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LAWS OF
PHYSICAL SCIENCE

A REFERENCE BOOK

11
EDWIN F. NORTHRUP, Ph.D.

PALMER PHYSICAL LABORATOKY, PRINCETON UHIVSRSITY, PRINCETON, N. ].

PHILADELPHIA AND LONDON

J. B. LIPPINCOTT COMPANY
s

COPYRIGHT, 1917
BY J. B. LIPPINCOTT COMPANY

Electrotyped and Printed by J. B. Lippincott Company.
The Washington Square Press, Philadelphia, U. S. A.

this collection of nature 8 laws is
lovingly dedicated to an able and most
honorable exponent of human laws

My Father
HON. ANSEL JUDD NORTHRUP

PREFACE

Exact knowledge consists of accumulated facts and seta
of formulated propositions respecting facts. Data, Mathe-
matical relations and Physical laws constitute the three
firm supports of Physical science and Engineering.

The data of physical science are readily accessible in
several published tables of physical constants. The mathe-
matics used in physical science has been summarized, classi-
fied and formulated, for ready reference, in many published
books. The author is not aware, however, of any hand-book
or reference work which contains a full list of the general
propositions or laws of science.

Such reference lists are not without value, and this book
has been prepared to fill an obvious gap in the literature of
Physical Science. Furthermore, it appears to the author
that students in any of the branches of Natural Science will
not only find guidance, but will also derive inspiration by
having before them under a single view the very epitome of
the world's heritage of the fundamentals of its knowledge
and wisdom. None will question that the fundamentals of
science are its laws, principles, theorems and precise state-
ments of the general properties of matter; but it is not
always easy for students in one branch of science to find and
to know the literature on important principles and facts in
an entirely different, or even in closely allied branches of
science. The author hopes that what has been here gath-
ered together and classified will help such students in their
search and give them the means to broaden their view.

We have chosen for a title, " Laws of Physical Science "
but many general propositions, theorems and mere state-
ments of important facts have been included which perhaps,

vi , PREFACE

if Strictly considered, could not be discriminated as laws.
Indeed, it was found impossible, in many cases, to decide if
certain propositions possess sufficient generality and validity
to deserve the title "law." When, however, such doubts
existed, a policy of inclusion has been followed in preference
to one of exclusioui

For convenience and system the general statements (in
all 480 with title) have been classified in six sections : I— Me-
chanics; II— Hydrostatics, Hydrodynamics and Capillarity ;
III— Sound; IV— Heat and Physical Chemistry; V^ -Elec-
tricity and Magnetism; VI — Light.

Each law, proposition or general statement is charac-
terized by giving it a heading or title. Each proposition
covered by a title is followed by one, and in many £ases by
several references to easily accessible text-books, standard
treatises, and, in a few cases, tb original articles or papers,
where one may find the propositions stated in different forms
and additional information concerning them of authorita-
tive character.

WhUe many laws of Physical Science have had their
origin with individual investigators, the perfected form of
statement they now possess has been in the main reached
by a process of intellectual growth in which many have
taken part. It has seemed, therefore, wiser tb make most of
the references to treatises and text-books on physics, physi-
cal chemistry and chemistry, rather than to papers written
by the authors of the propositions. Moreover, original
papers, beside containing much extraneous matter, are not
usually readily accessible as are text-books and treatises.

An alphabetically arranged bibliography of all books and
journals referred to and a very full index, with duplicated
references, to aid in the quick location of subject matter and
proper names, are included.

PREFACE '"

The author expresses his acknowledgment to Mr. H. A.
Frederick for assistance in collecting some of the material
used, and to Prof. K. T. Compton for his careful reading of
the manuscript. He further acknowledges with gratitud.e
the unfailing and valuahle assistance of his wife, Margaret
Stewart Northrup, in collecting material and in arranging
it for publication.

Paimeb Physical Laboratory The AUTHOR

Princeton, N. J.
January, 1917.

CONTENTS

PAOB

I. Mechanics 1

II. Hydrostatics, Htdhodtnamics and Capillarity 29

III. Sound 43

IV. Heat and Physical Chemistry 69

V. Electricity and Magnetism Ill

VI. Light 163

Bibliography and Index 191

I

MECHANICS

LAWS OF PHYSICAL SCIENCE
MECHANICS

NEWTON'S FIRST LAW OF MOTION.

Every body continues in its state of rest or of uniform
motion in a straight line, except in so far as it may be com-
pelled by force to change that state.

(Thomson and Tait, Treatise on Natural Philosophy,
Part I, art. 244.)

NEWTON'S SECOND LAW OF MOTION.

Change of motion is proportional to force applied, and
takes place in the direction of the straight line in which the
force acts.

(Thomson and Tait, Treatise on Natural Philosophy,
Part I, art. 251.)

NEWTON'S THIRD LAW OF MOTION.

To every action there is always an equal and contrary
reaction ; or, the mutual actions of any two bodies are always
equal and oppositely directed.

(Thomson and Tait, Treatise on Natural Philosophy,
P^rt I, art. 261.)

4 LAWS OF PHYSICAL SCIENCE

NEWTON'S LAW OF UNIVERSAL GRAVITATION.

All bodies attract eaeli other with a force proportional to
the product of their masses and inversely proportional to
the square of the distance between them.

Thus, F = G MMl-

G is called the Newtonian constant.

Cavendish (1797-1798) first measured G experimentally.
In 1874 C. V. Boys obtained 6.657e X lO"" on the C.G.S.
system for the value of G.

(Consult Scientific Memoirs, edited by J. S. Ames, Vol.
IX, The Laws of Gravitation. See p. 136.)

NEWTON'S LAW OF ATTRACTION FOR A SPHERE.

The gravitational attraction of a particle toward a
sphere, whose density m^y be a function of the distance from
the center, is the same as if the mass of sphere were concen-
trated in a particle at the center of the sphere.

The attraction by such a sphere on a particle within it is
due entirely to that portion of the sphere which lies inside
of a concentric surface through the particle. In other words,
the attractions due to all parts of the sphere which lie
farther than the particle from the center cancel each other.

(Consult Thomson and Tait, Treatise on Natural
Philosophy, Part II, arts. 462, 471 et seq.)

WEIGHT AND IIASS.

Bodies are constant in mass but variable in weight, and
weight is always proportional to mass.

Thus, W = Mg,
where W is the weight of the body, M its mass and g the
acceleration of gravity.

The value of g increases from the equator to the poles,
g = 978.00 cm. per sec. per sec. at the equator and g =
983.01 at the poles.

MECHANICS 6

(Jeans, Theoretical Mechardcs, pp. 29-31. For values
of g at various places consult Chwolson, Trcdte de Physique,
Vol. I, Part 3, p. 387. For accepted meam, value of g consult
Hering, Conversion Tables, p. 87.)

LAW OF FALLING BODIES.

The law by which bodies fall in vacuum is : The velocity
of descent is proportional to the time of falling, and the
distance of descent is proportional to the square of the time
of falling.

Thus, V oc t, and s oc t^
Or, V = gt and s= $. when the body starts from rest.
Or, ^-!| = g, which integrated gives,

s = J^ gt' +A, t + A,.

Here g is the constant acceleration of gravity. Ai and Aj
are constants of integration.

(Consult Mach, Science of Mechanics, p. 130 et seq. See
"Use of Analogy in Viewing Physical Phenomena" by
B. F. Northrup, Jour. Frank. Inst., p. 17, July, 1908.)

DESCENT ON AN INCLINED PLANE.

The acceleration along an inclined plane- AC is to the
acceleration along the vertical AB as the length AB is to
the length AC, or,

acceleration along incHned plane ^ AB _ ^^^ ^j^^^.^ ^
acceleration ot gravity At/

= the angle of inclination of the plane.

(Consult Mach, Science of Mechanics, pp. 137-138.)

« LAWS OF PHYSICAL SCIENCE

CONDITION OF EQUILIBRIUM FOR A SYSTEM.

When a system of particles is in equilibrium under the
action of any system of external forces, the sum of the
components of all these forces in any direction is zero ; and
the sum of the moments of all these forces about any line
is zero.

(Consult Jeans, Theoretical Mechanics, p. 64. For com-
plete treatment, see Thomson and Tait, Treatise on Natural
Philosophy, Part II, Chaps. VI and VII.)

BASIC EQUATIONS OF MECHANICS; COMMENT ON.

Let s = distance, t = time, v ^ instantaneous velocity,
a = acceleration of a uniformly accelerated motion, F =
ma = force and m = mass. Assuming the body starts from
rest the relations of these quantities may be placed in the
two groups ;

v = at ) ' mv = Ft

8 = 3^atni ms = J^Ft2

as=Hv' J Fs=J^mv2

Equations of group I contain the quantity a, and in addi-
tion two of the quantities s, t, v, as in table A, and equations
of group II contain the quantities m, F, s, t, v, each equation
containing m F and in addition to m F two of the quaai-
tities s, t, V, as in table B,

f V,t f V,

(A) a ] s, t (B) m, P ^ s, t

I s,v (. s,v

(Mach, Science of Mechanics, pp. 269-270.)

MECHANICS 7

GENERAI, MECHANICAL PRINCIPLE.

" A system always tends to move from rest in such a
way as to diminish the potential energy as much as possible,
and the force tending to assist a displacement in any direc-
tion is equal to the rate of diminution of the potential
energy in that direction."

(J. J. Thomson, Elements of Electricity and Magnetism,
p. 82.)

KEPLER'S FIRST LAW.

The planets move about the sun in ellipses, at one focus
of which the sun is situated.

(Consult Mach, Science of Mechanics, p. 187.)

KEPLER'S SECOND LAW.

The radius vector joining each planet with the sun de-
scribes equal areas in equal times.

(This second law, the law of areas, can be explained
simply if it be assumed, as a particular case, that the
acceleration toward the sun is constant.)

(Consult Mach, Science of Mechanics, p. 188.)

KEPLER'S THIRD LAW.

The cubes of the mean distances of the planets from the
sun are proportional to the squares of their times of revolu-
tion about the sun. This law may be stated mathematically,

r; r; r;

■^ = ';p"=S-=. . . . =a constant

I ! 1

Here Rj, Rj, R3, are mean radii and Tj, T^, T3, etc., the
respective times of revolution of the planets.

(Consult Mach, Science of Mechamics, pp. 188-189.
Also Laplace, Traite de Mecanaque Celeste, Book II,
Chap. I.)

8 LAWS OF PHYSICAL SCIENCE

LAW OF THE LEVER.

Any lever is in equilibrium when the algebraic sum of
the statical moments taken about the fulcrum equals zero.

By the statical moment is here meant the force acting at
a point multiplied by the perpendicular distance from the
line of support to the direction of the force.

(For a lucid discussion of the principle of the lever see
Mach, Science of Mechanics, Chap. I.)

COHPOSITION BY PARALLELOGRAM RULE.

If a parallelogram A B C D is so drawn that two of its
sides, as A B, A D, which meet in a point, represent as
regards both magnitude and direction, two displacements, or
two velocities, or two accelerations, or two forces to be com-
pounded, then the diagonal A C of this parallelogram will
represent, in respect to both magnitude and direction, the
resultant displacement, velocity, acceleration or force.

(Consult Jeans, Theoretical Mechanics, Chaps. I and
III.)

PARALLELOGRAM OF FORCES.

Two forces acting at the same time and for the same
time upon a particle produce accelerations which are in-
dependent of each other and proportional to the forces.
The resultant distance, direction, velocity or acceleration
is obtained by compounding by the parallelogram rule.

(Consult Jeans, Theoretical Mechanics, Chap. III.)

LAMI'S THEOREM.

" When a particle is acted on by three forces, the
necessary and sufficient condition for equilibrium is that
the three forces shall be in one plane and that each force
shall be proportional to the sine of the angle between the
other two."

(For proof see Jeans, Theoretical Mechanics, p. 40.)

MECHANICS 9

CENTRIPETAL ACCELERATION.

The centripetal acceleration of a body moving in a circle
is directed toward the center of the circle and is equal to the
square of its instantaneous linear velocity divided by its
distance from the center about which it rotates at the instant
considered.
For uniform rotation in a circle,

acceleration = ^^^°°'*y ' , or, a = ^ .
radius r

(Consult Laplace, Trwite de Mecandque Celeste, Book I,
art. 10.)

PRINCIPLES OF THE SCREW AND THE WRENCH.

Any motion can be reduced to a translation and a rota-
tion about an axis parallel to the translation. Such a com-
bination of a translation and a rotation is called a Screw.

Similarly any system of forces acting on a rigid body
may be replaced by a single force and a couple about the
line of the force. This combination of a force and a couple
is known as a Wrench.

Thus a screw and a wrench are the most general types
of motion and force respectively.

(Jeans, Theoretical Mechanics, pp. 91, 106-107. For
general treatment of motion in three dimensions, see Eouth.
Elementary Bigid Dynamics, Chap. V, pp. 184r^228.)

10 LAWS OF PHYSICAL SCIENCE

THEOREM OF PRINCIPAL AXES OF INERTIA.

In every rigid body there are three mutually perpen-
dicular axes of rotation intersecting at the center of gravity
of the body which are characterized by the property that,
if the body be rotated about any one of these axes, it will
continue thus to rotate unless acted on by an external force.
If rotating about any other axis the centrifugal forces tend
to change the axis of rotation and the rotation is therefore
unstable. The position of these axes is determined mathe-
matically by the fact that, if they be taken as the axes of
coordinates, the products of inertia.

.Jmxy, 2'myx, J'mzx

for the body wiU vanish.

(Webster, The Dynamics of Particles and of Rigid,
Elastic and Fluid Bodies, pp. 228-229. Also Routh, Ele-
mentary Rigid Dynamics, pp 12-13.)

MOMENT OF INERTIA AND ENERGY OF ROTATION.

The moment of inertia about any straight line through
a rigid body made up of a system of particles is the sum
of the products of the mass of each particle by the square
of its perpendicular distance from this straight line; its
axis of rotation.

Thu^, moment of inertia = i = Jmra where m is the
mass of a particle and r its perpendicular distance from its
axis of rotation.

The kinetic energy E^ of the rotating body is, E =
Yz I 0)", where w is its angular velocity. This is the analogue
of Bi = l^ mv^, where m is the mass of a body having a
rectilinear velocity v.

(Thomson and Tait, Treatise on Natural Philosophy,
Part I, art. 281.)

MECHANICS 11

RADIUS OF GYRATION, (i)

The radius of gyration of any body (made up of a
system of particles) about any axis is the perpendicular
distance from that axis at which, if the whole mass were
placed, it would have the same moment of. inertia as before.

Mathematically the radius of gyration is,

k=^

2m

The kinetic energy of rotation of the body when its
angular velocity is m is :

(Consult Thomson and Tait, Treatise on Natural Phi-
losophy, Part I, art. 281.)

RADIUS OF GYRATION. (3)

If the radius of gyration of any body revolving about a
line through its center of mass is k, then the radius of
gyration K about any parallel line a distance a from this
line is given by the relation,

K^ = k^ + a.\

(Consult Jeans, Theoretical Mechanics, p. 291.)

12 LAWS OP PHYSICAL SCIENCE

ANALOGUES IN TRANSLATION AND ROTATION.

Force is measured by the product, mass X linear acceler-
ation, or F = ma.

Moment of force is measured by the product, moment of
inertia X angular acceleration, or P, = lo.

One-half the mass X linear velocity = kinetic energy of
translation, or Ej = % iii"^^-

One-half the moment of inertia X the angular velocity
= kinetic energy of rotation, or E^ ^ % I cu^-

When no external forces are acting, the product, mass
X velocity is constant, or mv = linear momentum = a
constant.

Under similar conditions the product, moment of inertia
X angular velocity is constant, or I <o = angular momentum
= a constant.

(Ames, Theory of Physics, Chap. II.)

SIMPLE HARMONIC MOTION.

A body is said to oscillate with Simple Harmonic Motion
when its acceleration is always directed toward the middle
point of its path of oscillation and is proportional to its
displacement therefrom. ,

The acceleration is given by the formula,

a = -4ir'N»S,

where N is the frequency or number of complete oscillations
in the unit of time and S is the displacement of the point
from its middle position at any instant.

(Thomson and Tait, Treatise on Natural Philosophy,
Part I, art. 57,)

MECHANICS 13

' SIMPLE PENDULUM.

The oscillation of a pendulum may be regarded as Simple
Harmonic Motion when the amplitude of the oscillation is
small. The approximate time of complete vibration is,

— Vi-

Here 1 = the length of the simple pendulum and g =
the acceleration of gravity.

(Jeans, Theoretical Mechanics, pp. 259-262. Also
Ganot's Physics, art. 56.)

LAW OF THE COMPOUND PENDULUM.

If y is the length of a simple pendulum which oscillates
in the same time as a compound pendulum, then the prin-
ciple of the center of oscillation asserts that

^~ Zmr'

The compound pendulum executes isochronous oscilla-
tions when the amplitude of the oscillations is small. The
time of a complete oscillation is that of a simple pendulum
having this length y.

Here -Tmr is the sum of the products of the elements
of mass of the pendulum multiplied by their distances from
the point of support and S mr^ is the moment of inertia of
the pendulum about its axis of oscillations. Its complete
period for oscillations of small amplitude is

■=2-V-

Zmr'

g2inr
(Mach, Science of Mechanics, pp. 173-177.)

14 LAWS OF PHYSICAL SCIENCE

CONVERTIBILITY OF CENTER OF OSCILLATION AND
POINT OF SUSPENSION.

When the center of oscillation and the point of suspen- -
sion of a compound pendulum are interchanged the time of
oscillation remains the same.

, This principle was employed by Captain Kater for
determining the exact length of the seconds pendulum.
Hence the term "Eater's Pendulum."

(Mach, Science of Mechanics, p. 172 et seq. and p. 186.
Also Ganot's Physics, art. 81.)

CYCLOIDAL PENDULUM.

A particle subjected to the force of gravity and con-
strained to move in a cycloidal path will have a harmonic
motion which is strictly isochronous whatever its amplitude
of oscillation.

The period will be,

/ 2D

where D is the diameter of the rolling circle which generates
the cycloid. Thus the period is that of a simple pendulum of
length 2D.

(Jeans, Theoretical Mechanics, p. 267.)

MINIMUM POTENTIAL ENERGY.

When a system is in stable equilibrium its potential
energy is as small as possible ; that is, any small movement
imparted to the system, subject to its degrees of freedom^
will increase and not decrease its potential energy.

(For a mathematical discussion see Jeans, Theoretical
Mechanics, p. 174 et seq.)

MECHANICS 15

lUPACT BETWEEN TWO BODIES.

When two bodies meet in impact the impulse of restitu-
tion I' equals the impulse of compression I times a certain
coefficient e called the coefficient of elasticity or coefficient of
resilience.

Thus, I' = el.

The coefficient e is unity for perfectly elastic bodies and
zero for perfectly inelastic bodies.

Its value for actual bodies must be obtained by
experiment.

(Jeans, Theoretical Mechanics, pp, 238-241.)

HEWTON'S EXPERIMENTAL LAW OF IMPACT.

When two bodies meet in impact and their centers of
gravity lie, at the moment of impact, in a line through the
point of contact, then the normal component of relative
velocity of their centers of gravity after impact is equal to
the relative velocity before impact times the coefficient of
resilience, and is in the opposite direction, or

V - v' = - e (u - u')
where v, v' are the velocities of the two bodies after impact,
u, u' their velocities before impact and e the coefficient of
resilience.

(Jeans, Theoretical Mechanics, pp. 244, 245.)

RELATION CONNECTING VELOCITIES OF TWO BODIES
BEFORE AND AFTER IMPACT.

If two bodies of masses m and m' have velocities u, u'
before impact and velocities v, v' after impact, then by the
conservation of momentum

mv + mV = mu + m'u',
or, if U is the common velocity of the two bodies after impact,

U= "'"+'"'"' (called the rule of Wallis).
m + m'

(Jeans, Theoretical Mechanics, p. 245. Also Mach,
Science of Mechanics, p. 318.)

16 LAWS OF PHYSICAL SCIENCE

RELATION OF VELOCITIES AFTER AND BEFORE IMPACT.

The normal velocities v, v' of two impacting bodies of
masses m, m' after collision are given in terms of their
velocities u, u' before collision by the two relations,

_ mu + m'u' — em' (u — u')
^ ~ m + m' '

, mu + m'u' + em (u — uO

V = j -, '

m + m

where e is the coefficient of resilience.
(Jeans, Theoretical Mechanics, p. 245.)

IMPACT.

Relative velocity of approach and recession of impinging
perfectly elastic bodies is the same.

Also, mV + mv = m'u' + mu
and, mV^ + mv^ = m'u'^ + mu^,

namely, the quantity of motion before and after impact,
estimated in the same direction is the same, also, the vis
viva, or kinetic energy of the system.

(Consult Mach, Science of Mechanics, p. 322. See Thom-
son and Tait, Treatise on Natural Philosophy, Part I, arts.
300-301, for comments upon conditions which determine
loss of kinetic energy with impact.)

TIME OF ELECTRICAL CONTACT OF IMPACTING STEEL-SPHERES.

When two equal steel-spheres come together the time in
millionths of a second during which they are in contact
equals 74.7 times the diameter (in cm.) divided by the fifth
root of their velocity of approach.

Thus, T^ = 74.7 -^

yl/5

where T/i, = microseconds,

D = diameter of spheres in centimeters,

V = velocity in centimeters per second with

MECHANICS 17

whicli they approach each other at the moment of the
beginning of contact.

(This law was experimentally determined by E. F.
Northrup and A. E. Kennelly. See Jour. Frank. Inst.,
July, 1911.)

VELOCITY OF A DISTURBANCE IH AN ELASTIC MEDIUM.

The velocity with which any compressional disturbance
is propagated through an elastic medium is given by the
equation

v = V*

where E is the coefficient of volume elasticity and D is the
density of the medium. This law is due to Newton.
(Watson, A Text Book of Physics, pp. 362-365.)

BERTRAND'S PRINCIPLE OF SIMILITUDE.

If two systems differ only in geometrical magnitude,
the masses of corresponding parts being proportional to
each other, they will be mechanically similar if the forces
acting on and the velocities of respective parts of the systems
bear the relation

force proportional to mass X velocity .
linear dimensions

This principle gives the conditions under which two
systems which are geometrically similar may also be mechan-
ically similar and is important in testing small model
machines.

(Routh, Elementary Rigid Dynamics, pp. 234-297. For
a full discussion, see article by Bertrand in Cahier 32, of the
Journal de I' E cole Polytechnique.)

18 LAWS OF PHYSICAL SCIENCE

CHAITGE OF HOMENTTTM DUE TO AN IMPULSIVE FORCE.

When a body moves in such a way that its configuration
with respect to the force which acts upon it remains always
the same, the moving force is measured by the rate of
increase of the momentum.

If F is the force, p the momentum and t the time,

F = .f whence
at

p = jFdt.

This time-integral is called the Impulse of the Force.
(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 558.)

SLIDING FRICTION.

The tangential force required to slide one body over
another is independent of the extent of their surfaces in
contact and of the rate of sliding. It is proportional to the
normal force R pressing them together and to a constant
for the two bodies called the coefScient of friction /*. Thus
F = fiH.

Distinction is made between statical friction where a
relative motion of the bodies is just not produced by the
acting forces and kinetic friction where there is relative
motion of the bodies in contact. Kinetic friction is generally
less than the extreme force of static friction.

(Thomson and Tait, Treatise on Natural Philosophy,
Part II, arts. 450-452.)

ROLLING FRICTION.

Rolling friction acts much as does sliding friction. The
coefficient u is in this ease much smaller and the force
required to cause motion is again proportional to the normal
force R pressing the bodies together.

Thus F = uR.

(Oanot's Physics, art. 49.)

MECHANICS 19

HOOEE'S LAW.

Hooke's law states that within the elastic limit of any
body the ratio of the stress acting upon the body to the
strain produced is constant. This constant ratio of stress
to strain for any particular type of change in any body is
called its "coefficient of elasticity."

(Ames, Theory of Physics, pp. 103-104. Also see
Ganot's Physics, art. 85.)

YOUNG'S MODULUS.

By "Young's Modulus" is understood the force which
would be required to stretch a body of unit cross-section to
double its length if such lay within the elastic limit. It
is a constant for any one material.

Thus, Young's Modulus = M — ^^

ea

where L = length of body (usually a wire),
a = area of cross-section,
F^ stretching force applied and
e = total elongation produced by F.
Similar moduli hold for compression, flexure and torsion.
(Crew, General Physics, p. 131. Also Ganot's Physics,
art. 87. Also Thomson and Tait, Treatise on Natural
Philosophy, Part II, arts. 686-688.)

CONSERVATIVE SYSTEMS.

A system is called a conservative system when it is such
that the total work done in performing any series of dis-
placements which bring the system back to its original con-
figuration is algebraically zero.

(Consult Thomson and Tait, Treatise on Natural Phi-
losophy, Part I, art. 271 et seq.)

iO LAWS OF PHYSICAL SCIENCE

PRIHCIPLE OF WORK DONE BY A CONSERVATIVE SYSTEM.

When a conservative system is changed from one con-
figuration to another the work done is independent of the
manner in which the change is made and depends only on the
initial and final states of the system.

(Jeans, Theoretical Mechanics, p. 164.)

RAYLEIGH'S RECIPROCATION THEOREM.

If a system of bodies is struck successively at two dif-
ferent points a and b by impulsive forces Pi and Qj, each
blow wiU in general affect all the bodies of the system. Let
Ui, Uj be the velocities of the points a and b produced by
the blow Pi and Vj, Vj be the velocities produced by the
blow Qa- Kayleigh's Theorem states that

Pi Vi = Qa XL,.

(Routh, Elementary Rigid Dynamics, pp. 338, 339.
Many illustrations are given in Rayleigh's Theory of
Sound.)

PERPETUAL MOTION IMPOSSIBLE.

The impossibility of perpetual motion rests upon the
following consideration: Let a system be made to pass by
frictionless constraint from a configuration A to another
configuration B and return by another path to A. Assume
more work is done upon the system by the mutual forces when
it follows one path than when it follows the other. Let
this process be repeated over and over again forever. The
system wiU then be a continual source of energy without the
consumption of materials, something which all experience
shows to be impossible.

(Consult Thomson and Tait, Treatise on Natural Phi-
losophy, Part I, art. 272.)

CONSERVATION OF THE MOVEMENT OF THE CENTER OF GRAVITY.

The state of rest or movement of the center of gravity
of several bodies is not altered by the neciproeal action of

MECHANICS 21

these bodies. Or, the motion of the center of gravity of a
system of bodies is changed only by forces external to the
system.

(For extension of principle consult Laplace, Mecanique
Celeste, Vol. I, and Lagrange, Mecanique Analytique, Vol.
I, Part II, Section 1, art, 15, and Section 3, art. 1 et seq.)

PRINCIPLE OF THE MOTION OF THE CENTER OF GRAVITY.

The center of gravity of any system moves as if aU
the external forces were acting on the entire mass of the
system concentrated in a particle at the center of gravity
and all the external forces were applied to this particle.

(Jeans, Theoretical Mechanics, pp. 224-226.)

CONSERVATION OF MATTER OR MASS.

If a mass or quantity of matter of any kind be selected
for consideration, then the "principle of the conservation of
mass" asserts; that no changes of any kind which can occur
in the mass, whether brought about by the action of forces
internal or external to the mass, can alter in the slightest
degree its total quantity.

For the measure of the total quantity of mass considered,
may be taken, either its inertia or the gravitational attrac-
tion of the earth upon it when located at a particular place.

(Consult Ames, Theory^ of Physics, pp. 6, 7. Also
see definition under word conservation, New Century
Dictionary.)

LAW OF CONSERVATION OF MOMENTUM. (I)

When any system of particles moves without being acted
upon by any external force, the total momentum of the
system remains constant in magnitude and direction, and
the moment of momentum about any axis remains constant.

(Consult Mach, Science of Mechanics, p. 288.)

22 LAWS OP PHYSICAL SCIENCE

LAW OF CONSERVATION OF MOMENTUM, (a)

If all velocities in a given direction are reckoned as

positive and all ia the opposite direction as negative, then

the sum of the momenta of a system of bodies, uninfluenced

by forces external to the system, is preserved constant,

whether or not these bodies meet in impact and whether

or not they are elastic or inelastic.

dx
( For mathematical definition, 2in — = a constant, see

AppeU, Tra/ite de Mecanique Eationndle, "Vol. II, p. 20,
Consult Mach, Science of Mechanics, p. 326.)

LAW OF THE CONSERVATION OF MOMENT OF MOMENTUM.

Let a system of particles of masses mj, m.^, m.^, etc., have
velocities Ui, U2, U3, etc., in a plane or in planes which are
parallel. If be the position of any fixed line perpen-
dicular to these planes and r^, v^, r^, etc., the perpendicular
distances from this line to the directions in which the various
particles are moving at a given instant the products, m u r,
are "moments of momenta" and the law of the conservation
of moment of momentum states that the 2'mur = a con-
stant, if no forces external to the system act upon it. In
taking the sum of the products, the products are reckoned
positive or negative according as a particle viewed from
is moving toward the right or toward the left.

(Ames, Theory of Physics, p. 49. Consult also La-
grange, Mecanique Analytique, Vol. I., Part II, Section 1, .
art. 16, p. 260.)

CONSERVATION OF LIVING TORCES.

This principle states that the difference of the force-
functions (or work) at the beginning and at the end of the
motion of a system is equal to the difference of the vires

MECHANICS «a

vivcB (kinetic energies) at the beginning and the end of
the motion. Namely,

(For a formal development of this principle see Mach,
Science of Mechanics, p. 478 et seq.)

PRINCIPLE OF THE CONSERVATION OF AREAS
(D'ARCY'S STATEMENT).

According to d'Arcy, the principle states that the
sum of the products of the mass of each body by the area
which its radius vector describes about a fixed center on
any plane of projection is always proportional to the
time. Or,

SmA= Kt or Im ^'^y-y'^^ =K dt.

(See Laplace, Mecanique Celeste, Vol, I, Part I, Book I,
art. 21.)

PRINCIPLE OF THE CONSERVATION OF AREAS
(MACH'S STATEMENT).

Mach states this principle as follows: If from any
point in space radii be drawn to several masses and projec-
tions be made upon any plane of the areas which the
several radii describe, the sum of the products of these areas
into the respective masses will be independent of the action
of inteimal forces. I

(Mach, Science of MechoMtcs, p. 294.)

24 LAWS OF PHYSICAL SCIENCE

PRINCIPLE OF VIRTUAL VELOCITIES.

If the point of application of a force be displaced through
a small space, the resolved part of the displacement in the
direction of the force has been called its "Virtual Velocity."
Mathematically expressed the principle asserts that for a
body to be in equilibrium

Pp + P'p' + P"p"+ =0.

Here P, P', P", etc., are applied forces acting upon a
connected system at points A, B, C, etc., and p, p', p", etc.,
are the projections on the lines of the forces of small dis-
placements of the points. The projections are to be taken
positive -when they fall in the direction of a force and
negative when they fall in the opposite direction.

(See Mach, Science of Mechanics, Chap. I, Sec. IV, p.
49 et seq. for a clear and full discussion of this principle.)

PRINCIPLE OF LEAST ACTION.

In the movement of bodies which interact so that the
total energy remains constant, the sum of the products of
the masses by the velocities and by the spaces described
is a minimum. This principle was extended to systems of
masses by Lagrange who presented it in the form,

SSmCvda = 0.
(See Mach, Science of Mechanics, pp. 364-380 for a good
physical discussion of this rather obscure principle. It is
mathematically treated in Thomson and Tait, Treatise on
Natural Philosophy, Part 1, arts. 326, 327.)

KELVIN'S MINIMUM-ENERGY THEOREM.

If a material system, initially at rest, is set in motion by
impulses applied to particular points in such a way that
these points acquire specified velocities, the motion of the
entire system is such as to make the total kinetic energy of

MECHANICS 2S

the system less than it would be in any other possible motion
of the system consistent with the same velocity conditions.
(Houstoun, An Introduction to Mathematical Physics,
P.,67.)

STOKES' LAW FOR THE FALL OF A SPHERE THROUGH A
Viscous MEDIUM.

When a small sphere falls under the action of gravity
through a viscous medium it ultimately acquires a constant
velocity equal to

v = g ,

where a is the radius and di the density of the sphere, d^
is the density of the medium and k is its coefficient of
viscosity.

When great accuracy is required, correction factors must
be added to the above expression.

This formula has been of much service in determining
the charge on an electron.

(G. G. Stokes, Mathematical and Physical Papers, Vol.
Ill, p. 59. See also Campbell, Modern Electrical Theory,
p. 91.)

D'ALEMBERT'S PRINCIPLE, (i)

When forces act upon one or more rigidly connected
points of a system of masses, these forces, called the im-
pressed forces, may each be resolved into two components,
the equilibrated forces and the effective forces. The latter
only are operative in producing motion, while the former
form a system balanced by the connections. The sum of
the products of the effective forces by the elementary dis-
placements which they produce is equal to the element of
work performed upon the system.

(Mach, Science of Mechanics, pp. 335-343.)

86 LAWS OF PHYSICAL SCIENCE

S'ALEMBERT'S PSIITCIPLE. (])

All of the work performed on any system is performed
by the effective components of the impressed forces. When
no work is performed the system is in equilibrium. The
two mathematical forms in which D'Alembert's principle
is usually expressed are :

2'[(X-ma)«x+(Y-inb)«y+ (Z-mc) «z] = O,
Z (X«x + YSy + ZSz) = Sm (aSx + b«y + c«z).

Here X, Y, Z are the mutually perpendicular com-
ponents parallel to rectangular coordinates of every force P
impressed on the masses m. ma, mb, mc are the corre-
sponding components of every effective force W, where a,
b, c denote accelerations and Sx, 8y, 8z are displacements in
the directions of the coordinates.

(Consult Mach, Science of Mechanics, p. 342.)

GAUSS'S PRINCIPLE OF LEAST CONSTRAINT, (i)

Let the masses M, Mj, etc., be joined in any manner with
one another. If these masses were free they would describe
in the element of time under the action of forces impressed
on them, paths ab, &i\, etc. ; but in consequence of the
connections, they describe in the same element of time the
paths ac, ai Ci, etc. Gauss 's principle asserts that the motion
of the connected points is such that, for the motion actually
taken, the sum of the products of the mass of each particle
into the square of the distance of its deviation from the
position it would have reached if free, is a minimum.

(Consult Mach, Science of Mechanics, p. 350 et seq. for
a clear development of this principle.)

GAUSS'S PRINCIPLE OF LEAST CONSTRAINT. (3)

The motion of a system of material points intercon-
nected in any way and submitted to any influences, accords
at each instant as closely as possible with the motion the

MECHANICS 27

points would have if they were free. The actual motion
takes place so that the constraints on the system are the
least possible. For the measurement of the constraint,
during any element of time, is to be taken the sum of the
products of the mass of each point by the square of its
deviation from the position it would have occupied at the
end of the element of time, if it had been free.

PRINCIPLE OF LEAST CONSTRAINT; COMMENT ON.

Gauss's principle of "least constraint" gives equations
which when differentiated yield D'Alembert's principle.
The kernel idea is "that the work of the forces which
deviate the movement of the system from the paths it would
take if unconstrained is as small as possible under the con-
ditions." Gauss's principle includes both statical anid
dynamical cases.

(Clear expositions and analytical treatments of the
principle of least constraint are to be found in Appel,
Traits de Mecanique Bationnelle, and in Mach, Science of
Mechanics.)

HAMILTON'S PRINCIPLE.

This principle is expressed as follows:

"/t*' ^^ +'^ *^* " ^' °' /to* (*U + ^'^^ ^^ "^ ''■
where SU and ST denote the variations of the work and vis
viva vanishing for the initial and terminal epochs. ' ' Hamil-
ton 's principle is easily deduced from D'Alembert's, and,
conversely, D'Alembert's from Hamilton's." The prin}-
ciples, least action, least constraint, D'Alembert's, Gauss's
and Hamilton's, are not expressions of different facts, but
rather, are simply views of different aspects of the same
fact.

(Mach, Science of Mechanics, pp. 380-384. Also consult
Appell, Tradte de MecanAque Bationnelle, Vol. II, p. 422.)

28 LAWS OF PHYSICAL SCIENCE

PRINCIPLE OF THE CONSERVATION OF ENERGY.

In every modification of a material system, not affected
by forces foreign to the system, the sum of its potential
and kinetic energies remains constant.

Calling E the kinetic energy and P the potential energy
of the system, E + P = K, a constant.

(See Helmholtz's lecture delivered at Carlsruhe about
1862, "On the Conservation of Force," in Popular Scien-
tific Lectures, Vol. I. Also consult Thomson and Tait,
Treatise on Natural Philosophy, Part I, arts. 269-278.)

DIFFERENT STATES OF EQUILIBRIUM.

A body is in stable equilibrium when a slight movement
from its position will raise its center of gravity. It is in
unstable equilibrium when such movement will lower its
center of gravity. It is in neutral equilibrium when such
movement will neither raise nor lower its center of gravity.

{Ganot's Physics, art. 72.)

II

HYDROSTATICS, HYDRODYNAMICS AND
CAPILLARITY

HYDROSTATICS, HYDRODYNAMICS AND
CAPILLARITY

ARCHIMEDES' PRINCIPLE.

When a body is in equilibrium in a fluid the fluid exerts
an upward force on the body equal to the weight of the
displaced fluid and acts through its center of gravity.
Or a body immersed in liquid loses weight equal to the
weight of the displaced liquid.

(This principle furnishes the most convenient method
for the determination of the specific gravity of a body, in
terms of that of the fluid used.)

(Gemot 's Physics, art. 112.)

EQUILIBRIUM OF FLOATING BODIES.

1. The floating body must displace a volume of liquid whose

weight equals that of the body.

2. The center of gravity of the floating body must be in the

same vertical line with that of the fluid displaced.

3. The equilibrium of a floating body is stable or unstable

according as the metacenter is above or below the
center of gravity.
(Ganot's Physics, art. 114.)

PRESSURE PRODUCED IN LIQUIDS BY GRAVITY.

The pressure in each layer is proportional to the depth.

With different liquids and the same depth, the pres-
sure is proportional to the density of the liquid.

The pressure is the same at aU points of the same
horizontal layer.

(Ganot's Physics, art. 98.)

31

32 LAWS OF PHYSICAL SCIENCE

HYDROSTATIC PARADOX.

The total weight or downward force exerted by a vessel
containing liquid depends on the shape and size of the
containing vessel and may be greater or smaller than the
force which is applied to give the liquid its hydrostatic
pressure.

(Consult Ganot's Physics, art. 102. Also Ames, Theory
of Physics, pp. 112-114.)

PASCAL'S LAW.

The fluid pressure due to the reaction of the walls of
the containing vessel is the same at all points throughout
the fluid. Or pressure exerted anywhere upon a mass of
liquid is transmitted undiminished in all directions, and
acts with the same force on all equal surfaces and in a
direction at right angles to those surfaces.

(Kimball, College Physics, p. 111. Also Ganot's Physics,
art. 97.)

CONDITION OF THE EQUILIBRIUM OF LIQUIDS.

1. Its surface must be everywhere perpendicular to the

resultemt of the forces which act on the molecules of
the liquid.

2. Every molecule of the mass of the liquid must be subject

in every direction to equal and contrary forces.
(Consult Ganot's Physics, art. 103. For Clairaut's
mathematical statement of the general condition of liquid
equilibrium, see Mach, Science of Mechanics, p. 397.)

EQUILIBRIUM OF LIQUIDS IN COMMUNICATING VESSELS.

When two non-miscible liquids of different densities are
placed in communicating vessels their free surfaces will
stand at different heights above the surface of contact of the
two liquids. Neglecting any capillary action, the heights

HYDROSTATICS. HYDRODYNAMICS AND CAPILLARITY 38

of the free surfaces of the two liquids above their surfape
of contact are La inverse ratio tp their densities.

(Ames, Theory of Physics, p. 115. Also Chwolson, TraiU
de Physique, Vol. I, Part 5, p. 569.)

RESTTLTANT OF FORCES OF COHESIOIT AT THE SURFACE OF A
LIQUID.

At the sui-faee of a liquid all the forces of cohesion have
a resultant which is directed toward the interior of the
liquid normally to its surface.

(Chwolson, Traits de Physique, Vol. I, Part 5, p. 563.)

EFFLUX: TORRICELLI'S THEOREM.

The velocity of efiSux of a stream of liquid issuing from

a cistern is the velocity which a freely falling body would

have on reaching the orifice after having started to fall

from rest at the surface-level of the fluid. (Only strictly

true when friction and shape of the orifice are disregarded.)

The velocity is,

V = V'2gh

where g = acceleration of gravity, and

h = height of level of liquid above orifice.

(Ganot's Physics, art. 142. Also Mach, Science of

Mechanics, p. 402.)

QUANTITY OF EFFLUX: "VENA CONTRACTA."

The stream of water issuing from a cistern through a
circular, sharp-edged orifice of area A contracts after leav-
ing the orifice. The theoretical discharge would be :
1^ = ■^■\/2^, hut the actual discharge is found to be about
0.62 A-\/2^, where g equals acceleration of gravity and h
equals height of level of liquid above orifice.

The reduced velocity is due to the contraction of the
crose-section of the stream, called the vena contracta.

(Consult Ganot's Physics, art. 145. Also Chwolson,
Traits de Physique, Vol. I, Part 5, pp. 697, 698.)

M LAWS OF PHYSICAL SCIENCE

BERNOULLI'S THEOREM: FLOW OF LIQUIDS.

At any point in a tube, through, which a liquid is flowing,
the pressure plus the potential energy due to position plus
the kinetic energy remains constant (friction being disre-
garded). Or, along a horizontal stream-line, the relation
holds,

^ density X velocity + pressure == a constant.
The general mathematical statement of the theorem is,

p + g/oh + % pv^ = c, a constant,
where p = density of liquid,

g = acceleration of gravity,
V = velocity of flow,

h = distance above any horizontal plane of reference
to the point in the liquid considered and
p ^ the hydrostatic pressure.
(Consult Crew, General Physics, p. 149. Also Mach,
Science of Mechanics, p. 413 et seq. Also Chwolson, Traite
de Physique, Vol. I, Part 5, p. 690.)

FLOW OF WATER IBT PIPES.

1. The loss of head from friction is proportional to the

length of pipe.

2. It increases with the roughness of the interior surface.

3. It decreases as the diameter of the pipe increases.

4. It increases nearly as the square of the velocity.

5. It is independent of the pressure of the water.

(Merriman, Treatise on Hydraulics, p. 209.)

HYDROSTATICS. HYDRODYNAMICS AND CAPILLARITY 85
PRINCIPLE OP COHTINUITY.

In a state of steady flow, the quantity of fluid passing
any cross-section of the stream in a given time is the same
for all sections of the stream. Thus the product SV is
constant, where S is the cross-section and V the velocity
of the stream at any point.

The "Equation of Continuity," which expresses this
fact mathematically is,

X dS du , dv , dw _ J-.
S dt ■*" dx ■*" dy + dz '

where u, v, w are the velocity-components in the directions
of the X, y, z axes respectively at any point of the fluid,
8 is the density of the fluid and t represents time.

If the fluid is incompressible the first term of this
equation vanishes.

(Webster, The Dynamics of Particles and of Rigid,
Elastic and Fluid Bodies, pp. 496-499.)

RESISTANCE TO THE UOTION OF A SOLID THROUGH A FLUID.

The pressure against a solid moving through a fluid is
approximately proportional to the square of their relative
velocity.

(Values of the constant of proportionality for the
important case of wind against a sail or aeroplane are given
in Sndthsonian Physical Tables, p. 124.)

36 LAWS OF PHYSICAL SCIENCE

VORTEX MOTIOH.

Defimtions: A Vortex Line is a curve whose tangent at
every point coincides with the direction of the instantaneous
axis of rotation at that point. A space bounded by vortex
lines is called a Vortex Tube and the enclosed fluid is said
to have a Vortex Motion. The strength of such a tube at
any cross-section normal to its axis is defined as the product
of the angular velocity w and the cross-sectional area S of
the tube at that point.

Laws:

1. A vortex tube always contains the same elements of

fluid.

2. The strength wS of a vortex tube is the same at all parts

of the tube and does not change with time (in a
perfect fluid.)

3. Vortex. tubes are either closed surfaces or have their

extremities in the surface of the fluid.
(Webster, The Dynamics of Particles and of Rigid,
Elastic and Fluid Bodies, pp. 509-511. For a complete
treatment see Helmholtz, Uher Integrale der hydrodynam-
ischen Gleichungen, welche den Wirielhewegungen ent-
sprechen, Wissen.schaftliche Abhandlungen, Vol. I, p. 101.
For a photographic study of vortex motions in water see
"An Experimental Study of Vortex Motions in Liquids,"
by E. F. Northrup, Jour, of the Franklin Institute, Sept.
and Oct., 1911.)

HYDROSTATICS, HYDRODYNAMICS AND CAPILLARITY 37
FLOW THROUGH CAPILLARY TUBES. POISEUILLE'S LAW.

The volu,me of liquid V which will flow in unit time
through a capillary tube of length 1 and radius r, is given

by the formula v = |^- In C.G.S. units, p = pressure-
difference over the length 1 of the tube in dynes per
square centimeter and k is the coefficient of internal fric-
tion or viscosity. ,

The reciprocal of the viscosity, namely -r-, is called the
fluidity.

This law was discovered and investigated by M. Pois-
euille in 1843.

(Consult Poynting and Thomson, Properties of Matter,
pp. 207-209. Also Chwolson, Traite de Physique, Vol. I,
Part 5, p. 674.)

HYDROSYITAHICAL THEOREU.

"If the bounding surface of a liquid, primitively at
rest, be made to vary in a given arbitrary manner, the vis
viva of the entire liquid at each instant wiU be less than
it would be if the liquid had any other motion consistent
with the given motion of the bounding surface."

(Mathematical and Physical Papers, by Lord Kelvin,
Vol. I, p, 109.)

CAPILLARY ACTIOH: JURIN'S LAW.

For the same liquid and the same temperature, the mean
height of the ascent in a capillary glass-tube is inversely as
the diameter of the tube. Thus diameter X height ^ a
constant.

(Ganot's Physics, art. 131. Also consult Poynting axxd
Thomson, Properties of Matter, p. 140. Also, Chwolson,
Traite de Physique, Vol. I, Part 5, p. 616.)

38 LAWS OP PHYSICAL SCIENCE

LAW OF CAPILLARY ACTION. (l)

For various liquids and the same temperature and in
tubes of the same diameter the mean heights to which the
liquids will rise vary with the nature of the liquid. (Of all
liquids water rises the highest.)

{Gemot 's Physics, art. 131.)

LAW or CAPILLARY ACTION. (2)

For the same liquid and the same temperature, the mean
heights to which the liquid rises are independent of the form
of the capillary tube except at the meniscus. Provided the
liquid moistens the tube, neither the thickness of the tube
nor its nature has any influence on the height to which the
liquid rises.

{Ganot'a Physics, art. 131.)

LAW OF CAPILLARY ACTION. (3)

The height to which a liquid rises in a capillary tube
diminishes as the temperature increases. As the height
becomes less the meniscus becomes flattened. When the
sides of the tubes are not moistened Jurin's law holds
approximately for the depression of the liquid.

{Ganot's Physics, art. 131.)

CAPILLARY CORRECTIONS OF MERCURY COLUMNS.

' ' The height of the meniscus and the value of the capil-
lary depression depend on the bore of the tubtag, on the
cleanliness of the mercury and on the state of the walls of
the tube. The correction is negligible for tubes with diam-
eters greater than about 2.5 cms."

(See Kaye and Laby, Physical and Chemical Constants,
p. 17. Or see Smithsonian Physical Tables, p. 123.)

HYDROSTATICS. HYDRODYNAMICS AND CAPILLARITY 89
SURFACE TENSION AND WORE OF THE FORCES OF COHESION.

Every diminution in the extension of a liquid-surface is
associated with work done by forces of molecular cohesion.
Every augmentation of a liquid-surface is Eissociated with
work done by exterior forces; the result of which is an
augmentation of the store of potential energy in the liquid
and the quantity of this potential energy depends upon the
area of the liquid-surface.

(Chwolson, TraiU de Physique, Vol. I, Part 5, p. 590.)

NORMAL PRESSURE ON A LIQUID-SURFACE.

The forces of cohesion which act upon the molecules of
the superficial layer of a liquid result in producing a certain
normal pressure per unit area upon the surface of the liquid.
The magnitude of this pressure depends upon the form of
the surface. If K is its magnitude for a plane surface, the
normal pressure is greater than K on a convex surface and
less than K on a concave surface.

(Chwolson, Traits de Physique, Vol. I, Part 5, p. 590.)

FORM ASSUMED BY A LIQUID-MASS UNDER THE INFLUENCE
OF SURFACE TENSION ALONE.

A liquid-mass not subjected to any exterior force is in
equilibrium. The pressure exerted upon it by forces result-
ing from surface-tension is the same at all points of the
surface of the mass and the surface of the liquid assumes
at all points the same mean curvature.

(Chwolson, Traits de Physique, Vol. I, Part 5, p. 601.)

40 LAWS OF PHYSICAL SCIENCE

PRESSURE-DIFFERENCE OH THE TWO SIDES OF A SOAP-FILM.

The pressure-difference p on the two sides of a soap-
fllm is given by the relation,

Here T is the surface-tension of the film, and Rj, Rj are
the two radii of Principal Curvature of the surface of the
liquid-fllm at the point considered. The pressure on the
concave side always exceeds that on the convex side of the
film. In the formula above the convention must be made
that a radius of Principal Curvature is to be taken positive
or negative according as the corresponding center of curva-
ture falls on the side of the surface where the pressure is
greater or on the opposite side. For the case of a spherical
soap-film Rj = Rj and thus the normally acting inward
pressure on the inside of the film exceeds that on the outside
by au amount

T

where R is the radius of the spherical film.

(Poynting and Thomson, Properties of Matter, pp. 144r-
146. Also consult Chwolson, Traite de Physique, Vol. I,
Part 5, Chap. IV.)

VERTICAL DISTANCE BETWEEN TWO ELEMENTS OF A
LIQUID-SURFACE.

If two elements of the surface of an extended liquid-
mass resting on a plane surface be selected such that one
of these elements is horizontal and the other vertical, then
the perpendicular distance between the two elements is
equal to the square root of the capillary constant of the
liquid. Or,

where k is the capillary constant, T the surface-tension and
8 the density of the liquid.

HYDROSTATICS, HYDRODYNAMICS AND CAPILLARITY 41

(Chwolson, TraiU de Physique, Vol. I, Part 5, pp. 628-
630.)

ACTION ON LIGHT FLOATING BODIES OF SURFACE-TENSION.

Surface-tension will cause an at/traction between two float-
ing bodies both of which are wet if the curved portions of
the liquid surrounding each of them intersect ; the same is
true if both the bodies are not wet. If one is wet and the
other is not they will repel each other if the two curved
portions of the liquid intersect.

(For an account of various capillary phenomena see
Ganot's Physics, art. 136.)

WATER-WAVES, SPEED OF.

1. In water where the depth h is small compared with the

wave-length, the speed of propagation V of the wave-
crest is,

where g = the acceleration of gravity.

2. In deep water where the depth is large compared with
the waye-length 1, the speed is,

/^-

'27r

Thus a deep sea-wave 378 feet long travels with a speed of
44 feet per sec.

(Crew, General Physics, p. 188. Also Mathematical and
Physical Papers, by Lord Kelvin, Vol. Ill, p. 519.)

RIPPLES ON SURFACE OF LIQUIDS.

Very short waves (less than 1.6 cm.), or ripples, may
be considered to be propagated by the surface-tension of
the liquid alone. Ripples of short wave-length travel faster
than ripples of long wave-length which is just the reverse
of what happens with ordinary water-waves propagated
by gravity.

(For a general discussion of waves and ripples, consult
Chwolson, Traite de Physique, Vol. I, Part 5, pp. 706-709.)

■4

42 LAWS OF PHYSICAL SCIENCE

KIPPLES, SPEED OF.

1. Neglecting the effect of gravity upon the speed of short
waves or ripples, the velocity is,

-V

2irT
Ip ■

where T is the surface-tension of the liquid, p its
density and 1 is the wave-length.
When account is also taken of the acceleration g of
gravity

-V

2£T , jl_.
\p 2ir

(The slowest water-waves are found to have a length
of about 1.6 cm. Waves shorter than this are called
"ripples.")

(Crew, General Physics, pp. 191, 192.)

Ill

SOUND

SOUND

DEFINITIONS.

1. Sound in the physical sense is either the vibrations of its

source, or, of the elastic medium surrounding the
source.

2. Noise is a sound resulting from irregular and practically

unanalyzable vibrations.

3. Musical tones are distinguished by :

a. their force determined by the amplitude of the vibra-
tions,

b their pitch determined by the frequency of the vibra-
tions,

c. their quality determined by the harmonics present.

4. A periodic motion is one which constantly returns to the

same condition after equal intervals of time.

PROPAGATION OF SOUND.

Sound is propagated as a longitudinal wave in any
elastic medium. It cannot be transmitted through vacuous
space, as the presence of an elastic medium is essential.

{Ganot's Physics, arts. 224, 225.)

INTENSITY or SOUND, (i)

The intensity of sound is inversely as the square of the
distance of the sounding body from the ear. That is, sound
radiates from a point in a homogeneous medium so that the
wave-front is spherical in form.

{Ganot's Physics, art. 227.)

45

46 LAWS OF PHYSICAL SCIENCE

INTENSITY OF SOUND. (J)

The intensity of sound in a, physical sense is the quantity
of energy which traverses in the unit of time the unit of
area normal to the sonorous ray. In this sense the intensity
of a sound of given pitch is proportional to the velocity of
the sound, to the density of the medium and to the square
of the amplitude. In a formula the intensity is,

J = 27r=' N" a=8V,
where N = frequency, or number of complete vibrations per
second,

a ^ the amplitude of a vibration,
8 = density of medium and
V = the velocity of propagation.
(Chwolson, Traits de Physique, Vol. I, Part 7, p. 905.)

INTENSITY OF SOUND. (3)

The intensity of sound is modified by the motion of the
atmosphere and the direction of the wind and is strengthened
by the neighborhood of a sonorous body.

{Ganot's Physics, art. 227.)

SOUND-INTENSITY IN TUBES.

In a speaking-tube the intensity of sound does not de-
crease with the square of the distance.

According to experiments carried on by Regnault, the
distance to which a sound will carry in such a tube is
roughly proportional to its diameter.

( Ganot's Physics, art. 229.)

EXPANSIONS AND CONTRACTIONS IN SOUND-TRANSMISSION
ARE ADIABATIC.

In transmission of sound-waves, the expansions and com-
pressions of the medium occur so rapidly that the expansions
and compressions are adiabatic ; namely, they occur in such
maimer that no heat is gained or lost to the volume of gas
considered.

(Consult Ganot's Physics, arts. 231 and 507.)

SOUND 47

VEtOCITY OF SOUND, GENERAL PRINCIPLE.

The velocity of sound in any fluid equals the velocity
acquired by a body in falling through one-half the height
which represents the rate of variation of the pressure of the
fluid with its density during a sudden change of density.
Thus if V = velocity per second, g = acceleration of gravity,
D = density of fluid and p = pressure, in the gravitational
system of units,

-V

(Eankine, The Steam Engine, p. 321.)

VELOCITY OF SOUND IN AIR.

In dry air at 0° C. the velocity of sound is 331.7 meters,
or 1088 feet, per second.

This velocity increases as the square root of the absolute
temperature. Or, if T is the temperature in degrees centi-
grade, the velocity in meters per second at temperature T is,

V. = 331.7^ l+J^.

(Consult Ganot's Physics, art. 230. Also Chwolson,
Traite de Physique, Vol. I, Part 7, p. 923, and Smithsonian
Physical Tables, p. 102.)

VELOCITY OF SOUND AND AIR-DENSITY.

For the same temperature, the velocity of sound is in-
dependent of the density of the air and consequently of the
pressure and is also roughly independent of the intensity
and the quality of the sound.

{Ganot's Physics, art. 230.)

48 LAWS OF PHYSICAL SCIENCE

NEWTON'S FORMULA (modified by Laplace) FOR THE VEiOCITY OF
SOUND IN GASES.

The velocity of propagation of sound in a gas is directly
as the square root of the elasticity of the gas and inversely
as the square root of its density. The elasticity exceeds the
isothermal elasticity P (measure i by the pressure) by an
amount y which is the ratio of tl e specific heats of the gas
at constant pressure and at coi stant volume. Calling p
the density of the gas,

v.^-

For air y = 1.41 and the velocity of sound in air at 0° C.
is 331.7 meters per second.

(Consult Ganot's Physics, art. 281. Also Chwolson,
Trmte de Physique, Vol. 1, Part 7, pp. 92^-924.)

DOPPLER'S PRINCIPLE.

"When a sounding body approaches the ear the note
perceived is higher than the true one, but if the source
recedes from the ear, the note perceived is lower.

If n = frequency of sounding body, V = velocity of com-
pressional waves and v = velocity of body toward or from the

ear, the pitch heard is n' = , where the minus sign

VTv ■

is used for an approaching body and the plus sign for a
receding body.

Doppler's principle may be extended to any system of
waves in a medium.

(Ames, Theory of Physics, p. 160.)

VELOCITY OF SOUND IN LIQUIDS AND SOLIDS.

The rule of Newton that the velocity of propagation of

sound equals the / elasticity holds for liquids and solids
\ density

as well as for gases.

SOUND 49

For liquids the elasticity is the ratio of the pressure
applied to the compression produced.

For solids Young's modulus may be taken as the value
of the elasticity. The velocity, as in the ease of gases,
varies with the temperatilre.

{Ganot's Physics, arts. 234, 235. Also Chwolson, Traite
de Physique, Vol. I, Part 7, pp. 929-933.)

REFLECTION OF SOUND-WAVES: ECHOES.

1. The angle of reflection is equal to the angle of incidence.

2. The incident sonorous ray and the reflected ray are in

the same plane perpendicular to the reflecting surface.

An Echo is the repetition of a sound caused by its
reflection from some surface transverse to its line of
propagation.

{Ganot's Physics, arts. 236, 237.)

CHANGE OF PHASE AT REFLECTION.

When waves of sound pass from a less into a more
dense medium, a portion of the energy is reflected back
from the bounding surface without change of phase. If
the waves are passing from a more into a less dense medium,
the reflected wave undergoes a change in phase of one-half
wave-length at the reflecting surface.

(Poynting and Thomson, Sound, pp. 104-108.)

PRINCIPLE OF RESONANCE.

One vibrating system may resonate, or be set into
sympathetic vibration by another separate vibrating system
when their natural periods of vibration are nearly equal;
The more accurately they are tuned together the more
marked is the resonance.

(Poynting and Thomson, Sound, pp. 58-62. Also
Kimball, College Physics, pp. 204 et seq.)

50 LAWS OF PHYSICAL SCIENCE

REFRACTIOH OF SOUND. (I)

A sound-wave is refracted upon passing from a medium
of one density into a medium of a different density. Sound
may be deflected with a prism or focused with a lens.
Sound travels poorly against the wind, because its wave-
front is tilted upward.

{Oanot's Physics, art. 238.)

KEFRACIIOIT OF SOUND. (2)

When a sound-wave passes obliquely from one medium
to another in which its velocity is different, its direction
of propagation is changed. The laws of refraction are :

1. The normals to the incident and refracted wave-fronts

and to the plane-surface all lie in the plane of incidence.

2. The ratio of the sine of the angle of incidence to the

sine of the angle of refraction is constant for a given
form of matter and waves of definite wave-number.
It is entirely independent of the angle of incidence

itself. ^T^ = u is called the index of refraction,
sinr

(Consult Ames, Theory of Physics, pp. 424, 425. Also
Ganot's Physics, arts. 238, 546, 547.)

INTERFERENCE OF SOUND.

ooond-interference can occur between two sonorous
rays ; the interference being determined by a difference in
phase between the vibrations of the two wave-trains.

(Consult Chwolson, Traite de Physique, Vol. I, Part 7,
pp. 949, 950.)

DIFFRACTION OF SOUND.

Diffraction phenomena are manifested more markedly
with long than with short sound-waves and the length of
sound-waves is such that there scarcely exists anything of
the nature of a sound-shadow.

SOUND fll

Lord Rayleigh has experimentally demonstrated the
phenomena of the diffraction of sound.

(Chwolson, TraiU de Physique, Vol. I, Part 7, pp.
954, 955.)

VELOCITY OF A TRAHSVERSB WAVE ALONG A STRETCHED STRIlfG.

The velocity of propagation of a transverse disturbance
along a perfectly flexible stretched string or wire is given by

\ m

Here V may be taken as the velocity in cm. per second, the
tension of the string in dynes and m its mass in grams per
em. of its length.

(Poynting and Thomson, Sound, pp. 93-95. Also, Kim-
ball, CoUege Physics, p. 216.)

TRANSVERSE VIBRATIONS OF A CORD.

1. The number of vibrations per second made by a cord

under a given tension is inversely proportional to the
length of the vibrating segment.

2. In case of two cords of equal length, and equal mass

per unit length, the frequencies are proportional to
the square roots of the tensions.

3. If two cords have equal lengths and are under equal

tensions, their frequencies will be inversely propor-
tional to the square roots of their masses per unit
length.
Thus in a formula the frequency is,

where 1 equils length of cord or the distance between two
consecutive nodes, T equals the tension of the cord and m
equals the mass per unit length.

(Kimball, College Physics, p. 216. Also Ganot's Physics,
art. 268.)

62 LAWS OF PHYSICAL SCIENCE

KODES AND LOOPS IB AIT ORGAn-PIPE.

In a closed organ-pipe the top is always a node or point
of no vibration and the pipe, when the air column vibrates
to the fundamental note, is one-quarter wave-length long.
In the case of an open organ-pipe there is an antinode at
each end £ind the pipe, when the air column vibrates to
the fundamental note, is one-half wave-length long. Clos-
ing the end of a pipe lowers the tone one octave.

{Ganot's Physics, art. 275.)

irTTUBER OF VIBRATIONS PRODUCED BY A MUSICAL PIPE.

When the length L of the pipe exceeds 12 times its
diameter; for open pipes the frequency is,

2L '
and for closed pipes

n=(2Pi:L)v.

4L
Here p is any whole number, as 1, 2, 3, etc., and V is
the velocity of sound in air.
{Oanot's Physics, art. 277.)

LAW OF VIBRATION OF GEOMETRICALLY SIMILAR SYSTEMS.

When two vibrating systems are made of the same mate-
rial and are geometrically similar but of different size their
periods of vibration are in the same ratio as their linear
dimensions.

(Kimball, College Physics, p. 230.)

VIBRATION OF RODS AND PLATES.

The number of transverse vibrations made in a given
time by rods and thin plates of the same material is
directly as their thickness and inversely as the square of
their length.

{Ganot's Physics, art. 283.)

SOUND 68

VIBRATION OF PLATES.

In plates of the same kind and shape, and giving the
same system of nodal lines, the number of vibrations in a
second is directly as the thickness of the plates and in-
versely as their area.

{Ganot's Physics, art. 284.)

VIBRATION or BELLS.

Bells may be considered as curved plates. They do not
vibrate as a whole, but for the fundamental they vibrate
in four equal parts, these parts being separated by nodal
lines. They are also capable of vibrating in 6, 8, 10 or
12 parts, producing thus a series of overtones. The note of
a beU is higher in proportion as the surface is smaller and
the substance thicker.

{Ganot's Physics, art. 284.)

ACOUSTIC ATTRACTION AND REPULSION.

The vibrations of an elastic medium attract bodies which
are specifically heavier than itself and repel those which are
specifically lighter. Thus in air a balloon filled with car-
bonic acid gas is attracted toward the opening in a reso-
nance-box on which is a vibrating tuning fork, and a balloon
filled with hydrogen is repelled.

{Ganot's Physics, art. 292.)

NUMERICAL VALUE OF PRESSURE OF SOUND.

Lord Rayleigh has shown that acoustic vibrations when
they encounter the surface of a body must exert on it a
pressure p, which for a plane-wave and normal incidence
on a perfectly reflecting surface is,

2e

where e is the quantity of incident energy in unit time and
V is the velocity of the sound.

(Chwolson, Traits de Physique, Vol. I, Part 7, p. 909.)

64 LAWS OF PHYSICAL SCIENCE

LIMITS OF AUDIBILITY.

The limits between which the frequencies of vibrations
are audible vary considerably with different persons and
the results of different investigators vary through rather
wide limits. However, the lower limit seems to be between
16 and 24 vibrations per second, and the upper limit between
30,000 and 41,000 per second. Much depends also upon the
intensity and quality of the sound as to whether or not it
will be audible.

{Ganot's Physics, art. 244.)

AMPLITUDE IfECESSARY TO MAKE SOUND-WAVES AUDIBLE.

It is found that sound-waves are inaudible if the ampli-
tude of the sound-waves is less than about 8 X 10-° cm.
This limit is smaller as the pitch is higher.

(Poynting and Thomson, Somid, pp. 118, 119.)

COMBINATIONAL TONES.

When two pure tones are sounded simultaneously there
is often heard, in addition to these two tones, two others.
One of these has a pitch of frequency equal to the difference
of the frequencies of the original tones, and the other has a
pitch of frequency equal to their sum.

These tones are called Difference-Combinational tones
and Summation-Combinational tones. They have been
explained by Helmholtz.

(Helmholtz, Sensations of Tone. Also Encyclopedia
Britannica, 10th Ed., Vol. XXV, p. 56.)

PURE TONE.

A pure tone is due to the disturbances sent out by a
body vibrating with simple harmonic motion. Fourier
has shown that any periodic disturbance may be made up
of the resultant disturbances caused by a number of simple
harmonic motions.

(Consult Helmholtz, Sensations of Tone, Chap. 1.)

SOUND SS

LAW OF G. S. OHM.

' ' The human ear perceives pendular vibrations alone as
simple tones, and resolves all other periodic motions of the
air into a series of pendular vibrations, hearing the series
of simple tones which correspond with these simple vibra-
tions."

(Helmholtz, Sensations of Tone, p. 56.)

RESULTS OF VON HELMHOLTZ'S RESEARCHES.

Simple sounds, as those produced by a tuning fork with
a resonance-box, are soft and agreeable, without roughness
but weak and in deeper tones, dull. Musical tones accom-
panied by a series of harmonics, say up to the sixth, in
moderate strength are full and rich. They are grander and
more sonorous than simple tones. Such tones are produced
by the pianoforte.

(These tone-qualities are very fully discussed iu Chaps.
IV and V of Helmholtz 's Sensations of Tone.)

THE OCTAVE.

"A musical tone which is an octave higher than another,
makes exactly twice as many vibrations in a given time as
the latter."

(Helmholtz, Sensations of Tone, p. 13.)

EFFECT ON EAR OF A SYSTEM OF SOUND-WAVES.

' ' When several sonorous bodies in the surrounding atmos-
phere simultaneously excite different systems of waves of
sound, the changes of density of the air, and the displace-
ments and velocities of the particles of the air within the
passages of the ear, are each equal to the (algebraical) sum
of the corresponding changes of density, displacements, and
velocities, which each system of waves would have separately
produced, if it had acted independently."
(Helmholtz, Sensations of Tone, p. 28.)

66 LAWS OF PHYSICAL SCIENCE

ADDITIOIf OF iSIMPLE VIBRATIOKS.

"Any given regular periodic form of vibration can
always be produced by the addition of simple vibrations,
having pitch-numbers which are once, twice, thrice, four
times, etc., as great as the pitch-numbers of the given
motion. ' '

(Helmholtz, Sensations of Tone, p. 34.)

THE SUM OF PARTIAL TONES.

"Any vibrational motion of the air in the entrance to
the ear, corresponding to a musical tone, may be always,
and for each case only in one single way, exhibited as the
sum of a number of simple vibrational motions, correspond-
ing to the partials of this musical tone."

(Helmholtz, Sensations of Tonte, p. 34.)

THE PRINCIPLE OF MUSICAL SCALES.

In the major, or diatonic scale, the frequencies of the
notes bear the following ratios to that of the key note :

do re mi fa sol la si do
1, %, %, %. %, %.%2
This scale is built up of major triads, which have the
pitch relation of do, mi, sol, or the frequency relation of
4, 5, 6. The minor scale is built up of minor triads, or
notes with the frequency ratios 5, 6, 7%. The notes of
this scale bear to the key note the relations

1, %, %, %, %, %, % 2.
(Helmholtz, Sensations of Tone, p. 274.)

SOUND 57

HELMHOLTZ'S THEORY OF CONSONANCE AND DISSONANCE.

When two notes are sounded simultaneously they pro-
duce an agreeable sensation in proportion as their fre-
quencies form a simple ratio. Thus the octave, 1:2; the
fifth, 2 : 3 and the fourth, 3 : 4 are the most consonant com-
binations, in the order named. A ratio 8 : 9 or 7 : 11 would
be discordant. Helmholtz has shown that these facts are
due to the absence of rapid beats in the case of consonant
tones and their presence in the case of dissonant tones.

(Helmholtz, Sensations of Tom, pp. 228-330.)

IV

HEAT AND PHYSICAL CHEMISTRY

HEAT AND PHYSICAL CHEMISTRY

TEMPERATURE (Definition).

Temperature is a condition of matter. On the absolute
thermo-dynamie scale, temperature is a quantity which is
proportional to the mean kinetic energy E per molecule
of the molecules of an "ideal gas," also to the product of
the volume V and the pressure P of this gas. Thus,

pv
T=KE= -5—, where K and R are constants.

(Consult Maxwell, Theory of Heat, p. 51. Also Chwol-
son, Traits de Physique, Vol. Ill, Part 9, p. 7. Also article
by E. F. Northrup, "High Temperature Investigation and
a Study of Metallic Conduction." Journal of the Franklin
Institute, June, 1915.)

QUAHTITY OF HEAT (Definition).

Quantity of Heat is the total kinetic energy of the mole-
cules, or ultimate particles of a body. Thus every store
of heat-energy is expressed by the formula.

In this formula the unit of beat energy is to be taken the
same as the unit of mechanical energy, which is generally
chosen on a system of absolute units and is the same as the
unit of work. By m is to be understood the masses of the
smallest particles of the body which are moving at any given
instant with velocities v, different in general for each
particle.

(Chwolson, Traits de Physique, Vol. Ill, Part 9, pp. 2
and 18.)

61

62 LAWS OF PHYSICAL SCIENCE

TEMPERATURE EQUILIBRIUH.

If two bodies A and B are in temperature equilibrium
with a third body C, then A and B will be in temperature
equilibrium with each other. It does not follow from the
above that A, B, and C contain equal quantities of heat,
even if they are all of equal mass and the same material.
Thus A may be in grams of water at 0° C. and B in grams
of ice at 0° C. Then A and B will be in temperature equilib-
rium with C, which is in grams of ice at ° C, but A and
B, though in temperature equilibrium with each other, con-
tain different quantities of heat which differ by about 80
calories.

(Preston, Theory of Heat, Chap. I, Sec. II. See p. 20.)

EQUALIZATION OF TEMPERATURE.

When two bodies A and B are placed in contact, the
temperature of the one body being higher than the tem-
perature of the other body, the two bodies tend toward
equality of temperature. The equalization occurs or tends
to occur without oscillations of heat which have analogy
with the oscillations of electricity, as observed when a con-
denser is discharged. The progress or the rate of equaliza-
tion of temperature is a complex phenomenon which has
relations with specific properties.

HEWTON'S LAW OF COOLIlfG.

The rate at which a body cools is proportional to the
excess of its temperature above the walls of the enclosure

which surround it. Thus, log "17= -ait, * which gives

de
by differentiation, --^ = ae. Here, e is the excess of

temperature at time t, B the initial excess of temperature
and ai and a are constants.

This law expresses the facts only approximately.

(Poynting and Thomson, Heat, p. 245. Also Preston,
Theory of Heat, p. 528.)

HEAT AND PHYSICAL CHEMISTRY 63

SULONG AND PETIT'S CONCLUSIONS OH THE VELOCITY OF COOLING.

The cooling influence of gas surroundiag a body is not
affected by the nature of the surface of the body. The
nature of the surface is effective only on the emissivity
which would occur if the body were in a vacuum.

The empirical formula, which expresses the velocity of
cooling V, is

V = fc (j)P-aPo) + mp' (fl-O 1.233.

(For discussion of principle stated and interpretation
of formula, see Preston, Theory of Heat, pp. 530-540.)

ABSOLUTE (or LORD KELVIN'S) SCALE OF TEMPERATURE.

On the absolute, thermodynamic or Lord Kelvin's scale
of temperature any two temperatures bear to each other
the same ratio as the quantity of heat taken in at the higher
temperature bears to the quantity of heat ejected at the
lower temperature by a reversible engine working between
the two temperatures as source and condenser. Thus,

Ti _ Qi .

Tj Qj

The efficiency of the perfect reversible engine is,

Qi-Qi _ T1-T2
Qi T.

Here the fieat quantities are expressed by Q and the absolute
temperatures by T.

(Preston, Theory of Heat, p. 713. Also Eankine, The
Steam, Engine, p. 343.)

ABSOLUTE ZERO AND ABSOLUTE TEMPERATURE (Definitions).

The absolute zero (- 273.10° C.) on the gas-scale is the
temperature at which an ideal gas would theoretically exert
no pressure. It is numerically equal to the reciprocal of
the pressure-coefficient a^ of the gas at constant volume.

64 LAWS OF PHYSICAL SCIENCE

The absolute temperature is the temperature reckoned
from absolute zero. T (degrees absolute, now called degrees

Kelvin) = -A— +t (degrees centigrade).

(Consult Smithsonian Physical Tables, p. 247. Also
Gwnot's Physics, arts. 336, 337. Also paper by Arthur L.
Day and Robert B. Sosman, "The Nitrogen Thermometer
from Zinc to Palladium," Amet*. Jour, of Science, Vol.
XXIX, Feb., 1910. See pp. 100-102. Also Chwolson,
Traits de Physique, Vol. Ill, Part 9, pp. 14-17.)

CARNOT'S THEOREM.

All reversible heat engines working between two given
temperatures, and taking in and ejecting heat at the same
two temperatures, have the same efficiencies. This efficiency
is greater than that of any irreversible engine working
between the same two temperatures.

(Poynting and Thomson, Eeat, pp. 262-264; 265, 266.)

GAS-TEUPERATURE SCALE.

On the hydrogen or nitrogen gas-thermometer scale the
temperature t, in degrees centigrade, is given by the relation

■t lOO -t^o

where P^ is the pressure of the gas at 0° C, Pjoo its pressure
at 100° C. and Pt its pressure at the measured temperature,
the volume of the gas being maintained constant.

(Chwolson, Traite de Physique, Vol. Ill, Part 9, p. 23.
See also p. 17.)

HEAT AND PHYSICAL CHEMISTRY 65

TEMPERATURE BY PLATINUM RESISTANCE-THERMOMETER.

Call pj = ^^ — __^ 100 the "platinum temperature-/'

where R^ is the i-esistanee of the thermometer at t degrees
and Ro and Ri„o its resistance at 0° and 100° on the centi-
grade scale. Then the difference between the true tem-
perature and the platinum temperature is given by the
formula of Callendar,

*-P'=4-w + (w)]'

where 8 is a constant, of value 1.50 for pure and greater for
impure platinum.

(Northrup, Methods of Measwring Electrical Resistance,
p. 298. Also, Burgess and LeChatelier, Measurement of
High Temperatures, p. 197.)

EXPANSION or BODIES WITH HEAT.

Nearly all bodies expand when they receive an additional
quantity of heat. The expansion being slight, the coefficient
of cubical expansion can with small error be taken equal
to three times the coefficient of linear expansion.

For small ranges of temperature the expansion is very
nearly proportional to the rise in temperature of the body.

(For experimental values, consult Smithsonian Physical
TabUs, pp. 232-235.)

EXPANSION COEFFICIENTS OF ANISOTROPIC BODIES.

1. The sum of the coefficients of linear expansion along any

three directions mutually at right angles has a con-
stant value equal to the sum of the three principal
coefficients.

2. The coefficient of cubical expansion for an anisotropic

body is equal to the sum of the coefficients of linear
expansion along three directions mutually at right
angles.
(Chwolson, Traits de Physique, Vol. Ill, Part 9, pp.
110, 111.)

86 LAWS OF PHYSICAL SCIENCE

EXPANSION OF LIQUIDS.

Liquids in general expand when they receive an addi-
tional quantity of heat, but water between 0° and 4° C.
contracts, or increases in density, with increase in tempera-
ture or heat absorbed.

The real or absolute expansion of a liquid is the actual
increase in volume, while the apparent expansion is that
which is observed when a liquid contained in a vessel is
heated, and this is less than the real expansion, because of
the simultaneous expansion of the vessel itself.

{Ganot's Physics, arts. 322, 327, 331.)

DULONG AND PETIT'S LAW OF THERMAL CAPACITY.

For simple substances the atoms all have (approxi-
mately) the same thermal capacity, or the product of the
specific heat by the atomic weight is the same for all
elementary substances.

Regnault's mean value of this constant for 32 sub-
stances is 6.38.

(Preston, Theory of Heat, p. 294. Also Ganot's Physics,
art. 464.)

NEUMANN'S LAW.

F. E. Neumann has found that the product of the molecu-
lar weight and specific heat remains constant for all com-
pounds belonging to the same general formula and similarly
constituted, but that the product varies from one series to
another.

(Consult Preston, Theory of Heat, p. 296. Also Ganot's
Physics, art. 465.)

HEAT-FLOW: LAW OF FLOW FOR STEADY STATE.

The quantity of heat Q which passes through a homo-
geneous solid enclosed between two parallel infinite planes
at a distance d apart is expressed by,

HEAT AND PHYSICAL CHEMISTRY 67

where 9i = the higher temperature of the one plane,
62 = the lower temperature of the other plane,
A == the area through which the flow is reckoned,
t = the time the flow is measured and
K = a constant.
(Fourier, The Analytical Theory of Heat, Chap. I, Sec.
IV. For the steady flow of heat between opposite faces of
solids with certain geometric forms, see paper by Langmuir,
Adams and Meikle in Trans, of the Electrochem. Soc, Vol.
XXIV, 1913, pp. 53-84. Also paper by E. F. Northrup,
same Vol., pp. 85-106.)

GENERAL EQUATIOK FOR HEAT-FLOW.

The general differential equation for heat-flow is.

I. dx« '*' dy' "^ dz' y ~ "" dt '

■ d'fl . d'g \ _
'*' dy' "^ dz' )

which, for the steady state, becomes

d«9 , d'9 , d'fl „
• + j-» "T "^ns — o-

dx« ~Sy^ dz'
Here, K = thermal conductivity,

c = thermal capacity per unit volume = specific heat
X density.

6 = temperature and
t = time.
(Preston, Theory of Heat, art. 312. Also Fourier, The
Analytical Theory of Heat, p." 112 et seq.)

68 LAWS OF PHYSICAL SCIENCE

STEAD7 FLOW OF HEAT FROM A POINT-SOURCE IN AN INFINITE
ISOTROPIC MEDIUM.

1. The isothermal surfaces are concentric spherical shells

with the point-source as center.

2. The flow of heat is perpendicular to the isothermal sur-

faces.

3. .The total flow of heat across any isothermal surface is

the same as that across any other, or any heat that is
once within a tube of flow remains in it forever.

4. The flow per unit area through any cross-section of a

tube of flow varies inversely as the area of the section,
and hence inversely as the square of the distance from
the point-source.
(Preston, Theory of Heat, art. 313.)

FLOW OF HEAT IN AN INFINITE CRYSTALLINE MEDIUM.

In an infinite crystalline medium, if heat be introduced
at a single point, the isothermal surfaces, when the steady
state is reached, will be a system of concentric and similar
ellipsoids, the axes of any one of which are directly pro-
portional to the square roots of the three principal conduc-
tivities of the crystalline medium.

(Preston, Theory of Beat, p. 675.)

FIRST LAW OF THERMODYNAMICS.

When work is done (namely, when measurable forces act
through measurable distances or measurable electromotive
forces give rise to measurable currents or measurable cur-
rents pass through measurable resistances, etc.), there is an
equivalence between the work so done and the heat de-
veloped. This is expressed by the equation,
W = JH + w.

Here W is the total work done and H the heat developed.
J is the equivalent of the work done in producing heat, and
w is a quantity to express the processes occurring which

HEAT AND PHYSICAL CHEMISTRY 69

cannot be measured as heat, sucli as the production of

sound, radiant energy, etc. When w is zero,

W (in kilogram-meters) = 426.9 X H (iu kilogram-calories).

Or (according to Smithsonian Physical Tables, p. 237)
W (in ergs) = 4.181 X 10^ H (in, 20° C, gram-calories).
(Consult Hering, Conversion Tables, pp. 72 and 171.
Also Preston, Theory of Heat, art. 37. For a precise state-
ment, see Nemst, Theoretical Chemistry, pp. 7-10.)

SECOND LAW OF THERMODYNAMICS.

"Heat can never pass from a colder to a warmer body
without some other change, connected therewith, occurring
at the same time." (Clausius.)

{Clausius on Heat, p. 117. Also Preston, Theory of
Heat, p. 49. For a scholarly treatment of the fundamental
principles of thermodynamics, consult Chwolson, Traite de
Physique, Vol. Ill, Part 9, Chap. VIII, pp. 409-550.)

SECOND LAW OF THERMODYNAMICS (Ranklne's statement), (i)

' ' If the total actual heat of a homogeneous and uniformly
hot substance be conceived to be divided into any number
of equal parts, the effects of those parts in causiag work to
be performed are equal."

(Eankine, The Steam Engine, p. 306.)

SECOND LAW OF THERMODYNAMICS (Ranklne's statement). (2)

"If the absolute temperature of any uniformly hot sub-
stance be divided into any number of equal parts, the effects
of those parts in causing work to be performed are equal."

(Kankiue, Ih& Steam Engine, p. 307.)

70 LAWS OF PHYSICAL SCIENCE

TRANSFORMATION OF ENERGY.

' ' The effect of the presence in a substance, of a quantity
of actual energy, in causing transformation of energy, is
the sum of the effects of all its parts. ' '

(This general law was first enunciated at the Phil. Soc.
of Glasgow, Jan., 1853. Consult Eankine, The Steam En-
gine, p. 309.)

DIFFERENCE BETWEEN ABSORBED HEAT AND ENERGY.

The difference between the whole heat absorbed, and the
whole expansive energy exerted in any thermic operation
depends on the initial and final conditions of the substance
alone in respect to pressure and volume and not on the inter-
mediate process.

(Rankine, The Steam Engine, p. 304.)

CONVERTIBILITY OF ENERGY.

"All forms of energy are convertible. The total energy
of 4ny substance or system cannot be altered by the mutual
actions of its parts."

(Rankine, The Steam Engine, p. 299.)

INTRINSIC ENERGY.

The total intrinsic energy of a body or system of bodies
is never known. "When bodies mutually react it is only the
difference of tb^energy of each body in two states which is
considered. If a body has less energy in its actual than in
its standard state the expression for its energy is negative.

(Maxwell, Theory of Heat, p. 186. For a general treat-
ment of the fundamental energy relations, consult Nernst,
Theoretical Chemistry, pp. 7-10, 15-28,)

ENTROPY.

Entropy is a mathematical function introduced by
Clausius. The entropy S of a substance is defined.

HEAT AND PHYSICAL CHEMISTRY 71

dQ

■/■

T '

where dQ is an element of ^ quantity of heat and T is the
absolute temperature of the element of the body which con-
tains the element of heat dQ. In any change of condition
of a body the change in its entropy is

dQ

'-=/"^

where S^ is the value of the entropy in the final state of the
body and S^ the value in the original state of the body, and
dQ is an element of heat gained by the body at absolute
temperature T.

(For Clausius' original use of the term entropy see
Clausius on Heat, p. 357. Bankine called the same quantity
the "thermodynamic function." A clear explanation of
the rather elusive meaning of the term entropy is to be
found in Maxwell, Theory of Heat, pp. 162, 187, 189. Also
Preston, Theory of Heat, pp. 724r-726.)

CLAUSIUS' PRINCIPLE OF THE INCREASE OF ENTROPY.

All energy-changes in nature are such as to increase the
total entropy of the universe. The entropy of the universe,
therefore, tends toward a maximum. As the entropy in-
creases the store of energy capable of transformation into
useful work (available energy) diminishes, approaching a
value zero. In this condition the entire energy of the
universe will be in the form of heat and all bodies will be
at the same temperature.

The above statement is a hypothetical extension to the
universe of a principle recognized for the very limited por-
tion which can be studied.

(Poynting and Thomson, Heat, p. 277 et seq.)

72 LAWS OF PHYSICAL SCIENCE

HEAT PRODUCED BY RADIUM.

Radioactive matter continually evolves heat. Curie and
Laborde conclude from experiments that 1 gram of pure
radium emits 100 gram-calories per hour. Thus 1 gram
of radium emits per day nearly as much energy as will dis-
sociate 1 gram of water. .

(Rutherford, Badio- Activity, p. 159.)

FUNDAMENTAL LAWS OF GASES.

Gases obey approximately the following laws :

1. Boyle's law, that for a constant temperature the volume

of a gas diminishes in direct proportion to the pressure.

2. Gay-Lussac's law, that the volume of a gas at constant

pressure increases proportionally with the absolute
temperature.

3. Avogadro 's law, that equal volumes of different gases at

the same pressure and same temperature contain equal
numbers of molecules.

4. Dalton's law, that the pressure of a mixture of several

gases in a given space is equal to the sum of the pres-
sures which each gas would exert by itself if confined
in that space.

5. Joule's law, that gases in expanding do no interior work.

(For the laws which apply to a perfect gas, consult
Chwolson, Traite de Physique, Vol. Ill, Part 9, Chap. IX,
p. 551 et seq. Also Nemst, Theoretical Chemistry, pp.
38, 39.)

BOTLE'S (or MARIOTTE'S) LAW.

The temperature being constant, the volume of a given
quantity of gas varies inversely as the pressure which it
bears. Thus, PV = a constant, where P is the pressure and
V the volume of the gas.

This law, discovered by Boyle in 1662 and Mariotte in

HEAT AlSfD PHYSICAL CHEMISTRY 73

France in 1679, is only approximately true for actual gases,
and then only for low or medium pressures.

(Ganot's Physics, art. 181. Also, Chwolson, Trait e de
Physique, Vol. I, Part 3, p. 40 and Part 4, p. 422.)

VARIATIONS FROM BOYLE'S LAW.

1. No actual gas obeys Boyle's law rigorously. Tlie diver-

gence increases with the pressure.

2. Hydrogen is less compressible than Boyle 's law requires ;

all other gases are more compressible.

3. The divergence from the law is greater for the easily

liquefiable gases, such as carbonic acid, ammonia, etc.,
than for the gases formerly called permanent gases,
oxygen, nitrogen, methane, etc.
{Ganot's Physics, art. 182. For a precise treatment of

Boyle's and other gas-laws and their variations consult

Nernst, Theoretical Chemistry, pp. 37-53.)

TAN DER WAALS' FORMULA.

Van der Waals has proposed as an accurate expression,
relating the pressure and volume of a gas at any given
temperature, the formula,

f p -I ^ 1 ( v — b ) = a constant.

Here p is the pressure, v the volume and a and b are con-
stants which differ for each gas.

(Consult Chwolson, Traite de Physique, Vol. I, Part 4,
pp. 439-441. Also Ganot's Physics, art. 183.)

74 (LAWS OF PHYSICAL SCIENCE

GAY-LXTSSAC'S (or CHARLES') LAW.

The pressure of a gas being maintained constant its
volume varies directly with the absolute temperature. If
we define the coefficient of expansion a of a gas as the amount
by which the unit of volume of the gas at 0° centigrade
increases when the temperature is raised 1° C, the pressure
being kept constant, then the law gives,

Vt = v„ ( 1 + at ) ,
where v„ is the volume of the gas at 0° C. and v^ its volume
at t° C. We may take for air a = 0.003665. The law of
Charles holds for a wide variation in pressure.

(Maxwell, Theory of Heat, p. 29.)

VAN DER WAALS' FORUULA (combining tlie laws of Boyle and
CliaTles, witli corrections) :

(p+^)(v-b)=RT.

Here, T = absolute temperature,
p = pressure,
V = volume and
R= gas-constant,
a and b = constants which differ for different gases.

When for the unit of pressure is taken a column of
mercury of 1 meter and for the unit of volume, the volume
of 1 kilogram of gas at 0° C. under a pressure of 1 meter
of mercury, from Regnault's data:

for air a = 0.0037, b = 0.0026,
for CO2 a = 0.0115, b = 0.003,
for Hj a = 0.0000, b = 0.00069.
(Consult Chwolson, Traite de Physique, Vol. I, Part 4,
Chap. II, see p. 441.)

ADIABATIC EXPANSION.

The law of expansion of a perfect gas, without receiving
or emitting heat, gives for the relation between pressure p
and volume v,

HEAT AND PHYSICAL CHEMISTRY 75

p = — —, where K is a constant

For air, 7 = 1.4025 and for steam in the perfectly gaseous
state,

y = 1.33.
(Kankine, The Steam Engine, pp. 319, 320. For values
of y, see Smithsonian Physical Tables, p. 243.)

ADIABATIC RELATIONS.

The adiabatic relations between the pressure p and the
absolute temperature T, and between the volume v and the
absolute temperature T, are given by the two relations,
1-7
Tp 7 = a constant, and

Tyv-i = a constant, where y= ~~^ , the ratio of the

specific heat at constant pressure to the specific heat at con-
stant volume of the gas.

(Preston, Theory of Heat, p. 288.)

VARIATION OF PRESSURE WITH VOLUME IN A THERMALLY
NON-CONDUCTING VESSEL.

The rate of variation of the pressure with the volume,
when any fluid or gas is enclosed in a thermally non-
conducting vessel, exceeds the rate of variation when the
temperature is constant, in the ratio of the apparent specific
heat of the fluid at constant pressure to its apparent specific
heat at constant volume.

Symbolically expressed (according to Rankine) :
dp

dp = _ J, dT ^ which becomes for a perfect gas,
dv dv

dT

dv '^ V

(Rankine, The Steam Engine, p. 320.)

76 LAWS OF PHYSICAL SCIENCE

EQUIPARTITION OF ENERGY: BOLTZHANH-MAXWELL LAW.

"^ In- a medium (a gas is usually considered) consisting
of particles in motion the distribution of energy, throughout
a given volume, will be such that, on the average, every
mode of motion of its particles is equally favored, or, the
kinetic energy is uniformly distributed among the degrees
of freedom of the particles.

(Consult Campbell, Modern Electrical Theory, p. 229.
For mathematical treatment, consult Jeans, The Dynamical
Theory of Gases, pp. 67-69.)

LAW OF AVOGADRO.

When two gases are at the same pressure and tempera-
ture, the number of molecules in unit volume is the same
for both.

Let N, m, u be the number of molecules, the mass of
each molecule and the mean velocity of each molecule of the
one gas, and Nj, mi, u^ corresponding quantities for the
other gas. By the kinetic theory of gases i/^Nmu^ =
%Nimiu2 (where volume, pressure and temperature are the
same), but %mu^ = % mu^ (by equipartition of energy).
Henoe the law that, N = N^.

(Chwolson, Traite de Physique, Vol. I, Part 4, p. 491.
Also Nernst, Theoretical Chemistry, pp. 200, 201.)

LAWS OF THE MIXTURE OF GASeS (DALTON'S LAWS).

1. The mixture takes place rapidly and is homogeneous;

that is, each portion of the mixture contains the two
gases in the same proportion.

2. If the several gases and the mixture have the same tem-

perature and if the several gases and the mixture
occupy the same volume, then the pressure exerted by
the mixture will equal the sum of the pressures ex-
erted by the gases severally.
These laws are applicable to mixtures of gases and vapors.

HEAT AND PHYSICAL CHEMISTRY 77

(Preston, Theory of Heat, p. 71. Also Ganot's Physics,
wrt. 388. Also Chwolson, TraiU de Physique, Vol. I, Part
4, p. 468.)

JOULE'S LAW.

When an ideal gas expands in sueli a manner as not to
do any mechanical work its temperature does not change.

Joule's experimental test of this law shows that no in-
ternal work is done by a gas during expansion, or in other
words no molecular attractions have to be overcome.

Ordinary gases deviate slightly from Joule's law.

(Preston, Theory of Heat, p. 286. Also Nemst, Theo-
retical Chemistry, p. 42.)

SPECIFIC HEAT OF BASES.

It is concluded from experiments by Eegnault and others
that a gas has a specific heat which is independent of pres-
sure in proportion as it approaches a perfect gas.

(Preston, Theory of Heat, p. 281. For discussion, and
description of experiments, see Chwolson, Traite de Phy-
sique, Vol. Ill, Part 9, pp. 233-235.)

SPECIFIC HEAT OF A GIVEN VOLUME OF GAS.

The difference between the specific heats under constant
pressure and under constant volume, referred to the unit
of volume, is the same for all perfect gases taken at the
same pressure and at the same temperature.

(Chwolson, Traite de Physique, Vol. Ill, Part 9, p. 220.)

LAW OF DELAROCHE AND BERARD.

This law states that for all elementary diatomic gases
approximately in the perfect state, and for gaseous com-
pounds formed without condensation and approximately in
the perfect state, the product of the molecular weight and
the specific heat at constant pressure has the same value.

(New Century Dictionary under word law. Also con-
sult Nernst, Theoretical Chemistry, p. 42.)

78 LAWS OF PHYSICAL SCIENCE

< INTERNAL FRICTION OF A GAS.

It follows from the deductions of Maxwell that the in-
ternal friction or viscosity of any gas is a function of the
absolute temperature but is independent of the density of
the gas.

In respect to the density the statement is not rigorously
exact for actual gases.

(Chwolson, Traits de Physique, Vol. I, Part 4, pp. 504-
508. Also Poynting and Thomson, Heat, pp. 144-146.)

^ THEOREM OF CORRESPONDING STATES.

If the pressure, volume and temperature of any gas at
its critical point, be chosen for the unit values of these
quantities, then (assuming its accuracy for representing the
properties of a particular gas) Van der Waals' equation,
with uniform values of the constants, will apply to all gases.
In other words, all gases exhibit the same characteristics
when at pressures and temperatures which are proportional
to their critical pressures and temperatures.

(Bdser, Heat for Advanced Students, pp. 312, 313. Also
consult Nernst, Theoretical Chemistry, pp. 219-226.)

PRESSURE OF A GRAM-MOLECULE OF GAS.

The molecular weight of a chemical compound expressed
in grams is called a granv-molecule or mol. (A gram-mole-
cule or mol of Oj is 32 grams, of Hj 2 grams and of H2O
18 grams.)

The pressure exerted by one gram-molecule of any gas
which closely obeys the gas-laws, when at 0° C. and when
occupying a volume of one liter is 22.412 atmospheres.

(Nernst, Theoretical Chemistry, pp, 40, 41.)

HEAT AND PHYSICAL CHEMISTRY 79

THE GAS-CONSTANT.

For any gas which obeys the laws of an ideal gas

where p^ and v^, are the pressure and volume respectively
of the gas at 0° C. and p and v its pressure and volume
respectively at any absolute temperature T. The factor R,
called the gas-constant, is conditioned only by the units of
measurement chosen and is independent of the number of
atoms in the molecule and the chemical composition of
the gas.

If p is me£isured in atmospheres, v in liters and T in
centigrade degrees reckoned from absolute zero, the value of
E is 0.08204.

(Nemst, Theoretical Chemistry, p. 40. Also Smith-
sonian Physical Tables, p. 342.)

EQUATION OF CLAPEYRON.

For a perfect gas pv = ET, where p is the pressure and

V the volume of the gas, T is the absolute temperature and

E is a constant. The constant E is proportional to the mass

M of the gas and for equal masses inversely proportional to

the density 8. Thus,

„ M
Roc-^,

(Chwolson, Traits de Physique, Vol. I, Part 4, pp. 437,
438.)

80 LAWS OF PHYSICAL SCIENCE

CONSTANT OF CLAPEYRON AND THE WORK OF EXPANSION
OF A GAS.

The constant R in the equation, pv = RT, is numerically
equal to the work of the expansion of the gas when its
temperature is raised 1° C. under constant exterior pressure.
Thus,

R = ^^ Cp J, where 7 = tt-

is the ratio of the specific heats at constant pressure and
constant volume and J is the mechanical equivalent of heat.
(Chwolson, Traite de Physique, Vol. I, Part 4, pp. 493-
495.)

FUNDAMENTAL EQUATION OF THE KINETIC THEORY OF GASES.

The fundamental equation of the kinetic theory of
gases is,

pv = i Nmu^,
where v is the volume occupied by the gas, p the pressure
of the gas, N the number of molecules contained in the
volume v, m the mass of a molecule and ii^ the mean square
velocity of translation of the molecules.

(Chwolson, Trcdte de Physique, Vol. I, Part 4, p. 484.
Also Preston, Theory of Heat, pp. 68-71. See serial
article by Dr. Saul Dushmann in Oeneral Electric Review,
Oct., Nov., Dec, 1915, on " The Kinetic Theory of Gases,"
Vol. XVIII, pp. 952-958, 1042-1049, 1159-1168.)

PRESSURE AND ENERGY OF GAS.

The kinetic theory of gases states that the pressure of
a gas is equal to two-thirds the energy of translational
motion of the molecules which are contained in the unit of
volume of the gas, also that the energy of translational
motion of the molecules of a gas is proportional to the abso-
lute temperature of the gas. Thus, for the unit of volume,

3

where p is the pressure.

P= -g-Ei and RT= ^E,,

HEAT AND PHYSICAL CHEMISTRY 81

T the absolute temperature,
R a constant and

El the energy of translational motion in the unit of
volume.

Important relations of the kinetic theory of gases are i

pv = f NrnQ" = f Mu2 = RT = yE.

(Chwolson, Traits de Physique, Vol. I, Part 4, pp. 489,
490.)

VELOCITIES OF GAS-MOLECTTLES.

1. The velocity of the molecules of a given gas is propor-

tional to the square root of the absolute temperature
of the gas.

2. The velocities of the molecules of different gases, at the

same temperature, are inversely proportional to the
square roots of the densities of these gases. Or in
a formula.

u = ^3gR„^,

where g and R„ are constants, T absolute temperature and
8 the density of the gas.

(Chwolson, Traits de Physique, Vol. I,' Part 4, p. 490.)

MAXWELL'S LAW OF MOLECULAR VELOCITIES.

The components of molecular velocity (in a gas) are
distributed among the molecules according to the same law
as the errors are distributed among the observations in the
theory of eiTors of observations.

(Preston, Theory of Heat, pp. 71, 72.)

82 LAWS OF PHYSICAL SCIENCE

^ WORK PERFORMED WHEN TWO GASES MIX.

When two gases which exhibit no chemical interaction
become mixed by diffusion of each into the other no work
is performed if the volume of mixed gases remains constant.
If, on the other hand (with an arrangement which may be
realized experimentally), the volume of the first gas in-
creases from Vi to Vj + Y^ and the volume of the second gas
increases from V^ to Vj + ^u then the first gas in diffusing
will do external work:

Wi = niRT loge — iy , and the second gas will do external

work Wi = nzRT log, "^'^t^' ■

Here T is the absolute temperature, E the gas-constant,
and Ui and n2 are the number of gram-molecules concerned
in the diffusion of the first and second gases respectively.

The total external work done by the mixing of the two
gases is,

W W _L W T,rrf ^ Vl+Vi , , Vl-|-V» \

W = Wi -I- W2 = RT f ni logs y 1- ni! log, y 1 _

This formula, developed by Eayleigh and more thor-
oughly by Boltzmann, expresses a law which holds good
universally.

(Nemst, Theoretical Chemistry, pp. 96-100.)

NUMBER OF MOLECtTLES IN A GAS.

It is deduced from theory that, in a cubic centimeter
of air, or (according to the law of Avogadro) in every other
gas, there are contained in a gram-molecule of the gas about
4.5 X 10^^ molecules.

(Chwolson, Tra/ite de Physique, Vol. I, Part 4, p. 511.)

HEAT AND PHYSICAL CHEMISTRY 83

LAWS OF ABSORPTIOIT OF GASES BY LIQUIDS: HENRY'S LAW.

1. For the same gas, the same liquid and the same tem-

perature, the weight of gas absorbed is proportional to
the pressure, or, at all pressures, the volume dissolved
is the same. (Known as Henry's law.) The volume
absorbed varies with the gas. Thus, water dissolves
over fifty thousand times as great a volume of ammonia
as of nitrogen. The absorbing power also varies with
the liquid. Thus alcohol absorbs gases better than
water.

2. The quantity of gas absorbed decreases with increase of

temperature.

3. The quantity of gas which a liquid can dissolve is in-

dependent of the nature and of the quantity of other
gases which it may already hold in solution.
1 and 3 are only rigorously exact for gases which are
but slightly soluble and when the pressures do not exceed a
few atmospheres.

{Ganot's Physics, art. 190. Also Walker, Introduction
to Physical Chemistry, pp. 57, 58.)

f' SOLUTION IN LIQUID OF MIXED GASES: DALTON'S LAW.

When a mixture of gases dissolves in a liquid, each com-
ponent of the mixture dissolves proportionally to its own
partial pressure; or when a liquid acts to dissolve mixed
gases, each gas dissolves as if all the others were absent.
(Known as Dalton's law.)

Both Dalton 's and Henry 's law hold well only when the
gases are slightly soluble and the pressures do not exceed a
few atmospheres. The divergencies are large for very sol-
uble gases and great pressures.

(Walker, Introduction to Physical Chemistry, p. 58.)

M LAWS OF PHYSICAL SCIENCE

jt LAW OF PARTIAL PRESSURE.

In a mixture of liquids the vapor emitted by the liquids
will in general have the same components as the liquid
mixture remaining behind. The ingredients exert partial
pressures the sum of which is the vapor-pressure of the
mixture. The law holds good universally and is of funda-
mental importance, that: the partial pressure of each com-
ponent of a mixture of liquids is always less than the vapor-
pressure of a component in the free or unmixed state, the,
temperature being the same.

(Nernst, Theoretical Chemistry, pp. 105-107.)

^ WORK DONE BY EVAPORATIOH.

A liquid which evaporates against the constant external
pressure of its saturated vapor performs external work and
absorbs heat. The external work performed by the evapora-
tion of one gram-molecule of any simple liquid is indepen-
dent of the nature of the liquid and is directly proportional
to the absolute temperature T at which the evaporation
takes place. Thus the external work performed in the
evaporation of one gram-molecule of any liquid is,
W = p(V-V') = RT - pV.

Here p is the external pressure and V the volume of the
vapor. V is the volume of the one gram-molecule of the
liquid before evaporation started and R is the gas-constant.
When, as usual V is negligible and p is in atmospheres and
V in liters we have,

W=0.0821T liter-atmospheres.
(Nernst, Theoretical Chemistry, p. 57.)

"^ ABSORPTION or GASES BY SOLIDS.

The surfaces of solids by exerting an attraction on the
molecules of gases become covered with a layer of condensed
gas, and porous solids, which present an extended surface to

HEAT AND PHYSICAL CHEMISTRY 85

the gas, tend to absorb it, many solids absorbing a con-
siderable quantity of gas. This absorption takes place with-
out any chemical change. The absorption is in general
greater in the case of the more easily liquefiable gases.
(Ganot's Physics, art. 194.)

^ OCCLUSION OF GASES.

At a high temperature, platinum and iron allow hydro-
gen to traverse th^m quite readily. Some metals will absorb
gases when cooling and give them off when heating. This
property is most marked with palladium, which not only
absorbs hydrogen while being cooled after being heated but
even when cold. This may cause a palladium wire to
lengthen quite perceptibly.

{Ganot's Physics, art. 195.)

LAW OF DIFFUSION OF GASES (GRAHAM'S LAW).

The quantity of a gas which passes through a porous
diaphragm in a given time is inversely as the square root
of the molecular weight of the gas (or its density).

{Ganot's Physics, art. 191. Also consult Poynting and
Thomson, Heat, p. 327.)

^ EFFUSION OF GASES.

When gas passes through a small aperture, about 0.013
mm. in diameter, from a region where its pressure is h
(expressed in terms of the height of a column of the gas
which would exert the same pressure as that of the efQuent
gas), into a region where its pressure is h' the velocity of
efflux, or the rate of effusion, is,

v=^2g(h-h'),

where g = the acceleration of gravity.

Or we can say: the velocities of efflux, or the rates of
effusion of various gases, are inversely as the square roots
of their densities.

{Ganot's Physics, art. 192.)

86 LAWS OF PHYSICAL SCIENCE

BOILING.

1. The temperature of ebullition, or the boiling-point, in-

creases with the pressure.

2. For a given pressure boiling begins at a certain tem-

perature, which varies for different liquids, but which,
for equal pressures, is always the same in the same
liquid.

3. Whatever be the rate of input of heat into the liquid, as

soon as boiling begins the temperature of the liquid
remains stationary.
(Gemot 's Physics, art. 366. For cases of "Superheat-
ing" see Preston, Theory of Heat, p. 360. Also consult
Nernst, Theoretical Chemistry, pp. 63, 64.)

jt BOILING AND VOLATILIZATION.

If a substance is liquid at a temperature at which the
pressure of its vapor equals the pressure on its surface, the
substance will liquefy and boil, but, if the substance is
solid at a temperature at which its vapor has the pressure
of the pressure on its surface, the substance changes directly
from a solid to a vapor ; namely, it will volatilize or sublime.

Water is an example of the first, arsenic and carbon are
examples of the second.

(See Preston, Theory of Heat, p. 370. Also Walker,
Introduction to Physical Chemistry, p. 82. Also Nernst,
Theoretical Chemistry, pp. 70, 476.)

LATENT HEAT OF VAPORIZATION: TROUTON'S LAW.

For different liquids the latent heat of vaporization mul-
tiplied by the molecular weight is approximately propor-
tional to the absolute temperature at which vaporization
occurs; or the molecular latent heat is approximately pro-
portional to the absolute temperature.

Thus calling w the molecular weight of the vapor, L the
latent heat of vaporization of the liquid and T the absolute
temperature,

HEAT AND PHYSICAL CHEMISTRY 87

r- = a constant.
(Preston, Theory of Heat, p. 391. Also consult Nernst,
Theoretical Chemistry, pp. 272-274.)

f, VAPOR-PRESSURE IN COMMUHICATING VESSELS AT DIFFERENT
TEMPERATURES.

"When two vessels containing the same liquid, but at
different temperatures, are connected, the pressure is identi-
cal in both vessels, and is the same as that corresponding
to the lower temperature. ' '

The liquid distils from the vessel at higher temperature
to the vessel at lower temperature.

(Ganot's Physics, art. 364.)

VAPOR-PRESSURE OF MIXED LIQUIDS.

1. Liquids which do not mix : the vapor-pressure equals the

sum of the vapor-pressures of the constituents.

2. Liquids which mis partially: the vapor-pressure is less

than that of the sum of the pressures of the constituents.

3. Liquids which mix in all proportions: the diminution

of vapor-pressure is still greater.
(Preston, Theory of Heat, p. 406.)

/i. VAPOR FORMATION IN A VACUUM.

If any simple volatile liquid is brought into a vacuous
space the liquid evaporates with great rapidity and evapor-
ation contiuues until the vapor formed exerts a certain
definite maximum pressure. If, when this pressure is
reached, some liquid still remains in the space originally
vacuous, the maximum pressure obtained will be the so-
called vapor-pressure of the liquid at the particular temper-
ature of the experiment. When the temperature increases
the vapor-pressure increases, and usually very rapidly.

(Nernst, Theoretical Chemistry, pp. 56, 57. Also Ganot's
Physics, art. 355.)

88 LAWS OF PHYSICAL SCIENCE

*- CONDENSATION OF SATURATED VAPOR.

If air be perfectly free from dust particles it may be
considerably supersaturated without condensation in the
form of a cloud.

If air be ionized condensation is accelerated, the negative
ions being more effective nuclei than the positive ions. This
fact has been utilized in counting the number of ions in a gas.

(Poynting and Thomson, Heat, pp. 16S-172. Also
Thomson, Conduction of Electricity Through Gases, pp.
163-187.)

IL VAPOR-PRESSURE.

The pressure of a vapor in contact with its own liquid
depends only upon the temperature and is independent of
the relative proportions of the liquid and vapor.

The pressure is some function of the temperature but
not a linear function.

When the pressure upon the surface of a liquid is due
to the atmosphere, the liquid will boil at the moment when
its vapor-pressure just exceeds that of the atmosphere. A
diminution of 1° C, in the boiling-point of water, corre-
sponds to an ascent of about 1080 feet.

(Preston, Theory of Heat, pp. 148 and 408.)

^ LAW OF MASS-ACTION (Credited to Guldberg and Waage).

As stated by the authors of the law, the rate of chemical
action is proportional to the active mass of each of the re-
acting substances. It may be thus stated: when any sub-
stance in solution enters into a chemical reaction, the amount
of reaction in the unit of time is proportional to the active
mass of the substance, namely, to the number of gram-
molecules of the substance contained ia unit volume of the
solution.

HEAT AND PHYSICAL CHEMISTRY 89

(See Walker, Introduction to Physical Chemistry, p.
277 et seq. Also consult Nemst, Theoretical Chemistry, pp.
443-446, for an analytical treatment of this very funda-
mental law of chemical kinetics.)

LAW OF RELATIVE PROPORTIONS IN EQUILIBRIUM.

The condition of physical or chemical equilibrium in a
heterogeneous system is independent of the relative mass
of each phase present in the system.

Thus, at a given pressure and temperature the physical
equilibrium between water and its vapor is undisturbed by
an increase in the mass of either phase; and the chemical
equilibrium between CaCOajCaO and CO'a is undisturbed
by a change in the quantity by weight of any of the
substances enumerated.

(Nemst, Theoretical Chemistry, p. 471 et seq.)

f-LAVT OF DISTRIBUTION, AMONG SEVERAL MOLECULAR SPECIES.

"When several molecular species [as acetic acid, giving
in both vapor form and in solution the single molecules
CHjCOaH and the double molecules (01X300311)2] evapo-
rate at constant temperature from a common solvent (as
benzene) into a fixed vapor-space the ratio of the concentra-
tion in the vapor-space, of any one molecular species, to its
concentration in the solvent is constant.

This distribution in the quantity of a molecular species
between the solvent and the vapor-space is independent of
the presence of other molecular species, even when these
latter are chemically reactive with the former.

(Nemst, Theoretical Chemistry, p. 491. For application
of the "distribution law" to dilute solutions see "Washburn,
Principles of Physical Chemistry, pp. 148-150.)

90 LAWS OF PHYSICAL SCIENCE

f^ THE LAW OF HESS.

The total quantity of heat, disengaged in the passage of
a group A of substances to a group B, is independent of
the nature of this passage, namely, of the character and of
the order of intermediate reactions, provided the physical
state (in the wide sense of this word) of the groups A and
B is the same in all cases.

This law is a basic principle of thermochemistry.

(Consult Chwolson, Traite de Physique, Vol. Ill, Part
9, p. 284.)

f- EFFECT OF TEMPERATURE ON BALANCED CHEMICAL ACTION.

If a direct chemical action gives out a certain quantity
of heat per gram-molecule transformed, the reverse reaction
will absorb an exactly equal quantity of heat ; and rise of
temperature always affects chemical equilibrium in such a
manner that the displacement of the point of equilibrium
takes place in the direction which wiU determine absorption
of heat.

(See Walker, Introduction to Physical Chemistry, p.
293. Also Nemst, Theoretical Chemistry, pp. 673-676.
Note statement of LeChatelier on p. 676.)

*■ PROGRESS OF CHEMICAL DECOMPOSITION.

The phenomena of chemical decomposition of a body in
a confined space go on, if one of the elements of the decom-
position is gaseous, until a certaiu pressure is attained when,
for a particular temperature, the decomposition ceases.

Deville used the word "Dissociation" which has analogy
under the above conditions with vaporization of liquids.

(Roscoe and Schorlemmer, Treatise on Chemistry, Vol.
II, pp. 129-132.)

HEAT AND PHYSICAL CHEMISTRY 01

CRITICAL TEMPERATURE.

There is for every substance a critical temperature above
which the substance cannot be liquefied with pressure; or
there is for each gas a particular or critical temperature to
whfch the gas must be cooled in order to liquefy it with any
pressure.

(Preston, Theory of Heat, p. 450 et seq. See COj curves,
pp. 490, 491. Also consult Nernst, Theoretical Chemistry,
pp. 64^66.)

i MOLECITLAR SURFACE-ENERGY: LAW OF EOTVOS.

According to Eotvos, the work required to form the
surface of a spherical gram-molecule of a liquid varies with
the temperature in the same manner for aU liquids. This
law is stated in the expression,

yv2/3 = k(T - T„),
where v is the volume occupied by one gram-molecule of
a liquid and 7 is its surface tension. T^ is a temperature
taken not far from the critical, T the temperature of the
liquid, and k is a constant, independent of the nature of
the liquid. The quantity yv^/s is proportional to the
molecular surface-energy of a sphere formed from one gram-
molecule of the liquid in question.

The law of Eotvos has importance in determining the
molecular weight of liquids.

(Nemst, Theoretical Chemistry, pp. 275-277.)

CURIE'S LAW.

In paramagnetic substances the magnetic susceptibility,
or the ratio of the intensity of magnetization to the
magnetizing force, is inversely proportional to the absolute
temperature.

(Richardson, The Electron Theory of Matter, p. 378.)

92 LAWS OP PHYSICAL SCIENCE

LAWS OF DIFFUSION IN LIQUIDS.

1. The rate at which the diffusion of any substance goes on

is' proportional to the rate of variation of the strength
of that substance in the fluid as measured along the
line in which the diffusion takes place.

2. The rate of diffusion varies with the temperature.
The law of diffusion of matter has exactly the same

form as that of the diffusion of heat by conduction.

(See Maxwell, Theory of Heat, Chap. XIX, and p. 276.
Also consult Ganot's Physics, art. 140. Also Nemst, Theo-
retical Chemistry, p. 151 et seq.)

OSMOSE, OSMOSIS OR DIOSMOSE.

When two liquids which will mix are separated only by a
porous membrane there is a movement of the liquids in both
directions through the membrane. The greater movement
is usually from the less dense to the more dense liquid so
as to cause the level of the more dense liquid to rise above
that of the less dense. This action increases with the tem-
perature and is proportional to the strength of the solution.

{Ganot's Physics, art. 139. Also Chwolson, Traite de
Physique, Vol. I, Part 5, pp. 661-668. Also Nemst, Theo-
retical Chemistry, pp. 125-127.)

OSMOTIC PRESSURE AN ANALOGUE OF GAS-PRESSURE.

A principal feature in the analogy between a dissolved
substance and a gas consists in the correspondence between
the energy content of each, this energy content being in-
dependent, at any fixed temperature, of the volume occu-
pied by a given mass of either. Thus, the osmotic pressure
of a dissolved substance is exactly the same as the gas-
pressure which would be exerted if the solvent were removed
and the dissolved substance in gaseous form were left behind
to occupy the same volume at the same temperature. Thus
the gas-law for a dissolved substance inay be written
PV = ET = 0.0821T liter-atmospheres.

HEAT AND PHYSICAL CHEMISTRY 93

Here P denotes the osmotic pressure in atmospheres of
a solution which contains one gram-molecule of the substance
dissolved in V liters of solvent, and T the absolute tempera-
ture in degrees centigrade.

(Nemst, Theoretical Chemistry, p. 144.) *

f. RAOULT'S LAW ON THE LOWERING OF VAPOR-PRESSURE.

"The relative lowering of vapor-pressure experienced
by a solvent on dissolving a foreign substance is equal to
the quotient obtained by dividing the number of dissolved
molecules n, by the number of molecules N, of the solvent. ' '
Thus in symbols

p-p' ^ n .

p' N

(Nernst, Theoretical Chemistry, pp. 144, 145. See also p.
263 et seq.)

f GAS-LAWS APPLIED TO SOLUTIONS.

1. The osmotic pressure is, at constant temperature, pro-

portional to the concentration of the solution, or
inversely proportional to the volume occupied by a
given quantity of the dissolved substance (analogue of
Boyle 's law) .

2. The osmotic pressure is proportional to the absolute tem-

perature (analogue of Gay-Lussac's or Charles' law).

3. Equal volumes of isotonic solutions — solutions which

exercise the same osmotic pressure — when under the
some pressure and at the same temperature contain
the same number of molecules. This number of mole-
cules is the same as that of a gas under like conditions
of volume, pressure and temperature (analogue of
Avogadro'slaw).
(Chwolson, Traits de Physique, Vol. I, Part 5, p. 666.

See also Walker, Introduction to Physical Chemistry, p.

184. Also Nemst, Theoretical Chemistry, p. 141 et seq.)

94 LAWS OF PHYSICAL SCIENCE

^ LAW OF KOHLRAUSCH FOR DILUTE SALT-SOLTJTIOlfS.

The molecular conductivity of a solution (namely, its
electrical conductivity times the volume of the solution
which contains one gram-molecule of the dissolved sub-
stance) is independent of the concentration of the solution
for very dilute salt-solutions. This constant value of the
molecular conductivity of an electrolyte, at infinite dilution,
is the sum of two numbers, one of which depends solely upon
the speed of migration of the positive ions and one solely
upon the speed of migration of the negative ions.

(Walker, Introduction to Physical Chemistry, pp. 251,
256. Also Nemst, Theoretical Chemistry, pp. 365-367.)

'♦' ADDITIVE PROPERTY OF DILUTE SOLUTIONS.

In a sufficiently dilute electrolytic solution there is a
complete dissociation into ions of the dissolved substance
and it is a fundamental law that : "the properties of a salt-
solution are composed additively of the properties of the
free ions."

The law, to hold true, presupposes complete dissociation.

(Nemst, Theoretical Chemistry, p. 384.)

■f- SIMILARITIES IN BEHAVIOR OF IONS AND MOLECULES.

It may be stated, as a theorem, that: the ions of dis-
sociated substances in solution exhibit all the properties of
neutral molecules and some additional properties due solely
to the electrical charge of the former. Thus ions, like
molecules, diffuse, exert pressure, distribute according to
the law of equipartition of energy, show additive properties,
etc., and in addition, in virtue of their electric charge,
transport electricity.

(Nernst, Theoretical Chemistry, pp. 392, 393.)

HEAT AND PHYSICAL CHEMISTRY 05

OSTWALD-g LAW OF UOLECULAR CONDUCTIVITY.

This law is expressed by the formula,

■-7Z T — = k, a constant.

(1 — m) V

k is called the dissociation constant, v is the dilution and m

represents the degree of ionization and is equal to

where „„ is the molecular conductivity at dilution v and
/ice is the molecular conductivity at infinite dilution.

(Walker, Introduction to Physical Chemistry, Chap.
XXV, see pp. 262-265.)

r FORMATION OF " HYDRION."

Aqueous solutions of various acids possess one property
in common, on the dissociation theory: they form "hy-
drion, ' ' namely, yield hydrogen atoms in the ionic state.

The electrical conductivity of weak solutions of acids is
due almost wholly to the hydrion they contain, and measur-
ing the relative strengths of acids by measuring the conduc-
tivity of the solutions has practically superseded all other
methods, especially for the weaker organic acids.

(Walker, Introduction to Physical Chemistry, p. 313.)

i. PROCESS OF NEUTRALIZATION.

It is shown by conductivity measurements of nearly pure
water that hydrogen ions and hydroxyl ions can only exist
beside each other in mere traces, if at all. Hence when we
bring together in a water-solution two electrolytes (an acid
and a base) which yield hydrogen ions and hydroxyl ions
these unite to form water according to the equation

H -1- OH = H2O.
This process is called neutralization.
(Nemst, Theoretical Chemistry, pp. 517, 518.)

96 LAWS OF PHYSICAL SCIENCE

HETTTRALIZATION OF AN ACID AND A BASE.

When a strong acid is mixed with a strong base the
"hydrogen ions and the hydroxyl ions unite almost completely
to form molecules of water and the general result follows,
that: "the neutralization of a strong acid by a strong base
must always exhibit the same heat of reaction " — about
13,700 gram-calories.

(See Nemst, Theoretical Chemistry, pp. 578-610. Also
Washburn, Principles of Physical Chemistry, p. 243. Also
see for values of heats of neutralization, Smithsonian
Physical Tables, p. 213.)

^ RELATIVE AVIDITY OF ACIDS.

Two weak acids share between themselves a base for
which both are competing in a certain definite ratio. The
ratio of the avidities of the two acids is (under specified
conditions) equal to the square root of the ratio of their
dissociation-constants. Thus,

r^ = V'^''

where x is the amount of

one of the two acids neutralised by the base and k and k'
are the dissociation-constants of the two acids.

(Walker, Introduction to Physical Chemistry, pp. 314-
316. Also Nemst, Theoretical Chemistry, pp. 521-523.)

CHEMICAL COMBINATION OF THE ELEMENTS.

" Combination always takes place between certain defi-
nite and constant proportions of the elements or between
multiples of these."

The composition of a chemical compound can be learned
by (1) bringing the component elements into combination
under favorable conditions: synthesis, or (2) by separating
it into its component elements : analysis.

(Eoscoe and Schorlemmer, Treatise on Chemistry, Vol.
I, p. 72. Also Nemst, Theoretical Chemistry, pp. 30, 31.)

HEAT AND PHYSICAL CHEMISTRY 97

^ PERIODIC SYSTEM OF CHEMICAL ELEMEHTS.

The properties of the chemical elements are periodic
functions of their atomic weights.

This is shown by means of a systematic arrangement of
the elements in a tahle known as "MendelejefE's Periodic
System of the elements."

(See table, Nemst, Theoretical Chemistry, p. 180. Also
see table published July, 1915, in General Electric Beview,
by the Research Laboratory of the General Electric
Company.)

LAW OF GAY-LUSSAC AND HUMBOLDT ON COMBINATIONS OF
GASES BY VOLUME.

The volumes in which gaseous substances combine chem-
ically bear a simple relation to one another and to the
volume of the resulting product.

(Roscoe and Schorlemmer, Treatise on Chemistry, Vol.
I, p. 75.)

f RICHTER'S LAW.

When any two neutral salts undergo double decomposi-
tion by interchange of their acid and basic constituents the
two new salts resulting from such interaction are also neutral
in character.

(Roscoe and Schorlemmer, Treatise on Chemistry, Vol.
I, p. 33.)

^ " MOLECULAR ROTATION " OF AN OPTICALLY ACTIVE LIQUID.

A plane-polarized ray in passing through a liquid
optically active is rotated proportionally both to the length
of the path of the ray through the liquid and to the density
of the liquid. Hence when the actual observed rotation is
divided by the density and the length of the path the specific
rotary power of the liquid is obtained, and when this is
multiplied by the molecular weight the molecular rotation
is expressed.

(Walker, Introduction to Physical Chemistry, p. 158.)

98 LAWS OF PHYSICAL SCIENCE

5» OUDEMAN'S LAW OF OPTICAL ROTATIOH.

The molecular rotation of plane-polarized light by salts
of an optically active acid or base always tends to a definite
limiting value as the concentration of the solution dimin-
ishes. This regularity is known as Oudeman's law.

(Walker, Introduction to Physical Chemistry, p. 174.
Also Nernst, Theoretical Chemistry, p. 391.)

* RAOULT'S LAW FOR DEPRESSIOIf OF FREEZING-POINT.

If the same number of molecules of different substances
be dissolved in a given number of molecules of the solvent
the depressions of the freezing-points of the solutions are
equal.

(New Century Dictionary under word law. Also
Walker, Introduction to Physical Chemistry, p. 210 et seq.
Also Nernst, Theoretical Chemistry, p. 146.)

f' INFLUENCE OF PRESSURE ON THE MELTING-POINT.

The melting-point of a substance (as ice) which con-
tracts on liquefaction, is lowered by increase of pressure
and if the substance expands during liquefaction the
melting-point is raised by increase in pressure.

(Preston, Theory of Heat, pp. 341, 342. Also Nernst,
Theoretical Chemistry, p. 67 et seq.)

DISSOLVED SALTS RAISE BOILING-POINT.

The boiling-point of a liquid is raised by dissolving in
it a salt. Thus, a saturated solution of common salt in water
boils at about 102° C. and water saturated with calcium
chloride boils at about 179° C.

The boiling-point is lowered when a gas is dissolved in
the liquid.

(Walker, Introduction to Physical Chemistry, pp. 195,
196. Also Ganot's Physics, art. 368.)

HEAT AND PHYSICAL CHEMISTRY 99

FREEZING-POINT OF A SOLUTION (BLAGDEN'S LAW).

The freezing-poipt of a substance is lowered by dissolv-
ing in it some foreign matter. The change is usually
proportional to the amount of dissolved-substance in the
solvent, although sometimes the change is abnormally great.
This is the case with substances which have an abnormal
osmotic pressure.

(Consult "Walker, Introduction to Physical Chemistry,
pp. 65, 195 and 406.)

^ FUSION: OF CRYSTALLINE SOLIDS AND PURE METALS.

1. For a given pressure the temperature of fusion is fixed,

and is the same as that of solidification.

2. While fusion or solidification is taking place the tempera-

ture of the whole mass remains constant.

3. Duriug fusion heat is absorbed by the substance and an

equal quantity of heat is disengaged during solidifica-
tion.
(Preston, Theory of Heat, p. 336.)

« THE CRYOHYDRIC TEMPERATURE.

The cryohydric temperature is the lowest temperature
that can be produced by mixing salt and ice together. No
solution of salt in water can exist in a stable state below this
temperature.

(For full and accurate information on this matter, con-
sult Walker, Introduction to Physical Chemistry, Chap.
VIII.)

y GIBBS' CRITERIA OF THERMAL EQUILIBRIUM.

For the equilibrium of any isolated system it is neces-
sary and sufScient that, if the total energy of the system be
constant, the variation of its entropy shall either vanish or
be negative; or, if the total entropy of the system be con-
stant, the variation of its energy shall either vanish or be
positive.

(Gibbs, "Oil the Equilibrium of Heterogeneous Sub-
stances," Trans. Connecticut Acad., Vol. 3, p. 109.)

100 ^ LAWS OF PHYSICAL SCIENCE

THE PHASE-RULE.

The Phase-Rule, due to Gibbs, generalizes the condi-
tions which determine the equilibrium of any system. Let
K represent the number of components (salt, water, etc.)
and let i be the total number of phases in which these com-
ponents are present (crystalline, liquid, vapor, etc.). The
number of degrees of freedom, or independent ways in
which the system can be changed, is given by
n = K + 2 - i.

{New Century Dictionary under words phase exjijE.
Also Jones, Elements of Physical Chemistry, p. 489 et
seq. Also Walker, Introduction to Physical Chemistry,
Chap. XI, pp. 103-117. Also Gulliver, Metallic Alloys, pp.
157-164.)

*" LAW OF THE MUTUALITY OF PHASES.

"If two phases, respecting a certain definite reaction,
at a certain temperature, are in equilibrium with a third
phase, then at the same temperature and respecting the
same reaction, they are in equilibrium with each other. ' '

(Nernst, Theoretical Chemistry, p. 672. See also pp.
132, 137, 495 for applications of law.)

I ACCELERATIOH OF CHEMICAL REACTIONS WITH ELEVATION
OF TEMPERATURE.

It is a general principle of chemical kinetics that the
velocity with which a chemical system proceeds toward its
state of equilibrium increases very greatly with increase in
temperature.

(Ncmst, Theoretical Chemistry, pp. 679, 680. Also
Walker, Introduction to Physical Chemistry, p. 300 et
seq.)

HEAT AND PHYSICAL CHEMISTRY 101

HEATS OF REACTION: LAW OF CONSTANT HEAT-SUMMATION.

In processes which occur in nature the associated energy
changes may be discriminated as :

1. Production or absorption of heat.

2. Performance of external work.

3. Variation of the internal energy of a system.

In^a chemical system the sum of the heat produced in a
reaction and the external work performed is called the "heat
of reaction. ' ' This heat of reaction (also heat of formation)
may be either positive or negative. It represents the change
in the total energy of the chemical system.

The total heat generated in a chemical reaction is entirely
independent of the steps followed in passing from initial to
final state of the system, and this principle — "the law of
constant heat-summation" — makes it possible to calculate
heats of formation for steps which are chemically
impracticable.

(Nemst, Theoretical Chemistry, pp. 592, 597 et seq.
Also Smithsonian Physical Tables, p. 212.)

'*' HEAT OF FORMATION.

By "heat of formation of a chemical compound is meant
the quantity of heat which is given off in the formation of
the compound from its component elements." The sum of
the heats of formation of the substances formed by chemical
reaction minus the sum of the heats of formation of the
substances used up, is equal to the heat of reaction.

(Nemst, Theoretical Chemistry, p. 606. See also 607,
608, where it is shown how heats of formation may be ob-
4;ainied from heats of combustion.)

102 LAWS OF PHYSICAL SCIENCE

X CATALYSIS.

Many chemical reactions are observed to take place at
an accelerated rate when they occur in the presence of
certain substances which themselves suffer no chemical
change. Berzelius gave to this phenomenon the name catal-
ysis. The name means, " an increase in velocity of reaction
caused by the presence of substances which do not take part
in it (or only to a secondary extent) although the reaction
is capable of taking place without their presence."

A substance which produces catalysis is called a cator-
lyser.

(Nernst, Theoretical Chemistry, p. 581. Also Wash-
bum, Principles of Physical Chemistry, pp. 274, 275.)

^ CRYSTALLOIDS AND COLLOIDS.

Investigations of the phenomena of diffusion show that
substances can be divided into two classes, "crystalloids"
and "colloids." The former diffuse more rapidly and as a
rule are obtained in crystalline form while the latter are
amorphous. Graham named the process for separating the
two classes by means of an animal membrane, "dialysis."

(Consult Walker, Introduction to Physical Chemistry,
p. 233. Also Ganot's Physics, art. 140.)

f. ABSORPTION OF RADIANT HEAT.

It is a general principle that bodies absorb radiant heat
which proceeds from heated bodies of the same kind. Also,
"any substance is particularly transparent to radiation
which has already been sifted by a plate of that substance."

(Preston, Theory of Heat, p. 549.)

HEAT AND PHYSICAL CHEMISTRY 103

ABSOLUTE EMISSIVE POWER (Definition).

The absolute emissive power of a body for a particular
wave-length is the energy at that wave-length radiated per
second by unit surface at temperature 1° absolute to sur-
rounding enclosure at absolute zero.

(Preston, Theory of Heat, p. 588.)

^ MONOCHROMATIC EMISSIVE POWER.

The monochromatic emissive power of a body is the ratio
of the energy, having wave-lengths lying between A and A,
+ dA, which it radiates at absolute temperature T to the
energy of the same wave-lengths which a perfectly black
body radiates at the same temperature and under exactly
similar circumstances.

Call J/i' the energy of these wave-lengths radiated by a
unit surface of the body in the unit of time and call JA the
energy radiated by a black-body under the same circum-

K

stances. Then y- :3=e;^, the monochromatic emissive power

A

of the body. It is in general a function of the absolute
temperature T, the wave-length A and varies with the nature
of the body.

(Consult Preston, Theory of Heat, p. 587. Also Chwol-
son. Traits de Physique, Vol. II, Part 8, pp. 55, 56. For
experimental methods and values of emissivity, see Bulletin
of the Bureau of Standards, Vol. II, p. 41, p. 591 and p. 607 ;
articles by Burgess, "Waltenberg and Foote.)

104 LAWS OF PHYSICAL SCIENCE

ABSORPTIVE POWER AND DEFINITION OF A BLACE-BODY.

Let radiation of a given wave-length \ and of energy B
per unit volume fall upon a body. Denote this radiant
energy by E ,. Then if a portion E ' of this radiant
energy is absorbed by the body, the absorptive power of the
body for wave-length A is,

a, -

' \

If this fraction is unity for all values of X the body is
called, according to Kirchhoif, a perfectly "black-body."

No body has been found which is perfectly black as
Kirchhoflf defines black. If, however, any body is placed
in an enclosure the walls of which are at a uniform tempera-
ture, the body will finally assume the temperature of the
enclosure and when it does it will emit in quantity and
quality {i.e., wave-length) as much radiation as it receives,
It will then be indistinguishable to the eye from the neigh-
boring bodies and is said to be at "black-body" temperature.

(Consult Burgess and LeChatelier, Measurement of
High Temperatures, Chap. VI.)

KIRCHHOFF'S LAW REGARDING RADIATION.

For radiations of the same wave-length and the same
temperature, the ratio of the emissive and absorptive powers
is the same for all bodies and is equal to the emissive power
of a perfectly black-body. In symbols,

''' a

where e^ is the ■emissive power of the body for wave-length
A, and a^ is its absorptive power for wave-length A, and C^
is the emissive power of a black-body.

(Preston, Theory of Heat, p. 588. Also Burgess and

HEAT AND PHYSICAL CHEMISTRY 106

LeChatelier, Measurement of High Temperatures, pp. 243-
245. Also Chwolson, Traite de Physique, Vol. II, Part 8,
pp. 57-59.)

Propositions deducible from kirchhoff's radiation law.

1. The emissive power of an absolutely black-body is the

greatest emissive power possible.

2. Every body absorbs the rays which it emits at a given

temperature — but every body does not necessarily emit
all rays which it absorbs at a given temperature.

3. Every body can absorb rays which it emits at a given

temperature, and it can also. absorb other rays, pro-
vided that among these latter there are rays which a
black-body emits at the given temperature.

4. Every body which emits, at a given temperature and

■under given conditions in a particular direction (under
a given angle with the normal), rays of wave-length A.
and of a definite type of vibration (character of
polarization), absorbs, at the same temperature and
under the same conditions, rays of the same wave-
length and the same type of vibration which fall upon
it in the same direction.

5. The ratio of the emissive to the absorptive power, which

is the same for all bodies for the same given values of
temperature and wave-length, does not depend upon
the kind of vibration; namely, upon the character of
the rays emitted and absorbed, in respect to their
polarization.

6. The law of Kirchhoif applies to any composite flux of

calorific energy (where the wave-lengths are comprised
between any arbitrary limits Aj and k^) if the integral
absorption be referred to a flux, which has for its
source an absolutely black body taken at the same
temperature as the bodies which are to be intercom-
pared.

106 LAWS OF PHYSICAL SCIENCE

7. Kirchhoff 's law holds true for a composite fliix when two

given bodies are at the same temperature and when
each of them acts as source of the flux, the integral
absorption of which, by the other body, is measured.

8. In a closed space all parts of which are at the same

temperature all bodies inside and the walls of the
enclosure itself produce definite radiation which is
identical with the radiation from an absolutely black
body.
(For full explanation and proof of the above proposi-
tions, consult Chwolson, TraiU de Physique, Vol. II, Part
8, pp. 59-70.)

INTENSITY OF RADIANT HEAT.

The quantity of heat proceeding from a point-source of
heat which is received on a unit surface in the unit of time
may be called the intensity of radiant heat. It varies with
the temperature of the source and is inversely proportional
to the square of the distance from the source.

{Ganot's Physics, art. 420.)

^ HEAT-RADIAT^ON AT AN OBLIQUE ANGLE.

The intensity of oblique rays of radiant heat is propor-
tional to the cosine of the angle which these rays form with
the normal to the surface. This ' ' law of the cosine ' ' is not,
however, general.

Radiant heat is only one section of the spectrum which
extends from wave-lengths shorter than those of ultraviolet
light to the longest waves observed. Hence, the laws of
reflection and refraction are the same for rays of radieint
heat as they are for light.

(Fourier, The Analytical Theory of Heat, p. 34. Also
Ganot's Physics, art. 420.)

HEAT AND PHYSICAL CHEMISTRY 107

STEFAN-BOLTZMANN RADIATION LAW.

The energy radiated in unit time by a black-body is
proportional to the fourth power of the absolute tempera-
ture, or

E=K (T4-T4),

where E is the total energy radiated by the body at absolute
temperature T to the walls of an enclosure at absolute
temperature T„ and K is a constant.

(Preston, Theory of Heat, p. 590, and pp. 596-598. Also
Bulletin of the Bureau of Standards, Vol. I, p. 198. Also
Chwolson, Traite de Physique, Vol. II, Part 8, p. 71 et
seq.)

^ PRESSURE OF RADIATION.

When radiant energy is incident perpendicularly on a
plane-surface which is absolutely black, it exerts a pressure
on the surface equal to the density of the energy of radia-
tion. If the body is perfectly reflecting the pressure is
twice as great.

(Maxwell, Electricity and Magnetism, Vol. II, art. 792.
Also Chwolson, Traite de Physique, Vol. II, Part 8, p. 84.)

X WIEN'S DISPLACEMENT LAW (ist statement).

"If radiation of a particular wave-length corresponding
to a definite temperature is adiabatically altered to another
wave-length, then the temperature changes in the inverse

ratio," Or,

a constant
T= A

(Consult Preston, Theory of Heat, p. 600. Also p. 602
for confirmatory experiments.)

108 LAWS OF PHYSICAL SCIENCE

WIEK'S DISPLACEMENT LAW Und statement).

"When the temperature increases, the wave-length of
every monochromatic radiation diminishes in such a way
that the product of the temperature and the wave-length is
a constant." Or,

AT = A<,T„.
Hence for the wave-length of maximum energy
A^T == a constant.
{Bulletin of the Bureau of Standards, Vol. I, No. 2, p.
202.)

WIEN'S LAW OF RADIATION OF WAVE-LENGTH OF MAXIMUM
ENERGY.

The energy radiated from a black-body source which
corresponds to the wave-length having a maximum energy
is proportional to the fifth power of the absolute tempera-
ture. Or,

E„„ = BT%

max. '

where T is the absolute temperature and B is a constant.
{Bulletin of the Bureau of Standards, Vol. I, p. 202.
Also Preston, Theory of Heat, pp. 601, 610. Also Chwol-
son, Trwite de Physique, Vol. II, part 8, p. 73.)

WIEH'S LAW or SPECTRAL DISTRIBUTION OF ENERGY,
J = Ci X"^e NT

where J — the energy corresponding to wave-length X,

T = absolute temperature of the radiating black-body,
e = base of the natural system of logarithms and
Ci and Cj are constants.
{Bulletin of the Bureau of Standards, Vol. I, p. 204.

For use of the equation in pyrometry, see p. 210.)

HEAT AND PHYSICAL CHEMISTRY 109

FLAirCE'S LAW OF SPECTRAL DISTRIBUIIOH OF ENERGY.

1

J = Ci X"

e^T-1

where J is the energy corresponding to wave-length X, T is
the absolute temperature, e the base of the natural system
of logarithms and Cj and C2 are constants.

Planck's law agrees with experiment better than Wien's
and holds well for a wide variation in X.

{Bulletin of the Bureau of Standards, Vol. I, p. 206.)

PREVOST'S THEORY OF EXCHAHGES.

If two bodies are associated in such a manner that one
receives the radiation of the other, each radiates indepen-
dently and the temperature of either wiU fall if it radiates
more energy than it absorbs.

{Ganot's Physics, art. 421.)

THE QUAHTUm HYPOTHESIS.

The only hypothesis which has, satisfactorily accounted
for the laws of radiation from black bodies and many
phenomena, such as the emission of electrons from illumi-
nated metals, has been based on the supposition that radiant
energy is absorbed or emitted or both not continuously, but
by discontinuous, discrete units of magnitude proportional
to the frequency of the radiation.

(Planck, Vorlesungen uber die Theorie der Warme-
strahlung.)

LAW OF PHOTO-CHEMICAL REACTION.

Abundant research has led to the result that when a
photo-chemical system is illuminated the resultant action
depends both on the light-intensity and the time the illumi-
nation continues. Hence the law: "When light of the same
kind is used, the photo-chemical action depends solely on the
product of the intensity and the duration of the exposure. ' '
This law applies in photography and for X-Ray exposures.

(Nernst, Theoretical Chemistry, p. 786.)

110 LAWS OF PHYSICAL SCIENCE

i CRYSTALS: THE LAW OF INTERFACIAL ANGLES.

A crystal is a homogeneous body the various physical
properties of which are differently manifested when con-
sidered along different lines radiating from any point in
the body.

The fundamental law of crystallography states that,
"the inclination of two definite crystal planes to each other,
for the same substance, and measured at the same tempera-
ture, is constant and independent of the size and develop-
ment of the planes." This law is not, however, quite rigid.

(Nemst, Theoretical Chemistry, pp. 72, 73. Also
Spencer, The World's Minerals, p. 15.)

•f NEUMANN'S LAW OF CRYSTAL-ZONES.

A set of planes of a crystal which intersect in such
manner that the lines of intersection are parallel to one
another is called a zone and this common direction is called
the "zonal axis." Otherwise stated a crystal has a girdle
of faces (a zone) the edges of which form parallel lines.

It is a fundamental law of crystallography that all planes
which can occur on a crystal are related to each other in
zones or from any four planes, no three of which lie in any
one zone, all crystal planes can be derived by means of
zones.

(Nemst, Theoretical Chemistry, p. 73. Also Spencer,
The World's Minerals, pp. 17, 18.)

V

ELECTRICITY AND MAGNETISM

ELECTRICITY AND MAGNETISM

ELECTRIFICATIOH PRODUCED BY FRICTION.

If two unlike substances which are insulators are rubbed
together and then separated they are found to be electrified
with equal quantities of electricity, the one substance with
positive, or vitreous, and the other with negative, or resinous,
electricity.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, (frt. 27.)

ELECTRIFICATION PRODUCED BY INDUCTION.

When an electrified body is suspended in a hollow con-
ducting vessel without touching it, the outside of the vessel
acquires electrification of the same sign as the electrified
body, and the inside of the vessel acquires electrification of
the opposite sign. The electrification of the vessel thus
produced is called electrification by induction.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 28. Also Ganot's Physics, art. 764.)

ELECTRIFICATION BY CONDUCTION.

When an originally unelectrified and insulated metal
body is connected with an electrified body by means of a
conducting wire, which is itself insulated, the first body
becomes electrified by a passage of electricity over the wire.
This passage is called electrification by conduction.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 29.)

113

114 LAWS OF PHYSICAL SCIENCE

CONDUCTORS AND INSULATORS.

At low and at ordinary temperatures materials are
separable into two fairly distinct classes : those which readily
conduct electricity, called conductors, and those which con-
duct electricity very slightly, or not at all, called insulators.
Above 1200° or 1500° C. the distinction between the two
classes disappears rapidly as the temperature increases.

(See " Methods, Data and New Apparatus for Measuring
Electrical Conductivity above 1500° C. of Vapors at Normal
Pressure," by B. P. Northrup, Jour, of the Franklin Insti-
tute, March, 1915. Also Ganot's Physics, art. 743.)

EQUALITY OF POSITIVE AND NEGATIVE ELECTRICITY.

When electrification is excited by any means the quan-
tities of positive and negative electricity produced, or re-
vealed, are always equal.

Modern theory asserts that the ultimate unit of negative
electricity is a definite quantity of electricity called an elec-
tron. Millikan concludes from experiments that the charge
carried by an electron is 4.774 X 10-^" electrostatic unit.
The ratio, 1 , of the charge (electrostatic units) to the

mass (grams) of an electron is 5.31 X 10^', provided the
velocity of the electron is a small fraction of the velocity
of light.

(Thomson, Elements of Electricity and Magnetism, p.
10, See also Campbell, Modern Electrical Theory, pp.
'25-28, 78-80. Also Smithsonian Physical Tables, p. 342.)

LAW OF REPULSION OF ELECTRIC CHARGES.

When two charged bodies are at a distance r apart, r
being very large compared with the greatest linear dimen-
sions of either of the bodies, the repulsion between them is
proportional to the product of their charges and inversely
proportional to the square of the distance between them.

ELECTRICITY AND MAGNETISM 116

The repulsion between two charges Qi and Qj in air is,
in electrostatic units, F =^i-|?^ dynes.

(Thomson, Elements of Electricity and Magnetism, p.
12. Also, for simple proof, see Lommel, Experimental
Physics, p. 285. Also Ganot's Physics, art. 753.)

POTENTIAL.

The work which must be done by an external agent to
bring a unit of positive electricity, by any path, from an
infinite distance (or from a place where the potential is
zero) to a given point in space is called the potential at
the point.

A body charged positively always tends to move from
places of greater to places of less positive potential, and a
body charged negatively always tends to move in the oppo-
site direction.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, arts. 45, 70.)

ELECTROMOTIVE FORCE.

The difference of potential between two conductors or
between two points in space or between two points on a
body is equal to the electromotive force (e.m.f.) between
them. The e.m.f. between the two points in a circuit is equal
to the product of the current and the ohmic resistance
between the two points.

When a charge e moves along a given path from a point
A to a point B and work W is done by the electric force,
such that W = Ee, the quantity E is called the total elec-
tromotive force acting between the points A and B. In
electrostatics B and Vi-V,, the potential difference between
the points, are identical quantities.

(Maxwell, Treatise on Eledtricity and Magnetism, Vol.
I, arts. 45, 49, 69, 241.)

116 LAWS OF PHYSICAL SCIENCE

POTENTIAL DUE TO A SYSTEM OF POINT-CHARGES.

When any number of electrified points having charges
61, e^, 63, etc., are distributed through space, then if
Ti, Tj, T3, etc., are the distances of these points respectively
from a point P in space, the potential at P due to the
system is,

=-'(^)

(Maxwell, Treatise on Electricity and Magnetism, Vol. I,
art. 73.)

FORCE BETWEEN CHARGED BODIES VARIED BY THE MEDIUM.

When the charges are given, the mechanical forces on
bodies in an electric field are diminished by the interposi-
tion of a medium with a larger specific inductive capacity.
Thus the force between two point-charges Qi and Q2, a
distance r apart in a medium of specific inductive capacity
K, is,

(Thomson, Elements of Electricity and Magnetism, p.
129.)

ELECTRIC EQUILIBRIUM.

A conductor can only be in electric equilibrium when
every point in it is at the same potential. This potential is
called the Potential of the Conductor.

(Maxw«ll, Treatise on Electricity and Magnetism, Vol. I,
art. 45. Also consult, for an extended and clear exposition
of the facts and laws of electricity in equilibrium, Chwolson,
Traits de Physique, Vol. 4, Part 10, Chap. I. See p. 81.)

ELECTRIC ABSORPTION.

The phenomenon of electric absorption is not an actual
absorption of electricity, for if a condenser is in the interior

ELECTRiaTY AND MAGNETISM 117

of a hollow electric conductor there is no alteration in its
surface electrification by the "absorption" taking place in
the condenser. Or, as stated by Faraday : ' 'It is impossible
to charge matter with an absolute and independent charge
of one kind of electricity."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, arts. 53, 54.)

ELECTRIC INTENSITY INSIDE AN INCLOSED CONDUCTING SURFACE.

There is no electric intensity or lines of electric force on
the inside of a surface which is conducting and which com-
pletely incloses a portion of space, however highly this
surface is charged, if there is no charged body in the space.

(Thomson, Elements of Electricity and Magnetism, pp.
21, 22, 29. Also Ganot's Physics, art. 765.)

GATTSS' THEOREM.

"The total normal electric induction over any closed
surface drawn in the electric field is equal to 47r times the
total charge of electricity inside the closed surface."

(Thomson, Elements of Electricity and Magnetism, p.
14.)

COULOMB'S LAW.

The electric intensity of a point p close to the surface
of a conductor surrounded by air is at right angles to the
surface, It is equal to 4ir o- where a is the surface density of
the electrification. If the surface of the conductor is in
contact with a dielectric of specific inductive capacity K,
then the electric intensity at the point p is,

in
R = ^.-

(Thomson, Elements of Electricity and Magnetism, pp.
36, 122.)

118 LAWS OF PHYSICAL SCIENCE

ENERGY OF A SYSTEM OF CONDtTCTORS.

The potential energy of a system of charged electric
conductors placed in an electric field which arises from the
charges on the conductors of the system is equal to one-half
the sum of the products obtained by multiplying the charge
on each conductor by its potential, or,

E = y^SQV,
where Q is a charge and V the potential of that charge.

(Thomson, Elements of Electricity and Magnetism,
p. 37.)

ELECTRICITY AND AN INCOMPRESSIBLE FLUID COMPARED.

"The motions of electricity are like those of an incom-
pressible fluid, so that the total quantity within an imaginary
fixed closed-surface remains always the same."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 61.)

WORK DONE IN A DISPLACEMENT OF AN ELECTRIFIED SYSTEM.

The work done by the electric forces during the displace-
ment of an electrified system, when the potentials are main-
tained constant, is equal to the increment of the electric
energy. The work done, therefore, by a battery which
maintains the potentials constant is twice the work done by
the system during its displacement.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 93c.)

POTENTIAL ENERGY-CHANGES.

If any small displacement of a system of electrified con-
ductors takes place the diminution in the electric energy of
the system, when the charges are kept constant, is equal to
the increase in the potential energy when the same displace-
ment takes place and the potentials are kept constant.

(Thomson, Elements of Electricity and Magnetism,
p. 54.)

ELECTRICITY AND MAGNETISM 119

MECHANICAL FORCE AT THE SURFACE OF A CHARGED CONDUCTOR.

The surface of every charged conductor is subject to a
mechanical force which acts outward along the normal. The
value of the force per unit area of the surface of the con-
ductor for any dielectric surrounding the conductor, is,
F = % Eo-, and, if the dielectric surrounding the conductor
is air,

Here R is the electric intensity and a is the density of the
surface electrification. The maximum value of F in air at

10*
normal pressure and 15° C. is about -^ dynes per square

centimeter, which is a pressure of about 0.3 mm. of mercury.
(Thomson, Elements of Electricity and Magnetism, pp.
58, 59.)

PASCHEN'S LAW FOR SPARKING POTENTIALS IN A GAS.

The sparking potential between electrodes in a gas de-
pends on the length of the spark-gap and the pressure of
the gas in such a way that it is directly proportional to the
mass of gas between the two electrodes. Or we can consider
the sparking potential as a function of the pressure X the
density of the gas.

(Thomson, Conduction of Electricity Through Gases,
pp. 451^60.)

STATE OF ELECTRIC STRESS IN A MEDIUM.

When bodies in a dielectric medium are electrified, "at
every point of the medium there is a state of stress such
that there is tension along the lines of force and pressure
in all directions at right angles to these lines, the numerical
magnitude of the pressure being equal to that of the tension,
and both varying as the square of the resultant force at the
point. ' '

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 109.)

liW LAWS OF PHYSICAL SCIENCE

ELECTRIC DISPLACEMENT.

The whole quantity of electricity Q, which is displaced
through a unit area, when the electric intensity E is normal

rr

to the area, is Q'= "T^Rj where K is the specific inductive

capacity of the dielectric.

(Maxwell, Treatise on Electricity arid Magnetism, Vol.
I, art. 62. Compare with statement in Thomson, Elements
of Electricity and Magnetism, p. 122.)

ENERGY OF ELECTRIC FIELD.

The electric field represents a certain store of energy in
the medium. If E is the intensity of the field the stored

energy per unit volume is E = -5— 'B?, where K is the specific

inductive capacity of the medium.

(Thomson, Elements of Electricity and Magnetism, pp.
70-72, and 123.)

SPECIFIC INDUCTIVE CAPACITY.

The charge of electricity which a condenser will hold
when charged to a given potential is dependent upon the
nature of the dielectric. The ratio of the charge which the
condenser holds when a given dielectric is used, to the
charge it holds when air (or a perfect vacuum) is the
dielectric, is called the "specific inductive capacity" of the
dielectric. This ratio is either equal to or greater than
unity.

(See Thomson, Elements of Electricity and Magnetism,
Chap. IV. Also Ganot's Physics, art. 783.)

CONDUCTORS AND DIELECTRICS IN AH UNUNIFORM
ELECTRIC FIELD.

A conductor placed in an ununiform electric field tends
to move from the weak to the strong parts of the field ; so

ELECTRICITY AND MAGNETISM 121

likewise does a dielectric surrounded by one of smaller
specific inductive capacity.

(Thomson, Elements of Electricity and Magnetism,
p. 128.)

ELECTRIC INTENSITY INSIDE AND OUTSIDE CONDUCTORS
AND DIELECTRICS.

The electric intensity inside a conductor placed in an
electric field vanishes, and just inside a dielectric of greater
specific inductive capacity than the surrounding medium the
electric intensity is less than that just outside.

(Thomson, Elements of Electricity and Magnetism,
p. 128.)

EARNSHAWS THEOREM ON STABILITY.

If a charged body is placed in an electric field and is
altogether free to move, it is always in unstable equililbrium
in respect to translational motion.

(For proof, see Maxwell, Treatise on Electricity and
Magnetism, Vol. I, art. 116. Also Jeans, Electricity and
Magnetism, p. 165.)

CAPACITY OF CONDENSERS, JOINED IN PARALLEL AND IN SERIES.

The total capacity C of a number of condensers joined in
parallel is equal to the sum of the individual capacities
Ci, Cj, C3, etc., and the reciprocal of the total capacity of a
number of condensers, joined in series, is equal to the sum
of the reciprocals of the individual capacities. Thus for
parallel combination

C = Ci + C2 + C, + etc.
and for series combination

l = l + J-+l+etc.
C Ci .Cj C3

(Thomson, Elements of Electricity and Magnetism, pp.

110-113. Also C^anot's Physics, art. 799.)

122 LAWS OF PHYSICAL SCIENCE

UAGRETISM (DeflnlMons).

1. The magnetic moment of a magnet is, M ^ ml where m

is the strength of its pole and 1 the distance between its
poles.

2. The intensity of magnetization of a magnetizable sub-

stance, uniformly magnetized, is I = ^, where V is

its volume and M is its magnetic moment.
(Consult Jeans, Electricity and Magnetism, pp. 355,
358.)

3. A magnetic shell is a thin sheet of magnetizable substance

magnetized at each point in the direction of the nor-
mal to the sheet at that point.

4. The strength of a magnetic shell is the intensity I of its

magnetization times t its thickness. Thus,

1* = It = -^= magnetic moment per unit area.

5. The magnetic potential at a point due to a magnetic shell

of uniform strength is V = 6oi, where w is the solid
angle at the point subtended by the contour of the shell.
(Consult Jeans, Electricity and Magnetism, pp. 365,
366.)

FORCE ACTING BETWEEN MAGNETIC POLES.

The force between two magnetic poles in air is in the
straight line joining them, and is numerically equal to the
product of the strengths of the poles divided by the square
of the distance between them.

Thus, F = ^- ,

where m and m' are the strengths of the poles and r is the
distance between them. If m and m' are of unlike sign the
force is an attraction, if of like sign the force is a repulsion.
The law as here stated assumes that the strength of each
pole is measured in terms of a unit, the magnitude of which
is deduced from the terms of the law.

ELECTRICITY AND MAGNETISM 123

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, 'arts. 373, 374. For Gauss' proof of law, see Thomson,
Elements of Electricity and Magnetism, p. 206. Also
Ganot's Physics, arts. 716-720.)

POTENTIAL DUE TO A MAGKETIC SOLENOID.

A "Magnetic Solenoid" is a filament of magnetic matter
so magnetized that its strength is the same at every trans-
verse section. The Magnetic Potential due to a magnetic
solenoid depends only on its strength and the position of its
ends, called its poles.

Thus,V = m(J--J-),

where m is the strength of its poles and r^ and r^ are
distances from the positive and negative poles respectively
to the point where the potential is V.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 407.)

TOTAL CHARGE OF MAGNETISM.

The total charge of magnetism reckoned algebraically on
any magnet is zero.

(Thomson, Elements of Electricity and Magnetism, p.
190. Also Maxwell, Treatise on Electricity and Magnetism,
Vol. II, art. 377, Also Ganot's Physics, art. 697.)

124, LAWS OF PHYSICAL SCIENCE

MAGNETIC FORCE DUE TO A MAGNET.

If two points P and Q be taken equidistant from the
center of a bar-magnet, the point P being in the line of its
axis and the point Q in the equitorial plane at right angles
to the axis of the magnet, then the magnetic force at P is
tivice the magnetic force at Q. Thus, if OP is the distance to
P and OQ is the distance to Q from the center of the magnet,

„ 2M ,„ M
Hp =-^^5 and Hq =

DP- ' W

where M is the magnetic moment of the magnet.

(Thomson, Elements of Electricity and Magnetism, pp.
195-197.)

MAGNETIC FIELDS DUE TO A MAGNETIZED SPHERE AND A
SMALL MAGNET COMPARED.

A "uniformly magnetized sphere produces the same
effect outside the sphere as a very small magnet placed at
its center, the axis of the small magnet being parallel to
the direction of magnetization of the sphere, while the
moment of the magnet is equal to the intensity of magneti-
zation multiplied by the volume of the sphere."

(Thomson, Elements of Electricity and Magnetism, p.
224.)

MAGNETISM INDUCED BY A MAGNETIC FIELD.

All substances which are measurably diamagnetic or
paramagnetic when placed in a magnetic field become
charged with magnetism, and the diamagnetic substances
tend to move toward weaker portions of the field and the
paramagnetic substances toward stronger portions of the
field.

(Thomson, Elements of Electricity and Magnetism, p.
242. Also Ganot's Physics, arts. 700, 701.)

ELECTRICITY AND MAGNETISM 126

MAGNETIC IlfDTJCTIOH.

Magnetic induction is a vector quantity and is the sum
of two vectors, the magnetic force H and 4ir times the
magnetization I.

Thus, B = H + 4,rl.

Tubes of magnetic induction are always continuous and
closed tubes.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 400. Also Thomson, Elements of Electricity cmd
Magnetism,, pp. 244-247. Also Jeans, Electricity and
Magnetism, p. 373.)

SOME RELATIONS OF MAGNETIC QUANTITIES.

Let H = magnetic force,
B = magnetic induction,
I = magnetization,
/i. ^m.agnetic permeability and
k ^magnetic susceptibility, then
B = H + 4^1 = (1 +4^k)H=/iiH,

H= B - 4,rl = , ^. . ■ = —and

1 + 4irK M

/t = 1 + 4,i.k.

(Thomson, Elements of Electricity and Magnetism, p.
247. Also MaxweU, Treatise on Electricity and Magnetism,
Vol. II, art. 438.)

126 LAWS OF PHYSICAL SCIENCE

MAGNETIC HYSTERESIS.

Magnetic hysteresis is a phenomenon which results from
the transformation of magnetic energy into heat when the
magnetization of a substance is changed. It is a species of
molecular friction and is always exhibited when cyclical
reversals of magnetic flux in a magnetizable substance are
produced. It is analogous with friction in mechanics.

C. P. Steinmetz gives for the loss of energy due to
hysteresis in ergs per cycle per em',

W = vBi-6,
where B is the maximum magnetic induction per cm^ and i
is the "coefScient of hysteresis." Steinmetz gives ^ =
0.0025 as a fair average value for selected steel.

(Steinmetz, Alternating Current Phenomena, Chap. X.
See p. 116.)

MAGNETIC AND ELECTRIC ANALOGUES.

There is a complete analogy between the disturbance in
the distribution of an electric field produced by the presence
of uncharged dielectrics and the disturbance of a magnetic
field produced by paramagnetic or diamagnetic bodies in
which the magnetism is entirely induced. In this analogy
the magnetic force H is equivalent to the electric intensity
K and the magnetic permeability /* is equivalent to the
specific inductive capacity K.

(Thomson, Elements of Electricity and Magnetism, p.
258.)

ELECTROMOTIVE-FORCE SERIES.

Two unlike metals when immersed in an acid or salt
solution have acting between them an electromotive force.
The potential-difference between the two metals when im-
mersed in a given solution varies with the nature of the
metals.

Metals may be arranged in an electromotive-force series

ELECTRICITY AND MAGNETISM 127

in which the most electropositive metal begins and the most
electronegative metal ends the series.

{Ganot's Physics, art. 817. See also Smithsonian Phys-
ical Tables, p. 267.)

GALVANIC POLARIZATION.

When a chemical system, originally in equilibrium, is
electrolysed the decomposition caused by the electrolysis
produces a displacement of equilibrium, and it follows
necessarily that the current passed through the system has
to overcome an opposing electromotive force. The develop-
ment of this opposing electromotive force in a galvanic
cell is termed "polarization."

(Nemst, Theoretical Chemistry, p. 739. Also Walker,
Introduction to Physical Chemistry, pp. 374, 375.)

CONSERVATION OF ENERGY IN ELECTROLYSIS: THOMSON'S
THEOREM.

' ' The electromotive force of an electrochemical apparatus
is in absolute measure equal to the mechanical equivalent of
the chemical action on one electrochemical equivalent of
the substance."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, arts. 262, 263. For a scientific treatment of the principles
of electrochemistry and an analysis of the above statement,
consult Nemst, Theoretical Chemistry, Chap. VII, pp. 731
et seq.)

ELECTROLYSIS LITTLE AFFECTED BY PRESSURE.

If the products of an electrolysis are gases which obey
Boyle's law, the product of their pressure and volume will
be constant at a given temperature and the e.m.f. required
for electrolysis is nearly independent of the pressure.
Electrolysis of dilute acids, therefore, cannot be checked by
confining the gases of decomposition in a small space.

(Maxwell, Treatise on Electricity and Magnetism, art.
263.)

128 LAWS OF PHYSICAL SCIENCE

FARADAY'S FIRST LAW OF ELECTROLYSIS.

" The quantity of an electrolyte decomposed by the
passage of a current of electricity is directly proportional
to the quantity of electricity which passes through it. ' '

(Thomson, Elements of Electricity and Magnetism, p.
282. Also Ganot's Physics, art. 939.)

FARADAY'S SECOND LAW OF ELECTROLYSIS.

' ' If the same quantity of electricity passes through dif-
ferent electrolytes the weights of the different ions deposited
will be proportional to the chemical equivalents of the ions. ' '

(Thomson, Elements of Electricity and Magnetism, p.
282. For a full account and a bibliography, consult Chwol-
son. Traits de Physique, Vol. IV, Part 10, Chap. V. See p.
617 for statements of the two laws of electrolysis.)

A PRINCIPLE OF ELECTROLYTIC DECOMPOSITION.

An electrolytic decomposition can only proceed when
the loss of energy in the battery which supplies current to
an electrolytic cell is greater than the gain of energy in
the electrolyte of the electrolytic cell. An action contrary
to the above would violate the principle of the conservation
of energy.

(Thomson, Elements of Electricity and Magnetism,, p.
301.)

THE " FARADAY."

The Faraday is the name now commonly used to denote
the "quantity! of electricity associated with a chemical
equivalent in any electrochemical change."

The deposition of silver from a solution of silver-
nitrate has been most extensively investigated. G. W. Vinal
and S. J. Bates writing in the Bulletin of the Bureau of
Standards, Jan. 2, 1914, give as the value of the Faraday
for silver,

ELECTRICITY AND MAGNETISM 129

OOOinSOO "" ^^^^^ international Coulombs.
(Nemst, Theoretical Chemistty, pp. 727, 728.)

OHH'S LAW.

The current which flows in a metallic conductor or in an
electrolyte is proportional to the difference of potential at
the extremities of the conductor. Thus, if E = e.m.f., R =

resistance and I = current, I = -5-. The quantity E has
been experimentally proved to be independent of the
strength of the current (the temperature being the same)
to at least 1 part in 100,000.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, "Vol. I, art. 241. Also Thomson, Elements of Elec-
tricity and Magnetism, p. 284. Also Northrup, Methods
of Measuring Electrical Resistance, art. 105, p. 16.)

RESISTANCES IIT SERIES.

When several resistances are joined in series the total
resistance is equal to the sum of the individual resistances.

From coils which have resistance-values 1-1-4-3 or
1-3-3-2 (or any multiple of these values) the successive
values to 9 can be obtained by moving a single plug-
connector.

(Northrup, Methods of Measuring Electrical Resistance,
art. 503, p. 82.)

130 LAWS OF PHYSICAL SCIENCE

RESISTANCE OF A WtTMBER OF CONDUCTORS ARRANGED
IN PARALLEL.

When a number of resistances, Rj, Rj, R3, etc., are joined
together in parallel combination the reciprocal of the total
resistance R is equal to the sum of the reciprocals of the
individual resistances.

Thus, ^ = 4- + -4- + 4-+

Or, calling the reciprocal of any resistance a cotidiictance,
the total conductance = the sum of the conductances of
individual conductors when joined in parallel combination.

If n resistance-units are used singly, and joined in all
possible series combinations, 2" - 1 resistance-combinations
can be obtained. The total number N of combinations
possible for n units used singly and joined in series and
parallel combinations is,

N = 2n+i - (n-f 2).

{Ganot's Physics, art. 853. Also Northrup, Methods of
Measuritig Electrical Resistance, art. 501, p. 80.)

CONDITION FOR A DEFINITE RESISTANCE.

The conductor must be considered as having its surface
divided into three portions:

1. a portion over which the potential is maintained constant,

2. a portion over which the potential is held constant but

higher or lower than the first and

3. a remaining portion, which is impervious to electricity.
Only when the conductor is in approximately the above

condition can its resistance be said to be definite.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
T, art. 306.)

RESISTANCE OF A CONDUCTOR CHANGES WHEN SPECIFIC
RESISTANCE CHANGES.

If the specific resistance of any portion of a conductor
is increased, that of the remainder being unchanged, the

ELECTRICITY AND MAGNETISM 131

resistance of the whole conductor will be increased, and if
the ^specific resistance of any portion of it is decreased, that
of the remainder being unchanged, the resistance of the
whole conductor will be decreased.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 306.)

FOUR-TERMINAL CONDUCTORS.

"The generalized four- terminal conductor is a mass of
conducting material of any size or shape and has four
limited portions of the surface arbitrarily selected and
adapted for making electrical connection to other con-
ductors. ' '

This definition (given by Frank Wenner, Bulletin of the
Bureau of Standards, Vol. 8, p. 560) covers those standards
of resistance having two current and two potential ter-
minals, which are used in fall of potential methods and with
the Kelvin-double bridge, in the measurement of resistance.

EIRCHHOFF'S THEOREM ON INTERCHANGE OF ELECTRODES.

' ' In any conductor or system of conductors having four
terminals, 1, 2, 3, and 4 selected in any way, the drop in
potential from 1 to 2 caused by a current entering at 3
and leaving at 4, is equal to the drop in potential from 3
to 4 caused by an equal current entering at 1 and leaving
at 2."

This theorem, given by Kirchhoff in 1847, is of impor-
tance in connection with the measurement of resistances
by fall of potential methods.

(See Bulletin of the Bureau of Standards, Vol. VIII,
p. 563.)

132 LAWS OF PHYSICAL SCIENCE

JOULE'S LAW OF GENERATION OF HEAT IN A CONDUCTOR.

The heat produced by the passage of an electric current
through a solid metallic conductor is proportional to the
product of the resistance of the conductor, the square of
the current and the time, or to the product of the applied
e.m.f. the current and the time.

Thus, JH = RPt = EIt,
where J is Joule's dynamical equivalent of heat, H the
number of units of heat, R the resistance of the conductor,
I the current, t the time during which the current flows and
E the applied e.m.f. When practical units are used,
H = 0.24 Rl^t in gram-calories, approximately.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. I, art. 242. Also " High Temperature InvestigSr
tion and a Study of Metallic Conduction," by E'. F.
Northrup, Jour, of the Franklin Institute, June, 1915, pp.
650-652.)

CONVERSION OF MECHANICAL ENERGY INTO HEAT
IN A CONDUCTOR.

The mechanical work done by electromotive force in
driving electricity through a solid metallic conductor is
entirely converted into heat. If, however, the metallic con-
ductor is liquid (molten) some power is spent in circulating
the liquid metal.

(Consult paper by E. F. Northrup, "A New Type of
Ammeter, etc." Proc. of the American Electrochemical
Society, May 7, 1909, pp. 303-329.)

MINIMUM HEAT CONDITION.

"In any system of conductors in which there are no
internal electromotive forces the heat generated by currents
distributed in accordance with Ohm's law is less than if the
currents had been distributed in any other manner con-

ELECTRICITY AND MAGNETISM 133

sistent with the actual conditions of supply and oiitflow of
the current."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 284. Also Thomson, Elements of Electricity and
Magnetism, pp. 310-313.)

THE WIEDEMANN-FRANZ RATIO.

The best conductors of electricity are also the best con-
ductors of heat. The ratio of the thermal to the electrical
conductivity for all good conductors at the same temperature
has the same value ; the value of this ratio is proportional ,
to the absolute temperature. This law holds well for pure
metals, less accurately for alloys and not at aU. for poor
conductors.

(Campbell, Modem Electrical Theory, pp. 66-68. For
experimental illustrations see Richardson, The Electron
Theory of Matter, pp. 410-413.)

STJPERCONDUCTIVITY.

When the temperature of a very pure metal is reduced
to near the absolute zero of temperature (to less than 4°
K.) its electrical resistance vanishes or its conductivity
becomes practically infinite.

This is an experimental result of the researches of
Kamerlingh Onnes and has been termed "Superconduc-
tivity."

(Consult, "Electrical Conductivity at High Tempera-
tures and Its Measurement," by E. F. Northrup, Trans, of
the Amer. Electrochem. Sac, Vol. XXV, 1914, p. 377. For
data see Smithsonian Physical Tables, p. 280.)

134 LAWS OF PHYSICAL SCIENCE

RESISTANCE-TEMPERATURE RELATIONS FOR METALS.

The variation in electrical resistance per unit of resis-
tance per degree is called the "resistance-temperature
coefficient" of a substance. If B.^ is the ohmic resistance of
a sample of the substance when at temperature t then its
resistance-temperature coefficient at this temperature is

1 dRt
"« ~ R. dt ■

A few general relations between resistance and tempera-
ture for several pure metals and some alloys have been
obtained experimentally. Thus:

1. All pure metals when in the solid state increase in resist-

ance when the temperature is increased and the
coefficient a^ is approximately the same for all pure
metals and has the same order of magnitude as the
coefficients of expansion of gases.
(Northrup, Jour, of the Franklin Institute, June, 1915,
p. 636 et seq.)

2. Several pure metals (Ag, Au, Cu, Pb, Al, Sn, Sb, Bi)

when in the molten state increases in resistance linearly
with the temperature, namely, the coefficient u.^ is
strictly a constant over a considerable range of tem-
perature. The same is true for some alloys ; Sn + Bi
in particular.
(Northrup, Jour, of the Franklin Institute, June, 1915,
Figs. 1 and 2, pp. 638, 639.)

3. If we call/?t=v~jf^the coefficient of volume-expansion of

a molten metal, then the ratio -^ is nearly the same

quantity (lying within the limits 3.48 and 5.19) for at
least six pure metals (Na, K, Sn, Hg, Pb, Bi).
(Northrup, Trans, of the Amer. Electrochem. Sac, Vol.
XXV, 1914, p. 388)

ELECTRICITY AND MAGNETISM 135

4. The resistance of most metals (Na, Al, K, Cu, Zn, Cd, Sn,

Sb, Bi, Au, Hg, Pb, have been studied) upon changing
from the solid to the molten state approximately
doubles. Antimony and bismuth are exceptions to the
general rule, the resistance of these decreasing when
fusion occurs.
(See reference under 3)

5. The change in the resistivity per degree C. of a sample of

copper is 0.00681 microhm per centimeter cube, or the
conductivity of copper is strictly proportional, over
ordinary ranges of temperature (as 10° to 100°G.),
to its resistance-temperature coefficient. Thus either
may be deduced from knowledge of the other.
(J. H. Bellinger, " The Temperature Coefficient of

Eesistance of Copper, ' ' Bulleiin of the Bureau of Standards,

Vol. 7, pp. 83, 84.)

6. When the resistivity is given, at any temperature, for

pure tin and pure bismuth — both being in the molten
state — then the resistivity of any molten alloy of
known proportions of these two metals may be calcu-
lated, because the resistivity of the alloy bears a
strictly linear relation to the percentage of gram-atoms
in which either constituent is present.
(Northrup and E. G. Sherwood, "New Methods for
Measuring Kesistivity of Molten Materials ; Results for Cer-
tain Alloys," Jour, of the Franklin Institute, Aug., 1916.)

7. At temperatures which exceed about 1452° C, the melt-

ing-point of nickel, all known substances in either the
solid or liquid state are more or less electrically con-
ducting and above this temperature it is impossible by
any means to obtain, even approximately, good elec-
trical insulation.
(Northrup, Jour, of the Franklin Institute, March, 1915,
p. 352.)

136 LAWS OF PHYSICAL SCIENCE

A GEMERAL RELATION BETWEEN CONDUCTANCE AND CAPACITY.

Let two perfect conductors which serve as electrodes be
immersed in an electrically conducting homogeneous medium
of electrical conductivity o- and call G the electrical con-
ductance between the two electrodes. Then substitute for
the electrically conducting medium a dielectric medium of
specific inductive capacity K and call C the electrostatic
capacity between the same two electrodes. Under these
circumstances the general relation holds

C K ■

(Northrup, " Use of Analogy in Viewing Physical Phe-
nomena," Jour, of the Franklin Institute, July, 1908, pp.
31, 32. Also consult Jeans, Electricity and Magnetism, pp.
339, 340.)

A RELATION BETWEEN CAPACITY AND RESISTANCE.

In every portion of an electric circuit where electrostatic
lines and lines of current-flow pass through the same
medium, the product of the resistance and the capacity,

however dimensions are varied, is constant and equal to ^.

47r

Here p is the resistivity of the medium and K its specific
inductive capacity. This proposition is of importance in the
electrical measurement of high resistances.

(Northrup, Methods of Measuring Electrical Resistance,
art. 902, pp. 186-189.)

ELECTROMOTIVE FORCES IN SERIES.

Electromotive forces add in series as scalar quantities
but obey the law of algebraic signs.

Thus, E = ei -t- 62 -f ea -t- ( -Cj) + ( -es) + ee H .

(Consult Jeans, Electricity and Magnetism, pp. 298,
299.)

ELECTRICITY AND MAGNETISM 187

CONTACT ELECTRICITY.

When two (Jifferent metals are in contact there is in
general an e.m.f. acting from one to the other. Let C be
taken as a standard metal : then if the potential of a metal
I in contact with C at zero potential is i and that of a metal
Z in contact with C at zero potential is z, the potentijil of
Z in contact with I at zero potential is z - i.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, arts. 246-249.)

LAW OF VOLTA.

When any number of conductors, which conduct elec-
tricity without electrolytic dissociation, are joined together
to form a closed chain, and all are at the same temperature,
the total electromotive force, or sum of the contact-differ-
ences of potential at the surfaces of union of pairs of
elements, is zero.

Thus, for the three elements a, b, c, which form a closed
chain,

Va.+ V,„+V,,= 0,

where V^^ is the contact-difference of potential between the,
pair of elements a, b, and V^„ V^^, have similar meanings.

This result is known as Volta's Law.

(Consult Jeans, Electricity and Magnetism, p. 298. For
a very full treatment, see Chwolson, Traite de Physique,
Vol. IV, Part 10, pp. 198-200.)

138 LAWS OF PHYSICAL SCIENCE

KIRCHHOFF'S LAWS.

1. The algebraic sum of the currents which meet at any

point is zero.

2. In any closed circuit the algebraic sum of the products

of the current and resistance in each of the conductors

in the circuit is equal to the electromotive force in the

circuit.

(Thomson, Elements of Electricity and Magnetism, pp.

304, 305. For an illustration of the application of Kirch-

hoff 's laws, see Northrup, Methods of Measuring Electrical

Resistance, art. 301, pp. 44, 45.)

STEIHMETZ'S EXTENSION OF KIRCHHOFF'S LAWS TO
ALTERNATING CURRENTS.

(a) "The sum of all the e.m.fs acting in a closed circuit
equals zero, if they are expressed by complex quan-
tities, and if the resistance and reactance e.m.f.s are
also considered as counter e.m.f.s. ' '
(&) "The sum of all the currents flowing towards a dis-
tributing point is zero, if the currents are expressed
as complex quantities."
(Steinmetz, Alternating Current Phenomena, art. 31,
p. 40.)

RESOLVED ELECTROMOTIVE FORCE AND CURRENT.

(a) "The sum of the components, in any direction, of all
e.m.f.s in a closed circuit, equals zero, if the resist-
ance and reactance are considered as counter
e.m.f.s."

(6) "The sum of the components, in any direction, of all
the currents flowing to a distributing point equals
zero."
(Joule's law and the energy-equation do not give a

simple expression in complex quantities because power is a

quantity of double the frequency of the current or e.m.f.

wave.)

ELECTRICITY AND MAGNETISM 139

(Steinmetz, Alternating Current Phenomena, art. 31,
p. 41.)

STEINMETZ'S EXTENSION OF OHM'S LAW TO ALTERNATING
CURRENT.

E E

E — ZI, I = 2 I Z — J ,

where E, I and Z are e.m.f., current and impedance, ex-
pressed in complex quantities.

(For a full explanation of these symbols and their
relations, consult Steinmetz, Alternating Current Phe-
nomena, Chap. V. See art. 30, p. 40.)

WORK DONE BY ELECTROMOTIVE FORCE.

The work done by an e.m.f. is measured by the product
of the e.m.f. into the quantity of electricity which crosses
a section of the conductor under the action of the e.m.f.
This work is the same as the work done by an ordinary force
and both are measured by the same standards or units.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 570.)

POWER IN A CIRCUIT WHEN THE CURRENT IS ALTERNATING.

The average power in a circuit when the current is
alternating (sine-waves assumed) is,

P = EI cos <p,
where E is the effective value of the electromotive force, I
the effective value of the current and ^ the phase angle
between the current and the impressed electromotive force.

The instantaneous power in a circuit is,
p = ei,
where e and i are the instantaneous values of the electro-
motive force and the current respectively.

(Steinmetz, Alternating Current Phenomena, art. 5, p. 6.
Also consult Christie, Electrical Engineering, art. 80, pp.
113-118.)

140 LAWS OF PHYSICAL SCIENCE

LAW OF ACTION OF AN ELECTRIC CURRENT ON A MAGNET.

When current passes through a straight wire, a straight
magnetized needle held above the wire tends to place itself
at right angles to the wire and to the perpendicular let
fall from the center of the magnetized needle to the wire.
The direction of the current in the wire and the direction
of the lines of magnetic force which encircle the wire are
related as are the forward thrust and rotation of a right-
hajided screw.

(Consult Ganot's Physics, art. 835.)

ELECTRIC CIRCUIT AND A MAGNETIC SHELL COMPARED.

"A current flowing in any closed circuit produces the
same magnetic field as a certain magnetic shell known as
the 'equivalent magnetic shell.' This shell may be taken
to be any shell having the circuit for its boundary, its
strength being uniform and proportional to that of the
current. ' '

(See Jeans, Electricity and Magnetism, p. 415.)

AMPERE'S LAW FOR THE MAGNETIC FIELD DUE TO ANY
CLOSED LINEAR CIRCUIT.

At any point P, not in the wire of a closed circuit carry-
ing an electric current, the magnetic force due to the current
can be derived from a potential fl where Q ^a, constant X
the current X the solid angle subtended by the circuit at P.
When electromagnetic measure is used the constant is unity,
and Q = i oj, i being the current flowing round the circuit
and <u the solid angle subtended by the circuit at the point P.

(Thomson, Elements of Electricity and Magnetism, p.
325.)

FORCE ON A UNIT POLE EXTERIOR TO A LINEAR CONDUCTOR.

The force exerted by a linear conductor, carrying a
current I on a unit magnetic pole exterior to the con-
ductor, is perpendicular to the plane through the axis of
the conductor and the pole.

ELECTRICITY AND MAGNETISM 141

When electromagnetic units are used this force is equal
to twice the intensity of the current divided by the perpen-
dicular distance r from the pole to the axis of the conductor.

Thus, T„ = ^ ■
r

Here, T„ is the force in dynes which would act on a unit
magnetic pole, or it is the intensity of the magnetic field,
at a distance r from the axis of the conductor.

(Consult "Some Newly Observed Manifestations of
Forces in the Interior of an Electric Conductor," by E. F.
Northrup, Physical Review, June, 1907, p. 478. Also Max-
well, Treatise on Electricity and Magnetism, arts. 4:77-^19.)

MAGNETIC FORCE IN THE INTERIOR OF A CONDUCTOR OF
CIRCULAR CROSS-SECTION.

The lines of magnetic force form circles about the axis
of a linear conductor of circular cross-section. They vanish
at the axis and are a maximum at the circumference of the
conductor and increase uniformly with the distance from
axis to circumference.

If electromagnetic units are used, the intensity of the
magnetic field in the substance of the conductor is,

where I is the current in the conductor,
R the radius of the conductor and
r the distance from the axis to where the intensity of
the magnetic force is T,.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. II, art. 683.)

142 LAWS OF PHYSICAL SCIENCE

A SMALL ELECTRIC CIRCUIT COMPARED WITH A MAGNET.

"The magnetic action of a small plane-circuit at dis-
tances which are great compared with the dimensions of
the circuit is the same as that of a magnet whose axis is
normal to the plane of the circuit, and whose magnetic
moment is equal to the area of the circuit multiplied by
the strength of the current."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 482.)

MAGNETIC POLE AND EQUIPOTENTIAL SURFACES.

"The force acting on a magnetic pole placed at any
point of an equipotential surface is perpendicular to this
surface, and varies inversely as the distance between con-
secutive surfaces."

(See Maxwell, Treatise on Electricity and Magnetism,
Vol. II, art. 487.)

LAW OF THE INTERNAL PRESSURE PRODUCED BY AN ELECTRIC
CURRENT IN A CONDUCTOR.

In every conductor which carries an electric current, a
pressure is produced within the substance of the conductor
which results from the mutual attractions of all the current-
carrying elements of the conductor^

When the conductor has a circular cross-section, the
pressure is directed toward its axis. Its value per unit
area is a maximum at the axis and decreases to zero at the
circumference. When the radius of the conductor is R, and '
the current it carries is I, the pressure g per unit area at
any radial distance r from the axis is,

g=^.(R-r^).

When I is in electromagnetic measure and R and r are in
centimeters g is in dynes per cm".

(See "Some Newly Observed Manifestations of Forces

ELECTRICITY AND MAGNETISM 143

in the Interior of an Electric Conductor," by E. F.
Northrup, Physical Review, June, 1907.)

LONGITUDINAL MOTION IN AN ELECTRICAL CONDUCTOR.

When by any geometrical disposition whatever of an
electric circuit, in which the conducting material is a fluid
capable of free motion, normally acting electrodynamie
forces arise in any section of the circuit which vary in
magnitude from one point to another over a length meas-
ured along the axis of the conductor, there also arise
hydrodynamic forces which can impress motions on the
fluid substantially parallel to the longitudinal axis of the
conductor.

(This generalization is drawn from investigations by
E. F. Northrup which are not yet published.)

WORK DONE IN MOVING A MAGNETIC POLE ROUND
A CLOSED CURVE.

If there exists a magnetic field due to electric currents
and a closed curve is drawn in this field, the work done in
moving a magnetic pole of strength m round the closed
curve is zero if the closed curve does not thread an electric
circuit, and 4irm times the current in any circuit which the
closed curve threads once.

Thus, W = 47rlm is the work done when the closed curve
threads once a circuit which carries a current I, and W^ =
47rlmn when the closed curve threads the circuit n times.
The value of the line-integral 47rl is independent of the
medium in which the closed curve is drawn.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, arts. 480, 498, 499.)

144 LAWS OF PHYSICAL SCIENCE

FUNDAMENTAL LINE-INTEGRALS.

The two propositions:

1. The line-integral of the magnetic force round any

closed curve is equal to 4ir times the total current
flowing through the closed curve and

2. The line-integral of the electric force round any closed

curve is equal to the time-rate of diminution of the
total magnetic induction included by the closed curve,
are basic in many mathematical investigations of
alternating currents.
(For application of these principles, see "The Skin
Effects and Alternating Current Resistance," by B. F.
Northrup and John R. Carson, Jour, of the Franklin Insti-
tute, Feb., 1914, p. 141 et seq.)

VECTORIAL ADDITION OF MAGNETIC AND ELECTRIC FORCES.

Magnetic and electric forces or intensities in a homo-
geneous medium must add vectoriaUy, giving a resultant
intensity.

By combining this principle with certain elementary
laws of electrostatics and magnetism, a large number of
theorems have been demonstrated.

(Consult Jeans, Electricity and Magnetism, pp. 26,
358.)

INTERACTION OF MAGNETS AND ELECTRIC CURRENTS.

The mechanical action of currents on magnets is equal
and opposite to the action of magnets on currents.

It is shown by theory and experiments that a bar-
magnet magnetized to have a North pole at each end and a
South pole at its center, when freely suspended in a hori-
zontal position in a vertical, conducting, liquid column of
circular cross-section will rotate with continuous rotation
when the liquid carries an electric current. Reversing the
direction of the current reverses the direction of rotation.

ELECTRICITY AND MAGNETISM 145

(Ganot's Physics, arts. 888, 889. Also see article by
E. F. Northrup, Physical Review, June, 1907, p. 480.)

ELECTRIC AND MAGNETIC ANALOGIES.

Electric system,.

1. The line-integral of the electric force round any closed

curve passing through the battery is B, while round
any other closed curve it vanishes.

2. The lines of flow of electric current are closed curves

which pass through the battery.

3. The density of the electric current is o- times the electric

force, where o- is the conductivity of the medium
carrying the current.
Magnetic system.

1. The line-integral of the magnetic force round any closed

curve which threads a magnetizing circuit is 47rl, while
round any other closed curve it vanishes.

2. The lines of magnetic induction are closed curves which

thread the magnetizing circuit.

3. The magnetic induction is /* times the magnetic force,

where /* is the magnetic permeability.
(For amplification and applications of these analogies,
see Thomson, Elements of Electricity and Magnetism, pp.
348-350.)

LAW OF MAGNETIC INDUCTION. (l)

"When the number of lines of magnetic induction which
pass through the secondary circuit in the positive direction
is altered, an electromotive force acts round the circuit,
which is measured by the rate of decrease of the magnetic
induction through the circuit"— provided the integrity of
the original circuit is preserved.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
11, art. 531).

X46 LAWS OF PHYSICAL SCIENCE

LAW OF MAGNETIC INDUCTION. U)

Another, and to many, a preferable statement of this law
is : "Whenever a real or imaginary line in space is being cut
at right angles to itself by tubes of magnetic induction an
e.m.f . acts along this line which is proportional to the rate
of cutting.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. II, art. 541.)

LEIfZ'S LAW.

When a circuit is moved in a magnetic field in such
a way that a change takes place in the number of tubes of
magnetic induction passing through the circuit, a current
is induced in the circuit and a mechanical force is set up
such that this force tends to stop the motion which gave
rise to the current.

(Thomson, Elements of Electricity and Magnetism, p.
440. Also Maxwell, Treatise on Electricity and Magnetism,
Vol. II, art. 542.)

LINE OF MAGNETIC INDUCTION DEFINED.

Maxwell defines, negatively, a line of magnetic induction
in four ways:

1. If a conductor be moved along it parallel to itself it will

experience no electromotive force.

2. If a conductor carrying a current be free to move along

a line of magnetic induction it will experience no
tendency to do so.

3. If a linear conductor coincides in direction with a line

of magnetic induction and be moved parallel to itself
in any direction, it will experience no electromotive
force in the direction of its length.

4. If a linear conductor carrying an electric current co-

incide in direction with a line of magnetic induction
it will not experience any mechanical force.

ELECTRICITY AND MAGNETISM 147

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 597.)

ELECTROMOTIVE FORCE INDEPENDENT OF THE NATURE
OF THE CONDUCTOR.

The intensity of an electromotive force which results
from electromagnetic induction is entirely independent of
the nature of the substance of the conductor in which the
electromotive force acts, and also of the nature of the
conductor which carries the inducing current.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 534.)

MECHANICAL FORCE ACTS ON THE CONDUCTOR, NOT ON THE
CURRENT.

The mechanical force which urges a conductor carrying
a current across lines of magnetic force, acts, not on the
electric current but on the conductor which carries it.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 501.)

MECHANICAL FORCE ACTING ON A CONDUCTOR REPRESENTED
BY A PARALLELOGRAM.

The mechanical force which acts upon unit length of a
conductor carrying a current is numerically equal to the
area of a parallelogram, two sides of which are drawn
parallel to the conductor and proportional to the strength
of the current at any point, the other two sides being drawn
parallel and proportional to the magnetic induction at the
same point. The mechanical force is normal to the plane
of the parallelogram so drawn. If a right-handed screw
be turned from the direction of the current to the direc-
tion of the induction the direction in which the mechanical
force acts coincides with the direction of forward motion
of the screw.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 500.)

148 LAWS OF PHYSICAL SCIENCE

MAGNETIC ENERGY IS POTENTIAL ENERGY.

The energy of any strictly magnetic systcHi may be
considered as potential energy and if so considered this
energy is always diminished when the parts of the system
yield to the magnetic forces which act on them.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 638.)

MUTUAL ACTION BETWEEN TWO CIRCUITS IS DEPENDENT UPON
A SINGLE QUANTITY.

All phenomena of the mutual action of two circuits,
whether the induction of currents or the mechanical force
which acts between them, depend upon the value of the
coefficient of mutual induction between the circuits, a
coefficient which depends only upon the geometrical rela-
tions of the circuits.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 584. Very exact formula for calculating coefficients
of self induction and mutual induction are to be found in a
series of papers by Dr. E. B. Rosa in volumes 2, 3, and 4
of the Bulletin of the Bureau of Standards.)

VECTOR RELATIONS OF VELOCITY, INDUCTION AND
ELECTROMOTIVE FORCE.

"The magnitude of the electromotive force is repre-
sented by the area of the parallelogram, whose sides rep-
resent the velocity and the magnetic induction, and its
direction is the normal to this parallelogram, drawn so that
the velocity, the magnetic induction and the electromotive
force are in right-handed cyclical order."

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 599. Also see art. 594, where is shown how a volume
may be made to represent the increment in the electro-
kinetic momentum of a secondary circuit.)

ELECTRICITY AND MAGNETISM US

INDUCTION COEFFICIENTS.

In every electric circuit there is a certain quantity L,
called the coefficient of- self induction of the circuit. The
magnitude of this quantity is dependent only upon the
geometrical dimensions of the circuit. If the currents are
constant, changes in L always give rise to e.m.f .s, and if L
is constajit changes in the currents give rise to e.m.f.s. If
two or more circuits are in the neighborhood of each other,
there is for each of the circuits a self-induction coefficient
and in addition other quantities M, M^, Mj, etc., called
coefficients of mutvMl induction which depend only upon
the geometrical relations of the circuits. If there is a
steady current in one of the circuits, and there are changes
in L, Li, Lj, etc., or in M, M^, Mj, etc., e.m.f.s are always
produced. Or if these quantities are constant, changes in
the current produce e.m.f.s.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. II, arts. 540, 578, 579.)

COIL TO GIVE MAXIMUM SELF INDUCTION.

When the weight or length of a wire is given, the form
in which to wind this wire in a channel of square cross-
section, in order to form a coil of maximum self induction,
is obtained by making the mean radius r of the coil equal
1.85 times a side of the section of the channel.

The self ruduction in henrys is then,
L = 19.347m2 lO-^,

where n is the number of turns of wire.

(MaxweU, Treatise on Electricity and Magnetism, Vol.
II, art. 706. Also Bulletin of the Bureau of Standards,
Vol. 2, p. 108.)

ISO LAWS OF PHYSICAL SCIENCE

EZPRESSIOH FOR KINETIC ENERGY OF TWO CIRCUITS.

Tlie kinetic energy of a system formed of two circuits
when currents' in the first circuit induce currents in the
second circuit is given by the expression,

T = HIjL + IiIi,M+i^l2N,

where L is the coefSeient of self induction of first circuit
and N of the second circuit, and M is the coefficient of
mutual induction between the two circuits, and Ii and 1^
are currents in the first and second circuits respectively.
(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 581.)

MAGNETIC ENERGY COMPARED WITH ELECTROKINETIC ENERGY.

It is always possible to make an arrangement of infi-
nitely small electric circuits which shall correspond in all
respects to any magnetic system, provided that in calcu-
lating the potential we avoid passing through any of these
small electric circuits with a line of integration.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 637.)

VECTOR-POTENTIAL.

The vector-potential is a quantity which represents in
direction and magnitude, the time-integral of the intensity
of the electromotive force which a particle placed at a
point in a magnetic field would experience if the current
to which the magnetic field is due were suddenly stopped.
It is identical with the electrokinetic momentum at the
point.

All lines of magnetic induction through a closed curve
in being removed are shown by Maxwell to equal the line-;
integral of the resolved part of the vector-potential taken
round the curve. This line-integral, physically interpreted,
is the total electrokinetic momentum of the closed curve
or circuit, and when the line-integral is extended round a

ELECTRICITY AND MAGNETISM 151

primary circuit it is numerically equal to the product of
the self induction of the circuit by the current in the circuit.
It is the analogue of mass X velocity, or momentum in
mechanics.

(Maxwell, Treatise on Electricity and Mdgnetism, Vol.
II, arts. 405, 590, 592.)

ELECTROMOTIVE FORCE IMPRESSED OK A CIRCUIT.

If an e.m.f. is momentarily impressed on any electric
circuit three cases may exist:

1. The movement of electricity is retarded by an ohmic

resistance only.

2. It is retarded by ohmic resistance and opposed by

magnetic inertia.

3. It is retarded by ohmic resistance and opposed by

magnetic inertia and by a counter e.m.f. which varies
as a function of the impressed e.m.f.
In the 3rd case when the damping by ohmic resistance
is negligible the electricity always tends to oscillate upon
the sudden removal of the impressed e.m.f. and the period
is T = 27r V LC, where L is the self induction and C the
capacity of the circuit.

(Consult Bedell and Crehore, Alternating Currents,
Part 1.)

AN ELECTROMOTIVE FORCE ACTS ONLY ON ELECTRICITY.

An electromotive force has of itself no tendency to
cause the mechanical motion of any body, but acts only to
cause a movement of electricity.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, art. 535.)

162 LAWS OF PHYSICAL SCIENCE

DECAY OF INTERNAL CHARGES IN DIELECTRICS.

If in the interior of a mass of homogeneous poorly con-
ducting material, there exists at any point an electric charge
it will tend to die away. Neither its formation nor the rate
at which it dies away is influenced by the application of
external e.m".f .s which do not lead to disruptive discharges.

(Maxwell, Treatise on Electricity and Magnetism, Vol.

I, art. 325.) •

GENERAL PRINCIPLE OF MECHANICAL ACTION OF CURRENTS.

AU mechanical actions of electric currents depend upon
the strength of the currents and not upon their rate of
variation and all mechanical actions of currents remain the
same when all the currents are reversed in direction.

(Maxwell, Treatise on Electricity and Magnetism, Vol.

II, art. 574.)

MUTUAL RELATIONS OF CURRENTS.

1. If a current traverses a wire and returns by a tube

surrounding the wire there is no magnetic field ex-
ternal to the tube.

2. The external action of a crooked wire (bent like a row

of saw-teeth) upon a neighboring wire is the same as
that of a straight wire.

3. If a wire carries a current no external magnetic force

can so act upon the wire as to tend to make it move
in the direction of its length.

4. The force acting between two elements of two electric

circuits is inversely proportional to the square of the
distance between them.
(Maxwell, Treatise on Electricity and Magnetism, Vol.
II, arts. 505-508.)

INTERACTION OF ELECTRIC CONDUCTORS.

1. Two conductors which are parallel and carry currents
in the same direction attract one another, and when

ELECTRICITY AND MAGNETISM 163

they carry currents in opposite directions they repel
one another.
2. Two rectilinear current-carrying conductors, when their
directions are such that they form an angle with each
other, attract one another if the currents in both con-
ductors approach or recede from the apex of the angle,
and they repel one another if one current approaches
and the other recedes from the apex of the angle.
{Ganot's Physics, arts. 880-882.)

LAW OF RESOLUTION OF CURRENTS.

The law of the resolution of electric currents is the'
same as that of velocities, forces and all other vectors.

(Maxwell, Treatise on Electricity and Magnetism, Vol.
I, art. 286.)

MAGNETOMOTIVE FORCE, AND MAGNETIC INDUCTION
IN A SOLENOID.

In a solenoid of infinite length having n turns per unit
length, the magnetic force is uniform over the cross-section
of the solenoid and its value is H = 4jrni, where i is the
current in electromagnetic measure. Or, if I is the current
in amperes the magnetomotive force is

H = 1.2566nl gilberts.

The induction per cm^ in the solenoid is,

B := 1.2566nl/t gausses, where /* is the permeability

of the medium within the solenoid.

(Consult Christie, Electrical Engineering, p. 58 et
seq.)

154 LAWS OF PHYSICAL SCIENCE

A FUNDAMENTAL ENGINEERING EQUATION.

The following equation is of fundamental importance in
electrical engineering :

Ee„= y/~2-^n ¥>N10-' = 4.44n *.NK)-' volts.
Here, E^,, = effective e.m.f., n = total number of turns in
the circuit, ^ = the total maximum flux through the
circuit and N = the frequency or number of complete
cycles per second of the meignetizing current, sine waves
being assumed.

(Steinmetz, Aliernating Current Phenomena, Chap. III.
See p. 17.)

THERMOELECTRIC CURRENTS.

If an electric circuit is made up of two unlike metals and
one junction of the two metals is maintained at a higher
temperature than the other junction, an electric current wiU
flow in the circuit, and as the result of this current heat will
be transferred from the hotter toward the colder junction.

(Thomson, Elem&nts of Electricity and Magnetism, p.
501. Also Chwolson, Traite de Physique, Vol. IV, Part 10,
p. 737.)

THERMOELECTRIC LAW, THERMOELECTRIC POWER.

If Ci is the e.m.f. of a bimetallic circuit when the cold
junction is at temperature t„ and the hot Junction at tj and
if Cj is the e.m.f. when the cold junction is at tj and the hot
junction at tj, then, when the cold junction is at t^ and the
hot junction is at t^ the e.m.f. is Ci + e^. Or, the e.m.f. E
round a circuit whose junctions are at temperatures tj and

tjj is E =:J t^ Qdt, where Qdt is the e.m.f. round the circuit
when the temperature of the cold junction is t- y^ <it and
the temperature of the hot Junction is t + I/2 dt. The quan-
tity Q is known as the thermoelectric power of the circuit
at temperature t.

ELECTRICITY AND MAGNETISM 15S

(Thomson, Elements of Electricity and Magnetism, p.
506. Also consult Cliwolson, Traite de Physique, Vol. IV,
Part 10, p. 744. For values of thermoelectric power see
Smithsonian Physical Tables, pp. 268-270.)

THERUOELECTRIC LAW FOR DIFFERENT PAIRS OF METALS.

If E^„ is the e.m.f . round a circuit formed of the pair of
metals A, C, and E^„ the e.m.f . round a circuit formed of the
pair of metals B, C, then E^^ - E,,^ is the e.m.f. round a
circuit formed of the metals A and B; all these circuits,
being supposed to work between the same limits of
temperature.

(Thomson, Elements of Electricity and Magnetism, p.
506.)

PELTIER EFFECT.

When current flows across the junction of two unlike
metals it gives rise to an absorption or liberation of heat.
If the current flows in the same direction as the current at
the hot junction in a thermoelectric circuit of the two metals
heat is absorbed; if it flows in the same direction as the
current at the cold junction of the thermoelectric circuit
heat is liberated. This phenomenon is known as the Peltier
Effect.

(Thomson, Elements of Electricity and Magnetism, pp.
501, 502. Also Ganot's Physics, art. 878. Consult also
Smithsonian Physical TaJbles, p. 271.)

156 LAWS OF PHYSICAL SCIENCE

MEASURE OF PELTIER EFFECT.

The Peltier Effect equals (the thermoelectric power)
X ( the absolute temperature). Or, P = QT, where T =
absolute temperature.

(Thomson, Elements of Electricity and Magnetism, p.
510. Also Maxwell, Treatise on Electricity and Magnetism,
Vol. 1, art. 249.)

The Peltier Effect, the closely associated Thomson Effect
and the Hall Effect are thought by many to be explained
with fair satisfaction on the modern electron-theory. For
advanced views on this matter consult Campbell, Modern
Electrical Theory, pp. 71-74.)

ACTION OF CURRENT FLOWING ALONG AN UNEQUALLY HEATED
CONDUCTOR, CALLED THE " THOMSON EFFECT."

"When an electric current flows along an unequally
heated metallic conductor it tends, in the case of copper, to
diminish the inequality of temperature and in the case of
iron to increase this inequality.

(Thomson, Elements of Electricity and Magnetism, p.
505. Also, Maxwell, Treatise on Electricity and Magnetism,
Vol. I, art. 253. Also, for full account, see Chwolson, Traite
de Physique, Vol. IV, Part 10, pp. 752-756.)

RICHARDSON'S LAW OF ELECTRONIC EMISSION.

' 'The number of electrons emitted at different tempera-
tures T is governed by the formula,

n = at\-''/\

A, X and b are constants." A varies greatly with the sub-
stance, X js not far different than unity and b in equivalent
volts is always comparable with five.

(Richardson, The Electron Theory of Matter, p. 441.
Also 0. W. Richardson, PhU. Trans. (A), Vol. CCI, p. 543,
1903.)

ELECTRICITY AND MAGNETISM 157

LAW OF MAGNUS.

If a circuit is formed of a single metal no current will
be formed in it however the section of the conductor and
the temperature may vary in different parts of the circuit.

(Maxwell, Treatise on Electricity and Magnetism, Vol. I,
art. 251.)

THERHOELECTRIC INVERSION.

Call E the e.m.f. acting round a thermoelectric circuit
of two metals. Then, when the difference of temperature
of the two junctions is twice the difference of temperature
for which E is a maximum, E becomes zero and a further
increase in the temperature-difference, the lower tempera-,
ture of the one junction being held constant, causes E to
change direction. This is known as thermoelectric inversion.

(See Chwolson, Trait e de Physique, Vol. IV, Part 10, p.
739. Thermoelectric phenomena are very adequately and
fully treated in Chap. VI, pp. 728-761.)

RATIO OF THE ELECTROMAGNETIC TO THE ELECTROSTATIC UNIT
OF A QUANTITY OF ELECTRICITY.

The ratio of the electromagnetic to the electrostatic unit
of a quantity of electricity is a velocity (whatever units of
length and time are chosen) and this velocity is experi-
mentally shown to be the velocity of light. The ratio is
called V and, very nearly,

V = 3 X lO^o cm/sec.

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. II, Chap. XIX. Also Thomson, Elements of
Electricity and Magnetism, p. 470 et seq.)

158 LAWS OF PHYSICAL SCIENCE

RELATION OF SPECIFIC INDUCTIVE CAPACITY AND INDEX
OF REFRACTION.

According to Maxwell 's electromagnetic theory of light,
the dielectric capacity (specific inductive capacity) of a
transparent medium is equal to the square of its index of
refraction for electromagnetic radiation.

Thus, K = ii".

(Consult Maxwell, Treatise on Electricity and Magnet-
ism, Vol. II, arts. 788, 789.)

ROTATION OF THE PLANE OF POLARIZATION OF LIGHT.

In 1845 Faraday discovered that if a substance which
ordinarily will not rotate plane-polarized light be placed in
a strong magnetic field it acquires this property. In looking
from North to South along a line of magnetic force the
rotation is clockwise. On reversing the direction of magneti-
zation, the direction of rotation is reversed.

The angular rotation of the plane of polarization of a
plane-polarized ray of light, which is parallel to a magnetic
field, is numerically equal to the amount by which the
magnetic potential increases in passing from the point where
the ray enters the medium, (in which the rotation takes
place) to the point where the ray leaves it, multiplied by a
coefficient. This coefficient is generally positive for dia-
magnetic media.

(See Preston, The Theory of Light, p. 431. Also Max-
well, Treatise on Electricity and Magnetism, Vol. II, art.
808. For formulae and numerical values of Verdet's con-
stant see Smithsonian Physical Tables, pp. 326-330.)

PRESSURE OF RADIANT ENERGY,

In a medium, in which radiant energy is propagated as
light, heat or electromagnetic waves, there is a pressure in
the direction normal to the wave front. This pressure is
numerically equal to the energy in unit volume of the
medium.

ELECTRICITY AND MAGNETISM 159

If K is the specific inductive capacity and /* the permear
bility of the medium, P the maximum electromotive force
and p the maximum magnetic force (the direction of which
is at right angles to both the direction of P and the direc-
tion of the propagation of the wave) then,

2^1^ = ^(3*= mean energy in unit volume of the
medium.

(See Maxwell, Treatise on Electricity and Magnetism,
Vol. II, arts. 792, 793. For description of Nichols and
Hull's experimental proof of the above, see Wood, Physical
Optics, p. 466.)

HALL EFFECT.

When a thin rectangular sheet of metal carrying an
electric current flowing in the direction of its length is sub-
jected to a powerful magnetic field normal to the sheet, the
current streamlines are deflected toward one edge of the
sheet. This is called the Hall Effect.

{Gam)t's Physics, art. 900. Also Campbell, Modem
Electrical Theory, pp. 76-78.)

ELECTRO-OPTICAL EFFECT IN DIELECTRICS.

When certain dielectrics are subjected to electric strain
they become doubly refracting. Kerr states the law as
follows :

" The strength of the electro-optical action of a given
dielectric, that is the difference in the path of the ordinary
and extraordinary rays, for unifthickness of the dielectric,
varies directly as the square of the resultant electric force."

{Qanoi's Physics, art. 997,)

160 LAWS OF PHYSICAL SCIENCE

LAW OF SAEASIN AND DE LA RIVE, OF " MULTIPLE RESON AHCE."

The distance between two nodes on a resonator changes
in changing the resonator but not in changing the oscillator.
This distance (the intemode) is the half wave-length of
the free oscillations of the resonator only. (These results
are connected with the rapid damping of the waves which
usually occurs on the oscillator.)

(Vreeland, Maxwell's Theory of Wireless Telegraphy,
p. 62.)

LAW OF DISTRIBUTION OF ELECTRO-MAGNETIC RADIATION, (i)

At any point at a distance from the center of disturbance
there is an electric and magnetic disturbance at right angles
to the line drawn from the center of disturbance. The elec-
tric force is on a tangent to a great circle of a sphere with
the oscillator at its center and which has its poles at the
intercepts of the axis of the oscillator produced: and the
magnetic disturbances lie on tangents to circles parallel
to a plane perpendicular to the oscillator.

(Vreeland, Maxwell's Theory of Wireless Telegraphy,
pp. 76, 77.)

LAW OF DISTRIBUTION OF ELECTRO-MAGNETIC RADIATION. (2)

The two vibrations (electric and magnetic) are trans-
verse, as in light, and perpendicular to the direction of
propagation of the wave-front. The amplitude of these
vibrations varies inversely as the distance and their in-
tensity varies inversely as the square of the distance from
.the oscillator. The vibrations maintain a constant direction
as do those of polarized light.

(Vreeland, Maxwell's Theory of Wireless Telegraphy,
pp. 76, 77.)

ELECTRICITY AND MAGNETISM 161

RELATION OF MAGNETIC FORCE AND A MOVING FARADAY TUBE.

A Faraday tube in motion perpendicular to its direction
always gives rise to a magnetic force, the direction of motion,
the Faraday tube and the magnetic force being mutually at
right angles, when the medium is isotropic.

(Thomson, Elements of Electricity and Magnetism, pp.
479, 490-492.)

POYNTING'S LAW.

When a conductor carrying a current is in an electro-
static field the transfer of energy takes place through the
dielectric along paths which are the intersections of the
equipotential surfaces of the electrostatic field with the
equipotential surfaces of the electromagnetic field due to
the current.

(Gray, A Treatise on Magnetism and Electricity, p.
421.)

GENERALITY OF LAW OF INVERSE-SQUARES.

It may be stated generally that the intensity of an
effect, which emanates from a center and is transmitted
equally in all directions, is, in an isotropic medium, inversely
proportional to the square of the distance from the source.

(New Century Dictionary under word law.)

II

VI

LIGHT

LIGIHT.

A GENERALITY IN RADIATION.

Every fact of experience and every consideration of
theory go to prove that radiant energy of all wave-lengths,
whether called electric, heat-, or light-radiation, travels in
space free from ordinary matter with the same velocity and
obeys the same laws of propagation. This is a velocity which
would he determined on an elasticsolid theory by the equation

= ^J'^'

where e is the elasticity and d the density of the ether of
space.

(Consult Chwolson, Trmte de Physique, Vol. II, Part 8,
pp. 2 and 94 et seq. Also Preston, The Theory of Light,
pp. 28-30.)

LIGHT DEFINED; VELOCITY OF.

Light consists of radiant energy, propagated in free
space with the velocity common to all radiant energy but
ha,ving wave-lengths such that it affects the human eye.

Calling A. the wave-length and N the frequency of the
transverse vibrations the velocity is very approximately
T = XN = 3 X 10^" cms. pet second = 300,000 kilometers
per see. The wave-length of visible radiant energy lies
within the limits X. = 0.76/i and \ = 0.4/t, and the frequency
lies in the limits N = 4 XlO" and N = 7.5 X 10". (/n =
0.001 millimeter.)

(Consult Chwolson,- Trait e de Physique, Vol. II, Part 8,
p. 29. For methods of determining the velocity of light,
see Preston, The Theory of Light, Chap. XIX, p. 489 et seq.)

165

166 LAWS OF PHYSICAL SCIENCE

RECTILIKEAR PROPAGATION OF LIGHT.

Light travels in straight lines through a homogeneous
medium if the rays are not compelled to pass through any
very small openings. When it passes through media of
different kinds, it does not, in general, travel in the same
straight line through them all.

(Crew, General Physics, p. 428. Also Ganot's Physics,
art. 513.)

INTENSITY OF RADIATION.

The quantity of energy which traverses in the unit of
time unit surface normal to the ray is here chosen to define
the intensity of radiation. This intensity is inversely pro-
portional to the square of the distance to a point-source of
the radiation. The intensity of radiation received on an
oblique surface is proportional to the cosine of the angle
which the ray makes with the normal to the illuminated
surface.

(Chwolson, Traits de Physique, Vol. II, Part 8, pp.
24, 25. Also Ganot's Physics, art. 519.)

LAWS or REFLECTION.

1. The angle of reflection is equal to the angle of inci-
dence.

2. The incident and the reflected ray are both in the
same plane which is perpendicular to the reflecting surface.

{Ganot's Physics, art. 522. Also Preston, The Theory
of Light, p. 74. For extensive treatment consult Chwolson,
Traits de Physique, Vol. I, Part 2, p. 168 and Vol. II,
Part 8, p. 112 et seq.)

REFLECTION FROM A PLANE-HIRROR.

" The image of a point in a plane-mirror lies on the
perpendicular let fall from the point to the mirror, and
lies as far behind the mirror as the point lies in front of
the mirror."

LIGHT 167

(Crew, General Physics, p. 441. Also Ganot's Physics,
art. 524.)

SEAL AND VIRTUAL IMAGES.

Real images are those formed by the reflected rays them-
selves, and virtual images are those formed by their pro-
longations. Real images can be received on a screen, virtual
images cannot. Plane-mirrors give rise to virtual images
only.

(Ganot's physics, art. 525.)

LIGHT REFLECTED FROM A ROTATING MIRROR.

If a ray of light falls on a plane mirror and the mirror
is rotated through any angle about an axis which lies in
the plane of the mirror, the reflected ray is rotated through
twice the angle.

(Kimball, College Physics, p. 563. Also Ganot's Physics,
art. 530.)

LAW OF FERMAT, OR PRINCIPLE OF LEAST TIME. ^

When light passes from any point P to another point
P' by reflection at a point p on a surface, the path P p P'
is that which will be traversed by the ray in the least time
in passing from P to P' by reflection at the surface. A
similar law applies for the paths of refracted rays, so that
when light travels from one point to another the ray pur-
sues that path which requires the least time.

(Preston, The Theory of Light, p. 95. Also Chwolson,
Traite de Physique, Vol. II, Part 8, pp. 114, 115.)

168 LAWS OF PHYSICAL SCIENCE

REFLECTIONS FROM PORTIONS OF THE SURFACE OF A SPHERE.

The formulas for calculating the location of images
formed by spherical mirrors are simple only when the
spherical surface is but a small portion of the surface of
a sphere. In this case the distance s from a point-source
to a point on the surface of the mirror, the distance s' from
the image to the same point on the surface, and the radius
r of the spherical mirror bear the relations,

8_ ^ s-r i. 1 Jl ^ 2^
s' r— s' ' ""^ s s' r ■
These general relations hold for both concave and con-
vex surfaces which are small portions of a sphere, if, calling
the center of the mirror-surface the origin, distances to the
right are reckoned positive and to the left negative.

(Chwolson, Traits de Physique, Vol. II, Part 8, pp. 115-
122. Also Crew, General Physics, pp. 445, 446.)

RELATIVE SIZE OF OBJECT AND IMAGE.

For a spherical mirror,

the linear dimensions of the image _
the linear dimensions of the object

the distance from a point on the mirror to the image
the distance from the same point on the mirror to the object

This relation holds whether the image be real or virtual and
for convex as well as concave mirrors.
{Ganot's Physics, art 540.)

THE CAUSTIC.

When the aperture of a spherical mirror much exceeds
10 degrees the rays from a point-source reflected by the
mirror suffer spherical aberration by reflection. Every
reflected ray cuts the one adjacent to it and their points
of intersection form in space a curved surface which is
called the caustic by reflection.

(Ganot's Physics, art. 542.)

LIGHT 169

REFLECTIOIT FROM A PARABOLIC MIRROR.

In reflection from a parabolic mirror all rays parallel
to its axis, after reflection, meet at the focus of the mirror ;
and conversely, when a point-source of light is placed at the
focus, the rays incident on the mirror are reflected exactly
parallel to the axis, and their intensity tends to remain
constant at all distances.

{Gimot's Physics, art. 544.)

REFLECTION FROM MAT SURFACES: LAMBERT'S LAW.

Surfaces from which the diffusion of light is uniform
are said to be mat

The reflection, from such surfaces, of light incident
under a given angle follows the law of cosines, often called
"Lambert's law." According to this law the intensity
of the light diffusely reflected from a mat surface is pro-
portional to the cosine of the angle between the direction
of the diffused rays under consideration and the normal to
the surface. However, if the angle of incidence be varied,
the light reflected at a constant angle to the normal is not
proportional to the angle of incidence.

(See New Century Dictionary, under title cosine law,
and the sub-head, Lamiert's Law of Cosines.)

SELECTIVE REFLECTION.

Selective reflection is reflection in which the incident and
reflected rays differ in composition. Many otherwise trans-
parent bodies have absorption-bands in their spectra, and
waves of light having a wave-length to which the body is
opaque are more completely reflected than the others.

(JVew Century Dictionary, under word eepijECTion.
Consult Wood, Physical Optics, pp. 352, 353.)

170 LAWS OF PHYSICAL SCIENCE

REFLECTION FROM AH ELEMENT OF A NON-SPHERICAL SURFACE.

A bundle of rays, incident normally on an infinitely small
area of a non-spherical surface, give in reflection two in-
finitely small right-angled rectilinear focal lines, parallel
to the elements of lines of curvature of this surface. A
bundle of such rays is called astigmatic.

(Chwolson, Traite de Physique, Vol. II, Part 8, pp. 125,
126. See also Preston, The Theory of Light, p. 109.)

VELOCITY OF LIGHT IN ORDINARY MATTER.

Light travels more slowly in any kind of ordinary matter

than in vacuum. Thus for red rays,

velocity in air , „„. , , ,,

— ; — 7^. — = 1.329 (about)

velocity in water

, velocity in air i aio /• v, i\

and — ; — rr^ r 7~- — , , ., = 1.612 (about)

velocity in carbon bisulphide ,

I {Ganot's Physics, art. 561. Also Ames, Theory of

Physics, pp. 424^426.)

LAWS OF REFRACTION: SNELL'S LAW.

" The incident and refracted rays are in the same plane
with the normal to the surface ; they lie on opposite sides
of it, and the sines of their inclinations to it bear a con-
stant ratio to one another."

Denoting the angles of incidence and refraction by
i and r respectively, the relation between them is given
by the formula,

sin 1 _
sinr

The constant ratio (i is called the index of refraction.

(Preston, The Theory of Light, p. 88.)

RELATION OF VELOCITIES AND INDEX OF REFRACTION.

For any particular medium the index of absolute refrac-
tion /* varies inversely as the velocity of light in that
medium.

LIGHT 171

Denoting by i and r the angles of incidence and refrac-
tion respectively, and by v and V the velocities in ether
and the medium respectively,

BJni _y__
sin r v' '''

(Preston, The Theory of Light, pp. 89, 90.)

RELATIONS OF VELOCITIES IIT MEDIA OTHER THAN ETHER.

If the velocities in two media are Vj and v^ while the
velocity in ether is v, then w = — and mj = " ", or,

Vi Vj

Vi _ w .
Vl in

Also III sin i = ju,2 sin r.

For any number of media the continued product of the
relative refractive-indices of n substances is equal to the
ratio of the absolute refractive-index of the n"" substance
to that of the first. Or in a formula,

/^

(Preston, The Theory of Light, pp. 90, 91. Also Ames,
Theory of Physics, p. 425.)

ATHOSPHERIC REFRACTION.

The refractive index of the earth's atmosphere decreases
as we ascend and for this reason all light which reaches
us from stars not in the zenith travels in curved paths.
The effect of this refraction is to apparently raise the stars
toward the zenith. The mirage results from the double view
of an object given by rays reaching the eye by two paths,
one nearly direct and one concave upward, giving the effect
of reflection.

{Ganot's Physics, art. 551. Also Wood, Physical Optics,
pp. 69, 70.)

172 LAWS OF PHYSICAL SCIENCE

TOTAL REFLECTION AND CRITICAL ANGLE.

Let Hi and /i^ be the indices of refraction of two media,
fii being greater than fi^- Then, if a ray of light in passing
through the medium of greater refraction comes to the
boundary of the two media, it can only pass out into the
medium of less refraction when the sine of the angle of
incidence (angle between ray and normal to surface of

separation of the two media) is less than — .

For all values of the angle of incidence greater than
this the ray is totally reflected from the plane of separation
of the two media.

The maximum value of the angle of incidence which
does not give total reflection is known as the '* critical "
angle.

. (Ames, Theory of Physics, p. 427. Also Ganot's Phys-
ics, art. 550.)

REFRACTION AT A SINGLE SPHERICAL SURFACE.

When light passes from a point-source in a medium of
refractive index /j.^ into a medium of refractive index /ij
bounded by a spherical surface, then.

'"(T-T)='^(T-i'>

where r = radius of curvature of refracting surface — r
being taken positive when convex side of surface is toward
incident ray — s ^ distance from apex of surface to point-
source — s being taken negative when on same side of sur-
face as point-source — and s' = distance from apex to image
— where s' is taken negative or positive according as it is
on the same or opposite side of the surface aa the point-
source.

(Crew, General Physics, p. 458.)

LIGHT 17S

POSITIOIT OF IMAGE FORMED B7 REFRACTION AT A
SPHERICAL SURFACE

The position of the image when rays pass from air into
a substance bounded by a spherical surface is given by the
formula, ^_

^ - -1 = ilzl.

s' B r '

where n = index of refraction of substance used, s' = dis-
tance from apex of surface to image, s = distance from apex
of surface to source and r = radius of curvature of spheri-
cal surface.

(This formula comes from the one just preceding by
placing /Ai = 1 and /i^ = /*)

(Crew, General Physics, p. 460. Also Ganat's Physics,
art. 552.)

GENERAL EQUATION FOR THIN LENSES.

— = (n-1) [ — — j is given in Ganot's Physics,

art. 563, as a general formula applicable to all cases both of
convex and concave lenses.

The point where the refracted ray cuts the axis when
the incident ray is parallel to the axis is called the principal
focus. Its distance from the lens is the focal length of the

lens. The power of a lens equals j^s focal length" "^^^
unit of power is the ' ' dioptric ' ' which equals the power of
a lens of focal length of one meter.

(See above reference for interpretation of this formula,
or consult Crew, Gmeral Physics, p. 466.)

174 LAWS OF PHYSICAL SCIENCE

CHROMATIC ABEREATION.

The index of refraction for a given substance is not the
same for all wave-lengths, or colors of light. Thus,

li Red light, Yellow light, Green light,
for, C. D. F.

FUnt glass 1.630 1.635 1.848

Crown glass 1.527 1.530 1.536

This failure of a lens to bring all colors to the same focus
is known as ' ' Chromatic Aberration. ' '

(Crew, General Physies, p. 470. Also Ganot's Physics,
art. 594.)

DEVIATION OF A RAY IN PASSING THROUGH A PRISM.

The angle of deviation 8 of a ray passing through a
prism is the angle between the incident and emergent rays.

8=(ii.-rJ + (i2-rJ.

Here i, = angle of incidence (angle between normal to
first face of prism and incident ray),

ij = angle of emergence (angle between normal to second
face of prism and emergent ray),

Tj = angle of refraction at first face and

Tj = angle of refraction at second face. Also angle of
prism is, a = ri + t^.

The angle of deviation 8 is a minimum when the in-
cident ray makes the same angle with the first face of the
prism as the emergent ray does with the second face of
the prism.

(Crew, General Physics, p. 471.)

VELOCITY OF LIGHT IN MEDIA DEPENDS UPON WAVE-LENGTH.

The velocity of light is less the more highly refracting
the medium, and as light of short wave-lengths is more re-
fracted than light of long wave-lengths, the velocity of the
former is less than that of the latter in all media of greater
refractive index than unity.

LIGHT 176

(Consult Preston, The Theory of Light, Chap. XIX,
pp. 513, 516. Also Wood, Physical Optics, pp. 15, 16.)

DISPERSIVE POWER.

The transformation of a beam of parallel rays of white
light into a divergent pencil of light of different colors is
knows as ddspersiow.

The dispersive power of a prism is the ratio of the angle
of separation produced of two selected rays to the mean
deviation (angle between incident and emergent rays) of
the two rays,

d ~ n-l'

Here n^ is the refractive index of the material of a prism
or lens for violet rays, n^ the refractive index for red rays,
n the refractive index for mean rays, and du,dj, and d are
the corresponding deviations.
{Ganot's Physics, art. 577.)

VARIATION OF REFRACTIVE INDEX WITH DENSITY (GLADSTONE
AND DALE'S LAW).

When a substance is compressed or its temperature
varied the density changes. This is accompanied by a cor-
responding variation in the refractive index such that,

refractive index — 1 . .

; r; = a constant

density

(Preston, The Theory of Light, p. 131 et seq.)

176 LAWS OF PHYSICAL SCIENCE

BRIOT'S FORMULA.

Briot has determined the form of the function, which
expresses the variation of the index of refraction /i with
wave-length A. His formula is,

l=KV + A+^ + g+ ,

where A, B, C, etc., are constants depending on the nature
of the medium and diminishing rapidly as we proceed to
higher terms, and K is another constant.

(Preston, The Theory of Light, pp. 487, 488.)

DOUBLE REFRACTION: OPTIC AXES.

In all transparent crystals, of which the fundamental
form is not a cube, a black dot seen through the crystal
appears double for most positions of the crystal. There
are, however, in some crystals one and iu others two direc-
tions, along which the dot being viewed appears single.
This direction or directions constitute the optical axis or
axes of the crystal.

This phenomenon, called double refraction, is very
marked in the uniaxial crystal, Icelajid spar (calcium
carbonate.)

(Preston, The Theory of Light, p. 299. Also Ganot'i
. Physics, art. 654.)

DOUBLE REFRACTION IN AN UNIAXIAL CRYSTAL: GENERAL

STATEMENT.

Whatever be the plane of incidence, the ordinary ray
always obeys the two general laws of single refraction.

In every section perpendicular to the optic axis, the
extraordinary ray follows the laws of single refraction.

In every principal section the extraordinary ray follows
the second law of refraction only, but the ratio of the sines
of the angles of incidence and refraction is not constant.

{Ganot's Physics, art. 656.)

LIGHT 177

DOUBLE REFRACTION IN BIAXIAL CRYSTALS: GENERAL
STATEMENT.

When a ray of light enters a biaxial crystal and traverses
it in any direction not coinciding with an optic axis, it
bifurcates and generally both rays are extraordinary rays.
However, in a section of the crystal at right angles to the
medial line one ray follows the laws of ordinary refraction,
and in a section at right angles to the supplementary line
the other ray follows the laws of ordinary refraction.

{Ganot's Physics, art. 658.)

CONICAL REFRACTION.

This is of two kinds, Mernal conical refraction and
external conical refraction. For the former, an unpolarized
ray entering a crystal along an axis of single wave-velocity
diverges within the crystal as a cone and emerges as a
hollow cylinder of rays. These can be received on a screen
as a ring of light of constant diameter. In the latter case,
if a conical pencil of rays enters a crystal with the apex of
the cone on the face of the crystal and with its axis chosen
parallel to the axis of single wave-velocity the emergent
pencil of rays (superfluous rays being screened off) fonns
a hollow cone. This gives on a screen a ring of light which
increases in diameter with the distance of the screen from
the crystal. '

("Wood, Physical Optics, pp. 257-259.)

POLARIZATION EFFECTED BY A CRYSTAL.

In traversing an uniaxial crystal the ordinary ray is
polarized in the plane of the principal section, which in-
cludes this ray and the optic axis. The extraordinary ray,
is on the contrary, polarized in a plane perpendicular to the
plane of the principal section passing through the ray and
the optic axis. The polarization is complete for both ordi-
nary and extraordinary rays.

(Chwolson, Traits de Physique, Vol. II, Part 8, pp. 760,
761.)

12

178 LAWS OF PHYSICAL SCIENCE

POLARIZATION BY REFLECTION.

When a ray of ordinary light falls at an angle upon a
reflecting surface of a transparent substance the reflected
ray is more or less polarized. For a particular angle of
incidence the polarization of the reflected ray is most com-
plete and this angle is called the polarizing angle for the
substance.

{Ganot's Physics, art. 667, or Wood, Physical Optics,
Chap. IX. See p. 231.)

ANGLE OF POLARIZATION: BREWSTER'S LAW.

In polarization by reflection, the polarizing angle of a
substance is that angle of incidence for which the reflected
polarized-ray is at right angles to the refracted ray.

Brewster's law states that : The tangent of the polarizing

angle for a substance is equal to the index of refraction of

that substance. This law is expressed by the formula,

sin i

fi = tan 1 = 7-

cos 1

As -: — = M (Snell's law),
sin r

where i is the angle of incidence of the reflected ray and r
the angle which the refracted ray makes with the normal
to the surface of separation of the two media, it follows
that i + r = 90 ° ; namely, the reflected and refracted rays
form a right angle.

( Ganot's Physics, art. 668. Also Wood, Physical Optics,
p. 231. Also Preston, The Theory of Light, pp. 303, 304.)

LAW OF MALUS.

When a pencil of light, polarized by reflection at one
plane-surface, is allowed to fall upon a second plane-surface
at the polarizing angle, the intensity of the twice-reflected
beam varies as the square of the cosine of the angle between
the two planes of reflection.

LIGHT 179

(Preston, The Theory of Light, p. 305. Also Wood,
Physical Optics, p. 235 et seq.)

POLARIZATION OF REFRACTED LIGHT.

The relation between the polarized light in the refracted
pencil and that in the reflected beam was discovered by
Arago and is stated thus : ' ' When an unpolarized ray is
partly reflected at, and partly transmitted through, a trans-
parent surface, the reflected and transmitted portions eon-
tain equal quantities of polarized light, and the planes of
polarization are at right angles to each other."

(Preston, The Theory of Light, p. 304.)

LAWS OF INTERFERENCE OF POLARIZED LIGHT.

' ' 1. Two rays of light polarized at right angles do not inter-
fere destructively under the same circumstances as
two rays of ordinary light.

2. Two rays polarized in the same plane interfere like

two rays of ordinary light.

3. Two rays polarized at right angles may be brought to

the same plane of polarization without thereby
acquiring the quality of being able to interfere with
each other.

4. Two rays polarized at right angles, and afterwards

brought to the same plane of polarization, interfere
like ordinary light if they originally belonged to the
same beam of polarized light. ' '
(Preston, The Theory of Light, p. 308.>

180 LAWS OF PHYSICAL SCIENCE

THE PLANE OF POLARIZATION.

' ' The plane of polarization is defined as the particulax
plane of incidence in which, the polarized light is most
copiously reflected."

When a ray of light is incident at the angle of polariza-
tion on one mirror and the reflected ray is received on a
second mirror at the polarizing angle, the ray is most
copiously reflected from the second mirror when the two
mirrors are parallel. In this case the plane of reflection
coincides with the plane of polarization. The vibrations
of plane-polarized light are, however, in a direction at
right angles to the plane defined as the plane of polarization.

(Wood, Physical Optics, p. 233.)

ELLIPTICAL POLARIZATION.

Light may be plane, circularly or ellipticaUy polarized.
Plane and circular polarization may be treated as special
cases of elliptical polarization. Circular polarization results
from the simultaneous presence at a point of two rectangu-
lar vibrations of the same period but differing in phase
by a quarter period. In elliptical polarization both the
phase and amplitude of the vibrations differ.

(Wood, Physical Optics, Chap, XI, p. 266 et seq. Also
Preston, The Theory of Light, pp. 417, 418.)

ROTATION OF PLANE OF POLARIZATION.

Rotation of the plane of polarization occurs when plane-
polarized light is transmitted through quartz in the direc-
tion of its optic axis, /and this property is also possessed
by many other substances, including many liquids and
vapors. Some rotate the plane to the right (looking along
the direction of propagation of the light) and are called
" Dextrogyrate," some to the left and are called " Levo-
gyrate."

(Preston, The Theory of Light, p. 425. Also Ganot's
Physics, art. 687.)

LIGHT 181

BIOT'S LAWS.

The amount of rotation of plane-polarized light passed
through a quartz crystal or other rotating substance is
proportional to the thickness traversed by the ray. The
rotation effected by two plates is the algebraic sum of the
rotations produced by each separately. The rotation aug-
ments with the refrangibility of the light, and is approxi-
mately proportional to the inverse square of the wave-
length.

(Preston, The Theory of Light, p. 426. Also Wood,
Physical Optics, pp. 384, 385.)

POLARIZATION BY EMISSION AND DIFFUSION.

Rays emitted at an oblique angle from incandescent
platinum are partially polarized perpendicularly to the
plane of emission. Rays falling on a surface which is not
absolutely mat, in their diffuse reflection are partially
polarized.

(Chwolson, Traits de Physique, Vol. II, Part 8, p. 740.)

RELATION OF INTENSITY AND AMPLITUDE.

If the periods of two vibrations are the same, then the
intensities of the rays are in the ratio of the squares of
the amplitudes of the vibrations. Thus,

Ji af'

where J, Ji are intensities and a, 3,^ are amplitudes.

(On the electromagnetic theory of light, amplitude of
vibration would correspond to the maximum potential-
difference between opposite ends of the oscillator giving
rise to the radiant energy.)

(Consult, Preston, The Theory of Light, pp. 43, 44.)

182 LAWS OF PHYSICAL SCIENCE

A NATURAL RAY REPLACED BY TWO RAYS POLARIZED AT
RIGHT ANGLES.

Two rays reetilinearly polarized, which replace a natu-
ral ray, have amplitudes continually variable and, in gen-
eral, unequal at each instant ; the mean value of the square
of the amplitude determines the quantity of radiant energy
(light intensity) in the two rays.

Thus, A^ = %J, where J is the intensity of the natural
radiation and A^ is the mean value of the square of the
amplitude of the component rays.

(Chwolson, Traite de Physique, Vol. II, Part 8, p. 696.)

LAW OF LAMBERT FOR EMISSION FROM A SURFACE OF RADIANT

ENERGY.

The quantity of radiant energy emitted in the unit of
time, by an element of the surface of a body, in any given
direction, is proportional to the cosine of the angle between
this direction and the normal to the surface of the radiating
body. Thus, J^ = J coscp where J is the total quantity
of energy emitted normally and 3(f the quantity emitted
in the direction making the angle <p with the normal.
(It is because of this law that a sphere heated to be uni-
formly luminous appears equally bright at the central point
of its disc and at its boundary.)

(Chwolson, Traite de Physique, Vol. II, Part 8, p. 36.)

LAW OF ABSORPTION OF LIGHT.

When a ray enters a homogeneous medium the quantity
of light of a given wave-length which is absorbed is pro-
portional to the thickness of the medium traversed, and
the amount of light which passes through a number of
equal layers, diminishes in geometrical progression as the
number of layers increases in arithmetical progression.

(Consult Preston, The Theory of Light, Chap. XVIII.
See pp. 469-471. Also Wood, Physical Optics, Chap. XIV.
See pp. 350, 351.)

LIGHT 183

DOPPLER-nZEAU PRINCIPLE.

If a source of light and an observer approach, the fre-
quency of the disturbance, as it passes the observer, is
increased and the wave-length diminished; if they recede-
the reverse is true. On approach the spectrum lines are
shifted toward the violet and on recession they are shifted
toward the red.

(Wood, Physical Optics, Chap. I. See p. 19.)

HUYGENS' PRINCIPLE.

The wave-front in a train of light- waves is a surface of
disturbance which results from and envelops {i.e., is
tangent to) the secondary waves sent out by each particle
lying in the wave-front at an earlier instant.

(Wood, Physical Optics, Chap. II, p. 21 et seq. Also
see Ames, Theory of Physics, pp. 401-403. Also Preston,
The Theory of Light, p. 60.)

HUYGENS' PRINCIPLE OF SUPERPOSITION.

Any number of separate disturbances (light-waves) may
be propagated through one another in the same portion of
the medium. Each emerges from that portion as if it had
not been encountered by others. Rays of light from all
objects round about cross each other's paths in aU sorts
of ways, but each travels on as if the others did not exist.

(Preston, The Theory of Light, p. 45.)

STOKES' LAW.

The effect or intensity of an elementary wave at an ex-
ternal point varies as (1 + cos fl), where d is the obliquity,
or angle between the wave-normal and the line joining the
point to the center of the elementary wave. Thus the effect
only vanishes for 6 ^ tr, that is, for points' directly behind
the wave.

(Preston, The Theory of Light, p. 62.)

184 LAWS OF PHYSICAL SCIENCE

PRESSURE OF RADIANT ENERGY; GENERAL CASE.

When radiant energy (of any wave-length) falls nor-
mally on a perfectly black surface the pressure exerted on
unit area is numerically equal to the total quantity of radi-
ant energy contained in the unit of volume. If the surface
is perfectly reflecting the pressure is twice as great. If the
radiation falls on the surface making an angle ^ with
the normal to the surface then the pressure, per unit area is,

p = e (1 -)- a) cos'' <p,

E
where e = ^ is the energy in the unit of volume and a

is the fraction of the energy reflected, a = 1 for a perfectly
reflecting and a = o for a perfectly black surface.

(Chwolson, Trait e de Physique, Vol. II, Part 8, p. 84.)

LAW OF KIRCHHOFF-CLAUSmS ON THE RELATION BETWEEN
EMISSIVE POWER AND THE MEDIUM.

The emissive power of perfectly black bodies is propor-
tional to the square of the index of refraction [i of the sur-
rounding medium.

Thus, e = iu^E,
where E is the emissive power of a perfectly black body in
vacuum and e its value in any medium.

(Chwolson, Traits de Physique, Vol. II, Part 8, p. 83.)

DIFFRACTION OF LIGHT.

In passing through a very narrow aperture in a screen
a ray of light spreads out on either side of the line of
rectilinear propagation. This phenomenon is known as
diffractioii.

It is also observed under other circumstances, as when
light passes the edge of a body, in which case luminous rays
are bent into the shadow.

{Oanot's Physics, arts. 660, 661. For a detailed treat-
ment, see Wood, Physical Optics, Chap. VII, pp. 150-211.)

LIGHT 185

INTERFERENCE OF LIGHT-RAYS.

When light passes through two small openings in a
screen which are adjacent and the two transmitted rays
meet each other at any point under a small angle the two
trains of waves either annul or strengthen each other,
giving dark and light bands on a screen. In the former
case the distances from any point to the two openings
differ by a half wave-length or a multiple of this, and in
the latter case by a whole wave-length or a multiple. This
reciprocal action of two wave-trains is called interference.

Michelson's Interferometer is an instrument for the
measurement of lengths by means of the phenomena result-
ing from the interference of two rays of light. The instru-
ment permits the introduction of any relative retardation
between interfering-pencils of light and allows observation
to be made of interference bands corresponding to a large
difference of path.

{Ganot's Physics, art. 659. Also Preston, The Theory
of Light, p. 202. Wood, Physical Optics, Chap. VIII, pp.
212-229. Also Ames, Theory of Physics, pp. 395-397.)

INTERFERENCE IN THIN FILMS.

If two plates of glass are put together so as to form a
thin wedge of air, destructive interference of any mono-
chromatic light wiU take place where the thickness of the
wedge equals an even number of quarter wave-lengths and
reenforcement occurs where this thickness equals an odd
number of quarter wave-lengths.

(Crew, General Physics, pp. 502-505. For detailed
account of these phenomena, see Wood, Physical Optics,
Chap. VI, p. 100 et seq.)

186 LAWS OF PHYSICAL SCIENCE

ANOMALOUS DISPERSIOK.

In transparent substances in whieli the dispersion is
normal, the refractive index increases as the wave-length
decreases, but in substances which show selective absorp-
tion, the refractive index for short waves on the blue side
of an absorption band is often less than the index for red
light on the other side of the band. This phenomenon has
been named " anomalous dispersion."

(Wood, Physical Optics^ Chap. V. See p. 95.)

KUNDT'S LAW IN ANOMALOUS DISPERSION.

This law states that on approaching an absorption band
from the red side of the spectrum the refractive index is
abnormally increased by the presence of the band, while
if the approach is from the blue side the index is abnormally
decreased.

(Wood, Physical Optics, Chap. V. See p. 96.)

FRESNEL'S LAW.

"In crystals the velocities of the two light-waves are
proportional to the largest and smallest radii vectors of
the oval section of the wave-surface, made by a plane
through the center of the surface and parallel to the wave-
front."

{New Century Dictionary, under word law. For a
detailed treatment of wave-velocity in crystals, consult
Wood, Physical Optics, Chap. X. See pp. 246-257.)

NON-REVERSIBLE VISION.

If two Nicols are mounted with their directions of vibra-
tions at an angle of 45° and between them is a medium in
a magnetic field of such strength that the plane of rota-
tion is turned 45°, light will be stopped by the second
Nicol in going one way and be wholly transmitted in going

UGHT 187

the reverse way. It is thus possible to produce an arrange-
ment whereby we can see without being seen when mono-
chromatic light is used.

(Wood, Physical Optics, p. 401.)

NEWTON'S RINGS.

When a lens of large radius of curvature R is pressed
upon a plane-plate of plate-glass and viewed by reflected
monochromatic light, alternate dark and bright rings are
produced. If t is the thickness of air-film between the
lens and plate and r is the radius of a ring, then,

With sunlight the rings are many colored.

(Wood, Physical Optics, Chap. VI. See p. 131. Also
Ganot's Physics, art. 664.)

THE ZEEMAN^EFFECT.

When a source of monochromatic radiation is placed in
a strong magnetic field the lines of the spectrum are widened
and many spectral lines are broken up, in the Zeeman-
effect, into multiple lines.

(For description and explanation of the "Zeeman-
effect " see Wood, Physical Optics, pp. 403-410.)

STOKES' LAW REGARDING FLUORESCENCE.

Stokes' law asserts that the wave-length of the light
emitted by a fluorescent body always exceeds that of the
exciting light.

(Wood, Physical Optics, Chap. XVIII. See p. 434.)

188 LAWS OF PHYSICAL SCIENCE

DRAPER'S LAW OF VISIBILITY.

Draper considered that all bodies begin to become visible
by self-emission of light at the same temperature. He
stated that this temperature is 525° C. ; but the beginning
of visibility is somewhat dependent upon the condition of
the eye. The first visual sensation received from heated
bodies is now known to be a sense of brightness to which
definite color cannot be assigned. The first color-sensation
received coincides with the region in the spectrum of
maximum luminosity — a yellow green.

Lummer and others have shown that Draper's law does
not hold true.

{New Century Dictionary under word law. See the
interesting account of the supposed law of Draper in Chwol-
son, Traite de Physique, Vol. II, Part 8, pp. 33, 34 and
80, 81.)

LE CHATELIER'S LAW OF RADIATION.

The empirical law of Le Chatelier for the intensity of
radiation of red light is represented by the formula,

3210

I = 10=7. T~^f",

where T is the absolute temperature of the radiating body
and I is the intensity.

(Consult Burgess and Le Chatelier, Measurement of
High Temperatures, p. 302.)

TALBOT'S LAW.

iHelmholtz states this law thus: If any part of the
retina is excited with intermittent light, recurring periodi-
cally and regularly in the same way, and if the period is
sufficiently short, a continuous impression will result, which
is the same as thaA which would result if the total light
received during each period were uniformly distributed
throughout the whole period.

{Bulletin of the Bureau of Standards, Vol. II, p. 1.)

LIGHT 189

ASTRONOMICAL ABERRATIOH.

The phenomenon known as the astronomical aberration
of light is the apparent displacement of a star due to the
resultant effect of the velocity of light and the motion of
the earth.

This phenomenon was discovered and an explanation
of it given by Bradley in 1728. The maximum displacement
of a star from this cause is 20.51" of arc.

(Preston, The Theory of Light, pp. 12 and 518-520.
See treatment of this problem in Lorentz, The Theory of
Electrons, art. 155. For value of the " Constant of Aber-
ration," see Smithsonian Physical Tables, p. 109.)

THE ZONE-PLATE.

The zone-plate is a flat surface which will bring light,
transmitted through it, to a focus in the manner of a convex
lens. It is made by describing on a glass plate a system of
circles having a common center with radii which increase
proportionally to the square roots of the natural numbers,
and blackening all odd numbered rings. The smaller the
zones the shorter is the focal length.

(See illustration and description of this interesting
optical dcAdce in Wood, Physical Optics, p. 31.)

THE NICOL'S PRISM.

The Nicol's prism is a most valuable device for studying
polarized light. It is constructed from a rhombohedron of
Iceland spar which has been cut in two along a parallel
plane and the two pieces rejoined in their original position
with a layer of Canada balsam or air between. It transmits
the extraordinary ray only and polarizes light completely.

(Oanot's Physics, art. 674.)

BIBLIOGRAPHY AND INDEX

BIBLIOGRAPHY

American Journal of Science.

Ames, J. S., Theory of Physics, 1896.

Appell, TraAte de MScanique Ratiormelle, 1902-1904.

Bedell and Crehore, Alternating Currents, 1893.

Bulletins of the Bureau of Standa/rds.

Burgess and Le Chatelier, Measurement of High Temperature, 1912,

3rd Ed.
Campbell, Norman R., Modern Electrical Theory, 1913, 2nd Ed.
Christie, C. V., Electrical Emgi/neering, 1913.
Chwolson, Traits de Physique, 1906-1914, five volumes.
Clausius, E., On Heat, edited by Hirst, 1867.
Crew, Henry, Gerteral Physics, 1909.
Edser, Heat, 1908.
Encyclopedia Britannioa, 10th Ed.
Fourier, Joseph, The Analytical Theory of Heat. Trans, by Freeman,

1878.
Ganot's Physics, 1905, 17th Ed.
General Electric Review.

Gray, Andrew, A Treatise on Magnetism and Electricity, 1898.
Gulliver, G. H., Metallic Alloys, 1913, 2nd Ed.
Helmholtz, Popular Scientifio Lectures. Sensations of Tone.
Hering, C, Conversion Tables, 1904.
Houstoun, An Introduction to Mathematical Physics.
Jeans, J. H., Electricity and Magnetism,, 1908. The Dynamical Theory

of Gases, 1904. Theoretical Mechanics, 1907.
Journal of The Franklin Institute.
Kaye and Laby, Physical and Chemical Constants, 1911.
Kimball, A. L., College Physics, 1912.
Lagrange, Mecamque AnaVytique.
Laplace, P. S., TraAte de M6camique Celeste.
Lommel, Einperimental Physics, 1899.
Lord Kelvin, Mathematical amd Physical Papers.
Lorentz, H. A., The Theory of Electrons, 1909.
Mach, E., Science of Mechanics, 1902.
Maxwell, James Clerk, Theory of Heat, 1904. Treatise on Electricity

amd Magnetism, 1881, 2nd Ed.

193

194 BIBLIOGRAPHY

Merriman, M., Treatise on Hydraulics, 1903.

Nernst, Walter, Theoretical Chemistry, 1911.

New Century Dictiomary.

Northrup, E. F., Methods of Measuring Electrical Resistance, 1912'.

Planck, Vorlestmgen uber die Theorie Der Warmestrahlung.

Poynting and Thomson, Beat, 1904. Properties of Matter, 1902.
Sound.

Preston, Thomas, Theory of Heat, 1904.

Eankine, W. J. M., The Steam Engine, 1882.

Richardson, O. W., The Electron Theory of Matter, 1912.

Roscoe and Schorlemmer, Treatise on Chemistry, Vol. I, 1911; Vol.
II, 1907.

Routh, Elementary Rigid Dynamics, 1905, 7th Ed.

Scientifio Memoirs, edited by J. S. Ames.

Smithsonian Physical Tables, 1916, 6th Revised Ed.

Spencer, L. J., The World's Minerals, 1911.

Steinmetz, C. P., Alternating Current Phenomena, 1900.

Stokes, G. G., Mathematical and Physical Papers.

Thomson, J. J., Elements of Electricity and Magnetism, 1904. Corn
duction of Electricity Through Gases, 1906, 2nd Ed.

Thomson and Tait, Treatise on Natural Philosophy, Part I, 1886:
Part II, 1883.

Tramsactioms American Electrochemical Society.

Transactions Connecticut Academy.

Vreeland, Maaywell's Theory and Wireless Telegraph, 1904.

Walker, James, Introduction to Physical Chemistry, 1913.

Washburn, Edward W., An Introduction to the Principles of Physi-
cal Chemistry, 1915.

Watson, A Text-look of Physics, 1899, 1st Ed.

Webster, The Dynamics of Particles and of Rigid, Elastic and Fluid

Bodies, 1904, 2nd Ed.
Wood, R. W., Physical Optics, 1905, 1st Ed.

INDEX

(Numbers refer to pages)

Aberration, chromatic, 174

of light, 189
Absolute scale of temperature, 63

zero, definition of, 63
Absorption, electric, 116
of gases by liquids, 83

by solids, 84
of light, 182
Absorptive power, 104
Acceleration, centripetal, 9

of chemical reactions with
temperature, 100
Accelerations, compounded by par-
allelogram rule, 8
Acid and base, neutralization, 96i
Acids, relative avidity of, 96
Acoustic attraction and repulsion,

53
Action, principle of least, 24
Addition of simple sound-vibra-
tions, 56
Additive property of dilute solu-
tions, 94
Adiabatic expansion, 74

relations, 75
Alternating current power, 139

currents, Kirchhoflf's laws

applied to, 138
Ohm's law applied to, 139
Ampere's law for the magnetic

field, 140
Amplitude, necessary to make
sound-waves audible, 54

Analogies, electric and magnetic,
145

Analogue between osmotic press-
ure and gas-pressure, 92

Analogues in translation and rota-
tion, 12

Anomalous dispersion, 186

Kundt'a law on, 186

Archimedes' principle, 31

Areas, conservation of, D'Arey'?
statement, 23
Mach's statement, 23

Astigmatic rays, 170

Astronomical aberration, 189

Atmospheric refraction, 171

Attraction and repulsion, acoustic,
53

Audibility, limits of, 54

Avogadro, his gas-law, 76

Avidity of acids, 96

Bells, vibration of, 53
Bernoulli's theorem, 34
Bertrand's principle of similitude,

17
Berzelius, gave name catalysis, 102
Black-body, definition of, 104
Blagden's law on depression of

freezing point, 99
Boiling, 86

and volatilization, 86
point, dissolved salts raise, 98
Boltzmann-Mazwell, on equipar-

tition of energy, 76

195

196

INDEX

Boyle's law, for gases, 72

variations from, 73

Boys, C. v., his value of Newton-
ian constant, i

Bradley, on aberration, 189

Brewster's law, regarding polariz-
ing angle, 178

Biot's laws, on rotation of plane-
polarization, 181

Briot's formula, 176

Callendar, his formula, 65
Capacities, parallel and series com-
binations of, 121
Capacity and conductance, general
relation between, 136
and resistance, a relation be-
tween, 136
Capillary action, Jurin's law,'37
law of, 38
corrections of mercury col-
umns, 38
Carnot's theorem, 64
Catalysis, 102
Caustic, the, 168

Cavendish, measurement of New-
tonian constant, 4
Centripetal acceleration, 9
Charged bodies, force between

varied by medium, 116
Charges, decay of in dielectrics,

152
Charles' law, for gases, 74
Chemical action, balanced, 90

decomposition, progress of, 90
combination of elements, 96
reactions, acceleration of with
temperature, 100
Chromatic aberration, 174

Circuits, kinetic energy of two, 150

mutual action between, 148
Clapeyron, his constant and work
of expansion of a gas, 80
his equation for gas-constant,
79
Clausius, on entropy-increase, 71
Coefficients of induction, 148
Coil to give maximum self-induc-
tion, 149
Coils, resistance-values from, 129
Colloids and crystalloids, 102
Cooling, Dulong and Petit's con-
clusions on, 63
Newton's law of, 62
Combinational tones, 54
Composition by parallelogram

rule, 8
Compound pendulum, law of, 13
Condensers, combined in parallel

and series, 121
Condensation on nuclei of vapor,

88
Conductance and capacity, general

relation between, 136
Conduction, electrification by,

113
Conductor, force at surface of a
charged, 119
magnetic force within a, 141
-resistance changes with spe-
cific resistance, 130
Conductors and dielectrics, in uu-
uniform field, 120
and Insulators, 114
four-terminal, 131
interaction of electric, 152'
Conductivity, Onnes on, 133

Ostwald'a law of molecular,
95

INDEX

197

Conservation of areas, lyAicy's
statement, 2'3
Mach's statement, 23
of energy, principle of the, 28
of living forces, 22
of matter or mass, 21
of moment of momentum, 22
of momentum, 21, 22
of movement of center of grav-
ity, 20
Conservative system, work done
by, 20
systems, 19
Consonance and dissonance, Helm-

holtz on, 57
Constant heat-summation, law of,
101
Newtonian, 4
Constraint, comment on principle
of least, 27
Gauss's principle of least, 26
Cosine, law of the, in radiation,

106
Contact difference of potential, 137

electricity, 137
Continuity, principle of, 35
Conversion of energy in a conduc-
tor, 132
Convertibility of energy, 70
Cord, transverse vibrations of, 51
Corresponding states, 78
Coulomb's law of electric intensity,

117
Critical temperature, 91
Cryohydric temperature, 99
Crystalloids and colloids, 102
Crystals, law of interfacial angles

for, 110
Crystal-zones, Neumann's law of,
110

Crystal, polarization by, 177

Curie's law of magnetic suscepti-
bility, 91

Currents, law of resolution of, 153
mechanical action of, 152'
mutual relations of, 152
thermoelectric, 154 ,

Cycloidal pendulum, 14

D'Alembert's and Gauss' princi-
ples compared, 27
principle, 25, 26

Dalton, his law for solution of
mixed gases, 83

Dalton's laws of mixture of gases,
76

D'Arcy's principle regarding areas,
23

Definite resistance, condition for,
130

Definition, of a quantityof heat, 61
of temperature, 61

Definitions, for absolute zero and
temperature, 63
of sound, noise, etc., 45

Delaroche and Berard, on molecu-
lar heat, 77

Deville, his use of word " disso-
ciation," 90

Dielectric constant, or specific
inductive capacity, 120

Dielectrics, decay of charges in, 152

Diffraction, of light, 184
of sound, 50

Diffusion in liquids, 92

Dilute solutions, Kohlrausch on, 94

Diosmose, 92

Dispersive power, 175

Displacement, electric, 120

Distribution, law of, 89

Doppler-Fizeau principle, 183

198

INDEX

Boppler's principle, applied to

sound, 48
Double refraction, 176

in biaxial crystals, 177
in uniaxial crystals, 176
Draper on visibility, 188
Dulong and Petit, on velocity of

cooling, 63
Dulong and Petit's law of thermal

capacity, 66

Ear, effect of sound-waves on, 55
Earnshaw's theorem on stability,

121
Echoes, 49
EflBux, quantity of, 33

Torricelli's theorem, 33
Effusion of gases, 85
Elastic medium, velocity of dis-
turbance in, 17
Elasticity, coefficient of, 15
Elements, chemical combination
of, 96
periodic system of, 97
Elliptical polarization, 180
Electric absorption, 116

charges, law of repulsion of,

114
circuit and magnetic shell
compared, 140
and magnet compared, 142'
current, action of a magnet

on, 140
displacement, 120
equilibrium, 116
field, effects of ununiform, on
conductors, 120
energy of, 120
intensity at a surface, 117
inside a conductor, 117

Electric intensity, inside and out-
side conductors and dielec-
trics, 121
stress in a medium, 119
Electrical conductor, longitudinal
motion in, 143
and magnetic analogies, 145
Electricity, compared with incom-
pressible fluid, 118
positive and negative, 114
Electrification, by conduction, 113
by friction, by induction, 113
Electrified system, work done in

displacement of, 118
Electrodes, Kirchhoff on inter-
change of, 131
Electrolytic decomposition, a prin-
ciple of, 128
Electrolysis, conservation of energy
in, 127
Faraday's first law of, 128
Faraday's second law of, 128
little affected by pressure, 127
Electromagnetic radiation, distri-
bution of, 160
Electromotive force, 115

acts on electricity only,

151
and current, resolved, 138
impressed on a circuit, 151
independent of nature of

conductor, 147
-force series, 126
work done by, 139
forces in series, 136
Electron, unit of negative electric-
ity, 114
Electronic emission, Richardson's

law of, 156
Electro-optical effect in dielectrics,
159

INDEX

199

Emissive power, definition of abso-
lute, 103
Kirchhoff-Clausius on, 184
monochromatic, 103
e/m, value of, 114
Energy, conservation of in electro-
lysis, 127
conversion of mechanical into

heat in conductor, 132
oonvertilibility of, 70
equipartition of, 76
intrinsic, 70
Kelvin's theorem of minimum,

24
magnetic and electrokinetic

compared, 150
minimum potential, 14
of a system of conductors, 118
of electric field, 120
of rotation, 10
Poynting's law on transfer of,

161
pressure of radiant, 158
principle of the conservation

of, 28
transformation of, 70
Engineering equation, a funda-
mental, 154
Entropy, 70

Clausius on increase of, 71
Eotvos, law of, 91
Equation, fundamental engineer-
ing, 154
of continuity, 35
Equations, of mechanics, basic, 6
Equilibrium, condition of tempera-
ture, 62
for a, system, 6
for liquids, 32
electric, 116

Equilibrium, Gibbs' criteria of
thermal, 99
law of relative proportions in,

89
of a system, D'Alembert's con-
dition for, 25, 26
of floating bodies, 31
of liquids in communicating

vessels, 32
three states of, 28
Equalization of temperature, 62
Equipartition of energy, 76
Equipotential magnetic surfaces,

142
Evaporation, work done by, 84
Exchanges, Prevost's theory of, 109
Expansion, adiabatic, 74

of a gas, Clapeyron's constant,

80
of anisotropic bodies, 65
of bodies with heat, 65
of liquids, 66
Extension, Young's modulus, 19

Falling bodies, law of, 5
Faraday, thei, 128

on impossibility of absolute

charge, 116
on rotation of plane of polar-
ized light, 158
-tube, relation of magnetic
force and moving, 161
Faraday's first law of electrolysis,
128
second law of electrolysis, 128
Fermat, his principle of least time,

167
Floating bodies, action of surface
tension on, 41
equilibrium of, 31

200

INDEX

Flow of heat, for steady state, 66
from point-source, 68
general equation for, 67
, in crystalline medium, 68
of water in pipes, 34
through capillary tubes, 37
Films, light-interference in, 185
First law of thermodynamics, 68
Fluid, incompressible, and electric-
ity compared, 118
resistance to motion of solid
in, 35
Fluorescence, Stokes' law on, 187
Force, action of impulsive, 18

between charged bodies varied

by medium, 116
between magnetic poles, 122
-functions, 22

on magnetic pole exterior to
conductor, 140
Forces, compounded by parallelo-
gram-rule, 8
conservation of living, 22
equilibrated and effective,
D'Alembert's principle of,
25, 26
equilibriium — condition for

three, Lami's theorem, 8
of cohesion and surface ten-
sion, 39
vectorial addition of magnetic
and electrostatic, 144
Form assumed by liquid mass, 39
Fourier, on compounding harmonic

motions, 54
Four-terminal conductors, 131
Freezing point lowered, Raoult's
law, 98
points, depression of by press-
ure, 98

Fresnel's law, 186

Friction, electrification by, 113

sliding, 18

rolling, 18

statical and kinetic, 18
Fusion, of solids and metals, 99

Gas — constant, the, 79
for perfect gas, 79
Gas-laws, applied to solutions, 93
Gas, Joule's law regarding, 77
internal friction of, 78
-molecules, velocity of, 81
number of molecules in a, 82
pressure and energy of, 80
a gram-molecule of, 78
specific heat of given volume

of, 77
-temperature- scale, 64
work of expansion of, 80
Gases, absorption of by solids, 84
basic equation and kinetic

theory of, 80
combinations of, by volume, 97
Dalton's laws of mixture of, 76
diffusion of, Graham's law, 85
effusion of, 85

five fundamental laws of, 72
Graham's law of diffusion of,

85
Henry's law of absorption of,

83
occlusion of, 85
solution in liquid of mixed, 83
specific heat of, 77
work performed when two,

mix, 82
Gauss' and D'Alembert's principle

compared, 27
principle of least constraint,

26
theorem, 117

INDEX

201

Gay-LusBac and Humboldt's law,

97
Gay-Lussac's law, for gases, 74
General principle, mechanical, 7
Gibbs, his phase-rule, 100

on thermal equilibrium, 99
Graham, named crystalloids and

colloids, 102
Graham's law of diffusion of gases,

85
Gladstone and Dale's law, on re-
fractive index, 175
Gravitation, Newton's law of uni-
versal, 4
Gravity, acceleration of, 4

conservation of movement of

center of, 20
pressure produced by, in

liquids, 31
principle of motion of center
of, 21
Guldberg and Waage, on mass-ac-
tion, 88
Gyration, radius of, 11

Hall effect, 159
Hamilton's principle, 27
Harmonic motion, simple, 12'
Heat, absorption of radiant, 102
and energy, difference between

absorbed, 70
condition for minimum pro-
duction of, 132
definition of quantity of, 61
flow, general equation for, 67
law of, for steady state, 66
of, from point-source, 68
of, in crystalline medium,
68
Hess on disengagement of, 90

Heat, in a conductor. Joule's law
for, 132
intensity of radiant, 106
of formation, 101
produced by radiiun, 72
radiation at oblique angle, 106

Heats of reaction, 101

Helmholtz, on consonance and dis-
sonance, 57
on Talbot's law, 188
results of his researches on
sounds, 55

Henry's law, of gas-absorption, 83

Hess, on disengagement of heat, 90

Hooke's law, 19

Huygens' principle, 183

of superposition, 183

Hydrion, formation of, 95

Hydrodynamieal theorem, 37

Hydrostatic paradox, 32

Hysteresis, magnetic, 126

Image, by refraction at spherical
surface, 173
relative size of object and, 168
Images, real and virtual, 167
Impact, between two bodies, 15
for perfectly elastic bodies, 16
Newton's law of, 15
velocities after and before, 16
before and after, 15
Impulse of a force, 18
Inclined plane, descent on, 5
Index of refraction, 170
Induction, coefficient of mutual,
148
coefficients, 148

coil to give maximum self, 149
electrification, produced by,
113

202

INDEX

Induction, law of magnetic, 145,
146
magnetic, 125
Inertia, moment of, 10

principal axes of, 10
Insulators and conductors, 114
Intensity and amplitude, relation
of, 181
of sound, 45, 46
of radiation, 166
Interaction of currents and mag-
nets, 144
Interference in films, 185
of light-rays, 185
of polarized light, 179
of sound, 50
Internal pressure in a conductor,

law of, 142
Intrinsic energy, total not known,

70
Inverse square law, generality of,

161
Ions and molecules, similarity in
behavior of, 94

Joule's equivalent of heat, 132

law for heat in conductors, 132
respecting a gas, 77
of capillary action, 37

Kater's pendulum, 14
Kepler's first law, 7
second law, 7
third law, 7
Kelvin, his absolute temperature-
scale, 63
his minimum energy-theorem,
24
Kerr effect, 159

Kinetic energy of two circuits, 150
theory, fundamental equation
in, 80

Kirchhoff-CIausius, on emissive

power, 184
Kirchhoff's black-body, 104
law of radiation, 104

propositions deduclble

from, 105
for electric currents, 138
Steinmetz's extension of,
138
Theorem on interchange of
electrodes, 131
Kohlrausch, on dilute salt-solu-
tions, 94
Kundt's law of anomalous dis-
persion, 186

Lambert's law of light — emission,
182
of reflection, 169

Lami's theorem, 8

Latent heat of v'aporization, 86

Least action, principle of, 24
constraint, comment on, 27
Gauss' principle of, 26
time, for passage of light-i:ay,
167

LeOhatelier's law of radiation, 188

Lenses, formula for image by, 173

Lenz's law, 146

Lever, law of, 8

Light, absorption of, 182
defined, 165
defraction of, 184
-rays, interference of, 185
rectilinear propagation of, 166
reflection of, from rotating

mirror, 167
rotation of plane of, 158
velocity of, in matter, 170

Limits of audibility, 54

Line-integrals, fundamental, 144

INDEX

203

lane of magnetic^ induction de-
fined, 146
Liquid, boiling point raised by-
salt, 98
-mass, form assumed by, 39
rotation by optically active, 97
-surface, distance between two
elements of, 40
normal pressure on, 39
Liquids, condition of equilibrium
of, 32
diffusion in, 92
equilibrium of, in communi-
cating vessels, 32
expansion of, 66
forces of cohesion of, 33
theorem of Bernoulli regard-
ing, 34
vapor-pressure of mixed, 87
Iibngitudinal motion in an electri-
cal conductor, 143
Lummer, on Draper's law, 188

Mach's statement of conservation

of areas, 23
Magnet, action of current on, 140
compared with small electric

circuit, 142
magnetic force due to a, 124
Magnetic and electric analogues,
126
and electro-kinetic energy com-
pared, 150
energy is potential energy, 148
field. Ampere's law for, 140

magnetism induced! by, 124
fields due to a sphere and mag-
net compared, 124
force and a moving Faraday
tube, 161

Magnetic force, derived from a
potential, 140
due to a magnet, 124
exterior to a linear con-
ductor, 140
within a conductor, 141
hysteresis, 126
induction, 125

law of, 145, 146
line of, defined, 146
pole and equipotential sur-
faces, 142
work done by, 143
poles, law of force between, 122
quantities, relation between,

125
shell, compared with electric

circuit, 140
susceptibility, Curie's law on,
91
Magnetism, five definitions, 122
induced by magnetic field, 124
total charge of, 123
Magnetized' sphere and magnet

compared, 124
Magnetomotive force and induc-
tion in solenoid, 153
Magnets and electric currents, in-
teraction of, 144
Magnus, law of, 157
Mains, law of, 178
Mass-action, law of, 88
Mass, its relation to weight, 4
Mat surfaces, reflection from, 169
Matter or mass, conservation of, 21
Maxwell, law of molecular veloci-
ties, 81
on definition of line of mag-
netic induction, 146

204

INDEX

Maxwell, on K = m', 158

on internal friction of a gas,
78
Mechanical action of currents, gen-
eral principle, 152
force acts on conductor not
current, 147
at surface of charged con-
ductor, 119
parallelogram representa-
tion of, 147
principle, general, 7
Mechanics, basic equations of, 6
Medium, effect of, on force bet-

tween charged bodies, 116
Melting point, effect of pressure on,

98
Mercury columns, capillary correc-
tions for, 38
Mariotte's law, for gases, 72
Metals, resistance-temperature re-
lations of, 134, 135
Michelson's interferometer, 185
Millikan, on electronic charge, 114
Minimum heat condition, 132

potential energy, 14
Mirage, the, 171

Mirror, light reflected from rotat-
ing, 167
reflection from parabolic, 169
Molecular heat, Delaroche and Ber-
ard on, 77
rotation, 97
species, distribution among

several, 89
surface-energy, 91
Molecules, number of in a gas, 82'
Moment of inertia, 10
Momentum, changed by an im-
pulsive force, 18

Momentum, conservation of, 21, 22

moment of, 22
Motion, continuous, produced on
magnets, 144
of center of gravity, 21
Newton's first law of, 3
second law of, 3
third law of, 3
perpetual, impossible, 20
simple harmonic, 12'
vortex, 36
Musical pipe, vibrations produced
by, 52
scales, principle of, 56
tones, Helmholtz on, 55
Multiple resonance, 160
Mutuality of phases, 100

Natural ray replaced by polarized

ray, 182
Neumann's law of crystal zones,
110
on molecular specific heat,
66
Neutralization of acid and base, 96

progress of, 95
Newton, attraction due to a sphere,
4
experimental law of impact, 15
first law of motion, 3
his rule for velocity of sound,

48
law of cooling, 62
law of universal gravitation, 4
on velocity of sound in gases, 48
rings, 187

second law of motion, 3
third law of motion, 3
velocity of disturbances in
elastic medium, 17

INDEX

205

Nicol's prism, 189
Nodes and loops in organ-pipes, 52
Nuclei, effect of on vapor^on-
densation, 88

Occlusion of gases, by solids, 85

Octave, the, 55

Ohm, G. S., law of, for sound, 55

Ohm's law,

respecting sound- vibra-
tions, 55
Steinmetz's extension of,
139

Onbes, on supercondluctivity, 133

Optic axes, 176

Optical rotation, Oudeman's law
of, 98

Organ-pipes, nodes and loops in, 52

Oscillation, convertibility of points
in a pendulum, 14

Osmose, 92

Osmotic pressure and gas-pressure
compared, 92

Ostwald's law of molecular conduc-
tivity, 95

Oudeman's law of optical rotation,
98

Paradox, hydrostatic, 32'
Parrallelogram rule, 8
Partial pressure, law of, 84

of vapor, 84
Pascal's law, 32

Paschen's law for sparking poten-
tial, 119 '
Peltier effect, 155

measure of, 156
Pendulum, convertibility of points
of suspension of, 14
cycloidal, 14
law of compound, 13

Pendulum, simple, 13
Periodic system of chemical ele-
ments, 97
Perpetual motion impossible, 20
Phase, change of in sound-reflec-
tion, 49
rule, 100
Phases, law of the mutuality of,

100
Photo-chemical reaction, law of,

109
Pipes, flow of water in, 34
Plane, descent on an inclined, 5
of polarization, 180
mirror, reflection from, 166
Planetary motion, Kepler's three

laws of 7
Planck's law of spectral distribu-
tion, 109
Plates, vibration of, 52
Platinum-resistance thermometer,

65
Poiseuille's law of flow, 37
Polarization, angle of, 178
by a crystal, 177
by emission and diffusion,

181
by reflection, 178
elliptical, 180
galvanic, 127
of refracted light, 179
plane of, 180
rotation of plane of, 180
Polarized light, laws of interfer-
ence of, 179
rays and natural rays com-
pared, 182
Positive and negative electricity,

114
Potential, definition and meaning
of, 115

206

INDEX

Potential, due to a magnetic sole-
noid, 123

due to a system of point-
charges, 116

energy changes, 118
Power in a circuit, 139
Poynting's law on transfer of en-
ergy, 161
Pressure and energy of a gas, 80

-diflference on two sides of
soap-film, 40

law of partial, 84

little afi'ects electrolysis, 127

of gram-molecule of gas, 78

of radiant energy, 107, 184

of radiation, 107

of sound, 53

produced in liquid, 31

Bankine on variation of, 75

within an electric conductor,
142
Prevost's theory of exchanges, 109
Principal axes of inertia, 10

focus, 173
Prism, deviation of ray by, 174
Propagation of sound, 45
Pure tone, 54

Quantities, relation between mag-
netic, 125
Quantum hypothesis, 109

Radiant energy, Lambert's law
for, 182
pressure of, general case,
184
heat, absorption of, 102
-intensity of, 106
Kadiation, distribution of electro-
magnetic, 160

Radiation, intensity of, 166
Kirchhoff's law of, 104
-law, propositions deducible

from, 105
IieChatelier's law of, 188
of heat at oblique angle, 106
-pressure of radiant energy,

107
Stefan-Boltzmann law of, 107
velocity of, in ether, 165
Radium, heat produced by, 72
Radius of gyration, 11
Rankine, on absorbed heat and en-
ergy, 70
on second law of thermody-
namics, 69
on variation of pressure, 75
Raoult, his law on lowering of

vapor-pressure, 93
Raoult's law on depression of

freezing point, 98
Ratio of units of electricity, 157
Rayleigh, on pressure exerted by
sound-waves, 53
reciprocation theorem, 20
Reaction, heats of, 101
Reciprocation theorem, Rayleigh's,

20
Reflection from mat surfaces, 169
non-spherical surface, 170
parabolic mirror, 169
plane mirror, 166
spherical surface, 168
laws of, of light, 166
phase-change in sound-, 49
light polarized by, 178
eelective, 169

total and critical angle of, 172
Refracted light, polarization of,
179

INDEX

207

Refraction, at a single spherical
surface, 172
atmospheric, 171
conical, 177
double, 176

double, in biaxial crystals, 177
in uniaxial crystals, 176
image by, at spherical sur-
face, 173
index of, for sound, 50
laws of, 170
of sound, 50
(Refractive index, variation of with

density, 175
Regnault, constant of thermal
capacity, 66
his data for specific heat of

gases, 77
his data on gases, 74
on sound-intensity, 46
Relative proportions in equili-
brium, 89
Resistance and capacity, general
relation between, 136
and specific resistance,

changes, 130
combinations, total obtain-
able, 130
condition for a definite, 130
temperature-coefficient, defi-
nitions and applica-
tions, 134
relations for metals, 134,
135
to motion through fluid, 35
-values obtainable from coils,
129
Resistances in parallel, 130

in series, 129
Resonance, principle of in sound, 49
Resultant forces of cohesion at
surface of liquid, 33

Richardson's law of electronic

emission, 156
Richter's law, on interchange of

constituents of salts, 97
Ripples, on surface of liquids, 41

speed of, 42'
Rods and plates, vibration of, 52
Rolling friction, 18
Rotation of plane of polarization,

180, 158

Sarasin and de la Rive on multi-
ple resonance, 160
Scale of gas-thermometer, 64
Scales, principle of musical, 56
Screw and wrench, principle of, 9
Second law of thermodynamics, 69
Selective reflection, 169
Similar systems, vibration of, 52
Similitude, Bertrand's principle

of, 17
Simple harmonic motion, 12

pendulum, 13
Size of object and image, 168
Sliding friction, 18
Snell's law of refraction, 170
Solenoid, induction in, 153

potential due to, 123
Soap-film, formulae relating to, 40
Solution in liquid of mixed gases,

83
Solutions, additive property of
dilute, 94

gas-laws applied to, 93
Sound, defraction of, 50

intensity of, 45, 46
of, in tubes, 46

interference of, 50

pressure of, 53

propagation of, 45

refraction of, 50

208

INDEX

Sound, resonance in, 49

transmission, expansions and

contractions in, 46
velocity of, general principle,
47
of, in air, 47

of, in liquids and solids,
48
-waves, amplitude to be audi-
ble, 54
effect on ear of a system

of, 55
reflection of, 49
Sparking potential, Paschen's law

for, 119
Specific heat of gases, 77

of a given volume of gas,

77
molecular, 66
inductive capacity, 120
inductive capacity and index
of refraction, 158
Spectral distribution, Planck's law
of, 109
Wien's law of, 108
Sphere, fall of small, in viscous
medium, 25
reflection from surface of, 168
Stability, Earnahaw's theorem on,

121
States, theorem of corresponding,

78
Steel spheres, time of contact when

impacting, 16
Stefan-Boltzmann's radiation law,

107
Steinmetz, extension of Kirchhotf's
laws, 138
of Ohm's law, 139
his law for hysteresis, 126

Stokesi' law for fall of sphere, 25
on fluorescence, 187
on light intensity, 183
Stress and strain, Hooke's relation
for, 19
state of electric, 119
Sum of partial tones, 56
Super-conductivity, 133
Surface-tension, action of, on
floating bodies, 41
work of forces of cohesion
in, 39
System, definition of conservative,
19
work done by a conservative,
20

Talbot's law, 188

Temperature, absolute scale of, 63

by resistance-thermometer, 65

by gas-thermometer, 64

critical, 91

definition of, 61

efl'ect of, on balanced chemical
action, 90

-equilibrium, 62

equalization of, 62

of fusion, 99

the cryohydrie, 99
Theorem of Carnot on eSiciency, 64
Thermal capacity, Dulong and

Petit's law of, 66
Thermodynamic temperature-
scale, 63
Thermodynamics, first law of, 68

second law of, 69
Thermoelectric law, for pairs of
metals, 155

power and law of, 154

currents, 154

inversion, 157

INDEX

209

l*homBon effect, 156

theorem on electrolysis, 127

Time of contact of impacting
spheres, 16

Tone, pure, 54

Tones, combinational, 54
the sum of partial, 56

Torricelli's theorem, 33

Total reflection, 172

Transformation of energy, Rank-
ine's statement, 70

Translation and rotation, ana-
logues in, 12

Transverse vibrations of a cord, 51

Trouton's law, on latent heat of
vaporization, 86

Tubes, sound-intensity in, 46

V, the ratio of units of electricity,'

157
Van der Waals' equation, general-
alized, 78
formula, 73

combining Boyle's and
Charles' law, 74
Vapor, condensation of saturated,
88
-formation in vacuimi, 87
-pressure, in communicating
vessels, 87
of mixed liquids, 87
statements regarding, 88
Eaoult's law on lowering
of, 93
Vaporization, latent heat of, 86
Vector relations, 148
Velocities and relation of index of

refraction, 170
Velocities, compounded by paral-
lelogram rule, 8
in media other than ether, 171

Velocities, Maxwell's law of mo-
lecular, 81
of gas-molecules, 81
principle of virtual, 24
Velocity of disturbance in elastic
medium, 17
of light, 165

dependence of, on wave-
length, 174
in matter, 170
of mass-action, 88
of sound and air-density, 47
in air, 47

general principle, 47
in liquids and solids, 48
Newton on, 48
of transverse wave along cord,
51
Vectorial adidition of magnetic

forces, 144
Vector-potential, 150
Vena contracta, 33
Vertical distance between elements

of liquid-surface, 40
Vibration of bells, 53
of plates, 53
of rods and plates, 52
Vibrations, addition of simple
sound, 56
number produced by musical

pipe, 52
of a cord, 51
of similar systems, 52
Vinal and Bates, their value for

the " Faraday," 128
Vires vivae, 22

Virtual velocities, principle of, 24
Visibility, Draper on, 188
Vision, non-reversible, 186
Volatilization, 86

210

INDEX

Volta, on contact-difference of po-
tential, 137
Vortex motion, 36

Water-waves, speed of, 41
Wave-length of maximum energy,

Wien'a law of, 108
Wave, velocity of transverse, along

cord, 51
Waves, speed of water, 41
Weight and mass, proportionality
between, 4
loss of, by immersion in fluid,
31
Wenner, on four-terminal conduc-
tors, 131
Wiedemann-Franz ratio, 133
Wien's displacement law, first
statement of, 107
second statement of, lOS

Wien's law of spectral distribu-
tion, 108
of wave-length of maxi-
mum energy, 108
Work done by electromotive force,
139
by magnetic pole in
threading a circuit, 143
by evaporatioli, 84
in displacement of electrified

system, 118
performed when two gases
mix, 82
Wrench and screw, principle of, 9

Young's modulus, 19

Zeeman-effect, 187
Zero of absolute temperature, defi-
nition of, 63
Zone-plate, 189