinitely small, and u and v are infinitely
great, the integrals / udz and / vdz are the quantities of electricity
192 ELECTRO-MAGNETISM. [CHAP. ix.
which flow in a surface. The integrals on the right side vanish
simultaneously, and we obtain the value given in (d) for the difference
between the components of magnetic force on the opposite sides of
the surface.
SECTION LXXXII. THE ACTION OF ELECTRICAL CURRENTS ON
EACH OTHER.
The work which must be done upon two conductors A and E
(Fig. 96), carrying the currents i and i', in order to bring them nearer
each other, can be determined in the
following way : Let the position of the
current A be fixed, while B is brought
toward it from an infinite distance.
B may be replaced by a magnetic shell
whose surface-density is o- and whose
thickness is e. Suppose ODE to be
FIG. 96. a straight line, normal to the shell,
which cuts it on its negative face at
C and on its positive face at D. A surface element dS' at C carries
with it the quantity of magnetism - <rdS'. If V represents the
potential which A produces at C, the work done upon the element
dS' at C is - Fa- . dS'. Setting CD = dv, the work done upon the
element dS' at D is (F'+'dF'/'dv .dv).a- .dS'. Hence the work arising
from this portion of the shell is 'dF'j'dv .dv.a-. dS'. But dv . a- = i', and
hence the work done upon the whole shell is (a) W=i . \\dVfov. dS'.
The force acting in the direction CE is -'dF'/'dv. If CE makes
angles with the axes whose cosines are /', m', ri, we have
(b) W= - i' . Jf (I'a + m'p + riy)dS'.
The quantities a, (3, y are determined in LXXVIII. (c), and may
be put in the form
a = i\(dr~l/dy . d£/ds - 'dr^/dz . drj/ds)ds,
ft = i \(dr-lldz . dg/ds - dr~l/dx . d£/ds)ds,
y = tj(3r- 1 fa . dri/ds - Vr~lfdy . dg/ds)ds.
From the theorem of VI. (f) we have
\(X . dx/ds' + Y. dy/ds' + Z . dz/ds')ds'
z) + m'(dX/dz - dZ/dx) + n'(dY/dx - dX/dy)]dS'.
SECT. LXXXII.] MUTUAL ACTION OF CURRENTS. 193
We now set
X = Jii'/r . dg/ds .ds, Y= J«'/r . dij/ds .ds, Z= Jii'/r . dQds . ds,
and obtain
ii'\\(d£/ds . dx/ds + drj/ds . dy/ds' + dflds . dz/ds')dsds'/r
= f J jM'PX^-Vfy • dt/ds - Vr-ifdz . drj/ds)
+ m'@r-lfdz . dg/ds - dr^fdz . d£/ds)
+ n'(dr-i/-dx . drj/ds - 3r~V3y . d£/ds)]dsdS',
or ii'\\(dg/ds . dx/ds' + drj/ds . dy/ds' + d£(ds . dz/ds')dsds'/r
= \\i'(l'a + m'p + riy)dS' = - W.
Hence from (b)
(c) W= - ti\\(d£lds . dx/ds' + drjjds . dy/ds' + dflds . dzlds')dsds' /r,
where ds is an element of one conductor and ds' of the other.
If we represent the angle between two elements ds and ds' by c,
we obtain F. E. Neumann's expression for the potential energy of two
electrical currents, (d) W= -w'JJcose/r. dsds'.
If we represent the magnetic force which is perpendicular to an
arbitrary surface containing the circuit B by ^), we have from (a),
(e) W= - i' . J^)' . dS' = -i'N, if N represents, in Faraday's nomen-
clature, the number of lines of force enclosed by the conductor.
Therefore the potential energy of a current equals the negative pi'oduct of
the current-strength and the number of lines of force enclosed by the con-
ductor. Hence it follows that a current always tends to move so
that the number of lines of force enclosed by it shall become as great
as possible. The positive direction of the lines of force is the direc-
tion in which a north pole moves under the action of the current
[cf. LXXIV.] The above theorem has only been proved on the
assumption that the magnetic force ^p is due to another current.
But because currents and magnets are equiv-
alent, the law is true generally. Since the
surface containing the circuit B may be
arbitrarily chosen, the energy W depends
only on the contour of the circuit.
The force which acts on an element
AB = ds of the current ABO (Fig. 97)
may be determined in the following way :
Suppose that the conductor ABC moves so
that AB is displaced to A'B' in such a
manner that A A' and BB' are perpendicular to AB. If AA' = dp,
N
194 ELECTEO-MAGNETISM. [CHAP. ix.
the area contained by the conductor increases by ds.dp. If the
magnetic force at A is equal to £), and if its direction makes the angle
a with the normal" to the surface ABB' A', the component K of the
force normal to the surface is K=^>cosa. Hence the increment of
the potential energy of the conductor is, from (e), dW '= -iK. ds.dp,
if i is the current-strength. In order to cause the motion here described,
a force X must act on ds in the direction A A', which is determined
by X.dp = -iK. ds.dp, X= -iK.ds. Hence the force -IK.ds acts
on the current-element in the aforesaid direction, and if the current-
element is free to move, the direction of its motion is perpendicular
to the direction of the magnetic force as well as to its own direction.
The direction of the motion is determined by laying the right hand
on the current [cf. LXXVIII.].
It follows further that the force which acts on an element ds of
the current i is perpendicular to the plane determined by the current
and the direction of the magnetic force Q. If we represent the angle
between the direction of the force and the direction of the current
by <£, this force will equal $£>i . ds . sin <£.
SECTION LXXXIII. THE MEASUREMENT or CURRENT-STRENGTH OR
THE QUANTITY or ELECTRICITY.
(a) Constant Currents.
To measure constant currents we generally use a galvanometer con-
sisting of parallel circular conductors carrying the current whose
strength is to be determined. A magnet whose dimensions are small
in comparison with the radius of the coils, is suspended in the centre
of the apparatus, which is so placed that the coils are parallel to
the magnetic meridian. The current sets up a magnetic force whose
value is Gi perpendicular to the direction of the earth's magnetic force,
whose horizontal component is called H. G depends on the con-
struction of the galvanometer. If G is constant in the region in which
the magnet moves, the angle <£ by which the magnet is turned from
its position of rest by the current is determined by
(a) tg<t> = GilH, i = H/G.tg<j>,
that is, the current-strength is, in this case, proportional to the tangent of
the angle of deflection.
SECT. LXXXIII.] MEASUREMENT OF CURRENT-STRENGTH. 195
(b) Variable Currents.
It is very difficult to determine the strength of currents of short
duration at any instant. We may, however, easily measure the total
quantity of electricity Q which flows through the conductor. From'
LXXI. (d) the moment which tends to turn the magnet about a
perpendicular axis is - SCfta sin 0, if 9JJ is the magnetic moment of
the magnet, a the magnetic force, and 0 the angle between the direc-
tions of 9Ji and a. Setting a = Gi, where G is the galvanometer
constant, and assuming Q = ^TT, the directive J wee exerted by the current
on the magnet is equal to £0J6ri. The total moment caused by the
current is therefore \^llGi.dt = ^lG.Q, if we write Q = \i.dt. Q is the
quantity of electricity which passes through the conductor during
the discharge.
If .7 is the moment of inertia of the magnet, and <o its angular
velocity, Ju> will be its moment of momentum. We thus obtain the
equation (b) *$lGQ = Jo>. If the period of oscillation of the magnet
is called T, we have, by LXXI. (f),
(c), (d) T = Tr.Jj/WH and therefore Q = Hr^/Gir2.
The kinetic energy which the magnet receives from the impulse
given to it by the current is |/w2, in consequence of which it turns
through the angle 6. Its potential energy thereby increases from
-WlH to -WlHcosQ; the work done on it is 9Ji#(l -cos0). We
therefore have iJo)2 = 29JlH"sin2(0/2), or, if 6 is very small, (e) G-TW/TT
and Q = Hr/TrG . 0, that is, if there is no damping action on the magnet,
and if its angular displacement is small, the quantity of electricity flowing
through a section of the conductor is proportional to the angular displace-
ment of the magnet.
(c) Damping Action.
The oscillations of the magnet generally diminish rather rapidly
in consequence of what is called damping or damping action. Damping
arises from resistance of the air and the action of currents induced
by the motion of the magnet in neighbouring conductors. If there
is no damping, we have from LXXI. (e) and (f), when the oscillations
are small, d'2Q/dt2 = - 7r2/r2 . 0, T is therefore the period of oscillation
of the undamped magnet. We may assume that the damping action
is proportional to the angular velocity dQ/dt. Taking the damping
into account, we have, to determine the deflection 0, the differential
equation (f) 0 + 2?w0 + 7r2/r2 .0 = 0. The factor m depends on the size
and character of the oscillating magnet, on the density of the air,
196 ELECTRO-MAGNETISM. [CHAP. ix.
and on the size, character, and position of the masses of metal in
which currents are induced. If we set TT/T = n and 6 = eat, we have
a2 + 2ma + n- = 0 and a = - m ± \/n2 - m?*J - 1
in which it is assumed that n > m. Setting (g) n2 -m2 — n-2/Ti2> we
have 6 = (A sin (rtfa) + B COS^/T^) . e~mt. If 0 = 0 at the time / = 0,
we obtain Q — A . e~mt . sin (irf/Tj). dQ/dt = <*> at the time / = 0, and
therefore 9»r1«*/«>.tf^"*..nn(vl/T1), To find the magnitude of the
deflection we set dQ/dt = Q, and obtain (h) tg^/Tj) = 7r/wrr If TO is
the smallest root of this equation, the successive roots are TO + TJ,
TQ + 2rv .... The oscillations are therefore isochronous. If we repre-
sent the deflections by 0U 62, 03, ..., we have
.e-mr<>. sin (^TO/TJ),
. e - m(To+Ti> . sin (TT^/TJ ),
~ «<To+2Ti) . sin (*-TO/TJ ).
If the position of equilibrium is designated by A0, the first point
of reversal by Av the second by A.2, etc., the ratio between the
oscillations AA* and A.A is
(ej-e^^Og-e,) and (e1-e2y/(e3-e2)=e"-r'.
We set w-r^A, and obtain (i) A = log nat [(ex - 02)/ '(03 - 62)]. A is
the logarithmic decrement, which can be very exactly determined from
a series of oscillations. From (g) the period of oscillation TI} is
(k) TJ = T . Vl + A'2 --. Therefore the period of oscillation is increased
by the damping action. If we set T = TO in equation (h), we have
tg(7rro/Ti) = vlmri and 7rTo/Ti = arctg(ir/X), mr0 = A/V . arctg(7r/ A),
sm(irr0/T1) = l/Vl+A2/ir2.
Hence we have further 0: = T^/TT . g-V*-.arctg(ir/A) , i/^/j + xay^-' and
(1) fc) = TrOj/T! . Vl + A*/JT* . «W»" • arctg (x/X),
We obtain from (d) and (k) Q = Hr^jGir- . w/(l + A2/jr2), and using
equation (1), (m) # = ei.fir1/GV.«x/r-1TO»*0r/x>. l/Vr+A^.
In order to determine the quantity of electricity sent through a
conductor by an electrical current, whose duration is small in com-
parison with the period of oscillation of the magnet, we must determine
the logarithmic decrement and the period of oscillation of the magnet.
Q is then determined from these quantities, if we know in addition
the intensity of the earth's magnetism and the constant of the
galvanometer.
SECT. LXXXIII.] MEASUREMENT OF CURRENT-STRENGTH. 197
Setting arctg 7r/X = ^7r-x we have tg.r = A/7r. If A is very small,
we have x = X.jir and arctg (jr/A) = \TT — A/TT. If the damping action
is insignificant, we can neglect higher powers in the series in which
the exponential may be developed, and obtain
2 and Q =
SECTION LXXXIV. OHM'S LAW AND JOULE'S LAW.
We have up to this point assumed the existence of the electrical
current and have not discussed the question of the way in which it
is started and maintained. This mode of treatment in many respects
lacks clearness. We will therefore state such facts as are well established
by observation. The so-called galvanic elements can establish and
maintain an almost constant current. In order to maintain a constant
current in a conductor, an electromotive force must act in the direction
of the current. If u is the quantity of electricity which flows in unit
time through a unit of surface of the #>/-plane in the direction of
the .r-axis, we can set w=C.X, if C is the conductivity and X the
component of the electromotive force in the direction of the o-axis.
C depends on the nature of the conductor, and may be supposed to
have the same value in the conductor in all directions. If the com-
ponents of current and of force in the other two directions are v, w,
and Y, Z respectively, we have (a) u = CX, v=CY, w=CZ. Hence
'fafdx + 'dordy + 'du;l'dz=C('dX[dx + 'dY[dy + 'dZ[dz). If the steady state
of the electrical current has been reached, the left side of the equa-
tion equals zero, and hence the right side is also equal to zero. If
the electromotive forces have a potential V, we have (b) V2F~=0.
This equation states that no free electricity is present within the conductor-
as soon as the current becomes steady. The electromotive forces must
therefore arise from the free electricity on the surface of the conductor.
Suppose ABC (Fig. 98) to be an electrical conductor. We will
consider a portion of it which is bounded by the infinitely small
cross-sections A A ' = S and BB' = S, separated
from each other by the distance I, which is ______ — ^r- 1 £— __
also infinitely small. If AB is parallel to ^ ]b — -7-
the ./--axis, the component of current u equals
CX, and therefore the quantity of electricity FlG- 98'
i = uS=CX.S flows through the cross-section S. If V and V" are
the potentials at A and B respectively, we have i = C . S . ( V-
198 ELECTRO-MAGNETISM.. [CHAP. ix. SECT. LXXXIV.
and further (c) t = (V- P)/(//(7S) = (F- V")/K The resistance I! is
directly proportional to the length of the conductor and inversely propor-
tional to its cross-section, and to the conductivity of the substance constituting
the conductor. The difference of potential between A and B is V - V.
Equation (c) contains Ohm's law, according to which the current-
strength is directly proportional to the difference of potential and inversely
proportional to the resistance.
The quantity of electricity i . dt flows through the cross-section A A'
in the time dt and passes from A to B under the influence of the
electromotive force X. The work done is therefore
The work done in this part of the conductor by the electromotive
forces in unit time is (d) A = i(V- V) = i'2R This work is trans-
formed into heat in the conductor. Therefore the quantity of heat
developed in a conductor is proportional to the square of the current-strength
and the resistance of the conductor. This theorem was proved experi-
mentally by Joule and deduced theoretically by Clausius.
CHAPTER X.
INDUCTION.
SECTION LXXXV. INDUCTION.
FARADAY was the first to demonstrate that a current is set up in
a conductor if a magnet or a conductor carrying a current is moved
in its neighbourhood. F. E. Neumann discovered the laws of these
induced currents. Faraday himself afterwards described a method
of determining the strength and direction of the induced current
which possesses great advantages, because it makes it possible to
visualise the process. Suppose ABO to be a closed conductor
(Fig. 99), and DE, D'E', etc., the lines of force
enclosed by it. Let us designate an element
of a surface bounded by the conductor by *"* —
dS, the components of the magnetic force
(cf. LXXI.) by a, /?, y, and the angles which
the normal to dS makes with the axes by I,
m, n. An electromotive force arises in the
conductor if the interal
(a)
changes its value. If magnetizable bodies are enclosed by the circuit
which have a greater permeability for lines of force than air, the
components of force must be replaced by the components of magnetic
induction. An electromotive force then arises in the conductor if
the integral (cf. LXXIV.) N= \(al + bm + cn)dS changes its value. An
induced current arises if the number of lines of f wee enclosed by the con-
ductor is changed. If the change in the number of lines of force
is an increase, the induced current tends to diminish the number of
the enclosed lines of force, for the direction of the induced current
is such that its own lines of force are opposite to the lines of force
formerly existing. If the direction indicated in the figure by the
200 INDUCTION. [CHAP. x.
arrow is taken as positive, the induced electromotive force acts in
a negative direction.
According to Lenz's law, the current induced by the motion of a circuit
tends, by its electrodynamic action, to oppose the motion, by which it is
induced.
In order to determine the magnitude of the induced electromotive
force, we suppose that the current-strength at a given instant is
equal to i. If we move the conductor in the magnetic field, we
must, by LXXXIL, do the work -i.dN, and, at the same time, the
quantity of energy Ei2 . dt is transformed into heat. "We have at
once - i . dN= EP . dt, and therefore, because Ei equals the electro-
motive force e, (b) e = - dN/dt, that is, the induced electromotive force
is equal to the decrease in unit time of the number of lines of foi'ce enclosed
by the circuit.
The induced electromotive force depends on the value of the
magnetic induction, whose components are a, b, c, not on the magnetic
force, whose components are a, /3, y. If there is no magnet near,
and if the coefficient of magnetization of the region is & = 0 (cf.
LXXVL), the induction and the magnetic force have the same value,
and a, b, c may be replaced by a, /?, y.
For example, if the circuit ABC at the time t carries a current of
strength i, the number N of lines of force passing through the circuit
in the positive direction is N=Li. L is the number of lines of force
if the current-strength is unity. It is called the coefficient of self-
induction. If the current diminishes there arises an electromotive
force (c) e = — d(Li)/dt. According to Ohm's law we have e = 7iV. if we
represent the resistance by E, and therefore
(d) Ei = - d(Li)/dt = - L . flijdt,
provided that the coefficient of self-induction L is constant. This
coefficient depends on the permeability of the region and also on
the form of the conductor.
If i0 is the current-strength at the time / = 0, we have (e) i = i0 . e~R!L-'.
The current-strength therefore diminishes the more rapidly the greater the
resistance and the smaller the coefficient of self-induction.
We obtain from (c)
J"eidt = -Zp.di = Ui02.
From LXXXIV. (d), the left side of this equation is an expression for
the work done, which appears as heat in the conductor. Hence we
obtain for the electro-kinetic energy T of a conductor whose self-induction
SECT. LXXXV.] INDUCTION. 201
is L and which carries a current of strength i, (f) T=^Li2. The
electro-kinetic energy of the circuit is therefore equal to half the product of
the coefficient of self-induction L and the square of the current-strength i.
If ABC and A'B'C' (Fig. 100) are two
conductors carrying currents whose respec-
tive strengths are ^ and i2, the current
^ sets up a number Llil of lines of force
which pass through the conductor ABC.
The current i<, also sets up lines of force,
and the number of them which pass
through ABC may be represented by
M2li2. The total number of lines of force
enclosed by ABC is therefore
(g) N^L^ + Mrf*
The number of lines of force enclosed by A'B'C' is (h) N2 =
From LXXXII. and the discussion at the beginning of this section,
we have, in the usual notation,
= * COS er ' dS^
nlc2)dS1 = i2\ cos e/r .
The integral with respect to dS2 equals i^f^, that with respect to
dS1 equals i2M2l. From LXXXII. we have
M12 = M2l = | cos e/r . ds^dsy.
M12 = M2l is the coefficient of mutual induction of the two circuits.
If R1 and R0 are the resistances of the conductors ABC and A'B'C'
respectively, we have
Rfr = 6l= - dNJdt = - Ll . dijdt - M2l . di2/dt,
Rt,i2 = «2 = - dN2/dt = - L.2 . dijdt - M12 . dijdt.
Hence we have, for the electro-kinetic energy T of a system of two con-
ductors which carry the currents zt and i2,
T = J^i, + e2i2)dt = SLj* + Ml2ij2 + \L^.
For the electro-kinetic energy T of any system of conductors, we
find in the same way,
(i) T=i(Llil2 + L2i2^ + Lsi32+ ... +2M,2ili2 + 2Ml3ili3 + 2M23i2i,+ ...).
If Nv N2, N3 ... denote the number of lines of force enclosed
respectively by the conductors 1, 2, 3... so that, for example,
N1 = Lli1 + M2li2 + M3li3 + ..., the expression for the electro-kinetic
energy T becomes (k) r=£(JV1*'1 + A~2»'2 + JV3«3+ ...) = pM, that is, the
202
INDUCTION.
[CHAP. x.
electro-kinetic energy of a system of currents is equal to the sum of the pro-
, ducts of the number of lines of force enclosed by each conductor' and the
strength of the current present in the conductor.
Electrical currents arise not only if the neighbouring currents change
in strength, but also if they change their position, so that 7l/12, M13 . . .
vary, and also if the conductor itself changes its form. In all cases
the induced current is determined by the change in the number of
lines of force enclosed by the conductor.
SECTION LXXXVI. COEFFICIENTS OF INDUCTION.
In the investigation of variable electrical currents flowing in wire
coils, the coefficients of induction between different turns in any one
coil and between separate coils are of great
importance. The calculation of these co-
efficients is in most cases very difficult ; we
will consider only one simple case. We
suppose two circular conductors whose radii
are r: and r.2 (Fig. 101). They have a
common axis, and are separated by the
distance b. Suppose r2>r1. We have to
calculate the integral
FIG. 101.
. cos e.
We first evaluate the integral m,
m = ^dsjr . cos e.
We have r- = b2 + r22 - 2r1?'2 . cos c + 1\2. If
p is the shortest and q the longest distance between points of the
two conductors, we have
and m = 2 I ra . cos e . dtj*ffP + (q* - p'z) sm2£«.
If a is a small angle, so chosen that qa is very great in comparison
with p, we can set
m = 2 . deJP + .J/i + 2 f. *, . df . (1 - 2 sin2|e)/? sin it.
-a
f»«/2 sin £e - f sin ^« . efeT
. [Iog(2r1a/p) - log(a/4) - 2],
. [log(8r1//>) - 2] = 2[log(8r1/J9) - 2].
SECT. LXXXVI.] COEFFICIENTS OF INDUCTION.
203
With this value for m, we obtain Ml.2 = 47rr2(\og(8rl/p) - 2). On the
assumptions which have been made we may set rl = r.2 = R, so that
(a) Mn = ^E(\og(SE/p)-2).
The coefficient of induction between two coils, the number of whose
turns is n^ and ?i.> respectively, and for which the mean value of
logp is expressed by logP, is given by
(b) ^12 = 4rJl%7rJ??(log(8^/P)-2).
On the same assumptions the coefficient of self-induction L of a
single coil, if n is the number of turns, is given by
(c) L = 4irn2E(\og(8B/P) - 2).
We will not go further into the calculation of coefficients of induction ;
in most cases they are determined experimentally by one of the
following methods :
Methods of Determining the Coefficients of Induction.
(a) If the coefficient of mutual induction M of two coils Zx and L.2
is known, we may determine in the following way the coefficient
of mutual induction M' for two other coils Z/ and L.2'.
Let Zj and L.2 (Fig. 102) be the coils whose coefficient is known,
and L± and L.2 those which are to be investigated. A current is
FIG. 102.
passed through the coils Z/ and L.2 from the voltaic cell E. The
coils Z2 and L.2 are joined by conductors, and conductors are joined
from the points a and b to the galvanometer G. If the current /
which passes through L^ and Z/ is suddenly broken, electromotive
204 INDUCTION. [CHAP. x.
forces e and e' arise in L.2 and L.2'. If /2 and J.2' are the strengths
of the currents induced in L.2 and L.2\ we have
e = - d(L^J2 + MJ)/dt, e = - d(L.2'J.2 + M'J)/dt,
where L.2 and Z-2' represent the coefficients of self-induction. Applying
KirchhofTs laws to the circuit L.2G and L2G, it follows that
- d(L.2 /2 + M J)/dt = R2 J2 + G(J2 - J2'),
- d(L2J2 + M'J)/dt = R2'J.2r - G(J2 - J2'),
if the resistance of the galvanometer is designated by G, and the
resistances of the coils L2 and L2 by E.2 and R2 respectively. We
multiply these equations by dt and integrate from t = 0 to t = T, where
T is a very small time-interval. If the current / is broken at the
instant t = 0, a current is induced in the circuit L.y'GL2 which, in
the time T, sets in motion in the circuit L.2 the quantity of electricity
C2, in L2 the quantity C2, and therefore in the galvanometer the
quantity C2-C2. At the time t = Q, J=J and J.2 = J.2' = C2 = C2' = 0,
and at the time t = r, J=0 and the induced current has also vanished,
so that J2 = /2' = 0. Hence we have
MJ= E2C, + G(C2 - G,'), M'J= R2'C2 - G(C2 - C2).
C2 - C2 = J(M/E2 - M'/R2)/(1 + G/R2 + G/R2).
C2 - C2 is the induced quantity of electricity which flows through
the galvanometer, that is, the total current. The galvanometer shows
no deflection if the resistances satisfy the equation M'/M=R2'/R2.
(b) The comparison between two coefficients of self-induction can
be carried out in the following way : Let A BCD (Fig. 103) be a
FIG. 103.
Wheatstone's bridge- with a galvanometer inserted in the arm BD.
L^ and L2 are two coils inserted in the arms AB and BC, whose
coefficients of self-induction are to be compared. The current entering
at A and passing out at C distributes itself in the conductors, and
causes a deflection of the galvanometer needle. Let the resistances
SECT. LXXXVI.] COEFFICIENTS OF INDUCTION. 205
&L, Ry Sj_ and S2 in AB, EG, AD and DC respectively, be so adjusted
that no current passes through the galvanometer ; we then have
When the circuit is broken at E, an electromotive force arises
by induction in £: and L2, in consequence of which a current of
strength g flows through the galvanometer. Let ilt «'„ yv y.2 respec-
tively be the current-strengths in the conductors AB, BC, AD, DC.
They are connected by the relations il = yv «2 = 7o- We then have
- d(L&)ldt = (Rl + Sfo + Gg, - d(L.2i2)/dt = (E2 + S2)i2 - Gg.
At the instant £ = 0, when the circuit is broken at E, the same
current i0 was in AB and BC ; whence
LJQ = (Rl + SJ f\ .dt+GTg.dt; L,i0 = (Rz + S2) l\ .dl-oTg. dt,
Jo Jo Jo >>o
where r denotes a very short time. No deflection is caused in the
galvanometer by the current g if I g.dt = Q.
Since ^ = i.2 + g we have i: = i2, if no current flows through the
galvanometer, and hence in that case LJL2 = (El + <S'1)/(-K2
By the use of the relation Rl:fi.2 = Sl: S2, it follows that
that is, the coefficients of self-induction of the coils are in the same ratio
as the resistances of the two arms in which the coils are introduced.
If therefore the needle of the galvanometer is not deflected either
by a constant or a variable current passing through the circuits Z:
and L2, we have a means of determining the ratio between the
coefficients of self-induction L and £.,.
SECTION LXXXVII. MEASUREMENT OF RESISTANCE.
The strength of the electrical current in a conductor is determined
by its magnetic action ; the electromotive force is determined by
the change in the number of lines of force contained by the con-
ductor. The resistance in the conductor may then be determined
by the help of Ohm's law. Many methods of measuring resistance
are in use. We will confine ourselves to the description of some of
the simplest.
In one of these, used by W. Weber, the essential part of the apparatus
is a wire coil which can rotate about a vertical axis. This coil is
206
INDUCTION.
[CHAP. x.
set perpendicular to the magnetic meridian and is then turned through
180°. The coil is in connection with a galvanometer; the total
resistance of the circuit is R. If the plane of the coil makes an
angle <£ with the magnetic meridian at a particular instant, the
number of lines of force passing through the coil is SH sin <£. if <S'
represents the area enclosed by the coils and H the horizontal inten-
sity of the earth's magnetism. If L denotes the coefficient of self-
induction of the coil and galvanometer and i the current-strength,
we have (a) -d(SHsin tfi/dt -d(Li) dt = Ri.
Since </> changes, during the rotation of the coil, from + i~ to — ^TT,
and since i is zero both at the beginning and at the end of the
motion, we have (b) 2SH= RQ, where Q denotes the total quantity
of electricity which flows through the conductor. If Q is measured
by the method described in LXXXIIL, we have
The absence of H from this expression shows that it is not necessary
to know the intensity of the earth's magnetism in order to determine
the resistance.
Sir William Thomson's (Lord Kelvin's) Method.
If the coil above described turns with a constant angular velocity w,
we have by (a) - SHw cos <£ - L . di/dt = Ri. The integral of this equa-
tion is i = i0. e~KiL - A . cos(^-a). If the rotation is continued for
a considerable time, the exponential term vanishes and need be no
longer considered. To determine A and a, we have
(c) A=SHt»/(Rcosa + Lwsma); tga^Lu/R,
and therefore A = SHu/Rjl + LW/R* = SHu/R . cos a.
It thus appears that the self-induction appar-
ently increases the resistance.
If ON (Fig. 104) is the magnetic meridian
and if the line OM is perpendicular to it,
the coil acts on a magnetic needle at its
centre with the force Gi, whose direction is
that of the line OP perpendicular to the
plane of the coil. The components of this
FIG. 104. force are
OM= a = Gi . cos <£ and
SECT. LXXXVII.] MEASUREMENT OF RESISTANCE.
207
Let aa and bl denote the mean values of these forces. We then have
a, = 1 /2ir . / Gi . cos <f> . d<k ; ft, = 1 /2-n- . I Gi.sin<f>. d<f>.
Now * Jo
I cos (<f> — a) . cos <£ . d(f> = TT . cos a, / cos (<f> — a) . sin </> . d<f> = TT . sin a,
and hence flx = - \GA . cos a, 1^=- \GA.sm a.
The magnet at the centre of the coil turns from the meridian in
the same sense as that in which the coil rotates. If its angular
displacement is represented by B, we have
tg 6 = - «!/(# + bj) = GA cos a/(2H- GA sin a),
or, introducing the value of A, tg 6 = GS<» cos2a/(27i - GSw sin a cos a).
This equation, in connection with (c), serves to determine the resistance
R. If a is very small we have B=GSot/'2tgQ.
L. Lorenz's Method.
Suppose that a metallic disk ABC (Fig. 105), whose radius is a,
turns with constant velocity about an axis passing perpendicularly
through its centre. Around the rim of the disk, and concentric with
FIG. 105.
it, let there be placed a coil EF, through which flows an electrical
current of strength i, arising from the voltaic battery H. This current
sets up a magnetic force, whose component perpendicular to the
plane of the disk may be set equal to mi, where m is a function
of the distance from the centre of the disk 0. If the disk turns
from B to A, and if the current flows in the same direction, the
electromotive force induced in the disk is directed from the centre
to the periphery. A spring is placed at the point B, and is connected
208 INDUCTION. [CHAP. x.
by the conductor BGDEO with a rod which touches the disk at its
centre. DE is the conductor whose resistance E is to be determined.
If the current from the battery also flows through the conductor
DE, we may so adjust the angular velocity of the disk that no
current flows through the conductor which connects E with the
disk. When this condition is attained, the galvanometer needle
shows no deflection. If the electromotive force induced in the
disk is represented by e, we then have e = Ei, where i denotes the
current flowing through the resistance. To determine e, we consider
the disk replaced by a ring BAC and a straight conductor OA.
The circuit OAGDEO is divided into the two circuits OAB and
BGDEOB. The number of lines of force passing through the latter
circuit is not changed during the motion. On the other hand, the
number passing through the circuit OAB increases in unit time by
/ mino . dr = iw I mr . dr.
Jo JQ
This change in the number of lines of force gives the induced
electromotive force. We therefore have
,-a -a
Ei = i<ol mr.dr, E=wl mr.dr.
Jo Jo
If n is the number of revolutions made by the disk in one
second, we have o> = 'lirn and
n . 2irr . dr.
The integral gives the coefficient of mutual induction M between
the coil EF and the disk ABC, or the number of lines of force
which pass through the disk if a unit current is flowing in the
coil. Hence we have E = nM. Therefore to measure the resistance
we determine the number of revolutions per second which must be
given to the plate in order that no current shall flow through G.
SECTION LXXXVIII. FUNDAMENTAL EQUATIONS OF INDUCTION.
We have hitherto determined the electromotive force induced in
the conductor s by the change in the number N of lines of force
which are enclosed by s. N represents the number of lines of force
which pass through an arbitrary surface containing the conductor S ;
it is therefore determined by the conductor alone. Hence three
quantities, F, G, H, can be so determined that the line integral
l(F . dx/ds + G . dy/ds + H. dz/ds)ds
SECT. LXX xvm.] EQUATIONS OF INDUCTION. 209
is equal to the surface integral N=\\(al + bm + cn}dS. It is necessary
for this, by VI. (f), that
(a) a = a#/3y-3fiya?, b = 'dF/'dz--dH/'dx, c
We may also obtain these equations
of condition by the assumption that,
for example, dS is equal to the surface-
element di/dz, which is represented by
OB DC (Fig. 106). The line integral
then becomes
, dz)dy - Hdz
= (dH/dt/-'dG!'dz)dydz. FIG. 106.
Since the surface integral in this case is a . dydz, we obtain the first
of equations (a).
If these equations are supposed solved for F, G, H, we have
(b) N= ^(F. dx/ds + G . dy/ds + H. dz/ds)ds.
If the part of the region considered is at rest, the induced electro-
motive force e is determined by
(c) e = - dNjdt = - \(dF/dt . dx/ds + dG/dt . dy/ds + dH/dt . dz/ds)ds.
If we set (d) P= -dF/dt, Q = -dG/dt, R= -dH/dt, we may consider
P, Q, E, as the components of the electromotive force (£, and if
the integration is extended along the whole conductor, we obtain
as an expression for the total induced electromotive force
(e) e = $(P.dx+Q.dy + R.dz).
These components may also be determined directly by variation of
the value of the magnetic induction, whose components are a, b, c.
We thus obtain from (a) and (d)
- da/dt = ?>R[dy - VQfdz,
(f) - db/dt
Suppose that the electrical current, as has been remarked in
LXXXI., is made up of two parts, namely, the current of conduction,
which is proportional to the electromotive force, whose components
are p, q, r, and the current due to the changes in the electrical
polarization, whose components are /, g, h. Then for the components
of the total electrical current, we obtain
u =p + df/dt, v = q + dg/dt, w = r + dh/dt.
o
210 INDUCTION. [CHAP. x.
If C represents the conductivity, we have (g) p = CP, q=CQ, r=CR
and (h) ' f=kP/4v, g = kQ''±ir, h = kR/4Tr, where k is the dielectric
constant measured in electromagnetic units. Hence we have
(i) u=CP + k/4ir.dP/dt, v=CQ + k/±ir.dQ/dt, w=CR + kjlir .dR/dt.
From LXXVI. the components of the magnetic induction 33 and the
components of the magnetic force ^p are connected by the equations
(k) a = p.a, & = ju/3, e = /ry, where p. is the magnetic permeability of
the substances.
We have already found the following equations connecting the
magnetic force and the components of current (cf. LXXX.) :
(1) 47Ttt = 3y/3y-30/3s, 4™ = 'dafiz - "dy/'dx, 4-n-w = 3/3/3a; - "daffy.
SECTION LXXXIX. ELECTRO-KINETIC ENERGY.
By LXXXV. (k), the electro-kinetic energy of any system of con-
ductors is expressed by T=^Ni. By using LXXXVIII. (b) we
obtain T=^\(Fi.dx/ds + Gi .dy/ds + Hi. dz/ds)ds. If u, v, w, are the
components of current, a current i, in a conductor whose cross-section
is A, may be expressed by i = A. -Ju2 + v* + w2, and we have also
i.dx/ds = uA, etc. If we set dxdydz = A.ds, we obtain
(b) T= $ J J JCFw + Gv + Hw}dxdydz.
If u, v, w are here expressed by the components of the magnetic force
[LXXXVIII. (1)], we have
T= 1/87T . J Jf |7(3y/3y - 3/3/3«) + GCdafdz - Vy/dx)
+ HCdfi/'dx - 'da/'dy)] . dxdydz.
If the separate terms are integrated by parts, and the integration
is extended over the whole infinite region, it follows, since at the
boundary of this region a, (3, y are infinitely small of the third
order, that
f \\H . 3a/3?/ . dxdydz = - f J fa . ^Hffy . dxdydz
and Jf f G . Vafdz . dxdydz = - J J J a . VGfdz . dxdydz.
Analogous expressions hold for the other integrals.
By reference to LXXXVIII. we therefore obtain
(c) T= I/Sir . f f f (oo + /3b + yc)dxdydz.
If no magnets or no bodies which can acquire an appreciable
magnetization are present in the region, we have a = a, b = f3, c = y,
and (d) T= I/Sir. Hl
SECT, xc.] ABSOLUTE UNITS. 211
SECTION XC. ABSOLUTE UNITS.
In Physics we generally take the centimetre, (/mm and second as
units of length, mass and time respectively. These are the units which
are used in the theory of electricity. We will now proceed to
express in terms of them the most important electrical and magnetic
quantities. They are designated by the symbols L, M, T respectively
(cf. Introduction).
(a) The Electrostatic System of Units.
In electrostatics the force £y, with which two quantities of electricity
^ and e.2 act on each other, is expressed by (cf. LIII.) ^ = «1«2/r2»
where r is the distance between the quantities. If «1 = e2 = «, we have
e = rJ$, and hence the dimensions of a quantity of electricity e are
[e] = [LL*M*T-*] = [tfM*T-*].
The electrical force F, which acts on a unit quantity of electricity,
has the dimensions of the quantity e/r2, and therefore
TJie electrostatic potential ¥ (cf. LIV.) has the dimensions of the
quantity e/r, and therefore [¥] = [L^M^T^/L] = [L*M*T~1].
The capacity C [cf. LV. (g)] has the dimensions of the quantity «/¥",
and therefore [£] = [£]. The capacity, therefore, has the dimensions
of a length.
The surface-density cr (cf. LIV.) is the quantity of electricity present
on the unit of surface, and therefore [<r] = [Liflf* T~l/Lz] = [L-*M*T-*].
The dimensions of surface-density are the same as those of electrical
force [cf. LV. (e)]. Since the electrical displacement or polarization
(cf. LXV.) is the quantity of electricity which passes through unit
of surface in the dielectric, its dimensions are also [lT^M^T~1^, The
ratio between the electrical displacement ® and the electrical force
is expressed by KJ^TT, where K is the dielectric constant. K is there-
fore a mere number.
The electrical energy W [cf. LXI. (a)] is measured by the product
of the difference of potential and the quantity of electricity. Hence
[JF] = [/AMT~2]. These are also the dimensions of all other forms
of energy.
(b) The Electromagnetic System of Units.
Two magnet poles which contain the quantities of magnetism /^
and f*2, repel each other with the force F [cf. LXVIII. (a)], /'=/*1/*2/r2.
212 INDUCTION. [CHAP. x.
Hence quantity of magnetism has the same dimensions in the electro-
magnetic system as quantity of electricity in the electrostatic system.
This relation holds throughout between the two systems for corre-
sponding quantities in electrostatics and magnetism ; we will therefore
not consider each case separately.
The dimensions of the strength of the electrical current are deter-
mined from Biot and Savart's Law (cf. LXXVIIL). According to
this law the force K, with which a current -element ds acts on a magnet
pole containing the quantity of magnetism p., is K=p. ids. sin0/V2.
Since K is a mechanical force, the dimensions of the current-strength
» are
Since the quantity of electricity q may be considered as a product
of current strength and time, we have [q] = [ZA/lf*].
The electromotive force e which arises in a closed conductor has been
defined [cf. LXXXV. (b)] by e= -dNjdt, where N=\\(al + bm + cm)dS.
Since magnetic induction, by definition, has the same dimensions
as magnetic force, the dimensions of electromotive force are
In this system the electromotive force per unit of length of the
conductor may be considered as the measure of the electrical force,
whose components are P, Q, R. Hence the dimensions of electrical
According to Ohm's law the resistance is equal to the ratio
between the electromotive force in the conductor and the current-
strength ; the dimensions of resistance are therefore
that is, the dimensions of resistance are the same as those of velocity.
Surface-density of electricity and electrical polarization have the dimen-
sions of a quantity of electricity divided by an area, and are therefore
By LXXXVIII. (h) the dielectric constant k in this system has
he same dimensions as the ratio between the dielectric polarization
and the electrical force, or [L-*M*/L*M*T-*] = [L^-T2].
SECT, xc.] ABSOLUTE UNITS. 213
(c) Comparison of the Two Systems.
If we measure a quantity of electricity electrostatically by Coulomb's
torsion balance and electromagnetically by a galvanometer, we have
two values for the same quantity. In the first system this quantity
is expressed by e . [I$M*T~l], in the second system by q . [L^M?] ; the
ratio V between these expressions is V= e/q . [LT~1].
This ratio is therefore a velocity. It was first measured by Weber
and Kohlrausch. Its value, as found by them, is F=3,1.1010.
which is very closely the velocity 3,0. 1010 of light in air. Subsequent
experiments have made it probable that V is actually the same as
the velocity of light. It is thus shown that an electromagnetic unit
of electricity is equal to V electrostatic units.
If a certain quantity of electricity flows through a portion AB of
a conductor it produces heat in the conductor, which, considered as
energy, must be independent of the system of measurement employed.
The energy in electrostatic units is e . ^,, in electromagnetic units
q*Pmi where ""P, represents the difference of potential between A and B
in electrostatic units, "*Pm the same difference of potential in electro-
magnetic units. Hence we have e^f, = cflf m. We have shown that
e= Vq, and therefore ^rm = ^ft. V; that is, an electrostatic unit of potential
is equal to V electromagnetic units of potential.
If Ave designate electrical force in the electrostatic system by Fn
in the electromagnetic system by Fm, the difference of potential
between two points, which are distant from each other by dx, is in
the first system *Pt = Ft. dx, in the second system *Pm — Fm . dx. Since
"*Pm = ty, . V we have Fm = V . Ft. Hence one unit of electrostatic force
is equal to V units of electromagnetic force.
The dielectric polarization !£) is connected with the force F. by
the equation LXV. (d) (£)=K/'±ir. Fn if the electrostatic system is
used. In the electromagnetic system this equation takes the form
/= k/4ir . Fm. Since © and / are quantities of electricity divided by
areas, and since e = qpr, we have f£) = Vf. Since Fm is an electrical
force measured in electromagnetic units, we have
k = tirf>Fm = 47r£)/ V*F. = K! V2.
Hence in the electromagnetic system the equations connecting the
components of dielectric polarization with the electrical force are
We thus obtain ground for the assumption that V\<J]L is the velocity
of light in a medium whose dielectric constant is K.
214 INDUCTION. [CHAP. x. SECT. xc.
(d) Practical Units.
In practical work these absolute units are often discarded in favour
of others, called practical units. The unit of current-strength in this
practical system is the ampere, equal to 10"1 electromagnetic units
of current. The unit of resistance is the ohm, equal to 109 absolute
units of resistance. The ohm is nearly equal to the resistance of a
column of mercury, whose cross-section is 1 sq. mm. and whose length
is 106,3 cm. The unit of electromotive force then follows from Ohm's
law; it is called the wit, and is equal to 10s absolute units of electro-
motive force.
The unit of quantity of electricity is the quantity which flows
in one second through any cross-section of a conductor in which the
current-strength is an ampere. This unit is called a coulomb. The
capacity of a condenser, one of whose coatings is charged with one
coulomb when the difference of potential between its coatings is one
volt, is called a. farad; it is equal to IQ~1/IQ8= 10~9 absolute units
of capacity. A body whose capacity is unity in the absolute electro-
magnetic system must be charged with unit quantity of electricity
in order to reach unit potential. These quantities are the same as
the quantity of electricity V and the potential \/V in the electro-
static system. The electrostatic capacity of the body is therefore
V-. It follows from this that a farad is equal to F^/IO9 electro-
static units of capacity. Since this capacity is very great, the
millionth of a farad or a microfarad, is generally used as the practical
unit of capacity.
CHAPTER XL
ELECTRICAL OSCILLATIONS.
SECTION XCI. OSCILLATIONS IN A CONDUCTOR.
IF a conductor is traversed by alternating currents, that is, by currents
which reverse their directions at regular intervals, we say that electrical
oscillations exist in the conductor. Such alternating currents may be
produced by induction, as in the well-knoAvn experiments of Feddersen.
Let AB (Fig. 107) be a condenser whose plates are joined by con-
ducting wires to two small metallic spheres C and
D. If the condenser is so charged that A has the
potential ¥ and B the potential zero, and if the
distance CD is sufficiently diminished, a spark will
pass from C to D. Careful investigation has proved
that this spark consists of a series of sparks, which
correspond to currents in opposite directions. Hence
electrical oscillations will be set up by the discharge.
But if the current-strength i varies in the conducting
wires AC and DB, an electromotive force will be
induced in them, which, from LXXXV., is equal to
— L . di/dt, where L is the coefficient of self-induction.
If we represent the electrical resistance in the conducting wires and
in the spark-gap by r, the current-strength i is given by
(a) V-L.di/dt = r.i.
If c is the capacity of the condenser in electromagnetic units, the
charge q of the condenser at the time t is c*P; at the time t + dt, it
is c*P + c . d^/dt . dt. Hence we have
(b) c¥ = c¥ + c . d^jdt .dt + idt; i = - c . dV/dt.
From (a) and (b) it follows that Lc . d-i/dt2 + cr . di/dt + i = 0. An
integral of this equation is i = A. emit + B . e"^\ where ml and m2 are
215
216 ELECTRICAL OSCILLATIONS. [CHAP. xi.
the roots of the equation Lcm2 + crm +1 = 0. The roots of this
equation are m= - r/2L ± */r2/4:L- - l/Lc. If r is sufficiently small, the
roots m will be imaginary and we will have, neglecting r2/4Z2,
Using the real part and the real coefficient of the imaginary part
as particular solutions we can then express i by
i = e-*»L . [A1 . sin(//VZ<J) + V • cos(*/VZc)].
Hence the current-strength changes periodically and diminishes with the
time. This equation shows that the period T of the oscillation is
given by T/jLc = 27r) and hence T=2irsjLc. Therefore as the capacity
of the condenser is diminished, the period T diminishes also.
The amplitude of the oscillations is proportional to e~rt/2i ; it there-
fore diminishes continually as the time increases. The ratio between
the amplitudes of two successive oscillations, the so-called damping
coefficient, is e~rt/2Z : g -•*+*)/« = erT/'2L. The damping is therefore increased
as the resistance of the connections is increased, and with a given
period it is diminished as the self-induction L is increased. If we
substitute for T its value T= 27rx/Lc, the damping coefficient becomes
eirrVw.. Hence the effect of damping on the oscillations is diminished
when the capacity of the discharging conductor is diminished. Instead
of making observations on the damping coefficient itself, we generally
deal with its natural logarithm, the so-called logarithmic decrement 8.
We have 8 = rT/2L = irr . Jc/L. H. Hertz obtained very rapid oscilla-
tions by the use of the apparatus represented in Fig. 108. It con-
/^~x.4 C T) B s^sts °^ two ^ar§e sPneres °f equal size at
( QO ^ j A and B, which are fastened on the ends
^—^ of the copper rods AC and BD. The
other ends of the rods AC and DB carry
small spheres, separated by a distance of
about 1 centimetre. The spheres are
FlG- m charged by the induction coil EF '• the
discharge occurs in the gap between the spheres C and D. If at
a definite instant a current of strength i passes from A to B, and
if the potentials of A and B have respectively the values '*P1 and
¥"2, then ^rl - ^2 - L . di/dt = ri. If the capacity of each of the large
spheres is represented by c, we have cl . d^PJdt = -i, cl . d^2/dt = i,
or by using c = \cv i= -c. d(tirl - ^2)/dt. The current-strength is
therefore given by the same differential equation as before, and we
obtain from it the same expression for the period T.
SECT, xcn.] CALCULATION OF THE PERIOD. 217
SECTION XCIL CALCULATION OF THE PERIOD.
In order to determine the period, we must first determine the
coefficient of self-induction. The method of determining coefficients of
self-induction for closed conducting circuits has already been given.
In the present case we have to deal partly with actual currents in
the cylindrical conductors, partly with polarization or displacement
currents in the surrounding dielectric. The principal effect must
be due to the induction in the conductor itself, since in it the distance
between the inducing and the induced currents is least. The full
treatment of the question would be very difficult, because the current-
strengths in the different parts of the cross-section are not equal.
We will therefore neglect these differences in the subsequent dis-
cussion, and calculate the coefficient of self-induction L in a cylinder
on the assumption that the current-strength in all parts of the cross-
section is the same. According to F. Neumann, the electromotive
force E induced in a conductor s' by the action of a variable current
i flowing through another conductor s, is determined by the variation
of the integral Li, where L = ^cose/r.dsds'. In this integral, which
is to be taken over all the elements of both conductors, e denotes the
angle between ds and ds', and r the distance between ds and ds.
Let AB and CD be two parallel lines (Fig. 109) which together
with AC and BD form a rectangle. We set AB=CD = l, and CF=s'.
f-
fJs'
1
1
>;>';" i* ""G
i
A
fc b, G 12
FIG. 109.
In the present case the integral which is to be evaluated becomes
/ | ds.ds'/r, because (Fig. 109) cose = l. In order to find first the
-'o ^o
value of the integral P= I ds/r we draw FG perpendicular to CD
and AB, and write FG = a. If AG = bv BG = b9 FA = r1, FB = r.2,
we have P = \ ds/r + I ds/r, where s is the distance of any point on
AB from G. Now since
\dsfr = Ids/Jat + s* = log nat(s/a + Jl+sz/az)
we obtain P = \ogna,t[(rl + bl)(r2 + b2)/a2']
= log nsA[(AF + AG)(BF+ BG)/a?].
218 ELECTRICAL OSCILLATIONS. [CHAP. xi.
If a is very small in comparison with I, we may write
AF=AG=*' and FB = GB = l-s',
and have />=lognat[4s'(^ -s')/«2]. We then calculate the value of
the integral \P.ds and obtain
P.ds' = 2l. [log nat(2//a) - 1].
We will now calculate the coefficient of self-induction of a wire with
circular cross-section. Let AB (Fig. 110) be the cross-section of a
cylindrical conductor, of length / and radius
R. Let the current-density u be constant
throughout the cross-section. The current-
strength i is then given by i = r-iru. We
will first consider the inductive action due
to a filament D whose cross-section is ds;
this filament is supposed to act on a line
which is parallel to the axis of the cylinder,
FIG. 110. an(j passes through the point c. If DC=a,
the inductive action is obtained by the variation of the integral
u.dS. 2/(log nat(2//a) -l) = u.dS. (M+Nlog nat a),
where, for the sake of conciseness, we use M and N as symbols
for the quantities Jl/=2£(lognat2i-l) and N=-'2l. We must
distinguish between two cases : OD = r may be either greater or less
than OC=rv First let r>rr The elements dS may be taken so
as to form the surface of a ring whose area is 2irr . dr. From the
demonstration of XIII. log a equals the logarithm of half the sum of
the greatest and least values which a can take. These values are
respectively r + rl and r - rr The mean value required is therefore
logr. Hence we have the integral
*'2irr . dr . (M+ N\og nat r) = mt{M(Sr - rtf
+ N[R2 log nat R - r* log nat rx - $(R2 - r-f)] }.
For that part of the cylinder whose distance from the axis is less
than 1\ the mean value of the greatest and least values of a will
equal rr Hence the integral for this part is
u I Jfcw . dr . (M+ N\og nat r:) = Tru(Mi\2 + Ni\z . log nat rx).
The sum of both integrals is
[
-'
SECT, xcn.] CALCULATION OF THE PERIOD. 219
In order to obtain the mean value of this quantity for all the filaments
composing the cylinder, we need only find the mean value of rx2,
since all the other quantities are constant. But since
it follows that the mean value sought is iruR2(M + .ZV(log.E - J)}. If
we introduce into this equation the values
M= 2J(log 21-1) and N= - 21
and set iruR2 = i, we obtain for the quantity, the variation of which
gives the self-induction, 2/i(log(2?/^) - f ). We therefore obtain for the
quantity L, L= '2l(\og(2l/E) - f). In Hertz's investigation, 1=150,
.# = 0,25 and therefore Z=1902, where all lengths are expressed in
centimetres.
In order to calculate the period of oscillation, we will next determine
the capacity of a sphere with a radius of 15 cm., such as Hertz used.
If Q is its charge and >F its potential, the capacity C in electrostatic
units is C=Q/y. Eepresenting the charge and the potential in
electromagnetic units by Q' and ¥" respectively, and using V=3. 1010,
the velocity of light in vacuo, we have Q=7(^ and ¥ = ¥7^". The
capacity c in electromagnetic units is therefore c=Q'lW = C/V2.
The period of oscillation is then given by T=2irjLC/V.
Using the symbol introduced at the end of XCL, we have c = ^cv
where Cj is the capacity in electromagnetic units of each of the two
large spheres. Hence we must set £=15/2, and obtain r=2,5/108
seconds. The corresponding wave length in air is
2,5.10-8.3.1010
SECTION XCIII. THE FUNDAMENTAL EQUATIONS FOR ELECTRICAL
INSULATORS OR DIELECTRICS.
Maxwell shows that it follows, as a consequence of his theoretical
views of the nature of electricity, that a change in the electrical
polarization of the dielectric can set up electrical oscillations. The
results which he obtained are so important that we will consider
some of them here. For this purpose we will follow Hertz in sub-
stituting in the fundamental equations of LXXXVIII. electrostatic
units for the electrical quantities, while the magnetic quantities shall
be measured in electromagnetic units. The quantity of electricity
which is displaced by the electrical force at a point in the dielectric
220 ELECTRICAL OSCILLATIONS. [CHAP. xi.
through a surface-element which stands perpendicular to the direction
of the force, is, according to LXV., equal to K!±TT multiplied by
the magnitude of the force F. Representing the components of the
electrical displacement by /, g, h, and the components of the electrical
force by X, Y, Z, we have f=KXJ^ g = KY/4ir, li = KZ^. If the
component X increases by dx in the time dt, the quantity of electricity
df flows through unit area in the direction of the z-axis ; the com-
ponent u of the current-strength in the dielectric is eqiial to dfjdt,
and we have
(a) u = K/±ir.'dXI-dt, v=K/4ir.c)YI'dt, w = K/4ir . -dZfdt.
The equations LXXX. (a) express the fact that the work done by
the magnetic forces in consequence of the movement of a unit pole
about the current is equal to the current-strength multiplied by 4?r.
If the current-strength is measured in electrostatic units, we have
(b) 4irM/F=3y/3y-a)8/as, 4ir»/F=3a/3«-3y/aB, 4irw/F=3/3/az-'9a/3y,
since the electromagnetic unit of quantity of electricity equals V
electrostatic units.
The electromotive force induced equals - dNjdt, if N represents the
number of lines of force enclosed by the circuit. Since the electro-
motive force in electromagnetic units equals the electromotive force
in electrostatic units multiplied by F, we have from LXXXVIII.
(f) and (k)
f - >*/ F. da/3/ = ?)Z /-dy - -dY/-dz ;
- *
(c) - ,*/ F . 3(3^ = -dXfdz - 'dZ/'dx •
( - /*/ V.
From (a) and (b) we obtain
KIV. 'dX/'dt
Jf/F. -dYpt = 'da/'dz -
If we now set J='dX/'dx + ?>Y/'dy + ~dZfdz, we obtain from (c) and (d)
fj.K/F'2.d2X/'dl2 = V2X-'dJ/ox. If we are dealing with a region in
which K is constant, K/4ir . /= ?//3.c + 'bgj'dy + "dhfdz. If there is no
electrical distribution in the region, we have, from LXVI. (d), J=0;
and hence
(e) pKIF*.&XIW=V*X', pK/r*.&Y/df2 = V*Y; nK/r2.VZ/-dP = VZ.
These equations, in connection with equations (c) and (d), give
(f) fiK/F^.^a/'df = V2a- (JiK/F2.-d2pfdt* = r-p; nKj V*- . 32y/^2 = V2y.
at the same time 3o/daf-f dj9/3y+dy/d*««0 from LXXVI., if u is
constant.
SECT, xciv.] PLANE WAVES IN THE DIELECTRIC. 221
From LXVII. (g), the electrical energy W is expressed by
(g) W= l/8ir . J Jf K(X* + Y* + Z*)dxdydz.
The electrokinetic energy T, according to LXXXIX. (c), is
(h) T=l/8ir. J{jM(a2 + p2 + y2)dxdydz.
SECTION XCIV. PLANE WAVES IN THE DIELECTRIC.
We will now investigate the movement of plane waves in a dielectric.
Let the plane waves be parallel to the ys-p\&ne. The components
of the electrical force are then functions of .r only, and from the
equations XCIII. (e), we have
At the same time also 'dXj'dx + VYfdy + 'dZ/'dz = 0. Since Y and Z
are independent of y and zt we have 'dX/'dx = 0, and since, in this
case, the only forces which occur are periodic, X=0. The direction
of the electrical force is therefore parallel to the plane of the wave. By
a rotation of the coordinate axes we can make the y-axis coincide
with the resultant of the components Y and Z. We therefore need
to discuss only the equation /JT/F2. 327/9^ = 32F/3a;2. The integral
of this equation is (a) Y=bsin[2Tr/T . (t - a;/o>)], where T is the period
of oscillation and w the velocity of propagation. The differential equation
is satisfied if w= F'/Jp-K. For vacuum /*= 1, K= 1 ; hence V is the
velocity of propagation of plane electrical oscillations in vacuo. For
ordinary transparent bodies, /*=!. The velocity of propagation in
such bodies is therefore V/\/K. Maxwell assumed that electrical
oscillations are identical with light waves. It has been shown by
experiment that o> = V/N, where N represents the index of refraction
of the dielectric. The electromagnetic theory of light gives w = Vj^K;
hence we have K=N2, that is, the specific inductive capacity of a medium
is equal to the square of its index of refraction. The fact that this
theorem holds for a large number of bodies is a strong confirmation
of Maxwell's hypothesis. From this hypothesis almost all of the pro-
perties of light can be deduced.
According to XCIII. (c) we have, under the above conditions,
a = 0, 13 = 0, and
(b) /*y = 7b/u> . sin [27T/J . (t - ar/w)] = Nb sin [lirjl . (t - x/u)].
The direction of the magnetic force is therefore parallel to the plane of
the wave and perpendicular to the direction of the electrical force.
222 ELECTRICAL OSCILLATIONS. [CHAP. xi.
We will supplement this discussion by the following examination
of the relation between the electrical and the magnetic forces. Let
an electrical force act in the y^-plane, parallel to the axis Oy (Fig. Ill),
and suppose it to increase uniformly from
& ,£7 zero to YQ in one second. In consequence
of this an electrical current v is set up in
the same direction, and because 'dY[dt= Y0,
we have from XCIII. (a),
This electrical current will set up magnetic
forces, which are parallel to the ^-axis.
r IG. 111. ___ .__
We Avill assume that the magnetic force
increases uniformly from zero to y0. In consequence of this an
electromotive force will be induced in the surrounding region. We
will assume that the electrical and magnetic actions advance in one
second over the distance O.r = w (Fig. 111).
The magnetic force decreases uniformly from ;c = 0 to x = u; the
same statement holds for the electrical force OD=Y0. The electrical
current, on the other hand, has the same strength everywhere between
0 and x. This is explained by remarking that the electrical force
at a point F, whose distance from x equals l/n. Ox, has acted only
during Ijn seconds, and has, during this time interval, increased from
0 to l/n. Yn; its increase in one second is therefore equal to F0.
Let a unit pole move in the rectangular path OzBxO (Fig. 111).
The magnetic force y0 acts only in the path Oz and acts in the
direction of motion ; hence the work done by the magnetic forces is
equal to y0. Oz. The quantity of current, measured in electromagnetic
units, which the unit pole has encircled, is v/F~. Oz. Ox. From LXXX.
we have therefore y0 . OZ = ±TTV. Oz . Ox/F~, or because OX = (D, we
obtain (d) Fy0 = 47rz;w.
The electromotive force, measured in electromagnetic units, which
is induced by the motion about a closed path, is e = - dNjdt, if A7"
represents the number of lines of force enclosed by the path. We
have therefore JV= — ^e . dt. The mean value of the electromotive
force in the direction Oy is | . F0 . V, in electromagnetic units. The
value of the^ before-mentioned integral, extended over the rectangular
path 0>/Cx, is |F0F. Oy. The mean value of the magnetic force per-
pendicular to the surface OyCz is Jy0; hence the mean value of the
magnetic induction is \ . fj.y0. We therefore have the equation
SECT, xciv.] PLANE WAVES IN THE DIELECTRIC. 223
and hence (e) FT0 = /ry0w, NYQ = p.y(), if the index of refraction N is
substituted for F'/w. In this connection it must be noticed that the
wave is propagated in the direction of the .r-axis, that on our assump-
tion the electrical force acts in the direction of the y-axis, and that
then the magnetic force acts in the direction of the £-axis. Hence if
the right hand is held so as to point in the direction in which the wave
is propagated, with the palm turned toward the direction of the electrical
force, the thumb will point in the direction of the magnetic force. If we
represent the magnetic force by M and the electrical force by F,
NF= pM. From (c) and (d) it follows that Fy0 = -KT0w. Substituting
in (e) the value of y0, we obtain V1 = /JTu>2. Hence the velocity of
propagation is w = VjJp.K. From (a) and (b) it follows that the
relations (e) between the electrical and magnetic forces hold also
in the case of plane waves. In vacuo both forces have the same
numerical value.
SECTION XCV. THE HERTZIAN OSCILLATIONS.
H. Hertz succeeded in producing very rapid oscillations in a straight
conductor, which also caused oscillations in the surrounding dielectric.
We can form some idea of the nature of these oscillations in the
following manner, due to Hertz :
Let the middle point of a conductor coincide with the origin of
coordinates, and let the oscillations take place along the 2-axis. The
magnetic lines of force are then circles, whose centres lie on the
£-axis. The electrical lines of force have a more complicated form.
We start with the differential equation XCIII. (f) for the magnetic
forces. For the sake of conciseness we set VjKp- = w.
We first investigate an integral of the differential equation
(a) l/o>2.c>2w/3/2 = V%,
on the hypothesis that u is a function of t and of r = >Jx2 + y2 -j- z*.
We have then, from XV. (1), V% = 1/r . 32(™)/3r2, and therefore
l/o>2.32(rM)/3/2 = 92(?-tt)/3r2. If we set k=2Tr/T and J = 2jr/7w, where
I7 is a constant, then (b) u = a/r . sin (kt - lr) is a particular integral
of the differential equation. The function u, as well as its differential
coefficients taken with respect to x, y and z, therefore satisfies the
differential equations XCIII. (f) for the components of magnetic
force a, /?, y. In the case under consideration y = 0, and hence we
have 'da/'dx + 'dft/'dy = 0. As the simplest solution of the differential
224 ELECTRICAL OSCILLATIONS. [CHAP. xi.
equation we obtain (c) a = - fPu/'dfdy, /3 = ^ufdfdx, where the differ-
entiation with respect to t is introduced for use in the subsequent
calculation. From (c) we obtain a = - 'd-uj'dfdr .yjr; (3 = 'd-u/'dfdr . x/r.
The resultant magnetic force is therefore
M= a*M/a/3r . Jx2 + f/r = Wupfdr . sin 6,
where 0 is the angle between the radius vector from the origin and
the s-axis. The force M is perpendicular to the plane which contains
the point considered and the .e-axis.
If we set kt-lr = <f>, we have M = ka(l. sin <£/r - cos <£/r2) .sin 6. If
r is very small in comparison with l/7 = o>r/27r, we have
that is, the force is determined by Biot and Savart's law, the oscillations
in the conductor acting like a current element.
For greater distances the magnetic force is
(d) M= 4*-%/rW . sin [2-rr/T . (t - r/o,)] . sin 6.
Hence the magnetic waves proceed forward in space with the velocity
of light.
We will now calculate the electrical forces. From (c) and XCIII.
(d) we obtain
Since u depends only on r and f, the carrying out of the differentiations
, xz/r2 ;
(e)
KZI F= 32t*/3r2 . (r2 - z2)/7-2 + cto/'dr . (r2 + z2
The electrical force E in the direction r is (Xx +Yy + Zz)jr, and hence
from (e), KRj V= 2/r . 'duf'dr . cos 6 = - 2a(l cos <£/r2 + sin ^/r3) cos G. If
# = 0, the electrical force is tangent to a sphere whose radius is
determined from the equation tg(kt — Ir) — — Ir. Let the next spherical
wave have the radius r' ; then tg(kt - Ir) = - Ir'. From this it follows
that tgl(r' -r) = l(r' - r)/(l + I2rr'). If the radius of the waves is very
large we may set l(r' - r) = ir. But since / = 277/X, where A. = T<a, we
have r' - r = |X. We thus obtain at last equidistant spherical waves.
SECTION XCVI. POYNTING'S THEOREM.
Let there be an electrical current i flowing from A to B through
a long cylindrical conductor (Fig. 112) of circular cross-section. There
is then a magnetic force M acting at every point in the region around
SECT, xcvi.] POYNTING'S THEOREM. 225
the conductor, given by the equation 2?rr . M— 47ri, where r = OC,
the distance of the point from the axis of the cylinder. We therefore
have M=2i/r. The equipotential surfaces of the electrostatic field
of force within the conductor are planes perpendicular to the axis
of the conductor. Outside the conductor they s
are likewise perpendicular to the axis, at least
in the vicinity of the conductor. The equi-
potential surfaces of the magnetic field of force
are planes which, like the plane OF, contain
both the direction of the electrical force and
the axis of the conductor. Let the electrical
force in the surface of the conductor and in its
vicinity be F'. If we designate by S the
cross-section of the conductor, and by C its
conductivity, we have from Ohm's law i/S=CF'.
The heat produced in the conductor during one A
second is determined as follows : If the quantity ^IG- 1^.
of electricity i flows through the conductor, whose length we may
call /, the electrical force does work equal to F'il. If / represents
the mechanical equivalent of heat, the quantity of heat thus developed
equals F'iljJ. The work done by the electromotive force is therefore
F'il = \.MrFl,
Now Poynting assumes that this quantity of energy enters the
conductor through its surface. That portion of the surface which
is to be considered is equal to 1-xrl. The quantity of energy which
enters the conductor through unit area on its surface is therefore
IjlTT.F'M.
This quantity of energy moves in the direction CO, which is deter-
mined by the intersection of the electrical and magnetic equipotential
surfaces. The relation of the direction in which the energy is pro-
pagated to the directions of the electrical and magnetic forces is
determined in the same way as that given for the propagation of
waves in XCIV.
The electrical force is here measured in electromagnetic units; if
we express it in electrostatic units, and represent it by F, then
F' = VF. The quantity of energy which enters in one second through
a unit area of the surface, which is parallel to the directions both
of the electrical and of the magnetic forces, equals V/lir.F.M.
We will now treat a more general case. Eepresent the magnetic
force at a point in the region by M, the electrical force at the same
point by F, and the angle between the two forces by (M, F). We
226 ELECTRICAL OSCILLATIONS. [CHAP. xi.
assume that the quantity of energy passing in one second through
unit area, which is parallel to the directions of M and F, is
The direction in which the energy flows makes angles with the
coordinate axes whose cosines are /, m, n. We have then
I = (yy- pZ)/MFsm (MF) ; m = (aZ- 7X)/MF sin (MF) ;
n = ((3X - aY)/MFsin (MF),
if a, p, y, are the components of the magnetic force M, and X, Y, Z,
the components of the electrical force F. From these equations
it follows that la + m/3 + ny = (); lX + mY+nZ=0; l'2 + m2 + n2=l.
Hence the energy flows in a direction which is perpendicular to the
directions both of the magnetic and of the electrical forces.
If we represent by Ex, E^ Ea the components of the flow of
energy in the directions of the coordinate axes, we have
Ex = F/47T . MFsin (MF) . I
Hence we obtain the equations
Ex=T^7r.(yY-^Z}; Ey = F/47T . (aZ - yX) ; E._ = V\hr . ((BX-aY).
The energy present in a parallelepiped, whose edges are dx, dy, dz,
increases, in the time dt, by an amount equal to
- (dEJVx + VEJ-dy + VEJ-dz)dxdydzdt.
Hence the increase of energy A which a unit of volume receives
in the unit of time, is A= -(dEJ'dx + 3EJ'dy + 'dEt/'dz). If we sub-
stitute in these equations the values previously given for E& E^ E,,
we have
A = T/47T . [X(dyfdy - 3/3/3*) + Y(dafdz - 3y/3a;)
o/3y)] - F/47T . [a(dZfdy - 'dY/'dz)
- VZ/'dx) + yCdY/Vx - aX/3y)].
By the help of equations (c) and (d) of XCIIL, we obtain from this
A = KjSir . d(X2 +Y2 +
On comparing this expression with those given in (g) and (h) XCIIL
for the electrostatic and electrokinetic energies, we see that A repre-
sents the total increase of energy which the unit of volume receives
in the unit of time. Poynting's Theorem is thus proved, provided
the dielectric is not in motion. The demonstration can easily be
extended to the conductor if we use the developments of LXXXVIII.
SECT, xcvi.] POYNTING'S THEOREM. 227
and remember that a part of the energy absorbed by the conductor
is transformed into heat.
We will now apply Poynting's theorem to a simple case. According
to XCIV., we may, in the case of vibrations in a plane, express the
electrical force F, there designated by Y, by F= b . sin [2ir/jP. (t - z/w)],
and the magnetic force M, there designated by y, by
M= Fb/nw . sin [2ir/r . (t - */»)].
During any complete vibration there passes through unit area the
quantity of energy
F^/^TT/tw . rsin2[2;r/r. (t - x/o>)]dt = F^r/fywrw.
-0
We may set F=o> and /* = !, and obtain for the quantity of energy
which passes in one second through a unit area perpendicular to
the plane of the wave, the quantity F62/87r.
The quantity of heat which a square centimetre receives in one
minute from the light of the sun is equal to about three gram-calories.
This quantity of heat corresponds to the energy 3 . 4,2 . 10"/60 de-
veloped in one second. If we now set F=3. 1010, we have 6 = 0,04.
Since the unit of electrical force in the electrostatic system equals
300 volts, we obtain for the maximum electrical force of sunlight
12 volts per centimetre. The maximum magnetic force is 0,04, and
amounts therefore to a fifth of the horizontal intensity of the earth's
magnetism in middle latitudes.
The experimental basis for the mathematical treatment of electro-
statics was given by Coulomb. Poisson handled a number of problems
in electrostatics and gave the general method for their solution. Sir
William Thomson (Lord Kelvin) also treated the same problems in
part by a new and very ingenious method ; his papers are specially
recommended to the student (Reprint of Papers, 2nd Ed., 1884).
Faraday (1837) developed new views of electrical polarization or dis-
placement. On the foundation of these concepts, Maxwell constructed
his development of the theory of electricity (Treatise on Electricity
and Magnetism, 1873). Helmholtz treated electrostatics in a different
way and solved new problems. His papers may be found in
Wiedemann's Annalen.
The theory of magnetism advanced parallel with the theory of
electrostatics. The same authors, and sometimes even the same
works, deal with both subjects.
228 ELECTEICAL OSCILLATIONS. [CHAP. xi. SECT. xcvi.
Ampere in his Thforie MatMmatique des Phe"nomenes Electrodynamiques,
Paris, 1825, discussed the theory of electrical currents. This work
forms the foundation for all the recent development of that theory.
The new concepts of the magnetic and inductive actions of electrical
currents, developed by Faraday, were given a mathematical form
by Maxwell in his work : Treatise on Electricity and Magnetism, 1873.
We have followed Maxwell's methods in dealing with these topics.
On the other hand Gauss, W. Weber, F. E. Neumann, Kirchhoff, and
Lorenz, proceed from Ampere's theory.
We are indebted to William Thomson and G. Kirchhoff (Poggendorff's
Annalen, 121) for the theory of electrical oscillations in conductors.
Maxwell and Lorenz showed that electrical oscillations may also exist
in the dielectric. By the investigations of H. Hertz, the theory of
electrical oscillations has been given such extension and significance,
that no one can predict the consequences to which it may lead.
CHAPTER XII.
REFRACTION OF LIGHT IN ISOTROPIC AND
TRANSPARENT BODIES.
SECTION XCVII. INTRODUCTION.
As the number of facts discovered by the study of light increases,
and as additional relations are found between light and other natural
phenomena, it becomes increasingly difficult to construct a theory
of light. According to the emission theory, which in its main features
may be attributed to Newton, and which was handled mathematically
by him, energy is transferred by minute bodies, called light corpuscles,
which pass from the luminous to the illuminated body. It was
supposed that these light corpuscles carried with them not only
their kinetic energy but also another kind of energy, to which the
luminous effects were due. In the last century the emission theory
was sufficient to explain the phenomena then known. But its develop-
ment could not keep pace with the advances of experimental knowledge;
this became evident early in this century in connection with the
great discoveries in optics which were made by Young, Fresnel and
Malus. In opposition to this theory, Fresnel developed his first form
of the wave theory, which originated with Huygens ; in this form of
the theory, the light waves were supposed to be longitudinal.
According to the wave theory, the space between the luminous and
illuminated bodies is filled with a material medium. By the action
of the particles of this medium on each other, the energy emanating
from the luminous body is propagated from particle to particle
through this medium to the illuminated body. Hence energy resides
in the medium during the transfer of light from one body to the
other. The wave theory has many advantages over the emission
theory. Notably the phenomena of interference are explained by it
in a perfectly natural way. It succeeds also in explaining some of
the phenomena of double refraction. But the explanation of the
229
230 REFRACTION OF LIGHT. [CHAP. xn.
polarization of light by this theory offered difficulties which could be
overcome only by the assumption that the direction of the light vibra-
tions are perpendicular to the direction of the rays. Since Fresnel
retained the idea that the medium in which the light vibrations
are propagated, the ether, is a fluid, he encountered an obstinate
resistance to his new form of the wave theory ; Poisson rightly
maintained that transverse vibrations can never be propagated in a
fluid. Although the wave theory, in its original form, was some-
what open to criticism, and in many respects was insufficient, in
that, among other matters, it could not explain the dispersion of
light, yet it was a decided advance on the emission theory.
Since the phenomena of optics cannot be explained on the
assumption that light is due to vibrations in an elastic medium,
not even when this medium is supposed to be a solid, we must
endeavour to explain them in another way. Among recent efforts
in this direction, the electromagnetic theory of light, developed by
Maxwell, has special advantages. In Maxwell's view, light is also
a wave motion, but it consists of periodic electrical currents or dis-
placements, which take the place of the etherial vibrations of
Fresnel's theory. Maxwell determined on this assumption the
velocity of light in vacuo and in transparent bodies, and reached
conclusions which agree very well with the facts. Polarization
and double refraction can also be readily explained by Maxwell's
theory, and it has even been applied successfully to the study of
dispersion.
Since Fresnel's formulas are of great importance for our subsequent
study, we will develop them at the outset. We may here recall
briefly the principal laws of light which hold for isotropic and
perfectly transparent bodies. The knowledge of these laws is neces-
sary for the deduction of Fresnel's formulas, but is not sufficient.
I. Light is propagated in any one medium with a velocity which
depends on the wave length of the light, but not on its intensity.
The velocity of light has different values in different media.
II. If a ray of light falls on a plane surface, separating two
different media, both refraction and reflection occur at this surface.
All three rays — that is, the incident, the refracted, and the reflected
— lie in the same plane, which is perpendicular to the refracting
surface. If a represents the angle of incidence, ft the angle of
refraction, and y the angle of reflection, we have
7 = 0. and sin a/ sin (3 = N.
The index of refraction N is constant for homogeneous light.
SECT, xcvn.] INTRODUCTION. 231
III. If co represents the velocity of light in the medium con-
taining the reflected ray, and w' its velocity in that containing the
refracted ray, we have JV= to/to', and therefore sin a/sin /2 = w/w'.
IV. Light can be considered a wave motion in a medium, called
the ether. It is a matter of indifference whether we here consider
the bodies themselves, or an unknown substance, or perhaps changes
in the electrical or magnetic condition of the bodies. We wish
only to indicate that the luminous motion may be expressed by one
or more terms of the form a cos (2irt/T +<(>), where a is the ampli-
tude, T the period of vibration, <f> the phase, and i the variable time.
The intensity of light is then expressed by a2.
V. The motion of the ether is perpendicular to the direction of
the ray of light; that is, the vibrations are transverse. Either the
motion takes place always in the same direction, in which case the
ray is rectilinearly polarised, or two or more simultaneous rectilinear
motions may give the ether particles a motion in a curve, which is
in general an ellipse. Rays of light of this sort are said to be
elliptically polarized. If the path of the ether particle is a circle,
the light is circularly polarised. Fresnel's conception of natural light
was that its vibrations were also perpendicular to the direction of
the ray and rectilinear, but that they changed their directions
many times, and on no regular plan, in a very short interval of
time.
SECTION XCVIII. FRESNEL'S FORMULAS.
Suppose the plane surface OP (Fig. 113) to be the surface of
separation of two transparent isotropic media. We represent the
velocity of light in the medium above the surface of separation
OP by w, and that in the medium below the surface by w'. If N
represents the index of refraction of the ray of light in its passage
from the first to the second medium, we have w = JVw'. We select
the point 0 in the bounding plane as the origin of a system of
rectangular co ordinates, and draw the z-axis perpendicularly upward
and the y-axis in the plane of incidence, that is, the plane passed
through the normal to the surface at the point of incidence and
the incident ray SO. The 2-axis is therefore perpendicular to the
plane of incidence. Further, let SO be the incident, OT the reflected,
and OB the refracted ray. We designate the angle of incidence
by a, the angle of refraction by ft. The amplitude of the vibrations
232
KEFRACTION OF LIGHT.
[CHAP. xn.
of the incident ray may be called wa, and that of the vibrations of
the refracted and reflected rays u2 and u3 respectively. The planes
of vibration of these rays make angles with the plane of incidence,
which are represented by <f>v <£2, <f>3 respectively. The components
of motion along the coordinate axes are £1? tjv {v for the incident
ray ; £2, rjy £2, for the refracted ray ; £3, ?;3, £3, for the reflected ray.
It is further advantageous to introduce symbols for the components
of motion which lie in the plane of incidence and are perpendicular
to the direction of the rays. These components of motion for the
three rays are designated by sv sy ss, respectively. We then obtain
the following equations :
(a)
= -s3sma, 773 = s3cosa, VV + Cii tg'fcj = Wsv
In order to express s^ s3, and £2, £3, in terms of sl and ^, we
must make certain assumptions with regard to the behaviour of the
light at its passage from one medium to another.
I. Fresnel assumed first, that no light is lost by reflection and refraction,
or, the sum of the intensities of the reflected and refracted light is equal
to that of the incident. This law is only a statement of the law of the
conservation of energy in a particular case, it being merely the assertion
that the kinetic energy of the incident ray is equal to that of the
reflected and refracted rays. Let OPSS' (Fig. 113) be a cylinder,
the area of whose base OP is A, and whose slant height SO is equal
to w, the velocity of light. If we represent the density of the vibrating
SECT, xcvm.] FEESNEL'S FOEMULAS. 233
medium by p, the kinetic energy Lv of the light contained in the
cylinder considered is L^ = J . p . o> cos a . A . u^.
After the lapse of a second this kinetic energy is divided between
the reflected and refracted rays. The kinetic energy L3 in the
reflected ray is L3 = \ . p . w cos a . A . u32, and the kinetic energy L2
in the refracted ray is L2 = \p . o> cos ft . A . u22, when p represents
the density of the vibrating medium below the bounding surface.
Therefore, on the assumption made by Fresnel, we have
Ll = L2 + L3 or /3w(w12 — w32) cos a = pV . u22 . cos (3.
Taking into account the relations <a = N.to and sin a = N. sin /?, we
may give this equation the form
{b) p . (u-^ — u32) . sin a . cos a = p'u22 . sin /3 . cos f3.
If the vibrations of the ray lie in the plane of incidence, ul = sv
and we obtain
{c) p . (s^ - s32) . sin a . cos a = p . s22 . sin (3 . cos /?.
But if the vibrations of the ray are perpendicular to the plane of
incidence, we have u^ = £v and hence
(d) p(£i2 - £32) sin a . cos a = p . £22 . sin (3 . cos p.
II. Fresnel assumed, secondly, that the components of the vibra-
tions, which are parallel to the bounding surface, and on either side of it,
are equal. If the vibrations lie in the plane of incidence, we have,
on this assumption, ^ + ^3 = ^2' or
(e) (sl + s3) cos a = $2 . cos [I.
But if the vibrations are perpendicular to the plane of incidence,
we obtain (f) Ci + Cs = &• Ife follows from (c) and (e) that
s2 = sl . 2p . sin a . cos a/(p . sin a . cos (3 + pr . cos a . sin [3),
s3 = s^p. sin a. cos (3 - p. cos a. sin /3)/(p. sin a. cos (3 + p'. cosa. sin f3),
and from (d) and (f) that
i = d • -p • sin a . cos a/(p . sin a . cos a + p' . sin (3 . cos (3),
. = d . (p- sin a. cos a-p'. sin/3, cos (3) /(p. sin a. cosa + p. sin (3. cos (3).
III. Since the relation between p and p is entirely unknown, Fresnel
was compelled to make a third assumption, so he assumed that the
elasticity of the ether is everywhere the same, but that its density differs
in different media. On the other hand, F. E. Neumann assumed that
the density of the ether is the same in all media, but that its elasticity is
different in different media. Fresnel's assumption was natural, because
he considered the ether as a gaseous body, but this is not justified,
as has already been remarked. He further assumed that w and w'
234 REFRACTION OF LIGHT. [CHAP. xn.
can be expressed in the same way as in the theory of elasticity
[cf. XXXV. (k)], and therefore set w = V/Vp, w/ = vV'//3'- Now, on
Fresnel's assumption, p. = p, and hence, (i) p'/p = <a2/<a"2 = N2. By the
use of the third assumption the equations (g) and (h) can be given
the form
s.2 = sl . 2 cos a . sin /3/(sin(a + (3) cos (a -
I & = Ci • 2 cos « . sin /3/ sin (a
These formulas are due to Fresnel.
Experiment alone can decide as to the value of these formulas.
It follows, from the expression for s3, that $3 = 0, if a + ft = ^ir or if
tga. = N. Brewster showed that the light which is polarized per-
pendicularly to the plane of incidence, according to the definition
of Malus, is not reflected when tga = N. This value of the angle a
is called the angle of polarization. It is that at ivhich the reflected and
refracted rays are at right angles to each other. This conclusion agrees
with experiment, if we make the assumption that the vibrations of
polarized light are perpendicular to the plane of polarization. On the
whole, Fresnel's formulas agree very well with the results of experi-
ments on the intensity of the reflected light.
In the notation already employed, the plane of vibration of the
incident ray makes the angle <^ with the plane of incidence, the
corresponding angle for the reflected ray is (f>3. Brewster found that
tg<£3 = tg<£: . cos(o, - /i?)/cos (a + (3). This follows from Fresnel's formulas,
since tg<£3 = £3/ss = fi cos(a ~ P)/s\ cos(a + /*) = fcg^i • cos (a " /*)/cos (a + /*)•
This agreement argues for the correctness of Fresnel's formulas.
Fresnel assumed that the elasticity of the vibrating medium is
the same on both sides of the refracting surface. We have seen
that this assumption is to some extent arbitrary. On the other
hand F. E. Neumann assumed that p = p. We obtain on this latter
assumption from (g) and (h)
f s9 = 2 sin a . cos a . Sj/sin(a + /3),
(1) J £2 = 2 sin a. cos a. £/ sin (a + /3) cos (a -J3),
[53 = sin(a-/3).5l/sin(a + ^, & = tg (a-/3). ^/tg (a + /3).
These equations agree with the results of experiment, on the
assumption that the vibrations occur in the plane of polarization, as well
as those of Fresnel.
We will now consider the components of motion along the normal
during the passage of light from one medium to the other. This
SECT, xcviu.] FEESNEL'S FORMULAS. 235
component above the bounding surface is £x + £3, that below it is gy
It follows from (a) and (g) that
£j 4- £3 = 2/>'. sin a . cos a . sin ft . s^(p . sin a . cos (3 + p . cos a . sin /3),
£2 = 2p . sin a . cos a . sin )6 . s-^Kp . sin a . cos (3 + p. cos a. sin (3).
From these equations it follows that £i + gs = i2- P/P- If we
assume with Neumann that p = p, we have £x + £3 = £2 i ^a^ ig> ^e
components of vibration perpendicular to the bounding surface are equal
above and below it. On the other hand we obtain from Fresnel's
assumption (m) £x + £3 = N2 . £,.
Fresnel's equations agree fully with experiment only when the
index of refraction is about 1,5; it is in this case only that s3 = 0
at the angle of polarization. In other cases S3 is a minimum, but
does not vanish. Several attempts have been made to explain this
fact. Thus, for example, Lorenz assumed that the passage of the
light from one medium to another occurs through an extremely thin
intervening layer of varying density, and that therefore the change
of density is not discontinuous.
SECTION XCIX. THE ELECTROMAGNETIC THEORY OF LIGHT.
In XC1V. it was proved that electrical vibrations are propagated
in vacuo and in a great number of insulators with the velocity of
light. This fact suggests the assumption that light consists of
electrical vibrations. It was also shown that, when the wave is
plane, the electrical and magnetic forces lie in the wave front. In
the most simple case the electrical force F is perpendicular to the
magnetic force M. If the permeability //, of the medium is equal
to 1, which is approximately the case for most dielectrics, we have
(XCIV.), (a) M=NF, where N is the index of refraction.
We will now develop the usual expressions for the reflected and
transmitted light, considering first the boundary conditions. Since
no free magnetism is present, and since the electrical current
strength is everywhere finite, the magnetic force varies continuously
during the passage from one medium to another. "\Ve take the
refracting surface as the y^-plane, and the z-axis a$ the normal to it.
Hence if a, /3, y are the components of the magnetic force on one side
of the refracting surface and a', /?', y those on the other side, we have
(b) a = a',P = (3',y = y,
236
REFRACTION OF LIGHT.
[CHAP. xn.
The electrical force arises partly from induction and partly from
the free electricity on the refracting surface. Let a- denote the
density on this surface, X, Y, Z the components of the electrical
force immediately above, and X', V, Z those immediately below
this surface. We then have, from LXVL,
4ir<r = X-KX', Y=Y', Z=Z.
The components of the electrical force which are parallel to the
refracting surface vary continuously during the passage from one side
of the surface to the other.
Let SO, OB, and OT (Fig. 114) be the directions of the incident,
refracted, and reflected rays respectively. Suppose the direction of
the magnetic force to lie in the plane of incidence, and therefore the
electrical force to be perpendicular to that plane. The direction of the
FIG. 114. FIG. 115.
electrical force may be found by the rule given in XCIV. Since
the magnetic force varies continuously, we obtain, by referring to the
directions indicated in Fig. 114.
(d) Ml . cos a - Ma . cos a = Jl/2 . cos (3,
(e) Ml . sin a + Ms . sin a = M2 sin /3.
Since the electrical force also varies continuously, we have
(f) Fl + F3 = F2.
But from equation (a) M^ = Fv M3 = F3, and M2 = NF2, if N = sin a/ sin ft
is the ratio between the velocities in the first and second media.
SECT, xcix.] ELECTROMAGNETIC THEORY OF LIGHT. 237
Hence equations (e) and (f) are identical. We obtain from (f) and (d)
/ F2 = Fl . 2 cos a . sin /3/sin (a + (3) •
\JP3=-JF1sin(a-/3)/sin(a + /?).
Now let the electrical force be parallel to the plane of incidence. If the
electrical forces are positive in the directions indicated in Fig. 115,
the positive directions of the magnetic forces may be obtained by
the rule given in XCIV.
The boundary conditions are
(h) Fl . cos a + F3 . cos a = F2 . cos /?,
(i) Ml-Ms = M2.
In this case also Ml = Fl, M3 = F3, MZ = NF^, so that (k)
JFo = ^1.cosa.sin/3/sin(a + /3).cos(a-/3) and F3= -Frtg(a-p)/tg(a + (3).
Equations (g) and (k) correspond to Fresnel's equations XCVIII. (k).
Hence, the electromagnetic theory of light leads to the same results as those
which are contained in Fresnel's formulas, provided that the electrical force
is parallel to the direction of the vibrations assumed by Fresnel.
We can further show, by the help of Poynting's theorem (XCVL),
that the energy which in a given time is transported to the refracting
surface by the incident ray is equal to that which is carried away
from it in the reflected and refracted rays. Since in this case the
electrical and magnetic forces are perpendicular, the energy passing
through unit area equals VMF\±Tr. The bounding surface S receives
the quantity of energy 1/47T. PrMlFl . S cos a. in unit time. We may
write similar expressions for the energies of the reflected and trans-
mitted rays, and have the relation
1 , 4vr . VM^ . S . cos a = l/4;r . VM.2F^ .S.cos(B + l/47r . VM3FS .S.cos a,
or (M^ - M3FS) . cos a = M.2F2 . cos /3. By reference to the relations
between the electrical and magnetic forces, we obtain
(^2 _ ^CQS a = NF* . cos /3.
It appears from equations (g) and (k) that this equation is satisfied
if the electrical force is either perpendicular or parallel to the plane
of incidence.
SECTION C. EQUATIONS OF THE ELECTROMAGNETIC THEORY OF
LIGHT.
If we at first consider only bodies in which there is no absorption
of light, and in which the velocity of light is the same in all directions,
238
EEFRACTION OF LIGHT.
[CHAP. xii.
we have, from XCIII. (e), the differential equations of the electrical
force,
t l/o>2 . WXfdt* = VIST, 1/w2 . 32F/9£2 = V-Y,
\ l/w2 . wzrdP = V2Z, -dXfdx + ^Yj-dy + VZ/Vz = 0.
The boundary conditions are obtained by remarking that the com-
ponents of the electrical and magnetic forces parallel to the bounding
surface are equal on both sides of it. Therefore, if the z-axis is
perpendicular to the refracting surface, we have
<b) Y=Y', Z=Z'; /? = /?', y = y'.
The last two conditions (b) may be put in the form [XCIII. (e)],
/ -dXj-dz - -dZfdx = VX'fdz -
FIG. 116.
Let us suppose that a plane wave moves in a direction which
makes angles with the axes whose cosines are I, m, n. Let the elec-
trical force / at the origin be expressed by f=F.cos(2irt/T). The
electrical force at a point whose coordinates are x, y, z is then
/= F. cos[27r/r(^ - (Ix + my + n«)/o»)].
If the direction of the electrical force makes angles with the axes
whose cosines are A., p., v, we have X=\f, Y=pf, Z=vf. These
expressions satisfy the equations (b) ; that they may also satisfy the
last of equations (a) we must have l\ + mp. + nv = 0, that is, the direc-
tion of the electrical force is perpendicular to the direction of propagation.
Let OP (Fig. 116) represent the refracting surface, and SO, OB
and OT the incident, refracted, and reflected rays respectively. The
SECT, c.] EQUATIONS OF ELECTROMAGNETIC THEORY. 239
system of coordinates is drawn in the same way as in XCVIII. For
the incident wave, in which the direction of the electrical force lies in
the plane of incidence, we can set
I = - cos a, m = sin a, n = 0 ;
X= sin a, ju, = cosa, v = 0.
We have, for the reflected rays,
/ = cos a, m = sin a, n = 0 ;
A=-sina, fji = cos a, i/ = 0,
and for the refracted rays,
I = - cos (3, m = sin /?, n = 0 ;
A = sin /?, fi = cos f3, v = 0.
If the direction of the electrical force is perpendicular to the plane of incidence
we have A = 0, M = 0, v = 1 .
If Fv F2, F3 are the electrical forces in the incident, refracted, and
reflected rays, when they lie in the plane of incidence, and if Zv Zz, Zz
are the electrical forces in the same rays when they are perpendicular
to the plane of incidence, we have
/ X=Fl.sina.cosTl, Y= F1 . cos a. cos Fv Z = Zl.cosF'l,
\ F! = [27T/T .(t-(-xcosa + ysin a)/a>)].
We obtain for the refracted ray, if w' is the velocity of light in the
second medium,
= ^2.sin/3.cosF2, F= F2 . cos /3 . cos T2, Z=Z.cosT
(d) l rr, = [27r/r.(^-(-a;cos^ + ysm/3)/o>')].
For the reflected ray we have
r=^3.cosr3,
1 ^ j = [2-ir/T . (t - (x cosa + y sin a)/o>)].
To simplify the calculation we replace the trigonometrical form
(f) cos£(^- ( - z cos a + y sin a)/o>)
by the expression (g) ^(M-scosa+ysineO/a^ where i = \/- 1 and k=%Tr/T'
In the final result we use only the real part of (g), namely (f). Both
expressions satisfy the same differential equation, and therefore in
calculations one of them may be replaced by the other.
If the refraction occurs at a plane surface, we may replace the
expressions (c), (d) and (e) by the following :
X=Fl.$ina.
(b)
*,<
240 EEFEACTION OF LIGHT. [CHAP. xn.
, X=Fa.s\n(3.eki(t-<--*coal3+ys
(i)
{X= — F3 . sin a . «**{*-(*
Y=F3.cosa.eki(t-<-XC(iS
£ — £ gki(t-(xcosa+ysin
3 *
a+y sin
These equations express the components of the electrical force for
the incident, refracted, and reflected rays.
In order to satisfy the conditions (b) and (c) it is necessary that
sin a/a> = sin /3/w'. Since the velocities of propagation w and w' are
constant, we can set N= sin a/sin (3, where N is the index of refraction.
From equations (b) we have
(1) (Fl + F3)cosa = F.2.cos(3; Z1 + Zs = Zy
From (c) we obtain
(m) (Fl-F3)siu/3 = F2.sina- (Zl - Z3)cosa. sin/? = ^2. sin a. cos (3.
From (1) and (m) we obtain Fresnel's equations [XCVIII. (k)] for
the reflected and refracted waves. The problem is solved when (3
is not imaginary. /3 becomes imaginary when sin /?> 1, and therefore
when sin a > N. In this case we must use the complete expressions
(i) and (k).
If the electrical force is perpendicular to the plane of incidence, the
reflected wave is determined by the real part of the expression
(n) - Zl . Sin(a - /3)/sin(a + /?) . ei-i(<-(xcosa+ysina)/W)>
We get this expression by the use of the last of equations XCVIII.
(k). But since cos(3 = Jl - siri2a/N'\ and therefore
(o) Ni . cos /3 = Jsirfa-N12
we have
- sin (a - (3) /sin (a+(3) = (cos a + i*Jsin?a - N2)/(cos a - K/sin-a - ,/V2).
If we now set
(p) cos a = C . cos Jy, J sin2a - N'2 = C . sin |y,
we obtain
(q) tg | y = N/ sin'2a - N2/cos a, C?=l-N2;
(r) -sin(a-/3)/sin(a + £) = ^.
Hence the real part of the expression (n) is
(s) Z^ . cos [k(t -(xcosa + y sin a)/w) + y].
In this case the reflection is total, since the component Zl appears
in the expression for the incident as well as in that for the reflected
SECT, c.] EQUATIONS OF ELECTROMAGNETIC THEORY. 241
wave. But while, in the case of ordinary reflection, no difference
of phase arises between the two waves, we have in this case a difference
of phase 7, which may be determined from (q).
If the electrical forces far the incident wave are parallel to the plane of
incidence, we determine the real part of the expression
(t) - Fl . tg(a - /3)/tg(a + /2) . ett(«-Cscosa+3/sina)/w).
Using equation (o), we have
tg(a - /2)/tg(a + /3) = (TV2. cosa
If we set
(u) JV~2.cosa
so that (v) tg £8 = Vsin'2a - N2/N2 cos a,
we obtain (x) tg(a- /3)/tg(a + p) = eiS. Hence the real part of the
expression (t) is (y) -F1cos[k(t- (zcosa + ysina)/w) + S].
The reflection is therefore total. To determine the difference of phase
8 between the reflected and incident waves, we may use equation (v).
We obtain from (q) and (v) tg |(8 - y ) = V sin2a - N2/ sin a tg a. If a0
is the limiting angle of total reflection or critical angle, we have N = sin <x0,
and hence (z) tg|(8- y) = N/sin(a + a0). sin(a-a0)/smatga.
Since 8 and y are not equal, a linearly polarized ray of light, in
which the vibrations make any angle with the plane of incidence,
is elliptically polarized after reflection.
If the electrical force is perpendicular to the plane of incidence, the
transmitted light is determined by the real part of the expression
(a) Z±. 2cosa
Referring to (p), we have
2 cos a sin /?/ sin (a + /?) = 2 cos a/ C .
and ( - x cos /? + y sin fB)/<o' = (ix-Jsin^a -N2 + y sin a)/o>.
Hence the real part of (a) is
(/3) 2 cos a/(7 . etov/sini!a-^/w . zi . cos [k(t - y sin o/u) + ly].
Since C2 = 1 - JV2, we obtain 4 cos2a/(l - JV2) . Z* . (**#/***>•- W\t for
the intensity of the transmitted light, where X denotes the wave length.
The expression shows that, in this case also, a motion exists which
corresponds to the refracted ray in the case of ordinary reflection;
it is, however, appreciable only within a very small distance from
the refracting surface.
Similar results are obtained in the investigation of the refracted
ray if the electrical force of the incident light is parallel to the plane of
Q
242 REFRACTION OF LIGHT. [CHAP. xn.
Remark: In order to obtain the real part of an expression of the
form (n) we may use the following method. The expression (n) is
thrown into the form
(A + Si) . 0* = (A + Bi)(cosV + i sin ¥).
The real part of this is R--=A . cos^- B. sin^. Now if we set
A = C.cosy, B = C.siuy, we have (y) li= C. cos fF + y), where 0
and y are determined by
f
|tgy=
The expression (n) then takes the form
Z^ . (cos a + iv/sin2a - JV2)/(cos a - ijsin'2a - N'2).
In the case considered, therefore,
A+Bi = Zl. (cos a + i\/sin2a _ JV2)/(cos a - z'v/sin-a - N'2).
If + i and — i are interchanged, AVC obtain
A - Bi = ^(cosa - iv/sin-a — N'2)/(cos a + iv/sin2a-
By multiplication of the two expressions we obtain C~
From (8) it further follows that
tg y = 2 cos o\/sin2a — N2/(co& 2a - sin 2a + N'2) .
This equation may be also obtained from (q).
SECTION CI. REFRACTION IN A PLATE.
We will consider the case of a plane wave of light falling on a
plane parallel glass plate, whose thickness is a and whose index of
refraction is N. We can determine the intensities of the reflected
and transmitted light in the following way. We choose one surface
A of the plate as the y^-plane, and draw the positive a-axis outward
from this surface. Let a represent the angle of incidence, /5 the angle
of refraction, w and a/ the velocities of the light inside and outside
the plate. A part of the refracted ray is reflected toward the surface
A at a point E of the surface B. This part is again divided at the
surface A, a part of it passing through that surface in the direction
FG, while the other part is again reflected toward B. The light
is thus reflected within the plate repeatedly. Since the plate is
bounded on both sides by the same medium, the angle of exit is
equal to the angle of incidence a. Now plane waves which move
in the same direction may be compounded into a single plane wave.
SECT. CI.]
REFRACTION IN A PLATE.
243
Besides the incident wave we have to consider four others, namely,
the wave reflected from A, the wave passing through B, and two
waves in the plate itself.
I. We will first consider the case in which the electrical force of the,
incident wave is perpendicular to the plane of incidence. The component
of the electrical force outside A is expressed as in the former para-
graph by
FIG. 117.
Similar expressions hold for the component Z' of the electrical
force in the plate, which are obtained by replacing a by /?, u> by to',
and introducing the new constants Z^ and Z±. Thus we obtain
Z' = Z2 . gW(«-(-*«M|8+y8ii»j8)/w') + Z^ . gKft-fccos/S+if sinjSyW).
We have for the component Z" of the transmitted ray
Z" = Z-3 . el'l(l - ( - x cos °-+y sin a>/w).
The boundary condition Z=Z' when x = Q, gives (a) Zl + Z3 = Z2 + Z4.
Similarly Z' = Z" when x= -a, or
Z2.e~ kia • cos jS/w + Z± . ekia • cos /3/w/ = Z6.e~ kia • cos a/w.
Now if we set ka . cos /5/w' = ?/, ka . cos a/w = v, we can write the last
condition in the form (b)
+ Z± . eui = Z5 . e~vi.
We have, further, when x = 0, ?>Z'/'dx = 'dZ/'dx, and similarly, when
x = -a, 'dZ'/'dx = 'dZ" ' fdx. These equations of condition give
(c) (Zl - Z3) cos a . sin /? = (Z2 - Z4) sin a . cos /3
and (d) (Z2 . e~ui - Z± . eui) sin a . cos ft = Z5 . e'". cos a . sin /?. It follows
from (b) and (d) that
ZJZ.2 = e-2i". sin (a - /3)/sin(a + /5),
244 EEFEACTION OF LIGHT. [CHAP. xn.
or if sin(a-/3)/sin(a + /3) = €, ZJZ^ = t . e~*ui. From (a) and (c) we
have also Z3/Zl = (- eZ2 + Z4)/(Z2- fZj, and therefore
(e) Z3/Z, = - («* - O/(V« • *ui - e • O-
If N is greater than 1, u is always real, and we may set
ZJZi = - 2w . sin «/[(! - €2)cos M + (1 + e2)f . sin «].
Designating the intensity of the reflected light by C2, we obtain,
by the method indicated at the end of C.,
C* = Zl*. 4e2. sin2«/[(l - e2)2 + 4e2. sin2tt].
But because k = 2-!r/T = -2ino/\ and u = 2-nr . N. cos /3 . a/A, it follows that
/f) (CZ = Z*. 4e2. sin2(27rJV. Cos /3 . a/A)/[(l - £2)2
\ + 4«2. sin2(27rAr. cos 0 . a/A)].
Hence no light is reflected if 2-n-N . cos /3 . a/A =pr, where p is a
whole number. This result is of special importance in the study
of Newton's rings.
On the other hand, if N<1 and at the same time sma>N, [3 will
be imaginary. In this case we can no longer use equation (f). We
then have [C. (o)] Ni . cos (3 = >/sin2a - N'2, and hence
ui = 2ira/X. . *Jsiri2a-N2.
If we set m = ui and e= -e+{7 [C. (r)], equation (e) takes the form
Z3/Z, - (e-- e-™)/(e>"-v - «-*+*);
Designating by C2 the intensity of the reflected light, we obtain in
the same wajr as before,
(g) C2 = Z*. 1/[1 + 4 sin2y/(em - e-™)2],
where tg |y = >/sin2a - 7V2/cos a and m = 27ra/A. . Vsin2a - JV2.
The relations which we have here considered occur in the case
of two transparent bodies which are separated by a layer of air. If
the thickness of the layer of air is very much greater than the
wave length of the light, total reflection will occur. This is in accord
with equation (g), which in this case gives C2 = Zl2. On the other
hand, if a is small in comparison with the wave length, all the light
passes through the layer of air. In consequence of this a black spot
is seen if the hypotenuse of a right-angled glass prism is placed on
the surface of a convex lens of long focus. If the angle of incidence
a in the glass prism is less than the critical angle, a dark spot appears
surrounded by coloured rings ; but if the angle of incidence is greater
than the critical angle, the rings disappear while the spot remains.
SECT, ci.] KEFRACTION IN A PLATE. 245
The spot is larger for red than for blue light. This result is contained
also in the expression for the intensity of the reflected light. The
transmitted light is complementary to the reflected light.
II. If the direction of the electrical force of the incident light is parallel
to the plane of incidence, the disturbance outside the surface A is
determined by
X=Fl . Sin a . ei-i(«-(-*«>sa+y8ina)/W)
- Fs . Sin a . gK(t-(*coea+y sinetf/w),
Y= Fl . COS a . eki(t - ( - * cos a+y sin o)/w)
4 7^3 . cos a . eki(l ~ (x cos a+y sln a)M.
The disturbance inside the plate is given by
X' = FZ. Sin ft . ««(«-(-* cos p+y sin 0)/w)
- F4 . Sin ft . 6«('-(* cos/3+y sin/3)/W))
F = 1*2 . COS 0 . €**(« -<-* c°s/3+y sin/3)/u>')
+ /^ . COS /3 . ««(*-(* cosp+y sin £)/&>') .
and outside the surface .5 by
X" = FK . Sin a . gti{«-(-arcoso+y sinoyw^
Y" = F-0 . COS a . ^('-(-^cosa+J/sinaVw).
We must now determine the constants ^ J^, jF4, Fb. When
x = 0 the boundary conditions give F= F, or
(h) (^ + .F3) cos a = (F2 + F4) cos /8.
Similarly, when z= -a,
,F2 . cos ft . e~kia • co80/w + ^4 . cos ft . ekia • ™spl"' = F5.cosa. e~kia • cos a/w.
Using the same notation as before, we have
(i) F2 . cos ft . e~ui + F± . cos ft . eui = f5 . cos a . e-"'.
We have, further, when # = 0,
or (k) (^ - F3) sin ft = (F2- Fj sin a.
The same condition holds when x = - a, or
(1) f^ . sin a . e-"1 - .P4 . sin a . e'" = ^ . sin ft . r«.
We obtain from equations (i) and (1) FJF2 = e~-ui . tg(o-j8)/tg(a + )8).
But if we set tg(a-/3)/tg(a + /3) = e', we will have FJF2 = ^.e~M.
It follows from (h) and (k) that ^/^ = ( - e' + e' . e-a"«)/(l -£'2.e-2'»),
or (m) ^3^= -(e!(i-e-ui)/( 1 /«'•«"* -e'.e~Mi)- We thus obtain the
intensity Z>2 of the reflected light in the same way as we obtained
the expression (f) from (e),
™ „„ ..
1 ' ( I - e' 2)2 + 4e a . sina(2^V cos /3a/ A)'
246 REFRACTION OF LIGHT. [CHAP. xn.
If sin a > N and if (3 is therefore imaginary, we obtain the in-
tensity of the reflected light in the following way : We have
€ = tg(a- /?)/tg(a + f3) = e8i, if, as in C. (u), we set
JV2 . cos a = Z) . cos JS, >/sin2a -N'2 = D. sin |3.
If we further set ui = m, it follows that
The intensity D2 of the reflected light is then
(o) Z>2 = FS . l/( 1 + 4 sin28/(«w - e~mf-),
in which expression
tg |<5 = x/sin2u - N'2/N2 cos a, m = '2-n-a/X .
The expressions (n) and (o) for the intensity of the reflected light
when the direction of the electrical force is parallel to the plane of
incidence, lead to essentially the same results as equations (f) and
(g), which hold when the direction of the electrical force is perpen-
dicular to the plane of incidence. We only remark that, from
equation (n), D- vanishes if e' = 0 or (a + /2) = |TT. In this case the
angle of incidence is equal to the angle of polarization.
SECTION CII. DOUBLE REFRACTION.
Up to this point we have assumed that the value of the dielectric
constant K is independent of the direction of the electrical force.
Boltzmann, however, has shown that the dielectric constant of crystals
has different values in different directions, and depends on the
direction of the electrical force. Let Kv K2, JT3 represent the value
of the dielectric constant in three perpendicular directions which
are those of the coordinates a;, y, z. Then in place of equations
XCIII. (a), we use
(a) u = Kl/4v.'dXI'dt, v = K2/4ir .'dY/dt, w = K3/4ir . 'dZ/'di.
Equations XCIII. (d) and (c) become
KJ V. 'dX/ot = 3y/3y - 3/2/3,2, A'2/ V. 3 Yfdt = 3a/3^ -
K3/ V. 'dZj'dt = 3£/3* - 3a/3y,
and if we set the magnetic permeability /* = 1, we have
- I/ V. 3a/3* - *dZfdy - 'dY/'dz,
-\\V. 3/?/3* = 3JST/32 -
-\\V. 3y/3* = 3F/3.C -
SECT, cn.] DOUBLE REFRACTION. 247
Further, if we set
we obtain
f I/a2 . -
(b) J 1/42. 327/9*2
We will consider a plane wave moving through a body to which these
equations apply. Its direction of propagation is determined by the
angle whose cosines are I, m, n; the direction of the electrical force
/ is determined by the angle whose cosines are A, /*, v. We then
have
(c) X=\f, Y=rf, Z=vf, f=F.cos[2ir/T.(t-(lx + my + nz)/u>)'].
F is constant, and the velocity of propagation w depends only on
the direction in which the wave is propagated. It follows from (c)
that V2A"= - 47T2A//T2<o2, and if cos 8 = /A + m/* + nv, we obtain
(d) /= L>7r/ro) . F . cos 8 . sin [2ir/r . (t - (Ix + my + ««)/«)].
We obtain from the first of equations (b), (d') A - I. cos 8 — w2 . A/a2.
This equation and the two similar to it take the forms
(e) (a2 - <o2)A = a2/ . cos 8, (b2 — or)/* = b2m . cos 8, (c2 — «2)v = c2n . cos 8.
We use these equations to obtain the physical meaning of the
magnitudes a, b, c. If w = a we have either 1 = 0 or cos 8 = 0. In
the latter case //, = v = 0 and A= ±1 and therefore also 1 = 0. Hence
a plane wave parallel to the x-axis is propagated with the velocity
a when the electrical force is parallel to the same axis. The meaning
of the magnitudes b and c is obtained in a similar way. By the
optical axes of elasticity we mean the three directions in a body
which have the property that a plane wave, in which the electrical
force or the direction of vibration is parallel to one of the axes,
for example a, is propagated with the velocity a in all directions
perpendicular to the axis a.
We can find the velocity of propagation and the direction of the
force from equations (e) and (d), in connection with the relation
A2 + p.2 + v2 = 1. If equations (e) are multiplied by /, m, n, respectively,
and added, it follows from (d) that
a2Z2/(a2 - w2) + 62m2/(Z>2 - w2) + c%2/(c2 - w2) = 1.
For brevity we will write for this equation 2a2£2/(a2 - w2) = 1 . But
because a2 = a2 - w2 + w2, we also have
- w2) = 2/2 + 2a>2/2/(a2 - a>2) = 1.
248
REFRACTION OF LIGHT.
[CHAP. xn.
Since 2Z2 = 1, it follows that
(f ) I2 /(a2 — (o2) + m2/(i2 - o>2) + %2/(c2 - or) = 0.
This equation is of the fourth degree in w. Since two of its roots
are numerically equal to the other two but of opposite sign, the
electrical wave has two velocities of propagation, u^ and 0)3.
We may give equation (f) the form
(g) <»*-(l2(b2 + c2) + m2(a2 + c2) + n2(a2 + b2))u2 + I2b2c2 + mW + n2a2b2 = 0.
If 1 = 0, that is, if the plane wave is parallel to the z-axis, we have
The roots of this equation are w1 = a, o>2 = *Jm2c2 + n2b2.
This result can be represented by drawing lines in the yz-plane
from the point 0 (Fig. 118), which are
proportional to the velocities of propaga-
tion. The ends of these lines then lie
on two curves, one of which is given by
w1 = a, and is a circle; the other is given
by w2, and is an oval. If a > b > c, the
minor semi-axis c of the curve given by
o>2 lies in the y-axis, and its major semi-
axis b in the £-axis. The relations of the
" plane waves which are parallel to the y-
and £-axes respectively, are given in Figs.
119 and 120. The relation in the axz-plane is especially peculiar
(Fig. 119). In that case, we have, for m = 0, <al = b, o>2 = v7/-c- + 7i2a2
c
FIG. 118.
O tax.
FIG. 119.
and l2 + n?=l. The direction of propagation in which the two
velocities Wj and o>2 are equal is given by
/= ±J(a2-b2)/(a2-c2) ; n =-- ± J(b2 - c'2)/(a2 - c2).
SECT, cm.] VELOCITIES OF PROPAGATION. 249
SECTION CIII. DISCUSSION OF THE VELOCITIES or PROPAGATION.
If Wj and o>2 represent the two velocities of propagation of the
same plane wave, we have [CIL (g)]
(a) 2
{ Wl2 . w22 = We2 + m2a?c2 + n2a2b2,
and (Wl2 - o>22)2 = (I2(b2 + c2) + m2(a2 + c2) + n2(a2 + b2))2
- 4(/W + m2a2c2 + n-a2b2).
If we multiply the last term on the right side of this equation by
I2 + m2 + ri2 = 1 we obtain
1 - c2)2 + m4(c2 - a2)2 + 7i4(a2 - b2)2 + 2m2n2(a2 - b2)(a2 - c2)
' - c2)(b2 - a2) + 2l2m2(c2 - a2)(c2 - b2).
If a > b > c, it follows that
(M^ - o>22)2 = l*(b2 - c2)2 + m*(a2 - c2)2 + n\a2 - b2)2 + 2m2n2(a2 - b2)(a2 - c2)
- 2l2n2(b2 - c2)(a2 - b2) + 2l2m2(a2 - c2)(b2 - c2),
or
(Wl2 - o>22)2 = (I2(b2 - c2) + m2(a2 - c2) + n2(a2 - b2))2
(b)^ -^v-^xs2-^2),
Hence the two velocities ^ and w2 are equal for certain directions
of the wave normals. This equality exists when m = 0 and either
Ijb2 - c2 + nja?^b'2 = 0, or ijb2^2 - W«2 - 6s = 0.
These conditions are satisfied by m = Q and Z/»= ±J(a'2-b-)/(b2 - c2).
These equations represent four directions, which are parallel to the
rc^-plane and perpendicular to the axis of mean elasticity b. If we
represent the cosines of the angles made by these directions with
the coordinate axes by 10, m0, w0, we have
(c) m0 = 0, I0=±j(a2-b2)/(a2-c2), n0 = ± J(b2 - c2)/(a2 -I2).
We call the directions in the crystal, defined by equations (c), the
optic axes. There are two such axes, since each of these equations
represents two opposite directions.
If Oa and Oc (Fig. 121) represent the axes of greatest and least
elasticity a and c, and if OAl is one of the directions in which wx
and w2 are equal, they are equal not only in the opposite direction
OB but also in the directions OA2 and OB2, if OA2 makes the same
angle with Oa as that made by OAr
250
REFRACTION OF LIGHT.
We will now express the velocity of propagation in any direction in
terms of the angles made by this direction with the optic axes OAl
and OA 2. The cosines of the angles which the direction of propagation
of the plane wave makes with the axes are Z, m, ?<. We then have
f cos El = / . V(a2 - 6*)/(tt2 - c2) + n . V(&2 - c2)/(a2 - c2),
(c')
From this it follows that
(21 = (cos El + cos Ez) . >/(a2 - c2)/(a2 -
1 2« = (cos El - cos E2) . J(tf^
and
2 - c2).
FIG. 121.
If we eliminate m from equation (a) by the help of the equation
P + m2 + n'2=l,w6 obtain o^2 + o>22 = «2 + c- - l'2(a2 - b-) + n-(b- - c2), from
which we obtain by use of equation (d),
(e) <a* + o>22 = a2 + c2 - (a2 - c2) . cos E1 . cos ^2.
From equations (c') we obtain
(a2 - c2) . sin2^ = a2 - c2 - Z2(a2 - ft2) - w2(62 - c2) -
(a2 - c2) . sin2£2 = a2 - c2 - /2(a2 - 62) - n2(&2 - c2) + 2/n .
Further, since Z2 + m2 + ri2 = 1 , we also have
Z2(62 - c2) + m2(a2 - c2) + n2(rt2 - i2) = a2 - c2 - ^(a2 - ft2) - rc2(62 - c2).
By help of these relations we obtain from the first of equations (b)
(f ) w/2 - w22 = ± (a- - c2) . sin El . sin E2, and from (e) and (f ),
/ 2Wl2 = a2 + c2 - (a2 - c2) . cos(^1 - E2),
\ 2o>22 = a2 + c2 - (a2 - c2) . cos (E^ + E.2).
The greatest value of the velocity of propagation is a and the least c.
This follows if we set El = E2 and El + E2 = 7r. If the normal to
the waves is parallel with one of the optical axes, for example OAlt
we have El = 0 and cos \E.2 = /0, and hence Wl = o>2 = b. The velocity
of propagation is then equal to the axis of mean elasticity.
SECT, civ.] THE WAVE SURFACE. 251
SECTION CIV. THE WAVE SURFACE.
Suppose a plane wave to start from the origin of the system of
coordinates, in the direction in which its normal makes angles with
the axes whose cosines are I, m, n. After the lapse of a unit of time,
the distance of the wave from the origin is co. If about each of the
points of the plane wave we construct a wave surface as it would
appear after the lapse of unit time, the plane wave thus propagated
is the envelope of all the wave surfaces. If x, y, z are the coordinates
of a point of the plane wave in its new position, we have
(a) Ix + my + nz = (o,
and co is determined by
(b) Z2/(a2 - co2) + m2/(b2 - co2) + n2 (c2 - co2) = 0.
AVe have, further, (c) l2 + m2 + n2=l. If I, m, n, and co vary, the
planes (a) envelope a surface, which is called the wave surface. Hence
if we consider all possible plane waves passed through a point, and if we
determine the position of the same waves after unit time, the wave surface
is the envelope of all the plane waves thus determined. We will now
investigate the equation of this wave surface. AVe obtain from
(a), (c), and (b),
(d) x . dl + y . dm + z . dn = rfto,
(e) I . dl + m . dm + n . dn = 0,
(f ) I . dl/(a2 - co2) + m . dm/(b2 - co2) + w . dn/(c2 - to2) + Fu . da> = 0,
where (g) F= Pj(az - co2)2 + m2/(&2 - to2)2 + 7i2/(c2 - co2)2.
If we eliminate e?co by means of (d) from equation (f), we have
[Ftax + //(a2 - a>2)]d/ + [F<ay + m/(b2 - u2)]dm + [Fva + n/(c2 - ta2)]dn = 0.
We add, to the left side of this equation, equation (e) multiplied by
a factor A. Since dl, dm, dn may be considered as arbitrary quantities,
we have
00
If these equations are multiplied in order by /, m, n respectively and
added, we obtain, by reference to (a) and (b), A = - Fur. Therefore
ra2 _ W2j = Fu(lu - x), m/(b'2 - to2) = Fn>(mo) - y),
n/(c2 - to2) = Fwtyw - z).
If we square both sides of these equations, add them, and use equation
(g), we have 1 = /V(co2 - 2co(£c + my + nz) + r2), in which r- = :
252 REFRACTION OF LIGHT. [CHAP. xn.
Further, by reference to (a) we obtain (k) F<^(r2- w2) = l. F may
now be eliminated from equations (i) by means of k, and we have
a1 - o>2) = Z<o(a2 - r2), y(V- - to2) = mo>(Z>2 - r2),
These equations enable us to determine the point of contact between
the wave surface and the plane wave, and therefore the direction of
propagation of the ray. The plane wave moves in the direction
determined by Z, m, n.
If we multiply both sides of equations (1) by x, y, z respectively,
and add, we have
z2(a2 - w2)/(a2 - r2) + yz(b* - o>2)/(62 - r2) + z2(c2 - w2)/(c2 - r2) = o>2,
since by (a) Ix + my + nz = w. This equation may be written in the
abbreviated form 2z2(a2 - w2)/(a2 - r2) = w2, or in the form
Sr2(a2 - a>2)/(a2 - r2) = 2*2(a2 - r2 + r2 - a>2)/(a2 - r2)
But since in this notation 2z2 = x2 + y2 + z2 = r2, we have finally
(r2-w2)(l+2z2/(a2-r2)) = 0.
?%e equation of the wave surface is therefore
- r-) + 1-0.
But because
Sa^/r2 = 1 and 2(x2/(a2 - r2) + z2/r2) = 2a2.c2/(a2 - r2) = 0
we may also write the equation of the wave surface in the form
(m) aV/(a2 - r2) + &y/(62 - r2) + c%2/(c2 - r2) = 0.
We can easily transform this equation into
(n) (aV + jy + c2*2)r2-(62 + c>V- (a2 + c2)&V-(a2 + 62)c252 + «2i2c2 = 0.
7%g equation of the wave surface is therefoi'e of the fourth degree. In
order to investigate this equation we set x=fr, y = gr, z = hr. By
substitution of these values in the equation of the wave surface,
it becomes
2,-2(a2/2 + bY + c%2) - [(fr2 + c>2/2 + (a2 + c2)62^2 + (a2 + 52)c2A2] = ± R,
From which we get
B* = [(a2 - c2)&V2 + (afJW^c* +
x [(a2 - c2)6V + (a/v/^^c2 -
Hence a straight line drawn from the origin of coordinates cuts
the surface in two points, which coincide when R = Q or when
(o) .9 = 0 and f/h = ±c/a. J(a'-b'2)/(b'2 -c2).
SECT. CIV.]
THE WAVE SURFACE.
253
In this case
f=±e/b. V(«2 - &2)/(tt2 -~c~2), h = ± ajb . *J(b2 - c2)/(a2 - c2).
There are therefore four such points in the wave surface, all of
which lie in the o^-plane. Hence the wave surface is a surface of
the fourth degree with two nappes. The four points which the
two nappes have in common are called umbilical points.
To exhibit the form of this surface we will determine the curves
formed by the intersection of the wave surface, and the coordinate
planes yz, xz, yx. If, for this purpose, we set in equation (n) z = 0,
y = 0, 2 = 0 successively, we obtain
- Z>2c2) = 0,
2c2) ^ Q,
W) = 0,
Hence the curves formed by the intersection of the wave surface
with the coordinate axes are circles and ellipses, as represented in
figures 122, 123, and 124. The curves in the o^-plane are of
9
FIG. 122.
b
FIG. 123.
c b
FIG. 124.
special interest. The equation zz + x2 = b2 represents a circle of
radius b. The equation c2£2 + a?x2 - a?c2 = 0 represents an ellipse
whose semi-axes are a and c. On the assumption that a>b>c, the
circle and the ellipse intersect at a point P, and this point is one
of the umbilical points.
Equations (1) and (h) serve to determine the coordinates of the
point of contact between the wave surface and a plane wave which
moves in a direction determined by I, m, n.
The case in which the wave is propagated in the direction of one of
the optic axes is of special interest. In this case [CIIL], the velocity
equals b, and the direction of propagation is given by the equations
since we here consider only that optic axis which lies between the
positive directions of the z- and a:-axes. Equations (1) then become
z(a2 - b2) = lb(a2 - r2), z(b* - c~) = nb(r2 - c2).
254 REFRACTION OF LIGHT. [CHAP. xn.
If we introduce in these equations the values for / and n given
above, we have
(p) xj(a2 - 62)(a2 - c2) = b(a2 - r2), W(&2 - c'2)(a* - c-) = b(r* - c2).
These equations represent two spheres, in whose lines of intersection
lie the points of contact of the wave plane and the wave surface,
therefore in this case the plane of the wave touches the wave surface in
a circle.
We may also obtain this result in the following way. By the
use of equations GUI. (c), we give (p) the form
(q) x = b(a*-x2-f-z*)/l0(a2-c>-), z = b(x* + y2 + z'2-c2)/n0(a2-c2).
The curve represented by these two equations is a plane curve
because (r) xl0 + zn0 = b.
We now introduce a new system of coordinates with the same
origin ; suppose the ?/-axis to coincide with the y-axis, while the
£axis coincides with the optic axis. To effect this, we set
(s) z = £ra0 + $o> y = >?, z= -S/o + fV
The equation (r), which represents a plane, then becomes (t) £=b,
that is, the plane of the curve of intersection is perpendicular to the
direction of the optic axis and passes through its end point. The first
of equations (q), by the use of (s) and (t), takes the form
(u) f + ^Vo(rt2-<;2)/Z' + '?2 = 0•
This represents a circle, which passes through the point £ = 0,
*7 = 0, and £=&, or through the end point of the optic axis. The
radius r of the circle is r = >J(b2 - c2)(a2 - b'2)j 2b, and the coordinates
of its centre are £=—?', t] =• 0, £= b. Thus the circle is determined
in which the plane perpendicular to one of the optic axes at its
end point touches the wave surface.
SECTION CV. THE WAVE SURFACE (continued).
Let ON (Fig. 125) be the normal to a plane wave; the direction
of the normal is determined by the cosines /, m, n. Let OPl and
OP2 be the two velocities of propagation of the wave considered.
Let Ql and Q2 represent the points of contact between the plane
wave and the wave surface. We then have OQl = rl and 0$0 = ?V
We represent the coordinates of the points Ql and Q2 by xv yv z^
and x2, y2, z2 respectively. If QlPi=pl and Q.f^p^ are the per-
pendiculars let fall from the points of contact on the directions of
propagation, we have
SECT. CV.]
THE WAVE SURFACE.
255
A
The connection between the direction of the normal and the points
of contact is given by equations (1) CIV. We will investigate more
particularly the directions of the lines p1 and p2. The projection
of PQ on the a-axis is ul-x. If we
represent the cosines of the angles which
p makes with the axes by A.', /*', v', we
will have
A' = (wZ - x)/p, p! = (<am - y)/p,
v' = (wn-z)/p.
Introducing in this equation the values of
x, y, z, given in CIV. (1), we obtain
(a) A' = lo>p/(a? - w2), fjf = mupf(b2 - to2),
v' = n<apj(c2 - w2).
In order to find the angle between PlQl
and P2Q.2, we determine its cosine
=
FIG. 125.
But because [CIL (f)]
2/2/(a2 - wj2) = 0 and 2/2/(a2 - o>22) = 0,
we also have (wx2 - w22) . 2/2/(a2 - m^)(az - u>22) = 0.
Hence, if the values of co3 and to2 are different, we have cos(PlQlP2Q2)
equal to zero, and the angle between P^ and P2Q2 a right angle.
But if w1 equals o>2, the points Pl and P2 coincide, as we saw in CIV.
In this case, there is an infinite number of points of contact which
lie on a circle passing through the wave normals.
If the lines P1Tl and P2r2 are drawn from Pl and P2 perpendicular
to OQl and OQ2 respectively, and if we set PlT1 = ql and P27T2 = j2,
we have q:p = a)-.r, and therefore q=p(ojr. Further, OT^aPjr. If
A, /j., v are the cosines of the angles which q makes with the coordinate
axes, we have A = (coZ - OT . x/r )/q, etc., and hence [CIV. (1)],
(b) A(a2 - co2) = Ia2pjr, p.(b2 — co2) = mb2p/r, v(c2 - co2) = nc2pjr.
If we compare this result with the expressions in CIL (e), which
determine the direction of the electrical force F, whose components
are X, Y, Z, we see that the electrical force is parallel to q. If
we introduce in the equation CIL (d) the values for A, /*, v given
above, and notice that Ix + my + nz = to, we have cos 8 —p/r. Since
there are two directions of q, namely ql and q2, there are two direc-
tions, ql and q2, of the force in any plane wave. These lie in two
256 EE FRACTION OF LIGHT. [CHAP. xn.
planes perpendicular to the plane wave. There are two values for 8,
namely, ^OQ1P1 and L.OQ.,PZ; these angles are equal to Ll^P-fl and
t-T^P^O respectively.
The electrical forces X, Y, Z cause an electrical polarization, whose
variation may be looked on as an electrical current. The components
of current [OIL (a) and (c)] are
« = KJtir . VX^i = Kj_ A/47T . 9 Upt, etc.
If A.0, /x0, v0 are the cosines of the angles made by the axes with
the direction of the current, we have A0 : /*0 : v0 = A/a2 : /z/52 : v/c2.
But we obtain, by the help of OIL (e),
A0 : pQ : v0 = //(a2 - w2) : m/(b* - a>2) : n/(c2 - w2).
Hence the current has two directions, corresponding to the two values
of w. From equation (a) the same ratio holds between the cosines of
the angles which p makes with the axes as between the cosines
determining the directions of the current. Hence the two directions
of the current are parallel to p{ and p.2 respectively.
In order to determine the direction of the electrical force and the
current, we proceed in the following way. If a plane wave moves
in the direction determined by the normal ON, we construct two
planes which touch the wave surface and are parallel to the plane
wave. These planes are those constructed at Ql and Q2. We then
draw QlPl and Q2P2 perpendicular to the wave normal. The electrical
currents, which are in the wave planes, are parallel to $iA and Q2P2.
The corresponding velocities of propagation are OPl and OP2. There
are two directions of current in every plane wave, which are perpen-
dicular to each other. The electrical forces, which are connected with
these directions of current, are parallel to P^T^ and P2T2,
SECTION CVI. THE DIRECTION OF THE RAYS.
When a plane wave is propagated in an isotropic medium, the
direction of the normal to the wave coincides with the direction of
the ray. In doubly refracting media, the direction of the ray is in
general different from the direction of the wave-normals. We will
now determine the direction of the ray. Let MN (Fig. 126) be the
surface of a doubly refracting body on which the cylinder of rays
KOPL falls perpendicularly. By Huygen's principle, the separate
points in the bounding surface OP may be considered as centres of
luminous disturbance. The luminous disturbance is propagated
THE DIRECTION OF THE RAYS.
257
through the body in such a way that, after unit time, it reaches the
wave surfaces which are constructed about the separate points of
the bounding surface OP. Therefore, if the wave surfaces RA, SO,
etc., are constructed about 0, P, and the intervening points, we obtain
a plane ES which touches every wave surface and is congruent to
and similarly situated with OP. The direction OR or PS is then the
direction of the rays. If from the point 0 we let fall a perpendicular
OB on the plane US tangent to the wave surface RA, OB = u is the
velocity of propagation of the wave. If /, m, n represent the direction
cosines of the normal to the wave surface, o> is determined by
equation GIL (f)
(a) /2/(a2 - "2) + m2l(W - w2) + n2/(c2 - co2) - 0.
-fl
M 0
Li
P JT
J,
•</
/
R jE
f 5
FIG. 126.
The position of the point of contact of the plane US and the wave
surface RA is given [CIV. (1)] from the equations
(b) x(a2 -
- r2), y(bz -
where x, y, z are the coordinates of the point desired, and OR = r is
its distance from the origin of coordinates. OB represents the velocity
of propagation of the wave, OR the velocity of propagation of the ray.
Instead of the wave surface itself we may sometimes use to
advantage another surface, called the reciprocal wave surface. Let 0
be the centre of the wave surface, AR a part of the surface itself, and
BR a plane which touches the wave surface at the point R. From
the point 0 we let fall the perpendicular OB on the tangent plane.
The point B', in the perpendicular OB produced, is determined in
such a way that (c) OB' = r' = s2/«>, where <i> = OB, and s is constant.
The reciprocal wave surface is then the locus of the points determined
by (c). This surface, like the wave surface, is a surface of two
nappes. Its equation is obtained in the following way. If I, m, n
258
EEFRACTION OF LIGHT.
[CHAP. xii.
represent the direction cosines of OB = to, and x', y', zf the coordinates
of the point B1, we have (d) x' = lr\ y' = mr', d = nr'. But because
l2/(a? - w2) + w2/(62 - w2) + n2/(c2 - w2) = 0,
it follows by (c) and (d) that
x'2/(a2r'2 - s4) + 2rY(6V2 - s4) + *'2/(cV2 - s4) = 0.
We set (e) a' = s2/a, b' = s2jb, c' = s2/c, and obtain the equation of the
reciprocal wave surface in the form
(f ) a' 2x'2/(a'2 - r"2) + b'2y'2j(b'2 - r'2) + c'-Y2/(c'2 - r'2) = 0.
This surface differs from the ordinary wave surface [cf. CIV. (m)]
only in that its constants a', I', c' are the reciprocals of the constants
a, b, c of the wave surface.
If we draw through the point B' (Fig. 127) a plane tangent
to the reciprocal wave surface FA', we can show that the plane
B'R is perpendicular to the prolongation of OR, and that therefore
OR is perpendicular to B'R. Further, if OR = u>, Ofi = r, we have
(g) w' = s2/r. This follows by the same method by which we have
passed from one surface to the other. We can also prove it directly.
If the direction OR is determined by the cosines I', m', n', we have
[CIV. (1)] (h) z'(a'2-to'2) = rto'(a'2-'/2) etc.
The equations (h) determine w', I,' m', n'. Setting u/ = s'2/r and
l' = xfr, m' = y/r, ri = z/r,
and using equations (c), (d), (e), (g), equation (h) takes the form
x(a? — w2) = Zo(a2 - r2), etc. Since these equations are identical with
those in CIV. (1), it follows that the point of intersection of OR
SECT, cvi.] THE DIRECTION OF THE RAYS. 259
and the wave surface is the point at which the tangent plane
touches the wave surface. It follows further from (e) and (g) that
(i) r'to = ra> or OB .OB' = OR .OR. In order to determine the direction
of a ray from the reciprocal wave surface, we produce the wave normal
until it cuts that surface. The direction of the ray is then perpendicular
to the tangent plane at the point of intersection.
SECTION CVII. UNIAXIAL CRYSTALS.
If two of the constants a, b, c are equal, for example if b = c, the
equations become much simplified. The bodies for which this relation
holds are called uniaxial crystals. In order to find the velocities of
propagation w1 and o>2, we apply equation CII. (g) which is trans-
formed into (a) to4 - [b2 + r-b2 + (l- /2)a2>2 + b2[l2b2 + (1 - I2)a2] = 0.
From this equation we obtain (b) o^2 = b2, o>22 = I2b2 + (1 - l'2)a2. Hence
the velocity wx is constant ; the velocity o>2 depends on the direction
of the wave normal, or on the angle which the wave normal makes
with the axis of elasticity a. This axis is called the optic axis; it
coincides with the principal axis of the crystal. In the direction
of this optic axis there is only one wave velocity, arid therefore also
only one ray velocity. If we designate the angle between the wave
normal and the optic axis by e, we have (c) o>22 = «2sin2€ + 62cos2e.
Hence, a plane wave, on its passage from an isotropic to an
uniaxial medium, is divided into two waves, one of which is propagated
with a velocity o^, which is independent of the direction of the wave
normal. This wave is called the ordinary wave. The velocity of the
other or extraordinary wave changes with the direction of the wave
normal.
AVe obtain the equation of the
wave surface for uniaxial crystals
from CIV. (n), by setting b = c. We
thus obtain
<d) (r2 - b2)(a2x2 + b2(y2+z2) - aW) - 0.
Hence, the wave surface consists of
a sphere whose radius is b, and an
ellipsoid of revolution whose polar and
equatorial axes are 26 and 2a respec-
tively; the sphere and the ellipsoid FlG-
touch at the extremities of the polar axis. In Fig. 128,
represents the polar or optic axis, AE1A1 a plane section through
260 EEFRACTTON OF LIGHT. [CHAP. xn.
the sphere and AR0A1 a plane section through the ellipsoid. Let
OB^ be the normal to the plane wave POQ, B^D and B^R.2, two
planes tangent to the wave surface, which are both perpendicular
to the wave normal. OR^ and OB.2 are the velocities of propagation
in the direction of the wave normal. We call such a plane section,
which contains the optic axis as well as the wave normal, the
principal section. The direction of the rays of the extraordinary
wave is represented by OR.2, if the plane B2R.2 touches the ellipsoid
at the point R.2. The direction of the electrical force is given by
B2U.» which is perpendicular to OR.2. The direction of the rays and
the wave normal of the ordinary wave coincide, and the direction
of the electrical force is perpendicular to the plane of the figure.
The polar axis AAl = 2b (Fig. 128) is greater than the equatorial
axis 2a ; the crystals for which this occurs are called positive crystals.
If a>b, the crystal is called negative. The sphere can enclose the
ellipsoid or inversely ; crystals of the first kind are called positive, those
of the second kind negative. Iceland spar is a negative crystal,
quartz is a positive crystal.
If we set o> = 5 in OIL (e), we obtain (e) \1 = Q and SI = ^TT, that
is, the direction of the electrical force in the ordinary wave is perpendicular
to the optic axis as well as to the wave normal : it is therefore perpendicular
to the principal section.
In order to obtain the direction of the electrical force in the
extraordinary waves from CIL (e), we introduce in it the value
for <o2 given in (b), and notice that I = cos e, m = sin e, n = 0. We
then obtain
^ _ a2cos62 _ &2cos8.2 v _Q
/2~(a2_^)COS£» **2- (a* -b'2) sine V~~
Hence the direction of the electrical force in the extraordinary wave is
parallel to the principal section. It follows from the last equation
that l/cos282 = (a4sin2e + &4cos2e)/(a2-&2)2sin2ecos2€, and hence
(f ) tg 82 = ± (a2sin2e + 62cos2e)/(a2 - &2)sin e cos
We thus obtain the equations
±a2sine/Va
(g)
, = 0.
SECT. CVIH.] DOUBLE REFRACTION OF A CRYSTAL. 261
SECTION CVIII. DOUBLE KEFRACTION AT THE SURFACE OF A
CRYSTAL.
When a ray of polarized light falls on the plane surface of a
doubly refracting medium both reflection and refraction occur. Let
the x-axis of the system of coordinates be parallel to the normal
to the surface drawn outwards, and the £-axis perpendicular to the
plane of incidence ; the ?/-axis is then parallel to the line of intersec-
tion between the plane of incidence and the refracting surface. For
the components of the electrical force of the incident ray we have,
as in C.,
(a) X, = A/;, Yt = pji, Zt = v/i, ft = fjCospir/T. (t - ( - xcos a + y sin a)/0)],
or, using only the real part, (b)/=JP«^-(-*wa+»atoBJWl. In these
equations a is the angle of incidence and fi the velocity of the light
outside the crystal. In addition to these equations we have the
condition that the electrical force is perpendicular to the direction
of the ray. Hence, since the direction of the incident ray makes the
angles TT - a, |-TT - a and |TT with the axes, we have
(c) - AjCOS a + /AjSin a = 0.
In the corresponding notation we have for the reflected ray
(d) A; = Xrfr, Yr = /zr/r, Zr = i>rfr, fr = Fr. ew - (irx+™ry+»r*)/n].
That the electrical force shall be perpendicular to the direction of
the ray, we must have (e) Xrlr + p,rmr + vrnr = 0. Finally, for the re-
fracted ray, we have
(f) A>A6/6, Yb = nJb, Zb=vJM (g)ft = F^-«**+™»+«»W.
w depends on lb, mb, nb, or on the direction of the propagation of the
refracted wave. The boundary conditions are the same as those of
isotropic bodies. We have everywhere in the bounding surface, for
which x = 0, Yt+Yr=YM or
Since this equation must hold for all values of y and z, we have
(h) sin a/0 = mr/0 = mb/o> and (i) 0 = nr/£l = «6/w. By the last equation
0 = 7ir = 7iM that is, the wave noi-mals of the reflected and refracted waves
lie in the plane of incidence. It follows from (h) that wr = sina, that
is, the angle of reflection is equal to the angle of incidence. Therefore
the direction of the reflected ray is determined in the same way as
in the case of reflection by an isotropic body.
262
REFRACTION OF LIGHT.
[CHAP. xn.
If (3 represents the angle of refraction, we have
lb = - cos ft, mb = sin j3, nb = 0,
therefore from (h) sin a/12 = sin /3/w. If we determine the direction
of the wave normal by the cosines of the angles which it makes with
the axes of elasticity, we have, to determine w, the equation
(1) /2/(a2 - a)2) + TO2/(62 - w2) + rc2/(c2 - w2) = 0.
If (xa), (ya\ etc., denote the angles between the axes of elasticity
and the coordinate axes, we have l = lt>cos(xa) + mbcos(ya) + nl>cos(za).
Introducing here the values for lb, mt, etc., given above, we obtain
c I = - cos (3 . cos(za) + sin /3 . cos(ya)
(m) -j m = - cos (3 . cos(a$) + sin /3 . cos(yb)
\. n= -cos/?, cos(zc) + sin ft. cos (yc).
By the help of equations (m) and (1), w can be expressed in terms
of (3. The equation thus obtained in connection Avith (k) determines
the angle of refraction. In general we obtain two values for f3, one
or both of which may be imaginary ; if this is the case the reflection
is total.
We can find the direction of the wave normal and that of the ray
by a construction given by Huygens. About the point 0 (Fig. 129)
as centre construct the sphere PD,
whose radius is OD = Q, where 12
denotes the velocity of light in air.
If the incident ray is produced, it
meets the sphere at the point D.
The plane which the sphere touches
at D cuts the refracting surface in
a straight line, whose projection
on the plane of the figure is Q.
The plane QR containing this line
is drawn tangent to the wave sur-
129. face FR, whose centre is at 0.
The perpendicular OB = i» is let fall from 0 on the tangent plane
QR. The normal to the refracted wave is then OB and L'OB = (3,
if LOL' is the normal to the surface. Now OB=<a is the velocity
of propagation in the direction OB, and also OQ = OD/sina = OI>lsin(3,
or fl/sina = w/sin/:J, so that equation (k) is also satisfied. OB is the
direction of the wave normal of the refracted wave, and OR the
direction of the corresponding ray. Since the wave surface in general
has two nappes, two planes tangent to the wave surface can be
SECT, cvni.] DOUBLE EEFEACTION OF A CEYSTAL. 263
drawn through Q. The construction therefore determines two wave
normals and two ray directions.
This construction really serves only as a representation of the
refraction; it cannot be used for the determination of the direction
of propagation so long as the construction is confined to the plane,
because the point of contact R does not lie in the plane of incidence ;
we can, however, obtain the direction of the wave normal by a
construction in the plane of incidence given by MacCullagh.
If we draw through D (Fig. 129) the line DE perpendicular to
the refracting surface, the point of intersection B'. of DE and the
wave normal OB is so situated that OB . OB = OD2, for we have
OB = OQ.sinf3, OB' = OE/sin(3, and further, as may easily be seen
from Fig. 129, OQ. OE=OD2. From this follows the relation
OB.OB' = OD\
But we have OB = u, OD = Q, and if we set OB' = r', it follows that
(n) r' = 12'2/w. Therefore the point B' lies on the reciprocal wave
surface whose equation is [CVI. (f)]
a'2x2/(a'2 - r2) + b'2f/(b'2 - r2) + c'2z2/(c'2 - r2) = 0
if the coordinate axes are parallel to the axes of elasticity. In this
equation a' = fl2/a, b' = ^2/b, c' = 02/c. If we set
JV^fi/a, ^2 = ^/6, JV3 = 0/c,
and choose as the unit of length the velocity of light fi in the
surrounding medium, it follows that
( o) N^KN^ - r2) + N2y*l(N2 - r2) + N32z2/(Ns2 - r2) = 0.
This is the equation of the reciprocal wave surface. It follows
further from the discussion of CVI. that the direction of the rays
OR is perpendicular to the plane tangent to the reciprocal wave
surface at the point B.
We can therefore construct the wave normal in the following way.
About the point 0 as centre with unit radius we construct the circle
PD ; about the same point we draw the curve of intersection between
the plane of incidence and the surface (o). This curve is represented
in (Fig. 29) by B'F'. W e then produce the incident ray to the
point D lying on the circle, and draw the straight line DB' perpen-
dicular to the refracting surface and cutting F'B' at B'. OB is then
the direction of the wave normal, while the direction of the ray
OE is perpendicular to the plane tangent to the surface F'B at
the point B'. We can easily derive the condition for total reflection
from this construction. It can also be applied to the reflection of
light within the crystal itself.
264 REFRACTION OF LIGHT. [CHAP. xn.
SECTION CIX. DOUBLE REFRACTION IN UNIAXIAL CRYSTALS.
Using the same notation as in CVIIL, we have, to determine the
angle of refraction of the wave normal, the equation
(a) sin a/0 = sin /?/a>.
In the case of uniaxial crystals w has the values w: and w2 which are
[CVII. (b), (c)] o)12 = i'2, w92 = a'2sin2e + £>2cos2e. In the first case the
angle of refraction /31 is obtained from the equation
sin a = JV0 sin /^ where NQ = &/(OV
the so-called ordinary index of refraction. The corresponding direc-
tion of the electrical force is perpendicular to the principal section.
If the second wave normal makes the angle /32 with the normal to
the surface, we have sin a/fl = sin /?2/w.,. If we represent the angles
made by the axis of the crystal with the coordinate axes by (xa),
(ya), (za) and notice that TT - (3y |z- - jS^ |TT, are the angles made
by the refracted ray with the coordinate axes, we have
cosc= -cos(««)cos/?2 + cos(?/a)sin/32.
Hence, for the calculation of /32, we have the equation
fl2sin2/32/sin2a = a2 - (a2 - Z>2)(cos(.ra)cos /32 - cos (ya)sin /32)2.
The corresponding direction of the electrical force is parallel to the
principal section. If the optic axis lies in the plane of incidence we
set cos(:ra) = cos \f/, cos (ya) = sin ^, and then obtain
J22sin2£2/sin2a = a2 - (a2 - 62) . cos2(^ + /32).
If ^ = a2sin2r/'+62cos2^, JB = a2cos2^ + 62sin2^, (7=(«2-62)sin ^cos f,
we have A£-C2 = aW2 and I22/sin2a = A cotg2/?2 + 2(7 cotg /3.2 + H.
From this follows
(b) A cotg /32 = - C + v
If the axis of the crystal is perpendicular to the plane of incidence, we
have (xa) = (ya) = ^TT, from which sin a = JV^sin /?2, where Ne = Q.ja, is
the extraordinary index of refraction. If a and b are expressed in
terms of N, and N0, we have from (b)
r (A^sin2^ + JV^cosV) cotg /32 = - (N02 - N*) sin ^ cos $
I + JV0JVeX/sin--'a( JV^sin2 ^ + ^Ve-'cos2 ^) - 1 .
In order to obtain the equation of the reciprocal wave surface, we
set fl = l, and substitute N, for a, N0 for b, in the equation for the
wave surface. Thus we obtain [CVII. (d)],
(d) (r> - N*%N& + NQ %2 + *2) - NJN*] = 0,
SECT, cix.] DOUBLE REFRACTION IN UNIAXIAL CRYSTALS. 265
FIG. 130.
as the equation for the reciprocal wave surface referred to the axes of
elasticity as coordinate axes. We obtain the same result from
CVIIL (o), if we set N^N. and N2 = N3 = N0.
In Fig. 130, OP is the refracting surface, and OA the optic axis,
supposed to lie in the plane of incidence ; AMl and AM.2 are the
curves in which the plane of incidence cuts the reciprocal wave
surface. AM^ is a circle with
radius NQ, AM2 an ellipse whose
semi-major axis OA equals N0,
and whose semi-minor axis OM2
equals Ne. We draw a circle of
radius OD = 1 , which cuts at D
the prolongation of the incident
ray. The line ED, perpendicular
to the refracting surface, cuts the
reciprocal wave surface at the
points Bl and B.2. The normals to the refracted waves are then
OBl and OB%. For the ordinary wave the direction of the ray
coincides with the wave normal OBl ; for the extraordinary Avave it
is perpendicular to the plane tangent to the ellipsoid at the point B.2.
If the crystal is immersed in a fluid whose index of refraction is
greater than that of the crystal, the circle PD is replaced by another
circle of greater radius, for example PD'. If this circle cuts the
prolongation of the incident ray at D', the directions of the wave
normals are determined by the point of intersection between the
reciprocal wave surface and the line UE', perpendicular to the
refracting surface. In this case total reflection can occur. If HE'
does not cut the reciprocal wave surface there will be no refraction;
if UE' cuts only one curve, there is only one refracted ray. If, as
in Fig. 130, D'E' touches the ellipse at a point C, refraction will
occur ; the direction of the ray is parallel to the bounding surface OP.
Our presentation of optics is based on Maxwell's conception of light
as an electrical vibration. A more extended discussion on this same
basis has been given by H. A. Lorenz. Glazebrook published a
discussion of the most important optical theories in the Report for
1885 of the British Association for the Advancement of Science,
von Helmholtz has lately given a theory of the dispersion of light in
which he employs the electromagnetic theory of light.
CHAPTER XIII.
THERMODYNAMICS.
SECTION CX. THE STATE OF A BODY.
IF the particles of a system are in motion and exert force on one
another, the system possesses a certain energy U. The energy of
a system of discrete particles is made up of their kinetic and potential
energies. The former depends on the velocities of the particles at
any instant, the latter on their distances apart, or on the configuration
of the system ; together they determine the state of the body. Thus
the energy at any instant depends only on the state of the system
at that instant, and is independent of its previous states. The
principle of energy has been proved only for a system of discrete
particles ; we make the assumption in the mechanical theory of
heat, that the same principle or a corresponding one holds for all
systems of particles.
A certain amount of energy is inherent in every body. This we
call its internal energy, since we take no account of that part of its
energy which arises from its mutual actions with other bodies. By
the possession of this internal energy the body is in a condition
to do work ; thus variations occur in its form, volume, temperature
etc. The energy is determined solely by the state of the body ;
if the body in a certain state possesses the energy U, and if it is
subjected to any variations of form, magnitude, etc., and finally
returns to its original state, the internal energy will be again
equal to U.
To determine the internal energy of a body it is necessary to
know the quantities which determine its state. From Boyle's and
Gay-Lussac's laws the state of an ideal gas is completely determined
by its pressure and volume. The temperature is given if these two
quantities are known. Boyle's and Gay-Lussac's laws furnish an
equation which expresses the relations between pressure, temperature,
CHAP. xni. SECT, ex.] THE STATE OF A BODY.
267
aud volume ; we call it the equation of state of a gas, because it
enables us to determine the state of an ideal gas under any con-
ditions, if it is known under definite conditions, for instance, at
0° C. and 760 mm. pressure. The behaviour of real gases cannot
be accurately represented by an equation embodying Boyle's and
Gay-Lussac's laws, but conforms to other equations which include
these laws as a limiting case. The state of a fluid is in general
determined by the same quantities ; it depends to some extent on
the form of the surface and the nature of the bodies in contact
with it. The actions of electrical and magnetic forces may come
into play in both gases and fluids. As a rule the knowledge of a
great number of quantities is required to express the state of a
solid, especially if it is subjected to the action of forces. The
equation which unites all quantities which determine the state of
a body is called the equation of state.
Since the state of a gas only depends on the pressure p and the
volume v, it may be represented by a point in a plane with the
coordinates p and v ; a series of such points, or a curve, represents
a series of successive states. The t'-axis of this system (Fig. 131)
is drawn horizontal ; and the
^?-axis vertical. We represent
the volume and pressure of the
gas in its original state by r0
and p0 ; its state is then given
by the point A. If the gas ex-
pands under constant pressure,
its state is represented by a
horizontal line AB, parallel to
the v-axis. This is called the
curve of constant pressure. The
curves of constant volume are
vertical straight lines. If heat
is communicated to the gas whose original state is given by A, at
constant volume r0, the variation of the state of the gas is represented
by the straight line AC, and its pressure increases. If the tem-
perature of a gas remains constant during its successive states, we
have, from Boyle's law, v.p = const. Hence the curves of constant
temperature or the isothermal lines are rectangular hyperbolas whose
asymptotes are the coordinate axes. In order to change the state
of a gas in such a way that its temperature remains constant,
there is required either compression with abstraction of heat or expansion
J)
B
FIG. 131.
268 THERMODYNAMICS. [CHAP. xin.
with communication of heat. If a gas whose original state is represented
by A is subjected to compression with abstraction of heat, or to
expansion with communication of heat, in such a way that its
temperature remains constant, its successive states will be represented
by the hyperbola DAE. We may suppose the gas enclosed in a
receptacle put in connection with an infinitely great source of heat,
whose temperature is equal to that of the gas at the point A. If
we change the volume of the gas, the source of heat sometimes
takes up heat and sometimes gives it out, but the gas retains the
temperature of the source. If the gas is enclosed in an envelope
through Avhich heat cannot pass, it is heated by compression so
that its temperature rises, or cooled by expansion so that its tem-
perature falls. In this case the changes of state are called adiabatic
and the curve which represents them is called an adiabatic or isentropic
curve.
The state of a solid cannot in general be represented in a plane,
since it depends on more than two variables.
A series of changes by which the state of a body is altered in any
manner, and which is such that the body finally returns to its original
state, is called a cyclic process. If a body goes through a cyclic process
the energy which it receives from surrounding bodies is equal to that
which it gives up to them. The steam-engine is a system of bodies
which periodically returns to the same state. It appears from the
action of the steam-engine, that heat and work are similar or equivalent
quantities, which can be transformed into one another, and are both,
therefore, forms of energy. This conclusion has been established by
accurate experiment. The quantity of energy produced in the one
form is always proportional to that applied in the other form. This
law of the equivalence of heat and energy was first formulated by K.
Mayer (1842). The later observations of Joule and others have shown
that the quantity of work which is equivalent to a unit of heat,
or to the quantity of heat which will raise the temperature of a gram
of water by 1° C. is equal to 4.2.107 absolute units of work (C.G.S.).
This result is called the first law of thermodynamics. It may be thus
stated : Heat and work are equivalent ; work can be obtained from
heat and heat from work. The work equivalent or the mechanical
equivalent of the unit of heat is designated by J.
If the quantity of heat dQ is communicated to a body it receives the
energy / . dQ. This goes partly to increase the internal energy U
of the body, partly to do the work dW. We then have
(a) J.d
SECT. CX.]
THE STATE OF A BODY.
269
This equation is called the first fundamental equation. We will apply
it to the case in which the work dW is done by expansion against
external pressure.
We consider the body ABC (Fig. 132), which is subjected to the
hydrostatic pressure p at every point on its surface. When the body
expands its volume becomes A'B'C'. The normals AA', BB' are
drawn from the surface- element AB = dS to the new surface. We set
A A' = v and obtain for the work done by the body,
\vp . dS=p^v . dS—p . dv,
where dv denotes the total increase in volume of the body. Equation
(a) then becomes (b) J .dQ = dU+p .dv. If the state of a body is
determined by the independent variables p and v, the definite values p^
FIG. 132.
JL £'
FIG. 133.
and vl correspond to a point A (Fig. 133). Suppose the body to
pass through a series of states represented by the curve ACB; the
values p. 2 and v.2 correspond to the point B. We then have from (b)
(c) JQ=Ut-U1 + £p.dv.
Q is the quantity of heat introduced during the change of state,
£/2 - ZTj the increase of the internal energy, and / p . dv the work done.
The increase U2 - Ul is determined by the initial and final values of p
and v, or by the position of the points A and B. The external work
is measured by the area of the figure A' ABB' A ; this work therefore
depends on the process by which the change from one state to the other is
effected. This holds also for Q. Since U is a function of p and v, we
obtain (d) J.dQ = 'dU/'dp.dp + ('dUj'dv+p)dv. If the function U is
known, it is possible to find the quantity of heat necessary to produce
any change in the state of the body. U is determined from equation
(c), by measuring the quantity of heat received by the body and the
quantity of work done by it. Our knowledge of the quantity U is
still very limited.
270 THERMODYNAMICS. [CHAP. xm.
SECTION CXI. IDEAL GASES.
Clement and Desormes and subsequently Joule showed that the
temperature of a gas which expands without overcoming resistance,
that is, without doing work, remains unchanged.* The initial and
final states of a gas which expands without doing work lie on the
same isothermal, that is, the internal energy of a gas is a function of
its temperature only, and is therefore independent of its volume if
the temperature remains constant. If we take the temperature 0
and the volume v of the gas as independent variables, we have
J.dQ = 'dUJW .de + 'd Ufdv . dv +p . dr.
Now 3 Ufdv = 0 and therefore J.dQ = 'd U/W .d6+p. dv. If the mass
of gas contained in the volume v is equal to unity then ?>U/'dO = Jcn
where c, denotes the specific heat of the gas at constant volume, that
is, the quantity of heat which must be communicated to its unit of
mass in order to raise its temperature one degree in such a way
that, while its pressure changes, its volume remains constant. If the
specific heat of the gas at constant volume is constant, its internal energy
must be a linear function of its temperature.
For ideal gases the equation giving the relation between pressure,
volume and temperature is pv = R6, where R is a constant. If 0 and v
are the independent variables of the gas, we have (a) J.dQ = Jc,.d6+p.dv.
From the observations of Regnault, c, is independent of the pressure
and temperature of the gas. If 6 and p are chosen as the independent
variables, v must be considered as a function of them, so that
dv = "dvfde . dO + 'dvj'dp . dp,
-and substituting this in equation (a) we have
/. dQ = (Jc, +p . 'dv/Wfie +p . "dvfdp . dp.
From the equation pv = E6, it follows that
p . 'dv/W - R and p . 'dv/dp = - v, and J.dQ = (Jc, + B)dO - v . dp.
In order to obtain the specific heat cp at constant pressure, that is, the
quantity of heat which must be communicated to the unit of mass
of the gas to raise its temperature one degree, in such a way that,
while its volume changes, its pressure remains constant, we set dp = 0
and obtain cp = c, + ElJ, (b) J.dQ = Jcp.dO-v.dp. If p and v are
*More exact measurements show that the gas, in these circumstances, is
slightly cooled. From this it follows that there are attractive forces between
its separate particles.
SECT, cxi.] IDEAL GASES. 271
chosen as the independent variables, we have dO = 'dOj'dp . dp + Wfdv . dv.
It follows from pv = R0 that
R.o0/'dp = v, R.Wj?)v=p, R.d0 = v.dp+p.dv,
and from (b) that (c) R.dQ = c,v.dp + cpp . dv. If therefore the specific
heat cp and the constant R are known, the equation (c) enables us to
determine the specific heat for any change of state in the vp-p\a.ne.
The specific heat has an infinite number of values for a given state
in the t^-plane, depending on the direction in which this change of
state takes place.
The expressions (a), (b), (c) show a noteworthy peculiarity. If
one of them, say (a), is divided by 0, we obtain by the use of the
fundamental equation pv = R9, J .dQ/0 = Jc,.d0/9 + R.dv/v. If, for
example, the gas passes from the state A (Fig. 133) to the state B,
and if the temperatures and volumes at these points are 6V vl and
00, v2 respectively, we have by integration,
(d) /. \dQjO = J.c.. log^flj) + R . log(V»i)-
Therefore, while the integral ^dQ depends on the path on which the gas
passes from one state to another, the integral ^dQ/d does not depend on
this path.
Clausius called the quantity S = J .\dQjO the entropy. This concept
is of great importance in the theory of heat. If a body passes from
one state to another the change of the entropy is determined by the coordinates
of the initial and final points. This theorem is here proved only for
a gas, but holds also for all bodies.
If the change of state of a gas occurs along an isothermal curve, we
have from (a) J.dQ=p.dv. Using the equation of state and inte-
grating, we obtain
(e) JQ = £p . dv = Rd . log(^1).
All the heat communicated is therefore used in keeping the tempera-
ture constant. If we set v2 equal to fivv fj?vv p?vv etc., in succession,
where ^ is any number, the corresponding values for Q are
Q = R0/J.logp, 2R6/J. logp, SRe/J.logp, etc.
If the change of state occurs along an isothermal curve, and ii the
quantities of heat introduced are in arithmetical progression, the
volumes, according to equation (e), are in geometrical progression ; at
the same time the pressure changes proportionally to the density.
If the change of state occurs along an admbatic curve, we have from
(c) cvlogp + cplog v = cr Setting cp/c, = k we obtain (f ) pv* = c, where c is
272 THERMODYNAMICS. [CHAP. xm.
constant. The equation (f) is the equation of the adiabatic curves. Com-
bining this with the equation pv = B6, we have from (f) Bdvk~l = c.
If we introduce in this formula the density 8 = M/v of the gas, where
M denotes its mass, it follows that its temperature is proportional
to the (&-1) power of the density when the state of the gas changes
along an adiabatic curve.
Further we obtain the relation [CX. (b)] fp.dv=Ul- U.2. The
work is therefore done at the expense of the internal energy, if the
change of state is adiabatic.
SECTION CXII. CYCLIC PROCESSES.
A simple reversible cycle is one in which all changes occur in such
a way that if reversed they may be effected under the same circum-
stances. The body which performs the cycle is called the working
body. In the performance of a simple reversible cycle the working
body must be associated with two others, one which communicates
heat to it, and another which receives heat from it. In a gas engine
the working body is the gas in the cylinder ; in a steam engine it is
the water or steam. The gases of the fire and the walls of the boiler
give up heat, the water in the condenser receives heat. The gas or
steam passes through a series of states and, at least in some machines,
returns to its original state; it is then in condition to repeat the
same process. Since the value of the internal energy U at the
beginning and end of the process is the same, we have [CX. (a)]
(a) JQ=W,
where Q is the difference between the heat received and the heat
given up.
The quantity of heat received by the working body and not given
up to the colder body is the equivalent of the work done. If the
working body is a gas, we have for the cycle JQ = ^pdi: The
entropy of a gas depends only on the coordinates and therefore has
the same value at the beginning and end of the process. If S} denotes
the entropy at the starting point, the entropy at any instant during
the process is equal to Sl + \dQjQ. If the integration is extended over
the whole cycle, the entropy returns again to its value Sv and we
have therefore \dQ/B = 0.
We will discuss more particularly a special case, the so-called
Carnofs cycle, which is of great importance in the theory of heat.
SECT. CXII.]
CYCLIC PROCESSES.
273
Suppose a gram of gas to be in the state represented by the point B
(Fig. 134) in the vp-plane. The curve representing the cycle is in this
case composed of two isothermal curves BC and ED and of two
adiabatic curves CD and BE. The gas first expands at the constant
temperature Or This is accomplished by keeping it in contact with
the infinitely great body Ml at the temperature 0lt and by so
regulating the external pressure on the gas
that it passes to the state C along the path
BC. During the change of state BC the
quantity of heat Ql is absorbed and the
work represented by the surface BCC'B1 is
done. The gas then expands adiabatically,
in the manner represented by the adiabatic
curve CD, and its temperature falls to 02.
Then the gas is brought in contact with an I - — 4 » i — if
FIG. 134.
infinitely great body Me, at the temperature
02 and compressed ; during this process it
gives up to M.2 the quantity of heat Q. Its state is represented by E.
Finally the gas is further compressed without communication of heat
until it returns to the original state B. The integral ^p . dv, extended
over the whole cycle, equals the area BCDE, and represents the work
done by the gas. We have [CX. (a)] (c) J(Ql - Q2) = W.
When the gas expands from B to C, its entropy is increased by
QI/OI ; it remains constant along the path from C to D, is diminished
by Q.2/Q-2 along the path from D to E, and again remains constant along
the path from E to B. Since the gas on its return to B has the
same entropy as at the outset we have
From (c) and (d) it follows that
Therefore the work done by this cyclic process is proportional to the
quantity of heat Q} absorbed and to the difference of temperature
Ql - 6.2, and is inversely proportional to the absolute temperature Olt
at which the heat is absorbed. The heat received from the source Ml
is not wholly transformed into work, but is divided into two parts, one
of which is transformed into work, and the other transferred to M0.
The efficiency £ of the Carnot's cycle is the ratio of the heat trans-
formed into work to that communicated to the gas. We have
274 THERMODYNAMICS. [CHAP. xin.
Hence the efficiency depends only on the temperatures dl and 0.2 of
the sources of heat. If we consider the reversed cycle, the state of the
gas first changes along BE ; along the path ED it receives a certain
quantity of heat from M*, and has a certain quantity of external work
done upon it along the path DC. The heat received and the work
done transformed into heat are given up by the gas to the body M^.
along the path CB. In the case of the cyclic process first considered
heat is transformed into work ; in the reverse process work is trans-
formed into heat.
SECTION CXIII. CARNOT'S AND CLAUSIUS' THEOREM.
It was shown in the preceding section that ^dQ/d = Q for any
reversible cycle, when the body describing the cycle is a gas. We
will now see if this theorem holds when any other body is used
as the working body instead of a gas. Let us take the simple
case in which the process is carried out along two isothermal lines
BC and ED (Fig. 134), and two adiabatic lines CD and BE. Suppose
the change of state to take place in the sense given by the letters
BODE. If the body at the temperature Ol expands from B to C,
it receives the quantity of heat Q^ ; when it is compressed from
D to E it gives up the quantity of heat Q.2. Along the paths CD
and EB heat will neither be received nor rejected. In this process
the total quantity of heat received by the body is Ql - $„• Since
it returns to its original state, the quantity of heat Q1 - Q2 is
equivalent to the work done, which is therefore J(Ql - Q.2).
S. Carnot published, in 1824, a work on the motive power of
heat, in which he proposed an important theorem on the connection
between heat and work. He was of the opinion that heat was a
fundamental substance whose quantity remained invariable in nature.
Applying this view to explain the action of the steam-engine, he
supposed that the steam gave up a quantity of heat Q1 at the higher
temperature 6V that this heat was transferred to the condenser at
the lower temperature 02, and that the motive power of the heat
was due to its passage from the higher to the lower temperature.
The work thus done by this passage of heat from a higher to a
lower temperature was considered analogous to that done by n
falling fluid or by any falling body. This latter is proportional to
the weight of the falling body and to the distance which it falls.
Hence for the work done by the heat Carnot proposed the expression
SECT, cxin.] CAKNOT'S AND CLAUSIUS' THEOREM. 275
KQl(Ol - 02) where K is a function of the absolute temperatures Ol
and 02. This conclusion of Carnot was confirmed by experiment,
but did not agree with the mechanical theory of heat in so far as
it regarded heat as an invariable quantity. If for the present we
disregard this error, we have for the cycle just described
Since K must be independent of the nature of the body doing the
work we have, if the body is a gas [CXIL (e)], (b) K^J/0^ It
therefore follows that (Ql - Q^/^ - 02) = Q1/6l and hence Ql/0l = Q2/02,
that is, if a body traverses a Carnot's cycle any number of times, by being
placed alternately in contact with two infinite sources of heat, the quantities
of heat which it receives from one source and gives up to the other are
in the same ratio as the temperatures of the sources.
There can be no doubt that this theorem holds for a cycle of
the kind considered. The application of the theorem in many
departments of physics and chemistry has led to no results which
are as yet contradicted by experiment. Several attempts were made
to give a direct proof of the theorem, the first and most important
of which is due to Clausius, whose method may be presented in
the following way :
Suppose a gas to traverse the cycle BCDE (Fig. 135) composed
of the isothermal curves BC and DE, which correspond to the
absolute temperatures Ol and 02, and of
the adiabatic curves CD and BE. During
its expansion from B to C, the gas takes
the quantity of heat Ql from an infinitely
great source M-^ whose temperature Ol is
constant. It then expands from C to D
without communication of heat. It is
then brought in communication with the
infinitely great source of heat M.> whose
temperature 00 is constant, and by com-
://
pression is made to give up to it the ' FIG 135
quantity of heat Qy Finally it is
brought back to its original state B. During the cycle the gas
has received from the source Ml the quantity of heat Qv which
is divided into two parts. One of these parts is transferred as a
quantity of heat Q2 to the source M.2, the other is transformed
into work and is represented in amount by, the area BCDE.
Suppose B'C' and E'D' (Fig. 135) to be the two isothermal curves
corresponding to the temperatures 6l and 9.2 for another body, say
276 THERMODYNAMICS. [CHAP. xni.
for water vapour. C'D' and B'E' are two adiabatic curves so chosen
that the surface BC'D'E equals the surface BODE. If the water-
vapour is subjected to a process similar to that just described for the
gas, the heat which it will receive while in contact with the source
Ml is Ql + q, and during its passage from H to E', while in contact
with the source M2, it gives up to that source the quantity of heat
Q2'. The work done by the vapour is equal to that done by the gas,
because the surface BCDE is equal to the surface B'C'DE', and we
therefore have Ql + q - Q.2' = Ql - Q2, and therefore Q2 =Q.2 + q. The
vapour in expanding along B'(J receives the quantity of heat Ql + q,
and gives up the quantity Q2 + q along the path D'E'.
The cycle described can also be performed in the opposite sense.
For example, the water-vapour can expand along the isentropic curve
B'E'; it may then be brought in contact with the source of heat
M2, and expand from E' to D', during which expansion the quantity
of heat Q2 + q is taken from M2. It may then be compressed along
the isentropic curve Z^C", and lastly along the path (7-6', while in
contact with the source of heat Mv During this compression it
gives up to Ml the quantity of heat Ql + q. To carry out this pro-
cess, a quantity of work must be done which is equivalent to the heat
this work is represented in Fig. 135 by the surface B'C'HE'.
We consider finally two engines, one of which is a gas engine,
in which the gas performs the cycle BCDE, and the other a steam-
engine, in which the steam performs the reversed cycle B'C'D'E'.
The work done by the one is equal to that supplied to the other,
if we neglect friction and other resistances. The gas engine in
each revolution takes from the source Ml the quantity of heat Ql
and gives up to the source M.2 the quantity Q.2 ; at the same time
the steam-engine takes from M2 the quantity of heat Q2 + q and
gives up to M^ the quantity Ql + q. Hence, in these circumstances,
the source of heat at higher temperature receives during each revolu-
tion the quantity of heat q, while the source at lower temperature
M2 gives up the same quantity of heat ; this transfer of heat q from
the lower to the higher temperature being effected without the doing
of work.
By this process, therefore, heat can be transferred from a colder
to a hotter body. This Clausius declares to contradict experience.
While heat invariably tends to flow from hotter to colder bodies,
in the process described above the opposite occurs. The objection
has been raised to this conception of Clausius that a thermoelectric
SECT, cxin.] CARNOT'S AND CLAUSIUS' THEOREM.
277
circuit, in which one junction is at the temperature 100° and the
other at 0°, can produce a current which will heat a platinum wire red
hot, so that heat passes from a colder to a hotter body, that is, to
the red hot platinum. Clausius answered this objection by asserting
that this transfer of heat to a higher temperature is' compensated
for by the heat generated at the points of contact.
Clausius therefore proposed this theorem : Heat can never pass
from a colder to a hotter body without the expenditure of work or the
occurrence of some change of state. Hence, by this principle of Clausius,
2 = 0, and therefore for any cycle of the sort described, whatever
body is used in it, we have (a) Ql/0l = Q2/Q2.
We can now show that a similar theorem holds for a cycle of
any sort. Suppose that the change of state of a body proceeds from
B along the curve EG (Fig. 13G). If the isothermal curve BD passes
through B and the adiabatic curve CD through C, we may replace
the path BC by the path BDC, that is, the body may first expand
at constant temperature along BD and then at constant entropy, that
is, without communication of heat, along DC. If BC is infinitesimal,
BD and DC are so also, and the change of state BC may be replaced
by the two changes BD and DC. On both paths the body receives
FIG. 136,
FIG. 137.
the same increment dU of internal energy. The external work is
in the one case BCC'B, in the other BDCC'B. But since B'C' = dv
is infinitely small, while B'B=p remains constant, the surface BDC
vanishes in comparison with the surface BCC'B. If dQ represents
the quantity of heat supplied, we have J.dQ = dU+p.dv along the
path BC, as well as along the path BDC.
Let BCPDEQ (Fig. 137) be any cycle, Be, Cc', Ed, Dd' isothermal
curves, and BE, CD, etc., adiabatic curves. Let the body receive
278 THERMODYNAMICS. [CHAP. xm.
the quantity of heat dQ along EC, and give up the quantity dQ.2
along DE. As we have already seen, the body would receive
by a change of state along Be the same quantity of heat dQv and
give up along dE the same quantity dQ.2. Therefore we have for the
cycle BcdE, dQl/6l = dQ.2/62, if 6l and 02 are the absolute temperatures
corresponding to the isothermals Be and Ed. In the same way we
have for Cc' and Dd', etc., dQ^/O^dQ^/0^ dQ1a/0l" = dQ/l02"i etc.
If Q and P denote the points at which the cycle touches two isothermal
curves, we have by addition (b) \dQ1/0l = Je^/02, where dQ1 is the
quantity of heat received along an element of QBCP and 6l the corre-
sponding temperature, dQ.2 the quantity given up along an element
of PDEQ and 0., the corresponding temperature. If the heat received
is considered positive and that given up negative, tJie sum of all infini-
tesimal quantities of heat received during the performance of a reversible
cycle, each divided by the absolute temperature at which it is received, equal*
zero, that is, (c) ^dQ/0 = 0. This is the second law of thermodynamics.
The theorem (c) which Clausius first expressed in this form may
be given in another way. Let ABCD (Fig. 138) represent a cycle,
so that f dQ/6 = Q. We divide the integral into two parts,
J A BCD A
of which the first is extended from A
over B to C. the second from C over
D to Aj and have
If, therefore, the body passes from the
state A to the state C, the value of
FlG 13g the integral \dQ/6 is independent of the
path. If 6l and i\ are the coordinates
of the point A, and 62 and v2 those of the point C, we have
Clausius introduced a special symbol for the entropy by setting
dS = J. dQ/e, from which jCdQ/6 = S2- Sr The function S represents
the entropy of the body ; it depends only on the state of the body
at any instant, and is independent of all previous states.
SECT, cxiv.] APPLICATION OF THE SECOND LAW. 279
SECTION CXIV. APPLICATION OF THE SECOND LAW.
We have already obtained [CX. (a)] the equation
(a) J.dQ = dU+p.dv.
If the state of the body is determined only by the independent
variables 6 and v, equation (a) may take the form
(b) /. dQ = (VUfdff). . dO + ((3 Ufdv)e +p)dr,
where the indices attached indicate that the quantities which they
represent remain constant during differentiation. If S denotes the
entropy, we have
dS = J. dQ/6 = 1/6 . (3 UfiO), .dO+(l/e. (d Upv)g +pi'B)dv.
Since S is here a function of v and B, we may set
(3S/30), = 1 JO . (3 £7/30), ; (dSf'dv), = 1/6. (dUfdv), +p/0.
But for the same reason we have also
and further 3(3S/30),/3t;=l/0. 3(3 tf/30),/ 30,
3(3S/3»)9/30= 1/0. 3(3C7/30)e/30- 1/02.
Whence it follows that (c) (3Z7/30)e = 02. 3(p/0),/30, since U is also
a function of 0 and v only, and
The internal energy must satisfy the differential equation (c). The
second law furnishes the means of determining the internal energy.
It follows from equations (c) and (b) that
(d) J.dQ = (3£//30X . d6 + (<BFd(pl&)Jde +p)dv.
Hence if the equation of state and the specific heat c,= l/J. (dU/W)f
are known, the quantity of heat required for a given change in the
state of the body may be determined by equation (d).
The quantity of heat which a body has received is not determined
by the state of the body at any given instant, and therefore cannot be
considered as a function of the coordinates. We can, however, set
(e) J(dQ[d0). = @Uj-dO)n J(-dQ/-dv)e = (-dU/-d-v)e+p,
since (dQ;"dO), . dO is the quantity of heat which is used in raising the
temperature by dd, while the volume remains constant, and (dQj'dv)e is
the quantity which is absorbed during an increase in volume by dv
at constant temperature. But 'd-QjWdv is not equal to
From equation (c) we obtain
- l/J.
280 THERMODYNAMICS. [CHAP. xm.
The differential equation (c) is applied to the relations of an ideal
gas, for which p/0 = E/v. From this relation (dUj"dv)e = Qt which agrees
with the results in CXI.
If the energy of a body at constant temperature is independent of
its volume, its equation of state, from (c), will have the form
P/e=f(v).
SECTION CXV. THE DIFFERENTIAL COEFFICIENTS.
As a rule the equation of state of a body is unknown. There are,
however, many bodies for which, within narrow limits, we know
approximately the relations of volume, pressure, and temperature.
Within these limits, therefore, an equation of state may be constructed
of the form (a) f(v, p, 0) = 0. If the pressure p is constant, we have
f(p, v + dv, 6 + d6) = 0 and 3//3» . dv + 3//30 .d8 = Q.
The ratio between dv and dO is written in the form (c3f/3$)p. The
volume v of the body is generally given in the form
where t=0-273. Hence we have
(St;/30), = i;(a + 2/3(0 -273) + ...)/(! +a(<9-273)+ ...).
If /3 is very small, we obtain (b) (dv/c)6)p = va. This formula can be
used even when a is a function of 6.
If the temperature 0 is constant in equation (a), we have
"dffdp . dp + 'df/'dv .dv = 0.
From this equation the change of volume due to change of pressure
at constant temperature can be determined. Since the volume always
diminishes when the pressure increases, (dv/'dp)g is negative. From
the theory of elasticity (cf. XXIX.) we have found that
XVv/v = -'dp and 6 = "dv/v = - (1 - -2k)3?>p/E,
in the case of fluids and solids respectively. Therefore
(c) (3r/3p)a=-r/X.
It follows from the equation of state of an ideal gas that
(dv[dp)e= -v/p,
and hence \=p.
If the volume of the body is constant, the pressure is increased by dp
by the rise of temperature dO ; we obtain in this way a third quantity
Besides these differential coefficients we must also notice
SECT, cxv.] THE DIFFERENTIAL COEFFICIENTS.
281
three others, Wj'dv, ty/'dv, and 30/3p, which are connected with those
already mentioned by the following relations :
(d) (B»/30),. (30/3^=1, W9p)..(3p/3»).= l, (3p/30)..(a0/3p).= l.
Let LM and NP (Fig. 139) represent two isothermal curves cor-
responding to the temperatures 9 and 6 + d6. We then have
(3j9/3#)fl = tg AEv,
if AE is the tangent at the
point A of the isothermal curve
whose parameter is 0. If BAD
is perpendicular to Ov, and ^46'
parallel to Ov, we have
tgAEv = -AD/DE= -AS/AC,
and hence
(3p/3r)fl = -AB/AC.
Further, we have
and hence we obtain (e) (3p/3w)8 . (dvfd6)p . (d0/ty). = -1.
Equations (d) and (e) show that if we know two of these differential
coefficients which are independent, the others are also known. Equa-
tion (e) may be derived in the following way : If p is considered
a function of v and 6, we have dp = (3p/3v)e . dv + (dp/W), . d8. Assum-
ing the pressure constant, so that dp = 0, we have
dv/d6 = - (3p/30)./(3p/di;)fl.
The quantity dv/dQ in this equation is that which has already been
designated by (dv/?>6)p • we therefore again obtain equation (e).
For gases we have (dppv)e = -p/v; (dvpO)p = E/p ; (deity), = v/R
These values satisfy equation (e).
For liquids and solids we have (dvj^>Q\ = av, (dvfty)e = — v/X, and
hence by equation (e), (f) (ty/W)v = aX.
SECTION CXVI. LIQUIDS AND SOLIDS.
If 6 and v are the independent variables, we have [CXI.]
(a) J.dQ = (3 U/W). .dO+((d U/dv), +p) . dv.
From the second law of thermodynamics we obtain as in CXIV. (c),
(b) (dU/c;v)g = 62 . 3(^/0),/30 = 0 . (ty[d6),-p. Hence equation (a)
takes the form (c) J.dQ = (pU[b9\. d6 + 6.(dp/W),. dv. If we designate
282 THEEMODYNAMICS. [CHAP. xm.
the specific heat at constant volume by cv we have (d) /. cv = (dU/'d0')v.
Equation (c) then becomes [CXV. (f)] (e) dQ = ca.d0 + 6aX . dvfJ. It
follows from equations (b) and (d) that
(f ) /. 3c,/dt> = 32 Upv'dO = 6 . 3(3p/30X/30,
and therefore from CXV. (f) that (g) .7.3e,/30=0.3(aA.)/30. This
may be obtained also from equation (e) by the use of the second
law. Equation (g) shows that c, is independent of the volume if
aA. is independent of the temperature.
In order to express the dependence of the quantity of heat com-
municated to a body on its pressure and temperature, we set
dv = (dvfdff)p . dO + (do[dp)9 . dp,
and then obtain from (c)
j.dQ={(d Z//30), + e . (dppe), . (dvpe)p}d0 + e . (dppo), . (3^/3/4 . dp,
or [CXV. (e)], (h) J.dQ= {Jcv- 0.(3?>/30)*/(30/3p)e}<Z0- 6.(dvjW)p.dp.
If cp is the specific heat at constant pressure, we have
Now since 'dv/'dp is always negative, cp is greater than cv, so long as
30/30 is not equal to zero; the case in which 30/30 = 0 is exhibited
by water at 4° C.
Introducing the values for the differential coefficients found in
CXV., we obtain
(i) J.dQ={Jc, + a?X.v8}d6-av6.dp and (k) cp = c, +
The temperature of a fluid or of a solid is changed by compression.
If we set dQ = 0 in equation (i) the rise of temperature dd due to
the increase of pressure dp is dO= +av6{cpJ.dp, that is, the tempera-
ture rises with increasing pressure if a is positive, that is, if the body
expands when heated ; if a is negative the temperature falls ivith increasing
SECTION CXVII. THE DEVELOPMENT or HEAT BY CHANGE OF
LENGTH.
If the pressure p is exerted on each unit of surface of the ends of
a solid cylinder, each unit of length of the cylinder is shortened by
p/f, where e is a constant. If / denotes the original length of the
SKCT. cxvii.] THE DEVELOPMENT OF HEAT. 283
cylinder at 0° C., its length L at the temperature 0 and under the
pressure p, if the limits of elasticity are not exceeded, is
(a) L = l.(l-p/t). (1 +£(0-273)),
where (3 is the coefficient of expansion.
If the length of the body increases by dL and its temperature
rises by dO, the quantity of heat dQ must be supplied to it ; the work
done by the elongation is A . p . dL, if A is the cross-section of the
cylinder. If M represents the mass of the body when the tempera-
ture is 6 and the length L, we have
(b) J.dQ = M.'dU/W.d8 + (M . 3 Z7/3L + Ap) . dL.
Applying the second law to this expression we obtain
-d(M/0 . 3 £7/30)/3£ = -d(M/6 . 3 UfdL + Aplff)fdO
or M . (d U/3L)g = 0*-.A. "d(pl8)JdO.
Hence J.dQ = M.(dUfd8)L.d6+8.A. (op/W)L . dL.
If 6 and p are taken as the independent variables, we have
dL = (dL/W)p . d6 + (3Z/3p)fl . dp and
J.dQ = [M. (dUfd8)L + B . A .
Since the deformation is very small, we have, representing by cp
the specific heat at constant pressure,
(c) J.dQ = JMcp .dO-BA. (dL/W)p . dp,
since by analogy with CXV. (e) (dpfd6)L . (30/3^ . (3£/3/>)e = - 1 .
If the pressure on the ends is increased by dp, so that the total
pressure is A.dp = P, and if there is no communication of heat, the
temperature of the body increases by d9 = 6PL^!JMcp, or, if m is
the mass of unit length, by dd=B/3P/Jmcp. If the cylinder is
stretched by the force P a corresponding cooling will occur.
SECTION CXV III. VAN DER WAAL'S EQUATION OF STATE.
The equation of state of an ideal gas is pv = B&, and its isothermal
curve is therefore a rectangular hyperbola. Real gases, however,
at low temperatures and under high pressures, do not conform to
this equation. Suppose that a certain quantity of gas at a given
temperature has the volume OC' (Fig. 140) and is under the pres-
284
THERMODYNAMICS.
[CHAP. xni.
PG
sure CO'. If the pressure is increased while the temperature remains
constant, the volume will be diminished. At last the space in which
the gas is contained becomes saturated with it ; let the corresponding
pressure be DD'. DD' is then the pressure of the saturated vapour or
the vapour pressure at the given temperature. If the volume is still
further diminished, the pressure remains constant, while a part of
the vapour passes over into the liquid state. At last all the vapour
is transformed into liquid; let the corresponding volume be OF.
So long as the vapour and liquid are in the same space, the isothermal curve
is a straight line parallel to the axis Ov. If the volume of the liquid
is now diminished, the pressure increases very rapidly ; the corre-
sponding isothermal curve is represented by FG (Fig. 140). Andrews
found, by experimenting with carbon-dioxide, that, as the temperature
rises, the line DF becomes shorter,
and that, at a certain temperature,
which he called the critical tempera-
ture, it disappears altogether. If
the temperature of the carbon-
dioxide remains constant, its state
changes along curves which are
represented in Fig. 141. The
abscissas represent volumes, the
ordinates pressures. For example,
let us examine the isothermal
0 — ^~> — TV ~£, — curve ABCD, which corresponds
to the temperature 15'1°C. ; at
the point A the carbon-dioxide is
still in the gaseous state; at B it may be considered as saturated
vapour. If the compression is continued, condensation begins, and
the pressure remains constant until the substance has become liquid,
that is, until its state is represented by C. From C on, the pressure
increases very rapidly as the volume is diminished. At the tem-
perature 21-5°C. the condensation begins at B, and the horizontal
part of the curve is shorter. At 31'1°C. the horizontal part of the
isothermal curve vanishes ; the critical temperature has now been
reached. Isothermals corresponding to higher temperatures are
continuous curves ; it is therefore impossible to reduce carbon-dioxide
to the liquid state at a temperature higher than 31-1°C. At higher
temperatures than this there is no apparent difference between its
liquid and gaseous states. The liquid, at the critical temperature,
has the same density as the saturated vapour. A gas can be reduced
SECT, cxvni.] VAN DEE WAAL'S EQUATION OF STATE.
285
to the fluid state by compression only when its temperature is lower than
the critical temperature.
James Thomson substituted for the isothermal curve here described
a continuous curve CDHEJFG (Fig. 140); the part DHEJF cor-
responds to an unstable state. It appears from various investigations
on the relations between vapours and their liquids at the boiling
point, that it is possible to obtain a vapour in the states represented
FIG. 141.
by DH and FJ, while the states represented by HEJ are always
unstable, since in these states the pressure and volume change in
the same sense.
J. Clerk Maxwell called attention to an important peculiarity of
these isothermals which may be deduced by applying the laws of
thermo-dynamics. If a gas traverses the cycle FEDHEJF, in passing
along the straight line from F to D it receives the quantity of heat
286 THERMODYNAMICS. [CHAP. xm.
L, and in passing along the curve DHEJF gives up the quantity
L' ; the temperature is the same along both paths. Since the gas
has traversed a complete cycle, we have \dQ/0 = L/0 - L'/O = 0, and
therefore L = L'. Therefore, since no heat is used in this cycle, no
work can be done, and hence the surface FJE is equal to the surface
DEE. Thus, if the isothermal curve CDHEJFG is given, the
maximum pressure of the vapour can be determined, by determining
the line FD so that the surfaces FJE and DHE are equal.
Van der Waals has proposed an equation of state for gases, which
represents, more exactly than the simple one, the behaviour of the
gas and which permits the calculation of the critical temperature.
The volume of a gas is determined not only by the external
pressure but also by the attraction of its molecules ; we may think
of this attraction as replaced by a pressure p added to the external
pressure p. Since the attracting and attracted molecules approach
one another as the density increases, p' must be directly proportional
to the square of the density, and therefore inversely proportional
to the square of the volume. Hence we set p' = a/v*, so that the
total pressure of the gas is p + a/v2. Further, the molecules are not
free to move everywhere in the region v, for they themselves occupy
part of the region. Van der Waals assumed that the volume of a
fluid cannot fall below a certain limit without the particles losing
their freedom of motion. In place of the apparent volume v, he used,
as the true or effective volume, v - b, where b is a very small quantity,
though much greater (about 4-8 times) than the volume of all the
molecules of the gas. We thus obtain the equation of state
If the volume v is very great, this equation becomes the equation
of state of ideal gases.
The positions of the points H and / (Fig. 140), at which the
tangents to the isothermal curves are parallel to the r-axis, are
obtained from the equation dpfdv — Q or (b) p + a/iP — 2a(v— fc)/t'3 = 0.
This is the equation of the curve which passes through all points
at which the tangents to the isothermal curves are parallel to the
axis Ov. All these isothermal curves correspond to temperatures at
which the body can be either liquid or gaseous. When the two
points coincide we reach the critical state. But since the two coin-
cident points must have a line joining them parallel to the axis Oi;
we introduce the condition for the critical state by setting dp/dv,
obtained by differentiating the foregoing equation (b), equal to zero, or
•<c) 6rtv - bv* - 4rtV3 = 0.
SECT, cxvui.] VAN DER WAAL'S EQUATION OF STATE.
287
If vl denotes the critical volume, that is, the volume of unit mass
of the gas or liquid at the critical temperature, we obtain from the
last equation (d) vl = 3b. The critical temperature 6V and the critical
pressure plt which must exist in order that the fluid shall not boil
at a temperature which is lower by an infinitesimal than the critical
temperature, are (e) 01 = l/E.8a/27b, pl = a/2762. These values are
obtained by introducing the value of t\ in (b) and (a).
Choosing vv 0:, and pl as units of volume, temperature, and
pressure, we may set
v = V*i = V. 3b, p = PPl = P. fl/27ft2, e = T6l = T. 8aj27bR,
and thus give to the equation of state the form
(f) (P + 3/r2)(3F-l) = ST.
From this equation we obtain the following law : If the critical pressure
is taken as the unit of pressure, the critical volume as the unit of volume,
and the absolute critical temperature as the unit of temperature, the iso-
•thermals of all bodies are the same. We will now consider some applica-
tions of formula (f).
(a) Equation (f) may be written in the form
The following table shows how the product PV depends on P
when the temperature is constant :
1
F
r=l-0
r=i-5
r=2-o
r=2-5
T=3'0
PV
P
PV
p
PV
P
PV
P
PV
P
o-o
2-67
o-oo
4-00
o-oo
5-33
o-oo
6-67
o-oo
8-00
o-oo
0-2
2-26
0-45
3-69
0-74
5-11
1-02
6-54
1-31
7-97
1-59
0-4
1-88
0-75
3-42
1-37
4-96
1-98
6-50
2-60
8-03
3-21
0-6
1-53
0-92
3-20
1-92
4-87
2-92
6-54
3-92
8-20
4-92
0-8
1-24
0-99
3-06
2-44
4-87
3-90
6-69
5-35
8-51
6-81
1-0
1-00
1-00
3-00
3-00
5-00
5-00
7-00
7-00
9-00
9-00
1-2
0-85
1-01
3-07
3-68
529
6-35
7-51
9-01
9-73
11-68
1-4
0-80
1-12
3-30
4-62
5-80
8-12
8-30
11-62
10-80
15-12
1-6
0-91
1-46
3-77
6-03
663
10-60
9-49
1538
12-34
19-75
1-8
1-27
2-28
4-60
8-28
7-93
14-28
11-27
20-28
14-60
26-28
2-0
2-00
4-00
6-00
12-00
10-00
20-00
14-00
28-00
18-00
36-00
The results of this table may be represented by a figure, in which P
is the abscissa and PV the ordinate ; in the neighbourhood of the
288
THERMODYNAMICS.
[CHAP. xin.
critical temperature T=l, PV is very variable, and the departures
from the ordinary laws are in consequence very considerable. As
the temperatures rise the relations change rather rapidly, PV becoming
nearly constant. The product PV has a minimum value, which in
some cases is reached only when the pressure is negative. This
minimum is determined by dPVIdF=3/V*-8T/(3V- 1)2 = 0. PV
is therefore a minimum if 3F-1/FW8T/3.
The corresponding values of P and F", which we may designate
by Pm and Vm, are, by the use of the equation of state, equal to
Vm = l/(
Pm = 3(2v/8773 - 3)(3 -
and hence PmFm=6x/8f/3 - 9.
Hence we obtain lor
T=l-0
Pw=l-09
1-5
3-00
3-00
2-0
3-35
4-86
2-5
2-72
6-49
3-0.
1-37.
7-97.
As an example of the application of the relations here developed,
the following table has been calculated for 18°C. or 0=291 :
Crit. Temp.
ei
Crit. Press.
P,
T
Pm P=Pmpi
Hydrogen,
33
7-3
neg. neg.
Nitrogen,
127
33
2-29
3-08 ! 102
Carbon -monoxide, -
132
36
2-20
3-19 | 115
Oxygen, -
155
50
1-88
3-37
169
Nitrogen-monoxide,
179
71
1-63
3-21
228
Methane,
191
55
1-52
3-05
168
Ethylene,
263
51
I'll
1-68
86
The product of pressure and volume therefore diminishes as the
pressure increases, if the pressure is less than 102 atmospheres. If
the pressure is greater than this the product increases with the pressure.
(b) The coefficient a,, which is called the coefficient of pressure
(formerly called the coefficient of expansion at constant volume), and
which denotes the change of pressure for a rise of temperature of
1° C. at constant volume of the gas, is determined in the following
way : We have
a. = 1/p . -dp/W = 1/^P . VPfdT= 1/0! . 8/(3PF- P)
SECT, cxvin.] VAN DER WAALS' EQUATION OF STATE. 289
For very great values of V, a, = 1/6*, corresponding to an ideal gas.
Further a, . 6l = l/(T- f . 1/F+ 1 . 1/F2). Van der Waals defines car-
responding states as those in which the volumes, temperatures and
pressures of both gases are in the same ratio to the same quantities
in the critical state, that is, in which
v/v, - *//< - r, p/Pl =P'iplf = P, BIOI = &/e{ = T,
where the quantities v', »/, p', etc., refer to the second gas ; thus we
have <i,6l = a,' 6^ or a,/a,' = &1'/Bl. Hence the coefficients a, and a,' in
corresponding states are inversely as the critical temperatures.
(c) The coefficient of expansion ap, which represents the increase
of volume for a rise of temperature of 1°C. at constant pressure, is
defined in the following way :
With increasing values of F", ap approaches the value ar If the bodies
are in corresponding states, we have apdl = ap'6l'. If the basis for
this method is sound it should find application in the expansion of
liquids by heat. Since changes of pressure have only slight effect on
the volume of liquids, it is sufficient to compare the coefficients of
expansion at corresponding temperatures. The calculations made
by van der Waals have shown that this law is in essentials correct
for liquids whose critical temperatures are known.
(d) The pressure of saturated vapours. — To determine the pressure of
saturated vapours we use the above-mentioned theorem of Maxwell,
which states that the surface F'FEDD' (Fig. 140) is equal to the
surface F'FJEHDH, that is, setting
OF'=TV OD'=Vv and F'F=D'D = PV
we have P F! - F = *P . d V.
Effecting the integration and using the equation of state (f) it follows
that
(g) fs
For the points F and D the following equations hold :
(Pl + 3j F12)(3 ri-l) = 8T and (Pl + 3/ F22)(3 F2 - I ) = 8T.
If F! and F2 are eliminated from these equations, a relation is obtained
between the pressure Pl of the saturated vapour and the tempera-
ture T. We may therefore conclude with van der Waals that : If
for different bodies the absolute temperature is the same multiple of the
critical temperature, the pressure of their saturated vapours -is also the same
multiple of their critical pressures. Similar laws hold for the relation
290 THERMODYNAMICS. [CHAP. xin.
between the volume of the saturated vapour and its pressure and
temperature.
On the basis of van der Waals' investigations, Clausius has pre-
sented a slightly altered form of the equation of state, viz.,
This equation represents the actual relations better than the other,
but leads to essentially the same results. Starting from the equation
(h) J.dQ = (dUfdO)je + ((dU/'dv)e+p)dv, and applying the second
law we obtain the relation [CXIV. (c)], (3 U/^ = P . 3(^/0)./30.
From Clausius' equation p/0 = E/(v -&)- a/ (P(v + {$}-, and hence
(i) (dU/Vv)6=2a/6(v + W.
If the temperature increases by dd and the volume by dv, the internal
energy increases by
(k) dU=Jc,.dO + 2a/e(v + py2.dv,
where Jc, = (3 UfdO), is an unknown function of 6 and v. If the
changes of temperature are slight, c, may be considered constant, as
is shown by observation. If the temperature increases from 0 to
0 + A0, while v increases from ^ to v.2, the increment A?7 of the
internal energy is approximately equal to
(1) A U= Jc, . A0 + 2a/B .(Ifa- l/«2).
If the gas expands without resistance the internal energy remains
constant. Hence, if we set A £7 = 0 in equation (1), we have
(m) A0 = - 2a/Jc,0 . (Ifa - l/v2).
In this case the temperature falls.
SECTION CXIX. SATURATED VAPOURS.
If the volume of a gram of a certain substance, or its specific volume,
is called vl when it is a vapour and v.2 when it is a liquid, the volume
v occupied by a gram of liquid and vapour is (a) v = vlx + v2(\-x),
if the volume contains x grams of vapour and therefore (1 -x) grams
of liquid. If U± is the internal energy of the vapour, U2 that of
the liquid, the internal energy of the mixture of the two is
(b) U=UlX+UJl-x).
So long as the vapour is saturated, its internal energy and pres-
sure depend only on the temperature; the pressure and volume
of the liquid are also determined by it. Hence, for a mixture of
SECT, cxix.] SATUEATED VAPOUKS. 291
liquid and saturated vapour, we have (c) p=f(Q), where p is the pres-
sure of the saturated vapour at the temperature 0, and consequently 6 is
the boiling point of the liquid under the pressure p.
That quantity of heat which is needed to transform one gram of
liquid into vapour at constant temperature 6 and under the corre-
sponding pressure p is called the heat of vaporization L. This heat
is partly used in increasing the internal energy, partly in doing
external work. If U.2 denotes the energy of the liquid, U-^ that of
the vapour, at the temperature 6, the internal energy is increased by
£/! - U.-, by the transformation. Since the pressure p is constant,
the external work is I pdv =p(vl - v2). The work needed to evaporate
a gram of liquid is (d) J.L=Ul- U^+p^-v^.
If a mixture of liquid and vapour receives heat and also changes
its volume, the increase of internal energy, since 0 and x both
vary, is
<e) dU=(U1- U.2)dx + (
Since, from equation (a),
(f ) dv = (i\ - v.2)dx + (x .
it follows from the equation J. dQ = dU+p. dv that the heat imparted
to the mixture is determined by
( /. dQ = {x(dUl/W+p . S^/30) + (1 - x)(dU2/W+p .
\
Vi- U.2-pv.2}dx.
If in this equation we set dd = Q, and integrate from x = 0 to x=\,
we obtain equation (d). If equation (g) is brought into the form
J.dQ = Q.dd + X. dx, it follows, from the Carnot-Clausius theorem,
(JT/0)/30. But we have
= 1/0. (3 UJV0 +p .
c>(Z/0)/c>0 = 1/0 . (3 UJW +p .
Hence it follows that (h) U^ - U^(v^ - »2) . 0 . 3p/30 -p(v1 - »2), and
thus the difference between the internal energy of the vapour and
that of the fluid is determined.
We obtain from equations (d) and (h), (i) JL = (vl -vz)B . 'dp/W.
Now we know by observation that ^ > v0, so that 3p/30 is positive.
The boiling point is therefore higher, the higher t)ie pressure.
We may also apply equation (i) to the process of melting. In
that case i\ denotes the volume of the liquid, vz that of the solid.
We must here distinguish between two kinds of substances, those
292 THERMODYNAMICS. [CHAP. xm.
like wax, whose volume increases during melting, and those like ice,
whose volume diminishes during melting. For the former i\ > r.2,
and therefore 'dpI'dB is positive; for the latter i'l<r.2, and therefore
3^/30 is negative. For those substances whose volume increases during
melting, the melting temperature rises as the pressure increases. For those
substances whose volume diminishes during melting, the melting temperature
falls as tlie pressure increases.
If the volume is always filled with saturated vapour only, it
follows from (g), since «=1, that (k) J.dQ = (dU1l'd6+p.'drlfd6)d0.
Hence dQ is the quantity of heat which must be imparted to the
vapour that its temperature shall increase by dO while it remains
saturated.
From equation (d) we have L\- U^ = JL~p(vl-v2). If c denotes
the specific heat of the liquid, and if k is a constant, we may set
If we consider v2 as constant, it follows that
-p .
From equations (i) and (k) we then have dQ = (dL/d8- L/6 + c)d8.
Designating by h the quantity of heat which must be used in raising
the temperature of the vapour by 1°C. while it remains saturated,
we have (1) h = dL/dO - L/6 + c.
SECTION CXX. THE ENTROPY.
The methods which have here been applied to the discussion of
the equilibrium of a fluid and its vapour may be used to advantage
in many other cases, especially in connection with chemical problems.
All the methods are based on the equation Jd$/0 = 0 for a cyclic
process. M. Planck has given general formulas by which treatment
of such questions is much facilitated. The bodies whose chemical
equilibrium is to be investigated are contained in the volume V at
the temperature 6, and are subjected to an external pressure. A
change in the chemical composition, or in the proportions of the
mixture, is accompanied by a change of volume </Fand a change of
temperature dO, and at the same time the quantity of heat dQ is
received from surrounding bodies. If S denotes the entropy and V
the internal energy of the system, we have (a) dS=(dU + P.dV)jO^
The state of the system of bodies is determined by the pressure
P, the temperature 6, and certain other variables n, n^ n.» etc. If,
for example, the space contains water and saturated water vapour
SECT, cxx.] THE ENTROPY. 293
and if the whole mass equals M, we may call the quantity of vapour
Mn and the quantity of liquid Mnr where n + n1 = l. If we are
dealing with a case of dissociation, we may use n for the number of
molecules of the original gas, while n^ and n2 are the numbers of
the dissociated molecules. Hence the state of a system depends
generally on the quantities 0, P, n, nv n2... , and we have
(b) dV= -3VIW . dO + -dVj-d
[ dS = 3S pB .dd + 'dS /oP .dP + 'dS fdn .
From the definition (a) of the entropy we have
l/e.(dU/W +P.
l/0.(dUfdF+ P .
since 0 and P are independent. It follows from (a) and (b), since
6 and P do not depend on n, nv n.2..., that
'dS/'dn .
If we set (c) & = S-(U+PF)/Q, this equation takes the form
(d ) 3*/9n . dn + 'd^fdnl . dn^ + ?&l'dnz . dn,2 + . . . = 0.
If the quantities n, nv n.2... are independent of each other, we have
3^/d/i = 'dSj'dn -l/6.(d Uj'dn + P . 3F/3w)
and analogous equations, which may also in this case be obtained
directly from (a). In general, there will be some relation among
the quantities n, nv n.^ ____ As an example of this method, we will
consider the problem of the change of state. If a quantity of vapour
Mn and a quantity of liquid Mn^ are enclosed in a given volume,
we have as above n + itl = l. Then the following equations hold
S = Mns + Mn^, U = Mnu + Mn^i^ V= Mnv + Mn^,
where $, it, v denote the entropy, the internal energy and the volume
of the vapour respectively, while sv uv and i\ denote the same
quantities for the liquid. From equation (c) we then have
<£ = Mn(s - (u + Pv)/0) + Mnfa - (wa + PvJ/6)
and 0 = M(s - (u + Pv)/0)dn + ^/(^ - (MJ + Pvl}l6)dnr
In addition, we have dn + dn^ = 0. Hence equilibrium exists between
the vapour and the liquid if (e) sO - u -Pv = s^ - ul - Pvr Since
the quantities in this equation depend only on P and 0, it may take
the form P=f(0). Hence the equation (e) states the way in which
the pressure of the saturated vapour depends on the temperature.
294 THERMODYNAMICS. [CHAP. xin.
If the unit mass of the substance is transformed from liquid to
vapour at the temperature 0, the entropy increases by (f) s-sl = JL/0.
We therefore obtain from equation (e) JL = u-ul + P(v-vl), as in
CXIX. (d). If equation (e) is differentiated with respect to 0, and
P considered as a function of 6, and if we use the equations
= 1/0 . (3tt/30 + P . 30/30) ;
= 1/0 . (3MJ/30 + P . 3rj/30)
we obtain the relation J.L = (v-v1)6.'dP/W.
We may consider the method here described as having its basis
in a certain tendency in nature. Almost all natural processes are
accompanied by the development of heat ; hence energy seems to
be especially inclined to assume the form of heat, and heat tends
to pass from bodies of a higher to those of a lower temperature.
In all such transformations the total energy remains unchanged, but
it loses more and more the capacity of transforming itself into kinetic
energy. A body which contains a quantity of heat Ql and whose
temperature is 0P while the temperature of all surrounding bodies
is 02, may yield [CXII. (e)] the kinetic energy ^(0a - 02)/0r Hence
the lower 01? the less the kinetic energy yielded, if 02 remains
constant.
Clausius expressed this principle in the statement that the entropy
always increases and tends toward a maximum. For example, if a quantity
of heat Q passes by conduction or radiation from a body at the
higher temperature 6l to one at the lower temperature 02, the increase
of entropy is &S=Q/d2-Q/Or The entropy remains unchanged only
in the case of a cycle, in which the bodies receiving the heat have
the same temperature as those giving it up, and in which the whole
system is in neutral equilibrium, since after the performance of this
cycle \dQ/6 = 0 or A£=0. This is also the case at any instant
during the cycle, if we take into account not only the entropy of
the working body, but also that of surrounding bodies ; if the first
receives the quantity of heat dQ, the second gives up the same
quantity; since the temperature of both bodies is the same, the
entropy remains unchanged. This holds, however, only for ideal
cycles; in any actual movement of heat, the heat passes from a
higher to a lower temperature, and the entropy must therefore
increase.
Hence the condition for a change in the state of a system is an
increase of the entropy; changes in which the entropy diminishes
are impossible. The state of a body in equilibrium is such that if
SECT, cxxi.] DISSOCIATION. 995
it undergoes a small change, the entropy will either increase or
remain constant.
We thus return to the conditions of equilibrium (a). By the
communication of the quantity of heat dQ the entropy of the body
considered diminishes by dS; hence the increase of the total entropy
hdS-dQ/6, and this must be equal to zero. Since dQ = dU+P.dV
we obtain dS-l/0.(dU+P.dF) = 0.
SECTION CXXI. DISSOCIATION.
If a compound gas is separated into two or more constituent gases,
either by heating or by diminution of pressure, it is said to be dis-
sociated ; the extent of the dissociation depends on the pressure and
temperature. In order to determine it, we must determine the
function (a) 3? = S - ( U +PV}jQ. If n denotes the number of molecules
of the original gas, n1} n2... the numbers of molecules of the pro-
ducts of dissociation, we have, as an expression for the total internal
energy U, U=nu + nlul + n2u2 + ..., where u, uv u2 denote the energies
of the separate molecules. If m is the mass of a molecule, and c,
its specific heat at constant volume, we may set the internal energy
at 6° C. equal to Jmc,0 + /t, when A is a constant. If we represent
Jmc, by c, we have (b) U=n(cO + h) + nl(clQ + h1) + .... If v denotes
the volume of a molecule of a gas and p its pressure, we have pv = E6,
where R does not depend on the nature of the gas. For the sake
of simplicity Planck sets R=\. Since the gases are uniformly dis-
tributed throughout the whole volume, we have nv = nlv1 = ... = T,
and therefore pV= n6, plV= r^O, etc., (c) PV= (n + 1^ + n2+. . .)(?. By
this equation V is given as a function of P, 0, n, nl....
The entropy of the system is equal to the sum of the entropies of
all the gases, so that S=ns + nlsl + n2s2+ ..., if s denotes the entropy
of the molecule. Using c with the meaning given above, the entropy
of a molecule equals [CXI. (d)] c . log 0 + log v + k. Here nv = V, and
therefore by the use of equation (c),
s = c . log e + log(e/P . (n + n1 + n2 + . ..)/») + k.
If we set
n.2 +...), etc., (7+ C^
we have (d)
296 THERMODYNAMICS. [CHAP. xin.
By the use of equations (b), (c), and (d), (a) takes the form
n[(c + l)(log 8-I)-\ogP-logC + k- h/ff]
f *
I + njfa + l)(log 0 - 1) - log P - log C, + jfcj - V*] + . . -
But we have 3(«. logC+i^ AogC1 + ...)/3/i = log<7, and further
(f ) cX£/cto = (c + 1 )(log e~l)-logP-logC+k- h/e.
Similar expressions hold for the other differential coefficients. Intro-
ducing these values in the condition of equilibrium
3*/3» . dn + t&rdn^ . dn^ + . . . = 0,
it follows that the problem may be solved if we know the relations
existing among the quantities n, nlt n2 —
If we investigate, for example, the dissociation of hydriodic acid
into iodine and hydrogen, the proportions of the gases in the
mixture can be represented in the following way : nJH, n-^H^ ».*/.>.
By dissociation two molecules of hydriodic acid form one molecule
of hydrogen and one of iodine, hence the ratio between dn, dnv dn.2
is - 2 : 1 : 1. If we set generally dn : dn-^ : dn.2 : ... = v : vl : v2 : . . . , the
condition of equilibrium becomes
(g) v . oQ/dn + Vl . 3*73^ + v2 . 3*/3»3 + . . . = 0.
From equations (f) and (g) we obtain
(h) 2[v(c + l)(log 6 - 1) - v log P - v log C+ vk - vh/ff] = 0.
To simplify the calculation Planck assumed that the atomic heat
is constant even in the compound gas, and that the molecular heat
is equal to the sum of the atomic heats ; experiment shows that this
is approximately true in all cases. If a, a1? a2 denote the number
of atoms in the molecule of each gas, we may set
C = ya, Ct = yav c.2 = ya.2....
Since the whole number of atoms is unchanged by dissociation, the
sum na + nlal + n2a2+ ... is constant, and hence
a.dn + al. dn^ + a2 . dn.2 + . . . = 0.
Consequently also va + v^ + v2a2 4- . . . = 0 and vc + vlcl + v.2c.2 + . . . = 0.
Further if we set
v + vl + v.2 + v3+ ... =VQ; vh + v1hl + vji2+ ... =
it follows from (h) that
SECT, cxxi.] DISSOCIATION. 297
Hence for hydriodic acid we have
v0 = 0, and ClCzICz = k0.h™.
If no hydrogen or iodine is present except that set free by dis-
sociation, we have nl = n2, and therefore Cl = C.2 and C1/C=-Jk0.h01'e.
Now (71/C'=n1M. Hence the degree of dissociation is independent
of the pressure, but increases with the temperature. In this case,
however, the dissociation can never become complete, since, for 0 = o> ,
we have Cl/C=»Jk^.
From equation (i) the pressure has no influence on the degree of
dissociation if the total volume remains unchanged by the dissociation.
This is the case when v0 = 0. If, on the other hand, the volume
increases during the dissociation, any increase of the pressure will
lessen the degree of dissociation. This occurs in the case of nitrogen-
dioxide, N004, in which one molecule is broken up by the dissociation
into two molecules NO2. Hence v= - 1, Vj = 2, and therefore
c^/c=k0.h^.e/p.
This equation, together with Cl + C=\, determines the degree of
dissociation.
In order to occasion the dissociation determined by the quantities
dn, dnv dn2, at constant temperature and at constant pressure, the
quantity of heat dQ is required, which is determined by
or, from equations (b) and (c), by J . dQ = 2(c0 + h)dn + 6 . *2dn. Since
-ir = 0, the quantity of heat required for the dissociation determined
by the quantities v, vv v2... is determined by
(k) J.Q = vh + v1hl + vJi2+... + v00=v08-logh0.
We reach the same result from the equation J.dQ=B.dS together
with the relations (d) and (h).
CHAPTER XIV.
CONDUCTION OF HEAT.
SECTION CXXII. FOURIER'S EQUATION.
IF the temperatures of the different parts of a body are different,
a gradual change goes on until the temperature of all parts of the
body is the same, that is, until equilibrium of temperature has been
reached. In this statement it is assumed that the body neither
receives heat from surrounding bodies nor gives up heat to them.
The rate at which the condition of equilibrium is reached depends
upon the facility with which the body conducts heat. Without
making any assumptions on the nature of heat, we may say that
heat flows in a body until a state of equilibrium is reached. We
define the rate of flmv of heat in any direction, as that quantity of
heat which passes in unit time through unit area perpendicular to
that direction.
Hence, if Q represents the rate of flow of heat through an area
dS within the body, the quantity of heat which will pass through
that area in the time dt is Q . (IS . dt. If Z7, V, W are the components
of flow in the directions of the coordinate axes, the quantities of
heat which pass through the elementary areas dy . dz, dx . dz, dx . dy,
in the time dt, are U.dy.dz. dt, V . dx . dz . dt, and W.dx.dy.dt
respectively.
Using the general equations (XIV.) of fluid motion, we obtain
for Q (a) Q = lU+mV+nllf', where /, m, n are the direction cosines
of the normal to the elementary area dS.
Let 00' (Fig. H2) be a rectangular parallelepiped, whose edges
OA = a, OB = b, and 00 = c are parallel to the coordinate axes. If
U, V, W represent the components of flow at the point 0, those
at A are U+'dU/'dx . a, F+^Vj'dx.a, W+'dWJ'dx.a respectively,
supposing a, b, c so small that only the first terms in the expansion
CHAP. xiv. SECT, cxxii.] FOUEIEE'S EQUATION.
299
need be retained. The parallelepiped receives the quantity of heat
U.dt.bc in the time dt through the surface OB AC, and loses the
quantity (U+'dU/'dx. a)bc. dt, which flows out through the surface
AC'O'B' in the same time. The parallelepiped gains, on the whole,
the quantity - 9 U/'dx . a.bc.dt. If we take account of the other
surfaces, the quantity of heat which remains in the parallelepiped
is -(Bl7/aa;+3F/3y+a#7a»).a6e.&J or, if we set a.b.c = dv,
- (d U/dx + "d Vj-dy + 3 Wfiz)dvdt.
This quantity of heat raises the temperature of the parallelepiped
by d6, which, if c denotes the specific heat of the body, and p its
density, is determined by the following equation,
(b) cp . dd = -
This equation holds only if the heat received is used solely in
causing change of temperature and does not produce any change
in the state of aggregation or any chemical change. Sometimes,
too, heat exists in the interior of a body which has not penetrated
into it in the form of heat, but is produced by friction or by an
electrical current in the body ; and to this the above equation does
not apply.
The components of flow U, V, W depend on the distribution of
heat in the body and on the nature of the body. If the body
conducts heat equally well in all directions, that is if it is isotropic,
we may determine the rate of flow in the following way. Let
A and B be two points within the body infinitely near each other,
in which the temperatures are respectively 9 and &, If dv denotes
the distance between the points A and B, and k the conductivity of
the body for heat, the rate of flow of heat in the direction AB is
300 CONDUCTION OF HEAT. [CHAP. xiv.
given by Q = k(0- &)/dv. Hence the condi.i<-tint>/ is the quantity of
heat which flows in unit time through unit area of a surface in the
body, parallel with and between two surfaces whose temperatures
differ by 1°, and which are distant from each other by one centimetre.
Now since & = 6 + ddjdv . dv, we have also (c) Q= -k.dOldv. We
obtain in like manner for the components of flow U, V, IV, the
expressions (d) U= -k.W/Vx, V= -k.'dBj'dy, fT= -k.'dO/'dz. In
actual cases the conductivity k is a function of 0 • but for the sake
of simplicity we will assume that k is constant. We obtain from
(b) and (d)
(e) cp . Wfit = k(d-epx2 + 320/9/ + 320/3z2).
This equation was first given by Fourier, and is therefore called
Fourier's equation. The specific heat c, the density p, and the con-
ductivity k are functions of 6; we will, however, consider them
constant. Fourier's equation may also take the form
(f) 30/ctf = K2(920/9z2 + 320/3/ + 920/^2),
where (g) Kz = k/cp.
In the following table the values of k and K for several metals at
the temperatures 0° and 100° C. are given from the experiments of
L. Lorenz :
N>
*m
*0
AT100
Copper, . .
0,7198
0,7226
0,909
0,873
Tin, . . .
0,1598
0,1423
0,392
0,344
Iron, . . .
0,1665
0,1627
0,202
0,179
Lead, . . .
0,0836
0,0764
0,242
0,222.
SECTION CXXIII. STEADY STATE.
The state of the body with respect to heat is called steady, if the
temperatures of the different parts of the body are different, but
do not change with the time. In this case each particle gives up
on the one side as much heat as it receives on the other, and the
temperature is independent of the time t and dependent only on
the coordinates x, y, z. For the steady state, equation CXXII. (f)
becomes (a) 320/d.c2 + 320/3/ + 32^2 = V2# = 0. The components of
flow are expressed by equations CXXII. (d).
Flow of lieat in a phite. — We will consider a thin plate whose faces
L and M are parallel to the yz-plane. The temperatures of the faces
are respectively 6l and 02. This being so, the flow of heat is parallel
to the .r-axis, and the temperature 0 in the vicinity of the .T-axis
SECT, cxxin.] STEADY STATE. 301
depends only on x, so that from (a) we have d20jdx2 = 0. Hence
0=px + q. If the distances of the faces L and M from the y^-plane
are a and b respectively, we have 01=pa + q, 02=pb + q, and further
6 = (bO} - a0z)f(b -a)- (61 - 82)x/(b - a).
If we represent the distance b - a between the faces by e, the rate
of flow of heat U between them is (c) 17=^-0^/6.
Every integral of equation (a) corresponds to a steady state of heat.
If 8=f(x, y, z) is an integral of (a), 01=f(x, y, z) and 02=f(x, y, z)
are the equations of two surfaces of constant temperatures, or of two
isothermal surfaces, where Ol and 02 are constant. If the body is
bounded by the surfaces which are determined by 0l and 6.2, and
if 6 is a temperature which lies between 0l and 02, 0=f(x, y, z) is
the equation of any isothermal surface.
The flow of heat in a sphere. — If m and c are constant, and if
r2 = x2 + y2 + z2, 0 = m/r + c
is a solution of (a). Therefore setting 0l = m/rl + c, 09 = m/r.2 + c, we
>-^-^
as the equation of the system of isothermal surfaces, which in this
case are spheres. For the rate of flow of heat U in the direction r,
we have (e) U= -k. d0/dr = k(0l - 02)rlr2/r2(r<i - 1\). The temperature
and flow of heat in a hollow sphere, whose internal and external
surfaces are at the temperatures 0l and 02 respectively, are also
given by equations (d) and (e). The total quantity of heat which
flows out through the hollow sphere is ^Trr2U=4irk(0l- 0.2)rlr2/(r2 - ra).
The flow of heat in a tube. — If c and c are constants and if r2 = x2 + y-,
we have, from XV., 0 = clogr + c' as an integral of (a). Therefore, if
we set 6l = c.\ogrl + c', 02 = c . log r.2 + c', we obtain
(f ) 8 = (0l- 02)log /-/(log r, - log rz) + (^log r2 - 02log r1)/(log rz - log r,}.
The rate of flow U in the direction r is U=k(8l- 82)/r(\ogr2-\ogrl).
The quantity of heat which flows out through a unit length of the
tube is (g) 2irrCT=2^(01-02)/(log»3-logr1).
SECTION CXXIV. THE PERIODIC FLOW OF HEAT IN A GIVEN
DIRECTION.
If the temperature of the body depends only on one coordinate,
say on x, Fourier's equation becomes (a) 'dO/'dt = K2 . 'd20/'dx2. We will
hereafter investigate in what way this equation can be integrated.
302 CONDUCTION OF HEAT. [CHAP. xiv.
For the present we will consider special integrals which correspond
to simple yet important cases.
The temperature of the earth changes during the year ; it rises
and falls with the temperature of the air. The time at which the
maximum or minimum temperature at any point is reached is later
as the point lies further below the surface. In the following dis-
cussion we will not take into account the internal heat of the earth.
If the temperature at the earth's surface is given by (b) 0 = sin at,
we may express the temperature at a point in the interior of the
earth by (c) 6 = P . sin at + Q . cos at, where P and Q are functions of
the distance x of that point from the earth's surface. If we substitute
for 9 in (a) the expression (c) we have
Pa. . cos at-Qa. sin at = K2(sin at . d^P/^x2 + cos at . d^Q/dx2).
Hence we must have *c2 . d2P/dx2 = - Qa and K*.d2Q/dxz = Pa. Now
if we set e2 = a/*2, we have (d), (e), d*P/dx* = - e4P and Q = - 1/c2 . d2P/dx?.
In order to integrate equation (d) we set P = Aepx, and obtain
p = f$J ' - 1. The integral of equation (d) then takes the form
Since 0 must equal 0 when x = GO , we have A=B = Q, and hence
We obtain from equation (e)
Q = (C£-1+V^*X1V*- J)e(-l-
But from equations (b) and (c) we have P — 1 and Q = 0 when x = 0,
and therefore C=D = . Hence we obtain
p = e - ex/v/r ^ cos (^2) ; Q=-e- «/^ . sin
and 6 = e- «/^ . sin (at - ee,>/2).
Substituting for e its value, we have
6 = e -*•£/« . sin (at - */£«/*).
The difference between the highest and lowest temperatures at the
depth x below the surface is therefore 2e~xVWK. This difference
depends on the value of a. The faster the temperature changes
at the surface the smaller the influence of this change on the tem-
perature in the interior. For example, if we set the temperature
at the surface equal to 0 = 8in(27r2/r), the difference between the
highest and lowest temperatures is equal to '2e-x^^lK, and this is
very much greater when T is a year than when it is a day.
SECT, cxxv.] A HEATED SURFACE. 303
The temperature relations within the earth are actually different
from those here described, because the temperature at the surface
cannot be expressed in any simple way. The main features of the
phenomena, however, are similar to those deduced in this discussion.
SECTION CXXV. A HEATED SURFACE.
Let the temperature in an infinite body at the time t = 0 be every-
where zero, except in a plane in which each unit of area contains
the quantity of heat o-. Fourier showed that at the time t the
temperature 6 at a point at the distance x from the heated plane is
given by
~
where k is the conductivity and K the quantity defined in CXXIL (g).
We will now examine whether this expression for 6 satisfies all the
conditions of the problem. We will first consider the differential
equation W/dt = K2 . 320/3x2. From (a) we obtain
(b), (c),
(d)
It follows from (b) and (d) that the differential equation is satisfied.
Since the function ze~^ approaches zero as its limit if z becomes
infinitely great, it follows that for / = 0 we have 0 = 0, for all values
of a-, with the exception of the value x = Q. If 6 is determined by
the equation (a), we can further show that each unit of area of the
heated surface S contains the quantity of heat cr at the time t = 0.
The total quantity of heat which is present in the body is given
by the expression
But because (e) I e~**dq =
the quantity of heat present at any time must be So-, and since this
quantity is present on the infinite surface 8 at the time 2 = 0, the
unit of surface at that time must contain the quantity o-.
It follows from (a) that 6 = 0 for t = 0 as well as for t = <x> ; there-
fore there must be a certain time at which 6 is a maximum. This
time is found from (f) 0 = 0, which gives t = xz/2K2. The corre-
sponding value of 0 is (g) 6 = \/\/2ire . a-fcpx. It appears from equation
304 CONDUCTION OF HEAT. [CHAP. xiv.
(a) that the heat is propagated with an infinite velocity, since 0 is
everywhere different from zero as soon as t has a finite value.
We will now determine the temperature at any time in a region
in which the original distribution of heat depends only on one of
the coordinates. Let 0=f(a) when / = 0, where a is the distance
from the y^-plane. The part S of the region which is bounded by
two parallel planes for which x = a and x = a + da, contains the quantity
of heat So- = S.da. pc ./(«•)• Therefore the quantity o- which is present
in the unit of area of this sheet is a- = da . pc ./(«).
If the temperature of the rest of the region is zero, the heat flows
out from this sheet on both sides, and at a point whose distance
from the f/2-plane equals x, and which therefore is at the distance
x-a from the shell, the temperature by (a) is
,70 -JL
^fr
All other similar sheets emit heat according to the same law, and
we therefore have
If we set (i) q = (a-x)/2K,Jt, we have
6 = 1 /N/TT . [+
The expressions (h) and (i) contain the complete solution of the problem
before us. If we set t = 0 in equation (i) and make use of (e), it
follows at once that 6=f(x).
For example, if the initial temperature is constant and equal to
00 within the portion determined by -I <x< +1, but equal to zero
outside these limits, the integration in (h) is effected between these
limits, so that
SECTION CXXVI. THE FLOW OF HEAT FROM A POINT.
Let us suppose that, at the time 1 = 0, the temperature in an
infinitely great body is everywhere equal to zero, except at one
point, in which the quantity of heat m is concentrated. We will
investigate the distribution of heat in the body at any subsequent
SECT, cxxvi.] THE FLOW OF HEAT FROM A POINT. 305
time t. This problem was first handled by Fourier, who found that
the temperature 8 at a point whose distance from m is r is given by
(a) 0 = mK2/k . ( IjlK^tY . e ~ 5*/4*\
We can show that this expression satisfies all the conditions of the
problem. If the point which contains the quantity of heat m is at
the origin of coordinates, Fourier's equation
Wfdt - /c2(320/3a;2 + 320/3y2 + 320/as«)
takes the form (XV.) (b) 30/3< = *2(320/3r2 + 2/r . 30/3r), because 0
is a function of r only. This equation may be given the form
(c) -d(r6)pt = K*d*(rO)pr2.
We obtain from (a)
( 3(r0)/3< = ( - 3/2t + r2/4jc2/2) . r0,
(d) J 9(r0)/3/- = (l/r-r/2/c20.r0,
( 3*(r0)/3r3 = ( - 3/2*2£ + r2/4K4*2) . r0,
and by the use of these values prove first that Fourier's equation
is satisfied. Further, 0 = 0 when t = 0. The quantity of heat originally
present is ///, since the total quantity of heat at any time is given by
/ "
Jo
r2 . dr . pcd = 4*1* . dr . ™(l
Jo
If we set q = r/'2i<^Jf, the integral takes the form
We find by integration by parts, and by the use of CXXV. (e), that
the value of the integral is m.
The time /, at which B reaches its maximum value, is obtained
from the equation 0 = 0, and is from (d), t = r2j6K2. The corresponding
maximum value of 0 is 0=(l/\A|7re)3.'m/e/w3.
SECTION CXXVII. THE FLOW OF HEAT IN AN INFINITELY
EXTENDED BODY.
We will now investigate, with the aid of the results already
obtained, the flow of heat in an infinitely extended body, when the
distribution of heat at a particular time is given. Let 0=/(a, b, c)
at the time / = 0, where a, b, c are the coordinates of a point referred
to a system of rectangular axes. The quantity of heat contained
by a volume-element da . db . dc is dm =f(a, b, c) . k/K2 . dadbdc.
u
306 CONDUCTION OF HEAT. [CHAP. xiv.
If this quantity is propagated through the body, it produces the
rise of temperature [CXXVI. (a)],
dO=(ll2K,JrtFe-Kx-<tf+<y-by+<*-eW*lft.f(a, b, c)dadbdc.
If we take the sum of all increments of temperature which arise
from the distribution of heat considered, we obtain for the tempera-
ture 0 at the point x, y, z,
(a) 6 = (1/2/cv/^)3 f+" f+* [+g-[(*-«)2^-*>)2+(e-c)W2< .y(a> ^ c)dadbdc.
J— QO J-oo J-oo
This expression for B is an integral of the differential equation
(b) Wf-dt = K2(320/3.r2 + 320/3/ + 320/3z2).
We notice that the integration of this equation depends on that of
the simpler equation (c) 'dX/'dt = K-32JT/3x2. For if X is a function
of x and tt which satisfies equation (c), and if Y and Z are functions
of y, t and z, t respectively, which satisfy equations analogous to
(c) for y and z, the equation 0 = XYZ satisfies (b). We have
YZX+ XZY+ XYZ= K\YZ . 92JT/3z2 + XZ . ^Y/^f + XY. ^Z/^).
It follows from (c) and the analogous equations for y and z that
this equation is satisfied, from CXXV. (a), by X= l/Jt. g-C*-«W,
hence the expression
\IJt.e-b- «W*^ .Ijjt.e-to- *> W .\IJt.e-to- c^K'2f
is an integral of equation (b). Therefore, also,
e = Cf/+"I+"T"{1^3 ' e-{(x-a?+(>J-b?+(*-c™KU -/(a, b, c,)dadbdc
is an integral of equation (b). C is a constant, and /(a, b, c) an
arbitrary function of a, b, c. If we set
a = (a-x)l-2K^t, /3 = (b-y)/2KJt, y = (c-z)!'2K,Jt,
it follows that
0 = (2K)3Cf+Xf^f^e ~ at~^-^f(x + Znajt, y + '2><PJt,
z + 2K-yJt)dad/3dy.
If we now assume t = Q, we obtain by the help of CXXV. (e),
e = (2K)*.C(j7r)*f(x,y,z).
If f(x, y, z) is an expression for the temperature when t = Q, we
set C'=l/(2K>/7r)3, and obtain
The expressions (a) and (d) are identical, as may be shown by the
substitution already employed.
SECT, cxxvin.] THE FORMATION OF ICE. 307
SECTION CXXVIII. THE FORMATION OF ICE.
Suppose that the temperature of a mass of water is everywhere
0 = 0, and that the surface of the mass is in contact with another
surface whose temperature is - 00. 60 may be either constant or
variable, but must be always below zero. A sheet of ice will be
formed under this surface, whose thickness e is a function of the
time t. The temperature Q of the mass of ice is itself a function
of t and of the distance x from the surface. For x = e, 0 is always
equal to zero. The equation (a) Wfftt**K*&Ofdx* holds everywhere
within the mass of ice. New ice will form continually on the bound-
ing surface of the ice and water. The quantity of heat which flows
outward through unit area of the lowest sheet of ice is given by
k~dd/'dx.dt. During the same time a sheet of ice, whose thickness
is de, is formed, and the quantity of heat set free thereby is Lpde,
where L represents the heat of fusion of ice arid p its density. When
x = e, we have
(b) kWfdx = Lpd€/dt, or
We may write for 6 the expression
As may easily be seen, this expression satisfies equation (a). It also
satisfies the condition that 0 = 0 when x = e. In order to find whether
it satisfies the condition contained in (b), we differentiate (c) with
respect to x, and obtain
When x = e this becomes equation (b).
Since, at the surface, 6= — 60, it follows from (c) that
If the thickness of the sheet of ice is given as a function of the
time t, #0 may be easily determined ; on the other hand, if 00 is
given, it is in general difficult to determine e.
*This solution was communicated to the author by L. Lorenz. See also
Stefan, Wied. Ann., Bd. XLIL, S. 269.
308 CONDUCTION OF HEAT. [CHAP. xiv.
If 00 is constant, the right side of equation (d) must also be con-
stant. This condition is fulfilled if €2//c2 = 2p2t, where p is constant.
From (d) we then obtain the equation
which serves to determine p. In order to put the series in (e) into
a finite form, we form from (e)
and thus obtain d(cd0/Lp)/dp = 1 + cBJL. The integral of this equation
is (f) cejL^pTe-^-^^da. If the thickness of the sheet of ice
increases in direct ratio with the time t, that is, if € = JK/, where q
is a new constant, it follows from (d) that
rAft n6fS
^o/£ = ^+fVr 2 3 + "'' or (g) cOJL = f-\.
If e is very small, we obtain from (d) c60/L= 1/2/c2. d(?/dt, and
hence (h) e2 = 2k/ Lp . f 60dt. This result also follows if we set the
flow of heat upward equal to kdjt, in which, however, we assume
that the temperature in the ice increases uniformly from its upper
surface downward. On this assumption, the quantity of heat kO^.dtje.
flows upward through the ice in the time (It. In the same time a
sheet of ice, whose thickness is dt, is formed, and the quantity of
heat Lp.de is set free. Hence we have kOQ . dt/t — Lp . de. This
equation leads to the result we have already obtained. If 0(} is con-
stant, it follows that (i) f = j2
SECTION CXXIX. THE FLOW OF HEAT IN A PLATE WHOSE
SURFACE is KEPT AT A CONSTANT TEMPERATURE.
It is in general very difficult to determine the variations of tem-
perature in a limited body. We will discuss a few cases in which
it is possible to solve this problem. Suppose that the temperature
in the interior of a plate bounded by parallel plane faces is 6 =/(«),
where x denotes the distance of the point considered from one of
the faces of the plate. From the time t = 0 on, the surfaces are
supposed to be in contact with a mixture of ice and water, or to
be so conditioned that their temperature is kept at zero. The law
SECT, cxxix.] THE FLOW OF HEAT IN A PLATE. 309
according to which the temperature changes in the interior of the
plate is to be determined. Designating the thickness of the plate
by a, we have
for t = 0, 0=f(x) ; for t = oo , 0 = 0;
for x = 0, 0 = 0; for x = «, 0 = 0.
The rate at which the temperature changes at the surface is infinitely
great; just outside the surface it is equal to zero, while just within
it, at one face, it is equal to /(O). At the other face the temperature
outside the plate is also zero, and within it f(a). The function 0
must satisfy not only these conditions, but also the differential equa-
tion (b) 'dd/'dt = K2320/2te2. An integral of this equation is
(c) 6 = e~ m'K2t(A sinmx + Bcosrnx).
From (a) B = 0, so that (d) 0 = Ae~ mVt sin mx. This value of 0 satisfies
not only equation (b), but also vanishes for x = 0. Since 0 is also
zero when x = a, we must have sin ma = 0, and therefore ma= ±pir,
where p is a whole number. Hence we have
(e) 0 = Ae-#***«*l<* . sin (pxirja).
If we notice further that 6=f(x) when ^ = 0, we have
(f ) f(x) = A sin (pTTX/a).
In general, the function /(x) can not be represented by this expres-
sion. To solve the problem we use the following method.
Since the expression (e) is an integral of Fourier's equation, the
complete integral is obtained by taking the sum of the similar expres-
sions, which are obtained by giving p all values between 1 and oo .
The terms which correspond to a negative value of p differ from
those terms for which p is positive only in sign, and can therefore
be considered as contained in the latter. Hence we set
(g) 8 = Al sin fax/a) . e - **&!«* + A2 sin (2vx/a) . e ~ 2ViA/a* + _
When t = Q, we have 0 =/(«), so that for 0<x<a
(h) f(x) = Al $in(Trx/a) + A2 sin (2Trx/a) + ... .
We will now investigate whether a function f(x), which is arbitrary
within the given limits, can be represented by a trigonometrical
series of this form. For this purpose we choose instead of the
infinite series (h) another series with (n - 1) coefficients Av A^... AH_V
which coincides with /(x) at (n-}) points, namely, at the points
x — a/n , x = 2ajn, ... x = (n-l )a/n.
310 CONDUCTION OF HEAT. [CHAP. xiv.
We then obtain the following n - 1 equations :
f((n - l)a/n) = Al sin((w - I)TT/TI) + A2 sin((n - 1)2*» + . ..
+ An_lsin((n- l)(n -!)«•/»).
If we multiply the first of these equations by sin(ir/»), the second
by sin(27r/?i), etc., and add the right and left sides of the equations
thus obtained, we have
/») . sin(27i» + ... +f((n - l)a/»)sin((w - l)ir/w)
/w) + . . . + sin2((n - l)w/»)]
+ sin ((n - 1 )w/») sin ((TO - 1 )2ir/»)] + . . . .
Now
sin2a + sin22a + . . . + sin2((?i - 1 )a)
= jf"n-l -(cos2a + cos4a+ ... + cos((2/i-2)a))1.
But because
we have
sin2a + sin22a + ... +sin2((%- l)a) = |[w- 1— cos%a. sin(%- l)a/sina].
Substituting for a its value irjn, we have
sin2(7r/-n) + sin2(27r» + . . . sin2((n - 1 )ir/n) = w/2.
Now we can give the factor of A2 the form
|[cos(7r/?i) - cos(37r/n) + cos(27r/n) - cos(6ir/»)...
+ cos((n- l)w/n) - cos((n - 1)3»>)].
Applying the above given summation formula, we find that this
factor is equal to zero. In the same way the factors of A3, Av
etc., vanish, and we obtain finally
Al = 2/n. \_f(a/n) sin (77/71) +/(2a/n) . sin ( 2ir/n) + . . .
+f((n- l)a/7i)sin((n- l)»r/»)].
In general we have, for 0<m<w,
j Am= 2/n . \_f(a/n) . sin(mir/n) +/(2a/») . sin(2m7r/w)
1 +...+/((»-! )o/n) . sin ( (» - 1 )wir/»)].
SKCT. cxxix.] THE FLOW OF HEAT IN A PLATE. 311
Hence it is possible to so determine the coefficients Av Av ...
that f(x) and the trigonometrical series coincide for (n - 1) values
of x between 0 and a. The greater the value of n, the more values
will the two functions have in common, and when n = GO , one function
may be replaced by the other between the limits considered. The two
functions are not, however, necessarily identical, for their differential
coefficients may be entirely different. One of them is related to
the other in the same way as a straight line to a zig-zag line,
whose irregularities are infinitely small.
We will now assume that ?i=oo and write the equation (i),
Am = 2/7T . Tr/n . [f(a/n) . sin (rmr/n ) +f(2a/n) sin
+f(rajri) sin (rimr/n) + ...].
Setting r-rr/n = y, ir/n = dy, rafn = ay ITT, it follows that
(k) Am = 2/7T . f/(ay/7r) . sm(my)dy.
-0
Further, if we set x = ay/ir, we have
(1) Am = 21 a . /fa) sin (rmrxla)dx.
The same result is obtained in another way in XXXVII. (c).
Therefore, within given limits, we can replace the function f(x] by
a trigonometrical series, and set, for 0<£<a,
f(x) = 2/a . [sin(n-x/a) .
+ sin(27ra;/a).
o
Introducing the values for Av A2... contained in (g), the problem
is solved, and we obtain
[ \aQ = sin (iTx^e-****"* . [ /(or) sin (irx/a)dx
(n)
For example, if the initial temperature of the plate is constant
and equal to 00, we have
I 60sin(nnrx/a)dx = (l - cos mrfaOJmir,
and therefore
(o) \TrO = 60 . siu(7rxla)e-(*Kla}*-t + £00 . sin(37ne/a) . g- <"*/•)*•'+ ....
When t = 0, we obtain, for 0<x<a,
(p) ^TT = sin (irxja) + ^ . sin (3irx/a) + ^ . sin (5irx/a) + ...
312 CONDUCTION OF HEAT. [CHAP. xiv.
SECTION CXXX. THE DEVELOPMENT OF FUNCTIONS IN SERIES OF
SINES AND COSINES.
As shown in the foregoing paragraph, we may always set
(a) f(x) = Alsin(Trx/a) + A2sin('27rx/a) + ... ,
in which [CXXIX. (1)]
(b) Am = 2!a . I /(a) sin (mira/a)da,
.'o
where x is replaced by a. This development holds only for 0 < x < a ;
it does not hold for the limits 0 and a, except when f(x) itself is
equal to zero for these limits. The right side of (a) is an odd function,
which changes its sign with .r. The series (a) holds then within
the limits -a<a;<0, when/(z) is also odd. Setting /(.r) = .r, we have
Am = 2/a . I a sin (iwra/a)da = — 2a cos (mir)fmir,
and further
(c) | . irx/a = sin (irxja) - i . sin (2irx/a) + J . sin (3-.r/«)
Since x is an odd function, the series holds for negative values of x
if it holds for positive values. Further, since the series holds for
x=Q, it holds within the limits -a<x< + a. Setting Trxja = y, we
have for —ir<y< + TT,
(d) \y = sin y - £ . sin 2y + J . sin 3y - ....
Further, if we set (e) f(x) = E0 + Bl cos (irx/a) + B.2 cos (2irar/« )+...,
multiply both sides of this equation by cos(rmrx/a), and then integrate
from 0 to a, it follows, if in and n are whole numbers, that
[ cos (mirx/a) cos (mrx/a)dx = 0 and j eos2(nnrx/a)dx = \a.
Hence for m>0 we obtain
(f ) Bm = 2/a . I af(x) . cos (mirxla)dx and £0=l/a. ["/(zjdx.
We obtain BQ by multiplying both sides of (e) by dx and integrating
from 0 to a. If f(x) is an even function, the series holds within the
limits -a<x<a, since the cosine series does not change its sign
with x. But if /(.r) is an odd function, the series (e) holds only
within the limits 0 and a.
We therefore obtain the result (g)
\a.f(x) = sin(«/a) f/(a)sin(;ra/rt)r/a + sin(27r.c/fl) ^ /(a)sin(27ra/rt)f/a + . . . ,
SECT, cxxx.] THE DEVELOPMENT OF FUNCTIONS. 313
.f(x) = J . I f(a.)da + cos(Trx/a) . |/(a)cos(?ra/a)dfa
(h) ,
+ cos(2;r:c/a) | f(a)cos(2Tra/a)da
An arbitrary function f(x) can also be developed in a series of
sines and cosines, so that the development holds within the limits
-a<x<a. To effect this, we set
f(x) = i . [f(x] +f( - x)] + \ . [f(x] -/( - x)],
in which %[/(%)+ f( -x)~] is an even function, because it remains un-
changed when x is replaced by -x. This function §[f(x)+f( -x)]
can therefore be represented by a cosine series. The coefficient of
cos(mirx/a") is
= I • / f(a)cos(mira/a)da + £ . / /( - a) cos (nnra/a)da.
If in the last integral a is replaced by - a, the integral is transformed
into - i • / /(a)cos(w7ra/rt)rfa, and the coefficient sought becomes
i/ /(a)cos(w7ra/a)e?a. Hence we obtain
( !« • [/(*) +/( -«)] = £• f +/(«)^« + cos(^/a) f+f(a)CO&(ira/a)da
/•\ I -'-a J-a
<J) 1 r+«
+ cos(27rx/a) / /(a)cos(2iro/a)rfa+ ....
On the other hand, the function J . [/(x) -/( - x)] is an odd function.
because it changes its sign with x; therefore, by using (g), we can
represent this function by a sine series. The coefficient of sin(imrx/a)
is
2 • / l/(a) -/( - a)]sin(w7ra/a)da
= ^ . I /(a) sin (rmra/a)da - | . j f( - a)sm(mira/a)da.
If we replace — a by a in the last integral, it is transformed
into -i. I f(a)sin(mTra/a)da, and the coefficient becomes
r + i
J.j f(a)sm(nnrala)da.
We therefore obtain
f i« • [/(*) -/( - *)] - sin(7r.r/«) . f /(<z)sm(7ra/«ya
<k)
+ sin (2mc/a) / /(a) sin (2ira/a)da + ....
314 CONDUCTION OF HEAT. [CHAP. xiv.
By the addition of equations (i) and (k) we obtain finally
( a .f(x) = 1 . i +f(a)da + (+f(a)CosU(x - a^da
0)
[ +fj(a)COS(2Tr(x-«)/a)da+...,
or, for -a<z<a,
(m) f(x) = 1 /a . £**[£ + cos (ir(x - a) /a) + cos ( -lir(x - a)/a) + . . .]/(a)da.
This series is due to Fourier. It may also be expressed in the form
(n) f(x) = l/2a.+f(a)da+l/
We may now ascribe any value to a. If a is infinitely great, and
if / f(a)da. is finite, the first term on the right side of the equation
(n) vanishes, and we obtain
f(x) = 1/ir . 2>/o • /(a)cos(m7r(2 - a)/aVo.
m=l J-a
Now setting wi7r/a = A, and therefore 7r/a = dX, it follows that
(o) /(«) = I/TT . jf dx|_+/(a)cos( A(a; - a))^,
where -co < a; < ao . Instead of this equation we may often use
one of the two which are obtained from (g) and (h). The general
term in (g) is
sin (mirx/a) . f /(a) sin (mira/a)da,
and hence
f(x) = 2/7T . IT I a . ^ sin (mirx/a) I /(a) sin (mTra/a)da.
Now if we set mir/a = A, and therefore via = d\, we have for 0 < x < oo ,
(P) /(*) = 2/7T . l"d\. . sm(Xx)j /(a)sin(AaXa.
From (h) we obtain in the same way for 0 < x < oo ,
(q) /(*) = 2/7T . {™d\ cos(Aa;) . [7(a)cos(Aa)^a.
SECT, cxxxi.] APPLICATION OF FOURIEK'S THEOREM. 315
SECTION CXXXI. THE APPLICATION OF FOURIER'S THEOREM
TO THE CONDUCTION or HEAT.
If the temperature in a certain region depends only on the x-co-
ordinate, the temperature 6 must satisfy Fourier's equation
(a) 30/3* = K2320/3a;2.
From CXXIX. (c) 6 = e~x*K2:(A sin Xx + Bcoskx) is an integral of
equation (a) where A, A, B are constants. We may also give the
expression for 9 the form 6 = e~^K2t cos (\(x - a)) . /(a), where /(a)
is an arbitrary function of a, and A. and a are constants, which may
take all possible values. Any sum of such terms satisfies the equation,
and as the integral
(b) 0 = I/TT . J^Ae-^'[+7(a)cos(A(z - a))da
is such a sum, it will also satisfy the equation. But when / = 0,
we have
6>=1/7T. {°°d\f*f(a)cos(\(x-a))da,
from which, by comparison with CXXX. (o), we obtain 6=f(x).
The formula (b) contains the solution of the problem, to determine
the temperature in a body at any time t, when the temperature is
given, at the time £ = 0, by 6=f(x). This problem has already been
solved in another way in CXXV. (h) and (i). We proceed to show
that the solution hero given is identical with the former one.
Since (c) 0 = 1 /«• . f+J(a)da /" V *** cos ( X(x - a))d\,
we first determine the value of the integral
If we develop cos (A(z- a)) in a series, this integral is represented by
It follows by integration by parts that
/ VXU%A*V*A. = (2n - l)/2*2£
and by continued reduction
316 CONDUCTION OF HEAT. [CHAP. xiv.
But because j e~fdq = \,Jirt
we obtain
The value of the integral sought is therefore
or «-«-
2*^
If we replace the integral in (c) by this value, we obtain
This expression for 6 is identical with that given in CXXV. (h).
We will now apply Fourier's theorem to find the law of penetra-
tion of heat into a body. For this purpose we will consider the
simple case in which the body is in contact over a plane bounding
surface F with another body whose temperature 00 is constant and
given. Let the original temperature of the cold body be zero.
If we proceed as before and use CXXX. (p), we obtain
B = 00 + 2/?r . I "dX sin ( Aa-)<rX2K2< . /"/(a) sin ( Aa)da.
This expression for 6 satisfies all conditions if only we have, when t = 0,
0 = 00 + 2/7r. f°<Usin(Aa:). [ /(a)sin(Xo)da.
-0 -0
This condition is fulfilled [CXXX. (p)] when /(a) = - 00. Hence the
solution of the proposed problem is contained in
(e) 6 = 00 - 200/ir . f"d\ sin ( \x)e~x^' .' J'sin (\a)da.
Using the same method of reduction as that by which (c) is trans-
formed into (d), we obtain
from which B = 00 - -^ • -! (V*W? - l"e " *2dq } ,
VTT lJ-*tevi Jxftni/t
and therefore 6 = 00 { 1 - 1 /Jir . f^e^'dq } .
J-XpKVt
SECT, cxxxr.] APPLICATION OF FOURIER'S THEOREM. 317
Since e~q~ is an even function, we obtain
or by the help of CXXV. (e),
(f) e
Let A and B be two points within the body, whose distances from
the surface F are xl and x2 respectively. The temperature & which
A attains after the lapse of the time /, is & = '2BJJir . I e~g2dq. B attains
Jr^-'KVti
the same temperature after the lapse of the time /.„ given by the
f "
equation 0' = 20JJir . I e~g2dq. Comparing the two integrals, it appears
/VbV«1
that Xil*Jtl = xzl,Jtt> or t.2/tl = x22/xl2, that is, the times required for two
points to attain the same temperature are proportional to the squares of
the. distances of the points from the heated surface F.
AVe will now determine the quantity of heat which flows into
the cooler body through unit area in unit time. For this purpose
we give the equation (f) the form 0 = 20J,jTr. [f(ao ) -f(x/2K,Jt)], from
which follows, by (f), - kW/'dx = kOJitj^t . e~zt:^. Setting x = 0, we
find the quantity of heat U desired, (g) U = k60/ K-Jiri.
By the help of equation (g) we may solve an important problem.
Two bodies L arid L' are in contact over a plane surface, the tem-
perature of one of these bodies being T, that of the other T. If
the two bodies are brought in contact, one of them is heated and
the other is cooled. We can also determine the temperature 2\
of the surface of contact. Assuming that T0 is constant, the quantity
of heat which L receives in unit time is, from (g), given by
U=k(T0~T)lKj*t,
In the same time, L' receives the quantity of heat
U'=k'(T,-T')/K'^t,
where k' and K' have the same meaning for L' as k and K for L.
But since the infinitely thin bounding surface can contain no heat,
U+U' must equal zero, or k/n . (T0- T) = k'/K . (T - T0), from which
follows (h) * T0 = (Tjkcp + T*JJMji)l(Jkcp + */kVfi). It is thus shown
that the assumption is correct, that the temperature in the bounding
surface between two bodies which meet in a plane surface is constant.
Strictly speaking, the bodies in contact must both be infinitely large,
* L. Lorenz, Lehre von der Warme. S. 178. Kopenhagen, 1877.
318 CONDUCTION OF HEAT. [CHAP. xiv.
but the formula (h) may also be applied to small bodies if we only
consider them shortly after they are brought in contact. We may
show from (h) that the temperature of a heated solid is very little
diminished by contact with the air ; this holds for the metals and
for good conductors in general. It follows from equation (h) that
If T represents the temperature of the solid, p is always very much
greater than p. Hence T0-T is very much greater than T - T0,
especially since k is also greater than k', while c and c' are not very
different from each other.
SECTION CXXXII. THE COOLING OF A SPHERE.
Let us suppose that the temperature at a point in the interior
of a sphere depends only on the distance of that point from the
centre of the sphere. In this case [CXXVI. (c)] Fourier's equation
takes the form (a) 3(r0)/a* = K232(r0)/3r2. If m, A, B are arbitrary
constants, an integral of equation (a) is
rO = e-"™ . (A sin (mr) + B cos (mr)).
But since this equation leads to the conclusion that 0 = oo when r = 0,
B must equal zero, and we obtain as the integral of equation (a)
<b) r0 = A.e-a**.Bm(mr).
We will first consider the case of the sphere immersed in a mixture
of ice and water, or so situated that its surface is kept at the tem-
perature 0° by any means. If we represent the radius of the sphere
by R, we have 0 = 0 when r = E, and therefore sin(w7?) = 0. Hence,
if p is an arbitrary whole number, we must have mli = ±p-. We
•can now set
The constants Av A2...are determined by the help of the tem-
peratures of the different parts of the sphere at the time t = 0. Let
these temperatures be given by/(r). We then have
r .f(r) = A^ sin (lerfR) + A2 sin (2wr/.B) + . . . .
From CXXIX. (1) we obtain for Am
<d) Am = 2/fi. 'rf(r)aiu(mirrjE)dr.
SECT, cxxxn.] THE COOLING OF A SPHERE. 319
If the temperature is constant and equal to #0 at the time t = 0, we
have
R
rsin(mirr/R)dr, aud hence Am= - 200R/mir . cos (rmr).
Using these values we obtain finally
I 6=2R60/irr. [sin(7rr/R) . e-<
\ - i . sin (27rr/R)e-(*"«™ + £ . sin (STr
The mean temperature & is
(f ) & = 600/,r2 . {«-<«/*)* + | .
This equation may be applied to a thermometer which is immersed
in a fluid cooler than itself. The temperature of the thermometer
is then given, to a close approximation, by the first term of the above
equation. The rate of cooling is (g) - dO'/dt = Tr2kO'/cpR2.
We will now consider another important case, that of a sphere
in vacuo losing heat by radiation. We suppose the temperature of
the region, or rather of its boundary, to be 0°. We suppose the
radiation to take place according to Newton's law, and therefore to
be proportional to the temperature on the surface of the sphere.
From (b) the integral takes the form (h) rO — ^Ame~mt>ft . sin (mr).
If E represents the coefficient of radiation, the quantity of heat
which radiates in unit time from an element dS of the surface is
dS. E6. We will assume that E is constant. The same surface-
element dS receives from the interior of the sphere in the same time
the quantity of heat -k.dS.dO/dr. Since the quantity of heat
which dS receives must be equal to that which it emits, we have
(i) -k.dO/dr = E6 or -d6jdr = h6, where, for brevity, we set h = E/k.
Hence, for r = R,
2^TO<rmV2< . (m cos (mR)/R - sin (mR)/R?) = - KSAne~aAft sin (mR)/R,
or ^Ame-m^f . [mRcos(mR) - (1 - hR)sm(mR)] = 0.
If this equation is to hold for every value of t, we must have
(k) mR . cos (mR) = (l-hR). sin (mR).
This equation must be solved for m. We set mR = x and obtain
(1) tg z = z/(l -hR). If we further set yl = tgx, yl and x may be con-
sidered as the rectangular coordinates of a curve (Fig. 143). This
curve has an infinite number of branches, oa, irb, 27rc..., to which
the straight lines X = ^TT, x = %ir... are asymptotes. Further, if we
set i/.2 = x/ (I - hR) this equation represents a straight line, such as
opq, which passes through the origin of coordinates.
320
CONDUCTION OF HEAT.
[CHAP. xiv.
The constant h is positive, and its value must lie between 0 and GO .
First consider the case where h = 0 ; then y.2 = x, and this equation
represents the straight line opq which touches the curve oa at the
point o, and cuts irb at p, "2-rrc at q, etc. The abscissas 0, xv x2... are
roots of equation (1). We have further
ir<xl<3ir/2 ; 2?7 < z2 < 577/2 ; ...nir<xtt<(n + ^)ir.
As n increases xn approaches the superior limit (?i + i)ir. Besides
its positive roots equation (1) has also negative roots, which are equal
in absolute value to the positive.
FIG. 143.
Next suppose that 0<h<l/R, so that 0<l-hlt<l, and hence
y2>x. This represents a line such as oa/3 (Fig. 143) which cuts the
curve at o, a, /3. The abscissas 0, x^, x2', x3' ... of these points are
roots of equation (1), and we have
Q<Xl'<^ir; 77 <£.,'< 377/2; 2ir <xs'<5Tr/2 ... (n - l)ir<xn' <(n - $)TT.
As h increases, the angle aoir approaches the angle ITT, and the
roots approach their superior limits. If h = l/R, we have x = Q, and
the roots are 0, \ir, 3^/2, 5:7/2,.... Now if l/E<h<<v, we have
?/2= -xj(hR-\). The straight line then has the position ofi'y'. If
in this case we represent the roots of equation (1) by 0, a^", x2"...,
we have ^•jr<xl"<ir- 37T/2 <z./<27r.... As h increases, the roots
approach their superior limits. And if h = oo , we obtain ,TI" = TT ;
SECT, cxxxn.] THE COOLING OF A SPHERE. 321
We may now consider the roots of equation (1) as known and
determine m from them. The values of m corresponding to the
several roots are 0, mv m.2,.... We may neglect the negative roots,
because the terms in (h) corresponding to them may be considered
as included in those arising from the positive roots. We therefore
set (m) rd = A-ie~mi'*"'*t sin (m^r) + A2e~m*'K* sin (m.-,r] + .... If the tempera-
ture at the time t = 0 is given by 6=f(r), we have
(n) r.f(r) = Alsii\(mlr) + A.2sin(m2r) + ....
Now let ma and mb be two roots of equation (k), and multiply both
sides of equation (n) by sin(mar). It follows by integration from 0
to R that
,-R [R
(o) / r.f(r)sin(mar)dr = 'S,Ab sm(mbr)sin(mar)dr.
Now
ra
I sin (mbr) sin (mar)dr
= \ sin [(mb - ma)R]/(mb - ma) - \ . sin [(m. + ma)R]l(mb + ma)
I = [ma sin (mbR) cos (maR) - mb cos (mbR) sin (maR)]/(mb2 - wa2).
But because we have from (k)
maR = (1 - A.R) tg (maR), mbR = (l-hR)tg (m^R),
and therefore ma tg (mbR) = mb tg (maR),
or 7??a . sin (?%Z?) . cos(maR) = mh . sin (wa.R) . cos (mbR),
/K
sin (m,br) . sin (mar)dr = 0,
whenever ma and 7W6 are different from each other. But if they are
equal, (p) is indeterminate. We find the value of the expression (p)
in this case by setting mb = ma + e, where e is a small quantity. We
reach the same result more simply if we investigate the value o
the integral
r = 2 • L [i "~ cos(2mar)]dr = %[R - sin(2maR)/2mn].
We thus obtain
[R
Aa=2/R. I r.f(r)sin(mar)drl[I -sin(2mnfi)/2maR].
Jo
Hence the complete solution of the problem is contained in
(RrO= sin(w1r)g-""w
«-T« /*« *.\«— w
322 CONDUCTION OF HEAT. [CHAP. xiv.
In the simple case in which the initial temperature of the sphere
is everywhere equal, we have f(r) = 00, and then find
00 . FT sin (mr)dr = 6Jm? . [sin (mR) -mE cos (mR)],
or, by the use of (k),
09f*r sin (mr)dr = hR00sin (mE)^.
We therefore get the result
( \ 4. ti-hRfl ii 22
O'^m^m^-sin^m^R)] m.2[2m2R - sin(2m2R)]
If the coefficient of radiation E and therefore also h are very small,
or if the radius of the sphere is small, the product hR is a small
quantity. In this case, if we neglect the higher powers when the
sine and cosine are developed in series, we obtain from (k),
1 - 1 . m^R2 = (1 - hB)(l - 1 . m*fl*),
from which follows ml2 = 3h/R.
The other values of m are so very much greater that the corre-
sponding terms in (r) vanish in comparison with the first term. We
therefore obtain 0 = 60. e~3Hlcl<'B, or, substituting the value of h,
(s) 0=00e-3JB/<"*
We can derive this formula more simply. The quantity of heat
which the sphere radiates in the time dt is 4irR~E6dt. Thus the
temperature of the sphere increases by — dd ; and the quantity of
heat given up is - 4?r/3 . R?cpdd. Hence we have
= - 4;r/3 .
from which follows 6 = 00e~3£t'cf>K, since the temperature of the sphere
is On at the time t = 0.
SECTION CXXXIII. THE MOTION OF HEAT IN AN INFINITELY
LONG CYLINDER.
Let the cross-section S of the cylinder be so small that its tem-
perature 6 is constant, and let A arid B be two cross-sections separated
by the distance dx. The quantity of heat - Sk . Wfdx . dt flows through
A in the time dt, and the quantity - SktfO/'dx + 320/3a;2 . dx)dt flows
through B in the same time. Hence the part of the cylinder
between A and B receives the quantity of heat Sk . 320/3z2 . dxdt. A
SECT. CXXXIIL] THE MOTION OF HEAT. 323
part of this heat is given up to surrounding bodies by conduction
or radiation. If P is the perimeter of the cylinder, E a constant,
and if the temperature of the medium around the cylinder is 0, the
heat given up by conduction or radiation is PEO . dxdt. Another
portion of the heat received serves to heat the cylinder ; this portion
is S . dx . cp . dO. Hence we obtain the equation
Sk.-d*-efdx'2 = Spc.'d6/-dt + PEe, or (a) 90/3/ = K2. c>20/3x2- A0,
if we set K* = k/cp and h = PE/SPc.
If the state of the cylinder or rod has become steady, we have
30/3/ = 0, and equation (a) takes the form *2 . 320/3z2 = h0. From
this it follows that (b) 6 = A<?**K + Be-xVJtl\ If the temperature of
the rod is given at two points, we obtain from (b) the temperature
of the intermediate points. We will assume that the temperature
of a certain point in the rod is 00, and that at a very great distance
from this point the temperature of the rod is 0. Then, for x = oo ,
we will have 0 = 0 and therefore (c) 0 = 00 . «~*v'*'1'. But if the rod is
not in a steady kstate the equation (a) must be used. If we sub-
stitute in this equation 0 = u . e~ht, we obtain "dufdt = K2 . 3%/Bo;2. The
integral of this differential equation has already been given. By the
help of equations (c) and (d) we can determine the cooling of any
rod which is heated in any manner.
We will here consider only the case in which one cross-section S
of the rod has the temperature 00, while the temperature of all other
parts of the rod is zero. The heat flows from S toward both parts
of the rod, and after an infinitely long time the temperature at the
distance x from S is given by 0 = 00 . e~'yjtlK. On the other hand, the
temperature at the same place at the time t is given by
<e) 6 = 60.e-'yv< + u.e-'*.
In this case u must satisfy the following conditions :
(1) It must satisfy the equation dw/3* = K2 . 3%/Sx2 ;
(2) When t = 0 the equation holds 0 = 00 . «-***/« + u ;
(3) When x = 0 we must have u = 0.
Conditions (1) and (3) are satisfied by
% = 2/7T . l~dX sin (\x) (/(a)sin(Aa)e-X2K2<</a.
Jo ^0
And as u also satisfies (2) we have
+ 2/7T. / <Usin(Az)//(a)sin(AaX/c
324 CONDUCTION OF HEAT. [CHAP. xiv.
It is requisite for this that /(a) = -00e~aVr/K. Hence the solution of
the proposed problem is
8 = eo . e-*^'" - 200/7T . e~M f*dX sin(Az) . f"s
As in CXXXL, we can give the expression
2/7T . f°sui(Aa;) . si
Jo
the form
and this latter expression equals 1/2*7^7. (e-(*-w* _ g-(»f«w*«^ Sub-
stituting this expression in the equation for 6, it follows that
0 = eXe-**" - -^= . fV"-0**"' - e-^w**) . <r«v»> rfal
2/Cx/7T< -/O
To simplify this expression, we set p = (a- x)/2i<Jt, a = x + 2i<pjL It
then follows that
_ g_« _ ,
Vft/K ,00
_ r e_^
TT -V^ - z/2K VF
We obtain in the same way
2*
and therefore
(f)
A careful examination of this expression shows that it represents the
flow of heat through an infinitely long rod. For t = 0 the lower limit
of the first integral equals - oo , and the value of the integral itself
is then equal to JTT ; the lower limit of the second integral is in
the same case oo , and therefore the value of the second integral
equals zero. Hence for t = 0 we also obtain 0 = 0, which should be
the case for all cross-sections of the rod, except for the heated
section. Both integrals have the same value for x = Q, and there-
fore 6 = 00. Both integrals vanish when t = ao , and for the steady
state of heat we have the evidently correct result 0=QQ. e~*^lK.
Because h = PE/Scp, h is infinitely small, if the cross-section of the
SECT, cxxxiu.] THE MOTION OF HEAT. 325
rod is infinitely great, or if the coefficient of radiation E is infinitely
small. Setting h = 0 in (f) we come back to a case already treated,
This result is also found in CXXXI. The expression (g) gives the
temperature in an infinitely extended body, having, at the time t = 0,
the temperature 6 = 0 at all points, with the exception of the points
on the surface ce = 0, for which 0 = 60.
The solution (f) holds only for positive values of x; and that it
shall hold for those parts of the rod which correspond to negative
values of x, x must be replaced by - x in (f ).
SECTION CXXXIV. ON THE CONDUCTION OF HEAT IN FLUIDS.
Up to this point we have treated only the motion of heat in
solids. The results which have thus been obtained cannot in
general be applied to fluids, because any difference of temperature
which causes a different expansion in different parts of the fluid,
occasions so-called convection currents. In general, differences of
temperature are more quickly equalized by these currents than by
conduction alone. The relations are therefore very complicated. We
will confine ourselves to developing the general equations of motion
which will be applied in some simple cases.
We use the notation of hydrodynamics. The equation of con-
tinuity, which expresses that the quantity of matter is constant,
becomes [cf. XLI. (d)]
The momentum received by the unit of volume in the unit of
time is equal to the force acting on that unit of volume. We have
therefore from XLI.
A = p(du[dt + u'du/'dx + vdufiy + wduj'dz)
= 'dXJ'dx + VXj'dy + 'dXJ'dz + pX,
B = P(dvfdt + u'dv/'dx + vdvfdy + wdvfiz)
= VYJ'dx + ^YJ-dy + -dYfiz + PY,
C = p(dwfdt + u'dw/'dx + vdwfdy + vfdw/'dz)
= 'dZJdx 4- aZ,/3y + 'dZfiz + pZ.
The symbols A, B, C are introduced on account of the use to be
subsequently made of them.
326 CONDUCTION OF HEAT. [CHAP. xiv.
Suppose the fluid which is here considered to be a liquid and
incompressible. In this case it contains energy only in the form
of kinetic energy or heat. If the body, on the other hand, i&
gaseous, we suppose it to be an ideal gas, which conforms to the
law of Boyle and Gay-Lussac. Such a gas can indeed be com-
pressed, but the work done by the compression is transformed into
heat, so that the energy contained by the gas is independent of
the volume, and is determined only by its kinetic energy and
temperature.
A volume-element of the fluid d<a = dxdydz contains, at the time
/, a quantity of energy which is the sum of the kinetic energy and
the quantity of heat contained in it. We multiply the latter by
the mechanical equivalent / of the unit of heat. If E is the unit
of volume, and if the unit of mass receives the quantity of heat 0,
we have, designating the velocity by h, E = ^ph? + JpQ. During the
time dt the volume element d(a receives the quantity of energy
(c) dE/dt . dtda, where dEfdt = $d(ph*)/dt + J. d(pQ)/dt.
The increment of energy which the element dw receives in the time
dt proceeds from the following causes :
1. From the work done by the accelerating forces X, Y, Z.
2. From the kinetic energy which, in consequence of the flow of
the fluid, passes into the volume-element do> through its surface.
3. From the work done by the surface forces X^ Y^ ... on that
part of the fluid which is situated on the surface of the element dw.
4. From the heat contained by that part of the fluid which flows
through the surface-element d<o.
5. From the heat which passes into the element da by conduction.
We will designate these quantities of energy in order by e^wdt,
e^dudt, e.jdwdt, e4d(adt, and e.0da>dt; e1 is therefore the quantity of energy
received by the unit of volume in the unit of time only through
the influence of the accelerating forces. We will now investigate
the values of e^ &>, . . . .
We determine the work done by the accelerating forces in the
time dt in the following way. The volume element contains the
mass pdo> and moves in the time dt through the distance udt in the
direction of the ar-axis. Thus the force X does the work pd<o . Xudt.
The work done by the forces Y and Z is determined in the same
way. The work considered is therefore p(uX+vY+u-Z}dwdt. We
have represented this quantity of work by e^-wdt and hence obtain
(d) el
SECT, cxxxiv.] CONDUCTION OF HEAT IN FLUIDS. 327
The kinetic energy which Jw receives from that part of the fluid
which flows in the time dt through the element d<a, is determined
thus. The mass which flows through the surface-element dydz in the
time dt is p.u.dt. dydz ; the kinetic energy of this mass is therefore
^.p.itdt.dydz.h?. But if we set U=^puh'2, U is the component
of flow of the kinetic energy in the direction of the a-axis. Let
the corresponding components of flow with respect to the y- and
z-axes be V and W respectively ; we then have
F=|.M2, W=\.po®'
By a method similar to that used in XIV., it may be shown that
in the time dt the volume-element da receives the quantity of energy
-(dU/'dx + 'dV/'dy + 'dW/'dz)d(adt. We represent this quantity by
e0db)dt and hence obtain
- pu^idufdx + v'dv/'dx + wdwfdx)
- pv (u'du/'dy + vdvj'dy + wdwj'dy)
— pw(udufdz + vdv/'dz + wdwj?)z).
It follows from this by the help of equations (a) and (b) that
e2 = 1 h2 . dp/dt + p(u'du/'dt + vdvfdt + wdw\"t t) -(Au + Bv + Cw),
or more simply
e, = $hMp/dt + IpdWldt - (Au + Bv + Cw),
(e) «2 = 2 • d(ph2){dt - (Au +Bv+ Cic).
The quantity of energy which the surface forces Xx, Yy, ... impart
to the element du, may be determined in the following way. The
force - Xxdydz acts on the surface-element dydz which bounds d<D
on the side lying in the direction of the negative .r-axis, in the
direction of the #-axis. The fluid particles which flow in the time
dt through the element dydz traverse the path udt in the direction
of the a-axis. Thus the force - Xx does the work - Xxdydz . udt. But
the fluid particles in the surface-element have also tangential motions.
Thev traverse the path wit in the direction of the y-axis under the
influence of the force - Yxdydz, by which the work - Yxdy'dz . vdt is
done. The same particles also move in the direction of the 2 axis,
so that the work - Zxdydz . ivdt is done. The total work done by
the forces in the time dt on the element dydz is therefore
Yxv + Zxw)dydzdt.
328 CONDUCTION OF HEAT. [CHAP. xiv.
We will designate this flow of energy in the direction of the z-axis
by U'dydzdt; let the corresponding flow in the direction of the
y- and £-axes be V'dxdzdt and Wdxdydt respectively. We then have
U'=- (Xxu + Yjo + Zxw\ V=- (Xyu + Yyv + Z,w),
W' = ~(X2u+Yzv + Z,w}.
The quantity of energy which d<a thus receives is determined as
in the foregoing case, and is equal to
e^udt = - (d U'j'dx + Wfiy + -d!P'rdz)d<adt.
Hence we obtain
Yyv + Zju^fdy
But using equations (b) it follows that
f e3 = X^u/Zx + Ypvfdy + Z&cfiz
(f ) + Z,(dwfiy + Vvpz) + XtCdufdz +
[ + YjCdvfdx + 'dul'dy} + (A- pX)u + (B- pY)v +(C- pZ)w.
The quantity of heat which the separate parts of the fluid contain
is transferred with them by the flow. During the time dt the
mass pudt.dydz enters the element dw through the surface dydz,
and brings with it the quantity of heat pudydzdtQ or the energy
JpudydzdtQ. We determine in the same way the quantities of heat
which enter the element d<a through the other bounding surfaces.
If we use the method given above and set
the quantity of heat e4d<adt received by the parallelepiped d<a in the
time dt, is given by
or e4=
Hence, by use of equation (a), we have
(g) e4 = /. -d(pQ)pt - Jp(de/dt + udQrd
Finally the element do> receives heat by conduction. The com-
ponents of flow of heat are [CXXIL]
-Jk.VOfdx, -Jk.Wfiy, -Jk.-ddj-dz.
If we set the quantity of energy thus received by da> in the time
dt, equal to e^d^dt, and assume the conductivity constant, we have
(h) e.a = J .
SECT, cxxxiv.] CONDUCTION OF HEAT IN FLUIDS. 329
The increase in energy which d<a receives in the time dt is given
by [%.d(ph*)/dt + Jd(pQ)/dt]dudt. At the same time the quantity
of energy («1 + «2 + «s + «4 + «6) dwdt enters the element do>, and we
therefore have
(i) \ . d(fW)jdt + Jd(PQ)/dt = e1 + e2 + e, + e,+ ey
Introducing in this equation the values found for ev e.2, e3, ...it
follows that
If internal friction exists in the fluid, we have from XLVII. (h)
Xx = -p + 2p. . 'duj'dx - ^(dufix + Zv/dy + "die fix)
ZJ, = fu(dwl'dy + 'dvl'dz), etc.
By the help of these relations we may give equation (k) the form
Jp(dQj
r
\ = - p(du/dx + dv
+ (dz;/^)2 + (dw/-dz)* -
I
2 + (du/oz
For the determination of the motion and temperature of the fluid
we have the five equations given under (a), (b), and (1). These five
equations are not sufficient to determine the seven unknown quantities
u, v, w, p, p, 9, and 6. We obtain two other equations in the follow-
ing way. The total quantity of heat 9 contained by the unit of
mass must depend on 0, and we assume that (m) 9 = c0, where c
is the specific heat, a constant. If the fluid considered is gaseous,
c denotes the specific heat of constant volume.
The second equation must express the relation between density,
pressure, and temperature. In the case of liquids, we may set approxi-
mately (n) p = pJ(l+aB), where pQ is the density when 0 = 0 and
a is a constant. But for gases, if V is the volume of the unit of
mass at pressure p and temperature 6, V$ the volume of the same
mass at pressure p0 and temperature 0°, we have ^F"=j90F0(l +a0).
Since Vp=\ and J>0=1, we have (o) p/p =pJpQ . ( 1 + a0). The
equation (o) in connection with (a), (b), (1), and (m) serves to deter-
mine the unknown quantities. The complicated equations which
determine temperature and motion in a fluid are very hard to integrate,
so that up to this time no case has been completely solved.
330 CONDUCTION OF HEAT. [CHAP. xiv.
SECTION CXXXV. THE INFLUENCE OF THE CONDUCTION OF HEAT
ON THE INTENSITY AND VELOCITY OF SOUND IN GASES.
We have the following equations [CXXXIV.] for the determina-
tion of motion in a gas in which the temperature is variable :
1. The equation of continuity [CXXXIV. (a)], which may take
the following form :
'dp/'dt + p(dufdx + 'dv/'dy + 'dw/'dz) + udpj'dx + vdpfiy + wdpfdz = 0.
2. The equations of motion [CXXXIV. (b)]. We replace in these
equations the forces Xa Xy, ...by the values found in XL VII. (b)
and (h), and obtain [cf. XLVIIL (a)]
p(dufdt + uduj'dx + vdufdy + w'duj'dz)
= pX- 'dpj'dx + ft V2^ + If^^ufdx + 'dv/'dy + 'dwfdz)l'dx,
and analogous equations for y and z.
3. The condition for the conservation of energy [cf. CXXXIV. (1)].
4. The connection between the heat contained in the body and
the temperature [cf. CXXXIV. (m)].
5. The Boyle-Gay-Lussac law [cf. CXXXIV. (o)].
Let the velocity and change of temperature be very small quantities ;
the same is then true of such differential coefficients as 'dp/'dx, 'dQ/'dx,
etc., and we will therefore neglect the product of these quantities,
that is, terms of the form udpj'dx, udu/~dx, itdQjdx, etc. The equa-
tions 1 — 5 are then very much simplified. We obtain
(a) 3(log p)j?>t + 'du/'dx + 'dv/'dy + 'dwj'dz = 0.
.If we further set /t/p = //, equation (2), by use of (a), takes the form
(b) 'du/'dt + 1 Ip . 'dp/'dx = // V2« - £/*' . 32(logp)/ar3i5.
Similar equations hold for u and v, if x is replaced by y and s
respectively.
Eliminating 0 in equation CXXXIV. (1) by means of the relation
Q = cO and introducing the heat equivalent A of the unit of work
for l/J, it follows that (c) cp.Wj'dt-k\7-d = Apd(\o«p)l'dt. We have
further the equation [CXXXIV. (o)], (d) p/p=p0fp0.(l + a8). We
consider ///, k, and c as constants. We substitute p0 for p, if p or
l/p occurs as a coefficient; we also substitute p0 for p in (c). In
these substitutions we neglect only infinitely small quantities of the
second order. Setting p = Po(l+a-)) we obtain (e) log p = log p0 + <ry
because o- is a small quantity. Hence equation (d) takes the form
P=Po(\ +°")(1 +«#)> °r> because 0 is also a small quantity,
(0 7>=/>
SECT, cxxxv.] VELOCITY OF SOUND IN GASES. 331
Equation (c) now becomes Wfdi - k/cp0 . V2# = -dpo/cPo • d^/cM, and
if we set K- = k/cp0 and 9 =^ cpQB/Ap0, we obtain from the last equation
(g) Wfdt - K2 V20 = ftr/3*.
By the use of (e) and (f) we may transform equation (b) into
'du/'dt +PQ/PQ . 'do-fa +pQa/p0 . 'dB/'dx = /*' . V2« - $/*' • ^(r/'dt'dx.
Introducing in place of 6 the quantity 0 already defined, we have
'dv/'dt + b2. 'da-f'dy + (a2 -
The heat required to raise the temperature of a gram of air
under constant pressure from 6 to 0 + dO is equal to C . dd, if C is
the specific heat at constant pressure. A part of this heat, namely
c . dO, is used in raising the temperature, the other part is used in
overcoming resistance during the expansion, by which the woAp*dP
is done. We have therefore C.dO = e.dO + Ap.dT. It follows from
the equation pF=p0F'0(l + ad), because p is here constant, that
p.dV=pJPr<ft.dOt and therefore (i) C=c + 4p0a/pQ, since rop0=l.
Finally, if we set a2=p0G/p0c and b2=p0/p0, the equations (a), (b),
and (c) take the following forms :
'da-j'dt + 'dul'dx + *dvfdy + *dw/oz = 0,
(k)
These equations are due to Kirchhoff.* Eeference may be made to
KirchhofF s work for the application of these equations to the more
difficult cases of the transmission of sound. We will here investigate
only the influence of conduction and friction on the motion of plane
sound waves. First, however, the physical significance of the con-
stants (a) and (b) must be determined.
If there is neither conduction nor friction in the air, we have K = 0
and p.' = 0 ; further, if the vibrations occur in the direction of
the z-axis, we also have v = w = 0. Under these circumstances the
equations (k) become
If the second of these three equations is differentiated with respect
to /, we obtain
•Kirchhoff, Pogrj. Ann., Vol. 134. 1868.
332 CONDUCTION OF HEAT. [CHAP..XIV.
It thus follows, by the use of the first and last of these equations,
that 3%/322 = a2 . 32w/3a;2. An integral of this equation is
and this expression represents a wave motion which proceeds with
the velocity (1) a = »Jp0C/p(>c = b-JC/c. This value for the velocity of
sound was found by Laplace. It differs from the value calculated
in XXXV., which was originally found by Newton, and which in
our present notation is b = JpQ/p0. The difference between the two
formulas is due to the fact that in the first we have taken into
account the heating of the air by compression and its cooling by
expansion. Since the ratio C/c has been determined by direct experi-
ment, the true velocity of sound in the air may be calculated. For
atmospheric air at 0° C,, C/c = 1,405 ; hence a = 33815 cm. This value
agrees very well with experiment.
Suppose that a plane-wave is propagated in the direction of the
z-axis, and that K and /*' are not zero. The vibrations are parallel
to the x-axis, so that v = Q and w = Q. Since u, 0, and cr are then
functions of x and t alone, equations (k) become
-dufit + 62 . "do-fix + (a2 - V)
39/3* - /c2 . 920/3.s2 --= fa fit.
The unknown quantities u, 0, and o- are periodic functions of t.
We will represent by h a real magnitude, and by u', 0', and o-' three
magnitudes which are functions of x alone. It is then admissible
to make the assumptions (n) u = u' . e*", 0 = 6'. ehit, a- = ar' . e*", where
i = -J - 1. By the help of these equations we obtain from (m)
hi<r' + du'jdx = 0,
hiu' + b'2 . da-'/dx + (a2 - tf)dQ'/dx = /*' . d2u'/dx* - ^'hi . do-'/dx,
We eliminate a-' from these equations, and then have
J _ A v + fci(a2 _ j2) . dQ'/dx = (b2 + $p'hi) . tPu'/da?,
\ du'/dx = K2d'2Q'/dx2 - hiQ'.
If the first of these equations (o) is differentiated with respect to x,
u' may be eliminated, and we obtain the following differential equation :
(p) K2(&2 + *'p'hi) . d*&/dx* + (h*K'2 + ^V - hazi) . d*Q'/dx* - h*i& = 0.
Since this equation is linear, we set 0' = emx, and obtain
(r) K-'m4(Z»2 + * X/w) + TO2(A2*2 + |/*'A2 - hcM) - hsi = 0.
SECT, cxxxv.] VELOCITY OF SOUND IN GASES. 333
We will determine the exponent in only in the case in which
the conductivity as well as the internal friction is very small. If
« = 0 and /*' = (), we have from (r) m= -hi/a. If therefore we set
m = (-hi + $)/a, where 8 is a small quantity whose higher powers
may be disregarded, we have from (r), if the terms K2//, *28, etc., are
neglected, (s) 8 = - [^'h2 + (1 - 6'2/a2)K2fr2]/2a2. But it follows from
(n) and (q) that one value for 0 is 0 = e8x/a . ehi(t~xM • the other is
obtained by substituting - 1 for i, which gives 0 = eSx/a . «-**"-"/•». Half
the sum of the two values of 0 satisfies the conditions and is at
the same time real, since
(t) 0 = e-WaM-*»+ (i - tw* *n • *w m cos tyy _ xja^
From the exponent of e we see that the changes of temperature in
the wave diminish the further it travels ; at the same time u also
diminishes. The sound, therefore, becomes weaker as the wave travels
further. If T is the period of vibration and n the number of vibra-
tions, we have h = 27r/T—2mr.
By using this value of h it follows, from equation (t), that the
higher tones lose their intensity more quickly than the lower ones.
The mathematical treatment of conduction is principally due to
Fourier, who not only developed the partial differential equation
which is at the foundation of the treatment of conduction, but also
gave us methods for the solution of a great number of problems.
His principal work is : Thdorie Analytique de la Chaleur, Paris, 1822.
Of the later works on this subject we mention Eiemann, Partielle
Differentialgleichungen, edited by Hattendorf, Braunschweig, 1876.
INDEX.
Acceleration, 2.
Centripetal, 11.
Resultant, 4.
Action and Reaction, 48.
Adiabatic Curves, 268.
Amplitude, 90.
Attraction, Universal, 30.
Biot and Savart's Law, 184.
Bodies, Structure of, 50.
Rigid, 58.
Equilibrium of, 59.
Motion of, 59.
Rotation of, GO.
Boyle's Law, 267.
Capacity, Electrical, 131, 149.
of Coaxial Cylinders, 153.
of Condenser, 141.
of Spherical Condenser,140,152.
of Parallel Plates, 151.
Specific Inductive.
(Dielectric Constant), 156.
Relation to Index of Refrac-
tion, 221.
Capillarity, 121.
Capillary Constant, 121.
Tubes, 125.
Carnot's Cycle, 272.
Theorem, 274.
Clausius's Equation of the State of a Gas,
290.
Theorem, 275, 277.
Collision, 49.
Condenser, Cylindrical, 153.
Parallel Plate, 151.
Spherical, 152.
Conductivity for Electricity, 197.
for Heat, 300.
Conductors, System of, 147.
Work done on, 150.
Contact Angle, 125.
Continuity, Equation of, 105.
Cooling of a Sphere by Conduction, 318.
by Radiation, 319.
Corresponding States, 289.
Critical Temperature, 284.
Current, Electrical, Continuity of, 191.
Force of, 184.
Force of Linear, 190.
Measurement of Con-
stant, 194.
Measurement of Vari-
able, 195.
Potential of, 185.
Potential Energy of,
193.
Currents,Electrical,MutualActionof,192.
Potential Energy of,
193.
Systems of, 186, 190.
Cycle (Cyclic Process), 268, 272.
Carnot's, 272.
Efficiency of Carnot's, 273.
Damping Action, 195.
Deformation, 74.
Relation of, to Stress, 79.
Density, 35.
336
INDEX.
Descartes's Explanation of Mutual Ac-
tions, 49.
Dielectric, 155.
Equations of, 219.
Plane Waves in, 221.
Dielectric Constant, 156.
in Crystals, 246.
Displacement, 156.
Dilatation, 76.
Linear, 77.
Principal Axes of, 78.
Volume, 77.
Dirichlet's Principle, 136.
Dissociation, 295.
Dyne, 9.
Earth, Temperature of, 302.
Elastic Body, Equilibrium of, 82.
Motion of, 89.
Potential Energy of, 96.
Elasticity, Coefficient of, 79.
Modulus of, 81.
Electrical Convection, 161.
Displacement, 191.
Distribution, 128.
on a Conductor, 130.
on Conductors, 139.
on an Ellipsoid, 133.
on a Plane, 135.
on a Sphere, 132, 137.
Double Sheets, 160.
Energy, 145.
Force, Law of, 128.
Lines of, 143.
Images, 135.
Oscillations, 215, 223.
Polarization, 191.
Potential, 128.
of a Conductor, 131.
near a Surface, 129.
Electricity, Theories of, 127.
Electrified Body, Force on, 141.
Electro-kinetic Energy, 201, 210.
Electromagnetism, 184.
Equations of, 188.
Electrometer, Quadrant, 154.
Thomson's Absolute, 142.
Energy, Conservation of, 58.
Kinetic, 14.
of a System, 56.
Potential, 57.
Entropy, 271, 272, 278, 292, 294.
Entropy Criterion of Equilibrium, 294.
Equilibrium, 58.
Conditions of, 58.
of Fluid Surfaces, 12 .
Equipotential Surfaces, 23.
Construction of, 132.
Equivalence of Heat and Energy, 268.
Equivalent, Mechanical, of Heat, 268.
Ether, 230, 231.
Fresnel's Assumption Concerning,
233.
Neumann's Assumption Concern-
ing, 233.
Euler's Equations of Motion of Fluids,103.
Falling Bodies, Laws of, 5.
Flexure, 87.
Flow of Fluid, 108.
Through Tube, 119.
Flux of Force, 42.
Force, 8.
Centripetal. 11.
Components of, 9.
Line of, 23.
Measure of, 6, 9.
Normal, 13.
Tangential, 13.
Tubes of, 145.
Unit of, 9.
Forces, Conservative, 20.
External, 53.
Internal, 53, 62.
Fluid, Conduction of Heat in, 325.
Elasticity of, 81.
Equilibrium of, 62, 99.
Motion of, 103.
Viscous, 118.
Motion of Sphere in, 109.
Steady Motion of, 118.
Fourier's Theorem, 312, 315.
Fresnel's Formulas for Light, 231.
Failure of, 235.
Friction of Fluids, 115.
Coefficient of, 115.
Galileo's Laws of Falling Bodies, 5.
Gas, Elasticity of, 82.
Ideal, 270.
Specific Heats of, 270.
INDEX.
337
Gauss's Theorem, 41.
Gravity, Acceleration of, 5.
Centre of, 50.
Gyration, Radius of, 60.
Heat, Conduction of, 298.
in Fluids, 325.
Flow of, between two Bodies, 317.
from a Point, 204.
from a Surface, 303, 316.
in a Cylinder, 322.
in an Infinite Body, 305.
in a Plate, 308.
Fourier's Equation of, 298.
Steady Flow of, in a Cylinder, 323.
in a Plate, 300.
in a Sphere, 301.
in a Tube, 301.
Helmholtz's Transformation of Euler's
Equations, 106.
Hertz's Apparatus, 216.
Form of Max well's Equations, 219.
Huygens's Construction, 262.
Ice, Formation of, 307.
Impulse, 8.
Measure of, 9.
Inclined Plane, 24.
Induction, Coefficients of Magnetic, 149.
of Electrical Currents, 199.
Coefficients of, 202.
Equations of, 208.
Law of, 200.
Measurement of Coefficients
of, 203.
Mutual, 201.
Self-, 200.
Inertia, Moment of, 60.
Principle of, 6.
Isentropic Curves, 268.
Isothermal Curves, 267.
Joule's Law, 197.
Kepler's Laws, 27.
Lagrange's Equations of Motion of Fluids,
111.
Laplace's Equation, 45.
Application of, 46.
Lenz's Law, 200.
Light, Electromagnetic Theory of, 230,
235, 237.
Emission Theory of, 229.
Principal Laws of, 230.
Wave Theory of, 229.
Lines of Force, 23.
Electrical Force, 143.
Logarithmic Decrement, 196.
MacCullagh's Construction, 263.
Magnet, Constitution of, 163.
Forces acting on, 169.
Oscillation of, 170.
Potential Energy of, 171.
Potential of, 166.
Magnetic Axis, 166.
Induction, 173, 177, 179.
Coefficient of, 180.
Moment, 165.
Permeability, 180.
Poles, 163.
SheU, 180.
Strength of, 180.
Force, 166.
due to Electrical Cur-
rent, 184.
Law of, 163.
Lines of, 174, 178.
Tubes of, 174.
Magnetism, 163.
Distribution of, 165.
Magnetization, Intensity of, 165.
Magnetized Sphere, Potential of, 168.
Material System, 53.
Maxwell's Electromagnetic Theory of
Light, 230.
Theory of the Action of a
Medium, 72.
Melting, 291.
Moment, of Force, 56.
of Inertia, 60.
of Momentum, 55,
Momentum, 49.
Moment of, 55.
Motion, Constrained. 24.
Curvilinear, 3.
Equations of, of a Particle, 10.
In a Circle, 11.
Oscillatory, 12.
Periodic, 1, 12,
338
INDEX;
Motion, Uniform, 1.
Variable, 1, 2.
Newton's Law of Attraction, 30.
Ohm's Law, 197.
Optic Axes, 249.
Optic Axes of Elasticity, 247.
Oscillatory Motion, 12.
Path, 1.
Equation of, 4.
Pendulum, 25, 61.
Period, 90.
Points, Electrical Action of, 144.
Poisson's Equation, 45.
Application, 46.
Polarization, Angle of, 234.
Potential, 20, 21.
Coefficients of, 148.
Difference of, 22.
of a Circular Plate, 38.
of a Cylinder, 39.
of a Solid Sphere, 37.
of a Spherical Shell, 36.
of a Straight Line, 39.
of a System, 34.
Poynting's Theorem, 224.
Pressure, 64.
Hydrostatic, 62, 63.
Principal Section, 260.
Projectiles, 7, 11.
Kays, Direction of, in Crystals, 256.
Velocity of Propagation of, 257.
Reflection of Polarized Light, 232, 237,
239
Total, 240, 241.
Refraction, Double, 246.
at the Surface of a
Crystal, 261.
in Uniaxial Crystals,
264.
In a Plate, 242.
Index of, 230.
of Polarized Light. 233, 237,
239.
Resistance, 198.
Measurement of, 205.
Resistance, Measurement of —
Lorenz's Method, 207.
Thomson's Method, 206.
Shear, 77.
Solenoid, 186.
Solid, Internal Forces in, 62.
Sound, Velocity of, in Gases, 330.
Spherical Shell under Pressure, 83.
Sphondyloid, 143.
State of a Body, 266.
Diagram Representing, 267.
Equation of, 267.
Stateof a Gas, Clausius's Equation of ,290.
Van der Waal's Equation
of, 286.
Stokes's Theorem, 21.
Stress, 64.
Components of, 65.
Equilibrium under, 67.
Principal, 69.
Strings, Vibrating, 95.
Surface Tension, 122.
Tension, 64.
Thermodynamic Relations, 280.
PropertiesofBodies,281.
Thermodynamics, First Law of, 268.
Equation embodying, 269.
Second Law of, 170.
Application of, 279.
Torsion, 85.
Coefficient of, 86.
Trigonometrical Series Representing
Arbitrary Functions, 309, 312.
Tubes of Force, 145.
Uniaxial Crystals, 259.
Wave Surface in, 259.
Units, Absolute, 2, 211.
Derived, 2.
Practical, 214.
Van der Waal's Equation of State, 283.
Vaporization, Heat of, 291.
Vapour, Saturated, 290.
Specific Heat of, 292.
Vector, 55.
Velocity, 1, 2.
Angular, 1.
INDEX.
339
Velocity, Components of, 3.
Resultant, 4.
Velocity Potential, 106.
Vibrations, Longitudinal, 91.
Transverse, 91.
Vortex Motion, 107.
Filament, 109.
Wave Surface, 251.
Reciprocal, 257.
"Waves on the Surface of a Fluid, 112.
Waves, Plane, 90.
in Dielectric, 221.
Spherical, 93.
Stationary, 93.
Torsional, 93.
Velocity of, 91, 92.
in Crystals, 248, 249, 257.
Weight, 9.
Work, 14.
Done in Closed Path, 15, 16.
QC 20 C462E 1897
L 006 580 2
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