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SCIENTIFIC PAPEES
EonDon:
C. J.
CLAY AND
SONS,
CAMBRIDGE UNIYEESITY PRESS WAREHOUSE, AVE MAEIA LANE. ©laBsofa: 263,
ILeipjig:
ARGTLE STREET.
F. A.
BEOCKHAUS.
THE MACMLLLAN COMPANY. eombaB: E. SEYMOUR HALE.
gorli:
SCIENTIFIC PAPBES
BY
JOHN WILLIAM STEUTT, BARON RAYLEIGH, D.Sc, F.R.S.,
HONORARY FELLOW OP TRINITY COLLEGE, CAMBRIDGE, PROFESSOR OP NATURAL PHILOSOPHY IN THE ROYAL INSTITUTION.
VOL.
I.
1869—1881.
CAMBRIDGE AT THE UNIVERSITY PRESS. 1899 [All Rights reserved.]
Ki?
^> \^?>x^2.
©amttiSgc: PRINTED BY
J.
AND
C.
P.
CLAT,
AT THE UNIVERSITY PRESS.
PREFACE. nnHE -*-
papers of the present Collection are reprinted very nearly as
they originally appeared, and with a few partial exceptions in order
Obvious misprints have been corrected,
of date.
aid of the ficance
are
matter
is
manuscript.
original
indicated
introduced
to
principal results
Some
Other alterations of the
by the use of square brackets
In a few cases, where
reproduce a paper is
collections
I
of short and
slightest signi-
while additional
in
full,
a
it
has not been thought
brief
statement
of
the
given.
short papers of a rather
These may serve
[ ],
with the
with the proper date in the form of footnotes or
at the end of a memoir.
worth while
in several cases
slender character have been included.
to mitigate the general
have usually
felt
severity.
even more grateful
In consulting similar for
the i-eproduction
often rather inaccessible notes than for the lai'ger and better
known memoirs.
Terling Place, Witham, Ocloher 1899.
Tlie tvorhf
of the Lord are great,
Sought out of
all
them
tluit
liave pleasuj'e
therein.
..
CONTENTS. ART. 1.
On some
.......
Electromagnetic Phenomena considered in connexion
with the Dynamical Theory
Mag. xxxviii.
[Phil.
2.
On an
Electromagnetic Experiment [Phil.
3.
On
pp.
Mag.
x.xxix. pp.
the values of the Integral
I
1—15,
1869.]
..... 428—435,
14
1870.]
QuQu'dfi, Q„, Q„' being Laplace's
Jo Coefficients of the
orders
n,
n, with an application to the
Theory of Radiation
21
[PhU. Tram. CLX. pp. 679—590; read .June 1870.]
4.
..... ........
Remarks on a paper by Dr Sondhauss Postscript
{Phil.
5.
On
Mag.
XL. pp.
211—217,
1870.]
the Theory of Resonance
33 33
Introduction
Part
I.
.
26
32
.
.
37
.
Several Openings
39
Double Resonance
41
Open Organ-pipes Long Tube in connexion with
45 Reservoir
Lateral Openings
Part II
Long Tubes
....
Simple Apertures Cylindrical Necks Potential on itself of a uniform Circular Disk
48 50 51 51
52 53 55
— CONTENTS.
Vlll
P-^-GE
AET. 5.
On
the Theory of Resouauce
...
continued.
Nearly Cylindrical Tubes of Revolution
...
Upper Limit
Application to straight
.
Tube
of Revolution
whose end
62 62
.
.
lies
on
.64
two infinite Planes Tubes nearly Straight and Cylindrical but not necessarily .
.
of Revolution
.
Tubes not nearly Straight
.... ...
Part HI.
Experimental [Phil.
6.
Trans. CLXI. pp.
.
.
On
[Phil.
On
III.
Mag.
1870.]
in Verdet's Legoiis
On Double
On
.
pp.
267—269,
1871.]
.... ...
.
234—237,
pp.
XLI. pp.
Mag.
Refraction
.
8.5
264, 265, 1871.]
its Polarization
and Colour
87
.96
107—120, 274—279, 1871.]
XLI. pp.
Mag.
79
447—454,
.
XLi. pp.
.
104
.
Ill
1871.]
.
.
519—528,
.
1871.]
the Reflection of Light from Transparent Matter [Phil.
Oq
ill.
76
the Scattering of Light by small Particles
[Phil.
12.
Read Nov.
.......
[Phil.
11.
67
.67
.
;
.
the Light from the Sky,
Appendix
10.
64
.
77—118
Soc. Proc.
Some Experiments on Colour [Nature,
^9.
.66
...
Yellow
8.
.
.
Note on the explanation of Coronas, as given d'Optique Physique, and other works [London Math.
7.
•
.
Mag.
XLil. pp.
81—97,
a Correction sometimes required in
120
1871.]
Curves professing to
represent the connexion between two Physical Magnitudes [Phil.
13.
On
Mag.
135
—
444, 1871.]
....... — ........
the Vibrations of a Gas contained within a Rigid Spherical
Envelope
.
[Londmi Math. 14.
XLII. pp. 441
Soc. Proc. iv.
i}p.
93
.
Investigation of the Disturbance produced by a Spherical Obstacle on the Waves of Sound
[London Math. Soc. Proc.
iv.
pp. 253
138
103, 1872.]
— 283,
1872.]
139
CONTENTS.
IX
.....
ART.
15.
Notes on Bessel's Functions [Phil.
16.
On
Mag. XLIV. pp. 328—344,
.
.
.
[Phil.
....
.
Mag. xmi. pp. 321—338,
140
Opaque 141
1872.]
Preliminary note on the Reproduction of Diffraction-Gratings by
means
Photography
.
157
the Application of Photography to copy Diffraction-Gratings
160
of
.
.
.
.
[Proceedings of the Royal Society, xx. pp. 414
18.
I'AGE
1872.]
the Reflection and Refraction of Light by Intensely
Matter
"17.
.
Od
—417,
1872.]
[British Association Report, 1872, p. 39.]
19.
On
....
the Diffraction of Object-Glasses [Astron. Soc. Month. Not. xxxill. pp. 59
20.
An Experiment
— 63,
163
1872.]
an Electric
to illustrate the Induction on itself of
167
Current [Nature, vi. p. 64, 1872.]
21.
Some General Theorems Section
relating to Vibrations
.
.
.
170
I.
The natural periods of a conservative system, vibrating freely
about a configuration of stable equilibrium,
the stationary condition Section
On
.
.
.
.
.
.
170
.176
IH
179
[London Math.
22.
fulfil
.
II.
The Dissipation Function Section
...
Soc. Proc. iv. pp.
357—368,
1873.]
the Nodal Lines of a Square Plate
182
[PhU. Mag. XLVL pp. 166—171, 246, 247, 1873.]
23.
Note on a Natural Limit
to the Sharpness of Spectral Lines
183
[Nature, vill. pp. 474, 475, 1873.]
24.
On
the Vibrations of Approximately Simple Systems [Phil.
25.
On
Mag. xlvl pp.
357—361, 1873; xLvm.
pp. 258—262,
the Fundamental Modes of a Vibrating System [Phil.
Mag. XLVL pp. 434—439,
1873.]
.
.
185
.
186
1874.]
.
CONTENTS.
X
PAOE
ART.
26.
Vibrations of
Membranes
[London Math.
27.
Harmonic Echoes
.
188
vm.
pp. 319, 320, 1873.]
.....
Note on the Numerical Calculation of the Roots of Fluctuating Functions
.
[London Math.
29.
Soc. Proc. v. pp. 9, 10, 1873.]
.
[Nature,
28.
A
Soc. Proc. v. pp.
By
Todhunter,
I.
&
(London, Macmillan
On
v.
Insects
A
F.R.S.
.
Two Volumes.
....
pp. 176, 177, 1874.]
and the Colours of Flowers
Statical
199
.
xi.
p.
.
.
222
.
1874.]
6,
Theorem
223
[Phil. Mag. xlviil pp. 452
Mr
196
Mag. XLVii. pp. 81—93, 193—205, 1874.]
[JS'aticre,
33.
119—124, 1874.]
the Manufacture and Theory of Diffraction-Gratings [Phil.
32.
M.A.,
Co., 1873.)
[The Acadsmy,
31.
190
.
History of the Mathematical Theories of Attraction and the Figure of the Earth from the time of Newton to that of Laplace.
30.
.187
.
.
—456,
1874
;
XLIX. pp. 183—185, 1875.]
Hamilton's String Organ
230
[Nature, XL pp. 308, 309, 1875.]
34.
General Theorems relating to Equilibrium and Initial and Steady-
Motions
232
.
[Pha. Mag. XLIX. pp. 218—224, 1875.]
35.
On
the Dissipation of Energy
[Ron. Instit. Proc. vii. pp.
36.
On
the
Work
that
386—389, 1875
may be gained
[Phil.
37.
.......
Mag. xlix.
;
during the Mixing of Gases
242
pp. 311—319, 1875.]
Vibrations of a Liquid in a Cylindrical Vessel [Nature,
238
Nature, XL pp. 454, 455, 1875.]
xil. p.
251, 1875.]
.
.
.
250
CONTENTS. ART.
38.
On Waves The
PAGE
...
Solitary
Periodic
in
251
.
Wave
Waves
xi
256
.
Deep Water
.
Oscillations in Cylindrical Vessels [PhU. Mag.
39.
On
pp.
i.
...
.
257—279,
265
1876.]
the Approximate Solution of Certain Problems relating to
the Potential
....
.
\_London Matlu Soc. Proc.
40.
.261
.
Our Perception
pp. 70
vii.
—75,
.
.
272
1876.]
of the Direction of a Source of
Sound
.
277
{Nature, xiv. pp. 32, 33, 1876.]
41.
Questions from Mathematical Tripos Examination
January January January January January January January
—12 1^ —4 9 — 12 1^ — 4 1| — 9 — 12
9
6.
6.
17.
19. 20. 21. 21.
4
1^
—4
... ... ... .
.
_.
1876 .
280 .
....
.
.
for
.
.
.
.
.
.
.
280 281 282 283 284 285 286
[Cambridge University Calendar, 1876.]
42.
On
[Phil.
43.
....
the Resistance of Fluids Mag.
II.
pp.
430—441,
.
.
287
1876.]
Notes on Hydrodynamics
297
The Contracted Vein
297 302
Meeting Streams [PkU. Mag. IL pp. 441—447, 1876.]
44.
On
the Application of the Principle of Reciprocity to Acoustics [Proceedings of
45.
On
the
Royal
Society, xxv. pp. 118
— 122,
305
1876.]
a Permanent Deflection of the Galvanometer-Needle under
the influence of a rapid series of equal and opposite Induced
310
Currents [Phil.
Mag.
III.
pp.
43—46,
1877.]
G
CONTENTS.
Xll
PAGE
ART.
46.
Acoustical Observations.
I.
.
314 314
.
Perception of the Direction of a Source of Sound
The Head
as
Reflection of
an Obstacle to Sound
.315
.
31
Sound
Audibility of Consonants
.317
.
Interference of Sounds from two unisonant Tuning-forks
317
Symmetrical Bell.
317 318
... •
Octave from Tuning-forks
.
Open End
Influence of a Flange on the Correction for the of a Pipe
The Pitch
of Organ-pipes
.
.
319
.
.
320
.
.
.
.
.
[PIM. Mag. in. pp. 4.56—464, 1877.]
47.
On
Progressive
Waves
[London Math.
48.
On
.
...
.
Soc. Proc. ix. pp. 21
the Amplitude of Sound- Waves
Absolute Pitch
On Mr
...
51.
On
On
Soc. Proc. ix. pp. 61
Mag.
vi.
A
vii.
.
pp. 14
— 16,
vii. p.
of certain Acoustical
Uniformity of Rotation
342
.
347
.
69, 1878.]
Phenomena
.
.
.
p.
348
Nature, xviii. pp. 319—321, 1878.]
........
[Nature, xvill.
344
1877.]
simple Proof of a Theorem relating to the Potential
The Explanation
338
1878.]
the Irregular Flight of a Tennis-Ball
[Roy. Inst. Proc. vni. pp. 536—542, 1878;
56.
—64,
.......
[Messenger of Matlieviatics,
55.
336
.
pp. 270, 271, 1878.]
[Messenger of Mathematics,
54.
1877.]
pp. 409, 410, 1877.]
Note on Acoustic Repulsion [Phil.
53.
II.
.331
the Relation between the Functions of Laplace and Bessel [London Math.
52.
— 14,
Venn's Explanation of a Gambling Paradox [Mind,
328
.
pp. 248, 249, 1877.]
... [Nature, xvii. pp. 12
50.
322
1877.]
.
XXVL
[Proceedings of the lioi/al Society,
49.
— 26,
Ill, 1878.]
355
.
.
CONTENTS.
Xlll
PAGE
ART.
On
57.
..........
Determination of
the
Harmonium
Absolute Pitch by
Common
the
357
[Nature, xix. pp. 275, 276, 1879.]
On
58.
the Instability of Jets [London Math.
The
59.
.
.861
.
Sac. Proc. x. pp.
4
—
13,
1879.]
Water Drops
Influence of Electricity on Colliding
[Proceedings of the Royal Society, sxvill. pp. 406
On
60.
—409,
372
1879.]
....
the Capillary Phenomena of Jets
377
Appendix 1 Appendix II
396 400
[Proceedings of the Royal Society, xxix. pp. 71
Acoustical Observations.
61.
II.
.
— 97,
1879.]
....
.
Pure Tones from Sounding Flames Points of Silence near a Wall from which a Pure Tone .
reflected
Sensitive Flames
402 402
.
....
is
403
.
406 407 408
.
.
.....
Low Pitch maintained by Flames Notes on a large scale Mutual Influence of Organ-Pipes nearly in Unison Kettledrums.
Aerial Vibrations of very Rijke's
.........
409
.
411
The iEolian Harp [Phil.
413 Mag.
vil.
pp.
149—162, 1879.]
........
Investigations in Optics, with special reference to the Spectro-
62.
scope §
1.
63.
On
... ...
Rectangular Sections
§ 3.
Optical Power of Spectroscopes
.
§ 4.
Influence of Aberration
.
§
5.
On
§
6.
The Aberration
§
7.
Aberration of Lenses and Prisms
Mag.
.
415 418
.
423 428
.
the Accuracy required in Optical Surfaces
The Design viii.
415
.
Power of Optical Instruments
§ 2.
§ 8. [Phil.
Resolving, or Separating,
pp.
of Oblique Pencils
of Spectroscopes
.
Reflection of Vibrations at the Confines of two is
[London Math.
ix. pp.
.
.
440 444
40—55,
453 1880.]
Media between
Gradual Sac. Proc.
.
.
261—274, 403—411, 477—486, 1879;
which the Transition
436 .
460 XL
pp. 51
—56,
1880.]
XIV
CONTENTS.
ART.
64.
PAGE
On
the
Minimum
Aberration of a Single Lens for Parallel Rays
[Cambridge Phil.
65.
Soc. Proc.
.
of Siren
.
The Acoustical Shadow Mag.
\_Pha.
66.
On
On
On
.
.
the Resultant of a large
\_Pha.
Note on the Theory
Mag.
1880.]
488
116—119, 1880.]
pp.
number X.
472
474
.....
the Resolving-Power of Telescopes
Pitch and of arbitrary Phase
69.
.
57—70,
Soc. Proc. si. pp.
s.
468 468
278—283, 1880.]
ix. pp.
[PhU. Mag.
68.
.
of a Circular Disk
the Stability, or Instability, of certain Fluid Motions [Lmidon Math.
67.
1880.]
III.
Intermittent Sounds
Form
373—375,
... ... ..... .471
Acoustical Observations.
A New
pp.
III.
466
.
.
73—78,
pp.
same
of Vibrations of the
.
.
.
.
.
491
1880.]
of the Induction Balance
.
497
[British Association Report, Swansea, pp. 472, 473, 1880.]
70.
On
a
New Arrangement
for Sensitive
71.
The Photophone
....
Flames
...
[Cambridge Phil. Soc. Proc.
iv. pp.
17,
18,
500
1880.]
501
[Nature, xxni. pp. 274, 275, 1881.]
72.
On Copying
Diffraction-Gratings,
and on some Phenomena con-
nected therewith [Pha. Mag.
73.
On Images
.
.
xi.
pp.
.
504
196—205, 1881.]
formed without Reflection or Refraction
513
[PhU. Mag. XL pp. 214^218, 1881.]
74.
On
the Electromagnetic Theory of Light [Phil.
75.
On
Mag.
the Velocity of Light
xii.
.
pp.
.
.
.
518
81—101, 1881.]
-
[Nature, xxiv. pp. 382, 383
.
;
xxv.
.
p.
52,
1881.]
.
537
CONTENTS.
XV PAGE
ART.
76.
On
a Question in the Theory of Lighting
.
.
.
.541
[British Association Report, 1881, p. 526.]
77.
Experiments on Colour
........
[Nature, xxv. pp.
78.
On
64—66,
1881.]
the Infinitesimal Bending of Surfaces of Revolution [London Math.
542
Soc. Proc. xill. pp. 4
—
16,
1881.]
.
551
1.
ON SOME ELECTROMAGNETIC PHENOMENA CONSIDERED CONNEXION WITH THE DYNAMICAL THEORY. [Phil.
IN
Mag. xxxviii. pp. 1—15, 1869.]
It is now some time since general equations applicable to the conditions of most electrical problems have been given, and attempts, more or less complete, have been made to establish an analogy between electrical
phenomena and those
of ordinary mechanics. In particular. Maxwell has given a general dynamical theory of the electromagnetic field*, according to which he shows the mutual interdependence of the various branches of
the science, and lays
any
down equations
sufficient for the theoretical solution of
He
problem.
has also in scattered papers illustrated the solution of special problems by reference to those which correspond with, electrical
them
(at
least
in their mathematical conditions) in ordinary mechanics.
There can be no doubt,
mind
I think, of the value of
more
such illustrations, both as
what takes
place, and to a rough quantitative result which is often of more value from a physical point It is because the of view, than the most elaborate mathematical analysis. dynamical theory seems to be far less generally understood than its im-
helping the
to a
vivid conception of
portance requires that I have thought that some more examples of electrical problems illustrated by a comparison with their mechanical analogues might not be superfluous.
As a simple case, let us consider an experiment first made by De la Rive, which in a battery (such as a single Daniell cell) whose electromotive force is insufficient to decompose water, becomes competent to do so by the Thus, let the primary wire of a intervention of a coil or electromagnet. Ruhmkorff coil be connected in the usual manner with the battery, and the electrodes of the voltameter (which dilute sulphuric acid into
may
consist of a test-tube containing
which dip platinum wiies) with the points where
* Philosophical Transactions for 1865.
^
K.
I.
1
ON SOME ELECTROMAGNETIC PHENOMENA
2 in
the ordinary use of the instrument the contact
There
thus be always a complete
will
voltameter; but
when the contact
made and broken.
is
conducting
[1
circuit
made the voltameter
is
through the
will
be shunted,
and the poles of the battery joined by metal. Now when the shunt is open the battery is unable to send a steady current through the voltameter, because, as has been shown by Thomson, the mechanical value of the chemical action in the battery corresponding to the passage of any quantity of electricity is less
than that required
When, however, the shunt
in the voltameter. itself
the decomposition of the water
for
gradually in the
coil,
where there
is
is
closed, a current establishes
no permanent opposing electro-
and after the lapse of a fraction of a second reaches its full value as given by Ohm's law. If the contact be now broken, there is a momentary current through the voltameter, which causes bubbles of gas to appear on the electrodes, and which is often (but not, I think, well) Allowing the rheotome to act freely we get a called the extra current.
motive
force,
steady evolution of gas.
To
this
electrical
apparatus
Montgolfier's
hydraulic
ram
closely
is
analogous. The latter, it will be remembered, is a machine in which the power of a considerable quantity of water falling a small height is used to raise a portion of the water to a height twenty or thirtj' times as great.
The body
of water from the reservoir flows
down a
closed channel to the
When
place of discharge, which can be suddenly closed with a valve.
takes place, the
moving mass by
momentum
its
is
this
able for a time to
overcome a pressure many times greater than that to which it owes its motion, and so to force a portion of itself to a considerable height through a suitably placed pipe. Just as the electromotive force of the batter}' is unable directly to overcome, the opposing polarization in the
own
voltameter, so of course the small pressure due to the
down by a
fall
cannot
lift
a valve
But when an independent passage is opened, water (or electricity) begins the to flow with a motion which continues to accelerate until the moving force is balanced by friction (resistance), and pressed
greater.
then remains steady.
At
the
moment
the discharge- valve
in the electrical problem, the shunt-contact inertia, tends to continue
is
is
closed (or,
broken), the water, by
moving, and thus the pressure instantly
its
rises to
the value required to overcome the weight of the great column of water.
The second up.
Now
valve
is
accordinglj' opened,
and a portion of the water
is
forced
the electrical current, in virtue of self-induction, can no more
be suddenly stopped than the current of water and so in the above experiment the polarization of the voltameter is instantly overcome, and a ;
quantity of electricity passes. If no
second means of escape
hydraulic ram, the pipe would in
were provided all
for
the water in the
probability be unable to withstand
CONSIDERED IN CONNEXION WITH THE DYNAMICAL THEORY.
IJ
3
the shock, and in any case could only do so by yielding within the limits of its elasticity, so
as gradually, though of course very quickly, to stop the flow
The bursting
of the pipe may properly be compared to the passage of a spark at the place where a conductor carrying an electric current is opened. Just as the natural elasticity of the pipe or the com-
of water.
pressibility of the air in a purposely connected air-vessel greatly diminishes
the electrical spark may be stopped by connecting the breaking-points with the plates of a condenser, as was done by Fizeau in the induction-coil. Contrarj' to what might at first sight have been expected, the fall of the primary current is thus rendered more sudden, and the power the strain, so
many
of the instrument for
equally prevented
Of course the spark is are connected by a short
purposes increased.
when the breaking-points
conducting
circuit, as in our experiment by the voltameter. In fact the energy of the actual motion which exists the moment before contact is broken is in the one case transformed into that of the sound and heat of
the spark, and in the other has
its
equivalent partly in the potential energy
of the decomposed water, partly in the heat generated by the passage of the
momentary current
The experiment voltameter by a
in the voltameter branch. will
be varied in an instructive manner
if
we
replace the
(with or without soft iron), according to the resistance
coil
and self-induction of the latter. In order to know the result, we must examine closely what takes place at the moment when contact is broken.
The
on account of
original current,
At
continue.
its
self-induction or inertia, tends to
the same time the inertia in the branch circuit tends to
prevent the sudden
rise of
A force is thus
a current there.
produced at the
breaking-points exactly analogous to the pressure between two bodies, which
we
will
suppose
pressure
or
inelastic,
continues
opposed by a force of the nature of
Whether
or current.
to
rest.
The
vary until the velocities or
All this time the motion of each body or current
currents become equal. is
one of which impinges on the other at
tension
electrical
friction proportional to the velocity
this resistance will affect the
currents (or velocities) at the
moment they become
common
equal, will
value of the
depend on
its
magnitude as compared with the other data of the problem. There
is
for
every conducting circuit a certain time-constant which
determines the rapidity of the rise or fall of currents, and which is proportional to the self-induction and conductivity of the circuit. Thus, to use Maxwell's notation,
if
L
and
R
be respectively the coefficient of self-induction
and the resistance, the time-constant is LjR = t. If the current c exist at any moment in the circuit and fall undisturbed by external electromotive Any action force, the value at any time t afterwards is given by a; = c e""''. .
which takes place in a time
by
much
smaller than r will be sensibly unaffected
resistance.
1—2
ON SOME ELECTROMAGNETIC PHENOMENA
We
see, then,
that
we may
neglect the
efifects
[1
of resistance during the
time of equalization of the currents, provided that the operation pleted in a time much smaller than the time-constants of either
is
com-
circuit.
suppose to be the case. The value of the common current the or velocity at the moment the impact is over will of course be given by unchanged. is or ordinary, electromagnetic momentum, condition that the and be the coefficients of self-induction for the main and branch
And
If
this I shall
N
L
circuits
respectively,
X
x and
the
analytical expression of the above condition
{L It is here
and required currents, the
original is
L
+ N)X = Lw,
^^ZTW-
or
supposed that there
is
no sensible mutual induction between
the two circuits.
the result of the excess of the one current over the other, and lasts until its cause is removed. Its mechanical value is the difference between that of the original current in the main circuit and that of the
The spark
is
current in the combined circuit, and
initial
expressed by
is
^Laf-i{L + N)X^; or if the value of
X
be substituted,
^ LNa? ^
Exactlj' the collisiou'^
ordinary actual energy, at least state
may be
for the heat produced during the which is necessarily equal to the loss of if the permanent change of their molecular
same expression holds good
of the inelastic bodies,
neglected.
From
the value
X
the current gradually increases
or diminishes to that determined according to
may be seen from branch may be varied
of the combined circuit.
the resistance of the
provided always that to
it
It
is
branch
law,
by the resistance
without affecting the spark,
not so great in relation to the self-induction as
make the time-constant comparable
the spark.
Ohm's
the expression just found that
in
magnitude with the duration of
The spark depends only on the comparative self-induction of the being small when this is small, and when this is great ap-
circuit,
proximating to
its
full
These results are
value ^Laf.
easily illustrated experimentallj'.
I
have two
coils of
thick wire belonging to an electromagnet, which for convenience I will call
A
Each consists of two wires of equal length, which are coiled These may be called Ai A^, B^ B^. When A^ A^ are joined consecutively, so that the direction of the current is the same in the two wires, we have a circuit whose self-induction is four times that of either and B.
together.
*
[1898
— An
erratum
is
here corrected.]
CONSIDERED IN CONNEXION WITH THE DYNAMICAL THEORY.
1]
5
But if, on the contrary, the current flows opposite two wires, the self-induction of the cu-cuit becomes quite
wire taken singly.
ways
the
in
insensible.
The main
circuit
may be composed
of the wire A^ (A.^ remaining open)
into which the current from a single Daniell cell
is led, and which can be opened or closed at a mercury cup. One end of the branch circuit dips into the mercury while the other communicates with the wire whose entrance or withdrawal from the cup closes or opens the main circuit. In this way the coils of the branch may be said to be thrown in at the break.
If the branch is open, is
If the wire
^Lx'.
A be
approximately the value
we obtain thrown
at break the full spark, whose value
in,
the spark
N = L.
ILoc^, for
And
that the currents are parallel, the spark
by
X
\Loi?
now
|.
N=0;
the branch
is
But
is
considerable, having
B^ B^ are thrown
greater and
is
in, so
measured
the currents are opposed, the spark disappears, because
doubled, diminishes the spark. is
if
still
so that the addition of the wire
our calculation the branch
if
is
is still
JSj,
whereby the resistance of
It is true that to this last case
not properly applicable, inasmuch as the time-constant of
But
so exceedingly small.
not
it is
a case (where the self-induction of the branch
difficult to see
may be
that in such
neglected) the tension
more accurately the difference of potential between them, cannot exceed that of the battery more than in the proportion of the
at the breaking-points, or
and main circuits, so that it could not here give any sensible spark. Soft iron wires may be introduced into the but solid iron cores would allow induced coils in order to exalt the effects currents to circulate which might interfere with the result. resistances of the branch to
rise
;
In this form of the experiment there was no sensible mutual induction between the coils A and B. Should there be such, the result maj' be For instance, let the wire A^ be thrown at the considerably modified. break into the circuit of A^ and the battery. This may happen in two ways.
If the connexions are so
made
there will be no sensible spark;
that the currents are parallel in A-^ A^,
but
if
opposed, the spark appears equal to the
And same
condition as before,
M being the is
full
coefficient of
spark ^Lx'.
The current
this is in accordance with theory.
is
given by the
Mx = {L + 2M + N) X,
mutual induction between the two
circuits.
The
therefore T^
T,—
M
\La?-\{L + 2M + N)X^ = ^'^~^—,
Now
X
which leads to the equation
Lx +
spark
the directions of the currents are
in the first-mentioned connexion
M= — L;
M=L
so that the observed sparks are just
bb
N = L.
very nearly, and in the second
what theory
requires.
ON SOME ELECTROMAGNETLC PHENOMENA
6
With regard
to those electrical
induction of two circuits,
\l
phenomena which depend on the mutual
may be remarked
that it is not easy to find exact analogues in ordinary mechanics which are sufficientl}' familiar to be rough idea of the reaction of of much use as aids to conception. it
A
may be had from the consideration whose ends forces may be apijlied. If when
neighbouring currents a heavy bar to
one end
is
of the motion of
the bar
is
at rest
forwards in a transverse direction, the inertia of
suddenly pushed
the material gives the centre of gravity in some degree the properties of a This corresponds fulcrum, and so the other end begins to move backwards. to the inverse If the motion
wave induced by the be supposed
rise of
a current in a neighbouring wire.
through a sensible angle, the kinetic energy
^
(a^
+
that the body never turns
infinitely small, so
F-) «=
+i
(&"-
+ 1-) y" +
proportional to
is
{ah
- ¥) xy,
where a and h are the distances of the driving-points (whose velocities are x and y) from the centre of gravity, 1
