heaviside oliver - electrical papers vol 1

ELECTRICAL PAPERS
BY
OLIVEK HEAVISIDE
IN TWO
VOLUMES
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PKEFACE.
This Eeprint of my Electrical
Papers

comes
about
by
the
union
ot
a variety of reasons and
circumstances.
First, there was
a
demand
for certain
of my
papers,
especially
for
a set
relating
to Electromagnetic
Waves.
Although
I
distributed
49
copies in a collected form,
I was asked
for
more,
and
also

received
assurances that a republication
of my papers in
general
would
be
useful. But this demand
was too small
to lead
to
an
immediate
supply.
Secondly, however, at the beginning
of 1891 it
was
proposed
to
me by the
publisher of
The Electrician that my
articles
on
"Electro-
magnetic Theory," then commencing and
now
continuing
in
that
journal, should

be
brought out later in
book form.
This
was
satis-
factory so far as
it went,
but it brought the
question
of a
reprint
of
the
earlier papers to
a
crisis. For,
as the later
work
grows
out
of
the earlier, it seemed
an
absurdity to leave
the
earlier
work
behind.
Thirdly, the experimental work

of
Hughes in
1886, furnishing
the
first
evidence
(in the sense
ordinarily
understood, though
other
evidence
was
convincing to a
logical mind) of
the truth of the theory
of
surface
conduction along wires
under
certain circumstances, first
advanced
by
me a
year
previously; followed in
1887-8
by the experimental
work
of
Hertz and Lodge on electrical vibrations

and electromagnetic
waves,
still
further confirming the above,
and also broadly confirming
the
truth
of
the
theory of the propagation of
disturbances along
wires
I
had
worked
out
on the
basis of
Maxwell's
doctrine of the
ether
in
its
electromagnetic
aspect, and the
correctness of
Fitzgerald's
ideas
concerning
electrical

radiation,
and of
the nature of
the
energy-flux
developed
by Poynting and myself from
Maxwell's
theory," were
the
means
of stirring up an amount of interest
in
this theory that was
quite
wonderful to witness. That
electrical
disturbances were pro-
pagated
in time through
a
medium was raised
from a
highly probable
yl
ELECTRICAL
PAPERS.
speculation
to
an

established
fact.
A
careful
study by
electrical
physicists
of
Maxwell's
development
of
Faraday's
ideas
became
im-
perative,
especially
on
the
Continent,
where
Maxwell's
work
had
hitherto
met
with a
singular
want
of

appreciation,
arising,
I
believe,
mainly
from
misconception
of
his
theory
of
electrical
displacement.
This
misconception,
I
think,
exists
even
now,
since
some
writers
apply
to
Maxwell's
theory
ideas
and
processes

which
seem to
me
to
be
thoroughly
antagonistic
to
his views.
But
even in
England
the
theory
had
licen
much
neglected.
For
one
thing,
much
attention
was
being
devoted
to the
dynamo.
Then
again,

the
form in
which
Maxwell
presented
his
theory
did not,
I
think, display
its merits
in a
manner
they
deserve,
and
suited
for
legitimate
development.
Moreover,
the
contrast
between
the
old
notions
of electricity
and
Maxwell's

was so
great
that
mere
natural
conservatism
stood in
the way.
A
stimulus
was
wanted
in
favour
of a
theory so
ill-understood
and
(apparently)
so
far
removed
from
actual
observation.
But
the
experimental
stimulus
having

come,
the
result
has been
a tiood
of other
experimental
work,,
mostly
tending
to
confirm
the
general theory.
A
work,
therefore,
like
the
present,
which
is,
in
the main,
devoted to
the
elucidation
and
extension
of

Maxwell's
theory,
and
of the
mathematical
methods
suited
to
it,
should
have
a
legitimate
place amongst
others.
Though
it was
nearly
all
done
before
the
electrical
"boom" began,
it may
not
be
out
of
date,

and
may
perhaps be,
in some respects,
ahead.
Fourthly,
it
had
been
represented to me
that I
should
rather
boil
the
matter
down
to
a
connected
treatise
than
republish in
the
form
of
detached
papers.
But a
careful

examination and
consideration
of
the
material
showed
that it already
possessed, oh the
whole,
sufficient
continuity
of
subject-matter
and
treatment, and even
regularity
of
notation,
to
justify its
presentation in the original
form. For,
instead
of
being, like
most
scientific
reprints, a collection of
short
papers on

various
subjects,
having
little coherence from the
treatise point of
view,
my
material
was all upon
one subject (though with many
branches),
and
consisted mostly of
long articles, professedly
written
in a
connected
manner,
with uniformity
of
ideas and
notation. And
there
was so
much comparatively elementary
matter (especially in
what has
made the first
volume)
that the work

might
be
regarded not
merely as a
collection of papers for reference purposes, but also as
an
educational work for students of theoretical electricity.
As
regards the question, "Will it pay?" httle need
be said. For,
fifthly,
however absurd
it
may seem, I do in all seriousness hereby
PREFACE.
vii
declare that I am animated
mainly by
philanthropic motives.
I desire
to
do good to my
fellow-creatures, even to
the Cui bonos.
Having thus
justified
the existence of this
reprint, it
remains for
me to indicate

the
general nature of the
contents,
and, in
doing so,
I
will
imagine myself (usually) to be
addressing
an
intelligent and
earnest student, who means business. The
first twelve
articles,
pp.
1
to
46,
are
on matters
dealing mainly
with telegraphy,
and
are but
loosely
connected. But
a
sort of
continuity
then begins,

for the
next
eight
articles,
up to
p.
179,
deal
mainly with the
theory of
the pro-
pagation
of variations
of
current along
wires, beginning
with
applica-
tions
of
the simple electrostatic theory of
Sir W.
Thomson
(1855)
to
cables
under different circumstances
(terminal
resistances,
condensers,

etc.,
intermediate
leakage, etc.), and
folloAved by
extensions to
include
self-induction,
or the influence of the
inertia
of the
magnetic
medium,
and
the
mutual
influence, both
electrostatic
and
magnetic,
of
parallel
wires.
The
last of
this set.
Art.
xx., has
not been
printed
before.

It
is,
however, in
its
right place, having
been
written in 1882
as
a
sequel
to
the
papers preceding it.
It may
be
found
useful to
those
who
are
interested in
the
subject as
an
intermediate
between
the
papers
of this
set and the later

series in
the second
volume,
wherein
the
subject
is treated
from a
more
comprehensive
point of
view,
viz.,
Maxwell's
theory of the ether as a
dielectric.
There is no
conflict.
The
later
investigations
are
generalizations of
the
earlier,
or the
earlier
are
specializations
of the later;

and
I
can
recommend
the
earnest
student
to read the earlier set
first, before
proceeding
to
the
more
advanced
treatment
in the
later
set.
We
next come
to a
series of
papers
published
in The
Electrician
between
the
autumn of
1882 and

the autumn
of
1887,
when
under
the
editorship
of Mr.
C.
H. W.
Biggs,
to
whom
I
desire
to
express
my
obligations
for the opportunity
he
gave
me
of
exercising my
philanthropic
inclinations, in the face,
as
I
afterwards

learnt, of
con-
siderable
opposition. These
papers
extend
over
about 500
pages,
mostly
in this, partly in the second
volume,
and
are
usually
long
articles,
with
continuity.
They
relate
to
electrical
theory
in
general.
Beginning
with the abstract relations
of
the

electrical
quantities,
and
the
mathematics of
the subject
in vector
form
(of
an
elementary
kind),
including
a
general
theory of
potentials
and
connected
quantities
expressed
in the rational units I
introduced,
we
pass
on to
the
con-
sideration
of

the energy of the
electric
and
magnetic
fields,
and
the
transformations
concerned in the
phenomenon
of
the
electric
currents
viii
ELECTRICAL
PAPERS.
including
an
account of
Sir
W.
Thomson's
theory
of
thermo-electricity.
Next
comes a
pretty
full

study
of the
theory
of
the
propagation
of
induction and electric
current in
round
cores, to
which I
was led
by my experiments
with
induction
balances,
in the
endeavour
to
explain
certain
phenomena
observed.
The
analogy with the
motion
of
a
viscous

liquid
is also
introduced
and
developed.
Lastly
we come
(1885)
to a
more
comprehensive
treatment of
electromagnetism, based
upon
Maxwell's
theory,
in
"Electromagnetic
Induction
and its
Propagation,"
of
which the
first
half is
in this
volume.
I here
introduce a new
method

of treating
the
subject
(to
which
I was led by
considering
the
flux of
energy),
which may
perhaps
be appropriately termed the Duplex
method,
since its
main
character-
istic is
the exhibition of
the electric,
magnetic,
and
electromagnetic
equations in
a
duplex form, symmetrical
with respect
to
the electric
and magnetic

sides, introducing a new
form
of
fundamental
equation
connecting magnetic current with electric force, as a
companion to
Maxwell's well-known equation connecting magnetic
force
and electric
current.
The duplex method is eminently suited
for
displaying
Maxwell's theory, and
brings to light many
useful relations
which
were formerly
hidden from
view
by
the intervention
of the
vector-
potential and
its parasites.
There is considerable
difficulty in
treating

electromagnetism
by means
of Maxwell's equations of
propagation
in
terms
of these quantities,
as
presented in
his
treatise.
The
difficulty
is greatly
increased, if not
rendered practically insuperable, when we
pass
to more advanced
cases involving
heterogeneity
and eolotropy and
motion
of the medium supporting
the fluxes.
Here
the duplex method
furnishes what is
wanted in
general investi
cations,

and is the basis of
"
Electromagnetic Induction
"
and
of
the whole of the second volume.
The electric
and magnetic
forces
(or fluxes)
and
their
variations
are
the
immediate
objects of attention
in the duplex method, whilst
potentials
are treated
as auxiliary
quantities Avhich
do
not
possess
physical significance
as
regards the
actual state

of the
medium,
though
they may
be useful
for calculating
purposes.
Towards
the end
of this
volume
the electric
and magnetic
stresses
are considered.
The
treatment
was
interrupted,
but a later
paper,
"On
the Forces,
Stresses,
etc.," in
the second
volume contains
what
was to have
been its

continuation,
and
developments
thereof.
The
reason of the
break
was that
the
interest
excited
by Professor
Hughes's
188G
experiments
made it
desirable
that I
should
at once publish
other
matter long
in hand,
namely,
developments
of
the views relating
to
PREFACE.
ix

the functions of wires and
of the
dielectric surrounding
them,
explained
in Section ii. of
"Electromagnetic Induction."
These
developments
are
contained in the second
half of that article
(Art.
XXXV.,
vol. ii.)
and
in the article "On
the
Self-Induction of Wires
"
(Art.
XL., vol.
ii.),
published
in
the Philosophical
Magazine in
1886-7.
The
reader

is re-
commended to read the
former first,
as
it is much
more elementary
than the latter, which contains
mathematical
developments
and
ex-
aminations
unsuited to The Electrician.
The subject
is the
diffusion
of electrical waves into wires from
their boundaries
and the
propaga-
tion of waves along the wires
through the insulator
surrounding
them,
supplying the wires
themselves with the
energy they absorb.
Also
the
self-induction of various

arrangements of
apparatus, and
the
theory of induction
balances.
But in
the year 1887
I came, for a time, to a dead
stop, exactly
when
I
came
to
making practical applications in detail
of my theory,
with
novel
conclusions
of
considerable practical
significance
relating
to
long-distance
telephony (previously partly published),
in
opposition
to
the views
at that

time officially advocated.
On
the official
side
the
electrostatic
theory was upheld, with full application
of the re-
tardation
law of
the
inverse-squares
to
telephony
; inertia being
regarded
as a disturbing factor, assumed to be of
a
harmful
nature,
but
argued
to be quite
negligible in long copper-circuits,
because
telephony through
such circuits of
low resistance
was so
successful.

On the
other side was my theory asserting that owing to the .rapidity
of
telephonic
changes of current
inertia was
not
negligible,
that
it
was
often important, and sometimes, as in the case of wires of low
resistance,
even a
dominating
factor.
Furthermore,
that it was
not
harmful,
but was, on the contrary,
beneficial
in
its efi'ects, which
was,
in
fact,
the very reason why
long-distance telephony was successful.
Then,

as regards the measure of the
inductance, it was asserted
on
the
official side that the inductance per
centim.
of a copper suspended
circuit
was (in electromagnetic
units) only
a
minute fraction of unity
;
whilst
on
the other side it was declared to be
some hundreds of times
as
big,
say from
10
to
20 per centim. of circuit. Here was
the most
complete
possible antagonism between
my
views and official
views,
both

in
principle and in
detail,
and a
careful
consideration and
dis-
cussion
of the matter was desirable. Yet
I found
it next to impossible
to
ventilate
the matter.
First of all, I was
prevented by
circumstances
which
need not
be
mentioned from
bringing the
matter before the
S.
T.
E. and
E. in
the
spring of
1887 (Art.

xll,
vol. ii.).
Next,
a
little
X
ELECTRICAL
PAPERS.
later,
the editor
of the
Philosophical
Magazine
could no longer
aflPord
space
for the
continuation of my
article on "The
Self-induction
of
Wires,"
Part
VIII., dealing with the
non-distortional
circuit and
telephony
(p.
307,
vol. II.).

Thirdly, after a
partial exposition
in Sections
XL.
to
XLVI.
of
"
Electromagnetic
Induction,"
a
change of
editor occurred,
and
the new editor asked me to
discontinue. He politely
informed me
that
although he had made
particular enquiries amongst
students who
would
be likely to
read
my papers,
to
find if anyone
did
so, he
had

been
unable to discover
a
single one. Fourthly, he
returned
a
short
article
(Art. XXXVIII., vol. II.) on the same subject of
long-distance telephony,
which
pointed out official
errors
in detail, and directed
attention to
the contrary
results
indicated
by my theory,
this
paper
having been
in
official
hands. And lastly, three
other
journals declined the
same,
for reasons
best known to themselves.

Perhaps
it was thought that official views were so much more
likely
to
be right
that it
was
safe
to
decline the discussion of
novel views
in such striking
opposition thereto. There seemed also to be
an idea
that official
views, in virtue
of their official nature,
should not be
controverted
or criticized.
But there seems something wrong
here,
as the
above facts, and
the later evidence
in
support
of
my views,
have

shown. For
what other
object have scientific men than to get
at the truth,
and how is
it
to
be done without free discussion
1
The student is
particularly
recommended
to read the articles referred
to, not merely
on account
of the
telephonic
application,
but because
of the simplicity
of treatment
which the
distortionless circuit
allows,
and as
a preliminary
to the study
of Electromagnetic
Waves, to
which

it
supplies
a royal
road.
The
action of leakage in
promoting
quick
signalling
is
treated
of in the
early
set in this volume; now the
inductance
of
the circuit has
also
a beneficial
eff'ect
; and the two
together conspire
to
annihilate
the
distortion
which the resistance
of
the circuit produces.
The

same
occurs
(approximately)
without
the
leakage,
by the action
of
self-induction,
if the
frequency of
alternation
be
sufficiently
rapid, and
the
wires of not
too great resistance.
Now in the
theory of
electromagnetic
waves there is
a similar
pro-
perty, which
throws
considerable
light
upon
the subject

of
waves
in
general.
I
had
introduced,
in
1885,
for
purposes
of symmetry,
the
fictitious
quality of
magnetic
conductivity.
When
its effects
upon
the propagation
of waves
in
a real
conducting
dielectric
are
enquired
into, it is found
to

act
contrary
to the real
conductivity,
so that
the
distortion
due to
the latter
can
be
entirely
removed
by having
duplex
PREFACE.
xi
conductivit3^
How this strange
result comes to
pass
may
be readily
understood
in detail by studying
the theory of the distortionless circuit,
in
which
the leakage conductance and
the resistance

of
the circuit
act
oppositely
in respect to
distortion.
The
remainder of the second
volume
consists
of
investigations
growing out of
"Electromagnetic Induction,"
viz.,
the set relating to
electromagnetic
waves ; the
electromagnetic wave-surface
;
propaga-
tion in
a
uniform conducting
(duplex) dielectric,
with
the application
to
plane
waves, either

free or along straight
wires
;
the connected
theory of
convection currents; the
theory of resistance and conduct-
ance
operators ; with a few
miscellaneous
papers concerning
propaga-
tion in moving media;
finishing with an article
discussing
the forces
and
stresses concerned in the
electromagnetic field.
Acting under advice, I have not
carried out my original design
to
make large additions.
Limitations of space
prevented this,
and
T
have confined myself to
an occasional small
addition

or footnote.
These are
put in
square
brackets, all such signs
in
the original papers
being cancelled. For the rest, I have
corrected misprints and obvious
slips,
and
have made verbal
improvements and omitted
occasional
redundant matter.
The scientific reader may
therefore refer
to this
work as to the
original
papers.
Their dates, etc.,
are given
at the
commencement
of
the articles.
I
have introduced
uniformity in the notation

connected with vectors,
though
there
was little
change
to be
made except to
put all
vectors
into
Clarendon
black type, as in some
of the later of
the
original
papers.
The
vector-algebra, I should
mention, is of
a
rudimentary
kind,
and has nothing to do
with quaternions ; first,
only addition
and
the scalar product are used,
whilst later on
the
vector

product
is
introduced and freely
employed.
On
the vexed
question of
vectors, the
conclusions
to
which
I have
gradually
settled down are
as
follows
:
—The notorious
difficulty
of
understanding
and
working
Quaternions will always be a bar to
their
serious
practical use
by
any but
mathematical

experts.
But, on the
other
hand, a vector
algebra
and analysis
of
a
simple kind, independent
of
the quaternion,
and readily
understandable and workable, can with
great
advantage take the
place of much
of the usual cumbrous
Cartesian
investigations,
and be
made
generally useful in
all physical
mathematics
concerning
vectors, and be
employed,
comparatively
speaking, by
the

multitude.
It should
obviously
be
harmonized
with the
Cartesian
mathematics. The quaternionic
system is defective
in this respect;
xii
ELECTRICAL
PAPERS.
in its
very
nature it
cannot be
thus
harmonized.
The system I
recommend
is
fully
explained in
"Electromagnetic
Theory,"
chapter in.
{The
Electrician, Nov.
13,

1891,
and
after).
The
numerous
letter
prefixes
of
the
quaternionic
system,
which greatly
contribute to
the
difficulty
of reading
quaternionic
investigations,
are
abolished,
retaining
only
the
symbol
V
before
a
vector
product.
Another

difficulty is
in the
scalar
product of
Quaternions being
always the
negative
of the
quantity
practically
concerned.
Yet
another is the
unreal nature
of
quaternionic
formulae.
The
terms do not stand
for
physical
quantities. Again,
in
most
physical mathematics, the
quaternion
does not
even
present
itself for

consideration, or,
at
any
rate, may be
readily
dispensed
with.
Lastly, the
establishment of
vector-algebra
on
a
quaternionic
basis is
very
hard to understand, as
chapter II.
of
Professor Tait's
treatise
shows.
These troubles are
obviated
by
the
method I
follow,
basing
the
whole upon the definition

of
a
vector, and
of the
scalar
and
the vector product
of a pair of vectors.
The notation is
harmonized
with
Cartesians and
transition is
readily made.
We
may,
indeed,
regard a
vector
investigation, from this point of
view,
as a
systemati-
cally
abbreviated
Cartesian
investigation, and the
latter
as
the full

expansion of the
former.
And, considering that
the
bulk of special
investigations
are
necessarily scalar, it seems to me that we should
keep
in
touch
with them as far as
possible, and
not try to abolish
the
Cartesian
method, but
make it
a
useful auxiliary
to the vector
method.
That
quaternionic experts may do valuable work
is un-
doubted,
but how can the
bulk of mathematicians
possibly under-
stand it ?

Lastly, on
the question of
units,
it is not, I think, generally
under-
stood
that
the ordinary
electrical units
involve an absurdity
similar
to
what
would
be
introduced
into the metric system
of common
units
were
we to
define
the
unit area to be the area
of
a
circle
of unit
diameter. A
rational system

of
units founded
upon
a rational defini-
tion of
a pole
(electric or
magnetic),
associating
the
unit pole with
one line
of the
corresponding force or flux instead
of with iir,
was
employed
by
me in
some of the earlier
papers
(1882-3),
but was
not
carried out
further because I
believed
that
a
reform

of the
electrical
units was
impracticable.
Now, I
had
commenced
"
Electromagnetic
Theory"
in January,
1891,
with rational
units merely
to
exhibit
the
theory in a
fitting manner, intending to transform
later
to the
common
units. But 1
came afterwards to the definite
conclusion
that
a thorough
reform
of the
electrical units is practicable

and
perhaps
indeed
inevit-
PREFACE. xiii
able, and
shall
therefore
continue
the use
of
the
rational
units.
But
this
decision
was
only
arrived at
after
a
considerable
portion of
this
volume
was
in type.
I
have,

therefore,
not
altered
to
rational
units
throughout,
as
I
should have
preferred;
though,
on
the
other
hand,
the
long
article
Lii. at
the end
of
the
second
volume
remains
as
it
was
written,

in
rational
units.
But
we
are,
in the
opinion of
com-
petent
judges,
within a
measurable
distance
of a
reform
of
the
ordinary
heterogeneous
British
units, by
adoption
of
the
metric
system.
I
hope
and

believe
that the
smaller
reform I
advocate
will
be
determined
upon
by
electricians.
Paignton,
Devon,
June
16,
1892.
CONTENTS
OF VOL.
I.
Paqr
Art. 1.
COMPARING
ELECTROMOTIVE FORCES.
-
- -
- 1
Art.
2.
VOLTAIC

CONSTANTS. 2
Art. 3.
ON
THE BEST
ARRANGEMENT OF
WHEATSTONE'S
BRIDGE
FOR
MEASURING A GIVEN RESISTANCE
WITH
A GIVEN
GALVANOMETER AND BATTERY.
-
3
Art. 4.
SENSITIVENESS
OF
WHEATSTONE'S
BRIDGE.
- - 8
Art. 5.
ON
AN
ADVANTAGEOUS
METHOD OF
USING THE
DIFFERENTIAL
GALVANOMETER FOR
MEASURING
SMALL

RESISTANCES.
13
Art. 6.
ON
THE
DIFFERENTIAL
GALVANOMETER.
15
Art.
7.
ON
DUPLEX
TELEGRAPHY
(Part L).
18
Art.
8.
ON
DUPLEX
TELEGRAPHY
(Part II.).

24
Art. 9.
NOTES ON
MR.
EDISON'S
ELECTRICAL
PROBLEM.
-

34
Art. 10.
ON
THE RESISTANCE
OF
GALVANOMETERS.

38
Art. 11. ON
A
TEST
FOR
TELEGRAPH
LINES.

41
Art.
12.
ON
THE
ELECTROSTATIC
CAPACITY OF
SUSPENDED
WIRES.
42
Art.
13.
ON
TELEGRAPHIC
SIGNALLING

WITH CONDENSERS.
-
47
Art.
14. ON
THE
EXTRA
CURRENT.
-
-
53
Art.
15. ON
THE
SPEED
OF
SIGNALLING
THROUGH
HETERO-
GENEOUS
TELEGRAPH
CIRCUITS.
. -
- -
61
Art.
16.
ON
THE
THEORY

OF
FAULTS IN CABLES.

71
Art.
17.
ON
ELECTROMAGNETS,
ETC.
95
xvi
ELECTRICAL
PAPERS.
Paois
Art. 18.
MAGNETO-ELECTRIC
CURRENT
GENERATORS.
- -
11'2
Akt.
19.
ON INDUCTION
BETWEEN
PARALLEL WIRES. - - lliJ
Art. 20. "CONTRIBUTIONS
TO
THE
THEORY
OF THE PROPAGA-

TION
OF
CURRENT
IN WIRES. 141
Art. 21. DIMENSIONS
OF
A MAGNETIC POLE. - - - -
179
Art. 22.
THEORY OF
MICROPHONE
AND
RESISTANCE OP
CARBON CONTACTS.
181
Art. 23.
THE
EARTH AS A RETURN CONDUCTOR. - -
-
190
Art.
24. THE
RELATIONS
BETWEEN MAGNETIC
FORCE
AND
ELECTRIC CURRENT.
Section
1. The Universal Relation between a
Vector

and its Curl.
195
Section 2. The
Potentials of Scalars and
Vectors.
- - -
201
Section
3. Connected General Theorems in
Electricity
and
Magnetism.
:
206
Section 4.
The Characteristic Equation of
a
Potential,
and
its
Solution.
-
213
Section 5.
Relations
of Curl
and Potential,
direct and inverse.
Scalar
Potential

of
a
Vector.
218
Section 6.
Magnetic
Force
of Return Current through
the
Earth,
and AUied
Matter.
224
Art.
25.
THE
ENERGY OF
THE ELECTRIC
CURRENT.
Section 1.
The Mutual Potential Energy of
Magnetic
Shells
and
Linear
Currents.
231
Section
2.
Variation of the Energy with

the Size
of the
Systems.
The
Mutual Energy of
any
two Distributions
of
Current.
-
237
Section
3.
The Self-Energy of
a
Current
System.
- -
242
Section
4.
Probable Localisation of the Energy.
Division
of
any
Vector
into a
Circuital and
a Divergent
Vector.

-
- -
247
Art.
26.
SOME
ELECTROSTATIC AND MAGNETIC
RELATIONS.
§
1.
Comparison
of
Divergent
and Circuital
Vectors.
-
- -
255
§
4.
Extension of
Electrostatic Properties.

258
§
7.
Complete
Scheme
of
Potentials.

262
§
10.
Energy
Properties.
264
§
15.
The
Operator
V
and its Application.
268
S
18.
Disjjlacement
and Fluid Motion
Analogies.
-
- -
273
CONTENTS.
xvii
Page
Art. 27.
THE ENEEGY
OF
THE ELECTEIC
CUEEENT.
Section 5a. The

Induction
of
Electric
Currents.
-
- - -
277
Section
5/>.
Transference of
Energy.
Ohm's Law.
-
- -
282
Section
5r.
Ohm's Law and Eolotropy. The
Eotational Property.
286
Section
6a.
The
Conservation of Energy.
291
Section 66.
Application of
Conservation
of Energy to a Steady
Current. 297

Section
7.
The Minimum
Heat Property
in Conductors, Linear
or Continuous.

303
Section 8.
Thermo-electric
Force. Peltier
and Thomson effects. 309
Section 9a.
The First
and Second
Laws of Thermo-dynamics.
-
315
Section
9b.
Application of
the
Second
Law
to Thermo-electricity. 318
Sectiox 10. The Thermo-electric
Diagram
and
its Theory.
-

-
321
Section 11. The Thermo-electric
Theory of Clausius, and Objec-
tions thereto.

327
Section
12.
On
Speculation and
Explanation
in
Physical Ques-
tions.
331
Section
13.
Chemical Contact Force.
337
Section 14. Contact
Force
and Helmholtz's
Electric Layers
- 342
Section 15. Electric
Layers
do not imply
Electrification.
-

-
346
Impressed Force
and Potential.
- - . .
349
Art.
28.
THE INDUCTION
OF CUEEENTS
IN COEES.
-
-
-
353
§
2.
Geometrical and Electrical Data.
354
§
3.
Inductance of Coil-circuit.
355
§
4. Eesistance
of Coil.
356
§
5. Magnetic
Force and Current in Core.

357
§
6.
Electric Force and Current in
Core. 357
§
7.
Electric
Force and Magnetic Force in Core. -
-
-
357
§
8.
Coil-Current and Core Magnetic Force.
. . . .
355
§
9.
E.M.F. in the Coil-Circuit.
358
§
10.
Oscillatory Currents.
359
§
11. Waves of
Magnetic Force.
361
§

12.
Amphtude of Magnetic Force. 362
§
13. Heat in Core and
in
Coil. 363
§
14, Examples,
and
Eemarks
on
Variable
Permeability.
- -
365
xviii
ELECTRICAL
PAPERS.
Page
367
369
370
§
15.
Coil-Current
in terms
of
E.M.F.
§
16.

First
Approximation
to
Effect
of
Core-Currents
in
Altering
Amplitude
and
Phase
of
Coil-Current.
-
- - -
§
17.
Fuller
Examination
of
Reaction
of
Core
on the
Coil. -
§
18.
Induction
in a
Divided Core.

374
§
19.
Transmission
of
Energy
into a
Conducting
Core.
- -
377
§
20.
Comparison
of
Induction
in
a
Core
with a Case
of
Fluid
Motion.
378
§
21.
Normal
or
Harmonic
Distributions

of
Magnetic
Force.
-
384
§
22.
Example
1.
Coil-Circuit
Interrupted.
. . - -
388
§
23.
Note
on
Earth-Currents.
-
389
§
24.
Determination
of
Consta'nts.
Conjugate
Property.
- -
389
§

25.
Special Case. Hq
=
constant.
391
§
26.
Magnetic
Energy
and
Dissipation,
391
§
28.
Remarks
on
Normal
Systems.
,
-
392
§
29.
Example
2.
Coil-Circuit Closed.
Coil
of
Negligible
Depth.

394
§
30.
Description
of Fig.
3.
Subsidence
of
Induction in
Core.
-
397
§
31.
Telegraph
Cable
Analogue.
399
§
32.
Example
3.
Coil of any
Depth.
400
§
33.
Two
Coils,
with

Cores,
in Sequence.
402
§
34.
Three
similar
Coils
and
Cores in Sequence. - - -
405
§
35.
Any
number
of
Coils in
Sequence.
406
§
36.
Equal
Coils
with Cores,
in
Parallel.
406
§
37.
mj

Coils
in
Sequence
with
mo
Coils in
Parallel. -
- -
408
§
38.
Any
Combination
of Equal
Coils, with
Cores. - - -
410
§
39.
Dissimilar
Coils.
Characteristic
Function of
a
Linear
System
of
Conductors,
and
Derivation of the Differential

Equation.
-
- - - 412
A.RT. 29.
REMARKS
ON
THE
VOLTA FORCE, ETC.
-

416
Abt. 30.
ELECTROMAGNETIC
INDUCTION
AND ITS PROPAGA-
TION.
(FiKST
Half.)
Section 1.
Rough
Sketch of
Maxwell's
Theory.
-
- - -
429
Conductivity,
Capacity,
and
Permeability.

-
-
- 429
Sectiox 2.
On
the
Transmission
of Energy
through Wires
by
the
Electric Current.

434
CONTENTS.
Page
Section 3.
Eesumption of Kough Sketch.
Extensions.
- -
441
Real
Transient,
and
Suggested
Dissipative
Magnetic
Current.

441

Effect of gr in
a
Closed Iron
Eing.

441
First Cross- Connection of Magnetic
and
Electric
Force.
443
Magnetic Energy of Moving Charged Spheres.
-
- 446
Section
4.
Completion of Rough Sketch. 447
Second
Connection between Electric
Force and
Magnetic Force 447
The
Equation of Energy and its Transfer.
-
- -
449
Differential Equations of E and H.
-
- - -
450

r.
and jx
Self-Conjugate
; k
not necessarily so.
- - 461
Section 5.
Impressed Magnetic
Force. Intrinsic Magnetisation.
451
Magnetic Energy. Double
Work of Magnet.
- -
455
Section 6. The
Mechanical Forces and
their Potential Energy.
-
457
Section
7.
Work
done by
Impressed Forces
during Transient
States.
-
- -
-
462

Section 8.
Electric
Energy. Circuital
Displacement. -
-
466
Simple
Example of Closed
Displacement.
-
- -
468
Section 9.
Impressed
Electric Force
in
Dielectrics. - - -
471
Section 10.
Dielectric
Displacement
and
Absorption.
-
- -
476
Section 11. The
Principle
of Thermal
Resistance. - - -

481
Section 12.
Electrisation and
Electrification.
Natural
Electrets.
488
Section 13.
Simultaneous
Conduction
Current and
Elastic Dis-
placement.

494
Section 14. Conduction
and
Displacement
(continued).
- -
499
Section 15. Conduction
and
Displacement
(conclusion). - -
504
Various
Expressions
for the
Electric

Energy.
- -
506
Section
1G.
Magnetic
and
Electric
Comparisons.
-
- - -
509
Section
17. The
Magnetic Field due
to
Impressed
E.M.F. -
-
516
Section 18.
Normal
Electromagnetic
Systems.
Energy
Conjugate
Properties.
520
Section 19.
Remarks

on
Normal
Electromagnetic
Systems.
Con-
ditions
of
Possibility of
Oscillatory
Subsidence.
Equal Roots,
and
their
Effects. - - -
525
Section 20.
Some Cases
of
Subsidence
of
Displacement. - -
&31
Retardation
in a
Medium
in
which /x
=
0,
cjk

= constant.
532
Section 21. A
Network of
Linear
Dielectric
Conductors,
or
of
Shunted
Condensers.
•''36
ELECTRICAL
PAPERS.
?AGK
Section
22. The
Mechanical Forces
and Stresses.
Preliminary.
The
Simple
Maxwellian Stress.
.
- -
-
542
First
Electromagnetic
Application.

-
. -
-
545
Section
23. The
Mechanical Action
between
two Regions.
-
-
548
Summary
of some
Results of
Vector Analysis.
-
-
548
Limitation
to a
Bounded
Region.
. - -
-
549
Internal
and External Energies.
- 550
Mechanical

Force between two
Regions.
- - -
551
The External
P
in terms
of the
Surface P^^.
- -
553
Annihilation of the
Surface Current. -
- -
554
Annihilation of the
Surface Matter, when possible.
-
554
P
in terms
of the Surface Hq.
554
A in terms
of the Surface Hq.
555
Remarks on these
Formula.
. . . . . 555
Section 24. Action

between a
Magnet and
a
Magnet, or
between
a
Magnet
and
a
Conductor
supporting an
Electric
Current.
The Closure of the Electric Current.
Its
Necessity.
-
-
556
CORRECTIONS.
2h-\
,
2h\^
p.
44,
8th line from end,
for
-
—
J

read
—
1
.
p.
99,
last
formula,
for
L^
read
L,
and
for
i?2
read
B.
p.
267,
27th line,
for
E.3 read S
B.,.
p.
415, for
§
39
read
§
40.

p.
555,
equation
(39rt), for
//^
read Eq
I
ELECTPJCAL
PAPERS.
I—
COMPAEING
ELECTROMOTIVE
FORCES.
[English Mechanic, July 5th,
1872,
p.
411.]
The
following null
arrangement for
comparing
electromotive
forces
is,
as far as I
am
aware, original
:
—
Join

up the
two
batteries
E^ and
E^
with a
galvanometer, as
in the diagram, so that
their
currents
go through
it in opposite
directions. Also insert resistances
E and r.
Let
x and
y
be
the
unknown
resistances of the bat-
teries, and
ip i.p
%,
the
currents
in the
three
branches.
Then we have

'
; ^1
—
^2 — ^3
~
^'
Now, by
altering the
resistance
B,
bring
the needle
to zero. Then i^
=
0,
and
^-^
=
i^,
therefore
E^_ R +
x
El
r
+
y'
Here we
have
the
unknown

resist-
ances, X and
y,
in our
result
;
but
by
taking another
value
of
E,
say
i?', and
finding
the
corresponding
value
of r,
say
?•',
we get
the
simple result
El-
the
ratio
of a
difference in
the value of

i?
to a
difference
in the
value of
r. This method,
involving no
calculation, as only two differences have
to
be
observed,
and being
perfectly
independent
of the
resistances of
the batteries and
galvanometer, gives very
good results.
A
further
advantage is
that,
as i-^
=
i.2 and
no current passes
through
the
galvano-

meter, each battery is being
worked to exactly
the same degree.
Thus
they
are compared
under
similar
conditions,
which is not the
case in
Poggendorff
's
and other
methods.
E-E'
AE
r,
or
r-
,
r
-
r
A?-
H.E.P.
—
VOL.
T.
/'".

ELECTRICAL
PAPERS.
II.—
VOLTAIC CONSTANTS.
[Telegraphic
Journal,
May 15th,
1873, p.
146.]
This
journal for
April 15th
contains an article
on
a "New
Method of
Determining
Voltaic
Constants." It
is new, inasmuch as it is not
to
be
found in any
electrical
books, as far as I am
aware, but it is not
entirely
new.
The
method

to
which the
diagrams
6,
7,
and
8
refer was devised
by
me
about
three
years
ago, and
it will
be
found in the English
Mechanic for
July
5,
1872,
p.
411. A description
and- proof will be
found there.
I
arrived at
it nearly as M. Emile
Lacoine has,
by

considering the
potentials
of the
different points of a circuit containing
two
electromotive
forces
of the same sign.
Perhaps
a
few remarks
as
to
the value
of
this
method
may not
be
unacceptable. It gives very
different
results
from
Poggendorfi''s, and with reason. Poggendorff's
method
—in
which
the
battery
having

the lesser electromotive force is
not
allowed
to
work
—
especially as improved on
by
Latimer Clark,
most certainly is
an
exceedingly
accurate way of comparing
the electro-
motive
forces of
elements
when not in action, which may
be
then
very
well
called their
jjoteiitiah ; but
it
is a notorious
fact
that these potentials
fall more or
less,

generally more, when the batteries are
called upon to
make
themselves
useful.
The new
method
in
question compares
the
working
electromotive
forces of batteries when in action
through any
desired
resistance,
and
can
on that
account be of
some
value in practice.
(What is the use of
a
battery
having
a
very great potential
if it is only
while -sleeping

'?)
Suppose
we
compare
a
number of
Daniell's with an
equal number of
Leclanche's by
the
new
method.
Eeferring
to the
figure, let
the
right-hand
battery be the
Leclanche's with
the big and
lazy potentials,
and
the left the Daniell's
with the
smaller but
more industrious
potentials.
Then
D
Ab

expresses
their
relative
electromotive forces.
Now
we may
watch
the
behaviours of these
batteries in an instructive
manner
by
commencing
with very
high
values of B and b,
and for
convenience
we
may make
Ab
constantly
100
ohms. At first AB
will
be
found
much
higher, say
150,

showing
that tlie electromotive
force
of the
Leclanche's
is
at that
moment
50
per cent, higher
than
the
Daniell's;
but
by
constantly
taking
100
ohms away
from
b, the
corresponding
difference in
B,
namely AB,
becomes smaller
and smaller,
and
if
we

go on for
a
little
while
(for the
Leclaiiche's soon
get tired) A^
will
become
actually less than 100 ohms,
and,
if the
batteries
be left
working,
may
fall
much
lower.
We may
reverse the
process,
but AB
will
not
become
150
again unless we give
the
Leclanche's

a good
rest.
No
two
series
of trials agree, however.
The
meaning of
all this
is that
the
electromotive
force of the Leclanche
element,
for
continuous
working,
is
anything
between
nothing
and
1
-2
BEST ARRANGEMENT OF WHEATSTONE'S BRIDGE.
3
or
1'3
times that
of

Daniell's. I have even seen the current
of
a
Leclanche
cell reverse itself after a few hours' hard
work,
but it
partially
recovered
after a rest.
Ill—ON THE BEST ARRANGEMENT OF WHEATSTONE'S
BRIDGE
FOR MEASURING A
GIVEN RESISTANCE
WITH
A GIVEN GALVANOMETER
AND
BATTERY.
[Phil.
Mag., Feb.
1873,
S.
4,
vol.
45.]
In the figure,
a, b,
c, and d are the four
sides of the electrical arrange-
ment

known
as Wheatstone's bridge or
balance, e
the
galvanometer,
and
/
the
battery branch.
Throughout this
paper
d
is
supposed
to
be the
resistance
to be
measured, and e and
/
both known.
The
problem
is to find what resistances should be given
to the sides
a, b,
and c
(which
we are able to vary), so that
the galvanometer may

be
affected
the
most by any
slight
departure
from the
balance
which occurs
when
a:b
=
c:d. The
nature of
this problem
may be
more
easily
understood
from
the
following
considerations
:
—
1.
If
6,
c,
d,

e, and
/
are
given, then
there is only one value
of a
be
which will produce a balance, viz.,
a
=—.
2.
But if
c,
d,
e, and
/
are
given,
but not b, then there is an
infinite
number of
pairs
of
values
of a and
b
which will produce
a balance
by
satisfying

the
relation
a:b
=
c:d; and one
particular
pair will
constitute
the best arrangement,
by
which
is meant that the
galvanometer will
be
most sensitive
to
any slight
departure from
the
equality of
j
and
-
when
those particular
values
of a and b
are
used.
4

ELECTRICAL PAPERS.
3.
And if only d, e,
and
/
are given, then for
any value we give to
c
there is
a
pair of values of a
and h which constitutes the best
arrange-
ment for that value of
c ;
and there
will be
a
particular value of c
which,
with the corresponding values of a
and
h,
will
be
the best
arrangement
for the given values of d, e, and
/.
In order to find what

functions a,
b,
and
c
must be of d, e,
and
/
to
constitute
the best
arrangement, it
will
be
first necessary to find
the
best values of a and h
when c,
d, e,
and
/
are given. This I
now
proceed to do.
It is well
known, and may be
easily
proved by
Kirchhoff''s laws, that
the
current

passing through
the galvanometer
is represented
by
_
H
X
{a + h + c + d)(ad-hc)
,-,^
^^
~
{{(^
+
^)(c
+
d)
+
{a
+
b
+
c
+
d)e}{{a
+
c){b
+
d)
+
{a

+
b
+
c
+
d)/}'
"
in
which
E
is the electromotive force of the battery, (ad
-
be) may be
positive, negative, or
nothing, in which last
case
u
=
0,
and a balance is
obtained, no current passing through the galvanometer.
Dividing both numerator
and
denominator of
(1)
by
(a
+
b
+

c
+
d)^,
it becomes
ad
-
be
u
=
:Ex
a
+
b
+
c
+
d
.
,2^
( {a
+
b){c
+
d
)^^(
{a
+
e){b
+
d)

^.Y
^
^
\a
+
b
+
c
+
d j\a-{-b
+
c
+
d
J
from the form
of
which
it may easily
be seen that
the best
value of the
resistance
of the
galvanometer e, when
a balance is obtained and the
other resistances
are
fixed, is,
as Schwendler

has shown in the
Philoso-
phical
Magazine
for May,
1866,
{a
+
b){c
+
d
)
_.
c + d
,
.ox
(a
+
b
+
c
+
d)
'
b
+ d'
^
^
that
is, the

resistance of the
galvanometer
should
equal the resistance
external to
the galvanometer,
being
the
joint resistance
of the two
parallel
branches
{a
+
b) and (c +
d). Also
it may
be
proved
that the
best arrangement
of the
battery is
obtained
when
its resistance
equals
the
external
resistance, that is,

.
(
a
+ c){b + d
)
^
^
^
6_+^
a
+
b
+
c
+
d
'
c
+ d'
^
the joint
resistance
of
the two
parallel
branches
(a
+ c)
and
(b

+
d).
(In passing,
I may
notice that
Schwendler,
in the
paper
above
referred to,
and
also in
a
later
one in
the
Philosophical
Magazine
for
January,
1867,
has
assumed it
to be
necessary
for
the
battery
resistance
to

be very
small,
in
order that
the
relation
exhibited
in
equation
(3)
may be
satisfied.
This
appears
to me
to
be totally
unnecessary
;
for
the resistance
external
to the
galvanometer
when
a balance is
obtained
is quite
independent
of

/,
the
battery
resistance.
In
fact,
the
proper
resistance
for
the
battery
when it
is
to
be most
advantageously
used is
given
by
equation
(4).)
BEST
ARRANGEMENT OF
WHEATSTONE'S BRIDGE.
5
As in
the present
paper we
are only concerned with

such vahies of
a, b, c,
and d
as produce a
balance, or nearly
so,
one
of these four
resistances
may
be
eliminated at once.
Let it
be a. Then
{a
+ b){c + d)_j
c
+ d
a
+ b + c + d
'
b
+
d'
(a
+
c){b
+ d)_
b
+

d
a
+
b
+
c
+ d
c
+ d'
and
a + b
+
c
+
dJ-t:^)^.
d
Substituting these in equation
(2),
we get
(ad
-
bc)d
b
+
d
J
\
c
+
d

-^
j
=
Edx
'.^ih
(5)
{bc
+
ef){b
+
d){c
+
d)
+
ce{b
+
df
+
bf{c
+
df
^
'
Now c, d, e,
and
/
being
fixed, and
b the
variable, we have

to make u
a maximum. As
Ed is
constant, it may
be
dismissed.
As to
the
numerator
{ad
-
be),
it
vanishes
when at
a
balance
;
but
of course
such
a
thing
as an exact
balance is
unattainable. Let
c?±A be
the real
value
of

the
resistance
we are measuring,
d
being
the calculated
value
h(*
—,
and
A
a
small
difference,
then
a
a{d±A)-bc=
±«A,
Therefore the
numerator
varies as
a
or
as
b, since
in the
present case a
and b vary together.
Hence we may write b for {ad
-

be). Thus
b
{be
+
ef){b
+
d){c
+
d)
+
ce{b
+
d)^
+
bf{c
+
d)^'
By differentiation and putting
^^
=
0,
we obtain
clb
{be
+
ef}{b
+
d){c
+
d)

+
ce{b
+
d)^
+
bf{c
+
d)'^
=
bc{b
+
d){c
+
d)
+
b{bc
+
ef){c
+
d)
+
2bce{b
+
d)
+
bf{c
+
d)'^
;
therefore

ef{b
+
d){c
+
d)
+
ce{b
+
d)^
=
b{bc
+
ef){c
+
d)
+
2bce{b
+
d),
def{c
+
d)
+
ce{b
+
d)^
=
bh{c
+
d)

+
2bce{b
+
d),
bh'.{c
+
d
+
e)
=
de{cd
+
df+fc),
which
gives
the relation sought,
and as a
=
~,
therefore
d
&=Jl^l±f±/^.;
(6)
\
c c
+
d
+
e
Vc

cd +
df+fc
/-TV
-,
^—
^ .
e
(7)
d c+d+e
6
ELECTRICAL
PAPERS.
These
values of a and
h will be
found to
make
—
-
negative
;
there-
at"'
fore
they give the most
sensitive
arrangement
for the fixed
values
of

c, d,
e,
and/.
If
b vary
from nothing
upwards, it
will be
found
that u
rapidly
increases up
to
its
maximum value
and then
slowly
decreases,
from
which it
may be
concluded that
it is better to
use too
large
values
of a
and
b than too
small.

In
case c
=
d, formulae
(6)
and
(7)
become
ce'i±^.
(8)
As a
numerical example
of these
formulse, suppose
the
resistance to
be
measured d
=
1,000
ohms, the galvanometer e
=
500 ohms, the
battery
resistance
/=
100
ohms,
and we make c= 1,000
ohms;

then the
best
values for a and b will
be
found
to be
n/240,000=
100
x/24,
or
nearly
500
ohms.
Having thus
determined
the relations of
a
and b to c, d,
e, and
/,
the
latter resistances being fixed,
we now proceed to the
second
part of the
problem, to
determine
the best values of
a, b,
and c when

only
d,
e,
and
/
are given.
This
is the
case which occurs so often
in practice, when
we have a
battery,
a galvanometer,
and a
resistance to be
measured,
and
three sides of a
bridge to which
we maj'
give any
values
we
choose
(within certain limits).
Insert the values of a and
b,
as
given in equations
(6)

and
(7),
in
equation
(5)
; then, after some
reductions,
we
obtain
,_
ad
-
be
2de{cd
+
df+fc)
+
{
(c
+
d
+
e){cd
+
df+fc)
+
cde
] J-
.
^A±^fie

We must now
consider
c the independent variable,
a and
b being
dependent
variables,
(ad
-
be) still
varies
as
a. It does
not, however,
vary as b, but as the
product
be or
ad,
since
d is constant.
Therefore
we
may put the known
value
of be in the numerator
instead
of (ad
-
be).
Thus

4
aie
'd +
df+fc
c
+
d
+
e
2de{cd
+
df+fc)
+
{{e
+
d
+
e)(ed
+
df+fe)
+
cde}
J-
.
'A±M±f^e
\
c c
+
d
+

e
Multiply numerator
and
denominator
by
a
/4-
•
^"'"
,
and
we
have
M
de
cd
+
df+Jc
2
Jcde{c
+
d
+
e){ed
+
df+fe)
+
{c
+
d

+
e){cd
+
df+fc)
+
cde
which has to be made
a maximum.
Differentiating
and putting
—
=
0,
dr
BEST ARRANGEMENT
OF WHEATSTONE'S BRIDGE.
;
2
slcde{c
+
d
+
e){cd
+
df+fc)
+
(c
+
d
+

e)(cd
+
df+fc)
+
cde
_
c
Jcde{cd
+
df+fc)(c
+
d
+
e)
X
{
cde{cd
+
df+fc)
+
cde{c
+
d
+
e){d
+/)
+
de(c
+
d

+
e) {cd
+
df +fc)
}
+
c{c
+
d
+
e){d
+/)
+
c{cd
+
df+fc)
+
cde.
Therefore
2
sjcdeic
+
d
+
e){cd
+
df+fc)
+
df{d
+

e)- c%d
+f)
^
cde{c(cd
+
df+fc)
+
c(c
+
d
+
e)(d
+/)
+
(c
+
d
+
e){cd
+
df+fc)
}
Jcdeicd +
df+fc)
{c
+
d
+
e)
Multiplying both sides

of this equation
by
the
denominator
on the
right
hand side and reducing,
we get
{df{d
+
e)- c\d
+/)}
Jcde{c
+
d
+
e){cd
+
df+fc)
=
cde { c\d
+f)
-df(d
+
e)},
which
is satisfied
by
df(d
+

e)-c%d+f)
=
0,
which gives the required
relation,
-^i/
irr
(^'
that is, c equals the
square root of
the
product of the joint resistance
of
the
battery and the
resistance to be measured, into the sum of the resist-
ance
of the galvanometer
and the resistance to
be
measured. Inserting
this value
of c in
(6)
and
(7),
we
find
the values of a and b to be
a=

J'ef,
(10)
*=V^f+{-
*")
In using the
Wheatstone's bridge
for
measuring very high
resistances,
as, for instance,
the insulation resistances of (good)
telegraph lines, the
battery
resistance
is usually
very
small in comparison
with that of
the
line:
hence
-;
-^
,.
will
be very little different from
/'.
When this is
the
d+f

case,
formula
(9)
becomes
c=
Jf{d
+
e).
If
also the
galvanometer resistance is
small
compared
with the
resistance
to
be measured, then these
equations are
sufficient
for the
determination
of
b
and
c,
b=
side,
c=
Jdf
As

a
numerical example of
these
formulte,
suppose
/=
100
ohms,
e=1000
ohms, and d is
known to be
about
1,000,000
ohms.
Then
by
(10),
a=
^100,000
=
316 ohms.